Effect of particle exchange on the glass transition of binary hard spheres
aa r X i v : . [ c ond - m a t . s o f t ] D ec Effect of particle exchange on the glass transition ofbinary hard spheres
Harukuni Ikeda , Francesco Zamponi Laboratoire de physique th´eorique, D´epartement de physique de l’ENS, ´Ecolenormale sup´erieure, PSL University, Sorbonne Universit´e, CNRS, 75005 Paris, FranceE-mail: [email protected]
Abstract.
We investigate the replica theory of the liquid-glass transition for a binarymixture of large and small additive hard spheres. We consider two different ans¨atzefor this problem: the frozen glass ansatz (FGA) in whichs the exchange of large andsmall particles in a glass state is prohibited, and the exchange glass ansatz (EGA), inwhich it is allowed. We calculate the dynamical and thermodynamical glass transitionpoints with the two ans¨atze. We show that the dynamical transition density of theFGA is lower than that of the EGA, while the thermodynamical transition density ofthe FGA is higher than that of the EGA. We discuss the algorithmic implications ofthese results for the density-dependence of the relaxation time of supercooled liquids.We particularly emphasize the difference between the standard Monte Carlo and swapMonte Carlo algorithms. Furthermore, we discuss the importance of particle exchangefor estimating the configurational entropy. ffect of particle exchange on the glass transition of binary hard spheres
1. Introduction
The relaxation time of supercooled liquids increases dramatically upon decreasingtemperature or increasing density, and eventually exceeds the experimentally accessibletime scale, giving rise to the glass transition [1, 2]. Despite decades of studies, theunderlying mechanisms that cause the glass transition have yet to be fully understood.One of the biggest problems is how to define a proper order parameter for the transition,because a typical configuration of the glass is essentially as random as a standard liquidat slightly higher temperatures. The replica liquid theory (RLT) considers m replicasof the original system in order to circumvent the problem [3, 4]. In the liquid phase,the m replicas move independently, while in the glass phase, the m replicas are confinedaround their center of mass and behave like a molecule [5, 6]. Thus, one can usethe correlation function of the m replicas as a thermodynamic order parameter, whichphysically corresponds to the long-time limit of the time correlation of a single replicain the glass phase.The RLT was first developed for one-component systems [5, 7] and later extendedto binary mixtures [8, 9, 10]. In the latter case, the simplest ansatz corresponds toassuming that all replicas in a molecule are of the same species. From the physicalpoint of view, this assumption is tantamount to prohibit the exchange of particles in aglass state [8, 10], which we hereafter refer to as the frozen glass ansatz (FGA). However,the FGA-RLT displays unphysical behavior in the one-component limit [8]. The entropyof the glass, as predicted by the FGA, remains larger than that of the one-componentsystem by the mixing entropy, s mix = − X µ x µ log x µ , (1)where N µ is the number of particles of the µ -th species, N = P µ N µ , and x µ = N µ /N ,even in the limit where all particles are identical.As discussed by Coluzzi et al. [8], the above discrepancy originates from the Gibbsfactor of the replicated system, i.e. the number of exchanges of molecules that leave theconfiguration of the system invariant. In Fig. 1, we show the schematic configuration ofa replicated system for m = 2, N = 12 and N = 8. For one-component systems,all molecules are identical, and the Gibbs factor is G = ( N + N )!. For binarymixtures, if one assumes the FGA, only the molecules of the same species can beexchanged, leading to the Gibbs factor G = N ! N !, and to an entropy difference∆ S = − ∆ log G = log( N + N )! − log( N ! N !) = N s mix with respect to the one-component system. Note that ∆ S only depends on the particle numbers, and it thusremains constant even in the one-component limit.To resolve this contradiction, one should allow the exchange of replicas of differentspecies (i.e. the dissociation of molecules), see the right panel in Fig. 1. In this case,the particle species in a molecule change with time. After averaging over the time, themolecules are indistinguishable, meaning that the Gibbs factor is G = ( N + N )!, andthe one-component result is recovered. Hereafter, we shall refer to this as the exchange ffect of particle exchange on the glass transition of binary hard spheres Figure 1.
