Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013 (11–18)

Theorization and Modeling Ⅲ(Review)

Crystal Nucleation in Supercooled Liquid Metals
Kenneth F. Kelton
Abstract
It is becoming increasingly clear that nucleation processes in liquids and glasses are more complicated than previously thought, often
coupling to other phase transitions and ordering processes. Experimental and theoretical studies show the development of icosahedral shortrange order in many supercooled transition metal and alloy liquids, which in some cases extends beyond nearest neighbor distances. Thisatomic and chemical ordering couples to the nucleation barrier, and may play a role in glass formation in some cases. Select experimental
results are presented to demonstrate these points. These are discussed in light of nucleation theories, including the commonly used Classical
Theory of Nucleation, diffuse interface theories, and coupled-flux theory, which takes account of the interaction between interfacial processes
at the surface of the crystal nuclei and long-range diffusion fluxes.
Keyword(s):1.

nucleation, supercooling, liquid structure, glass formation

Introduction

Phase transitions are ubiquitous, ranging from thesolidification of liquids, to solid-state transformations, to theprecipitation of kidney stones, and even to changes in the early
universe. Most of these phase transitions are initiated by anucleation step. In this step large changes in some orderparameter, which characterizes the difference between the initial
and transformed phases, occurs within spatially small regions.
Liquid solidification is often used for the investigation of
nucleation processes because strain effects, which complicate
nucleation studies in solids, are effectively absent. Under theright conditions liquids can be maintained for long periods oftime at temperatures below their equilibrium melting (orliquidus) temperatures, i.e. in a supercooled state. That this ispossible indicates the existence of a barrier to the formation of
the stable crystal phases, generally called the nucleation barrier.
As will be shown in this article, even in metallic liquids
nucleation is complicated, challenging current understanding.
The thermodynamic model that underlies the most commonlyused model for nucleation, the Classical Theory of Nucleation,is questionable when applied to the small clusters involved in
the nucleation step. Inherent chemical and topological orderingin the volume of the liquid, and ordering near the interface withthe growing crystal and near surfaces, are not fully included innucleation theories, although some progress been has madeusing density functional approaches. The nucleating phase often
has a chemical composition that is different from that of theparent phase, making long-range diffusion effects potentially
important. To treat this problem correctly, the stochastic
interfacial processes described by most nucleation theories must
be coupled with the stochastic diffusion field, which is only
rarely done. Finally, there is a great deal of interest in metallic

glass formation, which requires that crystallization beeffectively bypassed. An understanding of the kinetics of crystal
nucleation during rapid cooling and how nucleation depends on
the structural evolution of the liquid is needed to treat thisproblem. These points are briefly surveyed in this article.

2.

Nucleation Theory

Before looking at experimental data, it is useful to discuss
briefly the current state of nucleation theory.

2.1

Classical Theory of Nucleation

In homogenous nucleation, small regions of crystalline orderarise spontaneously by spatially and temporally independent
fluctuations within the supercool liquid. In heterogeneous
nucleation these fluctuations are catalyzed at specific sites.
Both types of nucleation are most often analyzed within theClassical Theory of Nucleation (CNT).1) In homogeneous CNT,
small regions of the crystal phase (clusters) develop randomly inspace and time by a series of kinetic reactions in which single
atoms or molecules (monomers) attach and detach from thecluster interface. Assuming spherical clusters, negligible strain,
and a sharp interface between the crystal and amorphous phase,the reversible work of formation of a cluster of n monomers is:

Wn  n   (36 )1/3 v 2/3 n 2/3 Here  is the Gibbs free energy of the crystal phase less thanthat of the glass phase per monomer, v is the monomer volume,and  is the liquid-crystal interfacial free energy per unit surface
area. The competition between the volume free energy favoringcluster formation and the surface free energy opposing it leads
to a maximum in the work of formation, Wn*, for a criticalcluster size, n*. Clusters are assumed to evolve slowly by aseries of bimolecular reactions, where the rate of change of the

Department of Physics, Campus Box 1105, Washington University, One Brookings Drive, St. Louis, Missouri 63130 USA(E-mail: kfk@wustl.edu)

11

(1)

