Recasting a model atomistic glassformer as a system of icosahedra

1.2 MB·

by  Rhiannon Pinney

Recasting a model atomistic glassformer as a system of icosahedra
Rhiannon Pinney
HH Wills Physics Laboratory, Tyndall Avenue, Bristol, BS8 1TL, UK and
Bristol Centre for Complexity Science, University of Bristol, Bristol, BS8 1TS, UK
Tanniemola B. Liverpool
School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK
C. Patrick Royall
6 HH Wills Physics Laboratory, Tyndall Avenue, Bristol, BS8 1TL, UK
1 School of Chemistry, University of Bristol, Cantock Close, Bristol, BS8 1TS, UK and
0 Centre for Nanoscience and Quantum Information, Tyndall Avenue, Bristol, BS8 1FD, UK∗
2
Weconsider a binary Lennard-Jonesglassformer whose super-Arrheniusdynamics arecorrelated
n
with the formation of icosahedral structures. Upon cooling these icosahedra organize into meso-
a
J clusters. We recast this glassformer as an effective system of icosahedra which we describe with a
population dynamics model. This model we parameterize with data from the temperature regime
1
accessible to molecular dynamics simulations. We then use the model to determine the population
1
of icosahedra in mesoclusters at arbitrary temperature. Using simulation data to incorporate dy-

namics into the model we predict relaxation behavior at temperatures inaccessible to conventional
]
t approaches. Ourmodelpredictssuper-Arrheniusdynamicswhoserelaxationtimeremainsfinitefor
f
o non-zerotemperature.
s
.
t
a I. INTRODUCTION transition. Clearly, any discussion of the nature of the
m
glass transition itself requires extensive extrapolation of
- Among the challenges of the glass transition is to un- data. Significantly,thelimitofthisaccessible4-5decades
d
derstandhowsolidityemergeswithlittleapparentchange corresponds roughly to the Mode-Coupling temperature
n
o in structure. Indeed, whether the glass transition has a TMCT which in d = 3 leads to a crossover to a regime
c thermodynamic (implying structural) or dynamical ori- where relaxation is believed to occur in a qualitatively
[ gin remains unclear [1, 2]. It has been proposed that differentfashionthroughcooperativebehavior[2,26,27].

upon cooling, icosahedral arrangements of atoms might Recentlyithasbecomepossibletoaccesscertainprop-
1
v forminsupercooledliquids[3]andthatdynamicalarrest ertiesofthisdeeplysupercooledregime. Oneapproachis
4 may be related to a (geometrically frustrated) transition to vapor deposit onto a substrate cooled below the tem-
6 to a phase of such icosahedra [4]. It is now possible to peratureatwhichthesystemcanusuallybe equilibrated
4 identify geometric motifs such as icosahedra and related [28, 29]. Alternatively, by immobilizing or “pinning” a
2 locally favored structures (LFS) using computer simula- subset of particles a transition to an “ideal glass” can
0
tion[5–15]andparticle-resolvedstudiesincolloidexperi- be induced that is accessible to simulation [30, 31] and
.
1 ments[16–19]. Furtherevidenceofincreasingnumbersof experiments with colloids [32]. Other methods include
0 LFS upon cooling is also found in metallic glassformers the observation of a transition in distributions of over-
6
[20, 21]. laps in configuration space [33, 34] and so-called large
1
The discovery of dynamic heterogeneity, i.e. locally deviations where trajectory sampling of moderately su-
:
v fastandslowregions[22,23]hasspurredattemptstocor- percooled liquids indicates a dynamical transition to a
Xi relate LFSsuch asicosahedrawith the dynamicallyslow state rich in LFS with very slow dynamics [35, 36]. It is
regions. This hasmetwithsomesuccess[7,8,11–14,24] alsopossibletodecomposethe systemintoarangeofge-
r
a however such correlation does not by itself demonstrate ometricmotifs. Suchanapproachindicatedthatthereis
a mechanismfor arrest[24,25] andin anycase is depen- no thermodynamic transition [9, 10]. However obtaining
dent on the model system under consideration [14]. A dynamical information in the deeply supercooled regime
key limitation here is that direct detection of LFS and beyondthataccessibletosimulationremainsachallenge.
