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Development of Novel Binary and Multi-Component Bulk Metallic Glasses
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Xu, Donghua
(2005)
Development of Novel Binary and Multi-Component Bulk Metallic Glasses.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/XD2M-WW51.
Abstract
Bulk Metallic Glasses (BMGs) have been drawing increasing attention in recent years due to their scientific and engineering significance. A great deal of effort in this area has been devoted to developing BMGs in different alloy systems. BMGs based on certain late transition metals (e.g. Fe, Co, Ni, Cu) have many potential advantages over those based on early transition metals. These include even higher strength and elastic modulii, and lower materials cost, to name a few, which are highly preferable for a broad application of BMGs as engineering materials. Nevertheless, these ordinary-late-transition-metal-based BMGs generally have quite limited glass-forming ability (GFA). In particular, for the Ni-based and Cu-based alloys reported prior to this research, the maximum casting thickness allowed to retain their amorphous structures is only ~2 mm (or lower) and ~5 mm (or lower), respectively.
During this research it was first found that certain quinary Ni-based alloys in the Ni-Cu-Ti-Zr-Al system can be cast into 5 mm diameter amorphous rods. This critical casting thickness is the highest for any reported Ni-based BMG’s indicating that these alloys are the easiest metallic glass formers based on Ni discovered to date. Secondly, certain binary alloys in the Cu-Zr and Cu-Hf systems were found to form bulk glasses with casting thickness as high as 2 mm. The discovery of these binary BMGs was very surprising since it had been widely considered that only multi-component (containing at least three elements) alloys could form bulk metallic glasses. These new binary BMGs provide interesting subjects for future theoretical studies such as molecular dynamics simulations since they possess both the simplicity of binary alloys and the good GFA of multi-component BMGs. In fact, these binary BMGs have led to the third but perhaps the most significant progress during this research, i.e., the discovery of a family of Cu-based BMGs in the Cu-Zr-Al-Y system which possess a critical casting thickness up to 1 cm. These quaternary Cu-based alloys, together with some complicated Fe-based alloys reported by two other groups during the course of this research, are the first centimeter level BMGs based on the ordinary late transition metals.
This thesis first reviews the fundamentals related to BMG development, then reports in detail the formation and properties of the above-mentioned binary and multi-component BMGs based on Ni and Cu. A generalized geometric model for the critical-value problem of nucleation developed in this research is also presented.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
Amorphous alloys; bulk metallic glasses; Cu alloys; mechanical properties; Ni alloys; nucleation
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Johnson, William Lewis
Thesis Committee:
Johnson, William Lewis (chair)
Bhattacharya, Kaushik
Conner, Robert Dale
Fultz, Brent T.
Goddard, William A., III
Nicolet, Marc-Aurele
Ravichandran, Guruswami
Defense Date:
17 May 2005
Funders:
Funding Agency
Grant Number
Defense Advanced Research Projects Agency (DARPA)
DAAD19-01-1-0525
Record Number:
CaltechETD:etd-05272005-160315
Persistent URL:
DOI:
10.7907/XD2M-WW51
ORCID:
Author
ORCID
Xu, Donghua
0000-0001-5018-5603
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2158
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31 May 2005
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24 May 2024 18:36
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DEVELOPMENT OF NOVEL BINARY AND MULTI-COMPONENT
BULK METALLIC GLASSES

Thesis by
Donghua Xu

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

California Institute of Technology
Pasadena, California
2005
(Defended May 17, 2005)

Acknowledgements

Caltech is such an enthralling place. During my five years as a student here, so many
times, lying in bed, I could not fall asleep, being either too excited about my new results
or overly eager to figure out the solutions to the problems in my research. Being stingy
with my time spent beyond research, I found myself deeply indebted to my family (my
wife, parents, brother, sister etc.) for their great understanding and love. It is to them that
go my deepest acknowledgements.
This thesis could not have become a reality if Professor Bill Johnson had not offered me
the opportunity to come to Caltech. Bill’s erudition and insights, enthusiasm and
optimism provided me with such an endless source of knowledge and energy. His real
scholarly generosity made it possible for me to freely enjoy my research. Bill is the very
first person at Caltech who deserves my full gratitude.
Among many colleagues, Haein Choi-Yim, Jan Schroers, Sven Bossuyt, Paul Kim, Andy
Waniuk, Dale Conner, Boonrat Lohwongwatana (my long time officemate), Greg Welsh,
Chris Veazey, Sundeep Mukherjee, Gang Duan, Mary Laura Lind and Jin-Woo Suh in
the Johnson group; Seung-Yub Lee, Robert Rogan and Can Aydiner in the Ustundag
group; Jason Graetz and Olivier Delaire in the Fultz group; and Shiming Zhuang and
Theresa H. Kidd in the Ravichandran group are all acknowledged for their kind
assistance in experiments and/or for their friendly company in the laboratories.
My gratitude also goes to the many faculty and staff members in the department who
helped me in various ways. For example, Carol Garland helped me take TEM images of

iii
many of my samples, and Robin Hanan and Elizabeth Welsh offered me their patient
assistance in purchasing materials required for my research.
Although I did not socialize a lot, I did make some good friends whose company and
friendship made my life in California easy and colorful. These include Boonrat
Lohwongwatana, Seung-Yub Lee, Ling Li, Xin Yu, Bingwen Wang, Tingwei Mu,
Yajuan Wang, Hao Jiang, Yong Hao, and Peng Xu at Caltech, and Dawei Ren and Qi
Yang at the University of Southern California.
Most of this thesis work was supported by the Defense Advanced Research Projects
Agency, Defense Sciences Office, under ARO Grant No. DAAD19-01-1-0525.

Abstract

Bulk Metallic Glasses (BMGs) have been drawing increasing attention in recent years
due to their scientific and engineering significance. A great deal of effort in this area has
been devoted to developing BMGs in different alloy systems. BMGs based on certain late
transition metals (e.g., Fe, Co, Ni, Cu) have many potential advantages over those based
on early transition metals. These advantages include even higher strength and elastic
modulii, and lower materials cost, to name but a few that are highly preferable for a broad
application of BMGs as engineering materials. Nevertheless, these ordinary-latetransition-metal-based BMGs generally have quite limited glass-forming ability (GFA).
In particular, for the Ni-based and Cu-based alloys reported prior to this research, the
maximum casting thickness allowed to retain their amorphous structures is only ~2 mm
(or lower) and ~5 mm (or lower), respectively.
The first important finding during this research was that certain quinary Ni-based alloys
in the Ni-Cu-Ti-Zr-Al system can be cast into 5 mm diameter amorphous rods. This
critical casting thickness is the highest for any reported Ni-based BMG’s, indicating that
these alloys are the easiest metallic glass formers based on Ni discovered to date.
Secondly but more interestingly, certain binary alloys in the Cu-Zr and Cu-Hf systems
were found to form bulk amorphous samples with casting thicknesses as high as 2 mm.
The discovery of these binary BMGs was very surprising since it had been widely
considered that only multi-component (having at least three elements) alloys could form
bulk metallic glasses. These new binary BMGs have not only challenged the traditional
concept about bulk metallic glass formation, but also provided interesting subjects for

future theoretical studies such as molecular dynamics simulations since they possess both
the simplicity of binary alloys and the good GFA of multi-component BMGs. As a matter
of fact, these binary BMGs have also led to a third and perhaps most significant
discovery during this research: the family of Cu-based BMGs in the Cu-Zr-Al-Y system
that possesses a critical casting thickness up to 1 cm. These quaternary Cu-based alloys,
together with some complicated Fe-based alloys reported by two other groups during the
course of this research, are the first centimeter-level BMGs based on the ordinary late
transition metals.
This thesis first reviews the fundamentals related to BMG development, then reports in
detail the formation and properties of the above-mentioned binary and multi-component
BMGs based on Ni and Cu. A generalized geometric model for the critical-value problem
of nucleation developed in this research is also presented.

Contents
Acknowledgements.................................................................................................ii
Abstract..................................................................................................................iv
Contents…………………………………………………………………………..vi
List of Figures…………………………………………………………………....ix
List of Tables………………………………………………………………...….xii
Chapter 1: Introduction……………………………………………………….....1
1.1 Basic concepts about glasses and metallic glasses………………………….1
1.1.1

Glass and glass transition……………………………………….....1

1.1.2

Glass formation and glass-forming ability…………………..…….4

1.2 History of metallic glass research and motivation for this thesis…………7
1.3 Thermodynamics and kinetics related to glass formation and TTT
(Time-Transformation-Temperature) diagram……………….………….10
1.3.1

Thermodynamics of an undercooled liquid……………………...10

1.3.2

Kinetics of an undercooled liquid…………………………..……12

1.3.3

Classical theory for crystal nucleation and growth from
an undercooled liquid and TTT diagram……..…………………17

1.4 Frequently used criteria for the development of BMGs………….………26
1.4.1

Reduced glass transition temperature………………………...….27

1.4.2

Multi-component rule (confusion principle)………………....….31

1.4.3

Atomic size mismatch…………………………………………...32

1.4.4

Chemical interactions among constituent elements……………..34

1.4.5

Considerations based on phase diagrams……... ………………..35

vii
References……………………………………………………………………….41

Chapter 2: Formation and properties of Ni-based BMGs in
Ni-Cu-Ti-Zr-Al system……………………………………………...45
2.1 Introduction…………………………………………………………………45
2.2 Experimentals………………………………………………………….……46
2.3 Results and Discussion…………………………….…………………..……47
2.3.1 Ternary Ni45Ti20Zr35 alloy………………………………………….47
2.3.2 Quaternary Ni45Ti20Zr35-xAlx alloys………………………………..50
2.3.3 Quinary NixCua-xTiyZrb-yAl10 alloys (a~b~45)……………………..51
2.3.4 Effect of small Si additions………………………………………...54
2.3.5 Mechanical tests……………………………………………………55
2.4 Conclusions………………………………………………………………….57
References……………………………………………………………………….58

Chapter 3: Formation of bulk metallic glasses in binary Cu-Zr and
Cu-Hf systems……………………………………………………………60
3.1 Introduction………………………………………………………………....60
3.2 Experimentals……………………………………………………………….61
3.3 Results and discussions……………………………………………………..61
3.3.1 Glass-forming abilities……………………………………………..61
3.3.2 Thermal analyses with DSC………………………………………..65
3.3.3 Mechanical properties of the three best glass formers……………..69
3.4 Conclusions………………………………………………………………….71
References…………………………………………………………………….…72

viii

Chapter 4: A generalized model for the critical-value problem
of nucleation….……………….……………………………………….…74
4.1 Introduction………………………………………………………………....74
4.2 Model construction……….…………………………………………...…….77
4.3 Model solution and interpretation…………………………………………77
4.4 Conclusions……………………………………………………………….…85
References……………………………………………………………………….87

Chapter 5: Centimeter size BMG formation in Cu-Zr-Al-Y system……...…..88
5.1 Introduction………………………………………………………………....88
5.2 Experimentals……………………………………………………………….89
5.3 Results and discussions……………………………………………………..90
5.4 Conclusions………………………………………………………………...100
References………………………………………………..…………………….101

Chapter 6: Concluding Remarks………….………………………………………102

List of Figures
Chapter 1
1.1 Plots of viscosity data scaled by values of Tg for different glass-forming liquids…...14
1.2 Plots of the three terms in Eq. (1.13) vs. nucleus radius r………………………...…19
1.3 Nucleation rate I v and crystal growth rate u as a function of temperature for the
BMG alloy Pd40Cu30Ni10P20……………………………………………………….…22
1.4 TTT (Time-Temperature-Transformation) diagram of Pd40Cu30Ni10P20 calculated
using a crystallized volume fraction f = 10 −6 ……………………………………….24
1.5 Logarithm of nucleation rate (in cm-3s-1), log I v , vs. the reduced temperature, Tr ,
calculated at different values of the reduced glass transition temperature Trg ………30
1.6 Binary phase diagram of Zr-Be system ……………………...………………………36
1.7 Binary phase diagram of Ti-Be system..……………………………………………..36
1.8 Binary phase diagram of Zr-Cu system…………………………………….………..37
1.9 Binary phase diagram of Zr-Ni system……………………………………………....37
1.10 Binary phase diagram of Zr-B system……………………………………..………39

Chapter 2
2.1 XRD patterns of selected ternary and quaternary alloys taken with a Co Kα
source……………………………………………………………….…………….…..49
2.2 DSC scans of selected ternary and quaternary alloys at a heating rate of 0.33 K/s…..49
2.3 XRD patterns of selected quaternary and quinary alloys taken with a Co Kα
source………...………………………….……………………………………………52
2.4 Electron diffraction pattern taken from the transverse cross section of a 5mm

thick Ni40Cu5Ti16.5Zr28.5Al10 strip…………………………………………………….52
2.5 DSC scans of selected quaternary and quinary alloys at a heating rate of 0.33 K/s.…53
2.6 Effect of adding a small amount of Si …………………………….………………....53
2.7 Compressive stress vs. strain curves of two selected alloys:
(a) Ni40Cu5Ti16.5Zr28.5Al10; and (b) Ni45Ti20Zr25Al10………..………………………..56

Chapter 3
3.1 Binary Cu-Zr phase diagram (reproduced from Ref. [17]) ….…....………………….62
3.2 Binary Cu-Hf phase diagram (reproduced from Ref. [17])….……………………….62
3.3 X-ray (taken with a Cu Kα source) and electron diffraction patterns of
Cu46Zr54 (A1), Cu64Zr36(A2) and Cu66Hf34 (A3)…………….……………………….63
3.4 XRD patterns taken from 0.5 mm thick strips of Cu100-xZrx (x=34, 36, 38.2, 40
at.%) using a Cu-Kα source………….......…………………………………………..66
3.5 XRD patterns taken from the cross sections of the 2mm thick cast strips of
Cu100-xZrx (x=34, 36, 38.2, 40 at.%) using a Cu-Kα source……………………….…66
3.6 DSC scans of the 0.5mm thick strips of Cu60Zr40, Cu61.8Zr38.2, Cu66Zr34, and
the 2mm thick strip of Cu64Zr36 obtained at a heating rate of 0.33K/s……………….68
3.7 Variations of ∆T and Trg with respect to Zr content x in alloy series
Cu100-xZrx (x=34, 36, 38.2, 40 at.%)………………………………………………….68
3.8 Compressive stress vs. strain curves of the three best glass formers in Cu-Zr
and Cu-Hf systems obtained at a strain rate of ~4x10-4 s-1 at room temperature….…70

Chapter 4
4.1 The geometric factor as a function of contact angle θ in the large-wall
heterogeneous solution…………………………………………………………….…76

xi
4.2 (a) the geometric construction for the generalized nucleation model; (b) an
illustration of the mechanical equilibrium at point S in part (a)……..……………….78
4.3 (a) 3D image of the bivariate function g ( R, θ ) ; (b) 2D plots of g ( R, θ ) vs.

R / rc at different values of θ ; (c) 2D plots of g ( R, θ ) vs. θ at different values
of R / rc ……………………………………………………………………………....83

Chapter 5
5.1 (A) Pictures of three cast samples of Cu46Zr42Al7Y5, with different diameters:
S1, 10mm; S2, 12mm; S3, 14mm; (B) XRD patterns obtained from 10mm
(S1); 12mm (S2) and 14mm (S3) diameter rods of Cu46Zr42Al7Y5; and from
3mm (M1) and 4mm (M2) diameter rods of the matrix alloy Cu46Zr47Al7 ………...91
5.2 DSC scans of selected alloys at a constant heating rate of 0.33K/s. The
upward arrows refer to the glass transition temperatures and the downward
arrows refer to the onset of the first crystallization events. The inset at the
lower right corner is the isothermal DSC profile of the 10mm diameter rod of
Cu46Zr42Al7Y5 at a constant temperature of 739K …...………………………………93
5.3 Melting behaviors of selected alloys measured at a heating rate of 0.33K/s.
The arrows refer to the liquidus temperatures…………………..……………………95
5.4 TEM image (a), Cu Kα1 X-ray dot map image (b) and Y Kα1 X-ray dot map
image (c), of as-cast Cu46Zr42Al7Y5. The ripples and scratches in the images
were caused by the ultramicrotomy sample preparation method…………...………..98

xii

List of Tables
Chapter 2
2.1 Examples of the new Ni-based amorphous alloys developed in this work (Tg
and Tx1 were measured with DSC at a heating rate of 0.33K/s)……………………...48
2.2 Some measured mechanical properties of selected alloys …………………………...55

Chapter 3
3.1 Thermal properties of three best glass formers in Cu-Zr and Cu-Hf systems ……….67
3.2 Mechanical properties of three best glass formers in Cu-Zr and Cu-Hf systems ……69

Chapter 5
5.1 A list of representative alloys and selected properties……………………………….92

Chapter 1
Introduction
In this chapter, I will first explain several basic concepts about glasses and metallic
glasses. Then I will give a brief summary on the history of metallic glass research and the
motivation for this thesis. In the third section, I will go over the thermodynamics and
kinetics related to metallic glass formation, including D.R. Uhlmann’s TTT (TimeTemperature-Transformation) analysis. In the last section, I will review several
frequently used criteria for bulk metallic glass development which have been proposed in
the past years.

1.1

Basic concepts about glasses and metallic glasses

1.1.1

Glass and glass transition

A glass is a disordered (or amorphous) solid which does not possess the long range
periodicity as present in a typical crystal. As continuous refinement of crystal grains is
being achieved these days, the boundary between a glass and a nanocrystalline solid with
very fine grains is getting blurred. For most practical purposes, however, if no long range
order can be detected beyond a 1-2 nm scale, a solid can be called a glass.
It should be noted that the definition of a glass is made only based on its disordered
structure†, regardless of its chemical composition. In fact, a glass may chemically have
any of the available types of bonding: metallic, covalent, ionic, hydrogen and van der

Some researchers prefer not to use the word ‘structure’ on a glass. However, in this thesis I will still use
the word, considering the existence of short range ordering in a glass.

