Freeze Casting of Ceramics: Pore Design

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Freeze Casting of Ceramics: Pore Design from Solidification Principles
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Arai, Noriaki
(2021)
Freeze Casting of Ceramics: Pore Design from Solidification Principles.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/3rmr-cz93.
Abstract
Freeze casting is a porous material processing method which allows the creation of directionally aligned pores by the solidification process. Pores are generated by sublimation of solidified crystals which reject suspending particles or dissolved solutes during freezing. Although freeze-cast ceramics have been identified for applications such as filtration and bioceramics, the lack of understanding of the process often results in a discrepancy between the desired pore structure and the fabricated structures.
Since solidification is the foundation upon which freeze casting is built, this work seeks to understand the solidification process, especially the growth and time evolution of dendrites. To understand the dendritic growth process, two solidification parameters, freezing front velocity and temperature gradient, are independently controlled to investigate the effects of each parameter. Dendritic pore size changes with solidification parameters and shows good agreement with dendrite growth theory. The theory of constitutional supercooling serves as a guide to control pore morphology between dendritic pores and cellular pores. Furthermore, dendrite growth under the effects of the gravitational force is investigated by changing the solidification direction with respect to the gravity direction. Convection changes the degree of constitutional supercooling, and results in different pore sizes as well as pore morphology.
Time evolution of dendrites through isothermal coarsening is investigated. During the coarsening of dendrites, they are transformed to cylinder-like crystals, which yield honeycomb-like structures. Moreover, dendrite size changes linearly with the cube root of coarsening time. Both findings are well-established phenomena in alloy solidification. Further comparison with alloy systems are achieved with tomography-based analysis where similar microstructural evolution with alloy system is demonstrated.
Based upon the understanding of underlying solidification principles in freeze casting, three applications are explored. First, the freeze-cast structure is designed to improve shape-memory properties. Processing variables are controlled such that shape-memory porous zirconia can enable martensitic phase transformations and shape deformation without fracture. Other applications utilize unique pore space. Dendritic pores are investigated for size-based filtration to preferentially capture small particles. Flow-through experiments and in-situ observation by confocal microscopy confirm that pores created by secondary dendrites capture small particles. Finally, honeycomb-like structures are filled with functional microgels to create a ceramic/polymer composite as an application for membrane chromatography. The fabricated composite demonstrates advantages such as mechanical stability during the fluid flow.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Porous ceramics, solidification, freeze casting
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Faber, Katherine T.
Thesis Committee:
Fultz, Brent T. (chair)
Kornfield, Julia A.
Johnson, William L.
Faber, Katherine T.
Defense Date:
26 October 2020
Funders:
Funding Agency
Grant Number
NSF
DMR-1411218
NSF
CBET-1911972
Rothenberg Innovation Initiative
UNSPECIFIED
Jacobs Institute for Molecular Engineering for Medicine
UNSPECIFIED
Record Number:
CaltechTHESIS:11062020-163041829
Persistent URL:
DOI:
10.7907/3rmr-cz93
Related URLs:
URL
URL Type
Description
DOI
Article adapted for Appendix A.
DOI
Article adapted for portion of Chapter 6.
ORCID:
Author
ORCID
Arai, Noriaki
0000-0002-3040-2997
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
13994
Collection:
CaltechTHESIS
Deposited By:
Noriaki Arai
Deposited On:
11 Nov 2020 22:44
Last Modified:
02 Nov 2021 00:03
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Freeze Casting of Ceramics:
Pore Design from Solidification Principles

Thesis by

Noriaki Arai

In Partial Fulfillment of the Requirements for the
Degree of
Doctor of Philosophy

CALIFORNIA INSTITUTE OF TECHNOLOGY
Pasadena, California

2021
Defended October 26, 2020

Noriaki Arai
ORCID: 0000-0002-3040-2997

iii

To Shiori and Luna.

ACKNOWLEDGEMENTS
First and foremost, I am sincerely grateful to my advisor, Professor Katherine
Faber, for your patience and support during my graduate study. For five years,
your continuous guidance have shaped me as a researcher, and your encouragement
cultivated my entrepreneurial mindset to ask important research questions and to
pursue originality. Joining the Faber group was one of the best decisions I have
made at Caltech.
To Professor Brent Fultz, Professor Julia Kornfield, and Professor William Johnson:
thank you for serving as committee members in both my candidacy examination and
defense, and providing invaluable support and advice on my research. Professor
Kornfield, it was exciting to work with you on different projects. Your advice always
let me look at researches from different perspectives and drove me to explore new
ideas.
I would like to thank all the people I have worked with. Professor Peter Voorhees,
thank you for the insightful discussions. Your broad and deep knowledge on solidification led us to a new research and a collaboration with Dr. Tiberiu Stan, Sophie
Macfarland, Dr. Nancy Senabulya, and Professor Ashwin Shahani. Thank you all
for your guidance, patience and all the fruitful discussions. I also would like to
thank Professor Paolo Colombo for your support and advice regarding preceramic
polymers. I also enjoyed the visit to the University of Padova.
At Caltech, I had the privilege to work with Orland Bateman, on a multiple projects
for almost three years. I enjoyed and learned from you a lot. I also enjoyed the time
I spent together with your families outside researches. I would like to thank Dr.
Mamadou Diallo for your guidance and weekly tutorial sessions.
To the Faber group, Matthew Johnson, Maninpat Naviroj, Neal Brodnik, Claire Kuo,
Xiaomei Zeng, Benjamin Herren, Rafa Cabezas Rodríguez, Celia Chari, Vince Wu,
Laura Quinn, Natalie Nicolas, Carl Keck, Christopher Long, thank you for all your
comments and feedback during the group meeting and for reviewing my papers.
I also would like to thank the administrative staff, Angie Riley, Christy Jenstad,
Celene Gates and Jennifer Blankenship for making sure that I can focus on my
research, travel for the conferences safely, and stay safe during this pandemic.
I would like to acknowledge Kenji Higashi, Dr. Hitoshi Ohmori, and Dr. Koji

Ishibashi, who encouraged me to pursue this career. I would not be able to come
this far without your understanding and support.
This five-year journey would not have been possible without unconditional support
from my entire family in Japan: my father, my mother, Michiko, Hatsuko, Tomoe,
Tsutomu, Yuko, Soma, and Masaki. My studies abroad have not been possible
without your support.
Lastly, but most importantly, thank you so much, Shiori and Luna, for your love,
support and dedication. The achievements at Caltech belong to all of us. I am
looking forward to what awaits us in the future.
- Nori

ABSTRACT
Freeze casting is a porous material processing method which allows the creation
of directionally aligned pores by the solidification process. Pores are generated
by sublimation of solidified crystals which reject suspending particles or dissolved
solutes during freezing. Although freeze-cast ceramics have been identified for
applications such as filtration and bioceramics, the lack of understanding of the
process often results in a discrepancy between the desired pore structure and the
fabricated structures.
Since solidification is the foundation upon which freeze casting is built, this work
seeks to understand the solidification process, especially the growth and time evolution of dendrites. To understand the dendritic growth process, two solidification
parameters, freezing front velocity and temperature gradient, are independently controlled to investigate the effects of each parameter. Dendritic pore size changes with
solidification parameters and shows good agreement with dendrite growth theory.
The theory of constitutional supercooling serves as a guide to control pore morphology between dendritic pores and cellular pores. Furthermore, dendrite growth
under the effects of the gravitational force is investigated by changing the solidification direction with respect to the gravity direction. Convection changes the degree
of constitutional supercooling, and results in different pore sizes as well as pore
morphology.
Time evolution of dendrites through isothermal coarsening is investigated. During
the coarsening of dendrites, they are transformed to cylinder-like crystals, which
yield honeycomb-like structures. Moreover, dendrite size changes linearly with
the cube root of coarsening time. Both findings are well-established phenomena
in alloy solidification. Further comparison with alloy systems are achieved with
tomography-based analysis where similar microstructural evolution with alloy system is demonstrated.
Based upon the understanding of underlying solidification principles in freeze casting, three applications are explored. First, the freeze-cast structure is designed to
improve shape-memory properties. Processing variables are controlled such that
shape-memory porous zirconia can enable martensitic phase transformations and
shape deformation without fracture. Other applications utilize unique pore space.
Dendritic pores are investigated for size-based filtration to preferentially capture

vii
small particles. Flow-through experiments and in-situ observation by confocal microscopy confirm that pores created by secondary dendrites capture small particles.
Finally, honeycomb-like structures are filled with functional microgels to create a
ceramic/polymer composite as an application for membrane chromatography. The
fabricated composite demonstrates advantages such as mechanical stability during
the fluid flow.

viii

PUBLISHED CONTENT AND CONTRIBUTIONS

[1] Noriaki Arai and Katherine T. Faber. “Freeze-cast Honeycomb Structures via
Gravity-Enhanced Convection”. In Preparation, 2020
N. Arai performed experimental design, sample fabrication, characterization,
and data analysis.
[2] Noriaki Arai and Katherine T. Faber. “Gradient-controlled freeze casting of
preceramic polymers”. In Preparation, 2020
N. Arai performed experimental design, sample fabrication, characterization,
and data analysis.
[3] Noriaki Arai, Tiberiu Stan, Sophie Macfarland, Peter W. Voorhees, Nancy
Senabulya, Ashwin J. Shahani, and Katherine T. Faber. “Coarsening of dendrites in freeze-cast ceramic systems”. In Preparation, 2020
N. Arai designed experiments, fabricated samples, performed SEM imaging
and pore size measurement by mercury intrusion porosimetry, and analyzed
the data.
[4] Orland Bateman, Noriaki Arai, Julia A. Kornfield, Mamadou S. Diallo, and
Katherine T. Faber. “Freeze-cast SiOC/mixed matrix PVDF membrane composite for chromatography for monoclonal antibody polishing”. In Preparation, 2020
N. Arai and O. Bateman contributed equally to this work. N. Arai fabricated
and analyzed freeze-cast ceramics. N. Arai also performed SEM imaging and
water flux measurement.
[5] Noriaki Arai and Katherine T. Faber. “Hierarchical porous ceramics via twostage freeze casting of preceramic polymers”. In: Scripta Materialia 162
(Mar. 2019), pp. 72–76. issn: 13596462. doi: 10.1002/adem.201900398.
url: https://doi.org/10.1016/j.scriptamat.2018.10.037.
N. Arai performed experimental design, sample fabrication, characterization,
mechanical and permeability test, and data analysis.
[6] Xiaomei Zeng, Noriaki Arai, and Katherine T. Faber. “Robust Cellular ShapeMemory Ceramics via Gradient-Controlled Freeze Casting”. In: Advanced
Engineering Materials 21.12 (Dec. 2019), p. 1900398. issn: 1438-1656. doi:
10 . 1002 / adem . 201900398. url: https : / / onlinelibrary . wiley .
com/doi/abs/10.1002/adem.201900398.
X. Zeng and N. Arai both contributed to this work equally.

TABLE OF CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . viii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi
Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . 2
Chapter II: Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Review of the porous ceramic processing method . . . . . . . . . . . 3
2.2 Freeze casting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Solidification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4 Polymer-derived ceramics (PDC) . . . . . . . . . . . . . . . . . . . 19
Chapter III: Gradient-Controlled Freeze Casting . . . . . . . . . . . . . . . . 31
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Chapter IV: Freeze-cast Honeycomb Structures via Gravity-Enhanced Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Chapter V: Coarsening of Dendrites in Freeze-Cast Systems . . . . . . . . . 57
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.2 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3 Analysis of XCT images . . . . . . . . . . . . . . . . . . . . . . . . 61
5.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Chapter VI: Application of Freeze-Cast Structure: Microstructural Engineering of Material Space for Functional Properties . . . . . . . . . . . . . . 85
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.2 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 90

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Chapter VII: Applications of Freeze-Cast Ceramics: Pore Space Design for
Filtration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.1 Size-based filtration by dendritic pores . . . . . . . . . . . . . . . . 105
7.2 Ceramic/polymer composites for membrane chromatography . . . . 119
Chapter VIII: Summary and Future Work . . . . . . . . . . . . . . . . . . . 129
8.1 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . 129
8.2 Suggestions for Future work . . . . . . . . . . . . . . . . . . . . . . 131
Appendix A: Hierarchical Pore Structure . . . . . . . . . . . . . . . . . . . . 136
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
A.2 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . 137
A.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 139
Appendix B: Freezing conditions . . . . . . . . . . . . . . . . . . . . . . . . 149
Appendix C: Comparison of the conventional freezing and the gradient controlled freezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Appendix D: Influence of preceramic polymer concentration . . . . . . . . . 152

LIST OF ILLUSTRATIONS

Number
Page
2.1 A schematic showing three different types of porous material processing methods [1]. This figure is reproduced with permission. . . . 3
2.2 Micrographs showing porous materials fabricated by the replica
method (LiCoO2 cathode by wood templating) [3], the sacrificial
template method (macroporous SiC fabricated by silica template) [6]
and (c) direct foaming (porous SiOC) [7]. Figures are reproduced
with permission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 A diagram showing the freeze casting process [19]. . . . . . . . . . . 6
2.4 Pictures during the directional solidification of (a) suspension and (b)
solution [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 SEM images showing freeze-cast SiOC with dendritic pores from (a,
b) solution and (c, d) suspension [19]. . . . . . . . . . . . . . . . . . 10
2.6 SEM images showing three different freeze-cast structures: (a) isotropic
structure from cyclooctane, (b) dendritic structure from cyclohexane,
and (c) lamellar structure from dimethyl carbonate [19]. The solutions were frozen with a freezing front velocity of 15 µm/s. . . . . . . 10
2.7 Pore size distribution of dendritic pores showing the effect of preceramic polymer concentration [18]. . . . . . . . . . . . . . . . . . . . 11
2.8 Longitudinal SEM image of (a) a 20 wt.% polymer concentration and
XCT image of (b) a 5 wt.% polymer concentration in cyclohexane [18]. 11
2.9 A sample freeze-cast (a) and (b) without a polydimethylsiloxane
(PDMS) wedge frozen with (b) multiple nucleation site and a single
vertical temperature gradient. It shows short-range lamellar pores in
(c) the SEM image. A sample freeze-cast (d) with a PDMS wedge
frozen (e) with a confined nucleation site and a dual temperature
gradient: vertical temperature gradient and horizontal temperature
gradient. It shows long-range lamellar pores in (f) the SEM image.
From ref. [36]. Reprinted with permission from AAAS. . . . . . . . 13

xii
2.10

A schematic of the constitutional gradient during solidification and
the liquidus temperature gradient ahead of the freezing front. The ap𝑑𝑇𝑞 (𝑧)
plied temperature gradient, 𝐺 = ( 𝑑𝑧
) 𝑧=0 , is lower than the liquidus
𝑑𝑇𝐿 (𝑧)
temperature gradient, 𝐺 𝑐 = ( 𝑑𝑧 ) 𝑧=0 , resulting in the constitutional
supercooling (cross-hatched region) [24]
Credit:W. Kurz and D. J. Fisher, Fundamentals of solidification, Third
edition, Trans Tech Publication, 1992. . . . . . . . . . . . . . . . . .
2.11 Illustrations showing (a) small perturbation grow and (b) small perturbation disappear [24].
Credit:W. Kurz and D. J. Fisher, Fundamentals of solidification, Third
edition, Trans Tech Publication, 1992. . . . . . . . . . . . . . . . . .
2.12 A schematic showing the stability of the interface as a function of
wavelength for Al-2wt.%Cu [24].
Credit:W. Kurz and D. J. Fisher, Fundamentals of solidification, Third
edition, Trans Tech Publication, 1992. . . . . . . . . . . . . . . . . .
2.13 Images showing (a) non-faceted crystals (dendrites) and (b) faceted
crystals [47]. This figure is reproduced with permission. . . . . . . .
2.14 Free energy curve as a function of adatom coverage with different
values of the Jackson 𝛼 factor [48]. This figure is reproduced with
permission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.15 Optical micrographs showing freezing microstructures of preceramic
polymer solutions with (a) cyclooctane, (b) cyclohexane, (c) tertButanol, and (d) dimethyl carbonate [18]. . . . . . . . . . . . . . . .
2.16 Si-based preceramic polymer with different backbones [54]. This
figure is reproduced with permission. . . . . . . . . . . . . . . . . .
2.17 A model for the nanodomains in SiOC [68]. This colored image was
taken from [54]. This figure is reproduced with permission. . . . . .
3.1 Stability-microstructure map showing the independent control of
freezing front velocity and temperature gradient allows one to change
crystal morphology (a). Modified from ref. [9]. (b) Illustration of
cells and dendrites. . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 A photograph showing gradient-controlled freeze-casting setup. . . .

18
18

20
21
23

32
34

xiii
3.3

3.4

3.5

3.6

3.7
4.1

SEM images showing a control sample at (a) low and (b) high magnifications and a gradient-controlled sample at (c) low and (d) high
magnifications in transverse direction. Yellow arrows in low magnification and high magnification images indicate primary pores and
tertiary pores, respectively. (e) Pore size distribution of control sample and gradient-controlled sample. . . . . . . . . . . . . . . . . . .
SEM images of the sample frozen with V = 17 µm/s and G = 2.2
K/mm in (a) transverse and (b) longitudinal direction, the sample
frozen with V = 1.8 µm/s and G = 2.4 K/mm in (c) transverse and
(d) longitudinal direction, and the sample frozen with V = 1.5 µm/s
and G = 5.0 K/mm in (e) transverse and (f) longitudinal direction.
(g) A stability-microstructure map showing examined conditions by
colored marker. (h) Pore size distributions of corresponding samples.
SEM images showing dendritic structure from (a) 20 wt.% solution
and (b) 10 wt.% solution in transverse direction. (c) Corresponding
pore size distribution from MIP. . . . . . . . . . . . . . . . . . . . .
SEM images showing a SiOC from cyclohexane crystals (20 wt.%
polymer solution) in (a) transverse and (b) longitudinal directions,
from cyclohexane crystals (10 wt.% polymer solution) in (c) transverse and (d) longitudinal directions, and from dioxane crystals in (e)
transverse and (f) longitudinal directions. . . . . . . . . . . . . . . .
Plots of (a) Primary pore size as a function of V with different G and
(b) secondary pore size as a function of cooling rate. . . . . . . . . .
Freeze-casting setup of (a) conventional freezing and (b) convectionenhanced freezing. (c) Freezing front position as a function of time
with images of (d) the freezing front in conventional freezing, and
in convection-enhanced freezing at (e) t = 45 min and (f) t = 47 min
(Red dashed line indicates the freezing front), and (g) the associated
freezing front velocity and temperature gradient as a function of
freezing front position. . . . . . . . . . . . . . . . . . . . . . . . . .

40
41

xiv
4.2

4.3

4.4

5.1
5.2

5.3
5.4

5.5

SEM images of conventional freeze-cast samples showing transverse
images at (a) FFP is ∼1.6 mm and (b) FFP is ∼5 mmfrom nucleation face, and (c) longitudinal image. SEM images of convectionenhanced freeze-cast sample showing transverse images (d) FFP is
∼1.6 mm and (e) FFP is ∼5 mm from nucleation face, and (f) longitudinal image. Yellow arrows indicate freezing direction, v, and
gravity direction, g. Red lines in (c) and (f) indicate the nucleation face 49
Pore size distribution data from (a) nucleation section and (b) middle
section from samples from conventional freezing and convectionenhanced freezing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
Illustrations showing temperature and concentration variation in (a)
conventional freezing and (b) convection-enhanced freezing. (c) An
illustration showing convective flows in liquid phase in convectionenhanced freezing. (d) Stability-microstructure map. (e) Pore size
distribution of conventional freeze-cast sample frozen under 0.7 µm/s
and 4.9 K/mm. (f) Porosity difference between top section and three
sections (middle-top, middle-bottom, and bottom). Three samples
were investigated for each freezing direction. . . . . . . . . . . . . . 51
Schematic of the gradient-controlled freeze casting setup . . . . . . . 59
Cross-section of XCT data from (a) a control sample, (b) a sample
coarsened at 2 ◦ C for one hour, and (c) a sample coarsened at 4 ◦ C
for three hours. Scale bar: 200 µm. . . . . . . . . . . . . . . . . . . 61
A map of interfacial shapes of patches for the Interfacial Shape Distribution (ISD). This is a modified figure from ref. [20]. . . . . . . . 62
SEM images showing (a, b) control sample, and sample coarsened at
(c, d) 2 ◦ C for one hour, (e, f) 2 ◦ C for three hours, (g, h) 4 ◦ C for one
hour, and (i, j) 4 ◦ C for three hours. Inset images in (a) and (b) show
primary pore and secondary pores, respectively, as indicated by red
arrows, (scale bar: (a) 60 µm and (b) 40 µm). Transverse images and
longitudinal images show cross-sections perpendicular and parallel
to the freezing direction, respectively. . . . . . . . . . . . . . . . . . 64
SEM images showing longitudinal direction of (a) the control sample
and (b) the sample coarsened at 4 ◦ C for one hour. Flat surface and
circular surface are indicated by red arrows in (a) and (b), respectively. 66

xv
5.6

5.7
5.8
5.9
5.10

5.11

5.12

5.13

5.14

6.1

SEM images of the samples coarsened at 4 ◦ C for three hours (a:
Transverse image, b: Longitudinal image) and five hours (c: Transverse image, d: Longitudinal image). . . . . . . . . . . . . . . . . . 66
Pore size distribution data of samples coarsened for 30 minutes and
one hour at (a) 2 ◦ C and (b) 4 ◦ C (including three hours). . . . . . . . 67
Primary pore fraction as a function of coarsening time. . . . . . . . . 68
Plots of (a) Primary pore size and (b) secondary pore size as a function
of the cube root of coarsening time at different coarsening temperatures. 68
Illustration showing four different coarsening models for secondary
arm coarsening: (1) radial remelting, (2) axial remelting, (3) arm
detachment, and (4) arm coalescence. Based on ref. [27]. . . . . . . 70
Pore size distribution from samples coarsened at 2 ◦ C for three hours
and 4 ◦ C for one hour (a). SEM images showing a sample coarsened
at (b, c) 2 ◦ C for three hours, and (d, e) 4 ◦ C for one hour. (Red
arrows indicate some of the thin solid tubes). . . . . . . . . . . . . . 73
3D XCT reconstructions and subsections for the (a, d) control sample, (b, e) the sample coarsened at 2 ◦ C for one hour, and (c, f)
sample coarsened at 4 ◦ C for three hours. The sides of the solid-pore
interfaces that face the dendritic pores are colored according to the
normalized mean curvature (H/SS ), as indicated by the color bar in
(c). White arrows in (e) show secondary pores with positive curvature
caps, while the red arrow indicates a ligature with negative curvature. 74
Interface Shape Distributions (ISDs) for the (a) control sample, (b)
sample coarsened at 2 ◦ C for one hour, and (c) sample coarsened at
4 ◦ C for three hours. (d) Map of the interface shapes possible in an
ISD where P is pore and S is solid. This is a modified figure from
ref. [20]. Sections of the 2 ◦ C coarsened sample cylindrical patches
colored in red (e) and porous caps colored in pink (f). . . . . . . . . 77
Interface Normal Distributions (INDs) for the (a) control sample, (b)
sample coarsened at 2 ◦ C for one hour, and (c) sample coarsened at 4
◦ C for three hours. The green arrow in (a) corresponds to the green
patches in (d). The purple arrow in (b) corresponds to purple patches
in (e). The blue arrow in (b) corresponds to the blue patches in (f). . 79
A schematic showing shape-memory effect and superelastic effect
[2]. This figure is reproduced with permission. . . . . . . . . . . . . 86

xvi
6.2

6.3

6.4
6.5

6.6

6.7

6.8
6.9
6.10

Stability-microstructure map based on constitutional supercooling
of a solid–liquid interface controlled by freezing front velocity and
temperature gradient (modified based on Rettenmayr and Exner [13]).
Schematic illustration of b) dendrites and c) cells. . . . . . . . . . . .
The proposed shape-memory effect in a unidirectional cellular structure during uniaxial compression and heat treatment. The red highlights represent transformed grains within the cellular walls. . . . . .
Plots showing (a) freezing front velocity and (b) temperature gradient
as a function of frozen height. . . . . . . . . . . . . . . . . . . . . .
Stability-microstructure map based on measured freezing front velocity and temperature gradient of cyclohexane, with the corresponding
longitudinal microstructures of freeze-cast zirconia-based ceramics. .
Microstructure of freeze-cast cellular zirconia-based ceramics viewed
from (a) the transverse (the inset image shows an off-axis view of
pores) and (b) the longitudinal directions. Oligocrystalline cellular
walls from (c) the transverse and (d) longitudinal directions. (e)
Pore size distribution within the measurement range of 100 nm–80
µm from mercury intrusion porosimetry, with inserted sample image
after machining. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stress–strain behavior of the cellular structure (v = 1.43 µm s− 1),
transitional structure (v = 3.87 µm s− 1), and dendritic structure (v =
11.57 µm s− 1) under a compressive stress of 25 MPa (a). (b) The
evolution of phase content on compression and after heat treatment,
with inserted XRD patterns of cellular structure corresponding to
each condition. (c) Stress–strain curves of the transitional structure
tested consecutively at stresses from 10 to 40 MPa. (d) The change
in the monoclinic content of all samples after compression as a function of applied stress, with inserted XRD patterns of the transitional
structure in between each compression test. . . . . . . . . . . . . . .
XRD spectrum of a sample (a) after machining, and (b) after annealing without experiencing mechanical compression. . . . . . . . . . .
The stress-strain curve of the sample used for the shape recovery
measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Stress-strain curves showing (a) five loading-unloading cycles. (b)
Monoclinic composition after each five cycles and after each anneal. .

88
89

93
94
95
98

xvii
6.11

6.12
6.13
6.14
7.1
7.2

7.3

7.4

7.5
7.6
7.7
7.8
7.9

7.10

Slope of stress-strain curves as a function of applied stress (a). Each
data represents the slope of the 5th loading cycles from each set of
five loading-unloading cycles. (b) Magnified plateau region. . . . . . 98
Stress-strain curves showing five loading-unloading steps at 10 MPa,
20 MPa, and 24 MPa. . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Slopes of stress-strain curves as a function of applied stress(a). XRD
peak before and after compression (b). . . . . . . . . . . . . . . . . . 101
Slope of stress-strain curves as a function of the applied stress. The
material was compressed to 24 MPa for 5 times, and above. . . . . . 101
A graph showing patient survival rate and patients with effective
antibiotic therapy [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . 106
Illustration of (a) elasto-inertial based particle focusing and separation [7] (Reproduced under Creative Commons) and (b) larger cells
enter into vortices due to the larger net-force acting on larger cells [8]
(Reproduced with permission.) . . . . . . . . . . . . . . . . . . . . . 107
An illustration showing fluid flow in the dendritic pores. Large blood
cells flow through the primary pores while small pathogens enter a
recirculating flow in secondary pores. . . . . . . . . . . . . . . . . . 108
Cooling profiles for top and bottom thermoelectric plates to create a
dual structure. The red-shaded region creates dendritic pores and the
green-shaded region creates cellular pores. . . . . . . . . . . . . . . 109
A picture of the flow-through experimental setup. . . . . . . . . . . . 111
A picture of the confocal microscope setup. . . . . . . . . . . . . . . 112
An SEM image and pore size distribution of a membrane used in the
flow-through study. . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Pictures of freeze-cast SiOC pyrolyzed under (a) Ar and (b) Ar with
water vapor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
Pictures showing SiOC pyrolyzed under Ar and H2 O atmosphere with
pores filled with (a) air, (b) DI water, and (c) canola oil (n: refractive
index). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
Overlay of bright field and fluorescence micrographs from laser scanning confocal microscope. The series of micrographs shows a 2 µm
particle (indicated by the red arrow) flowing along the main channel
and being captured at the side cavity after 45 seconds. . . . . . . . . 116

xviii
7.11

7.12

7.13

7.14

7.15

7.16
7.17

8.1

8.2
8.3

8.4

SEM images showing transverse direction of dendritic structure after
flow-through experiment at (a) low magnification and (b) high magnification. SEM images showing transverse direction of honeycomblike structure after flow-through experiment at (c) low magnification
and (d) high magnification. Some of the 2 µm and a group of the 0.3
µm particles are indicated by yellow and red circles, respectively. . . 117
SEM images showing (a) longitudinal direction and transverse direction of (b) cellular pore region, and (c) dendritic pore region. (d)
Pore size distribution of a dual structure. . . . . . . . . . . . . . . . 118
A schematic of (a) permeability setup. A figure taken from [20]. (b)
A picture of an acrylic fixture. (c) An illustration of side view of the
acrylic fixture holding a composite. . . . . . . . . . . . . . . . . . . 122
SEM images showing a composite without gel layer ((a) transverse
and (b) longitudinal direction) and a composite with gel layer ((c)
transverse and (d) longitudinal direction) . . . . . . . . . . . . . . . 123
An SEM image showing a PVDF membrane, PEI gel layer, and SiOC
wall. Yellow dashed lines indicate boundaries between a gel layer
and SiOC wall. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
An SEM image showing a thickness of around 1.5 mm composite. . . 125
A plot of (a) water flux and pressure drop as a function of time. (b)
Water flux at different pressure drops as a function of time (from the
study by Kotte et al. [13]). This figure is reproduced with permission. SEM images of (c) inlet and (d) outlet side after permeability
measurement with sample pictures as insets. . . . . . . . . . . . . . 126
SEM images of freeze-cast structures using cyclooctane as a solvent in
longitudinal direction. As the higher temperature gradient is applied,
the directionality of pores improved. Left image is taken from a study
by Naviroj et al. [1] . . . . . . . . . . . . . . . . . . . . . . . . . . 131
SEM images showing freeze-cast lamellar structures with (a) 5 minutes and (b) 6 hours of stirring after adding the cross-linking agent. . 132
Compressive strength and permeability constants of different structures. Data for "Lamellar 15 µm/s" and "Dendritic 15 µm/s" are taken
from the work by Naviroj [2]. . . . . . . . . . . . . . . . . . . . . . 133
A stability-microstructure map with an arrow indicating an increase
of diffusion coefficient results in change in stability criterion. . . . . . 134

xix
A.1
A.2

A.3

A.4

A.5

A.6
A.7
A.8

A.9

C.1
C.2

Freezing solution by (a) conventional unidirectional freezing and (b)
conventional conditions coupled with mold heating. . . . . . . . . . 137
SEM images of a plane perpendicular to the freezing direction from
(a) single-stage freeze casting with 20 vol% polymer concentration,
(b) two-stage freeze casting with 5 vol% polymer concentration at
the second stage, (c) two-stage freeze casting with 10 vol% polymer
concentration at the second stage. (d) Schematic illustration showing
bridge formation during the second stage. . . . . . . . . . . . . . . . 139
Compressive strength by single-stage freeze casting and two-stage
freeze casting (a). (b) Load displacement curve of single-stage freezecast sample. (c) Load displacement curve of two-stage freeze-cast
sample. The insets show samples after compression. Note the difference in y-axis scales in (b) and (c). . . . . . . . . . . . . . . . . . . 140
Example of a domain boundary in (a) single-stage freeze-cast sample
(20 vol.%), and (b) two-stage freeze-cast sample (5 vol.% at the
second stage). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Load-displacement curve of the two-stage freeze-cast sample which
exhibited noticeable low strength. The inset shows sample after
compression. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Permeability constants of samples by single-stage freeze casting and
two-stage freeze casting. . . . . . . . . . . . . . . . . . . . . . . . . 143
Compressive strength and permeability constants compared to the
Naviroj study on lamellar and dendritic pore structures [11]. . . . . . 144
SEM images of two-stage freeze-cast SiOC using DMC as the solvent
in the first stage and cyclohexane at the second stage. (a) Transverse
image (a plane perpendicular to freezing direction) and (b) longitudinal image (a plane parallel to freezing direction). . . . . . . . . . . 145
SEM images of the hierarchical pore structure in two-stage freezecast SiOC using cyclohexane as the solvent in the first stage and
cyclooctane at the second stage at (a) low magnification and (b) high
magnification. A grain-selection template [25] was used at the first
stage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Freezing profile of Conventional freezing (V = 15 µm/s) and gradientcontrolled freezing (V = 15 µm/s, G = 2.6 K/mm) . . . . . . . . . . . 150
Pore size distribution from three different sections. . . . . . . . . . . 151

xx
D.1

SEM images showing (a, b) a control sample, and (c,d) a sample
coarsened at 3 ◦ C for 1 hour. (e) Pore size distribution from 30 wt.%
preceramic polymer solution. . . . . . . . . . . . . . . . . . . . . . 152

xxi

LIST OF TABLES
Number
Page
5.1 The slope of linear fit from Figure 5.9 . . . . . . . . . . . . . . . . . 70
5.2 Metrics from the three XCT datasets. SS −1 is the inverse specific
interface area, calculated as the total pore volume divided by the total
solid-pore interface area. . . . . . . . . . . . . . . . . . . . . . . . . 76
6.1 Sample height and diameter before compression, after compression,
and after heat treatment; associated residual and recovered displacements used to establish recovered strain. . . . . . . . . . . . . . . . . 96
7.1 Particles captured in the flow-through experiments. . . . . . . . . . . 113
A.1 Average porosity of single-stage freeze-cast samples and two-stage
freeze-cast samples. . . . . . . . . . . . . . . . . . . . . . . . . . . 138
A.2 Bridge density of two-stage freeze-cast samples. . . . . . . . . . . . 140
B.1 List of freezing front velocities and temperature gradients used in
Chapter 3 for 20 wt.% polymer-cyclohexane solution. . . . . . . . . . 149
C.1 List of the peak pore diameters for primary and secondary pores from
Figure C.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

