Structural correlations and cooperative dynamics in supercooled liquidsMurari Singh, Manish Agarwal, Debdas Dhabal, and Charusita Chakravarty Citation: J. Chem. Phys. 137, 024508 (2012); doi: 10.1063/1.4731705 View online: http://dx.doi.org/10.1063/1.4731705 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v137/i2 Published by the American Institute of Physics. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 137, 024508 (2012)

Structural correlations and cooperative dynamics in supercooled liquidsMurari Singh,1 Manish Agarwal,2 Debdas Dhabal,2 and Charusita Chakravarty2,a)

1School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110067, India2Department of Chemistry, Indian Institute of Technology–Delhi, New Delhi 110016, India

(Received 29 February 2012; accepted 14 June 2012; published online 12 July 2012)

The relationships between diffusivity and the excess, pair and residual multiparticle contributionsto the entropy are examined for Lennard-Jones liquids and binary glassformers, in the context ofapproximate inverse power law mappings of simple liquids. In the dense liquid where diffusivitiesare controlled by collisions and cage relaxations, Rosenfeld-type excess entropy scaling of diffu-sivities is found to hold for both crystallizing as well as vitrifying liquids. The crucial differencesbetween the two categories of liquids emerge only when local cooperative effects in the dynamicsresult in significant caging effects in the time-dependent behaviour of the single-particle mean squaredisplacement. In the case of glassformers, onset of such local cooperativity coincides with onset ofdeviations from Rosenfeld-type excess entropy scaling of diffusivities and increasing spatiotemporalheterogeneity. In contrast, for two- and three-dimensional liquids with a propensity to crystallise, theonset of local cooperative dynamics is sufficient to trigger crystallization provided that the liquid issufficiently supercooled that the free energy barrier to nucleation of the solid phase is negligible. Thestate points corresponding to onset of transient caging effects can be associated with typical values,within reasonable bounds, of the excess, pair, and residual multiparticle entropy as a consequenceof the isomorph-invariant character of the excess entropy, diffusivity and related static and dynamiccorrelation functions. © 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4731705]

I. INTRODUCTION

Simple liquids may be defined as those for which thehard-sphere fluid is a reasonable zeroth-order model.1–3 Thekey role of short-range repulsions in controlling the struc-ture and thermodynamics of liquids is well-illustrated bythe success of Weeks-Chandler-Andersen (WCA) perturba-tion approach in computing liquid-state properties.3 Morerecently, the work of Dyre and co-workers shows that a num-ber of useful insights can be obtained by defining simple liq-uids as “strongly correlating” liquids, where configurationalcontributions to the virial and potential energy are linearlyproportional.4–8 Many of the properties of such strongly cor-relating fluids can be mapped onto inverse power-law (IPL)fluids for which pair interactions vary as r−n, where r is theinterparticle distance. Such fluids are characterised by iso-morphs along which the scaling variable ργ /T is constant,where ρ is the density, T is the temperature, and γ is thedensity-scaling exponent that will depend on the specific in-teractions in the system. In appropriate reduced units, a num-ber of properties can be classed as isomorph-invariant, such asthe excess entropy, the static and dynamic correlation func-tions, and the transport coefficients. While the relationshipsbetween alternative definitions of simple liquids may be sub-ject to some argument in special cases, such as the discon-tinuous hard-sphere system,9 the concept of approximate IPLfluids provides a relatively straightforward explanation of thedensity-scaling of relaxation times observed in a large num-ber of glass-forming liquids and polymers, and suggests that

a)Author to whom correspondence should be addressed. Electronic mail:charus@chemistry.iitd.ac.in. Tel.: (+)91-11-2659-1510. Fax: (+)91-11-4716-5550.

many molecular and atomic liquids will conform to this defi-nition of simple liquids.10, 11

Since many liquids can be approximately mapped ontoIPL fluids, we focus on understanding the extent to whichsuch systems will conform to two results for IPL fluids basedon the work of Rosenfeld: first, exponential scaling of trans-port properties with the excess entropy and second, the ex-istence of a characteristic threshold value of the entropy ofthe fluid at freezing.12–15 Both the predictions are based onarguments related to the principle of corresponding states,and therefore should be viewed as semiquantitative predic-tions, rather than very rigid numerical bounds; in particu-lar, one may expect IPL fluids with n ≤ 3, and thereforewith true long-range interactions, to be qualitatively differ-ent from those with n > 3.16 The framework of approximateinverse-power law picture of simple liquids is therefore an in-teresting one within which to examine the relationships be-tween structural correlations, entropy, and diffusivity. We fo-cus specifically on the relationship between the diffusivity (D)and the excess, pair and residual multiparticle contributionsto the entropy, denoted by Se, S2, and �S, respectively, fortwo- and three-dimensional Lennard-Jones liquids and binaryglassformers. These are systems that conform to the defini-tion of simple liquid in terms of both the applicability of ahard-sphere zeroth-order model as well as approximate in-verse power law character.

The excess entropy (Se) is defined as the difference be-tween the total entropy (S) and the ideal gas entropy (Sid) atthe same temperature (T) and density (ρ) and can be expressedin terms of a multiparticle correlation expansion

Se = S2 + S3 + . . . , (1)

0021-9606/2012/137(2)/024508/10/$30.00 © 2012 American Institute of Physics137, 024508-1

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024508-2 Singh et al. J. Chem. Phys. 137, 024508 (2012)

where Sn is the entropy contribution due to n-particle spatialcorrelations and is always negative.17–19 The pair entropy (S2)is always negative and is typically the dominant contributionto the excess entropy. It is defined as

S2/NkB = −2πρ∑α,β

xαxβ

∫ ∞

0{gαβ(r) ln gαβ(r)

