Superprotonic Phase Transitions in Solid

Superprotonic Phase Transitions in Solid Acids: Parameters Affecting the Presence and Stability of Superprotonic Transitions in the MHₙXO₄ Family of Compounds (X=S,Se,P,As; M=Li,Na,K,NH₄,Rb,Cs) - CaltechTHESIS
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Superprotonic Phase Transitions in Solid Acids: Parameters Affecting the Presence and Stability of Superprotonic Transitions in the MHₙXO₄ Family of Compounds (X=S,Se,P,As; M=Li,Na,K,NH₄,Rb,Cs)
Citation
Chisholm, Calum Ronald Inneas
(2003)
Superprotonic Phase Transitions in Solid Acids: Parameters Affecting the Presence and Stability of Superprotonic Transitions in the MHₙXO₄ Family of Compounds (X=S,Se,P,As; M=Li,Na,K,NH₄,Rb,Cs).
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/FYTW-7G64.
Abstract
The present work attempted to uncover the structural and chemical parameters that favor superprotonic phase transitions over melting or decomposition in the MHXO₄, MH₂ZO₄, and mixed MHXO₄-MH₂ZO₄ classes of compounds (X=S, Se; Z=P, As; M=Li, Na, K, NH₄, Rb, Cs) and to thereby gain some ability to "engineer" the properties of solid acids for applications. Three approaches are described. First, the general observation that larger cations enable superprotonic transitions was investigated in both the isostructural M₂(HSO₄)(H₂PO₄) and non-isostructural MHSO₄ family of compounds. The results of these studies confirmed and explained such a cation size effect, and also supplied a crystal-chemical measure for determining the likelihood of a compound undergoing a phase transition. Second, the entropic driving force behind the transitions was explored in the mixed CsHSO₄-CsH₂PO₄ system of compounds. From these investigations, a general set of rules for calculating the entropy change of a superprotonic transition was established and the role of entropy in the transitions illuminated. Finally, the superprotonic phase transition of CsHSO4 was simulated by molecular dynamics, with which means the transition was probed in ways not possible through experimental methods. A sufficiently general approach was utilized so as to be applicable to other (as yet un-synthesized) compounds, thereby speeding up the process of discovering novel superprotonic solid acids. All three approaches increase the fundamental understanding of which chemical/structural features facilitate superprotonic transitions and should aid attempts to create new solid acids with properties ideal for application.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
order-disorder transition; proton conductor; solid acids; superprotonic conduction
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Haile, Sossina M.
Thesis Committee:
Haile, Sossina M. (chair)
Ustundag, Ersan
Gray, Harry B.
Goddard, William A., III
Johnson, William Lewis
Defense Date:
13 December 2002
Record Number:
CaltechETD:etd-01292003-150309
Persistent URL:
DOI:
10.7907/FYTW-7G64
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ID Code:
398
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03 May 2021 21:42
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Superprotonic Phase Transitions in Solid Acids:
Parameters affecting the presence and stability of
superprotonic transitions in the
MHnXO4 family of compounds
(X=S, Se, P, As; M=Li, Na, K, NH4, Rb, Cs)

Thesis by

Calum Ronald Inneas Chisholm
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy

California Institute of Technology
Pasadena, California
2003
(Defended December 13th, 2002)

 2003
Calum Ronald Inneas Chisholm

iii

Acknowledgements

I must first say that I have had a truly wonderful time here at Caltech. I can only
hope that the carefree lifestyle and true intellectual forum Caltech offers to its graduate
students remain available to future generations. I count myself doubly blessed to have
been accepted at this most distinguished institution.
I owe, of course, the greater part of my success and happiness at Caltech to my
professor and mentor, Sossina Haile, without whose care and guidance I might never
have discovered my love of research. I feel my critical thinking and ability to express
ideas has been incalculably increased by our interactions. It has truly been a pleasure and
honor to have worked with Professor Haile, and I am overjoyed to have the opportunity
to collaborate with her for post-doctoral research.
Needless to say (but I certainly must), I thank my mother, who, after all, typed up
my graduate applications and kept me from sending them in late. She has been a constant
source of comfort and encouragement. Thanks, Momma. To my father, I am ever grateful
to him for being my hero and role model, not just in the sciences, but in all of life’s
pursuits. His love for his work and his words of wisdom regarding a career in science
convinced me long ago that my life would be fuller following the path I now tread.
Cheers, Dad.
This work would not be possible without the efforts of other co-researchers. First
and foremost, Dane Boysen, without whose efforts, assistance, and collaboration I might
never have stepped into the lab, much less have arrived at the conclusions of this study. I
must also single out Ryan Merle, who helped me immensely by doing all the “dirty
work” and who has been a constant sounding board for new ideas. For the simulations
work, I owe a huge debt of gratitude to Yun Hee Jang. Without her ever thoughtful
assistance I could not have begun or finished that study.
There are a whole host of others that I have to thank for their time and effort,
including: Sonjong Hwang, Dr. Ma Chi, Dr. Chuck Strouse, Lisa Cowen, Liz Miura,
Prof. Rossman, Prof. Goddard, Lan Yang, Prof. Grubbs, and Prof. Fultz. To Dr. Chuck
Witham, who was my first friend at work and always helped me both on and off campus,
I wish you were here laughing at my defense, buddy. I must also thank my friends all
over the states that have always supported me in my endeavors.
Financial support was provided by The National Science Foundation and various
Caltech fellowships, making my life very easy here.
And finally, I must thank Erica, with whom I have shared this entire experience.
You make me a better person who looks to the future while still enjoying the present.

Abstract
The present work attempted to uncover the structural and chemical parameters
that favor superprotonic phase transitions over melting or decomposition in the MHXO4,
MH2ZO4, and mixed MHXO4-MH2ZO4 classes of compounds (X=S, Se; Z=P, As; M=Li,
Na, K, NH4, Rb, Cs) and to thereby gain some ability to “engineer” the properties of solid
acids for applications. Three approaches are described. First, the general observation that
larger cations enable superprotonic transitions was investigated in both the isostructural
M2(HSO4)(H2PO4) and non-isostructural MHSO4 family of compounds. The results of
these studies confirmed and explained such a cation size effect, and also supplied a
crystal-chemical measure for determining the likelihood of a compound undergoing a
phase transition. Second, the entropic driving force behind the transitions was explored in
the mixed CsHSO4-CsH2PO4 system of compounds. From these investigations, a general
set of rules for calculating the entropy change of a superprotonic transition was
established and the role of entropy in the transitions illuminated. Finally, the
superprotonic phase transition of CsHSO4 was simulated by molecular dynamics, with
which means the transition was probed in ways not possible through experimental
methods. A sufficiently general approach was utilized so as to be applicable to other (as
yet un-synthesized) compounds, thereby speeding up the process of discovering novel
superprotonic solid acids. All three approaches increase the fundamental understanding
of which chemical/structural features facilitate superprotonic transitions and should aid
attempts to create new solid acids with properties ideal for application.

Contents
Acknowledgements

iii

Abstract

1 Introduction

1.1 Overview …………………………………………………………………..

1.2 Ionic Conductivity …………………………………………………………

1.3 Structural Features of Solid Acids ………………………………………...

1.3.1 Characterization of hydrogen bonds ……………………………….

1.3.2 Hydrogen-bonded networks in solid acids …………………………

1.3.3 Common structures found in the low temperature phases of
solid acids ………………………………………………………….

1.4 Protonic Conduction ………………………………………………………

1.4.1 Mechanisms of proton transport …………………………………...

1.4.2 Room temperature proton conduction in solid acids ………………

1.4.3 High temperature proton conduction in solid acids ……………….

2 Experimental Methods

2.1 Synthesis …………………………………………………………………..

2.2 X-Ray Diffraction …………………………………………………………

2.3 Neutron Diffraction ……………………………………………………….

2.4 Thermal Analysis ……………………………………………………………

2.5 Chemical Analysis …………………………………………………………..

2.6 Optical Spectroscopy / Microscopy …………………………………………

2.7 NMR Spectroscopy …………………………………………………………. 42
2.8 Impedance Spectroscopy …………………………………………………...

2.8.1 Complex impedance ………………………………………………...

3 Cation Size Effect on the Superprotonic Transitions of MHnXO4
Compounds (M = Cs, Rb, NH4; X = S, Se, P, As; n = 1-2)

3.1 Introduction ………………………………………………………………...

3.2 Mixed Cation Sulfate Systems………………………………………………

3.2.1 Synthesis and characterization techniques ………………………….

3.2.2 Resulting phases of the mixed system investigations ……………….

3.2.3 Conclusions from mixed system investigations …………………….

3.3 M2(HSO4)(H2PO4) Compounds ……………………………………………..

3.3.1 Structures of the M2(HSO4)(H2PO4) compounds ……………………

3.3.2 Synthesis of the M2(HSO4)(H2PO4) compounds …………………….

3.3.3 Characterization of the M2(HSO4)(H2PO4) compounds ……………..

3.3.4 What exactly is the effect of the cation size? ………………………..

3.3.5 Conclusions and interpretations of the cation/anion effect ………….. 88

4 Mixed Cesium Sulfate-Phosphates: Driving Force for the
Superprotonic Transitions of MHnXO4 compounds (M = Cs, Rb,
NH4; X = S, Se, P, As)

vii

4.1 Introduction ....................................................................................................

4.2 Characterization of Mixed Cesium Sulphate-Phosphates ………………….

4.2.1 Synthesis of the compounds …………………………………………

4.2.2 Structural features of room temperature phases …………………….

4.2.3 Structural features of high temperature phases ……………………… 104
4.2.4 Key features of the superprotonic phase transitions …………………

107

4.3 Introductory Comments on Entropy Rules ………………………………….

117

4.4 Entropy Rule for Room Temperature Structures ……………………………

118

4.4.1 Entropy of CsHSO4 and Cs6(H2SO4)3(H1.5PO4)4 ⇒ ZERO! ………..

120

4.4.2 Entropy evaluation for CsH2PO4 −the disordered hydrogen bond ….

121

4.4.3 Entropy of Cs3(HSO4)2.50(H2PO4)0.50 and Cs3(HSO4)2.25(H2PO4)0.75 −
partially occupied hydrogen bonds …………………………………..

125

4.4.4 Room temperature entropy of Cs3(HSO4)2(H2PO4) ………………....

129

4.4.5 Room temperature entropy of Cs5(HSO4)3(H2PO4)2 ………………..

130

4.4.6 Room temperature entropy of Cs2(HSO4)(H2PO4) ………………….

135

4.4.7 Summary of entropy evaluations for the room temperature phases …

137

4.5 Entropy Rules for the High Temperature Phases …………………………..

138

4.5.1 Plakida’s theory of the superprotonic phase transition in CsHSO4 …..

138

4.5.2 Ice rules type model for superprotonic transitions …………….........

140

4.6 Calculated Transition Entropies for the CsHSO4-CsH2PO4 System of
Compounds ……………………………………………………………….

145

4.6.1 Entropy calculations for CsHSO4 ……………………………………

146

viii

4.6.2 Entropy calculations for CsH2PO4 ………………………………….

148

4.6.3 Entropy calculations for pure cubic phases .........................................

150

4.6.4 Entropy calculations for mixed tetragonal/cubic compounds ……….

151

4.6.5 Entropy calculations for Cs6(H2SO4)3(H1.5PO4)4 …………………….

155

4.6.6 Summary of entropy calculations for high temperature phases……...

158

4.6.7 ∆Strans and comparison with experimental ∆Strans …………………...

159

4.6.8 Application of the adjusted ice rules to other superprotonic transitions

163

5 Superprotonic Phase Transition of CsHSO4 : A Molecular Dynamics
Simulation Study with New MSXX Force Field

170

5.1 Introduction …………………………………………………………………

170

5.2 Characterization of CsHSO4 ……………………………………………….

171

5.2.1 Crystal structures of CsHSO4 ……………………………………….

171

5.2.2 Nature of the superprotonic transition of CsHSO4 …………………

175

5.3 MD simulation of superprotonic transition of CsHSO4 ……………………

176

5.3.1 Overview …………………………………………………………….

176

5.3.2 Calculation details: Force fields …………………………………….

176

5.3.3 Calculation details: Simulations …………………………………….

184

5.4 Results ………………………………………………………………………

185

5.4.1 Phase II at room temperature: Calculation vs. experiment ………….

185

5.4.2 Phase transition: Cell parameters ……………………………………

189

5.4.3 Phase transition: Volume and energy change across Tsp …………….

193

5.4.4 Phase transition: X-ray diffraction …………………………………..

196

5.4.5 Vibrational spectrum of Phase I CsHSO4 ……………………………

197

5.4.6 Orientation of the HSO4 groups ……………………………………..

199

5.4.7 Reorientation of the HSO4 groups …………………………………...

204

5.5 Parameters Effecting the Phase Transition Temperature …………………… 208
5.5.1 Oxygen charge distribution ………………………………………….

211

5.5.2 Hydrogen bond strength ……………………………………………..

214

5.5.3 Torsional barrier height ……………………………………………...

224

5.6 Summary and Conclusions ………………………………………………….

230

6 Conclusions

234

Appendix

238

A.1 Chapter 3 …………………………………………………………….

238

A.1.1 β-CsHSO4-III ……………………………………………………… 238
A.1.2 Cs2Li3H(SO4)3*H2O ……………………………………………….

243

A.2 Chapter 4 …………………………………………………………….

248

A.2.1 Causes for discrepancies in experimental data between published

248

values and those reported in Chapter 4 …………………………….

A.2.2 CsHSO4 ……………………………………………………………..

249

A.2.3 Cs3(HSO4)2.50(H2PO4)0.50 …………………………………………...

251

A.2.4 Cs3(HSO4)2.25(H2PO4)0.75 ……………………………………………

252

A.2.5 Cs3(HSO4)2(H2PO4) …………………………………………………

254

A.2.6 Cs5(HSO4)3(H2PO4)2 ………………………………………………

256

A.2.7 Cs2(HSO4)(H2PO4) ………………………………………………….

257

A.2.8 Cs6(H2SO4)3(H1.5PO4)4 ……………………………………………

259

Bibliography

263

List of Figures
1.1 Effects of symmetry on strong, medium, and weak bonds

1.2 Simple hydrogen-bonded networks found in solid acids

1.3 Room temperature structures for CsHSO4 and CsH2PO4

1.4 Tetragonal structure of KH2PO4

1.5 Structure of monoclinic K3H(SO4)2

1.6 Vehicle mechanism of proton transport

1.7 Grotthuss mechanism of proton transport

1.8 Representation of the Grotthuss mechanism in ice

1.9 Normal and interstitial hydrogen bonds proposed for room temperature
phases of KH2PO4 and MH3(XO4)2 classes of compounds

1.10 Possible conduction paths for proton vacancy/interstitial defects in H-bond
zigzag chains

1.11 Structure of CsHSO4 above its superprotonic phase transition

2.1 Equivalent circuit for a dielectric material between two electrodes

2.2 Separation of bulk, grain boundary, and electrode resistances

2.3 Realistic impedance plot

3.1 Measurements on the α, β, γ-CsHSO4-III compounds

3.2 Crystal structures of the mixed Cs/Na compounds

3.3 DSC and conductivity measurements on CsNa(HSO4)3 and CsNa2(HSO4)3

3.4 Characteristic lengths of MHSO4 compounds as a function of average cation
radius

xii

3.5 Average hydrogen bond length versus mean cation radius

3.6 Structure of Cs2(HSO4)(H2PO4)

3.7 Cubic phase of Cs2(HSO4)(H2PO4)

3.8 Thermal analysis of the M2(HSO4)(H2PO4) compounds

3.9 Conductivity measurements along the b-axis of the M2(HSO4)(H2PO4)

compounds
3.10 Cation size effect on the room temperature conductivities

3.11 Compensation law for M2(HSO4)(H2PO4) compounds and other solid acids

3.12 Characteristic distances for the M2(HSO4)(H2PO4) compounds

3.13 Changes in the characteristic distances of Cs2(HSO4)(H2PO4), CsHSO4 and
CsH2PO4 across superprotonic transitions

3.14 Average thermal parameters of the oxygen atoms versus distances

3.15 Cation radius versus polarizability

3.16 Schematic representation of the potential wells for oxygen atoms

4.1 Room temperature structures for CsHSO4 and CsH2PO4

102

4.2 X-ray powder diffraction patterns of the mixed cesium sulfate-phosphates at
room temperatures (~25˚C)

104

4.3 X-ray powder diffraction patterns of the mixed cesium sulfate-phosphates
above superprotonic phase transitions

105

4.4 Proposed superprotonic structures for CsHSO4 and CsH2PO4

106

4.5 Protonic conductivity of the mixed cesium sulfate-phosphate compounds

108

4.6 Various transition properties versus phosphate percentage

112

4.7 Arrangement of the hydrogen-bonded tetrahedra

120

4.8 Disordered hydrogen bonds in CsH2PO4

121

xiii

4.9 Room temperature structure Cs3(HSO4)2.50(H2PO4)0.50

126

4.10 Structure of Cs3(HSO4)2(H2PO4)

129

4.11 Structure of Cs5(HSO4)3(H2PO4)2

131

4.12 Probable effect of local order in the mixed S/P sites on neighboring
disordered hydrogen bonds

134

4.13 Structure of Cs2(HSO4)(H2PO4)

135

4.14 Local variants in the structure of Cs2(HSO4)(H2PO4)

136

4.15 Disordering of protons across the superprotonic transition

139

4.16 Hexagonal ice

141

4.17 Possible configurations of the sulfate tetrahedral in the superprotonic phase

147

4.18 Cubic structure of CsH2PO4

149

4.19 Configurational entropy loss due to a proton transfer from the cubic to
tetragonal phase

152

4.20 Possible source of extra entropy in the cubic phase of
Cs6(H2SO4)3(H1.5PO4)4

157

4.21 Measured versus calculated transition entropies

160

4.22 Transition volume versus enthalpy

168

5.1 Crystal structure of CsHSO4, as proposed by Jirak

173

5.2 Possible configurations of the sulfate tetrahedral in the superprotonic phase

174

5.3 Structures used to adjust the Dreiding parameters

179

5.4 Potential energy curves for an HSO4- ion with fixed O(1)-S-OD-H torsional
angles

184

5.5 Cell parameters as a function of temperature (MD simulations)

191

5.6 Potential energy and volume as a function of temperature from MD
simulations

195

xiv

5.7 Calculated X-ray powder diffraction patterns

197

5.8 Calculated IR spectra for MD simulations

198

5.9 Probability distribution functions for the S-O vectors

201

5.10 Probability distribution functions for all S-O vectors in phase I

203

5.11 Orientation/reorientation of an HSO4- ion defined by its S-OD vector

205

5.12 Autocorrelation functions for all 32 tetrahedra

207

5.13 Average angular velocity of all 32 S-OD vectors versus temperature

208

5.14 Results of equal oxygen charge MD simulations

211

5.15 Calculated X-ray diffraction patterns for equal oxygen charge MD
simulations

213

5.16 Simulation results with the binding energy of the H-bonds increased by 150
and 200 percent

215

5.17 Autocorrelation functions for original simulation and simulations with
increased H-bond strength

219

5.18 X-ray diffraction patterns for simulations with increased H-bond strength
below and above the low temperature transitions

222

5.19 Rearrangements of the sulphate tetrahedral across the low temperature
transitions of the simulations with the increased H-bond strength

223

5.20 Results of the simulations with lowered torsional barrier

225

5.21 Results for simulations with torsional barrier 10 times value in original FF

227

5.22 Autocorrelation functions for simulations with decreased and increased
torsional barrier heights

229

A.1 Crystal structure of β-CsHSO4-III

239

A.2 H+ NMR measurements on β-CsHSO4-III and true CsHSO4-III

242

A.3 Structure of Cs2Li3H(SO4)3*H2O

243

A.4 TGA and DSC measurements on Cs2Li3H(SO4)3*H2O

247

A.5 Conductivity of Cs2Li3H(SO4)3*H2O

247

A.6 PXD patterns of CsHSO4 taken at various temperatures

250

A.7 PXD patterns of Cs3(HSO4)2.50(H2PO4)0.50 taken at various temperatures

251

A.8 PXD patterns of Cs3(HSO4)2.25(H2PO4)0.75 taken at various temperatures

253

A.9 PXD patterns of Cs3(HSO4)2(H2PO4) taken at various temperatures

255

A.10 PXD patterns of Cs5(HSO4)3(H2PO4)2 taken at various temperatures

256

A.11 PXD patterns of Cs2(HSO4)(H2PO4) taken at various temperatures

258

A.12 PXD patterns of Cs6(H2SO4)3(H1.5PO4)4 taken at various temperatures

259

xvi

List of Tables
1.1 Correlation between hydrogen bond strength, Od··· Oa, and Od—H distances

1.2 Hydrogen-bonded networks in solid acids by their H:XO4 ratio

3.1 Superprotonic phase transitions for MHXO4 class of compounds

3.2 Compounds synthesized in the mixed Cs-KD/Na/Li systems

3.3 Hydrogen bond parameters for the MHSO4 compounds

3.4 Cystallographic data for the M2(HSO4)(H2PO4) compounds

3.5 Successful synthesis conditions for the M2(HSO4)(H2PO4) compounds

3.6 High temperature transition parameters for the M2(HSO4)(H2PO4) compounds 75
3.7 Activation energy and pre-exponential term for the proton conduction in the
room temperature phases of the M2(HSO4)(H2PO4) compound

3.8 Cation/anion radius ratios for the M2(HSO4)(H2PO4) and MH2PO4
compounds

4.1 Synthesis of the mixed cesium sulfate-phosphates

4.2 Structural parameters of the mixed cesium sufate-phosphates in their room
and high temperature phases

4.3 Thermodynamic parameters of the superprotonic phase transitions

115

4.4 Values for the configurational entropy of the room temperature structures

137

4.5 Calculated entropies for high temperature phases

159

4.6 Calculated experimental transition entropies

162

4.7 Application of the ice rules to other solid acid superprotonic phase transitions

164

5.1 Force field for CsHSO4

177

5.2 Force field parameters for CsHSO4

178

xvii

5.3 ESP charges for CsHSO4

181

5.4 Phase II at room temperature: calculation vs. experiment

185

5.5 Atomic positions for MD simulation at 298 K

186

5.6 Hydrogen bond comparison between MD and experiment in phase II

187

5.7 HSO4 group arrangement: QM and FF calculations vs. MD and experimental
values in phase II

188

5.8 MD vs. experimental parameters for tetragonal phase I CsHSO4

192

5.9 MD vs. experimental atomic positions for Cs and S in phase I CsHSO4

193

5.10 Characteristic values of the superprotonic phase transition in CsHSO4

194

5.11 Proposed librations in CsHSO4 phase I compared to simulation results

204

A.1 Atomic coordinates and equivalent displacement parameters (Å2) for

238

β-CsHSO4-III. Ueq = (1/3)Tr(Uij)
A.2 Anisotropic thermal parameters (Å2) for β-CsHSO4-III

240

A.3 Data collection specifics for β-CsHSO4-III, (CsHSO4)3

240

A.4 Atomic coordinates and equivalent displacement parameters (Å2) for

244

Cs2Li3H(SO4)3*H2O. Ueq = (1/3)Tr(Uij)
A.5 Anisotropic thermal parameters (Å2) for Cs2Li3H(SO4)3*H2O

244

A.6 Data collection specifics for Cs2Li3H(SO4)3*H2O

245

A.7 Variation of the transition enthalpy for CsHSO4 from pure and mixed cation
solutions

249

A.8 Results of rietveld analysis on CsHSO4 PXD patterns taken at various

250

temperatures
A.9 Results of rietveld analysis on Cs3(HSO4)2.50(H2PO4)0.50 PXD patterns

252

xviii

taken at various temperatures
A.10 Microprobe data on Cs3(HSO4)2.25(H2PO4)0.75.

253

A.11 Results of rietveld analysis on Cs3(HSO4)2.50(H2PO4)0.50 PXD patterns
taken at various temperatures

254

A.12 Results of rietveld analysis on Cs3(HSO4)(H2PO4) PXD patterns taken at
various temperatures

255

A.13 Results of rietveld analysis on Cs5(HSO4)3(H2PO4)2 PXD pattern in cubic
high temperature phase

257

A.14 Results of rietveld analysis on Cs2(HSO4)(H2PO4) PXD patterns taken at
various temperatures

258

A.15 Results of rietveld analysis on Cs6(H2SO4)3(H1.5PO4)4 PXD patterns taken
at various temperatures

260

A.16 Atomic coordinates and equivalent displacement parameters (Å2) for
Cs6(H2SO4)3(H1.5PO4)4

260

A.17 Anisotropic thermal parameters (Å2) for Cs6(H2SO4)3(H1.5PO4)4

261

A.18 Data collection specifics for Cs6(H2SO4)3(H1.5PO4)4

261

Chapter 1

Introduction

1.1 Overview
Solid acids, or acid salts, are a class of compounds with unique properties arising
from the incorporation of “acid” protons into a crystalline structure: e.g.,
½ Cs2SO4 + ½ H2SO4 Æ CsHSO4. Initial research into these compounds focused on the
ferroelectric properties that many solids acids, such as KH2PO4, express below room
temperature due to ordering of the protons within their potential wells1. Near room
temperatures, the structural proton leads to protonic conductivity on the order of Log(σ) ~
-6 to –9 in most solid acids. This conductivity is due to local defects in the structure and
subsequent protonic hopping2.
While most early studies focused on low temperature behavior of solid acids, in
1981 it was observed that CsHSO4 had a first-order phase transformation at 141° C3. Not
long after, it was discovered that as a result of this solid-solid phase transformation, the
protonic conductivity increased by over three orders of magnitude4. Since then, there has
been increasing interest in the high temperature properties of these compounds and their
first-order phase transformations.
Most solid acids with superprotonic phase transitions have monoclinic symmetry
in their room temperature phase5. Above the phase transition temperature, the symmetry
of the compounds increases and to accommodate the higher symmetry, the oxygens
become disordered. The partial occupancy of the oxygen sites gives a nearly liquid-like
nature to the protons as the previously static hydrogen bonded system becomes highly

dynamic6,7. In this dynamic system, the XO4 groups librate at ~1011 Hz with intertetrahedra hopping of the proton occurring at ~109 Hz8. This fast reorientation of the
tetrahedra in conjunction with proton translations leads to the jump in conductivity across
the phase transition and the “superprotonic conduction” many solid acids exhibit in their
high temperature phases.
With the discovery of superprotonic conductivity in CsHSO4, other known solid
acids were investigated to reveal if they exhibited similar properties, resulting in many
new superprotonic conductors being found. The three most extensively studied families
of solid acids with superprotonic phase transitions have formulas M3H(XO4)2, MHXO4,
and MH2YO4 (X=S, Se; Y=P, As; M=Li, Na, K, NH4, Rb, Cs). The high temperature
behavior of these compounds has been analyzed by a myriad of techniques including, but
not limited to, X-ray diffraction, impedance spectroscopy, thermal calorimetry and
gravimetric analysis, infrared and Raman spectroscopy, acoustic absorption, neutron
diffraction and scattering, and NMR spectroscopy. There is therefore a good physical
understanding of these phase transition. They are order-disorder transitions of first order
that are entropically driven. Below the transition, the protonic transport is of the intrinsic
type due to proton defects in the static hydrogen bonded network. Above the transition,
symmetry increases and proton transport is due to fast reorientations of the XO4
tetrahedra in combination with proton translation along a “dynamically disordered
network of hydrogen bonds”9.
However, in spite of this plethora of theoretical and experimental data, there were
only generalities for the question of which chemical and structural properties influence
superprotonic phase transitions in solid acids. The objective of this thesis work was,

therefore, to uncover the chemical and structural parameters that favor superprotonic
phase transitions over melting or decomposition in the MHXO4, MH2YO4, as well as the
new mixed MHXO4-MH2YO4 classes of compounds and to thereby gain some ability to
“engineer” the properties of solid acids for applications.

1.2 Ionic Conductivity10-12
In general, the total ionic conductivity of a material, under the influence of an
electric field, will be the sum of the conductivities of each mobile species in the material.
The conductivity of each ionic species is directly proportional to the number density of
the ions, their mobility and the charge per ion. The total conductivity is then

σ total = ∑ σ i = ∑ N i ez i µ i

(1-1)

where i refers to the species, N i is the number of mobile ions per unit volume, µ i is their
mobility, and ezi equals the ion’s charge (charge per electron times valence of ion). For a
pure ionic conductor in which the current is carried by only one type of ion, the total
conductivity simplifies to

σ i = N i ez µ i

(1-2)

The evaluation of the conductivity can then be reduced to calculating N i and µ i .
In the calculation of N i we must consider the mechanism of ion mobility. For any
crystalline material, the diffusion of atoms will be caused by the presence of defects in its

structure. These defects allow the atoms of the material to move on an otherwise fixed
lattice. The intrinsic defect concentration can be evaluated as follows.
Thermodynamically, the very presence of these defects is due to the increase in
entropy when some number of defects are added to the material. For an ideal solution,
this increase in entropy per mole can be written

S mix = − R[(1 − N1 ) ln(1 − N1 ) + N1 ln( N1 )]

(1-3)

where here, N 1 is now the mole fraction of defects. The increase in entropy per mole of
material, ∆S mix , due to a small addition to the mole fraction of defects, ∆N1 , is then

∆S mix =

 N1 
dS mix
∆N1
∆N1 = − R ln
dN1
1 

(1-4)

As this function shows, the initial increase in entropy per vacancy added is
extremely large: ∆S mix → ∞ as N 1 → 0 . Therefore, for a material at equilibrium there
will always be a finite number of defects. To calculate this equilibrium number, we can
use the fact that the change in Gibbs free energy, G, of a system in equilibrium is zero for
any small displacement. The change in the Gibbs free energy, ∆G, with the addition of
∆N 1 defects to a mole of crystal already containing a concentration N 1 of defects is

∆G = H d

∆N1
∂S ∆N1
−T
∂N1 N

(1-5)

where N is Avagadro’s number, and H d / N and (∂S / ∂N 1 )(1 / N ) are the increase in
enthalpy and entropy, respectively, in the crystal per defect added. The enthalpy increase
is due to local distortions to the atomic and electronic configuration resulting from the
incorporation of a defect into the crystal structure. The entropy term includes the ideal
entropy of mixing given in Eq. (1-3) plus another term due to the change in vibrations of
the atoms when a defect is included, S d / N . Substituting these entropy terms into Eq. (15) gives

N1  ∆N1
∆G =  H d − TS d + RT ln
1 − N1  N

(1-6)

Now, if the concentration of defects is very low, they are unlikely to interact and
H d and S d should be independent of N 1 . This equation then is most appropriate when
N 1 << 1 . This is certainly true for most metals and ionic solids, where

N 1 < 10 −4 (Shewman, p70, 160). Using N 1 << 1 , Eq. (1-6) becomes

∆G = [H d − TS d + RT ln N1 ]

∆N 1

(1-7)

Since ∆G = 0 for any small displacement, ∆N 1 , from equilibrium, we can write the

equilibrium concentration of N 1 as
N 1equil = exp( S d / R) exp(− H d / RT )
or,

(1-8)

N 1equil = exp(−Gd / RT )

(1-9)

where Gd = H d − TS d is the free energy change to the crystal per one mole of defects
added, on top of the entropy of mixing.
For ionic solids, the dominant defect will be a vacancy or interstitial10. Also,
defects must leave the material charge neutral. These two facts lead to two types of
disorder in ionic solids: Schottky and Frenkel. For Schottky disorder, an equal number of
anion and cation vacancies are formed. It is found in materials where the energies of
formation of a defect on either lattice are similar and the motion of both defects can be
measured. However, experimentally it is observed that the mobility of cation vacancies is
often much greater than that of anion vacancies, due mostly to the fact that cation
vacancies are usually smaller than their companion anion vacancies. Cation vacancies are
therefore responsible for most ionic conductivity by the Schottky defect mechanism.
Using the results from Eq. (1-9), we find that at equilibrium in an ideal solution where
defects do not interact,
( N vaequil )( N vcequil ) = exp[−(Gva + Gvc ) / RT ]

(1-10)

where Gva = H va − TS va and Gvc = H vc − TS vc are the molar free energy of formation of
an anion and cation vacancy, respectively. Now, N vaequil = N vcequil , so we can write Eq. (110) as
N vaequil = exp[−(Gva + Gvc ) / 2 RT ] = exp[−G S / 2 RT ]

(1-11)

where GS is the molar free energy to form the pair of vacancies. Schottky defects are
found in alkali halides, e.g., NaCl and CsCl.
If the Frenkel type of disorder is dominant, the molar free energy to form a cation
interstitial, Gic , is much less than that to form an anion vacancy, Gva . Cation vacacies
will then be charged balanced by cations jumping to interstitial sites rather than anion
vacancies. Hence, the equilibrium concentration of cation interstitials, N icequil , and cation
vacancies, N vcequil , will be equal. As opposed to the Schottky mechanism where only the
cation vacancies were significantly mobile, for Frenkel defects, cation mobility is due to
both the vacancy and interstitial mechanism. If these defects are randomly located (i.e.,
do not interact), we can again use Eq. (1-9) to get
( N icequil )( N vcequil ) = exp[−(Gic + Gvc ) / RT ] = exp(−G F / RT )

(1-12)

or, since N icequil = N vcequil ,
N icequil = exp[−(Gic + Gvc ) / 2 RT ] = exp(−G F / 2 RT )

(1-13)

where G F is the molar free energy of formation for the cation interstitial-vacancy pair.
This type of disorder is found in AgCl and AgBr.
We will now look at the other unknown in Eq. (1-2), the mobility of the charge
carrier, µ i . Because the movement of an ion under the influence of an electric field is
governed by the same atomistic mechanisms as diffusion of atoms due to a concentration

gradient, the ionic mobility can be directly related to the ion’s diffusivity. This is
expressed by the Nernst-Einstein equation:

µ ion
ez

k BT

(1-14)

where k B is Boltzmann’s constant (8.62x10-5 eV/K), T is the temperature in degrees
Kelvin and D is the diffusion coefficient. Regardless of the method of transport, if we
assume that the jumps of the ion are uncorrelated and random, then the diffusion
coefficient is equal to

D = γ a o2 Γ

(1-15)

where γ is a geometric constant derived from the structure, a o is the jump distance and
Γ is the jump frequency. As the jumping of the atom necessarily involves some amount
of energy, we can give the jump frequency an Arrhenius-type temperature dependence:

Γ = ν o exp( − G ojump / k b T )

(1-16)

where ν o is the attempt frequency, and G ojump = H ojump − TS ojump is the jump activation
energy per ion. The exact value of ν o is difficult to determine from theory. It is
comparable to a phonon frequency and is often approximated by the Debye frequency.
The attempt frequency can be measured directly by experiments; usually by neutron
scattering, nuclear magnetic resonance, or light scattering techniques. The exponential in
(1-16) represents the probability that any given oscillation will cause a jump. Replacing
Γ in Eq. (1-15) with the right side of Eq. (1-16) gives

D = γ a o2ν o exp( − G ojump / k b T )

(1-17)

If we set Do = γao2ν o , then Eq. (1-17) becomes
D = Do exp(−G ojump / k bT )

(1-18)

Using Eqs. (1-18) and (1-14) we can now express the conductivity as

N (ez ) D
σi = i
k bT

(1-19)

 − G ojump 
N i (ez ) D o
σi =
exp 
k bT

(1-20)

 − G ojump 
N i (ez ) γ a o2ν o
σi =
exp 
k bT

(1-21)

with Do = γao2ν o .
From Eq. (1-9), we can solve for N i since
N i = N 1equil N o = N o exp(−Gd / RT )

(1-22)

where N o is the number of lattice sites of the mobile ion per molar volume of the crystal.
We can convert the exponential in Eq. (1-22) to the same units as the exponential of Eq.
(1-21) by substituting
R = kb N A

(1-23)

where N A is Avagadro’s number = 6.023x1023. Eq. (1-22) then becomes
 − Gdefect
 − Gd / N A 
 − Gd 
exp
exp
N i = N o exp
 kT 
 b A 
 b

(1-24)

10
The term Gdefect
represents the free energy of formation for one defect. So, Eq. (1-24)

simply states that the equilibrium concentration of defects will equal the number of
possible defects sites per unit volume times the probability (Boltzman) that a defect will
exist. Substituting the right side of Eq. (1-24) into Eq. (1-21) for N i gives
 − G defect
 − G ojump 
ez ) γ a o2ν o
 exp 
σ =
N exp 

k bT

 S defect
+ S ojump   − ( H defect
+ H ojump ) 
N o (ez) γao2ν o
 exp
σi =
exp
kbT
 

(1-25)

(1-26)

using the fact that G ojump = H ojump − TS ojump and G defect
= H defect
− TS defect

Eq. (1-26) can be further simplified to:

σi =

−Q 

exp 
 b 

(1-27)

where
 S defect
+ S ojump 
N o (ez ) γ a o2ν o
A=
exp
kb

(1-28)

and
Q = H defect
+ H ojump

(1-29)

The parameters A and Q will vary from material to material, but are independent of
temperature. Eq. (1-27) then relates the underlying structural and thermodynamic
properties of an ionic solid to its conduction. Experimentally, we find A and Q from an
Arrhenius plot of Eq. (1-27) in the form of ln(σ i T ) versus 1/T. Q is often called the

11
activation energy for migration: the energy required to both form ( H defect
) and move

( H ojump ) a defect to an adjacent lattice site. These two terms can be resolved by either
calculations of the electronic energy change due to defect formation/migration or by
experimentation.
For ionic solids, these experiments usually involve doping of the sample. If a
material is heavily doped, then there will exist some low temperature range where the
number of defects due to doping (Ndp), which is fixed, will be greater than the number of
intrinsic defects ( N vaequil , N icequil , and N vcequil ), which increases/decreases exponentially with
increasing/decreasing temperature. In this low temperature region, then, the concentration
of defects is independent of temperature and a plot of ln(σ i T ) versus 1/T will give a
slope proportional to Q = H jump . The conductivity of the material in this region is called

“extrinsic” as it depends on the dopant concentration and not the inherent properties of
the crystal.
For sufficiently high temperatures, the number of intrinsic defects will be much
greater than the number of defects due to doping. For this high temperature range, the
concentration of defects will vary with temperature according to Eqs. (1-10) and (1-12),
for Schottky and Frenkel defect mechanisms, respectively. The slope of ln(σ i T ) versus
+ H ojump , as the conductivity will be
1/T in this region will be proportional to Q = H defect

due to both defect formation and migration. Not surprisingly, the conductivity of this
region is labeled “intrinsic.” The change in slope going from the extrinsic to intrinsic
regions should then be equal to H defect

12
The only variable not yet evaluated in Eq. (1-27) is then S total
= S defect
+ S ojump . A

direct measurement of the entropy of ionic conduction is difficult and since it is
independent of temperature, it is often simply lumped into σ o and forgotten. However, it
and S ojump are due to changes in the vibrational
can be estimated from theory. Both S defect

spectrum when a defect is created or an ion jumps, respectively. We can calculate this
change by using the harmonic oscillator approximation of the vibrational partition
function13:

Zvib = ∑e

− (εi / kbT)

= ∑e

− [(i+1/ 2)hν / kbT]

e−1/ 2(hν / kbT)
= − hν / kbT
1−e

(1-32)

where the partition function is over all possible states of one harmonic oscillator. We can
then use the definition of the Helmholtz free energy, relative to that at absolute zero, to
get

F = −kbT ln Z vib = −kbT ∑ ln
−hν i / kbT 
1− e

(1-33)

where the summation is now over all frequencies of the crystal (i.e., system of harmonic
oscillators). We can substitute Eq. (1-33) into the thermodynamic equation

S = −(∂F / ∂T)V

(1-34)

and get

 
 hvi 

S = −kb ∑ ln
 −hν / k T 
−hν i / kbT 
 kbT  e i b − 1 
i  1− e
or for temperatures well above the Debye temperature, where k bT >> hvi ,

(1-35)

S = −kb ∑ ln(hvi / kbT )

(1-36)

The entropy change as the crystal is perturbed from its ideal structure is then:

S = −kb ∑ln(vii / vip )

(1-37)

where vii and vip are the vibrational modes of the ideal and perturbed crystal, and the
summation is over all vibrational modes of the crystal lattice. This summation is not
easily done, but by dividing the crystal into nearest neighbors, elastically stressed
neighbors, and the rest of the lattice, one can arrive at a good approximation.
It is possible then, at least in theory, to completely describe the ionic conductivity
in terms of the structural and thermodynamic properties of a material. This is done
experimentally most often by measuring not only the ionic conductivity of a material, but
also its diffusion coefficient or attempt frequency, so that the other parameters in Eq. (127) can be resolved. More generally, Eq. (1-27) is used to calculate the activation energy
of ion transport and the overall structural parameter. These are then compared to like
compounds and the method of ionic transport is inferred from known conduction
mechanisms.

1.3 Structural Features of Solid Acids
Before describing the mechanisms of proton transport, which differs quite
significantly from that of other ions, it is necessary to describe the structural features of
solid acids which influence proton conduction. Very generally, the structures of solid
acids are similar to that of other ionic solids in that the compounds are made up of two
lattices, one for cations and one for anions. However, the incorporation of the acid proton
leads to the fundamental structural difference between solid acids and their analogous
salts: the presence of hydrogen bonds. These hydrogen bonds link the anions together and
the conduction of protons will be greatly effected by both the types of and particular
arrangement of the hydrogen bonds found in a solid acid. Therefore, to properly explain
protonic conduction in solid acids, it is necessary to first describe the types of hydrogen
bonds and hydrogen-bonded networks that are found in them.

1.3.1 Characterization of Hydrogen Bonds
A hydrogen bond is said to exist if two electronegative species X and Y are
connected to each other through bonds to a hydrogen atom, H. Usually, one bond will be
stronger, written X―H, and is called the normal X―H bond while the weaker bond,
written H···Y, is termed the hydrogen bond. The X and Y atoms are termed the donor and
acceptor atoms, respectively. The dissociation energy of the X―H···Y complex is equal
to the strength of the H···Y bond or the hydrogen bond strength14. Hydrogen bond
strengths run in the range of 2 to 15 kcal/mole, which is significantly greater than other
intermolecular forces (e.g., van der Waals forces with energies in the range of 0.1 to 2

15
kcal/mole for smaller molecules), but much less than intra-molecular covalent bonds (30
to 230 kcal/mole)15.
This bond energy is not entirely due to the electrostatic attraction between the
electronegative atoms and the hydrogen, but also involves a certain amount of covalent
character arising from the overlap of lone pair electrons from the Y acceptor atom with
those of X―H bond. As the hydrogen-acceptor distance decreases, the amount of
covalency increases and so does the hydrogen bond strength16. This correlation between
bond strengths and bond distances leads to hydrogen bonds being very generally labeled
as strong, medium, or weak depending upon their donor-acceptor and donor-hydrogen
lengths. As shown in Table 1.1 for X,Y = O, the hydrogens of strong hydrogen bonds are
more equally shared between the donor and acceptor oxygens (Od and Oa, respectively),
resulting in smaller Od···Oa and bigger Od―H distances when compared to weak
bonds17,18.
Table 1.1 Correlation between hydrogen bond
strength, Od··· Oa, and Od—H distances14,17,18
Bond Strength

dOd··· Oa(Å)

dOd—H(Å)

Strong

2.4 to 2.6

1.3 to 1.0

Medium

2.6 to 2.7

1.02 to 0.97

Weak

2.7 to ~3

Below 1.0

Which strength hydrogen bond favors protonic conductivity depends on a
material’s mechanism of proton transport. If protons are transported on a mobile species,
then strong hydrogen bonds would decrease the mobility of the carrier. Hence, weak

16
hydrogen bonds are preferable for such a mechanism. However, if the mechanism of
proton transport requires both the translation of protons along hydrogen bonds and the
breaking of hydrogen bonds, then medium strength hydrogen bonds are preferable. This
is due to the trade-off between the energy required for proton translation, which decreases
with increasing hydrogen bond strength (decreasing Od···Oa distance), and the energy
required to break the bond, which by definition decreases with decreasing hydrogen bond
strength (increasing Od···Oa distance)19.
The local crystallographic symmetry also effects proton conduction as the
geometry of a hydrogen bond is partially determined by its site symmetry. This can be
seen in Figure 1.1 for an O―H···O bond. For two oxygen atoms related by a center of
symmetry and bound by a strong hydrogen bond, the potential energy of the proton will
have a single minimum exactly between the two oxygens; the dual nature of each oxygen
is signified by the label Oa/d. If the oxygen atoms are not related by symmetry, the
hydrogen will reside slightly closer to one oxygen, but the hydrogen will be strongly
bound near the center of the Od···Oa complex.
In hydrogen bonds with medium Od···Oa distances, two minima exist in the proton
potential and the presence or absence of local symmetry influences the relative
population of each minima by the proton. For the symmetric case, the proton will be
found with equal probability in either minima and therefore can be considered “locally
disordered” as the proton will hop between the two sites17. When the potential well of the
proton is asymmetric, the proton will preferentially occupy one minimum over the other.
However, the proton will still occasionally hop to the second minimum, thus dramatically
increasing its intra-hydrogen bond mobility.

Hydrogen Bonding
bond type

strong

medium

weak

symmetric
Energy

~ 2.4Å

~1.2Å

~1.3Å

not generally observed

Energy

~2.0Å

~ 2.6Å

1 or 2

asymmetric
Energy

~1.3Å
~ 2.4Å

~1.3Å

Energy

~ 2.6Å

~2.2Å

~1.0Å

~1.8Å

~1.2Å

> 2.9Å

Figure 1.1 Effects of symmetry on strong, medium and weak hydrogen bonds18.

Weak hydrogen bonds are almost always asymmetric, with only one minimum in
potential energy of the proton. Moreover, as stated earlier, a weak hydrogen bond
requires large energies to transfer the proton to the other side of the potential well (~2 to
10 times that of medium and strong hydrogen bonds, respectively)19. Consequently, the
local symmetry of the hydrogen bond has a much greater effect on the proton transport
properties of medium strength hydrogen bonds than it does on those of weak or very
strong hydrogen bonds. The hydrogen bonds found in solid acids are ~2.5 to 2.7 Å20 and
are thus, for both symmetric and asymmetric examples, of the moderately strong to
medium strength type of hydrogen bonds. Hydrogen bond energies associated with O···O
distances of 2.5 to 2.7 Å are ~14 to 6 kcal/mole, respectively21.