Chapter 1

INTRODUCTION
1.1

Motivation

Porous ceramics can be found in many industrial applications such as filtration, catalyst supports, bioceramics, insulators, etc., and demands for engineered porosity
have been driving porous ceramic processing research. Each application requires
its own specification of pores, including pore concentration, size, morphology, and
connectivity. Hence, it is essential to not only know the desired pore characteristics
and properties, but also to have a deep understanding of the processing methods to
manipulate pores. The present work was established based upon such needs. Directional freeze casting creates directionally aligned pores by using solidifying crystals,
which push particles or segregated preceramic polymer aside, and act as sacrificial
templates. Subsequent sublimation removes the frozen crystals, leaving pores in
their place. Compared to other porous ceramic processing methods, this method
indirectly controls the resulting pores through solidification. The long history of
theoretical and experimental alloy solidification research provides a foundation to
apply to freeze casting method to achieve control of pore characteristics. The majority of freeze casting studies involve suspensions, in which particles and dissolved
additives (binders and dispersants) present complexities in thermal fields and solute fields in the liquid phase during solidification. Alternatively, solutions, which
consist of only a solvent and solute, allows the application of alloy solidification
principles to better understand the relationship between processing variables and
the resultant freeze-cast structure. Hence, this is the focus of the present work.
1.2

Objectives

There are two objectives in the present work. The first objective is to understand the
freeze casting process from the standpoint of fundamental solidification principles.
Solidification parameters such as freezing front velocity and temperature gradient
are independently manipulated so as to investigate their influence on pore structure.
Moreover, the effects of the ubiquitous external force, the gravitational force, on
dendrite growth was investigated. Because solidification microstructures are not
only determined by the crystal growth process, but also the solidified crystals’
evolution with time, the coarsening process is also explored. The morphological

changes and the relationship between coarsening time and pore size are investigated.
Tomography-based curvature analysis is used to reveal the mechanism of coarsening.
Taking what has been learned in solidification studies, the second objective is to
create porous ceramics for engineering applications. Honeycomb shape-memory
ceramics are created to mitigate intergranular cracking, unavoidable in bulk shapememory ceramics. While this is an example of engineering porous microstructures
to improve the functional properties of the material space, other examples are to
utilize the pore space. Two examples are demonstrated. Size-based filtration is
examined with the ultimate goal to isolate pathogens from the bloodstream. It is
shown that small particles can be preferentially captured by dendritic pores. A
ceramic/polymer composite for membrane chromatography is also examined, and
enhanced mechanical stability is demonstrated.
1.3

Thesis organization

Following this chapter, Chapter 2 provides background on porous ceramic processing methods, with much attention to the freeze casting process. It further includes
a discussion of solidification principles and polymer-derived ceramics. In Chapter
3, fine tuning of the freeze casting process is explored through gradient-controlled
directional solidification and the resultant pore structures are discussed. Chapter 4 discusses the effects of the gravitational force in gradient-controlled freeze
casting. For the first time, an in-depth study of coarsening during freeze-casting
is performed. These are reported in Chapter 5. The thesis then turns to freezecasting examples which may show promise in engineering applications. Chapter 6
explores the creation of porous shape-memory ceramics and demonstrates the shapememory effect. Chapter 7 explores two other applications: size-based filtration and
ceramic/polymer composites for membrane chromatography, highlighting the diversity of pore structure and functionality. Finally, Chapter 8 summarizes the works
and proposes future research directions. One additional example which provides
hierarchical pore structure by two-stage freeze casting is discussed in Appendix A.

Chapter 2

BACKGROUND
2.1

Review of the porous ceramic processing method

Ceramics has been of interest for various applications due to hardness, chemical
inertness, high temperature resistance, and low electrical and thermal conductivity. Combining these properties with engineered porosity, porous ceramics can be
used in various applications such as insulators, bio-medical implants, and filtration.
Since each application requires different pore characteristics and networks, there
is growing research on porous ceramic processing techniques. The processing of
porous ceramics can be divided into three types as shown in Figure 2.1 [1] [2].

Figure 2.1: A schematic showing three different types of porous material processing
methods [1]. This figure is reproduced with permission.

Replica method
This method replicates the porous structure of cellular materials by infiltrating a
ceramic suspension or precursor solution. The process is followed by drying and
removing template materials by firing, leaving the porous structure resembling the
template, but made of the desired materials. Cellular materials, which serve as
templates in this method, can be synthetic templates such as polymeric sponge or
natural templates such as coral, diatoms, and wood. Synthetic templates have been
used to create filters in industry due to its simplicity. The natural template is an
alternative option when templating porous structure or pore size is challenging to
synthesize with existing techniques. Typically, natural materials possess hierarchical
structures which can be used in applications such as battery electrodes (Figure 2.2
a) [3] and photocatalysts [4]. The disadvantage of this method is that struts of the
structure are often prone to flaws such as cracks and pores during the burning step
of the replica materials, leading to degraded mechanical properties.
Sacrificial template method
In this method, the sacrificial phase is dispersed in a ceramic matrix or ceramic
precursor, and removed by heat treatment or sublimation. The sacrificial template
include synthetic organics, natural organics, and liquid. Figure 2.2 b shows a
macroporous SiC fabricated by using a silica template. Unlike the replica method,
the removal of the sacrificial phase does not introduce flaws in struts, so the processed
materials by the sacrificial template method possess superior mechanical properties.
Although this method offers better tailorability in porosity, pore size distribution,
and pore morphologies, the main disadvantage of this process is the time-consuming
step needed to remove the sacrificial phase either by heat treatment or sublimation.
Direct foaming
This method directly introduces air into suspension or liquid, consolidates the material with pores by a setting agent, and sinters to produce porous solids. The porosity
will be controlled by the amount of air incorporated. A critical step in this method is
the stabilization of pores in the liquid by surfactant to avoid undesirable coalescence
of the incorporated pores. The porous SiOC fabricated from this method is shown
in Figure 2.2 c. Although this method is simple, inexpensive, and environmentally
benign, due to the nature of introducing pores, pore morphologies are typically
spherical. Furthermore, although pore size as small as ∼ 40 µm is possible through

an efficient surfactant and a rapid set-in process [5], the methods mentioned above
can produce smaller pore sizes.

Figure 2.2: Micrographs showing porous materials fabricated by the replica method
(LiCoO2 cathode by wood templating) [3], the sacrificial template method (macroporous SiC fabricated by silica template) [6] and (c) direct foaming (porous SiOC)
[7]. Figures are reproduced with permission.

2.2

Freeze casting

Freeze casting is one of the sacrificial template methods, and uses solidified crystals
as the sacrificial phase to template pores. Freeze casting was originally used to
produce a preform of refractory powders for subsequent infiltration of metals to
create near-net-shape turbosupercharger blades at NASA [8]. Nearly 50 years later,
freeze casting started to gain attention from materials scientists [9, 10, 11]. Due to
broad ranges of applications of this process, a number of review articles are available
[12, 13, 14] and an open data repository was launched recently [15].
While the majority of studies focuses on freeze-casting of ceramic powders (referred to here as suspension-based freeze casting), this study primarily focuses on
freeze-casting of preceramic polymers (referred to as solution-based freeze casting).
The pioneering work of solution-based freeze casting was done by Yoon in 2007
[16], who investigated polycarbosilane/camphene solution and produced porous
SiC with dendritic pores. Naviroj studied solution-based freeze casing in detail by
exploring different solvents, polymer concentrations and solidification parameters,
and demonstrated tailorability in pore size and pore morphology using solidification theory [17, 18]. Freeze casting of preceramic polymers not only expands the
phase space for porous ceramics, but also makes use of fundamental solidification
principles to provide a powerful tool to control pore characteristics. This section introduces the freeze-casting process, highlights differences between suspension and
solution routes, and reviews processing variables.

2.2.1

Process overview

Figure 2.3 shows the freeze casting process. First, a suspension or solution is
prepared. A suspension contains a dispersion medium, typically water, the ceramic
powders, and additives such as binders and dispersants. A solution is prepared
by dissolving preceramic polymer in a solvent; a cross-liking agent is added before
solidification to ensure the mechanical integrity during the pyrolysis. The suspension
or solution is then directionally frozen such that the growing crystals reject or
segregate suspending particles or dissolved polymers. The subsequent sublimation
step removes sacrificial solvent crystals, leaving pores in the materials. Sintering
or pyrolysis yields porous solids. Hence, templating the pores is accomplished by
the solidified crystals. Although solidification induces the phase separation in both
routes, the mechanisms of the phase separation processes are different as discussed
below.

Figure 2.3: A diagram showing the freeze casting process [19].

Rejection of particles in suspension
In suspension-based freeze casting, when a liquid phase freezes, the freezing front
rejects the particles in suspension and they are pushed into the interdendritic regions.

A particle pushed ahead by the freezing front was modeled by Korber and Rau,
considering two counteracting forces, a viscous drag force and Van der Waals forces
[20]. A viscous drag force is an attractive force acting on the particles toward the
freezing front. The viscous drag force in a case of flat freezing front can be expressed
by the following expression [21]:
𝐹𝜂 = 6𝜋𝜂𝑣𝑟 2 /𝑑
where 𝜂, v, r, and d are the viscosity of suspension, freezing front velocity, the particle radius, and the distance between the particle and the freezing front, respectively.
Van der Waals forces come from the interfacial energy difference. The thermodynamic criterion for the rejection of the particles can be expressed by the following
expression:

Δ𝜎0 = 𝜎𝑠𝑝 − (𝜎𝑠𝑙 + 𝜎𝑙 𝑝 ) > 0
where 𝜎𝑠𝑝 , 𝜎𝑠𝑙 , and 𝜎𝑙 𝑝 are the surface free energy of solid-particle, solid-liquid,
and liquid-particle, respectively. Using Δ𝜎0 , the repulsive force can be expressed
by the following equation:

𝐹𝑅 = 2𝜋𝑟Δ𝜎0 (

𝑎0 𝑛

where 𝑎 0 is the average molecular distance in the liquid film between the particle
and freezing front, and n is the exponent. This exponent is the correction to the
repulsive force, and can vary, for example, with particle size [20, 21, 22]. Equating
the attractive force and repulsive force, the critical freezing front velocity, 𝑣 𝑐 , is:

𝑣𝑐 =

Δ𝜎0 𝑎 0
3𝜂𝑟

Above 𝑣 𝑐 , the particles will be engulfed, whereas particles will be repelled below
𝑣 𝑐 . In suspension-based freeze casting, it is desirable for the majority of particles
to be repelled by ensuring that the critical freezing front velocity is not exceeded so
the pores are templated by the growing crystals. However, this equation also serves
as a guide to deliberately engulf large particles while the small particles are rejected
by adjusting freezing front velocity to improve mechanical properties. Ghosh et al.
demonstrated a strengthening strategy for freeze-cast materials by engulfing platelet
particles and repelling equiaxed particles [23].

Segregation of solutes in solution
In solution-based freeze casting, a phase separation between solutes and solvent
crystals is a result of segregation due to the equilibrium solubility difference between
the liquid and solid phases. The relation between concentration in liquid and solid
(C 𝐿 and C𝑆 , respectively) is expressed using the equilibrium distribution coefficient,
k0, also known as the chemical segregation coefficient or the partition coefficient:

𝑘0 ≡

𝐶𝑆
𝐶𝐿

During solidification, if the freezing front velocity is slow enough to assume that the
local thermodynamic equilibrium holds at the solid-liquid interface, the equilibrium
distribution coefficient can be used to assess the redistribution of solute between
solids and liquid. Similarly to suspension-based freeze casting, the solute should be
segregated by the solid phase so the pores are templated by the crystals. Ideally, the
equilibrium distribution coefficient should be as small as possible so as to segregate
the majority of the solute. However, when the freezing front velocity is sufficiently
high such that the atoms have no time to rearrange themselves at the solid-liquid
interface, the solute will be frozen with the same composition as they arrive from
the melt, an effect known as solute trapping [24]. In such a case, the distribution
coefficient approaches unity, meaning that there is no segregation.
Advantages of solution-based freeze casting

Figure 2.4: Pictures during the directional solidification of (a) suspension and (b)
solution [18].

Naviroj et al. investigated both suspension- and solution-based freeze casting with
different solvents and reported the differences between the two routes [19]. In
suspension-based freeze casting, one has a variety of materials choices, ranging
from metals [25, 26, 27], ceramics [12], and even polymers [28] as long as a the
stable suspension can be prepared. Thus, the majority of the reports in the literature
focus on suspension-based freeze casting. In contrast, solution-based freeze casting
requires a solute which is soluble in solvent. Although the material choice is limited
compared to the suspension route, it offers a few advantages. In suspensions (Figure
2.4a), due to opacity, one has to use a method such as X-ray radiography to observe
the freezing front [29]. On the other hand, since the solution is transparent (Figure
2.4b), the measurement and the control of freezing front velocity is possible using
a camera so that the resulting pore size is easily tailorable. Second, suspensions
contain additives such as binders and dispersants, in addition to particles, which
make the system more complex. It was observed by Naviroj et al. that particle
suspensions disrupt the solidification microstructure in suspension-based samples
[19]. The freeze-cast microstructures by solution and suspension routes differ
as shown in Figure 2.5. While solution-based freeze-cast samples clearly show
dendritic morphology, the suspension-based freeze-cast samples lack fine dendritic
features and anisotropy. The in-situ microtomography study of a metal alloy system
also revealed that the presence of particles modifies the dendrites to hyperbranched
morphologies through multiple splitting, branching, and curving of the secondary
arms of the dendrites [30], which likely resulted from the local variation of solute
content caused by the particles during the crystal growth. In solution-based freeze
casting, however, such a complexity does not exist. As a result, the unique templated
pore morphology results in ceramics microstructures appropriate for filtration for
medical devices [31]. Finally, processing time and cost are longer and expensive
in suspension route. Preparing suspensions require a time-consuming ball-milling
process whereas a solution can be prepared within 30 minutes. Moreover, pyrolysis
temperatures (∼1300 ◦ C or lower) are lower than sintering temperatures (∼1700 ◦ C
or higher), saving cost and energy.
2.2.2

Processing variables

Since this study primarily focuses on solution-based freeze casting, processing
variables of solution-based freeze casting are mainly highlighted.

Figure 2.5: SEM images showing freeze-cast SiOC with dendritic pores from (a, b)
solution and (c, d) suspension [19].

Figure 2.6: SEM images showing three different freeze-cast structures: (a) isotropic
structure from cyclooctane, (b) dendritic structure from cyclohexane, and (c) lamellar structure from dimethyl carbonate [19]. The solutions were frozen with a freezing
front velocity of 15 µm/s.
Solvent
The choice of the solvent is an essential part of freeze casting, and a few important
points are as follows. First, the solvent must be chosen such that the preceramic
polymer can be dissolved. Second, the cross-linking process in a solvent should be
slow enough to allow solidification without gelation, but fast enough so the preceramic polymer has mechanical integrity after sublimation to survive the pyrolysis
step. Third, the solvent must be compatible with the freeze casting process. The
solvent needs to have a sufficiently high freezing point so that the solution can be
completely frozen. In addition, the solvent also needs to be sublimable at pressures

11
of the freeze dryer to remove the solvent crystals. Finally, a solvent must be chosen to achieve desired pore structure since the microstructure is dependent upon
solvent crystallography and solidification parameters. Figure 2.6 shows three pore
structures freeze-cast from different solvents.
Solids loading

Figure 2.7: Pore size distribution of dendritic pores showing the effect of preceramic
polymer concentration [18].

Figure 2.8: Longitudinal SEM image of (a) a 20 wt.% polymer concentration and
XCT image of (b) a 5 wt.% polymer concentration in cyclohexane [18].
Solids loading is an important parameter as it determines pore characteristics such
as porosity, pore size, and pore network. As the solid loading increases, the volume

12
of growing crystals, the sacrificial phase, decreases, resulting in lower porosity.
Solids loading also modifies the pore size as shown in Figure 2.7 because the
solids loading controls the space for the growing crystals. A hybrid system which
contains preceramic polymer and ceramic particles were studied by Naviroj et al.
[19] and Schumacher et al. [32]. These composites are useful to control not
only mechanical integrity but also surface characteristics. Solids loading is also
important for control of the pore network. Figure 2.8 shows freeze-cast structures
from cyclohexane with different polymer concentrations: 20 wt.% and 5 wt.%.
Both have dendritic structures, but possess different pore networks. Because there
are enough preceramic polymers segregating into interdendritic regions in 20 wt.%
solution, each dendritic pore is isolated and not connected to neighboring pores.
In contrast, when the concentration is decreased, the ceramic wall became much
thinner and the dendritic pores are highly interconnected.
Freezing conditions
Freezing conditions are also critical as they determine the pore morphology, pore
size, and pore directionality. Two important solidification variables are usually controlled: the freezing front velocity and temperature gradient.
Freezing front velocity determines the microstructural length scale of growing crystals. The rapidly growing crystals tend to exhibit sharp tips and fine features
while slowly growing crystals show blunt tips and coarse features. Fine features of
fast-growing crystals increase the relative surface area, which enables crystals to efficiently transport heat or solute (the so-called point effect of diffusion), so the small
crystals are favored in fast freezing front velocities. Consequently, faster freezing
front velocity leads to small pore size and slower freezing front velocity leads to
large pore size. In freeze casting, the typical pore size of freeze-cast structures
ranges from around 300 nm to 500 µm (macropores) [33]. Smaller pores can be
achieved by quenching a solution. If the solution is quenched, the solvent transitions
to the glassy state. Subsequently, the temperature is slowly increased to initiate the
crystallization of the solvent and phase separation. This process creates nanocrystals, and after the solvent extraction, the material is left with large free surface areas
exceeding 300 m2 g−1 and small pore radii as low as 1.9 nm [34]. This surface area
is significantly larger than those of typical freeze-cast solids which are less than 1
m2 g−1 [18, 35].
In contrast, the temperature gradient has an impact on the directionality of the

Figure 2.9: A sample freeze-cast (a) and (b) without a polydimethylsiloxane (PDMS)
wedge frozen with (b) multiple nucleation site and a single vertical temperature
gradient. It shows short-range lamellar pores in (c) the SEM image. A sample
freeze-cast (d) with a PDMS wedge frozen (e) with a confined nucleation site and a
dual temperature gradient: vertical temperature gradient and horizontal temperature
gradient. It shows long-range lamellar pores in (f) the SEM image. From ref. [36].
Reprinted with permission from AAAS.

pores. Although the majority of the solidification in freeze casting were conducted
from bottom to the top by the vertical temperature gradient, there are a number of
studies controlling the direcionality of temperature gradient or combining several
temperature gradients. In suspension-based freeze casting, Bai et al. used a mold
with the copper cold finger rod placed in the center so that the ceramic slurry was
frozen radially to mimic the structure of the bones [37]. In another study by Bai et
al., a polydimethylsiloxane wedge was used to limit the nucleation site and control
the growth direction by creating a dual temperature gradient to attain long-range
order alignment of lamellar pores [36] (Figure 2.9). The temperature gradient not
only affects the pore directionality but also their morphologies.
Freezing front velocity and temperature gradient are variables during the crystal
growth, however, the solidification microstructure can be controlled through nucleation process or post-crystal growth process. To control the nucleation process,
Munch et al. modified the surface pattern of the cold finger to control the orientation of lamellar pores [38]. Naviroj et al. controlled the nucleation process by
applying grain selector templates to align the dendritic pores with improved permeability [39]. Post-crystal growth processes such as coarsening were also studied.
Pawelec et al. investigated low-temperature ice annealing in a collagen suspension,

14
and observed coarsened microstructures after twenty hours of annealing. Liu et al.
examined coarsening of camphene crystals in freeze casting of bioactive glass to
obtain a controllable pore diameter, ranging from 15 µm to 160 µm. However, both
were restricted to pore size measurement and a qualitative image analysis. Chapter 5 reports a quantitative study of morphological evolution of dendritic pores by
coarsening using a tomography-based analysis.
Rheology of solution
Rheological property of solution is another parameter to control freeze casting, and
the polymer solution viscosity can be controlled in a simple way. Xue et al. changed
the rheological properties of the solution by increasing the cross-linking agent
for polycarbosilane, and demonstrated improved mechanical robustness by tuning
cross-linking agent concentration. It is also possible to change the rheological
properties by changing the molecular weight of preceramic polymer by thermal
curing, which will be discussed in Chapter 8. Typically, a cross-linking agent
is introduced prior to the solidification. As a result, viscosity changes over time
and the solution eventually gels, which limits the solidification time. However,
recent work by Obmann et al. demonstrated that photopolymerization is possible at
temperatures below -10 ◦ C after the solidification [40]. This work not only offers
different processing avenues for solution-based freeze casting, but also provides
more flexibility in solidification time. Because the cross-linking step can be carried
out after the solidification, longer solidification is feasible. Longer solidification
time in suspension-based freeze casting poses an issue due to the sedimentation of
the suspended particles. In suspension-based freeze casting, controlling rheological
properties requires additives such as glycerol [41], polyethylene glycol [42], or
gelatin [43].
2.3

Solidification

As the crystal templates the pores in freeze casting, an understanding of the solidification is fundamental. In this section, these solidification principles are reviewed.
2.3.1

Constitutional supercooling and interface instability

The recognized concept for understanding interfacial instability leading to cellular
growth is constitutional supercooling, which was originally conceived by Rutter and
Chalmers [44] to describe the breakdown of the stable planar solid-liquid interface
into cellular morphologies in directional solidification. It was reported that cellular

Figure 2.10: A schematic of the constitutional gradient during solidification and the
liquidus temperature gradient ahead of the freezing front. The applied temperature
𝑑𝑇𝑞 (𝑧)
) 𝑧=0 , is lower than the liquidus temperature gradient, 𝐺 𝑐 =
gradient, 𝐺 = ( 𝑑𝑧
𝑑𝑇𝐿 (𝑧)
( 𝑑𝑧 ) 𝑧=0 , resulting in the constitutional supercooling (cross-hatched region) [24]
Credit:W. Kurz and D. J. Fisher, Fundamentals of solidification, Third edition, Trans
Tech Publication, 1992.
crystal growth resulted from the instability of the interface, which was caused by
the concentration gradient in the liquid ahead of the freezing front (Figure 2.10).
As the solid-liquid interface advances, the solute is segregated from the interface
and the segregation of solutes creates a concentration gradient. This concentration
gradient in the liquid phase can be converted to the liquidus temperature gradient,
𝑑𝑇𝑞 (𝑧)
) 𝑧=0 ,
using the phase diagram. If the temperature gradient in the melt, 𝐺 = ( 𝑑𝑧
𝑑𝑇𝐿 (𝑧)
is lower than the liquidus temperature gradient, 𝐺 𝑐 = ( 𝑑𝑧 ) 𝑧=0 , at the solid-liquid
interface, there exists a zone of constitutional supercooling as shown in the hatched
region (Figure 2.10). Later, this was mathematically formulated by Tiller, Rutter,
and Jackson using the steady-state diffusion equation [18]. The condition for stable
planar front can be expressed by the following equation,

𝐺=

𝑚𝐶0 𝑣 𝑘 0 − 1
𝑘0

where G is the temperature gradient, m is the slope of the liquidus, 𝐶0 is initial
concentration of solute in liquid, v is freezing front velocity, D is diffusivity of

16
solute in liquid, and 𝑘 0 is the equilibrium distribution coefficient (𝑘 0 ≡ 𝐶𝑆 /𝐶 𝐿 ).
Chapters 3 and 6 explain the strategies based upon this concept that tailor the pore
morphologies.
Although the theory of the constitutional supercooling could successfully show the
conditions for the breakdown of the planar freezing front, the drawbacks of this
analysis include the following: (i) it does not take the surface tension of the interface into account, (ii) it cannot predict the size scale of the morphologies after
the breakdown. To overcome these drawbacks, Mullins and Sekerka considered a
case where the interface is slightly disturbed and analyzed the development of this
perturbation [45]. In this analysis, a sinusoidal perturbation, 𝛿, is introduced into
the planar front. These perturbations can be insoluble particles, temperature fluctuations, or grain boundaries in the melts. The equation known as Mullins-Sekerka
instability criterion is expressed as:
𝑉𝜔{−2𝑇𝑀 Γ𝜔2 [𝜔∗ −(𝑉/𝐷) (1−𝑘 0 )]−(𝑔 0 +𝑔) [𝜔∗ −(𝑉/𝐷) (1−𝑘 0 )]+2𝑚𝐺 𝑐 [𝜔∗ −(𝑉/𝐷)]}
𝛿¤
(𝑔 0 −𝑔) [𝜔∗ −(𝑉/𝐷) (1−𝑘 0 )]+2𝜔𝑚𝐺 𝑐

and
(𝜅 𝑆 𝐺 0 + 𝜅 𝐿 𝐺)
𝜅𝑆 + 𝜅 𝐿
𝑔0 − 𝑔 =
(𝜅 𝑆 𝐺 0 − 𝜅 𝐿 𝐺)
𝜅𝑆 + 𝜅 𝐿
𝑉 2
𝜔∗ =
+ [(
) + 𝜔2 ] 2
2𝐷
2𝐷
𝑔0 + 𝑔 =

where 𝛿 is the amplitude of the perturbation, 𝜔 is a frequency of a sinusoidal
perturbation, 𝑉 is the freezing front velocity, 𝑇𝑀 is the melting temperature, Γ
is a capillary constant which involves the solid-liquid interfacial free energy and
the latent heat, D is the diffusion coefficient of the solute in the liquid, 𝑘 0 is
the equilibrium distribution coefficient, 𝐺 𝑐 is the solute concentration gradient at
the interface, and 𝜅 𝑆 and 𝜅 𝐿 are thermal conductivities of the solid and liquid,
¤ is positive, the perturbation will grow
respectively. This analysis shows that if 𝛿/𝛿
(Figure 2.11a). If negative, it will disappear (Figure 2.11b).
The Mullins-Sekerka instability criterion can be used to estimate the size-scale of the
¤ for Al-2wt.%Cu
growing interfaces for any particular systems. In Figure 2.12, 𝛿/𝛿
alloy under the specified solidification condition (V = 0.1mm/s, G = 10 K/mm) is
plotted as a function of wavelength, 𝜆 = 2𝜋/𝜔 [24]. The wavelength range within

Figure 2.11: Illustrations showing (a) small perturbation grow and (b) small perturbation disappear [24].
Credit:W. Kurz and D. J. Fisher, Fundamentals of solidification, Third edition, Trans
Tech Publication, 1992.
¤ being positive gives a rough estimate for the perturbed morphology. Kurtz and
𝛿/𝛿
Fisher used this wavelength to further estimate the dendrite tip radius [46].
2.3.2

Crystal morphology

Each material exhibits its characteristic solidified morphology. Figures 2.13 a and b
show two different crystal morphologies: non-faceted crystals and faceted crystals.
This difference in crystal morphologies can be explained by the atomic attachment
kinetics. The non-faceted crystals, also called as dendrites, are often observed in
metals. The atomic attachment kinetics are independent of crystallographic planes
so they are solidified with an atomically rough solid-liquid interface, where the
atom can easily attach to the solid phase. In contrast, faceted crystals, a morphology
seen in intermetallic compounds or minerals, have a preferential atomic attachment,
depending on crystallographic planes. Hence, the interfaces tend to be flat with
faceted morphologies. The analysis of the equilibrium configuration at the solidliquid interface was performed with a two-layer interface model proposed by Jackson

Figure 2.12: A schematic showing the stability of the interface as a function of
wavelength for Al-2wt.%Cu [24].
Credit:W. Kurz and D. J. Fisher, Fundamentals of solidification, Third edition, Trans
Tech Publication, 1992.

Figure 2.13: Images showing (a) non-faceted crystals (dendrites) and (b) faceted
crystals [47]. This figure is reproduced with permission.

[47]. Although this analysis considers only the first nearest neighbors at the solidliquid interfaces, it serves as a useful guide to predict the crystal morphologies. In
this model, a parameter now known as the Jackson 𝛼 factor, was proposed:

𝛼=

𝜂 𝐿
𝑍 𝑘 𝐵𝑇𝑚

19
where 𝜂 is the number of nearest neighbors adjacent to an atom in the plane of the
interface, Z is the total number of nearest neighbors in the crystal, L is the latent
heat of fusion, 𝑘 𝑏 is the Boltzmann constant, and 𝑇𝑚 is the material’s melting point.
The Jackson 𝛼 factor assesses the change in free energy of the adatoms to join the
solid phase. Consequently, the location where the maximum or minimum of the
free energy curve occurs changes (Figure 2.14). Above the critical value of 𝛼 = 2,
the free energy curve finds its minimum at either near 𝜉 = 0 or 𝜉 = 1. The physical
meaning of this is that the interface is occupied by few adatoms or fully occupied
with few vacancies, indicating that the interface is atomistically flat. In case of
𝛼 < 2, the minimum of the free energy curve is at 𝜉 = 0.5. There are almost
equal number of adatoms and vacancies, indicating the interface is atomistically
rough. The Jackson 𝛼 factor is thereby a guide to judge whether a crystal exhibits
rough or flat interfaces at specific crystallographic orientations of the material. In
most cases, the crystallographic term of the Jackson 𝛼 factor, 𝜂/𝑍, is challenging
to know for all the crystallographic planes, but the thermodynamic term, 𝐿/𝑘 𝐵𝑇𝑚
is relatively easy to estimate. Since the crystallographic term is always less than
one, but greater than 1/4, the thermodynamic term is used to estimate if the 𝛼 factor
is less than 2. Figure 2.15 shows the freezing microstructures of different solvents
along with the value of the thermodynamic term [18]. As the thermodynamic term
is increased, the anisotropy of the frozen crystals increases. The frozen crystals
turn from round-shaped seaweed-like to dendritic, then to prismatic, and finally to
lamellar.
2.4

Polymer-derived ceramics (PDC)

A preceramic polymer is a precursor which can be converted into ceramics by
pyrolysis. The resulting ceramics are known as polymer-derived ceramics (PDCs).
PDCs have brought a technological breakthrough in ceramic processing by achieving
the development of ceramic fibers and ceramic coatings with impressive hightemperature properties such as resistance to crystallization and creep. A brief
introduction of PDCs as well as their structures and properties are highlighted in
this section.
2.4.1

Overview of polymer-derived ceramics

The first notable achievement for PDCs was done by Yajima et al. who developed
silicon carbide fibers with high tensile strength [49], which eventually resulted in
Nicalon fibers manufactured by Nippon Carbon. While traditional ceramic process-

Figure 2.14: Free energy curve as a function of adatom coverage with different
values of the Jackson 𝛼 factor [48]. This figure is reproduced with permission.