dynamic heterogeneity is only possible in the first 4-5 Here,weconsiderabinaryLennnard-Jonesglassformer
decadesofdynamicslowingaccessibletoparticle-resolved whosedynamicsarestronglycorrelatedwithLFS,which
experiments on colloids [16] and computer simulation. areicosahedra[8,13,14]. Inparticular,theoccurrenceof
This compares to 14 decades of slowing corresponding super-Arrhenius dynamics coincides with the emergence
to the glass transition in molecular systems and diver- of a population of icosahedra (Figs. 1 and 2) [8, 13].
gence of relaxation time at a putative thermodynamic We build on this observation and use the population of
icosahedra and its dynamics to predict the behavior in
the regime beyond that accessible to computer simula-
tion. To do this, we introduce a stochastic model for
∗ [email protected] the population dynamics of icosahedra which we param-
2
a
1010
105
100
0.5 1 1.5 2
FIG. 2: (Color online) “Angell” plot of relaxation time as a
b
function of inversetemperature. Circles; simulation data.
Dashed red line; Arrhenius. Dashed green line; population
dynamics model prediction. Solid navyline; as described by
Geometric Frustration [4]. Thinner, solid pale blueline;
MCT fit to thedata across theregion 0.58≤T ≤1. Thin,
solid red line; VFT fit to data T ≤1. Inset: proportion of
particles identified as being part of an icosahedra as a
function of temperature. Circles; simulation data. Solid red
line; fitted model. Simulations are limited toT &0.56 (blue
shaded region indicates inaccessibility). Black dashed lines
corresponds to TVFT =0.456 and TMCT =0.57.
ulation data, however it indicates there is no divergence
of the relaxation time at finite temperature.
Our model captures the super-Arrhenius behavior of
FIG. 1: (Color online) Snapshots showing thesimulated
the system with reasonable accuracy. The largest dis-
system. Large, colored particles indicate those identified as
crepancy between the predictions of the model and the
being part of icosahedra. Small, gray particles are all other
simulation data occurs during the first few decades of
particles. Sizes are not to scale. (a) At high temperatures
(T =1.0), we see very few, small mesoclusters and (b) at super-Arrhenius dynamics around 0.7& T &0.6. These
low temperatures (T =0.58) a network of very large first few decades of arrest are well-described by Mode-
mesoclusters forms (red particles). These mesoclusters Coupling theory. MCT takes as its input two-point cor-
percolate at around T =0.6. relations. These are entirely neglected in our analysis
which focuses on higher order correlations, and we at-
tribute the discrepancy predominantly to our neglect of
MCT.
This paper is divided as follows: we discuss the sim-
eterize in the regime accessible to simulation. We then
ulation details in section II, and describe our model in
use the model to obtain the population of icosahedrafor
sectionIII. Results are shownin sectionIV, and we con-
all temperatures. In the simulation accessible regime we
clude with a summary and discussion in section V.
show that the population and lifetime of the domains of
icosahedrawhichwetermmesoclusters iscorrelatedwith
thesuper-Arrheniusdynamics. Usingthecalculatedpop-
II. SIMULATION DETAILS
ulation of icosahedra we predict the dynamical behavior
ofthe systematalltemperatures. Inparticularwemake
the following assumptions: the dynamical behaviour of We simulate the Wahnstro¨m equimolar additive bi-
the system is democratically represented by particles in nary Lennard-Jones model [37]. The size ratio is 5/6
differentsizedmesoclustersoficosahedraandthosenotin and the well depth between all species is identical.
icosahedra. The dynamics of particles not in icosahedra The mass of the large particles is twice that of the
we assume to be Arrhenius. The dynamics of particles small. We use molecular dynamics simulations of N =
in mesoclusters we determine from parameterisation of 1372,10976,87808 particles. We equilibrate for at least
mesocluster lifetimes using simulation. The model pre- 100τ in the NVT ensemble before sampling inthe NVE
α
dictsasuper-Arrheniusdynamicscomparabletothesim- ensemble. Hereτ isthestructuralrelaxationtimewhich
α
3
is determined from a stretched exponential fit to the in- 101
termediate scattering function. We identify icosahedra a Data
withthetopologicalclusterclassification(TCC)andcon- ∝ n0.52
siderthosewhichlastlongerthan0.1τ (thedistributions
α
of whichcanbe seen inFig. 3(a)) to suppressthe effects
ofthermalfluctuations. Oursimulationandanalysispro-
R
tocol are detailed in [13, 38].
