Roles of liquid structural ordering in glass transition, crystallization, and water's anomalies
←
→
Page content transcription
If your browser does not render page correctly, please read the page content below
Roles of liquid structural ordering in glass transition, crystallization,
and water’s anomalies
Hajime Tanaka
Research Center for Advanced Science and Technology, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8904, Japan
Department of Fundamental Engineering, Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan
arXiv:2201.02334v1 [cond-mat.soft] 7 Jan 2022
Abstract
The liquid state is one of the fundamental and essential states of matter, but its physical understanding is far behind
the other states, such as the gas and solid states, due to the difficulties associated with the high density causing many-
body correlations and the lack of long-range order. Significant open problems in liquid science include glass transition,
crystallization, and water’s anomalies. Austen Angell has contributed tremendously to these problems and proposed
many new concepts of fundamental importance. In this article, we review how these concepts have influenced our
work on liquid physics, focusing on the roles of liquid structural ordering in glass transition, crystallization, and
water’s anomalies.
Keywords: glass transition, crystallization, water’s anomalies, glass-forming ability, many-body correlations,
angular order
1. Introduction towards the glass-transition point T g in terms of liquid
fragility. Fragile liquids show super-Arrhenius-type dy-
Liquids are one of the fundamental states of matter namic slowing down upon cooling, whereas strong liq-
around us, and a physical understanding of the liquid uids exhibit Arrhenius-type one. I was fascinated by this
state is of vital importance. The physics of an equilib- classification and became very interested in what phys-
rium state of simple liquids like hard spheres has been ical factors control the fragility of liquids. This is how
rather well established [1]. However, there are many I got started with the research of liquid science. I first
open problems concerning liquids not that simple. They met Austen and his family with excitement in the In-
include (1) metastable liquids below the meting points ternational Discussion Meeting on Relaxations in Com-
(i.e., supercooled liquids) [2] and its dynamic arrest (i.e, plex Systems held at Vigo, Spain, in 1997. Since then,
glass transition) [3, 4, 5, 6, 7, 8, 9, 10], (2) liquid-to- I have been very fortunate to keep communicating with
crystal transition [11, 12], (3) physical properties of liq- him closely over the years.
uids with complex directional interactions such as wa-
ter [13, 2, 14, 15, 16, 17, 18, 19], and (4) liquid and
amorphous polymorphs [20, 21, 22, 23].
Austen Angell has carried out a number of pioneering This article will review our research on glass transi-
studies in these topics and has been at the forefront of re- tion, crystallization, and water’s anomalies in connec-
search moving forward over the years. I mainly studied tion to many new concepts and ideas brought by An-
critical phenomena and soft matter physics until 1995, gell. Since I recently wrote a lengthy review article on
and my primary interests were rather far from these top- liquid-liquid transition and polyamorphism [23], I will
ics. When I first saw the so-called Angell plot, which not discuss this topic much in this article. In Sec. 2, we
classifies the types of slowing down of liquid dynamics discuss our study on glass transition initiated from my
interest in the Angell plot. In Sec. 3, we discuss crystal
∗ Correspondingauthor
nucleation from a supercooled liquid and glass-forming
Email address: tanaka@iis.u-tokyo.ac.jp (Hajime ability. In Sec. 4, we discuss the physical origin of wa-
Tanaka) ter’s anomalies. In Sec. 5, we summarize.
Preprint submitted to Journal of Non-Crystalline Solids January 10, 2022
2. Glass transition (a) liquid
(b)
liquid
Fragility and glass-forming ability: Two-order- Tc Tm 〜Ton
parameter model. Motivated by Angell’s classification supercooled liquid supercooled liquid
of glass-forming liquids, I wonder what controls the liq-
uid fragility. I came up with the idea that the frustration
on crystallization may be a critical factor. Tcry
The role of local icosahedral ordering in frustrating
crystallization
crystallization was first pointed out by Frank [24]. Sub-
sequently, the concept of geometrical frustration asso- Tg Tg
ciated with icosahedral ordering has been popularised glass transition glass transition
in the community of structural glasses because of its
single-order-parameter model
analogy to frustration in spin glasses and has become internal frustration
a mainstream theoretical concept. For example, Stein- (spin-glass-based model)
hardt et al. proposed a theoretical model based on the
two-order-parameter model
icosahedral ordering, which suffers from geometrical frustration against crystal ordering
frustration of the 6-fold bond orientational order param- (competing orderings)
eter Q6 [25]. Similarly, Tarjus and Kivelson and their
coworkers [26, 27] proposed a frustration-limited do-
main theory in which the critical point of structural or- Figure 1: (a) Single-order-parameter model of glass transition based
on spin-glass-like scenarios, where frustration is internally embedded
dering is avoided by internal frustration in the order pa- in the order parameter itself. In this scenario, the key temperatures are
rameter. In both approaches, the frustration originates the mode-coupling critical temperature T c , the glass transition tem-
from the “internal” geometrical frustration of the order perature T g , and the ideal glass transition temperature T 0 . (b) Two-
order-parameter model of glass transition based on frustration against
parameter itself. These approaches can be regarded as
crystallization (or compering orderings), where frustration is between
a single-order-parameter description of glass transition, a part of the potential compatible with the crystal and the other part
similarly to the spin-glass theory described by a spin incompatible with it. In this scenario, the key temperatures are the
variable alone (see Fig. 1(a)). Spins on a triangular lat- onset temperature of the cooperativity T on , which is located near the
melting point T m , T g , and T 0 .
tice, for example, suffer from intrinsic geometrical frus-
tration. The link between a structural glass and a spin
glass model, particularly Potts glass [28], was also the
basis of the random-first-order transition (RFOT) the- causing vitrification (see Fig. 1(b)). In other words, we
ory [29, 30, 8, 31]. Its connection to the mode-coupling focus on the frustration embedded in the free energy of
theory [32] was also be shown [33]. We may say that a system [9]: A part of the free energy tends to form a
all these models are based on frustration in a spin-glass- crystal, whereas the other part tends to prevent it. Our
like sense (see Fig. 1(a)). model based on “frustration against crystallization” or
In the above models based on a single order param- “competing orderings” is essentially different in physics
eter, crystallization is out of consideration, i.e., the ki- from the other frustration models mentioned above, al-
netic avoidance of crystallization is implicitly assumed. though it might not have been recognized so clearly in
Thus, particular forms of free energy were introduced the glass community. For example, our model can dis-
to describe glass transition alone. Contrary to this, I cuss the ability to form glass depending on the strength
thought that the same free energy should describe crys- of the frustration, whereas the models based on internal
tallization and vitrification and that for modeling glass frustration of a single amorphous order parameter can-
transition, it is necessary to consider squarely that a not because crystallization is out of consideration.