Figure 2.15: Optical micrographs showing freezing microstructures of preceramic
polymer solutions with (a) cyclooctane, (b) cyclohexane, (c) tert-Butanol, and (d)
dimethyl carbonate [18].
ing typically requires sintering at temperatures higher than ∼1700 ◦ C with sintering
additives, pyrolysis can be conducted at ∼1300 ◦ C or lower. In addition, preceramic polymers can be processed by polymer-forming techniques such as injection

21
molding [50], fiber drawing [51], extrusion [52], or stereolithography [53]. Shaping of preceramic polymers before pyrolysis can eliminate machining, mitigating
or avoiding tool wear or material brittle fracture. Due to the significant shrinkage,
by-product gas release, and formation of porosity during the pyrolysis, dimensions
of PDC components are limited to a few hundred micrometers or smaller (fibers,
coating, etc.), otherwise they are prone to cracking. Although porosity has been
viewed as a source of flaws in ceramics, producing a porous structure with wall
thickness within the length scale via freeze casting opens up new opportunities for
applications of PDCs.
Preceramic polymer type

Figure 2.16: Si-based preceramic polymer with different backbones [54]. This
figure is reproduced with permission.
A variety of preceramic polymers is available, which result in binary compounds
such as Si3 N4 , SiC, and BN, ternary compounds such as SiCN and SiOC, and even
quartenary compounds such as SiBCN and SiAlCO. Si-based preceramic polymers

22
have been studied extensively as promising precursor ceramic applications beyond
fibers, such as ceramic heating elements and ceramic brake disk [55]. As shown
by Figure 2.16 [54], the different compositions of the backbone resulted in various
classes of ceramics. The functional groups (denoted as "R" in Figure 2.16) control
the carbon content in the resulting ceramics [56], which affect properties such as
thermal and mechanical properties (resistance to crystallization and creep). Moreover, the addition of metal elements such as Al [57] and Ti [58] is possible. It
was demonstrated that introducing Al improves high temperature stability and the
solid remains crack-free at 1400 ◦ C and up to 1700 ◦ C [59]. In addition to Si-based
preceremic polymers, other preceramic polymer systems have been studied such
as B-based [60] and Al-based polymers [61]. Although the ceramic yield from
preceramic polymers is still limited compared to powder routes, there is continued
work to expand the material space. Recent achievements include the development
of polymer-derived refractory ceramics by Unites States Naval Research Laboratory
(NRL). This is particularly attractive as metal carbide powders are, for example,
produced by carbothermal reduction at 2000 ◦ C, followed by high pressure sintering
at 2000 ◦ C under over 1 GPa [62], which impose challenges in cost and scalability.
To address this challenge, NRL developed 1,2,4,5 tetrakis(phenylethynyl)benzene
(TPEB), which acts as a carbon source for carbothermal reduction [62], and demonstrated a novel polymer-derived boron carbide (B4 C) monolith. This process can
be used to fabricate components as large as 15 × 15 cm2 panels which are over 1
cm thick. With preceramic polymers, the refractory ceramics can be processed at
much lower temperatures and shorter time than powder processing. NRL further
demonstrated the technology with other refractory ceramics such as titanium carbide
[63], tungsten carbide [64], and tantalum carbide [65].
Processing of preceramic polymer
PDCs are manufactured or fabricated from preceramic polymers by the following
three steps [66]:
• Preceramic polymer synthesis from monomer or oligomer precursor with
desirable rheological properties for shaping.
• Shaping of preceramic polymer by plastic forming methods such as injection
molding, extrusion, fiber drawing, etc., and thermal curing at 150◦ C - 250◦ C
to set structural integrity for pyrolysis.

23
• Pyrolysis in an inert atmosphere (Ar or N2 ) at temperature ∼1300 ◦ C or lower.
A preceramic polymer can be either a liquid or a solid at ambient temperatures, and
must have functional groups so that it can form thermoset and retain its shape during
the pyrolysis. The cross-liking can be undertaken by thermal cross-liking typically
below 200◦ C. In case of polysiloxane, this would be the condensation of the silanol
group (Si-OH). A catalyst can be also added to facilitate the cross-linking process.
The degree of cross-linking needs to be carefully controlled for desired rheological
properties such that plastic forming techniques can be employed. During pyrolysis,
the by-product gas will be released [56]. For the pyrolysis of polysiloxane, from
100◦ C to 420◦ C, thermal cross-linking gas such as water and alcohol as well as
oligomers will be released. From about 420◦ C to 850◦ C, the decomposition process
will start by releasing hydrocarbons such as methane, and result in amorphous
ceramics, but can be crystallized by heating at high temperatures [67].
2.4.2

Structures and properties of silicon oxycarbide (SiOC)

Figure 2.17: A model for the nanodomains in SiOC [68]. This colored image was
taken from [54]. This figure is reproduced with permission.
Saha et al. proposed a model of the SiOC as shown in Figure 2.17, showing silica

24
nanodomains encased with mixed SiOC bonds and a sp2 carbon. Pyrolysis leaves
a significant amount of carbon from organics which is insoluble in silica. Hence,
it was postulated that the carbon is rejected as the silica nanodomains coarsen
during the pyrolysis. As a result, it forms continuous SiOC mixed bonds and a
sp2 carbon network, which inhibits further growth of silica domains and creation of
silica nanodomains. The presence of these nanodomains were confirmed by using
solid-state NMR, micro-Raman, SAXS, XRD, and HRTEM [69].
This microstructure results in unique electronic, magnetic, optical, thermal, and
mechanical properties. A few properties are highlighted here. First, Si-based
ceramics are known to have resistance to crystallization up to 1400◦ C. The nucleation
of silica crystallites require embryos of a critical size. However, due to the presence
of SiOC mixed bonds and the sp2 carbon network acting as diffusion obstruction,
crystallization is prohibited. Varga et al. also explains this high temperature stability
from a thermodynamic standpoint and attributes the energetics of domain walls,
which is constituted from sp2 carbon and mixed SiOC bonds, as the source of
stabilizing amorphous phase [70]. PDCs are also known to possess remarkable
creep resistance due to their high viscosity, two order of magnitude higher than
vitreous silica at 1400◦ C [71]; viscosity increases with increasing carbon content
[72]. This creep resistance can also be attributed to the presence of excess carbon.
Because the sp2 carbon creates scaffolding, which enables load transfer from the
silica phase to sp2 carbon network, it would be difficult to deform viscously [73].
Lastly, PDCs also have chemical stability. Soraru et al. studied the chemical
durability of SiOC with varied carbon content in alkaline and HF. SiOC exhibits
greater chemical durability than silica glass due to the Si-C bonds and the presence
of the carbon network, which impedes the local transport of the reactant [74].
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31
Chapter 3

GRADIENT-CONTROLLED FREEZE CASTING
3.1

Introduction

Because the solidifying crystals template pores, fundamental solidification principles can guide control of pore space in freeze casting. Although the fundamental
understanding of the solidification in freeze casting is still incomplete [1], some studies the highlighted importance of solidification theory as a useful guide for tailoring
pore size and morphologies. Miller et al. reported a study of freeze casting which
predicts freezing front velocity from cooling conditions of suspension. The work
shows agreement between solidification theory predictions and dendrite or lamellar
spacing [2]. Naviroj et al. explored different pore morphologies freeze-cast with
various solvents, and correlated the resulting structures to the Jackson α-factor, a
parameter representing crystal’s anisotropy. In-situ imaging by confocal microscopy
of particles being rejected at the freezing front by Dedovets et al. illustrated the
importance of controlling the temperature gradient and growth rate to influence
crystal morphology [3]. These demonstrations motivate the further exploration of
fundamental solidification principles to manipulate freeze-cast structures.
This study focuses on another aspect of solidification, constitutional supercooling.
The notion of constitutional supercooling was originally conceived by Rutter and
Chalmers [4], and reported that cellular crystal growth resulted from the instability
of interface, which was caused by the concentration gradient in the liquid ahead of
the freezing front. Later, this was mathematically formulated by Tiller, Rutter and
Jackson using the steady-state diffusion equation [5]. The condition for a stable
planar front can be expressed by the following equation,
𝐺 𝑘 0 − 1 𝑚𝐶0
𝑘0

(3.1)

where G is the temperature gradient, V is the freezing front velocity, 𝑘 0 (= CS /CL )
is the equilibrium distribution coefficient, m is the slope of liquidus, 𝐶0 is the initial
concentration of solute in liquid, and D is the diffusion coefficient of the solute in
the liquid. Eqn. 3.1 defines the critical ratio, G/V, to ensure that no constitutional
supercooling occurs [15][6]. As shown in a stability-microstructure map (Figure

32
3.1(a)), Eqn. 3.1 represents the boundary between cellular crystals and a stable
planar front. Dendrites form when G/V is far away from this condition while cells
form in the narrow region close to planar front in the stability-microstructure map
(illustrations of cells and dendrites are shown in Fig. 1(b)). Hence, controlling
the degree of constitutional supercooling by G and V allows crystal morphology
to be adjusted, and therefore, serve as a useful guide to achieve the desired porous
microstructure. One example which uses this constitutional supercooling theory to
modify crystal morphology is additive manufacturing of metals [7, 8] as equiaxed
dendritic grains are favored over columnar grains to avoid hot cracking.

Figure 3.1: Stability-microstructure map showing the independent control of freezing front velocity and temperature gradient allows one to change crystal morphology
(a). Modified from ref. [9]. (b) Illustration of cells and dendrites.
Although numerous freeze-casting investigations have demonstrated freezing-front
velocity (V) control, there is a paucity of studies which focus on the effect of
temperature gradient (G) in freeze-cast structures. Zheng et al. showed independent
control of V and G to create axially homogeneous pore structure [19][10]. The first
study to achieve cellular pores based on the stability-microstructure map was done by
Zeng et al. [11], which reported the morphological transition of columnar dendrites
to cells as V is decreased at constant G and demonstrated improved shape-memory
properties. However, in suspension-based freeze casting, it is challenging to analyze
the morphological change of dendritic structures in detail as the suspending particles
destabilize and break down the dendrite tips, leading to less anisotropy in the porous

33
structure [12]. As a result, the pore size distribution exhibits a unimodal distribution
instead of bimodal distribution, characteristic of dendritic structures.
In this study, freeze-cast structures are created through control of G and V based on
the underlying theory of constitutional supercooling, with the goal of manipulating
pore size and pore morphology. We use preceramic polymer solutions as the
freeze-casting medium since distinct dendritic structures with bimodal pore size
distributions can be achieved via this route. Hence, for the first time, a more
quantitative analysis of the effects of G and V are possible. Dendritic structural
evolution as a result of a change in V at fixed G and a change in G at fixed V is
investigated using image analysis and pore size measurement by mercury intrusion
porosimetry, establishing an effective tool for pore morphology control.
3.2
3.2.1

Experimental methods
Preceramic polymer solution

A polymer solution was prepared by dissolving polymethylsiloxane preceramic
polymer (Silres®MK Powder, Wacker Chemie, Munich, Germany), in cyclohexane
(Sigma-Aldrich, St. Louis, MO, USA) with the polymer concentration of 10 wt.%
(6.5 vol.%) and 20 wt.% (13 vol.%). Such concentrations result in porosities
of 89 % and 78 %, respectively. A solution with dioxane (Sigma-Aldrich, St.
Louis, MO, USA) is also prepared with the same volume concentration as 20 wt.%
cyclohexane solution. A cross-linking agent (Geniosil®GF 91, Wacker Chemie,
Munich, Germany) was added at a concentration of around 1 wt.% to all polymer
solutions and stirred for 5 min. Before freezing, the solutions were degassed for 10
min to avoid air bubbles during freezing.
3.2.2

Freeze casting

The solution was poured into the glass mold (h = 12.5 mm or 20 mm, Ø = 24 mm)
and the mold was placed on a thermoelectric plate which is continuously cooled
by a circulating refrigerated silicone oil. A second thermoelectric plate was placed
on top of the mold to control the temperature of the top side (Figure 3.2). Due
to shrinkage during solidification, the copper plate was designed to be inserted 5
mm into the glass mold, creating a reservoir for the solution to avoid detachment
of the solution from the top cold finger. The temperature of both thermoelectric
plates was controlled by a PID controller. Two solidification parameters, V and G
were measured from images taken by camera with the intervalometer. Images were
taken at different intervals ranging from 30 seconds to 10 minutes, depending on V.

Figure 3.2: A photograph showing gradient-controlled freeze-casting setup.

Images were analyzed using ImageJ (National Institutes of Health) to determine V.
G was also determined from images by the following equation:

𝐺=

𝑇𝑡𝑜 𝑝 − 𝑇 𝑓 𝑟𝑜𝑛𝑡

(3.2)

where Ttop , Tfront , and d are the temperature of the top thermoelectric plate, the
temperature of freezing front, and the distance between the copper plate and freezing
front, respectively. The temperature of the freezing front is assumed to be the
liquidus temperature of the solution, the value of which is taken from the study by
Naviroj [13]. A mold with a different height was chosen to alter G. The molds with
12.5 mm and 20 mm heights result roughly in temperature gradients of 5.0 K/mm
and 2.5 K/mm for cyclohexane solution, respectively. Appendix B summarizes all
of the examined freezing front velocities and temperature gradients. For reference,
a sample with no prescribed temperature gradient was made. In this control sample,
the top thermoelectric was removed and the top surface was kept open to the ambient
atmosphere, as is performed in conventional suspension- or solution-based freeze
casting. See Appendix C for a detailed comparison between conventional freezing
and gradient-controlled freezing.
After freezing, samples were placed into a freeze drier (VirTis AdVantage 2.0, SP
Scientific, Warminster, PA, USA), where the solvents were completely sublimated.

35
After sublimation, the green bodies were pyrolyzed at 1100 ◦ C in argon for 4 hours
with a ramp rate of 2 ◦ C/min to convert polymethylsiloxane into silicon oxycarbide
(SiOC). Sample porosity was determined using the Archimedes method.
3.2.3

Characterization

The porous microstructure was imaged using scanning electron microscopy (SEM;
Zeiss 1550VP, Carl Zeiss AG, Oberkochen, Germany) in two different directions:
the transverse direction (a cross-section perpendicular to freezing direction) and
the longitudinal direction (a cross-section parallel to freezing direction). The pore
size distribution was measured by mercury intrusion porosimetry (MIP; Auto Pore
IV, Micromeritics, Norcross, GA, USA). For MIP, the samples were core-drilled
to a diameter of ∼13 mm to remove edges. Specimens for imaging and pore size
measurements were sectioned from locations where V and G remain reasonably
constant.
3.3
3.3.1

Results
Temperature gradient effect

Figure 3.3 shows SEM images and pore size distributions of the sample freezecast from 20 wt.% polymer solution with cyclohexane. Figure 3.3a is the SEM
image showing the transverse direction, a cross-section perpendicular to the freezing
direction, of the control sample frozen under V = 15 µm/s. It shows the characteristic
feature of the dendritic pores having the primary pores, secondary pores, and even
tertiary pores, also reported by Naviroj et al. [12]. Figure 3.3b shows a magnified
image of dendritic pores, where the black contrast outlines the primary pore and
four secondary branches. In some instances, tertiary pores also branch out from
the secondary pores (tertiary pores are indicated by yellow arrows in Figure 3.3b).
Figure 3.3c shows the analogous images of the sample frozen with V = 15 µm/s
and G = 2.6 K/mm (gradient-controlled sample). In contrast to Figure 3.3a, the
growth of the secondary and tertiary pores is limited, leading to smaller dendritic
pore spacing and higher primary pore concentrations (a tertiary pore is indicated
by yellow arrows in Figure 3.3d). Figure 3.3e displays the pore size distribution
data from MIP; both samples show bimodal peaks showing large primary pores and
small secondary pores compared to the control. The primary pore size for both
samples is approximately 20.3 µm. In contrast, the peak secondary pore size differs
by only 1 µm (13.7 µm from the control sample and 12.7 µm from the gradientcontrolled sample). More compelling, the volume associated with primary pores

Figure 3.3: SEM images showing a control sample at (a) low and (b) high magnifications and a gradient-controlled sample at (c) low and (d) high magnifications
in transverse direction. Yellow arrows in low magnification and high magnification images indicate primary pores and tertiary pores, respectively. (e) Pore size
distribution of control sample and gradient-controlled sample.

and secondary pores has changed significantly as shown by the change in peak
height. Summing the incremental intrusion of each type of pores gives 6 vol.%
for primary pores in the control sample whereas 28 vol.% for primary pores in the
gradient-controlled sample, in agreement with SEM images, an increase of more
than four-fold by applying G.

37
3.3.2

Change in freezing front velocity and temperature gradient

Figures 3.4 a-e show SEM images of samples freeze-cast from 20 wt.% polymer
solution in transverse direction and longitudinal direction. Figures 3.4a-d show SEM
images comparing freeze-cast structures frozen under different velocities (17 µm/s
sample and 1.8 µm/s) at nearly the same temperature gradient. This corresponds to
moving from the orange to blue marker horizontally in the stability microstructure
map in Figure 3.4g. When V is decreased about an order of magnitude, pores are
still dendritic, but both primary and secondary pore sizes increase. This is further
confirmed by pore size distributions in Figure 3.4g. The calculated primary pore
volume fractions increased from 24 to 30 vol.% by decreasing V. Figures 3.4e and
f show SEM images of freeze-cast structures frozen under V of 1.5 µm/s and G of
5.0 K/mm. The comparison between Figures 3.4c and e reveal the effect of G at
constant V, which corresponds to moving from the blue to green marker vertically
in Figure 3.4g. The transverse images show the primary pore spacing, λd (indicated
by the yellow arrows), decreased as G is increased. Pore size distributions show
that both structures have similar primary pore sizes although secondary pores size
of higher G is smaller. In addition, as shown in pore size distribution, primary pore
volume fraction increased by increasing G. The calculated primary pore volume is
increased from 30 to 48 vol.% when G is increased at similar V.
3.3.3

Change in polymer concentration

The preceramic polymer concentration change from 20 wt. % to 10 wt.% at similar
V and G is also investigated. SEM images and pore size distributions are shown in
Figure 3.5. SEM images show that 10 wt.% preceramic polymer solution yielded
dendritic structures with larger pore size, further confirmed by pore size distribution
data. Both primary and secondary pore sizes are larger from 10 wt.% polymer
solutions even though both samples are frozen under similar V and G. Moreover, the
primary pore volume fraction increased from 30 to 51 vol.% by decreasing polymer
concentration from 20 wt.% to 10 wt.%, indicating that more than half of the pore
volume is attributed to primary pores from the 10 wt.% polymer solution.
3.3.4

Morphological change from dendrites to cells

The stability-microstructure map suggests that cellular growth is possible with low V
and high G. In order to achieve cellular pores, V was significantly reduced to 0.6 µm
/s while G was fixed at 5 K/mm. Slower velocities for this 20 wt.% polymer solution
would result in the gelation of the solution before freezing was complete. As shown

Figure 3.4: SEM images of the sample frozen with V = 17 µm/s and G = 2.2 K/mm in
(a) transverse and (b) longitudinal direction, the sample frozen with V = 1.8 µm/s and
G = 2.4 K/mm in (c) transverse and (d) longitudinal direction, and the sample frozen
with V = 1.5 µm/s and G = 5.0 K/mm in (e) transverse and (f) longitudinal direction.
(g) A stability-microstructure map showing examined conditions by colored marker.
(h) Pore size distributions of corresponding samples.

Figure 3.5: SEM images showing dendritic structure from (a) 20 wt.% solution and
(b) 10 wt.% solution in transverse direction. (c) Corresponding pore size distribution
from MIP.
in the Figures 3.6a and b, although the transverse image would indicate cellular
or honeycomb-like morphologies, the longitudinal image shows the presence of
secondary pores and the cellular morphologies limited to only a portion of the solid.
When the polymer concentration is reduced to 10 wt.% and frozen under similar V
and G, SEM images reveal a larger portion of cellular morphologies (Figures 3.6c
and d). In contrast, dioxane, also investigated as a solvent in this study and known to
yield dendritic structures [12], has a significantly longer gelation time of the solution
(4 to 5 days) and higher boiling point. Hence, the solution can be frozen with slower
velocities and higher temperature gradient. Figures 3.6e and f show the images of
the sample freeze cast with dioxane where V = ∼0.2 µm/s and G = 12 K/mm. The
transverse and longitudinal images show the complete cellular morphologies.
3.4

Discussions

3.4.1

Pore size control

Because this setup allows independent control of V and G, the effects of the two
parameters on pore structure can be analyzed separately. Figure 3.7a shows the peak
values1 of primary pores in pore size distribution as a function of V for two values
of G in 20 wt.% polymer solution. The primary pore size decreases with increasing
1 Some pore size distributions contain outliers which are probably due to the fracture of samples

during the intrusion of mercury. These pore size distribution data were fitted using the software,
Fityk [14], and the peaks of fitted curves were plotted in Figures 3.7a and b.

Figure 3.6: SEM images showing a SiOC from cyclohexane crystals (20 wt.%
polymer solution) in (a) transverse and (b) longitudinal directions, from cyclohexane
crystals (10 wt.% polymer solution) in (c) transverse and (d) longitudinal directions,
and from dioxane crystals in (e) transverse and (f) longitudinal directions.

V. This is expected since freezing front velocity affects the dendrite size, and is
consistent with other studies on dendritic pores [2, 13]. It also shows that primary
pore sizes from two values of G, 2.5 K/mm and 5 K/mm, follow the same trend,
indicating that the primary pore sizes do not strongly depend on G. Alternatively,
secondary pore sizes can be described using a model for secondary arm spacing.
While secondary arm spacing measures center-to-center spacing of neighboring
secondary arms, which includes the secondary arm diameter and the interdendritic
phase, the secondary pore size reported here is a measure solely of secondary arm
diameter. Since the polymer concentration remains the same, it was assumed that the
secondary arm diameter and the interdendritic phase increase their sizes at the same
rate. Hence, secondary arm spacing model can be applied to analyze secondary

Figure 3.7: Plots of (a) Primary pore size as a function of V with different G and
(b) secondary pore size as a function of cooling rate.

pore size. The secondary arm spacing is known to depend on both V and G based
on the model by Feurer and Wunderlin [15]:
𝐶𝑚
 1/3
𝐷Γ ln 𝐶𝐿0
 𝑡 1/3
𝜆 2 = 5.5 
 𝑚(1 − 𝑘 0 )(𝐶0 − 𝐶 𝐿 ) 

𝑡𝑓 =

Δ𝑇 0
𝐺𝑉

(3.3)

(3.4)

where CL m is often equal to the eutectic composition, tf is the local solidification
time, and ΔT’ is the difference between the temperature at the tip of the dendrites
and the melting point of the last interdendritic liquid. The local solidification time is
the time the liquid phase and solid phase coexist at a fixed point, and is determined
by dividing ΔT’ by cooling rate, VG. Hence, the secondary arm spacing depends
on the cube root of 1/VG. Figure 3.7b shows the peak value of secondary pore size
in the pore size distribution as a function of cooling rate, VG. The exponent from
curve fitting is -0.34, in excellent agreement with the model.
One other dimension in dendrites, which has been studied in solidification of alloy
extensively but rarely studied in freeze casting, is a dendrite spacing, 𝜆 1 . Kurz and
Fisher reported a model for the dendrite spacing in alloys [16]:

∗ 1/2

𝜆 1 = 4.3(𝑇𝐿 − 𝑇𝑒 − Δ𝑇 )

𝐷Γ
(𝑇𝐿 − 𝑇𝑆 )𝑘 0

1/4

𝑉 −1/4 𝐺 −1/2

(3.5)

where TL is liquidus temperature, TS is solidus temperature, Te is eutectic temperature, Δ𝑇 ∗ is tip undercooling, D is diffusion coefficient of solute, Γ is the Gibbs
Thomson parameter, and k0 is the equilibrium distribution coefficient. This model
predicts that 𝜆 1 decreases when G is increased at constant V, assuming all the other
parameters remain constant. This is consistent with observations made in Figures
3.3a and c, and Figures 3.4c and e, which show that primary pore spacing tends to
decrease with increasing G. Since pore size distribution shows that primary pore
size does not change significantly by increasing G, the results imply that the lengths
of secondary or even tertiary pores decrease, another dimension controlled through
G.
Finally, when the preceramic polymer concentration is decreased from 20 wt.% to
10 wt.% (Figure 3.5), the primary and secondary pore sizes increased. This is also
expected because the preceramic polymer concentration determines the space for
the growing crystals as reported by Naviroj [13]. As a result, lower concentration
yields larger pore sizes since more space is available for crystals to grow. Moreover,
secondary arm spacing also directly depends on C0 in Eqn. 3.3; decrease in C0
would increase the secondary arm spacing, as shown in Eqn. 3.5. Hence, an
increase in both primary and secondary pore sizes is expected with decreasing the
preceramic polymer concentration.
3.4.2

Cellular growth

In addition to manipulation of pore size, independent control of V and G allows
one to explore the stability-microstructure map in detail, especially with an aim to
achieve cellular growth. As a parameter to represent transition from dendritic to
cellular pores, primary pore volume fraction from pore size distribution is calculated
and reported here. As solidification conditions are changed such that the conditions
approach the cellular growth regime, the pore size distribution evolves from bimodal
to unimodal and the primary pore fraction reaches 100%. In the case of decreasing
V at constant G, one would move horizontally in the stability-microstructure map
(Figure 3.4g). As expected, the primary pore volume fraction increased (from 24
vol.% to 30 vol.% for 20% polymer solutions) as the cellular growth regime is
approached. With the increase in G at constant V, one would move up vertically in
the stability-microstructure map. Again, by advancing toward the cellular growth

43
regime, an increase in primary pore fraction is expected, and was demonstrated
experimentally (from 30 vol.% to 48 vol.%) in the same 20% polymer solution. As
shown in Figures 3.4c and e, the pore structure starts to exhibit honeycomb-like
structures when G is increased.
The conditions for stable planar growth front expressed by Eqn.3.1 provide guidance
on how to achieve cellular growth in addition to controlling V and G. For example,
the concentration of the solute, C0 , can be reduced, resulting in a decrease of the
slope of the boundary in the stability-microstructure map. As a result, the critical
conditions for V and G to establish a stable planar front become less stringent, and
the structures tend to have cellular-like morphologies even if V and G are similar.
This explains why reducing preceramic polymer concentration increases primary
pore fraction from 30 to 51 vol.% even though V and G are similar (Figure 3.5c).
Due to the short gelling time of the cyclohexane solution and the boiling point of
cyclohexane (80.7 ◦ C), the slowest V and the highest G examined in this study was
not sufficient to achieve long-range cellular growth (Figure 3.6b). To overcome
this challenge, the preceramic polymer concentration was reduced to 10 wt.% and a
solution was frozen with similar V and G. Although the pore structure still exhibits
secondary pores, Figure 3.6d displays a larger portion of cellular pores. This
demonstrates that the constitutional supercooling theory can be applied successfully
to freeze casting and can be used to change dendritic pores to cellular pores through
optimization of solidification parameters along with solution parameters such as
solute concentration.
3.4.3

Anisotropy of cellular pores

Since the dioxane solution has a much longer gelling time, slower velocities were
able to be examined. As a result, long-range honeycomb-like structures were yielded
from the dioxane solution (Figure 3.6d). Also noteworthy is that the pores from
cyclohexane crystals show more circular shapes, while the pores from dioxane
crystals are elongated. This difference can be attributed to the anisotropic nature of
the crystal growth. Naviroj et al. showed freeze-cast structures from cyclohexane
and dioxane exhibit dendritic structures, but dioxane-derived dendritic structures
exhibited a linear and two-dimensional configuration due to its higher Jackson α
factor (1.16 for cyclohexane and 5.21 for dioxane) [12]. While the planar freezing
front follows the direction of the temperature gradient, primary dendrites and the
dendritic arms grow along preferred crystallographic directions. Cells grow under

44
the condition close to the limit of the constitutional supercooling of the planar
interface, and are intermediate morphology of the plane and dendrites [15]. Thus, the
cells are still expected to exhibit some crystallographic features, and the anisotropic
honeycomb pore morphology from dioxane solution is expected.
3.5

Conclusion

Directional solidification with controlled freezing front velocity and temperature
gradient was conducted in solution-based freeze casting, and the relationship between solidification parameters and pore structures was investigated. Solidification
theory explains dendritic pore size dependence on solidification parameters well,
and constitutional supercooling theory can be successfully used to control pore
morphologies.
While the freezing front velocity is the major solidification parameter to control
primary pore size, temperature gradient does not significantly change the primary
pore size in the temperature gradient range between 2.5 K/mm and 5 K/mm. Alternatively, secondary pores are determined by the cooling rate, the product of
temperature gradient and freezing front velocity, and the experimental data in this
study agree well with the theoretical models. The benefit of controlling temperature
gradient and freezing front velocity is not only to control pore size but also pore
morphology by changing the degree of constitutional supercooling. The cellular or
honeycomb-like structures are observed in systems with cyclohexane by manipulation of freezing front velocity, temperature gradient, and polymer concentration,
although there are still noticeable regions of dendritic pores. In contrast, the honeycomb structures are observed in systems with dioxane as the solvent, which gelled
sufficiently slowly and has high a boiling point so as to permit slow freezing front
velocities and high temperature gradients. Finally, similarly to dendritic pores, cellular pores also exhibit crystallographic features of solvent crystals. These concepts
can be extended to other solvents or dispersion media in other freeze-casting systems
for fine-tuning pore networks.
References
[1] Sylvain Deville. “The lure of ice-templating: Recent trends and opportunities
for porous materials”. In: Scripta Materialia 147 (2018), pp. 119–124.
[2] Sarah M Miller, Xianghui Xiao, and Katherine T. Faber. “Freeze-cast alumina
pore networks: Effects of freezing conditions and dispersion medium”. In:
Journal of the European Ceramic Society 35.13 (2015), pp. 3595–3605.