Waals. Here in this thesis, we mainly focus on glasses with metallic bonding, i.e.,
metallic glasses.
Due to the existence of crystalline counterparts which have lower free energy, a glass is
not a thermodynamically stable form of solids even though it may possess excellent
metastability [1] (e.g., silicate glasses). For this reason, a glass always has a tendency to
transform into more stable crystalline forms by a crystallization process. The
crystallization may be induced by heating or mechanical deformation, and may proceed
either rapidly or slowly as we will discuss in more detail in the following sections.
Upon continuous heating, a glass with good metastability undergoes a glass transition
before crystallization occurs. This glass transition is manifested by a rather rapid increase
in both heat capacity (Cp) and coefficient of thermal expansion (CTE)†. At the same time,
the sample’s mechanical response to external stresses also changes rapidly from solidlike to liquid-like as described by a significant drop in viscosity (η) (typically by several
orders of magnitude). Because this transition spans a range of temperatures, there are
different ways to define a characteristic temperature for this transition, usually called
glass transition temperature (Tg) [2]. The definition of Tg adopted in this thesis is the
onset temperature of the Cp increase upon heating at a constant rate of 0.33 K/s. Another
frequently used definition of Tg is the temperature at which the equilibrium viscosity of
the heated glass becomes 1013 poise (i.e., 1012 Pa s). This specific value of viscosity was
chosen rather arbitrarily to distinguish a viscous liquid from a solid. Whichever definition
one uses, the Tg should always be regarded as the boundary between the pre-transition

Other second order derivatives of energy such as compressibility and elastic modulli also experience an
abrupt change (either an increase or decrease) during the glass transition.

and the post-transition states of the glass. Due to its liquid-like feature and its
thermodynamic metastability, a glass in the post-transition state is also called an
undercooled liquid.
Although the above description of glass transition is based on the continuous heating of a
glass, the glass transition also takes place during the continuous cooling of a liquid, if no
crystallization interferes. Moreover, the glass transition is roughly reversible upon
cooling and heating if the cooling rate ( T& C ) is about equal to the heating rate ( T& H ) in
magnitude. If T& H >> T& C , however, the glass transition occurs at a higher temperature
during heating ( T gH ) than during cooling ( TgC ). On the other hand, if T& H << T& C , the
glass transition appears at a lower temperature during heating than during cooling. In
addition, the larger the mismatch in the two rates, the more obvious the discrepancy
between T gH and TgC . These phenomena are all related to the structural relaxation of the
glass which is a kinetic process. In the former case above ( T& H >> T& C ), the glass does
not have as much time as during cooling to relax when it passes through TgC during
heating and therefore, it retains its solid configuration to higher temperature until it
reaches T gH . In the latter case ( T& H << T& C ), the glass has more time to relax during
heating (than during cooling) so that it readily gives up its solid configuration at a lower
temperature than TgC .

1.1.2

Glass formation and glass-forming ability

Glasses, including metallic glasses, can be formed through different routes starting from
different initial states [2]. Perhaps most frequently, however, glasses are formed through
the continuous cooling of liquids from above their thermodynamic melting temperature
(Tm) to below their glass transition temperature Tg.
In order to eventually form a glass upon continuous cooling, a liquid has to ‘successfully’
suppress crystallization. Although crystallization is favored by thermodynamics below
the melting temperature, it is subject to the control of the kinetics of crystal nucleation
and growth such that it requires a certain amount of time to proceed. Apparently, if a
liquid were cooled instantaneously from Tm to Tg using an infinitely high cooling rate,
there would be no time for crystallization to proceed and the liquid would be directly
frozen into a glass by going through a glass transition around Tg. In actual cases, however,
the cooling rate required to form a practical glass does not need to be infinitely high
because the crystallization does not have to be completely restrained. As long as the
crystallized volume fraction in the resulted solid is beyond the detection limit of the
characterization instruments, the resulted solid is considered as a glass for practical
purposes. This limiting crystallization volume fraction, f c , is often taken as 10-6 (a value
chosen rather arbitrarily) for all practical glasses. Hence, corresponding to this f c , there
is a finite critical value of the cooling rate which is normally called the critical cooling

rate Rc for each liquid. A liquid can form a glass if and only if the actual cooling rate is
higher than its Rc . The critical cooling rate depends on the thermodynamics and kinetics
of crystallization (as we will discuss in more detail in Section 1.3) and may vary a lot

from one

liquid to another. For example, the liquid of multi-component

Zr41.2Ti13.8Cu12.5Ni10Be22.5† alloy has an Rc around 1.4 K/s [3], while the liquid of binary
Zr65Be35 alloy has an Rc around 107 K/s [4]. The critical cooling rate is the ultimate
judgment factor for the glass-forming ability (GFA) of a liquid. Obviously, a liquid with
a lower Rc has a higher GFA.
There are many different methods to cool a liquid into a glass. As for metallic glasses, the
available cooling methods include melt spinning, splat quenching, metal (usually copper)
mold casting, water quenching and others [5], among which copper mold casting is the
one most heavily utilized in this thesis.
Consider the heat transfer during the cooling of a liquid alloy within a copper mold. If the
liquid solidifies into a glass, then no latent heat due to crystallization needs to be
considered. If one dimension (we call it thickness, l) of the slot holding the liquid is
significantly lower than the other two, the problem can be described by the one
dimensional Fourier heat flow equation along the thickness direction:
∂T
∂ 2T
=κ 2
∂t
∂x

(1.1)

where x is spatial coordinate ( 0 ≤ x ≤ l ), t is time, T = T ( x, t ) is temperature, and κ is the

thermal diffusivity of the liquid alloy. The initial condition is T ( x,0) = Tl where Tl is the
liquidus temperature of the alloy. The boundary conditions are T (0, t ) = T ( x, t ) = Tr
where Tr is the room temperature (we treat the copper mold as a heat reservoir and

All the compositions in this thesis are given in atomic percentage.

neglect the temperature change in the mold). The exact solution to this problem can be
easily found:

T ( x , t ) = Tr +

4(Tl − Tr ) ∞

∑ 2n − 1 sin[

n =1

(2n − 1)π
κt
x] exp[−(2n − 1) 2 π 2 2 ] .

(1.2)

We would like to know how fast the center ( x = l / 2 ) of the liquid is cooled down to Tg
since this determines whether crystallization would occur. Using the first order
approximation to Eq. (1.2), we get the time τ it takes the center to reach Tg:

τ=

π κ

ln(

4 Tl − Tr
π T g − Tr

(1.3)

Therefore, the cooling rate at the center averaged in the temperature interval between Tl
and Tg is:

Tl − Tg

π 2κ

(Tl − Tg ) / ln(

4 Tl − Tr
π Tg − Tr

(1.4)

It can be seen that the cooling rate depends on several parameters: glass transition
temperature Tg , liquidus temperature Tl, thermal diffusivity κ, and sample thickness l.
For a given alloy, R is inversely proportional to the square of l. In order to successfully
suppress crystallization, R has to be larger than the critical cooling rate Rc of the alloy
liquid, which requires that l be smaller than a critical value (usually called the critical
casting thickness) lc. If the contribution of other factors, i.e, Tg, Tl, and κ, to the cooling
rate R does not vary much from one alloy to another, according to Eq. (1.4), the critical
casting thickness of the alloys can then be used to distinguish their glass-forming ability.

Eq. (1.4) can be used to calculate either one of the critical cooling rate and the critical
casting thickness if the other is known. For extremely good metallic glass formers like
Pd40Cu30Ni10P20 [6] and Zr41.2Ti13.8Cu12.5Ni10Be22.5 [3], it is very easy to measure the
critical cooling rate (e.g., via DSC—Differential Scanning Calorimetry), but difficult to
directly measure the critical casting thickness by repeated castings† of large quantities of
materials. For moderately good metallic glass formers like the ones developed in this
thesis, Ni59.35Nb 34.45Sn6.2 [7], Cu64Zr36 [8], Cu46Zr54 and Cu46Zr42Al7Y5 [9], and
Ni40Cu5Ti16.5Zr28.5Al10 [10] to name but a few, it is easy to find out the critical casting
thickness via repeated castings, but difficult to directly measure the critical cooling rate
using DSC.

1.2

History of metallic glass research and motivation for this thesis

The production of an amorphous flake (~10 µm in thickness) by rapid cooling of the
molten Au75Si25 alloy in 1960 [11] is generally considered the beginning of the era of
metallic glasses, even though prior to that, disordered metallic materials had been made
through other routes such as vapor condensation on substrates at liquid helium
temperature [12]. In the long period that followed until a breakthrough occurred in the
late 1980’s and early 1990’s with the discovery of several multi-component metallic
glasses with very low critical cooling rates, the formation of metallic glasses in many
simple alloy systems (mainly binary and ternary systems) was discovered and the
properties of the resulted metallic glasses were studied (see Ref. [13-18] for examples of
these early works). Many of these studies showed that metallic glasses possess several

To find out the critical casting thickness, multiple values of sample thickness have to be tested; each value
requires a casting and subsequent inspection with characterization instruments like an X-ray diffractometor.

unique properties that are superior to the conventional crystalline metals and alloys, such
as very high strength (close to theoretical limits for metals and alloys), high elastic strain
limit, high hardness, high corrosion resistance, and very good soft magnetism (with very
low coercivity and low hysteretic losses). These properties indicated that metallic glasses
could be much better than conventional metals and alloys in many structural and
functional applications. Nevertheless, the real applications of metallic glasses in most
cases could not be realized because of the barriers in the formation of these materials.
The critical cooling rates of these early metallic glasses were so high (on the order of 105107 K/s) that the samples had to be very small (typically tens of microns or less) in at
least one dimension in order to form a glassy structure and that special techniques (e.g.,
melt spinning) had to be utilized to acquire such high cooling rates. A logical way to
solve these problems would be to find some new metallic glasses with low critical
cooling rates which can be formed into bulk samples†. In 1984, one such alloy,
Pd40Ni40P20, was found to form 10 mm thick glassy samples using a very low cooling rate
(<1 K/s) [19]. Unfortunately, this alloy could not be used as a practical metallic glass
because it was based on palladium, a very expensive component.
A breakthrough came in the late 1980s and early 1990s when several metallic glasses
based on practical elements with very low critical cooling rates were discovered (e.g.,
Zr41.2Ti13.8Cu12.5Ni10Be22.5 [20] and La55Al25Ni20 [21]). These alloys could easily be
fabricated into bulk glassy samples by the regular mold casting method or water
quenching. On one hand, the discovery of these bulk metallic glasses (BMGs)
significantly enhanced the promise of metallic glasses as practical materials. Particularly,

The boundary between ‘bulk’ and ‘thin’ is generally taken as 1 mm (in the smallest dimension) by
researchers in this area.

the Zr-based BMGs have already been fabricated into commercial parts and articles for
packaging, medical, and sporting purposes. On the other hand, the discovery of these
BMGs also made it possible to perform fundamental studies on metallic glasses and their
undercooled liquids that had been forbidden by the formerly limited sample size or
limited resistance to crystallization.
However, there are still limitations to these early BMGs. For example, their glass
transition temperatures are usually quite low, which prevent them from high temperature
applications, and their material costs are relatively high, leaving space for further
reduction. Meanwhile, the limited number of BMGs prevents a complete understanding
of metallic glasses as a whole category of solids. It is therefore necessary to develop more
BMGs, from either a technological or scientific point of view. As part of a unified
endeavor in the academic community, this thesis research is aimed at facilitating the
variety within the family of BMG alloys.
BMGs based on certain late transition metals (e.g., Fe, Co, Ni, Cu) have many potential
advantages over those based on early transition metals (e.g., Zr, La). These advantages
include even higher strength and elastic modulii, and lower material costs, to name but a
few that are highly preferable for broad applications of BMGs as engineering materials.
Nevertheless, these ordinary-late-transition-metal-based BMGs generally have quite
limited glass-forming ability. In particular, for the Ni-based and Cu-based alloys reported
prior to this research, the maximum casting thickness allowed to retain their amorphous
structures is only ~2 mm (or lower) and ~5 mm (or lower), respectively [22-25]. In this

10
thesis, attention is focused on the development of novel Ni- and Cu-based BMGs with
higher glass-forming ability.

1.3

Thermodynamics and kinetics related to glass formation and TTT
(Time-Transformation-Temperature) diagram

1.3.1

Thermodynamics of an undercooled liquid

A liquid (l) has the same Gibbs free energy (G) as its crystal (x), i.e., Gl = G x (or
∆G m = Gl − G x = 0 ) at the melting temperature Tm. Also at Tm, the liquid has more
entropy (S) than its crystal by the amount of ∆S m = S l − S x = ∆H m / Tm , where ∆H m is
the heat (i.e., enthalpy) of fusion. When temperature T is below Tm, the liquid is
undercooled and has a higher Gibbs free energy than the crystal. The energy difference

∆G = ∆G m + ∫ (− ∆S )dT = − ∫ ∆SdT acts as the driving force for crystallization of the
undercooled liquid. The entropy difference between the two states is described as

∆C p

∆S = ∆S m + ∫

dT , where ∆C p = C p ,l − C p , x (>0) is the heat capacity difference

between the liquid and the crystal. Since the liquid has a higher heat capacity, it loses its
entropy faster than the crystal upon cooling. As the temperature drops continuously, there
would appear a point at which the entropy of the liquid became equal to that of the crystal.
This tendency was first pointed out by Kauzmann [26] and therefore, it is now known as
the ‘Kauzmann Paradox.’ The isentropic point is called Kauzmann temperature, TK. The
Kauzmann temperature is considered as the limit to which a liquid can be cooled with the

11
argument that a liquid can’t have lower entropy than its crystal†. Therefore, before
reaching TK, either glass transition or crystallization has to occur in order to terminate the
liquid state. In either case, the heat capacity experiences an abrupt drop.
In principle, both C p ,l (between Tm and Tg) and C p , x can be measured experimentally
using calorimetric method [27] or ESL (ElectroStatic Levitation) technique [28],
although the measurement of C p ,l requires high resistance of the undercooled liquid to
crystallization. In fact, this has been done on some extremely good metallic glasses (e.g.,
Zr41.2Ti13.8Cu12.5Ni10Be22.5 [27, 28]). The melting temperature Tm and the heat of fusion

∆H m can easily be measured using high temperature calorimetry. With the C p ,l , C p , x
(and hence ∆C p ), Tm and ∆H m known, the Gibbs free energy difference between the
undercooled liquid and the crystal as a function of temperature can be calculated as
follows:

∆G =

T ' ∆C p (T ' ' )
∆H m
(Tm − T ) − ∫ dT '∫
dT ' '
Tm
Tm
T ''
Tm

(1.5)

It should be noted that the thermodynamic parameters are well defined functions of
temperature T and pressure P, and do not depend on the cooling rates or the heating rates
exploited in their experimental measurements.

This would happen if the liquid were cooled to below TK.

12
1.3.2

Kinetics of an undercooled liquid

Viscosity η is perhaps the most important kinetic parameter of an undercooled liquid
since other kinetic parameters can usually be obtained from η. For example, relaxation
times, τ, including shear stress (Maxwell) relaxation time and internal viscosity
equilibration time, are directly proportional to η, although the proportion coefficient may
be different for different times [29]. Diffusivity D is inversely proportional to η as
formulated by the Stokes-Einstein equation:

k BT
3πlη

(1.6)

where k B is the Boltzmann constant and l the average atomic diameter. Therefore, it is a
key issue to study the viscosity of an undercooled liquid as a function of temperature.
There are several models to describe the experimentally measured temperature
dependence of the equilibrium viscosity (fully relaxed) of an undercooled liquid. Perhaps
the two most frequently used are the Vogel-Fulcher-Tamman (VFT) model [2, 30-32]
and the free volume model [33].
In the VFT model (slightly modified by C. A. Angell [2]), the viscosity is expressed as:

η (T ) = η 0 exp(

DT0
T − T0

(1.7)

where η 0 , D and T0 are three constants from fitting of experimental data. Physically, η 0
refers to the viscosity extrapolated to the infinite temperature, D is a ‘strength’ parameter

13
(the higher D, the stronger the liquid; see below for more details), and T0 is an
extrapolated temperature (usually called the VFT temperature) at which the viscosity
would diverge.
Experimental data have shown that most glass-forming liquids show a deviation from
Arrhenius behavior (corresponding to T0 =0 and D = ∞ in the VFT model) in their
viscosity. As can be seen in Fig 1.1 (reproduced from [2]), the log10 η vs. Tg / T plots of

the many glass-forming liquids are curved rather than linear (corresponding to Arrhenius
behavior). The higher the curvature, the more the liquid deviates from Arrhenius behavior,
or in more fashionable terms, the more ‘fragile’ or less ‘strong’ is the liquid. If η 0 and T0
are fixed in the VFT model above, a higher D will correspond to a lower curvature (i.e., a
strong liquid). Hence, the D is normally known as the strength parameter of a liquid.
Another way to describe the deviation from the Arrhenius behavior [34] is the slope of
the log10 η vs. Tg / T plot at the glass transition temperature Tg

d log10 η
dT g / T

= 0.434
Tg

DT g T0
(T g − T0 ) 2

(1.8)

The larger the slope m, the more fragile is the liquid. Therefore, m is called the ‘fragility’
of the liquid. Usually, fragile liquids have m ≥ 100 and strong liquids have m in the range
of 16~30 [34]. For some good† metallic glasses such as Zr- and Pd-based BMGs, the
fragility typically values from 32 to 66 [35], while for some poor metallic glass-formers
like Al-based alloys, the fragility may be higher than 200 [36]. Due to its purely kinetic

in terms of glass-forming ability

Fig 1.1 Plots of viscosity data scaled by values of Tg for different glass-forming liquids.
The inset is the heat capacity change during the glass transition for these liquids
(reproduced after Ref. [2]).

