Statistical Mechanics and Dynamics of a 3-Dimensional Glass-Forming System
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b Statistical Mechanics and Dynamics of a 3-Dimensional Glass-Forming System
Edan Lerner, Itamar Procaccia and Jacques Zylberg Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel (Dated: November 2, 2018)In the context of a classical example of glass-formation in 3-dimensions we exemplify how toconstruct a statistical mechanical theory of the glass transition. At the heart of the approach is asimple criterion for verifying a proper choice of up-scaled quasi-species that allow the constructionof a theory with a finite number of ’states’. Once constructed, the theory identifies a typical scale ξ that increases rapidly with lowering the temperature and which determines the α -relaxation time τ α as τ α ∼ exp( µξ/T ) with µ a typical chemical potential. The theory can predict relaxation timesat temperatures that are inaccessible to numerical simulations. Introduction : Among the best studied models of theglass transitions are those employing point-particles witha soft binary potential. Some repeatedly studied ex-amples are the Kobb-Andersen model [1], the Shintani-Tanaka model [2], the Dzugutov model [3] and variousversions of binary mixtures with purely repulsive poten-tials, see for example [4, 5, 6]. While easy to simulateon the computer, these models are challenging for theo-rists due to the fact that it is extremely hard to evaluatestatistical-mechanical partition-function integrals in con-tinuous coordinates. It is therefore very tempting to finda reasonable up-scaling (coarse-graining) method thatwould define a discrete statistical-mechanics with parti-tion sums rather than integrals, with the sum running ona finite number of quasi-species which have well charac-terized degeneracies and enthalpies. Indeed, in a numberof examples in 2-dimensions it was shown that such adiscrete statistical-mechanics is possible [7, 8, 9, 10, 11]and quite advantageous [12, 13] in providing a successfuldescription of the statistics and the dynamics of systemsundergoing the glass transition. In this Letter we offera general criterion for the selection of up-scaled quasi-species and demonstrate it, for the first time, in the con-text of a 3-dimensional model system undergoing a glasstransition.
Model : We employ here a version of a much studiedmodel consisting of a 50:50 mixture of N point-particlesin 3-dimensions ( N = 4096 in our case), interacting via abinary potential. We refer to half the particles as ’small’and half as ‘large’; they interact via a pairwise potential U ( r ij ): U ( r ij ) = ǫ (cid:20)(cid:16) σ ij r ij (cid:17) α − (cid:16) σ ij r ij (cid:17) β + a (cid:21) , r ij ≤ r c ( i, j )0 , r ij > r c ( i, j )(1)Here, ǫ is the energy scale and σ ij = 1 . , . r c , whichis calculated by solving ∂U/∂r ij | r ij = r c = 0 which trans- t F k ( t ; T ) T = 1 T = 0 . T = 0 . T = 0 . T = 0 . T = 0 . T = 0 . T = 0 . T = 0 . T = 0 . T = 0 . /T τ α FIG. 1: Color online: Time dependence of the correlationfunctions (2) for a range of temperatures (decreasing fromleft to right) as shown in the figure. The inset shows therelaxation time τ α in a log-lin plot vs 1 /T , compared to anArrhenius temperature dependence. lates to r c = ( α/β ) α − β σ ij . The parameter a is cho-sen to guarantee the condition U ( r c ) = 0. Below weuse α = 8 and β = 6, resulting in r c = p / σ ij and a = 0 . F k ( t ; T ) ≡ * N N/ X i =1 exp { i k · [ r i ( t ) − r i (0)] } + . (2)In Fig. 1 we show these correlation functions for k =5 . σ − and for a range of temperatures as indicated inthe figure. We see the usual rapid slowing down thatcan be measured by introducing the typical time scale T h C s i ( T ) Small Particles n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 T h C ℓ i ( T ) Large Particles n = 6 n = 7 n = 8 n = 9 n = 10 n = 11 n = 12 n = 13 n = 14 FIG. 2: Color online: Temperature dependence of the concen-trations of the various quasi-species. Symbols are simulationdata and the lines are a guide to the eye. τ α that is determined by noting the time where F k ( t = τ α ; T ) = F k (0; T ) /e ≡ /e . The relaxation times areshown in the inset of Fig. 1 as a function of 1 /T in a log-lin plot to stress the non-Arrhenius dependence at lowertemperatures. Statistical Mechanics : Our aim is to provide a sta-tistical mechanical theory that captures the structuralchanges upon lowering the temperature such that therewill pop-up a typical scale that can be used to predictthe relaxation time τ α . To this aim we need to up-scale (coarse-grain) from particles to quasi-species thatcan be characterized by their enthalpy and degeneracy.Up-scaling can be done in various ways and there is nounique algorithm to select a-priori a ‘best’ up-scaling.Here we offer a criterion to validate a chosen up-scaling.We choose to work with particles and their nearest neigh-bors, where ‘neighbors’ are defined as all the particles j around a chosen central particle i that are within therange of interaction r c ( i, j ). In the interesting range oftemperatures we find 8 quasi-species with one ‘small’ cen-tral particle and 3 , . . .