45
[3] Dmytro Dedovets and Sylvain Deville. “Multiphase imaging of freezing particle suspensions by confocal microscopy”. In: Journal of the European Ceramic Society 38.7 (2018), pp. 2687–2693.
[4] JW. Rutter and B. Chalmers. “A prismatic substructure formed during solidification of metals”. In: Canadian Journal of Physics 31.1 (1953), pp. 15–
39.
[5] Kenneth A. Jackson. “Constitutional supercooling surface roughening”. In:
Journal of Crystal Growth 264.4 (2004), pp. 519–529.
[6] W.A. Tiller et al. “The redistribution of solute atoms during the solidification
of metals”. In: Acta metallurgica 1.4 (1953), pp. 428–437.
[7] Fuyao Yan, Wei Xiong, and Eric J. Faierson. “Grain structure control of additively manufactured metallic materials”. In: Materials 10.11 (2017), p. 1260.
[8] John H. Martin et al. “3D printing of high-strength aluminium alloys”. In:
Nature 549.7672 (2017), pp. 365–369.
[9] Martin Eden Glicksman. Principles of solidification: an introduction to modern casting and crystal growth concepts. Springer Science & Business Media,
2010.
[10] Tao Zheng et al. “Implementing continuous freeze-casting by separated control of thermal gradient and solidification rate”. In: International Journal of
Heat and Mass Transfer 133 (2019), pp. 986–993.
[11] Xiaomei Zeng, Noriaki Arai, and Katherine T. Faber. “Robust Cellular ShapeMemory Ceramics via Gradient-Controlled Freeze Casting”. In: Advanced
Engineering Materials 21.12 (2019), p. 1900398.
[12] Maninpat Naviroj, Peter W. Voorhees, and Katherine T. Faber. “Suspensionand solution-based freeze casting for porous ceramics”. In: Journal of Materials Research 32.17 (2017), pp. 3372–3382.
[13] Maninpat Naviroj. “Silicon-based porous ceramics via freeze casting of preceramic polymers”. PhD thesis. Northwestern University, 2017.
[14] Marcin Wojdyr. “Fityk: a general-purpose peak fitting program”. In: Journal
of Applied Crystallography 43.5-1 (2010), pp. 1126–1128.
[15] W. Kurtz and D.J. Fisher. Fundamentals of solidification, Trans Tech. 1998.
[16] W. Kurz and D.J. Fisher. “Dendrite growth at the limit of stability: tip radius
and spacing”. In: Acta Metallurgica 29.1 (1981), pp. 11–20.

46
Chapter 4

FREEZE-CAST HONEYCOMB STRUCTURES VIA
GRAVITY-ENHANCED CONVECTION
4.1

Introduction

Gravity is known to have a significant influence on materials processing. In float
glass processing, a glass ribbon is produced by flowing molten glass on a molten
tin bath [1]. With the help of gravity and surface tension, a flat glass with high
surface quality can be fabricated. While this is an example where the presence
of gravity is advantageous in material processing, some processing is negatively
influenced by the gravitational force. In colloidal suspensions, sedimentation of
particles by gravitational forces must be mitigated with suspension agents [2]. In
another example, directional solidification for producing semiconductor crystals or
nickel-based single crystals, gravity influences the convective flow in the melt. Since
convection in the melt creates defects known as freckles in casting [3], solidification
under microgravity [4] or with magnetic damping [5] has been explored to alleviate
convection.
Motivated by studies in directional solidification of metal alloys with convective
flow [6, 7, 8], this chapter focuses on the effect of the convective flow induced
by gravity during freeze casting. In alloy systems, depending on the density of
the composition in alloys, convective flow may be present during the directional
solidification [9, 10]. For instance, in what has been labeled downward freezing
(in the same direction as the gravitational force) if the solute is denser than the
solvent, the segregated solute creates a denser fluid region ahead of the freezing
front, and enhanced convective flow. In general, however, convection is limited to
regions near the mold-alloy interface in upward freezing. Although the effect of
gravity is actively studied in alloy solidification, a limited number of studies have
examined gravity effects in freeze casting. Scotti et al. investigated the effect of the
freezing direction with respect to the gravity. Microstructures were found to contain
tilted lamellar walls, ice lens formation and radial micro-segregation, caused by
the convective flow [11]. Another study demonstrated that different gravitational
forces (micro-, lunar and Martian gravity) affect the lamellar spacings [12]. Both
however, were based on freeze-casting suspensions. Since suspensions are made up

47
of particles and additives such as binders and dispersants, they are more complex
systems compared to solutions, which contain only solutes and solvents. In this
study, solidification was performed with solutions in both the conventional set-up,
where the gravitational force is opposite in direction than the freezing direction, and
a convection-enhanced set-up, where the gravitational force is in concert with the
freezing direction. The effect of the enhanced convection induced by the gravitational force on the solidification and resulting porous structures are examined. In
particular, the freezing front velocity and temperature gradient are compared between two freezing conditions, and the resulting pore morphologies and pore sizes
are investigated.
4.2

Experimental methods

A preceramic polymer, polymethylsiloxane (Silres®MK Powder, Wacker Chemie,
CH3 -SiO1.5 , Munich, Germany), was dissolved in cyclohexane (Sigma-Aldrich,
St. Louis, MO, USA) at a concentration of 20 wt.%. A cross-linking agent (Geniosil®GF 91, Wacker Chemie, Munich, Germany) was added at a concentration
of 1 wt.% with respect to the solution and stirred for 5 min. The polymer solution
was degassed for 10 min to avoid air bubble formation during freezing. Directional
freezing was conducted using a gradient-controlled freeze-casting setup [13]. The
polymer solution was poured into a cylindrical mold placed on a thermoelectric plate.
A second thermoelectric plate was placed on top of the mold, enabling the control
of freezing front velocity and temperature gradient. The sample was frozen in two
different directions: one against the direction of gravity, referred to as conventional
freezing and one along the direction of gravity, referred as to convection-enhanced
freezing (Figures 4.1ab). The cooling profiles for two thermoelectric plates were
programmed such that solidification took place with a freezing front velocity of
1.8 µm/s and a temperature gradient of 2.5 K/mm in conventional freezing. For
the convection-enhanced freezing, the cooling profiles were switched between the
upper and lower thermoelectric plates so the freezing proceeded from top to the
bottom. Images of freezing front were captured once each minute by a camera with
an intervalometer. Image analysis was performed using ImageJ (National Institutes
of Health) to determine the freezing front velocity. The temperature gradient was
defined by:
𝑇ℎ𝑜𝑡 − 𝑇 𝑓 𝑟𝑜𝑛𝑡
𝐺=
where Tℎ𝑜𝑡 , T 𝑓 𝑟𝑜𝑛𝑡 , and d are the temperature of the thermoelectric plate toward
which the crystals are growing, the temperature at the freezing front, and the distance

48
between points where Tℎ𝑜𝑡 and T 𝑓 𝑟𝑜𝑛𝑡 , respectively. The freezing front was assumed
to be at the liquidus temperature reported by Naviroj [14]. The frozen samples were
placed in a freeze drier to completely remove solvent crystals, and then pyrolyzed
under argon at 1100 ◦ C for 4 hours, resulting in porous silicon oxycarbide (SiOC).
Porosity was measured using the Archimedes’ method. Porous structures were
imaged using scanning electron microscopy (SEM; ZEISS 1550VP, Carl Zeiss
AG, Oberkochen, Germany), and pore sizes were determined by mercury intrusion
porosimetry (MIP; AutoPoreIV, Micromeritics, Norcross, GA, USA).
4.3

Results

Figure 4.1: Freeze-casting setup of (a) conventional freezing and (b) convectionenhanced freezing. (c) Freezing front position as a function of time with images of
(d) the freezing front in conventional freezing, and in convection-enhanced freezing
at (e) t = 45 min and (f) t = 47 min (Red dashed line indicates the freezing front),
and (g) the associated freezing front velocity and temperature gradient as a function
of freezing front position.
Figure 4.1c shows the freezing front position (FFP) from the nucleation face as a
function of time. Conventional freezing shows a nearly linear increase with time
indicative of constant freezing front velocity, and the freezing front is planar (Figure

49
4.1d). In contrast, in convection-enhanced freezing, the FFP gradually increases
for approximately 50 min, followed by the sudden increase in slope, representing a
distinct increase in freezing front velocity, and the planar freezing front is deformed
(Figure 4.1e). Between 45 min and 47 min, the freezing front even retracts, indicative
of re-melting of the frozen solid. This behavior is shown in Figure 4.1e f. Figure 4.1g
illustrates freezing front velocity and temperature gradient as a function of FFP from
the nucleation face. Both remained nearly constant in conventional freezing, while
in convection-enhanced freezing, shows a large variation. Specifically, the freezing
front velocity during its first four millimeters slows to the point of arresting and
then becomes negative, where the freezing front re-melts, with the average freezing
front velocity being ∼0.7 µm/s. After four millimeters, the freezing front velocity
suddenly increases, and exceeds the average freezing front velocity of conventional
freezing.

Figure 4.2: SEM images of conventional freeze-cast samples showing transverse
images at (a) FFP is ∼1.6 mm and (b) FFP is ∼5 mmfrom nucleation face, and
(c) longitudinal image. SEM images of convection-enhanced freeze-cast sample
showing transverse images (d) FFP is ∼1.6 mm and (e) FFP is ∼5 mm from nucleation
face, and (f) longitudinal image. Yellow arrows indicate freezing direction, v, and
gravity direction, g. Red lines in (c) and (f) indicate the nucleation face

Figure 4.2 displays SEM images in transverse directions, a cross-section perpendicular to the freezing direction, and longitudinal directions, a cross-section parallel
to the freezing direction. The transverse images were taken from two different
regions: a cross-section with FFP of ∼1.6 mm and ∼5 mm from the nucleation
point. The longitudinal images of the conventional freeze-cast sample and the
convection-enhanced freeze-cast sample are shown in Figures 4.2c and Figure 4.2f,

50
respectively, with the nucleation face indicated by red lines. In the conventional
freeze-cast samples (Figure 4.2a-c), the pore morphologies are mainly dendritic
structures, which consist of primary pores templated by primary dendrites and secondary pores templated by dendritic secondary arms, and the pore size is relatively
consistent between the two regions (Figure 4.2a and 4.2b). In the longitudinal image,
the first several hundred micrometers consist of a cellular region but the remaining
pores are dendritic. In stark contrast, the convection-enhanced freeze-cast sample
shows cellular pores, which result in honeycomb-like structures, in the slow freezing
region (FFP = ∼1.6 mm, Figure 4.2d) while the fast freezing region (FFP = ∼5 mm)
exhibits dendritic structures (Figure 4.2e). The longitudinal image shows that the
majority of pores (over more than 2 millimeters) are cellular pores. The comparison
of images between Figures 4.2a-c and Figures 4.2d-f exhibits that the pore size in
the convection-enhanced freezing is smaller than in the conventional freezing.

Figure 4.3: Pore size distribution data from (a) nucleation section and (b) middle
section from samples from conventional freezing and convection-enhanced freezing.
Two specimens for MIP were sectioned from each sample and imaged: one near the
nucleation region (FFP is around from 0.8mm to 3.3mm in Figure 4.1g – referred as
nucleation section), and another from the mid-section (FFP is around from 4.5mm to
7mm in Figure 4.1g – referred as middle section). Figure 4.3a compares the pore size
distributions of the specimens sectioned from nucleation section from two freezing
conditions. A bimodal distribution can be observed in the conventional freeze-cast
sample. Larger and smaller pores correspond to primary pores and secondary pores,
respectively. In contrast, the convection-enhanced freeze-cast sample demonstrates
a unimodal distribution, consistent with the SEM images (Figures 4.2d and 4.2f)
showing a honeycomb-like structure. Figure 4.3b shows pore size distributions of

51
specimens sectioned from middle section. In this region, both samples display
bimodal distributions as seen in SEM images (Figures 4.2b and e). Furthermore,
the pore sizes are larger for conventional freeze-cast samples, again consistent with
the SEM observations.
4.4

Discussions

Figure 4.4: Illustrations showing temperature and concentration variation in (a)
conventional freezing and (b) convection-enhanced freezing. (c) An illustration
showing convective flows in liquid phase in convection-enhanced freezing. (d)
Stability-microstructure map. (e) Pore size distribution of conventional freeze-cast
sample frozen under 0.7 µm/s and 4.9 K/mm. (f) Porosity difference between top
section and three sections (middle-top, middle-bottom, and bottom). Three samples
were investigated for each freezing direction.
As shown in Figures 4.1d and 4.1e, the conventional freezing yielded a planar
freezing front while convection-enhanced freezing reveals a protruded freezing
front. The difference can be attributed to the convective flow caused by the density
variation in the liquid phase. This is a result of the concentration gradients of
preceramic polymer and temperature gradient in the liquid phase. Figures 4.4a and
4.4b are schematic of temperature and concentration variation in the liquid region.
In conventional freezing, the preceramic polymer is segregated just ahead of the
freezing front, resulting in a higher concentration of preceramic polymer at the

52
freezing front, decaying into the liquid region. Since the segregated preceramic
polymer (∼1.26 g/cm3 ) is denser than cyclohexane (∼ 0.78 g/cm3 ), the underlying
liquid is heavier than that above. In addition, the region near the freezing front is
colder than the overlying liquid. Since the density of the underlying liquid is higher,
the conventional freezing is a convectively stable configuration (Figure 4.4a). In the
case of convection-enhanced freezing, the density gradient is reversed, therefore,
leading to the convective flow ahead of the freezing front (Figure 4.4b). Under such
conditions, upwelling and downwelling currents are created, shown schematically in
Figure 4.4c. The preceramic polymer is depleted above the upwelling current which
gives rise to a crest, whereas the preceramic polymer is rich above downwelling
current, which produces a trough, similar to the case reported by Drevet et al. [9].
As a result, the freezing front is deformed, consistent with the observation in Figure
4.1e. Remelting of the freezing front (the crest) was also observed (Figure 4.2f). A
possible explanation is that the continuous convective flow transporting heat from
the bottom to the top could remelt the frozen region.
While conventional freezing yielded nearly constant freezing front velocities and
temperature gradients, that was not the case for convection-enhanced freezing. This
discrepancy in freezing front velocity between conventional freezing and convectionenhanced freezing can be explained by constitutional supercooling of the solution.
Due to convective flow, the solute is transported away from the solid-liquid interface
and heat is transported toward the solid liquid interface in convection-enhanced
freezing, decreasing the degree of constitutional supercooling and lowering the
driving force for crystal growth. As a result, the freezing front velocity initially
remains slow (Figure 4.1g). However, as shown in the same figure, as the temperature
gradient continues to decrease, degree of the constitutional supercooling increases.
This leads to a larger driving force for dendritic growth, and the freezing front
velocity increases.
It is important to note that convective instabilities change both pore size and morphology. In conventional freeze-cast sample, the freezing front velocity and temperature gradient remain nearly constant, such that the pore structures and pore
sizes remain similar between nucleation section and middle section (Figures 4.3a
and b). Since pore structures in nucleation section and middle section are dendritic
(Figures 4.2a,b), the pore size distributions are bimodal. It is worth noting that the
primary pore volume fraction in nucleation section, represented by a larger incremental intrusion for primary pores, is larger than the one in middle section. This is

53
likely due to the presence of cellular pores found at the nucleation site and several
hundred micrometers onward, as shown in longitudinal image (Figure 4.2c). These
pores are templated by cellular growth which is expected in the initial stages of the
dendrite growth. As the freezing front advances as a flat interface, the interface is
destabilized by the Mullins-Sekerka instability [15], leading to the transition from
a flat interface to cells and eventually to dendrites. This transition is also observed
in another freeze-casting study by Deville et al. [16]. In contrast, convectionenhanced freezing leads to the long-range cellular regions as shown in Figure 4.2f.
The theory of constitutional supercooling is a useful tool to explain cellular growth
in convection-enhanced freezing. Figure 4.4d shows a stability-microstructure map,
which shows that cellular morphologies are formed only in a narrow region of slow
freezing front velocities and high temperature gradients. Two possible factors are
considered to explain the cellular morphology. The first is slower velocities (0.6∼0.7
µm/s at 1.9 K/mm) as shown in Figure 4.1g. With the slower freezing front velocity,
one would advance to the left in the stability-microstructure map in which cellular
growth are expected (indicated by a orange arrow in Figure 4.4d). This is demonstrated by Zeng et al. who observed cellular pores by decreasing the freezing front
velocity at constant temperature gradient [13]. However, this could not be the sole
factor for the formation of the cells because the sample frozen under 0.7 µm/s and
4.9 K/mm with conventional freezing still exhibit dendritic pores with a bimodal
pore size distribution (Figure 4.4e). A second consideration is the effect of convective flow on constitutional supercooling. Convective flow in convection-enhanced
freezing drives the solute transport away from the freezing front, and this effect on
constitutional supercooling can be described using a stability criterion for a stable
planar freezing front [17]:
𝐺 𝑚𝐶0 1 − 𝑘 0
𝐷 𝐿 𝑘0
where G, 𝑣, m, 𝐶0 , and 𝑘 0 are temperature gradient, freezing front velocity, liquidus slope, concentration of the solution, and equilibrium distribution coefficient,
respectively. This equation defines the critical ratio, G/V, which ensures no constitutional supercooling, and defines the boundary between stable planar front and
cellular growth in Figure 4.4d. Based on this equation, a higher diffusion coefficient
provides a less stringent criterion to achieve a planar front. While it is only diffusion which transports solutes away from the freezing front in conventional freezing,
convection further enhances the transport of solute in convection-enhanced freezing. As a result, the critical ratio for convection-enhanced freezing becomes less
stringent, which makes cellular growth easier to attain, and cells crystallize instead

54
of dendrites. Furthermore, convective flow leads to temperature homogenization
in the liquid phase, which might make the actual temperature gradient larger than
the measured temperature gradient in Figure 4.1g at the solid-liquid interface. This
would also contribute to a reduction in the degree of constitutional supercooling [4]
and cells are more likely to grow.
To assess the length scale of the preceramic polymer transport by convection, the
porosity of the conventional freeze-cast samples and convection-enhanced freezecast samples were measured. Four specimens each with a thickness of ∼1.9 mm
(corresponding to ∼2.5 mm in the liquid phase) were sectioned from each pyrolyzed
sample (top, middle-top, middle-bottom, and bottom). In order to show porosity
variations along the direction of gravity, porosity of top section was subtracted
from porosity of three sections (middle-top, middle-bottom, and bottom) and these
differences are shown in Figure 4.4f. The differences are approximately ±1 %, and
no consistent trend can be observed in either freezing direction. It is likely that the
variation in porosity is due to the measurement error in the Archimedes’ method.
This implies that the distance over which the preceramic polymer is transported by
convection during convection-enhanced freezing is less than 2.5 mm in the solution.
Even though transport of the solute appears to be limited to the near vicinity of
the freezing front rather than throughout the entire liquid phase, constitutional
supercooling is known to take place just ahead of the solid-liquid interface. Hence,
even this local solute transport reduces the degree of the constitutional supercooling,
resulting in morphological and size changes of dendritic pores.
4.5

Conclusion

The effect of freezing direction with respect to the direction of the gravitational force
was investigated in solution-based freeze casting. Two freezing directions were
examined: conventional freezing, against the gravitatonal force, and convectionenhanced freezing, in concert with it. While conventional freezing allows a convectively stable configuration in the liquid phase, convection-enhanced freezing leads
to convective instability. Convection in the liquid phase gives rise to transport of
the preceramic polymer as well as heat in the vicinity of the solid-liquid interface.
Due to the reduced degree of constitutional supercooling in convection-enhanced
freezing, a long-range honeycomb-like pore structure results and the pore size decreases. Hence, the understanding of convective flow in the liquid phase during
freeze casting allows further control of pore morphology and pore size.

55
References
[1] Lionel Alexander Bethune Pilkington. “Review lecture: the float glass process”. In: Proceedings of the Royal Society of London. A. Mathematical and
Physical Sciences 314.1516 (1969), pp. 1–25.
[2] Jennifer A. Lewis. “Colloidal processing of ceramics”. In: Journal of the
American Ceramic Society 83.10 (2000), pp. 2341–2359.
[3] S.M. Copley et al. “The origin of freckles in unidirectionally solidified castings”. In: Metallurgical transactions 1.8 (1970), pp. 2193–2204.
[4] R. Jansen and P.R. Sahm. “Solidification under microgravity”. In: Materials
Science and Engineering 65.1 (1984), pp. 199–212.
[5] P.J. Prescott and F.P. Incropera. “Magnetically damped convection during
solidification of a binary metal alloy”. In: Journal of Heat Transfer (1993).
[6] Jose Eduardo Spinelli, Ivaldo Leao Ferreira, and Amauri Garcia. “Influence of
melt convection on the columnar to equiaxed transition and microstructure of
downward unsteady-state directionally solidified Sn–Pb alloys”. In: Journal
of Alloys and Compounds 384.1-2 (2004), pp. 217–226.
[7] José E. Spinelli et al. “Influence of melt convection on dendritic spacings of
downward unsteady-state directionally solidified Al–Cu alloys”. In: Materials
Science and Engineering: A 383.2 (2004), pp. 271–282.
[8] Natalia Shevchenko et al. “Chimney formation in solidifying Ga-25wt pct In
alloys under the influence of thermosolutal melt convection”. In: Metallurgical and Materials Transactions A 44.8 (2013), pp. 3797–3808.
[9] B. Drevet et al. “Solidification of aluminium–lithium alloys near the cell/dendrite
transition-influence of solutal convection”. In: Journal of crystal growth
218.2-4 (2000), pp. 419–433.
[10] José Eduardo Spinelli, Otávio Fernandes Lima Rocha, and Amauri Garcia. “The influence of melt convection on dendritic spacing of downward
unsteady-state directionally solidified Sn-Pb alloys”. In: Materials Research
9.1 (2006), pp. 51–57.
[11] Kristen L. Scotti et al. “The effect of solidification direction with respect to
gravity on ice-templated TiO2 microstructures”. In: Journal of the European
Ceramic Society 39.10 (2019), pp. 3180–3193.
[12] Kristen L. Scotti et al. “Directional solidification of aqueous TiO2 suspensions
under reduced gravity”. In: Acta Materialia 124 (2017), pp. 608–619.
[13] Xiaomei Zeng, Noriaki Arai, and Katherine T. Faber. “Robust Cellular ShapeMemory Ceramics via Gradient-Controlled Freeze Casting”. In: Advanced
Engineering Materials 21.12 (2019), p. 1900398.
[14] Maninpat Naviroj. “Silicon-based porous ceramics via freeze casting of preceramic polymers”. PhD thesis. Northwestern University, 2017.

56
[15] William W. Mullins and R.F. Sekerka. “Stability of a planar interface during
solidification of a dilute binary alloy”. In: Journal of applied physics 35.2
(1964), pp. 444–451.
[16] Sylvain Deville, Eduardo Saiz, and Antoni P. Tomsia. “Ice-templated porous
alumina structures”. In: Acta materialia 55.6 (2007), pp. 1965–1974.
[17] W.A. Tiller et al. “The redistribution of solute atoms during the solidification
of metals”. In: Acta metallurgica 1.4 (1953), pp. 428–437.

57
Chapter 5

COARSENING OF DENDRITES IN FREEZE-CAST SYSTEMS
The work was done in collaboration with Tiberiu Stan, Sophie Macfarland, Peter
W. Voorhees, Nancy Senabulya, Ashwin J. Shahani, and Katherine T. Faber. N.
Arai designed the systems for study, fabricated and characterized freeze-cast ceramics using SEM and mercury intrusion porosimetry, and wrote the majority of the
manuscript. N. Senabulya and A Shahani performed X-ray computed tomography
(XCT). T. Stan and S. Macfarland analyzed XCT datasets. K. Faber and P. Voorhees
supervised this work.
5.1

Introduction

In this chapter, our focus extends to the morphological evolution of the frozen crystals
over time. Coarsening, also known as Ostwald ripening, is a phenomenon which
occurs in two-phase systems such as alloys and metal oxides [1], and this is driven
by the reduction of interfacial energy to minimize the free energy of the system.
The total interfacial area is decreased through mass transport, which is driven by the
concentration gradient resulting from a large interfacial mean curvature to a small
interfacial mean curvature due to the Gibbs-Thomson effect:

𝐶 𝐿 = 𝐶∞ + 𝑙𝐶 𝐻
where
𝐻=

(5.1)

(𝜅1 + 𝜅2 )

and C 𝐿 is the composition of liquid at the solid-liquid interface, C∞ is the composition of the liquid at flat solid-liquid interface, l𝑐 is the capillary length, and H is the
mean curvature of interfaces. H is determined by the two principle curvatures, 𝜅1
and 𝜅2 . Coarsening of alloys has been extensively studied in systems ranging from
simple spherical geometries [2] to complex interconnected structures such as dendrites [3, 4]. Coarsening studies span from theory [5] to modeling [6, 7] to in-situ
and ex-situ experimental studies [8, 9, 10, 11]. Two important results on coarsening
of dendrites are highlighted here. First, Bower et al. found that secondary dendritic
arm spacing, 𝜆 2 , increases with coarsening time as:

58
𝜆 2 ∼ 𝑡 1/3

(5.2)

where t 𝑓 is local solidification time [12]. Second, Kammer et al. reported that the
dendritic structures turned into cylinders or cylindrical-like shapes after coarsening
Pb-Sn alloys and Al-Cu alloys for four days and three weeks, respectively [4]. These
two observations motivate this work to apply coarsening to freeze casting in order
to control the morphology and size of dendritic pores.
Studies of coarsening in freeze-cast systems are limited. Pawelec et al. investigated
low-temperature ice annealing in a collagen suspension, and observed coarsened
microstructures after twenty hours of annealing [13]. Liu et al. examined coarsening
of camphene crystals in freeze casting of bioactive glass to obtain a controllable
pore diameter, ranging from 15 µm to 160 µm [14]. Both were restricted to pore size
measurements and qualitative image analysis. Hence, there remains a gap between
these observations and what is understood at a fundamental level in alloy systems.
Furthermore, these studies were conducted using suspension-based freeze casting,
where suspended colloids or powders and dissolved additives such as dispersants
and binders make a comparison to alloy systems challenging and complex.
This study focuses on solution-based freeze casting and investigates the evolution
of dendrites during isothermal coarsening and its effects on dendritic pore morphology and size. By varying time and temperature, coarsening phenomena were
explored using scanning electron microscopy and mercury intrusion porosimetry.
To gain further insight into the coarsening processes in freeze-cast systems in three
dimensions, X-ray computed tomography enabled us to quantitatively analyze morphologies and directionality by Interfacial Shape Distributions (ISD) and Interfacial Normal Distributions (IND). By coupling images, pore size distributions with
tomography-derived ISDs and dendritic pore directionality through their INDs, our
studies provide new understanding into coarsening in freeze-cast systems, allow
comparisons with coarsening behavior of alloy system, and offer an additional
means for pore network tailorability.
5.2
5.2.1

Experimental methods
Processing

A polysiloxane (CH3 -SiO1.5 , Silres®MK Powder, Wacker Chemie) preceramic polymer was dissolved in cyclohexane (C6 H12 , Sigma-Aldrich), with compositions of
preceramic polymer of 20 wt.% and 30 wt.%. After a homogeneous solution was

Figure 5.1: Schematic of the gradient-controlled freeze casting setup

obtained by stirring, a cross-linking agent (Geniosil®GF 91, Wacker Chemie) was
added in concentrations of 1 wt.% and 0.75 wt.% in 20 and 30 wt.% solutions,
respectively, and stirred for an additional 5 minutes. Subsequently, the polymer solution was degassed for 10 minutes to prevent air bubbles during freezing. Freezing
was done using gradient-controlled freeze-casting setup as described in Chapter 3
(Figure 5.1). All samples were frozen at freezing front velocities of 15 µm/s for
20 and 30 wt.% solutions, and temperature gradients of ∼2.6 K/mm to maintain
homogeneous pore structures.
To induce coarsening after freezing was completed, the top and bottom thermoelectrics were set to temperatures close to the liquidus temperature of the solution
(2 ◦ C or 4 ◦ C for 20 wt.% solution and 3 ◦ C for 30 wt.% solution) and held for up to
5 hrs. To determine time for frozen samples to reach the prescribed temperature, a
type K thermocouple was used to measure temperature of the samples during coarsening1. After coarsening, the samples were cooled to -30 ◦ C to re-freeze. Once
frozen, the samples were placed in a freeze drier where the solvent crystals were
completely sublimated. After freeze drying, the polysiloxane green bodies were
pyrolyzed in argon at 1100 ◦ C for four hours with a 2 ◦ C /min ramp rate to convert
the preceramic polymer into silicon oxycarbide (SiOC). This resulted in a porosity
of ∼77% for the 20 wt.% solution and 64% for 30 wt.% solution. The resulting
sample dimensions were approximately 9.5mm in height and 18mm in diameter.
1 Generally, type K thermocouples have an accuracy of ± 1.1◦ C or larger [15].

60
5.2.2

Characterization

Pore structures were observed using scanning electron microscopy (SEM). Longitudinal and transverse cross-sections were prepared using a diamond saw and imaged.
Pore size distributions were measured using mercury intrusion porosimetry (MIP).
All samples for MIP were machined with a core drill (∅ = 15.9 mm) to remove the
edges, and a ∼1.8 mm disk was sectioned from the center of the sample.
X-ray computed tomography (XCT) was performed on selected samples to quantitatively measure the morphological evolution of dendrites via Absorption Contrast
Tomography (ACT) on a laboratory X-ray microscope (XCT; Zeiss Xradia Versa
520, Carl Zeiss AG, Oberkochen, Germany) at the Michigan Center for Materials
Characterization. Three samples (h = ∼5 mm, ∅ = ∼1.2 mm) were chosen for this
analysis: a control sample without coarsening, one coarsened at 2 ◦ C for one hour,
and another coarsened at 4 ◦ C for three hours. During the ACT measurement, each
sample was positioned 5.1 mm in front of a polychromatic X-ray source tuned to 40
kV, 3 W, and 75 µA. The X-ray beam interacted with a sample volume of 1025 µm
x 1132 µm x 1090 µm. A series of 1601 X-ray projection images was collected at
0.2◦ intervals while the sample rotated through 360◦ at exposure times of 1.1s per
projection. A scintillator downstream from the sample converted the X-ray projection images into visible light images and a 4X objective lens magnified the visible
light image before coupling it to the 2k x 2k CCD detector placed 23.5 mm away
from the sample. With the CCD operating at a pixel binning of 2, a scan pixel size
of 1.2 µm/voxel was achieved. The collected projection images were reconstructed
using a filtered back projection algorithm in the Scout and Scan software provided
by Zeiss Xradia Inc. to create a virtual 3D volume of the sample. Worth noting
is that phase retrieval [16] was not necessary because there was sufficient contrast
between the SiOC matrix and the pore network in the traditional absorption-based
images. The SiOC matrix is a light gray and the pore network is a dark gray (Figure
5.2).
The control sample (without coarsening) was segmented using Otsu’s method [17]
in MATLAB. Although Otsu’s method is computationally straightforward and the
preferred segmentation approach, it was not successful on the 2 ◦ C and 4 ◦ C coarsened datasets due to the presence of debris and bright spot artifacts at random
sections throughout the reconstructions. The coarsened datasets were instead segmented using a convolutional neural network (CNN) machine learning approach as
described by Stan et al. [18, 19]. First, 35 representative slices were selected from

Figure 5.2: Cross-section of XCT data from (a) a control sample, (b) a sample
coarsened at 2 ◦ C for one hour, and (c) a sample coarsened at 4 ◦ C for three hours.
Scale bar: 200 µm.
each reconstruction to include sections of debris and bright spots and split into three
categories: 20 images for training, 10 images for validation, and 5 images for testing.
Each image was then segmented using a combination of thresholding and manual
cleaning using the GIMP software. These ground truth segmentations (along with
the original images) were used to train CNNs with the SegNet architecture using the
PyTorch framework. Each CNN was trained for 100 epochs on the Quest supercomputer at Northwestern University. The CNNs each achieved 99.4% segmentation
accuracy when applied to test images from the 2 ◦ C and 4 ◦ C coarsened datasets.
MATLAB was used for all post-segmentation analysis. It was found empirically that
120 µm-thick sections (100 z-slice images) of each XCT dataset were large enough
to capture the defining morphological features, yet small enough to be computationally manageable. All three segmented datasets were meshed and smoothed using
the “smoothpatch” function. The control and 2 ◦ C datasets were smoothed for 5
iterations, while the coarser 4 ◦ C dataset was smoothed for 15 iterations. Principle
curvatures (𝜅1 and 𝜅2 ) and normal vectors were calculated at each of the triangular patches. Their respective frequencies within the microstructures are plotted as
interface shape distributions (ISD) and interface normal distributions (IND).
5.3
5.3.1

Analysis of XCT images
Interfacial Shape Distribution (ISD)

The quantitative analysis of morphological evolution of dendrites (or resulting pores)
was carried out by measuring the curvature of the interfacial patches. First, two
invariants of the curvature tensor, 𝜅𝑖 𝑗 , were measured. With this measurement, the
mean curvature, 𝐻 is established:

Figure 5.3: A map of interfacial shapes of patches for the Interfacial Shape Distribution (ISD). This is a modified figure from ref. [20].