15
nature, however, the fagility parameter is not expected to fully describe the glass-forming
ability of a liquid as we will see in the next section depends on both thermodynamics and
kinetics of the undercooled liquid.
While the viscosity of some metallic glass forming liquids such as Pd40Cu30Ni10P20 [37]
can be described well by the VFT model, others (e.g., Zr41.2Ti13.8Cu12.5Ni10Be22.5 [29])
require another: the free volume model. In this model, the viscosity is described as:

η=

νm

exp(

bν m

νf

(1.9)

where ν m and ν f are the molecular (atomic in the case of metallic glass) volume and the
mean free volume per molecule, respectively, and h is Planck’s constant. This model
assumes that the change in viscosity of a liquid is caused by the reduction (upon cooling)
or expansion (upon heating) of the mean free volume ν f . The temperature dependence of
the free volume is fitted as follows:

ν f = c1 [T − T0 + (T − T0 ) 2 + c 2T ]

(1.10)

where c1, c2 and T0 (all positive) are three fitting parameters. Eq. (1.10) prevents the
divergence of viscosity at a finite temperature. At high temperature, ν f ~ 2c1 (T − T0 ) and
at low temperature, ν f ~ c1T . Therefore, this model describes a transition from VFT
behavior (at high T) to Arrhenius behavior (at low T) that has been observed in
Zr41.2Ti13.8Cu12.5Ni10Be22.5 [29].

16
It should be noted that the above models are only for equilibrium viscosity. The
instantaneous viscosity at a temperature (especially around the glass transition
temperature) may be quite different from the equilibrium viscosity, depending on the
history (initial condition), time, and temperature. For example, if a liquid is maintained at
a constant temperature, its viscosity tends to evolve with time towards the equilibrium
value determined by the above models. This is a kinetic relaxation process which can
usually be described by a first-order reaction law as follows:
∂η
= k (η ∞ − η )
∂t T

(1.11)

where k is the rate constant (the inverse of relaxation time τ) and η ∞ is the equilibrium
viscosity (corresponding to t = ∞ ) at the temperature T. Both k and η ∞ depend on
temperature only. Therefore, the instantaneous viscosity during this isothermal relaxation
process can be derived from Eq. (1.11) as:

η (t ) = η ∞ + (η 0 − η ∞ ) exp(−kt )

(1.12)

where η0 is the initial viscosity at t = 0 . It can be seen that the time evolution of the
instantaneous viscosity takes an exponential form. Also note k = 1 / τ = G / η ∞ where G is
the shear modulus at temperature T for the internal relaxation process (which may be
different from the shear modulus for external stress relaxation within the Maxwell’s
model of viscoelasticity [29]).

1.3.3 Classical theory for crystal nucleation and growth from an
undercooled liquid and TTT diagram
When a liquid is cooled below its melting temperature Tm, crystallization tends to occur
by crystal nucleation and growth.
For a crystal to nucleate, the Gibbs free energy difference between the liquid and the
crystal acts as the driving force. On the other hand, nucleation involves the creation of an
interface between the two phases which tends to increase the system energy. Assuming a
spherical shape of the nucleus and a homogeneous† manner of the nucleation process (the
other type, heterogeneous nucleation, will be discussed in Chapter 4), the total energy
change caused by the formation of the nucleus is:
∆E = 4πr 2σ − πr 3 ∆G

(1.13)

where r, σ and ∆G are the radius of the nucleus, the interfacial energy per area, and the
Gibbs free energy difference per volume between the two phases, respectively. It can
been seen from Fig 1.2 that there is a maximum for ∆E corresponding to a critical
nucleus radius rc . If a nucleus has a radius lower than rc , it can’t grow spontaneously
because that would cause the system energy to increase. In contrast, a nucleus with a
radius larger than rc can grow spontaneously because that will cause the system energy

Homogeneous nucleation means there are no extrinsic nucleating agents.

to drop. By setting

d∆E
= 0 , one can calculate the critical nucleus radius rc and its
dr rc

corresponding critical energy barrier ∆E c :
2σ
∆G
16π σ 3
∆E c =
3 ∆G 2

rc =

(1.14)

The classical theory of nucleation (e.g., Ref. [38]) depicts the nucleation rate†, I v , as the
product of one kinetic term and one thermodynamic term as follows:
∆E c
Av
16πσ 3
Iv =
exp(−
)=
exp(−
k BT
3k B T∆G 2
Av

(1.15)

where Av is a constant of the order of 1032 Pa s/(m3 s) for homogeneous nucleation, η is
the viscosity, and k B is the Boltzmann constant. The viscosity has a temperature
dependence as described by the kinetic models in Section 1.3.2. Here, we choose to use
the VFT model (i.e., η (T ) = η 0 exp(

DT0
) ). The Gibbs free energy difference per
T − T0

volume between the liquid and the crystal (i.e., ∆G ), also has a temperature dependence
as given by Eq. (1.5) in Section 1.3.1. For simplicity, we take a first-order approximation
(i.e., ∆G = ∆S m (Tm − T ) ; ∆S m is the entropy of fusion per volume).
The classical theory describes the crystal growth rate‡ also using the product of one

defined by the number of critical nuclei that are formed within a unit volume per second
defined by the derivative of the crystal grain (nucleus) radius with respect to time, thus in units of m/s

4pi*r *sigma

delta(E)c

delta(E)

4/3*pi*r *delta(G)

Fig 1.2 Plots of the three terms in Eq. (1.13) vs. nucleus radius r.

20
kinetic term and one thermodynamic term as follows:

k BT
n∆G
[1 − exp(−
)]
k BT
3πl η

(1.16)

where l is the average atomic diameter and n is the average atomic volume. The first
(kinetic) term was originally expressed using atomic diffusivity. The form of Eq. (1.16)
has utilized the Stokes-Einstein relation between the diffusivity and the viscosity which
was given by Eq. (1.6) in Section 1.3.2.
For the best metallic glass former Pd40Cu30Ni10P20, the constant parameters have been
determined to be: η 0 = 9.34 × 10 −3 Pa s, D = 9.25 , T0 = 447 K [37], Tm = 823 K,

∆S m = 9.344 × 10 5 J/(m3 K) [39], σ = 0.067 J/m2, Av = 4.4 × 10 31 Pa s/(m3 s),
n = 1.52 × 10 −29 m3, and l = 3.1 × 10 −10 m [40]. Using these parameters, one can plot both
the nucleation rate and the crystal grow rate as a function of temperature for this alloy as
shown in Fig1.3.
From Fig1.3, one can see that both I v and u exhibit a maximum below Tm . Further, the
maximum of the nucleation rate occurs at a lower temperature than that of the growth rate.
This is generally true for any liquid. The physical reason for this is that the influence of
the viscosity factor (the kinetic term) to the nucleation rate is not as significant as to the
growth rate. Since the viscosity increases as temperature decreases from Tm , it hinders
the increase in the growth rate more effectively than it does the increase in the nucleation
rate. Therefore, the increase in the growth rate is stopped earlier by the viscosity than the

21
increase in the nucleation rate, which results in a higher peak temperature for the growth
rate than for the nucleation rate.
Having known I v and u, one can calculate the volume fraction, f, of the crystallized part
of the undercooled liquid as a function of time at an early† stage of crystallization by an
iterated integral as follows:
f = ∫ Iν (t ' )dt ' π [ ∫ u (t ' ' )dt ' ']3

(1.17)

where t ' and t ' ' are the two time coordinates for the nucleation and for the growth,
respectively.
For an isothermal crystallization process with the viscosity equilibrated (no relaxation
with time), I v and u are independent of time. Then, Eq. (1.17) can be simplified as:

f = πIν u 3 ∫ dt ' (t − t ' ) 3 = πIν u 3 t 4

(1.18)

Therefore, the time required to crystallize a certain volume fraction of the liquid is given
by:

t=(

3 f 1/ 4
πIν u 3

(1.19)

In Section 1.1.2, we introduced a critical value of f, i.e., f c = 10 −6 as the boundary

which means that the growing nuclei are not affected by each other and that the liquid matrix is not
changed much by crystallization so that Eq. (1.15) and (1.16) can apply.

Crystal growth rate u (m/s)
0.0

-8

5.0x10

-7

1.0x10

-7

1.5x10

-7

2.0x10

-7

2.5x10

-7

3.0x10

3.5x10

4.0x10

800

Temperature T (K)

750

700

650

600

550

500
0.0

2.0x10 4.0x10 6.0x10 8.0x10 1.0x10 1.2x10 1.4x10 1.6x10

Nucleation rate Iv (1/(m s))

Fig 1.3 Nucleation rate I v and crystal growth rate u as a function of temperature for the
BMG alloy Pd40Cu30Ni10P20.

23
between ‘crystallized’ and ‘not crystallized’ for practical glasses. Here, we can calculate
the time required to crystallize an f c fraction of the liquid at different temperatures using
Eq. (1.19). We still use the above Pd40Cu30Ni10P20 as a sample system. The calculation
result for this alloy is plotted in Fig 1.4. This certain type of plot as in Fig 1.4 is called
Uhlmann’s TTT (Time-Temperature-Transformation) diagram since Uhlmann first
utilized this to analyze the glass-forming ability of different substances [41].
From Fig 1.4 one can see that a TTT diagram has a C shape with a nose at an
intermediate temperature between the Tm (we use liquidus temperature Tliq for an alloy)
and Tg . Crystallization (of a small volume fraction, here f c = 10 −6 ) takes the shortest
incubation time, t n , at the nose temperature Tn . Here, from Fig 1.4, we find t n = 32 s,
and Tn = 671 K. According to early analysis by Uhlmann [41], the dotted line passing
through the nose of the TTT curve in Fig 1.4 gives the critical cooling rate RcTTT −iso †
required to bypass significant crystallization and form a practical glass:

RcTTT −iso =

Tm − Tn
tn

(1.20)

Here we find RcTTT −iso = 4.75 K/s for Pd40Cu30Ni10P20. Note that there is some small
difference in the t n , Tn and RcTTT −iso values‡ determined here from those determined in
Ref. [40] due to the different values for f utilized in the two calculations. Further, one

The superscript denotes that this value is obtained from isothermal TTT diagram.

In Ref. [40], t n = 50 s, Tn = 680 K, Rc

TTT −iso

= 2.86 K/s.

850
825

Tliq = 823 K

800
775
750

T (K)

725
700
675
650
625
600

Tg = 582 K

575
550

100

150

200

250

300

t (s)

Fig 1.4 TTT (Time-Temperature-Transformation) diagram of Pd40Cu30Ni10P20 calculated
using a crystallized volume fraction f = 10 −6 . The dotted line passing through the nose
of the diagram indicates the critical cooling rate required to form a glass upon cooling.

25
notices that the actual critical cooling rate for this alloy measured by continuous cooling
experiments [40] is only ~0.33 K/s -- one order of magnitude lower than RcTTT −iso . This is
because the calculation method used in Eq. (1.20) implicitly assumes that with a given
time length, the degree of crystallization is the same in the isothermal process as in the
continuous cooling. This is apparently not true since the continuous cooling process goes
through a set of temperatures at which the kinetics of crystallization differs a lot.
Uhlmann also realized this and made a correction by constructing continuous cooling
curves following the approach of Grange and Kiefer [42]. By doing so, he found that a
given degree of crystallinity (i.e., a given f) develops at lower temperatures and longer
times during the continuous cooling than during the isothermal process, as a result of
which the nose of the TTT diagram is shifted to the right (longer time) and the critical
cooling rate to bypass the nose is thus lower than the one predicted by Eq. (1.20) which is
solely based on isothermal TTT diagram.
Later Uhlmann and co-workers solved the continuous cooling problem directly, without
the aid of isothermal TTT diagram [43]. By plugging the time dependence (equivalent to
temperature dependence since dT = − Rdt , where R is the cooling rate assumed to be
constant) of I v and u into Eq. (1.17), one can find f =

4π T
dT
u (T ' ' )dT ' ']3
4 ∫T
3R liq

where Tliq is treated as the initial temperature where the crystallization starts. Since glass
formation means that the degree of crystallinity (crystallized volume fraction) developed
between Tliq and Tg is no larger than the critical value f c , the critical cooling rate
required for glass formation can then be calculated as follows:

Rc = {

Tg
4π Tg
Iν (T ' )dT '[ ∫ u (T ' ' )dT ' ']3 }1 / 4
T'
3 f c Tliq

(1.21)

The value obtained using this equation for Pd40Cu30Ni10P20 is 0.63 K/s ( f c = 10 −6 ) or
0.11 K/s ( f c = 10 −3 ), much closer to the measured 0.33 K/s than the RcTTT −iso based on
isothermal TTT diagram.
The rest of the difference between 0.63 K/s (or 0.11 K/s) and 0.33 K/s is almost
negligible, considering the possible error (passed through experimental measurements of
viscosity, isothermal TTT diagram measurement, and the related data fitting) in the
parameters used for the calculation. Another possible source of the difference may be the
approximation for the Gibbs free energy difference between the liquid and the crystal (i.e.,
∆G = ∆S m (Tm − T ) ). If we know the temperature dependence of the heat capacity
difference between the liquid and the crystal, we can then use the full description of ∆G
(i.e., Eq. (1.5)), which may bring about an even better agreement between the calculated
and the measured critical cooling rates.

1.4

Frequently used criteria for the development of BMGs

Even though the classical theory of crystal nucleation and growth as demonstrated in the
last section may describe well the crystallization kinetics of a glass-forming liquid and
may even calculate the critical cooling rate for the glass formation to a very high
precision, it is obviously not a convenient way to predict the best glass-forming
compositions among a large number of candidate alloys since it requires a large amount
of experimental data from each of the candidates. Therefore, ever since the first discovery

27
of metallic glass in 1960, continuous efforts have been devoted to the establishment of a
simple and universal criterion to be used in the development of BMGs. As a result, quite
a number of such criteria have been proposed to date. Although these criteria generally
provide some guidance to the alloy development, there are always exceptions to every
single one of them, such that excessive reliance on these criteria sometimes causes the
negligence of very good metallic glass-formers. Here, however, we should still do a brief
review of the most frequently used criteria. The limitations of each criterion will be
discussed.
1.4.1

Reduced glass transition temperature ( Trg )

Shortly after the discovery of the first metallic glass (Au75Si25) in 1960, Turnbull
proposed that a glass tends to form easily from a liquid with a high reduced glass
transition temperature (defined as Trg =

Tg
Tm

where Tg and Tm are the glass transition and

melting temperatures, respectively [44]).
Turnbull based his argument on the assumption that a glass would form if the nucleation
rate is so low that no nuclei can form virtually on the cooling time scale. He used a
different form of Eq. (1.15) in Section 1.3.3 as follows:

Iv =

exp[−

bα 3 β
Tr (∆Tr ) 2

(1.22)

where k n is a constant specified by the model, b is a constant determined by the nucleus
shape ( b = 16π / 3 for a spherical nucleus), α and β are dimensionless parameters defined

28
as α = ( NV 2 )1 / 3 σ / ∆H m (N is the Avogadro’s number, V is the molar volume, σ is the
interfacial energy between the liquid and the crystal, ∆H m is the molar heat of fusion);
and β = ∆S m / R ( ∆S m is the molar entropy of fusion, R is the universal gas constant). Tr
and ∆Tr are the reduced temperature and reduced undercooling, respectively, defined as:
Tr = T / Tm and ∆Tr = (Tm − T ) / Tm = 1 − Tr . In the above equation, he also utilized the
first order approximation for the Gibbs energy difference between the liquid and the
crystal (i.e., ∆G = ∆S m (Tm − T ) ). Further, he simplified the VFT expression (Eq. (1.7))
for the viscosity as:

η = 10 −3.3 exp(

3.34
Tr − Trg

(1.23)

in which he equated Tg with the VFT temperature T0 and reduced the temperatures with
Tm . By substituting the values for the other parameters, namely, k n = 10 35 Pa s/(m3 s)†,
b = 16π / 3 , αβ 1 / 3 = 1 / 2 , he obtained the plots of log I v vs. Tr based on different values

of the reduced glass transition temperature Trg . His result is reproduced here in Fig 1.5.

From these plots, Turnbull argued that the larger the Trg , the smaller the magnitude of the
nucleation rate and the narrower the time window available for nucleation. Further, when
Trg approaches 2/3 or higher, he argued, the nucleation rate is so low that within the
cooling time scale in laboratory, no nuclei can form virtually (i.e., the number of nuclei

Interestingly, in Turnbull’s original paper [44], he assigned a wrong unit to this constant and this small
mistake was followed by others for years. Here, both the magnitude and the unit have been corrected in
order to reproduce his result exactly. The small mistake does not affect his conclusion, on the other hand.