− 11 −

Crystal Nucleation in Supercooled Liquid Metals

time-dependent population density as a function of the cluster
size, n, and time, t, is

dN n,t N n1,t kn1
 N n,t kn  kn   N n1,t kn1
,
dt

(2)

where kn is the rate of monomer addition to a cluster of size n
and kn is the rate of monomer loss. Assuming interface-limited
kinetics (strictly valid only for nucleation processes where theinitial and final phases have the same chemical composition)kn  4n 2/3  n . For spherical clusters, the first term (4n2/3) is thenumber of attachment sites on the surface of the cluster, and,
 n 

6D

2

  Wn  where D is the diffusion coefficient in theexp  
 2kBT 

liquid,  is the jump distance, kB is Boltzmann’s constant, T is
the temperature and Wn = Wn+1  Wn. The nucleation frequency,
In,t, is the time- and size-dependent flux from clusters of size nN n1,t ).
to ones of size n+1 ( I n,t kn N n,t kn1cluster distribution ( N n,t  N nst ), the steady-state nucleation rate
is
(3)

3

where the critical size, n*, is 32 3 / 3 v g , the critical workformation,Wn*(calledthenucleation216  / 3v g (where g / v ),3

barrier)

Wn* 

4
b2
gil  32
3
1 b

(4)

where  is the interfacial width, b 2 1  gsl / gil  , gsl is thefree energy difference of the liquid and solid phases ( gs  gl ) ,and gil is the free energy difference between the liquid andinterface ( gi  gl ). By Equating this value for Wn* with thatobtained from the CNT, an expression for the interfacial freeenergy is obtained,  ls    gil gsl / 2 . A key result fromthis model is that  ls is predicted to have a positive temperature

1/2

 W 
24D (n* )2/3 N A   I  exp   n*  ,
*
2
6

k
Tn
 k BT 
BstDiffuse Interface Theories

The CNT assumes that the interface between the volume andsurface contributions to the work of cluster formation can becleanly divided (Eq. (1)), requiring a sharp interface between the
nucleating cluster and the parent liquid phase. However, thisinterface is actually diffuse 5)-7), with a width that is of the orderof the radius of the nucleation clusters when the driving freeenergy is large. The regions of the liquid near the cluster are,therefore, more ordered than expected within CNT.
A phenomenological model to account for the ordering was
proposed independently by Gránásy8)-10) and Spaepen11).
Ordering in the liquid ahead of the advancing cluster interface
implies that the Gibbs free energy g(r) will change continuously
on crossing the cluster boundary. To lowest order, this can be
described by the step-function shown in Fig. 1, allowing thework of cluster formation to be readily obtained,

Except for

extremely rapid cooling rates (see Section 3.3), time-dependent
effects are generally not important for crystallization from theliquid, although they can play a very prominent role in glass
crystallization2),3).
To a lowest approximation, the nucleation rate is the forward
flux of clusters past the critical size.
To obtain a morequantitative expression, assuming a time-invariant (steady-state)

of

2.2

is

and NA is the total

number of monomers in the system, typically taken to be
Avogadro’s number per mole (see Ref. 1 for more details).
Frequently, solid impurity sites, such as a container wall, lower
the value of Wn*, which from Eq. (3) will increase the nucleation
rate. Within CNT, this is taken into account by multiplying Wn*

dependence in agreement with experiment 12). Here, a simplistic
profile was considered. The precise profile of g(r) can beobtained from a density-functional theory (DFT) approach interms of one or more order parameters that characterize theinitial and transformed phases.
In principle a DFT formalism can also be used to account forcoupling of nucleation processes to other phase transitions in theparent phase (such as chemical or magnetic ordering transitions).
For illustration, consider the semi-empirical density functional

by a factor f() that varies between 0 and 1 depending on thecontact angle,  , between the crystal phase and the solid (seeRef. 1 for a detailed discussion of heterogeneous nucleation).
Equation (3) predicts a sharp increase in the nucleation ratewith decreasing temperature due to the increasing driving freeenergy, followed by a decrease at lower temperatures due to theslowing atomic dynamics. This predicted behavior is in good
agreement with experimental data1),4). A fundamental problem
with CNT is that the magnitude of the nucleation rate isextremely sensitive to the value of the interfacial energy, whichis generally only known from fits to nucleation data.