− [gαβ(r) − 1]}r2dr, (2)

where gαβ(r) is the atom-atom radial distribution function(RDF) between atoms of type α and β, N is the total num-ber of particles, xα is the mole fraction of component α in themixture, and kB is the Boltzmann constant. The residual mul-tiparticle entropy (RMPE), �S = Se − S2, reflects the effectof three-body and higher order correlations. Se, S2, �S, andD will all be isomorph invariants, i.e., in suitably chosen re-duced units, they will be identical for all state points with thesame value of ργ /T. Therefore, it is expected that for a liq-uid obeying IPL scaling, data from different state points forthe diffusivity plotted against any excess entropy contributionwill fall on a unique curve. Interestingly, based on results forinverse-power law and Yukawa fluids, Rosenfeld predicted aspecific excess entropy scaling relationship for transport prop-erties of the form

X∗ = A exp(αSe), (3)

where X* are dimensionless transport properties with eithermacroscopic (Rosenfeld) or microscopic (Dzugutov) reduc-tion parameters.12, 13, 20, 21 The relationships are quasiuniver-sal, corresponding states type of relations where the scalingparameters, A and α, are identical for systems with iden-tical functional forms for the interparticle interactions. Forsimple liquids with pair-additive potentials, it is found thatA ≈ 0.6 and α ≈ 0.8 when excess entropy is expressed inunits of the Boltzmann constant kB and dimensionless dif-fusivities are obtained by using the macroscopic Rosenfeldreduction factor, i.e., D∗

R = Dρ1/3/(kBT /m)1/2, where m isthe mass of the particle. In keeping with the correspond-ing states character of Rosenfeld-scaling, these values for Aand α hold approximately for simple liquids but do showsome variation even within the family of short-range IPLpotentials.22 Overall, however, we can expect that all short-range IPL fluids, regardless of the system-specific exponentγ , would have very similar scaling parameters for the ex-cess entropy scaling of diffusivities. The exponential relation-ship between excess entropy and transport properties has beenfound to be semiquantitatively obeyed by a much wider vari-ety of dense liquids in the stable and supercritical regime thanwas originally assumed, including liquid metals,21 molecu-lar fluids,23–26 ionic melts,27–29 core-softened fluids,27, 30–33

chain fluids,34 and room temperature ionic liquids.35 This sug-gests that Rosenfeld-scaling may be a more general feature ofatomic and molecular liquids where a single length scale dom-inates both the static structure and the cage relaxation dynam-ics. The extension of such entropy-transport scaling relation-ships to mesoscopic or colloidal fluids is currently an activearea of research.36, 37

The exponential excess entropy scaling of the diffusiv-ity is expected to be valid in the dense liquid regime where

diffusion occurs as a result of binary collisions and cage re-laxations. It can be intuitively justified by noting that thebinary collision contribution can be approximately factoredout by using macroscopic reduction parameters based on ele-mentary kinetic theory, provided that an effective hard-sphereradius can be assigned to the constituent particles. The fre-quency of cage relaxations will then be proportional to thenumber of accessible configurations, exp (αSe), since config-uration space connectivity is high in the stable liquid phase.The state-point independent character of the scaling relation-ship implies that the exponential parameter α controlling thenumber of accessible configurations is determined only bythe interaction potential and will be otherwise state-point in-dependent. This is an assumption that is plausible for sim-ple liquids, though not necessarily so for complex fluids.38, 39

With decreasing temperature and increasing density, one ex-pects cooperative effects to become important for single-particle diffusivity, resulting in deviations from Rosenfeld-scaling. A convenient signature for onset of cooperativedynamics in liquids on supercooling is the “local caging”effect associated with a plateau region in the double loga-rithmic plot of the mean square displacement (MSD) againsttime.40–43 Deviations from Rosenfeld-type excess entropyscaling of diffusivities in a number of liquids with decreas-ing temperature are correlated with onset of local cooperativedynamics though there has been no detailed study of modelglassformers designed to be frustrated in such a way as toprevent crystallization.23, 35, 44–47 The cage formation and co-operative behaviour may be seen as a minimal prerequisitefor a liquid to crystallise; we examine in this study whetherthis condition has any relationship with the freezing pointand associated thermodynamic parameters for a model sim-ple liquid. This also allows us to understand when differencesemerge between entropy-diffusivity relationships in liquidswith a propensity to crystallise as opposed to vitrify.

The entropic quantities (Se and S2) that correlate stronglywith transport properties have also been suggested as the ba-sis for freezing criteria for both three- and two-dimensionalfluids.14, 15, 48, 49 Rosenfeld demonstrated that the excess en-tropy per particle had a characteristic value of −4kB at freez-ing for inverse-power law and Yukawa fluids, where kB is theBoltzmann constant. This is of obvious interest in the con-text of approximate IPL scaling definition of simple liquidssince the melting line is expected to be parallel to neighbour-ing liquid isomorphs.8 The relationship between the RMPEand diffusivity has not been examined before though theRMPE, despite its numerically smaller magnitude comparedto the pair entropy, has been shown to carry significant in-formation on particle ordering in a fluid. Moreover, the zero-RMPE criterion has also been suggested as a signature foronset of an ordering transition, including freezing, in a num-ber of fluids.50–56 We note that freezing criteria based on thepair correlation function have been widely used for colloidalsystems, though the use of the pair entropy criteria is lesscommon.48, 49, 57

To summarise, simple liquids show semiquantitativeagreement in the entropy scaling of transport properties aswell as threshold values for excess entropy and related en-tropic quantities at freezing. These relationships need to be

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024508-3 Singh et al. J. Chem. Phys. 137, 024508 (2012)

placed in the context of approximate inverse power law defi-nitions of isomorphic behaviour of simple liquids. It is alsoexpected that structure-entropy-transport relationships mustchange qualitatively as cooperative dynamics sets in withsignificant differences emerging between liquids that have apropensity to vitrify as opposed to crystallise. Here, we studythe relationships between diffusivity and various structuralcontributions to the excess entropy for well-studied model flu-ids to identify any possible relationships between structuralcorrelations and cooperative behaviour.