𝐻 = tr{𝜅𝑖 𝑗 } =

(𝜅1 + 𝜅2 )

(5.3)

where the two principle curvatures, 𝜅1 and 𝜅2 , the minimum and maximum principle curvatures, respectively, can be determined to construct the interfacial shape
distribution (ISD). The ISD is presented as a contour plot to map the probability of
finding a patch with a given pair of principal curvatures (Figure 5.3). Since 𝜅2 is the
maximum principle curvature of the patches, the entire distribution must reside to
the left of the 𝜅 1 = 𝜅2 line. The plot can be divided into four regions. For dendritic
porous materials:
• Region 1 represents positive 𝜅1 and 𝜅2 and the interface patches are concave
toward the solid (SiOC walls).

63
• Regions 2 and 3 represent 𝜅1 < 0 and 𝜅2 > 0, and interface patches are
saddle shaped. Region 2 embodies interface patches which are strongly
curved toward the pores whereas region 3 signifies interface patches which
are strongly curved toward the solid.
• Region 4 represents negative 𝜅1 and 𝜅2 and interface patches are convex toward
the solid.
All the principle curvatures were normalized with respect to the specific interface
area, S𝑠 , which is the total surface area of the interface divided by the volume of
the dendrites, or equivalently the volume of pores. This normalization is necessary
for mapping probability distributions such that microstructures with different coarsening conditions can be compared and inspected for self-similarity. One hundred
slices of images, which represent 120 µm of the sample in freezing direction with a
diameter of roughly 1.2 mm, were used for analysis. Since the samples were frozen
under constant freezing front velocity and temperature gradient and other 100 slices
from different section show similar ISD, 100 slices are assumed to be sufficient to
represent the whole structures.
5.3.2

Interfacial Normal Distribution (IND)

The Interfacial Normal Distribution (IND) is a contour plot which shows the probability distribution of the orientation of normals to interfacial patches, and is useful in
determining the directionality of dendrites, or in this study, directionality of pores.
First, the orientation of the interfacial normals to patches are determined and stored
in a unit reference sphere, in which their origins sit in the center of the sphere and
their ends sit in the surface of the sphere. Then, they are projected on a 2D plane,
which is tangent to the sphere and, in this case, perpendicular to the direction of the
freezing. The projection used in this study is an equal-area projection. Two simple
cases can be considered as examples. If the porous structure has perfectly spherical
shapes, the orientation of the normals is isotropic, which results in a uniform probability distribution in the IND. In contrast, in the case of cylindrical pores perfectly
aligned along [001] direction, the probability distribution in the IND concentrates at
the outer rim of the projection. For off-axis aligned pores, an arc-like band appears
across the IND.

Figure 5.4: SEM images showing (a, b) control sample, and sample coarsened at (c,
d) 2 ◦ C for one hour, (e, f) 2 ◦ C for three hours, (g, h) 4 ◦ C for one hour, and (i, j)
4 ◦ C for three hours. Inset images in (a) and (b) show primary pore and secondary
pores, respectively, as indicated by red arrows, (scale bar: (a) 60 µm and (b) 40 µm).
Transverse images and longitudinal images show cross-sections perpendicular and
parallel to the freezing direction, respectively.

65
5.4
5.4.1

Results and discussion
Pore structure

Figure 5.4 shows a series of SEM images of dendritic pores as a function of coarsening treatment beginning with the control sample as-cast and pyrolyzed (Figures
5.4a and b). Since cyclohexane dendrites template the pores, the pores (appearing
black in SEM images) are the negatives of dendrites [21, 22]. The transverse image
(perpendicular to the solidification direction) in Figure 5.4a shows primary pores
templated by primary dendrites (red arrows), and secondary pores templated by
secondary dendrite arms. Tertiary pores are occasionally observed in regions where
primary interpore spacings are large. The four-fold symmetry of dendritic pores is
consistent with the cubic structure of cyclohexane crystals [23]. The longitudinal
image (approximately parallel to the solidification direction) (Figure 5.4b) shows
the cutaway view of dendritic pores, where the red arrows in inset image indicate
secondary pores. When the dendrites are coarsened at 2 ◦ C for one hour, there is
an increase in both primary and secondary pore sizes as shown in Figures 5.4c and
d. After three hours of coarsening at 2 ◦ C, the transverse image shows larger domains of honeycomb-like structures (Figure 5.4e) although the secondary pores are
still present as noted in the longitudinal image (Figure 5.4f). When the coarsening
temperature is increased to 4 ◦ C, morphological evolution proceeds at a higher rate
(Figures 5.4g-j). Coarsening for one hour yields larger domains of the honeycomb
structure in the transverse direction (Figure 5.4g) while secondary pores are still
noted in the longitudinal image (Figure 5.4h). After three hours of coarsening at
4 ◦ C, the majority of secondary pores disappear in the longitudinal image (Figure
5.4j), producing a largely honeycomb-like structure. The morphological evolution
of dendritic pores observed in this solution-based freeze casting agrees well with
what has been reported in coarsening of dendrites in alloys [10, 4], where dendrites
evolve into cylindrical morphologies. In addition to the overall morphological
change from dendritic pores to cellular pores, these SEM images further reveal the
morphological change of primary pores and secondary pores. The transverse image
of the control sample shows four-fold symmetric primary pores (Figure 5.4a). When
the structures are coarsened, primary pores evolve to circular-like shapes. See also
secondary pores in the longitudinal images in Figure 5.5 comparing the control sample and the sample coarsened at 4 ◦ C for one hour as an example. While the sides
of secondary pores exhibit curvature, the top and bottom faces of secondary pores
are nearly flat. After coarsening, these flat surfaces disappear, and the secondary
pores became circular in cross-section. Longer coarsening time (five hours) at 4

66
◦ C was also investigated, but there were only minor morphological changes (Figure

5.6). These minor changes can be attributed to the decreasing diffusion coefficient
of preceramic polymer as the gelation of the solution started around 5-6 hours. The
influence of diffusion coefficient can also be demonstrated by changing the polymer
concentration in the solution, as described in Appendix D.

Figure 5.5: SEM images showing longitudinal direction of (a) the control sample
and (b) the sample coarsened at 4 ◦ C for one hour. Flat surface and circular surface
are indicated by red arrows in (a) and (b), respectively.

Figure 5.6: SEM images of the samples coarsened at 4 ◦ C for three hours (a:
Transverse image, b: Longitudinal image) and five hours (c: Transverse image, d:
Longitudinal image).

Figure 5.7: Pore size distribution data of samples coarsened for 30 minutes and one
hour at (a) 2 ◦ C and (b) 4 ◦ C (including three hours).

5.4.2

Pore size distribution

Pore size distributions, which illustrate pore diameters, obtained from MIP for each
coarsening temperature are shown in Figure 5.7. Figure 5.7a shows the pore size
distribution from samples coarsened at 2 ◦ C compared to the control sample. All
samples exhibit a bimodal distribution, which can be attributed to large primary
pore diameters and small secondary pore diameters. As the samples are coarsened,
primary pores and secondary pores become larger with the distributions shifting
to the right, consistent with SEM images. Dendritic structures typically have a
secondary pore volume that exceeds the primary pore volume because of the large
number of secondary arms that grow from each primary dendrite [24]. In contrast,
in the current coarsening studies, not only does the pore size distribution shift to
larger pores, but also the primary pore volume eclipses the secondary pore volume.
For coarsening at 4 ◦ C, the same trend can be observed (Figure 5.7b). For the sample
coarsened at 4 ◦ C for one hour, the distributions for primary and secondary pores
begin to overlap; this is more significant in the sample coarsened for three hours.
Distinct bimodal distributions disappear in favor of a unimodal distribution. This
corresponds well with Figures 5.4i and j in which the majority of secondary pores
disappear, yielding the honeycomb-like structure. The primary pore volume can be
established, calculated by using the software Fityk [25], and is plotted as a function
of coarsening time in Figure 5.8. The primary pore fraction increased by 146% and
160% after coarsening for one hour at 2 ◦ C and 4 ◦ C, respectively.
To compare the coarsening behavior of freeze-cast systems with typical alloy systems (Eqn. 5.2), primary and secondary pore sizes are plotted as a function of the

Figure 5.8: Primary pore fraction as a function of coarsening time.

Figure 5.9: Plots of (a) Primary pore size and (b) secondary pore size as a function
of the cube root of coarsening time at different coarsening temperatures.
cube root of coarsening time2, t1/3 , in Figures 5.9a,b. In this plot, the peak values
2 Coarsening time is defined as the time interval over which the thermoelectric plates are at
the prescribed coarsening temperature adjusted by the time it takes the frozen sample to reach the
equilibrium coarsening temperature.

69
of the pore size distribution were plotted as representatives of primary pore size and
secondary pore size. The t1/3 dependence is obeyed for both coarsening temperatures, consistent with coarsened dendrites in alloys. Typically, only the dependence
of the secondary arm spacing on t1/3 is reported [12], but it was found in the current
study that the diameters of primary dendrites have a similar dependence. The slopes
of the linear fit are summarized in Table 5.1. As expected, the slopes increase as
coarsening temperature increases, in agreement with the observations of Chen and
Kattamis who studied Al-Cu-Mn dendrite coarsening [26]. For both coarsening
temperatures, the slopes of primary dendrites are larger than those for the secondary
arms of dendrites. Specifically, increasing the coarsening temperature increased the
slope for primary pores by a factor of 1.6, whereas the slope increased for secondary
pores by a factor of 1.3, indicating that primary dendrites coarsen at a faster rate.
We attribute this difference to active coarsening mechanisms for primary dendrites
and secondary arms (Figure 5.10). As shown in Figure 5.10, the secondary arm
coarsening can be explained by 4 models: radial remelting, axial remelting, arm
detachment, and arm coalescence. Radial remelting could occur by radial dissolution
of small arms surrounded by a larger arm and diffusion of the material to adjacent
larger arms. If radial remelting is a dominant mechanism for coarsening in freezecast systems, the secondary pore diameter would show a decrease in size in the
pore size distribution. However, in all the pore size distributions, the secondary
pore diameter is larger than that of the control samples, indicating that the radial
remelting is not the dominant mechanism. Another mechanism is axial remelting.
Axial remelting takes place by melting at the tip of the arms and solidifying at the
root of the arms. Since this would require remelting of the arms, the secondary
pore size would not increase from this mechanism. Since arm detachment would
yield closed pores and pores formed by this mechanism are not measurable by
MIP, arm detachment is not considered here. As a result, it appears that arm
coalescence could be a major mechanism to increase secondary pore diameter in
a freeze-cast system. On the other hand, primary dendrite could coarsen by two
mechanisms - the coalescence of primary dendrites and axial remelting. Since
axial remelting takes place by solidifying the root of the arms, it will increase
the primary dendrite diameter, resulting in the increase of primary pore diameter.
Hence, primary dendrites coarsen by two mechanisms while secondary arms coarsen
by arm coalescence, which could qualitatively explain the faster coarsening rate for
primary pores.

70
Table 5.1: The slope of linear fit from Figure 5.9
Coarsening temperature

Primary dendrites

Secondary arm of dendrites

2 ◦C

5.0

3.5

4 ◦C

8.0

4.6

Figure 5.10: Illustration showing four different coarsening models for secondary
arm coarsening: (1) radial remelting, (2) axial remelting, (3) arm detachment, and
(4) arm coalescence. Based on ref. [27].

It is worth mentioning that coarsening times in this study are much shorter than
those in alloy studies. In the latter, dendrites were coarsened from a few to several
days to observe the significant morphological change from dendrites to cylinders,
while coarsening requires only a few hours in solution-based freeze casting. Here,
the freeze-cast system is compared with Sn-rich Pb-Sn alloys studied by Kammer
and Voorhees [10], where the dendrites evolve into cylinders after coarsening for
two days. The model by Kirkwood [28] for secondary arm coarsening provides
some insight to explain the faster morphological changes in freeze-cast system:

71
𝜆=(

128𝐷𝜎𝑇𝑚 1/3 1/3
) 𝑡
𝐿𝑚𝐶 𝐿 (1 − 𝑘)

(5.4)

where 𝜆 is secondary arm spacing, D is diffusion coefficient of solute in liquid, 𝜎
is solid-liquid interface tension, T is absolute melting temperature, L is volumetric
heat of fusion, m is liquidus slope, 𝐶 𝐿 is mean composition in the liquid region, k
is distribution coefficient, and t is coarsening time. Here the assessment of a few
known variables, T, L, and m, are possible. Even though liquidus temperature, T,
of Sn-rich Pb-Sn alloy (from 456 K to 504 K) is higher than that of the freezecast system (278 K for 20 wt.% solution), this contribution to the coarsening rate
remains small. On the other hand, there are significant differences in m and L
between two systems. The absolute value of the liquidus slope, m for Sn-rich Pb-Sn
alloy is ∼ 1.4 while that of freeze-cast system (0 wt.% - 40 wt.% preceramic polymer
concentration) ranges from around 0.07 to 0.16 [29], which is an order of magnitude
different. Furthermore, due to the higher density of Sn-Pb alloy, the heat of fusion,
L of eutectic Pb-Sn alloy is around 300 J/cc, while pure cyclohexane’s is around 25
J/cc, which is also an order of magnitude difference. Although only three variables
are examined here, an order of magnitude difference in m and L is consistent
with higher coarsening rate in freeze-cast system based on the Kirkwood model.
Finally, although the mean composition, 𝐶 𝐿 , is not known during the coarsening
process in freeze-cast systems, one can look at the homologous temperature at the
coarsening temperature to provide insight into the liquid fraction. For example, in
Kammer’s study [10], the coarsening of Pb-80 wt.% Sn alloy, which at the coarsening
temperature of 185◦ C (2◦ C above the eutectic temperature), consists of 51% Sn-rich
dendrites and 49% Sn-lean liquid . The homologous temperature is 0.953 at 185◦ C.
By contrast, in the freeze-cast system, the homologous temperatures are 0.992 and
0.999 for 2 ◦ C and 4 ◦ C, respectively, based on the liquidus temperature reported by
Naviroj [29]. Hence, it is likely that the liquid fraction during the coarsening in the
freeze-cast system is greater than the alloy system, or equivalently 𝐶 𝐿 of freeze-cast
system is lower, resulting in faster morphological changes.
5.4.3

Tailoring pore morphology and network by coarsening temperature

Coarsening time and coarsening temperature are two major parameters to tailor pore
size and morphology. In the previous section, it was demonstrated that primary and
secondary pore sizes linearly depend on the cube root of coarsening time. Here, the
effects of coarsening temperature in the resulting freeze-cast structure are examined
in detail.

72
It was demonstrated above that a higher coarsening temperature could accelerate the
morphological transition from dendrites to honeycomb-like structures along with the
increase of pore size. Again, this acceleration of coarsening at higher temperature
can also be explained by Kirkwood’s model (Eqn. 5.4). Changing the coarsening
temperature can alter the diffusion coefficient. However, if the diffusion coefficient,
D, follows Arrhenius behavior [30] or, for the case of long-chained polymers, a
reptation model, the temperature dependence of D is sufficiently small (less than 1%
increase) that it cannot account for the 70% and 40% increase in coarsening rate for
primary pores and secondary pores, respectively. Instead, the enhanced coarsening
rate with temperature is likely due to the increase in liquid fraction by increasing
coarsening temperature, causing 𝐶 𝐿 to decrease. Furthermore, a decrease in 𝐶 𝐿 is
expected to increase D [31]. Hence, it can be hypothesized that the changes in 𝐶 𝐿
and D as a result of a higher coarsening temperature give rise to acceleration of the
coarsening process.
In addition to the aforementioned acceleration, different coarsening temperatures
could further lead to different freeze-cast structures. In order to highlight this
difference in structure as a function of coarsening time and temperature, Figure 5.11
presents pore size distributions and SEM images for samples coarsened at 2◦ C for
three hours and 4◦ C for one hour. As demonstrated by MIP data, both samples
have nearly identical pore size distributions, with marginally larger secondary pores
present in the sample coarsened at 2◦ C for three hours. The pore morphologies
shown in SEM images, however, reveal distinct differences. In the transverse
direction, some of the primary pores in the sample coarsened at 2◦ C retain four-fold
symmetry (Figure 5.11b), whereas primary pores in the sample coarsened at 4◦ C lose
such symmetry and are more cellular-like in shape (Figure 5.11d). In the longitudinal
direction, additional differences can be observed. First, primary pores connect to
neighboring primary pores by coalescence of secondary arms at 2 ◦ C (Figure 5.11c).
Second, a closer look at secondary pores shows some large elliptical-shaped pores.
Since the major axis of the ellipse is along the dendrite growth direction, it is
likely that these large elliptical pores are the result of coalescence of secondary
pores in the dendrite growth direction, as indicated by red arrows in Figure 5.11c.
These large elliptical pores likely give rise to a slight shift of the secondary pore
peak in the sample coarsened at 2◦ C (Figure 5.11a). It is hypothesized that this
morphological difference can be attributed to the difference in liquid fraction during
coarsening. When samples were coarsened at 4◦ C, close to the melting point of the
solution, a sufficient fraction of the liquid phase is present to surround the dendrites,

Figure 5.11: Pore size distribution from samples coarsened at 2 ◦ C for three hours
and 4 ◦ C for one hour (a). SEM images showing a sample coarsened at (b, c) 2 ◦ C
for three hours, and (d, e) 4 ◦ C for one hour. (Red arrows indicate some of the thin
solid tubes).

providing pathways for mass diffusion. This allows the structure to coarsen all the
parts of the dendrites, hence, both primary pores and secondary pores change their

74
morphologies. In contrast, if the coarsening temperature is lower, the liquid phase
still exists, but there are regions rich in liquid phase and those poor in liquid phase.
This creates a large discrepancy in coarsening rates within an individual dendrite,
and results in disparate coarsening behavior. This demonstrates that the change in
coarsening temperature gives one further tool in tailoring pore morphology and pore
network.
5.4.4

Coarsening mechanisms in solution-based freeze casting

Figure 5.12: 3D XCT reconstructions and subsections for the (a, d) control sample,
(b, e) the sample coarsened at 2 ◦ C for one hour, and (c, f) sample coarsened at 4 ◦ C
for three hours. The sides of the solid-pore interfaces that face the dendritic pores
are colored according to the normalized mean curvature (H/SS ), as indicated by the
color bar in (c). White arrows in (e) show secondary pores with positive curvature
caps, while the red arrow indicates a ligature with negative curvature.
Since the coarsening proceeds as a result of the Gibbs-Thomson effect, the mean
curvature of the dendritic pores provides insight on the coarsening mechanism.
Figure 5.12 shows three-dimensional reconstructions of the control sample (Figure
5.12a), the sample coarsened at 2 ◦ C for one hour (Figure 5.12b), and the sample
coarsened at 4 ◦ C for three hours (Figure 5.12c). The datasets are plotted such that
the solidification direction is pointing out of the page. Subsections from each of the
three reconstructions are shown in Figs. 7d-f. The interface sides that face the solid

75
SiOC are colored dark gray. The interface sides that face the dendritic pores are
colored according to their normalized mean curvature H/SS . The specific interface
area (SS ) is a characteristic microstructural length scale and is calculated as the
total interface area divided by the total pore volume in the dataset. Normalizing
curvatures by Ss is used to facilitate visual comparison between coarsening datasets
and to check for self-similarity.
Comparing Figures 5.12a, b and c, a significant change in pore size and pore morphology is observed, consistent with earlier SEM images and pore size distributions.
The control sample (Figures 5.12a and d) has patches with large positive mean curvature (yellow) mainly located at the tips and sides of the secondary pores, and
patches with small and negative mean curvature (purple) primarily present at the
roots of secondary pores. The interfaces with nearly zero mean curvature (light blue
and light green) are at the flat sections along secondary pores. (See top and bottom
faces of secondary pores in Figure 5.5a). There are two distinct domains of dendritic
pores in the control sample that vary by dendritic pore spacing. While the domain
with smaller spacing (bottom right) is well aligned along the temperature gradient,
that with large spacing is slightly misaligned. This is consistent with the other observations [32, 33, 34], which show that dendrite spacing generally increases with
misorientation.
Morphological changes in the dendritic pore network and secondary arms are evident
when comparing the three XCT datasets. Most secondary pores in the control sample
have capped ends such that each dendritic pore was isolated from adjacent dendritic
pores, as shown in the SEM image (Figure 5.4b). In the sample coarsened at 2
◦ C for one hour, some of the secondary pore caps remain (white arrows in Figure
5.12e). However, some caps are lost during coarsening resulting in connections
between secondary pores and formation of ceramic ligatures (red arrow in Figure
5.12e). The sample coarsened at 4 ◦ C for three hours no longer contains secondary
pores and the microstructure is instead primarily composed of larger channels with
nearly-flat sides, as indicated by the green and light-blue coloring in Figure 5.12f.
Areas of higher curvature (yellow stripes in Figure 5.12f) are present where the
flatter sections intersect.
The color-coded 3D reconstructions can explain why dendritic morphologies evolve
to honeycomb-like structures through coarsening. Large positive mean curvatures
are preferentially found at the tips and sides of secondary pores. During coarsening,
these regions contain high solvent content and equivalently low preceramic polymer

76
concentration due to the Gibbs-Thomson effect. In contrast, small and negative
mean curvatures are found at the roots of the secondary pores, which are high in
preceramic polymer concentration. Hence, the preceramic polymer diffuses from
the roots of secondary arms to the tips and sides of secondary arms, which melts the
tips and sides, but solidifies the roots. As a result, secondary arms will disappear,
resulting in honeycomb-like structures.
5.4.5

Quantitative microstructure analysis

It is challenging to quantitatively compare highly complex microstructures using
only 2D SEM images and visualizations of the 3D XCT reconstructions. A major
advantage of the XCT technique is the ability to measure volumetric and interfacial
properties. Metrics from the three XCT datasets are reported in Table 5.2. The pore
volumes and volume fractions are similar between the datasets, and consistent with
MIP measurements. As indicated by the interface area measurements, the control
sample contains ∼17 times more interface area than the 2 ◦ C coarsened sample, and
∼25 times more interface area than the 4 ◦ C coarsened sample. The inverse specific
interface area (SS −1 ) is found to be equal to roughly half of the secondary pore size.
The control and 2 ◦ C datasets have SS −1 = 6.1 and 10.7 µm, and MIP-measured
secondary pore sizes of 12.7 and 25.7 µm, respectively. The tilt angle reported
in Table 5.2 is a measurement of the angle between the average dendrite growth
direction and the solidification direction (discussed in detail in Subsection 5.4.6).
Table 5.2: Metrics from the three XCT datasets. SS −1 is the inverse specific interface
area, calculated as the total pore volume divided by the total solid-pore interface
area.
Pore volume

Pore volume fraction

Interface area

SS −1

Tilt angles

(×107 µm3 )

(%)

(×106 µm2 )

(µm)

(◦ )

Control

4.1

67.0

6.1

∼5, ∼28

2 ◦ C, 1 hr

4.3

4.0

10.7

∼35

4 ◦ C, 3 hrs

4.2

2.7

15.6

∼15, ∼20, ∼30

Sample

Interface Shape Distributions (ISDs) are also used to quantitatively compare the
complex microstructures. For example, freeze-casting has been used to fabricate
structures which mimic bone for medical implants, and the morphology of freezecast foams have been characterized by ISD to investigate the similarity with bone [3].
In this section, in addition to quantitatively defining the structures, ISDs are used to

77
compare to an alloy systems to investigate if the similar morphological evolution is
observed.

Figure 5.13: Interface Shape Distributions (ISDs) for the (a) control sample, (b)
sample coarsened at 2 ◦ C for one hour, and (c) sample coarsened at 4 ◦ C for three
hours. (d) Map of the interface shapes possible in an ISD where P is pore and S
is solid. This is a modified figure from ref. [20]. Sections of the 2 ◦ C coarsened
sample cylindrical patches colored in red (e) and porous caps colored in pink (f).
Figures 5.13a-c show the interfacial shape distributions (ISDs) and Figure 5.13d
shows a map of interfacial shapes of patches for the ISD, the same figure as Figure
5.3. The three ISDs span the range of interfacial shapes, although the shapes
of the distributions differ, implying that all of the structures are not self-similar,
despite following the t1/3 power law. For the control sample, since the probability
distribution of the cylindrical shaped region and cap shaped region extends to large
values of 𝜅1 /SS and 𝜅2 /SS , the interfacial patches are predominantly cylindrical in
shape and cap-shaped (Figure 5.13a). This is expected since the sides of secondary
pores are cylindrical and the tips of secondary pores are cap shaped. The probability
distribution also extends to the origin, indicative of a flat interface region, consistent
with SEM observations (Figure 5.5a) and the 3D reconstruction of the control
sample in Figure 5.12a. This is in stark contrast with the ISD of freeze-cast lamellar
structures reported by Fife et al. [3], which shows a well-defined peak near the origin,

78
resulting from the flat interfaces of plate-like structures. For the sample coarsened
at 2 ◦ C for one hour (Figure 5.13b), a few changes can be highlighted. First, the peak
shifts away from the origin which represents the flat interface and a larger fraction
of the distribution is now located in the saddle-shaped region. Second, compared
to the control sample, the distribution no longer extends along 𝜅1 /SS = 0. This is
consistent with the SEM images of decreased secondary pore length. For the sample
coarsened at 4 ◦ C for three hours, the probability distribution shifts to the region
near 𝜅1 /SS = 0, indicating that the cylindrical patches are the dominant morphology
(Figure 5.13c) consistent with the honeycomb structures viewed via SEM and 3D
reconstructions.
Figures 5.13e and f show the same section of the sample coarsened at 2 ◦ C for 1 hour
as in Figure 5.12e, but the structures are colored according to interfacial shapes of
interest. Patches with cylindrical shapes were isolated from the red-box region in
the ISD in Figure 5.13b and displayed on the microstructure in Figure 5.13e. The
majority of these cylindrical features are primarily found along the walls of primary
pores, but some patches are also present along the walls of secondary pores. The
porous caps in the pink region of Figure 5.13b are shown on the reconstruction in
Figure 5.13f. As expected, these high-curvatures features are mostly present at the
tips of secondary pores.
Since the dendrites in freeze casting undergo similar morphological evolution to
cylinder-like shapes as alloy system, the ISDs for control sample and the sample
coarsened at 4 ◦ C for three hours are compared to the dendrite coarsening in Pb-Sn
alloys reported by Cool and Voorhees [35]. The ISD of the least-coarsened dendrites
(10 min coarsening) in the Cool and Voorhees study looks similar to the ISD of the
control sample. Both ISDs show significant peaks in the cylindrical shaped region
and cap shaped region as both structures contain a significant number of interfacial
patches from secondary arms or secondary pores. When the structures are coarsened
to cylindrical shapes, the resulting ISDs for the freeze-cast system and the Pb-Sn
alloy look similar, too. Hence, the general trends of ISDs, such as the shapes of
the probability distributions and their changes after coarsening, are similar to those
seen in dendrite coarsening in Pb-Sn alloys.
5.4.6

Directionality of dendritic pores

Directionality is of great interest because it influences the transport [36] and mechanical properties [37] of freeze-cast solids. As clearly observed in 3D reconstruction

Figure 5.14: Interface Normal Distributions (INDs) for the (a) control sample, (b)
sample coarsened at 2 ◦ C for one hour, and (c) sample coarsened at 4 ◦ C for three
hours. The green arrow in (a) corresponds to the green patches in (d). The purple
arrow in (b) corresponds to purple patches in (e). The blue arrow in (b) corresponds
to the blue patches in (f).

of the control sample (Figure 5.12a), not all dendrites grow along a temperature
gradient. Interface Normal Distributions (INDs) are used to quantitatively measure
the averaged dendritic growth directions. The [001] stereographic projections are
presented as INDs in Figure 5.14 for the (a) control sample, (b) sample coarsened at
2 ◦ C for one hour, and (c) sample coarsened at 4 ◦ C for three hours. The IND center
where the white lines intersect is the [001] direction, parallel to the temperature
gradient. The colormap used for all three INDs is at the right side of Figure 5.14c.
The IND for the control sample (Figure 5.14a) is uniformly blue except for one
spot (green arrow) at ∼28◦ from the IND center (green arrow). Some of the
interfacial patches which contribute to this IND spot have been highlighted on the
microstructure section in Figure 5.14d. The green patches are primarily present
on flat top faces of secondary pores. These interfaces have normal vectors that
point in roughly the same direction, thus yielding a spot on the IND. Owing to the
cubic symmetry of the system, the plate-like pores are roughly perpendicular to the

80
primary pore growth direction. Thus, the primary dendrites grew at ∼28◦ from the
temperature gradient direction.
The IND of the sample coarsened at 2 ◦ C for one hour (Figure 5.14b) contains a spot
(purple arrow) at ∼35◦ from the center. The microstructure section in Figure 5.14e
has purple patches corresponding to the spot in the IND. As in the control sample,
these interfaces are mostly present at secondary pores. The IND also has an arc-like
band, marked by four red arrows and one blue arrow at the highest-intensity spot.
Figure 5.14f shows a section of the microstructure where patches that contribute
to the IND spot are highlighted in blue. These areas are primarily found on the
flatter regions of primary pores. The primary pores in the 2 ◦ C coarsened sample
have walls with normal vectors that point in many directions, all of them nearly
perpendicular to the primary dendrite growth direction. This is manifested as an
arc-band in the IND (red arrows in Figure 5.14b). The band is tilted at ∼55◦ away
from the temperature gradient direction, and ∼90◦ away from the secondary arm
spot (purple arrow in Figure 5.14b). Together, these observations indicate that the
average dendrite growth misalignment for the 2 ◦ C coarsened sample is ∼35◦ .
The IND of the sample coarsened at 4 ◦ C for three hours (Figure 5.14c) contains many
light-blue bands. The bands appear smeared largely because the honeycomb-like
structure is composed of large channels, each with a slightly different orientation.
Three main orientations are identified (red, yellow, and pink arrows), indicating
misorientations of ∼20◦ , ∼15◦ , and ∼30◦ , respectively. In contrast, freeze-cast
lamellar pores identified from INDs by Fife et al. show two peaks located 180◦
apart due to the plate-like pores [3].
The misalignment of dendritic pores is expected since dendrites were randomly
oriented at nucleation. As the dendrites grow, misoriented dendrites tend to impinge
on aligned dendrites and stop growing. However, some fraction of off-axis dendrites
are retained. Although the misalignment of ∼35◦ affects transport properties of
freeze-cast solids, nucleation control by a grain selector can be used in freeze casting
to align dendritic pores and has been shown to improve the Darcian permeability
constant more than 6-fold [36].
5.5