1.3.2 Hydrogen-Bonded Networks in Solid Acids
Very generally, hydrogen bonds can link together molecules into structures of 0,
1, 2, 3 dimensions. Some simple hydrogen-bonded networks found in solid acids are
shown in Figure1.2. More complicated networks involve tetrahedra linking in such a way
as to give three-dimensional structures (e.g., branched chains or layers) or true 3-D
networks that run through out the crystal. To a certain degree, the type of hydrogenbonded network found in a solid acid is predetermined by the average number of
hydrogen-bonded oxygens per XO4 tetrahedra. This number is itself a function of the
compound’s H:XO4 ratio. Table 1.2 shows the hydrogen-bonded structures for an average
of 1, 2, 3, and 4 bonded oxygens per XO4 group. The example structures in this table
were selected from solid acids of interest to this work.

Cyclic Dimers

Linear Dimers

Rings

Chains

Layers

Figure 1.2 Simple hydrogen-bonded networks found in solid acids.
Hydrogen Bonds are indicated by
lines.

Table 1.2. Hydrogen-bonded networks in solid acids by their H:XO4 ratio22
Type of
Examples from MHxXO4 and
H:XO4 ratio Number of oxygens
involved in hydrogen hydrogen-bonded M3H(XO4)2 compounds bonds per XO4
{ref}
network
1:2
Dimer
K3H(SO4)2 - 23
1:1
Cyclic dimer
KHSO4 [2] - 24
Rings
Cs2Na(HSO4)3 - 25
Chains
CsHSO4 - 26
3:2
Layers
Cs2HSO4H2PO4 - 27
2:1
Layers
CsH2PO4 - 28
3-dimensional
KH2PO4 - 29

1.3.3 Common Structures Found in the Low Temperature Phases of
Solid Acids
The zigzag chains of Figure 1.2 are a very common feature in solid acids. Straight
chains are also observed, but more often than not a chain of tetrahedra will zig and zag as
will the surrounding cations to a give a “checker-board” appearance to the arrangement
of anions and cations30. This arrangement of atoms is shown in Figure 1.3 for the stable
room temperature structure of CsHSO4 (phase II)26.

r r
Figure 1.3 Room temperature structures for CsHSO4 and CsH2PO4: a) the b x c plane of
CsHSO4 showing zigzag chains of sulfate tetrahedra parallel to c and b) view down the
c-axis revealing the checkerboard arrangement of cation and anion chains The rectangles
represent unit cells.

21
These zigzag chains are often found in solid acids with higher dimensional
networks where they are cross-linked by other hydrogen bonds to form planes and threedimensional hydrogen-bonded networks 31,31,32. The zigzag chains found in CsH2PO4 are
cross-linked to form planes of hydrogen-bonded tetrahedra perpendicular to the [001]
direction32. In KH2PO4, each tetrahedron is connected to two zigzag chains at right angles
to each other to form a hydrogen-bonded network of tetrahedra that runs throughout the
structure, Figure1.4.

Figure 1.4 Tetragonal structure of KH2PO4 projected down a) the [001] and b) the [100]
directions. Chains of hydrogen-bonded PO4 tetrahedra extend along the [100] and [010]
directions in a) and the zigzag nature of these chains can be seen in b). Circles represent
K atoms and tetrahedra PO4 groups.
Another common structural type is found in solid acids belonging to the
M3H(XO4)2 family of compounds (M=Cs, Rb, NH4, K, Na; X=S, Se). In the ordered
(room temperature) phases of these compounds, the tetrahedra are hydrogen-bonded
together into dimers. These dimers and their companion cations are arranged in such a
way as to form planes with almost trigonal symmetry perpendicular to the [001]

22
direction, the true symmetry being monoclinic, space group A2/a. For the compound
K3H(SO4)2, this arrangement of sulfate dimers and K+ ions is shown in Figure 1.532.

Figure 1.5 Structure of monoclinic K3H(SO4)2 projected down a) the c*-axis and b)
down the b-axis.

1.4 Protonic Conduction

1.4.1 Mechanisms of Proton Transport
Although the general concepts of ionic conduction apply to protonic conduction,
there is a fundamental difference between the two due to the fact that H+ is the only ion
with no core shell of electrons. It must therefore be solvated by the electrons of another
atom or atoms33,34. For nonmetallic materials, and in particular ionic solids, the proton
will be coordinated by only one or two atoms35. Due to the positive charge on the proton,
the coordinating atom is usually the most electronegative atom around: F,O, N and
sometimes Cl and S15,34.

23
For most protonic conductors, the coordinating atom(s) is an oxygen. If the
oxygen is well separated from other oxygen atoms, the proton-oxygen pair forms an O-H
bond ~ 1 Å in length. As the distance to other electronegative species lessens, a hydrogen
bond, O-H···O, will be formed, with O···O distances in the range of 2.4-3 Å long20. As the
proton can never be free from the electron density of its neighbors, it must move by a
method where it is bonded to at least one atom during the entire process36. This restraint
leads directly to the two main methods of proton conduction: the vehicle and Grotthus
mechanisms2,37.
In the vehicle mechanism, the proton is attached to a mobile species (e.g.,
H2O+H+⇒H3O+). Protonic conductivity is then achieved by the diffusion of the vehicle
and counter-diffusion of unprotonated vehicles (here, H2O), as shown in Figure 1.6.
Clearly, in this mechanism the diffusion rate of the vehicle will determine the overall
conductivity of the proton38. This mechanism is responsible for the protonic conductivity
in oxonium β-alumina, hydrogen uranyl phosphate, and hydrated acidic polymers (e.g.,
NAFION)37,39.

H+

Figure 1.6 Vehicle mechanism of proton transport. Protons are carried to the left while
empty vehicles travel to the right37.
In contrast, the Grotthuss mechanism has the chemical species to which the
proton is attached remain translationally stationary on the timescale of proton transport.
By transfer of the proton within a hydrogen bond and subsequent structural relaxation
(i.e., a structural or dipole reorientation of the new carrier), the proton can diffuse through
the material40. This process requires that the protonic carriers have significant local
dynamics. The relevant rates for this mechanism are then that of proton transfer and
structural relaxation. Some materials that conduct protons by the Grotthuss process are
ice37, concentrated aqueous solutions and hydrates of acids (e.g., H3PO4, H2SO4, HCl,
etc.) 41,42 43, fused phosphoric acid44, the solid proton conductor HClO445, and solid acids
in both their low and high temperature phases 35,46,47. A schematic description of the
Grotthuss mechanism for ice is shown in Figure 1.7.

H+

Figure 1.7 Grothhuss mechanism of proton transport. Proton jumps to an adjacent
vehicle, a), which then reorients, b), to form a new hydrogen bond, c).
The ideal structure of normal (hexagonal) ice, first described by Bernal and
Fowler, has each oxygen atom tetrahedrally coordinated by four other oxygens at a
distance of 2.76 Å. Associated with each oxygen will be exactly two protons. Each
proton will form a hydrogen bond with OH and OH···O distances of 0.95 and 1.81 Å,
respectively, resulting in each oxygen being involved in four hydrogen bonds48.
According to the prevailing theory, there are two pairs of defects responsible for protonic
conduction in ice. The first pair is created by reorientation of the water molecule, which
causes doubly occupied and empty hydrogen bond sites: D and L defects, respectively49.
This reorientation cannot be definitively labeled as a structural reorientation (with the
molecule rotating around an axis of symmetry) or a dipole orientation (with the proton
hopping from one site to another). Thermodynamically, both mechanisms must be
present to some degree, but which one dominates the structural relaxation involved in

26
ice’s protonic conduction is difficult to determine. Intra-hydrogen bond translation of the
proton results in the formation of the second defect pair: the hydroxyl (OH−) and
hydronium ions (H3O+)50. Note that both steps require the protons to move from one to
another crystallographic proton site.
It is necessary for both types of defect pairs to exist for true translation of a proton
as each pair, alone, moves the protons in only a coordinated way, leaving the hydrogenbonded system “polarized” in the direction of proton transport. By traveling along the
same hydrogen-bonded system, the alternative pair can “unpolarize” this chain of
hydrogen bonds. In particular, a D defect traveling in the same direction as a hydronium
ion (and vice versa) will “unpolarize” the hydrogen-bonded system, as shown in Figure
1.8 c, d, and e. Similarly, an L defect following a hydroxyl ion (and vice versa) will allow
for a continuation of proton migration in the same direction37. Proton conduction in ice
then requires both proton transfer along hydrogen bonds and a reorientation of the proton
carriers, and hence occurs by the Grotthuss mechanism.

d)
L defect

D defect

e)
Figure 1.8 Representation of the Grotthuss mechanism in ice. Intra-hydrogen bond
transfer of the proton, a), leads to the formation of hydroxyl (OH-) and hydronium
(H3O+) ions, b). Reorientation of a water molecule, c), results in an L/D defect pair, d),
with further reorientations removing the defects and leaving the chain able to continue
proton conduction to the right, e).
For completeness, it should be mentioned that some materials exhibit mixed
vehicle and Grotthuss mechanisms of proton transport. This occurs when there is both a
high mobility for the proton carriers and a significant amount of proton transference
between carriers. Dilute aqueous solutions of acids and bases and solid acid hydrates with
high water content have mixed mechanisms of proton transport39. However, in general,

28
these two mechanisms operate exclusively of each other. The presence of continuous
hydrogen-bonded pathways is essential to proton conduction via the Grotthuss
mechanism, but an extensive hydrogen-bonded network hinders the translation of mobile
species necessary to the vehicle mechanism51.

1.4.2 Room Temperature Proton Conduction in Solid Acids
Proton conduction in solid acids is similar to that in ice for the low temperature,
low symmetry, and (for the most part) ordered phases of solid acids that exist near room
temperatures. Migration of protons again requires both transfer of the protons along
hydrogen bonds and reorientations of the tetrahedral anions. As with ice, it is unclear as
to whether the necessary structural reorientations occur by a physical rotation of the
tetrahedra or by proton hopping leading to dipole reorientations. The transfer of protons
along hydrogen bonds will result in the formation of tetrahedra that are negatively and
positively charged when compared to average charge on the tetrahedra (e.g., 2(HSO4−) →
SO4−2 + H2SO4).
For solid acids with all tetrahedral oxygen atoms involved in hydrogen bonds, the
formation of D and L defects would seem essential for proton conduction and the
mechanism of proton conduction to be nearly identical to that found in ice. Indeed,
models proposed by Murphy52, O’Keeffe53 and Pollock54 relate the intrinsic conductivity
to the formation of D and L defects in KH2PO4 type solid acids. In KH2PO4, all oxygens
are involved in crystallographically symmetric hydrogen bonds (with O···O distances of
2.491 Å), and the proton resides in a symmetric double minimum potential well55.

29
Evidence of this symmetric double minimum over a single minimum is found in
KH2PO4’s low temperature ferroelectric properties56. The distance between the minima is
~ 0.37 Å, a distance much shorter than the van der Waals radius for hydrogen (r =
1.2 Å)15. D defects will therefore have a large electrostatic repulsive energy associated
with them1. To avoid the formation of such high energy defects, a model put forth by
Sharon57 involves the synchronous reorientations of multiple tetrahedra. Disadvantages to
this model are that it requires the breaking of multiple hydrogen bonds as well as the
coordinated rotation of multiple tetrahedra57.
All the above models require the proton to move from one normal
(crystallographic) site to another normal site; similar to the mechanism proposed for ice.
In contrast, Baranov suggests a mechanism of proton conduction where the protons hop
between normal and interstitial (i.e., non-crystallographic) sites, similar to a Frenkel
defect mechanism2. This mechanism requires neither the formation of D defects nor
rotations of the tetrahedra, since the protons jump between the normal and interstitial
sites without aid of tetrahedral reorientations 2. The necessary structural relaxation for a
Grotthuss proton conduction mechanism is accomplished by dipole reorientations in this
model. Intra-hydrogen bond transfer of the protons will still lead to tetrahedral
equivalents of hydroxyl and hydronium ions, but the equivalent of the D, L defect pair for
this model is a proton vacancy (L defect) and interstitial pair. The free energy of
formation of the proton vacancy/interstitial pair will be lowest for interstitial sites that
reform a hydrogen bond. Proton conduction is then possible by the migration of protons
within and between the normal hydrogen-bonded network and the instantaneous network
of interstitial hydrogen bonds.

30
In KH2PO4, Baranov suggests probable interstitial sites with O(1)···O(2') distances
of 3.16 Å, which could form a weak to medium strength hydrogen bond after structural
relaxation 46. A representation of this interstitial site is shown in Figure 1.9 a. An
equivalent interstitial site in ice is not found as the next-nearest neighbor oxygens for
each oxygen atom are ~ 4.5 Å distant, too far away to form a hydrogen bond58. Baranov
states this mechanism of proton conduction seems even more likely for solid acids with
oxygens that are not structurally involved in hydrogen bonds, such oxygens acting as
“built-in” interstitial sites. The direction(s) of the interstitial hydrogen bond(s) are then
determined by finding the nearest oxygen atoms. In the M3H(XO4)2 compounds, the
pseudo-trigonal symmetry of the room temperature phases results in two interstitial sites
per tetrahedron and is schematically depicted in Figure 1.9 b46. For a solid acid
containing infinite chains of hydrogen-bonded tetrahedra similar to those found in
CsHSO426, a vacancy/interstitial proton conduction mechanism is shown in Figure 1.10.
The ideal structure of CsHSO4-II has O(3) and O(4) atoms not involved in hydrogen
bonds. An interstitial hydrogen bond between these two oxygens is proposed by Baranov
as the O(3)·· O(4) distance is only 3.2 Å46.

Figure 1.9 Normal and interstitial hydrogen bonds proposed for room temperature
phases of a) KH2PO4 and b) the M3H(XO4)2 class of compounds. Solid and dashed lines
denote normal and interstitial hydrogen bonds, respectively.

Figure 1.10 Possible conduction paths for proton vacancy/interstitial defects along and
in between hydrogen-bonded zigzag chains of anion tetrahedra. Notice the formation of
potential water molecules (i.e., an oxygen with two hydrogen bonds) is not a necessity for
proton transport.

32
Although there is agreement in the literature that proton conduction occurs by the
Grotthuss mechanism in the low temperature phases of solid acids, in the end there is no
definitive proof of the particulars of the mechanism. Hence, there is a question as to
which pair is formed: D and L or vacancy/interstitial defect pairs. Similarly, it has not
been determined if the structural relaxation necessary for protonic conduction in these
low temperature phases occurs by actual rotations of the tetrahedra or simply by dipole
reorientations resulting from proton jumps.

1.4.3

High Temperature Proton Conduction in Solid Acids
For the high temperature superprotonic phases of solid acids, the mechanism of

proton conduction is not in dispute. A highly disordered state leads to fast local dynamics
of the anion tetrahedra and subsequent proton translation via the Grothhuss mechanism35.
It has been determined that the tetrahedra are librating much faster (1011 Hz) than protons
are being transferred (109 Hz) which indicates that the structural relaxation essential to
the Grotthuss mechanism is due to the physical reorientations of the tetrahedra in these
phases59,60. The increase in symmetry across the phase transition (typically monoclinic →
rhombohedral, tetragonal, or cubic) results in disorder on the oxygen sites, which are then
free to vibrate and librate between crystallographically identical positions. This nearly
free rotation of the tetrahedra creates many more crystallographically equivalent proton
sites than there are protons, resulting in a “dynamically” disordered hydrogen-bonded
network9.

33
In contrast to protonic conduction in the room temperature phases of solid acids, it
is then possible for proton conduction through only “normal,” crystallographic proton
sites. The combination of fast tetrahedral dynamics and proton translations along
hydrogen bonds of a disordered network results in high protonic conductivity.
Superprotonic conduction is therefore a result of the ideal structure rather than intrinsic
defects60. In terms of Equation 1-1, this superprotonic conductivity is a product of the
increase in the proton’s mobility and the increase in the number of mobile protons (all of
them).
The structure proposed by Jirak for CsHSO4 in its superprotonic phase is given in
Figure 1.1161. It should be mentioned that there is some disagreement in the literature
over the exact position of the oxygen atoms, and hence the protons. This structure was
chosen as it gives the most realistic arrangement and length to the hydrogen bonds, as
well as its overall fit to experimental data (to be discussed in Section 4.6.1).

H+
SO4

Figure 1.11 Tetragonal structure of CsHSO4 above its superprotonic phase transition
projected along the [100], c), and [010], d), directions. Two orientations of the tetrahedra
result in partially occupied proton sites and a disordered network of hydrogen bonds
(dashed lines).

The room temperature phase of CsHSO4-II is monoclinic, space group P21/c,
comprising zigzag chains of hydrogen bonded SO4 tetrahedra alternating with zigzag
rows of cesium atoms (Figure1.3). There are four crystallographically distinct oxygens,
two of which are involved in asymetric hydrogen bonds with O(1)···O(2) distances of
2.63 Å26. On the contrary, after transforming to the superprotonic tetragonal phase (space
group I41/amd), the oxygens become crystallographically identical and all oxygens
participate in hydrogen bonds. There are two possible orientations of the tetrahedra,
resulting in ½ and ¼ occupancy of the oxygen and proton sites, respectively. Hydrogen
bonds of average length 2.78 Å connect the oxygens61. Other proposed structures have a

35
different number of tetrahedral orientations, hydrogen bond lengths and hydrogen bond
orientations. However, regardless of the exact configuration of oxygens and protons in
the superprotonic phase of CsHSO4, the method of proton conduction remains the same:
rapid reorientations of the SO4 group forming a dynamically disordered network of
hydrogen bonds through which protons can jump from one tetrahedron to the next.
This mechanism of proton transport is responsible for the high conductivity in all
superprotonic phases of solid acids, with any differences between their conductive
processes attributed largely to the relative symmetry of the specific material. For
example, CsHSO4, being tetragonal, shows a small anisotropy in its conductivity parallel
and perpendicular to its 4-fold axis62. In contrast, the compound Cs2(HSO4)(H2PO4),
which transforms to a cubic structure (space group Pm 3 m) exhibits isotropic
conductivity in the superprotonic phase63. Nevertheless, on a very local scale, the process
of proton transfer and reorientation is considered to be very similar in all superprontic
phases and conclusions reached for one compound should apply at least to structurally
related compounds, if not to the whole class of solid acids.

Chapter 2. Experimental Methods
2.1 Synthesis
The solid acids analyzed in these studies were all grown by slow evaporation of
an aqueous solution containing high purity metal carbonates and the appropriate mineral
acids:
− H 2O

M2CO3 + HnXO4 + H2O 
→ → single crystals
where M = Cs, Rb, NH4, K, Na, Li and X = S, P. Most crystals were grown at room
temperatures, but some compounds were found to grow only at elevated/lowered
temperatures. The compounds discussed in this work are primarily mixed cation sulfates
(Chapter 3) and cation sulfate-phosphates (Chapter 4). First attempts at their synthesis
were carried out in solutions with total metal to anion (M:XO4) ratios of 1:1. Therefore,
unless otherwise noted, it is safe to assume a compound was synthesized at ~25°C with a
solution M:XO4 ratio of 1:1.
After the formation of crystal samples, they were collected by either removing
individual crystals directly from solution, or by filtration over a porous ceramic (since the
solutions are still quite acidic and would eat through normal filters). If necessary, the
samples were washed with acetone or isopropanol to remove any excess solution clinging
to the crystals. Deliquescent compounds were placed in desiccated containers, while most
other samples were stored in ambient conditions.
For large quantities of a desired phase or to force the synthesis of a compound not
found to grow by the above method, organic solvents were used to precipitate powder
samples. The most common solvents used were acetone, methanol, and isopropanol. The

37
powders were filtered from solution and washed with the precipitating liquid on ceramic
filters. Powder samples were stored in sealed containers to limit surface water absorption.

2.2 X-ray Diffraction
X-ray diffraction methods were almost exclusively used to identify the phases of
crystals grown as above. For the most part, the diffraction measurements were performed
on single crystal samples so as to provide very accurate phase determinations, single
crystal samples for other measurements, and the possibility of orienting the samples. If it
was necessary to analyze the abundance of different phases grown from the same
solution, a random sampling of crystals was finely ground together and a powder X-ray
diffraction (PXD) measurement taken.
Single crystal X-ray diffraction (SCXD) measurements also provided the data for
determining the structures of any novel compounds. SCXD samples were cut from single
crystals and shaped into rough cubes, on average ~ 0.15 mm a side. The small crystallites
were then attached to the top of a thin glass fiber by a common two-part epoxy and the
glass fiber mounted in a cylindrical brass holder. The brass holder was then placed in a
goniometer and the cube aligned in the center of the X-ray beam. Diffraction intensity
data for the aligned samples were obtained on a Syntex four-circle diffractometer using
Mo Kα radiation (λ = 0.71073 Å). Decay and absorption corrections were applied as
necessary and structural refinements performed on the resulting F2 data for the collected
reflections. The SHELXS86 and SHELXL93 (or SHELXL97) programs were used for
structure solution and refinement, respectively64,65. Visual inspection and depiction of the
structures were accomplished with the ATOMS program66.

38
Unfortunately, the relatively large (~ 1 to 2 %) volume changes typical of
superprotonic transitions causes single crystal samples to turn polycrystalline and become
useless in SCXD measurements. Hence, the high temperature structures were determined
from PXD measurements taken above the phase transition temperature. Also, from
Reitveld refinements of PXD patterns taken at elevated temperatures the thermal
expansion coeffiecients for both low and high temperature phases were measured
allowing for accurate calculations of the transition volume changes. The program Rietica
was used in such refinements67. PXD measurements were also used to confirm the phase
purity of solvent precipitated samples. Calculated patterns were generated from published
data using the Micro-Powd program and then compared to the measured PXD patterns
with the program JADE68,69. Unless otherwise stated, the PXD measurements reported in
this work were taken on a Siemens D500 diffractometer with Cu Kα radiation (λ =
1.5418 Å).

2.3 Neutron Diffraction
Neutron diffraction on both single crystal and powder samples was performed to
take advantage of its sensitivity to light elements and atoms with similar atomic numbers.
Due to the difficulty in taking such measurements, a nuclear reactor or scintillation
source being required, neutron diffraction measurements were only taken when analysis
of the X-ray diffraction data failed to definitively resolve a crystal’s structure. For
compounds in this work, any ambiguity in their structures usually resulted from the
inability to accurately locate H/D atoms or to differentiate between sulfate and phosphate
groups. Both problems are a direct result of the fact that the scattering lengths for X-rays

39
increases monotonically with atomic number. Hence, H/D atoms are fairly transparent to
X-rays, while SO4 and PO4 groups will scatter X-rays almost identically70.
Conversely, in neutron diffraction H and D atoms are easy to discern as their
coherent scattering lengths are -3.739 and 6.671 fm, respectively, making both atoms
strong neutron scatterers71. Also, the scattering lengths for S and P are quite different for
neutrons (5.13 and 2.847 fm, respectively), and it is usually straightforward to distinguish
between the two atoms or even determine their individual occupancies on a mixed S/P
site72. In particular, neutron diffraction was used in this work to resolve the H/D and O
positions for atoms involved in disordered hydrogen bonds in the otherwise ordered room
temperature phases. For the superprotonic phases, the H/D and O positions resulting from
the fast reorientations of the tetrahedra were also investigated, along with the possible
existence of superstructures do to potential ordering of the tetrahedra in mixed sulfatephosphate compounds. Measurements used thermal neutrons with wavelengths ~ 1 Å
generated from both reactor and spallation sources.

2.4 Thermal Analysis
The behavior of compounds with increasing temperature was probed by two main
techniques: differential scanning calorimetry (DSC) and thermal gravimetric analysis
(TGA). The presence and characterization of phase transitions both above and below
room temperature were accomplished by DSC measurements. A compound’s response to
heating was examined with a Perkin-Elmer DSC 7 calorimeter in a flowing nitrogen
environment. The most common heating rates were 5, 10, and 20°C/min. For low
temperature measurements, an in house apparatus was used which essentially consisted of
a Perkin-Elmer DSC 7 calorimeter immersed in a helium environment that had been cool

40
by liquid nitrogen. A sample was therefore cooled to ~ -150°C (the limit for LN2) and
heated, at rates varying from 1-5°C, in 30°C intervals to limit instrument drift.
The onset of decomposition in a sample was probed by a Perkin-Elmer TGA 7 or
Neztsch STA 449 analyzer under flowing nitrogen and argon, respectively. Again, the
most common heating rates were 5, 10, or 20°C/min. The Neztsch system can
simultaneously take DSC and TGA data, but for consistency’s sake all DSC
measurements were taken on the Perkin-Elmer machines.

2.5 Chemical Analysis
The compositions of any new compounds were measured using a JEOL JXA-733
electron microprobe. Single crystal samples were mounted in an epoxy resin, polished
and then coated in carbon by evaporation. The polishing of single crystal samples was not
trivial as the compounds are water soluble. For some compounds, even best attempts at
polishing still resulted in poor quality surfaces for microprobe measurements. Hence,
pressed powder pellet samples (from ground up single crystals) were also analyzed, with
the advantage that the surfaces of the pellets were already flat and needed only to be
carbon coated. Microprobe data were taken at a minimum of seven points on a sample for
statistical averaging. High quality samples (single crystals or pellets) of compounds with
a like, but known, nature were used as standards. Measured X-ray peak intensities were
converted to elemental weight percentages using the CITZAF program73. For most
compounds (new/known solid acids and single crystal/pellet samples alike), visible beam
damage was observed during data collection. This damage is most likely the dehydration
of the surface when excited by the electrons in the beam. For this reason, larger rather
than smaller areas were scanned in the measurements.

2.6 Optical Spectroscopy/Microscopy
Superprotonic phase transitions of some compounds were further investigated by
infrared (IR) spectroscopy and polarized light microscopy. The vibrational spectrum of
compounds pressed into optically transparent KBr pellets (sample:KBr mass ratio of
1:300) were measured on a Nicolet Magna 860 FTIR spectrometer in flowing nitrogen.
An in house heating stage was employed to heat the pellets and observe the changes in
their spectrums with temperature. Most attention was given to the changes with
temperature in the bending and stretching modes of the tetrahedrons (in the range of 450
to 1100 cm-1) as well as the OH stretching modes within the O-H···O bonds. These modes
show up as three broad peaks ~ 1700, 2400, and 2800 cm-1 and are often referred to as the
“ABC bands” of hydrogen bonds14. From the behavior of the tetrahedral modes, the
increase in symmetry associated with almost all superprotonic transitions could be
observed, hopefully validating the assigned symmetry taken from PXD data.
Observations of the ABC bands not only confirmed the presence of hydrogen bonds, but
also revealed the general effect of a phase transition on these bonds.
Polarized light microscopy was most often used to judge the quality of single
crystal samples. This was accomplished by observing the sample under extension
conditions on a transmission Leica DMLB microscope. For a single domain crystal
without inclusions or attached crystallites the perceived image should be homogeneous.
By attaching a single crystal sample to a heating stage, high temperature transitions could
be observed. In most cases, single crystal samples would become completely opaque
above a superprotonic transition due to the high symmetry of the phases. This technique

42
was also an easy way to determine if single crystal samples would turn polycrystalline or
not when undergoing a transition, which influenced other measurements.

2.7 NMR Spectroscopy
Pulsed Fourier transform H+ NMR measurements were performed on a finely
ground sample to characterize the proton environment of the compound. Specifically, the
number of crystallographically distinct hydrogen atoms and their relative amounts were
investigated. Also, the percent deuteration of a compound was accurately measured by
taking the ratio of the integrated intensities of deuterated and fully protonated samples.
All measurements were taken on either a Bruker DSX 500 MHz or a Bruker AM 300
MHz NMR spectrometer. The chemical shifts of the samples were referenced to
tetramethylsilane (TMS). Magic angle spinning (MAS) was employed to reduce the
proton-proton dipole broaden of the signal lines resulting from the local interactions of a
proton’s magnetic moment with the dipole fields generated its neighbors.
For most measurements, a 12 kHz spinning rate was used in conjunction with a 4
µs, 90° pulse. The spin-lattice relaxation time, T1, was on the order of 1000 s for all
compounds measured, revealing that the excited H+ nuclei in these solid acids interact
weakly with their surrounding lattices. The observed chemical shifts for the
crystallographic protons were ~ 10-12 ppm, typical values for protons residing in
medium strength hydrogen bonds74. There was often a very sharp peak seen at ~ 6 ppm
that was attributed to absorbed water based on its disappearance with heating and a
comparison to measurements on calcium phosphates, where similar peaks were observed
and assigned to surface water75.

2.8 Impedance Spectroscopy
The conductivity of a compound was measured by a.c. impedance spectroscopy
using a 4284 LCR (inductance-capacitance-resistance) meter. Conductivity
measurements were taken on cut and polished single crystal samples along known
crystallographic directions (as determined by SCXD methods), while polycrystalline
samples were made from finely ground single crystals that had been uni-axially pressed
into pellets. Silver paint (Ted Pella cat. no. 16032) served as the electrode material.
Samples were prepared so as to have a large area to length ratio (A/L) with respect to the
direction of the applied field. Such a geometric ratio is desirable as it decreases a
sample’s effective resistance and gives better signal resolution. Measurements were made
over the frequency range of 20 Hz to 1 MHz with an applied voltage of 1 V under either
inert (dry argon or nitrogen) or ambient atmospheres. Heating and cooling rates were
0.5°C/min (unless otherwise noted). For most samples, the impedance spectra exhibited a
single arc in the Nyquist representation. The effective d.c. resistivity, ρ, was determined
by fitting such an arc to an equivalent (RQ) circuit using the least squares refinement
program EQUIVCRT76. The effective resistivities (ρ = R) were then converted into
geometry independent conductivities, σ = 1/(ρ*A/L), and plotted in an Arrhenius form to
facilitate the extraction of informative parameters from the data (see section 1.2). Since
this impedance spectroscopy method is probably the least well known technique used in
this work, its basic theory will be described below.

2.8.1 Complex Impedance (from ref 77-79)
The simplest model for an electrode-sample system under an applied voltage is a
capacitor and resistor in parallel, Figure 2.1 a. The capacitor is a result of the sample’s
geometry, while the resistor represents the resistivity of the bulk. For such a circuit, the
response to an applied voltage,

V (t ) = Vo eiω t

(2-1)

will be a current in the resistor,

iω t
V e
V (t )
I = o

(2-2)

and a current in the capacitor,

dQ(t )
dt

d (CV (t ))
dt

iω t 
iω t
 dV e
=C
= iω CVo e
= iω CV (t )
dt

(2-3)

The total current in the circuit is then

total

V (t )

+ iω CV (t )

(2-4)

Exactly like the conventional impedance, Z, the complex impedance is defined as the
ratio between the voltage and current, which is here:

V (t )
V (t )
+ iω CV (t )

+ iω C

The impedance can be separated into its real, Z′, and imaginary, Z′′, parts to give

(2-5)

1
  + (ω C )2
R

−i

ωC

1
  + (ω C )2
R

= Z ′ − iZ ′′

(2-6)

A plot of Z′ vs. -Z′′ (as parametric functions of ω) will result in a semicircle of radius R/2
in the first quadrant, Figure 2.1 b. The time constant of this simple circuit is defined as

τ o = RC =

ωo

(2-7)

and corresponds to the characteristic (dielectric) relaxation time of the sample.
Substituting ω from Eq. (2-7) into Eq. (2-6) gives Z′ = R/2, Z′′ = R/2, so that the

characteristic frequency lies at the peak of the semi-circle. A plot of Z′ vs. -Z′′ is often
called a Nyquist plot.

Figure 2.1 Equivalent circuit for a dielectric material between two electrodes, a): Rb and
Cb represent the bulk resistance and capacitance, respectively. This circuit gives a semicircle in the complex impedance plot of Z′ vs. -Z′′. The frequency increase from right to
left and the characteristic frequency of the electrode-material system lies at the peak of
the semicircle.

46
One of the major advantages of complex impedance spectroscopy over single
frequency or DC techniques is its ability to resolve the electrode, bulk, and grain
boundary (for polycrystalline samples) contributions to the resistance. In an ideal sample,
the impedance plot would show three semicircles and would be modeled as three (RC)
circuits in series, Figure 2.2. This type of impedance plot is often seen for pure ceramics
such as ZrO2, but almost never seen for the proton conducting compounds of this work.
Instead, only one (single crystal samples) or two semi-circles (polycrystalline samples)
were usually present at low temperatures, representing the bulk and grain boundary
responses to the applied voltage. This type of impedance plot was modeled by two (RC)
circuits in series, i.e., the first two circuits in Figure 2.2 a.

Figure 2.2 Separation of bulk, grain boundary, and electrode resistances is possible by
impedance spectroscopy if a sample’s complex impedance plot shows three separate
semi-cirlces, a), by fitting the data to a three element RC circuit, b).

47
At elevated temperatures, the second arc (due to grain boundaries) virtually
always disappeared, which can be attributed to the grain boundaries having a higher (than
the bulk) activation energy for proton conduction. The total resistance of the grain
boundaries would then decrease much faster than the bulk, represented in the Nyquist
plots by an ever shrinking second arc with increasing temperature. In its place, a nearly
straight line was usually seen, caused by a variation of the effective resistance and
capacitance of some element(s) in the circuit with frequency, Figure 2.3 a. This variation
comes from a distribution of relaxation times in the sample as a result of inhomogeneties
in the material and/or when the diffusion of an uncharged (or effectively uncharged)
species responding to a chemical potential becomes the rate controlling step. For solid
electrolytes, the later situation typically refers to the mobile species diffusing through the
electrodes, which have an effective potential gradient of zero due to the presence of
majority electronic carriers. Such a process results in a straight line at 45° degrees to the

-Z’’ (Ω)

Z′ (real) axis as proven by Warburg and Macdonald78,80.

ωο =1/RbCb

Rb /2

CPE

Rb /2

Z’ (Ω)
a)

Figure 2.3 Realistic impedance plot showing a depressed semi-circle with center below
the real axis and straight line at low frequencies, a). Both effects are due to the
distribution of characteristic frequencies in the sample and are modeled with a constant
phase element (CPE), b).

48
However, lines observed in this work usually deviated from 45 degrees, attesting
to true physical inhomogeneities in the samples. The most common cause of such
inhomogeneous behavior is rough electrode/electrolyte interfaces, which causes the
microscopic resistivities and capacitances near the interface to be “distributed” around
the mean macroscopic values. Distributed relaxation times are also caused by variations
of local composition and/or structure. As well as the appearance of lines at low
frequencies, these inhomogeneities also result in depressed semi-circles with centers
below the real axis, Figure 2.3 a. Both these distributed effects are modeled by
introducing a constant phase element (CPE) with impedance

CPE

= A(iω ) −ψ

(2-8)

The CPE equivalent of the normal RC circuit then has an impedance of

V (t )
V (t )
V (t )
+ A(iω )
−ψ
A(iω )

(2-9)

The CPE reduces to an ideal capacitor for Ψ = 1 and to a resistor for Ψ = 0, and thus can
model the distribution of microscopic capacitors and resistors in a material.
For compounds with superprotonic transitions, the conductivity increases by ~
102-103 across the transition. Not surprisingly then, the semi-circle in Figure 2.3 a
disappears, usually leaving only a straight line visible in the Nyquist plots of the
superprotonic phases. The resistance of the bulk (and therefore the materials
conductivity) was then estimated by the intercept with the real axis of a least squares

49
refinement on the line. This estimation was necessary as the frequencies associated with
superprotonic conduction exceeded the upper limit of our impedance meter (1 MHz), but
nevertheless gave highly reproducible values that also compared well with those in the
literature, and so was deemed acceptable.

Chapter 3.

Cation Size Effect on the

Superprotonic Transitions of MHnXO4
compounds (M = Cs, Rb, NH4; X = S, Se, P, As;
n = 1-2)
3.1 Introduction
The effect of alkali ion substitution on solid to solid phase transitions in the
MHnXO4 class of compounds has been well documented. The initial investigations of
these solid acids focused on the low temperature behavior of the MH2XO4 (M = Cs, Rb,
NH4, K; X = P, As) compounds, looking for ferroelectric transitions similar to that
discovered in KH2PO4 at 123K1. It was found that the K, Rb, and Cs phosphates and
arsenates all exhibited ferroelectric transitions with the average change in the transition
temperatures upon isovalent substitution being81:
⊕14 K

⊕ 36 K

K → Rb → Cs
(3-1)
− 40 K

PO4 → As O4
From this it seems clear that larger cations inhibit the ferroelectric transitions in this class
of compounds.
After the discovery of the superprotonic phase transition in CsHSO4 at 142°C, the
high temperature properties of the entire class of compounds began to be examined3. In
contrast to the results of the low temperature transitions, increased cation size was found
to lower the superprotonic phase transitions, which were observed only in the compounds

51
with the largest cations, Table 3.1. The explanation for this phenomenon was generally
held to be that the increased ionic radius of the cations resulted in larger X-X distances
(X = S, Se, P, As), thereby creating more room for the nearly free rotations of the
tetrahedra observed in the superprotonic phases39. The phosphate and arsenate
compounds are not listed on Table 3.1 as only the Cs compounds undergo superprotonic
phase transitions at 232 and 162°C for CsH2PO4 and CsH2AsO4, respectively46. Also, the
TlHSO4 compound is reported to have a superprotonic transition at 115°C, but its room
temperature structure has not yet been reported and hence the coordination of the Tl
cations is not known82. Since the ionic radius of the Tl ions varies from 1.76 to 1.60 in
going from a coordination of XII to VIII, respectively, it is not appropriate to compare
the properties of TlHSO4 to those of the other MHSO4 compounds until its structure is
known.

Table 3.1 Superprotonic phase transitions for MHXO4 class of compounds. The ionic
radii of the cations are based on their average coordination (superscripted Roman
numerals) in these materials. For the central ion of the tetrahedra, the covalent radii are
given for a four-fold coordination83.
Ref
Radius of M/X
Se IV 0.43
S IV 0.26
(Å)
84
Tsp = 142 °C
Tsp = 128 °C
85
1.81
Cs
Mono, P21/c →
Mono, P21/c →
Tetra, I41/amd
Tetra, I41/amd
86
Tsp = 227 °C at 0.31 GPa
Tsp = 174 °C
85,87
1.61
RbVIII
Mono, P21/c →?
Mono, B2 →
88
At 1 atm, Tmelt = 203 °C
Mono, C2h?
89
Tsp = 177 °C at 1.77 GPa
Tsp = 144 °C
90
NH4VIII 1.59
Mono, B21/a →?
Mono, B2 →
91
At 1 atm, Tmelt = 146 °C
Mono, P21/b?
As was seen in the ferroelectric transitions, substitution of a larger central ion in
the tetrahedra lowers the superprotonic transition temperatures. This effect is evident in

52
the fact that under ambient conditions the Rb and NH4 selenate compounds have
superprotonic phase transitions before melting, whereas in the analogous sulfate
compounds pressure must be applied to raise Tmelt above that of Tsp86. Also, in the Cs
phosphate and arsenate compounds, the transition temperature drops 60 degrees when the
PO4 groups are replaced by the larger AsO4 tetrahedra. These results are at odds with the
statement that larger cations increase the volume in which the tetrahedra reorient since
one would then expect the bigger tetrahedra to require coordination by proportionally
larger cations for a superprotonic transition to be feasible. However, exactly the opposite
result is measured. The underlying cause for the observed behavior was therefore not
clearly understood with the limited number of data points given in Table 3.1, although the
overall effect of increasing the size of the cation and/or the tetrahedral ion is clearly to
promote superprotonic phase transitions in these compounds.
This work was carried out to better explain this connection between
cation/tetrahedral ion size and the presence of superprotonic transitions. The approach
taken was to synthesize compounds with mixed M+1 ions and thereby vary the average
cation size. Unfortunately, attempts to grow selenate compounds categorically failed; the
normal (and even abnormal) synthesis routes resulting in, almost exclusively, the M2SeO4
salts. Also, the phosphate and arsenate compounds were avoided due to the known
instability of the superprotonic phases of the pure cesium compounds92,93. Hence,
attention was focused on mixed cation sulfate compounds.

3.2 Mixed Cation Sulfate Systems
Attempts to deduce the correlation between a compound’s average cation size and
the presence/absence of a superprotonic phase transition started with investigations into

53
the mixed Cs/K, Cs/Na, and Cs/Li systems. The emphasis on Cs is for the obvious reason
that CsHSO4 has a known superprotonic transition. On the other hand, the K, Na, and Li
hydrogen sulfate compounds all melt/decompose without transforming to a highly
conductive phase94-96. Therefore, replacing some of the Cs atoms with the smaller alkali
cations in CsHSO4 was hoped to have quite dramatic and quantifiable effects on the
superprotonic transition. Mixed Cs/Rb compounds were not explored as the Cs/Rb
system had already been investigated resulting in two new compounds, Cs0.9Rb0.1HSO4
and Cs0.1Rb0.9HSO4, which can be considered as structural modifications of end members
CsHSO4 (phase II) and RbHSO4, respectively97,98. The high temperature properties of
these compounds are nearly unchanged from those from which they were derived,
namely the cesium rich compound has a superprotonic transition ~ 142°C, while the
rubidium rich compound exhibits no high temperature transition before melting ~
177°C99. These compounds then confirm that a larger average cation radius encourages
superprotonic transitions, but do not further illuminate the fundamental correlation
between the two parameters since the structures and properties are nearly identical to
those of the end-member compounds.
Mixed Cs/NH4 compounds were avoided as the presence of the NH4 cations is
known to cause markedly different properties in solid acids. For example, the
(NH4)3H(SeO4)2 compound has a superprotonic phase transition at 27°C, whereas the
isostructural K and Rb compounds have transitions at 115° and 185°C, respectively100,101.
Also, note that the NH4HSeO4 compound transforms to the superprotonic phase 30
degrees lower than the RbHSeO4 compound, Table 3.1. This anomalous behavior is
attributed to the fact that the hydrogen atoms of the ammonium ions often form hydrogen

54
bonds to the tetrahedral oxygen atoms and that the NH4 groups typically show some
degree of disorder at room temperatures102-104. The bonding of the ammonium cations
will therefore be highly directional and/or highly variable when compared to the purely
electrostatic interactions of the spherical alkali metal cations. Analysis of any mixed
Cs/NH4 compounds would be complicated by such considerations, it being difficult to
resolve the cation size effect from the ammonium ion effect on a phase transition, and
therefore their synthesis was not attempted.
For the above reasons, only the mixed Cs- K/Na/Li systems were investigated.
These systems also had the additional advantage in that there is a large difference
between ionic radius of Cs versus K, Na, and Li. It was hoped that this difference would
highlight the essential structural properties associated with large cations and
superprotonic phase transitions. It should be mention here that the synthesis and
characterization of all the mixed systems Cs/M+1 mentioned here have been reported by
other researchers (primarily Mhiri et al.). The published results suggest that solid
solutions of the mixed cations are possible and that often the high temperature properties
gradually change from those of CsHSO4 to those of the MHSO4 compound in question.
This is in complete disagreement with the results of the present work and seems highly
implausible as none of the other MHSO4 compounds are isostructural to CsHSO4.
Moreover, except for work on crystals whose structure had been determined (e.g.,
Cs0.9Rb0.1HSO4 and Cs0.1Rb0.9HSO4), these investigations analyzed powder samples
created by grinding together crystals grown by aqueous synthesis99,105,106 107. It is
therefore not very surprising that they found very smooth changes in properties as the
percentage of substitutant M+1 cation in the solutions was increased. Also, the techniques

55
used to characterize the powders measured only the average properties of the samples:
powder X-ray diffraction, Raman spectroscopy, differential scanning calorimetry, and
conductivity measurements of pressed powder pellets. For these reasons, this work will
not refer to these investigations.

3.2.1 Synthesis and Characterization Techniques
Crystals examined in these mixed cation systems were synthesized by mixing the
appropriate amounts of the metal carbonates (Alpha Aesar puratonic, assay 99.999%) and
sulfuric acid (98% aq. sol.) in an aqueous solution, followed by slow evaporation at room
temperatures:
~ 25o C
(1-x)*Cs2CO3 +(x)*M2CO3 + H2SO4 + H2O → →single crystals

where M = K, Na, or Li and the total cation to anion ratio, (Cs+M):SO4, was held at 1:1.
This process was carried out in 10% molar increments of the secondary cation, M, except
where the discovery of new compounds merited a smaller increment of 5%.
The phases of the resulting single crystals were identified by single-crystal X-ray
diffraction (SCXD) techniques. Differential scanning calorimetry (DSC) and thermal
gravimetric analysis (TGA) were used to measure the thermal properties of the singlecrystals at elevated temperatures. Finally, conductivity measurements on single crystals,
or single crystals ground-up and pressed into pellets, were performed to confirm the
presence/absence of a superprotonic transition and to compare with the conductivity of
CsHSO4. The emphasis here is that whenever possible, only single-crystal samples were
grown and only single-crystal samples were analyzed.

3.2.2 Resulting Phases of the Mixed System Investigations
The above synthesis route resulted in the compounds listed in Table 3.2. At very high
cesium percentages, slight modifications to CsHSO4-III, the meta-stable phase of
CsHSO4 that grows out of aqueous solutions, were discovered for all three systems. The
evidence for incorporation of the smaller cations into CsHSO4-III was first seen in the
SCXD measurements. The lattice constants of the modified structures were nearly
identical to that of CsHSO4-III, in an alternative primitive cell, but with the length of the
c-axis tripled compared to the pure compound. This primitive cell is transformed into the
crystallographically correct cell of undoped CsHSO4-III by the transformation: a′ = a –
½*c, b′ = b, c′ = 2*c. The amount of K, Na , and Li incorporated into CsHSO4-III’s
structure is quite small as full data collections were not able to locate the ions although
they did confirm the tripling of the c-axis. It would appear that the smaller cations are
substituted on the Cs sites where they are hidden by cesium’s much larger scattering
factor for X-rays108. In a similar manner, electron microprobe measurements were unable
to observe the lighter cations.
For the Na compound, both Na+ and H+ NMR measurements were performed. A
very small peak in the Na+ NMR measurement was observed, but it was impossible to
rule out small amounts of Na contaminants as the cause of this peak. The proton NMR
measurements were more conclusive, in that two distinct peaks of significant magnitude
were observed for the doped sample, whereas the scan of the reference, undoped,
CsHSO4 sample showed only one peak (see appendix A). The sodium ions then again

Table 3.2 Compounds synthesized in the mixed Cs-K/Na/Li systems. The average cation radius was calculated using both the ratio and particular
coordination of the cations in a compound.
System

Compound
Obtained

All Systems

Solution
Composition
% M2CO3

CsHSO4−KHSO4

CsHSO4−NaHSO4

Lattice
Parameters

CsHSO4-III

Space
Group or
Symmetry
P21/n

10-30

α-CsHSO4-III

P21(?)