Computational studies of the following model liquids arereported here. As examples of model structural glasses, weconsider the three-dimensional binary Lennard-Jones (BLJ)glassformer and two related systems with purely repulsive in-teractions, one constructed using the WCA approximation58

which we refer to as BLJ-WCA and a second one correspond-ing to the IPL approximation, which we refer to as the BLJ-IPL model.6 Though the three models have different equationsof state, the static and dynamic correlation functions areessentially identical in the relatively high-temperature orlow-density regime. In the supercooled or viscous regime,relative to the BLJ system, the BLJ-WCA model shows devi-ations while the BLJ-IPL system reproduces the correlationfunctions remarkably well. As an aside, in view of the Hen-derson theorem stating that the pair interaction required toreproduce a given pair correlation function must be unique,59

the ability of the repulsive IPL potential to reproduce thestatic and dynamic correlations of the BLJ system with a fairdegree of accuracy is quite remarkable. For understandingcooperativity in the neighbourhood of the melting transition,we consider the two- and three-dimensional Lennard-Jonesliquids, referred to as the 2DLJ and 3DLJ fluids, respectively.The 2DLJ is examined in addition to the 3DLJ since thereare known to be qualitative differences in various aspects ofthe solid-liquid phase transition in two and three dimensions;for example, true long-range order is not supported in twodimensions and melting of a two-dimensional crystal isexpected to proceed by a two-stage continuous transition.57

The 2DLJ system also serves as a limiting case of progressiveconfinement of the three-dimensional liquid in a parallel slitpore geometry. While diffusion in the two-dimensional liquidis ill-defined because of the presence of long-time tails in thevelocity autocorrelation function,16, 60 we decided to examinethe behaviour of the mean square displacement and the asso-ciated diffusivity in part because of the analogous behaviourof the local order, mobility, and local pair entropy observed ina recent study.61 We also note that two-dimensional randomwalks on a discrete lattice are well-defined and a continuumlimit would lead to a measure of the diffusivity, implicitlyneglecting large length scale hydrodynamic effects. Thecomputational details for simulations of the different systemsare given in Sec. II. Results are discussed in Sec. III andconclusions are presented in Sec. IV.

II. COMPUTATIONAL DETAILS

A. Model glassformers

The model glassformer that we primarily focus on is theKob-Andersen binary Lennard-Jones mixture with an 80:20

TABLE I. Computational details of molecular dynamics simulations ofBLJ, BLJ-IPL, and BLJ-WCA systems. Each system has been simulatedalong 7 isochores ρσ 3 = 1.0, 1.05, 1.10, 1.125, 1.5, 1.20, and 1.25. In thehigher temperature, from T = 5.0 to 1.0, simulations were performed at tem-perature intervals of 0.25 and, for T < 1.0, simulations were performed atintervals of 0.05. Equilibration and production run lengths are reported inreduced units of time and are denoted by teq and tprod, respectively.

Model ρrange Trange teq tprod

BLJ 1.0–1.15 5.0–0.55 927 37071.20–1.25 5.0–1.5 927 3707

1.25–0.85 4634 37070.80–0.55 9269 3707

BLJ-IPL 1.0–1.25 5.0–0.55 927 3707BLJ-WCA 1.0–1.25 5.0–0.55 927 3707

ratio of particles A and B interacting via Lennard-Jones pairinteractions.41, 62 In units of the Lennard-Jones size and lengthparameters for the A-A interactions, labeled by σ AA and εAA,respectively, the other potential parameters are εAB = 1.5, εBB

= 0.5, σ AB = 0.8, and σ BB = 0.88. The masses of A andB are taken to be equal. We use a quadratic cut-off for theLennard-Jones interactions with cutoff distance correspond-ing to 2.5σ AA.63, 64 The potential used for the BLJ-WCA sys-tem corresponds to that studied in Ref. 58. The potential pa-rameters for the BLJ-IPL model are taken from Ref. 6.

Molecular dynamics simulation of 256 particles in a cu-bic simulation cell, with 204 particles of type A and 52 parti-cles of type B, have been carried out in the canonical (NVT)ensemble, using the DL_POLY software package65 for theBLJ, BLJ-WCA, and BLJ-IPL models. The Verlet leapfrogalgorithm with a time step of 0.00092τ , where τ =

√mσ 2/ε,

was used to integrate the equation of motion. The Berend-sen thermostat, with a time constant 4.63τ , was used to main-tain the desired temperature for the production run. The statepoints simulated, and the associated MD equilibration andproduction run lengths, are given in Table I.

The thermodynamic excess entropy, Se, for all the modelglassformers was evaluated following the procedure given inRef. 66. The entropy was initially evaluated at a high tem-perature (T = 5) and low reduced density of ρ = 0.0005σ 3,where the system can be assumed to behave as an ideal gas.Entropies at any other state points, relative to this referencestate point, can be evaluated using a combination of isochoricand isothermal paths, ensuring that no phase transitions occuralong the path. The entropy change along an isothermal pathis given by

S(T , V ′) − S(T , V )= U (T , V ′) − U (T , V )

T+

∫ V ′

V

P (V )

TdV

(4)and, along an isochoric path, it is given by

S(T ′, V ) − S(T , V ) =∫ T ′

T

1

T

(∂U

∂T

)V

dT . (5)

The thermodynamic properties of the BLJ system availablein the literature were reproduced.62 The pair entropy can beevaluated from simulation data at a given state point andthe RMPE can be obtained as the difference of Se and S2.

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024508-4 Singh et al. J. Chem. Phys. 137, 024508 (2012)

Diffusivities were evaluated using the Einstein relation andthe results available in the literature were reproduced.

B. Three-dimensional Lennard-Jones liquid

Molecular dynamics simulation of 864 particles ina cubic simulation box were carried out in the canonicalensemble (NVT) for the truncated Lennard-Jones potentialusing the DL_POLY software package. The pair interactionswere truncated at a distance of 2.5σ and long-range correc-tions for the potential energy and pressure were applied.67

Equations of motions were integrated using Verlet leapfrogalgorithm with a time step of 0.00092τ , where τ =

√mσ 2/ε.