Conclusions

This study demonstrates that coarsening processes can be applied to solution-based
freeze casting to give rise to changes in both pore morphology and pore size.
Coarsening temperature and coarsening time were explored. Two important findings

81
were reported. First, the morphological evolution from dendrites to cylindrical-like
crystals was demonstrated in solution-based freeze casting, and ultimately resulted
in honeycomb-like structures. Second, dendritic pore size, both primary pore size
and secondary pore size, was found to scale with the cube root of coarsening time.
Both findings are well-known in dendrite coarsening in metal alloy systems.
While many studies in freeze casting have focused on controlling crystal growth,
to best of our knowledge, this is the first study to use X-ray tomography to quantitatively explore morphological evolution during coarsening of freeze-cast systems,
specifically with interfacial shape distributions and interfacial normal distributions.
In freeze casting, the characterization of pore morphologies previously has been
limited to the qualitative interpretation of 2D or 3D images, which would cause
the characterization of coarsened dendritic structures to be challenging. However,
curvature analysis by ISDs, in this case, were used quantitatively to determine that
non-coarsened and coarsened pore structures are not self-similar, the same findings
as coarsened dendrites in alloys [10, 35]. INDs were used further to elucidate the
preferential direction of pores, which is important for mechanical and transport
properties of porous solids. Since the dendritic structures can be obtained by a
variety of solvents, ISDs and INDs provide a useful platform to investigate morphological evolution of other dendritic structures. Finally, morphological evolution by
coarsening in freeze casting was found to be similar to those in alloy systems. Following other freeze-casting studies to apply solidification theory, the current study
validated that even post-crystal growth processes, coarsening, in alloy systems can
be applied to freeze casting, offering an additional strategy to control pores in freeze
casting.
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Chapter 6

APPLICATION OF FREEZE-CAST STRUCTURE:
MICROSTRUCTURAL ENGINEERING OF MATERIAL SPACE
FOR FUNCTIONAL PROPERTIES
This chapter is based on the work from the journal article, "Robust cellular shapememory ceramics via gradient- controlled freeze casting" by X. M. Zeng, N. Arai,
and K.T. Faber. X. Zeng and N. Arai both contributed to this work equally. This
article has been published in Advanced Engineering Materials.
Zeng X, Arai N, Faber KT. Robust cellular shape-memory ceramics via gradientcontrolled freeze casting Advanced Engineering Materials. 2019;21(12):1900398.
6.1

Introduction

Shape-memory ceramics are ceramics which undergo martensitic (diffusionless)
phase transformations by the aid of heat or stress. Figure 6.1 shows two properties
of shape memory ceramics: the shape-memory effect and the superelastic effect.
In the former, the material is deformed upon application of stress, but recovers its
original shape only when it is heated. The deformation and shape recovery are the
result of a forward martensitic transformation and reverse martensitic transformation
between tetragonal and monoclinic phases. For the superelastic effect, the material
undergoes reverse martensitic transformation when the applied stress is removed,
hence, it will deform with a large recoverable strain (∼ 1.9 % [1]). Shape-memory
ceramics, however, are known to experience a volume change and shape change
which leads to intergranular cracking. As a result, the shape-memory performance
is historically limited to only a few cycles [1].
Recently, a new strategy to mitigate this intergranular cracking has been reported
by Lai et al. It was demonstrated that the shape-memory effect in micro-/submicroscale pillars and particles exhibited superelastic behavior with significant deformation, full recovery and over 500 load cycles [3]. Despite their promising potential in
applications like actuation and energy damping [4], shape-memory properties are
found to be limited to small volumes to accommodate mismatch stresses along grain
boundaries [5]. Though the microscale dimensions are convenient for elucidating

Figure 6.1: A schematic showing shape-memory effect and superelastic effect [2].
This figure is reproduced with permission.

the fundamentals of material behavior [6], the challenge remains to transfer such
shape-memory properties into desirable 3D bulk forms for practical applications.
Addressing this challenge, therefore, involves the design of a suitable bulk structure that locally mimics the characteristic features of oligocrystalline pillars and the
development of appropriate fabrication approaches to realize such structures. One
approach involving the scale-up of particles in a granular form, where each particle acts as a transformation site, has proven effective in demonstrating high-energy
damping capacity at a pseudo-bulk scale [7]. Additionally, Crystal et al. reported a
single crystal shape memory zirconia and demonstrated repeatable transformation
without significant damages compared to polycrystals [8]. Alternatively, a onepiece porous foam with thin oligocrystalline walls has been reported, showing that a
significant volume fraction of the porous material (>60%) could experience martensitic transformations under an applied stress [9]. These studies motivate the concept
that a high specific surface area with oligocrystalline features accommodates stress
during martensitic transformation of grains and the associated large deformation
in bulk form [10]. However, the full potential in shape-memory ceramics is characterized by their unique properties of large recoverable strain at high mechanical
stress, which are not present in the aforementioned investigations. We propose that

87
its realization relies on a desirable 3D geometry with the following properties: 1)
a homogeneous feature size comparable with microscale pillars for transformation
events to occur uniformly in the structure [11]; 2) a particular cellular configuration
that can resolve the applied uniaxial force into uniform compressive stress to trigger
the martensitic transformation without introducing tensile or bending stress [12];
and 3) sufficient strength to survive a large compressive transformation stress before
reaching the fracture stress [6].
The approach in this work is to develop zirconia-based ceramics with a directionally
aligned honeycomb-like cellular porous structure, afforded by the structural tunability offered by gradient-controlled freeze casting. As shown in Chapter 3, the effects
of temperature gradient on pore morphologies were demonstrated in solution-based
freeze casting. Here, the same method is applied to suspension-based freeze casting
to achieve cellular porous structure.

Figure 6.2: Stability-microstructure map based on constitutional supercooling of a
solid–liquid interface controlled by freezing front velocity and temperature gradient (modified based on Rettenmayr and Exner [13]). Schematic illustration of b)
dendrites and c) cells.
The cellular structure can be considered an intermediate structure between the stable
planar front and the dendrites [14]. Thus, cellular structures span a narrow region
on the stability-microstructure map, referred as stability map, (Figure 6.2a) and can

Figure 6.3: The proposed shape-memory effect in a unidirectional cellular structure during uniaxial compression and heat treatment. The red highlights represent
transformed grains within the cellular walls.

only be achieved with very limited conditions of low freezing front velocity and
high temperature gradient [15].
By precisely controlling freezing front velocity and temperature gradient, the resulting cellular crystals are expected to yield a homogeneous unidirectional cellular
pore morphology (Figure 6.3). The unidirectional cellular structure in principle
would have high strength in the out-of-plane direction [12] for the material to be
mechanically deformed to reach phase transformation stress prior to fracture. The
thin cellular walls would mimic the features of oligocrystalline pillars, offering a
feasible approach for exhibiting the shape-memory effect in a bulk structure. During
mechanical compression, grains with suitable crystal orientations can experience the
martensitic transformation that leads to large deformation, whereas those nontransformed grains serve as the framework to provide sufficient mechanical strength for
structural integrity. With such a design, shape recovery can be achieved through subsequent heat treatment to trigger reverse martensitic transformation to demonstrate
a full cycle of shape-memory effects. In this study, in addition to the shape-memory
effect, the superelastic effect was also examined.
6.2
6.2.1

Experimental methods
Suspension preparation

A ceramic powder suspension was prepared by mixing zirconia (ZrO2 ) nanopowders and ceria (CeO2 ) nanopowders (99.9%, Inframat Advanced Materials) with
cyclohexane (99.5%, Sigma-Aldrich). The suspension compositions were set to a
10 vol.% solid loading for a target porosity of 70 %. A powder mixture having
a composition of 12.5 mol% CeO2 –87.5 mol% ZrO2 and 14.5 mol% CeO2 –85.5
mol% ZrO2 was chosen so that the ceria-doped zirconia exhibits the shape-memory
effect and the superelastic effect, respectively. For shape memory, the composition

89
was deliberately selected to control the characteristic transformation temperature,
at which the thermally induced tetragonal/monoclinic phase transformation occurs,
to be in the vicinity of room temperature. For superelasticity, the composition was
selected to make the austenite finishing temperature to be below the room temperature. In the bulk, ∼13 mol.% is a superelastic composition [16] while ∼15 mol.% is
a superelastic composition for granular particles [7] due to the less constraint from
the matrix [17]. Hence, 14.5 mol.% was chosen for porous form as superelastic
composition. Among various suspension media used in freeze casting [18], cyclohexane was chosen in this study to produce dendritic/cellular pore structures. A
dispersant of Hypermer KD-4 (Croda Inc.) was added at a concentration of 7 wt.%
of solid powders. The mixture was ball milled for 48 h with zirconia milling balls
to achieve a homogenous suspension.
6.2.2

Gradient-controlled freeze casting

Figure 6.4: Plots showing (a) freezing front velocity and (b) temperature gradient
as a function of frozen height.
The suspension was freeze-cast in a glass mold with an inner diameter of 24 mm
and a height of 12.5 mm. The freezing front velocity of pure cyclohexane solvent
and dispersant with no ceramic powders (Figure 6.4a) was measured. The method
to measure freezing front velocity and temperature gradient can be found in Chapter
3. The temperature at the freezing front was assumed to be 6◦ C, the melting point
of cyclohexane. The melting point depression effect from the dispersant is not taken
into account when temperature gradient is calculated. Four different freezing front
velocity conditions at the constant temperature gradient were studied in this work

90
(Figure 6.4b). The frozen samples were placed in a freeze dryer (VirTis AdVantage
2.0; SP Scientific, Warminster, PA, USA) to fully sublimate cyclohexane. Finally,
the samples were sintered in air at 1500◦ C for 3 h at a ramping rate of 2◦ C/min, after
holding at 550◦ C for 2 h to burn out any residual organic compounds.
6.2.3

Characterization

The microstructures were observed using a scanning electron microscope (SEM;
Zeiss 1550VP, Carl Zeiss AG, Oberkochen, Germany). The pore size distribution
was characterized using mercury intrusion porosimetry (MIP; Auto Pore IV, Micromeritics, Norcross, GA, USA). The samples were uniaxially compressed along
the longitudinal direction (parallel to the freezing direction) with a universal testing
machine (Instron 5982, Norwood, MA, USA), with a displacement rate of 0.06
mm/min. An X-ray diffractometer (PANalytical X’Pert Pro, Cu K𝛼, I= 40 mA,
V=45 kV) was used to analyze the phase content before and after compression tests,
with 2𝜃 ranges between 25–35◦ and a scan rate of 1◦ /min.
6.3
6.3.1

Results and discussion
Morphological control

As can be seen in Figure 6.5, the chosen conditions allow one to horizontally shift the
locus on the stability map between dendritic and cellular regions, as evidenced by the
obtained microstructure corresponding to each condition. The secondary arms of the
dendrites (at v of 11.57 and 8.19 µm s−1 ) become shorter at a lower v of 3.87 µm s−1
to form a transitional structure with wavy surface cellular walls. At a low v of 1.43
µm s−1 , a cellular structure with well-aligned straight walls and no secondary arms
is developed. As ceramics are much stronger under compression than under tension
or bending [4], the cellular structure is considered critical to effectively constrain the
resolved applied force to be mainly compressive on the walls, instead of the complex
stress field expected in a dendritic structure which can easily lead to local fracture.
Even though similar freezing front velocities and temperature gradients were used to
freeze-cast the preceramic polymer solution in Chapter 3, the pore structure was still
dendritic. Three possible reasons can explain this difference. First, this could be due
to the dissolved preceramic polymer at higher concentrations, which would cause a
large degree of constitutional supercooling [18]. Although, in the suspension, the
interfacial undercooling is caused by the particulate consitutional supercooling and
solute constitutional supercooling, You et al. showed that interfacial undercooling
mainly comes from solute constitutional supercooling caused by solutes in the

Figure 6.5: Stability-microstructure map based on measured freezing front velocity and temperature gradient of cyclohexane, with the corresponding longitudinal
microstructures of freeze-cast zirconia-based ceramics.

92
solvent and particulate constitutional supercooling is minor based on quantitative
measurements [19]. In suspension, the dispersant can be considered as the solute
dissolved in small concentration, which makes it easier to achieve cells with the
conditions examined in this study. Second, the diffusivity of the solute also affects
the degree of constitutional supercooling. The preceramic polymer has a high
molecular weight to avoid volatilization during the pyrolysis, and low diffusivity of
preceramic polymer results in a larger degree of constitutional supercooling. Third,
it has been observed by Sekhar and Trivedi that the presence of particles changes
morphologies from dendritic to cellular due to solute accumulation between the
particle and the freezing front. This results in a smaller concentration gradient in
front of the interface, hence it leads to a small degree of constitutional supercooling
[20].

Figure 6.6: Microstructure of freeze-cast cellular zirconia-based ceramics viewed
from (a) the transverse (the inset image shows an off-axis view of pores) and (b) the
longitudinal directions. Oligocrystalline cellular walls from (c) the transverse and
(d) longitudinal directions. (e) Pore size distribution within the measurement range
of 100 nm–80 µm from mercury intrusion porosimetry, with inserted sample image
after machining.
The cellular structure obtained with the lowest freezing front velocity is homogeneous throughout the sample with a height of 3 mm and porosity of 70% (Figure
6.6a,b). The structure is honeycomb-like with an array of pores formed between thin
vertical walls that align along the freezing direction. The average grain size is 2 µm,
whereas the wall thickness is 2–4 µm, indicating that the walls are largely oligocrystalline with only one or two grains in the thickness direction (Figure 6.6c, d), thereby
successfully mimicking the oligocrystalline pillar structures. The pore size measured with mercury intrusion porosimetry shows a narrow unimodal distribution

93
around 20.3 µm (Figure 6.6e).
6.3.2

Compressive response of shape-memory system

Figure 6.7: Stress–strain behavior of the cellular structure (v = 1.43 µm s− 1),
transitional structure (v = 3.87 µm s− 1), and dendritic structure (v = 11.57 µm
s− 1) under a compressive stress of 25 MPa (a). (b) The evolution of phase content
on compression and after heat treatment, with inserted XRD patterns of cellular
structure corresponding to each condition. (c) Stress–strain curves of the transitional
structure tested consecutively at stresses from 10 to 40 MPa. (d) The change in the
monoclinic content of all samples after compression as a function of applied stress,
with inserted XRD patterns of the transitional structure in between each compression
test.
The mechanical response of porous ceramics with various microstructures was
studied by applying a uniaxial compressive force along the longitudinal direction; a
second set of mechanical tests was accompanied by a phase content study with X-ray
diffraction (XRD) between stress increments. Under monotonic loading to 25 MPa
(Figure 6.7a), linear elastic behavior was observed for all samples up to 20 MPa. The
major difference lies in their behavior above 20 MPa, where cellular structures experience a marked decrease in slope, reaching a maximum strain of 7.5% at 25 MPa.
Upon unloading, a residual strain of 3.9% persists, a magnitude comparable with
shape-memory pillars [5, 6]. The dendritic and transitional structures both exhibit
much smaller deformations with residual strains of less than 0.4%. The significant

94
variation of stress–strain behavior in cellular, transitional, and dendritic structures
supports the hypothesis that only with a precisely designed 3D cellular structure
can the shape-memory effect be observed in bulk form. The phase composition was
calculated based on the intensity ratio of XRD peaks: (111̄)𝑚 , (111)𝑡 and (111)𝑚
between 27 to 32° 2𝜃. All samples were composed of 2.7∼18.9% monoclinic phase
before compression (Figure 6.7b).

Figure 6.8: XRD spectrum of a sample (a) after machining, and (b) after annealing
without experiencing mechanical compression.
The monoclinic phase in the as-processed samples was introduced during the machining process to obtain a disk-like shape for compression tests (Figure 6.8). An
annealed sample after machining was determined by XRD to have no monoclinic
phase content. Cellular structures experienced a significant tetragonal → monoclinic phase change of 11.5% during compression, whereas the transitional and dendritic structures experienced only 6.6% and 3.0% transformation, respectively. All
samples remained intact after compressive tests without any noticeable macroscale
cracks. The typical abrupt stress drop in a brittle honeycomb structure that signifies
the beginning of the brittle fracture of cell walls [14] was not observed in any cellular
structures. No further mechanical tests were conducted on cellular structures since
the as-compressed samples were composed of a 24.3% monoclinic phase, which
we consider to be significant enough for shape deformation, whereas 75% of the
parent tetragonal phase would provide sufficient mechanical support to preclude
fracture. All compressed samples were annealed at 700°C for 2 h, after which
only the tetragonal phase was observed, suggesting a complete reverse phase transformation during heat treatment. To confirm the thermal-induced shape recovery
in the cellular structure, the dimensions of a second identically processed sample
were recorded before compression, after compression to a maximum strain of 6.4%,
and after heat treatment. The compression test was halted as soon a drop in load
was detected, which suggested the onset of structural failure (Figure 6.9). Hence,
we expected only partial strain recovery from heat treatment, measured here to be

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43–48% (Table 6.1).

Figure 6.9: The stress-strain curve of the sample used for the shape recovery
measurement.

The large residual strain on loading above 20 MPa, the XRD evidence of the stressinduced phase transformation, and the fully reversible phase transformation on
heating indicate that the cellular structures exhibited the shape-memory effect. The
critical stress of martensitic transformation of grains is highly orientation dependent, varying between 100 MPa and 2 GPa [11]. Therefore, the random distribution
of grain orientations in these cellular structures leads to a continual tetragonal →
monoclinic transformation at different stress levels and a marked decrease in slope,
instead of a single flat plateau in the strain, as observed in single-crystal pillars [6]
or a step-wise plateau, as in oligocrystalline pillars [21]. According to Gibson and
Ashby [12], for perfect cellular structures, walls are effectively compressed when
a compressive stress is applied, whereas more poorly aligned structures like foams
experience a complex stress field under compression. For the dendritic structure,
a complex stress field involving compression, tension, and bending is expected
and therefore limits the material fraction that participates in phase transformation
through compressive deformation. In the transitional structure, the walls are well
aligned but with high surface waviness, leading to an inhomogeneous compressive
stress distribution across the walls. Consequently, a smaller fraction of grains is

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Table 6.1: Sample height and diameter before compression, after compression,
and after heat treatment; associated residual and recovered displacements used to
establish recovered strain.
Before
compression

After
compression

After heat
treatment

Residual displacement

Displacement
recovery

2.43

2.33

2.39

0.10

0.06

2.44

2.33

2.36

0.11

0.03

2.43

2.35

2.38

0.08

0.03

2.43

2.33

2.38

0.10

0.03

2.44

2.32

2.37

0.12

0.05

Average

2.43

2.33

2.38

0.10

0.04

43%

Diameter
(mm)

Before
compression

After
compression

After heat
treatment

Residual displacement

Displacement
recovery

Recovered
strain

9.77

9.80

0.03

0.00

9.77

9.87

9.83

0.10

0.04

9.78

9.85

9.81

0.07

0.04

9.79

9.85

9.82

0.06

0.03

9.76

9.83

9.78

0.07

0.05

9.77

9.84

9.81

0.07

0.03

Height (mm)

Measurement

Average

Recovered
strain

48%

able to reach the critical transformation stress, resulting in a negligible change in
slope (Figure 6.7a and v = 3.87 µm s−1 ). This limited nonlinearity is reminiscent
of the stress–strain behavior of granular shape-memory powders [7], where transformation is limited by nonuniform stress distribution. The extent of this effect is
further evaluated by applying ascending stresses from 10–40 MPa to the transitional
structure (Figure 6.7c). The transitional structure survived a maximum stress of 40
MPa without any macroscale fracture, providing latitude for a significant volume of
the ceramics to experience transformation prior to fracture. The stress–strain curves
are plotted with the residual strain of each test accounted for; the total residual
strain of 2.5% lies between that of cellular and dendritic structures. The change
in monoclinic content in between each compression test is shown in Figure 6.7d,
together with those of cellular and dendritic structures. The slope of the change

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in monoclinic content against applied stress, an indication of the effectiveness in
triggering the transformation through compression, increases from dendritic to transitional to cellular structures. The general trend of the increasing slope with applied
stress is due to a nonlinear distribution of transformation stress over random crystal
orientations [6]. The high correlation between the change in monoclinic content
and residual strain further supports the idea that a homogeneous compressive stress
in the walls is most desirable for inducing shape-memory effect in ceramics. The
difference in the monoclinic phase introduced during the machining process in
the as-processed samples also qualitatively suggests the variation in difficulty in
triggering the deformation through shear cutting.
6.3.3

Cyclic experiments

Up to this section, only one cycle of forward and reverse martensitic phase transformation was demonstrated. Here, multiple cycles are demonstrated for both the
shape-memory effect and superelasticity. The materials were subjected to multiple
cycles of forward and reverse martensitic phase transformations to see if honeycomblike structures are robust. Microcrack formation of the materials were also investigated by analyzing the slope of the stress-strain curve.
All the cyclic experiments were performed on the samples frozen at velocities of 3.87
µms−1 (transitional structure) since these can sustain higher stresses, and therefore,
yield higher monoclinic contents, as can be seen in Figure 6.7(d). For the samples
with shape-memory composition, the samples were annealed after machining to
start with a fully tetragonal phase.
Shape-memory effect
When the sample was compressed at 35 MPa once, it was found that the monoclinic
phase after compression reaches only 3-4%, compared to the sample experiencing a
14% tetragonal phase transforming into a monoclinic phase at 35 MPa compression
(Figure 6.7c). Hence, having a residual monoclinic phase in the starting material
helps the nucleation and growth processed of the monoclinic phase. In order to
induce further phase transformation, the samples were compressed to 35 MPa five
consecutive times (Figure 6.10a). Specifically, the samples were tested by five
loading-unloading cycles, annealing at 700°C for 2 hours followed by XRD, and the
next set of five loading-unloading cycles resumed.
The first loading-unloading cycle has a larger hysteresis compared to other loading-

Figure 6.10: Stress-strain curves showing (a) five loading-unloading cycles. (b)
Monoclinic composition after each five cycles and after each anneal.

Figure 6.11: Slope of stress-strain curves as a function of applied stress (a). Each
data represents the slope of the 5th loading cycles from each set of five loadingunloading cycles. (b) Magnified plateau region.

unloading cycles (Figure 6.10a), which is due to the alignment of the sample.
After the first loading-unloading cycle, however, the other curves are consistent and
reproducible. After five loading-unloading cycles, the monoclinic phase increased
to a maximum of 12% (2nd set of in Figure 6.10b). This is similar to the monoclinic
fraction of shape-memory nanofiber yarns after bending five times [22]. As Figure
6.10b shows, the monoclinic composition gradually decreased after the second set
of five loading-unloading cycles. One possible explanation is due to the so-called
training effect, which is observed in both shape-memory ceramics [8] and shapememory alloys [23]. During the training effect, the material tries to find a favored
kinematic transformation pathway, and it takes some cycles for the material to

99
exhibit a consistent response [3, 8]. As a result, due to this effect, the transformed
monoclinic fraction might not be the same even if the same stress is applied.
In order to investigate microcrack formation during the test, the slope of the stressstrain curve during loading was plotted as a function of applied stress (Figure
6.11a). This is similar to the study of Gu and Faber to investigate stress-induced
microcracking [24]; they observed an elastic modulus decrease after a multiple
loading-unloading cycle of the specimen. The stress-strain curves from the 5th
cycle of each set of loading-unloading cycles were chosen as representative. As the
stress is applied, the slope continues to increase, and this is due to the non-linearity
caused by the realignment of the sample. At approximately 30 MPa, the slope
reaches a plateau; Figure 6.11b shows a magnified version of this region. The first
set has the highest slope, but the slope remains within similar values at subsequent
loadings. In addition to the possible microcrack formation at the first set, the fourth
set also has gradual change in the slope, indicating the microcrack formation.
Superelastic effect
Superelasticity was also demonstrated with compositions of 14.5 mol% CeO2 –85.5
mol% ZrO2 . Figure 6.12 shows five loading-unloading cycles at increasing compressive stress levels (10 MPa, 20 MPa and 24 MPa). Hence, except for the first
loading-unloading cycle at 10 MPa, which shows a large hysteresis due to the realignment of the sample, the loading-unloading cycles are consistent at all the stress
levels. With increasing stress, the size of the hysteresis increases, consistent with a
superelastic transformation. However, this is insufficient to determine if the material
underwent martensitic transformation. To provide further evidence of the transformation, the slope change in the loading curve was investigated. Shape-memory
ceramics exhibit slope changes in stress strain curves [1] due to the shape deformation resulting from the formation of detwinned martensite (monoclinic phase in
this study) [25] [26]. Hence, if a slope decrease is observed, it suggests that the
martensitic phase transformation takes place, although careful analysis is needed
to distinguish the transformation from microcrack formation and will be discussed
in the next paragraph. Figure 6.13 shows the slope of selected loading curves as a
function of applied stress. The slope shows a continuous increase when the sample
was compressed to 10 MPa. When the material was compressed to 20 MPa, the
slope reached the maximum values of ∼ 2.9 GPa at the stress of ∼ 18 MPa and
does not increase further. When the applied stress is increased to 24 MPa, the slope

100

Figure 6.12: Stress-strain curves showing five loading-unloading steps at 10 MPa,
20 MPa, and 24 MPa.

started to decrease, which can be indicative of the martensitic phase transformation.
The sample was further compressed to higher stress, but the sample failed. After
the compression test, the phase composition was analyzed by XRD, and confirmed
that the sample remained a tetragonal phase before and after the compression test.
To assess if the observed slope change is a consequence of the phase transformation or from microcrack formation, Figure 6.14 shows the change in slope of five
loading curves when the material was compressed to 24 MPa and above (denoted
as “Higher loading”). The slopes of the first four loading curves show consistent
and repeatable cycles although the slope is lower than others on the 5th loading,
indicating microcrack formation. At "Higher loading", the slope is significantly
lower than others, indicating further microcrack formation and possibly material
failure. Hence, it is possible that the material experienced the superelastic effect
accompanied by microcrack formation and failure after the 4th loading-unloading

101

Figure 6.13: Slopes of stress-strain curves as a function of applied stress(a). XRD
peak before and after compression (b).

Figure 6.14: Slope of stress-strain curves as a function of the applied stress. The
material was compressed to 24 MPa for 5 times, and above.

cycle. As a future direction, in-situ investigation of monoclinic phase evolution, for
example by neutron diffraction [27], during compression testing might be useful to
confirm the superelastic effect.

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6.4

Conclusions

In summary, with a precisely designed honeycomb-like cellular structure, the singleand oligocrystalline martensitic transformations have been successfully extended to
bulk-scale deformation to achieve the shape-memory effect in a 3D geometry. With
independent control of freezing front velocity and temperature gradient through
gradient-controlled freeze casting, the freeze-cast microstructures can be fine-tuned
into the desired cellular structure with feature sizes similar to those of shape-memory
ceramic micropillars. The resultant cellular structure can experience a significant
recoverable deformation of up to 7.5% under compression at a stress of 25 MPa. The
cyclic experiments were performed to assess shape-memory and superelastic effects.
The shape memory material had initially 100 % tetragonal phase and resulted in a 712 % tetragonal phase upon compressing to 35 MP. Cellular shape-memory zirconia
demonstrated four cycles of the shape-memory effect. The superelastic effect was
also studied by looking at the hysteresis and slope changes of stress-strain curves.
Both were consistent with the superelastic effect. After multiple loading steps,
however, a decrease in the stress-strain slope suggested that microcracks started to
form and ultimately led to the material failure.
References
[1] Patricio E. Reyds-Morel, Jyh-Shiarn Cherng, and I-Wei Chen. “Transformation plasticity of CeO2-stabilized tetragonal zirconia polycrystals: II, pseudoelasticity and shape memory effect”. In: Journal of the American Ceramic
Society 71.8 (1988), pp. 648–657.
[2] Katherine T. Faber. “Small Volumes Create Super (elastic) Effects”. In: Science 341.6153 (2013), pp. 1464–1465.
[3] Zehui Du et al. “Synthesis of monodisperse CeO2–ZrO2 particles exhibiting
cyclic superelasticity over hundreds of cycles”. In: Journal of the American
Ceramic Society 100.9 (2017), pp. 4199–4208.
[4] Xiaomei Zeng et al. “Enhanced shape memory and superelasticity in smallvolume ceramics: a perspective on the controlling factors”. In: MRS Communications 7.4 (2017), pp. 747–754.
[5] Zehui Du et al. “Size effects and shape memory properties in ZrO2 ceramic
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[7] Z. Yu Hang et al. “Granular shape memory ceramic packings”. In: Acta
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[8] Isabel R. Crystal, Alan Lai, and Christopher A. Schuh. “Cyclic martensitic
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[9] Xueying Zhao, Alan Lai, and Christopher A. Schuh. “Shape memory zirconia
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[10] Xiao Mei Zeng et al. “In-situ studies on martensitic transformation and hightemperature shape memory in small volume zirconia”. In: Acta Materialia
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[11] Zehui Du et al. “Superelasticity in micro-scale shape memory ceramic particles”. In: Acta Materialia 123 (2017), pp. 255–263.
[12] Lorna J. Gibson and Michael F. Ashby. Cellular solids: structure and properties. Cambridge university press, 1999.
[13] M. Rettenmayr and H.E. Exner. “Directional Solidification”. In: (2001).
[14] W. Kurtz and D.J. Fisher. Fundamentals of solidification, Trans Tech. 1998.
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[16] Edward L. Pang, Caitlin A. McCandler, and Christopher A. Schuh. “Reduced
cracking in polycrystalline ZrO2-CeO2 shape-memory ceramics by meeting
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[17] Paul F. Becher and Michael V. Swain. “Grain-size-dependent transformation
behavior in polycrystalline tetragonal zirconia”. In: Journal of the American
ceramic society 75.3 (1992), pp. 493–502.
[18] Maninpat Naviroj, Peter W. Voorhees, and Katherine T. Faber. “Suspensionand solution-based freeze casting for porous ceramics”. In: Journal of Materials Research 32.17 (2017), pp. 3372–3382.
[19] Jiaxue You et al. “Interfacial undercooling in solidification of colloidal suspensions: analyses with quantitative measurements”. In: Scientific reports 6.1
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[23] Carmine Maletta et al. “Fatigue properties of a pseudoelastic NiTi alloy:
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Chapter 7

APPLICATIONS OF FREEZE-CAST CERAMICS: PORE SPACE
DESIGN FOR FILTRATION
The work in Section 7.1 was done in collaboration with Orland Bateman. N.
Arai fabricated and analyzed freeze-cast ceramics. O. Bateman performed flowthrough experiments and analyzed the data. N. Arai performed SEM imaging on
the membranes after flow-through experiments. N. Arai and O. Bateman performed
in-situ observation of particle flows by laser scanning confocal microscope.
The work in Section 7.2 was done in collaboration with Orland Bateman. N.
Arai fabricated and analyzed freeze-cast ceramics. O. Bateman performed in-situ
polymerization with phase separation micromolding. N. Arai performed SEM
imaging and water flux measurement.
Up to this Chapter, it was demonstrated that from a single solvent (cyclohexane),
not only can pore size be controlled through solidification parameters, but also pore
morphology can be tailored from dendritic pores to cellular pores. In this chapter,
the unique pore space provided by dendrites and cylinder-like crystals are utilized
for filtration applications.
7.1
7.1.1

Size-based filtration by dendritic pores
Introduction

Sepsis is a life-threatening condition caused by the body’s response to an infection.
Each year, at least 1.7 million adults in USA have sepsis, and nearly 270,000 die
from sepsis according to the Centers for Disease Control and Prevention (CDC). It
is a medical emergency which requires a timely diagnosis and antibiotic therapies
since the patient survival rate drops significantly after 36 hours [1] (Figure 7.1).
Antibiotic therapies start with broad-spectrum antibiotics until the pathogens are
identified by methods such as blood cultures [2] for effective treatment. Although
blood cultures are considered to be the gold standard to determine pathogens in
bloodstream and can be detected as low as 1 colony-forming unit (CFU) of bacteria
in 10 ml blood, cultures can take up to 96 hours [3]. As a result, the fraction of
patients treated with the most effective treatment remains low at a point when the
survival rate is high.