Schematic examples of configurations of the m = 2 replicated system.The number of the first (red) and second (blue) species are N = 12 and N =8, respectively. The purple circles denote particles of either species. (Left) Aconfiguration of the one-component system. The configuration does not change underthe exchange of any pair of particles, and the Gibbs factor is ( N + N )!. (Middle) Aconfiguration of the FGA. The configuration is only invariant under the exchangeof molecules of the same species. The Gibbs factor is then N ! N !. (Right) Aconfiguration of the EGA. The Gibbs factor is the same as that of the one-componentsystem, G = ( N + N )!. glass ansatz (EGA). From the physical point of view, the EGA is tantamount to allowthe exchange of particles in a glass state; the exchange process leads to the dissociationof molecules. Note that the FGA corresponds to the extreme case of the EGA whenthe probability of exchanging particles of different species vanish. This is indeed thecase if, say, the size ratio of different species is sufficiently large. On the contrary, if thesize ratio is close to unity or particles have a continuous size distribution, the EGA isneeded to avoid unphysical behavior.The difference between the FGA and EGA also sheds light into the algorithmicdependence of the relaxation time of supercooled liquids [11, 12] (see [13] for analternative approach). Recently, it has been shown that the swap Monte Carlo (SMC)algorithm, which is nothing but a standard Monte Carlo algorithm (MC) combinedwith particle exchange moves, significantly accelerates the relaxation of supercooledliquids [14, 15]. The FGA and EGA give an effective description, within mean fieldtheory, of the metastable states accessible by MC and SMC [11]. Over a time scalesufficiently shorter than the relaxation time τ α , particles undergo vibrational motionaround their equilibrium position. The exchange of particle species hardly occurs overthis time scale in standard MC, meaning that the FGA gives a good description ofthe corresponding metastable state. On the contrary, the SMC exchanges the particlespecies as well as the particle positions. Particle exchanges thus frequently occur evenat time scales shorter than τ α , meaning that the corresponding metastable state wouldbe well described by the EGA. It is expected that the FGA and EGA give differentglass transition points, which provides an explanation for the difference of relaxationtime between the standard MC and SMC [11, 12]. ffect of particle exchange on the glass transition of binary hard spheres
2. A mean field theory of glass transition
Here, we briefly summarize the random first order transition (RFOT) theory, whichis a semi-phenomenological theory of the glass transition built on the analogy witha class of mean field spin-glasses [18, 19]. The key assumption of RFOT theory isthat the slow dynamics around the glass transition is due to the emergence of complexstructures within the free-energy landscape. The ROFT theory predicts that thedynamics of supercooled liquids changes qualitatively around two important points. Thefirst is the dynamical transition, at packing fraction ϕ d , at which long-lived metastablestates appear in the free energy landscape. Mean field theories of the glass transition,such as the mode-coupling theory (MCT), predict that the relaxation time divergesas τ α ∼ ( ϕ d − ϕ ) − γ [20, 21]. In finite dimensions, however, because of the thermalfluctuations, the system escape from a metastable state after a sufficiently long time.The activation events are driven by the configurational entropy Σ( ϕ ), which is thelogarithm of the number of metastable states [18, 19]. The second important point isthe Kauzmann transition, at packing fraction ϕ K , at which Σ( ϕ ) vanishes. For ϕ > ϕ K ,the system is permanently stuck in a free energy minimum, because it can no longeracquire entropy by visiting several other minima. In this regime, the system is referredto as an ideal glass , and it has a finite rigidity [22]. Below, we calculate the transitionpoints ϕ d and ϕ K , as well as the configurational entropy Σ( ϕ ), of binary hard spheresby using the RLT.