40
50-100

α-CsHSO4-III &
K3H(SO4)2
K3H(SO4)2

5-10
15-35
40

45-55

Average
Cation
Radius
1.81 Å

Phase Transitions
Above RT

Comments and
References

~ 62°C→CsHSO4-II;
142 °C→supeprotonic

109

a = 7.311(5) Å, b = 5.818(4) Å
c = 16.52(2) Å, β = 101.55(4)º

~ 1.81 Å

~ 67°C→CsHSO4-II?;
140 °C→supeprotonic

New modification
of CsHSO4–III

A2/a

a = 9.790(4) Å, b = 5.682(2) Å
c = 14.702(4) Å, β = 103.02(5)º

1.51 Å

190 °C→supeprotonic
190 °C→supeprotonic

23,110

β-CsHSO4-III

P21/m

a = 7.329(5) Å, b = 5.829(4) Å
c = 16.52(1) Å, β = 101.55(3)º

~ 1.81 Å

~73°C→CsHSO4-II?;
141 °C→supeprotonic

New modification
of CsHSO4–III

Cs2Na(HSO4)3

P63/m

a = 8.572(2) Å
c = 9.982(2) Å

1.55 Å

139°C→melt

a = 10.568(2) Å

1.28 Å

Cs2Na(HSO4)3
CsNa2(HSO4)3
CsNa2(HSO4)3

P213

a = 8.229(2) Å, b = 5.8163(9) Å
c = 9.996(3) Å, β = 106.46(2)°

a; b

new compound
25,111

125°C→melt

new compound
25,111

CsHSO4−LiHSO4

60-100
10

NaHSO4·H2O
γ-CsHSO4-III

20-80

Cs2Li3H(SO4)3
•H2O

90-100

Li2SO4 •H2O

P21(?)

a = 7.316(10) Å, b = 5.818(7) Å
c = 16.50(2) Å, β = 101.54(5)º

~ 1.81 Å

~108°C→CsHSO4-II?;
141 °C→supeprotonic

Pbn21

a = 12.945(3)
b = 19.881(4)
c = 5.111(1)

1.08 Å

105°C→slow
decomposition

a) Compound previously known.
b) High temperature properties not previously investigated.

New modification
of CsHSO4–III
new compound

58
revealed their presence indirectly through their effect on the surrounding structure, in this
case, the environment of the protons.
The incorporation of the K, Na and Li ions also showed up in the DSC
measurements. Upon heating the modified forms of CsHSO4-III, the transition to
CsHSO4-II (another monoclinic form) was observed to be systematically shifted to higher
temperatures as the size of the secondary cation decreased, Figure 3.1 a. Also, for the
Cs/Na compound, β-CsHSO4-III, two exothermic transitions instead of only one where
observed upon cooling, Figure 3.1 b. For this reason, the β-CsHSO4-III compound was
more extensively studied than the others. Conductivity measurements along the b-axis
revealed three, rather than two transitions, Figure 3.1 c. This discrepancy between the
DSC and conductivity results is probably due to sample size, i.e., very small crystals and
very large crystals were used in the DSC and conductivity measurements, respectively.
Low temperature DSC measurements also revealed an apparently second order transition
at -123.25°C not found in CsHSO4, Figure 3.1d112.
The temperature of the superprotonic phase transition, however, was not
significantly effected by the small amounts of K, Na, and Li present, Figure 3.1 a,
although the transition enthalpy was consistently lower for the mixed CsHSO4-III
compounds (see appendix A). These compounds, as was the case with Cs0.9Rb0.1HSO4,
do little to illuminate the cation size effect: their superprotonic phase transitions and
structures being essentially identical to those of CsHSO4. On the other hand, they do
reveal how sensitive these solid acids are to the addition of a secondary cation. In fact,
trace levels would appear to be the upper solubility limit for K, Na, and Li in CsHSO4

59
(and vice versa), the rest of the crystals synthesized being either line compounds or

2.0

1.6

1.8

1.4

Cs/Li

1.2

1.6

Cs/Li

1.0

Cs/Na

1.0

Cs/Na

0.8
0.6

Heat Flow (W/g)

Cs/K

0.8
0.6

Cs/K

0.4
0.2
0.0

-0.2

0.4

-0.4

0.2

0.0
40

Cs
80

100 120 140 160

-0.6

exo

Heat Flow (W/g)

1.4
1.2

Heat/Cool
rate = 5 C/min
under flowing N2
40

100 120 140 160

Temperature ( C)

Figure 3.1 (See figure caption on next page.)

endo

compounds with a single type of cation, Table 3.2.

Temperature ( C)
180 160 140

120

100

40
0.4

-1

L o g [ σ T ] Ω cm

-1

He a t F l o w ( mc a l / s e c )

heating
cooling

-2

-3

-4

-5

Heat/Cool rate = 0.5 C/min
under ambient atmosphere
2.2

2.4

2.6

2.8

3.0
-1

0.3

0.2

0.1

0.0

-6

Scan rate 0.1 C/min
Helium atmosphere

3.2

-125

-124

-123

-122

-121

1000 / T (K )

Temperature ( C)

Figure 3.1 Measurements on the α, β, γ-CsHSO4-III compounds. DSC curves upon
heating, a), and cooling, b) for all three modified forms of CsHSO4-III. Also, b-axis
conductivity and low temperature DSC measurements for the Na compound, c) and d),
respectively. Figure a) shows an increase in the phase III-II transition temperature with K
to Na to Li substitution. The difference of the Na compound from pure CsHSO4 is shown
in its two and three reverse transitions visible in the DSC, b), and conductivity data, c),
as well as the presence of a low temperature (apparently second order) transition, d).
Experimental parameters given on graphs.
For the Cs/K system, this insolubility phenomenon is particularly easy to see in
that only α-CsHSO4-III and K3H(SO4)2 crystals grew from the solutions. The K3H(SO4)2
compound belongs to another class of superprotonic conductors with general formula
M3H(XO4)2 (M = Cs, Rb, NH4, K, Na and X = S, Se). This compound had been
previously synthesized and its structure determined, but its high temperature properties
had not been sufficiently investigated23. Our studies revealed K3H(SO4)2 to have two high
temperature transitions before decomposition, both of which are superprotonic in nature

61
and neither of which are analogous to the superprotonic transitions found in the other
M3H(XO4)2 compounds110. Typically, these transitions involve very small structural
changes from pseudo-trigonal to trigonal unit cells, with superprotonic conduction
primarily in the basal planes100. The tetrahedra in the superprotonic phases do not
undergo true rotations, but simply librate around a site with C3 symmetry113. These
librations primarily effect the positions of basal plane oxygen atoms, hence the
anisotropic proton conduction of the phases. It is therefore not appropriate to compare the
superprotonic transitions of the M3H(XO4)2 compounds to those of the MHXO4
compounds, and so the results for K3H(SO4)2 will not be included in this work.
The Cs/Li system resulted in a new mixed compound, Cs2Li3H(SO4)3·H2O. DSC,
TGA and conductivity measurements show no evidence for a superprotonic transition
before the start of decomposition above 105°C (see appendix A). The lack of a
superprotonic transition is not surprising as the average radius for the four- and tenfold
oxygen coordinated lithium and cesium ions, respectively, is 1.078 Å83. Also, as this
compound is hydrated and has a cation to tetrahedra ratio of 5:3 (instead of the desired
1:1 ratio), any correlations between its structure and properties are not particularly
pertinent to the present discussion.
Fortunately, the Cs/Na system did produce two new mixed solid acids in the
MHXO4 family with chemical formulas of Cs2Na(HSO4)3 and CsNa2(HSO4)325. The unit
cell of Cs2Na(HSO4)3 is hexagonal while that of CsNa2(HSO4)3 is cubic, both novel
symmetries for the room temperature structures of the MHXO4 compounds. Moreover,
the single asymmetric hydrogen bond in both compounds links the SO4 groups into
unique three-membered (HSO4)3 rings. These rings are most likely due to the Na atom’s

62
preference for a 6-fold oxygen coordination, with the resulting NaO6 octahedra serving as
a template for the (HSO4)3 units22. The Cs atoms in both compounds reside in irregular
polyhedra with a coordination of 9 to 12 oxygens, depending on the upper limit one sets
for the Cs−O bonds. The rings in Cs2Na(HSO4)3 are linked together by NaO6 octahedra to
form infinite Na(HSO4)3 chains that extend along [001], Figure 3.2 a and b, while in
CsNa2(HSO4)3 the rings form a distorted cubic close-packed array. In this array, the Cs
atoms are located within the “octahedral” sites and the Na atoms within the “tetrahedral”
sites, Figure 3.2 c and d.

Figure 3.2 Crystal structures of the mixed Cs/Na compounds. The hexagonal structure
of Cs2Na(HSO4)3 is projected down [001]: a) unit cell contents from z = 0 to ½ and b)
from z = ½ to 1. Sodium atoms have elevations of z = 0 and ½, while those of the
cesiums are as indicated. Cubic structure of CsNa2(HSO4)3 projected along [100]: c) unit
cell contents from x = -¼ to +¼ and d) from x = +¼ to ¾. Elevation of cations as
indicated. Some oxygen atoms have been omitted for clarity25.
Neither of these compounds undergoes a superprotonic phase transition before
melting at 139 and 125°C for Cs2Na(HSO4)3 and CsNa2(HSO4)3, respectively, as

64
established by thermal analysis and visual inspection. The DSC curves for the
compounds are shown in Figure 3.3, along with conductivity measurements which show
the compounds to be fairly poor protonic conductors despite their high crystalline
symmetry.

CsNa2(HSO4)3

heat
cool

-1

exo

-2

110 130 150 170
Temperature ( C)

190

T, OC

100

Cs2Na a-axis
Cs2Na c-axis
CsNa2 pellet
CsHSO4 pellet

0.26 eV

-3

∆E σ = 0.66 eV

-4
-5
-6

∆E σ ~ 1.0 eV

-7
70

120

-1 ∆Eσ =

log(σT) [Ω-1cm-1K]

Heat Flow (W/g)

Cs2Na(HSO4)3 - cool

140

endo

Cs2Na(HSO4)3 - heat

160

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

-1

1000/T (K )

Figure 3.3 a) DSC and b) conductivity measurements on Cs2Na(HSO4)3 and
CsNa2(HSO4)3. Figure a) shows the melting transitions of the compounds beginning at
139 and 125°C, respectively. The cooling curve for CsNa2(HSO4)3 does not reveal a
solidification peak, which is in agreement with visual observations that the compound
solidifies as a glass upon cooling from the melt. Conductivity measurements revealed the
compounds’ protonic conductivity to be lower and activation energy higher than that of
CsHSO4’s room temperature phase. The observed curvature in the conductivity of
Cs2Na(HSO4)3’s a-axis is likely due to the onset of melting. DSC and conductivity
measurements taken under flowing N2 and dry argon atmospheres, respectively, with
heating/cooling rates of 10°C/min and 0.5°C/min, respectively.

3.2.3 Conclusions from Mixed System Investigations
Studies into the mixed CsHSO4-K/Na/LiHSO4 systems have resulted in six new
compounds, two of which are appropriate with the other MHXO4 compounds. Three of
the compounds (α, β, γ-CsHSO4-III) are slight modifications of CsHSO4-III, with

65
correspondingly small changes to the structural and superprotonic parameters of the
parent compound. Analysis of these compounds with respect to the cationsize/superprotonic-transition correlation tells us little we did not already know from
CsHSO4 itself. On the other hand, the new solid acid discovered in the Cs/Li system,
Cs2Li3H(SO4)3·H2O, has a very different structure from the other MHXO4 compounds.
However, this compound also has a cation to anion ratio of 5:3 and is hydrated, both
properties which any cation-size effect conclusions drawn from this compound unsuitable
for comparision with those of the MHXO4 family of compounds.
Hence, the only compounds synthesized of use to the present discussion are the
mixed Cs/Na compounds, Cs2Na(HSO4)3 and CsNa2(HSO4)3. Using the structures of
these two compounds as well as those of the other MHSO4 compounds, we can create a
graph that depicts the changes to the characteristic distances of the crystals as a function
of the average cation radius, Figure 3.4. These distances act as crystal-chemical measures
of the cations’ role in the presence or absence of a superprotonic phase transition. Such
distances include the mean S-S, M-S, M-M, and M-O distances and the effective length
per formula unit (taken as the cube root of the volume per MHSO4 unit). The trend of the
mean S-S distance with cation radius is of particular interest because, as mentioned
earlier, the general consensus is that large X-X distances are necessary to lower anionanion interactions and thereby promote the rapid XO4 reorientations of the superprotonic
phases35. The most salient feature of Figure 3.4 is then that the average S-S distance in
Cs2Na(HSO4)3 is larger than that of CsHSO4. This result suggests that either the XO4 to
XO4 interactions are not critical to superprotonic transitions, or that the distance

66
is not a useful measure of such interactions. The same can be said of the distance
as that of Cs2Na(HSO4)3 is also larger than that found in CsHSO4.
On the other hand, the average M-S, M-O and V1/3 distances all scale with the
mean cation radius. Of these three distances, the M-S distance varies much more across
the no-transition/transition line than the other two. This tends to confirm the observation
derived from the effects of substituting Se for S in the Cs, Rb, and NH4HXO4
compounds (Table 3.1) that the M-X distance reflects a truly critical crystal-chemical
parameter with respect to superprotonic phase transitions. Of course, these results do not
exclude the possibility that the M-O and V1/3 distances are of equal or more importance
than the M-X distance.

Distance [Angstroms]

5.5

No
Transition Transition

1/3

5.0
4.5
4.0
3.5

Cs2Na

Cs1Rb9

3.0

Rb
2.5

LiHSO4

2.0
0.8

1.0

CsNa2
1.2

1.4

Cs9Rb1

NH4

1.6

1.8

[Angstroms]
Figure 3.4 Characteristic lengths of MHSO4 compounds as a function of average cation

67
radius. Crystallographic data taken from the following sources: LiHSO4, 114; α-NaHSO4,
115
; CsNa2(HSO4)3, 25; KHSO4, 24; Cs2Na(HSO4)3, 25; NH4HSO4, 116, RbHSO4, 117;
Cs0.1Rb0.9HSO4, 98; Cs0.9Rb0.1HSO4, 97; CsHSO4-II, 26.
Another distance of possible significance is the average O-O distance of the
hydrogen bonds in these solid acids. This distance is proportional to the energy associated
with a compound’s hydrogen bonds. As the energy of a hydrogen bond exponentially
increases with decreasing O-O distance, shorter hydrogen bonds will require much more
energy to break than longer bonds21. One would then expect that the presence of longer
(weaker) hydrogen bonds would favor a compound transforming to a superprotonic
phase, where these bonds will have to be continually broken and reformed as the
tetrahedra reorient. Moreover, the hydrogen bonds of the superprotonic phases are nearly
always longer than those below the transition35. Therefore, shorter bonds at room
temperature should increase the required transition enthalpy to the high temperature
phase. Nevertheless, in plotting the mean hydrogen bond O-O distance versus average
cation radius, Figure 3.5, there is no apparent relationship between the hydrogen bond
lengths and the presence of a superprotonic transition. This is concluded from the fact
that LiHSO4, NaHSO4, CsNa2(HSO4)3, and Cs2Na(HSO4)3 all have longer HBOND
distances than CsHSO4 and Cs0.9Rb0.1HSO4.

Distance [Angstroms]

HBOND
Cs2Na

2.70

Na
2.65

2.60

Transition

2.75

No Transition

CsHSO4

CsNa2
LiHSO4

2.55

0.8

1.0

1.2

1.4

1.6

1.8

[Angstroms]

Figure 3.5 Average hydrogen bond length versus mean cation radius. The Li, Na, CsNa2
and Cs2Na compounds all have an average hydrogen bond length longer than that of
CsHSO4. This fact suggests it would require a smaller loss in hydrogen bond energy for
these compounds to transform to a superprotonic phase compared to CsHSO4.
There is, however, a problem to the above comparisons and the conclusions
drawn from them, which is simply that the structures of the compounds are not the same.
In particular, the mixed Cs/Na compounds are quite different from the other compounds.
First, the unit cell symmetries of CsNa2(HSO4)3 and Cs2Na(HSO4)3 are cubic and
hexagonal, respectively, versus the monoclinic or tetragonal cells found in the other
MHSO4 compounds. Second, the alternating rows of anions and cations almost always
observed in the MHSO4 compounds are replaced with either the channels or FCC-like
array of the mixed Cs/Na compounds31. Finally, the way in which the tetrahedra are
connected by hydrogen bonds in CsNa2(HSO4)3 and Cs2Na(HSO4)3, into three-membered

69
rings, is completely unique for the MHSO4 family of compounds. It is then possible that
the properties of the mixed Cs/Na compounds are due to their unique structures, making a
comparison with the other MHSO4 compounds less than ideal.
A rigorous application of this argument also invalidates the comparison between
the remaining MHSO4 compounds for all but RbHSO4 and NH4HSO4, which are
isostructural to each other118. This fact is particularly evident when evaluating the
hydrogen bond lengths, which are very much connected to the types of hydrogen bonds
(single or double minimum; fully or partially occupied) and hydrogen-bonded networks
(dimers, rings, or chains) present in the compounds. As seen in Table 3.3, the differing
structures of the MHSO4 compounds result in their having a broad distribution of
hydrogen bond types and networks, possibly explaining the seemingly random trend seen
in Figure 3.5.
Table 3.3 Hydrogen bond parameters for the MHSO4 compounds. Single, double and
partial hydrogen bond types refer to ordered single minimum, disordered double
minimum and partially occupied hydrogen bonds, respectively. Shaded cells denote
isostructural compounds.
Compound
Space
Mean O- Types of H-bonds
H-bonded
ref
Group O distance
present
networks present
114
LiHSO4
P21/c
2.644
Single
Chains
115
α-NaHSO4
2.642
Single, double,
Branched chains
P1
partial
25
CsNa2(HSO4)3
P213
2.674
Single
Rings
24
KHSO4
Pbca
2.596
Single, double
Dimers, chains
25
Cs2Na(HSO4)3
P63/m
2.728
Single, partial?
Rings
116
NH4HSO4
B21/a
2.556
Single, partial
chains
117
RbHSO4
P21/c
2.564
Single, partial
chains
98
Cs0.1Rb0.9HSO4
P21/c
2.541
Single, double
Chains
97
Cs0.9Rb0.1HSO4
P21
2.59
Single
Chains
26
CsHSO4-II
P21/c
2.636
Single
Chains

70
The conclusions drawn from this work on the mixed Cs-K/Na/LiHSO4 systems in
conjunction with the other MHSO4 compounds therefore concur with the initial
observations that the distance seems to be a critical crystal-chemical measure of
whether a compound will have a superprotonic transition or not, rather than the X-X or
other characteristic distances in the compounds. However, a plausible argument against
this result is that the underlying structural differences in the compounds may have a much
more important role in determining the presence/absence of a transition than either the
cation or anion size effect. To deconvolute any structural effects from the cation/anion
size effect, it is necessary to find a system of compounds that remained isostructural
while the size of the cation/anion is changed. The results from such a system would
complement those of the above mixed systems, the problem having being approach from
both top and bottom, so to speak. If the same trends were observed, it would conclusively
confirm the M-X distance as the critical parameter in predicting superprotonic phase
transitions. Luckily, just such a system exists.

3.3 M2(HSO4)(H2PO4) Compounds
The M2(HSO4)(H2PO4) compounds are isostructural for M = K, NH4, Rb, and Cs
(Table 3.4), the Cs compound being discovered during the investigations of the CsHSO4CsH2PO4 system (Chapter 4). Characterization of Cs2(HSO4)(H2PO4) showed it to have a
superprotonic phase transition in the range of 61 to 110°C63. On the other hand, the
properties of the other M2 compounds at elevated temperatures were not known. The
combination of these compounds being isostructural and having a known superprotonic
transition makes this system ideal for exploring the cation size effect irrespective of
structure.

3.3.1 Structures of the M2(HSO4)(H2PO4) compounds
These compounds crystallize in a monoclinic unit cell, space group P21/n, with
two formula units per cell. Their lattice parameters and other crystallographic data are
listed in Table 3.4. Each compound has six crystallographically distinct, non-hydrogen
atom sites: one for the M+1 cations, four for the oxygen atoms and one site on which the S
and P atoms are evenly distributed. The structures consist of loosely defined MOx
polyhedra and well-defined XO4 tetrahedra. For the K, NH4 and Rb compounds the
cations are coordinated by nine oxygens, while the Cs compound has CsO10 polyhedra.
These coordination numbers are not particularly well defined as they depend to a great
deal on the upper limit one puts on the M-O bonds. However, using the published
coordination numbers, the ionic radii of the cations are 1.55, 1.61, 1.63, and 1.81 Å, for
K, NH4, Rb, and Cs, respectively83. Here, the radius of the ammonium cations has been
scaled with those of the rubidium ions for the sake of consistency with the previous
section and because it is difficult to calculate a spherical radius for these cations due to
the presence of highly directional N-H-O hydrogen bonds119. Considering that the NH4
compound’s volume is slightly larger than the Rb’s, it might be closer to the truth if the
RNH4 > RRb, but as the difference between the compounds is minimal, setting RNH4 < RRb
should make little to no difference in the analysis.
Table 3.4 Crystallographic data for the M2(HSO4)(H2PO4) compounds
Radius ref
Cation Space a (Å) b (Å) c (Å) β (Å)
Vol. Z Dcalc
Group

(Å3)

(g/cm3) M2 (Å)

P21/n

7.856 7.732 7.827

99.92

468.3 2

3.261

1.81

P21/n

7.632 7.552 7.448

100.47 422.1 2

2.872

1.63

120

NH4

P21/n

7.723 7.540 7.482

101.32 427.2 2

1.789

1.61

119

P21/n

7.434 7.341 7.148

99.56

2.350

1.55

121

384.7 2

The M-O and X-O distances in these compounds are all quite regular. For
Cs2(HSO4)(H2PO4), the mean Cs-O distance is 3.27 Å, with a low of 3.055(4) and high of
3.622(4) Å, giving a calculated bond sum of 1.10. The bond valence sum at the X cation
site is 5.51, in good agreement with the value of 5.5 predicted from a site occupancy of
0.5 S6+ and 0.5 P5+. The tetrahedral angles for this compound range from 107.1(2) to
112.6(3)°, as expected for PO4 and SO4 anions27. The average X-O distance varies very
little with the nature of the cation: the K, NH4, Rb, and Cs compounds having values of
1.508, 1.506, 1.505, and 1.503 Å, respectively. These values all lie between those
typically encountered in PO4 and SO4 tetrahedra, ~1.52 and ~1.47 Å, respectively,
agreeing with the assignment of a completely mixed S/P occupancy on the X site83.
More confirmation of this mixing on the X site is evident in the fact that each XO4
group is involved in exactly three hydrogen bonds, the mean value of the two and four
bonds expected for HSO4 and H2PO4 tetrahedra, respectively. Two of these hydrogen
bonds connects the XO4 tetrahedra into zigzag chains running in the [010], while the third
bond cross-links the chains into sheets that lie in parallel (-101), Figure 3.6 a) for
Cs2(HSO4)(H2PO4). The hydrogen bonds of the chains are ordered (single minimum
potential), while the cross-linking bonds are disordered (double minimum potential). The
zigzag chains of hydrogen-bonded tetrahedra alternate with rows of M+1 cations to give a
checkerboard pattern, shown for the Cs compound in Figure 3.6 b.

Figure 3.6 Structure of Cs2(HSO4)(H2PO4): a) the sheets of hydrogen bonded tetrahedra
in the (-101) plane with zigzag chains running in [010] and cross-linking hydrogen bonds
connecting the chains in [100]. A projection down [010], b), shows the checkerboard
arrangement of anion and cation rows as well as the sheets of tetrahedra extending along
[101]27. A unit cell is outlined in each picture.
High temperature X-ray powder diffraction and infrared spectroscopy revealed
that the high temperature phase of Cs2(HSO4)(H2PO4) is cubic, with ao = 4.926(5) Å. The
suggested symmetry of the unit cell is Pm 3 m, in which the compound would take on a

CsCl structure, with Cs atoms at the corners of a simple cubic unit cell, and the XO4
groups at the center, Figure 3.763. The coordinates for the Cs and X atoms are therefore 0
0 0 and ½ ½ ½, respectively. The oxygen atoms were placed at ½ ¼ 0.323 based on X-O
and Cs-O distance considerations. The single, crystallographic oxygen resides on a 24l
site, resulting in 6 orientations of the XO4 tetrahedra. Rapid librations between these
orientations, facilitating proton transport between the tetrahedra, are thought to result in

74
the high conductivity of this phase, a nearly identical process being known to occur in the
superprotonic phase of CsHSO460.

Figure 3.7 Cubic phase of Cs2(HSO4)(H2PO4). Cs atoms reside at the corners and S/P
atoms in the center surrounded by the partially occupied oxygen sites.

3.3.2 Synthesis of the M2(HSO4)(H2PO4) compounds
These M2(HSO4)(H2PO4) compounds were synthesized by slow evaporation from
aqueous solutions using the following procedure:
10 − 25o C
M2CO3 + x*H3PO4 + y*H2SO4 + H2O   → single crystals
where x and y were varied from 1 as necessary to achieve the desired compound. The
reagents used were the same as those for the mixed Cs/M systems with the addition of
phosphoric acid (86% aq. sol.). Successful synthesis conditions varied slightly from
compound to compound, Table 3.5. Copious amounts of large wedged-shaped crystals
were easily grown for the K, NH4, and Rb compounds once the synthesis route was

75
perfected. On the other hand, the Cs compound was very difficult to grow, with only
small quantities of plate like crystals being produced after much perseverance. The
phases of the crystals were confirmed using SCXD techniques, at which time the crystals
were also oriented for directional conductivity measurements. All experimental results
presented here were performed on single crystals so identified.
Table 3.5 Successful synthesis conditions for the M2(HSO4)(H2PO4) compounds.
Cation M2CO3:H2SO4:H3PO4 Temperature (°C)
Resulting Phases
Cs
1:1:1
10
Cs2(HSO4)(H2PO4) &
Cs3(HSO4)2(H2PO4)
Rb
1:1:1
25
Rb2(HSO4)(H2PO4)
NH4
1:1:1
25
(NH4)2(HSO4)(H2PO4)
1:2:6
25
K2(HSO4)(H2PO4)

3.3.3 Characterization of the M2(HSO4)(H2PO4) compounds
The presence and type of phase transitions present in these compounds were
determined by thermal (DSC and TGA) and conductivity measurements. The results of
these measurements showed that only the Cs compound undergoes a superprotonic phase
transition, the other compounds melting at 110, 160, and 170°C, for the NH4, Rb and K
compounds, respectively. The melting of these compounds (instead of decomposition)
was determined by comparing the DSC scans with the TGA curves, Figure 3.8, and by
visual inspection of heated crystals under an optical microscope. The specifics of the high
temperature transitions of these compounds are given in Table 3.6.

Table 3.6 High temperature transition parameters for the M2(HSO4)(H2PO4) compounds.
Cation
Transition Type
Tonset (°C)
Tdecomp. (°C)
Cs
Superprotonic
60
190
Rb
Melt
160
193
NH4
Melt
110
143
Melt
170
183

2.0

Heat Flow (W/g)

NH4

1.5

Rb
1.0

0.5
0.0

SP Trans
50

100

Melting/Decomposition
150

200

250

300

Temperature ( C)
a)

103

Weight Percent (%)

Rb
102
101

100

99
98

NH4

97
75

100 125 150 175 200 225 250 275 300

Temperature ( C)
b)

Figure 3.8 Thermal analysis of the M2(HSO4)(H2PO4) compounds at elevated
temperatures by DSC, a), and TGA, b), measurements. The DSC scans show the
superprotonic and melting transitions of the compounds, while the start of decomposition
is indicated by arrows in the TGA curves. Both sets of measurements were taken at
5°C/min under flowing argon (DSC) or nitrogen (TGA) atmosphere.

From these results one can conclude that large cations are indeed necessary for
superprotonic transitions without any structural qualifications. This conclusion was
supported by conductivity measurements, which showed the K, NH4, and Rb compounds
to remain poor conductors up to the onset of melting, Figure 3.9.

T, C
180 160 140 120

100

Cubic Phase

-1
-1

Log[σ T] Ω cm K

-1

-3

Cs2HSO4H2PO4
Rb2HSO4H2PO4
(NH4)2HSO4H2PO4
K2HSO4H2PO4

-2

-4

Monoclinic Phases

-5
-6
-7
-8
2.2

2.4

2.6

2.8

3.0

3.2

1000/T

Figure 3.9 Conductivity measurements along the b-axis of the M2(HSO4)(H2PO4)
compounds. The measurements were taken in a dry argon atmosphere with a heating rate
of 0.5°C/min.
Looking at Figure 3.9, it is quite interesting to note that a larger cation size
facilitates the room temperature conductivity of a compound as well as its transition to a

78
superprotonic state. In fact, if one plots the logarithm of the conductivity versus cation
radius for the compounds, there is a highly linear relationship that becomes more linear
as the temperature approaches that of the Cs compound’s superprotonic transition, Figure
3.10. This effect on the room temperature conductivities is expected if one considers the
mechanisms thought to govern protonic conductivity in the room temperature phases as
precursors to those known to occur in the superprotonic phases.
In the room temperature phases of solid acids, protons are thought to conduct by
the formation and migration of defects such as doubly occupied and empty hydrogen
bonds (D and L defects), interstitial hydrogen bonds (Frenkel-like defects), and
positive/negative ion pairs (i.e., H2SO4+ and SO4- in CsHSO4)2,37,46. By either proton hops
or tetrahedral rotations, these defects move through the otherwise ordered structures of
the room temperature phases. Proton conduction by any of the proposed defect
mechanisms will therefore result in increased hydrogen bond and orientational (dipole or
tetrahedral) disorder. It is then quite logical that if larger cations favor the transition to a
state in which a disordered hydrogen-bonded network and rapid tetrahedral reorientations
are built into the structure, they should also facilitate the defect conduction mechanisms
of the room temperature phases. Exactly why the conductivity of the compounds
produces the effect seen in Figure 3.10 b is unclear, but it would seem that as the cation
size effect becomes more fully realized, possibly due to increased thermal vibrations of
the atoms, the possibility of transforming to the superprotonic state becomes open for
compounds with large enough cations.

-8.0

0.98

At 60 C
-8.5

0.94

-1

Rb
NH4

-9.0

-1

Log[σ ] Ω cm

0.96

0.92

R = 0.917
0.90
-9.5
0.88

-10.0

0.86
1.6

1.7

1.8

M Radius (Angstroms)

Temperature ( C)

Figure 3.10 Cation size effect on the room temperature conductivities: the logarithm of
conductivity versus cation radius, a), shows an almost linear relationship, which becomes
more pronounced as the temperature is increased, b).
Since the overall magnitudes of the room temperature conductivities appear to
scale with the size of a compound’s cations, it is quite interesting that neither the
activation energy nor pre-exponential function of the compounds follows such a trend.
The values for these parameters are given in Table 3.7 and show the K and Cs
compounds to have both higher activation energies and greater pre-exponential functions
than the Rb and NH4 crystals. The activation energy represents the energy required for a
successful proton migration step, while the pre-exponential function mainly reflects the
number density of proton conduction producing defects 122. Intuitively, one might guess

80
that if larger cations facilitate protonic transport, this effect would show up in either
smaller activation energies or larger pre-exponential terms as the cation size increased.
However, neither trend is evident.

Table 3.7 Activation energy and pre-exponential term for proton conduction in the room
temperature phases of the the M2(HSO4)(H2PO4) compounds determined from a fit of the
data to σ = (A/T)exp[Ea/kbT]. The conductivity of the crystals at 60°C is also given. The
crystal axis refers to the direction of the applied field with respect to the crystallographic
axes of the monoclinic phases.
Cation
Crystal Axis
Ea (eV)
σ (60°C) (Ω-1 cm-1)
Log[A] (Ω-1 cm-1 K)
Cs

0.43

0.82

6.4 x 10-9

0.30

-1.80

1.6 x 10-9

NH4

0.31

-1.84

9.1 x 10-10

0.71

3.57

1.4 x 10-10

If, on the other hand, one compares the relationship between activation energies
and pre-exponential terms (independently of cation size) one finds a strong correlation.
This can be seen by plotting the two terms against each other, which gives a nearly linear
relationship between the parameters, Figure 3.11 a. Such a phenomenon has been
observed for thermoactivated processes in general, and in particular, for solid acids by
Sinitsyn et al. who labeled it the compensation law for protonic conductors122. This law
correlates the activation energy required for proton transport with the entropy created by
the migration process. If we include the data points for the M2(HSO4)(H2PO4)
compounds with those of other solid acids, the R2 value of a linear fit increases from the
0.84 value reported by Sinitsyn to 0.90, Figure 3.11 b. However, in spite of the improved
fit to the data, the results of the M2(HSO4)(H2PO4) compounds actually contradicts one of
the paper’s main results: that for activation energies smaller than 0.5 eV room

81
temperature proton transport is impossible as the entropy of the conduction process
becomes negative below this value (by their estimations). As can be seen in Figure 3.11
b, this statement was supported by the data available at the time and it is possible that the
estimations taken to derive this limit were correct for the other solid acids, but not for this
family of crystals. In any case, this data makes it clear that activation energies as low as
0.3 eV are possible in the room temperature phases even though such values are usually
associated with superprotonic conduction 122.

-1

Pre-exponetial [Ω cm K]

R = 0.98

-2

NH4

-4

R = 0.90

-1

Pre-exponetial [Ω cm K]

Cs in S.P. phase

-6 Rb

10
M2 compounds
Other solid acids

-10

0.3 0.4 0.5 0.6 0.7

0.5

1.0

1.5

Activation Energy (eV)

Figure 3.11 Compensation law for M2(HSO4)(H2PO4) compounds, a), and the entire
family of solid acids, b), both show linear relationships between the activation energies
and pre-exponential terms derived from a fit of the data to σ = (A/T)exp[Ea/kbT]. Dotted
line in b) designates the cutoff activation energy of 0.5 eV calculated by Sinitsyn122.

To conclude, both the overall magnitude and particular parameters of these
compounds’ conductivities behave expectedly/understandably in the room temperature
phases: a larger cation radius enhancing the protonic conductivity in a manner
presumably similar to the size effect of the superprotonic transitions.

3.3.4 What exactly is the effect of cation size?
The previous section provided the evidence that large cations are essential to the
presence of superprotonic transitions, but what exactly is so crucial about the size of the
cations? A very simple guess might be that large cations are require to stabilize the high
temperature structure. Such a guess would lead to the critical ratio between the anion (R)
and cation (r) radius in the CsCl structure of the Cs compound’s high temperature phase.
Assuming hard spheres for the ions and using the eightfold coordination of this structure,
the critical r/R value becomes 0.732, below which the anion (which usually has the larger
radius) spheres will begin to overlap. This coordination is stable until the r/R value is
greater than one, at which point a twelvefold coordination becomes more energetically
stable123. Estimating the radius of the XO4 groups is much more speculative than those of
the cations, but if we use the reported values for the covalent radius of oxygen
coordinated by four atoms (1.24 Å) and the average of the S and P covalent radii in
tetrahedral coordination (0.26 and 0.31 Å, respectively), the average X+O radius is83:

0 .26 + 0 .31
+ 1 .24 + 1 .24 = 2 .765 Å

The X-O distance calculated this way is 1.525 Å, very close to the average value of ~
1.51 Å observed in the room temperature structures. The calculated (spherical) anion
radius of 2.765 Å would then seem to be a reasonable value for the XO4 groups. Using
this anion radius in the high temperature phase, the r/R value for the Cs (1.81 Å)
compound is 0.655, significantly below the critical value of 0.732. Of course the r/R
value is even farther away from the critical value for Rb, NH4, and K in the CsCl
structure, Table 3.8.
There is a major problem with this calculation, however, which arises from the
basic assumption that the tetrahedra act like hard spheres. This does not seem too
unreasonable for a superprotonic phase, where the tetrahedra undergo rapid
reorientations, but nevertheless leads to unrealistic consequences in the high temperature
CsCl-like structure of the Cs compound. In a qualitative way, making the tetrahedra into
hard spheres allows for the possibility of linear configurations like X-O-O-X and X-OCs, which are very unlikely 63. Quantitatively, anion spheres with a radius of 2.765 Å are
incompatible with the experimental findings as they would result in a lattice constant for
a CsCl cubic cell of 2*2.765 = 5.53 Å, based on the anion spheres just touching. Since
the value observed for the Cs compound is 4.926 Å, it would appear that although the
distance from the center of a tetrahedron to the outer edge of one of its oxygens is on the
order of 2.765 Å, the effective radius of the tetrahedra must be smaller than this number.
A very straightforward way of estimating an effective anionic radius is to simply
do the reverse of the above calculation and take the known lattice constant, 4.926 Å, and

84
divide it by two, giving a value of R = 2.463 Å. With such an effective radius for the XO4
groups and the critical r/R value of 0.723, it is trivial to calculate the minimum radius
required for the cations r = (r/R)*R = 0.723*2.463 = 1.803 Å. For such a cutoff, only the
Cs compound would be stable in the CsCl structure (i.e. r/R > 0.732), in good agreement
with the experimental findings, Table 3.8.
Taking the above estimation one step further, we note that the cesium radius is
slightly larger than 1.803 Å, and therefore the tetrahedra do not actually touch in the
cubic structure so that the true effective radius is even smaller than 2.463 Å. Assuming
that the anion and cation radii touch along the body diagonal, this structure determined
effective radius will simply be half the body diagonal minus the radius of a cesium ion:

* aO − rcesium = 0.866 * 4.926 −1.81= 4.266 − 1.81 = 2.456 Å

For this value, all the cation to anion radius ratios increase, but the Cs compound is still
the only crystal with a ratio above 0.732, Table 3.8.
The CsH2PO4 compound is also reported to have CsCl structure and a lattice
constant of 4.961 Å92. Calculating the different anionic radii as was done for
Cs2(HSO4)(H2PO4) results in ratios all below the critical value of 0.732 for Rb, NH4, and
K, Table 3.7. In the case of CsH2PO4, however, the r/R values evaluated using the anion
radii derived from the structure are close to the critical value, which agrees with the
nearly commensurate superprotonic phase transition and decomposition of the compound
at 232°C92. It should be noted that a Cs ionic radius of 1.81 Å assumes a coordination by
ten oxygen atoms (as opposed to the eight fold coordination of the Cs site), which is the
case the room temperature structures of Cs2(HSO4)(H2PO4) and CsH2PO4, but may not be
the case in the superprotonic phases27,28. In fact, assuming an average of 1½ oxygens

85
from each of the eight surrounding tetrahedra, a coordination number of 12 seems quite
possible, which would equate to a ionic radius of 1.88 Å for Cs83. There is then some
flexibility in the calculated ratios, although even with a 12-fold coordination, the Rb,
NH4, and K compounds would remain below the critical r/R ratio of 0.732 (i.e. RbXII =
1.73 Å)83.

Table 3.8 Cation/anion radius ratios for the M2(HSO4)(H2PO4) and MH2PO4
compounds in a CsCl structure. The cation radius is given for an eight-fold coordination
while the anion radius is fixed at the three values derived in text. The stability range for
eight fold coordination is 0.723 ≤r/R < 1.
Radius (Å)
Cs-1.81
Rb-1.63
NH4-1.61
K-1.55
XO4-structure
0.737
0.664
0.656
0.631
2.456
XO4-effective
0.735
0.662
0.654
0.629
2.463
XO4-spherical
0.655
0.590
0.583
0.561
2.765
PO4-structure
0.728
0.656
0.648
0.623
2.486
PO4-effective
0.730
0.658
0.649
0.625
2.480
PO4-spherical
0.649
0.585
0.577
0.556
2.790
One might now be tempted to conclude that the cation size effect on
superprotonic transitions is no more than the prerequisite that the high temperature
structures be energetically stable, which of necessity calls for large cations. However, just
as was the case with the initial observations on the MHXO4 family of compounds, this
logic leads to the conclusion that larger XO4 groups are detrimental to the presence of a
superprotonic transition as proportionally larger cations would be required to meet the
critical r/R value. This flies in the face of all available experimental evidence which
shows larger tetrahedral groups to facilitate superprotonic transitions. The cation size

86
effect, although undoubtedly linked to the discussed ratio rule considerations, must
therefore have more subtle effects as well.
To uncover such effects, let us look at the characteristic distances in these
compounds as we did previously for the MHSO4 compounds. A graph similar to Figure
3.4 reveals no critical parameters as all the distances scale with cation radius, Figure 3.12
a. Interestingly, the hydrogen bond lengths of the compounds do not show a particularly
strong dependence on the size of the cations, Figure 3.12 b. This analysis does not reveal
a critical crystal-chemical parameter for exactly the same reason that the results of these
compounds are so conclusive, namely the compounds are isostructural.

5.5

1/3

Distance [Angstroms]

5.0

2.550
2.545
2.540

4.5

2.535
4.0

2.530
NH4

3.5

2.525

2.520

3.0
1.6

1.7

1.8

O-O distances
Asymmetric
Symmetric

2.515
1.6

1.7

1.8

[Angstroms]

Figure 3.12 Characteristic distances for the M2(HSO4)(H2PO4) compounds, a), scale
with the cation radius, while the O-O lengths of the symmetric and asymmetric hydrogen

87
bonds found in the crystals do show a fairly random dependence with , b). If the NH4
compound is removed from consideration, however, the asymmetric bonds that link the
tetrahedra might be said to lengthen as increases. Crystallographic data taken from
the same sources as found in Table 3.3.
For an increased understanding of the cation size effect on superprotonic
transitions, it is therefore necessary to analyze the transition in which we have just
determined the cation size effect to play the dominant role in its presence, i.e., the
transition of Cs2(HSO4)(H2PO4). If we look the changes in the characteristic distances
across the transition, it is immediately clear that the Cs-X distance changes most, Figure
3.13. Furthermore, this fact appears to be true for CsHSO4 and CsH2PO4 as well (the only
other MHXO4 compounds with superprotonic transitions for which both the room and
high temperature structures are known), Figure 3.13. This is particularly interesting for
CsHSO4 as its superprotonic phase is tetragonal, space group I41/amd, and so is quite
different from the CsCl structure into which CsH2PO4 and Cs2(HSO4)(H2PO4) transform
at high tempertures61. The distance therefore again emerges as the single most
important crystal-chemical measure of a MHXO4 compound’s likely-hood to undergo a
superprotonic phase transition. Moreover, as the structures of Cs2(HSO4)(H2PO4),
CsHSO4 and CsH2PO4 vary significantly in both the low an high temperature regimes, the
distance would appear to be predictive irrespective of a MHXO4 compound’s
structure both below and above the transition.

Cs2(HSO4)(H2PO4)
CsHSO4
CsH2PO4

% change

-2

1/3

Figure 3.13 Changes in the characteristic distances of Cs2(HSO4)(H2PO4), CsHSO4 and
CsH2PO4 across their superprotonic transitions. The crystallographic data comes from the
following sourses: Cs2(HSO4)(H2PO4)27,63; CsHSO426,61,124,125; CsH2PO428,92. For
CsHSO4, the position of the oxygen atoms in the superprotonic phase is in dispute, so the
average of the distances from the three different structures proposed was used in
the figure.

3.3.5 Conclusions and interpretations of the cation/anion effect
This work has shown conclusively that large cations are necessary for
superprotonic transitions in the M2(HSO4)(H2PO4) family of compounds. From this it was
established that the average cation to tetrahedral anion, , distance surfaces as the
best measure of a MHXO4 compound’s probability for undergoing a supeprotonic
transition, agreeing with the generally observed behavior of the compounds. The

89
distance was found to be much less useful as a predictive measure of a superprotonic
transition, contrary to the proposed hypothesis that the main effect of increased cation
size was to create larger X-X distances and thereby allow freer rotations of the tetrahedra.
Having identified M-X distances as such a critical crystal-chemical measure, the
question is then what exactly does this distance do to the interactions of the atoms so as
to favor the presence of superprotonic transitions. In the present study, the
distance was modified by varying the radius of the cations, but as can be seen in the
MHSO4/MHSeO4 systems, varying the size of the tetrahedra has an equal, if not greater,
effect on superprotonic transitions, Table 3.1. As stated before, this anion size effect
contradicts the assumption that bigger X-X distances are the critical measure for
transitions, as bigger tetrahedra in an otherwise unchanged structure should cause more
steric hindrances between the oxygen atoms of the tetrahedra. Instead, increasing the size
of a tetrahedron, which is equivalent to increasing the distance, seems to decrease
these inhibiting interactions. For this reason, it is sensible to assume that the increased XO distances allow for a greater degree of freedom in the oxygen’s position as a
tetrahedron rotates/librates. Similarly, a larger cation radius equates to proportionally
larger distances, creating “floppier” MOx polyhedra. An increase in the
distance therefore causes both the XO4 tetrahedra and MOx polyhedra to loosen up, which
can be seen experimentally in the increasing thermal parameters of the oxygen atoms
with increasing distances, Figure 3.14. The good match between the two
parameters is particularly pleasing since this comparison includes all the compounds
presented in this work, plus all the published compounds from the MH2XO4 and mixed

90
MHYO4-MH2XO4 family of compounds (M = alkali metals and NH4; X = P, As; Y = S,
Se).

Oxygen (Angstroms )

Compounds:
without
with
Superprotonic Transitions
NH4HSeO4

R = 0.68
3.5

3.6

3.7

3.8

3.9

4.0

4.1

4.2

4.3

4.4

(Angstroms)

Figure 3.14 Average thermal parameters of the oxygen atoms versus distances:
the two parameters generally scale with each other. The dashed lines denotes the cutoff
between the with and without transition regions of the graph. It appears that either distances larger than ~ 4.1 Å and/or Oxygen parameters greater than ~ 3.0 Å2 in
a room temperature compound are likely to produce a superprotonic transition. Note that
one might not predict the NH4HSeO4 compound to transform from these criterion, but its
transition is probably facilitated by the presence of the highly directional ammonium
ions. Crystallographic data was taken from various sources.
The distance is then a measure of the overall mobility of the oxygen
atoms, an increase in which should lower any barriers to tetrahedral reorientations.