Int. J. Microgravity Sci. Appl. Vol. 30 No.1 2013− 12 −

Fig. 1 Step-function change of the free-energy density
through the interfacial region between the liquidand solid (from rs to rs+. (Adapted from Ref. 11,
copyright (1994), with permission from Elsevier.)

12

K. F. Kelton

approximation (SDFA), where a single order parameter, M, isassumed. In the liquid far from the cluster, M(r,t) is zero; it isequal to one for an infinitely large solid cluster. The free energy
can be written as a functional of M, G  M  r, t  , and the workof cluster formation can be computed in a similar way as fromthe CNT, W[M ]   g  M (r, t)    M (r, t) dr (see Refs. 1 andV13) for a detailed treatment). Figure 2 shows the computedorder parameter, M, as a function of distance from the center of , which is a scaled
the cluster for three different values of parameter that corresponds to the driving free energy. The
values for rs* , the corresponding scaled critical radii, are listed
in the figure. Like the CNT, increased magnitudes of thedriving free energy give smaller values for the critical size.

 the interface is diffuse. As shown
However, for all values of  becomes very small (= -0.05,
in the inset, only when corresponding to rs* 20 ) does the interface approach the sharpboundary assumed in CNT. Further, although M should be unity
in the solid phase, it does not reach that value even in the centerof the cluster, except when the driving free energy is very small.
Calculations also show that the CNT overestimates the work ofcluster formation for larger driving free energies. Taken together,
these results indicate that the CNT is only quantitatively correctwhen the departure from equilibrium is small, i.e. near themelting (or liquidus) temperature. Although not discussed here,studies show that the DIT and SDFA models fit nucleation rate
data in liquids and glasses better than the CNT, demonstrating
the importance of the ordering in the liquid near the cluster
interface1), 14).

Fig. 2

2.3

The nucleation theories previously discussed are interfacelimited theories, appropriate if the chemical composition of thenucleating phase is the same as that of the liquid (polymorphic
crystallization), or if the diffusion rates in the liquid are much
faster than the interfacial attachment kinetics. The latter canmake CNT particularly inappropriate for solid-state
precipitation 15),16), but it is generally not an important issue fornucleation from the liquid if the temperature is constant or
slowly varying. When the interfacial attachment rates to thecluster do become competitive with the diffusive transport rates
in the liquid, these two stochastic fluxes become coupled.
Following an approach first suggested by Russell 18), this may
be treated to lowest order by focusing attention on three regions(Fig. 3): the cluster, the immediate neighborhood around thecluster (the shell region), and the parent phase 17),19). In thisCoupled-Flux Model (CFM), the flux between the shell and theparent phase is coupled with that between the shell and thecluster. The cluster evolution underlying time-dependent
nucleation is determined by solving numerically a system ofcoupled differential rate equations that incorporate theinterfacial and shell/parent phase fluxes. Numerical solutions
show that the coupled-fluxes can significantly lower the
nucleation rate and increase the induction time for nucleation
beyond the predictions from the CNT. Surprisingly, a key
prediction of CFM is that for sub-critical clusters, the liquidcomposition is closer to that of the precipitating cluster, incontradiction with expectations based on the growth of large
clusters. These predictions have been confirmed in Kinetic
Monte Carlo simulations that probe the growth and shrinkage ofclusters in a simple lattice gas model, augmented withadjustable diffusion kinetics 20).
While coupling of the interfacial and diffusive fluxes
typically has little influence on nucleation from supercooled

Order parameter M as a function of the scaled
distance from the center of the critical cluster.
Profiles are shown for three different values of the . (Reprinted fromscaled driving free energy, Fig. 3 A schematic illustration of the fluxes in the CoupledFlux Model. (Reprinted from Ref. 17, copyright(2000) with permission from Elsevier.)

Ref. 13, copyright (1994), American Institute ofPhysics.)

13Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013

Coupled Flux Theory - Incorporation of LongRange Diffusion− 13 −

Crystal Nucleation in Supercooled Liquid Metals

liquids, it can have a significant effect for nucleation during a
rapid quench (see Section 3.3) and in glass devitrification 21).
Coupled-flux effects may also be important for nucleation underquiescent micro-gravity conditions, where stirring (which isalways present in a terrestrial environment) is suppressed.