liquid always crystallizes upon slow enough cooling. Based on this model, I speculated that stronger
There should be a link between crystallization and glass frustration against crystallization makes a glass-former
transition as long as crystallization does not involve stronger and increases the glass-forming ability [47].
other phenomena such as phase separation. Thus, un- This view is consistent with the link between the ki-
like models based on the internal frustration in a sin- netic and thermodynamic fragilities pointed out by
gle order parameter, I proposed a “two-order parame- Martinez and Angell [48]. For example, it was pre-
ter model” [41, 42, 43, 44, 45, 46], where frustration dicted that metallic glass-formers with a stronger ten-
against crystallization caused by another type of order- dency of icosahedral ordering should be less fragile
ing plays a critical role in avoiding crystallization and (i.e., stronger) and better glass-formers, as long as
2
(a) (b)
LFS of water LFS of water
D D
Li+ Cl-
Cl- Li+
hydrates
ice Ih (c)
competing ordering
high glass-forming ability
Figure 2: (a) V-shaped phase diagram of water-type liquids [34]. Stable crystals are S -type crystals with tetrahedral symmetry that expand upon
crystallization at low pressure, whereas they are ρ-type crystals that shrink upon crystallization at high pressure. We proposed that water-type
liquids include water, five atomic elements (Si, Ge, Sb, Bi, and Ga), and some group III-V (e.g., InSb, GaAS, and GaP) and II-VI compounds (e.g.,
HgTe, CdTe, and CdSe) [34]. The melting point has a minimum, T X , at PX , around which the S - and ρ-type orderings compete and the degree
of frustration on crystallization is maximized. (b) Schematic phase diagram of water/LiCl mixtures [35, 36, 37]. Locally favored structures have
symmetry consistent with ice crystals (S -crystal) but not with hydrate crystals (ρ-crystal). If we replace the salt concentration φ with pressure,
this figure represents the phase diagram of water or other water-type liquids (see (a)). Although both adding salt and applying pressure decrease
locally favored tetrahedral structures, their roles should differ since the former has local effects whereas the latter has global effects. (c) Schematic
representation of the φ dependence of the fraction of locally favored structures and hydration structures. We note that hydrated structures and locally
favored tetrahedral structures are the sources of frustration against tetrahedral crystals (S -crystal) and hydrate crystals (ρ-crystal), respectively.
Panels (b) and (c), which are courtesy of Mika Kobayashi, are reproduced from Fig. 30 of Ref. [38].
icosahedral ordering is not too strong to form quasi- ability (see Fig. 3(b)). This liquid forms an antifer-
crystals [49, 47]. I also focused on the fact that tetra- romagnetic crystal upon crystallization at high ∆, and
hedral liquids such as water, silicon, germanium, and the antiferromagnetic orientational ordering competes
bismuth, generally have a V-shaped temperature (T )- with the pentagonal spin ordering in a supercooled state.
pressure (P) phase diagram [34] (see Fig. 2(a)). A liquid We note that the increase in the anisotropic part of the
expands upon crystallization into a low-pressure crys- potential leads to the increase in the activation energy
tal, which we call S -type crystal. In contrast, above for particle motion at high temperatures above the on-
the triple point Pmin
m , it shrinks upon crystallization into set temperature T on below which cooperative motion
a high-pressure crystal, which we call ρ-type crystal. emerges, also making liquids stronger.
Our model predicts that the glass-forming ability and We also performed simulation studies on 2D weakly
fragility should be maximized near the triple point since polydisperse hard-sphere-like liquids, where the degree
the competition (i.e., frustration) between the two types of size polydispersity ∆ is a measure of the strength
of orderings (S -ordering and ρ-ordering) is maximized of frustration against crystallization into a hexatic crys-
there. tal [40]. This system has a φ-∆ phase diagram simi-
Later we performed two-dimensional (2D) numeri- lar to the above 2D spin liquids (see Fig. 3(c)), show-
cal simulation to confirm our claim by introducing a ing the generality of our frustration-based scenario. For
spin to each particle with an angle-dependent three- this system, we have also found that a liquid suffering
body potential (strength ∆) favoring pentagonal sym- stronger frustration against crystallization, i.e., larger ∆
metry in addition to an isotropic Lennard-Jones poten- becomes stronger and has a better glass-forming ability
tial [39, 50] (see Fig. 3(a)). Pentagonal local struc- (see Fig. 3(d)).
tures that cannot tile the space induce strong frustration Molinero, Sastry, and Angell [51] also performed nu-
against crystallization. We indeed found that a liquid merical simulations of modified Stillinger-Weber poten-
with a stronger tendency (larger ∆) of local pentagonal tial, which is made of an isotropic part and anisotropic
ordering (i.e., stronger frustration against crystalliza- part favoring tetrahedral symmetry. They found that the
tion) becomes stronger and has a better glass-forming glass-forming ability and the fragility are maximized
3
(a) 0.5 (b) 6 of the critical experiments that was the basis of devel-
Liquid 5 oping our two-state model. We also did experiments on
0.4
4 the same system (a water-LiCl mixture) to test the pre-
0.3
3 diction of a three-state model, which includes hydration
log10
T
Supercooled
0.2
Plastic crystal
liquid
2
structures as the third state to our two-state model of wa-
Glass ter (see Fig. 2(b)). This model phenomenologically de-
0.1 1
AFM scribes the dependence of the viscosity on the salt con-
0.0
crystal 0
0.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 centration [36, 37]. In the course of these works, Angell
g kindly provided us with many valuable comments.
(c) (d) Recently, we have used the model with tunable tetra-
4
0.50
I 0% Liquid 16% hedrality λ developed by Molinelo, Sastry, and An-
13%
I
11% gell [51] to study the link between the degree of compet-
Dynamically 9% ing orderings and the glass-forming ability in a system
log10
0.60 heterogeneous
m 3
g with a V-shaped phase diagram (see Fig. 2(a)) [54]. We
0.70
will discuss this problem and provide a more detailed
0
physical origin of the phenomenon in Sec. 3.
Glass 2
Crystal
0.80 0.85 0.9 0.95
0 5 10 15 Structural origin of slow glassy dynamics. One of the
(%) g
central questions of glass transition is the physical
origin of drastic slowing down of the dynamics to-
Figure 3: (a) The phase diagram of 2D spin liquid as a function of
wards the glass transition point. The extreme case is
T and the strength of energetic frustration ∆ [39]. An increase in ∆ the fragile limit according to the Angell classification.
changes the stable equilibrium crystal from a hexagonal plastic crys- Many simulation models of spherical particles belong
tal, where spins can rotate on the lattice, to an antiferromagnetic crys- to the fragile-limit glass formers [59]. Our two-order-
tal. For ∆ ≥ 0.6, the system forms glass with a specific cooling rate.