A Berendsen thermostat, with the time constant 4.63τ , wasused to maintain the desired temperature for the produc-tion run. The 3DLJ system has been simulated along 9isochores, ranging in density from 0.85 to 2.417 in reducedunits, for which melting temperatures were available in theliterature.68, 69 Equilibration run lengths were 926.9τ andproduction run lengths were 3707τ at each state point. Thetotal configurational energy (U) and total pressure (P) for3DLJ agree with those given in Ref. 70. Diffusivities wereevaluated using the Einstein relation. The excess entropy wasevaluated using the thermodynamic integration proceduredescribed for the BLJ system in Sec. II A.

C. Two-dimensional Lennard-Jones liquid

For the 2DLJ liquid, a truncated and shifted Lennard-Jones potential was used with a radial cutoff of 4.0σ ; the

cut-off is large enough to ensure a relatively small effectdue to truncation and shifting and we were able to repro-duce the Monte Carlo results for the system available in theliterature.67, 71, 72 Microcanonical (NVE) molecular dynam-ics simulation of 256 particles in a square simulation boxwere performed using the velocity Verlet algorithm with atime step of 0.002 reduced units. Equilibration run lengthswere 2 × 105 time steps and production run lengths were2 × 106 time steps at each state point. The 2DLJ systemhas been simulated along 8 isochores, ranging in densityfrom 0.788 to 0.95 in reduced units, for which the melt-ing temperature was known from the literature.71, 73 The totalconfigurational energy (U) and total pressure (P) for 2DLJagree with those given in Ref. 71. Diffusivities were eval-uated using the Einstein relation for state points for whichthe g(r) did not display any additional structuring indica-tive of crystallisation. The excess entropy was evaluated us-ing an analogous approach to the thermodynamic integra-tion procedure described for the BLJ system in Sec. II A.The pair and residual multiparticle entropies from the resultsgiven in Ref. 52 were reproduced. A set of NPT Monte Carlosimulations were also carried along the P = 5.40 isobar us-ing a truncated potential for which the melting temperature isknown to be T = 1.0 from the literature.72

III. RESULTS AND DISCUSSION

A. Binary glassformers

In Figure 1(a), we compare the behaviour of theRosenfeld-scaled diffusivities, D∗

R , as a function of the excess

10-3

10-2

10-1

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0

D* R

Se/NkB

Tonset (Se≈-4.3kB)

(a) BLJ

BLJ-WCABLJ-IPL

10-3

10-2

10-1

-0.5 -0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3

D* R

ΔS/NkB

(b)

Tonset

BLJ

BLJ-WCABLJ-IPL

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0

ΔS/N

k B

S2/NkB

Tonset

(c) BLJ

BLJ-WCABLJ-IPL

10-610-510-410-310-210-1100101102

10-4 10-3 10-2 10-1 100 101 102 103 104

⟨Δr2 (t

)⟩*

t*

(d) Tonset

FIG. 1. Three-dimensional binary glassformers: (a) Correlation of reduced diffusivity (D∗R) with excess entropy (Se). Rosenfeld scaling parameters are

A = 0.665, α = 0.865 for BLJ, A = 0.610, α = 0.824 for BLJ-WCA, and A = 0.625, α = 0.831 for BLJ-IPL. (b) Correlation of reduced diffusivities (D∗R) with

residual multiparticle entropy (�S). (c) Correlation of residual multiparticle entropy (�S) with pair correlation entropy (S2). (d) Mean square displacementsversus time at Tonset from different isochores of BLJ, BLJ-IPL, and BLJ-WCA. Black circles represent the state points corresponding to the onset temperatue(Tonset). Note that for each system, data from seven isochores with densities listed in Tables II and III are shown.

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024508-5 Singh et al. J. Chem. Phys. 137, 024508 (2012)

TABLE II. Comparison of Tonset for BLJ, BLJ-WCA, and BLJ-IPL; theonset temperatures were identified to an accuracy of 0.025.

ρσ 3 1.0 1.05 1.10 1.125 1.15 1.20 1.25

BLJ . . . 0.60 0.75 0.85 0.95 1.10 1.40BLJ-WCA . . . . . . . . . 0.60 0.70 0.90 1.15BLJ-IPL . . . 0.60 0.75 0.85 0.95 1.10 1.40

entropy, Se, for the BLJ, BLJ-WCA, and BLJ-IPL fluids. Therelationship between the excess entropy and diffusivity forall three glassformers is virtually identical, as expected forfluids obeying inverse power law scaling.8 The Se range ly-ing above −4kB can be identified as the Rosenfeld-scalingregime where ln(D∗

R) depends linearly on Se. The low-densitylimit for which Enskog theory applies is not studied here. ForSe < −4.3kB, the deviation from linear dependence of theln(D∗

R) versus Se plot becomes apparent. We did not studystate points with Se < −5.5kB due to the requirement of longproduction runs but the deviation from Rosenfeld-scaling be-haviour has been documented elsewhere.46, 47

The crossover from “normal” to “slow” dynamics or theonset of the “landscape-influenced” regime can be associatedwith a temperature Tonset along each isochore correspondingto onset of plateau behaviour in the double logarithmic plot ofthe single-particle mean square displacement (MSD or �2r)against time.43, 74 Table II compares the Tonset temperaturesfor the BLJ, BLJ-WCA, and the BLJ-IPL systems; for a givenisochore, Tonset values are identical within simulation error forthe BLJ and the BLJ-IPL glassformers. Figure 1(d) shows thatlog (�r2) versus log (t) plots at Tonset for the three glassform-ers are indistinguishable in reduced units. Our MSD-basedvalue for Tonset is typically about 10% higher than the temper-ature Trxn for onset of the landscape-influenced regime iden-tified from relaxation times by Sastry for the BLJ system.74

The dominant contribution to Se comes from the pair en-tropy, S2, and the D∗

R versus S2 plot is very similar to that ofSe except that the change in slope at Tonset is less pronounced.Figure 1(b) shows the correlation between D∗