29
formed is less than one). This argument later gained a lot of attention and became the
famous ‘2/3 law’ for glass formation from liquids.
Indeed, it has been found from practice that the easy glass-formers as a whole family
usually show higher Trg values than very poor glass-formers. Meanwhile, this rule can
also explain the high frequency of the occurrence of BMGs around deep eutectics. In a
given alloy system, the variation in Tg as the alloy composition changes is not as
significant as the variation in Tl , and the low Tl around the eutectic composition thus
results in a high Trg ( = Tg / Tl )†. However, when it comes to quantitatively comparing the
glass-forming ability of two easy glass formers or pinpointing the best glass-forming
composition in a certain system, there are many exceptions to this rule, including some of
the BMG systems developed in this thesis, where alloys with higher Trg exhibit poorer
glass-forming ability.
The uncertainty in this rule comes from several factors‡: 1). the contributions to I v from
other parameters such as α, β, T0 and D (the strength parameter in the VFT model of
viscosity) which may also have non-negligible dependence on the alloy composition
(compared to Trg ) have not been considered; 2). the reliability of the first order

The replacement of Tm in Turnbull’s criterion by the liquidus temperature Tl rather than the solidus
temperature Ts in practice has been justified in statistical studies on a large number of BMG systems. A
simple justification can be made using the fact that the primary equilibrium phase (different on different
sides of an eutectic) is in most cases, the competing phase for glass formation. As such, the temperature at
which this primary phase starts to form (i.e. the liquidus temperature of this phase) should be utilized for
the calculation of Trg .

Some of these factors were discussed in Turnbull’s paper, and some were not.

30
25

Trg = 0

20
15
10

log Iv

Trg = 1/2

-5
-10
-15

Trg = 2/3

-20
-25
-30
0.0

0.2

0.4

0.6

0.8

1.0

Fig 1.5 logarithm of nucleation rate (in cm-3s-1), log I v , vs. the reduced temperature, Tr ,
calculated at different values of the reduced glass transition temperature Trg (reproduced
after Ref. [44]).

31
approximation for ∆G used in Turnbull’s analysis may not be the same good for any
alloy, depending on whether or not the heat capacity difference between the undercooled
liquid and the crystal states of the alloy is negligible; 3). the assumption that the glassforming ability can be fully represented by the number of nuclei formed during cooling is
not well justified (i.e., the contribution from crystal growth to crystallization has not been
clarified in this rule).
1.4.2

Multi-component rule (confusion principle)

The sudden improvement in the glass-forming ability from the early poor metallic glass
formers (before the end of the 1980’s) to the first bulk metallic glasses (Zr-, La-, and Mgbased, found at the end of the 1980’s and the beginning of the 1990’s), and the historical
fact that the former were mostly binary alloys while the latter were invariantly multicomponent alloys have led to an impression that good glass-forming ability solely
belongs to multi-component systems. For a very long time, it has been considered a
‘must’ rule for BMG formation that the system contain at least three elements [45].
Further, a ‘confusion principle’ has been proposed for developing new BMGs which
states that ‘the more elements involved, the lower the chance that the alloy can select
viable crystal structures, and the greater the chance of glass formation’ [46].
While these rules make a certain amount of sense about the chemical complexity required
by BMG formation, their importance can’t be over-exaggerated, considering their purely
empirical nature. The effect of increased chemical complexity on the glass-forming
ability may well be twofold. On one hand, it may cause denser packing in the liquid and
lead to higher viscosity and lower atomic mobility; on the other hand, it may introduce

32
new competing crystalline phases with lower energy as well. Simple evidence for this is
that the addition of more randomly selected elements -- or even those selected with
caution (following other empirical rules such as those discussed below) -- to the two
known

easiest

metallic

glass

formers,

namely

Pd40Cu30Ni10P20

and

Zr41.2Ti13.8Cu12.5Ni10Be22.5, actually has a higher chance of ruining the superior glassforming ability of these alloys†.
As a matter of fact, with more and more BMG alloy compositions being reported, it
seems quite likely that the ideal number of components for most good metallic glass
formers sits between three and five, with few exceptions on either side of this range.
Moreover, it will be shown in Chapter 3 that BMG formation is not solely a privilege of
multi-component alloys. Instead, it may appear in as simple as binary systems with only
two components.
1.4.3

Atomic size mismatch

There are basically two different expressions for the requirements on atomic size
difference by BMG formation. One expression states that BMG formation requires, or at
least prefers, significantly different atomic sizes among main constituents [45]. The main
consideration underlying this expression is that a large difference in atomic sizes helps
destabilize the competing crystalline phase(s) by producing large lattice stress and thus
increasing the energy of the crystalline state [47]. This seems reasonable considering the
Hume-Rothery rule in classical metallurgy which states that the larger the atomic size

This has been manifested directly by the fact that no easier metallic glass formers based on these two, but
comprised of more components, have been discovered.

33
difference between solute and solvent elements, the smaller the solid solubility [48]. In
order to form a glass stable crystalline solid solutions have to be destabilized. A
quantitative model was established based on this consideration [47], which led to a
satisfying correlation between the hence calculated and the experimentally observed
minimum solute concentration in a large number of binary glass-forming alloys. However,
this atomic level stress model does not provide specific information about the optimal
solute concentration corresponding to the highest glass-forming ability in a given system,
especially a multi-component system.
The other expression of the atomic size criterion for BMG formation is based on the
topology of local atomic packing in the liquid state [49-51]. By analyzing the packing
geometry within the first coordinate shell around a solute atom, Miracle first found that
certain values of the atomic size ratio of the solute atom to the solvent atom can lead to
the maximum, 1, in the local packing efficiency† [49]. A survey on the reported metallic
glasses supported the idea that these values are preferred by metallic glass formation.
Later [51], Miracle extended his model by building a unit lattice cell in which the singleshell clusters (each has a layer of solvent atoms on the surface of a single solute atom)
occupy the vertices. He then applied his model to multi-component BMGs by introducing
a second and a third topological species to occupy the octahedral and the tetrahedral
interstices within the cluster lattice. He even suggested a quantitative way to calculate the
concentrations of each topological species of components. A reasonable agreement
between the hence calculated and the experimentally obtained optimal compositions of

The maximum local packing efficiency, 1, corresponds to a case in which the solvent atoms (surrounding
and touching a given solute atom) tightly touch each other with no spacing between two neighboring
solvent atoms.

34
many BMG alloys was demonstrated in his paper. Nevertheless, his model still has
limitations: 1). it has been based solely on the topology of local atomic packing within a
glass (or its liquid) without considering other factors such as the chemical interactions
among the constituent species; 2). the mechanism of how the topological factor
contributes to the thermodynamics, kinetics and the glass-forming ability of glassforming liquids is not clarified, or at least not quantified; 3). it provides no information
about the long range atomic packing which may actually be even more important than the
local packing to the kinetics of undercooled liquids, considering the free volume model of
viscosity. As a matter of fact, quite a number of the recently discovered BMGs, including
some developed in this thesis work, can’t be fully explained by this model in terms of
composition.

1.4.4 Chemical interactions among constituent elements
According to thermodynamics, the free energy change caused by mixing two different
species (A and B) is expressed as:
∆Gmix = ΩX A X B + RT ( X A ln X A + X B ln X B )

(1.24)

where X A and X B are the atomic concentrations, and Ω is proportional to the molar heat
of mixing (chemical interaction) between these two species†. Therefore, for a given X A
and X B , a negative heat of mixing (an attractive interaction) tends to lower the system’s
energy. If this mixing effect is more influential in the liquid (undercooled) state than in
the crystalline state, then the Gibbs free energy difference between these two states (i.e.,

The proportional coefficient is the coordination number which may differ from the liquid to crystal state.

35
the driving force for crystallization) can be lowered, and thus, the chance of glass
formation will be enhanced. This consideration has been proposed as another empirical
rule for BMG development [45] since many of the reported BMG systems appear to have
relatively large negative heat of mixing. However, the assumption made above (i.e., the
different effects of mixing on the liquid and the crystalline states) may not be well
satisfied in certain cases. As a result, excessive reliance on this empirical rule may not be
advisable.

1.4.5 Considerations based on phase diagrams
Phase diagrams are handy tools for BMG development because they provide important
information about both the liquid and the competing crystalline phases. Moreover, many
other empirical criteria such as the ones introduced in the previous sections are often
more or less reflected by phase diagrams.
Since high glass-forming ability usually appears around deep eutectics, it is often a good
choice to start with an alloy system whose phase diagram contains deep eutectics. Zr-Be
(Fig 1.6) is a good example of such systems. One can notice that there is a dramatic
decrease (of ~890 K) in the liquid temperature when 35% Be is added to pure Zr to form
the binary eutectic. This particular system has also another attractive feature: the eutectic
zone is far from both stable intermetallic compounds and terminal solid solution (Be in
Zr), which indicates that in order for the liquid (around the eutectic zone) to crystallize,
the local chemical composition inside the liquid has to undergo a severe change by
significant atomic rearrangement.

Fig 1.6 Binary phase diagram of Zr-Be system (reproduced from Ref. [52])

Fig 1.7 Binary phase diagram of Ti-Be system (reproduced from Ref. [52])

Fig 1.8 Binary phase diagram of Zr-Cu system (reproduced from Ref. [52])

Fig 1.9 Binary phase diagram of Zr-Ni system (reproduced from Ref. [52])

38
In order to further improve glass-forming ability, additional alloying elements can be
introduced to the starting system. Many studies have shown that the combination of
several simple eutectic systems often leads to an even deeper eutectic in the resulting
complex system. For example, both Zr-Be (Fig 1.6) and Ti-Be (Fig 1.7) binary phase
diagrams exhibit a deep eutectic on the Zr-rich side, and replacing part of Zr in Zr-Be
alloys (around its binary eutectic composition) with certain amount of Ti further lowers
the liquidus temperature and brings the alloys closer to an even deeper ternary eutectic.
Fig 1.8 and 1.9 show the binary phase diagrams of Zr-Cu and Zr-Ni systems, respectively.
One notices two important features on the Zr-rich side (Zr content > 50%) of both of
these two diagrams: 1). there exist a couple of deep binary eutectics; and 2). the
competing crystalline phases are the intermetallic compounds ZrM and Zr2M (where
M=Cu or Ni) † which do not have counterparts in Zr-Be system. Therefore, one expects
that by introducing Ni and/or Cu into Zr(Ti)-Be system, the eutectic temperature will be
further lowered and, meanwhile, the crystallization of the liquid will become more
difficult due to the increase in the number of competing crystalline phases with different
structures‡.
When adding more alloying elements to a base system, it is important to avoid
introducing very stable crystals. For example, when adding B into Zr-based alloys, the
amount of B has to be carefully controlled because otherwise a very stable compound
(ZrB2, as shown in Fig 1.10) may be encountered.

A closer look at the database for the compounds discloses that the structures of these compounds are even
different for Cu and for Ni, although their chemical formulae are the same [52].
Also noteworthy is that none of the newly introduced competing phases (i.e., ZrM and Zr2M, where
M=Cu or Ni) is much more stable than the original competing phases (i.e., ZrBe2 and Be-in-Zr solid
solution), judging from the melting temperatures of all these crystals.

Fig 1.10 Binary phase diagram of Zr-B system (reproduced from Ref. [52])

40
Like other empirical criteria, the application of phase diagrams in BMG development
also has limitations. On one hand, most of the available phase diagrams refer to only
thermodynamically stable phases and provide no information about metastable or even
unstable phases that may come up in rapid cooling of a liquid. On the other hand, the
phase diagrams for complex alloy systems are generally not available and have to be
conjectured from the phase diagrams of the sub-systems. The conjecture may not be very
reliable sometimes, especially when unknown crystalline phases are resulted from the
combination of sub-systems.

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Chapter 2
Formation and properties of Ni-based BMGs in Ni-Cu-Ti-Zr-Al system
2.1

Introduction

Although the early discovered BMGs such as Zr- [1,2], La- [3,4], Mg- [5] based alloys
provided for the first time an excellent combination of high glass-forming ability and
good mechanical properties, they still have certain limitations as candidates for structural
materials. For example, their glass transition temperatures are usually quite low, which
prevents them from high temperature applications; their relatively high material costs and
low elastic modulus also leave room for further improvement. Accordingly, there is a
growing interest in identifying processable amorphous alloys with greater strength,
elastic modulus, hardness, and lower material costs. Of particular interest are the alloys
based on common metals like Fe, Co, Ni, etc.
As to Ni-based alloys, although quite a few glass-forming systems with very high
strength (typically larger than 2 GPa, and some even approaching 3 GPa) have been
reported, the glass-forming abilities achieved so far are very limited. For example, NiNb-Cr-Mo-P-B alloys could only form 1 mm diameter amorphous rods [6]. The more
recent Ni-Ti-Zr-(Si,Sn) [7], Ni-Nb-Ti-Zr-Co-Cu [8] and Ni-Nb-Sn (co-developed by H.
Choi-Yim and the author at Caltech) [9] alloys could be cast to 2 mm diameter glassy
rods, but no larger. Further, some of these reported systems [6,7] comprise metalloids in
their chemical compositions which limit their manufacturability. For broader engineering

46
applications or scientific studies on Ni-based glasses, it is necessary to develop new Nibased alloys with higher glass-forming abilities and better manufacturability.
This chapter reports on the formation, thermal and mechanical properties of a new family
of Ni-based bulk glass-forming alloys with the formula NixCua-xTiyZrb-yAl10 (a~b~45, in
at.%) which possess a critical casting thickness ranging from 2mm up to 5mm†.

2.2

Experimentals

The ingots of the alloys studied in this work were prepared by arc melting mixtures of
ultrasonically-cleansed elemental metals having a purity of 99.5 at.% or higher. The arc
melting was performed in a Ti-gettered high purity Argon atmosphere. Each ingot was remelted in the arc melter at least three times aimed at obtaining chemical homogeneity.
The alloyed ingots were then re-melted under high vacuum in a quartz tube using an
induction-heating coil, and then injected through a small nozzle into a copper mold using
high purity argon at a pressure of 1-2 atm. The copper molds have internal rectangular
cavities with various thicknesses ranging from 0.5 mm to 5 mm. For comparison
purposes, very thin samples of thickness ~60 µm were also prepared using an Edmund
Buhler D-7400 splat quencher. Both the thin samples and the transverse cross sections of
the bulk cast samples (cut along a plane normal to the length of the samples) were
examined by X-ray diffraction (XRD), using a 120° position sensitive detector (Inel) and
a collimated Co Kα source. The amorphous structures of the bulk cast samples were
further confirmed by transmission electron microscopy (TEM) analyses performed on
their cross sections. The glass transition and crystallization behaviors of all samples were

These results have been published in Ref. [10]. A patent has been applied for these new Ni-based BMGs.

47
examined with a differential scanning calorimeter (Perkin-Elmer DSC 7) at a heating rate
of 0.33 K/s. Vicker’s hardness was measured on fully amorphous rectangular cast strips
using a Leitz micro-hardness tester. Young’s modulus, shear modulus and Poisson ratio
were obtained by measuring the longitudinal and shear sound velocities in the fully
amorphous strips with an ultrasonic device and substituting the velocities into a set of
formulas to be shown in Section 2.3.5. 2 mm (diameter) x 4 mm (height) cylindrical cast
samples were used to measure compressive mechanical properties of the alloys on an
Instron testing machine at a strain rate of ~4x10-4 s-1. Prior to the compression test, both
the top and the bottom of each specimen were examined with X-ray to make sure the
casting was successful and that no crystallization due to unexpected factors occurred.

2.3

Results and discussions

2.3.1 Ternary Ni45Ti20Zr35 alloy

Table 2.1 lists some examples of the newly discovered bulk metallic glasses in an order
that reflects the sequential optimization of successive alloy additions which resulted in
the improvement of the critical casting thickness for obtaining fully glassy samples.
The first phase of this work was the discovery of the ternary thin glass former,
Ni45Ti20Zr35, which produced a 0.5 mm thick partially amorphous strip using injection
mold casting. The XRD and DSC scans of this ternary alloy are included in Fig 2.1 and
Fig 2.2, respectively. Although there is some evidence of crystallinity on the XRD pattern
in Fig 2.1, the apparent diffuse background represents a large fraction of amorphous
phase in the sample. This is further confirmed by the DSC scan in Fig 2.2 which gives a

48
total exothermic heat release of 52 J/g caused by the crystallization of the amorphous
fraction of the specimen. The Tg and Tx1 (marked in the figure by arrows) of this alloy are
725 K and 752 K, respectively, and thus, the stable undercooled liquid range is ∆T=Tx1Tg=27 K.

Table 2.1 Examples of the new Ni-based amorphous alloys developed in this work
(Tg and Tx1 were measured with DSC at a heating rate of 0.33K/s)

Alloy Composition

Critical Casting

Tx1

∆T=Tx1-Tg

(at.%)

thickness (mm)

(K)

Ni45Ti20Zr35

~0.5

725

752

Ni45Ti20Zr27Al8

<0.5

761

802

Ni45Ti20Zr25Al10

773

818

Ni45Ti20Zr23Al12

<0.5

783

832

Ni40Cu6Ti16Zr28Al10

765

807

Ni40Cu5Ti17Zr28Al10

762

808

Ni40Cu5Ti16.5Zr28.5Al10

763

809

Ni39.8Cu5.97Ti15.92Zr27.86Al9.95Si0.5

768

815

Relative Intensity (a.u.)

Ni45Ti20Zr23Al12, 2mm

Ni45Ti20Zr25Al10, 2mm

Ni45Ti20Zr27Al8, 2mm
Ni45Ti20Zr35, 0.5mm

100

Two Theta (degree)
Fig 2.1 XRD patterns of selected ternary and quaternary alloys taken with a Co Kα
source.

Tx1

Normalized Heat Flow (W/g)
Exothermic

0.8

Ni45Ti20Zr23Al12

0.5mm

0.3

Ni45Ti20Zr25Al10

2mm

-0.2

Ni45Ti20Zr27Al8

2mm

-0.7

Ni45Ti20Zr35

0.5mm

-1.2

Tg
-1.7
-2.2

Heating Rate = 0.33 K/s

-2.7
-3.2
700

750

800

850

900

950

Temperature (K)

Fig 2.2 DSC scans of selected ternary and quaternary alloys at a heating rate of 0.33 K/s.