91
Specifically, the oxygen atoms would have more flexibility to avoid close contact with
the electrons of the cations by bending their respective M-O and X-O bonds. With respect
to the cation size effect, this flexibility in the M-O bonds could be restated in terms of the
higher polarizability of the cations as their radius increases, Figure 3.15. In this case, it
would be the electrons of the cations that are adjusting their positions, resulting in the
oxygen atoms having access to positions not available to them with smaller cations. Such
a phenomenon also explains the observed increase in room temperature protonic
conduction of the M2(HSO4)(H2PO4) compounds as the cation radius is enlarged (section
3.3.3), bigger movements of the oxygen atoms facilitating the formation and migration of
defects. Larger M-X distances then assist both room temperature conduction and
superprotonic transitions by enhancing the mobility of the oxygen atoms and thereby
reducing barriers to structural rearrangements.

Electric Dipole Polarizabilities (Angstroms )

3.5

3.0

R = 0.97

2.5
2.0

1.5

1.0
0.5

0.0

1.3

1.4

1.5

1.6

1.7

1.8

(Angstroms)

Figure 3.15 Cation radius versus polarizablity shows a nearly linear relationship
between the two parameters. Larger cations therefore lead to “floppier” MOx polyhedra.
Polarization data taken from calculated electric dipole polarizabilities of M+1 cations126.
This reduction of the barriers to tetrahedral reorientations can be visualized
energetically by considering that longer M-X distances will lead to weaker M-O and X-O
bonds. The potential wells in which the oxygen atoms reside will therefore become
increasingly shallow as M-X distances are lengthened. For such potentials, oxygen atoms
will have a larger range of motion and smaller transition energies when compared to the
deeper potential wells associated with smaller M-X distances, Figure 3.16. The
transitions under consideration here are distortions from the optimal arrangement of the
oxygen atoms due to the formation of defects and/or XO4 reorientations. This energetic
explanation of the cation/anion size effect then further illuminates the correlation
between the magnitude of a room temperature phase’s protonic conductivity and its
probability of having a superprotonic transition.

a)
b)

∆E1

Energy

D1
∆ E2

D 1 > D2
r1 > r 2
∆ E1 < ∆ E2

Distance
Figure 3.16 Schematic representation of the potential wells for oxygen atoms with a)
longer and b) shorter M-O/X-O distances. A shallow potential well associated with a long
M-X distance results in a large range of motion for an oxygen atom, but small transition
energy necessary to reach a distance outside of this range. In contrast, a shorter M-X
distance will result in the oxygen atom residing in a deeper well with a smaller range of
motion and bigger transition energy.
As all the above interpretations of the cation/anion size effect are quite general in
nature, larger M-X distances should facilitate tetrahedral reorientations, and thereby
superprotonic transitions, in a similarly general manner. However, this effect will be most
evident for superprotonic transitions in which almost freely rotating tetrahedra are
required. For the superprotonic transitions of the MHXO4 compounds, the
distance is then a good chemical-crystal measure with which to predict the presence of

94
superprotonic transitions, the known superprotonic phases for this family of compounds
having highly disordered tetrahedra. For other compounds, the anion/cation size effect
should still apply, but may not be the determining factor in the presence or absence of a
transition, other structural effects having a more dominant role (i.e., the M3H(XO4)
family of compounds). Even in such compounds, the results of these studies should help
reveal exactly what is the critical parameter, as any cation/anion size effects can be
examined in the manner shown here and removed from consideration if they are found
not to fully describe the situation. Moreover, as the stoichiometry of a compound can
often be used to guess its possible superprotonic structure (which in turn governs how the
tetrahedra will reorient), the search for new superprotonic conducting solid acids can be
narrowed to those most likely to have a transformation using the criteria described in this
chapter. This focused attention will hopefully speed up the process of synthesizing novel
solid acids with properties ideal for application.

Chapter 4.

Mixed Cesium Sulfate-Phosphates:

Driving Force for the Superprotonic Transitions
of MHnXO4 compounds (M = Cs, Rb, NH4;
X = S, Se, P, As)
4.1 Introduction
Preliminary investigations into the CsHSO4-CsH2PO4 system127 were conducted
in the hope of explaining why CsHSO4 exhibited a superprotonic transition at 141° C3
while CsH2PO4 was reported to decompose and/or transform to a cubic phase around
230° C46,93. This difference in high temperature properties was in spite of the structural
similarities of the compounds at room temperature with regards to the arrangement of the
Cs+ cations and tetrahedral anions 26,28. By making solid solutions of CsHSO4 and
CsH2PO4, it was anticipated that compounds with varying S to P ratios could be created.
Analysis of such compounds could help answer questions about the driving force behind
superprotonic transitions similar to that of CsHSO4. Questions such as what structural
features are necessary for a transition to occur? Does the presence of phosphorus
somehow hinder the transition? How is the transition effected by the density and
distribution of hydrogen bonds?
Unfortunately, solid solutions proved impossible to achieve in the initial and all
following studies into the CsHSO4-CsH2PO4 system. Fortunately, these studies did
discover many new line compounds with varying S:P ratios. And indeed, the analysis of

96
these compounds answered many of the initial questions: The presence of phosphorous
does not prohibit superprotonic phase transitions as all the mixed cesium sulfates to date
have superprotonic phases at elevated temperatures. There are no apparent structural
features essential to these transitions; the mixed compounds having a very diverse set of
room temperature structures. Neither the density of hydrogen atoms in the structure
(varying from a H:XO4 of 1:1 to 2:1), their distribution (linking the tetrahedra into 1-D,
2-D, or 3-D networks), or their local geometry(symmetric or asymmetric) are a
determining factor in the presence or absence of a transition. Nevertheless, the question
of what exactly was the driving force behind these transitions still remained unanswered.
The most obvious answer is entropy, since the superprotonic phases of these compounds
were found to have disordered oxygen atoms while the room temperature phases have
fixed oxygen positions. Until very recently, however, precisely how entropy was driving
the transitions was not clear. In fact, this chapter is dedicated to not only an in-depth
description of the structures and properties exhibited by the mixed cesium sulphatephosphates, but mainly to a theory that describes the change in transition entropy as the
S:P ratio is varied.

4.2 Characterization of Mixed Cesium Sulfate-Phosphates
As was the case with CsHSO4 and CsH2PO4, the basic arrangement of cations and
tetrahedra is quite similar for all the mixed compounds. However, the actual structures
and properties of the compounds can be quite different from each other. This section will
give a general comparison of the mixed compounds, as well as CsHSO4 and CsH2PO4, in
terms of their room and high temperature structures, conductivities, and thermal

97
properties. Such a comparison is in preparation for the entropy calculations that will
follow, in which the structures of each compound will be examined in more detail.

4.2.1 Synthesis of the Compounds
The synthesis of these compounds was not trivial. Although most of the
compounds can be grown from slow water evaporation of an aqueous solution, the
resulting crystals are extremely sensitive to the solution stoichiometry, synthesis method,
and solution temperature. Table 4.1 shows the particulars for the compounds used in this
work. The reagents for all compounds listed in consisted of high purity cesium carbonate
powder (99.999%), and aqueous solutions of sulfuric (98%) and phosphoric acid (86%).
It was essential to keep the solutions free from contaminants, particularly other metal
cations, for the properties of the compounds to remain consistent.

Table 4.1 Synthesis of the Mixed Cesium Sulfate-Phosphates
Compound
CsHSO4
Cs3(HSO4)2.50(H2PO4)0.50
Cs3(HSO4)2.25(H2PO4)0.75
Cs3(HSO4)2(H2PO4)
Cs5(HSO4)3(H2PO4)2
Cs2(HSO4)(H2PO4)
Cs6(H2SO4)3(H1.5PO4)4
CsH2PO4

Compound- Solution- Solution- Method
S:P
S:P
Cs:XO4
100:0
100:0
1:1
Slow H2O
evaporation
83:17
75:25
1:1
Slow H2O
evaporation
75:25
55:45
1:1.5
Methanol
precipitation
67:33
70:30
1:1
Slow H2O
evaporation
60:40
50:50
1:1
Agitated H2O
evaporation
50:50
50:50
1:1
Slow H2O
evaporation
43:57
45:55
1:1
Slow H2O
evaporation
0:100
0:100
1:1
Slow H2O
evaporation

Temp.
(°C)
25
25
25
25
60
10
25
25

The primary synthesis route attempted was the room temperature evaporation of
aqueous solutions with varying S:P ratios, but a fixed Cs:XO4 ratio of 1:1. High-quality
single crystals were not always attained by this process, requiring further
experimentation. The highlighted cells in Table 4.1 show the most significant departures
from the normal route. For both the Cs3(HSO4)2.25(H2PO4)0.75 and Cs5(HSO4)3(H2PO4)2
compounds, it was not possible to acquire the quantity and quality of single crystals
desired, so high quality powders were made. For all other compounds high quality, but
not always high quantity, single crystals were synthesized.

4.2.2 Structural Features of Room Temperature Phases
The room temperature structures of the mixed cesium sulfate-phosphates,
including CsHSO4 and CsH2PO4, are listed in Table 4.1. These room temperature phases
are comprised of isolated anion tetrahedra linked together by hydrogen bonds with the
Cs+1 cations arranged between the anions in loosely defined CsO10-12 polyhedra. The
polyhedra have an average Cs−O distance of 3.28 Å with a range of 3.02 to 3.72 Å, all
typical values for Cs coordinated polyhedra83. Not surprisingly, the bond valence sums
calculated using the Cs−O distances of these polyhedra give values very close to expected
value of 1.0128.
Tetrahedra found in these structures are quite regular with deviations from the
ideal O−X−O angle of 109.5° and expected X−O bond distances attributed to the presence
of hydrogen bonds. Hydrogen bonds are well known to cause an increase in the

Table 4.2 Structural parameters of the mixed cesium sulfate-phosphates in their room and high temperature phases.
Compound
CsHSO4-II

S:P
1:0

H:XO4
1:1

RT structure

H-bond network

HT (Superprotonic)

and type

structure

1-D, chains – 1 ordered

Tetragonal, I41/amd

26,61

Monoclinic,

3-D, cross-linked chains –

Cubic, Pm 3 m &

C2/c

2 ordered & 1 disordered

Tetragonal, I41/amd*

Monoclinic,

3-D, cross-linked chains –

Cubic, Pm 3 m &

C2/c*

2 ordered & 1 disordered

Tetragonal, I41/amd*

Monoclinic,

3-D, cross-linked chains –

Cubic, Pm 3 m*

P21/n

3 ordered & 2 disordered

Monoclinic,

2-D, cross-linked &

Cubic, Pm 3 m*

C2/c

branched chains – 2 ordered

Cubic, Pm 3 m*

Monoclinic,

refs

P21/c*
Cs3(HSO4)2.50(H2PO4)0.50

Cs3(HSO4)2.25(H2PO4)0.75

Cs3(HSO4)2(H2PO4)
Cs5(HSO4)3(H2PO4)2

5:1

3:1

2:1
1.5:1

1.16:1

1.25:1

1.33:1
1.4:1

& 3 disordered
Cs2(HSO4)(H2PO4)
Cs6(H2SO4)3(H1.5PO4)4

1:1
0.75:1

1.5:1
1.71:1

Monoclinic,

2-D, cross-linked chains –

P21/n*

1 ordered & 1 disordered

Cubic I 4 3d*

3-D, inter-connected XO4’s

Cubic, Pm 3 m?*

– 1 ordered
CsH2PO4
*this work

0:1

2:1

Monoclinic,

2-D, cross-linked chains –

P21/m

1 ordered & 1 disordered

Cubic, Pm 3 m

28,92

100
X−O distances of both donor and acceptor oxygen atoms while simultaneously decreasing
the remaining tetrahedral X−O bond lengths22. This effect gives rise to the 1.43 to 1.58 Å
range of X−O distances found in the room temperature phases of the compounds.
However, overall these deviations do not unduly distort the tetrahedra and the average
X−O distance for the whole group is 1.5 Å, in between the typical values of ~ 1.52 and ~
1.47 Å for PO4 and SO4 tetrahedra, respectively83. Moreover, bond valence sums
calculated using the P−O and S−O distances give values very close to the expected
numbers of 5 and 6 for phosphate and sulfate tetrahedra, respectively, while the same
calculation on tetrahedra with a mixed central cation results in intermediate values128.
Finally, the angles of the tetrahedra are quite normal for sulfates and phosphates, ranging
from a low of 102.3 to a high of 114.8°22.
Thermal displacements for the Cs, S, and P atoms are all rather isotropic, whereas
the oxygen atoms most often have the greatest thermal displacements in the direction
perpendicular to the X−O bond, as expected for such compounds. Both asymmetric and
symmetric hydrogen bonds are present in the room temperature phases of the mixed
cesium sulfate-phosphates. These bonds have the chemical and geometric features typical
of strong to medium strength hydrogen bonds with an average O···O distance of 2.54 Å
and a range of 2.45 to 2.64 Å5,17.
In some cases the symmetric hydrogen bonds have sufficiently short O···O
distances (< 2.47 Å ) to have a single minimum potential well; however, crystallographic
data and the similarities of the compounds strongly suggest that all the symmetric
hydrogen bonds in the room temperature phases have double-minima potential energy
wells. For example, CsH2PO4 is well known to have a ferroelectric transition at 159 K,

101
attributed to the protons ability to hop between the two minima of the paraelectric phase’s
symmetric hydrogen bond (O···O distance of 2.472(7) Å)28,81. The symmetric hydrogen
bonds in these room temperature phases are hence often called “disordered” since the
hydrogen resides equally on either side of the double-minima potential well.
Consequently, asymmetrical hydrogen bonds are frequently termed “ordered” hydrogen
bonds.
Despite the stoichiometry differences, all but the Cs6(H2SO4)3(H1.5PO4)4
compound have similar anion and cation arrangements in their structures. This underlying
configuration is best described as zigzag chains of hydrogen bonded tetrahedra that
alternate with similarly zigging and zagging rows of cesium atoms in a checkerboard
appearance. As the phosphorous content of these compounds increases, the chains of
anions become increasingly more cross-linked, resulting in the diverse set of hydrogenbonded networks found in these compounds. In Figure 4.1, it can be seen that both room
temperature phases of CsHSO4 and CsH2PO4 are comprised of zigzag rows of hydrogen
bonded tetrahedra, the chains being cross-linked in CsH2PO430. The zigzag chains of
sulfates and phosphates are clearly visible in Figure 4.1 a and c, respectively, while the
straight cross-linking chains in CsH2PO4 can be seen in Figure 4.1 d. The checkerboard
pattern to the chains and rows of anions and cations are evident in Figure 4.1 b and c, for
CsHSO4 and CsH2PO4, respectively. This arrangement is aligned down the zigzag chains
(c-axis) in the sulfate compound, but runs perpendicular to such chains in phosphate
compound, where the pattern is observable down the straight chains (c-axis), Figure
4.1 c.

102

r r
Figure 4.1 Room temperature structures for CsHSO4 and CsH2PO4: a) the b x c plane of
CsHSO4 showing zigzag chains of sulfate tetrahedra parallel to c and b) view down the
c-axis revealing the checkerboard arrangement of cation and anion chains26; c) in
CsH2PO4 hydrogen bonds connect the phosphate groups into zigzag chains along b and
d) cross-link the tetrahedra into straight chains running parallel to the c -axis28. The
rectangles represent the unit cells of the compounds.

103

The X-ray powder diffraction patterns for these compounds are shown in Figure
4.2. The structures of the first four compounds from the bottom to top are very similar,
with the phosphate tetrahedra being incorporated in ever increasing amounts to every
third tetrahedra down the zigzag chains of CsHSO4, until in Cs3(HSO4)2(H2PO4), these
sites are occupied exclusively by phosphate groups. This structural likeness results in the
similarity of the low angle peaks for these four X-ray powder diffraction patterns. The
variation of the top four patterns reveals that although these compounds have very similar
general structural features, they can be crystallographically quite different from each
other in there room temperature phases.
It should be noted here that the Cs6(H2SO4)3(H1.5PO4)4 compound is quite unique
for the mixed compounds in terms of both its arrangement of cations and anions and
hydrogen bonded network. This difference is most apparent in the cubic symmetry of its
room temperature phase instead of the otherwise universal monoclinic symmetry of the
other seven room temperature phases. Its uniqueness is undoubtedly due to the fact that it
is the only compound not to have a Cs:XO4 ratio of 1:1 (it has 6:7), with a proton taking
the place of a Cs+1 cation, resulting in a structure quite distinctive among these
compounds.

104

30000

CsH2PO4
25000

Intensity

Cs6(H2SO4)3(H1.5PO4)4
20000

Cs2HSO4H2PO4
Cs5(HSO4)3(H2PO4)2

15000

Cs3(HSO4)2(H2PO4)
10000

Cs3(HSO4)2.25(H2PO4)0.75
Cs3(HSO4 )2.5(H2PO4 )0.5

5000

CsHSO4

2 Theta (Degrees)
Figure 4.2 X-ray powder diffraction patterns of the mixed cesium sulfate-phosphates at
room temperatures (~ 25°C).

4.2.3 Structural Features of High Temperature Phases
Although some of the compounds were previously known and their high
temperature structures previously investigated, the high temperature diffraction patterns
of all the compounds (with the exception of CsH2PO4) were collected to permit direct
comparisons, Figure 4.3.

105

40000

(110)
(100)

35000

(111)

(200)

CsH2PO4 @ 237 C
(210)

(211)

(220)

Intensity

30000

140 C

133 C

25000

140 C

20000

140 C

15000

10000

140 C

5000

(004)
(103)

(101)

140 C

(105) (211) (204) CsHSO @150 C
(112) (200)
(213)
(116)
(312)

2 Theta (Degrees)

Figure 4.3 X-ray powder diffraction patterns of the mixed cesium sulfate-phosphates
above their superprotonic phase transitions. The order of the patterns is the same as that
shown in Figure 4.2. Peaks for the tetragonal and cubic structures of CsHSO4 and
CsH2PO4, respectively, have been indexed. The pattern for CsH2PO4 was calculated from
the published structure92. The * indicates the position of K-beta peaks.
It is quite clear from Figure 4.3 that these high temperature diffraction patterns
resemble that of CsHSO4, CsH2PO4, or a combination of the two. These X-ray diffraction
results, as well as that from preliminary neutron diffraction, advocate that these
compounds exhibit only two structural types at elevated temperatures: a tetragonal body
centered structure and a cubic structure similar to that of CsCl, Figure 4.4. The tetragonal
structure has space group I41/amd as determined by X-ray and neutron diffraction
measurements on the superprotonic phase of CsHSO4. High temperature X-ray powder

106
diffraction measurements on CsH2PO4 (under water saturated atmosphere) revealed that
this phase’s space group is most likely Pm 3 m.

Figure 4.4 Proposed superprotonic structures for CsHSO4 and CsH2PO4. The tetragonal
phase of CsHSO4 (Jirak’s) is shown along its twofold axes a), b) and the cubic phase of
CsH2PO4, c), viewed down [100]. Dashed lines represent the dynamically disordered
hydrogen bonds. The two closely spaced oxygen atoms of the sulfate tetrahedra and
hedgehog appearance of the phosphate groups are a result of tetrahedral disorder of the
tetragonal and cubic phases (2 and 6, respectively)61,92. Rectangles represent the unit
cells.
The tetrahedra in these structures are distributed over crystallographically
identical orientations, the direction and number of which is of some debate in the
literature. For the tetragonal phase of CsHSO4, three distinct structures have been
proposed, exemplified by those of Jirak, Merinov, and Belushkin61,124,125. These
published structures are in agreement with respect to the lattice parameters and symmetry
of the unit cell, as well as the position of the cesium and sulfur atoms, but in marked

107
disagreement in the position of the oxygen atoms. This disagreement is undoubtedly
caused by the disorder of the oxygen atoms (i.e., tetrahedral reorientations), which make
it extremely difficult to determine their exact positions. For the purpose of this paper, the
disagreement boils down to there being either 2 (Jirak and Merinov) or 4 (Belushkin)
orientations for the sulfate groups. Unfortunately, without further experimentation it is
impossible to favor one published structure above the others. The entropy of the high
temperature phases exhibiting this tetragonal structure will therefore need to be
calculated with both 2 and 4 orientations of the tetrahedra. For CsH2PO4, and hence the
other cubic phases, it is quite clear that the tetrahedral groups have six orientations63,92.
The 2/4 versus 6 orientations of the tetrahedra is then the most relevant difference
between the tetragonal and cubic structures, respectively, with regards to evaluating the
configurational entropy of the high temperature phases. The X-ray diffraction peaks
arising from the tetragonal and cubic structures are labeled for CsHSO4 and CsH2PO4 in
Figure 4.3. From a comparison of the patterns, it is evident that the high temperature
forms of the Cs3(HSO4)2.5(H2PO4)0.5 and Cs3(HSO4)2.25(H2PO4)0.75 compounds consist of
a combination of the tetragonal and cubic phases, whereas the other compounds are
purely cubic. It should then be possible to calculate the entropy of these compound’s
high temperature forms once an entropy model for the tetragonal and cubic structures has
been worked out and the amount of each structure in a phase is determined.

4.2.4 Key Features of the Superprotonic Phase Transitions
Although the X-ray diffraction patterns are quite convincing evidence that the
high temperature phases are superprotonic, this assumption was not confirmed until the
ionic conductivity was measured in these phases. Figure 4.5 shows conductivity for this

108
whole group of compounds, all of which exhibit a 2-3 order of magnitude jump in their
conductivity from the low to high temperature phase. The disorder of the oxygen atoms,
or alternatively, reorientations of the tetrahedra are responsible for the phenomenon of
superprotonic conduction in solid acids35. The combination of X-ray diffraction and
conductivity data then justifies assigning the disordered structures proposed for CsHSO4
and CsH2PO4 to the rest of the high temperature phases of the mixed cesium sulfatephosphates.

240

160 140

120

100

CsHSO4
Cs3(HSO4)2.50(H2PO4)0.50
Cs3(HSO4)2.25(H2PO4)0.75
Cs3(HSO4)2(H2PO4)
Cs5(HSO4)3(H2PO4)2
Cs2(HSO4)(H2PO4)
Cs6(H2SO4)3(H1.5PO4)4

-1

Log [σT] Ω cm K

40 C

-1

-2
-3
-4
-5
-6
1.8
1.8 2.0

CsH2PO4
2.4

2.6

2.8

3.0
-1

1000/T [K ]
Figure 4.5 (See caption on next page.)

3.2

3.4

109

-1

Log [σT] Ω cm K

-2
-3
-4
-5
-6
1.8

2.0

2.4

2.6

2.8

-1

1000/T [K ]

3.0

3.2

3.4

Figure 4.5 Protonic conductivity of the mixed cesium sulfate-phosphate compounds
measured by a.c. impedance spectroscopy: a) upon heating of fresh (never heated)
samples and b) upon cooling. All experiments were performed on pressed powder
samples with heating/cooling rates of 0.5°C/min under dry argon, except for the CsH2PO4
compound. The heating data for CsH2PO4 was taken on a large single crystal sample to
decrease the effect of surface dehydration46while the cooling data used pressure (1GPa)
to inhibit decomposition of a pressed pellet sample129. The dashed lines represent cuts in
the temperature axis and the jump in the cooling conductivity for CsH2PO4 in its low
temperature phase is an artifact of the cut.
Before discussing the trends in these data, we note that there are moderate to large
differences between the published transition enthalpies and those reported in this work
(see Table 4.3). The discrepancy in the values could be due to many factors including (1)
use of powder vs. single crystal samples, (2) difficulty in obtaining large quantities of

110
high quality crystals for many of the phases, and (3) sensitivity of material properties to
very slight impurity concentrations. See appendix A for the specifics of these effects.
The greatest cause of variation between measurements is quite possibly the use of
single crystal versus powdered samples. It is well documented that measurements on
powdered samples of these compounds can give highly varying results due to surface
interactions with water92,130. Powdered samples also tend to dehydrate sooner and it was
likely just this effect that gave the erroneous ∆H of 7.6 J/mol for CsH2PO4’s
superprotonic transition131. Therefore, single crystal samples were used wherever
possible and powder samples only when there was no alternative or because a
measurement required it (e.g., powder X-ray diffraction experiments). Regardless of
these potential sources of error, the values presented in this work should be internally
consistent as they were obtained using exactly the same instruments, procedures and
experimental parameters, and executed by the same individual.
Looking at Figure 4.5, it would appear that some properties correlate with
phosphorous content, in particular the superprotonic transition temperature, whereas
others show only a mild or even, erratic correlation. A summary of properties taken from
conductivity, DSC, TGA, and PXD measurements is provided in Table 4.3, most of
which are plotted versus phosphorous percentage in Figure 4.6. It can be seen in Figure
4.6 a, that from CsHSO4 to Cs2(HSO4)(H2PO4), the onset (σ) of the transitions drop from
144° to 78°C, respectively. This observed trend of falling transition temperatures with
rising phosphorus content was the impetus behind this search for the entropic driving
force of the transitions. Of course, from Cs2(HSO4)(H2PO4) to CsH2PO4 the transition

111
temperature increases from 78° to 228°C, a fact that any entropic theory would also need
to explain.
The transition range (T[σ]final – T[σ]onset) on heating also seems to show some
dependence on phosphorous content changing from 6° to 31°C as the S:P ratio decreased
from 1:0 to 1:1, respectively, and then back down to 3°C for a ratio of 0:1, Figure 4.6 b.
An exception to the range-of-transition trend is found in Cs5(HSO4)3(H2PO4)2, which
transforms much faster than the compounds with similar S/P ratios. This is probably due
to its room temperature structure, being composed of alternating layers of CsHSO4 and
CsH2PO4 like layers, which is quite unique among the compounds (see section 4.4.5).
It is interesting to note that these increased transition ranges appear to be a
thermodynamic, rather than kinetic, phenomenon. This conclusion was derived from the
fact that the X-ray powder diffraction patterns measured in the transition regions showed
a reproducible mixture of the room and high temperature phases. Moreover, these
diffraction patterns confirmed the onset temperatures and ranges observed in the
conductivity measurements even though the powders were held above the transition
temperatures for ~ 2 hrs/pattern. Presumably, if the width of a transition upon heating is
due to a kinetic process, the heating rate of the measurement would have a large effect on
this width. However, thermal analysis on these compounds gave results data very similar
to those of the conductivity and diffraction experiments. Experiments with nominal
heating rates of ~ 5, .5, and 0.05°C/min (DSC, σ, PXD rates, respectively) then all gave
roughly the same values and therefore, the spans of the transitions should truly be a result
of the thermodynamics of the phase.

112

TONSET - DSC
TONSET - σ
TPEAK - DSC
TFINAL - σ

240
220

Temperature ( C)

200
180
160

CHS

β'

140

Cs5

120

CDP

Cs6
Cs2

100
80

100

% PO4

b) 34
32

26
24
22

20
18
16
14

12
10

TransitionRange (heating)
Hysteresis (cooling)

% PO4
Figure 4.6 (See caption on next page.)

100

Log [Hysterisis] (seconds)

Transition Range ( C)

30
28

-7

∆ H transition

∆ V transition

Enthalpy Change (kJ/mol of CsHXO4)

c) 16

Volume Change (m /mol of CsHXO4 x 10 )

113

100

% PO4

∆ Htransition

EHbond

12
60
10

H-bond Energy (kJ/mol of CsHXO4)

Enthalpy Change (kJ/mol of CsHXO4)

d) 16

100

% PO4
Figure 4.6 Various transition properties versus phosphate percentage: transition a)
temperature upon heating by various definitions, b) range and hysteresis upon heating
and cooling, respectively, c) enthalpy and volume change, and d) enthalpy compared to
H-bond energy of RT structures. The range and hysteresis values taken from conductivity
data collected at 0.5°C/min in ambient atmospheres. DSC data obtained at 5°C/min under
flowing N2. Transition volume changes calculated from PXD pattern refinements (see
appendix A for measurement specifics). H-bond energies calculated using H-bond energy
vs. O-O distance correlation21 and RT structures (references on Table 4.2).

114

Upon cooling, the presence of phosphate groups has perhaps an even more
dramatic effect, with some of the higher phosphorus content compounds revealing no
reverse transition in both the temperature and time scale of the conductivity
measurements, Figure 4.5 b. This effect has been quantified as the hysteresis, temperature
difference between the end of the transition on heating and beginning of the transition on
cooling, and is shown as function of PO4 percentage in Figure 4.6 b. As measured by
powder X-ray diffraction, this transition hysteresis from a high to low temperature form
can last from days to months (see Appendix A). However, this trend does not directly
correlate to the increasing phosphate content of the compounds going from CsHSO4 to
CsH2PO4. Even if the results of the Cs6(H2SO4)3(H1.5PO4)4 compound are excluded
because of its unusual Cs:XO4 ratio, recent experiments on CsH2PO4 have definitely
shown a very fast superprotonic transition to occur above 230°C with a hysteresis effect
upon cooling similar to that found in CsHSO4129.
Furthermore, the behavior of the transitions for the Cs3(HSO4)2.5(H2PO4)0.5,
Cs3(HSO4)2.25(H2PO4)0.75, and Cs3(HSO4)2(H2PO4) compounds on cooling seem to be
reversed with the higher phosphorus content compound transforming rapidly from the
high to low temperature phase, while the compound with the least amount of
phosphorous transforms over a range of 38°C, Figure 4.5 b. These compounds also show
a strange behavior to the onset temperature of the reverse transitions, with a hysteresis of
22, 16, and 53 degrees for the Cs3(HSO4)2.5(H2PO4)0.5, Cs3(HSO4)2.25(H2PO4)0.75, and
Cs3(HSO4)2(H2PO4) compounds, respectively. Again, the phosphate content of the three
compounds does not obviously relate to the observed trend. These mixed compounds

115

Table 4.3 Thermodynamic parameters of the superprotonic phase transitions. The sometimes large errors of the values are due to
variations between measurements. These variations were particularly noticeable for the compounds with drawn-out transitions and did
not decrease significantly with the number of measurements. Numbers in [ ] brackets are the published values.
Compound
(Sample type)-abbreviation

Tsp – Onset,
DSC (ºC)

Tsp – Onset,
σ (ºC)

Tsp – Peak,
DSC (ºC)

Tsp – Final,
σ (ºC)

Tmelt/decomp
(ºC)

∆Hsp
(kJ/mol )

∆Vsp
(m3/mol)*x 10-7

refs

CsHSO4-II

142(2)

144(1)

147(2)

150(1)

Melt-200(3)

6.2(2)

2.1(13)

4,59,130

(Single crystals)-CHS

[142(2)]

[139]

[145(2)]

[141]

[205]

[5.5]

[1.7]

Cs3(HSO4)2.50(H2PO4)0.50

123(1)

131(1)

143(1)

137(1)

Melt-175(3)

7.4(2)

5.1(11)

(Single crystals)-β

[130.8]

[119(2)]

[140.2]

[129(2)]

Cs3(HSO4)2.25(H2PO4)0.75

116(6)

117(3)

141(4)

137(1)

Melt-150(2)

8.3(5)

10.9(8)

Cs3(HSO4)2(H2PO4)

104(6)

106(3)

141(1)

137(1)

Melt-148(2)

10.7(2)

13.5(11)

30,127

(Single crystals)-α

[139]

[111]

[143]

[125]

[150]

[9.2]

Cs5(HSO4)3(H2PO4)2

96(3)

89(4)

117(4)

107(1)

Decomp-180(2)

9.2(7)

N/A

133

(Powder)-Cs5

[110(2)]

Cs2(HSO4)(H2PO4)

85(2)

78(2)

94(1)

109(1)

Decomp-185(2)

8.3(2)

7.6(24)

27,63

(Single crystals)-Cs2

[61]

[75(2)]

[65]

[110(1)]

[187]

[10.1(5)]

Cs6(H2SO4)3(H1.5PO4)4

101(8)

90(5)

124(9)

120(1)

Decomp-200(3)

15.1(6)

25.3(8)

CsH2PO4

230(2)

228(1)

239(4)

231(1)

Decomp-203(3)

11.3(5)

[10.8(14)]

(Single crystals)-CDP

[230]

[229(1)]

[233(1)]

[232(1)]

[175]

[7.6]

132

[6.9(2)]

(Powder)- β′

[116(2)]

[10.2]

(Single crystals)-Cs6
46,92,93,134

*Values are given per molar unit of CsHXO4; the Cs6(H2SO4)3(H1.5PO4)4 compound was assigned a value of 6.4 molar units (based
on MW ratios, interatomic distances, and simple geometric considerations) when converting the ∆H and ∆V from the measured units.

116
then have a very complicated set of reverse transitions. Any possible overarching
explanation for the behavior of these transitions with respect to the phosphorus content of
the compounds is likely to be similarly complex. At this time and with the limited data
available, no such model presents itself which sufficiently explains the varying behavior
of these compounds upon cooling.
A look at Figure 4.6 c shows that the change in transition enthalpies with
phosphate percentage is closely mimicked by the volume change of the transitions. This
would suggest that the amount of energy required to transform into a high temperature
phase is directly linked to the increase in volume necessary to achieve that phase’s
structure. However, the work required for even the substantial transition volumes of these
compounds is inconsequential (at ambient atmosphere). For the Cs6(H2SO4)3(H1.5PO4)4
compound, with the largest volume change, the reversible work done by the crystal
would only be P∆V ≅ (1x105 Pa)*(25.3x10-7 m3/mol-CsHXO4) ≅ 0.25 m3Pa/mol =
2.5x10-4 kJ/mol, an insignificant value when compared to the transition enthalpies. It
must then be the internal energies of the compounds which are changing across the
transitions.
This change in the internal energy is most easily attributed to the loss of ionic and
hydrogen bond energies, resulting from increased interatomic distances and the dynamic
behavior of the high temperature phases. With respect to the hydrogen bond energy, this
statement is supported by Figure 4.6 d, which shows a fairly good correlation between the
transition enthalpies and their RT structure’s mean hydrogen bond energy. This mean
energy was calculated using the published O-O distances and multiplicities of the
hydrogen bonds and the energy versus O-O distance function given by Lippincott et al.21.

117
The observed correlation is quite logical since the more hydrogen bond energy a
compound has at room temperature, the more it has to lose when transforming to a higher
volume structure, which will necessarily show-up in the transition enthalpy. This
statement is particularly true for compounds which have similar high temperature
structures (i.e., the pure cubic phases). The Cs5(HSO4)3(H2PO4)2 compound, as with the
range-of-transition trend, is the odd man out, which is again attributed to its unusual
structure.
Of course there must be an energetic advantage to a transition or it will not
happen. This energetic benefit is most easily modeled through a configurational entropy
change across a compound’s transition which we can compare to the experimental value,
derived from the measured transition enthalpy and temperature via ∆G = 0 ⇒∆S = ∆H/Tc.
It is therefore the experimental entropies that these calculations will aim to duplicate.

4.3 Introductory Comments on Entropy Rules
The following two sections will set down the rules used in calculating the
configurational entropy of each compound’s room and high temperature structure, and
thereby, the entropy change (disregarding other non-structural forms of disorder) of its
superprotonic transition. That two different sets of rules are required for evaluating the
entropy of the low and high temperature phases is probably not surprising. However, the
fact that the entropy rules for the “static” low temperature structures are actually more
complicated and subjective than those that describe the “dynamically disordered”
superprotonic phases, was indeed unexpected (at least to this researcher). After all, at

118
room temperatures the exact positions of the compounds’ atoms are known. Any
configurational entropy should, as a result, be easily identified and accounted for.
Unfortunately, in appraising the entropic contributions from symmetric hydrogen bonds,
mixed S/P sites, and partially occupied hydrogen positions, many “best guess” estimates
will have to be made based on the results of relevant literature and its implications to the
particular structure in question.
In contrast, the rules applied to the highly disordered, but also highly symmetric,
superprotonic phases are quite universal; applying equally well to both the cubic and
tetragonal structures. The process for evaluating the entropy of these phases combines
Pauling’s ice rules for the calculation of the residual entropy of ice at 0 K with the
orientational disorder of the tetrahedra135. This approach gives a much better agreement
to experimental results than traditional methods which focus on only the disordering of
the protons or on tetrahedral reorientations, but not both136,137.

4.4 Entropy Rule for Room Temperature Structures
Contributions to the entropy of the room temperature structures come from two
basic sources: mixed S/P sites and hydrogen bond disorder. The mixed S/P tetrahedra
result in the well-known entropy of mixing for two species on one site:

S mix = − R[(1 − N1 ) ln(1 − N1 ) + N1 ln( N1 )]

(4-1)

where N1 and (1- N1) are the mole fractions of S and P (or vice versa) on the site and R is
the universal gas constant = 8.314 J/mol*K. However, in these compounds, the local
density of hydrogen around a tetrahedron should (very generally) be more nearly 2 for a

119
phosphate and 1 for a sulfate. An alternative phrasing for this fact is that the phosphates
are usually involved in four hydrogen bonds (1 per oxygen), while sulfate tetrahedron
typically participate in two hydrogen bonds. This general rule is followed in all the room
temperature phases of these compounds (except the Cs5(HSO4)3(H2PO4)2 and
Cs6(H2SO4)3(H1.5PO4)4 compounds) and agrees with the difference in pK values of
phosphoric and sulfuric acid138. Therefore, one might expect that the particulars of the
local structure will greatly affect the specifics of the local hydrogen bonds. There would
then be a direct correlation between the local occupancy of a mixed S/P site and the
second source of entropy in these room temperature phases, hydrogen bond disorder.
This disorder of the hydrogen bonds comes in two forms: partial occupation and
symmetric distribution. The presence of partially occupied hydrogen bonds occurs only in
the structures of Cs3(HSO4)2.50(H2PO4)0.50 and Cs3(HSO4)2.25(H2PO4)0.75, while all but the
CsHSO4 and Cs6(H2SO4)3(H1.5PO4)4 compounds, have symmetric hydrogen bonds
linking some of their tetrahedra. The entropy of either a partially occupied pair of
hydrogen bonds or a proton disordered over two sites within one hydrogen bond (per
CsHnXO4 unit) is
(Occupancy * 2) * R * ln(2)

S config = (# of CsH n XO units per H −bond pair )

(4-2)

where the occupancy of one bond is multiplied by two to give the occupancy of the pair
and the occupancy on a disordered proton site is defined as 1/2. The evaluation of Eq. (42) will best be done with a specific structure in mind. Moreover, the local ordering of the
S/P sites leads to a rather compound specific determination of how to apply Eq. (4-2).

120

4.4.1 Entropy of CsHSO4 and Cs6(H2SO4)3(H1.5PO4)4 ⇒ZERO!
For the CsHSO4 and Cs6(H2SO4)3(H1.5PO4)4 compounds, calculating their room
temperature entropy is trivial as both compounds have neither mixed S/P sites nor any
form of disordered hydrogen bonds, Figures 4.7.

Figure 4.7 Arrangement of the hydrogen-bonded tetrahedra for a) the sulfate chains of
CsHSO4 and b) the sulfate and c) phosphate groups of Cs6(H2SO4)3(H1.5PO4)4.
Asymmetric bonds connect all tetrahedra. Note the peculiar arrangement of the hydrogen
bonds in b) and c) where the sulfate and phosphate groups are involved in 4 and 3
hydrogen bonds, respectively.
By definition, CsHSO4 has no mixed S/P sites and as the hydrogen bonds linking
the sulfate groups into chains are ordered (Figure 4.7 a), there is only one
crystallographic configuration to the room temperature structure. CsHSO4 then has no
configurational entropy below the phase transition. The Cs6(H2SO4)3(H1.5PO4)4
compound owes its lack of entropy to the fact that it has only one crystallographic sulfate
and phosphate group with hydrogen bonds connecting only dislike tetrahedra, Figure 4.7
b and c. There are then no mixed S/P sites and since the hydrogen bonds are between

121
ordered tetrahedra of a different nature, they are necessarily asymmetric. This structure
consequently has only one configuration and zero configurational entropy.

4.4.2 Entropy evaluation for CsH2PO4 −the disordered hydrogen bond
Unfortunately, there are no such definitive statements concerning the entropy
associated with a symmetric, double-minimum hydrogen bond. This type of bond is
found in CsH2PO4 where such disordered hydrogen bonds link the tetrahedra into zigzag
chains along the b-axis, Figure 4.8.

Figure 4.8 Disordered hydrogen bonds in CsH2PO4 connect the phosphate groups into
zigzag chains along the b-axis139. Chains are cross-linked by asymmetric hydrogen
bonds, so the disordered chains are pseudo-one-dimensional with respect to ferroelectric
behavior.
Classically, a double-minimum potential hydrogen bond will have the proton
residing equally in each minimum with a resulting entropy contribution, using the
formulation of Eq. (4-2), of (1/2*2)*R* ln(2)/1 = R*ln(2) = 5.76 J/(mol*K), where here
the pair refers to the two equivalent proton positions. It turns out that for CsH2PO4 this
classical value is almost double that found experimentally, the possible cause of which
will be discussed here.

122
Upon cooling, CsH2PO4 exhibits a second-order ferroelectric phase transition at
159 K with spontaneous polarization along the b-axis (parallel to the symmetric hydrogen
bonds) 140. As well as being connected into zigzag chains by the symmetric hydrogen
bonds, each tetrahedra is also hydrogen-bonded by two asymmetric hydrogen bonds,
resulting in straight chains running perpendicular to the ferroelectric b-axis (see Figure
4.1 c and d). These cross-linking hydrogen bonds have a weak interaction with the
disordered protons of the zigzag chains. CsH2PO4’s ferroelectric transition is therefore
most often evaluated using a pseudo-one-dimensional Ising model. The classical
Hamiltonian of such a model in an external electric field, E, is

H = − ∑  J σ σ
i, j

i +1, j

i, j

+ J ⊥σ i , j +1σ i , j + µ E σ i , j 

(4-3)

where the two possible values of the pseudospin variable, σ = ± 1, represent the position
of the proton in the double minima potential well and µ is the electric dipole moment of
the spins141. J// and J⊥ are, respectively, the intrachain (from protons along the zigzag
chains) and interchain (from protons in the straight chains) interactions acting upon the
disordered protons of the zigzag chains. The weak interchain coupling is essential to
properly describe the ferroelectric phase transition as the exactly solvable onedimensional Ising Hamiltonian (which excludes the second term of Eq. 4-3) does not
produce a phase transition for any finite temperature142.
Using experimental data from heat capacity, dielectric, and solid state NMR
measurements, the ferroelectric transition of CsH2PO4 has been modeled using
Hamiltonians identical or very similar to that in Eq. (4-3)140,143,144. The ratio of weak
interchain to strong intrachain interactions, J⊥ / J//, is consistently found to be on the order

123
of 1:100145. Such models describe well the critical slowing down of the dielectric
relaxation time, as well as the heat capacity jump of and temperature of the transition.
However, anomalous behavior in the heat capacity and dielectric measurements were
observed significantly above and below the transition (Tc ± 60 K), which cannot be
described using Eq. (4-3). This property was universally attributed to abnormally large
polarization fluctuations which develop below the ferroelectric transition and continue
well into the paraelectric phase. Theoretical discussions of these polarization fluctuations
suggest they are due to anisotropic short range correlations, from the interaction of J//
and J⊥ , resulting in local order extending along the symmetrically bonded chains141.
Such fluctuations would be due to mainly the strong intrachain interaction and should
therefore be most evident for temperatures with kT < J// 146. Indeed, the anomalous
behavior of the heat capacity and dielectric constant measurements ends well before the
average value of J// / k ≈ 275 K.
The other inconsistency with theory based on the Hamiltonian of Eq. (4-3) comes
in the total transition entropy. Even including the entropy of the anomalous regions, the
measured entropy changes were in the range of 1.05(1)-3.2(1) J/(mol*K), much smaller
than the expected value of R*ln(2) = 5.76 J/(mol*K)143,145. Although it has not been
suggested in the literature, this rather large difference between the theoretical and
measured transition entropy would seem (to this author) to be related to the anomalous
behavior seen in the heat capacity and dielectric measurements credited to short range
order running “along the chain(s) for many fundamental chain lengths.”141 It seems
possible that the chains have some small amount of local order even outside the measured
anomalous temperature ranges found in the experiments. This is particularly likely as

124
these anomalous regions were defined by a baseline fit to the heat capacity curves, with
the result that the anomalous temperature range and total calculated entropy change were
quite dependent on the form of the baseline fit143. For temperatures with J// /kT < 1, the
length scale of any such ordering would be quite small, and yet even an average local
order involving only two hydrogen bonds would decrease the entropy of the transition to
1/2*R*ln(2) = 2.88 J/(mol*K), a value much closer to those measured. Such a very short
range ordering would be extremely difficult to measure by the heat capacity and
dielectric experiments due to the very large temperature and frequency ranges,
respectively, necessary to discern the effect.
There is then a dilemma as to the amount of entropy that should be associated
with a disordered symmetric hydrogen bond. For CsH2PO4, this value would seem to be
in the range 3.2 ≤ ∆S ≤ R*ln(2) = 5.76 J/(mol*K). However, as the superprotonic phase
transition occurs nearly 350, 290, and 230 K above the ferroelectric transition, the end of
the anomalous regions, and J// / k, respectively, a value closer to the full R*ln(2) appears
more likely. Hence, the “best guess” value for the entropy of CsH2PO4’s disordered
hydrogen bonds is exactly what one would expect a priori, SHbonds = R*ln(2). The last
three pages may therefore seem unnecessarily pedantic, but their purpose was to lay the
foundation for compound specific arguments on the entropic contribution of other
symmetric hydrogen bonds found in the room temperature phases, where the “best guess”
value may not be that derived from statistical mechanics. Specifically, the interchain and
intrachain interactions should vary significantly from structure to structure, which the
above discussion tells us will have a large effect on the behavior of protons in doublemimima potential wells.