3.

Nucleation Studies in Metallic Liquids

3.1

Experimental Techniques

Heterogeneous nucleation, on impurities in the liquid or on
the container walls, generally dominates the solidification ofliquid metals. Since the number of heterogeneous sites and theircatalytic efficiencies are generally unknown, a quantitative
analysis of such nucleation data is difficult or impossible. For
this reason, techniques have been developed to minimize theimpact of heterogeneous nucleation, so that homogeneous
nucleation may be studied (see Refs. 1 and 4 for a more
extended discussion than provided here). These include (i)isolation, where impurities are compartmentalized into a small
volume fraction of the liquid12),22),23), (ii) fluxing, where theliquid is coated with a material that dissolves or renders
ineffective impurities and protects it from the container wallsand (iii) containerless processing, where liquids are held
without containers in vacuum or a high-purity non-oxidizing
atmosphere. The most common containerless techniques are
based on aerodynamic levitation, electromagnetic levitation(EML) and electrostatic levitation (ESL) 1),24). Aerodynamic
levitation is achieved by a controlled gas flow through nozzlesof optimal design for the size and density of the samples ofinterest. However, the flowing gas makes temperature andpositioning control difficult, may lead to heterogeneous
nucleation on gas impurities, and cannot be used for quantitative
measurements of thermophysical properties. EML uses a highfrequency EM field to induce eddy currents in metallic samples,
providing levitation from Lenz’s law. However, since bothheating and levitation are coupled in EML24), it is not always
possible to supercool the liquid while maintaining sample
levitation under terrestrial conditions. Further, only metallic
materials, or semiconducting materials that become metallic in
the liquid phase (e.g. Si), can be studied. Electrostatic levitation(ESL) is the most versatile, offering several key advantages overaerodynamic and electromagnetic levitation: (1) non-metallic aswell as metallic systems can be studied; (2) the heating andpositioning power are decoupled, allowing measurements inmore deeply supercooled liquids; and (3) the rf coils required
for electromagnetic levitation limits the view of the sample
while ESL provides a wide range of access to the sample. In
ESL charged samples with a 2.0 – 3.0 mm diameter(approximately 30-70 mg mass) are levitated by Coulomb forcesin an electrostatic field (0 to 2 MV/m) under high vacuum(typically 10-7 – 10-8 torr) 25), 26). The samples are initiallyInt. J. Microgravity Sci. Appl. Vol. 30 No.1 2013

(a)

(b)

(c)
Ref: 8Fig. 4 (a) Photograph of the ESL at Washington University,
optimized for X-ray scattering studies. (b) A levitatedsphere; the different colors of light are from the highintensity LEDs used in the sample positioning
feedback algorithm. (c) Schematic diagram showing
the incoming and scattered X-rays in a transmission
geometry from a levitated liquid sample; thediffraction pattern is recorded on an area detector.
The vertical and two sets of side electrodes used forlevitation and positioning are also shown (c iscourtesy of N. A. Mauro).
charged by induction.
During processing the charge ismaintained with an external UV source at low temperatures andby thermionic emission at high temperatures. Three pairs oforthogonal electrodes and a robust control algorithm27),28)
maintain the sample position during processing to within 50-100μm, based on error signals from two orthogonal positionsensitive detectors, which are provided as input to the DC
amplifiers connected to the electrodes. The sample can then beheated to any temperature up to ≥ 3000 K using one or morelasers. Our liquid diffraction data, some discussed in this review,
were obtained using an ESL facility (WU-Beamline ESL, or
WU-BESL) (Fig. 4) that has been constructed at Washington
University and optimized for X-ray diffraction studies of thelevitated supercooled liquids 29). A schematic of the transmission
X-ray diffraction geometry is shown in Fig. 4.c. Complementary
physical property data, such as maximum supercooling, density,
surface tension and viscosity, can be obtained in ESL, allowing
the liquid structure to be linked with the crystal nucleation
barrier (Section 4) and with thermophysical properties.