R as a function of�S for the three glassformers. Interestingly, there is a negativecorrelation of diffusivity and the RMPE for lower diffusivitiescorresponding to S2 values below −2.5kB. From Figure 1(c),one can see that at high diffusivities, there is a weak positivecorrelation of the diffusivity with Se, S2, and �S, reflecting thefact that increasing localization due to enhanced pair correla-tions results in reduced RMPE and diffusivity. To understandthis change from positive to negative correlation of the diffu-sivity with the residual entropy (RMPE), Figure 1(c) showsthe correlation between �S and S2 for the three glassformers.At high values of Se or S2, corresponding to high tempera-tures and/or low densities, the RMPE is negative and variesrelatively slowly. Thermodynamic and dynamical propertiesare dominated by pair correlations and there is positive cor-relation between S2, �S, and the diffusivity. As Se decreasesdue to the dominant pair correlation contribution, the RMPEincreases and becomes more positive suggesting that the mul-tiparticle correlations act to enhance ordering and local freevolume, as with crystallisation of hard spheres, and therebyincrease the entropy. The crossover from positive to negative

TABLE III. Comparison of excess entropy (Se), pair correlation entropy(S2) and residual multiparticle entropy (�S) at Tonset for BLJ liquid.

ρσ 3 1.0 1.05 1.10 1.125 1.15 1.20 1.25

Se . . . −4.149 −4.185 −4.175 −4.180 −4.295 −4.245S2 . . . −3.993 −4.026 −4.014 −4.014 −4.159 −4.090�S . . . −0.156 −0.159 −0.161 −0.166 −0.136 −0.155

RMPE (the �S = 0 locus) lies at an S2 = Se = −4.8kB whilethe Tonset values for different isochores are found to corre-spond to a Se = −4.2kB, �S = −0.16kB, and S2 = −4.15.Table III provides a summary of the numerical results for theBLJ liquid; those for BLJ-WCA and BLJ-IPL are very similarin terms of values of entropy contributions at Tonset.

The results presented in this section for the BLJ glass-former and its repulsive analogues provide some new insights.The onset of cooperative “caging” dynamics clearly correlateswith the onset of deviations from Rosenfeld scaling, as sug-gested in earlier studies.47 The isomorph-invariant characterof the diffusivity, as well as the excess, pair, and residual en-tropies implies that this onset temperature will closely corre-late with specific values of the RMPE (�S ≈ −0.2), excessentropy (Se ≈ −4.2kB), and pair entropy (S2 ≈ −4.0kB) (seeTable III for exact values). One may reasonably expect bi-nary fluids that map onto the same IPL model to show similarvalues of the excess entropy, as well as structural estimatorsof the entropy, at the onset of cooperative dynamics. Thus, itshould be possible to devise structure-based predictors, par-ticularly the pair entropy (S2), to locate the onset of coopera-tive dynamics in glassforming fluids. This may prove partic-ularly useful for confocal video microscopy experiments oncolloidal glasses.75, 76

It is useful at this point to make some connections withrecent work on model glassformers. Simulations in such sys-tems provide evidence of increasing spatiotemporal hetero-geneity with supercooling beyond the onset temperature.77–79

They also show a strong correlation between local meansquare displacements, local order, and local entropy inglassformers.61 Our study looks at the ensemble-averagedanalogues of these quantities and our results suggest thatonly below the onset temperature, as “caging” times becomelonger, will the correlation between local mobility and localordering be sufficiently strong as to result in the emergenceof transient regions of relatively ordered, immobile particlesspatially separated from more mobile regions. The onset oflocal caging effects may therefore be seen as a minimal pre-requisite for onset of spatiotemporal heterogeneity.

B. Three-dimensional Lennard-Jones liquid

Figure 2(a) is the Rosenfeld-scaling plot showing the cor-relation of D∗

R with S2 as well as Se while Figure 2(b) showsthe correlation between D∗

R and �S. Below a threshold valueof Se corresponding to −4.5 ± 0.3kB, we could not equilibratethe system and obtain reliable diffusivities. Along any givenisochore, attempts to simulate at temperatures lower than thisthreshold temperature (Tthr) resulted in crystallization with nodiffusional regime detectable in the MSD versus time plots.

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024508-6 Singh et al. J. Chem. Phys. 137, 024508 (2012)

10-2

10-1

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0

D* R

(S2, Se)/NkB

Tfreeze

(a) S2Se

10-2

10-1

-0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

D* R

ΔS/NkB

Tfreeze

(b)

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0

ΔS/N

k B

S2/NkB

Tfreeze

(c) ρ 0.85

0.8810.9981.1441.7332.417

10-610-510-410-310-210-1100101102103

10-4 10-3 10-2 10-1 100 101 102 103 104

⟨Δr2 (t

)⟩*

t*

(d) ρ 0.85

0.8810.9981.1441.7332.417

FIG. 2. Three-dimensional Lennard-Jones liquid: (a) Correlation of reduced diffusivity (D∗R) with excess entropy (Se) and pair correlation entropy (S2).

Rosenfeld scaling parameters are A = 0.70, α = 0.818 with Se and A = 0.389, α = 0.697 with S2. (b) Correlation of reduced diffusivities (D∗R) with residual

multiparticle entropy (�S). (c) Correlation of residual multiparticle entropy (�S) with pair correlation entropy (S2). (d) Mean square displacements versus timeat the threshold temperature (Tthr) along various isochores. Black circles represent the state points corresponding to the freezing temperature (Tfreeze) taken fromRef. 69.

Table IV shows the isochores studied with the correspondingthermodynamic freezing temperatures (Tfreeze) and Tthr. Alsoshown are the corresponding values of Se, S2, and �S. Despitethe wide range in variation of Tfreeze by a factor of 100, the ex-cess entropy (Se) and the pair entropy (S2) show characteristicvalues of −3.8 ± 0.2kB and −3.7 ± 0.2kB, respectively, whichmay be compared with Rosenfeld’s prediction of Se ≈ −4kB

at freezing for inverse power fluids.14, 15 The zero-RMPE cri-terion for freezing is also reasonable since we consistently ob-tain �S ≈ −0.07kB at freezing. The Tthr values are associatedwith characteristic values of Se ≈ −4.6kB and S2 ≈ −5.1kB.