(b) The Angell plot for 2D spin liquids with various ∆. A liquid with parameter model predicts that a system only weakly
larger ∆ is less fragile, i.e., stronger, and has a better glass-forming frustrated against crystallization, which is sitting in the
ability [39]. (c) The phase diagram of 2D polydisperse hard-sphere- border between crystallization and vitrification, should
like liquid as a function of T and the variance of size polydispersity belong to the fragile-limit glass formers [9]. Based on
∆ (%) that is a measure of the frustration strength against crystalliza-
tion [40]. The stable equilibrium crystal can be formed up to ∆ = 9%, this idea, we studied weakly frustrated liquids and found
but above it, the system tends to form glass without crystallization. that crystal-like orientational order grows with an in-
(d) The Angell plot for liquids with various polydispersity ∆. A liq- crease in the degree of supercooling. We show such ex-
uid with larger ∆ is less fragile, i.e., stronger, and has a better glass-
amples of growing angular orders upon cooling in var-
forming ability [40], as in the 2D spin liquids. Panels (a) and (b) are
reproduced from Figs. 2 and 6b of Ref. [39]. Panel (d) is reproduced ious supercooled liquids, as shown in Fig. 4, indicat-
from Fig. 1(b) of Ref. [40]. ing the generality in fragile liquids. We emphasize that
two-body density correlators, such as the radial distri-
bution function g(r) and structure factor S (q), are hard
around the triple point, consistent with the prediction to detect these angular orders. This fact also implies
of our two-order-parameter model [34]. I vividly re- little coupling between the density and the angular or-
member a fascinating discussion on this topic with An- ders for fragile-limit liquids (note that this is not the
gell and Molinero in the International Discussion Meet- case for strong liquids, such as silica). In these liquids,
ing on Relaxations in Complex Systems held at Lille, steric repulsion is the primary source of structural order-
France, in 2005. Later, Angell and his coworkers even ing. Noting that an angular order requires at least three-
proved a high glass-forming ability of atomic germa- body correlations, we may say that glassy amorphous
nium near the triple point by performing high-pressure order must be intrinsically due to many-body correla-
experiments [52]. They successfully produced a metal- tions, casting doubt on scenarios based on the two-body
lic amorphous state of single-component atomic germa- correlation [9, 59, 60].
nium by temperature quench. We also found that these spatial fluctuations of static
A similar V-shaped melting point minimum is also structural order parameters are strongly correlated with
observed in binary mixtures. Such melting point min- the mobility fields, i.e., dynamic heterogeneities [39,
imum is widely known as a eutectic point. Angell 40, 56, 55, 57, 61]. These results not only support
also did pioneering experiments with Sare on the glass- a structure-dynamics link revealed by an isoconfigura-
forming ability of a water-salt mixture [53, 15]. It is one tional sampling method developed by Harrowell and his
4
(a) (b) (c) (d)
Y6 Y6 Q6
0.50 0.60 0.70 0.80 0.35 0.40 0.45 0.50 0.55 Y0 40 0 50 0 60 0 70 0 80
Figure 4: Spatial fluctuations of angular orders developed in various supercooled glass-forming liquids. We stress that these angular orders are not
coupled to the density field and thus cannot be detected by a two-body density correlator, such as the radial distribution function g(r) and structure
factor S (q) [55]. (a) Spatial distribution of hexatic order parameter Ψ̄6 time-averaged over τα in a 2D supercooled polydisperse hard-sphere-like
liquid (polydispersity ∆=9%) [40]. (b) Spatial distribution of hexatic order parameter Ψ̄6 time-averaged over τα in a 2D supercooled polydisperse
driven granular liquid (polydispersity ∆=9%) [56]. (c) Spatial distribution of 6-fold bond orientational order parameter Q̄6 time-averaged over τα
in a 3D supercooled polydisperse hard-sphere-like liquid (polydispersity ∆=6%) [55, 57]. (d) Spatial distribution of spin alignment order parameter
Ψ̄ time-averaged over τα in a 2D supercooled spin liquid (the frustration parameter ∆ = 0.6) [39].
(a) 102 (b) (c) 7
(2DPC) 5
(2DPC) ergodic
3D
(2DPC) aging ( =0.587)
6
(2DGL) 4
0)
(2DGL)
101 (2DBL) 5
(2DBL) 3 log
log10(
(2DSL) 4
(2DSL) 2
(3DPC)
100 (3DPC) 3 3D
1 aging ( =0.568)
(3DLJ) 3D
aging ( =0.553)
2
-2 -1 0 0 0 1 2 3 4 5
10 10 10 0 5 10
D (X-X 0 )/X 0 =D
2/d d/2 3/2
[( X- X0) /X0] 6
Figure 5: (a) Power-law relation between ξ/ξ0 and (X − X0 )/X0 [55]. X is the volume fraction Φ for 2D and 3D polydisperse hard-sphere-like
liquids (2DPC and 3DPC), 2D granular liquids (2DGL), and 2D binary hard-sphere-like liquids (2DBL), whereas the temperature T for 2D spin
(2DSL) and 3D Lennard-Jones liquids (3DLJ). ξ and ξ6 are the static correlation lengths, and ξ4 is the dynamic correlation length. ξ6 is the
correlation length of hexatic order Ψ6 for 2DPC and 2DGL, whereas the correlation length of 6-fold bond orientational order Q6 for 3DPC and
3DLJ. (b) Relation between log(τα /τ0 ) and DF (X − X0 )/X0 = DF (ξ/ξ0 )d/2 [55], where DF is the fragility index and d is the spatial dimensionality.
(c) Relationship of τα to ξ6 during the aging process of 3DPC (polydispersity: 6%) for the volume fractions Φ = 0.550, 0.568, and 0.587, together
with the relationship of τα to ξ6 in the equilibrium liquid as a function of Φ [58]. Note that during aging, both τα and ξ increase with the aging
time. Panels (a) and (b) are reproduced from Figs. 4a and b of Ref. [55], respectively, and panel (c) from Fig. 4(f) of Ref. [58].
coworkers [62] but also show the importance of angu- that the static correlation length grows with an increase
lar order, i.e., many-body interactions, at the origin of in the degree of supercooling, in proportion to the dy-
structure-dynamics correlation. namic correlation length ξ4 [39, 40, 56, 55, 57, 61].