3.2

Maximum Undercooling Results for Elemental
Liquids

Since, as demonstrated in studies of nucleation in silicate
glasses, the steady-state nucleation rate rises rapidly withsupercooling 4), and the growth velocities in the supercooled
liquid are large, the time scale for crystallization is dominatedby the time required to form a nuclei. Experimental studies
show that the reduced undercooling (Tr = (Tm – Tu)/Tm, where
Tm is the melting temperature and Tu is the minimum− 14 −

14

K. F. Kelton

1614

g (kJ/mol)

12
10
8
6
4
2
0

0

5

10

15

20

25

30

35

f (kJ/mol)

Fig. 6 Calculated homogeneous nucleation rate for
Au81Si19 as a function of temperature and quenchingrate. (Reprinted from Ref. 31, copyright (1986),
with permission from Elsevier.)

Fig. 5 Interfacial free energy obtained from maximum
supercooling studies of elemental metallic liquids,
shown as a function of the heat of fusion. (Reprinted
from Ref. 4, copyright (1991), with permission from
Elsevier.)
temperature to which the liquid can be cooled before
crystallizing) for pure liquid metals is in the range 0.1 to 0.4.
This indicates that the barrier to nucleation is large. An analysis
of compiled data (see Refs. 1 and 4) demonstrates that the gramatomic interfacial free energy (  lsM   ls V 2/3 N A2/3 , where  ls is
the interfacial free energy determined from fits to Eq. (3), V isthe molar volume and N A is Avogadro’s number), scales
linearly with the heat of fusion (  lsM  0.44 h f ) (Fig. 5),
suggesting a “bond-breaking-like” process to form the interface.

3.3 Nucleation and Glass Formation
Much of the current interest in the study of metallic liquids is
to gain a better understanding of glass formation. Ultimately,
this is a question of nucleation and growth, since if significant
crystallization can be avoided the liquid will transform to a glassat the glass transition temperature. Traditional silicate glasses
are formed at modest cooling rates, of order 1 K/s or less. Until
recently, much higher cooling rates, 105 – 1012 K/s, wererequired to form metallic glasses. Now there are many alloys inwhich glasses can be formed with cooling rates that are more
typical of silicate glasses.
The expected number of nuclei that form during the quench is
t

T2

0

T1

N   I dt  R 1  I T  dT ,(5)

where R is the quenching rate, and T1 and T2 are the upper andlower end of the temperature range over which nucleation issignificant.
Assuming steady-state nucleation and usingparameters obtained from crystallization studies of Au81Si19, oneof the first known metallic glasses, approximately 1016-1018

15Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013quenched-in nuclei/mol are predicted from Eq. (5) with aquenching rate of 106 K/s. While some number of quenched-in
nuclei is unavoidable, the large number predicted for Au81Si18,
when coupled with reasonable estimates of the growth velocity
would predict that at least 1% of the quenched sample would becrystallized. No evidence of crystallization is observed,
however. Greer first showed experimentally that the number ofquenched-in Fe-B glasses scaled as R-2 to 4, not to R-1 aspredicted from Eq. (5) 30). A likely reason for this is that theassumption of a steady-state nucleation rate during cooling isincorrect. During a rapid quench there is insufficient time tomaintain the steady-state distribution that is appropriate for eachtemperature. This is why transient nucleation effects areobserved in glass crystallization experiments; the cluster size
distribution obtained at the end of the quench is not theappropriate one at the annealing temperature, but is more similar
to the steady-state distribution at a higher temperature.
Numerical studies have shown that the nucleation rate during
the quench can be significantly lower than the expected steadystate rate (Fig. 6), making glass formation easier and increasing
the stability of the glass to crystallization 31). This becomes
particularly important if, as is usually the case, the chemicalcompositions of the crystal phases that can form during thequenching are different from that of the liquid, since thetransient times become much longer than for the case ofpolymorphic crystallization 17), 32).

4.