The Wanstro¨m mixture has been shown to crystallise
toaFrank-Kasperphase[39]. Indeedsomesimulationsof
N =1372particles crystallisedat temperaturesT ≤0.6;
100
clearly evidenced by a substantial increase in the popu-
lation of icosahedra at fixed temperature. These simula-

tions were discarded. No crystallisation was observed in 102 103
n
the larger systems.
We plot the structural relaxation time τα with an Ar- b 100
2.5
rhenius form for T & 1 and a Vogel-Fulcher-Tamman
2.0
(VFT) form for lower temperatures in Fig. 2. The VFT 10−1 1.5
formreadsτ =τ exp[D/(T−T )]wherethefragilitypa- 1.0
α 0 0
IrpnarmeFdeiitgcet.rs2Dawd=eiv0ear.l7sgo9e9nincaedniodcfattτheαetathteemaMpteeomrdaept-ueCrraeotuauptrleiwnThg0ict=hemo0pu.4er5rfia6t-. ≥P(l t)1100−−32 0000....9876
ture T =0.57, fitted across the region 0.58≤T ≤1. 0.59
MCT
0.58
10−4
III. POPULATION DYNAMICS MODEL
10−5
10−1 100 101 102 103 104
InFig. 1,weseedomainsoficosahedrawhichweterm t / τArr
α
mesoclusters. We define the size of a mesocluster, m,
by the number of icosahedra that comprise it; i.e. how FIG. 3: (Color online) (a) The radius of gyration for
many particles found at the center of anicosahedronare different size mesoclusters (here, size is determined by
numberof particles). Data (red crosses) fitted with
contained in the mesocluster; so here we only concern
ourselves with m≥1 (m= 0 refers to particles that are R2∝n1/d, where d is thefractal dimension, and thefitted
value 1/d=0.52; d=1.92. (b) Distribution of icosahedra
not in an icosahedron).
lifetimes for a range of temperatures, scaled by the
We make the assumption that any change in meso- Arrheniustimescale, τArr.
cluster size is a change in m of ±1. This assumption α
means that even large mesoclusters (which in practise
might break in two) can only decrease by incrementally.
Thusmesoclusterscission,orindeedcoalescence,isnota
The steady state solution for the master equation (so-
featureofourmodel. Theratesatwhichthemesocluster
lution when p˙ =0 for all m) is:
m
size increases or decreases are given by g and r respec-
tively. Themasterequationforthe populationdynamics
model reads: pm = gmr−1pm−1 = g1r·····g·mr−1p1 (2)
m 2 m
p˙1 =g0p0+r2p2−[g1+r1]p1 If gm,rm are assumed constant across all m (for a given
T),g =g andr =r,wecandenotetheratioasa“de-
p˙m =gm−1pm−1+rm+1pm+1−[gm+rm]pm cay pmarameter” am= g/r. This results in p = am−1p ,
p˙M =gM−1pM−1−rMpM (1) so all p can be determined from just twomparameter1s.
m
where p is the probability that a particle is in a meso- We impose M p = 1 and M p = φ(T)
m m=0 m m=1 m
cluster of size m. We set a limit at m=M which is the where φ(T) is the expected proportion of particles to be
P P
largestpossiblemesoclusterthatcanbeformedgiventhe in icosahedra at that temperature as shown in Fig. 2
relative populationof particles in icosahedra,φ. Consid- inset [and Fig. 5(b)]. In the high temperature Arrhe-
ering geometric frustration we expect φ < 1 and set a nius regime T ≥ 1, and at slightly lower temperatures
maximum value for the relative population φ =0.75. (T ≥ 0.7), the mesocluster size distribution is well de-
max
This in turn constrains the largestnumber of icosahedra scribed by a decaying exponential p = am−1p . The
m 1
in a mesocluster, M. While the limit of the parameter φ parametrisationis discussed in section IIIB.