3. Model
We now introduce the model used in this work. We consider a binary mixture of largeand small particles. The total interaction potential is V N = ,N X i 4. Derivation of the RLT equations Here, we derive the RLT equations for binary hard spheres. For this purpose, weextend the quantitative approximation scheme that has been developed in [24] for one-component hard spheres. Our starting point is the replicated partition function [5, 6, 16] Z m = ∞ X N =0 N ! m Y a =1 N Y i =1 X µ ai Z dx ai e − β P ma =1 P ,Ni 1, the FGA gives a pathologicalresult. In this limit, the interaction term converges to the one-component result − ρ R dr [ q ( r ) log q ( r ) − g ( r ) log( r )], while the liquid entropy becomes s liq = s oneliq + s mix .Thus, the difference between Σ of the one-component and two-component systems doesnot vanish, ∆Σ → s mix > 0. This unphysical behavior implies that the FGA must breakdown around r ∼ 1. In the next section, we show that this problem can be cured bytaking into account particle exchange. ffect of particle exchange on the glass transition of binary hard spheres q ( ∆ ) ∆ Figure 2. Dependence of the order parameter for particle exchange, q (∆), on thecoupling ∆. Here we derive the free energy within an ansatz that takes into account the effect ofparticle exchange in a glass state. For this purpose, we first map the particle species toa binary spin variable by introducing σ ( µ ), where σ (Large) = 1 and σ (Small) = − 1. Weassume that σ ( µ ) follows the same distribution as a mean field spin glass model [25, 11], g ( µ ) = C m (∆) − e H P ma =1 σ ( µ a )+ ∆22 P ,mab σ ( µ a ) σ ( µ b ) = C m (∆) − Z Dhe ( h + H ) P ma =1 σ ( µ a ) , (22)where Dh = dh √ π ∆ e − h , (23)and C m (∆) = R Dh (cid:16)P µ e ( h + H ) σ ( µ ) (cid:17) m is the normalization constant. The value of H fixesthe species concentration x µ , which is calculated as x µ = P µ g ( µ ) δ ( µ a , µ ). To simplifythe calculation, hereafter we consider the equimolal binary mixture with x L = x S = 1 / H = 0. The value of ∆ controls the correlation of the particlespecies among different replicas, q (∆) ≡ lim m → m ( m − X a
1) log(2 πA ) + d m − m ) + log C m (∆) − ∆ m + m ( m − q m (∆)) + X µ x µ log x µ . (25)Similarly, Eq. (8) reduces to Q µν ( r ) = − g µν ( r ) (1 − m ) /m x µ x ν Z DuDvC m e uσ ( µ )+ vσ ( ν ) × Z dr ′ γ A ( r + r ′ ) q ( r, u, v ) m − , (26)where Du, Dv are defined in Eq. (23) and q ( r, u, v ) = X µν e uσ ( µ )+ vσ ( ν ) Z dr ′ γ A ( r + r ′ ) g µν ( r ′ ) /m . (27)Substituting Eqs. (25) and (26) into Eq. (11), we obtain the replicated free energy. Onecan derive the self-consistent equations for A and ∆ from the extremization condition ∂ A log Z m = ∂ ∆ log Z m = 0. Because the derivation is straightforward, but the resultis cumbersome, we do not show the explicit expressions here. Using the Monasson’sformula, Eq. (20), we obtain the configurational entropy asΣ = s liq − d πA ) − d − f (∆)2 + ∆ q (∆)) − ρ Z dr ( e − ∆ Z DuDvq ( r, u, v ) [log q ( r, u, v ) − f (∆)] − X µν x µ x ν g µν ( r ) log g µν ( r ) ) , (28)where we have introduced an auxiliary function: f (∆) ≡ lim m → C m (∆) ∂C m (∆) ∂m = e − ∆22 Z Dh h ) log(2 cosh( h )) . (29)In the one-component limit, r → 1, one can show that ∆ → − f (∆) / →− log(2) = − s mix , which exactly cancels out the mixing entropy in s liq . Thus, werecover the configurational entropy of the one-component system obtained in previouswork [24]: Σ → s oneliq − d πA ) − d − ρ Z dr [ q ( r ) log q ( r ) − g ( r ) log g ( r )] . (30) ffect of particle exchange on the glass transition of binary hard spheres Here we summarize how to calculate A and ∆ numerically for a given ϕ and r . Ourtheory requires the liquid entropy per particle s liq and the pair correlation function g µν ( r )as input. Following previous work [9, 24], we use a binary version of