106

Figure 7.1: A graph showing patient survival rate and patients with effective antibiotic therapy [1].

However, these challenges are currently being addressed by the recent development of digital quantitative detection. Schlappi et al. developed a method to
capture and detect nucleic acid at zeptomolar concentration from MES (2-(Nmorpholino)ethanesulfonic acid) buffer with in-situ amplification in a short period
of time [4]. In addition, it was demonstrated that the digital detection can be applied
to antimicrobial susceptibility testing, reducing the time of the test to within 30
minutes [5]. Furthermore, a study by Rolando et al. demonstrated a phenotypic
antibiotic susceptibility test on urine samples from patients who were diagnosed
with urinary tract infections. Because this assay can be performed by using commercially available microfluidic chips and reagents and open-source components,
the advance is significant [6]. Although studies mentioned above are critical for
fast diagnosis for sepsis, the remaining challenge is the development of membranes
which can rapidly capture and concentrate pathogens from the bloodstream into a
small volume so that digital detection can be applied to complete the workflow to
diagnose and treat sepsis. Hence, the goal of this study is to develop a membrane
which captures pathogens in a sample of blood with a high capture efficiency and
concentrates them into a small volume in 30 minutes.

107

Figure 7.2: Illustration of (a) elasto-inertial based particle focusing and separation
[7] (Reproduced under Creative Commons) and (b) larger cells enter into vortices
due to the larger net-force acting on larger cells [8] (Reproduced with permission.)

Faridi et al. demonstrated the removal of red blood cells from blood to isolate
bacteria using elasto-inertial-based particle focusing and separation [7] (Figure
7.2a). Although the bacteria were captured with an efficiency of 76 % from blood,
the method was limited due to a slow flow rate (∼ 60 µL/h) and faced challenges with
scalability. Work by Hur et al. demonstrated that laminar vortices inside a cavity
can selectively isolate large cancer cells [8] (Figure 7.2b). This separation is based
on the net-force acting on the particles, which pushes the larger particles toward
the vortex centers in the cavity and traps them while small particles flow through
the channel. Although this method achieves high throughput with a processing rate
of ml/min scale, this method is particularly useful to selectively isolate larger cells
or particles. Hence, there is a need to develop membranes which can isolate and
concentrate small pathogens from the bloodstream with high throughput.
To achieve high throughput and isolate pathogens from the complex fluids, freezecast dendritic pores were examined in this work. Figure 7.3 shows an illustration
of how blood containing small pathogens flow in the dendritic pores. When fluid
flows through primary pores, secondary pores will exhibit recirculating flow which
is much slower than the flow in the primary pores. This phenomenon, sometimes
called "flow over cavities," has been investigated in a number of studies [9, 10].
Since primary and secondary pore sizes can be controlled through solidification
parameters during freeze casting, dendritic pores can be designed and fabricated such
that larger blood cells flow through primary pores while pathogens and platelets are
small enough to diffuse into recirculating flows in secondary pores. In this section,
this mechanism is referred to as hydrodynamic trapping. Capturing small pathogens
relies on diffusion, hence, slower flow velocity is essential to ensure sufficient time

108

Figure 7.3: An illustration showing fluid flow in the dendritic pores. Large blood
cells flow through the primary pores while small pathogens enter a recirculating
flow in secondary pores.
for diffusion of pathogens. While this concept might contradict the necessity of
high-throughput, high primary pore density (more than thousands of primary pores
per square centimeter) tunable by gradient-controlled freeze-casting would enable
high-throughput processing. For example, by increasing the primary pore density,
the throughput can be set to constant while the fluid flow velocity in each primary
pore can be decreased. In this chapter, the following results are reported:
• Preferential capture of small particles with dendritic pores
• In-situ observation of a particle captured by a secondary pore using confocal
microscopy
• Design of a dual structure to mitigate surface accumulation of particles.
7.1.2

Experimental methods

Fabrication characterization of freeze-cast membranes
The preceramic polymer (Silres®MK Powder) was dissolved in cyclohexane at a
concentration of 20 wt.%. A cross-linking agent (Geniosil GF 91) was added at a
concentration of 1 wt.% with respect to the solution and stirred for 5 min. Four
different freeze-cast structures were fabricated in this study. The first structure
was freeze-cast using a conventional freeze-casting setup, in which temperature is

109
controlled on only one side (bottom side) of the mold. The solutions were frozen
at a freezing front velocity of 15 µm/s. The second sample type was freeze-cast
with the coarsening process described in Chapter 5. The freezing front velocity
and temperature gradient were 15 µm/s and 2.6 K/mm, respectively, and the sample
was coarsened at 4 ◦ C for three hours. This resulted in a honeycomb-like structure.
These freeze-cast structures were used for flow-through experiments. The third
freeze-cast structure was fabricated for confocal microscope observation using the
gradient-controlled freeze-casting setup discussed in Chapter 3. The membrane was
freeze-cast with a freezing front velocity of 15 µm/s and a temperature gradient of
2.6 K/mm. As a fourth freeze-cast structure, a dual structure was fabricated with
the cooling profiles shown in Figure 7.4. This structure contains dendritic pores and
cellular pores, and will be described in detail later. Top and bottom temperatures
refer to the temperature of the top and bottom thermoelectrics, respectively.

Figure 7.4: Cooling profiles for top and bottom thermoelectric plates to create a
dual structure. The red-shaded region creates dendritic pores and the green-shaded
region creates cellular pores.
For a flow-through experiment, the samples were pyrolyzed under argon. For the
samples prepared for confocal microscopy and the dual structure, pyrolysis was
done in the presence of water vapor in addition to argon to remove free carbon in the
SiOC. A beaker containing water kept at 90 ◦ C provided the source for water vapor.
This was feed into the argon line flowing into the tube furnace. Argon gas flow rate

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was kept at 1.5 standard cubic feet per hour (SCFH). Freeze-cast membranes of 3.2
mm thickness were sectioned using a diamond saw. After flow-through experiments,
membranes were imaged by a SEM (ZEISS 1550VP, Carl Zeiss AG, Oberkochen,
Germany). Pore size of the dual structure was characterized by mercury intrusion
porosimetry (Auto Pore IV, Micromeritics, Norcross, GA, USA).
Flow-through experiments
In preparation of flow-through experiments, a glycerol solution was prepared by dissolving glycerol in water at a concentration of 30 vol.%. This solution was prepared
to eliminate density mismatch between particles and the suspending medium. The
ceramic membrane was immersed into the glycerol solution and left under in-house
vacuum (25 mmHg) overnight to fully infiltrate the solution in dendritic pores. The
particle suspension was prepared with 300 µL of glycerol solution and 20 µL of
poly(diallyldimethylammonium chloride) solution (Sigma-Aldrich, St. Louis, MO,
USA) added to 1,660 µL of water. Subsequently, 20 µL of 2 µm fluorescent particle
suspensions (Spherotech, Inc, Lake Forest, IL, USA) were added. After mixing, 20
µL of the 0.3 µm fluorescent particle suspensions (Spherotech, Inc, Lake Forest, IL,
USA) were further added and mixed thoroughly.
The flow-through experimental setup is shown in Figure 7.5. A syringe was filled
with 30 vol.% glycerol solution, which served as the working fluid. A microfluidic
device holding a membrane was connected to the syringe. A syringe pump was used
to drive flow of the working fluid at rates of 10 µL/min and 40 µL/min. After a steady
stream of droplets were obtained at the outlet tube, the flow rate was set to 10 µL/min,
and 300 µL of the particle suspension was added from the in-line injection connector.
Twenty-four aliquots, each with a volume of 200 µL were collected in a 96-well plate.
Then, the membrane was washed with several milliliters of working fluid using the
same setup1. Subsequently, the same flow-through experiment was performed on
the same membrane but with a flow rate of 40 µL/min. Three membranes were
tested using this procedure.
A plate reader (FlexStation®3 Microplate Reader, Molecular Devices, LLC, San
Jose, CA, USA) was used to measure the fluorescence signal from two particle
populations. To determine the background signal from the working fluid, the flu1 Several aliquots were collected and analyzed by the plate reader to see if the aliquots contained

any particles. It was confirmed that the fluorescent signal was within the error of the background
reading of the working fluid. Hence, it was assumed that the particles in the membrane were
irreversibly captured.

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Figure 7.5: A picture of the flow-through experimental setup.

orescent signal from glycerol solution was measured and subtracted. The total
particles retained in membranes after flow-through experiments were reported.
Confocal microscopy
A stock suspension (30 µL) of 2 µm polystyrene particles (Spherotech, Inc, Lake
Forest, IL, USA) was dried to remove the suspending medium. After being completely dried, particles were suspended in 20 mL of canola oil by sonication. Canola
oil was chosen as a suspending medium since the refractive index of white SiOC
and canola oil are similar so that the in-situ observation of particle flow in pores
is possible. The ceramic membrane was sliced into a 500 µm thick parallelepiped
and assembled into a device as shown in Figure 7.6. The sample was sandwiched
between a microscope slide and acrylic plates with Teflon tape to seal. Confocal
microscopy images were taken with a Zeiss LSM 710 (Carl Zeiss AG, Germany).
The setup for in-situ observation is shown in Figure 7.6. During the experiment, the
syringe pump was set to a flow rate of 10 µL/min.
7.1.3

Results and discussion

Flow through experiments
Figure 7.7 shows an SEM image of a dendritic structure in the transverse direction
and the corresponding pore size distribution. This membrane was chosen for flowthrough experiments since both primary (∼ 20 µm) and secondary pore sizes (∼ 14
µm) are larger than the particles used in this study. Hence, the capture of particles
due to clogging of pores is unlikely. Additionally, the secondary pore volume
fraction is sufficiently large so there is ample space for particles to be captured.

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Figure 7.6: A picture of the confocal microscope setup.

Figure 7.7: An SEM image and pore size distribution of a membrane used in the
flow-through study.

The flow-through experiments were conducted by flowing two different size particles
(0.3 µm and 2 µm) at two different flow rates, 10 µL/min and 40 µL/min, and the

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Table 7.1: Particles captured in the flow-through experiments.
10 µL/min

40 µL/min

0.3 µm

2 µm

0.3 µm

2 µm

Membrane 1

74.7%

57.1%

59.3%

40.7%

Membrane 2

74.7%

81.3%

55.8%

43.9%

Membrane 3

79.4%

72.4%

60.8%

42.6%

results are summarized in Table 7.1. Two important trends can be observed. First,
comparing the two different flow rates, a larger number of particles are captured with
a slower flow rate (10 µL/min). This is not surprising; particles have longer residence
time inside membranes, which allows diffusion into the secondary pores. Second,
membranes tend to preferentially capture 0.3 µm particles. Although membrane 2
captures more 2 µm than 0.3 µm at 10 µL/min, all other samples and flow rate show
that smaller 0.3 µm particles were captured preferentially. This is consistent with
Stokes-Einstein equation:

𝐷=

𝑘 𝐵𝑇
6𝜋𝜂𝑟

where D is the diffusion coefficient of spherical particle, 𝑘 𝐵 is Boltzmann’s constant,
T is the absolute temperature, 𝜂 is the dynamic viscosity, and r is the radius of
spherical particles. Smaller particles have a higher diffusion coefficient, hence, they
are captured by membranes with higher probability.
Although further investigations are necessary to understand discrepancies in capture
efficiency, one possible reason is air bubbles in the microfluidic device. Although
great care was taken to avoid air bubbles when the membrane was assembled in the
microfluidic device, sometimes air bubbles can be trapped in the device. Since the
fluorescent particles are hydrophobic and suspended with a surfactant, they could
remain at bubble/water interface. In such a case, the capture efficiency would be
overestimated.
Observation of particle flow inside the dendritic pores
In order to confirm if particles are captured by secondary pores, in-situ observation
of particle flow in dendritic pores was conducted. Although SiOC is black in color

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Figure 7.8: Pictures of freeze-cast SiOC pyrolyzed under (a) Ar and (b) Ar with
water vapor.
due to the presence of sp2 carbon, this carbon can be removed by introducing water
vapor during the pyrolysis, making the SiOC white in color [11]. Figures 7.8a and
b show pictures of freeze-cast SiOC pyrolyzed under argon and argon with water
vapor, respectively. As shown, when the preceramic polymer is pyrolyzed in the
presence of water vapor, SiOC turned white. This is necessary to observe particle
flow inside dendritic pores using the confocal microscope since white SiOC does
not absorb light. Thus, by refractive index matching, white SiOC can be transparent
as demonstrated in Figure 7.9. It was found that canola oil is a promising working
fluid to use in this experiment.
A series of confocal microscope images (left column: overlaid bright field and
fluorescent images, right column: fluorescent images) were shown in Figure 7.10.
The fluid flow direction is from right to left in the images. This image focuses on the
movement of one of the particles, which is indicated by a red arrow in each image.
From t = 0 s to t = 45 s, the particle travels along a primary pore. After t = 45s, the
particle was captured by a secondary pore, demonstrating that secondary pores are
essential for capturing particles.
Dual structure
One of the challenges which must be overcome is particle accumulation on the
surface of the membrane. The idea of hydrodynamic trapping is to capture small
particles (pathogens and platelets) by secondary pores while the large particles (white
blood cells and red blood cells) flow through primary pores. Hence, the design of a
membrane which allows the majority of particles to enter primary pores is essential

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Figure 7.9: Pictures showing SiOC pyrolyzed under Ar and H2 O atmosphere with
pores filled with (a) air, (b) DI water, and (c) canola oil (n: refractive index).

for this concept. Figure 7.11a shows the inlet surface of the dendritic structures
after flow-through experiments, and the majority of the surface consists of SiOC
walls. This leads to the accumulation of particles on the surface of the membrane
as shown in the magnified image (Figure 7.11b). On the other hand, Figure 7.11c
shows honeycomb-like structures with smaller areas of SiOC walls. As shown in
a magnified image in Figure 7.11d, the amount of particles accumulated on the
surface was significantly reduced. This motivated the design of a dual structure,
which contains both cellular pores and dendritic pores. This dual structure is
possible by controlling the freezing front velocity and temperature gradient afforded
by gradient-controlled freeze casting developed in Chapter 3. Figure 7.12a shows a
longitudinal image of dual structure revealing ∼400 µm of cellular pore region and a
large portion of dendritic pores. Figures 7.12b and c show transverse images of the
cellular pore region and dendritic pore region, respectively. This structure might be
an ideal structure to capture particles of interest by secondary pores. Cellular pores
act as funnels so that the majority of particles enter into primary pores. As particles
travel along the cellular pore region, they enter into the dendritic pore region where
particles are captured by secondary pores. Pore size distribution further confirmed
the presence of cellular pores in addition to dendritic pores (Figure 7.12d).
7.1.4

Summary

Dendritic pores were investigated to see if they could be used for size-based filtration.
Flow-through experiments demonstrate higher capture efficiency when the fluid flow
rate is decreased. Moreover, smaller particles are preferentially captured by dendritic
pores. Both results indicate that diffusion is an important mechanism in capturing
particles. A slower flow rate provides longer residence time in the membranes to
provide sufficient time for particle diffusion into secondary pores. Smaller particles
have higher diffusion coefficients so smaller particles diffuse faster to secondary
pores. Furthermore, capture of a 2 µm particle by secondary pores is confirmed by

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Figure 7.10: Overlay of bright field and fluorescence micrographs from laser scanning confocal microscope. The series of micrographs shows a 2 µm particle (indicated by the red arrow) flowing along the main channel and being captured at the
side cavity after 45 seconds.

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Figure 7.11: SEM images showing transverse direction of dendritic structure after
flow-through experiment at (a) low magnification and (b) high magnification. SEM
images showing transverse direction of honeycomb-like structure after flow-through
experiment at (c) low magnification and (d) high magnification. Some of the 2
µm and a group of the 0.3 µm particles are indicated by yellow and red circles,
respectively.

in-situ particle flow observation by confocal microscopy. Finally, a dual structure
was fabricated to facilitate particle capture in secondary pores by mitigating surface
capture.

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Figure 7.12: SEM images showing (a) longitudinal direction and transverse direction
of (b) cellular pore region, and (c) dendritic pore region. (d) Pore size distribution
of a dual structure.

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7.2

Ceramic/polymer composites for membrane chromatography

In the next example, honeycomb-like structures as developed by coarsening described in Chapter 5 will be applied to create ceramic/polymer composites for
membrane chromatography.
7.2.1

Introduction

The monoclonal antibody (mAb) therapeutic market is fast-growing and is expected
to generate revenue of $300 billion by 2025 [12]. However, the downstream processing such as mAb capture and impurity removal, also known as product polishing,
comprised 80% of the manufacturing cost. Membrane chromatography has been
receiving attention to replace conventional resin-based column chromatography to
reduce the cost, increase throughput, and reduce operating pressure. However, the
challenges which current membrane chromatography is facing are: (1) low protein
binding capacity and (2) costly pleating steps. Recently, Kotte et al. demonstrated
fabrication of mixed matrix polyvinylidene fluoride (PVDF) membranes with embedded polyethylenimine (PEI) particles by in-situ polymerization with phase inversion casting [13]. The significance of this work is that it demonstrates a high
binding capacity and selectivity for proteins. The work in this section is built upon
this development of PVDF membranes to overcome the challenges of the current
commercial membrane chromatography.

In this section, a ceramic/polymer composite is explored to demonstrate the following advantages of the composites. First, by combining phase separation micromolding [14] and in-situ polymerization [13], functional polymer microgels can fill the
honeycomb-like structures of ceramics, which would allow thicker membranes with
uniform thickness. Typically, membrane thickness is limited to several hundred
micrometers. As a result, the membrane requires a pleating process to maximize the
filtration area within a small volume, and this pleating process needs to be carefully
designed to produce optimal filtration performance [15]. However, if a composite
can be fabricated with thicker dimensions, it would provide the ability to configure
the composite into scalable modules without the pleating process. Additionally,
creating polymer membranes with uniform thickness has been a challenge and a
focus of research since membrane thickness variations were known to significantly
broaden the breakthrough curve (lower binding capacity) [16]. Hence, creating
thicker membrane with uniform thickness by phase separation micromolding would
be beneficial for improving binding capacity and module configuration. Second, ce-

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ramic stiffness will add mechanical integrity to the membranes. This is particularly
important since cross-linking density and the type of cross-linker affect mechanical
properties of polymeric membranes [17] as well as other functional properties such
as adsorption [18]. Hence, if superior mechanical properties of composites are
demonstrated, one can then explore and optimize polymer composition for functional properties such as binding capacity while mechanical properties are ensured
by the ceramic scaffold.
This section reports two results: (1) successful demonstration of in-situ polymerization with phase separation micromolding in freeze-cast ceramics to create thick
membranes and (2) superior mechanical stability of the ceramic/polymer composite
during fluid flow.
7.2.2

Experimental methods

Fabrication of freeze-cast ceramics
Freeze-cast ceramics were fabricated using the coarsening process, the details of
which can be found in Experimental Methods (Subsection 5.2.1) of Chapter 5. The
freeze-cast solution was prepared by dissolving a polysiloxane (Silres® MK Powder)
preceramic polymer in cyclohexane with a concentration of 15 or 20 wt.%. A crosslinking agent (Geniosil® GF 91) was added in concentrations of 1 wt.% and stirred
for an additional 5 minutes. Subsequently, the polymer solution was degassed for
10 minutes. Next, the solution was quenched to -30◦ C and coarsened at 4 ◦ C for 1
hour. After sublimation, the freeze-cast preceramic polymer was pyrolyzed at 1100
◦ C for 4 hours under argon and water vapor unless otherwise mentioned. Water
vapor was introduced in the same way as described in Subsection 7.1.2. SiOC has
a silanol group [19] and water vapor was introduced to remove carbon and expose
more silanol groups on the surface for functionalization of the SiOC. The freeze-cast
ceramics were core-drilled into ∼13 mm diameter cylinders. A disc with a thickness
of 1.5∼1.6 mm was sectioned from the midsection by a diamond saw prior to the
functionalization with polymers.
Functionalization with polymers
Polyvinylidene fluoride (PVDF; Kynar, Arkema, Inc., Colombes, France) was
dissolved in triethyl phosphate (TEP; Sigma Aldrich) at 80 ◦ C. Under nitrogen,
polyethylenimine (PEI; Polysciences, Inc., Warrington, PA) dissolved in TEP was
added to the PVDF solution. After producing a homogeneous solution, ∼500 µL of

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concentrated hydrochloric acid (HCl; EMD millipore, Burlington, MA) was added
to the solution. After mixing for 15 minutes, a crosslinker, epichlorohydrin (ECH;
Sigma Aldrich), was added to the solution. After 4 hours of the cross-linking reaction, 10 mL of TEP was added and the resulting solution was mixed for 30 minutes.
This solution is called a dope solution, and its composition is 12.42 wt.% PVDF,
5.39 wt.% PEI, 3.55 wt.% ECH, and 78.64 wt.% TEP. The dope solution was then
put under vacuum for 10 minutes in preparation for infiltration into the pores of the
functionalized ceramic membranes.
Next, the surfaces of the freeze-cast ceramics were functionalized with amine groups
by the following procedures. Freeze-cast ceramics were immersed in concentrated
sodium hydroxide (NaOH; Avantor, Radnor, PA) for 90 minutes. After washing
freeze-cast ceramics with water, they were incubated in a 0.1 M HCl solution for
30 minutes. The freeze-cast ceramics were washed with water again and dried
at 110 ◦ C for 1 hour. Then, they were immersed in a 2 vol.% solution of (3Aminopropyl)trimethoxysilane (ATMS; Sigma-Aldrich, St. Louis, MO, USA) in
isopropanol. After being incubated for 3 hours at 60 ◦ C, the samples were washed
with water, and then isopropanol. After washing, they were cured at 110 ◦ C for 30
minutes.
Next, the freeze-cast ceramics were coated with a PEI gel layer using the following
procedure. A solution for a gel layer was prepared by mixing 0.78g of PEI and
1.68mL of ECH in 3mL of IPA. The freeze-cast ceramics were immersed in the
solution and left in the solution overnight at room temperature to form a PEI gel
layer. Finally, dimethyl sulfoxide (DMSO; Sigma Aldrich) was added and heated to
80 ◦ C for 1 hour to remove excess PEI. The sample was then washed with isopropanol
and dried at room temperature. One sample was fabricated without a PEI gel layer
to show its effect on bonding between the ceramic and the PVDF membrane.
The dried ceramics were placed inside an infiltration device and infiltrated with the
dope solution using a syringe pump. During the infiltration, the flow rate of the
dope solution was maintained at 100 µL/min. To promote cross-linking between the
ceramic gel layer and amine groups in the dope solution, the infiltrated samples were
heated to 80 ◦ C for 1 hour. Following the incubation, the samples were removed
and placed in isopropanol for overnight incubation. The samples were immersed in
water to remove trace solvents before characterization and testing.

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Figure 7.13: A schematic of (a) permeability setup. A figure taken from [20]. (b)
A picture of an acrylic fixture. (c) An illustration of side view of the acrylic fixture
holding a composite.

Characterization
The mechanical stability of the composites was characterized by flowing deionized
water for 95 minutes using a voltage-controlled pump using the setup shown in
Figure 7.13a [20]. Both the pressure drop and water flow rate were measured
simultaneously using a pressure transducer and an electronic scale, respectively.
Samples were held by acrylic fixture shown in Figure 7.13b. This fixture has an
outer diameter of 19mm and inner diameters of 13.8 mm and 10 mm due to the
internal step. A schematic illustration of the side view of the acrylic fixture holding
the composite is shown in Figure 7.13c. The composite was stuck to the acrylic
fixture by adhesives and Silly Putty was filled in the spaces between the composite
and acrylic fixture to avoid water leaks. This fixture holding the composite was
placed and clamped in the setup.

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SEM images of composites were taken in both transverse and longitudinal directions.
To image the longitudinal direction, the composites were simply snapped in half and
imaged.
7.2.3

Results and discussion

Ceramic/polymer composites

Figure 7.14: SEM images showing a composite without gel layer ((a) transverse
and (b) longitudinal direction) and a composite with gel layer ((c) transverse and (d)
longitudinal direction)
Figure 7.14 shows SEM images of ceramic/polymer composites and compares the
effect of the PEI gel layer on bonding between the ceramic and the mixed matrix
PVDF membrane, referred to here as a microgel. Figures 7.14a and b show a
composite fabricated without PEI gel layer between ceramics and the microgel in
transverse and longitudinal directions. The bonding between the ceramics and the
microgel is poor and transverse image show debonded regions. This debonding is
likely due to the shrinkage of the microgel during the drying process prior to SEM
imaging. The resulting stress by the shrinkage caused the microgel to peel off from
the ceramic. The longitudinal image also shows evidence of poor bonding. The
microgel had peeled off from the ceramic walls, probably caused by the fracture of a

124
composite for imaging purposes. In contrast, Figure 7.14c and d show a composite
with PEI gel layer between the ceramics along with the microgel, and demonstrate
enhanced bonding in both transverse and longitudinal directions. A significant
difference can be seen in the longitudinal direction. When the PEI gel layer is
present, the microgel is torn rather than peeled from the ceramic walls as shown in
Figure 7.14d. Hence, the PEI gel layer helps to hold the microgel and the ceramic
together and prevents the microgel from peeling off.

Figure 7.15: An SEM image showing a PVDF membrane, PEI gel layer, and SiOC
wall. Yellow dashed lines indicate boundaries between a gel layer and SiOC wall.
Figure 7.15 shows a magnified image of one of the fabricated composites, which
demonstrates characteristic features of the PVDF membrane with embedded PEI
particles, consistent with those reported by Kotte et al. on the same composition
[13]. The structure contains a matrix of PVDF spherulites with a fibrous texture.
This image proves that in-situ polymerization and phase separation micromolding
inside the honeycomb-like structure of ceramics was successful. The microgel is
bonded to a thin layer of PEI gel, which adheres to the SiOC wall. The yellow dashed
lines in the figure indicate the boundary between the PEI gel layer and ceramic walls.
The PEI gel layer is effective at holding the ceramic and the microgel together due
to a sufficient amount of amine groups to bond functionalized amine-terminated
ceramics and PEI particles of PVDF membrane through the cross-linker, ECH.

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Figure 7.16: An SEM image showing a thickness of around 1.5 mm composite.

Figure 7.16 shows a composite with a thickness of ∼1.5mm. (This ceramic scaffold
was pyrolyzed without the presence of water vapor.) As demonstrated, in-situ
polymerization with phase separation micromolding was successfully used to create
a ceramic/polymer composite thicker than conventional polymeric membrane with
uniform thickness.
Mechanical stability during fluid flow
The mechanical stability of the composite was characterized by flowing water for
95 minutes through the composite membrane. Figure 7.17a shows water flux and
pressure drop as a function of time. During the experiment, the pressure drop
remained between 1 and 1.2 bar and measured water flux is around 2200 Lh−1 m−2 .
In contrast, Figure 7.17b shows the water flux measurement of the PVDF membrane
at a different pressure by Kotte et al. [13]. Although the porous structure by Kotte et
al. and resulting transport properties are different from the ones of the current study,
the microgels in both studies were made with the same materials and composition.
Hence, these data can be used to compare mechanical stability (Figure 7.17a). The
composite in this study is shown to have superior mechanical stability to the PVDF
membrane by Kotte et al. For the PVDF membrane, after 45 minutes of flowing
water at 1 bar, the water flux decreased by roughly 21% due to the compaction of

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Figure 7.17: A plot of (a) water flux and pressure drop as a function of time. (b)
Water flux at different pressure drops as a function of time (from the study by Kotte
et al. [13]). This figure is reproduced with permission. SEM images of (c) inlet and
(d) outlet side after permeability measurement with sample pictures as insets.