Furthermore, the correlation length ξ has been re- This suggests that the dynamic heterogeneity is a con-
vealed to diverge towards the ideal glass transition point sequence of the spatial fluctuation of static structural
X0 as [39, 40, 56, 55, 57, 61] order. Moreover, we found that the exponent ν is the
!−ν same as the critical exponent for the correlation length
|X − X0 |
ξ = ξ0 , (1) for the d-dimensional Ising universality class of critical
X0
phenomena [55]: ν ∼ 2/d. We show this relation sug-
where ν = 2/d, d is the spatial dimensionality, and X gesting the Ising criticality for various liquids is shown
is temperature T or volume fraction φ. We also found
5
in Fig. 5(a). Similar Ising-like criticality was also re- der thermal noise have the following relation to the sys-
ported by Mosayebi et al. [63] and Zheng et al. [64]. tem temperature (T ) or density (ρ) [68] (see Fig. 6(a)):
Furthermore, we obtained the following relation be-
tween ξ and the structural relaxation time τα (see (Ψ − Ψ0 )/Ψ0 = (T − T 0 )/T 0 = (ρ0 − ρ)/ρ0 , (3)
Fig. 5(b)):
where Ψ0 and ρ0 are the values of Ψ and ρ at the ideal
K(ξ/ξ0 )d/2 glass-transition point. This relation indicates that Ψ is
!
τα = τ0 exp , (2) an essential control variable of the slow glassy dynamics
kB T
not only locally but also globally [68]. For example. we
where τ0 is the microscopic time, K is related to the have confirmed a general relationship between Ψ and
fragility index DF as K = DF kB T (larger DF indicates τα :
a stronger character), and kB T is the thermal energy. BΨ0
!
This relation indicates that the activation energy for par- τα = τ0 exp , (4)
Ψ − Ψ0
ticle motion increases in proportion to (ξ/ξ0 )d/2 . Fur-
thermore, we found [55] that the dynamics of the struc- where B is a constant.
tural order parameter belongs to the dynamic universal- It has been known [75] that liquids interacting with
ity of the kinetic Ising model, i.e., model A [65, 66]. We the Weeks-Andersen-Chandler (WCA) and Lennard-
note that our angular structural order parameter is non- Jones (LJ) potentials, which have an identical g(r) but
conserved since it can change locally at each point in- very different dynamics at the same temperature. Re-
dependently from other points (as spin can flip indepen- cently, there have been some efforts to save this failure
dently without any constraint). This non-conserved na- of the two-body description of slow glassy dynamics by
ture of the order parameter is consistent with the model its modification [76, 77]. However, we have shown that
A-type dynamics. these liquids actually have very different local packing
capability Ψ at the same temperature, responsible for the
Generalized angular order parameter: Packing capa- different dynamics [69]. We have confirmed that both
bility Ψ. In the above examples showing the Ising crit- systems obey relation (4). This finding indicates that a
icality, angular ordering has a connection to the crystal two-body correlation such as g(r) is not enough to de-
symmetry (see, e.g., Fig. 4) because these systems suffer tect the amorphous order responsible for slow dynamics
from weak frustration against crystallization. However, of a supercooled state, and many-body angular corre-
such a connection is not expected for binary mixtures lations play an essential role in slow glassy dynamics
that involve phase separation upon crystallization. The [9, 59, 60].
phase separation cuts the link between the liquid-state
and crystal-state free energies. To overcome this diffi- Link between glassy order parameter and slow dynam-
culty, we have recently introduced a new structural or- ics. Now we consider how local liquid structures deter-
der parameter, i.e., local packing capability Ψ, which is mine slow glassy dynamics. For a weakly polydisperse
a measure of local vibrational entropy, i.e., local free en- system, the spatial correlation function of the coarse-
ergy. This structural order parameter Ψ can detect ster- grained crystal-like bond orientational order parameter,
ically favored topologies in an order-agnostic manner. a complex order parameter that has both amplitude and
It characterizes the angular order of the configuration phase information, can directly detect the correlation
of neighboring particles around a particle, thus intrinsi- length of growing structural order. Here, the coarse-
cally reflecting many-body correlations. We may regard graining up to neighboring particles is necessary to re-
Ψ as a generalized bond orientational order parameter move the disturbance due to icosahedral ordering in
that does not rely on specific symmetry. hard-sphere-like systems. Thus, it is hard to detect a
Using this order parameter, we have confirmed that proper correlation length without coarse-graining (see,
relations (1) and (2) hold universally for any particu- e.g., Ref. [78]).
late glass-formers, including binary mixtures, in which What is critical here is that the phase (angular) coher-
the interaction potentials are isotropic and additive [67, ence of the bond orientational order parameter plays an
68, 69]. Furthermore, we have found that a link between essential role in detecting the static correlation length
the amplitude of fast β motion and slow structural relax- ξ [40, 55, 57, 9] (see below). On the other hand, the
ation [70, 71, 72, 40, 73, 74] is through local structures amplitude correlation function fails in detecting ξ. For
identified as the local packing capability [67]. a scalar structural order parameter, the spatial coarse-
We have also found that the average local packing ca- graining over the correlation length ξ [67, 68] or the
pability defined in an instantaneous liquid state, Ψ, un- time averaging over τα [39, 40, 55, 57, 61] are crucial
6
(a) 0.16 Y (b)
1.0
0.14
P (3)+ P (4)
Packing capability Y(T)
y6
0.8
P (3)
P (N )
0.12
0.6
0.04 0.115 0.19
0.10
0.4
P (4)
0.08
T0 Tg Ton 0.2
0
0.06 0
0 0.001 0.002 0.003 0.004 0.68 0.70 0.72 0.74 0.76 0.78
T f
Figure 6: (a) Evolution of structural order Ψ in instantaneous liquid states (circles) in a 2D polydisperse harmonic systems (the polydispersity
∆ = 13%). The onset temperature (T on = 0.00276) and dynamical glass transition temperature for the slowest cooling rate (T g = 0.00135) are
indicated, allowing us to separate the simple-liquid (light red), supercooled and glass regimes (light blue). For instantaneous liquid states, the
temperature dependence of Ψ can be fitted with a linear function in the supercooled regime (Ψ − Ψ0 )/Ψ0 ∝ (T − T 0 )/T 0 (solid line; dashed line
below T g ). Note that smaller Ψ means higher order. The inset image is the spatial distribution of Ψ at T = 0.0018. See Ref. [68] for the details.
(b) φ-dependence of the fraction of triangles and squares, which are averaged over 10τα , for 2D polydisperse hard-sphere-like system (∆ =9%).