ischosen,ratherthandetermined,resultsobtainedinthe Upon cooling, at around T ≈0.6 the number of icosa-
range 0.6 ≤ φ ≤ 0.9 exhibit only slight quantitative hedra is sufficiently large that the mesoclusters form a
max
differences and have no impact on our conclusions. percolating network (see Fig. 3; the change in slope in-
4
dicates percolationat mesoclusters with &500 particles, a 0.04
whichcorrespondstom&70). Nowthispercolationdoes
not correspond to arrest, because the icosahedra have a T)
(10.02
limited lifetime [13]. Indeed, the Angell plot in Fig. 2 k
shows no significant feature when the LFS begin to per-
colate. Howeverpercolationleads to a peakin the meso- 0
1 1.5 2
clustersizedistribution,whichnecessitatessomeexplicit
b 20
considerations for the population dynamics model. We
introduce a Gaussian-like weighting function, Wm(T) to T)
accountfor the peak thatformsin the distributionwhen (010
k
percolation occurs. W (T) is a system-size dependent
m
parameter that controls the location and width of the 0
distributionpeakconstrainedsuchthatthelargestmeso- 1 1.5 2
1/T
cluster does not exceed M. The steady state solution in
the percolated regime is then pm =a(T)Wm(T)pm−1. FIG. 4: The parameters (a) k1 and (b)k0 that describethe
To describe the dynamics, given the population of slope and intercept (respectively) of themesocluster
icosahedra,we proceedas follows. From the mesocluster lifetimes in theT-dependentregime.
size distributions, we candetermine the super-Arrhenius
contribution to τ :
α
linearexpressions;aT-dependentexpressionforsmallm
τ =τArr l (T)p (T) (3) and a T-independent expression for large m.
α α m m
m
X
Here τArr is the relaxation time assuming Arrhenius be- 10k1(T)m+k0(T) for m≤m∗
l = (4)
haviour, extrapolated from the high-temperature T > 1 m (10h1m+h0 for m>m∗
value. Each icosahedron is categorised according to the
∗
Here, m is the point at which the two expressions are
largest mesocluster it joins during its lifetime. The life-
equal, i.e. the (feasible) solution to (k (T)m+k (T))−
time of an icosahedron is determined by the amount of 1 0
(h m+h )=0. (h andh arelistedintableI.andk (T)
(simulation) time that has elapsed between the first and 1 0 0 1 0
andk (T)aredescribedinthefollowingsection)Athigh
last instance of an icosahedron being identified by the 1
temperatures, the mesocluster lifetimes are dominated
TCC. The mesoclusterlifetimes, l , arethe averagelife-
m
by the temperature-dependent expression corresponding
times of the icosahedra in the corresponding size cate-
to m ≤ m∗. At low temperatures, the mesoclusters are
gory.
able to grow large enough to pass this “threshold” into
Theexpressionaboveisbasedonthefollowingassump-
a regime where the mesocluster lifetimes are no longer
tions: (1) the dynamics of each particle in the system
relatedtothetemperatureandarefunctionsofsizeonly.
is represented democratically and (2) particles not in
icosahedra have Arrhenius dynamics. We therefore at-
tribute all super-Arrhenius behavior to the emergence
A. Mesocluster lifetime parameters
of the icosahedra. As noted above this is motivated by
thecorrelationbetweenicosahedraandfragilityinmodel
[8, 15] and metallic glassformers [40]. Given the popu- Throughout the model descriptions, we utilise a
lations of the mesoclusters as a function of temperature smoothing function with the following form:
and extrapolating the dynamical trends we see, we pre-
dict τ at temperatures far beyond those accessible to
α B (j)=0.5 1+tanh[Y(j−X)] (5)
simulation (see Fig. 2). i
Since the emergence of the population of icosahedra (cid:16) (cid:17)
whereiisafunctionindicator,j isthevariableandX,Y
is associated with super-Arrhenius dynamics, we con-
are fitted values. All functions and values are listed in
sider the dynamical behavior of the system by compari-
table III.
sonwith anArrhenius relaxationtime τArr which we as-
α The temperature-dependent regime of the lifetime
sumethe systemwouldhavewerenoicosahedrato form.
model has two parameters described as follows:
Fig. 3(b)showsthelifetimedistributionofallicosahedra
whose lifetimes are ≥ 0.1τ across a range of tempera-
α
tures (0.58≤T ≤2.5)scaledby the Arrhenius timescale k (T)=d T−3+d T−2+d T−1+d (6)
0 3 2 1 0
τArr. The lifetime distributions collapse onto each other
α
at temperatures T &1 but spread out at T .1.