50
2.3.2 Quaternary Ni45Ti20Zr35-xAlx alloys

Fig 2.1 shows the structural effects of subsequent Al additions to the ternary alloy (in
replacement of Zr). In Fig 2.1, the quaternary samples used for taking XRD patterns are
all 2 mm-thick strips and the patterns were taken from the transverse cross sections of the
strips. The sample with 8% Al shows a weak broad background together with some
nanocrystalline-like peaks, indicating a two-phase (amorphous phase + crystalline phase)
partially crystallized structure. The sample with 10% Al only shows a series of broad
diffraction maxima without any observable crystalline Bragg peaks, indicating a fully
amorphous structure. The sample with 12% Al shows many crystalline peaks without any
noticeable diffuse background, indicating the formation of one or more complex
intermetallic compounds. Clearly, the alloy with 10% Al is the best glass former in this
quaternary alloy series.
Fig 2.2 presents the thermodynamic effects of the Al additions. As can be seen from these
DSC scans, as well as from Table I, the Tg, Tx1, and ∆T all increase monotonically as the
Al content increases from 8% to 10% and 12%. It was previously proposed [11] that the
glass-forming ability increases with ∆T or, in other words, alloys with higher ∆T values
tend to have higher glass-forming abilities. However, the diffraction results in Fig 2.1
clearly show that the highest glass-forming ability in the current quaternary alloy series
occurs at 10% Al which does not correspond to the highest ∆T value. The alloy with the
highest ∆T value (i.e., Ni45Ti20Zr23Al12) seems to have the lowest glass forming ability
among the three alloys, judging from diffraction patterns in Fig 2.1. A similar
discrepancy between glass forming ability and thermal stability (i.e., ∆T) has been

51
observed in other alloy systems [12-14]. For the Vitreloy series of BMGs [12], a
decomposition mechanism was used to explain why the best glass former does not have
the highest thermal stability upon heating at a constant rate. Further work including
SANS (Small Angel Neutron Scattering) experiments is needed to clarify if the same
mechanism is involved in the current alloys.
2.3.3 Quinary NixCua-xTiyZrb-yAl10 alloys (a~b~45)

Fig 2.3, 2.4, and 2.5 present further major improvements achieved by adding copper to
the above quaternary alloy Ni45Ti20Zr25Al10. With small amounts of copper and small
adjustments in compositions, thicker, fully amorphous samples have been successfully
prepared. Without Cu, the quaternary alloy is significantly crystallized at 3 mm thickness
as shown in Fig 2.3. The appropriate additions of Cu prevent the formation of the
intermetallic compounds yielding fully amorphous samples. The best glass forming
ability was achieved from Ni40Cu5Ti16.5Zr28.5Al10 (‘RAG2’, in the following) which has a
critical casting thickness above 5 mm. To the best of our knowledge, this is the highest
critical casting thickness ever obtained for Ni-based BMGs. To confirm the fully
amorphous structure of the 5mm thick strip of RAG2, TEM analysis was also performed
on its transverse cross section. From Fig 2.4, one can see its electron diffraction pattern
only comprises a series of diffuse halo rings. No distinct evidence of sharp crystalline

Relative Intensity (a.u.)

Ni40Cu5Ti16.5Zr28.5Al10, splat quenched
Ni40Cu5Ti16.5Zr28.5Al10, 5mm
Ni40Cu5Ti17Zr28Al10, 4mm

Ni40Cu6Ti16Zr28Al10, 3mm

Ni45Ti20Zr25Al10, 3mm

100

Two Theta (degree)

Fig 2.3 XRD patterns of selected quaternary and quinary alloys taken with a Co Kα
source

Fig 2.4 Electron diffraction pattern taken from the transverse cross section of a 5mm
thick Ni40Cu5Ti16.5Zr28.5Al10 strip

Tx1

Normalized Heat Flow (W/g)

Ni40Cu5Ti16.5Zr28.5Al10, splat quenched

Tg
Ni40Cu5Ti16.5Zr28.5Al10, 5mm

Ni40Cu5Ti17Zr28Al10, 4mm

-1
Ni40Cu6Ti16Zr28Al10, 3mm

-3
-5

Heating Rate = 0.33 K/s
-7
700

750

800

850

900

950

Temperature (K)

Relative Intensity (a.u.)

Fig 2.5 DSC scans of selected quaternary and quinary alloys at a heating rate of 0.33 K/s

(Ni40Cu6Ti16Zr28Al10)99.5Si0.5, 5mm

Ni40Cu6Ti16Zr28Al10, 4mm

Two Theta (degree)
Fig 2.6 The effect of small amount Si addition

54
rings was found anywhere across the specimen. Therefore, it is clear that the 5 mm strip
of RAG2 indeed has a fully amorphous structure. TEM analyses were also performed on
other bulk samples. The results, which are not shown here, are all in good agreement with
the XRD analyses.
Fig 2.5 shows the DSC traces of the quinary alloys. All these samples exhibit an
endothermic glass transition and a fairly wide undercooled liquid region, followed by one
or more exothermic events characteristic of crystallization. Their Tg, Tx1, and ∆T values
are listed in Table 2.1. For a comparison, a DSC trace taken from a ~60 µm thick splat
quenched sample of RAG2 is also included in Fig 2.5. Within the measurement range,
there is no appreciable difference in the Tg, Tx1 and ∆Hx (total enthalpy of crystallization)
values of the splat quenched sample and the 5mm thick strip of RAG2. This again
confirms the fully amorphous structure of the bulk cast sample.
2.3.4

Effect of small Si additions

A small amount Si addition also appears to provide an improvement as illustrated in Fig
2.6. Without Si, 4 mm thick Ni40Cu6Ti16Zr28Al10 strip shows an observable Bragg peak
superimposed on the broad, amorphous diffraction band, indicating that small
nanocrystals have precipitated from the amorphous matrix. However, with 0.5% Si, the
alloy (Ni40Cu6Ti16Zr28Al10)99.5Si0.5 is fully amorphous up to 5 mm, as shown by the
absence of any sharp crystalline peaks on the XRD pattern in Fig 2.6. The thermal
parameters of this Si-containing alloy are also included in Table 2.1, where it can be seen
that the small Si addition enlarges the undercooled liquid region ∆T by increasing the
crystallization temperature (Tx1) while leaving Tg almost unchanged. This enhancement

55
of glass forming ability and stability of the glassy state by adding small amounts of Si
agrees with previous reports for Zr-based BMG’s [16].
2.3.5

Mechanical tests

Vicker’s hardness and elastic modulii were measured using those cast strips confirmed to
be fully amorphous by both XRD and DSC. Selected results are shown in Table 2.2. The
modulii and Poisson ratio were obtained by measuring the sound propagation velocities
of plane waves (longitudinal and transverse, Cl and Cs, respectively) in the alloys, then
using the following relations (valid for isotropic materials such as glasses):
ν=(2-x)/(2-2x)=Poisson ratio, where x=(Cl/Cs)2
G=ρ*Cs2 = shear modulus, where ρ is density
E=G*2(1+ν) = Young’s modulus.
Table 2.2 Some measured mechanical properties of selected alloys
Alloy Composition

Vicker’s

Poisson

Shear

Young’s

Fracture

(at.%)

Hardness

Ratio

Modulus

Strength

(GPa)

(Kg/mm2)

Ni45Ti20Zr25Al10

791

0.36

114

2.37

Ni40Cu6Ti16Zr28Al10

780

0.361

40.9

111

2.18

Ni40Cu5Ti17Zr28Al10

862

0.348

49.7

133.9

2.3

Ni40Cu5Ti16.5Zr28.5Al10

800

0.355

45.2

122

2.3

Ni39.8Cu5.97Ti15.92Zr27.86Al9.95Si0.5

829

0.36

117

2.32

Compressive Stress (MPa)

2500
2000
(a)
1500
(b)
1000
500

0.5

1.5

2.5

Compressive Strain (%)

Fig 2.7 Compressive stress vs. strain curves of two selected alloys: (a)
Ni40Cu5Ti16.5Zr28.5Al10; and (b) Ni45Ti20Zr25Al10

57
Fig 2.7 presents the compressive stress vs. strain curves for two selected alloys. The
slopes have been calibrated using the Young’s modulus data measured from acoustic
experiments. The strength data obtained from these compression tests are included in
Table 2.2. These alloys have quite high fracture strength (~2.3-2.4 GPa). The quaternary
alloy, Ni45Ti20Zr25AL10, has a slightly higher strength than the quinary alloys (e.g.,
RAG2). This is associated with the small drop in Tg caused by the addition of Cu (see
Table 2.1). It is noteworthy that these alloys roughly obey the theoretical relation
between Vicker’s hardness and strength: σ ~3*Hv (Hv in Kg/mm2, σ in MPa) for isotropic
materials. Significantly premature failure known for silica glass and some Ni-based
BMG’s [17] does not happen to these alloys.

2.4

Conclusions

The formation and properties of a new series of Ni-based bulk glass-forming alloys with
formula NixCua-xTiyZrb-yAl10 (a~b~45, in at.%) are reported which have a critical casting
thickness ranging from 2 mm to 5 mm. The best GFA appears around x=40, y=16.5,
a=b=45. These new amorphous alloys exhibit high thermal stabilities (∆T ~40-50 K) and
excellent mechanical properties (e.g., σf ~2.3-2.4 GPa). Small amount Si-addition is
found to enhance the glass-forming abilities and the thermal stabilities of these alloys.
The GFA and ∆T of some quaternary alloys are found not to be in agreement with each
other. The glass forming abilities reported here may be the highest ever obtained for Nibased alloys. Meanwhile, the all-metallic compositions endow these present Ni-based
BMG’s with excellent manufacturability.

References
[1]

A. Peker and W. L. Johnson, Appl. Phys. Lett. 63, 2342 (1993).

[2]

A. Inoue and T. Zhang, Mater. Trans. JIM 37, 185 (1996).

[3]

A. Inoue, T. Nakamura, T. Sugita, T. Zhang and T. Masumoto, Mater. Trans. JIM
34, 351 (1993).

[4]

Z. P. Lu, Y. Li, S. C. Ng and Y. P. Feng, J. Non-Cryst. Solids 252, 601 (1999).

[5]

A. Inoue, A. Kato, T. Zhang, S. G. Kim and T. Masumoto, Mater. Trans. JIM 32,
609 (1991).

[6]

X. Wang, E. Yoshii, A. Inoue, Y. H. Kim and I. B. Kim, Mater. Trans. JIM 40,
1130 (1999).

[7]

S. Yi, J. K. Lee, W. T. Kim and D. H. Kim, J. Non-Cryst. Solids 291, 132 (2001).

[8]

T. Zhang and A. Inoue, Mater. Trans. JIM 43, 708 (2002).

[9]

H. Choi-Yim, D. H. Xu and W. L. Johnson, Appl. Phys. Lett. 82, 1030 (2003).

[10]

D. H. Xu, G. Duan, W. L. Johnson and C. Garland, Acta Mater. 52, 3493 (2004).

[11]

A. Inoue, Acta Mater. 48, 279 (2000).

[12]

T. A. Waniuk, J. Schroers, W. L. Johnson, Appl. Phys. Lett. 78, 1213 (2001).

[13]

Z. P. Lu, Y. Li and S. C. Ng, J. Non-Cryst. Solids 270, 103 (2000).

59
[14]

Z. P. Lu and C. T. Liu, Acta Mater. 50, 3501 (2002).

[15]

D. Turnbull, Contemp. Phys. 10, 473 (1969).

[16]

C. T. Liu, M. F. Chisholm, M. K. Miller, Intermetallics 10, 1105 (2002).

[17]

H. Choi-Yim, D.H. Xu, unpublished work. We found Ni-Nb-Sn alloys reported in
Ref. [9] fail prematurely; the apparent strength obtained from compression tests is
quite lower than expected values.

Chapter 3
Formation of bulk metallic glasses in binary Cu-Zr and Cu-Hf systems
3.1

Introduction

As we have discussed earlier (Section 1.4.2) in Chapter 1, the sudden improvement in the
glass-forming ability (GFA) from the early poor metallic glass formers (before the end of
the 1980’s) to the first bulk metallic glasses (Zr- [1,2], La- [3,4], and Mg- [5] based,
found at the end of the 1980’s and the beginning of the 1990’s), and the historical fact
that the former were mostly binary alloys while the latter were invariantly multicomponent alloys, have led to an impression that good glass-forming ability belongs
solely to multi-component systems. In fact, containing at least three elements has long
been considered a ‘must’ rule for BMG formation [6].
Why should binary alloys so distinctly differ from multi-component alloys in terms of
glass-forming ability? Inoue [6] did not give an answer because nobody (including
himself) even asked this question. It was just taken for granted that bulk glass formation
is solely the privilege of multi-component alloys. Is the answer to this question important?
Yes, because a ‘binary bulk metallic glass,’ if it exists, would be an excellent subject for
theoretical studies on the fundamental problem of glass formation, since it possesses both
the simplicity of binary alloys and the good glass forming ability of multi-component
alloys. On the other hand, from an engineering point of view, such a binary bulk metallic
glass might provide important guidance for the search for extremely good GFA and
might improve the current alloy developing efficiency considerably.

61
Aimed to find the answer to the above question, and to save possibly innocent binary
alloys from a possibly wrong conviction, I started alone the search for binary bulk
metallic glasses at Caltech. This search turned out to be fruitful, with two binary BMG
systems (namely, Cu-Zr and Cu-Hf) and three 2 mm thick BMG alloys (namely, Cu64Zr36,
Cu46Zr54, Cu66Hf34) discovered†.

3.2

Experimentals

The alloy compositions studied are Cu100-xZrx (34≤x≤75) and Cu100-xHfx (30≤x≤50) in
Cu-Zr and Cu-Hf system‡, respectively. The sample preparation and characterization
methods are basically the same as described in Chapter 2 (Section 2.2) except that a
Bruker AXS diffractometer with a Cu- Kα source was used for X-ray structural analyses
instead of the Inel with a Co Kα source. All of the studied alloys were subjected to
copper mold casting and subsequent X-ray and DSC scannings. The three best
compositions (with a critical casting thickness of 2 mm) were further examined using
TEM. The mechanical properties of these three best alloys were then measured.

3.3

Results and Discussion

3.3.1

Glass-forming abilities

Cu-Zr system was once among the most intensively studied binary metallic glass systems
before any BMG was reported (see, e.g., [10-13]). However, since the discovery of multi-

Part of these results has been published in Ref. [7-9].
In fact, quite a number of alloys in other binary systems (such as Ti-Cu, Zr-Ni, etc.) were also studied in
this research, but no bulk glasses (defined by a critical casting thickness of at least 1 mm) were found
except in the Cu-Zr and Cu-Hf systems to be discussed here.

Fig 3.1 Binary Cu-Zr phase diagram (reproduced from Ref. [17])

Fig 3.2 Binary Cu-Hf phase diagram (reproduced from Ref. [17])

Relative Intensity (a.u.)

A1
A2
A3

Two Theta (degree)

Fig 3.3 X-ray (taken with a Cu Kα source) and electron diffraction patterns of Cu46Zr54
(A1), Cu64Zr36(A2) and Cu66Hf34 (A3)

64
component BMG alloys in the late 1980’s and early 1990’s, this system -- like other
binary systems -- has received dramatically reduced attention, although occasionally, a
few studies on this system can still be found in the literature (e.g., [14-15]). The reason
behind this is that binary alloy systems including Cu-Zr were considered impossible to
form bulk metallic glasses and thus, were thought not likely to have any practical impact.
According to a recent paper [16] by Inoue and his colleagues, “it is well known that no
bulk glassy alloy is formed in Cu-Zr binary alloys by the copper mold casting method.”
Nevertheless, no systematic studies (directly by copper mold casting method) could be
found on the glass-forming abilities in this as well as other binary systems to support
Inoue’s conclusion. This present research is most likely the first study of this type.
The alloy compositions were selected around deep eutectics. According to Turnbull’s Trg
(=Tg/Tl) rule [18], the deep eutectic zone where high Trg values may occur (due to low Tl)
is probably also the place where good glass-forming ability would appear. On the other
hand, as was discussed in Chapter 1 (Section 1.4.1), the Trg rule may not be able to
exactly pinpoint the best glass forming composition. Therefore, one should examine a
range of compositions instead of the lowest liquidus point (i.e., x=38.2 in Cu-Zr system
and x=38.6 in Cu-Hf system) only.
Figure 3.3 shows the X-ray and electron diffraction patterns taken from the cross sections
of 2 mm thick cast strips of the three best glass formers in Cu-Zr and Cu-Hf systems
discovered in this study (namely, Cu64Zr36, Cu46Zr54, Cu66Hf34). It can be seen that the
diffraction patterns of these 2 mm strips only consist of diffuse maxima (X-ray) or diffuse
halo rings (electron diffraction) without any sharp Bragg peaks (or rings). Therefore, both

65
of these two diffraction methods prove that the 2 mm strips are all fully amorphous
within the instruments’ detection limits.
Varying the alloy composition by even 1 or 2 at.% on either side of these three best glass
formers results in a lower glass-forming ability. Take the alloy Cu64Zr36 as an example: to
its left side on the phase diagram (Fig 3.1), reducing Zr content from 36% to 34% leads
to a dramatic decrease in the critical casting thickness (from 2mm to below 0.5 mm). This
is evidenced by the XRD pattern taken from the 0.5 mm thick cast strip of the alloy
Cu66Zr34, which is shown in Fig 3.4. A couple of sharp Bragg peaks are superimposed on
the diffuse diffraction background of this alloy, indicating that the 0.5 mm thick strip of
Cu66Zr34 is already partially crystallized. Therefore, one expects that the critical casting
thickness required to get a fully amorphous structure of this alloy is below 0.5 mm. On
the right side of Cu64Zr36, all the studied alloys exhibit a fully amorphous structure at a
0.5 mm thickness, as can be seen from Fig 3.4. However, the 2 mm thick strips of
Cu61.8Zr38.2 and Cu60Zr40 are significantly crystallized, as evidenced by the fairly welldeveloped sharp Bragg peaks appearing on their XRD patterns (shown in Fig 3.5).
Apparently, the critical casting thickness of both Cu61.8Zr38.2 and Cu60Zr40 is lower than 2
mm (most likely ~ 1 mm, since their 0.5 mm thick strips are fully amorphous, according
to XRD patterns in Fig 3.4).
3.3.2

Thermal analyses with DSC

Thermal analyses were performed using DSC aimed to measure the thermal properties,
monitor the glass transition and crystallization behaviors of these binary BMGs, and to
understand the tendencies in their glass-forming abilities.