Squares (geometrical defects) decrease, or transform to compact triangles, with an increase in φ and tend to disappear around φ0 completely. The
inset image shows the correlation of the instantaneous hexatic order parameter, ψ6 , to geometrical defects (i.e., voids) at φ = 0.73. Bonds are shown
as thin white lines. ψ6 is evidently anti-correlated with the geometrical defects (black voids). See Ref. [55] for the details.
for seeing nearly one-to-one structure-dynamic correla- between the Debye-Waller factor and the slow dynam-
tion (ξ ∼ ξ4 ) in supercooled glass-forming liquids. This ics [80, 70, 72, 40, 73, 81, 82].
fact means that the dynamics of a particle cannot be Here we consider the role of the phase (angular) co-
determined locally but depends on its environment; in herence of the complex bond orientational order param-
other words, it is determined in a coarse-graining man- eter. Considering the above finding, we speculate that
ner [79, 67, 68]. the phase coherence controls how the mobility field sta-
tistically develops, following the structural order param-
We have shown that the size of the environment af- eter field. This fact again strongly suggests the critical
fecting the motion of a particle determines the static importance of angular structural order in determining
correlation length ξ as well as the dynamic correla- slow glassy dynamics. We stress that the angular co-
tion length ξ4 [67]. Using isoconfigurational averag- herence is tightly linked to the spatial extendability of
ing, we have also studied how the mobility field de- sterically favored topologies without voids.
velops from an inherent structure with time. We have Finally, we also emphasize that Eq. (2) holds even
found that particle motion starts to evolve from most in out-of-equilibrium situations during aging [58] (see
defective parts with low local packing capability (i.e., Fig. 5(c)) and for systems under spatial confine-
large Ψ(r)) and gradually spreads towards regions with ments [83]. These results strongly suggest that liquid
higher packing capability (smaller Ψ(r)), following the dynamics at a specific location is controlled by a struc-
Ψ(r) field. The strongest correlation is seen between tural order parameter of many-body origin there, irre-
the mobility field pattern at the maximum dynamic het- spective of whether the liquid is in equilibrium or out of
erogeneity (its characteristic length=ξ4 ) and the static equilibrium or whether it is confined or not confined.
order parameter field with the coarse-grained length ξ,
i.e., ξ4 ∼ ξ. Since this process proceeds statistically, The origin of the Ising-like criticality. These studies
there is no one-to-one relationship between the local indicate that structural fluctuation of particulate glass-
scalar order parameter such as Ψ(r) and the mobility formers interacting with isotropic potentials shows
field. Here we note that since Ψ(r) is inversely corre- Ising-like criticality, suggesting a discrete symmetry of
lated with the local Debye-Waller factor (solidity) (or, glassy structural ordering. Langer proposed an Ising
positively correlated with the amplitude of fast β mo- model of glass transition, focusing on the two-state fea-
tion) [67], this result is consistent with the correlation ture of sterically favored configurations [84, 85]. This
7
is an intriguing idea. We might be able to say that inset image of Fig. 6(b) illustrates the spatial distribu-
frustration generally changes the nature of the order pa- tion of geometrical defects (squares, pentagons, · · · ) for
rameter from the continuous to discrete symmetry [55] 2D polydisperse hard-sphere-like system [55], which is
since similar phenomena are also observed in spin sys- anti-correlated with the degree of hexatic order (i.e.,
tems [86, 87]. Below we consider this problem from a tiling with rather regular triangles). We also plot the
different angle. φ-dependence of P(3) and P(4) in Fig. 6(b). Triangles
In the above, we have emphasized the importance of become more and more dominant with an increase in φ,
the phase coherency of orientational order parameter and squares (i.e., geometrical defects) tend to disappear
and the spatial coherence of packing capability. Such completely at the ideal glass-transition volume fraction
local coherence of order between neighboring particles φ0 . The fraction of voids, which is roughly the fraction
is favored sterically because a particle with high orien- of squares, P(4), follows P(4) ∝ (φ0 − φ).
tational order (or packing capability) can gain an extra Interestingly, the T -dependence of voids, P(4), is es-
free volume (or vibrational entropy) if its neighbor also sentially the same as that of the packing capability, Ψ,
coherently has high order. This situation is essentially Eq. (3) (compare Figs. 6(a) and (b)). We stress that
the same as the favored parallel alignment of neighbor- since relation (3) holds in any systems interacting with
ing spins in the Ising model. isotropic interactions, including both 2D and 3D sys-
Here, we point out an essential difference in the char- tems [68], the behaviors of these two quantities charac-
acter of the order-parameter fluctuations between 2D terizing local packing are universal. The packing capa-
monodisperse and glass-forming polydisperse disks. In bility Ψ and the fraction of voids P(4) act as control pa-
the former, ordered patches with some phase coherence rameters of the dynamics as the temperature does. We
are formed in a liquid state, and its coherence length ξcoh may regard these quantities as ‘free volume’. This re-
(i.e., the patch size) exponentially toward the hexatic- sult supports the above speculation on the importance of
ordering point. On the other hand, in the latter, frus- frustration-induced disorder for the Ising-like criticality.
tration creates low ordered regions, leading to critical- However, since the discussion is speculative, identifying
like fluctuations of the order parameter amplitude in the origin of discrete symmetry remains a task for future
addition to the phase, as shown in Figs. 4(a) and (b). theoretical investigation.
The correlation length diverges, obeying the Ising-like
power law instead of the exponential growth. Similar Ising-criticality scenario or RFOT scenario. Our Ising-
behaviors are also observed in other systems, such as criticality scenario [55, 9] and the RFOT scenario [29,
2D spin liquid (see Fig. 4(d) [39] and glass-forming bi- 30, 8, 31] both predict the same relations for ξ and τα
nary mixtures [67, 68]. We speculate that the introduc- like Eqs. (1) and (2) (for the former, below the onset
tion of this disorder due to frustration and the resulting temperature T on , whereas for the latter, only near T 0 be-
order-parameter-amplitude fluctuations may be the ori- low the mode-coupling T c ). However, there is a fun-
gin of the discrete twofold symmetry, i.e., the Ising-like damental difference concerning the nature of the order-
criticality. parameter fluctuations.
In relation to the above speculation, we focus on the In our scenario, spatial fluctuations of the order pa-
nature of the disorder. The above order parameter Ψ is rameter are critical-like (i.e., obeying the Ornstein-
scalar, representing the local packing capability around Zernike correlation [66]) for our scenario, and, upon
a central particle. Thus, its decrease with decreasing T cooling, their size exceeds the particle size below the
or increasing φ indicates the decrease of the degree of onset temperature T on . Thus, T on marks a crossover
disorder. We also define voids for weakly 2D polydis- from the simple Arrhenius to Vogel-Fulcher-Tammann
perse hard-sphere-like systems as follows [55]. We first law (see. e.g., Ref. [45]). Below T on , the static (ξ) and
apply Delaunay triangulation to a spatial pattern of the dynamic (ξ4 ) correlation lengths grow in a coupled man-
centers of mass of particles. When the length of a side ner since the growth of structural ordering is the origin
of a triangle connecting particle i and j is larger than a of slow dynamics in this scenario (see Fig. 7(a)).