PVDF membrane. In contrast, there is only a 5% drop of water flux in 45 minutes
for the composite. The experiment was maintained for more than 90 minutes and
the total water flux drop reached only 10%. This demonstrates that the ceramic
scaffold prevented compaction, which resulted in more stable water flux. After the
water flux measurement, the inlet and outlet sides of a composite were imaged by
SEM (Figures 7.17c and d) to check if the microgels were still held by the ceramics.
For both sides, the microgel still remained within the ceramics scaffold, indicative
of robust bonding between ceramics and the microgels. The picture of the inlet side
shows a gray color possibly due to contaminants from the permeability setup (inset
image of Figure 7.17c), as confirmed by the SEM image. Although it is still possible
that compaction of microgels might take place in the composite, it is likely that the
water flux drop in the composite is due to clogging of pores by these contaminants.
Hence, the ceramic scaffold provides mechanical stability during fluid flow, and this

127
superior mechanical property would allow this composite to be used in conditions
where other polymeric membranes might collapse or fail.
7.2.4

Summary

Ceramic/polymer composites were successfully created by in-situ polymerization
with phase separation micromolding in freeze-cast pore structures. Composite
membranes, 1.5 to 1.6 mm thick, were produced with characteristic features of PVDF
membranes reported by Kotte et al. [13]. The PEI gel layer between the microgel and
ceramic walls were necessary to ensure the robust bonding between the ceramics
and the microgel. The mechanical stability of composites was demonstrated by
flowing water for 95 minutes. The water flux drop of the composite remained lower
than the polymer-only membrane, which implies that the composite can be used in
conditions where the polymer-only membrane will collapse or fail.
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Chapter 8

SUMMARY AND FUTURE WORK
8.1

Summary and conclusions

Directional freeze casting was investigated from the standpoint of alloy solidification
principles. The growth and time evolution of dendrites were studied to control pore
morphology and size. Pore controls through established solidification parameters
led to pore structure designs for applications such as robust shape-memory ceramics
and pharmaceutical filtration.
Although freezing front velocity had been a major solidification parameter to control
pores in freeze casting, temperature gradient had often been neglected. Through
gradient-controlled freeze casting, both temperature gradient and freezing front velocity were controlled independently and the resultant dendritic pore structures were
investigated. While freezing front velocity changed primary pore size, temperature
gradient did not significantly change primary pore size, but rather, primary pore
spacing. In contrast, secondary pore sizes were found to depend on cooling rate, the
product of freezing front velocity and temperature gradient. As the freezing front
velocity was decreased at a constant temperature gradient, secondary pores began to
disappear and the freeze-cast microstructure evolved to a honeycomb-like structure,
indicating the transition from dendrites to cells. As demonstrated by the stability
criterion for planar solidification front, the transition to cellular crystals were not
only determined by solidification parameters but also by other variables such as the
concentration of the preceramic polymer. It was shown that dendritic pores further
turned into honeycomb-like structures by reducing the concentration of preceramic
polymer. These observations were consistent with the solidification principles of
alloys. In addition to controllable solidification parameters, effects of a ubiquitous
external force, gravity, was investigated. It was found that gravity-induced convection reduced the degree of constitutional supercooling in the liquid phase and
yielded long-range cellular pores (∼2 mm of cellular pores).
Although dendrite size and morphology were manipulated by dendrite growth conditions, post-crystal growth processing also significantly were found to impact the
dendritic structures. Thus, coarsening of dendrites and resulting pore structure were
investigated. Two important results were reported. First, primary pore size and

130
secondary pore size were found to depend on the cube root of coarsening time.
Second, when the dendrites were coarsened at the temperature close to the liquidus
temperature of the solution, the pore structure evolved to honeycomb-like structures.
Both findings agreed well with the observation in alloy systems. Tomography-based
analysis on curvature of dendritic pores aided the understanding of the coarsening
mechanisms. Interfacial shape distributions (ISDs) and interfacial normal distributions (INDs) were used to quantitatively define the shape and directionality of
dendritic pores, respectively, and confirmed similar coarsening behavior as alloy
systems.
The above studies provided foundations for pore designs to create freeze-cast ceramics for specific applications. The first example was porous shape memory ceramic.
The pore structures were designed and fabricated such that the structure was a mechanically robust honeycomb structure and, most importantly, the wall thickness
was comparable to grain size to mitigate intergranular cracking during the martensitic phase transformation. The structure was sustained through the martensitic
phase transformation as well as the accompanying deformation. While this was
an example of improving functional properties by microstructural engineering of
material spaces, other explored applications focused on unique pore space. A dendritic structure with tailorable primary and secondary pore sizes was examined for
filtration of pathogens in the bloodstream. Particle capture by secondary pores were
demonstrated. Flow-through experiments showed that particle capture efficiency
improves with decreasing the flow rate due to the longer residence time for diffusing
into secondary pores. In addition, smaller particles diffused faster to secondary
pores, and hence, were captured with higher probability. In-situ observation by
confocal microscopy confirmed particle capture by secondary pores. These results
demonstrated a potential application of dendritic pores as size-based filtration. Finally, honeycomb-like structures were filled with functional microgels for membrane
chromatography. Ceramic scaffolds for infiltration provided a template for thicker
membranes with uniform thickness, which had been a challenge in conventional
polymeric membrane fabrication. This design would potentially eliminate a costly
pleating step for polymer-only membranes and offer an opportunity to readily configure scalable membrane modules. Additionally, ceramics added mechanical stability
during the fluid flow, which might broaden operating conditions where polymeric
membranes might collapse or fail.

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8.2

Suggestions for Future work

8.2.1

Different solvent systems

Figure 8.1: SEM images of freeze-cast structures using cyclooctane as a solvent
in longitudinal direction. As the higher temperature gradient is applied, the directionality of pores improved. Left image is taken from a study by Naviroj et al.
[1]
In this work, effects of temperature gradient and coarsening process in solutionbased freeze casting were studied in detail. However, only cyclohexane was studied
as a solvent, and there are other solvents which produce seaweed structures, lamellar
structures, and highly anisotropic, two-dimensional dendritic structures [1] which
deserve attention. Figure 8.1 shows freeze-cast structures templated by cyclooctane
crystals. Cyclooctane-based microstructure is seaweed-like [2] and the resulting
ceramic microstructure is isotropic and non-directional, as shown in the left image. However, as the higher temperature gradient is applied, pore directionality
is improved (the middle and right images were freeze-cast with a slightly lower
preceramic polymer concentration than the one from the left image).
8.2.2

Rheology

Rheology is another important parameter in solution-based freeze casting. In
suspension-based freeze casting, changing viscosity requires additives or higher
solid loading. However, in solution-based freeze casting, rheology can be controlled by changing the molecular weight of preceramic polymers through chemical
cross-linking or thermal cross-linking. This gives further control in pore structure
and material property, and should be studied further.

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Figure 8.2: SEM images showing freeze-cast lamellar structures with (a) 5 minutes
and (b) 6 hours of stirring after adding the cross-linking agent.

Enhanced mechanical properties
In most freeze-casting studies, the preceramic polymer solution is directionally
frozen right after a cross-linking agent is added. Figure 8.2 shows SEM images
of freeze-cast lamellar structures from dimethyl carbonate. Although they are
freeze-cast from the same solvent and preceramic polymer, they exhibit significantly
different morphologies due to the different cross-linking time, the time the solution
had been stirring after addition of the cross-linking agent. Figure 8.2a shows a
typical lamellar structure which was fabricated after five minutes of cross-linking
time. In contrast, Figure 8.2b shows a porous structure after six hours of crosslinking time. This porous structure has bridges between lamellar walls. Although
bridges are effective at enhancing the mechanical properties of lamellar structures,
the mechanism by which bridge formation occurs is unknown. Further investigations
are required to elucidate this morphology change. In addition, it would be interesting
to explore how viscosity affects different porous structures, such as dendritic and
isotropic-like structures, and the resultant mechanical and transport properties.
Precise control of pore structure
As shown in Chapter 3, the stability criterion for stable planar front can be expressed
as follows:
𝐺 𝑚 𝑘0 − 1
𝐶0 .
𝐷 𝑘0
In most examples of freeze-casting of preceramic polymers, a cross-linking agent is
added prior to solidification. Once the cross-linking agent is added, the cross-linking

133

Figure 8.3: Compressive strength and permeability constants of different structures.
Data for "Lamellar 15 µm/s" and "Dendritic 15 µm/s" are taken from the work by
Naviroj [2].

process starts and increases the molecular weight of preceramic polymer, resulting
in a change in the diffusion coefficient, D, as a function of time. This is one of
the reasons why it is challenging to achieve cellular pores in solution-based freeze
casting because the stability criterion becomes more stringent as the time proceeds
(Figure 8.4).
However, recent developments of photopolymerization-assisted freeze casting [3]
enables the precise control of the diffusion coefficient. With this photopolymerization route, one can start with desired molecular weight of the polymer or viscosity
of solution, freeze-cast the solution with fixed rheological properties of the solution, and then cross-link after solidification to ensure the integrity of the sample
for pyrolysis. With photopolymerization and gradient-controlled freeze casting,
freezing front velocity, temperature gradient and diffusion coefficient can be independently controlled, which allows to fine-tuning of the porous structure. Hence, it
would be interesting to investigate how primary pore spacing, primary pore size and

134

Figure 8.4: A stability-microstructure map with an arrow indicating an increase of
diffusion coefficient results in change in stability criterion.

secondary pore size change with the diffusion coefficient of preceramic polymers.
Additionally, one can explore how conditions for cellular growth can be altered with
a change of diffusion coefficient.
8.2.3

Porous shape-memory zirconia with precise dopant control

In Chapter 6, it was demonstrated that improved shape-memory properties were
possible by making honeycomb structures. However, the suspension contains both
zirconium oxide (zirconia) and cerium oxide (ceria) were mixed by ball-milling.
Because there are two different materials in suspension, one of them might sediment
faster during the directional solidification, leading to a variation in composition. As
demonstrated by Pang et al., the transformation-induced cracking can be mitigated
by tuning the composition of ZrO2 -CeO2 by manipulating the crystallographic phase
compatibility [4]. In their study, it was shown that a variation in composition as
small as 0.5 mol.% could have significant impact on crack-resistance properties.
Hence, preparing pre-doped powders [5] and freeze-casting them would be ideal to
avoid composition variation within samples, and warrants exploration.
References
[1] Maninpat Naviroj, Peter W. Voorhees, and Katherine T. Faber. “Suspensionand solution-based freeze casting for porous ceramics”. In: Journal of Materials Research 32.17 (2017), pp. 3372–3382.

135
[2] Maninpat Naviroj. “Silicon-based porous ceramics via freeze casting of preceramic polymers”. PhD thesis. Northwestern University, 2017.
[3] Richard Obmann et al. “Porous polysilazane-derived ceramic structures generated through photopolymerization-assisted solidification templating”. In:
Journal of the European Ceramic Society 39.4 (2019), pp. 838–845.
[4] Edward L. Pang, Caitlin A. McCandler, and Christopher A. Schuh. “Reduced
cracking in polycrystalline ZrO2-CeO2 shape-memory ceramics by meeting
the cofactor conditions”. In: Acta Materialia 177 (2019), pp. 230–239.
[5] A.L. Quinelato et al. “Synthesis and sintering of ZrO2-CeO2 powder by use
of polymeric precursor based on Pechini process”. In: Journal of Materials
Science 36.15 (2001), pp. 3825–3830.

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Appendix A

HIERARCHICAL PORE STRUCTURE
This chapter is based on the work from the journal article, "Hierarchical porous
ceramics via two-stage freeze casting of preceramic polymers," by N. Arai, and K.T.
Faber. This article has been published in Scripta Materialia.
Arai N., and Faber K. T. Hierarchical porous ceramics via two-stage freeze casting of
preceramic polymers. Scripta Materialia. 2019;162:72–76. https://doi.org/
10.1016/j.scriptamat.2018.10.037
A.1

Introduction

Recently, solution-based freeze casting of preceramic polymers have demonstrated
advantages over suspension-based freeze casting with more precise control over
the freezing process due to the homogeneity and transparency of solutions [1, 2,
3]. During freezing, phase separation between the solvent and preceramic polymer
occurs, analogous to suspension-based freeze casting, followed by sublimation of
the solvent, and pyrolysis of the preceramic polymer for conversion to the ceramics.
Zhang et al. demonstrated that freeze-cast silicon oxycarbide (SiOC) has suitable
anisotropic thermal properties for cryogenic wicking for space applications [4].
While both suspension- and solution-based freeze-cast solids have desirable porous
microstructures, enhancing the mechanical properties is also of great importance.
Various studies have reported improved strength of freeze-cast ceramics by reducing
pore size [5] and tuning the sintering temperature [6]. Another approach is to create
ceramic bridges between lamellae [7, 8, 9], which enhance the compressive strength
by limiting Euler buckling and crack propagation parallel to lamellae. Porter et
al. found changes in viscosity or pH influence the number of bridges and improve
compressive strength [8]. Work by Munch et al. demonstrated that additives such
as trehalose and sucrose changed interfacial tension and interparticle forces, creating bridges [7]. Another method by Ghosh et al. mixed large anisotropic ceramic
platelets with small isotropic ceramic particles, and engulfed ceramic platelets resulted in interlamellar bridges that improved compressive strength and stiffness [9].
It is important to note that all the bridge formation methods mentioned here were
developed for suspension-based freeze casting and, to the best of our knowledge,

137
comparable methods for solution-based freeze casting have not been explored. In
addition, among various pore morphologies, lamellar pores are of greatest interest
since they inherently possess high permeability and their highly anisotropic pore
structure results in anisotropic thermal properties which are useful in applications
such as insulation or cryogenic wicking [2,8][10, 4]. A recent study in solutionbased freeze casting by Naviroj showed that lamellar pore structures possess high
permeability, but low compressive strength ranging from around 0.5 MPa to 3 MPa
[11], motivating this study to improve strength while maintaining high permeability
by solution-based freeze casting.
In this chapter, two-stage freeze casting is explored to create a second set of lamellae
bridges between (and perpendicular to) lamellae in a hierarchical fashion by solutionbased freeze casting. In two-stage freeze casting, after freezing the solution and
freeze drying, a porous polymer green body is infiltrated with a second polymer
solution and frozen along the same direction with the aims to create a lamellar pore
structure in the first step and to form bridges between lamellar walls in the second.
Bridge formation is investigated by scanning electron microscopy (SEM) and image
analysis, and its effect on compressive strength and permeability is also studied.
A.2
A.2.1

Experimental methods
Two-stage freeze casting

Figure A.1: Freezing solution by (a) conventional unidirectional freezing and (b)
conventional conditions coupled with mold heating.
A polymer solution is prepared by dissolving preceramic polymer, polymethylsiloxane (Silres®MK Powder, Wacker Chemie, CH3-SiO1.5, Munich, Germany), in
dimethyl carbonate (DMC) (Sigma-Aldrich, St. Louis, MO, USA), followed by
the addition of 1 wt.% of a cross-linking agent (Geniosil®GF 91, Wacker Chemie,
Munich, Germany). The polymer solution was degassed at ∼30 kPa for 5 minutes

138
to avoid air bubbles during freezing. In the first stage of two-stage freeze casting, a
solution with 20 vol.% polymethylsiloxane was poured into a cylindrical glass mold
(h = 20 mm; 𝜙= 25 mm) placed on a PID-controlled thermoelectric plate which was
continuously cooled by silicone oil recirculated in a heat sink. In addition, another
thermoelectric plate was placed on top of the glass mold and maintained at 35 ◦𝐶 to
avoid crystal growth along the mold and achieve unidirectional solidification (designated as mold heating, Figure A.1). Freezing front velocity was adjusted to be in
the range of 12-14 µm/s to keep pore sizes homogeneous within the samples [12].
Once frozen, the samples were moved to a freeze drier (VirTis AdVantage 2.0, SP
Scientific, Warminster, PA, USA) where the solvents were completely sublimated
at ∼25 Pa. Subsequently, the samples were cured at 200 ◦𝐶 in air. In the second
stage, the cured green body was infiltrated with another solution with 5 vol.%, 7.5
vol.% or 10 vol.% of polymethylsiloxane by using vacuum. Samples were frozen in
the same direction as in the first stage, and the same cooling profiles (12-14 µm/s)
were maintained. After sublimation, the green body was pyrolyzed at 1100 ◦𝐶 in
Argon for 4 hours with a ramp rate of 2 ◦𝐶/min to convert polymethylsiloxane into
SiOC. Control samples were also prepared by the same first-stage process as above,
but pyrolysis followed instead of curing; this process is referred to as single-stage
freeze casting in this paper. Polymer concentrations with 20 vol.%, 25 vol.%, and
30 vol.% for single-stage freeze casting were selected. Porosity of the samples was
determined using the Archimedes method. The average porosity of all samples are
summarized in Table A.1.
Table A.1: Average porosity of single-stage freeze-cast samples and two-stage
freeze-cast samples.

Single-stage freeze casting

Two-stage freeze casting

Polymer concentration

Average porosity (%)

20 vol.%

25 vol.%

30 vol.%

Second stage polymer concentration

Average porosity (%)

5 vol.%

7.5 vol.%

10 vol.%

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A.3
A.3.1

Results and discussion
Two-stage freeze casting

Figure A.2: SEM images of a plane perpendicular to the freezing direction from (a)
single-stage freeze casting with 20 vol% polymer concentration, (b) two-stage freeze
casting with 5 vol% polymer concentration at the second stage, (c) two-stage freeze
casting with 10 vol% polymer concentration at the second stage. (d) Schematic
illustration showing bridge formation during the second stage.
Figure A.2 shows SEM images of a plane perpendicular to the freezing direction.
Figure A.2a shows an SEM image of SiOC prepared with single-stage freeze casting.
The lamellar structure characteristic of DMC solidification is clearly visible [3]. In
contrast, Figure A.2b and A.2c show SEM images of two-stage freeze-cast SiOC
produced with 5 vol.% and 10 vol.% polymer concentration at the second stage,
respectively. Some of the bridges, produced during the second stage, are indicated by
arrows in Figure A.2b. These bridges were created by crystals of infiltrated polymer
solution which nucleated and grew inside the pore generated during the first stage.
When more than two crystals grow inside a single pore, the polymethylsiloxane
is segregated between crystals which results in bridges (Figure A.2d). The interbridge spacing ranges from approximately 20 µm to 100 µm, consistent with the
pore size distribution achieved in single-stage freeze casting DMC; prior pore size
measurements by mercury intrusion porosimetry range from 10 µm to 90 µm for a
sample produced with a 15 µm/s freezing front velocity [3]. Another consideration

140
is the high anisotropy of DMC crystals. Similar to ice which produces lamellar
structures [13], DMC exhibits preferred growth directions. If a preferred growth
plane of the DMC during the second stage is perpendicular to the preferred growth
plane of DMC of the first stage, i.e., what produced the lamellar walls, one would
expect high bridge densities with small spacings. Alternatively, when a preferred
growth plane is parallel to the preferred growth plane existed during the first stage,
the bridge spacing tends to be larger. In order to measure the bridge density, twostage freeze-cast samples were infiltrated with low viscosity, low shrinkage epoxy
resin (EpoThin 2, Buehler, Lake Bluff, Illinois, USA), polished and imaged. The
number of bridges was evaluated over areas of at least 2 mm2 . The results show that
bridge density increased as polymer concentration at the second stage increased in
agreement with observations in Figs. 1b and 1c (Table A.2). Based on the result,
bridge density increased almost linearly with polymer concentration, implying that
the thickness of bridges did not significantly change. This further implies that
increasing polymer concentration created smaller crystals during freezing, which is
similar to observations of Kurtz et al. who found that dendrite tip radius decreases
with increasing solute concentration [14].
Table A.2: Bridge density of two-stage freeze-cast samples.
Second stage polymer concentration

5 vol.%

7.5 vol.%

10 vol.%

Average bridge density (mm−2 )

Figure A.3: Compressive strength by single-stage freeze casting and two-stage
freeze casting (a). (b) Load displacement curve of single-stage freeze-cast sample.
(c) Load displacement curve of two-stage freeze-cast sample. The insets show
samples after compression. Note the difference in y-axis scales in (b) and (c).

141
Compression tests were performed using an Instron 5982 universal testing machine
(Instron, Norwood, MA, USA). Cylindrical samples (approximately 7 mm height
and 13 mm diameter) were uniaxially compressed parallel to the freezing direction
with displacement rate of 0.05 mm/min. A low-shrinkage and high-hardness acrylic
system (VariDur 3003, Buehler, Lake Bluff, Illinois, USA) was applied to the top
and bottom surfaces of the sample to avoid contact fracture at the sample ends [15].
The compressive strength was calculated based on the peak load in the elastic region.
Figure A.3a shows compressive strength as a function of porosity of single-stage and
two-stage freeze-cast samples, along with representative load-displacement curves
for single-stage (Figure A.3b) and two-stage (Figure A.3c) freeze-cast samples typical of the compressive failure of porous brittle materials with a characteristic linear
elastic region followed by a sudden load drop corresponding to initiation of material failure, and a plateau region representing progressive failure [16]. A clear
trend is observed that the majority of two-stage freeze-cast samples demonstrate
higher compressive strengths than single-stage freeze-cast samples. The compressive strength of single-stage freeze-cast samples shows little dependence on porosity
over the range of porosities studied, contrary to other studies of water-based suspension freeze-cast yittria-stabilized zirconia and lanthanum strontium manganite
[17]. The inset picture in Figure A.3b shows evidence of shear failure, indicating fracture of struts connecting lamellar walls [11]. This is in contrast to failure
behavior in another study on compressive strength of suspension-based freeze-cast
ceramics with lamellar pores which showed buckling fracture or wall splitting [5,
17]. There are two possible reasons. First, in suspension-based freeze casting,
the bridges are said to be formed by tip splitting and subsequent healing of the
crystals [18], whereas polymer chains tend to be expelled from solvent crystals so
bridges are rarely formed. These bridges prevent shear failure and samples fail by
buckling instead. Second, observations in this study are more similar to a study
by Lichtner et al., who observed ceramic sliding along broken walls when samples with poorly oriented pores and walls are loaded [17]. Hence, the freeze-cast
SiOC in this study likely has relatively misaligned pores with respect to the freezing and loading axis. On the other hand, the compressive strength for two-stage
freeze-cast samples increases as bridge density increases (and porosity decreases).
Bridges between lamellar walls enhance compressive strength; an example of one
which failed by buckling is shown in the inset image in Figure A.3c. In addition to
bridges between lamellar walls, bridges were also observed in domain boundaries
in two-stage freeze-cast samples (Figure A.4). These observations imply that any

142
defects such as cracks or domain boundaries generated during the first stage will be
mitigated or eliminated at the second stage. However, two samples demonstrated
shear failure (noted by red circles in Figure A.3a). Their compressive strength falls
squarely among single-stage freeze-cast samples, where shear failure is likely due
to the significant misaligned pores. (Figure A.5)

Figure A.4: Example of a domain boundary in (a) single-stage freeze-cast sample
(20 vol.%), and (b) two-stage freeze-cast sample (5 vol.% at the second stage).

Figure A.5: Load-displacement curve of the two-stage freeze-cast sample which
exhibited noticeable low strength. The inset shows sample after compression.

143

Figure A.6: Permeability constants of samples by single-stage freeze casting and
two-stage freeze casting.

The permeability of single-stage freeze-cast and two-stage freeze-cast samples
were compared. During permeability measurements, deionized water was pumped
through the sample at pressures ranging from 6 kPa to 145 kPa by a voltagecontrolled pump; pressure and water mass flow were measured simultaneously
using a pressure transducer and an electronic scale, respectively. The permeability
of porous media can be determined from the Forchheimer equation [21–23][19, 20,
21],
Δ𝑃
= 𝑣 + 𝑣2
𝑘1
𝑘2
where Δ𝑃 is the pressure drop across the samples, 𝐿 is sample thickness along flow
direction, 𝑣 is flow velocity, 𝜇 is viscosity of liquid, 𝜌 is density of liquid, and 𝑘 1 and

144
𝑘 2 are Darcian and non-Darcian permeability constants. The second term on the
right hand side represents non-linearity in the pressure drop and flow velocity, which
corresponds to a high Reynolds number where the inertial force is non-negligible
as with gas flow [20]. In this experiment, since the pressure drop was linear with
flow velocity, the second term was ignored and only Darcian constants are reported.
Figure A.6 shows permeability constants of single-stage freeze casting and two-stage
freeze casting. As expected, permeability constants decrease as porosity decreases.
Similarly to compression testing results, the variation in permeability constant for a
given condition is likely due to variation in pore alignment. Additionally, it is notable
that two-stage freeze cast samples always have lower permeability than single-stage
freeze-cast samples. The reduction in permeability in two-stage freeze-cast samples
is likely due to higher pressure drop resulting from the additional surface area of the
bridges.

Figure A.7: Compressive strength and permeability constants compared to the
Naviroj study on lamellar and dendritic pore structures [11].
Figure A.7 shows compressive strength and permeability constants for single-stage
freeze-cast samples and two-stage freeze-cast samples along with reported val-

145
ues by Naviroj [11]. Two-stage freeze-cast samples have comparable compressive
strength to dendritic pore structures with similar permeability constants. The samples produced via two-stage freeze casting show a greater range of permeability and
compressive strength than freeze-cast SiOC reported by Naviroj [11]. This likely
can be attributed to mold heating employed in this study, which prevents nucleation
and growth from the mold, leading to fewer domains of different pore alignment;
compressive strength and permeability are very sensitive to a pore orientation. From
the result of compression and permeability measurements, control of pore alignment
is crucial to take further advantage of two-stage freeze casting. It has been shown
that significant misalignment of pore channels with respect to freezing direction resulted in shear failure and low compressive strength despite the presence of bridges,
and also low permeability. However, large-scale pore alignment is possible by introducing a wedge between the cold plate and the solution to control nucleation and
growth with a dual temperature gradient [22] or placing a grain selection template
to align pores by reducing off-axis crystals [23]. Two-stage freeze casting combined
with these large-scale pore alignment methods is expected to produce porous ceramics with high strength, high permeability, as well as highly anisotropic thermal
properties which will be useful in earlier mentioned applications.

Figure A.8: SEM images of two-stage freeze-cast SiOC using DMC as the solvent
in the first stage and cyclohexane at the second stage. (a) Transverse image (a plane
perpendicular to freezing direction) and (b) longitudinal image (a plane parallel to
freezing direction).
It should be noted that any appropriate solvent for preceramic polymers can be used
to tailor the pore network using two-stage freeze casting. For example, cyclohexane,
which forms dendrites, has been used as a second stage solvent to create bridges.
(Figure A.8). In addition, two-stage freeze casting enables hierarchical pore networks (Figure A.9). Cyclohexane as a first stage solvent creates dendritic pores,
while cyclooctane forms isotropic pores [3] during the second stage.

146

Figure A.9: SEM images of the hierarchical pore structure in two-stage freeze-cast
SiOC using cyclohexane as the solvent in the first stage and cyclooctane at the
second stage at (a) low magnification and (b) high magnification. A grain-selection
template [25] was used at the first stage.

In summary, hierarchical lamellar microstructures with interlamellar bridges were
created by two-stage freeze casting of a preceramic polymer. Unlike other bridge
formation methods developed in suspension-based freeze casting, bridge density can
be controlled simply by changing polymer concentration during the second stage. As
the bridge density increased, the compressive strength increased by nearly threefold
over those produced by single-stage freeze casting. It was shown that two-stage
freeze-cast samples tend to exhibit dendritic-like properties which show a stronger
correlation between compressive strength and permeability. Compressive strength
and permeability measurements show that misalignment of pores is not favorable
for either property, however, two-stage freeze casting coupled with pore alignment
methods [22, 23] would enable to tune compressive strength and permeability for
desired applications. Finally, the two-stage freeze casting method is potentially
applicable to suspension-based freeze casting as long as the green body at the first
stage can sustain infiltration of the suspension during the second stage, which is
possible by choosing a suitable binder.
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e1500849.
[23] Maninpat Naviroj et al. “Nucleation-controlled freeze casting of preceramic
polymers for uniaxial pores in Si-based ceramics”. In: Scripta Materialia 130
(2017), pp. 32–36.

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Appendix B

FREEZING CONDITIONS

Table B.1: List of freezing front velocities and temperature gradients used in Chapter
3 for 20 wt.% polymer-cyclohexane solution.
Freezing front velocity

Temperature gradient

1.1 ± 0.3 µm/s

2.3 ± 0.0 K/mm

1.8 ± 0.5 µm/s

2.4 ± 0.1 K/mm

5.7 ± 0.8 µm/s

2.4 ± 0.0 K/mm

8.4 ± 0.7 µm/s

2.7 ± 0.0 K/mm

15.5 ± 1.6 µm/s

2.6 ± 0.1 K/mm

17.2 ± 1.8 µm/s

2.2 ± 0.1 K/mm

0.6 ± 0.2 µm/s

5.2 ± 0.4 K/mm

0.7 ± 0.2 µm/s

4.9 ± 0.2 K/mm

1.5 ± 0.4 µm/s

5.0 ± 0.3 K/mm

2.9 ± 0.8 µm/s

4.6 ± 0.2 K/mm

8.4 ± 1.2 µm/s

5.3 ± 0.3 K/mm

10.3 ± 1.2 µm/s

5.1 ± 0.3 K/mm

150
Appendix C

COMPARISON OF THE CONVENTIONAL FREEZING AND
THE GRADIENT CONTROLLED FREEZING

Figure C.1: Freezing profile of Conventional freezing (V = 15 µm/s) and gradientcontrolled freezing (V = 15 µm/s, G = 2.6 K/mm)
Figure C.1 shows freezing profiles for a conventional freezing and a gradientcontrolled freezing. Since the temperature at the top side is not controlled in the
conventional freezing, a thermocouple was inserted in the solution to measure the
temperature during the freezing. The thermocouple measured a temperature at ∼17
mm from the bottom and the temperature is plotted as T17𝑚𝑚 in Figure C.1. As
shown, the temperature difference between two points gets larger as time proceeds.
In this conventional freezing, the temperature gradient cannot be controlled. In
contrast, the temperature difference between top and bottom sides are similar during
the gradient-controlled freezing. With the temperature control from both sides,
temperature gradient can be maintained at 2.6 ± 0.1 K/mm. Another advantage of
the gradient-controlled freeze casting is that one can change or control temperature
gradient by two ways. First, one can simply change the mold height, which was
demonstrated in Chapter 3. Another way is to change the temperature difference
between top and bottom sides. This demonstrates the advantages of controlling
freezing front velocity and temperature gradient by the gradient controlled freezecasting setup.
The homogeneity of the freeze-cast structure was investigated. A sample was
freeze-cast under freezing front velocity of 15 µm/s and temperature gradient of 2.6

151

Figure C.2: Pore size distribution from three different sections.
Table C.1: List of the peak pore diameters for primary and secondary pores from
Figure C.2
Primary pore diameter

Secondary pore diameter

Top section

22.0 µm

14.8 µm

Middle section

21.1 µm

14.3 µm

Bottom section

20.3 µm

13.7 µm

K/mm. The sample was pyrolyzed under the presence of water vapor in addition
to argon as described in subsection 7.1.2 (pyrolysis at 1100 ◦ C under a mixed
argon/water atmosphere created by flowing argon over a beaker of water at 85 ◦ C
before entering the furnace). The three specimens (top, middle, bottom section)
were sectioned from a sample and characterized by mercury intrusion porosimetry.
A pore size distribution and peak pore diameters are shown in Figure C.2 and Table
C.1, respectively. As shown, the peak primary and secondary pore sizes are similar
in the three sections, and the differences were found to be around 8% or smaller.

152
Appendix D

INFLUENCE OF PRECERAMIC POLYMER CONCENTRATION

Figure D.1: SEM images showing (a, b) a control sample, and (c,d) a sample
coarsened at 3 ◦ C for 1 hour. (e) Pore size distribution from 30 wt.% preceramic
polymer solution.
Figure D.1 displays SEM images and the pore size distribution of the control sample
and the sample coarsened at 3 ◦ C for 1 hour created from the 30 wt.% solution.
The morphologies of primary pores and secondary pores change similarly to the
20 wt.% solution (Figures D.1 a-d). In 20 wt.% solution, the volume fraction of
secondary pores decreased by coarsening (Figure 5.8), making the primary pores
dominant pores as shown by pore size distribution (Figure 5.7). SEM images also
demonstrated that the sample coarsened at 4◦ C for 1 hour exhibited honeycomb-like
structure (Figure 5.4g). This is, however, not the case in the 30 wt.% solution
as shown in the SEM image (Figure D.1c). In the 30 wt.% solution, although
the secondary pore size increases, the structure still remains dendritic, leaving a
large volume of secondary pores even after 1 hour coarsening (Figure D.1e). We
hypothesize that the presence of long secondary pores after coarsening can be
attributed to the low diffusion coefficient of polymer in the solution. With higher
polymer concentrations, the solution has higher viscosity and shorter gelling time,
which both contribute to the reduction of diffusion coefficient in the solution. This
could slow the shortening of secondary arms.
Chen and Kattamis proposed a model to describe the increase in secondary arm
spacing and decrease of secondary arm length [1], in which the secondary arm

153
spacings increase by mass diffusion from small secondary arms to large adjacent
secondary arms. Alternatively, a decrease in secondary arm length can occur
by the mass diffusion between the tip of the secondary arms and the root of the
secondary arms. This latter case requires a longer diffusion distance. Because
of low diffusivity in the 30 wt.% solution, the latter process was slowed down,
maintaining long secondary pores. As a result, a large volume of secondary pores
are still present in pore size distributions of materials with higher solute content.
References
[1] M. Chen and T.Z. Kattamis. “Dendrite coarsening during directional solidification of Al–Cu–Mn alloys”. In: Materials Science and Engineering: A
247.1-2 (1998), pp. 239–247.