Relative Intensity (a.u.)

Cu60Zr40 , 0.5mm
Cu61.8Zr38.2 , 0.5mm
Cu64Zr36 , 0.5mm

Cu66Zr34 , 0.5mm

Two Theta (degree)

Fig 3.4 XRD patterns taken from 0.5 mm thick strips of Cu100-xZrx (x=34, 36, 38.2, 40
at.%) using a Cu-Kα source

Relative Intensity (a.u.)

Cu60Zr40 , 2mm

Cu61.8Zr38.2 , 2mm
Cu64Zr36 , 2mm

Cu66Zr34 , 2mm

Two Theta (degree)

Fig 3.5 XRD patterns taken from the cross sections of the 2mm thick cast strips of Cu100xZrx (x=34, 36, 38.2, 40 at.%) using a Cu-Kα source

67
Table 3.1 lists the thermal properties of the three best glass formers: Cu64Zr36, Cu46Zr54,
and Cu66Hf34. The reduced glass transition temperatures (Trg) of these three alloys are all
on the high side (close to 2/3), according to Turnbull’s model [18]. Besides, these alloys
also exhibit a fairly wide undercooled liquid region (∆T) which means they are quite
stable against crystallization upon heating.
Nevertheless, neither of these two parameters can be used to explain the exact tendencies
in the glass-forming abilities in these two binary systems. Still, we take the alloys around
Cu64Zr36 (i.e., the alloy series Cu100-xZrx (x=34, 36, 38.2, 40 at.%)) as examples. Their Tg
and Tx1 values were all extracted from their DSC scans which are shown in Fig 3.6. Then,
both the ∆T and Trg† values are calculated and plotted against x as shown in Fig 3.7. One
can see that these two parameters have quite similar trends within this alloy series. They
both increase when x changes from 34 to 38.2, and decrease when x goes from 38.2 to 40,
but neither of them assumes its maximum at x=36 which corresponds to the best GFA.
This once again proves that solely relying on these empirical parameters (rules) may not
be a good choice, especially when fine optimization of glass-forming composition within
a given system is concerned.
Table 3.1 Thermal properties of three best glass formers in Cu-Zr and Cu-Hf systems
Alloy
Cu46Zr54
Cu64Zr36
Cu66Hf34

Tg
(K)
696
787
787

Tx1
(K)
746
833
841

Tl
(K)
1201
1233
1263

∆T= Tx1- Tg
(K)
50
46
54

Trg= Tg/Tl
0.58
0.64
0.62

Critical casting
thickness (mm)

For the calculations of Trg, the liquidus temperatures were taken directly from the phase diagrams (i.e.,
Fig 3.1 and 3.2, provided in Ref. [17]).

Heat Flow (W /g)

Exothermic

Tx1

Cu60Zr40 , 0.5mm

Cu61.8Zr38.2 , 0.5mm
Cu64Zr36 , 2mm

-1

Cu66Zr34 , 0.5mm

-3
-5
-7
Heating rate = 0.33K/s
-9
700

750

800

850

900

950

Temperature (K)

Fig 3.6 DSC scans of the 0.5mm thick strips of Cu60Zr40, Cu61.8Zr38.2, and Cu66Zr34; and
the 2mm thick strip of Cu64Zr36 obtained at a heating rate of 0.33K/s

0.69

0.68

0.67

delta T (K)

0.65

0.64

Trg

0.66

0.63

0.62

15
0.61

0.60

0.59
42

Zr content, x (at.%)

Fig 3.7 Variations of ∆T and Trg, with respect to Zr content x in alloy series Cu100-xZrx
(x=34, 36, 38.2, 40 at.%)

69
3.3.3

Mechanical properties of the three best glass formers

The mechanical properties of the three best glass formers -- Cu64Zr36, Cu46Zr54, and
Cu66Hf34 -- were measured using the methods described in Chapter 2 (Section 2.2). The
obtained data are summarized in Table 3.2. In addition, the compressive stress vs. strain
curves of these alloys obtained at a strain rate of ~4x10-4 s-1 at room temperature are
given in Fig 3.8. From both Table 3.2 and Fig 3.8, one can see that the two alloys rich in
Cu, namely, Cu64Zr36 and Cu66Hf34 both exhibit yielding strength exceeding 2 GPa. The
one rich in Zr (i.e., Cu46Zr54), although yielding at a lower stress, has better ductility as
evidenced by its distinct plastic strain ~1.1% on the stress-strain curve.

Table 3.2 Mechanical properties of three best glass formers in Cu-Zr and Cu-Hf systems
Alloy

Density
(g/cc)

Cu46Zr54
Cu64Zr36
Cu66Hf34

7.3
7.9
11.4

Vicker’s
Hardness
(Kg/mm2)
698
742
779

Young’s
Mod.
(GPa)
83
92.3
111

Shear
Mod.
(GPa)
31
34
40

Poisson
Ratio
0.35
0.34
0.37

Yielding
Strength
(GPa)
1.4
2.1

Fracture
Strength
(GPa)
1.7
2.1

Yielding
Strain
(%)
1.7
2.2
1.86

Plastic
Strain
(%)
1.1
~0
~0

2500

Compressive Stress (MPa)

Cu66Hf34

Cu64Zr36

2000

Cu46Zr54
1500

1000

500

0.5

1.5

2.5

Compressive Strain (%)

Fig 3.8 Compressive stress vs. strain curves of the three best glass formers in Cu-Zr and
Cu-Hf systems obtained at a strain rate of ~4x10-4 s-1 at room temperature

3.4 Conclusions
For the first time, glass-forming abilities of binary alloys in Cu-Zr and Cu-Hf systems
were systematically studied using the copper mold casting method. It was found that
some of the binary alloys can form bulk metallic glasses, even with a critical casting
thickness up to 2 mm. These results have proven that the previously well-accepted ‘three
component rule’ for BMG formation is actually wrong. There may not be a strict
boundary between the glass-forming abilities of binary alloys and those of multicomponent alloys, although more multi-component alloys have been seen to form BMGs.
It was also found that the two empirical factors for evaluating glass-forming ability (i.e.,
∆T and Trg), although being able to explain to some extent the good glass-forming
abilities of the two binary alloy systems as a whole, can’t yet be used to pinpoint exactly
the best glass-forming compositions in these systems. The thermal and mechanical
properties of the three best glass-formers in the two systems were measured and reported
in this chapter. Finally, the three best glass-formers in the two studied systems (namely,
Cu64Zr36, Cu46Zr54, and Cu66Hf34) provide an exceptional combination of chemical
simplicity and good glass-forming ability and thus, may become special subjects for
simulation or modeling of BMG formation.

References
[1]

A. Peker and W. L. Johnson, Appl. Phys. Lett. 63, 2342 (1993).

[2]

A. Inoue and T. Zhang, Mater. Trans. JIM 37, 185 (1996).

[3]

A. Inoue, T. Nakamura, T. Sugita, T. Zhang and T. Masumoto, Mater. Trans. JIM
34, 351 (1993).

[4]

Z. P. Lu, Y. Li, S. C. Ng and Y. P. Feng, J. Non-Cryst. Solids 252, 601 (1999).

[5]

A. Inoue, A. Kato, T. Zhang, S. G. Kim and T. Masumoto, Mater. Trans. JIM 32,
609 (1991).

[6]

A. Inoue and A. Takeuchi, Mater. Trans. 43, 1892 (2002).

[7]

D. H. Xu, B. Lohwongwatana, G. Duan, W. L. Johnson and C. Garland, Acta
Mater. 52, 2621 (2004).

[8]

G. Duan, D. H. Xu and W. L. Johnson, Metall. Mater. Trans. A. 36A, 455 (2005).

[9]

D. H. Xu, G. Duan and W. L. Johnson, Phys. Rev. Lett. 92, 245504 (2004).

[10]

J. Chevrier, Solid State Commun. 65, 1461 (1988).

[11]

A. Sadoc, Y. Calvayrac, A. Quivy, M. Harmelin and A. M. Flank, J. Non-Cryst.
Solids 65, 109 (1984).

[12]

E. Kneller, Y. Khan and U. Gorres, Z. Metallkd. 77, 43 (1986).

73
[13]

E. Kneller, Y. Khan and U. Gorres, Z. Metallkd. 77, 153 (1986).

[14]

B. C. Anusionwu and G. A. Adebayo, J. Alloys Comp. 329, 162 (2001).

[15]

M. H. Braga, L. F. Malheiros, F. Castro and D. Soares, Z. Metallkd. 89, 541
(1998).

[16]

A. Inoue and W. Zhang, Mater. Trans. JIM 43, 2924 (2002).

[17]

Binary Alloy Phase Diagrams, 2nd ed., T. B. Massalski, ed. (ASM International,
Metals Park, OH, 1990).

[18]

D. Turnbull, Contemp. Phys. 10, 473 (1969).

Chapter 4
A generalized model for the critical-value problem of nucleation
4.1

Introduction†

As we have seen in Chapter 1, crystal nucleation is an important issue related to (metallic)
glass formation. Nucleation is generally classified into two categories: homogeneous
nucleation and heterogeneous nucleation. The former refers to cases in which there are no
extrinsic nucleating agents while the latter is influenced by such agents. In Chapter 1, we
considered only homogeneous nucleation. However, in more cases, nucleation takes
place in a heterogeneous manner.
The existing theory for heterogeneous nucleation is based mainly on the large-wall
assumption in which the size of the extrinsic agent is taken as infinity. The solution to
this problem can be found in many textbooks (e.g., [1]) as follows:

2σ
16π σ 3 het
E chet =
f (θ )
3 G2
f het (θ ) = [2 − 3 cos θ + cos 3 θ ] / 4

rchet =

(4.1)

where θ is the contact angle between the nucleus and the extrinsic agent, σ is the
interfacial energy per unit area (or, the interfacial tension) between the new phase (i.e.,
the nucleus) and the parental phase, and G is the Gibbs free energy difference per unit

The content of this chapter has been accepted for publication in Phys. Rev. B (2005).

75
volume between the two phases. This solution states that in order for a small nucleus to
grow from the parental phase, the size of the nucleus (r) and the energy associated with
the formation of the nucleus (E) have to exceed the critical value rc, and Ec determined by
Eq. (4.1), respectively. By comparing this heterogeneous solution with the homogeneous
solution given in Chapter 1 by Eq. (1.14), one can find that although the critical size of
the nucleus rc is the same for the two cases, the critical energy barrier Ec differs by the
factor f het (θ ) , given as part of Eq. (4.1), i.e.,
E chet = E chom f het (θ )

(4.2)

By plotting f het (θ ) (as shown in Fig 4.1), one notices that f het (θ ) takes on a value
between 1 and 0, depending on the contact angle θ. If θ is 0, Echet is zero; if θ is π (i.e.,
180°), E chet = E chom ; and if 0 < θ < π , E chet < E chom . Therefore, unless the contact angle is
π, the critical energy barrier for large-wall heterogeneous nucleation is always lower than
for homogeneous nucleation. This model explains why heterogeneous nucleation is
preferred most of the time if a large extrinsic wall exists.
Nevertheless, in a large variety of cases such as the formation of rain droplets from
clouds or the nucleation of crystals from the interior of small-particle-bearing liquids, the
assumptions adopted in the derivation of the above two classical solutions (i.e., a super
clean parental phase for the homogeneous case, or a large extrinsic wall for the
heterogeneous case) are not well satisfied, and consequently, these solutions may not
provide precise descriptions of such cases.

1.0

0.6

0.4

het

/ Ec

hom

het

= f (theta)

0.8

0.2

0.0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

Theta

Fig 4.1 The geometric factor as a function of contact angle θ in the large-wall
heterogeneous solution

77
In the field of metallic glass, it has been experimentally observed that the glass-forming
ability (GFA) of an alloy upon cooling from its molten state is strongly influenced by the
nucleating effect of the finite-sized impurities buried in the alloy melt. When the
impurities are either fluxed [2,3] or deactivated [4], the undercooling and GFA of the
alloy can be improved dramatically. However, it is still not clear exactly how these finitesized impurity particles affect the nucleation process and how their effects can be
carefully controlled by processing methods such as fluxing and microalloying. To help
answer these questions, I present in this chapter a generalized geometric model for the
critical problem of nucleation based on a finite-sized nucleating agent, then derive the
exact solution and discuss its physical implications.

4.2

Model construction

Fig 4.2 (a) illustrates the geometric construction for the new model, where a nucleus (N)
forms at the interface between the parental phase (P) and a finite-sized nucleating agent
(A). O1 and O2 are the spherical centers of N and A, respectively. S is a joint where the
three phases, P, N, and A, meet each other. Fig 4.2 (b) is an illustration of the mechanical
equilibrium at S, in which σ, σPA and σNA denote the interfacial tensions between P and N,
P and A, and N and A, respectively. In both figures, θ is the contact angle between N and
A. Besides θ, we introduce another important angle ϕ, i.e., ∠SO2O1, to relate the radius
of A (R, i.e., SO2) with that of N (r, i.e., SO1).

4.3

Model solution and interpretation

With the above construction and denotations, it is trivial to obtain the following
expressions:

Fig 4.2 (a) the geometric construction for the generalized nucleation model; (b) an
illustration of the mechanical equilibrium at point S in part (a)

79
1. The interfacial area (I.A.) between P and N (from Fig 4.2 (a)):
I . A.PN = 2πr 2 [1 − cos(ϕ + θ )]

(4.3)

2. The interfacial area between N and A (from Fig 4.2 (a)):
I . A. NA = 2πR 2 (1 − cos ϕ )

(4.4)

3. The volume of N (from Fig 4.2 (a)):

r 3 [2 − 3 cos(ϕ + θ ) + cos 3 (ϕ + θ )] −

R 3 (2 − 3 cos ϕ + cos 3 ϕ )

(4.5)

4. The interconnection among r, R, ϕ and θ (from triangle SO1O2 in Fig 4.2 (a)):
sin ϕ sin(ϕ + θ )

(4.6)

5. The interconnection among σ, σPA,σNA and θ (from Fig 4.2 (b)):

σ PA − σ NA = σ cosθ

(4.7)

Therefore, the energy change associated with the formation of nucleus N is: (G as defined
earlier)
E = σ × I . A.PN + (σ NA − σ PA ) × I . A. NA − G × V
= σ × I . A.PN − σ cosθ × I . A. NA − G × V

(4.8),

= E1 − E 2
where,
E1 = 2πσr 2 [1 − cos(ϕ + θ )] − 2πσR 2 cosθ (1 − cos ϕ )

(4.9),

and
E2 =

Gr 3 [2 − 3 cos(ϕ + θ ) + cos 3 (ϕ + θ )] −

GR 3 (2 − 3 cos ϕ + cos 3 ϕ )

For a given system, we have fixed R and θ. From Eqn. (4.6), we get

(4.10).

80
dr
R sin θ
dϕ sin 2 (ϕ + θ )

(4.11).

From Eqn. (4.9), (4.6) and (4.11), we get
dE1
sin 2 ϕ
= 2πσR 2 { 2
[1 − cos(ϕ + θ )] − cosθ (1 − cos ϕ )}
dr
dr sin (ϕ + θ )
sin 2 ϕ
dϕ d
= 2πσR
− cosθ (1 − cos ϕ )]
dr dϕ 1 + cos(ϕ + θ )

sin 2 (ϕ + θ )
2 cos ϕ
sin ϕ sin(ϕ + θ )
= 2πσR
× sin ϕ × {
− cosθ }
sin θ
1 + cos(ϕ + θ ) [1 + cos(ϕ + θ )]2
1 − cos 2 (ϕ + θ )
2 cos ϕ
sin ϕ sin(ϕ + θ )
= 2πσr sin(ϕ + θ )
×{
− cosθ }
sin θ
1 + cos(ϕ + θ ) [1 + cos(ϕ + θ )]2
1 − cos(ϕ + θ )
= 2πσr
× {2 cos ϕ sin(ϕ + θ ) + sin ϕ[1 − cos(ϕ + θ )] − cosθ sin(ϕ + θ )[1 + cos(ϕ + θ )]}
sin θ
1 − cos(ϕ + θ )
= 2πσr
× {2[sin θ + sin ϕ cos(ϕ + θ )] + sin ϕ[1 − cos(ϕ + θ )] − cosθ sin(ϕ + θ )[1 + cos(ϕ + θ )]}
sin θ
1 − cos(ϕ + θ )
= 2πσr
× {2 sin θ + sin ϕ[1 + cos(ϕ + θ )] − cosθ sin(ϕ + θ )[1 + cos(ϕ + θ )]}
sin θ
1 − cos(ϕ + θ )
= 2πσr
× {2 sin θ − sin θ cos(ϕ + θ )[1 + cos(ϕ + θ )]}
sin θ
= 2πσr[2 − 3 cos(ϕ + θ ) + cos 3 (ϕ + θ )]
(4.12).
From Eqn. (4.10) and (4.6) we get
dE 2 ∂E 2 ∂E 2 dϕ
∂r
∂ϕ dr
dr
= πGr 2 [2 − 3 cos(ϕ + θ ) + cos 3 (ϕ + θ )] +

dϕ π
× { Gr 3 [3 sin(ϕ + θ ) − 3 cos 2 (ϕ + θ ) sin(ϕ + θ )] − GR 3 (3 sin ϕ − 3 cos 2 ϕ sin ϕ )}
dr
dϕ
= πGr 2 [2 − 3 cos(ϕ + θ ) + cos 3 (ϕ + θ )] + πG
× [r 3 sin 3 (ϕ + θ ) − R 3 sin 3 ϕ ]
dr
= πGr 2 [2 − 3 cos(ϕ + θ ) + cos 3 (ϕ + θ )]
(4.13).
Therefore, from Eqn. (4.8), (4.12) and (4.13), we have

81
dE dE1 dE 2
dr
dr
dr
= 2πσr[2 − 3 cos(ϕ + θ ) + cos 3 (ϕ + θ )] − πGr 2 [2 − 3 cos(ϕ + θ ) + cos 3 (ϕ + θ )]
= πr (2σ − Gr )[2 − 3 cos(ϕ + θ ) + cos 3 (ϕ + θ )]

(4.14).