critical value σcij , which we set σcij = 1.5(σi + σ j )/2, In contrast, mosaic-like metastable islands are a
we cut that bond. In this way, we can identify geo- prominent player for the RFOT scenario and emerge
metrical defects, which are polygons whose number of only below the mode-coupling T c , below which the
sides are more than 4 (squares, pentagons, · · · ). The free energy is supposed to have a metastable minimum.
driving force of geometrical ordering is steric repul- Thus, the static (ξ) and dynamic (ξ4 ) correlation lengths
sion, which maximizes entropy by gaining vibrational are strongly decoupled far above the Kauzmann (or
entropy while sacrificing configurational entropy. The ideal glass-transition) temperature T K = T 0 , more pre-
8
(a) (b) Static-dynamic
decoupling
4
= above Tc
Static-dynamic
coupling
below Ton
P S 4
TK Ton TK Tc Ton
T T
Figure 7: Comparison of the temperature dependences of the static (ξ and ξPTS ) and dynamic correlation lengths ξ4 between the Ising criticality
(a) and RFOT (b) scenarios. T on is the onset temperature, T c is the mode-coupling critical temperature, and T K is the Kauzmann temperature equal
to the ideal glass transition temperature T 0 .
(a) 7 (b)
growing static&dynamic lengths growing static&dynamic lengths
6 5
4
× 1.5
5 rm
4 Dyn
× 1.75
x
4
x
3 3
2
2 PTS
1
0.97 0.96 0.95 0.94 0.93 0.92 0.0006 0.0009 0.0012
T
Figure 8: (a) Various correlation lengths as a function of density ρ for 2D polydisperse hard-sphere-like liquid (polydispersity ∆ =11%) [88]: red
squares for bond orientational order ξ6 , blue triangles for the two-body correlations ξ2 , black circles for ξK for random pinning, green diamonds for
ξK for uniform pinning, black filled circles for ξPTS , and pink triangles for the dynamic correlation length ξ4 . The dynamical correlation length is
scaled with ξ40 ∼ 1.7 to ease the visual comparison with the static length scales. (b) Temperature dependences of the dynamic correlation length ξ4
and the static correlation length ξ for 3D soft binary mixture (T 0 = 3.5 × 10−4 ) [67]. The solid line is a power-law fit with ξ = ξ0 ((T − T 0 )/T 0 )−2/3 .
Here, we include the dynamic ξDyn and static PTS ξPTS lengths of the same system reported in Ref. [89] for comparison: the dynamic length
ξDyn × 1.75 (open circles) and the static length ξPTS × 4.6 (open diamonds). Panel (a) is made based on Fig. 1D of Ref. [88], and panel (b) is
reproduced from Fig. 5(f) of Ref. [67].
cisely, above T c (see Fig. 7(b)). The crucial point is there should exist no rapidly growing static length above
that T on is generally located much higher temperature T c , where mosaic structures are absent (see Fig. 7(b)).
than T c , and thus, the relation between the static and Thus, these results support our Ising-criticality scenario
dynamic correlation lengths in a temperature region be- (compare Figs. 7(a) and (b)).
tween T on and T c should provide critical information on
which scenario is more appropriate for the description We also note that the standard point-to-set length
of slow glassy dynamics. ξPTS , which was developed to pick up the mosaic length
based on the RFOT scenario [90, 91, 89], cannot detect
Our molecular dynamics simulations showed that the our angular order correlation length ξ [88, 67]. This fact
relations given by Eqs. (1) and (2) hold even in the tem- seems to indicate that the standard point-to-set (PTS)
perature region between T on and T c (or slightly below length that is inherently designed to detect translational
it). We have confirmed for 2D polydisperse hard-sphere order cannot pick up orientational order [88] unless it is
liquid (Fig. 8(a)) [88] and 3D soft binary mixture liq- specially designed to detect angular order [92] (note that
uid (Fig. 8(b)) [67] that the static correlation length ξ the PTS defined in this way is no more order-agnostic
is directly coupled with the dynamic correlation length since the knowledge of the order type is required in ad-
ξ4 even above T c . According to the RFOT scenario, vance). The ROFT theory is exact for hard-sphere liq-
9
uids at the infinite dimension (d = ∞), where many- this transitional behavior has been reported for various
body effects disappear entirely, and the two-body-level liquids such as silica [101], metallic glass-formers [102,
description becomes exact [93]. However, at low di- 103], and chalcogenide glasse-formers [104], and has
mensions such as d = 2 and 3 relevant for real systems, attracted considerable attention in the glass community.
many-body angular correlations that originate from en- It motivated us to consider the physical origin of the
tropy gain associated with high particle packing capa- fragile-to-strong transition of water [105]. This un-
bility lead to ‘entropy-driven structural ordering’ in a usual phenomenon has been claimed to be explained
supercooled liquid [9, 59, 60]. by a crossover from the power-law-type mode-coupling
The physics behind our scenario can be relatively anomaly to a strong (Arrhenius) behavior upon cross-
easily understood by noting that (1) crystallization of ing the Widom line of liquid-liquid transition [97]. This
hard spheres discovered by Alder and Wainwright [94] scenario has become very popular.
is purely entropically (or sterically) driven and (2) hex- I focused on the fact that the fragile behavior of wa-
atic ordering plays a critical role in the ordering of hard ter is observed even at a temperature almost twice of
disks in 2D; i.e., an intermediate hexatic phase with T g , which is usually thought free from the glass tran-
quasi-long-range orientational order emerges between sition or far above the onset temperature of coopera-
liquid and crystal states [95]. Thus, we may conclude tive slow glassy dynamics, T on . I proposed [105] that
that in liquids made of particles where interparticle this transition may not be related to glass transition,
steric repulsions play a critical role in structural order- but due to the crossover between the two states from a
ing, slow glassy dynamics is caused by entropy-driven high-temperature disordered liquid without locally fa-
angular ordering, i.e., enhancement of packing capa- vored structures (tetrahedral structures in the case of
bility. Recently, Han and his coworkers [64] showed water and icosahedral structures for metallic liquids),
that liquids made of hard ellipsoids also exhibits Ising- which we call “ρ-liquid”, to a low-temperature liquid
criticality near the glass transition, indicating that our full of locally favored structures, which we call “S -
scenario is valid as long as structural ordering is entrop- liquid”. Since ρ- and S -liquids have different activa-
ically driven. tion energies, a liquid with the fraction of locally fa-
We emphasize that entropically driven structural or- vored structures s should have the activation energy of
der can grow its correlation length without limitation E(s) = Eρ (1 − s) + ES s, where Eρ and ES are the ac-
upon cooling until geometrical frustration embedded in tivation energy for pure ρ-liquid (s = 0) and S -liquid
the structural order parameter prevents it. We note that, (s = 1), leading to an Arrhenius-to-Arrhenius crossover.
unlike crystalline order, amorphous order cannot tile the The assumption behind this simple additive form of
space thoroughly and thus intrinsically suffers from in- E(s) is that the lifetime of locally favored structures is
ternal frustration in the structural order itself. For ex- much shorter than τα [105], which was later confirmed
ample, polydispersity above a particular level prevents by numerical simulations [99]. It was predicted that the
infinite growth of crystal-like bond orientational order. structural relaxation time τα is described as a function
Thus, the critical-like power-law growth of the correla- of s by the following phenomenological relation [105]:
tion length ξ and the resulting VFT-like divergence of
τα may eventually stop upon cooling. Eρ (1 − s) + ES s
!