the Carnahan-Starling (CS) approximation [26] for s liq , and the Verlet-Weis approximation [27] for g µν ( r ), see Appendix A for details. To solve the self-consistent equations for A and∆, we first set A = A ini and ∆ = ∆ ini for the initial conditions, where we selected A ini = 10 − and ∆ ini = 10 − (and we confirmed that the final results are independentof this choice if A ini and ∆ ini are sufficiently small). Then, we solve the self-consistentequations using a standard iterative method. When ϕ < ϕ d , this iterative process doesnot converge, which indicates that A → ∞ . In practice, we stop the calculation when A exceeds unity during the iteration process. When ϕ > ϕ d , A and ∆ converge tofinite values. In this case, we stop the calculation when | ( A i +1 − A i ) /A i | < − . Bysubstituting ∆ into Eq. (24), one gets the correlation q (∆) of particle species. 5. Results In this section, we present numerical results for the order parameters, A and q (∆), andthe configurational entropy Σ. In Fig. 3, we show the numerical results for A and q (∆) for size ratios r = 1 . r = 1 . 4. For r = 1 . ϕ , A = ∞ and q = 0,meaning that there is no correlation between different replicas, and the system behavesas a standard liquid. A and q begin to have finite values at packing fraction ϕ = ϕ d ,corresponding to the dynamical transition. The dynamical transition point of the FGAis smaller than that of the EGA, ϕ FGA d < ϕ EGA d . As mentioned in the introduction, thismight explain the efficiency of the SMC reported in recent numerical simulations [15],if one identifies ϕ FGA d and ϕ EGA d as the dynamical transition points of the standard MCand SMC, respectively [11]. For the same ϕ , the value of A calculated with the FGAis smaller than that of the EGA, which is also consistent with numerical results [15].Above ϕ EGA d , q increases with ϕ . This is a natural result, because particle exchangeshardly occur at high ϕ . For very high ϕ , the effect of particle exchange is negligible, andthe values of A calculated with the FGA and EGA are similar. The difference between ϕ FGA d and ϕ EGA d increases with increasing r (right panels in Fig. 3). Above ϕ EGA d , theFGA and EGA give very similar values of A , which is consistent with the higher valueof q above ϕ EGA d . We calculate the configurational entropy Σ by substituting the order parameterscalculated in the previous section into Eqs. (21) and (28). Σ is well defined only for ffect of particle exchange on the glass transition of binary hard spheres A ϕ A ϕ q ϕ ϕ FGA d ϕ FGA K ϕ EGA d ϕ EGA K q ϕ Figure 3. Dependence of the order parameters on packing fration ϕ . (Top) Thecage size A . The results of the FGA and EGA are shown with blue and red markers,respectively. The dashed and full vertical lines indicate the dynamical and Kauzmanntransition points, respectively. (Bottom) The correlation of particle species q calculatedwith the EGA. Σ ϕ Σ ϕ Figure 4. Dependence of the configurational entropy Σ on packing fraction ϕ , forsize ratio r = 1 . r = 1 . ϕ > ϕ d , because otherwise the order parameters do not have finite values, indicatingthat there are no metastable states. In Fig. 4, we show the numerical results for smalland large size ratios, r = 1 . r = 1 . 4. For r = 1 . 1, the complexity calculated with ffect of particle exchange on the glass transition of binary hard spheres ϕ ϕ FGA d ϕ FGA K ϕ EGA d ϕ EGA K Figure 5. Phase diagram of equimolar binary hard spheres in three dimensions. Blueand red symbols represent the results of the FGA and EGA, respectively. Circlesrepresent the dynamical transition point ϕ d , while squares represent the Kauzmanntransition ϕ K . The solid lines are guides to the eye. the FGA has a higher value than that calculated with the EGA. This results can benaturally understood as follows. The complexity is the difference between the entropiesof the liquid and the (metastable) glass, Σ = s liquid − s glass . In general, s glass calculatedwith the EGA is larger than with the FGA, since the EGA includes extra degree offreedom (the species) in the glass description. As a consequence, the EGA predicts alower value of Σ. The Kauzmann transition point, ϕ K , is defined by Σ( ϕ K ) = 0. Dueto the higher value of Σ, the FGA predicts a higher value of the Kauzmann transitionwith respect to the EGA, ϕ FGA K > ϕ EGA K . A similar trend is observed for the larger sizeratio r = 1 . 