125
For example, the behavior of CsH2PO4 near its ferroelectric transition is quite
different from that of the related compounds KH2PO4, RbH2PO4, and PbHPO4 due to the
different dimensionality (3-D, 3-D, and more fully 1-D, respectively) of those compounds
hydrogen-bonded networks140,144. Moreover, for the compounds in question, the
disordered hydrogen bonds are almost always situated between the mixed S/P sites. The
average structure seen by X-ray diffraction methods could then be a compilation of
locally ordered structures distributed at random, depending on only the occupants of the
nearby mixed S/P sites. This was exactly the result found from a neutron diffraction
measurement taken on Cs3(HSO4)2.50(H2PO4)0.50 at 15 K, where the average structure
reported from X-ray diffraction methods at 298 K was resolved into two different, but
related, local structures (discussed below in section 4.4.3)72. It would then seem that the
entropy contribution from a disordered hydrogen bond will have to be evaluated
independently for each compound. Generally, the most reasonable conclusion reached for
these mixed compounds is that the local occupation of the S/P sites causes an effective
ordering of the protons, thereby greatly diminishing or completely negating the entropic
contribution from the disordered hydrogen.

4.4.3 Entropy of Cs3(HSO4)2.50(H2PO4)0.50 and Cs3(HSO4)2.25(H2PO4)0.75 −
partially occupied hydrogen bonds

The structure of Cs3(HSO4)2.50(H2PO4)0.50 is composed of zigzag chains of
hydrogen-bonded tetrahedra and Cs cations arranged almost identically to those in

126
CsHSO4 but with phosphate groups inserted into every third tetrahedral site of a chain,
Figure 4.9. The ensuing mixed tetrahedra have a S:P ratio of 1.

Figure 4.9 Room temperature structure of Cs3(HSO4)2.50(H2PO4)0.50. Averaged structure
(from X-ray) view down the b-axis, a), shows the altered zigzag chains of CsHSO4
running in the [10-1] direction while the symmetric hydrogen bonds between the mixed
tetrahedra form chains in the [001] direction32.Two variants (from neutron) of the
averaged structure along the zigzag chains, b) and c), showing the effect of local order on
hydrogen bonding72. The rectangle shows a unit cell and Cs atoms are absent for clarity.
With respect to the evaluation of the entropy of this compound, the addition of
phosphates to CsHSO4’s zigzag chains has three effects: the obvious introduction of
mixed S/P sites, formation of symmetric hydrogen bonds linking the mixed tetrahedra of
neighboring chains, and the creation of a symmetry related pair of partially occupied
hydrogen bonds between the adjacent SO4 groups of the zigzag chains, Figure 4.9 a. The
contribution of the mixed sites to the structures entropy can be evaluated by Eq. (4-1),
giving Smix = -(1/3)*R[(1/2*ln(1/2)+1/2*ln(1/2)] = (1/3)*0.69*R = 1.92 J/mol*K, where
the factor of 1/3 comes from the fact that there is only one mixed site for every three
moles of CsHXO4 unit.

127
The entropy contribution from the symmetric bonds can be evaluated with much
less work than that of CsH2PO4 due to very precise diffraction measurements and
structure refinements. The structure of Cs3(HSO4)2.50(H2PO4)0.50 has been studied by both
single crystal X-ray and neutron diffraction experiments at 298 and 15 K, respectively. In
the room temperature X-ray measurement, the symmetric bond between the mixed
tetrahedra was thought to be disordered. This assignment was based on the bond’s O−O
distance of 2.474(9) Å, which is above the lower limit of observed double-minimum
symmetric bonds, and its similarity to other disordered hydrogen bonds such as those
found in CsH2PO432. However, the low temperature neutron experiment found
“unsatisfying” thermal displacement parameters in the refinement using a disordered
hydrogen bond. Furthermore, the persistence of the center of symmetry at such low
temperatures indicates the lack of a ferroelectric transition72. The hydrogen of this bond
was therefore fixed at the symmetry position, thereby making the symmetric bond
ordered with no entropic contribution to the room temperature structure.
This leaves only the entropy of the partially occupied hydrogen bonds to evaluate.
Using the structure resolved by X-ray diffraction measurements, Figure 4.9 a, these
bonds have proton sites with 1/4 occupancy, so that only half of the SO4-SO4 neighbors
along the chains are joined by hydrogen bonds72. From Eq. (4-2), the entropy per
CsHnXO4 unit of these partially occupied proton sites is then SHbond = (1/3)*(1/4*2)*R
*ln(2) = 1/6*R*ln(2) = 0.96 J/mol*K, corresponding to only one of these sites being
occupied, on average, for every six adjoining tetrahedra down a chain.
From the neutron data, it was possible to resolve the average structure found by
the X-ray measurements into two variants that differed in the local order of the mixed

128
sites, leading to two different hydrogen bonded schemes, Figure 4.9 b) and c). For each
of these variants, the central hydrogen bond is 1/2 occupied, so that again on average
only one of these hydrogen bonds will exist for every six neighboring tetrahedra. The
entropy calculated using this locally ordered model is then the same as that calculated
using the averaged structure, SHbond = (1/6)*(1/2*2)*R*ln(2) = 0.96 J/mol*K. Although
the locally ordered (but globablly disordered) model seems to more accurately represent
the real structure of Cs3(HSO4)2.50(H2PO4)0.50, the equivalence between the entropy
contribution from both models is convenient as the difference between their refinement
residuals is very small, and therefore neither structure can be conclusively ruled out.
The Cs3(HSO4)2.25(H2PO4)0.75 compound is assumed to have a nearly identical
structure to that of Cs3(HSO4)2.50(H2PO4)0.50 based on the results of X-ray powder
diffraction measurements and the stated stoichiometry was taken from the results of
electron microprobe experiments (shown in appendix A). This ability to vary the molar
ratio of Cs3(HSO4)2.50(H2PO4)0.50 was proposed with the original structural determination
due to the observation that the S:P ratio was not fixed by the crystal structure, additional
phosphates simply decreasing the S:P ratio of the mixed tetrahedra132. For the
stoichimetry of this compound, the S:P ratio becomes 1:3. The entropy of mixing is then
Smix = -(1/3)*R[(1/4*ln(1/4)+3/4*ln(3/4)] = (1/3)*0.56*R = 1.56 J/mol*K.
The higher phosphate content will also increase the hydrogen content of the
compound with the most logical conclusion being that the proton occupancy of the
partially occupied hydrogen bonds increases from 1/4 to 3/8. This results in an entropy,
per CsHnXO4 unit, of SHbond = (1/3)*(3/8*2)*R*ln(2) = 1/4*R*ln(2) = 1.44 J/(mol*K).
Or, in terms of the locally ordered model, the proton occupancy for each structural

129
variant increases from 1/2 to 3/4, to which is associated an entropy of SHbond =
(1/6)*(3/4*2)*R*ln(2) = 1/4*R*ln(2) = 1.44 J/(mol*K). Again, the entropy contribution
from these partially occupied hydrogen bonds being equivalent for both the average and
locally ordered models.

4.4.4 Room Temperature Entropy of Cs3(HSO4)2(H2PO4)
In this compound every third tetrahedra in the zigzag chains is fully occupied by a
phosphate group, Figure 4.10.

Figure 4.10 Structure for Cs3(HSO4)2(H2PO4) with emphasis on the similarity to the
structures of Cs3(HSO4)2.50(H2PO4)0.50 and Cs3(HSO4)2.25(H2PO4)0.75. The fully occupied
PO4 site results in ordered hydrogen bonds between all tetrahedra except for the bonds
between the phosphate groups where two distinct disordered bonds link the phosphates
into chains along [001]30. Otherwise, the arrangement of anions and cations is nearly
identical to the two related compounds. The rectangle shows the unit cell and Cs atoms
have been removed for the purpose of clarity.

130
This compound’s only form of disorder in the room temperature phase comes
from two crystallographically distinct disordered hydrogen bonds connecting the
phosphate groups into chains down the [001] direction. These chains are structurally very
similar to those found in CsH2PO4, although here the disordered chains should have a
very small interchain interaction as they are separated by two sulfate tetrahedra, instead
being directly linked to each other as in CsH2PO4. The chains should therefore be more
one-dimensional than those of CsH2PO4 and it would be surprising to find a ferroelectric
transition except very close to 0 K. The entropy will therefore be taken to be the classical
value, per CsHXO4 unit, which gives an entropy of (1/6)*(1/2*2)*R*ln(4) = (1/3)*R
*ln(2) = 1.92 J/mol*K to the room temperature structure.

4.4.5 Room Temperature Entropy of Cs5(HSO4)3(H2PO4)2
This compound has a structure that has been described as being composed of
alternating layers of CsHSO4 and CsH2PO4 like regions due to the checkerboard
arrangement of hydrogen bonded chains and Cs rows when view down the c-axis, Figure
4.11 a. A comparison with Figure 4.1 b reveals the similarity of layer I to CsHSO4 when
one looks down the zigzag chains of the structures, while Figure 4.1 c shows the anions
and cations in CsH2PO4 distributed almost identically to layer II except for small
differences in the orientation of the tetrahedra and hydrogen bonds. Crystallographically,
there are two distinct mixed tetrahedra and three distinct symmetric hydrogen bonds in
the structure. The mixed sites, labeled as 1 and 2 in Figure 4.11, have site multiplicities of
4 and 8, respectively. Both mixed tetrahedra have an S:P ratio of 1:2 and the remaining
tetrahedral site is solely occupied by SO4 groups with a multiplicity of 8. There are then
20 tetrahedra in the unit cell with 12 mixed sites of (S,P) occupancy (1/3,2/3). The

131
entropy of mixing from these sites per CsHXO4 unit is then, Smix = -(12/20)*
R*[(1/3)*ln(1/3) + (2/3)*ln(2/3)] = (12/20)*R*[0.366 + 0.270] = 3.175 J/mol*K.

-b

b)
XO4

SO4

Figure 4.11 Structure of Cs5(HSO4)3(H2PO4)2 : a) shown to maximize the similarities of
the different layers, I and II, to the arrangement of cations and anions in CsHSO4 and
CsH2PO4 and b) to reveal the disordered hydrogen bonds connecting the mixed tetrahedra
into chains along the c-axis31. The rectangles illustrate the unit cell and the cesium atoms
have been removed in b) to call attention to the disordered chains.

132
The three distinct symmetric hydrogen bonds are found between the mixed
tetrahedra with two of them, label (a) and (b), linking the type 2 tetrahedra of layer II into
zigzag chains along the b-axis, Figure 4.11 a. The other symmetric bond, (c), connects
the type 1 tetrahedra of layer I into straight chains parallel to the c-axis, Figure 4.11 b.
The O−O distances are 2.589(18), 2.483(17), and 2.517(15) Å for bonds (a), (b), and (c),
respectively. The first bond, (a), is almost certainly disordered due to its long bond
length, large thermal parameters of its oxygen atoms, and bond sum considerations, but
the short X−O distances associated with the bond are inconsistent with such a designation.
For the (b) and (c) symmetric bonds, it is even more unclear whether the bonds have
single or double-minima potential wells. However, on the basis of the low temperature
neutron results for Cs3(HSO4)2.50(H2PO4)0.50, it seems likely that all three bonds are
ordered on a local scale, but globally disordered. This would explain the intermediate
behavior of the bond parameters, as they would pertain to the average structure recorded
by the X-ray diffraction measurement.
In describing this local ordering and evaluating the entropy associated with it,
there will be one assumption: the oxygen atoms of the sulfate groups in the disordered
chains do not act as donor oxygens. This assumption is quite reasonable from considering
structures of the two disordered chains. In layer II, each mixed tetrahedral site is bonded
by four hydrogen bonds, two disordered and two ordered. From the ordered bonds, each
tetrahedra will have one oxygen participate as a donor and one as an acceptor. For a
sulfate group thus bonded, its oxygen atoms are much less likely to participate as donor
oxygens in the disordered bonds compared to the oxygen atoms of a similarly bonded
phosphate group. Such an effect would follow from the difference in pK values of

133
phosphoric and sulfuric acid138. The disordered chains of layer I also have four hydrogen
bonds per tetrahedra, two of which are disordered and two ordered. However, for these
mixed tetrahedra, the oxygen atoms involved in the two asymmetric hydrogen bonds are
acceptors. In contrast to the disordered bonds of layer II , a sulfate on such a mixed site
would desire to have one of its remaining oxygens act as a donor atom. However, the
similarly bonded neighboring phosphate groups would be proportionally more hungry for
the proton than the sulfate groups. Therefore, the protons in layer I’s disordered chains
should not reside next to the sulfate groups, resulting in exactly the same local ordering
effect, although for a different reason.
It should be mentioned that the assumption used in these arguments (that the
oxygen atoms on the sulfate groups involved in disordered hydrogen bonds do not act as
donor oxygens) ignores the possibility of two sulfate groups residing next to each other.
In such a case, either the hydrogen bond between the sulfate groups would be truly
disordered or other local effects would favor one of the oxygen atoms over the other. The
local ordering arguments to date would tend to suggest that the later case happens more
often and in any case, the two SO4 groups should be neighbors only 1/3*1/3 = 1/9th of
the time. It therefore seems reasonable to ignore such configurations when calculating the
configurational entropy of this compound’s room temperature structure.

134

Figure 4.12 Probable effect of local order in the mixed S/P sites on neighboring
disordered hydrogen bonds: a) the average structure with disordered hydrogen bonds
connecting the mixed tetrahedra and b) the two variants due to the local arrangement of
the sulfate and phosphate groups.
With this assumption it becomes easy to evaluate the entropy of the disordered
bonds based on the average unit of two phosphates and one sulfate tetrahedra. As the
proton sites adjacent the sulfate group will be vacant along the disordered chains, the two
disordered bonds near it will become locally ordered, leaving only the hydrogen bond
between the phosphates with a choice to the position of its proton, Figure 4.12.

135
As there are twelve of these mixed S/P sites per twenty tetrahedra in the unit cell, the
entropy per mol of CsHXO4 will be SHbond = (12/20)*(1/3)*R*ln(2) = (1/5)*R*ln(2) =
1.15 J/mol*K.

4.4.6 Room Temperature Entropy of Cs2(HSO4)(H2PO4)
Cs2(HSO4)(H2PO4)’s room temperature structure is composed of zigzag chains of
tetrahedra cross-linked to neighboring chains to form a planar structure, Figure 4.13 a.

r r r
Figure 4.13 Structure of Cs2(HSO4)(H2PO4) in a) the b x (a + c ) plane showing the
sheets of XO4 tetrahedra made of ordered hydrogen bonded chains in the [010] direction
being connected by disordered hydrogen bonds in the [101] direction. The planes are also
visible when viewed down the b-axis, b), as is the checkerboard arrangement of anion
and cation rows. Rectangles represent the unit cell and Cs atoms were omitted in a) for
clarity.

A unique property of this phase is that there is only one crystallographic (S,P)
site, with a S:P ratio fixed by the stoichiometry of the compound at 1:1. This results in a
unique hydrogen-bond network where each tetrahedra is involved in three hydrogen

136
bonds: two ordered bonds (crystallographically identical) that link the tetrahedra into
chains running down [010] and one disordered bond in the [101] direction that connects
r r r
the chains into sheets in the b x (a + c ) plane, Figure 4.13 b). As the X site is evenly

occupied by S and P atoms, the entropy of mixing for this compound is the maximum
value of Eq. (4-1) Smix = -R*[2*(1/2)*ln(1/2)] = R*ln(2) = 5.76 J/mol*K.
There is some suggestion that the asymmetric hydrogen bond connecting the
tetrahedra into chains has a very unusual double-minimum asymmetric potential well
with minima of comparable energy. Since there is a lack of symmetry relating the
minima, which minimum is occupied would be fundamentally related to the whether the
adjoining tetrahedra were PO4 or SO4 groups. Any disorder in the proton site should then
be related to the disorder on the XO4 site. Hence, the S/P mixing entropy term is taken to
include any disorder associated with this bond. The entropy of the symmetric hydrogen
bond can be evaluated as where those of Cs5(HSO4)3(H2PO4)2. Since the S:P ratio for the
tetrahedra is 1:1 and there is one disordered hydrogen bond per two tetrahedra, there are
only two possible local variants to the structure, Figure 4.14. The entropy contribution
from the hydrogen bonds is then SHbond = (1/2)*R*ln(2) = 2.88 J/mol*K.

Local
OR
Order
a)

Figure 4.14 Local variants on the structure of Cs2(HSO4)(H2PO4) with the average
structure, a), being resolved into two distinct arrangements of the proton system, b).

137

4.4.7 Summary of entropy evaluations for the room temperature
phases
We have now finished the evaluation of the entropy related with the room
temperature phases. For the entropy of mixing, the values expected from statistical
mechanics were found acceptable. The evaluation of hydrogen bond disorder produced
some adjustments to the expected statistical mechanics’ values. These adjustments
accounted for the effect of local order (of mixed S/P sites) on the disordered hydrogen
bonds. The configurational entropy of each compound’s room temperature structure is
listed in Table 4.4. These values will be subtracted from the calculated entropies of the
high temperature phases to assess the transition entropy for each compound.
Table 4.4 Values for the configurational entropy of the room temperature structures
Compound
Smix
SHbond
Stotal
(J/mol*K)

(J/mol*K)

CsHSO4

Cs3(HSO4)2.50(H2PO4)0.50

1.92

0.96 (partial)

2.88

Cs3(HSO4)2.25(H2PO4)0.75

1.56

1.44 (partial)

3.00

Cs3(HSO4)2(H2PO4)

1.92 (disordered)

1.92

Cs5(HSO4)3(H2PO4)2

3.18

1.15 (disordered)

4.33

Cs2(HSO4)(H2PO4)

5.76

2.88 (disordered)

8.64

Cs6(H2SO4)3(H1.5PO4)4

CsH2PO4

5.76 (disordered)

5.76

138

4.5 Entropy Rules for the High Temperature Phases
Having described the sources of configurational entropy in the room temperature
structures, we can now proceed to the more satisfyingly general approach for calculating
the entropy of the highly dynamic superprotonic phases. However, before describing this
work’s model, it is appropriate to discuss the only other theoretical approach used in the
literature to explain the driving force behind these superprotonic phase transitions.

4.5.1 Plakida’s theory of the superprotonic phase transition in CsHSO4
This model focuses only on the “disordering” of the protons in the zigzag chains
of CsHSO4 across the superprotonic transition147. It is based on Landau’s theory of phase
transitions, and assigns an order parameter to the proton positions. The basic premise is
that in the room temperature structure a second chain of hydrogen-bonded tetrahedra is
possible, perpendicular to the existing zigzag chains, but with zero proton occupancy
below the phase transition, Figure 4.15. This assumption is arrived at not by actual
structural considerations, but more by a general comparison to CsH2PO4’s structure, in
which exactly such an arrangement of cross-linked zigzag chains is found. In this model,
the superionic transition results from a disordering of the protons along these two chains,
where disorder here refers to the occupancy of the perpendicular chains and not any intrahydrogen bond dynamical disordering136.

139

SO4

Superprotonic

SO4

Transition

SO4

Emtpy

Full

Half
b)

Figure 4.15 Disordering of protons across the superprotonic transition. Below Tc, a), the
hydrogen bonds in the zigzag chains are completely occupied and those of cross-linking
chains unoccupied. Above the transition, b), the protons are distributed with equal
probability.
The values obtained for the model’s parameters result in a first-order transition of
the Slater type with a jump in the order parameter at Tc. The calculated transition entropy
of this model is ∆S ≈ 0.52*R, which the author noted was much less than the
experimental data of the time, ∆S = 1.32*R3. This discrepancy between the calculated
and measured ∆S was attributed to the disregard for entropic contributions from other
degrees of freedom. In particular, they noted that the rapid reorientations of the tetrahedra
had been ignored. A later calculation that incorporated the disorder of the oxygen atoms
into the proton positions of this model and arrived at an entropy jump of ∆S = 1.1*R for
the transition, the difference between calculated and observed values again being related
to the neglect of additional degrees of freedom148.
Although it is certainly true that the protons are ordered in the room temperature
structure and disordered in the high temperature structure, this model does not seem to
adequately describe the whole transition, but only a sub-system within it. Moreover, for

140
CsH2PO4 this model (not including oxygen disorder) does not predict a transition as the
protons would be ordered in both the low and high temperature structures. There would
then be no entropic benefit to a superprotonic phase transition from a disordering of the
protons. This is probably why no attempt was made to apply this theory to the transition
of CsH2PO4, as well as the fact that the structure of the compound’s superprotonic phase
was not known at the time.
When this structure was published, the entropy of the transition was attributed
completely to the tetrahedral disorder and ignored any possible proton disorder. In this,
we might absolve the theorists as the measurement of the day gave ∆S = 1.821*R for the
superprotonic transition of CsH2PO4134. This value is very close to R*ln(6) = 1.792*R,
the entropy associated with the six orientations of the phosphate groups calculated from
diffraction measurements. There were then two fairly independent models for the
transitions of CsHSO4 and CsH2PO4, which stressed the increased entropy of the protons
and tetrahedra, respectively.

4.5.2 Ice rules type model for superprotonic transitions
In the early stages of formulating this work’s model, discussions with other
researchers led to the discovery that the evolving rules governing these calculations were
very similar to those used in the evaluation of the residual entropy of ice by Linus
Pauling135. Based on the observations of Bernal and Fowler, hexagonal ice (ice Ih) is
composed of oxygen ions and protons, with each oxygen atom coordinated by the four
closest oxygens residing on the corners of a regular tetrahedron. Hydrogen bonds connect
the oxygen atoms with O−O and O−H distances of 2.76 and 1.0 Å, respectively. Each

141
oxygen atom is surrounded by four potential proton sites, the distance between proton
sites on the same hydrogen bond being 0.76 Å, Figure 4.1648.

Figure 4.16 Hexagonal ice: each oxygen in tetrahedrally surrounded by four oxygens
and four potential proton sites48.
Bernal and Fowler also concluded that the structures of individual water
molecules in ice were not that different from the those in steam and therefore must satisfy
two rules48:
i)

two and only two protons are bonded to each oxygen

ii)

one and only one proton is allowed per hydrogen bond.

To these so called ice rules, Pauling added that135
iii)

the hydrogen bonds must be directed approximately towards two of the
four neighboring oxygen atoms

iv)

the interaction of non-neighboring water molecules does not energetically
favor one possible arrangement of protons with respect to other possible
configurations so long as they all satisfy i)-iii).

142
Using these four rules Pauling estimated the number of configurations for a
molecule to be
 # of


 protons 

Ω =  # of proton  *  probability a proton 
site is open
 configurations  

( )

(4-4)

( )

 4
= 6* 1
=3
=   * 2
 

giving ice a residual molar entropy of R*ln(3/2) = 3.37 J/(mol*K), in extremely good
agreement with the experimental data149. Now these rules were first applied to the
relatively static structure of ice, but others found that it equally well explains the increase
in entropy of order-disorder transitions in ice polymorphs, clathrate hydrates, and many
other water containing compounds such as SnCl2• 2H2O, Cu(HCO2)2• 4H2O, and
[H31O14][CdCu2(CN)7]150-153. This is in spite of the fact that the compounds vary greatly
in both the extent and dimensionality of their hydrogen bonded networks. The application
of Pauling’s rules to systems with reorientational disorder is then well documented.
The only remaining logical leap is to apply these ice rules to the tetrahedra found
in the compounds in question. A literature search showed that this exact step was
performed by Slater to describe the ferroelectric transition of KH2PO4154. This further
application of the ice rules would seem trivial, since each oxygen atom in ice is
tetrahedrally coordinated by four oxygen atoms and each phosphate in KH2PO4 is
similarly surrounded by four other phosphate groups. However, in ice the six allowed
configurations of the protons are crystallographically identical, whereas in KH2PO4 two
of the arrangements are different from the other four. The two special configurations

143
result in a dipole pointing either in the positive or negative c-axis (the preferred axis of
the crystal since KH2PO4 is tetragonal) while the other four give polarizations in the
plane perpendicular to the c-axis.
The two configurations aligned with c can therefore have different energies from

those perpendicular to c . This is actually the cause of the spontaneous polarization of the

ferroelectric phase, dielectric measurements having shown the c-axis aligned
configurations to have a lower energy than the other four. Thus, a crystal should be
completely polarized (ordered) at zero temperature and completely random at high
temperatures with the configurational entropy difference between the two states equal to
R*ln(3/2) as T ⇒∞154. It turns out that the measured entropy change for the ferroelectric
transition of KH2PO4 has an excess entropy when compared to R*ln(3/2). This was
recently explained by way of local excitation of phosphate defects (HPO4-2 and H3PO4),
the formulation of which was given by Takagi in 1948155. These defect pairs add
significantly to the entropy of both the ferroelectric and paraelectric phases and lead to
the 12 % increase in the measured transition entropy compared the ice rules value156.
The ice rules have then successfully described the entropy changes of both
disordered ice-like systems and compounds containing tetrahedral groups. This makes the
step of applying them to the disordered tetrahedra of the high temperature phases more
like a hop. However, this is certainly the first time they have been applied to the
superprotonic phases of solid acids, resulting in a very compelling description of the
entropic driving force for these transitions, found lacking in the current literature. The ice
rules applied to the compounds under consideration are very similar to those given by
Slater for KH2PO4, but with the additional complexity that there are now both sulfate and

144
phosphates in the structure. Besides adding the obvious entropy associated with mixing,
calculated with Eq. (4-1), this will also cause the average number of hydrogen atoms per
tetrahedron to change from compound to compound. This changes the first ice rule to:
i)

only one or two protons will be associated with a tetrahedron

(4-5a)

There will therefore be two types of tetrahedra in these disordered phases, differentiated
not by their central cation, but by the number of protons bonded to their oxygen atoms.
This will add to the entropy of these phases as there will be different possible
configurations associated with the ordering of the one and two proton laden tetrahedra.
The other rules will remain relatively unchanged:
ii)

only one proton per hydrogen bond

(4-5b)

iii)

hydrogen bonds are directed towards oxygen atoms of

(4-5c)

neighboring tetrahedra
iv)

interactions of non-adjacent tetrahedra do not effect the

(4-5d)

possible configurations of a tetrahedron and its protons
The reference to the configurations of a tetrahedron in Eq (4-5d) is necessary to include
the entropic contribution from the crystallographically identical orientations of the
tetrahedra.
With this formulation we can adjust Eq. (4-4) to calculate the entropy of these
high temperature phases:
# of
  probabilit y 
Ω =  proton  *  a proton 
 configurat ions  
  site is open 

 # of
 protons 

# of
  # of 
 
*  tetrahdera l  *  oxygen 
 arrangemen ts   positions 
 

(4-6)

145
where the first two terms are evaluated exactly as with ice, only here the coordination
need not be tetrahedral. The third term arises from the distinguishable arrangements of
one and two proton laden tetrahedra. For example, there are three distinguishable
arrangements of one HXO4 and two H2XO4 groups, ten distinguishable arrangements of
two HXO4 and three H2XO4 groups, and so on. The final term is caused by the librations
of the tetrahedra between their possible orientations that result in multiple oxygen
positions for the same hydrogen bond direction, hence increasing the number of
configurations. With Eq. (4-6) we are now ready to calculate the configurational entropy
of the high temperature phases!

4.6 Calculated Transition Entropies for the CsHSO4CsH2PO4 System of Compounds
In this section, the rules developed in section 4.5 will be applied to the high
temperature structures found in the cesium sulfate-phosphate compounds. As was
previously suggested, the calculation of the high temperature forms using Eq. (4-6) is
straight-forward compared to the evaluation of the entropy associated with the room
temperature phases. Since the structures of the high temperature forms are either
isostructural to or a combination of the superprotonic structures of CsHSO4 and CsH2PO4
(see section 4.2.2), we will start with evaluating the configurational entropy of these two
structures. It will then be possible to evaluate the entropy of the remaining superprotonic
phases by taking the right ratios of the two values.

146

4.6.1 Entropy calculations for CsHSO4
As was discussed in section 4.2.3, the high temperature structure of CsHSO4 is a
body centered tetragonal structure in which each sulfate anion is tetrahedrally surrounded
by four other SO4 groups. This leads to four possible directions for the dynamically
disordered hydrogen bonds that connect the tetrahedra. On average, the oxygen atoms of
each tetrahedra should be involved in two hydrogen bonds, so that the proton associated
 4
each sulfate will have   = 4 possible positions (i.e., the four hydrogen bond
1

directions), with each position having a 3/4th probability of being open. Each tetrahedra
should have only one proton, so the number of tetrahedral arrangements equals one. This
leaves only the knowledge of the number of tetrahedral orientations necessary for the
evaluation of Eq. (4-6). Unfortunately, as was mentioned before, this number is of
considerable debate in the literature. The three distinct structures proposed result in either
two or four orientations of the tetrahedra, Figure 4.17. There is no definitive reason to
prefer one model over the others and so the entropy for both 2 and 4 orientations of the
tetrahedra will be calculated.

147

2 orientations

O/OD

4 orientations
* O

Jirak

Merinov

Belushkin

Figure 4.17 Possible configurations of the sulfate tetrahedra in the superprotonic phase:
the structures by Jirak-a)61, Merinov-b)125, and Belushkin-c)124 have two, two, and four
orientations, respectively, which transform into each other by rotations of 32°, 30°, and
30°, respectively. The * designates one possible arrangement for the oxygen atoms of a
tetrahedra.
Since there are only four hydrogen bond directions, the number of tethradedral
orientations will equal the number of oxygen positions around each hydrogen bond.
Using Eq. (4-6), the average number of configurations for CsHSO4 in its tetragonal phase
is therefore
# of
  probabilit y 
Ω =  proton  *  a proton 
 configurat ions  
  site is open 
 4
*
 1

= 

()

 # of


protons

# of
  # of 
 
*  tetrahdera l  *  oxygen 
 arrangemen ts   positions 
 

(4-7)

3 1* (1)* (2 or 4) = 6 or 12

resulting in an entropy of
1.79*R 14.90 J / mol*K
Sconfig = R*ln(Ω) = R*ln(6 or 12) = 
= 
2.48*R 20.66J / mol*K

(4-8)

148
Now, since we are going to use one of these numbers as the entropy associated
with the tetragonal phases of the other compounds, we need to pick one or the other.
From section 4.2.4, this work measured the transition enthalpy for CsHSO4 as being 15.0
J/mol*K, a value higher than previously reported3,130. Trusting in this work’ s value and
the fact that all other published values are smaller, it seems pretty likely that the 14.90
J/mol*K value (corresponding to two orientations of the tetrahedra) better represents the
superprotonic structure of CsHSO4. An entropy of

Sconfig = R*ln(Ω) = R*ln(6) =1.79*R =14.90 J / mol*K

(4-9)

is then the value assigned to the tetragonal phase in these calculations.

4.6.2 Entropy calculations for CsH2PO4
The CsCl-like structure of CsH2PO4 has the PO4 anions at the center and Cs
cations at the corners of a cube. This arrangement allows hydrogen bonds to extend out
the six faces of the cube, Figure 4.18. There will be two protons per tetrahedra resulting
6
in   = 15 ways of positioning the two protons in the six possible directions of the
 2

hydrogen bonds. Two protons per tetrahedra also means that two hydrogen bonds will
enter the cube, giving the probability of a direction being open = 4/6 = 2/3. The
tetrahedra should all have an average of two protons, so the number of tetrahedral
arrangements equals one. And finally, each tetrahedra will have six orientations, a
number which is pleasingly not in dispute. Six orientations equates to 24 oxygen
positions spread out over the six faces, or 4 oxygen positions per hydrogen bond
direction.

149

Figure 4.18 Cubic structure of CsH2PO4: a) the shortest distance between oxygen atoms
of different tetrahedra extends out the faces of the cube, which results in six possible
directions for hydrogen bonding, b)92. Dashed lines represent disordered hydrogen bonds.
The number of configuration for CsH2PO4 in its cubic phase is therefore,
# of
  probabilit y 
Ω =  proton  *  a proton 
 configurat ions  
  site is open 
 6
*
 2

= 

 # of
 protons 

# of
  # of 
 
*  tetrahdera l  *  oxygen 
 arrangemen ts   positions 
 

(4-10)

( 64 )2 * (1)* (4)=15 * ( 94 ) * 4 = 26.6

from which comes an entropy of

Sconfig = R *ln(Ω) = R *ln(26.6) =3.28* R = 27.30 J / mol* K

(4-11)

This value is then the amount of entropy associated with CsH2PO4’s cubic phase. Unlike
the tetragonal phase of CsHSO4, however, this value will not be assigned to all the high

150
temperature cubic phases because of the varying proton content of the phases. For each
compound the first, second, and third terms in Eq. (4-6) will need to be evaluated before
the entropy of the compounds high temperature phase can be calculated.

4.6.3 Entropy calculations for pure cubic phases
With no further ado, we can now calculate the entropy associated with the high
temperature phases of the Cs3(HSO4)2(H2PO4), Cs5(HSO4)3(H2PO4)2, and
Cs2(HSO4)(H2PO4) compounds. We will need three CsHXO4 units to describe the cubic
phase of Cs3(HSO4)2(H2PO4), with one two-proton unit and two one-proton units. Eq. (46) then gives
# of
  probabilit y 
Ω =  proton  *  a proton 
 configurat ions  
  site is open 
 6  6 
=   
 2  1 

( )

* 14
18

 # of
 protons 

# of
  # of 
 
*  tetrahdera l  *  oxygen 
 arrangemen ts   positions 
 

(4-12)

4  3! 
 * (4) = 2.37 x103
* 
 2!1!

equal to an entropy per CsHXO4 unit of

Sconfig=1 3*R*ln(Ω) =1 3*R*ln(2.37x103) =2.59*R = 21.54 J / mol*K

(4-13)

Applying Eq. (4-6) to the Cs5(HSO4)3(H2PO4)2 compound results in
 6
Ω =  
 2

2  6 3
 
 
 1

( )

7  5! 
* 23 * 
* (4) = 3.03x105
30
 3! 2!

(4-14)

giving an entropy per CsHXO4 unit of

Sconfig=1 5*R*ln(Ω) =1 5*R*ln(3.03x105) =2.52*R = 20.99 J / mol*K

(4-15)

151

And finally, the Cs2(HSO4)(H2PO4) compound has
 6  6 

(12 ) 1!1!
3 

Ω =    * 9 *  2!  * (4) = 3.04 x10 2
 2  1 

(4-16)

configurations, which leads to an entropy per CsHXO4 unit of

Sconfig=1 2*R*ln(Ω) =1 2*R*ln(3.04x102) =2.86*R = 23.76 J / mol*K

(4-17)

This leaves only the entropy of mixing for the sulfate and phosphate tetrahedra to be
calculated. Using Eq (4-1) on the compounds gives
Smix[Cs3] = -R*[(1/3)ln(1/3)+(2/3)ln(2/3)] = 0.64*R = 5.29 J/mol*K
Smix[Cs5] = -R*[(2/5)ln(2/5)+(3/5)ln(3/5)] = 0.67*R = 5.6 J/mol*K

(4-18)

Smix[Cs2] = -R*[(1/2)ln(1/2)+(1/2)ln(1/2)] = 0.64*R = 5.76 J/mol*K
The total configurational entropies for these three compounds in their cubic structures is

Stotal[Cs3 ] = Sconfig + Smix = 21.54 + 5.29 = 26.83 J / mol * K
Stotal[Cs5 ] = Sconfig + Smix = 20.99 + 5.60 = 26.59 J / mol * K

(4-19)

Stotal[Cs2 ] = Sconfig + Smix = 23.76 + 5.76 = 29.52 J / mol * K
4.6.4 Entropy calculations for mixed tetragonal/cubic compounds
To evaluate the configurational entropy of these compounds, we need to know the
stoichiometry and mole fractions of the tetragonal and cubic phases. For the tetragonal
(CsHSO4-type) phases, the stoichiometry will be assumed that of CsHSO4. This
assumption is justified by the observation that the cubic phase is the preferred phase for a
wide compositional range, from an S:P ratio of 2:1 to 0:1, Figure 4.3. Therefore, for S:P
ratios ~ 2:1, equivalent to H:XO4 ratios of 1.33:1, the configurational entropy gained by

152
all tetrahedra being in the cubic phase balances any other entropy and energy bonuses
conveyed by the co-existence of the tetragonal and cubic phases.
The conjecture that each tetrahedron in the tetragonal phase has an average of
only one proton can also be justified on an atomistic level. Assuming an equilibrium state
in which both the tetragonal and cubic phases are present, taking a proton from a twoproton loaded tetrahedra in the cubic phase and moving it to a one-proton tetrahedra in
the tetragonal phase results in an entropy loss, the magnitude of which will depend on the
average H:XO4 value of each phase. For tetragonal and cubic phases with H:XO4 ratios
of 1 and 2, respectively, this entropy loss equals 0.39*kb per proton transferred, Figure
4.19. Of course, there are other entropy and energy terms associated with such a switch.
This is obvious from the very fact that there is a tetragonal phase at all. After all, from
entropy considerations alone, the cubic phase is preferred over the tetragonal phase for all
possible proton loadings.

Figure 4.19 The configurational entropy loss due to a proton transfer from the cubic to
tetragonal phase. No entropy is gained in making a two-proton from a one-proton
tetrahedra in the tetragonal structure, but entropy is lost by reducing a tetrahedra’s proton
loading from two to one in the cubic structure.

153
From the above listed arguments, the tetragonal phases are assumed to have an
average proton/tetrahedra value extremely close to 1 (i.e., it is pure CsHSO4), while the
minimum H:XO4 ratio for the cubic phases should be ~ 1.33:1, thus allowing the
evaluation of the high temperature forms that express both structural types.
The requirement that the tetragonal phase be pure CsHSO4 implies that not only
do the sulfate and phosphate tetrahedra migrate into different phases, but that they form
phase domains on the order of 1000 Å, as evidenced by the diffraction patterns of
Cs3(HSO4)2.5(H2PO4)0.5 and Cs3(HSO4)2.25(H2PO4)0.7 which show two distinct structures.
This might seem to give the tetrahedra an unreasonably high mobility, but even for the
fastest measurement of the transitions, 20 K/min, these transitions took ~ 2 min to
complete. From the simple diffusion equation, x = Dt , we can deduce a minimum
diffusion coefficient (assuming the tetrahedra to move 1000 Å in 2 min) of D ≅ 1x10-12
cm2/s. This value can be compared to that of phosphorous in fused phosphoric acid (i.e.,
the diffusion constant of PO4 groups), which at ~25°C is approximately 1x10-7 cm2/s44.
Since our minimum diffusion coefficient is 100,000 times smaller than this measured
value, the tetrahedral migration necessary to form a pure CsHSO4 tetragonal phase
certainly seems possible on the atomistic level.
To evaluate the entropy of these mixed superprotonic phases we now need only to
know the amount of cubic and tetragonal phase present in the compounds at elevated
temperatures. The required values were calculated from Rietveld refinements of high
temperature diffraction patterns for the two compounds (Appendix A). At 140°C, the
ratio of the cubic to tetragonal phase was 44(1):56(1) and 50(1):50(1) for the
Cs3(HSO4)2.5(H2PO4)0.5 and Cs3(HSO4)2.25(H2PO4)0.75 compounds, respectively.

154
Refinements at other temperatures above the onset of the transitions showed that the ratio
of cubic to tetragonal phase increased with increasing temperature, agreeing with the
cubic phase having the higher entropy of the two phases. The above listed ratios will be
used in the entropy calculations for these two compounds because the diffraction patterns
at 140°C were the first not to exhibit a monoclinic phase, suggesting the superprotonic
transitions had just completed. A temperature of 140°C for the final transition
temperature is also in agreement with the values deduced from the conductivity data
(Table 4.3).
A cubic to tetragonal ratio of 44(1):56(1) results in the cubic phase of
Cs3(HSO4)2.5(H2PO4)0.5 having a S:P ratio of 1.6:1, with a nominal stoichiometry of
Cs13(HSO4)8(H2PO4)5. The number of configurations for such a compound is
 6
Ω =  
 2

5  6 8
 
 
 1

( )

* 60
78

18  13! 
 * (4) = 5.84 x1013
* 
 8!5!

(4-20)

giving an entropy per CsHXO4 unit of

Sconfig=1 13*R*ln(Ω) =1 13*R*ln(5.84x1013) =2.44*R = 20.27 J / mol*K (4-21)
Using the entropy of the tetragonal phase, 14.9 J/mol*K, and the cubic phase, 20.27
J/mol*K, the calculated entropy of the superprotonic phase of Cs3(HSO4)2.5(H2PO4)0.5
becomes

Sconfig = Xcub*Scub+ Xtetra*Stetra=
= 0.44*(20.27) +0.56*(14.9) =17.26 J / mol*K

(4-22)

The entropy of mixing will be likewise weighted and using Eq. (4-1) gives an additional
entropy of
Smix[Cs3(HSO4)2.5(H2PO4)0.5] = 0.44*Smix(cubic) + 0.56*Smix(tetra) =

155
= 0.44*(- R*[(5/13)ln(5/13)+(8/13)ln(8/13)]) + 0.56*(0) =

(4-23)

= 0.44*(0.67*R) = 2.44 J/mol*K
The total entropy of the high temperature phase of Cs3(HSO4)2.5(H2PO4)0.5 is then

Stotal = Sconfig + Smix =17.26 + 2.44 =19.7 J / mol* K

(4-24)

For the Cs3(HSO4)2.25(H2PO4)0.75 compound a 50(1):50(1) cubic to tetragonal ratio
leads to the cubic phase having a S:P ratio of 1:1 and a stoichiometry of
Cs2(HSO4)(H2PO4). Therefore, taking the average of the calculated entropies for CsHSO4
and Cs2(HSO4)(H2PO4),

Sconfig= Xcub*Scub+ Xtetra*Stetra=
= 0.5*(23.76) +0.5*(14.9) =19.33 J / mol*K

(4-25)

gives us the configurational entropy, per CsHXO4 unit, for the high temperature phase of
Cs3(HSO4)2.25(H2PO4)0.75. The entropy of mixing for this phase is
Smix[Cs3(HSO4)2.5(H2PO4)0.5] = 0.5*Smix(cubic) + 0.5*Smix(tetra) =
= 0.5*(- R*[(1/2)ln(1/2)+(1/2)ln(1/2)]) + 0.5*(0) =

(4-26)

= 0.5*(0.69*R) = 2.88 J/mol*K
Making the total entropy of the high temperature phase of Cs3(HSO4)2.25(H2PO4)0.75 equal
to

Stotal = Sconfig + Smix = 19.33+ 2.88 = 22.21 J / mol* K

(4-27)

4.6.5 Entropy calculations for Cs6(H2SO4)3(H1.5PO4)4
The only configurational entropy calculation remaining is that for the black sheep
of the family, Cs6(H2SO4)3(H1.5PO4)4. The high temperature structure for this compound

156
is presumed to be similar to the other cubic phases based on the high temperature X-ray
diffraction pattern (Figure 4.3) and current neutron diffraction experiments which both
confirmed a primitive cubic space group. Of course, the Cs:XO4 ratio does not conform
to the 1:1 ratio implied by the CsCl structure. It is therefore assumed that there are Cs
vacancies in the Pm3 m structure. This conjecture is supported by the fact that this phase
has a negative thermal expansion, which can be most simply explained by Cs cations
vibrating more and more into the vacancy sites as temperature increases. If we take the
existence of these Cs vacancies as fact, we then must make some changes to the
assumptions used in calculating the other cubic phases entropy. First, instead of there
being only two types of proton burdened tetrahedra in the cubic phase, i.e., HXO4 and
H2XO4, there will now be a third type, H3XO4. From the stoichiometry of the compound,
we can estimate the ratio of these tetrahedral forms, H3XO4:H2XO4: HXO4, will be 1:3:3.
Using such a ratio and Eq. (4-6), the configurations for the proton system becomes
 6  6 
Ω =   
 3  2 

3  6 3
 
 
 1

( )

* 30
42

12  7! 
*
* (4) = 1.44 x108
 3!3!1!

(4-28)

equating to a configuration entropy of

Sconfig=1 7*R*ln(Ω) =1 7*R*ln(1.44x108) =2.68*R = 22.31 J / mol*K

(4-29)

The presence of Cs vacancies also changes the entropy of mixing calculation. There will
now be two entropy of mixing terms: one for the sulfate/phosphate groups and one for the
CsCs/VCs sites. Using the appropriate ratios from the stoichiometry of the compound and
Eq. (4-1) these entropy terms are evaluated as
Smix[XO4’s] = - R*[(3/7)ln(3/7)+(4/7)ln(4/7)]) = 0.68*R = 5.68 J/mol*K

(4-30)

Smix[Cs/V] = - R*[(1/7)ln(1/7)+(6/7)ln(6/7)]) = 0.41*R = 3.41 J/mol*K

(4-31)

157
The final contribution to this phase’s entropy comes from positional disorder of
the Cs cations, the support for which is again based on this phase’s negative thermal
expansion. It is postulated that the cations near a vacancy have an extra entropy
component, evaluated in terms of configurational entropy by allowing these cesium
atoms two positions: the lattice sites on which they should reside and dynamic positions
halfway between the cesium and vacancy sites. As the Cs:V ratio is 6:1 this can be
visualized by the vacancy being placed at the center of a regular octahedron with Cs
atoms at the vertices and the dynamic positions lying between the cesiums and the
vacancy, Figure 4.20. As the material is heated up, the cesium atoms should jump more
frequently to the intermediate position giving the observed negative thermal expansion.

Figure 4.20 Possible source of extra entropy in the cubic phase of
Cs6(H2SO4)3(H1.5PO4)4: Normal and dynamic position for the cesium atoms due to the
presence of vacancies on the cesium lattice.

158
The entropy of the arrangement shown in Figure 4.19 would increase as the two
positions for the cesium atoms became equally occupied at which point each Cs would
have an extra entropy of R*ln(2) associated with it. The entropy per CsHXO4 unit would
then be

Sconfig( Cs/V ) =1 7*R*ln(Ω) =1 7*R*ln(26) =0.59*R = 4.94 J / mol*K

(4-32)

Again, this is a configurational evaluation of the extra entropy associated with the
presence of cesium vacancies, the real entropy possibly being better described as
vibrational or translational. The total entropy of this phase is then the sum of Eqs.
(4-29), (4-30), (4-31), and (4-32):

Stotal = S config + S mix ( XO ) + S mix (Cs / V ) + S config (Cs / V ) =
= 22.31 + 5.68 + 3.41 + 4.94 = 36.34 J / mol * K

(4-33)

4.6.6 Summary of entropy calculations for high temperature phases
Table 4.5 details the calculated entropies for the tetragonal and cubic high
temperature phases of these compounds. For CsHSO4, CsH2PO4, and the pure cubic
compounds, the total entropy of the high temperature phases was calculated using only
the ideal entropy of mixing, Eq. (4-1), and applying the adjusted ice rules, Eq. (4-6). The
compounds which transform to both tetragonal and cubic phases at elevated temperatures
required an additional assumption to calculate their total entropy. This was that the
tetragonal phase consists of pure CsHSO4. The entropy of these compounds was then
straightforwardly appraised by Eq. (4-1) and Eq. (4-6). Finally, the
Cs6(H2SO4)3(H1.5PO4)4 compound obligated multiple assumptions, the most central of
which was that there exist cesium vacancies in the cubic high temperature phase. This

159
calculation is quite speculative due to the lack of data concerning this particular high
temperature structure and will need further experimental input to become more
conclusive.