10 neighbors, and 9 quasi-specieswith one ‘large’ central particle with 6 , . . .
14 neighbors,all in all 17 quasi-species. Other combinations have neg-ligible concentration ( < . C s ( n ) and C ℓ ( n ) with s and ℓ denoting the small or large centralparticle, while n denotes the number of neighbors. Wemeasured the mole-fractions h C s ( n ) i ( T ) and h C ℓ ( n ) i ( T )and the results are shown in Fig. 2. T F s ( T ) Small Particles n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 T F ℓ ( T ) Large Particles n = 6 n = 7 n = 8 n = 9 n = 10 n = 11 n = 12 n = 13 n = 14 FIG. 3: Color online: The approximate linear dependence ofthe free energies of the chosen quasi-species on the temper-ature. From the slope we read the degeneracy and from theintercept the enthalpies (up to normalization), cf. Eq. 4.Note that when the free energies are large we do not havedata: the concentrations become exponentially small and ina finite simulation box they disappear completely.
To decide whether this up-scaling provides a useful sta-tistical mechanics we now ask whether there exist freeenergies F s ( n ; T ) and F ℓ ( n ; T ) such that h C s ( n ) i ( T ) = e −F s ( n ; T ) /T P n =3 e −F s ( n ; T ) /T , h C ℓ ( n ) i ( T ) = e −F ℓ ( n ; T ) /T P n =6 e −F ℓ ( n ; T ) /T . (3)The free energies are found by inverting Eqs. (3) in termsof the measured concentrations. We then plot thesequantities as a function of the temperature, as demon-strated for the present case in Fig. 3. If F s ( n ; T ) and F ℓ ( n ; T ) can be well approximated as linear in the tem-perature, we can interpret F s ( n ; T ) ≡ H s ( n ) − T ln g s ( n ) , F ℓ ( n ; T ) ≡ H ℓ ( n ) − T ln g ℓ ( n ) , (4)where now the degeneracies g s ( n ) and g ℓ ( n ) (read fromthe slopes in Fig. 3 and enthalpies H s ( n ) and H ℓ ( n ) (readfrom the intercepts) are temperature-independent .This validates the choice of up-scaling. In other words,the approximate linearity of the inverted free energies inthe temperature means that we can write the concentra-tions as h C s ( n ) i ( T ) ≈ g s ( n ) e − H s ( n ) /T P n =3 g s ( n ) e − H s ( n ) /T , h C ℓ ( n ) i ( T ) ≈ g ℓ ( n ) e − H ℓ ( n ) /T P n =6 g ℓ ( n ) e − H ℓ ( n ) /T . (5)Then we can use these forms also as a prediction fortemperatures where the simulation time is too short toobserve the relaxation. The resulting degeneracies g s ( n )and g ℓ ( n ) can be easily modeled theoretically, given basi-cally by a Gaussian distribution around the most proba-ble number n mp of nearest neighbors for small and largeparticles respectively: g s ( n ) ≈ e − [( n − n s mp ) / σ s ] , n s mp = 4 . , σ s = 1 . ,g ℓ ( n ) ≈ e − [( n − n ℓ mp ) / σ ℓ ] , n ℓ mp = 7 . , σ ℓ = 2 . . (6)The comparison of the theoretical to the measured de-generacies is shown in Fig. 4, upper panel. The samefigure shows in the middle panel the enthalpies of thevarious quasi-species. One could model the enthalpiesas a linear function in n . These results are easily in-terpreted; we have high enthalpies when there are largefree volumes (few neighbors). The lowest enthalpies arefound when there are many neighbors and there is nomuch costly free volume. In other words, at the presentdensity and range of temperatures the pV term domi-nates the energy in the enthalpy. Using the theoreticaldegeneracies and the measured enthalpies we computethe concentrations of all our quasi-species and comparethem with the measurement in the lowest panel of Fig.4. The agreement that we have, especially consideringthe