The mesoclusterlifetimes aremodeledasafunctionof 0.039B (1) for k (T)<h
steizmep(enruamtubreers.oWf peafirttictihpeatminegsoiccolussatheerdlriafe)tifmoresawraitnhgetwoof k1(T)=(0 k1 T for k00(T)≥h00 (7)
5
The coefficients for di in the expression for k0 have dif- 0.25
ferent values for the two ranges T > 0.7 and T ≤ 0.7. a 1372
10976
These are listed in table II. 0.2
87808
1372
N h0 h1 10976
0.15
1372 0.0232 0.3733 T) 87808
10976 0.0033 0.8423 φ(
87808 0.00047 1.2 0.1
TABLE I: Fitted parameter valuesfor lifetime model (Eq.
0.05
4).
0
1 1.2 1.4 1.6 1.8 2
1/T
Region d0 d1 d2 d3 b 0.02
k0(T),T >0.7 -1.0256 0.5033 0.0733 0 1372
k0(T),T ≤0.7 -202.77 398.73 -262.4 57.8 10976
87808
0.015
TABLE II: Fitted parameter valuesfor lifetime model (Eq. 1372
6). 10976
T) 87808
(1 0.01
p
0.005
B. Mesocluster population dynamics parameters
0
Our model for φ is shown in Fig. 5(a), and described 0 0.5 1 1.5 2 2.5
1/T
as follows (coefficients d listed in table IV):
i
FIG. 5: (Color online) Data (points) and model
descriptions (lines) of (a) thepopulation of particles in
φ(T)=B 1 exp d T−2+d T−1+d icosahedra, φ and (b) p1. Note that these models are very
φ T 2 1 0 similar regardless of system size.
(cid:18) (cid:19) (cid:18) (cid:19)
1
+φ 1−B (8)
max φ
T
" (cid:18) (cid:19)#
where B (1/T) controls the transition to a plateau. Us-
ingsimulφationdataonthe numberofparticlesthatcom- a(T)=Ba(T)exp d2T−2+d1T−1+d0
priseatypicalmesoclusterof(arbitrary)sizem,weinfer (cid:18) (cid:19)
a linear relation between the proportion of particles in +0.6 1−B (T) (11)
a
icosahedra, φ, and the size of mesocluster that could be
(cid:16) (cid:17)
formedifalltheparticlesinicosahedraweretoaggregate where B (T) is a smoothing function (Eq. 5).
a
into a single mesocluster. We use this linear relation as The maximum in a(T) occurs at percolation. Beyond
our model for M: this (T . 0.6), the system is dominated by the large
percolatingmesocluster (largem) and small-mesocluster
(small m) effects become increasingly negligible. In this
1
M(φ)= Nφ−7.1 (9) simulationinaccessibleregime,we assigna fixedvalueto
7.66 a(T) for simplicity.
(cid:16) (cid:17)
The function W (T) has a Gaussian-like form:
where N is the total number of particles in the system m
(1372, 10976,87808).
Our model for p1 is shown in Fig. 5(b) and described (m−µ)2
below (coefficients di listed in table IV): Wm(T)=1+BW(m)Gexp − 2σ2 (12)
!
where µ and σ are temperature-dependent parameters
p1(T)=exp d3T−3+d2T−2+d1T−1+d0 (10) that control the location and width of the peak in the
(cid:16) (cid:17) distribution,andGischosenforeachT inordertosatisfy
M
The decay parameter a is plotted in Fig. 6(a) and de- p = φ(T). The parameters µ and σ only apply
m=1 m
scribed as follows (coefficients d listed in table IV): to T <0.7, and are only defined in this range.
i
P
6
a 10−1 Bi(j) N X Y
0.9 B (1/T) 1372 1.95 -30
φ
0.9 10976 1.95 -30
10−2 0.8 87808 1.9 -30
0.8
B (T) 1372 0.56 50
10−3 0.7 a 10976 0.54 50
0.7
m 87808 0.55 50
p 0.65
10−4 0.65 Bµ(φ) 1372 0.41 -15
0.62 10976 0.42 -12
0.62 87808 0.43 -10
10−5 B (φ) 1372 0.14 5.3
σ
10976 0.06 20
10−6 87808 0.07 17
0 20 40 60 80 100 B (m) All 10 0.1
m W
B (T) All 1.67 -17
g1
b 1 c 104
TABLE III: Fitted values for smoothing function (Eq. 5)
103 parameters relating to population model components.