Table 4.5 Calculated entropies for high temperature phases
Compound
Smix
Sconfig

Stotal

(J/mol*K)

CsHSO4

14.9

Cs3(HSO4)2.50(H2PO4)0.50

2.44

17.26

19.7

Cs3(HSO4)2.25(H2PO4)0.75

2.88

19.33

22.21

Cs3(HSO4)2(H2PO4)

5.29

21.54

26.83

Cs5(HSO4)3(H2PO4)2

5.6

20.95

26.55

Cs2(HSO4)(H2PO4)

5.76

23.76

29.52

Cs6(H2SO4)3(H1.5PO4)4

9.09

27.25

36.34

CsH2PO4

27.3

4.6.7 Calculated ∆Strans and comparison with experimental ∆Strans
The calculated transition entropies for these cesium sulfate-phosphate compounds
are then simply the values in Table 4.4 subtracted from those of Table 4.5. We can
compare these numbers to the measured entropies by dividing the experimental transition
enthalpies by the mean of the various transition temperatures listed on Table 4.3. The
results of this comparison are shown graphically in Figure 4.21 and listed in Table 4.6.
The first thing one should observe when viewing Figure 4.21 is the very satisfactory

160
agreement between the experimental and calculated transition entropies, which are quite

Transition Entropy (J/mol*K)

often within error of each other.

∆ Sexp.

∆ Scalc.

35
30
25
20
15

100

% PO4
Figure 4.21 Measured versus calculated transition entropies. The shape of the calculated
curve closely mimics that of the experimental. Note calculated and experimental values
are nearly identical for CsHSO4, for which the subjective evaluation of the room
temperature entropy was not necessary.
The sometimes large errors in the experimental entropies are due mainly to the
ambiguity in Tc caused by the large range over which some of the compounds transform.
From a thermodynamic perspective, one might expect that the onset temperatures,
Tonset(DSC) and Tonset(σ), would tend to underestimate ∆Htrans because the compound has

161
not actually reached equilibrium with respect to the high temperature phase until the
transition is complete. Conversely, the final temperatures, Tpeak(DSC) and Tfinal(σ), will
tend to overestimate ∆Htrans as the room temperature phase stopped being the most
energetically favorable phase at Tonset. For these reasons, the mean value of the transition
temperatures was taken as Tc for each compound, which led to large errors in the
experimental entropies for compounds with extend transition temperature ranges.
It is interesting that the calculated and measured transition entropies for CsHSO4
are very similar. This would tend to confirm not only the hypothesis that CsHSO4 has two
(rather than four) orientations in its tetragonal phase, but also justify the use of the mean
transition temperature for the following reason: this compound had zero entropy in its
room temperature structure and therefore the somewhat subjective entropy evaluation of
its room temperature phases was avoided. Consequently, the calculated entropy for
CsHSO4 should have the least amount of unaccounted for entropy. The nearly perfect
match of calculated and experimental values is then very reassuring. The systematically
lower values of the calculated, compared to experimental, entropies for the rest of the
compounds are probably a combination of the fact that the maximum reasonable amount
of entropy was assigned to the room temperature phases and that only the mixing and
configurational contributions to the transition entropy were evaluated. Even the
calculated entropies for the Cs6(H2SO4)3(H1.5PO4)4 compound have the right magnitude,
although this result must be taken with a large grain of salt considering the amount of
speculation that went into the entropy evaluation of this compound’s cubic structure.

162
Table 4.6 Calculated and experimental transition entropies.
Compound
Tc (mean)∆Hexp ∆Sexp = ∆Hexp / Tc
(K)
CsHSO4
Cs3(HSO4)2.50(H2PO4)0.50
Cs3(HSO4)2.25(H2PO4)0.75
Cs3(HSO4)2(H2PO4)
Cs5(HSO4)3(H2PO4)2
Cs2(HSO4)(H2PO4)
Cs6(H2SO4)3(H1.5PO4)4
CsH2PO4

(kJ/mol)

(J/mol*K)

∆Scalc -

(J/mol*K)

419(3)

6.2(2)

14.8(6)

14.90

407(7)

7.4(2)

18.2(8)

16.82

401(11)

8.3(5)

20.7(18)

19.21

395(17)

10.7(2)

27.1(17)

24.91

375(11)

9.2(7)

24.5(26)

22.22

364(12)

8.3(2)

22.8(13)

20.88

382(14)

15.1(6)

39.6(30)

36.34

505(4)

11.3(5)

22.4(12)

21.54

Finally, it should be noted that although the investigations into the entropic
driving force of these compounds were originally propelled by an apparent correlation
between phosphorous content and Tc (see Figure 4.6 a)), the final results deny any such
relationship. It was originally thought that the lowering of Tc with rising phosphate
percentage indicated that ∆H was remaining relatively constant while ∆S increased with
phosphorous content. However, as more data became available, it became clear that this
was not in fact the case. With the full data set available to us now, it would seem that
although there are undoubtedly very general effects to increasing the phosphorous
content, the particulars of the room temperature structures far outweigh any such effects.
This conclusion is quite evident in Figure 4.6 d), a plot of molar H-bond energy
versus %PO4, where one might have guessed a priori that the energy associated with the
hydrogen bonds would increase fairly linearly with phosphate, and therefore hydrogen,
content. In fact, starting with just the end members CsHSO4 and CsH2PO4, such a linear
relationship would have seemed justified as the hydrogen bond energy (per mole

163
CsHXO4) of CsH2PO4 is almost twice that of CsHSO4. The intermediate compounds,
however, fall far from the line connecting the two end members and it can only be said
very generally that increasing phosphate/hydrogen content correlates to higher molar
hydrogen bond energies.
It is then even more pleasing that Pauling’s ice rules, adjusted to properly
describe the superprotonic phases of these cesium sulfate-phosphate compounds, produce
transition entropies that compare very well with the measured values. Since these rules
combine the positional disorder of the proton system with the rotational disorder of the
tetrahedra, it should be applicable to any transition that involves a disordering of a
hydrogen-bonded network via disorder of the hydrogen carriers. This has already been
shown to be true in compounds where the hydrogen-bonded network is composed of
water molecules and would now appear to be true for systems containing hydrogenbonded tetrahedra.

4.6.8 Application of the adjusted ice rules to other superprotonic
transitions
There are other compounds for which these adjusted ice rules should apply. First
and foremost are the compounds CsHSeO4 and CsH2AsO4, which have superprotonic
transitions at (Tonset) 128 and 165°C, respectively157,158. The compounds are also
isostructural to CsHSO4 and CsH2PO4, respectively, in their superprotonic phases46,159.
CsHSeO4 has no configurational entropy below its transition as it is isostructural to
CsHSO4’s room temperature phase160. The CsH2AsO4 compound, however, is not
isostructural to its phosphate cousin, but has a tetragonal structure at room
temperatures161. The tetrahedra of CsH2AsO4 have all four oxygen atoms involved in

164
hydrogen bonds, similar to CsH2PO4, but here all bonds are disordered, so that
CsH2AsO4 will have an entropy of 2*R*ln(2) associated with its room temperature
structure. Using the calculated configurational entropy of the superprotonic tetragonal
and cubic phases (Eqs. (4-9) and (4-11), respectively), the transition entropies can then be
calculated. The resulting entropies match up very well with the measured values, Table
4.7. Having successfully applied these ice rules to the CsHSO4-CsH2PO4 system and to
the end members CsHSeO4 and CsH2AsO4, one would expect that they should apply
equally well to any mixed Cs-S-Se-P-As compounds. Some of these mixed compounds
have already been synthesized, such as Cs4(SeO4)(HSeO4)2(H3PO4),
Cs3(HSeO4)2(H2PO4), and Cs5(HSeO4)3(H2PO4)2, (NH4)2(HSO4)(H2AsO4), however their
properties have not been reported162,163.

Table 4.7 Application of ice rules to other solid acid supeprotonic phase transitions
Compound

Tc-mean
(K)

Scalc – RT
(J/mol*K)

Scalc – HT
(J/mol*K)

∆Scalc
(J/mol*K)

∆Sexp=∆Hexp/Tc
(J/mol*K)

ref

CsHSeO4

140

14.90

16.0(5)

157

CsH2AsO4

186

11.53

27.30

15.77

17.4(6)

158

K3H(SeO4)2

121

5.76

13.38

7.62

7.8(3)

165

CsHPO3H

140

30.67

30.1(11)

166

RbHSeO4

182

2.88

23.9(4)

167

NH4HSeO4

157

2.88 + 9?

15.1(5)

167

164

101

Until now, only compounds with Cs cations have been examined, but this theory
places no limitation on the type or number of cations present. The prevalence for Cs
cations is directly linked with the cation size effect discussed in Chapter 3, in that
superprotonic transitions are more often found in compounds with large cations. These

165
ice rules should then also be applicable to the superprotonic transitions of MHXO4
compounds (where M = Li, Na, K, NH4, Rb, Tl, Cs; X = S, Se, P, As). These compounds
could have varying M:XO4 ratios, mixed cations, or both, such as (NH4)4H2(SeO4)3,
Cs0.9Rb0.1HSO4, and Rb4LiH3(SeO4)4, respectively, all of which have reported
superprotonic transitions (without, unfortunately, the transition enthalpies or
entropies)168, 97,169. And, of course, the intersection of these two sets, compounds with
mixed anions and mixed cations, will be equally susceptible to having these ice rules
applied to any uncovered superprotonic phase transitions.
Also, the disordered network of hydrogen bonds need not be three-dimensional,
as with all the previous examples, for these rules to apply. The class of compounds
M3H(XO4)2 (M= Na, K, NH4, Rb, Cs: X = S, Se) exhibits superprotonic phase transitions
where the proton transport occurs within planes170. The compounds are pseudo-trigonal in
their room temperature phases and most of them transform into a trigonal phase at
elevated temperatures100. For the compounds with such transitions, these ice rules should
reproduce the measured transition enthalpies quite well, as can be seen for the
K3H(SeO4)2 compound in Table 4.6.
Finally, these ice rules also appear valid for compounds with alternative anion
chemistries, such as CsHPO3H, where one of the tetrahedral oxygens has been replaced
by a hydrogen atom. This compound exhibits a superprotonic phases transition at 137°C,
transforming into the same cubic CsCl like structure as the mixed cesium sulfate
phosphates166. Adjusting the ice like rules developed here for the dissimilarity of the
tetrahedra’s coordinating ions will cause two changes. First, no hydrogen bonds can be
formed to the tetrahedral hydrogens. The tetrahedral hydrogen then effectively acts as an

166
OH group and the probablility of a direction being open will be 4/6 rather than the normal
5/6 for a hydrogen to tetrahedron ratio of 1:1. Second, there will be three distinguishable
configurations of the two possible acceptor oxygen atoms and the tetrahedral hydrogen
for every configuration of the proton/donor oxygen system. This will cause an extra
factor of three. The number of configurations for this compound in its cubic phase is then
 6   4 1
Ω =   *   * 1 * 4 * (3) = 48
 1  6 

() ( )

(4-32)

which results in a calculated enthalpy very close to the measure value, Table 4.6.
This exposition of applications serves to prove the flexibility of these ice rules; a
flexibility that allows for a certain amount of prediction concerning poorly characterized
compounds or entirely new systems. For example, the high temperature structures of
RbHSeO4 and NH4HSeO4 are not well determined and so an evaluation of their entropy is
not possible87,91. However, the measured transition entropies of 24 and 15 J/mol*K for
RbHSeO4 and NH4HSeO4, respectively, and these ice rules indicate that the high
temperature phase cannot be the tetragonal phase of CsHSO4167. These compounds are
isostructural to each other in their room temperature phase with one disordered hydrogen
bond per two tetrahedra (S = 1/2*Rln(2) = 2.88 J/mol*K)171. The ammonium compound
also has orientational disorder associated with the SeO4 and NH4 ions, which most
probably accounts for the difference in transition entropies between the two
compounds102. There is then a considerable amount of entropy incorporated into the room
temperatures of these compounds and yet the transition entropies are both above the
calculated 14.9 J/mol*K maximal transition entropy for the tetragonal phase of CsHSO4.

167
It is also possible that entirely new systems of compounds with superprotonic
transitions will be discovered, systems with perhaps mixed M+2 and M+1cations or
including various other anion groups (i.e., SiO4, ClO4, PO3F, SiF6, COF3, etc.). It would
be very nice to estimate the probability of an order-disorder transition in such new
compounds so as to narrow the focus of experiments to those compounds most likely to
exhibit superprotonic conduction. After all, the entire purpose of this work is to better
understand what causes superprotonic phases to exist and to then apply that to making
materials more suited for application.
With this purpose in mind, it is suggested that a hypothetical transition
temperature could be derived from an observed correlation between the transition ∆V and
∆H, and liberal use of these ice rules in estimating a transition entropy. For the CsHSO4-

CsH2PO4 compounds, the correlation between transition enthalpy and volume is quite
clear, Figure 4.22. Since the variation of the data is so small, it seems possible that if one
estimated a transition volume from the predicted room and high temperature structures, it
would be possible to derive a fairly accurate transition enthalpy. Also, using the predicted
structures and these ice-like rules, a likely entropy could also be obtained. Taking the
ratio of these two values would then give an approximate transition temperature,
hopefully telling the investigator whether a compound was worth investigating or not. In
other systems, a similar relationship could be calculated from existing data, or perhaps
extrapolated from structurally and chemically related compounds. This process could
save a vast amount of experimental time as synthesis of even these water soluble
compounds was not trivial.

168

R = 0.96
Y = 5.4(7) + 0.39(5)*X

∆ H (kJ/mol CsHXO4)

14
12

CsH2PO4

Cs3(HSO4)2.25(H2PO4)0.75

-7

∆ V (m /mol CsHXO4 x 10 )

Figure 4.22 Transition volume versus enthalpy. The apparent correlation between the
two values suggests the possibility of estimating a transition enthalpy from a predicted
volume change.
It would be interesting to add the transition enthalpies and volumes of the other
known superprotonic conductors to Figure 4.22. Alas, even though the room and high
temperature structures have been measured for most of the known superprotonic
compounds, accurate thermal expansion coefficients are almost universally lacking. Since
the difference between the temperature at which the room and high temperature structures
are measured is usually in the hundreds of degrees, the expansion (or contraction) of the
phases with temperature would greatly effect the transition volumes. If and when more
accurate transition volumes become available, it will be very interesting to see if the

169
linear trend seen in Figure 4.22 holds for all the known superprotonic conductors, or if
different structural and chemical families of compounds require their own categorization.

170

Chapter 5.

Superprotonic Phase Transition of

CsHSO4: A Molecular Dynamics Simulation
Study with New MSXX Force Field
5.1 Introduction
This molecular dynamics (MD) study of the superprotonic phase transition of
CsHSO4 was undertaken with two aims: to determine whether the transition could be
simulated without allowing proton migration and to develop a procedure for creating MD
force fields (FF) applicable to other solid acids. The first objective was motivated by the
desire to know whether proton hopping or tetrahedra reorientations are the essential
ingredient in stimulating a transition from the ordered room temperature structure to the
highly disordered superprotonic phase. The latter goal comes from the desire to greatly
speed up the search for new superprotonic compounds with properties ideal for
application. It was hoped that a simple process could be developed to predict
superprotonic phase transitions of, as yet unknown, compounds without first synthesizing
the material, which can take untold time in the laboratory.
Success in simulating the transition of CsHSO4 gave sufficient confidence in the
new FF that the effects of changing various FF parameters on the transition were
investigated. The adjusted parameters included the charge distribution of the oxygen
atoms, hydrogen bond strength and torsional barrier height. In each case, a single
parameter was changed and the simulations re-run with all other FF and simulation

171
variables held constant. Thus, the superprotonic phase transition of CsHSO4 was probed
in a manner not possible by experimental methods.
The results of this chapter will then compliment those of the experimental
chapters (3 and 4) in that all three chapters aim to better our understanding of which
parameters favor superprotonic transitions. In particular, this chapter gives atomistic
information (you can even watch them if you like!) not available from physical
measurements. Also, the success of this chapter’s FF in simulating the superprotonic
transition of CsHSO4 suggests that the same procedure could be employed to generate
FF’s for other cations and anions. Combining these FF could then give us a powerful tool
for predicting novel superprotonic conducting solid acids.

5.2 Characterization of CsHSO4
Although both the structures and superprotonic phase transition of CsHSO4 have
been described multiple times in this text, for the sake of this chapter’s completeness, the
compound’s important characteristics will be detailed below.

5.2.1 Crystal structures of CsHSO4
The actual room temperature phase of CsHSO4, especially when it is obtained
from a mixture of equimolar Cs2SO4 and H2SO4 in aqueous solution, is CsHSO4-III
(phase III) not CsHSO4-II (phase II) that has been described throughout the text. That is,
there are three phases in the crystal of CsHSO4 in the temperature range from 123 to 420
K84,172:
415 K
phase III (P21 /c) 330
-370

→ phase II (P21 /c) 410
-
→ phase I (I41 /amd)

172
Whereas the II-to-I phase transition is quite reversible, the III-to-II phase
transition depends on the amount of the absorbed water in the sample. A water-free
powder sample (“dry” sample) remains at phase II on cooling down to 123 K. Only a
water-saturated sample (“wet” sample) becomes the initial phase III on cooling84.
Moreover, if CsHSO4 is deuterated to more that 30-40 %, only phases II and I are present
in the temperature range of 123-420 K172.
Thus, in the present work, we assumed that the room temperature phase of “dry”
CsHSO4 is the phase II rather than phase III and focused only on the II-to-I phase
transition. Phase II is monoclinic (space group P21/c) as determined by single crystal Xray diffraction at 298 K173, Figure 5.1 a. The lattice parameters are a=7.781(2) Å,
b=8.147(2) Å, c=7.722(2) Å, and β=110.78° . The hydrogen bonds configure so as to
form zigzag chains along the [001] direction (c-direction) and the O-H⋅ ⋅ ⋅ O bonds are
fully ordered with <(O-H⋅ ⋅ ⋅ O) = 174(6)° , d(O-H) = 0.94(4) Å, d(H⋅ ⋅ ⋅ O) = 1.70(4)
Å, and d(O⋅ ⋅ ⋅ O) = 2.636(5) Å.

173

Figure 5.1 Crystal structure of CsHSO4: a) monoclinic phase II 26and b) tetragonal phase
I as proposed by Jirak61. In b), each oxygen position has half occupancy and the
hydrogen atoms are placed in the middle of the disordered hydrogen bonds (dashed
lines).

The high temperaturephase (phase I) is tetragonal (space group I41/amd), Figure
5.1 b. There is considerable disorder in the orientation of the hydrogensulfate (HSO4)
groups (HSO4 libration) in this phase102, with some debate as to the actual number and/or
direction of the orientations possible for each tetrahedron. The multiple orientations for
each tetrahedron are a result of the high symmetry of this phase and each tetrahedron’s
need to conform to this symmetry. As the sulfur is centrally located in a tetrahedron, the
exact position of the oxygen atoms will then determine how many orientations are
necessary to achieve the desired tetragonal symmetry. The disagreement in the literature

174
about the number and/or direction of the orientations for each tetrahedron is then
equivalent to the proposed positions of the oxygen atoms, which were determined by
diffraction experiments, Figure 5.2. According to Jirak, who performed a powder neutron
diffraction study on CsHSO4 at a temperature slightly above 414 K, each tetrahedron
adopts one of two orientations and the phase has lattice constants of a = 5.718(3) Å and c
= 14.232(9) Å61. The structure proposed by Jirak is shown in Figure 5.1 b. The structure
put forward by Merinov from an single crystal X-ray diffraction study of CsDSO4 at 430
K has lattice constants of a=5.729(9) Å and c=14.21(1) Å with two orientations for each
tetrahedron125. A high-resolution neutron powder diffraction study by Belushkin, on
CsDSO4 at 448 K, gave lattice parameters of a=5.74147(9) Å and c=14.31508(26) Å with
four orientations for each tetrahedron124.

O/OD

2 orientations

4 orientations
* O

Jirak

Merinov

Belushkin

Figure 5.2 Possible configurations of the sulfate tetrahedra in the superprotonic phase:
the structures by Jirak-a)61, Merinov-b)125, and Belushkin-c)124 have two, two, and four
orientations, respectively, which transform into each other by rotations of 32°, 30°, and
30°, respectively. The * designates one possible arrangement for the oxygen atoms of a
tetrahedra.

175

5.2.2 Nature of the superprotonic transition of CsHSO4
Regardless of which structure you pick for CsHSO4 phase I, the basic nature of
the phase transition and mechanism of proton conduction remains the same. The
transition is of first order from the ordered, low symmetry phase to the disordered high
symmetry phase124. The increase in entropy due to this disorder is the energetic driving
force for the transition. The reorientations of the tetrahedra are then both energetically
and symmetrically required for this transition. As the protonic conductivity is a direct
result of these tetrahedral reorientations, the superprotonic conductivity of phase I is a by
product of the ideal structure2. Across the transition, the protonic conductivity increases
by 3-4 orders of magnitude from 10-6 Ω-1 cm-1 (phase II) to 10-3-10-2 Ω-1 cm-1 (phase I)4.
That the main contribution to the enhanced conductivity of phase I stems from the
mobility of protons is confirmed by both H+ NMR measurements and quasi-elastic
neutron scattering (QNS) experiments on phase I of CsHSO4. Both methods found a
proton diffusion constant, DH, equal to ~ 1x10-7 cm2/s at temperatures above 414 K6. Rfmicrowave dielectric measurements have confirmed that the sulfate tetrahedra are
undergoing fast reorientations (1012Hz) while proton transfer occurs more slowly
(109Hz)174.
The proton transfer process in phase I is therefore thought to consist of two steps:
the creation of hydrogen bonds between previously isolated tetrahedra by HSO4 rotations
and the translation of protons between the two equilibrium sites in the newly created OH⋅ ⋅ ⋅ O bond resulting in an H2SO4 defect35. The “doubly protonated” H2SO4 defect
may propagate rather fast by phonon-assisted tunneling or classical hopping of protons
between the two minima in the O-H⋅ ⋅ ⋅ O bonds along the H-bonded chains6. With

176
libration of the HSO4 groups the weak link H-acceptor disordered hydrogen bond is
broken, while the strong link H-donor is preserved. On account of rapid rotation of the
sulfate groups, the proton samples all possible crystallographic positions; with translation
of the proton along a newly formed two-minimum hydrogen bond occurring once in
about a hundred rotations of the tetrahedra. Thus, migration of protons is effected both
by their jumping between positions on the hydrogen bond and by rotation of HSO4
groups. Such a process is called a Grotthuss type mechanism proton conduction39.

5.3 MD Simulation of Superprotonic Transition of CsHSO4
5.3.1 Overview
The II-to-I superprotonic phase transition of CsHSO4 was simulated by the
molecular dynamics (MD) as temperature was increased from 298 K to 723 K in 25 K
steps. The force field for these MD simulations treated the hydrogen as bonded
exclusively to a single oxygen atom (donor oxygen, OD), with hydrogen bonds extending
to nearby oxygen atoms (acceptor oxygen, OA). Proton diffusion (i.e., proton jumps)
between oxygen atoms cannot occur with this kind of force field. Thus, the contribution
of proton jumps to the phase transition was removed and only the effects of the
orientation disorder of HSO4 groups were considered.

5.3.2 Calculation details: Force fields
The functional forms and parameters of the force field (FF) used in the simulation
are given in Tables 5.1 and 5.2. This FF is based on Dreiding FF 175. The off-diagonal

177
van der Waals (vdW) parameters (Cs-O, Cs-S, Cs-H, S-O, S-H, and O-H) are determined
by the standard combination rules175. No nonbonding interaction is considered for 1,2pairs (bonded atoms) and 1,3-pairs (atoms bonded to a common atom), because it is
considered that their electrostatic and vdW interactions are included in their bond- and
angle-interactions.

Table 5.1 Force fielda for CsHSO4.
E = E nonbond + E valence
E nonbond = E coulomb + E vdW + E H −bond
bond
angle
RR
θθ
torsion
E valence = E SO
+ EOSO
+ EOSO
+ E SOOO
+ EOSOH

Eijcoulomb ( R) = C 0

qi q j

εRij

 R 
6 

 6  ς  1− R0    ς  R0   
e
   
− 
( R ) = D0  
  ς − 6  R   
 ς − 6 

vdW
ij

H −bond
O −O

10
  R0 12
 R0  
( R) = D0 5  − 6  
  R 
 R  

bond
E SO
( R) =

K R (R − R0 )

angle
angle
EOSO
(θ ) = E SOH
(θ ) =

1 Kθ
(cosθ − cosθ 0 )2
2 sin θ 0

RR
EOSO
( R1 , R2 ) = K RR ( R1 − R0 )( R2 − R0 )

θθ
E SOOO
(θ 1 ,θ 2 ) =

torsion
EOSOH
(ϕ ) =

Kθθ
(cosθ 1 − cosθ 01 )(cosθ 2 − cosθ 02 )
sin θ 01 sin θ 02

K ϕ [1 + cos(2ϕ )]

The constants in ECoulomb are the dielectric constant (ε) and C0 = 332.0637 (the unit
conversion factor when atomic charges qi's are in electron units (|e|), the distance R is in
Å, and ECoulomb is in kcal/mol).

178

Table 5.2 Force field parameters for CsHSO4.a
Cs
Rob
4.1741i
Do c
EvdW

0.37i

18i

Rob

4.03h

Do c

0.344h

12.0h

3.4046h

Do c

0.0957h

13.483h

3.195h

Do c

0.0001h

12.0h

S-OD

Rob

1.6925i

Kb d

700.0h

S-OA

Rob

1.499i

Kb d

700.0h

OH-H

Rob

0.988i

Kb d

700.0h

OA-S-OD

θ oe

105.933i

Kθ f

350.0h

OA-S-OA

θ oe

115.2i

Kθ f

350.0h

S-OD-H

θ oe

109i

Kθ f

350.0h

ERR

OA-S-OA

Ro(OA)b

1.4856i

Ro(OA)b

1.4856i

KRR d

102.0h

ERR

OA-S-OD

Ro(OA)b

1.4856i

Ro(OD)b

1.65i

KRR d

102.0h

Eθθ

S-OA-OA-OA θο(OA,OA)e 112.8i

Kθθ f,g

72.5h

S-OA-OA-OD θο(OA,OD)e 105.933i

Kθθ f,g

72.5h

S-OA-OD-OA θο(OA,OA)e 112.8i

θο(OA,OD)e

105.933i Kθθ f,g

72.5h

Etorsion

OA-S-OD-H

Symmetry

ϕ(min.)

60°

EHbond

OD···OA

Ro b

3.0004k

Do c

0.2366k

Ebond

Eangle

Kϕ c

2.1669j

For functional forms, see Table 1.
In Å. cIn kcal/mol. dIn kcal/mol/Å2. eIn degrees. fIn kcal/mol/rad2.
In the current version of Polygraf (version 3.30), the divisor for angle-angle cross term
Eθθ is written as Eθθ (cos θ1 - cos θ10)(cos θ2 - cos θ20) where Eθθ = Kθθ/ sin θ01 sin θ02 =
81.5625 where Kθθ = 72.5. In Cerius 2 the input is in terms of Kθθ176.
From Dreiding FF175.
Adjusted to reproduce a CsHSO4 monomer ab initio structure, binding energy and
frequencies.
Adjusted to reproduce ab initio barrier height (in kcal/mol) for HSO4- ion in a dielectric
medium with a relative dielectric constant of 10.
Adjusted to fit ab initio O−O distance and binding energy of an H2SO4- H2SO4 dimer.

179
Dreiding FF values were adjusted by fitting the parameters to three separate ab
initio calculations at the B3LYP/LACVP** level 177-181 182 (set denotes basis sets of 631G** for H/O/S and LACVP for Cs) using Jaguar software183. The first calculation
was on a gas-phase CsHSO4 monomer, the second on a gas-phase (H2SO4)2 dimer, and
the third calculation on an HSO4- ion in a dielectric medium of relative dielectric constant
10 (Figure 5.3 a), b, and c), respectively). Adjustments to the Dreiding FF values were
made so that each chemical species would duplicate the results of the QM calculations
after a FF minimization to lowest potential energy. All FF energy minimizations were
carried out with the Newton-Raphson method on Cerius2 software176. A more detailed
explanation of how these QM calculations were used in altering the Dreiding FF values is
given below.

(1)O

O(2)

Figure 5.3 Structures used to adjust the Dreiding FF parameters: (a) CsHSO4 monomer,
(b)(H2SO4)2 dimer, and (c) HSO4- ion projected down S−O(H) bond. CsHSO4 monomer
used to adjust Cs vdW and all HSO4- FF parameters except for the hydrogen bond and OS-O-H torsional terms which were adjusted with b) and c), respectively.

180
I. CsHSO4 gas phase monomer:
All cesium (Cs) vdW parameters and FF values for HSO4-, except for torsional
and hydrogen bond values, were varied to reproduce a CsHSO4 monomer derived from
the ab initio calculation on gas-phase CsHSO4. Both the initial and final structures of this
calculation had the Cs near the three fold axis of symmetry of the tetrahedral face
opposite the hydrogen bonded oxygen, Figure 5.3 a). Cesium vdW parameters were fit to
the average Cs−O distance for the three non-hydrogen bonded oxygens, the binding
energy of the Cs+ + HSO4- ions, and the symmetric stretch frequency of CsHSO4
monomer. FF values for S, O, and H in HSO4-, except for torsional and hydrogen bond
parameters, were adjusted from Dreiding FF values to fit the ab initio structure (bond
lengths and angles) and frequencies of the HSO4- ion. This was a rather straightforward
process, with the added complexity of having two different types of oxygen atoms in the
FF: donor oxygens, OD and acceptor oxygens, OA. Such a segregation of the oxygen
atoms was a direct result of fixing the H atoms to particular oxygens.
Charges for all atoms were derived from the electrostatic-potential-fitted (ESP)
charges of the CsHSO4 monomer ab initio calculation184-186. The charges taken directly
from the ab initio calculation are shown in Table 5.3 along with the final adjusted charges
used in these simulations. Adjustment to the charges involved only the Cs and O atoms;
the final charges for S and H atoms being identical to those of the QM calculation. The
atomic charge for all Cs atoms was fixed at its formal charge +1.0|e|. An increase in
negative charge to balance the increased positive charge on the Cs atoms (+0.072|e|) was
distributed evenly over all oxygen atoms (i.e., -0.018|e| on each oxygen). The adjusted
charges of the O(1) and O(2) oxygen atoms (now, -0.648|e| and -0.654|e|, respectively)

181
were then averaged together giving the final charges of these atoms (-0.651|e| each). This
arrangement of oxygen charges was picked not only to conform with the ab initio values,
but also as such a distribution of charges seemed likely for the oxygen atoms of a
tetrahedron in phase II CsHSO4. In this phase, asymmetric hydrogen bonds connect the
SO4 tetrahedra into infinite chains and therefore every tetrahedron has a donor and
acceptor oxygen, and two oxygen atoms not involved in hydrogen bonds26.

Table 5.3 ESP charges for CsHSO4: from the ab initio QM calculation on the CsHSO4
monomer [B3LYP/LACVP**] and the final set used in the simulations.
Environment

qCs(|e|)

qS (|e|)

qO(1) (|e|)

qO(2) (|e|)

qO(3) (|e|)

qOD (|e|)

qH (|e|)

Gas-phase

0.928

1.045

-0.630

-0.636

-0.588

-0.523

0.404

Simulation

1.0

1.045

-0.651

-0.606

-0.541

0.404

There are then essentially three types of oxygen atoms in this force field when
both an oxygen’s FF type and charge are considered: non-hydrogen-bonded, donor and
acceptor oxygen atoms. The non-hydrogen-bonded atoms, O(1) and O(2), have OA FF
parameters and a charge of –0.651|e|, while oxygen acceptor atoms, O(3), have OA FF
parameters but a charge (–0.606|e|). Oxygen donor atoms, OD, have there own FF
parameters (OD values) and charge (–0.541|e|). This division of the oxygens represents
the fact that the S−O bonds are not equivalent in the HSO4- ion. The addition of a
hydrogen makes the S−O(H) bond rather like a single bond, whereas the S−O(1),O(2),O(3)
bonds behave more like multiple bonds with an average bond order of one and two thirds.
The O(3) atom was picked as the acceptor oxygen for the simple reason that its charge

182
was smaller than that of O(1) and O(2). This fact agreed with the premise that an acceptor
should have an average bond order less than one and two thirds, but more than one, and
so have a charge in between that of OD and the non-hydrogen bonded O(1) and O(2)
oxygens.
II. (H2SO4)2 gas phase dimer:
The ab initio calculation on the gas-phase (H2SO4)2 dimer was used to adjust the
Dreiding FF values of the hydrogen bond to those found in Table 5.2. Using the
previously optimized FF parameters for S, O, and H, the hydrogen bond Ro and Do values
were varied to reproduce the ab initio O−O distance (2.647 Å) and binding energy

18.569 kcal/mole) of the (H2SO4)2 dimer calculation. The charges for all atoms were set
to the ESP charges of the QM calculation without adjustment. This means that the FF
parameters for S, O and H atoms were those previously determined for the charges in
Table 5.3, but the charges used were not those found in the table. Also, each H2SO4
group had two OD FF type oxygen atoms instead of just one, as was used in the CsHSO4
monomer minimization. Hence, there was some distortion of the bond lengths and angles
of the H2SO4 tetrahedra from the QM structure when the dimer was minimized using the
adjusted FF. However, ignoring this distortion, the hydrogen bond Ro and Do values
were adjusted until the FF minimized O−O distance and binding energy of the (H2SO4)2
dimer matched those of the QM calculation.
III. HSO4- ion in dielectric medium:
The third QM calculation, on an HSO4- ion in a dielectric medium with dielectric
constant of 10, was used to adjust the hydrogen torsional barrier height. This adjustment

183
actually involved a series of QM calculations where the O(1)-S-OD-H torsional angle was
fixed from 60° to 0° , by steps of 7.5° , while the rest of the HSO4- ion was allowed to
relax. The initial input for these calculations was the optimized structure of the CsHSO4
monomer with the Cs atom removed and the O(1)-S-OD-H torsional angle fixed at 60° .
The result of the 60° calculation was then used as the input for the 52.5° calculation and
so on until the O(1)-S-OD-H torsional angle was optimized at 0° . The QM barrier height
was taken to be the difference between the minimum and maximum of the resulting
potential energy curve, Figure 5.4 a. Symmetry considerations allow the potential energy
curve to be plotted over a full 360° even though calculations were only performed from
0° to 60° .
The potential energy difference between the minimum (~ 52.5° ) and maximum
(at 0° ) of this curve is 1.6 kcal/mol. Analysis of impedance measurements on CsHSO4 in
phase II show the dielectric constant not to vary much from 103. The calculations were
therefore run with a dielectric constant of 10 to simulate the environment the HSO4- ion
would encounter while changing its torsional angle. In the optimized structures, the
oxygen atoms nearest the hydrogen had charges similar to O(1) and O(2), while the
oxygen farthest from the hydrogen had a charge similar to O(3). The barrier height of the
FF was therefore adjusted so that the HSO4- ion, minimized with a fixed O(1)-S-OD-H
torsional angle between 0° and 60° , had an energy difference between the minimum and
maximum of 1.6 kcal/mol. This procedure caused an asymmetry in the FF barrier height
due to the difference in the fixed oxygen charges, Figure 5.4 b.

184

1.6 kcal/mole

0.0

Potential Energy (kcal/mole)

0.0

-0.4

-0.8

-1.2

-1.6

-0.4
-0.8
-1.2
-1.6
-2.0

1.6 kcal/mole

1.8 kcal/mole

-2.4
-120

-60

120

180

240

O(1)-S-O-H Torsional Angle

-120

-60

120

180

240

O(1)-S-O-H Torsional Angle

Figure 5.4 Potential energy curves for an HSO4- ion with fixed O(1)-S-OD-H torsional
angles: a) optimized by QM and b) minimized with the adjusted FF.
Finally, it should be mentioned again that in this force field, the hydrogen was
treated as exclusively bonded to an oxygen atom with hydrogen bonds to other oxygen
atoms. Proton diffusion (or jumps) from one oxygen atom to another cannot occur with
this kind of force field. This FF does not correctly describe a proton in either CsHSO4
phase II or I, since in both phases individual protons migrate through the material
(requiring proton jumps between tetrahedra)35. However, by employing such a constraint,
we can separate out the contribution of proton transfer (diffusion or jump) to the
superprotonic phase transition.

5.3.3 Calculation Details: Simulations
The structure of the phase II of CsHSO4 was optimized with the Newton-Raphson
method with a periodic boundary condition. A 2× 2× 2 supercell including 32 CsHSO4
units was treated as a unit cell and a series of MD simulations were carried out at various
temperatures from 298 K to 723 K in 25 K steps. At each temperature, the Nosé-Hoover

185
(NPT) Rahman-Parrinello MD simulations187,188 were carried out at 1 atm for 300 ps with
a time step of 1 fs. Properties (potential energy, lattice constants, HSO4- orientation, etc.)
were calculated, after a 150 ps equilibration, from the average over the final 150 ps. This
same process was used on the secondary simulations where a particular parameter was
varied to quantify its effect on the phase transition. All the simulations were carried out
using the Cerius2 software176.

5.4 Results
5.4.1 Phase II at Room Temperature: Calculation vs. Experiment
The average structural parameters (density and cell parameters) obtained from the
MD simulation at 298 K are within a few percents from the experimental values, Table
5.4. Also, the atomic coordinates are almost all within error of the published values,
Table 5.5. This adjusted Dreiding FF has then well reproduced phase II CsHSO4 on both
the global (unit cell) and atomistic scale, which is very encouraging since the method
which developed it was quite general (i.e., did not use any phase II structure specific
information).
Table 5.4 Phase II at room temperature: calculation versus experiment.
Parameter

MD at 298 K ExperimentX-raya

ExperimentNeutronb

Error vs X-ray

density (g/cm3)

3.35(3)

3.338(3)

3.3429(1)

0.36%

a (Å)

7.93(5)

7.781(2)

7.78013(9)

1.91%

b (Å)

8.11(5)

8.147(2)

8.13916(2)

0.45%

c (Å)

7.74(5)

7.722(2)

7.72187(9)

0.26%

186
α (°)

90.0(6)

β (°)

113.7(7)

110.775(13)

110.8720(4)

2.6%

γ (°)

90.0(6)

From single-crystal X-ray diffraction on CsHSO4 at 298K26.

From the high-resolution neutron powder diffraction study of CsDSO4 at 300 K124.

Table 5.5 Atomic positions for MD simulation at 298 K.
Experiment at 293 Ka

Atom MD at 298 K
x/a

y/b

z/c

x/a

Deviation

y/b

z/c

0.12907(3)

0.20605(4)

1/ 2

∆ x2 +∆ y2 +∆ z2 

0.22(2) 0.12(2) 0.22(2) 0.21551(4)

0.009

0.75(2) 0.11(2) 0.27(2) 0.75214(14) 0.12727(12) 0.27996(14) 0.012

0.60(3) 0.20(3) 0.09(3) 0.5890(5)

0.2207(5)

0.1312(6)

0.025

OA(2) 0.65(3) 0.03(4) 0.37(3) 0.6647(5)

0.0700(4)

0.4079(5)

0.035

O(3)

0.87(4) 0.24(2) 0.88(3) 0.8947(5)

0.2536(4)

0.8594(5)

0.021

O(4)

0.83(4) 1.00(4) 0.20(4) 0.8062(6)

0.9960(4)

0.1867(5)

0.017

0.66(4) 0.25(3) 0.02(2) 0.625(8)

0.295(6)

0.057(7)

0.040

From single crystal X-ray diffraction on CsHSO4 at 298K26.

Looking at Table 5.5, it can be seen that the largest differences between the
experimental and calculated atomic positions (i.e., the deviations) occur for the atoms

187
which crystallographically participate in the hydrogen bonds (OD, OA, and H). This can
be attributed predominantly to the high “thermal vibrations” of the hydrogen atom, which
was observed to vary its position quite dramatically. However, it was also observed that
the non-donor oxygen atoms moved appreciably and in fact even rotated, with a 3-fold
like symmetry, around the S-OD bond. This motion was significantly activated even at
298 K and became more so with temperature. Not surprisingly, these motions had a
particularly dramatic effect on the hydrogen bond parameters, Table 5.6. It was difficult
to get these average values without either (1) influencing the results of the measurement
or (2) including the effects of atomic motions other than normal thermal vibrations.
Nevertheless, the listed values should reasonably well describe the average values
involved in the hydrogen bonds from which it is clear that they deviate significantly from
the average values determined by X-ray diffraction. This is particularly interesting for the
O-O distance as the FF was adjusted to a value of 2.647 Å, very similar to the
experimental value.
Table 5.6 Hydrogen bond comparison between MD and experiment in phase II
MD at 298 K ExperimentX-raya

ExperimentNeutronb

Error (%) vs.
X-ray

r(OD-H) (Å)

0.99 ± 0.07

0.94 ± 0.04

0.983 ± 0.005

r(H···OA) (Å)

2.03 ± 0.4

1.70 ± 0.04

1.667 ± 0.008

r(OD···OA) (Å) 2.77 ± 0.16

2.636 ± 0.005 2.633 ± 0.005

<(ODHOA) (°) 130 ± 24

174 ± 6

166.6 ± 0.6

From single crystal X-ray diffraction on CsHSO4 at 298K26.

From the high-resolution neutron powder diffraction study of CsDSO4 at 300 K124.

188
The simulation results also deviate from the measured values for the bond lengths
and angles of the HSO4 groups, Figure 5.7. However, in this case the simulation values
are very close to those with which the FF was optimized. These deviations are then more
a result of the procedure by which the FF was developed and than an artifact of the
simulations themselves.

Table 5.7 HSO4 group arrangement: QM and FF calculations versus MD and
experimental values in phase II
MD
Optimized FF Min.
CsHSO4 CsHSO4 (at 298 K)
Monomer Monomer

Experiment- ExperimentX-raya
Neutronb
(at 298 K)
(at 300 K)

r(S-O) (Å)

1.488,
1.487

1.484,
1.484

1.48(3),
1.48(3)

1.438(3),
1.433(3)

1.430(5),
1.435(9)

r(S-OD) (Å)

1.650

1.65(3)

1.573(4)

1.589(8)

r(S-OA) (Å)

1.479

1.487

1.49(3)

1.461(3)

1.472(5)

<(SODH) (°)

105.9

106(3)

114.7(4)

110.6(6)

<(OSOD) (°)

107.3,
106.6,
103.9

107.0,
105.3,
105.3

105(2),
104(2),
104(2)

107.4(2),
106.6(2),
101.5(2)

107.5(5),
106.9(4),

<(OSO) (°)

113.5,
113.1,
111.8

113.0,
113.0,
112.3

114(2),
114(2),
114(2)

113.6(2),
113.5(2),
113.1(2)

114.2(6),
113.4(5),
111.9(5)

53.1

60.6

60(15)

42.2(4),
79.7(3)

47.3(7),
75.6(7)

r( Cs − Onn )(Å) 3.218

3.217

3.16(27)

3.220(4)

3.218(6)

101.9(5)

From single crystal X-ray diffraction on CsHSO4 at 298K26.

From the high-resolution neutron powder diffraction study of CsDSO4 at 300 K124.

189
In conclusion, it can be said that these MD simulations have done a very good job
in reproducing the overall structure of phase II CsHSO4, in spite of the fact that the
hydrogen bonds parameters and internal structure of the tetrahedra deviate significantly
from experimental values.

5.4.2 Phase Transition: Cell Parameters
The average cell parameters, a, b, c, α, β, and γ were plotted as a function of
temperature in Figure 5.5. At 598 K, the cell parameters, a, b, c, and β, show dramatic
changes. These parameters are expected to change if the symmetry of the cell is to
increase from monoclinic to tetragonal. The exact nature of the changes is determined by
r r
relating the cell vectors of the monoclinic phase ( a m , bm and c m ) to those of the
r r
tetragonal phase ( at , bt and ct ) as follows125:

r 1 r
r 1 r
r r
at = a m + bm , bt = a m − bm , ct = a m + 2c m

(5-1)

Using this relation, we can compare the lattice parameters of the supercell at 623K to the
experimental values of phase I CsHSO4, Table 5.8, where a, b, c of the supercell have
r r
been divided by two to give a m , bm and c m , respectively. There is a good agreement

between the transformed lattice parameters of the simulation and the published values.
Thus, Figure 5.5 indicates the phase transition from the monoclinic phase II to the
tetragonal phase I CsHSO4 between 573 K and 623 K. Moreover, the atom positions of
the Cs and S atoms, transformed with Eq. 5-1, are extremely close to the measured values
for the tetragonal phase, Table 5.9.

190
Note that the average internal structure of the HSO4 groups remains basically
unchanged from that of the simulations at 298 K. It would then seem possible to predict
superprotonic phase transitions without reproducing the specifics of the tetrahedral
groups below or above the transition. This fact makes the search for new superprotonic
solid acids via computer simulations seem quite feasible.

191

a) 120
115

Cell Angles (Degrees)

110

alpha
beta
gamma

105
100
95
90
85
80
75
70
65
300

400

500

600

700

Temperature (K)

Cell Lengths (Angstroms)

17.50

a-axis
b-axis
c-axis

17.25
17.00
16.75
16.50
16.25
16.00
15.75
15.50
15.25
300

400

500

600

700

T em perature (K )

Figure 5.5 Cell parameters as a function of temperature (MD simulations): average
values calculated from the final 150 picoseconds of each 300 ps simulation. Lines
indicate the two transitions at 598 and 698 K.