τα (s) = τ0 exp , (5)
Finally, we note that the situation should be markedly kB T
different for energetically driven structural ordering
such as tetrahedral ordering in water and silica. Locally where τ0 is the microscopic time.
favored structures formed energetically by directional Recently, we have confirmed the validity of this phe-
bonds must accompany a sizeable entropic loss upon nomenological relation based on numerical simulations
the ordering (see below) [96, 59, 60]. Thus, these struc- of realistic classical water models [98, 99], but at the
tures cannot extend spatially and remain localized with- same time found that s is not the fraction of locally
out growing, increasing only their fraction upon cool- favored structures, but its spatial average up to neigh-
ing. bors sD . This is because the dynamics of a molecule
cannot be determined by itself but also affected by its
Fragile-to-strong transition. Angell and his cowork- neighbors. The crucial point is that diffusion occurs
ers [100] discovered that some liquids like water ex- via the exchange between neighboring molecules, and
hibit fragile behavior at a high temperature, whereas hydrogen-bond reconnections are necessary for the dif-
strong behavior near T g . This unusual behavior has been fusion of a water molecule. This consideration has led
widely known as “fragile-to-strong transition”. Later, us to develop a hierarchical two-state model [98, 99].
10(a) (b)
(c) 14 Arrhenius law
(d)
VFT function
12 Two-state model Two-state One-state regime
log (1/D) (106 sm-2)
water
10 silica
o-terphenyl
8
strong liquid
6
fragile liquid
4
2
0.4 0.5 0.6 0.7 0.8 0.9 1.0
Tg / T
Figure 9: Dynamical crossovers in TIP5P water and real water at 1 bar. (a) The Widom-line interpretation [97] of dynamical crossovers in
TIP5P water: a crossover from power-law above the Widom line (s = 1/2 line here) to Arrhenius law below the Widom line. This scenario
results in unphysically slow dynamics at T g . (b) Our hierarchical two-state scenario for dynamical crossovers in TIP5P water. In the pure ρ-state
regime, water shows an Arrhenius behavior. When entering the two-state regime upon cooling, water shows the first dynamic crossover from the
Arrhenius to non-Arrhenius (the so-called “fragile”) behavior; When leaving the two-state regime by further cooling, water exhibits the second
dynamic crossover from the non-Arrhenius to Arrhenius (the so-called “strong”) behavior. (c) Two-state fitting to the temperature dependence
of the experimentally measured diffusion constants of water and silica, together with a fit of our hierarchical two-state model. We also show the
data of a typical fragile liquid, o-terphenyl, with a fit of the Vogel-Fulcher-Tammann law. On the details of the experimental data, see Ref. [98].
(d) Breakdown of the Stokes-Einstein-Debye relation in TIP5P water. The solid and dotted lines represent the two-state and dynamic ρ-state
contributions, respectively, indicating that the growth of dynamic S -state upon cooling causes the breakdown of the Stokes-Einstein-Debye relation
in supercooled water. The effective hydrodynamic radius a = 1.3 Åwas estimated from the high-temperature data for TIP5P water. These figures
(a)-(d) are reproduced from Figs. 15(a) and (b) of Ref. [99], Fig. 5 and Fig. 4A of Ref. [98], respectively. See Refs. [98, 99] for the details.
Figure 9 shows the analyses of the dynamics of TIP5P formers such as CuZr, it was shown that local structural
water based on the Widom-line scenario (a) and our hi- orderings such as icosahedral ordering are linked to
erarchical two-state scenario (b), and the hierarchical the fragile-to-strong transition, although local structures
two-state model analysis of the dynamics of actual wa- may not be one type [103]. We note that Angell and
ter (c). We also note that our two-state model can de- his coworkers pointed out the importance of medium-
scribe the breakdown of the Stokes-Einstein-Debye re- range order and liquid-liquid transition in the fragile-
lation without any adjustable parameters (Fig. 9(d) (see to-strong transition in phase-change materials used for
Refs. [98, 99] on the details). This indicates that the memory [108, 109, 110].
breakdown of the Stokes-Einstein-Debye relation in wa-
ter does not come from glassiness but simply from the
Perspective. The above consideration has led us to clas-
two-state feature. We also discuss this problem in more
sify glass-forming liquids based on which of entropy
detail in Sec. 4.
or energy is dominant in structural ordering [59, 60].
We argue that this Arrhenius-Arrhenius crossover Liquids with entropy-driven structural ordering, such as
scenario based on the two-state model may be valid hard spheres, can have amorphous structural (angular)
also for silica, metallic, and chalcogenide liquids since order whose correlation length tends to diverge towards
these liquids tend to form distinct locally favored struc- the ideal glass-transition point but may eventually stop
tures. For silica, we have evidence for the two-state fea- growing due to geometrical frustration in the amorphous
ture, similarly to water [106, 107]. For metallic glass- order parameter. In these liquids, the packing capabil-
11ity is the essential order parameter. On the other hand, Crystal nucleation begins with increasing the phase
for liquids with energy-driven structural ordering, such (angular) coherence of the crystal-like bond orien-
as water and silica, the size of locally favored struc- tational order without accompanying density change.
tures cannot grow and be localized due to a significant Later, the density change, i.e., translational ordering,
entropy loss upon structural ordering. Lowering tem- comes into play only when the crystal nucleus becomes
perature increases only the fraction of locally favored large enough to have its spatial periodicity. Such crystal
structures (s) (without increasing their sizes), leading to nucleation behavior was directly confirmed by confocal
an Arrhenius-Arrhenius crossover discussed above. In microscopy experiments in colloidal systems by Tan et
these liquids, the fraction of locally favored structures al. [118]. We have found that preordering also plays
(s) is the order parameter. Finally, the intermediate be- a critical role in the polymorph selection by simula-
havior is expected for liquids in which both entropy and tions [116] and confocal microscopy experiments [119].
energy contribute to the free energy. This case of liquids Furthermore, we have confirmed that this scenario of
with intermediate fragility remains unclear, and further crystal nucleation is valid for soft spheres [120] and wa-
research is necessary to reveal the origin of the slow ter [121], indicating the universality of this scenario of
glassy dynamics in such liquids. crystal nucleation [117].