Developing Structure in Supercooled
Liquids – Impact on Phase Transitions

From the discussion in Section 2.2, experimental and theoreticalevidence indicate that the liquid is ordered close to the interfaceof the nucleating cluster. But does order inherently develop− 15 −

Crystal Nucleation in Supercooled Liquid Metals

within a supercooled liquid, independent of the presence of
crystal nuclei?
And does this order influence liquid
crystallization? In 1952, Charles Frank proposed that theanswer to both questions was yes 33). To explain the ability tosignificantly supercool liquid metals, first demonstrated by
Turnbull, Frank proposed that metallic liquids contain a large
amount of icosahedral short-range order (ISRO). Because thepoint-group symmetry of the icosahedron is incompatible withtranslational periodicity, Frank argued that this low energy
structure in the liquid was the source of the barrier to the
nucleation of the crystal phase, allowing the liquid to be deeply
supercooled.
Many MD simulations are consistent with Frank’s hypothesis.
One example of the results of an ab-initio MD simulation forbinary and ternary Cu-Zr based glass-forming liquids is shownin Fig. 7. A dramatic increase in the number of <0,0,12,0>
Voronoi polyhedron (reflecting ISRO) is observed between theliquidus temperature, Tl, and the glass transition temperature, Tg.
This is consistent with experimental results in a
Zr59Ti3Cu20Ni8Al10 glass-forming liquid 3).

4.1

Measurements of Short-Range
Liquids and Glasses

Order

in

Nucleation measurements provided the first experimental
evidence for icosahedral order in amorphous metals, with a verysmall interfacial free energy (il) measured between a metallic
glass and the primary crystallizing icosahedral quasicrystal (iphase) 35). This was subsequently confirmed by quantitative
time-dependent nucleation measurements of i-phase formation
in a Zr59Ti3Cu20Ni8Al10 metallic glass, obtaining il = 0.01 
0.004 Jm-2, much smaller than expected for a crystal phase.
Holland-Moritz and co-workers made the first systematic studyof the supercooling of liquids. They found that as the
icosahedral short-range order in the crystallizing phases

Fig. 7 The results of ab-inito MD calculations showing
the growth of the number of <0,0,12,0> Voronoi
polyhedral as a function of supercooling in Cu-Zr
based liquid alloys. (Reprinted with permission
from Ref. 34), copyright (2008), American Instituteof Physics.)

Int. J. Microgravity Sci. Appl. Vol. 30 No.1 2013

increased, the interfacial free energy decreased, as expectedfrom the previous glass studies 36), 37).
Over the past ten years there has appeared direct structuralevidence for ISRO from X-ray and neutron diffraction studies ofsupercooled liquids, using electromagnetic and electrostatic
levitation scattering techniques. The structure factor, S(q), iseasily obtained directly from the scattering data. The pairdistribution function, g(r), which give the angular-averaged
radial probability distribution of atom locations, is subsequently
obtained by taking the Fourier transform of S(q). The radial
distribution function, R(r), is related to g(r), R(r) 4 r 2 o g(r) ,where  o is the average number density. The utility of R(r) isthat an integration over its first peak gives the coordination
number. (see Ref. 38) for a full discussion of these points).
Waseda made the first extensive diffraction studies of liquids 39).
The first structural studies of liquids in a containerlessenvironment were neutron diffraction measurements made with
EML40)-42). A key observation in the data from those liquids is apeak on the high-q side of the second peak in S(q). Assumingthat to lowest order, the liquid structure can be modelled byassuming tightly bound, non-interacting, close-packed, clusters,
this split peak is consistent with developing ISRO, arisingbecause the central-to-vertex atomic distance is approximately
5% shorter than the vertex-to-vertex distance. Of course, theassumption of a single cluster to describe the liquid is incorrect.
The correct approach to more realistically analyse scattering
data from amorphous liquids and glasses is, however, hotly
debated. Reverse Monte Carlo (RMC), an inverse Metropolis
Monte Carlo method, offers one approach 43)-46). The limitationsof the RMC method must be recognized, however. It is a
maximum entropy approach, producing the most disorderedstructure that is consistent with the experimental scattering data;
it works best for pairwise additive potentials, but not forstrongly covalent systems 45). Based on our experience in liquid
metals, the structures produced by RMC fits are reasonable interms of characteristic metrics such as bond angle distributions
and average dominant atomic local topologies. However, it
must be emphasized that chemical information is unreliable,
unless the RMC fits are constrained by other experimental data,such as from neutron diffraction or EXAFS studies 45), or by
results from ab-initio MD calculations 47).
Based on extensive experimental and theoretical studies, it
now seems clear that ISRO is a dominant short-range order intransition metal element and transition alloy liquids, and even insome non-transition metal liquids 48). Chemical ordering
accompanies the topological short-range ordering, withevidence for a chemical ordering transition in some cases 49). In
addition there are emerging examples of medium-range order(MRO) in liquid (i.e. order extending beyond the nearestneighbor shell), manifest by a pre-peak in X-ray 47) and− 16 −