φ)
a(T)0.5 µ(102
101 Fctn. N d0 d1 d2 d3
0.2 0.4 0.6
0 d 104 φ(T) 110397726 --1144..790262 1101..504735 --11..699368
1 2
1/T 87808 -14.697 10.482 -1.627
n = 1372 σφ()102 p1(T) 1372 -12.551 1.588 9.737 -4.673
10976 -12.932 3.053 8.079 -4.109
n = 10976
n = 87808 100 87808 -13.366 4.678 6.330 -3.568
0.2 0.4 0.6 a(T) 1372 -10.718 13.104 -4.091
φ
10976 -10.642 12.914 -3.991
87808 -10.750 13.182 -4.129
FIG. 6: (Color online) (a) High temperaturedata fitted
µ(φ) 1372 18.546 -165.35 794.51
with an exponential decay as described in Eq. 2. Model
10976 16.780 -99.817 3439
description of (b) thedecay parameter, (c) thepeak location
87808 -173.14 2537 1013.9
(“mean” mesocluster size) parameter, µ and (d) peak width
σ(φ) 1372 4.5 0.18
(“standard deviation”) parameter, σ for different system
10976 5.3 0.17
sizes.
87808 7 0.1
TABLE IV: Fitted parameter values for all population
model components.
The parameters µ and σ are given by (coefficients d
i
listed in table IV):
behavior which is accompanied by the emergence of a
µ(φ)=B (φ) d φ2+d φ+d
µ 2 1 0 population of icosahedra found in mesoclusters which
(cid:16) 0.75N −7.1 (cid:17) grow upon supercooling [7, 8, 13]. These two dynami-
+ 1−Bµ(φ) (13) calregimesareindicatedintheAngellplotinFig. 2and
7.66
(cid:18) (cid:19)(cid:16) (cid:17) the population of icosahedra is shown in the inset.
σ(φ)=exp d B (φ)+d (14) Motivated by these observations, we propose that the
0 σ 1
(cid:18) (cid:19) correlation between icosahedra and super-Arrhenius dy-
namics continues to lower temperature. This gives us a
means to predict the relaxation behavior of the system
with smoothing functions Bµ(φ) and Bσ(φ), and N the at arbitrary temperature under the assumptions made
total number of particles in the system. in constructing the model (section III). The dynamical
behavior we predict from measurements of cluster life-
times. The global dynamical behavior is shown in the
IV. RESULTS Angell plot in Fig. 2 and discussed in section V. Be-
low we consider some further dynamical features of the
Our approachis basedonthe observationthatat high model.
temperaturethereisArrheniusbehaviorinthedynamics In Fig. 7(a) we compare the results of the population
and very few icosahedra, but at the onset temperature dynamics model (pm = a(T)Wm(T)pm−1) for the size
(T ≈ 1 [13]), there is a crossover to super-Arrhenius distributionofmesoclusterswithsimulationdata. Inthe
on
7
0
0
a 100 0.65 10−2 a 103 103 0.58
10−1 00..6652 10−3 00..5587 102 0.57
0.56 0.65
10−2 000...6662 1100−−54 00..5565 102 101 0.55 00..6652
pm10−3 00..5599 100 101 102 103 lm 10100 0 101 102 103 00..662
0.58 101 0.6
0.59
0.58
10−4 0.59
0.58
0.58
10−150 0 101 102 103 10100 0 101 102 103
m m
b 10−2 b
1372
1372
10976
87808 102 10976
87808
10−3
m
p m m*
l
10−4
101
10−150 0 101 102 103 104 10 0 101 102 103
m m
FIG. 7: (Color online) (a) Mesocluster probability FIG. 8: (Color online) (a) Mesocluster lifetimes.
distributions. Simulation data (points) fitted with the Simulation data (points) fitted with thelifetime model
probability model (lines) as descibed in equation 1. Inset: (lines) as described in equation 4. Small mesoclusters (lower
thepredicted distributions for some temperatures values of m) have highly T-dependentlifetimes, but at some
inaccessible to simulations. (b) The probability distribution large enough mesocluster size, the lifetimes become
for different system sizes at a fixed temperature (T =0.58). T-independentand are determined only bythe mesocluster
The decay parameter is almost unchanged across system size. Inset: the predicted mesocluster lifetimes for some
sizes, but µ,σ and consequently p have strong system size temperatures inaccessible to simulations. (b) The resulting
m
dependence. However, note that the distribution shapes are, lifetime model for a fixedtemperature, T =0.58, for
approximately,compressed/stretched versions of each other. different system sizes. The sudden increase in slope in (b)
occurs at thepoint m∗, where the T-dependentdescription
crosses to theT-independentcurve.
inset, we show the predicted mesocluster distributions
for temperatures inaccessible to simulations. The popu-
lationdistributionmodelresultsfordifferentsystemsizes between the functions for large and small m in Eq. 4
atafixedtemperature(T =0.58)areshowninFig. 7(b). described in section III.