The critical condition is

dE
= 0 . Hence, we get
dr rc
rc =

2σ

(4.15).

It is clear that the critical diameter of the nucleus does not depend on either the contact
angle θ or the nucleating-agent size R, and has the same value for the generalized case
and for the two classical cases (i.e., rc = rchom = rchet ).
To find out the critical energy barrier Ec, we first substitute rc into Eqn. (4.9) and (4.10),
and then into (4.8). We get
E1 r = 8π

E2 r =

σ3

[1 − cos(ϕ + θ ) −

sin 2 (ϕ + θ )
cosθ (1 − cos ϕ )]
sin 2 ϕ

(4.16),

8π σ 3
sin 3 (ϕ + θ )
cos(
cos
(2 − 3 cos ϕ + cos 3 ϕ )] (4.17),
3 G
sin ϕ

and
Ec = E1 r − E 2 r

16π σ
g ( R, θ )
3 G2

(4.18),

where
sin 2 (ϕ + θ ) sin 3 (ϕ + θ )
g ( R,θ ) = [1 − cos (ϕ + θ ) − 3 cosθ
(2 − 3 cos ϕ + cos 3 ϕ )]
1 + cos ϕ
sin ϕ

(4.19),

82
and ϕ = ϕ ( R,θ ) is determined by Eqn. (4.6), or explicitly,

ϕ = arc cot(

R / rc − cosθ
sin θ

(4.20).

Fig 4.3 (a), (b) and (c) represent the 3D image and some 2D projected curves of this
bivariate function g ( R, θ ) with R scaled by rc =

2σ
. From these figures as well as from

Eqn. (4.19) and (4.20), it can be seen that for any fixed contact angle θ (i.e., fixed type of
nucleating agent), as the agent size R goes to 0 [5], g ( R, θ ) goes to 1, corresponding to
Ec going to Echom =

16π σ 3
, which means nucleation occurs in a homogeneous manner
3 G2

in the limiting case where R = 0 . Also, for any fixed θ, as R goes to +∞, g ( R, θ ) goes to
a constant value g (+∞, θ ) . It is trivial to find the expression for g (+∞, θ ) through Eqn.
(4.19) and (4.20) [5]:
g (+∞,θ ) =

(1 − cos 3 θ − cosθ sin 2 θ )

(2 − 3 cosθ + cos 3 θ )
= f het (θ )

(4.21).

Therefore, the classical heterogeneous solution (i.e., Eqn. (4.2)) actually describes only
the limiting case of the current generalized model, where the agent size tends to infinity.
Although this limiting solution may be at the same time a good estimate for g ( R, θ )
when R is significantly larger than the critical nucleus diameter rc (by ~ two orders of
magnitude or more according to Fig 4.3(b)), it can’t be used to depict a large category of
nucleation processes occurring at low ‘undercoolings’ of a parental phase [6]. This is
because at low undercoolings, the Gibbs free energy difference between the parental

(b)

Theta=2/3Pi

0.9
0.8

g (R,theta)

0.7
0.6

Theta=1/2Pi

0.5
0.4
0.3

Theta=1/3Pi

0.2
0.1

Theta=1/6Pi

R/rc

100

g (R,theta)

0.7
0.6
0.5

R/rc=0.1
R/rc=0.4
R/rc=0.7
R/rc=1
R/rc=5
R/rc=100

0.4
0.3
0.2
0.1

0.5

1.5

2.5

3.5

Theta
Fig 4.3 (a) 3D image of the bivariate function g ( R, θ ) ; (b) 2D plots of g ( R, θ ) vs. R / rc

at different values of θ ; (c) 2D plots of g ( R, θ ) vs. θ at different values of R / rc .

84
phase and the new phase is very small, and thus, the critical nucleus diameter is very
large according to rc =

2σ
, as a result of which a finite nucleating agent can’t be readily

considered significantly larger than rc. Therefore, in such general cases, the present
model should be considered.
Besides the nucleating-agent size effect, this generalized model also releases new
information about the dependence of the critical energy barrier on the contact angle θ.
The classical heterogeneous solution (i.e., Eqn. (4.1)) predicts that the critical energy
barrier drops from Echom to 0 as θ decreases from π to 0. However, in the present
generalized model, the conclusion is somewhat different. For convenience sake, here we
consider another form of Eqn. (4.19):
g ( R,θ ) = [1 − cos 3 (ϕ + θ ) − 3( ) 2 cosθ (1 − cos ϕ ) + ( ) 3 (2 − 3 cos ϕ + cos 3 ϕ )]
rc
rc

(4.22)

since we fix the value of R in order to study the contribution of varying θ. As can been
seen from Fig 4.3 (c), as well as from Eqn. (4.20) and (4.22) [7], for any fixed R
(0 ≤ R < ∞) , as θ tends to π, g ( R, θ ) always tends to 1, and thus, nucleation always
tends to occur in a homogeneous manner. Nevertheless, as θ tends to 0, the value of
g ( R, θ ) depends on whether R ≥ rc or R < rc . If R ≥ rc , g ( R, θ ) always tends to 0,
meaning the nucleation energy barrier disappears at θ = 0 ; if R < rc , g ( R, θ ) tends to a
finite value g ( R,0) = 1 − 3( ) 2 + 2( ) 3 as determined by Eqn. (4.20) and (4.22). In the
rc
rc

latter case, the smaller the ratio

, the closer g (R,0) is to 1 (i.e., the closer the
rc

nucleation process is to the homogeneous case, even though θ = 0 ).
It is also noteworthy that the previously presumed relationship between the critical
energy barrier (Ec) and the volumetric Gibbs free energy of the critical nucleus ( E2 r ):

Ec =

E2 does not necessarily hold for a finite R in the present generalized model (see
2 rc

Eqn. (4.17-4.19)), although it is correct in the two limiting cases (i.e., when R = 0 or

R = +∞ ).

4.4

Conclusions

A generalized geometric model for the critical problem of nucleation has been established
to account for the size effect of an extrinsic nucleating agent. The classical solutions to
homogeneous and large-wall heterogeneous critical problems have been proven to be
limiting cases of this generalized model. Since in many cases, the limiting conditions
adopted in the derivations of the two classical solutions (i.e., a super clean parental phase
for the homogeneous case, or a large extrinsic wall for the heterogeneous case) are not
well satisfied, this present model is expected to provide a more complete and reliable
description for general nucleation phenomena. Although the quantitative preciseness of
this generalized model requires careful experimental verification in the future, yet this
model clearly proves: 1). heterogeneous nucleation always has a critical energy barrier no
larger (most of the time, lower) than does homogeneous nucleation regardless of the size

86
of the extrinsic nucleating agent; 2). the larger the extrinsic agent size (relative to the
critical nucleus size) the lower the critical energy barrier for heterogeneous nucleation.

References
[1]

D. A. Porter and K. E. Easterling, Phase Transformations in Metals and Alloys,
2nd Ed. (Chapman & Hall, London, 1992).

[2]

H. W. Kui, A. L. Greer and D. Turnbull, Appl. Phys. Lett. 45, 615 (1984).

[3]

D. M. Herlach, D. Holland-Moritz, T. Schenk, K. Schneider, G. Wilde, O. Boni, J.
Fransaer and F. Spaepen, J. Non-Cryst. Solids 250, 271 (1999).

[4]

Z. P. Lu, C. T. Liu and W. D. Porter, Appl. Phys. Lett. 83, 2581 (2003).

[5]

According to Eqn. (4.20), for a fixed θ ( 0 ≤ θ ≤ π ), R going to 0 is equivalent to

ϕ going to π-θ , and R going to +∞ is equivalent to ϕ going to 0.
[6]

Here, ‘undercooling’ refers to the deviation from the thermodynamic equilibrium
between the parental phase and the new phase such that the following discussion
not only applies to the crystallization-of-liquid case, but also to most other cases.

[7]

According to Eqn. (4.20), for any fixed R (0 ≤ R < ∞) , θ going to π is equivalent
to ϕ going to 0. For a fixed R ( R > rc ), θ going to 0 is equivalent to ϕ going to 0.
For a fixed R ( R < rc ), θ going to 0 is equivalent to ϕ going to π. If R = rc , θ
going to 0 is equivalent to ϕ going to π / 2 .

Chapter 5
Centimeter size BMG formation in Cu-Zr-Al-Y system
5.1

Introduction

In the last chapter, we saw that crystal nucleation may be enhanced by finite-sized
extrinsic agents inside an alloy melt. Previous experiments [1] have shown that some
oxide particles (e.g., Zirconium oxides) present in BMG alloys are among such effective
nucleating agents. Therefore, the removal or deactivation of such oxide particles is
expected to improve an alloy’s glass-forming ability (GFA). For Pd-Cu-Ni-P alloys, B2O3
has been used to remove the oxide particles† and clean the molten alloys by a fluxing
method [2]. However, this method can’t be applied to alloy systems containing species
(such as Zirconium) with higher oxygen affinity than boron because B2O3 may be
chemically reduced by such species and then lose its fluxing function. It was then found
that for such systems, heterogeneous nucleation may be restrained by chemically
transforming the active oxides into a certain type of deactivated oxides. For example, by
transforming detrimental Zirconium oxides into neutral Yttrium oxides, small amount
yttrium addition to Fe- (containing Zr) and Zr- based amorphous alloys can alleviate the
oxygen problems in these systems to a certain degree [3].
Meanwhile, in Chapter 3, we have reported the surprising discovery of bulk metallic
glasses in binary Cu-Zr and Cu-Hf systems. Cu46Zr54 is one of the three best glassforming compositions in these two systems which exhibit a critical casting thickness of

and maybe other detrimental particles

89
2mm. The discovery of these binary BMGs strongly suggests that even higher GFA may
be achievable in Cu-based alloys. On one hand, the further improvement of GFA may be
realized by appropriately introducing additional alloying elements. As a matter of fact,
Inoue et al. had reported earlier [4] that the critical casting thickness of certain ternary
Cu-based alloys in Cu-Zr-Al system is ~3 mm. On the other hand, the GFA may be
enhanced by deactivating detrimental Zr oxides. Based on these two considerations, I
systematically examined the effects of Y doping on a ternary alloy, Cu46Zr47Al7 (referred
to as ‘matrix alloy’ in the following context). The results† show that the consequent Cubased alloys, Cu46Zr47-xAl7Yx (0amorphous structure of a representative alloy, Cu46Zr42Al7Y5, can readily be obtained
even when the casting diameter exceeds 1cm. The physical mechanism underlying the
achievement of this unusual GFA is investigated by high temperature thermal analysis
and TEM (Transmission Electron Microscopy), combined with EDS (Energy Dispersive
X-ray Spectroscopy) and X-ray dot mapping techniques.

5.2

Experimentals

The samples were prepared by arc melting and subsequent copper mold casting, as
described in Chapter 2 (Section 2.2). The copper molds used here have internal
cylindrical cavities of diameters ranging from 2mm to 14mm. The transverse cross
sections of the as-cast samples were analysed with X-ray diffraction (XRD) method using
a Cu-Kα source. The glass transition and crystallization behaviors of amorphous samples
were analyzed with a Perkin-Elmer DSC7 (Differential Scanning Calorimeter) which was

A part of these results has been published in Ref. [5].

90
calibrated using Zn and Al standards.

The melting behaviours of the alloys were

analyzed with a Setaram DSC 2000K high temperature calorimeter at a heating rate of
0.33K/s. A Philips EM430 TEM operating at 300kV with an attached STEM unit and
EDX detector was utilized for imaging, microstructural and chemical analysis. The TEM
sections were prepared by ultramicrotomy and Mo grids were used to support the ultrathin sections.

5.3

Results and Discussion

Fig 5.1(A) shows the pictures of three as-cast samples (S1, S2 and S3) of a representative
alloy, Cu46Zr42Al7Y5, having a diameter of 10mm, 12mm and 14mm, respectively. Their
as-cast surfaces all appear smooth and lustrous. No apparent volume reductions can be
recognized on their surfaces, indicating there was no drastic crystallization during the
formation of these samples. The XRD patterns of S1, S2 and S3 are presented in Fig
5.1(B). It can be seen that the pattern of S1 consists only of a series of broad diffraction
maxima without any detectable sharp Bragg peaks, indicating that this 10mm diameter
sample is fully amorphous. Moreover, the 12mm (S2) and 14mm (S3) samples, even
though partially crystallized, still possess very large amorphous fractions judging from
the broad diffraction background on their XRD patterns. This implies that the growth of
the crystalline phase(s) in the supercooled liquid is quite sluggish, even when the sample
size exceeds the critical value (~10mm) for the formation of a fully amorphous structure
by ~40%. For a comparison, Fig 5.1(B) also presents the XRD patterns of 3mm (M1) and
4mm (M2) diameter rods of the matrix alloy, Cu46Zr47Al7, from which it can be seen that

(A)

Relative Intensity (a. u.)

(B)

S3
S2
S1
M2
M1
20

Two Theta (degree)

Fig 5.1 (A) Pictures of three cast samples of Cu46Zr42Al7Y5 with different diameters: S1,
10mm; S2, 12mm; S3, 14mm; (B) XRD patterns obtained from 10mm (S1), 12mm (S2)
and 14mm (S3) diameter rods of Cu46Zr42Al7Y5, and from 3mm (M1) and 4mm (M2)
diameter rods of the matrix alloy, Cu46Zr47Al7

92
the critical casting diameter of the matrix alloy is only ~3mm -- in agreement with the
report in Ref. [4].
Table 5.1 lists some representative alloys studied in this work, together with their
selected properties. The Tg and Tx values in the table were obtained from the DSC scans
which are shown in Fig 5.2. These alloys all exhibit a clear endothermic glass transition,
followed by a series of exothermic events characteristic of crystallization. As Y content
increases, the exothermic peaks tend to be broadened, indicating a possible slowdown in
the kinetics of nucleation and growth.

Table 5.1 A list of representative alloys and selected properties
Alloy

Critical casting

∆T= Tx-Tg

Trg

Composition

diameter

(K)

=Tg/Tl

(in at.%)

(mm)

Cu46Zr54

696

746

1201

0.58

Cu46Zr47Al7

705

781

1163

0.61

Cu46Zr45Al7Y2

693

770

1143

0.61

Cu46Zr42Al7Y5

672

772

1113

100

0.60

Cu46Zr37Al7Y10

665

743

1118

0.59

(B)
(C)
(D)
Tg

A: Cu46Zr37Al7Y10, 4mm dia rod
B: Cu46Zr42Al7Y5, 10mm dia rod
C: Cu46Zr45Al7Y2, 8mm dia rod
D: Cu46Zr47Al7, 3mm dia rod

600

700

Exotherm. (a.u.)

Exothermic (a.u.)

(A)

Cu46Zr42Al7Y5, 10mm dia rod
Isothermal annealing
at 739K

800

Time (min)

900

Temperature (K)

Fig 5.2 DSC scans of selected alloys at a constant heating rate of 0.33K/s. The upward
arrows refer to the glass transition temperatures and the downward arrows refer to the
onset of the first crystallization events. The inset at the lower right corner is the
isothermal DSC profile of the 10mm diameter rod of Cu46Zr42Al7Y5 at a constant
temperature of 739K.

94
It was pointed out by Chen and Spaepen [6] that isothermal calorimetric profiles can
distinguish truly amorphous materials from ‘microcrystalline’ materials that exhibit
similarly broad diffraction halos. Truly amorphous materials exhibit exothermic peaks
during isothermal scan, while ‘microcrystalline’ materials release monotonically
decaying heat flow signals. Hence, isothermal scanning was also performed on the
present alloys. The inset in Fig 5.2 represents the isothermal DSC profile of the 10mm
diameter as-cast sample of Cu46Zr42Al7Y5 annealed at a constant temperature of 739K.
The apparent exothermic peak characteristic of a nucleation-and-growth process confirms
the as-cast glassy structure of the alloy as concluded from X-ray diffraction.
In Ref. [3], Lu et al. reported the twofold effect of Y on the glass formation of Fe-based
alloys: 1). ‘Y adjusted the compositions closer to the eutectic, and thus, lowered their
liquidus temperatures’; 2). ‘Y improved the manufacturability of these alloys by
scavenging the oxygen impurity from it via the formation of innocuous yttrium oxides.’
Before discussing other contributing factors, we first confirm whether this ‘twofold
effect’ of Y also applies to the present Cu-based alloys.
The melting behaviors of these alloys were studied through high temperature calorimetric
scanning. The signals are exhibited in Fig 5.3, where the liquidus temperatures, Tl
(defined by the offset temperature of an entire melting process) are marked with arrows,
whose values are included in Table I. It can be seen that Y content significantly affects Tl
and the melting behaviors of these alloys. The ternary matrix alloy, Cu46Zr47Al7 has a
rather high Tl ~1163 K, although it is quite close to a ternary eutectic composition, as
indicated by the nearly-single event feature of its melting process (the ternary eutectic is

Endothermic (a.u.)