16

K. F. Kelton

neutron50) scattering data. While medium-range order is notuncommon in metallic glasses, that it occurs in the liquid, even
above the liquidus temperature, is surprising. Where MRO is
observed in binary liquids, the solute and solvent have strong
and negative heats of mixing, indicating strong bonding. In
such cases, it is likely present even when no pre-peak isobserved in the X-ray diffraction data, since observation
requires a sufficiently large difference in the form factors of theconstituent elements 51).

4.2

Coupling of Liquid
Nucleation Barrier

Structure

with

the

Frank’s hypothesis linking ISRO to the nucleation barrier was
first experimentally demonstrated in a Ti-Zr-Ni liquid 52). Insitu high-energy X-ray diffraction studies of an electrostaticallylevitated Ti37Zr42Ni21 liquid showed that the first recalescenceobserved was due to the primary nucleation of an icosahedral
quasicrystal (i-phase). The i-phase was not stable at this hightemperature 53), and within one to two seconds transformed tothe stable C14 Laves phase, producing a second recalescence toa higher temperature. Since the polytetrahedral C14 phase iseasy to nucleate 53) and has the larger driving free energy, thepreferential nucleation of the i-phase demonstrates that its
interfacial free energy with the liquid is less than that of the C14
phase, indicating that it was structurally more similar to theliquid.
The S(q) data for the Ti37Zr42Ni21 liquid show anincreasing prominence of the second shoulder of the second
peak, consistent with increasing ISRO; this was subsequently
confirmed from RMC fits to the scattering data 54) . It is ofinterest to note that at the temperature of primary recalescence,the coherence length for the ISRO in the supercooled liquid isapproximately 2.1 nm; the estimated critical size for nucleationfrom CNT at that temperature is approximately 3.5 nm. The
comparable values of these estimates suggest that the structureof the liquid plays an important role, acting as a template for the
nucleation of the ordered phase.
Since the ISRO in the liquid prevented the nucleation of thestable crystal, it might be imagined that it would favour glass
formation. However, as in the case for the Ti-Zr-Ni liquid, it
often hinders glass formation, instead leading to the nucleationof a metastable quasicrystal. Whether ISRO helps or hinders
glass formation, then, depends partially on the details of the freeenergy surfaces of all of the accessible phases. However, it iseven more complicated. The free energy of the liquid phase andthe atomic mobility depend on the structural ordering within theliquid, although how is currently an area of active investigation.
When, in addition, the coupled-flux effects of nonstoichiometric primary crystallization are included, it is clearthat the role of nucleation in glass formation remains a richsubject warranting further study.

17

Int. J. Microgravity Sci. Appl. Vol. 30 No. 1 2013

5.

Summary and Future Investigations

Nucleation is an area of active interest, for basic as well aspractical reasons. The CNT provides a reasonably quantitativedescription of nucleation processes in liquids for small
departures from equilibrium. However, it fails to take into
account ordering near the interface with the developing crystal
nucleus, which can have a significant influence on nucleation.
A simple single order parameter approach was discussed as anillustration of how this might be addressed. However, to more
fully account for the influence on crystal nucleation of thechemical and structural ordering that occurs within supercooled
liquids will require a multiple order parameter approach, such as
outlined by Tanaka 55). Little of the MD work has focused in aquantitative way on how ordering in the liquid, which impacts
all of the parameters important for nucleation, may impact
crystallization and hence glass formation. This would seem tobe a fruitful direction for future investigation.

Acknowledgement
This work was partially supported by the National Science
Foundation under grants DMR–08-56199 and DMR–12-06707
and NASA under grants NNX07AK27G and NNX10AU19G.

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(Received 26 Nov. 2012; Accepted 7 Jan. 2012)18