192

Table 5.8 MD vs. experimental parameters for tetragonal phase I CsHSO4
Parameter

Experiment- ExperimentNeutrona
Neutronb

ExperimentNeutronc

(at 414 K)

(at 430K)

(at 448K)

density (g/cm3) 3.13(5)

3.282(6)

3.27(2)

3.2366(2)

a (Å)

5.77(17)

5.718(3)

5.729(9)

5.74147(9)

b (Å)

5.77(17)

5.718(3)

5.729(9)

5.74147(9)

c (Å)

14.66(54)

14.232(9)

14.21(1)

14.31508(26)

α (°)

90(2)

β (°)

90(2)

γ (°)

90(1)

r(S-O) (Å)

1.65(4), 1.49(4), 4 x 1.48(2)

2 x 1.46(5),

2 x 1.504(10),

1.48(4), 1.48(4)

2 x 1.48(7)

2 x 1.570(5)

(at 623 K)

<(OSO) (°)

r(OD-OA) (Å)

114(4), 114(4),
114(4), 104(3),
104(3), 104(3)

2 x 111.9(8), 125.5(9), 95.5(8), 112.5(3), 112.5(3)

2.9(3)

8 x 2.79(4)

4 x 108.3(9)

2 x 116.8(8),

108.9(3), 108.9(2)

2 x 98.2(10),

107.2(2), 107.1(4)

4 x 2.84(6)

16 x 2.59(1),
8 x 2.806(18)

r( Cs − Onn ) (Å) 3.17(34)

3.17(2)

3.18(4)

3.248(7)

From the neutron powder diffraction study of CsHSO4 at 414 K61.

From the neutron powder diffraction study of CsDSO4 at 430 K125.

From the high-resolution neutron powder diffraction study of CsDSO4 at 448 K124.

193
Table 5.9 MD vs. experimental atomic positions for Cs and S in phase I CsHSO4
Experiments above 414 Ka

MD at 623 K
x/a

y/b

z/c

x/a

y/b

z/c

Deviation
((∆x2+∆y2+∆z2)/3)1/.2

0.501(16)

0.249(9) 0.124(17)

0.5

0.25

0.125

0.0010

0.002(11)

0.749(7) 0.125(6)

0.75

0.125

0.0015

From X-ray and neutron diffraction studies of CsHSO4 above 414 K61,124,125.

5.4.3 Phase transition: Volume and energy change across Tsp
The average potential energy and volume of the unit cell is shown as a function of
temperature in Figure 5.6. Both graphs display a jump in their values at 598 K and again
at 698 K. From the arguments of the previous sections, we can clearly associate the first
discontinuity with the transition of CHSO4 to its superprotonic phase. The second
discontinuity would appear to be melting as it involves a large volume change. A least
squares fit to the data in the ranges 298-573 K and 598-673 K allows for a comparison
between the simulation and experimental values of the volume and enthalpy changes
across the phase II→I, Table 5.10
The MD values for the volume and enthalpy change of the transition as well as
the stability range of the superpronic phase (Tm-Tsp) are in very good agreement with the
reported values. The transition temperature, however, is almost 200 K higher than that
measured experimentally. This is the first large discrepancy with the experimental data,
but is not particularly distressing as it most likely represents some of unrealistic

194
limitations placed on the system by the simulations. The parameters effecting the phase
transition temperature are investigated in more depth in section 5.5.

Table 5.10 MD vs. experiment: Characteristic values of the superprotonic phase
transition in CsHSO4.
∆(volume)sp
(%)

∆(enthalpy)s Tm
(K)
(kcal/mol)

Stability Range(Tm-Tsp) (K)

MD results 598

1.7

1.6

673 K

experiment 414(1)

0.9-1.9

1.43(12)

485(2) K

71(3)

61,109 109,109a

85,189

59,130

3,85

Tsp
(K)

reference

3,85

59,130

Calculated from structural data. Note thermal expansion coefficient for phase II
CsHSO4 given in 109is incorrect when compared to the printed data: listed as 0.056
cm3/deg, but a direct calculation gives 0.035 cm3/deg.

-7 3 0 0

Potential Energy (kcal/mole)

195

-7 4 0 0

Y = -8 2 1 1 + 1 .0 8 X
R = 0 .9 6 4

-7 5 0 0
-7 6 0 0
-7 7 0 0
-7 8 0 0

Y = -8 0 8 8 + 0 .7 9 X
R = 0 .9 9 7

-7 9 0 0
300

400

500

600

700

T e m p e ra tu re (K )

4300

Volume (Angstroms )

4200
4100

Y = 3 2 3 2 + 1 .0 9 X
R = 0 .9 5 1

4000
3900
3800
3700

Y = 3 4 8 4 + 0 .5 6 X
R = 0 .9 8 9

3600
300

400

500

600

700

T e m p e ra tu re (K )

Figure 5.6 Potential energy, a), and volume, b), as a function of temperature from MD
simulations: 298 K to 723 K.

196

5.4.4 Phase transition: X-ray diffraction
Perhaps the most direct and convincing way to determine if the MD simulations
have correctly predicted the superprotonic transition of CsHSO4 is to look at the
diffraction patterns calculated from the MD structures below and above the transition.
Thus, the time-averaged X-ray powder diffraction (XPD) pattern for each tempeture was
calculated using the instantaneous structure at each 0.1 ps of the final 150 ps and
averaging the 1500 XPD patterns generated using the Cerius2 program176. For
comparison, the XPD patterns of the experimentally determined crystal structures of
phase II, Figure 5.7 a, and phase I, Figure 5.7 d, were also calculated. It is clear that the
patterns below 598 K are characteristic of phase II, Figures 5.7 b and c, while the patterns
above 598 K are characteristic of phase I, Figures 5.7 e. This is quite conclusive
evidence that the II →I phase transition was obtained during the simulation. A X-ray
diffraction pattern from the MD simulation at 723 K was also generated, Figure 5.7 f. It
shows significantly less structure than the other patterns and reinforces the idea that the
second phase transition is associated with melting.

197

MD simulations
at 573 K

30
20

20
15
10

80
70
60
50
40
30
20
10

MD simulations
at 298 K
Intensity

MD simulations
at 623 K

50
40

30
20
10

100

Experimental
structure at 298 K

Experimental
structure at 414 K

Intensity

MD simulations
at 723 K

Intensity

40
20

2Θ (Degrees)

Figure 5.7 Calculated X-ray powder diffraction patterns. (a) Phase II CsHSO4 calculated
from the experimental structure26. (b) and (c) Phase II calculated from MD simulations at
298 K and 573 K, respectively. (d) Phase I CsHSO4 calculated from Jirak’s experimental
structure61. (e) Phase I calculated from MD simulations at 623 K. (d) X-ray diffraction
pattern calculated from MD simulations at 723 K, above the 2nd (melting) transition.

5.4.5 Vibrational spectrum of Phase I CsHSO4
It is quite apparent from sections 5.4.1 to 5.4.4 that the time averaged structures of
both phase II and phase I CsHSO4 have been well duplicated by these MD simulations.
To determine whether the dynamics these phases were equally well reproduced, the IR
spectra of the simulations at 298 and 623 K were calculated using the Cerius2
software176, Figure 5.8 a and b. These graphs were created by taking the average of 30 IR

198
spectra, calculated every 5 ps from 150 to 300 ps. The comparison with experimental
data is favorable for both phases of CsHSO4, Figure 5.8 c) and d).

Absorbtion (%)

b) IR spectrum at 623 K
60

Simulation Wavenumber (cm-1)

100

d) CsHSO4 Phase I
Exp1 vs Sim.
2500
Exp2 vs Sim.
3000

2000
1500
1000

Slope = 0.93(2)
R = 0.994

500

1000

2000

3000

Wavenumber (cm-1)

3500

Absorbtion (%)

a) IR spectrum at 298 K
60

1500

2000

2500

3000

3500

c) CsHSO4 in Phase II
Exp1 vs Sim.
Exp2 vs Sim.

3000
2500
2000
1500

Slope = 0.92(2)
R = 0.993

1000
500

Simulation Wavenumber (cm-1)

1000

Experimental Wavenumber (cm-1)

100

500

1000

2000

Wavenumber (cm-1)

3000

500

1000

1500

2000

2500

3000

3500

Experimental Wavenumber (cm-1)

Figure 5.8 Calculated IR spectra for MD simulations: for phase II and I of CsHSO4, a)
and b), respectively. Peaks with strongest absorption are marked with an asterix and the
frequencies at the peak maximums are compared to those of the strongest peaks found by
infrared spectroscopy on polycrystalline CsHSO4 107,190. In both phase II, c), and phase I,
d), the simulation and experimental peak positions are very similar (for perfect match ⇒
slope = 1). A plot of experiment 1 versus experiment 2 (not shown) results in a slope of
0.98(1), R2 = 0.999, as the measured IR spectra changes little over the phase II-I
transition.

199
The overall vibrational behavior of the simulations therefore closely mimics
reality. However, as the IR spectra for phase II and I are very similar (for the
experiments and simulations alike), further analysis of the simulation data is required to
discover if the dynamics of the two phases do indeed change across the phase transition
as expected: i.e., from oscillations around a fixed structure to librations/reorientations in a
disordered phase.

5.4.6 Orientation of the HSO4 groups
To confirm that the simulations contain the observed change from a static to
dynamic structure across the superprotonic phase transition, the orientation of the HSO4ions in the simulations was examined. In phase II, the tetrahedra have only one
crystallographically distinct orientation, whereas the HSO4- groups accommodate the
higher symmetry of phase I by librating between different orientations 84. The orientation
of a tetrahedron was defined as the vector between its S and OD atoms. If the simulations
correctly predicted these two phases, one would expect a particular tetrahedron to have
only one general orientation for all temperatures below 598 K, but that its S-OD vector
would begin to change direction significantly for T > 598 K. With this in mind, the
vector pointing from the S85 to O86 atom was measured every 0.1 ps from 150-300 ps at
each temperature step. This particular vector was chosen because it belongs to one of the
eight central tetrahedra not directly effected by the boundary conditions of the supercell
and should therefore librate (or not) the most freely. The orientation of this vector was

200
mapped onto a polar coordinate system with a particular direction (x,y,z) determining a
pair of angels ( θ , φ ) using the standard equations:
θ = a cos 

θ (x,y,z)

Laboratory (fixed) frame

and

x + y + z 

 if y > 0
φ = a cos
 x y 
 if y < 0
cos
 x +y 

These pairs of angles were then used to create the density maps shown in Figures
5.9 and 5.10, where each contour line represents a tenth of the maximum density of the
plot. In Figure 5.9 a, b, and c) it is clear that the S85-O86 vector had only one orientation
while in phase II (298-573 K). Immediately above the transition, however, this vector
began reorienting between four different directions, Figure 5.9 c). A plot of all
orientations taken by the four S-O vectors of the S85 tetrahedron from 598-673 K, Figure
5.9 e), shows the same four basic orientations, agreeing with the expectation that the
oxygen atoms swap positions in the superprotonic phase124. As stated before, this type of
dynamical behavior is essential to the phase II to I transition of CsHSO4 and it therefore
seems that these simulations have correctly predicted the dynamics as well as the
structure of phase I. Above the second transition at 698 K, the S85-O86 vector appears to
orient randomly, Figure 5.9 f), consistent with this phase’s designation as a liquid.

201

0.0

d) T = 598 K

0.5

1.0

1.5

2.0

2.5

0.5

Theta (radians)

1.0

1.5

2.0

2.5

3.0

0.0

a) T = 298 K

c) T = 573 K

Phi (radians)

0.5

1.0

1.5

2.0

Theta (radians)

1.0

2.5

3.0

0.0

1.5

2.0

2.5

3.0

Theta (radians)

0.0

0.5

Theta (radians)

Phi (radians)

0.0

3.0

f) T = 723 K

Phi (radians)

b) T = 498 K
Phi (radians)

Phi (radians)

e) 598-673K

0.5

1.0

1.5

2.0

Theta (radians)

2.5

3.0

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Theta (radians)

Figure 5.9 Probability distribution functions for the S-O vectors: a), b), c), d), and f)
were created using only the orientations of the S85-O86 (donor oxygen) at 298, 498, 573,
598, and 723 K, respectively, e) is a combination of the orientations for all four S-O
vectors of the S85 tetrahedron in the temperature range 598-673 K (phase I). The
directions of the S-O vectors were converted into the polar coordinates ( θ , φ ) and then
mapped onto the 2-D ( θ , φ ) space. A probability distribution function was created by
dividing up the full ranges of each variable (0≤θ < π, 0≤φ <2π) into a 20x20 matrix and
assigning each ( θ , φ ) pair to a particular cell.
A direct comparison of Figure 5.9 d) and f) shows a small but significant
difference in the positions of their respective peaks. To examine this discrepancy, the
data from 598 to 698 K was combined separately for each oxygen of the S85 tetrahedra.
It was found that the three oxygens not bonded to the hydrogen (i.e., OA(1), O(2), O(3))
had peaks positions that were indistiguishable from each other, while the donor oxygen
had peak positions very similar to those shown in Figure 5.9 d. This subtle difference in
peak positions is therefore an artifact of the FF which differentiated between donor

202
oxygens and all other oxygen atoms, as is evident in the O-S-O angles of the tetrahedra
(Table 5.7 and 5.8).
By plotting the orientations of the S-O vectors for all tetrahedra in the
simulations, another trait of the high temperature phase of CsHSO4 is duplicated.
Although there is only one crystallographically distinct sulphate tetrahedra in phase I, the
symmetry of the phase is such that the tetrahedra can be grouped into two types when the
S-O vectors are mapped onto the ( θ , φ ) plane. These two types of tetrahedra can be seen
in Figure 5.1 b, where layers of “up” and “down” pointing tetrahedra are arranged
perpendicular to the c-axis. A density map for all the orientations of all the S-O vectors
in phase I should then have eight distinct directions (4 “up” and 4 “down”). Indeed,
when all S-O vectors in the supercell are used to create a density map, eight general
directions are evident in the 598-673 K simulations (temperature range of phase I), Figure
5.10.
Again, the “up” and “down” tetrahedra and their corresponding orientations are
not crystallographically distinct and do not refer to the librations necessary to satisfy the
symmetry of phase I. The actual number and direction of the libration orientations are in
dispute, Table 5.11. We can compare these directions with those of the experimental
structures after properly adjusting the coordinate system of these structures to that of the
laboratory frame of the simulations. Such a comparison does not, unfortunately,
significantly favor one structure over another as all three have orientations distributed
around those found in the simulation. It does, however, confirm that the orientations of

203

Figure 5.10 Probability distribution functions for all S-O vectors in phase I (from 598673 K simulations). The eight distinct directions correspond fairly well to S-O directions
derived from the published structures.
the tetrahedra in the simulations are consistent with those determined experimentally.
The density map also helps to explain the difficulty of determining the exact position of
the oxygen atoms in phase I as the large deviation of each orientation in Figure 5.10
suggests an equally disperse electron density around the corresponding oxygen.
Quantitatively, the deviation of the orientations was evaluated by measuring the
FWHM of the eight positions and calculating the average, resulting in an average of
27(3)° and 39(7)° for θ and φ , respectively. The statistically larger average FWHM in
the φ direction is most easily explained by librations of the oxygen atoms. To compare

204
this result with the published structures, the average angular difference between the
experimentally determined libration directions (in the simulation frame) was calculated,
Table 5.11. The librations inherent to the structures of both Jirak and Belushkin result in
a larger angular difference in the φ compared to θ direction, in agreement with the
simulation results. The librations proposed by Merinov, however, result in a larger
angular difference for the θ direction.

Table 5.11 Proposed librations in CsHSO4 phase I compared to simulation results
Source
# of
Average ∆θ
Average ∆φ
∆φ – ∆θ ref
librations

between librations

Merinov

-10

125

Jirak

Belushkin

124

Simulation

N/A

27(3)

39(7)

12(10)

5.4.7 Reorientation of the HSO4 groups
The degree to which the 32 HSO4- ions reoriented in the simulation was also
evaluated. As it was necessary to know not just how much the tetrahedra were
reorienting, but also the type of reorientation (e.g., oscillation, libration, rotation), the
tetrahedra were examined by two methods. First, reference orientations for the tetrahedra
were defined as their orientation at 150 ps into each temperature step. Then the amount
each tetrahedra changed these reference orientations was measured. Second, the average
angular change per picosecond of these orientations was calculated for each temperature

205
equilibration. Both these methods continued to define the orientation of an HSO4- ion by
the vector pointing from its S to OD atoms and measured this vector, for all 32 tetrahedra,
from 150-300 ps in 1 ps steps, Figure 5.11.

θ (t')

t=0

t = t'

Figure 5.11 Orientation/reorientation of an HSO4- ion defined by its S-OD vector: a) an
HSO4- ion has the vector pointing from its S to OD atom initially aligned vertically; b)
after a time t′, the HSO4- ion has changed its orientation and the S to OD vector now
forms an angle θ(t′) with the vertical.
To measure the extent to which each tetrahedra changed its orientation with
respect to the reference S-OD vector (at 150 ps), the cosines of the angles formed by the
reference vector and the S-OD vectors at 151-300 ps (by 1 ps steps) were calculated. At
each temperature the average cosine value and its standard deviation were then computed
from the cosine values of all 32 tetrahedra to determine the overall variance of the HSO4ions’ orientation with increasing temperature, Figure 5.12 a. Looking at Figure 5.12 a,
we see a gradual decrease in the average cosine value as it approaches the transition
temperature of 598 K and a drop across the transition (from 0.76 to 0.48 at 573 and 598
K, respectively), indicating that the HSO4- groups are reorienting to a far greater degree
above the transition.

206
In fact, the data is misleading as an analysis of the average cosine value per
picosecond reveals that the tetrahedra reorient significantly away from the reference
direction at 573 K, but do not completely “forget” their original orientation, Figure 5.12
b. This behavior is consistent with a large oscillation or libration around a central
direction. However, from the previous section (Figure 5.9) we know that the simulation
tetrahedra do not librate in phase II (298-573 K) and so the form of Figure 5.12 b should
represent oscillations.
Above the superprotonic transition, the average cosine value steadily drops
toward zero, Figure 5.12 c): where zero represents a total randomization of the
tetrahedral orientations with respect to the reference directions. In fact, at 623 K the
average cosine value reaches zero by the end of the simulation, Figure 5.12 d). One can
then expect that the autocorrelation plot at 598 K, Figure 5.12 c), and all like it above 573
K, would approach zero with a significantly long equilibration time. This is consistent
with the description of near “free rotation” of the tetrahedra in phase I and the melt190.

207

1.0

0.8
0.4

Phase I
0.6
0.4

0.2

Melt

0.0

Average Cosine Values

Average Cosine Value

0.6

0.8

Phase II

0.2
0.0
-0.2

0.8
0.6
0.4
0.2
0.0

400

500

600

Temperature (K)

700

c) T = 598 K

0.8
0.6
0.4
0.2

300

d) T = 623 K

0.0

b) T = 573 K
160

180

200

220

240

260

280

300

Tim e (ps)

Figure 5.12 Autocorrelation functions for all 32 tetrahedra evaluated using their average
cosine values between the reference orientations (at 150 ps) and subsequent orientations:
a) average cosine value versus temperature; b), c), d) average cosine value at 573, 598,
and 623 K, respectively, versus time. Note the decrease in the average cosine value
across the phase II to I transition in a) and the fact that the average cosine values continue
to decrease above the transition, c) and d), whereas they equilibrate around 0.73 below
the transition, b).

To quantitatively describe the extent to which the sulfate tetrahedra were
reorienting in the simulations, the average angular change per picosecond for the S-OD
vectors were calculated for each temperature. This calculation involved measuring the
angular change between a HSO4- ion’s S-OD vector with respect to its orientation 1 ps
before (Figure 5.11 , t′ = 1 ps). Similar to the previous method, the angular velocities for
all 32 tetrahedra were then averaged at each temperature, Figure 5.13. Looking at the
results of this calculation, it is apparent that although the tetrahedra increase their angular
velocities at the phase II to I transition, this increase is not a large one. This again
emphasizes that the two phases are fairly similar in their overall vibrational spectra,

208
observed by IR and Raman spectroscopy 190, and that the manner in which they reorient is
the cause of the phases’ dramatically different properties (i.e., superprotonic conductivity
of phase I).

∆ angle / ∆ time (degrees/ps)

45
40
35
30

Phase II

Melt

20
15

Phase I

10
300

400

500

600

700

Temperature (K)

Figure 5.13 Average angular velocity of all 32 S-OD vectors versus temperature. Note
that the jump in this value at the phase II-I transition is only ~ 25 %. Therefore the
dramatic increase in protonic conductivity across the transition (10,000-100,000 %) is
due to a change in the nature of the reorientations (e.g., oscillation to libration/rotation)
rather than the degree of the reorientations.

5.5 Parameters Effecting the Phase Transition Temperature
As the previous results have described, these simulations have duplicated
extremely well the characteristics of CsHSO4 in its room and high temperature phases

209
including the necessary changes at the phase transition. However, the one property in
which the simulation results and experimental observations are glaringly at odds is the
phase transition temperature itself: 598 K versus 414 K, respectively. This contradiction
between very good and very poor agreement with experimental values suggests that in
general the force field parameters correctly define the interatomic interactions of CsHSO4
in both phase-II and I, but that some particular force field parameters have a large
retarding effect on the phase transition temperature. Of course, it is a distinct possibility
that the decision to prohibit proton hopping is responsible for the high transition
temperature. Proton hops between oxygen atoms in phase II CsHSO4 would necessarily
require breaking of crystallographic hydrogen bonds and/or rotations of the sulphate
groups. Both these actions would tend to destabilize the fixed structure of phase-II when
compared to the dynamically disorder phase-I, where the breaking of hydrogen bonds and
tetrahedral rotations are a must. Therefore, fixing the protons to their respective donor
oxygens will favor phase-II over phase-I, possibly to the tune of 175 degrees K.
However, as one of the purposes of these simulations was to determine if indeed the
superprotonic transition was possible without proton hopping (YES!), we must look for
other parameters effect the transition temperature.
A priori, one would suspect that parameters delaying the onset of the
superprotonic transition would include any that tend to favor a fixed over dynamic
structure, order versus disorder, low versus high symmetry; that is to say, parameters that
inhibit the “free” reorientation of the sulphate groups. One would hence tend to ignore
the parameters that deal with the internal atomic interactions of the HSO4- ions and focus
on the parameters that govern the HSO4- to HSO4- and HSO4- to Cs+ interactions. Such

210
parameters as the atomic charges, hydrogen bond strength, and hydrogen torsional
barrier height. Hence, the effect of these three areas on the transition temperture were
evaluated by re-running the MD simulations after the parameters effecting one (and only
one) of these three areas were changed, with all other FF parameters identical to that
found in Table 5.2.
Before exploring the results of the above experiments, it should be mentioned that
the effect of increasing the equilibration time was also investigated as this overarching
parameter effects all simulation runs. This investigation consisted of doubling the soak
time of each temperature step from 300 to 600 ps and re-running the original simulations.
Analysis of this series of simulations revealed the transition temperatures for both the
phase-II to I and phase-I to melt transitions to decrease by 25 K (one temperature step).
Hence, the equilibriation time does indeed have an effect on the temperature at which the
phase transition appears (as expected) and therefore it is possible the original set of FF
parameters would give a transition just above 414 K with a sufficiently long soak time.
This statement applies equaly well to the following simulation runs and in particular to
those with “transition zones” in which the simulations results do not conform to any of
the experimentally observed phases, as the timescale of most laboratory experiments are
in seconds if not minutes or hours. However, since it is not resonable to perform
simulations such as these with equilibration times much over 300 ps, a better
understanding of the FF parameters effecting the presence and temperature of phase
transitions is necessary. Of course, identifying the parameters that have the largest effect
on a phase transition also gives very useful insight into underpinings of the transition
itself.

211

5.5.1 Oxygen charge distribution
As the dominant binding energy of ionic solids comes from their Coulombic
potential energy191, the atomic charges were expected to have a large effect on the MD
simulations. Indeed, by far the largest change to the superprotonic phase transition
temperature (from 598 to 423 K, Figure 5.14) resulted from setting all the oxygens
charges to – 0.612 |e|, equal to the average of the four charges in Table 5.3. This
egalitarian distribution of the oxygen charges favors the superprotonic phase where all
oxygens are identical over time and should therefore have an equivalent average charge.

17.5

Transition
Zone

Lattice Constants
(Angstroms)

Phase II
a-axis
b-axis
c-axis

17.0

16.5

Phase I

16.0

Melt

15.5
300

400

500

600

700

500

600

700

Lattice Constants
(Degrees)

130
120
110
100
90

alpha
beta
gamma

80
70
60
300

400

Temperature (K)

a)
Figure 5.14 (See figure caption on next page.)

212

-7400

Transition
Zone

Potential Energy (kcal/mole)

-7300

Phase II

-7500

Phase I

-7600

-7700

Melt
Y = -8088 + 0.79X
R = 0.999
Y = -7918 + 0.89X
R = 0.994

-7800

-7900
300

400

500

600

700

Temperature (K)

b)
Figure 5.14 Results of equal oxygen charge MD simulations: a) plot of lattice constants
versus temperature reveals only the c-axis lattice constant to change significantly from
398 to 423 K and that lattice parameters do not achieve the values respective of phase I
until 523 K; b) graph of potential energy versus temperature shows little change until 523
K and then again at 673 K.
From Figure 5.14 a, b, and the analysis of section 3.2, it is clear that the transition
from phase II to phase I begins at 423 K. A close inspection of Figure 5.13 a shows that
only the c-axis lattice constant changes significantly from 398 to 423 K. This increase in
the c-axis is consistant with a straightening of the zigzag chains of hydrogen bonded
sulphate groups. A careful analysis of all possible radial distribution functions confirmed

213
such a straightening. The transition to phase I then begins at 423 K, but is not completed
until 523 K; at which temperature |c| > |a| = |b| corresponding to phase I. To confirm such
an analysis, the X-ray diffraction patterns were calculated at 398, 423, and 523 K, Figure
5.15. The simulation patterns at 398 and 423 agree with that calculated from the
experimental data on phase II26, while the pattern at 523 K matches up with the XPD
pattern calculated using the measured structure of phase I61.

Intensity

e) Sim. - 523 K
d) Exp. - 415 K
c) Sim. - 423 K
b) Sim. - 398 K
a) Exp. - 298 K
10

2 Theta (Degrees)
Figure 5.15 Calculated X-ray diffraction patterns for equal oxygen charge MD
simulations: a) from structure determined by single crystal X-ray diffraction at 298 K, b)
from simulation results at 398 K, c) from simulation results at 423 K, d) calculated from
structure determined by powder X-ray diffraction at 415 K, e) from simulation results at
523 K. Comparison of patterns in a), b), and c) show the simulations at 398 and 423 K to
be in phase II CsHSO4. Whereas, the diffraction pattern at 523 K matches the phase I
pattern calculated from experiment, e) and d), respectively.

214
Giving all the oxygen atoms equivalent charges then caused the onset temperature
of the transition to drop by 175 degrees, to the first MD simulation above the reported
transition temperature of 414 K. Although making all the oxygen charges equivalent is
not realistic, a force field that allowed for continually changing charges on the oxygens
based on their environment would be expected to give similar results. Such a force field
is currently under development at Caltech in the Goddard Simulations Group and these
simulations will be run again with this force field to judge the veracity of the previous
statement. This new force field can also allow the hydrogen atoms to hop between
oxygens which, combined with the continual re-evalutation of oxygen charges, should
give very realistic MD simulations, the results of which will make for an interesting
comparison with this study.

5.5.2 Hydrogen Bond Strength
To evaluate the effect of hydrogen bond strength on the superprotonic transition,
it was first necessary to define the strength of a hydrogen bond in these simulations. The
strength of a hydrogen bond usually refers to the energy required to disociate the ODH···OA complex which is highly correlate to the OD-OA distance: the smaller the donor to
acceptor distance, the greater this disociation energy and vice versa14,21. Hence to
increase/decrease the hydrogen bond strength should require shortening/lengthening the
equilibrium OD—OA distances in the simulations. However, for the purpose of a direct
comparison between the original and subsequent simulations, changing these distances
was not ideal. So, the hydrogen bond strength was defined as the binding energy of the
H2SO4 dimer used to determine the hydrogen bond parameters found in Table 5.2. Using

215
the same force field energy minimization techniques as described in section 5.3.2, the
hydrogen bond parameters (RO and DO) were adjusted to either increasing or decreasing
this binding energy, while maintaining the same equilibrium OD—OA distance of 2.647
Å. Unfortunately, such a definition did not allow for a significant decrease in the
hydrogen bond strength (due to competition with electrostatic energy) to merit re-running
the simulations. However, an increase in the hydrogen strength was possible and the
simulations were re-run with a 150% and 200% increase in the afore mentioned binding
energy; i.e., 1½ and 2 times the original hydrogen bond strength, Figures 5.16.

Phase I

Lattice Constants (Angstroms)

a)
a-axis
b-axis
c-axis

17.5

M elt

17.0

Phase II
16.5

16.0

15.5
300

400

500

600

Tem perature (K)

Figure 5.16 (See figure caption on following page.)

700

216

-7500

-7600

Y = -8289+0.92*X
R = 0.994

-7700

Melt

-7800

Phase I

Potential Energy (kcal/mole)

Phase II

-7900

-8000
300

400

500

600

700

Temperature (K)

a-axis
b-axis
c-axis

17.5

Transition

Lattice Constants (Angstroms)

17.0

Phase II
16.5

16.0

Melt

15.5

300

400

500

600

Tem perature (K)

Figure 5.16 (See figure caption on next page.)

700

217

d)
Potential Energy (kcal/mole)

-7 7 0 0

-7 8 0 0

Y = -8 4 0 9 + 0 .8 4 *X
R = 0 .9 9 7

-7 9 0 0

R = 0 .9 9 1
-8 0 0 0

-8 1 0 0

T ra n s itio n
a n d M e lt

P h a s e II
300

400

500

600

700

T e m p e ra tu re (K )

Figure 5.16 Simulation results with the binding energy of the hydrogen bonds increased
by 150 percent, a) and b), and by 200 percent, c) and d). Note the low temperature
transitions in the 323 to 373 K regions of a) and c) and the fact that overall, potential
energy of the simulations decreases by around 150 kcal/mole as the hydrogen bond
strength increases from 1 (see Figure 5.6) to 1.5 to 2 times that of the original simulation.
In agreement with previous arguments, the 50 and 100 percent increase in the
hydrogen bond strength favored the fixed structure of phase II over the disordered phase I
and delayed the transition by 25 K. Thermal experiments show the superprotonic
transition of CsHSO4 to be 3-5 degrees higher than that of CsDSO4, which has been
attributed to an increase in the OD-OA distance and subsequent decrease in bond strength
when deuterium is swapped for hydrogen2. This experimental result would seem to
concur with the above delays in the transition temperature. However, as there is no
statistical difference between the OD-OA distances of the two compounds26, it is

218
impossible to say whether the measured 3-5 degree shift is due to an increase in hydrogen
bond strength or, perhaps, the higher mobility of the proton versus deuterium atom.
The increase in hydrogen bond strength also had an effect on the phase I to melt
transition. For the simulation with a 50 percent increase in hydrogen bond energy, this
effect was to lower the transition by 25 K. Even more dramatically, the simulation with
hydrogen bond strength doubled never achieved phase I, but transformed continuously
from phase II through a phase I like region to the melt. A possible explanation for this
behavior involves the stronger hydrogen bonds limiting the “free rotations” of the
tetrahedra in phase I and the melt, but as the melt also has translational entropy, it
becomes more energetically stable compared to phase I. Creating autocorrelation
functions for these two simulations, as was done for the original simulation in section
5.4.7, it is apparent that as the hydrogen bond strength increases, the degree of tetrahedral
reorientation decreases significantly only in phase I and the melt, Figure 5.17.

219

1.0
0.8

Phase I

Phase II

0.6

Melt

0.0
1.0

a) 1 x H-bond B.E.
300

0.8

400

500

600

Phase II

0.6
0.4
0.2
0.0
1.0

300

400

500

600

700

Melt

Phase II

0.6
0.4

0.0

Melt

b) 1.5 x H-bond B.E.

0.8

0.2

700

Phase I

0.2

Phase I

Average Cosine Value

0.4

c) 2 x H-bond B.E.
300

400

500

600

700

Temperature (K)

Figure 5.17 Autocorrelation functions for a) original simulation and simulations
with b) 50 and c) 100 % increase in hydrogen bond strength. The average of the cosine
values in phase two increases from 0.27 to 0.37 to 0.55 going from a) to b) to c).
Similarly, in the melt this average value is 0.06, 0.15, and 0.45 for a), b), c), respectively.
In contrast, the averages of the cosine values in phase II are within error for all three
simulations: 0.91, 0.90, 0.92 for a), b), and c), respectively.
Although there is no way to experimentally confirm this conclusion, the related
compound CsH2PO4 does exhibit properties that collaborate this rotation limited theory.
This compound has all oxygens involved in strong hydrogen bonds with OD-OA distances
of either 2.54 or 2.47 Å, which connect the phosphate tetrahedra into planes28. As both
these hydrogen bonds are shorter than those found in phase II CsHSO4, this compound

220
should have significantly more than twice the amount of energy invovled in hydrogen
bonds than CsHSO4. At 505 K, CsH2PO4 has a superprotonic phase transition to a cubic
phase which is stable in a water saturated environment, but under ambient conditions
quickly decomposes to form Cs2H2P2O792. If the differences in the phase transitions of
CsHSO4 and CsH2PO4 are mainly due to the difference in the total hydrogen bond
energies of the two compounds (see section 4.2.4, Figure 4.6 d) , then the stronger and
twice as plentiful hydrogen bonds in CsH2PO4 are responsible for a 90 K delay in the
superprotonic transition and the instability of the superprotonic phase versus a second,
more entropic, state (in this case decomposition). Of course, as the two compounds are
neither isostructural below nor above their superprotonic transitions, phosphates have
been swapped for sulfates, and two strong hydrogen bonds have replaced one medium
strength bond, that is a very big if. Nevertheless, since the overall arrangement of atoms
in the two compounds is quite similar (both have zigzag chains of hydrogen bonded
tetrahedra, but those chains are cross-linked in CsH2PO4) and because of the chemical
similarity of phosphorus and sulfur (the S-O and P-O bond valence contributions are
nearly identical), this analogy does not seem too far-fechted31.
Another interesting feature of these simulations with increased hydrogen bond
energies, is the low temperature changes in their lattice constants which are suggestive of
the phase III to II transition of CsHSO4 (Figure 5.16 a and c, between 298 and 423 K).
These changes occur at 348 and 373 K with cell volume changes of -0.7 and -1 percent,
for the 1.5 and 2 times hydrogen bond strength simulations, respectively.
Experimentally, the phase III to II transition occurs around 330 K with an accompaning
volume decrease of ~ 1 percent109. The enthalpy change of the transition is small, ~ 0.12-

221
0.24 kcal/mol130 (which equates to ~1-2 kcal/mole for the supercell), and therefore the
lack of transition in the potential energy plots (Figure 5.16 b and d) further encourages
associating these changes with a phase III-II like transition.
Perhaps most importantly, the real phase III-II transition involves a lengthening of
the hydrogen bonds: the OD-OA distances are 2.54(1) and 2.636(5) Å for phase III and II,
respectively26,109. From experimental data, these two OD-OA distances can be related to
hydrogen bond energies of approxiamately 11 and 7.5 kcal/mole, respectively21. The
hydrogen bonds in phase III CsHSO4 should therefore have around 1 ½ times more
energy associate with its hydrogen bonds than phase II. Hence, the low temperature
transitions of these two simulations are probably due to the energetic stabilization of
phase III versus II as the hydrogen bond stength is increased. Indeed, for the force field
with 1 ½ times the original hydrogen bond energy, the low temperature transition occurs
at the first simulation step above 330 K. Whereas for the force field with twice the
hydrogen bond energies of the original simulation, the phase III to II like transition
occurs 25 K later at 373 K, consistent with increasing hydrogen bond strength stabilizing
phase III over phase II.
However, the average X-ray diffraction patterns show little change before and
after these transitions, Figure 5.18. Although this would seem to disprove the above
arguments, the fact that the equilibrium OD-OA distance remained at 2.647 Å in both
simulations pretty much rules out the atoms actually arranging themselves in a phase III
structure as this would require OD-OA distances of around 2.5 Å. Instead, the transition
seems to consist of a relaxation of the phase II structure, which was held fairly fixed

222
before the transition. This change can be seen in the X-ray diffraction patterns in the loss
of some of the smaller peaks and general broadening of the peaks across the transition.

140

S im . 348 K

S im . 373 K

S im . 3 23 K

S im . 3 48 K

E xp . 29 8 K

120

Intensity

100

2 T h e ta (D e g re e s)

2 T h e ta (D e g re e s )

Figure 5.18 X-ray diffraction patterns for simulations with increased hydrogen bond
strength below and above the low temperature transitions: simulation with 1.5 times, a),
and 2 times, b), the hydrogen bond energy of the original simulations.
Further proof that these transitions are similar to the phase III to II transition come
from Raman and infrared experiments. These vibrational measurements point to a
substantial increase in sulphate tetrahedra rearrangements across the phase III to II
transition190. By calculating the number of reorientations (rotations involving all four
oxygens of a tetrahedra) and 3-fold rotations of the simulation tetrahedra at each
temperature, it is clear that the number of 3-fold rotations increases significantly above

223
the two transitions, Figure 5.19. At the same time, the number of reorientations of the
tetrahedra remains essentially zero, which explains how the simulations could increase
their vibrational disorder without this increase showing up in the autocorrelation

1200

Number of Reorientations/Rotations

functions of Figure 5.17.

Reorientations
3-fold rotations

1000
800
600
400
200
300

320

340

360

Temperature (K)

380

400

Reorientations
3-fold rotations

200

150

100

300

320

340

360

380

400

Temperature (K)

Figure 5.19 Rearrangements of the sulphate tetrahedra across the low temperature
transitions of the simulations with increased hydrogen bond strength: a) 1.5 and b) 2
times the original hydrogen bond energies. Note the jump in 3-fold rotations at precisely
the same temperatures the lattice constants changed in Figure MM a) and c), while the
number of reorientations remains at or very near zero.
Not only are these results quite convincing evidence that these low temperature
phase transtitions mimic the phase III to II transition of CsHSO4, they also suggest that
the measured increase in orientational disorder of the suphate groups across the transition
is due to 3-fold rotations and not cyclic dimers as proposed in the literature190.
Particularly as the published structures disagree with the presence of cyclic dimers in
phase II26,124. It is encouraging that the total number of 3-fold rotations in the 2 times
simulation is much less than that of the 1.5 times simulation, as one would expect the

224
strong hydrogen bonds to hamper such rearrangements. These increased hydrogen bond
strength simulations are therefore in very good agreement with both the experimental
data and a priori knowledge concerning the hampering effect of increased hydrogen bond
strength on superprotonic phase transitions like that found in CsHSO4.

5.5.3 Torsional barrier height
One would expect the effect of decreasing the hydrogen torsional barrier height to
be an increase in mobility of the hydrogens, leading to an increase in their ability to break
and reform hydrogen bonds. Consequently, the HSO4- ions themselves would be more
free to vibrate, librate, and rotate. This greater vibrational character of the sulfate
tetrahedra should make phase I more energetically favorable when compared to phase II.
Hence, the transition to phase I should happen at a temperature lower than 598 K for a
simulation with a smaller hydrogen torsional barrier height than the original. To test such
assumptions, the value of the hydrogen torsion barrier was divided by ten, from 2.1699 to
0.21699 kcal/mole, and the simulations re-run with no other changes to the original FF.
Analysis of these simulations show that the phase II to I transition was indeed encouraged
by the lowered torsional barrier, beginning at 473 K and finishing at 523 K, 75 degrees
lower than the original simulations, Figure 5.20.

225

1 8 .0

a - a x is
b - a x is
c - a x is

1 7 .5

Transition

Lattice Constants (Angstroms)

1 7 .0

P h a s e II
1 6 .5

M e lt

1 6 .0

P hase I
1 5 .5

300

400

500

600

T e m p e r a tu r e ( K )

700

-7 5 0 0

Transition

Potential Energy (kcal/mole)

-7 4 0 0

P h a s e II

P hase I

-7 6 0 0

M e lt
-7 7 0 0

Y = -8 0 8 2 + 0 .7 6 * X
R = 0 .9 9 9 7

-7 8 0 0

300

400

Y = -8 1 1 4 + 0 .9 * X
R = 0 .9 9 0

500

600

700

T e m p e r a tu re (K )

Figure 5.20 Results of simulations with lowered torsional barrier: a) cell lengths and b)
potential energy versus temperature. Simulations undergo the superprotonic phase
transition from 473-523 K and melt at 673 K.

226
Although this simulation’s values for the potential energy in phase II are virtually
identical to the corresponding values of the original simulation, the transition enthalpy
change (measured at 523 K) was 5.5 kJ/mol, which is only 83 % of the original value.
Therefore, decreasing the torsional barrier height did energetically favor phase I over
phase II. The transition to the melt was also decreased by 25 K compared to the original
melt temperature. Just as the lowered torsional barrier favored the dynamic phase I over
the static phase II, the increased mobility of the HSO4- ions will favor the isotropic melt
over the dynamically disordered phase I, hence the lowering of the melt temperature.
X-ray diffraction patterns of the simulations at low temperatures show the atoms
to remain basically in phase II until 473 K. The low temperature changes in the lattice
parameters are then probably due to phase changes similar to the ones found in the
simulations with increased hydrogen bond strength (i.e., a relaxing of the atoms into a
more energetically stable structure nearly identical to phase II). This is not unexpected as
arbitrarily changing a parameter of the force field is bound to have effects in all
temperature regions as was seen in the hydrogen bond strength simulations.
The effect of increasing the hydrogen torsional barrier was explored by setting the
barrier height to 10 times the original value, from 2.1669 to 21.669 kcal/mole. The
simulations were then re-run with no other changes to the original FF parameters.
Following the arguments that a smaller barrier height lowers the superprotonic phase
transition, one might expect a bigger barrier height to raise the transition temperature.
The results of the simulation, however, are more complicated than that as the transition
starts at 523 K, but does not finish until 648 K, Figure 5.21. The onset temperature is

227
therefore 75 K lower, while the final arrival in phase I is 50 K higher than 598 K value of
the original simulations.

a - a x is
b - a x is
c - a x is

1 8 .0

T r a n s it io n
Zone

1 7 .5

Phase I

Lattice Constants (Angstroms)

2 2 .5

M e lt

P h a s e II

1 7 .0
1 6 .5
1 6 .0
1 5 .5
300

400

500

600

700

T e m p e ra tu re (K )

P h a s e II

T ra n s itio n

-7 5 0 0

-7 6 0 0

-7 7 0 0

Phase I

Potential Energy (kcal/mole)

-7 4 0 0

Y = -8 0 7 3 + 0 .7 5 *X
R = 0 .9 9 9

M e lt

-7 8 0 0

300

400

500

600

700

T e m p e ra tu re (K )

Figure 5.21 Results for simulations with torsional barrier 10 times value in original FF:
a) cell lengths and b) potential energy versus temperature. The superprotonic transition
begins at 523 and ends at 648 K, while melting ocurrs at 698 K.

228
These two effects would appear to be in contradiction. The lower onset
temperature suggests the higher torsional barrier to have energetically favored phase I
over phase II. Whereas the higher arrival temperature implies a greater degree of thermal
energy is required to transition to phase I. As with the lower barrier height simulations,
the potential energy values of this simulation in phase II are almost identical to those of
the original simulation. Energetically then, phase II should be very similar for the three
simulation runs with 1, 1/10 and 10 times the barrier height value of 2.1669 kcal/mole.
Measuring the enthalpy change for the current simulation gives 1.8 kJ/mole (at 648 K), a
value that is 114 % of the original. Therefore, increasing the barrier by 10 times
increased the transition enthalpy by 0.2 kcal/mole while decreasing the barrier by 10
times decreased the transition enthalpy by 0.3 kcal/mole. Although the values of 1.8, 1.6,
and 1.3 certainly within the error of each other, the almost linear decrease in ∆H as the
torsional barrier height is lowered strongly points to the stabilizing effect of a highly
mobile hydrogen on phase I. The explanation for the lowered onset temperature of the
superprotonic transition in the simulations with the largest torsional barrier should then
lie in the effect of this barrier on phase II and not phase I.
The lower onset temperature is most simply explained by a destabilization of
phase II due to a conflict between increasing tetrahedral vibrations and the rigidity
imposed on the hydrogen by the high torsional barrier. An autocorrelation function for
these simulations (Figure 5.22) shows a small but significant amount of tetrahedral
reorientations in phase II. The hydrogen bonds in phase II will become more and more
distorted as these reorientations increase because of the hydrogen atoms’ reduced ability
to cross the torsional barrier. The lower onset temperature is then possibly the result of a

229
compromise between the energy required to rearrange the atoms and the hydrogen bond
energy regained in the process.

1 .0

Melt

Phase I

0 .6

0 .4

P h a s e II

Transition

Average Cosine Value

0 .8

0 .2

0 .0
300

400

500

600

700

1 .0

Phase I

b)
Average Cosine Value

0 .8

Melt

T e m p e ra tu re (K )

0 .6
0 .4

P h a s e II
T r a n s itio n

0 .2
0 .0

300

400

500

600

700

T e m p e r a tu r e ( K )

Figure 5.22 Autocorrelation functions for simulations with a) decreased and b) increased
torsional barrier heights. The values are lower in a) and higher in b) when compared to
the original simulations Figure 5.12 a).

230

A comparison of the two autocorrelation functions for the lower and higher
torsional barrier simulations reveals the average cosine values of the lower barrier runs to
be consistently smaller than those of the higher barrier runs at all temperatures. This
systematic difference equates to a higher degree of tetrahedral reorientations for the lower
versus higher barrier simulations, as expected. The change with temperature of the
average cosine values above the onset transition is also quite different for the two
simulations; with a rapid fall versus a gradual decrease for the low versus high barrier
runs, respectively. This result agrees with the previous arguments as the low and high
torsional barriers should increase/decrease the rotational ability of the sulfate groups
generated by the structural changes.

