De-coupling of Exchange and Persistence Times in Atomistic Models of Glass Formers
Lester O. Hedges, Lutz Maibaum, David Chandler, Juan P. Garrahan
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec De-coupling of Exchange and Persistence Times in Atomistic Models of Glass Formers
Lester O. Hedges, Lutz Maibaum, David Chandler, and Juan P. Garrahan School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, UK Department of Chemistry, University of California, Berkeley, CA 94720-1460
With molecular dynamics simulations of a fluid mixture of classical particles interacting withpair-wise additive Weeks-Chandler-Andersen potentials, we consider the time series of particle dis-placements and thereby determine distributions for local persistence times and local exchange times.These basic characterizations of glassy dynamics are studied over a range of super-cooled conditionsand shown to have behaviors, most notably de-coupling, similar to those found in kinetically con-strained lattice models of structural glasses. Implications are noted.
Facilitated dynamics, as encoded in various kineticallyconstrained lattice models (KCMs),[1, 2] implies a gen-eral picture of structural glasses in which never-endingexcitation lines coalesce, branch and percolate through-out space-time.[3, 4] Particles cannot move except whenintersected by these excitation lines.[5, 6, 7] As such,principal signatures of glassy dynamics can be viewedas manifestations of the distribution of times for which aparticle must wait to first encounter an excitation, andthe distribution of times between subsequent encounters.These are what we have called the distributions of “per-sistence” times and “exchange” times, respectively.[5, 6,7] Definitions and behaviors of excitation lines, persis-tence and exchange are most obvious in idealized latticemodels. Nevertheless, these concepts have proved usefulfor interpreting intermittent and heterogeneous dynamicsand transport de-coupling[8, 9] in continuous force modelsystems and real systems,[5, 7, 10, 11] suggesting thatthese concepts are not limited to KCMs. Here, we showthat indeed, exchange and persistence are well defined interms of classical trajectories of the sort computed frommolecular dynamics or observed with microscopy,[12] andfor a specific continuous force model, the distributions forexchange and persistence times behave similarly to thosewe have previously gleaned from KCMs.[6, 13]Figure 1 illustrates our main result. It shows the dis-tributions of persistence and exchange times for particledisplacement events obtained from molecular dynamicssimulations of an atomistic model. A local dynamicalevent or excitation is defined as a displacement beyonda specified cutoff length (see below). As the liquid be-comes increasingly super-cooled the two distributions de-couple, and the typical persistence time becomes muchlarger than the typical exchange time. This de-couplingoccurs because excitations or mobility and their asso-ciated excitation lines become relatively sparse at lowtemperatures. As such, for low enough temperatures,the time extent of space-time regions devoid of excita-tion lines (regions that dominate persistence processes)is typically very long compared to that for regions bridg-ing the width of excitation lines (regions that dominateexchange processes). Thus, the origin of this de-couplingis the same as that of dynamic heterogeneity.[3, 4, 5] This de-coupling is reflected, for example, in observedbreakdowns [8] in mean-field transport relations like theStokes-Einstein inverse proportionality between diffusionconstant, D , and structural relaxation time, τ α .[14] Inparticular, when de-coupling occurs, diffusion is muchfaster than would be predicted from D ∝ /τ α becausediffusion, being an exchange process, has 1 /D propor-tional to the first moment of the exchange-time distri-bution while τ α is the first moment of persistence-timedistribution.[5] FIG. 1: De-coupling of exchange and persistence times in theWCA mixture. A local event is defined as a particle movinga distance larger than d . We show results for d = 0 .