TABLE IV. Freezing temperature (Tfreeze) and the temperature correspond-ing to the supercooling limit (Tthr) for the three-dimensional Lennard-Jones(3DLJ) system. Also shown are the values of excess entropy (Se), pair corre-lation entropy (S2), and residual multiparticle entropy (�S) at the associatedstate points.

Solid-liquid coexistence Liquid stability limit

ρσ 3 Tfreeze Se S2 �S Tthr Se S2 �S

0.850 0.694 −3.692 −3.627 −0.065 0.35 −4.564 −5.106 0.5420.859 0.725 −3.729 −3.648 −0.081 0.35 −4.663 −5.259 0.5960.881 0.809 −3.767 −3.693 −0.074 0.40 −4.718 −5.304 0.5860.913 0.946 −3.814 −3.752 −0.062 0.50 −4.698 −5.238 0.5400.998 1.443 −3.871 −3.811 −0.060 0.80 −4.729 −5.266 0.5371.144 2.739 −3.909 −3.849 −0.060 1.80 −4.533 −4.860 0.3271.339 5.593 −3.923 −3.854 −0.069 3.50 −4.644 −5.055 0.4111.733 17.12 −3.925 −3.854 −0.071 11.0 −4.617 −5.007 0.3902.417 68.49 −3.918 −3.846 −0.072 50.0 −4.408 −4.641 0.233

Rosenfeld-scaling holds for the entire range of temperaturesstudied for the 3DLJ liquid despite the fact that Tthr is 50%to 30% lower than Tfreeze. Note that lower bound on the su-percooling temperature and entropy is consistent with the sta-bility limit of the Lennard-Jones liquid expected from otherstudies48, 80–82 and is not likely to be strongly dependent onsystem size.

This lower bound on Se at Tthr is remarkably close tothe value at which the binary LJ glassformer begins to showdeviations from Rosenfeld-scaling due to the onset of localcooperative dynamics. In Figure 2(d), we show the double-logarithmic plot of �r2(t) at state points with Se = −4.5± 0.3kB, which make it evident that caging effects are

TABLE V. Freezing temperature (Tfreeze) and the temperature correspond-ing to the supercooling limit (Tthr) for the two-dimensional Lennard-Jones(2DLJ) system. Also shown are the values of excess entropy (Se), pair corre-lation entropy (S2), and residual multiparticle entropy (�S) at the associatedstate points.

Solid-liquid coexistence Liquid stability limit

ρσ 2 Tfreeze Se S2 �S Tthr Se S2 �S

0.8000 0.5495 −2.857 −3.343 0.486 0.420 −3.078 −4.932 1.8540.8213 0.6994 −2.851 −3.318 0.467 0.525 −3.112 −5.073 1.9610.8317 0.8355 −2.786 −3.162 0.376 0.602 −3.101 −5.304 2.2030.8477 0.9350 −2.816 −3.283 0.467 0.743 −3.050 −5.117 2.0670.8580 1.0000 −2.864 −3.418 0.554 0.796 −3.104 −5.638 2.5340.9000 . . . . . . . . . . . . 1.243 −3.062 −5.359 2.2970.9500 . . . . . . . . . . . . 1.903 −3.077 −5.515 2.438

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024508-7 Singh et al. J. Chem. Phys. 137, 024508 (2012)

comparable to those of the BLJ fluid at the onset temperature.While caging effects in the MSD are detectable at Tfreeze forthe high-density isochores, they are less pronounced for lowerdensities. Comparison with the BLJ glassformer discussedin Sec. III A suggests that for a structurally frustrated sys-tem, local caging and associated ordering results in onset ofspatiotemporal heterogeneity whereas for the one-componentsystem, this degree of local ordering and reduction in lo-cal mobility seems sufficient to trigger crystallisation. Effec-tively, for a one-component liquid, when supercooling is suf-ficient to remove the free energy barrier to crystallization, thelocal caging effect, which may be viewed as a minimal prereq-uisite for onset of either spatiotemporal heterogeneity or for-mation of a crystalline nucleus is actually sufficient to resultin crystallisation. Thus, in the one-component system with astrong propensity to crystallise, it is not possible to observethe gradual onset of deviations from Rosenfeld-scaling withsupercooling. For completeness, Figure 2(c) shows the corre-lation of �S with S2. Since S2, Se, and �S are all isomorphinvariants, there is essentially a unique curve correlating anytwo of these quantities. On the basis of arguments given inRef. 8, one expects both the freezing line as well as the stabil-ity limit of the supercooled liquid to be isomorphs, implyingthat characteristic values of all three quantities can be associ-ated with Tthr and Tfreeze. However, the threshold temperaturecorresponds to a state point at which there is a pronouncedsignature of local caging effects in the single particle dynam-ics which results in a qualitative change in the kinetics of crys-tallisation. This is clearly demonstrated in Figure 3, whichshows the S2 values as a function of temperature for the LJliquid along several isochores. This rapid crystallization oc-curs at temperatures well below the thermodynamic freezingtemperature and results in a very sharp drop in S2.

Since the pair correlation function is accessible from x-ray or neutron scattering experiments for atomic liquids andfrom confocal video microscopy for colloidal fluids, it is use-ful to compare our results for the S2 value at freezing for 3DLJwith that of other fluids with pair additive interactions, keep-ing in mind Rosenfeld’s criterion of Se ≈ −4kB for inversepower law fluids. Figure 4 shows a correlation plot of the pairentropy at the thermodynamic freezing point with the entropychange associated with the solid-liquid phase transition for

-14.0

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

S2/

Nk B

T

ρ 0.8500.8590.8810.9130.9981.144

FIG. 3. Pair correlation entropy (S2) as a function of temperature (T) alongdifferent isochores for the three-dimensional Lennard-Jones liquid. Black cir-cles along each isochore represent the thermodynamic freezing point takenfrom Ref. 69. Horizontal line corresponds to the S2 = −5.0kB.