4, see the right panel in Fig. 4. The difference of Σ calculated with theFGA and EGA decreases with increasing r , simply because large and small particle arehardly exchanged. In Fig. 5, we show the phase diagram predicted by our theory. We first discuss thedynamical transition points ϕ FGA d and ϕ EGA d . For r > 1, we always obtain ϕ FGA d < ϕ EGA d .This is consistent with recent results from mode coupling theory (MCT) [12]. However,MCT predicts that ϕ FGA d is almost independent of r , while our theory predicts that ϕ FGA d monotonically increases with r . This discrepancy may come from the approximationmade in Eq. (12), where we assumed that the cage size A does not depend on thespecies. In principle one can avoid this approximation and construct a more accuratetheory, but the calculation gets much harder [11]. We leave this investigation for future ffect of particle exchange on the glass transition of binary hard spheres ϕ FGA K and ϕ EGA K , is qualitativelydifferent from that of ϕ FGA d and ϕ EGA d . We observe that ϕ FGA K > ϕ EGA K for all r , which isa consequence of the behavior of the configurational entropy Σ described in Sec.5.2. Thedifference between ϕ FGA K and ϕ EGA K decreases with increasing r . This is a natural resultbecause large and small particle are hardly exchanged for large r , and in particular athigh density near ϕ K . We would like to stress that the FGA is metastable with respectto the EGA, because the EGA is a more general ansatz within a variational theory. Thismeans that the thermodynamically meaningful transition point is not ϕ FGA K but ϕ EGA K .One should then take into account the degrees of freedom associated to particle exchangewhen calculating the entropy of the glass state, otherwise the Kauzmann transition pointis overestimated. A numerical algorithm for this purpose has been recently proposedin [28]. 6. Summary and discussions In this work, we theoretically investigated a binary hard sphere mixture by using thereplica liquid theory (RLT). For this purpose, we constructed two different ans¨atze: thefrozen glass ansatz (FGA), where the replicas in similar position are constrained to beof the same species, and the exchange glass ansatz (EGA), where the replicas in similarposition can have different species. Using these ans¨atze, we calculated the transitionpoints for different size ratio r . We found that the dynamical transition point calculatedusing the FGA is smaller than that of the EGA, ϕ FGA d < ϕ EGA d . The opposite relationholds for the Kauzmann transition point, ϕ FGA K > ϕ EGA K . In the rest of this section, wediscuss possible implications of our results for experimental and numerical studies of theglass transition.As mentioned in the introduction, our theoretical results might give insight onthe increased efficiency of the swap Monte Carlo algorithm (SMC), as compared tothe standard Monte Carlo algorithm (MC) [15, 29]. For this discussion, it is usefulto introduce two timescales: the density relaxation time, τ α , and the typical timescaleto exchange large and small particles in a metastable glass state, τ ex . Over a timescale sufficiently shorter than the relaxation time, t ≪ τ α , the system is trapped in ametastable state where particles just undergo vibrational motion. In case of MC, largeand small