A: Cu46Zr37Al7Y10
B: Cu46Zr42Al7Y5
C: Cu46Zr45Al7Y2
D: Cu46Zr47Al7

(A)
(B)
(C)
(D)

950

1000

1050

1100

1150

1200

1250

1300

Temperature (K)
Fig 5.3 Melting behaviors of selected alloys measured at a heating rate of 0.33K/s. The
arrows refer to the liquidus temperatures.

96
located around Zr50Cu40Al10 according to Ref. [8]). When 2% Y is added, the quaternary
alloy, Cu46Zr45Al7Y2 shows a lower Tl (~1143 K), but multiple exothermic events which
indicate that this alloy is quite far from any quaternary eutectic composition. With 5% Y,
the alloy, Cu46Zr42Al7Y5 shows an even lower Tl (~1113 K) and a simpler melting
process consisted of one major exothermic event characteristic of a quaternary eutectic
reaction, followed by a minor secondary event corresponding to the melting of a lessconcentrated primary crystal. When Y content is further increased, the Tl tends to become
higher, reaching ~1118 K at 10% Y, and multiple events appear again during its melting
process. It is apparent that the alloy, Cu46Zr42Al7Y5 which has the highest GFA is the
closest to a nearby quaternary eutectic among this present alloy series.
Considering together Cu46Zr54, Cu46Zr47Al7, and the present Cu46Zr42Al7Y5, one can find
that all three of these bulk glass-formers are associated with, although not exactly at, the
eutectic compositions in their individual systems (binary, ternary and quaternary,
respectively). As the dimension of the alloy system (i.e., the number of components)
increases, the eutectic temperature is continuously lowered, and the GFA of the alloys is
improved as evidenced by the increased critical casting thickness using the same copper
mold casting method (refer to Table 5.1). This agrees with the ‘confusion principle’
proposed in Ref. [9] and the previous observation that high GFA often occurs around
deep eutectics [10]. Therefore, it is clear that the unusual GFA of the present quaternary
alloy series -- especially the alloy Cu46Zr42Al7Y5 – comes, in part, from the alloying
effect of Y which lowers the liquidus temperature of the matrix alloy and brings the
composition to a deeper eutectic.

97
To study the possible effect of Y on the oxygen impurities and examine the
microstructures of the as-cast alloys, TEM, EDS and X-ray dot mapping techniques were
utilized. Fig 5.4(a) presents the TEM image obtained from a typical area of the ultramicrotomed section of as-cast Cu46Zr42Al7Y5 alloy. It can be seen that there are some
small particles dispersed in a broad matrix. SAD (Selected Area Diffraction) shows the
matrix is amorphous and exhibits only a set of diffuse halo rings -- in agreement with
previous X-ray diffraction and thermal analyses. EDS and X-ray dot mapping show that
the small particles in the matrix are mainly composed of Y and oxygen. Fig 5.4(b) and (c)
represent as two examples the Cu Kα1 and Y Kα1 X-ray dot map images, respectively.
Zr and Al map images resemble the Cu map image; all three of these appear very dark
across the small particles, but appear bright in the matrix, indicating that the particles are
much depleted of Cu, Zr and Al. In contrast, the Y and O map images are very similar,
both showing greater brightness across the particles than in the matrix, thus indicating
that Y and O are the predominant constituent elements of the small particles.
According to the law of mass action in thermodynamics, the concentration of the
dissolved oxygen, [O], in the matrix upon the establishment of the following equilibrium:
xM + O ↔ M x O (M refers to a metal; O, oxygen; x may be either a fraction or an
integer), is proportional to exp(

∆G f
RT

) , where ∆Gf is the normalized Gibbs energy of

formation of oxide MxO. Given ∆G f = ∆H f − T∆S f (where ∆Hf and ∆Sf are the
normalized enthalpy and entropy of formation of MxO, respectively), one gets

Fig 5.4 TEM image (a), Cu Kα1 X-ray dot map image (b) and Y Kα1 X-ray dot map
image (c), of as-cast Cu46Zr42Al7Y5. The ripples and scratches in the images were caused
by the ultramicrotomy sample preparation method.

[O] ∝ exp(

∆H f
RT

∆S f

) . In the present concerned temperature range (several hundred

Kelvin) and in the present alloy system, we have

∆H f
RT

〉〉

∆S f

[11]. Therefore, larger

negative values of ∆H f will result in lower [O]. The normalized values of ∆H f
(corresponding to one mole of O atoms) for ZrO2, CuO, Al2O3, and Y2O3 are: -550.3, 157.3, -558.6, -635.1 kJ/mol [11], respectively. One can see that Y oxide has a very large
negative value of ∆H f , highest among all the four oxides. On one hand, by forming Y
oxide, the addition of Y to the matrix alloy can significantly lower the concentration of
oxygen dissolved in the matrix. On the other hand, it also greatly reduces the number of
oxygen atoms bonding with Zr, Cu and Al. Since Y oxides are ‘innocuous’ particles and
do not actively trigger heterogeneous nucleation [3], this change in the state of presence
of oxygen leads to an enhanced GFA.
Besides the above confirmed ‘twofold effect’ of Y, the particularly large GFA of the
present Cu-based alloys may have benefited from other factors. The present quaternary
alloys can be considered as close derivatives from a simple binary base alloy, Cu46Zr54
(reported in Chapter 3). The subsequent additions of Al and Y follow the ‘confusion
principle’ proposed by Greer [9]. The more uniformly distributed atomic sizes (Y: 1.8 Å;
Zr: 1.6 Å; Al: 1.43 Å; and Cu: 1.28 Å [12]) and the large negative heat of mixing among
the constituent elements (e.g., Y-Al: -31 kJ/mol; Y-Cu: -22 kJ/mol; and Zr-Cu: -23
kJ/mol [13]) help stabilize the supercooled liquids and thus, lead to a high GFA [7].

100
5.4

Conclusions

Unusual glass-forming ability of a family of Cu-based alloys (Cu46Zr47-xAl7Yx (0≤x≤10))
has been discovered. By injection mold casting, the amorphous structure of a
representative alloy, Cu46Zr42Al7Y5, can be readily obtained with a diameter above 1cm.
By using high temperature thermal analysis, TEM, EDS and X-ray dot mapping
techniques, the achievement of such an unusual GFA was found to be associated with the
‘twofold effect’ of Y as previously reported for Fe-based amorphous alloys [3].
Meanwhile, a bulk glass-forming binary alloy, Cu46Zr54, has provided an excellent basis
for the extraordinary success of Y-doping and the ‘confusion principle’ [9] in this
particular system.

101

References
[1]

X. H. Lin, W. L. Johnson and W. K. Rhim, Mater. Trans., JIM 38, 473 (1997).

[2]

A. J. Drehman and A. L. Greer, Acta Metall. 32, 323 (1984).

[3]

Z. P. Lu, C. T. Liu and W. D. Porter, Appl. Phys. Lett. 83, 2581 (2003).

[4]

A. Inoue and W. Zhang, Mater. Trans. 43, 2921 (2002).

[5]

D. H. Xu, G. Duan and W. L. Johnson, Phys. Rev. Lett. 92, 245504 (2004).

[6]

L. C. Chen and F. Spaepen, Nature 336, 366 (1988).

[7]

A. Inoue, Acta Mater. 48, 279 (2000).

[8]

Y. Yokoyama, H. Inoue, K. Fukaura and A. Inoue, Mater. Trans. 43, 575 (2002).

[9]

A. L. Greer, Nature 366, 303 (1993).

[10]

W. L. Johnson, Mater. Sci. Forum 225, 35 (1996).

[11]

D. R. Lide, Handbook of Chemistry and Physics, 81st ed. (CRC Press, Boca Raton,
2000).

[12]

C. Kittel, Introduction to Solid State Physics, 7th ed. (John Wiley & Sons, New
York, 1996).

[13]

F. R. de Boer, R. Boom, W. C. M. Matterns, A. R. Miedema and A. K. Niessen,
Cohesion in Metals (North-Holland, Amsterdam, 1988).

102

Chapter 6
Concluding Remarks
During this thesis research, my interest was mainly focused on the development of novel
bulk metallic glasses based on ordinary metals, particularly nickel and copper. For me,
alloy development is a tempting job that provides excitement once in a while, although it
can be quite boring between periods of those exciting moments. To find really useful
materials is not an easy job. Although there are several ‘wise’ rules proposed about how
to find BMG alloys (some of them discussed in Chapter 1), the most important factors for
success in alloy development are perhaps hard work and persistent interest.
Are the materials developed in this thesis useful? Yes, but only to some extent. They are
useful in the following sense: 1. NixCua-xTiyZrb-yAl10 (a~b~45, in at.%) (see Chapter 2)
alloys are quite strong, having

fracture strength (compressive) of ~2.2 - 2.4 GPa,

Young’s modulus of >110 GPa, Vicker’s hardness of >800 Kg/ mm2 and yet, not too
brittle (with no premature failure), together with fairly good glass-forming ability (up to 5
mm in critical casting thickness -- the highest value achieved so far for nickel based
BMGs); 2. Cu46Zr54, Cu64Zr36, Cu66Hf34 and other binary BMGs in Cu-Zr and Cu-Hf
systems are so far, perhaps the best combination of good glass-forming ability and
chemical simplicity which should qualify them as very good subjects for theoretical
simulation and modeling; they are also interesting for their own properties: the two rich
in copper are quite strong and the one rich in zirconium is quite ductile; 3. Cu46Zr47xAl7Yx (0

casting thickness up to 1 cm. They are, at the same time, not extremely useful in the sense

103
that: 1. NixCua-xTiyZrb-yAl10 (a~b~45, in at.%) and the binary BMGs in Cu-Zr and Cu-Hf
systems are still limited in glass forming ability; 2. Cu46Zr47-xAl7Yx (0although very good glass-formers, are not significantly different from early Zr-based
BMGs in terms of material costs and strength.
Finally, I would like to conclude this thesis with a very important thought: “Nothing
should be taken for granted in scientific research.” This has led me to the discovery of
binary bulk metallic glasses and should continue to benefit me in my academic career in
the future.

DONGHUA XU
Prof. William L. Johnson group, Materials Science Option
Division of Engineering and Applied Science, CALTECH, 138-78
1200 E. California Blvd., Pasadena, CA 91125
Office: 626-395-3571 Fax: 626-795-6132
E-mail: xudh@caltech.edu

EDUCATION
2005 PhD, Materials Science, Caltech (graduation expected: Jun. 2005) (advisor: William L Johnson)
2002 MS, Materials Science, Caltech (advisor: William L Johnson)
1998 BS, Materials Science, Jilin University, P. R. China
AWARDS AND HONORS
1. 1998-2000 Jilin Univ. fellowship for postgraduates, awarded by Jilin University (JLU)
2. 1998 privilege of entering graduate program exempted from admission exams, awarded by JLU
3. 1998 Honor of 'Outstanding Graduation From Jilin Univ.', awarded by JLU
4. 1997 Prize for 'Ten Best Students of Jilin Univ.', awarded by JLU
5. 1997 Baogang Prize for outstanding Chinese students, awarded by BaoSteel Co. (China’s largest steel
manufacturer)
6. 1994-1998 Jilin Univ. scholarship for undergraduates, and title of 'excellent students of Jilin Univ.',
awarded by JLU
7. 1994 Prize for outstanding admissions, first class, awarded by JLU.
PATENT APPLICATIONS
1. Ni-base bulk refractory glasses based on Ni-Cu-Ti-Zr-Al system, D.H. Xu and W.L. Johnson (in review by
US Patent Office, filed in Dec. 2003; licensed by Liquidmetal Technologies Inc., royalty payment received)
2. Bulk refractory metallic glasses based on Ni-Nb-Sn system, H. Choi-Yim, D.H. Xu and W. L. Johnson (in
review by US Patent Office, filed in Jul. 2003; licensed by Liquidmetal Technologies Inc., royalty payment
received)
RESEARCH EXPERIENCE (PhD Project Summary)
My PhD research work at CALTECH was mainly focused on the synthesis and characterization of novel
ordinary-metal-based bulk amorphous alloys (also known as bulk metallic glasses—BMG’s). During this
research, I worked on a large variety of binary or multi-component metal alloys. I gained first-hand
experience with most of the commonly-used elemental materials, including simple metals (such as Al, Sn,
Be, Mg, Ca), early transition metals (such as Ti, Zr, Hf, Y, La, V, Nb, Ta, Cr, Mo, W), late transition
metals (such as Fe, Co, Ni, Cu, Pd, Ag, Zn) as well as metalloids (such as B, C, Si, Ge, P). I acquired many
experimental techniques in materials synthesis, processing and characterization (see next page for detailed
Experimental Skills). I successfully developed, independently or together with others, two Ni-based and
several Cu-based BMG alloy systems, some of which have exhibited exceptionally high strength up to ~3.8
GPa, high micro-hardness up to ~12.8 GPa, or exceptionally large casting thickness up to 10 mm. These
results have been and are continuing to be published in highly-recognized peer reviewed journals including
Physical Review Letters. Two patents have been applied for some of my new BMG’s. Besides the synthesis
and characterization of these new alloys, my work has also made a major contribution to the understanding
of BMG formation. Until my work binary alloys were widely considered excluded from the family of
BMG’s due to their chemical simplicity. My results show that the glass forming abilities of certain Cu-Zr
and Cu-Hf binary alloys can be high enough to form 2 mm thick BMG samples. Not only have these binary
BMG’s challenged the previous concept about BMG formation, but they also provide a wonderful
opportunity for theorists to better simulate and model BMG’s since these alloys possess both the simplicity
of binary alloys and the good glass-forming abilities of multi-component BMG’s.
CURRENT RESEARCH INTERESTS
1. Synthesis/fabrication, processing, structure and properties of advanced engineering materials, including
amorphous/nanocrystalline alloys, light-weight metals/alloys, intermetallics, superalloys, metal composites,
ceramics, etc.
2. Physics, chemistry and metallurgy related to the above materials.

D.H. Xu, Curriculum Vitae

Page 2

5/27/2005

PUBLICATIONS
1. D.H. Xu, W.L. Johnson, ‘A geometric model for the critical-value problem of nucleation phenomena
containing the size effect of nucleating agent’ (accepted for publication in Physical Review B, 2005).
2. G. Duan, D.H. Xu, W.L. Johnson, Q. Zhang, G.Y. Zhang, T. Cagin, and W.A. Goddard, ‘Molecular
dynamics study of the binary Cu46Zr54 metallic glass motivated by experiments: Glass formation and
atomic-level structure’ (accepted for publication in Physical Review B, 2005).
3. G. Duan, D.H. Xu, W.L. Johnson, 'High copper content bulk glass formation in bimetallic Cu-Hf system',
Metallurgical and Materials Transactions A-Physical Metallurgy and Materials Science 36A, 455 (2005).
4. D.H. Xu, G. Duan, W.L. Johnson, ‘Unusual glass-forming ability of bulk amorphous alloys based on
ordinary metal copper', Physical Review Letters 92, 245504 (2004).
5. D.H. Xu, G. Duan, W.L. Johnson, C. Garland, ‘Formation and properties of new Ni-based amorphous
alloys with critical casting thickness up to 5mm’, Acta Materialia 52, 3493 (2004).
6. D.H. Xu, B. Lohwongwatana, G. Duan, W.L. Johnson, C. Garland, ‘Bulk metallic glass formation in binary
Cu-rich alloy series - Cu100-xZrx (x=34,36,38.2,40 at.%) and mechanical properties of bulk Cu64Zr36 glass’,
Acta Materialia 52, 2621 (2004).
7. H Choi-Yim, D.H. Xu, W.L. Johnson, 'Ni-based bulk metallic glass formation in the Ni–Nb–Sn and Ni–Nb–
Sn–X (X = B,Fe,Cu) alloy systems', Applied Physics Letters 82, 1030 (2003).
EXPERIMENTAL SKILLS
materials synthesis (ultrasonic cleaning, induction/arc/resistance melting, metal mold casting, water
quenching, vacuum, inert gas sealing, etc.), materials processing (heat treatment under vacuum/inert gas,
cold/hot/superplastic deformation, infiltration, condensation, grinding, polishing, etc.), and materials
characterization (X-ray Diffraction, Transmission Electron Microscopy, Scanning Electron Microscopy,
Energy Dispersive X-ray Spectroscopy, Differential Scanning Calorimetry, Differential Thermal Analysis,
Micro-hardness testing, ultrasonic wave speed measurement of elastic modulii, tensile/compression testing,
etc.).
PROJECT PARTICIPATED
DARPA (Defense Advanced Research Projects Agency) - SAM (Structural Amorphous Metals)
PROFESSIONAL MEMBERSHIP
American Chemical Society
LANGUAGES
English (proficient, both oral communication and writing)
Chinese (native)
SERVICE ACTIVITIES
1. 2001
Vice President of CaltechC (association of Chinese students and scholars at Caltech)
2. 1998-2000 President of the Graduates' Association of the Dept. of Mater. Sci., JLU
3. 1995-1996 Head of the Study Section of the Students' Assoc. of the Dept. of Mater. Sci., JLU
4. 1994-1998 study monitor of the class, Dept. of Mater. Sci., Jilin Univ.