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

S2/

Nk B

ΔSfus

(3-D Systems)Lennard-Jones

Soft-SpheresMetalsMorse

FIG. 4. Pair correlation entropy (S2) of liquid at freezing versus the en-tropy of fusion (�Sfus) for three-dimensional Lennard-Jones, soft-spheres,metals, and Morse systems. Data for pair correlation entropy (S2) and en-tropy of fusion (�Sfus) for LJ, soft-spheres and Morse systems are taken fromRefs. 48, 69, 83, and 84. For liquid metals represented by many-body poten-tials (Al, Cu, Pb) data have been taken from Refs. 85 and 87.

Morse, inverse power, LJ, and metallic systems. The Morsepotential has a range parameter, in addition to a size and well-depth parameter, that can be varied to reproduce the thermo-dynamic parameters at melting for a number of atomic liq-uids. We use the data available in the literature on the entropyof fusion (�Sfus) and pair entropy at freezing for range param-eter values appropriate for transition metals and van der Waalsliquids.48, 83 For repulsive soft spheres with 1/rn interaction,we use the available data for n = 10–20.48, 84 For metallic sys-tems, we also use results for a many-body potential.85 De-spite the significant variation in the entropies for the solid-liquid transition, �Sfus, with the range and softness of thepair interactions, the pair entropy of the liquid at solid-liquid coexistence remains remarkably constant (−3.5kB

± 0.3kB), and somewhat above the its value of −5.1kB as-sociated with stability limit of the supercooled liquid. Thisis consistent with the applicability of thermodynamic scal-ing and approximate-IPL behaviour of many liquids andsuggests a useful, experimentally accessible, structure-basedcriterion for freezing. We note that previous studies have dis-cussed that the IPL scaling framework with the melting linecorresponding to an isomorph provides a natural explanationfor the validity of many freezing rules for simple liquids8; ourresults here provide a demonstration for several simple liquidswith otherwise very variable thermodynamic parameters atmelting.

C. Two-dimensional Lennard-Jones liquid

Figure 5(a) shows the relationship between theRosenfeld-scaled diffusivity and the excess entropy (Se)as well as the pair entropy (S2) for the two-dimensionalLennard-Jones liquid. Data are plotted only for those statepoints where no splitting in the second peak of the pairdistribution function is seen, which has been identified as aprecursor to the freezing transition.86 Unlike in the case of thethree-dimensional liquids discussed in Secs. III A and III B,there is a qualitative difference in the scaling behaviour ofln D∗

R with S2 and Se. There is a distinct change in slope ofthe ln D∗

R versus Se curve below the point where Se = S2.Even though ln D∗

R remains linearly proportional to pair

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024508-8 Singh et al. J. Chem. Phys. 137, 024508 (2012)

10-3

10-2

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0

D* R

(S2, Se)/NkB

(a)

Tfreeze

S2Se

10-3

10-2

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

D* R

ΔS/NkB

Tfreeze

(b)

-0.5

0.0

0.5

1.0

1.5

2.0

2.5

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0

ΔS/N

k B

S2/NkB

Tfreeze

(c) ρ 0.80

0.82130.83170.84770.8580

0.900.95

10-610-510-410-310-210-1100101102

10-3 10-2 10-1 100 101 102 103 104

⟨Δr2 (t

)⟩*

t*

(d) ρ 0.80

0.82130.83170.8477

0.8580.900.95

FIG. 5. Two-dimensional Lennard-Jones liquid: (a) Correlation of reduced diffusivity (D∗R) with excess entropy (Se) and pair correlation entropy (S2). Rosenfeld

scaling parameters are A = 0.176, α = 1.17 with Se and A = 0.067, α = 0.809 with S2. (b) Correlation of reduced diffusivity (D∗R) with residual multiparticle

entropy (�S). (c) Correlation of residual multiparticle entropy (�S) with pair correlation entropy (S2). (d) Mean square displacements versus time at thethreshold temperature (Tthr) along various isochores. Black circles represent the state points corresponding to the freezing temperature (Tfreeze) with data takenfrom Refs. 71 and 72.

corelation entropy (S2). Unlike in the case of the binaryLJ glassformer, this change in slope is not accompaniedby a local caging effect in the mean square displacementand pair distribution functions shows no qualitative change.This is consistent with earlier observations that multiparticlecorrelations play a somewhat different role in creating localorder in two, compared to three dimensions; for example, inthe applicability of the zero-RMPE criteria to freezing in twoand three dimensions.52

The ln D∗R versus �S curves are very similar in two and

three dimensions, though the observed variation in �S ismuch greater in the two- than the three-dimensional fluid. Un-like in the 3DLJ fluid, the thermodynamic freezing points lieat �S = 0.5 ± 0.1kB, and do not satisfy the zero-RMPE cri-terion which is consistent with earlier studies on freezing ofhard-disks (see Table V).54 The isomorph invariant nature ofS2 and �S, however, implies that freezing is associated withan S2 value of −3.5 ± 0.2kB. As in the case of the 3DLJ liquid,once the local caging effect in the mean square displacementbecomes significant, the liquid crystallises rapidly in simula-tions, as indicated by a qualitative change in the pair corre-lation function, and reliable diffusivities for the metastable,supercooled fluid cannot be obtained.

The possibility of a pair entropy-based freezing criterionwith S2 ≈ −4.5kB for two-dimensional fluids was recentlysuggested by confocal video-microscopy experiments on N-isopropyl acrylamide (NIPA) colloidal spheres as well as sim-ulations of Yukawa and hard-sphere fluids.49 In Figure 6, weplot the data in Ref. 49 along with our simulation results for

the two-dimensional Lennard-Jones liquid under NPT condi-tions and show that our value of the pair entropy at the liq-uid stability limit is consistent with the S2 value at the on-set of spontaneous crystallisation identified in the previousstudy. As in the three-dimensional case, the existence of sucha freezing criterion can be understood as a consequence of ap-proximate IPL behaviour of the two-dimensional fluids con-sidered here.