We have also found that entropy-driven structural or- The above scenario suggests that preordering com-
dering in a supercooled liquid exhibits critical-like fluc- patible with the symmetry of a crystal to be formed
tuations, whose growth leads to the increased activation promotes crystal nucleation since preordering lowers
energy for particle motion. However, the origin of the the liquid-crystal interface tension due to wetting ef-
Ising-like criticality and its link to slow dynamics re- fects [111]. We emphasize that the wetting effects are
main elusive. This problem is a challenging but inter- different from those in a scalar order parameter [122]
esting subject for future theoretical study. since the relevant order parameter responsible for the
effects is orientational. The effects are a consequence
of symmetry matching between the precursor and crys-
3. Crystallization
tal structures, leading to the polymorph selection due to
preordering [116, 119]. Such preordering also plays a
critical role in heterogeneous crystal nucleation on the
Role of preordering of supercooled liquid in crystal nu- substrate [123]. As mentioned above, we pointed out a
cleation. Crystallization is another important topic in possibility that competition between two types of crys-
the field of liquid science [11, 12]. Angell also made tals leads to the minimum melting point at the triple
many pioneerinng experimental studies on crystal nu- point [124], making a liquid structure more disordered
cleation [112, 113, 114, 115]. As discussed above, we and increasing the frustration against crystallization and
view glass transition as a consequence of frustration thus glass-forming ability [47]. Angell and coworkers
against crystallization. Then, as described above, we indeed found that this is the case for liquids with a mod-
have found that in liquids suffering from weak frustra- ified Stillinger-Weber potential [51].
tion against crystallization, crystal-like orientational or-
der develops in a supercooled state [40, 55, 57, 61]. This Impact of the preorder-crystal similarity on glass-
finding led us to the idea that preordering in the form forming ability. Recently, we have studied the physical
of crystal-like orientational order acts as precursors for origin of enhanced glass-forming ability near the melt-
crystal nucleation [111, 57] (see Fig. 10(b)). ing point minimum, focusing on the degree of preorder-
On the other hand, in the classical nucleation theory ing in a supercooled state [54]. According to the classi-
and density functional theory of crystallization [2, 11], cal nucleation theory [11], the crystal nucleation rate I
a supercooled liquid has been regarded as homogeneous is obtained as
(see Fig. 10(a)), and thus, crystal nuclei have been as- D
sumed to be formed randomly in space. Contrary to I= f exp(−β∆F ∗ ) , (6)
D0
this, we have confirmed in hard spheres [111, 57, 116]
that crystal nuclei are always formed in regions of high where f is a numerical factor (which depends on terms
crystal-like bond orientational order formed in a super- like the Zeldovich factor [11]), D/D0 is the adimen-
cooled state [116, 9, 117]. We emphasize that this struc- sional diffusion constant, ∆F ∗ is the free energy barrier
tural ordering is an intrinsic feature of a supercooled that the system has to overcome in order to crystallize,
liquid state not suffering from strong frustration against and β = 1/kB T . Here we note that the crystallization
crystallization. kinetics is controlled by the diffusion constant D and
12(a) classical homogeneous (b) supercooled liquid with crystal nucleation growth of crystal nuclei
liquid picture bond orientational ordering in precursors
BOO
time t
Figure 10: (a) A homogeneous liquid state before crystal nucleation, assumed in the classical nucleation theory and density functional theory.
(b) A supercooled liquid state of a monodisperse hard-sphere-like liquid with transient high 6-fold bond orientational order (Q6 ) regions (red
particles) [111], which act as precursors for crystal nucleation (left panel). As times go on, crystal nuclei (green particles) are formed exclusively
in red precursor structures (middle panel) and grow with time (right panel). This figure is reproduced from Fig. 2 of Ref. [111].
not by the viscosity η, which are decoupled for T ≤ T m general two-order-parameter Ginzburg-Landau theory,
(see, e.g., Refs. [125, 9, 126]). It should be noted that where the interfacial tension is obtained from the gra-
the Stokes-Einstein relation is violated below the onset dients of the order parameter fields [9]:
temperature T on . The fact that crystal nucleation is con- !2 !2
trolled by D and not by η can also resolve Kauzmann’s
Z
dρ dQ
βγ = + KQ ,
dx K1
(9)
paradox [125, 9]. dx dx
Now we consider the crystal nucleation based on our
two-order-parameter model. The free-energy cost to where ρ and Q are density and structural order param-
form a crystal nucleus, of volume Vn and interface area eters, respectively, x is the coordinate perpendicular to
S n , can be expressed in the adimensional form by scal- the interface, and Kρ and KQ are positive coefficients
ing it with the thermal energy as follows [54]: associated with the spatial gradients of ρ and Q respec-
tively. Since Q is not scalar, the above relation is ap-
β∆F = −Vn (β∆µ) + S n (βγ), (7) proximate but may be enough for a qualitative expla-
where γ is the liquid-crystal interfacial tension, and nation. The above relation shows that not only |∇ρ|
∆µ = µliquid − µsolid is the chemical potential difference but also |∇Q| are essential in determining the interface
between solid and liquid, i.e., the driving force of crys- tension. It also directly shows that the order parame-
tallization. It is important to note that the terms compet- ter gradient, an intrinsic physical quantity characteriz-
ing in the dimensionless free-energy cost are β∆µ and ing the penalty associated with the structural difference
βγ, and not bare ∆µ and γ themselves (see Eq. (7)). between the two phases, is characterized by βγ and not
Thus, the nucleation rate should not be discussed by the by γ.
absolute values of ∆µ and γ but by those scaled by the The nucleation rate I is determined by D/D0 , β∆µ,
available thermal energy. This also allows us to make a and βγ. We have examined which of these three factors
unified description of the glass-forming ability for ther- is dominant in determining I and found that βγ is pre-
mal and athermal (hard-sphere) systems. In this picture, dominant [54]. Furthermore, βγ is largely determined
β∆F ∗ is expressed as by the degree of crystal-like preordering Q in a super-
cooled liquid (compare Figs. 11(a) and (b)). That is, the
β∆F ∗ = c(βγ)d /(β∆µ)d−1 , (8) suppression of crystal-like preorder due to competition
between the two types of crystal orderings is responsi-
where d is the dimensionality and c is a numerical factor ble for a higher nucleation barrier, i.e., a higher glass-
depending on the shape of the nucleus. forming ability near the melting-point minimum. In the
We have shown [54] that the macroscopic definition case of multi-component mixtures such as metallic alloy
of the interfacial tension can be directly linked to the glass-formers, the composition in preorder also strongly
structural difference between the two phases via the affects the liquid-crystal interface tension and thus the
13You can also read