Largersystemsallowforlargermesoclusters,resultingin Themesoclusterlifetimesdifferwithsystemsizewhich
the location of the peak having system-size dependence. we explainas follows. Letus assume thatinthe thermo-
However,the systemsstillpercolateatthe sametemper- dynamic limit, eachmesocluster size hasa fixedlifetime.
ature (T ≈0.6). We imagine that in a simulation box, percolating meso-
In the simulation accessible regime we can directly clusters of a given size correspond to a distribution of
measure the dynamical properties of the mesoclusters. larger mesoclusters in the thermodynamic limit, rather
These we show in Fig. 8(a) which plots the mean life- than being a fixed size. So in the thermodynamic limit
timeoficosahedraasafunctionofmesoclustersize. This these larger mesoclusters would have a distribution of
increases strongly with m, while retaining some temper- lifetimes, but here we assign a single lifetime for each
ature dependence. The mesocluster lifetime model [Eqs. sampled size. Crucially, this effect varies with system
4]for different systemsizes (for fixed T =0.58)is shown size.
in Fig. 8(b). This shows the behaviour of the model In Fig. 9 we show the structural relaxation time our
predictions. Note thecusp-likefeatureshowninthis log- modelpredictsfordifferentsystemsizesacrossthewhole
∗
log representation which is due to the meeting point m rangeofT (Eq. 3). Despitethestrongsystemsizedepen-
8
models in which a phase transition is encountered would
104
τArr leadtodynamicaldivergence[41]. Itisalsopossiblethat
α
furtherrefinementmayfittherelaxationtimedatabetter
VFT
103 1372 than our current approach[Fig. 2], which might provide
10976 further insight into the question as to whether there is a
87808 thermodynamicglasstransition. Atthisstageweobserve
τα 102 thatourmodelwhichpredictsnothermodynamictransi-
tion actually over-estimates the super-Arrhenius nature
ofthedynamics. Thiswouldlendsupporttotheobserva-
101
tionthat within this frameworkthere shouldbe no tran-
sitionasalsofoundinother treatmentsofLFS [4,9,10].
100 WealsoplotinFig. 2predicationsfromgeometricfrus-
1 1.2 1.4 1.6 1.8 2 2.2
1/T tration [4]. Here τα(T) = τ∞exp(∆E∗(T)+E∞/kBT)
where E∞ is the Arrhenius contribution. Below the on-
FIG. 9: (Color online) The α relaxation time as predicted
set temperature T the super-Arrhenius contribution
by thepopulation dynamics model across thethree different on
ψ
system sizes. The system size dependencein p and l ∆E = 0, for T < T , ∆E(T) = Bk T 1− T
m m on B c Ton
produce only very minor differences themodel τ .
α where B =650, ψ =8/3 and Tc =0.65. We s(cid:16)ee that t(cid:17)he
results, also predicated on icosahedra, seem to describe
thedynamicalbehaviorinasimilarwaytoourmodel. It
is possible that certain aspects of geometric frustration
dence in both the mesocluster population distributions
are captured by our approach.
and lifetimes, the resulting relaxation times are scarcely
One explanation for the rather strong increase in τ
effected by the system size. α
exhibited in Fig. 2 might be that our model doesn’t
include mesocluster scission or coalescence, because the
mesocluster size increases/decreasesonly by one. Larger
V. SUMMARY AND DISCUSSION changes in mesocluster size might lead to closer agree-
ment with simulation data, but would not change the
We havepresentedanapproachtopredictthe dynam- picture in a qualitative fashion. System sizes for simula-
ics of a glass forming liquid at arbitrary temperatures. tions which representcertain properties of deeply super-
This we have done by decomposing a model glassformer cooledsystemsaresmall[33,35,36],butitis tantalizing
intoaneffectivesystemofmesoclustersof locallyfavored toconsiderparameterizingthemodelwithsuchdata. Al-
structureswhosepopulationisdescribedbyapopulation ternatively one can consider how the network geometry
dynamicsmodelwhichweparameterizewithresultsfrom mightbeinfluencedbycertainscalingpropertiesnearan
simulations. Thelifetime ofmesoclustersofLFSarealso assumed transition [42].