5.6 Summary and Conclusions
The superprotonic phase transition (phase II →phase I; 414 K) of CsHSO4 was
simulated by 300 ps molecular dynamics as temperature increased from 298 K to 723 K
in 25 K-step. A Dreiding based force field was used in the simulation. The initial force
field parameters of S, O, and H were set to Dreiding default values, which were then
adjusted to reproduce the quantum mechanically derived structure and frequencies of a
gas-phase CsHSO4 monomer. Cesium vdW parameters were modified to duplicate the
quantum mechanical bonding energy, average CsO distance and symmetric-stretch
frequency of the monomer. Hydrogen bond parameters were adjusted to reproduce the ab
initio OD-OA distance and binding energy of a gas-phase (H2SO4)2 dimer. The hydrogen

231
torsional barrier height of the HSO4- groups was fit to a series of ab initio calculations on
a HSO4- ion in a dielectric medium of relative dielectric constant 10, where each
calculation fixed the O(1)-S-OD-H torsional angle and let the remaining structure relax to
lowest energy.
Such a process for adjusting the Dreiding FF parameters was picked not only for
its simplicity, but also because it could potentially be used to derive the FF for any other
MHnXO4 compound (M = Cs, Rb, NH4, K, Na, Li; X = S, Se, P, As). By analysis of
simulations similar to those presented here, one might then be able to explain why only
CsHSO4, CsHSeO4, CsH2AsO4 and RbHSeO4 exhibit stable superprotonic phase
transitions under ambient pressures85. An FF that combined the parameters of the
different cations and anions might also predict which new mixed compounds will have
superprotonic phase transitions.
In this force field, the hydrogen was treated as bonded exclusively to a single
oxygen atom (proton donor), with hydrogen bonds extending to other nearby oxygen
atoms (proton acceptors). Proton diffusion (i.e., proton jumps) between oxygen atoms
cannot occur with this kind of force field. Nevertheless, this series of simulations showed
a clear phase transition during the 300 ps simulation at 598 K. Evidence of the phase
transition was present in the change of: lattice parameters, X-ray powder diffraction
patterns, enthalpy and volume of the cell as well as the direction and degree of
reorientation of the HSO4 groups. The orientations of HSO4 groups were dramatically
randomized and the hydrogen bonds re-distributed above the transition temperature, in
agreement with other experimental and theoretical results that attribute the dramatic
increase of the proton conductivity to the nearly free rotation of the tetrahedra173,192.

232
These results show that proton diffusion is not essential to the existence of a
superprotonic phase transition, and that rotational disorder of HSO4 groups is a sufficient
condition to predict the presence of such transitions.
The importance of the hydrogen torsional barrier height, hydrogen bond strength,
and oxygen charge distribution to the transition temperature was probed by changing one
of these parameters and re-running the series of simulations. The results of these
secondary simulations are in agreement with a priori arguments that any parameter
inhibiting the rotations of the HSO4 groups will increase the temperature of the transition
and vice versa. Of particular interest were the results of the simulations run with all
oxygen electrostatic charges equivalent, where the transition temperature dropped from
598 to 423 K, immediately above the experimental value of 4l4 K3. This was expected as
the even charge distribution should favor the more symmetric and highly dynamic phase I
compared to the relatively fixed, monoclinic structure of phase II. As the transition
temperature changed so much with this variable, it will be interesting to compare these
results to those of future simulations which will use a reactive force field that constantly
adjusts the oxygen charges. Such a force field will also allow for proton migration
between the tetrahedra, the results of which could be compared to these results to
evaluate the effect of fixing the protons to a particular oxygen atom (as was done in this
work) on the transition temperature.
In conclusion, these simulations have convincingly reproduced both the structural
and dynamic properties of CsHSO4’s superprotonic phase transition using a FF derived
from Dreiding default values. A sufficiently general approach was utilized to adjust the

233
FF parameters so as to be applicable to other systems, suggesting that similar force field
calculations can be used to “discover” new superprotonic conducting compounds.

234

Chapter 6.

Conclusions

The present work attempted to uncover the structural and chemical parameters
that favor superprotonic phase transitions over melting or decomposition in the MHXO4,
MH2ZO4, and mixed MHXO4-MH2ZO4 classes of compounds (X=S, Se; Z=P, As; M=Li,
Na, K, NH4, Rb, Cs) and to thereby gain some ability to “engineer” the properties of solid
acids for applications. Three separate investigations were carried out.
First, the cation size effect on superprotonic phase transitions similar to that of
CsHSO4 was studied. Preliminary investigations attempted to create new, mixed cation
solid acids from the Cs/K, Cs/Na, Cs/Li systems and thereby vary the average cation size.
This work resulted in two new compounds pertinent to the question of the cation size
effect: Cs2Na(HSO4)3 and CsNa2(HSO4)3. Comparing the defining distances of these two
compounds as well as the other known MHSO4 compounds, the distance
surfaced as the likely critical crystal-chemical measure of whether a compound has a
superprotonic phase transition or not. This was in contrast to the predominant theory that
the distance was the critical parameter 39. However, it could not be ruled out that
these results were due to structural differences between the compounds, in particular as
the structures of Cs2Na(HSO4)3 and CsNa2(HSO4)3 were quite unique.
Therefore an investigation into the M2(HSO4)(H2PO4) family of compounds was
undertaken as these compounds are isostructural for M = Cs, Rb, NH4, K. In this system
only the Cs compound was found to have a superprotonic phase transition, so that the
cation size effect was conclusively confirmed. The distance was once again the
most salient crystal-chemical measure in predicting the superprotonic phase transition.

235
The importance of this distance was explained in terms of bigger M-O and X-O distances
giving “floppier” MOx polyhedra and XO4 tetrahedra, respectively, thereby lowering the
barriers to tetrahedral reorientations, which are inherent to superprotonic phase
transitions. One then has an a priori measure of a (known or unknown) compound’s
likelihood for undergoing a superprotonic phase transition.
Second, the entropic driving force behind superprotonic phase transitions was
probed by investigations into the CsHSO4-CsH2PO4 family of compounds. Three new
compounds were synthesized for this study: Cs2(HSO4)(H2PO4),
Cs3(HSO4)2.25(H2PO4)0.75, and Cs6(H2SO4)3(H1.5PO4)2. All the known mixed cesium
sulfate-phosphate compounds were synthesized and their properties, particularly those
involving their superprotonic phase transitions, were carefully analyzed. Detailed
analysis of these properties revealed that the transition enthalpy and volume change were
closely related.
The (configurational) entropy of the transitions was then modeled using two sets
of rules: one for the room temperature structures and one for the superprotonic structures.
The low temperature rules used statistical mechanics, adjusted to account for the probable
local ordering of protons near mixed S/P sites, to evaluate the entropy of the disordered
hydrogen bonds (symmetric and partial occupancy disorder) and mixed sulfate/phosphate
tetrahedra found in the room temperature structures. The set of rules used to calculate the
entropy of the superprotonic structures was based on Pauling’s approach to the residual
entropy of ice 135. His rules were adjusted to describe the highly dynamic tetrahedra and
disordered hydrogen bonds of the superprotonic structures, which resulted in the
following equation for evaluating the configurational entropy

236

# of
  probabilit y 
Ω =  proton  *  a proton 
 configurat ions  
  site is open 

 # of
 protons 

# of
  # of 
 
*  tetrahdera l  *  oxygen 
 arrangemen ts   positions 
 

Applying this equation to the superprotonic structures of the mixed cesium sulfatephosphate compounds allowed for the evaluation of the configurational entropy change
across their transitions. These values were found to be in excellent agreement with the
experimental data. The above equation was then applied to other solid acids with known
superprotonic transitions but different room/high temperature structures and was found to
give results that matched well the published data. It would then seem possible to predict a
potential compound’s transition entropy from predicted room and high temperature
structures. Since the transition volume change and enthalpy were closely related, it
should also be possible to estimate a transition enthalpy from the same predicted
structures. There is then the possibility of calculating a transition temperature for a
desired, but as yet unsynthesized, compound and thereby deducing if it is likely to
transform before decomposition or melting.
Third, the superprotonic phase transition of CsHSO4 was simulated by molecular
dynamics. The phase transition was successfully simulated and analysis of the data
showed both the structural and dynamic behavior of the superprotonic phase of CsHSO4
to have been reproduced. The importance of oxygen charge distribution, hydrogen bond
energy and the torsional barrier height was investigated through a series of secondary
simulations. Analysis of these simulations confirmed the a priori assumption that
superprotonic phase transitions are facilitated by the easy reorientations of the tetrahedra

237
and vice versa. Also, since this force field did not allow proton migration, it can be said
definitively that proton hopping is not essential to superprotonic phase transitions.
The approach used to generate the FF of the simulations adjusted Dreiding default
values (where available) to reproduce three ab initio calculations. The S,O, H, and Cs
vdW force field parameters were adjusted to reproduce the quantum mechanically
derived structure, binding energy and frequencies of a gas-phase CsHSO4 monomer.
Hydrogen bond parameters were tuned to reproduce the ab initio OD-OA distance and
binding energy of a gas-phase (H2SO4)2 dimer. And finally, the hydrogen torsional barrier
height of the HSO4- groups was fit to a series of ab initio calculations on a HSO4- ion in a
dielectric medium of relative dielectric constant 10.
Such a process for adjusting the Dreiding FF parameters was picked not only for
its simplicity, but also because it could potentially be used to derive the FF for any other
MHnXO4 compound (M = Cs, Rb, NH4, K, Na, Li; X = S, Se, P, As). It should then be
possible to develop a general FF that could be used to “discover” novel superprotonic
solid acids.
All three approaches were therefore successful in furthering the knowledge of
which structural and chemical features favor superprotonic phase transitions, and as such,
should be useful to future research on these compounds.

238

Appendix: Compendium of Experimental Data
In the interest of time and space, many of the measurements performed on the
compounds found in this work were alluded to but not explicitly shown. Those results are
shown here, in the order that they were discussed in the text.

A.1 Chapter 3

A.1.1

β-CsHSO4-III
The structures for α, β, γ -CsHSO4-III are nearly identical, and as much more

attention was given to β-CsHSO4-III, only its structure will be shown in this appendix.
The SCXD data collection was taken on a crystal from a 80:20 CsHSO4:NaHSO4
aqueous solution. The crystal structure is shown in Figure A.1, the atomic coordinates in
Table A.1, anisotropic thermal parameters in Table A.2, and the data collection
parameters in Table A.3. The compound has a formula of (CsHSO4)3 and crystallizes in
space group P21/m. The final residuals, based on 4032 independent reflections, were
wR(F2) = 0.2314 and R(F) = 0.0566. The data were weighted as described in Table A.3
and refinements were preformed against F2 values. Note the disorder of the protons in the
hydrogen bonds, which motivated the DSC experiment looking for an ordering of the
protons at low temperatures.
Table A.1 Atomic coordinates and equivalent displacement parameters
(Å2) for β-CsHSO4-III. Ueq = (1/3)Tr(Uij).
Atom
Cs1
Cs2

x/a
0.2888(2)
0.28799(18)

y/b
0.25
-0.75

z/c
0.66095(10)
0.99514(9)

Ueq
0.0313(4)
0.0250(4)

239
Cs3
S4
S5
S6
O1
O2
O3
O4
O5
O6
O7
O8
O9
H2
H3

0.7127(2)
0.2466(7)
0.7552(9)
0.7534(6)
0.392(2)
0.606(2)
0.608(3)
0.702(3)
0.121(2)
0.702(3)
0.308(2)
0.8779(15)
0.869(3)
0.04(3)
0.90(8)

-0.25
-0.25
0.25
0.25
-0.25
0.25
0.25
0.25
-0.460(2)
0.25
-0.25
0.455(2)
0.454(3)
-0.45(4)
0.52(12)

0.67268(9)
0.8177(4)
0.5168(3)
0.8496(4)
0.7744(12)
0.5624(11)
0.8978(13)
0.7602(10)
0.7893(9)
0.4287(10)
0.9056(11)
0.8777(7)
0.5405(12)
0.850(12)
0.55(4)

0.0309(5)
0.0242(10)
0.0272(11)
0.0251(11)
0.040(4)
0.042(5)
0.044(5)
0.038(4)
0.047(3)
0.039(4)
0.040(4)
0.032(3)
0.052(4)
0.05
0.05

Figure A.1 Crystal structure of β-CsHSO4-III: a) down the [001] and b) down the [010]
directions. Note the disordered hydrogen bonds connecting the tetrahedra into zigzag
chains along the [010] direction. Parallelepipeds represent unit cells.

240

Table A.2 Anisotropic Thermal parameters (Å2) for β-CsHSO4-III.
Atom
Cs1
Cs2
Cs3
S4
S5
S6
O1
O2
O3
O4
O5
O6
O7
O8
O9

U11
0.0269(7)
0.0249(7)
0.0290(7)
0.017(2)
0.041(3)
0.012(2)
0.033(8)
0.024(7)
0.038(9)
0.064(12)
0.062(8)
0.050(10)
0.021(7)
0.032(5)
0.053(9)

U22
0.0343(8)
0.0227(6)
0.0330(8)
0.025(2)
0.033(3)
0.031(3)
0.043(10)
0.065(13)
0.034(9)
0.029(8)
0.028(6)
0.037(9)
0.065(13)
0.034(6)
0.058(11)

U33
0.0329(8)
0.0278(7)
0.0317(8)
0.032(3)
0.009(2)
0.033(3)
0.053(11)
0.043(10)
0.064(13)
0.020(7)
0.061(8)
0.023(8)
0.030(8)
0.031(5)
0.050(8)

U23
-0.013(6)
-0.017(5)
-0.023(8)

U13
0.0068(6)
0.0061(5)
0.0084(6)
0.0081(19)
0.0075(19)
0.0057(19)
0.029(8)
0.019(7)
0.021(9)
0.008(8)
0.033(7)
-0.008(7)
0.000(6)
0.007(4)
0.022(6)

Table A.3 Data collection specifics for β-CsHSO4-III, (CsHSO4)3
#-----------------_chemical_formula_
_chemical_formula_weight

CHEMICAL

(CsHSO4)3
689.88

#-----------------_cell_length_a (Å)
_cell_length_b (Å)
_cell_length_c (Å)
_cell_angle_alpha (°)
_cell_angle_beta (°)
_cell_angle_gamma (°)
_cell_volume (Å3)
_cell_formula_units_Z
_cell_measurement_temperature (K)

UNIT CELL
7.329(5)
5.829(4)
16.525(13)

#------------------

CRYSTAL

_exptl_crystal_density_diffrn (g/cm3)
_exptl_crystal_density_method
_exptl_crystal_F_000
_exptl_absorpt_coefficient_mu
_exptl_absorpt_correction_type
#-----------------_diffrn_radiation_wavelength (Å)

INFORMATION

90
101.55(3)
90
691.7(9)
293(2)
INFORMATION

2.785
'not

measured'
536
5.442

none
DATA

COLLECTION
0.7107

U12
-0.013(6)
-0.015(5)
-0.026(7)

241
_diffrn_radiation_type
_diffrn_reflns_number
_diffrn_reflns_limit_h_min
_diffrn_reflns_limit_h_max
_diffrn_reflns_limit_k_min
_diffrn_reflns_limit_k_max
_diffrn_reflns_limit_l_min
_diffrn_reflns_limit_l_max
_diffrn_reflns_theta_min
_diffrn_reflns_theta_max
_diffrn_reflns_theta_full
_diffrn_measured_fraction_theta_max

MoK\a
4032
10
-8
-23
22
1.26
29.99
29.99

_diffrn_measured_fraction_theta_full
_reflns_number_total
_reflns_number_gt
_reflns_threshold_expression

2204
906
>2sigma(I)

#------------------

COMPUTER

PROGRAMS

_computing_structure_refinement

'SHELXL-97

(Sheldrick,

#------------------

REFINEMENT

INFORMATION

_refine_ls_structure_factor_coef
_refine_ls_matrix_type
_refine_ls_weighting_scheme
_refine_ls_weighting_details

Fsqd
full
calc

_refine_ls_hydrogen_treatment
_refine_ls_extinction_method
_refine_ls_extinction_expression
_refine_ls_extinction_coef
_refine_ls_number_reflns
_refine_ls_number_parameters
_refine_ls_number_restraints
_refine_ls_R_factor_gt
_refine_ls_wR_factor_ref
_refine_ls_goodness_of_fit_ref
_refine_ls_restrained_S_all
_refine_ls_shift/su_max
_refine_diff_density_max
_refine_diff_density_min

'calc
mixed
SHELXL

w=1/[\s^2^(Fo^2^)+(0.0781P)^2^+15.6726P]

Fc^*^=kFc[1+0.001xFc^2^\l^3^/sin(2\q)]^-1/4^
0.0148(17)
2204
107
0.0566
0.2314
1.135
1.135
0.487
1.583
-1.181

242
The NMR experiments on this compound revealed the presence of Na cations best in the
H+ NMR measurements where the proton signal was split into two peaks compared to the
references one, Figure A.2. The main peak was also shift ~ 1 ppm between the two plots.
a)
15000

β -CsHSO4-III

Intensity

10000

5000

PPM

b) 50000

Intensity

40000

CsHSO4

30000

20000

10000

PPM

Figure A.2 H+ NMR measurements on a) β-CsHSO4-III and b) true CsHSO4-III. The
two peaks in a) versus one in b) as well as the ppm shift in the main peaks is attributed to
the trace incorporation of Na cations into the structure of CsHSO4-III. Both
measurements taken on a Bruker DSX 500 MHz NMR spectrometer using the MAS
technique with spin rates of 12 kHz. Each measurement is a combination of 8 scans with
a D1 of 1000s.

243

A.1.2 Cs2Li3H(SO4)3*H2O
The compound Cs2Li3H(SO4)3*H2O is orthorhombic, crystallizing in space group
Pbn21, with lattice parameters a = 12.945(3), b = 19.881(4), c = 5.111(1) Å, as
determined by SCXD. The unit cell has four formula units and a volume of 1315.41(30)
Å3. The crystal structure is shown in Figure A.3, the atomic coordinates in Table A.4,
anisotropic thermal parameters in Table A.5, and data collection parameters in A.6.

Figure A.3 Structure of Cs2Li3H(SO4)3*H2O projected down a) the c-axis and b) the aaxis. The hydrogen atoms were not found in this structure and therefore the water
molecules appear as isolated oxygens. Rectangles show the unit cell.

244

Table A.4 Atomic coordinates and equivalent displacement parameters
(Å2) for Cs2Li3H(SO4)3*H2O. Ueq = (1/3)Tr(Uij).
Atom
Cs1
Cs2
S3
S4
S5
O1
O2
O3
O4
O5
O6
O7
O8
O9
O10
O11
O12
OH2
Li1
Li2
Li3

x/a
0.60461(5)
0.77140(5)
0.59253(16)
0.42422(18)
0.76667(16)
0.4570(6)
0.7444(7)
0.6139(8)
0.8613(7)
0.7778(7)
0.5050(10)
0.6813(8)
0.4158(7)
0.5750(7)
0.3270(8)
0.4986(8)
0.6816(7)
0.4702(11)
0.4591(12)
0.8958(15)
0.7944(14)

y/b
0.26657(3)
0.49948(3)
0.41951(11)
0.17708(12)
0.16184(11)
0.2465(4)
0.2333(4)
0.4904(4)
0.1407(4)
0.1488(5)
0.1298(7)
0.3781(5)
0.1608(6)
0.4126(5)
0.1617(6)
0.3961(5)
0.1187(5)
0.0273(7)
0.4267(8)
0.1640(9)
0.3232(9)

z/c
0.2722(3)
0.2711(5)
0.7661(13)
-0.2276(15)
0.7684(13)
-0.228(5)
0.779(5)
0.797(7)
0.8853(16)
0.4684(18)
-0.129(7)
0.8306(18)
-0.5363(16)
0.4742(17)
-0.1201(18)
0.8925(17)
0.8586(17)
-0.709(8)
0.239(4)
0.302(5)
0.778(11)

Ueq
0.02334(18)
0.03029(19)
0.0156(4)
0.0219(5)
0.0182(5)
0.0290(18)
0.040(3)
0.057(5)
0.0220(17)
0.029(2)
0.02(2)
0.036(3)
0.033(2)
0.0263(19)
0.031(2)
0.027(2)
0.030(2)
0.095(6)
0.014(4)
0.020(4)
0.029(4)

Table A.5 Anisotropic Thermal parameters (Å2) for Cs2Li3H(SO4)3*H2O.
Atom

U11

U22

U33

U23

U13

Cs1

0.0221(3)

0.0271(3)

0.0208(3)

0.0001(7)

Cs2
S3
S4
S5
O1
O2
O3
O4
O5
O6
O7
O8
O9

0.0269(3)
0.0105(9)
0.0135(9)
0.0120(9)
0.029(4)
0.032(4)
0.022(4)
0.019(4)
0.024(5)
0.019(5)
0.030(5)
0.024(5)
0.017(4)

0.0304(3)
0.0145(9)
0.0176(10)
0.0167(10)
0.017(3)
0.022(4)
0.015(4)
0.024(4)
0.041(6)
0.035(7)
0.042(5)
0.063(7)
0.040(5)

0.0336(3)
0.0218(10)
0.0345(12)
0.0259(11)
0.041(4)
0.065(7)
0.134(14)
0.023(4)
0.022(4)
0.00(7)
0.034(7)
0.011(4)
0.022(4)

0.0001(8)
0.0012(10)
0.006(3)
0.006(3)
-0.014(3)
0.011(9)
0.026(8)
-0.010(13)
-0.006(3)
-0.005(4)
0.043(18)
-0.007(4)
0.004(4)
-0.005(4)

0.0000(10)
0.008(2)
0.009(3)
-0.003(2)
0.002(9)
0.007(10)
-0.003(13)
-0.011(3)
-0.002(4)
0.023(16)
-0.004(4)
0.001(3)
-0.002(3)

U12
0.0002(2)
0.0044(2)
0.0009(7)
0.0005(8)
0.0028(8)
-0.001(3)
0.002(4)
-0.002(3)
0.011(4)
0.006(4)
0.005(5)
0.022(5)
0.000(5)
-0.002(4)

245
O10
O11
O12
OH2

0.021(4)
0.026(5)
0.018(4)
0.055(8)

0.043(6)
0.034(5)
0.035(5)
0.040(6)

0.029(4)
0.023(4)
0.036(5)
0.189(18)

-0.006(4)
-0.013(4)
0.013(4)
0.023(17)

0.002(4)
0.006(3)
0.005(3)
0.026(18)

-0.003(4)
-0.015(4)
0.004(4)
0.001(6)

Table A.6 Data collection specifics for Cs2Li3H(SO4)3*H2O
#------------------

CHEMICAL INFORMATION

_chemical_formula_
_chemical_formula_weight

Cs2Li2H(SO4)*H2O

#------------------

UNIT CELL INFORMATION

_cell_length_a
_cell_length_b
_cell_length_c
_cell_angle_alpha
_cell_angle_beta
_cell_angle_gamma
_cell_volume
_cell_formula_units_Z
_cell_measurement_temperature

12.95(3)
19.881(4)
5.1110(10)

#------------------

CRYSTAL INFORMATION

_exptl_crystal_density_diffrn
_exptl_crystal_density_method
_exptl_crystal_F_000
_exptl_absorpt_coefficient_mu
_exptl_absorpt_correction_type
#-----------------_diffrn_radiation_wavelength
_diffrn_radiation_type
_diffrn_reflns_number
_diffrn_reflns_limit_h_min
_diffrn_reflns_limit_h_max
_diffrn_reflns_limit_k_min
_diffrn_reflns_limit_k_max
_diffrn_reflns_limit_l_min
_diffrn_reflns_limit_l_max
_diffrn_reflns_theta_min
_diffrn_reflns_theta_max
_diffrn_reflns_theta_full

593.84

90
90
90
1315(3)
293(2)

2.999
'not measured'
1096
6.078
none
DATA COLLECTION
0.71073
MoK\a
5717
18
-10
27
-7
1.88
30
30

246
3833
2858

_reflns_number_total
_reflns_number_gt
_reflns_threshold_expression

>2sigma(I)

#------------------

COMPUTER PROGRAMS USED

_computing_structure_refinement

'SHELXL-97

#------------------

REFINEMENT INFORMATION

_refine_ls_structure_factor_coef
_refine_ls_matrix_type
_refine_ls_weighting_scheme

Fsqd
full
calc
w=1/[\s^2^(Fo^2^)+(0.0787P)^2^+11.8743P]
where P=(Fo^2^+2Fc^2^)/3'
mixed
SHELXL
0.0026(4)
3833
176
0.0628
0.1737
1.166
1.166
9.983
4.104
-2.833

_refine_ls_weighting_details
_refine_ls_hydrogen_treatment
_refine_ls_extinction_method
_refine_ls_extinction_coef
_refine_ls_number_reflns
_refine_ls_number_parameters
_refine_ls_number_restraints
_refine_ls_R_factor_gt
_refine_ls_wR_factor_ref
_refine_ls_goodness_of_fit_ref
_refine_ls_restrained_S_all
_refine_ls_shift/su_max
_refine_ls_abs_structure_Flack
_refine_diff_density_max
_refine_diff_density_min

This compound did not exhibit a superprotonic phase transition before
decomposition/dehydration at ~ 105°C, which was deduced from TGA, DSC, and
conductivity data, Figures A.4 and A.5.

247

101
0.8

TGA data

0.7
0.6

99
0.5
98

0.4
0.3

0.2
96

Heat Flow (W/g)

% Weight Loss

100

endo

0.1

DSC data

0.0

95
100

200

300

exo

-0.1
500

400

Temperature ( C)

Figure A.4 TGA and DSC data for Cs2Li3H(SO4)3*H2O. No superprotonic phase
transition is evident before the onset of decomposition/dehydration at ~ 105°C. Both
measurements taken under flowing N2 with heating rates of 5°C/min.

300

250

200

150

Cs2Li3H(SO4)3*H2O
along c-axis

-1

Log[σ T] Ω cm K

100

-2
-3

Decompisition
or Dehydration
begins

-4
-5

Onset of Melting

1.8

2.0

2.2

2.4

2.6

2.8

3.0

1000/T
Figure A.5 Conductivity of Cs2Li3H(SO4)3*H2O. Measurement taken on a single crystal
sample parallel to the c-axis under ambient air atmosphere at a heating rate of 0.5°C/min.

248

A.2

Chapter 4

A.2.1 Causes for discrepancies in experimental data between
published values and those reported in Chapter 4
It was mentioned in the text that there are some discrepancies between the
published values and those presented in Chapter 4, with particular emphasis on the
transition enthalpies. Some probable reasons for this were already expressed, but here we
will go into more detail. First, these differences are probably due mostly to the quality
and quantity of the samples as well as the measurement techniques used in those
measurements. For many of the mixed compounds, the samples measured were reported
to have liquid filled voids or be part of a powder mixture of different phases30,32,133.
Moreover, many of these compound are extremely hard to grow as large single crystals,
and so very small crystals would have to be identified from a multiple of phases, limiting
the number and type of measurements possible27,31. These limits to the quality and
quantity of the desired compound can only have had an adverse effect on the
measurement of the transition enthalpy.
By far the most time-consuming and laborious part of this work was preparing
adequate amounts of the mixed compounds with a high level of phase purity.
For the end members, CsHSO4 and CsH2PO4, high-quality large single crystals are not
difficult to grow. The minor discrepancy between the published and this work’s ∆H for
CsHSO4 (5.5 vs. 6.2(2) kJ/mol CsHXO4, respectively) may simply be to statistical error
or differences in measurement techniques. However, during this work it was found that
CsHSO4 samples from mixed cation solutions had reproducibly lower transition

249
enthalpies even though their lattice constants (measured by SCXD techniques) were
nearly identical to published values. For example, from mixed Cs/K or Cs/Na solutions
the average ∆H for seven different measurements was 5.2(5) kJ/mol, with a low of 4.5
and high of 5.5 kJ/mol, Table A.7. These values are to be compared to the average of
6.2(2) kJ/mol measured on samples grown from solutions containing only Cs cations.
This suggests that the purity of the initial reagents has a significant effect on enthalpy of
the transition. The effect of trace impurities also showed up in other properties of these
samples. To avoid any such obfuscating effects, only ultrahigh purity reagents were used
in making the compounds of this study.

Table A.7 Variation of the transition enthalpy for CsHSO4 from pure and
mixed cation solutions.
Tonset(Na/K)
∆H(Na/K)
Tonset(Cs)
∆H (kJ/mol)
(°C)
(kJ/mol)
(°C)
(kJ/mol)
Exp 1
142.8
4.5
6.0007
140.9
Exp 2
142.5
5.5
6.2491
141.2
Exp 3
143.5
5.1
6.2376
140.5
Exp 4
143.5
5.2
6.8241
140.5
Exp 5
139
5.3
6.0881
143.5
Exp 6
140.2
5.5
6.0858
143.8
Exp 7
138.8
5.0
37.4854
141.6
Average
141.5(19)
5.2(5)
141.9(16)
6.2(2)

A.2.2

CsHSO4
Powder X-ray diffraction patterns from which the thermal expansions of the room

temperature and high temperature phase of CsHSO4 were calculated are shown in Figure
A.6 and the results of the Rietveld analysis on these patterns in Table A.8

250

26000
24000

22000

h) cooled to 25 C

20000

g) 180 C

Intensity

18000

f) 165 C

16000
14000

e) 150 C

12000

d) 140 C

10000
8000

c) 80 C

6000

b) 25 C

4000
2000

a) Calc.

2 Theta (Degrees)
Figure A.6 PXD patterns of CsHSO4 taken at various temperatures (as shown). Data
taken in ambient atmosphere with a 3 second scan rate and 0.02° 2θ scan step.

Table A.8 Results of Rietveld Analysis on CsHSO4 PXD patterns taken at various
temperatures.
Temp

Volume

phase

(°C)

phase

Rwp

a (Å)

b (Å)

c (Å)

Beta (°)

Per CsHXO4

mono

25(1)

100

11.1

15.46

7.772

8.133

7.715

110.84

113.93(5)

mono

80(2)

100

9.44

13.77

7.811

8.156

7.729

110.972

114.94(5)

mono

140(2)

35.5(2)

14.03

22.29

7.881

8.16

7.743

111.32

116.0(2)

tetra

140(2)

64.5(3)

5.712

14.199

115.8(1)

tetra

150(3)

100

5.66

8.26

5.725

14.225

116.56(4)

tetra

165(3)

100

6.43

8.44

5.7277

14.233

116.73(4)

tetra

180(3)

100

8.33

12.74

5.73

14.262

117.06(6)

251

A.2.3 Cs3(HSO4)2.50(H2PO4)0.50

Powder X-ray diffraction patterns from which the thermal expansions of the room
temperature and high temperature forms of Cs3(HSO4)2.50(H2PO4)0.50 were calculated are
shown in Figure A.7 and the results of the Rietveld analysis on these patterns in Table
A.9

20000
17500

g) cooled to 25 C

15000

Intensity

f) 160 C

12500

e) 140 C

10000

d) 130 C

7500

c) 90 C

5000

b) 25 C

2500

a) Calc.

2 Theta (Degrees)
Figure A.7 PXD patterns of Cs3(HSO4)2.50(H2PO4)0.50 taken at various temperatures (as
shown). Data taken in ambient atmosphere with a 3 second scan rate and 0.02° 2θ scan
step.

252
Table A.9 Results of Rietveld Analysis on Cs3(HSO4)2.50(H2PO4)0.50 PXD patterns taken
at various temperatures.
phase
mono

phase
100

Temp
(°C)
25(1)

Rp
7.66

Rwp
12.13

a (Å)
19.930(3)

b (Å)
7.862

c (Å)
8.996

Beta (°)
100.16

Volume Per
CsHXO4
115.62(6)

mono

100

90(2)

4.41

5.72

19.991(1)

7.902

9.011

100.038

116.81(3)

mono

49(1)

130(3)

5.27

6.63

20.046(3)

7.933

9.01

100.028

117.58(5)

cubic

12(1)

130(3)

4.932(1)

4.932

119.97(7)

tetra

39(1)

130(3)

5.732(1)

5.732

14.167

116.37(9)

Cubic

44(1)

140(3)

4.9345(5)

4.9345

120.15(4)

tetra

56(1)

140(3)

5.7359(5)

5.7359

14.178

116.62(3)

Cubic

58(1)

160(3)

4.9423(5)

4.9423

120.72(4)

tetra

42(0.5)

160(3)

5.7411(6)

5.7411

14.211

117.10(4)

3.89
5.29

4.88
6.63

A.2.4 Cs3(HSO4)2.25(H2PO4)0.75

Microprobe data supplying the stoichiometry of the compound are given Table
A.10. Data taken at a beam voltage and current of 15 kV and 25 mA, respectively, on a
pressed powder sample. Standards were pressed powder pellets of CsHSO4 and CsH2PO4.
Visible beam damage occurred while taking data, which is probably responsible for th
low totals. Powder X-ray diffraction patterns from which the thermal expansions of the
room temperature and high temperature forms of Cs3(HSO4)2.50(H2PO4)0.50 were
calculated are shown in Figure A.8 and the results of the Rietveld analysis on these
patterns in Table A.11

253

i) after 7 hrs at 25 C
20000

h) 140 C

Y Axis Title

g) 130 C

15000

f) 125 C

e) 120 C

10000

d) 115 C

c) 110 C

5000

b) 25 C
a) Calc.

X Axis Title
Figure A.8 PXD patterns of Cs3(HSO4)2.25(H2PO4)0.75 taken at various temperatures (as
shown). Data taken in ambient atmosphere with a 3 second scan rate and 0.02° 2θ scan
step.
Table A.10 Microprobe data on Cs3(HSO4)2.25(H2PO4)0.75.
Cs
Exp 1
Exp 1
Exp 1
Exp 1
Exp 1
Exp 1
Exp 1
Exp 1
Exp 1
Exp 1
Exp 1
Average
SDeV

56.311
57.679
57.083
57.051
55.809
57.078
55.283
57.025
57.485
56.896
59.042
56.977
0.986

10.615
10.584
10.913
10.617
10.195
10.798
10.475
10.888
10.686
10.55
10.805
10.648
0.207

Sum
3.496
3.813
3.589
3.848
3.515
3.662
3.594
3.498
3.397
3.203
3.453
3.552
0.183

70.422
72.075
71.584
71.517
69.519
71.538
69.353
71.41
71.568
70.649
73.301
71.176
1.169205

254

Table A.11 Results of Rietveld Analysis on Cs3(HSO4)2.50(H2PO4)0.50 PXD patterns
taken at various temperatures.
phase
mono

phase
100

Temp
(°C)
25(1)

Rp
7.87

Rwp
11.17

a (Å)
19.913

b (Å)
7.853

c (Å)
8.999

Beta (°)
100.132

Volume Per
CsHXO4
115.44(5)

mono

100

110(2)

6.46

8.92

19.975

7.912

9.021

100.031

116.99(4)

mono

90(1)

115(2)

6.12

8.35

19.98

7.9175

9.02

100.063

117.08(4)

cubic

8(1)

115(2)

4.93

119.82(1)

tetra

2(1)

115(2)

5.424

17.535

129.0(8)

mono

93(1)

120(2)

19.978

7.9186

9.025

100.04

117.16(5)

cubic

6(1)

120(2)

4.91

118.4(4)

tetra

1(1)

120(2)

5.415

17.733

120.0(9)

mono

77(1)

125(2)

19.982

7.926

9.025

100.038

117.29(7)

cubic

5(1)

125(2)

4.907

118.2(1)

tetra

18(1)

125(2)

5.715

14.224

116.14(9)

mono

7(1)

130(2)

19.9552

8.002

9.03

99.65

118.5(1)

cubic

46(1)

130(2)

4.9519

121.43(4)

tetra

47(1)

130(2)

5.7359

14.173

116.57(4)

cubic

140(3)

4.9522

121.4492

tetra

140(3)

5.7362

14.183

116.6693

7.15

7.58

5.36

4.38

11.6

11.52

8.65

5.59

A.2.5 Cs3(HSO4)2(H2PO4)

Powder X-ray diffraction patterns from which the thermal expansions of the room
temperature and high temperature forms of Cs3(HSO4)2(H2PO4) were calculated are
shown in Figure A.9 and the results of the Rietveld analysis on these patterns in Table
A.12.

255

42000

g) after 3 hrs at 25 C
36000

f) 140 C

Intensity

30000

e) 130 C

24000

d) 110 C

18000

c) 70 C

12000

b) 25 C

6000

a) Calc.

2 Theta (Degrees)

Figure A.9 PXD patterns of Cs3(HSO4)2(H2PO4) taken at various temperatures (as
shown). Data taken in ambient atmosphere with a 3 second scan rate and 0.02° 2θ scan
step.
Table A.12 Results of Rietveld Analysis on Cs3(HSO4)(H2PO4) PXD patterns taken at
various temperatures.
phase
mono
mono
mono
mono
cubic
cubic

Temp
(°C)
25(1)
70(1)
110(2)
130(2)
130(2)
140(2)

phase
100
100
100
71(2)
29(1)
100

Rp
11.45
5.78
9.84
7.45

Rwp
19.47
7.42
16.41
10.3

4.93

6.43

a (Å)
19.527
19.603
19.627
19.689
4.9304
4.9336

b (Å)
7.871
7.9
7.93
7.93
4.9304
4.9336

c (Å)
9.162
9.167
9.172
9.159
4.9304
4.9336

Beta (°)
100.51
100.309
100.165
99.94
90
90

Volume
Per
CsHXO4
115.38(9)
116.39(3)
117.10(8)
117.38(6)
119.85(3)
120.09(3)

256

A.2.6 Cs5(HSO4)3(H2PO4)2

Powder X-ray diffraction patterns from which the hysteresis of the reverse
transition and lattice parameter for the high temperature phase of Cs5(HSO4)3(H2PO4)2
were calculated are shown in Figure A.10 and the results of the Rietveld analysis on these
patterns in Table A.13.

24000
22000
20000

18000

e) after 38 days at 25 C

Intensity

16000
14000

d) after 4 hrs at 25 C

12000
10000

c) 140 C

8000
6000

b) 25 C

4000
2000

a) Calc.

2 Theta (Degrees)

Figure A.10 PXD patterns of Cs5(HSO4)3(H2PO4)2 taken at various temperatures (as
shown). Data taken in ambient atmosphere with a 3 second scan rate and 0.02° 2θ scan
step.

257
Table A.13 Results of Rietveld Analysis on Cs5(HSO4)3(H2PO4)2 PXD pattern in cubic
high temperature phase.
Temp
(°C)
140(2)

phase
cubic

phase
100

Rp
5.89

Rwp
7.46

4.9378

Beta
90

Volume/CsHXO4
120.39(3)

A.2.7 Cs2(HSO4)(H2PO4)

Powder X-ray diffraction patterns from which the thermal expansions of the room
temperature and high temperature forms of Cs2(HSO4)(H2PO4) were calculated are
shown in Figure A.11 and the results of the Rietveld analysis on these patterns in Table
A.14. The hysteresis of the reverse transitions was also estimated from the after heating
patterns.

258

260000

240000

j) after 4 months at 25 C

220000

i) after 16 days at 25 C

200000

h) after 2 hrs at 28 C

Intensity

180000

g) cooled to 50 C

160000

140000

f) 133 C

120000

e) 119 C

100000

80000

d) 85 C

60000

c) 50 C

40000

b) 25 C

20000

a) Calc.

2 Theta (Degrees)
Figure A.11 PXD patterns of Cs2(HSO4)(H2PO4) taken at various temperatures (as
shown). Data taken in ambient atmosphere with a 3 second scan rate and 0.02° 2θ scan
step.
Table A.14 Results of Rietveld Analysis on Cs2(HSO4)(H2PO4) PXD patterns taken at
various temperatures.
c (Å)
7.827

Beta
(°)
99.92

Volume
Per
CsHXO4
117.1(2)

7.704

7.796

99.892

115.67(2)

7.86

7.743

7.838

99.907

117.47(5)

24.18

7.864

7.745

7.845

99.91

117.67(5)

45.13

54.79

7.884

7.774

7.845

99.8

118.4(6)

100

25.75

34.24

4.9291

119.76(3)

133(5)

100

25.38

34.21

4.9349

120.18(3)

50(1)

100

30.59

64.87

4.9186

118.99(3)

phase
mono

Temp
(°C)
25(1)

phase
100

Rp
3.48

Rwp
8.55

a (Å)
7.856

b (Å)
7.732

mono

25(1)

100

6.61

9.36

7.8196

mono

40(3)

100

18.58

23.82

mono

50(3)

100

mono

85(4)

100

cubic

119(5)

cubic
cubic

259

A.2.8 Cs6(H2SO4)3(H1.5PO4)4

Powder X-ray diffraction patterns from which the thermal expansions of the room
temperature and high temperature forms of Cs6(H2SO4)3(H1.5PO4)4 were calculated are
shown in Figure A.12 and the results of the Rietveld analysis on these patterns in Table
A.15. The hysteresis of the reverse transitions was also estimated from the after heating
patterns.

45000

h) after 11 days at 25 C
40000

g) after 3 days at 25 C

Intensity

35000

f) after 10 hrs at 25 C

30000

e) 180 C

25000

g) 140 C

f) 140 C

20000

e) 120 C
15000

d) 95 C

10000

c) 70 C

5000

b) 25 C

a) Calc.

2 Theta (degrees)
Figure A.12 PXD patterns of Cs6(H2SO4)3(H1.5PO4)4 taken at various temperatures (as
shown). Data taken in ambient atmosphere with a 3 second scan rate and 0.02° 2θ scan
step.

260

Table A.15 Results of Rietveld Analysis on Cs6(H2SO4)3(H1.5PO4)4 PXD patterns taken
at various temperatures.
c (Å)
14.5388

Beta
(°)
90

Volume
Per
CsHXO4
116.4(1)

14.541

116.46(2)

14.573

117.23(2)

11.62

14.582

117.45(5)

7.12

4.9908

124.31(3)

phase
cubic I

Temp
(°C)
25(1)

phase
100

Rp
1.64

Rwp

a (Å)
14.5388

b (Å)
14.5388

cubic I

25(1)

100

8.49

11.36

14.541

cubic I

70(2)

100

8.46

11.53

cubic I

95(2)

100

8.58

cubic P

120(3)

100

5.88

123.83(3
cubic P

140(3)

100

5.05

6.56

4.9844

cubic P

160(3)

100

5.98

8.08

4.9631

122.25(7)

cubic P

180(3)

100

5.98

7.83

4.952

121.4(1)

The structure of this compound was described in the text. Its atomic coordinates
are given in Table A.16, anisotropic thermal parameters in Table A.17, and data
collection parameters in A.18. The final residuals, based on 2719 independent reflections,
were wR(F2) = 0.0339 and R(F) = 0.0164. The data were weighted as described Table
A.3 and refinements were preformed against F2 values.
Table A.16 Atomic coordinates and equivalent displacement
parameters (Å2) for Cs6(H2SO4)3(H1.5PO4)4. Ueq = (1/3)Tr(Uij).
Atom

x/a

y/b

z/c

Ueq

Cs1

0.75

0.84924(2)

0.02458(12)

0.74779(6)

0.0193(3)

0.75

1.125

0.0201(3)

0.7014(2)

0.8384(2)

0.7815(2)

0.0245(7)

0.6722(2)

1.1833(2)

1.02869(19)

0.0275(7)

0.6891(2)

0.0291(13)

0.645(4)

0.840(4)

0.794(3)

0.04

261
Table A.17 Anisotropic Thermal parameters (Å2) for Cs6(H2SO4)3(H1.5PO4)4.
Atom

U11

U22

U33

U23

U13

Cs1

0.0259(2)

0.0273(2)

0.0205(2)

0.0193(3)

0.0168(5)

0.0268(9)

0.0168(5)

0.0218(14)

0.0196(14)

0.0322(17)

-0.0010(12)

0.0009(13)

0.0224(16)

0.0367(17)

0.0233(17)

-0.0040(13)

-0.0037(11)

0.0291(13)

-0.0035(13)

-0.0009(5)

0.00116(19)
-0.0009(5)

Table A.18 Data collection specifics for Cs6(H2SO4)3(H1.5PO4)4
#------------------

CHEMICAL INFORMATION

_chemical_formula_
_chemical_formula_weight

Cs6(H2SO4)3(H1.5PO4)4
459.87

#------------------

UNIT CELL INFORMATION

_cell_length_a
_cell_length_b
_cell_length_c
_cell_angle_alpha
_cell_angle_beta
_cell_angle_gamma
_cell_volume
_cell_formula_units_Z
_cell_measurement_temperature

14.539(6)
14.539(6)
14.539(6)
90
90
90
3073(2)
16
293(2)

#------------------

CRYSTAL INFORMATION

_exptl_crystal_size_max
_exptl_crystal_size_mid
_exptl_crystal_size_min
_exptl_crystal_density_diffrn
_exptl_crystal_density_method
_exptl_crystal_F_000
_exptl_absorpt_coefficient_mu
_exptl_absorpt_correction_type

0.4
0.3
0.3
3.976
'not
3328
9.977
none

#------------------

DATA COLLECTION INFORMATION

_diffrn_radiation_wavelength
_diffrn_radiation_type

0.71073
MoK\a

262
_diffrn_reflns_number
_diffrn_reflns_limit_h_min
_diffrn_reflns_limit_h_max
_diffrn_reflns_limit_k_min
_diffrn_reflns_limit_k_max
_diffrn_reflns_limit_l_min
_diffrn_reflns_limit_l_max
_diffrn_reflns_theta_min
_diffrn_reflns_theta_max
_diffrn_reflns_theta_full
_diffrn_measured_fraction_theta_max
_diffrn_measured_fraction_theta_full
_reflns_number_total
_reflns_number_gt
_reflns_threshold_expression

2719
15
-4
15
-4
15
3.43
29.98
29.98
0.755
0.755
483
466
>2sigma(I)

#------------------

COMPUTER PROGRAMS USED

_computing_structure_refinement

'SHELXL-97

#------------------

REFINEMENT INFORMATION

_refine_ls_structure_factor_coef
_refine_ls_matrix_type
_refine_ls_weighting_scheme
_refine_ls_weighting_details

Fsqd
full
calc
'calc
w=1/[\s^2^(Fo^2^)+(0.0138P)^2^+1.8917P] where
P=(Fo^2^+2Fc^2^)/3'
mixed
SHELXL

Fc^*^=kFc[1+0.001xFc^2^\l^3^/sin(2\q)]^-1/4^
0.00160(8)
483
36
0.0164
0.0339
1.167
1.167
0.001
-0.03(4)
0.415
-0.41

263

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