number of quasi-species and the simplicity of thetheory, is very satisfactory. Notice that the competitionbetween degeneracy and enthalpy explains the rather in-tricate temperature-dependence of the concentrations ofthe quasi-species, sometimes declining when the temper-ature drops, sometime rising, and sometime having non-monotonic behavior. Prediction of Relaxation Times : Finally, wewant to connect the structural theory to the dynami-cal slowing-down. To this aim we note that there are anumber of quasi-species whose concentration goes downexponentially (or maybe faster) when the temperaturedecreases, and that the relaxation time shoots up at thesame temperature range. We refer to these quasi-speciesas the ‘liquid’ ones; in this example the liquid concentra-tions are those with small particles with three and fourneighbors, and large particles with six, seven and eightneighbors. We sum up these concentrations and denotethe sum as h C liq i ( T ). The dependence of h C liq i ( T ) onthe temperature is shown in Fig. 5 Small Particles n g s ( n ) datamodel 6 8 10 12 14123 x 10 Large Particles n g ℓ ( n ) datamodel4 6 8 106810 n H s ( n ) n H ℓ ( n ) T h C s i ( T ) T h C ℓ i ( T ) FIG. 4: Color online: Upper panel: The degeneracies g s ( n )and g ℓ ( n ) read from the slopes of Fig. 3 (in circles) and thedegeneracies according to the gaussian model Eq. (4). Mid-dle panel: the measured enthalpies. Lower panel: compari-son of the measured concentrations of quasi-species to thosecalculated from Eqs. (5) using the model degeneracies andmeasured enthalpies. Here symbols are data and lines aretheoretical predictions. T h C li q i ( T ) n s = 3 n s = 4 n ℓ = 6 n ℓ = 7 n ℓ = 8sum FIG. 5: Color online: The temperature dependence of C liq ( T )is shown as the upper continuous line. The contributions ofthe various ‘liquid’ sub-species are shown with symbols whichare identified in the inset. −1 ξ ( T ) /T τ α ( T ) FIG. 6: Color online: The relaxation time τ α ( T ) in terms ofthe typical scale ξ ( T ). We show the excellent fit to Eq. (8)with µ = 0 .
04. Note that the intercept at T → ∞ is of theorder of unity as it must be. This concentration is used to define a typical scale ξ ( T ), ξ ( T ) ≡ [ ρC liq ( T )] − / ; (7)where ρ is the number density. This length scale hasthe physical interpretation of the average distance be-tween the ‘liquid’ quasi-species. It was argued before[9, 10, 14] that this length scale can be also interpreted asthe linear size of relaxation events which include O ( ξ ( T ))quasi-species. We can therefore estimate the growing freeenergy per relaxation event as ∆ G = µξ ( T ) where µ isthe typical chemical potential per involved quasi-species.This estimate, in turn, determines the relaxation time as τ α ( T ) = e µξ ( T ) /T . (8)The quality of this prediction can be gleaned from Fig. 6,where we can see that the fit is excellent, with µ ≈ . T → ∞ .A few points should be stressed. As we expect (cf. Ref[14]), in systems with point particles and soft potential,there is no reason to fit the relaxation time to a Vogel-Fulcher form [15] which predicts a singularity at finitetemperature. In our approach we predict that ξ → ∞ only when T →
0, and there is nothing singular on theway, only slower and slower relaxation. At some point thesimulation time will be too short for the system to relax,but we can use Eq. (8) to predict what should be thesimulation time to allow the system to reach equilibrium.
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