parameterized with simulations Under the assumptions Ourmodelunderestimatestheinitialincreaseinstruc-
aboveandthatthe super-Arrheniusdynamicscanbe at- turalrelaxationtime inthe dynamicalregimewhereitis
tributed to the population of particles in mesoclusters well described by MCT (Fig. 2). It is tempting to sug-
of icosahedra, our model predicts dynamical behaviour gestthatthis is relatedto ouremphasis onicosahedrain
at arbitrary temperature. In its present form the model describing the dynamical slowdown. In the temperature
predictsthatthereisnothermodynamicglasstransition. regimeinquestion,LFS(icosahedra)arerelativelyfewin
We have considered different system sizes and find that number and there is no percolating network. Moreover
while the population and lifetime components of the it is possible that in this regime, dynamical slowdown
model are strongly system size dependent, the resulting may be dominated by lower-order correlations such as
relaxation time is scarcely dependent on system size. the 2-body, 3-body etc. These arguments are supported
Whether or not there is a thermodynamic transition, by Banerjee et al. [43] and Nandi et al. [44] who have
in the sense of a divergence in relaxation timescales at suggestedthattwo-pointbasedrelaxationtimes strongly
finite temperature, boils down to whether the lifetime of increase before higher-order contributions. In Fig. 2 we
icosahedra in mesoclusters lm, diverges. In Fig. 8(a) we showtheMCTfitwhichdescribesthesimulationdataac-
see that it does not. Although our data are compatible curately for 0.7 & T & 0.6 but diverges at T = 0.57
MCT
with dynamical divergence of lm, — that is to say lm [45]. One imagines that better agreement might be ob-
can be fitted in the regime accessible to simulation such tained by including contributions from MCT in this dy-
that it diverges at finite temperature — better fits are namical range, noting that these can be systematically
obtained with non-divergent behaviour. extended at least to 4th order [46]. At deeper super-
Now since the population dynamics model itself does coolings where MCT diverges, our population dynamics
not exhibit a phase transition, perhaps one could argue model presumably captures other dominant relaxation
thatitisnaturalthatwedonotfindadivergenceinrelax- pathways absent from MCT [27], at least for the Wahn-
ationtime. Onemightimaginethatpopulationdynamics str¨ommodel. Thusitisinthedeeplysupercooledregime
9
beyondMCTwhose dynamicsremaininaccessibleto the terials are well known to exhibit correlations between
particle-resolved techniques of computer simulation and non-Arrheniusdynamicsandtheemergenceoficosahedra
colloid experiment where our approachhas most to con- [21]. These models also exhibit networks of icosahedra,
tribute. Other possibilities to explainthe discrepancy at whoseemergenceseemssimilarlyrelatedtothecrossover
weaksupercooling(T >T )include amorphousorder to super-Arrhenius dynamics as the Wahnstro¨m model
MCT
distinct from icosahedra. The correlation of the dynam- we consider [48, 49].
ics with the icosahedrais high in the Wahnstro¨mmodel,
butitisnotperfect[14]. Considerationsfromotherwork
[24, 47] suggest that other contributions from the struc-
Acknowledgements
ture may also contribute to the dynamics [20].
Before closing, we comment on the generality of our
results. Recently, [14] a number of models have been The authors would like to thank Andrea Cavagna,
compared. Of those, the Wahnstro¨m model considered Daniele Coslovich, Jens Eggers, Bob Evans, Rob Jack,
hereexhibitsthestrongestcorrelationsbetweenstructure Gilles Tarjus and Francesco Turci for many helpful dis-
anddynamics,soonemightexpectthis tobemostlikely cussions. CPR would like to acknowledge the Royal So-
to exhibit a structure-based transition at finite temper- ciety for financial support and the European Research
◦
ature. That our analysis hints towards no such tran- Council under the FP7 / ERC Grant agreement n
sition therefore suggests the same should hold at least 617266 “NANOPRS”. RP is funded by EPSRC grant
for the range of models considered [14]. Our approach code EP/E501214/1. This work was carried out using
may also be used to optimise metallic glassformers such the computational facilities of the Advanced Computing
as CuxZr1−x. Like the Wahnstro¨m model, these ma- Research Centre, University of Bristol.
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