particles are hardly exchanged in this timescale, implying that τ ex ≫ τ α ,thus the effect of the particle exchange is negligible. The metastable state is then welldescribed by the FGA. On the contrary, using the SMC with r close enough to unity,large and small particles are easily exchanged, implying that τ ex ≪ τ α . In this case, theEGA provides a good description of the metastable state. Within these assumptions,the efficiency of the SMC is explained by the larger value of ϕ EGA d with respect to ϕ FGA d .Note that it is known that the SMC works only for binary mixtures of sufficiently smallsize ratio r ≈ . r ≈ . 4, the SMC gives a compatible result with that of the MC [28]. ffect of particle exchange on the glass transition of binary hard spheres ϕ EGA d and ϕ FGA d does not vanish even around r ≈ . 4. The same result is also obtainedby MCT [12]. The inefficiency of SMC at large r should instead be attributed to the factthat the assumption τ ex < τ α does not hold for larger r , even for the SMC, because theexchange moves are never accepted. The EGA thus no longer gives a good descriptionof the metastable state. Unfortunately, because τ ex is a purely dynamical quantity, itcannot be calculated by a static theory such as the RLT. It would be interesting to extenddynamical theories of the glass transition, such as the mode-coupling theory (MCT), tocalculate τ ex and reconcile this discrepancy between theoretical and numerical results. Acknowledgments We thank M. Ozawa and G. Szamel for interesting discussions. This project hasreceived funding from the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation programme (grant agreement n.723955-GlassUniversality). Appendix A. Verlet-Weis approximation for binary mixtures Here we review the binary version of the Verlet-Weis (VW) approximation. We labelhere the two species by indices i, j ∈ { , } . As for a one-component system, we assumethat g ij ( r ) can be written as g VW ij ( r ) = θ ( r − σ ij ) h g PY ( ξr ) + ∆ g ij ( r ) i ,ξ = ϕϕ ∗ ! / ,ϕ ∗ = ϕ − ϕ , ∆ g ij ( r ) = A ij r e − b ij ( r − σ ij ) cos( b ij ( r − σ ij )) , (A.1)where A ij /σ ij = g CS ij ( σ ij ) − g PY ij ( ξσ ij , ϕ ∗ ) , (A.2)where g CS ij and g PY ij denote the results of the Carnahan-Starling approximation (CS) [26]and Percus-Yevick approximation (PY) [30], respectively. We determine b ij from theconsistency of the compressibility equations: ∂βP CS ∂ρ = 1 − X ij x i x j ρ Z d r c VW ij ( r ) = X ij h δ ij − ρ i c VW ij ( k = 0) i x j = X ij [ I + H (0)] − ij x j = X ij x i x j S − ij (0) . (A.3)where [ I + H ( k )] ij = δ ij + ρ i h ij ( k ) ,S ij ( k ) = δ ij x j + ρx i x j h ij ( k ) . (A.4) ffect of particle exchange on the glass transition of binary hard spheres S ij ( k ) denotes the structural factor and h ij ( k ) denotes the Fourier transform ofthe pair correlation function. For a binary mixture, we have S − = 1 D (cid:16) x + ρx h (cid:17) ,S − = S − = − D ρx x h ,S − = 1 D (cid:16) x + ρx h (cid:17) , (A.5)where D = ( x + ρx h )( x + ρx h ) − x x h . (A.6)At large ϕ , the compressibility has a large value, which implies thatlim k → D ( k ) = lim k → ( x + ρx h ( k ))( x + ρx h ( k )) − x x h ( k ) ≈ . (A.7)We shall determine the value of b ij from this condition. First, note that h ij ( k ) can bedecomposed as h ij ( k ) = Z dre ikr [ g ij ( r ) − 1] = h PY ij ( k ) + ∆ h ij ( k ) ,h PY ij ( k ) = Z dre ikr h θ ( r − σ ij /ξ ) g PY ij ( ξr ) − i , ∆ h ij ( k ) = Z dre ikr [ θ ( r − σ ij ) − θ ( r − σ ij /ξ )] g PY ( ξr )+ Z dre ikr θ ( r − σ ij )∆ g ij ( r ) . (A.8)Substituting the above equations into Eq. (A.7), we have D (0) = D PY (0) + O (∆ h ij (0)) . (A.9)Because D (0) and D PY (0) are expected to have small values, O (∆ h ij ) terms should alsovanish. The simplest condition is then∆ h ij (0) = 0 . 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