-16.0

-14.0

-12.0

-10.0

-8.0

-6.0

-4.0

-2.0

0.0

0.45 0.60 0.75 0.90

S2/

Nk B

ρ

(2-D systems)Hard-disk

YukawaNIPA-FreezingLennard-Jones

FIG. 6. Pair correlation entropy (S2) as a function of area fractionρ = ρnπσ 2/4, where ρn is the number density, for various two-dimensionalsystems, namely, NIPA colloidal spheres, hard-disk, Yukawa and Lennard-Jones liquid. Data for the NIPA colloidal spheres, hard-disk, and Yukawasystems are taken from Ref. 49. For 2D-Lennard-Jones system Monte-CarloNPT simulation has been performed along pressure P = 5.40 by varying tem-perature. Black circles along each curve represent the freezing point. The darkstraight line corresponds to S2 = −4.5kB.

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024508-9 Singh et al. J. Chem. Phys. 137, 024508 (2012)

IV. CONCLUSION

The relationship between diffusivity, excess, pair, andresidual entropy is found to be very similar in three-dimensional Lennard-Jones liquids and glassformers at statepoints preceding the onset of local cooperative dynamics. Theobserved exponential scaling of diffusivities with pair and ex-cess entropy observed in this regime can be explained by com-bining the approximate inverse power law mapping of suchfluids with Rosenfeld’s theoretical predictions, supported bysubsequent simulations, that fluids with inverse power law in-teractions will show very similar entropy scaling behaviour.The superimposition of the results from various state pointsfor the binary Lennard-Jones glassformer and its WCA andIPL approximants on a single curve in the Rosenfeld plotis particularly notable, given the difficulties of mapping theWCA model onto an IPL fluid.9 The Rosenfeld exponentialscaling parameter is very similar (within 5%) for the three-dimensional LJ liquids and glassformers, despite differencesin IPL scaling exponents. It should also be noted that whileexponential scaling of diffusivities is observed in many sim-ple liquids, the dependence of the precise values of the param-eters on the functional form of the interparticle interactionsrequires further investigation.

Deviations from Rosenfeld-scaling are expected with on-set of local cooperative dynamics as signalled by transientcaging or localization effects in the time-dependent meansquare displacement curves. Such caging effects may be re-garded as the minimal prerequisite for onset of spatiotempo-ral heterogeneity in binary glassformers and is associated withwell-defined values of the excess entropy, and its componentssuch as the pair and the residual multiparticle entropy. Despitethe differences in interactions and equations of state betweenthe BLJ, BLJ-WCA, and BLJ-IPL fluids, the scaled diffusiv-ity curves against the excess entropy are essentially identicalfor all three systems with a characteristic value of the entropywith onset of local cooperativity. This can be understood asa consequence of the isomorph-invariant character of the ex-cess entropy, the diffusivity and the static and dynamic struc-tural correlation functions associated with these two quanti-ties. While we cannot at this point make a causal connectionbetween the onset of cooperative effects in determining diffu-sivity, and the overall change in excess entropy scaling, thisthermodynamic marker of cooperative dynamics is itself ofinterest and requires further exploration.

In contrast to the binary glassformers, the one-component, three-dimensional Lennard-Jones liquid is notstructurally frustrated, and the onset of transient caging ef-fects is sufficient to result in crystallisation provided the sta-bility limit for the supercooled liquid has been reached. Thelowest temperature at which the Lennard-Jones liquid is foundto be stable in our simulations is consistent with thermody-namic analysis of supercooling and superheating limits pub-lished elsewhere. It should be possible to explore whether thisvanishing of the free energy barrier to crystallization can berelated to a size for the critical nucleus that is comparable tothe particle dimensions. Both the thermodynamic solid-liquidcoexistence line and the supercooling limit must be paral-lel to liquid state isomorphs, and therefore must be associ-

ated with well-defined values of the pair, excess, and residualmultiparticle entropy for an inverse power law fluid with agiven exponent. Rosenfeld’s prediction based on fundamentalmeasure theory suggests that all inverse power law fluids willhave essentially the same value of excess entropy ≈−4kB atfreezing. The isomorph picture would imply that there mustbe a corresponding characteristic value of the pair and resid-ual multiparticle entropy. Examination of available data fromthe literature on pair entropies of Lennard-Jones, soft-spheres,embedded-atom models for metals, and Morse liquids showthat a pair entropy value of −3.5 ± 0.5kB is associated withsolid-liquid coexistence.

The excess entropy scaling of transport properties hasnot been examined before in two dimensions. We find thedependence of the scaled diffusivity on the pair and excessentropy is somewhat different in two- and three-dimensionalLennard-Jones fluids, with the exponential scaling of diffusiv-ities with pair entropy observed for a significantly wider rangeof state points compared to the scaling with excess entropy.In the two-dimensional Lennard-Jones liquid, as in the three-dimensional case, the caging effect in the time dependence ofthe mean square displacement is consistently observed onlyat the stability limit of the liquid which again corresponds towell-defined values of the excess and pair entropies.

We show that the onset of cooperative local dynamics inboth two- and three-dimensional simple liquids, which maybe viewed a minimal prerequisite for either crystallisation orspatiotemporal heterogeneity, is associated with specific val-ues of the pair, residual, and multiparticle entropy as a conse-quence of the approximate IPL behaviour. In the case of one-component liquids, the freezing line and the stability limit ofthe supercooled liquid may be viewed as isomorphs while inthe glassformers, the state points for onset of local cooperativ-ity would appear play a similar role. While the connection be-tween one-phase freezing rules, the pair correlation functionand IPL behaviour has been discussed before, we show thatthese structural and entropy-related quantities can also serveas markers for onset of slow dynamics in model glassformers.

ACKNOWLEDGMENTS

This work was financially supported by Indian Instituteof Technology–Delhi. M.S. and D.D. are financially sup-ported from University Grants Commision through Seniorand Junior Research Fellowships, respectively. M.A. was fi-nancially supported by Indian Institute of Technology–Delhithrough a Senior Research Fellowship.

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