5. (Top)Distributions of exchange times and of persistence times forparticle species A and B for various temperatures T . (Bot-tom) Ratio of the average persistence time, τ p ≡ h t p i , to theaverage exchange time, τ x ≡ h t x i , as a function of T . Theinset shows τ p and τ x for both species as a function of 1 /T . The model [17] we study is a variation of the binaryLennard–Jones mixture of Ref. 18, which has been ex-tensively studied as a model super-cooled liquid (see e.g.Ref. 19). We modify this system by removing the attrac-tive part of the Lennard–Jones interaction by adoptingthe Weeks-Chandler-Andersen (WCA) separation of thepair-potential.[20] We consider a mixture of two particlespecies A and B in a cubic simulation box of side length L and volume V = L . The potential energy is the sum ofthe pairwise interactions between two particles of species µ, ν ∈ { A, B } , V µν ( r ) = 4 ε µν (cid:20)(cid:16) σ µν r (cid:17) − (cid:16) σ µν r (cid:17) + 14 (cid:21) (1)if their separation r is less than 2 / σ µν , and V µν ( r ) = 0otherwise. Following Ref. 18 we choose σ AA = 1 , σ BB =5 / , σ AB = ( σ AA + σ BB ) / ε AA = ε BB = ε AB = 1.The particle masses are m A = 2 and m B = 1. Lengths,times and temperatures are reported in units of σ AA ,( m B σ AA /ε AA ) / and ε AA /k B , respectively. The numberof particles of species µ is N µ , and the corresponding molefraction is N µ /N = N µ / ( N A + N B ).Due to the shortness of the interaction range, 2 / σ ≈ . σ , each particle interacts with only its nearest neigh-bors, significantly reducing the computational overheadof calculating forces. We find that this feature makes theWCA mixture up to one order of magnitude faster to sim-ulate than the corresponding Lennard–Jones mixture.[17]It helps us study a reasonably large system at signifi-cantly super-cooled conditions.Figure 2 demonstrates that the WCA mixture hasthe standard phenomenology associated with glass for-mation. The figure presents results from two-pointtime-correlation functions for varying temperatures rang-ing from the normal liquid regime to the super-cooledregime.[21] Figure 2(top) and Fig. 2(center) show themean-squared displacement h| ∆ r ( t ) | i and the self-intermediate scattering function F s ( k, t ) ≡ h e i k · ∆ r ( t ) i for k at the peak of the static structure factor k = k , respec-tively, as a function of time. The curves display the char-acteristic low temperature features of increasing slowingdown, plateaus and stretching observed in simulationsof similar systems.[18, 23] Static density correlations arestationary over the range of temperatures considered,which suggests nothing of thermodynamic significance ishappening throughout the simulation. The density cor-relation functions decay to their uncorrelated values atlengths much smaller than the simulation box length, andits structure factor shows no anomalies at small wavevectors. These facts indicate that the system remains aliquid mixture throughout the simulation and does notevolve towards a crystallized or phase-separated state.Dynamical quantities show typical liquid-like behaviorwhen measured over long enough time scales. For in-stance, the small wave vector incoherent scattering func-tion is fully consistent with diffusive relaxation. This factalso supports the conclusion that the system is not evolv-ing towards a crystallized or phase-separated state.[17] FIG. 2: (Top) Mean squared displacement h (∆ r ) i = h| ∆ r ( t ) | i for both particle species A and B as a functionof time t for various temperatures T . (Center) Incoherentscattering function F s ( k, t ) = h e i k · ∆ r ( t ) i , evaluated at thewavevector k = k of the first peak in the partial struc-ture factors, for both species at the same temperatures asabove. (Bottom) Violation of the Stokes-Einstein relation(dashed line) for decreasing temperatures. The inset showsthe corresponding relaxation times τ α , defined by the rela-tion F s ( k , τ α ) = 0 .
1, for each species as a function of inversetemperature 1 /T . Figure 2(bottom) shows the breakdown of the Stokes-Einstein relation at low temperatures. At the lowest tem-perature we have studied, the de-coupling between thediffusion constant and the structural relaxation time isabout an order of magnitude. This again is similar towhat is seen in similar systems.[8] A detailed study ofdynamic heterogeneity in the WCA mixture is given inRef. 17 and is left to a future publication.To obtain the distributions shown in Fig. 1, we monitorthe time series of events for each particle in the system,defining an event as a particle is displaced beyond a cutofflength d . Consider particle i . At the initial time t = 0,when we start the observation, its position is r i (0). Thefirst event time for that particle, t , is the first time thatparticle i has moved far enough so that | r i ( t ) − r i (0) | = d . A second event occurs for that particle at time t = t + t , when particle i manages to move again a distance d , this time from its position at t ; i.e., | r i ( t + t ) − r i ( t ) | = d . A third event takes place after a further wait t , and so on.This convention establishes a set of waiting times be-tween events for the i -th particle, { t , t , t , . . . } . No-tice, however, that t has an important physical differ-ence from t , t , . . . . The time t is the time for the firstevent to take place without condition on when the previ-ous event occurred. The times t , t , . . . are times between events. As noted, we call the time t a persistence time,and the times t , t , . . . exchange times.[24]Figure 1(top) shows the distributions of exchange andpersistence times for the two kinds of particles in theWCA mixture, for temperatures ranging from the normalliquid regime to the super-cooled one. We choose thecutoff length d to be comparable to the particle size, so asto probe particle motion relevant to structural relaxationand diffusion. As argued in previous work,[3, 4] dynamicfacilitation features emerge only after a suitable coarse-graining of short scale motion. The length d plays therole of a coarse-graining length. The distributions of Fig.1 are for d = 0 .
5. We have obtained similar results for d in the range d = 0 . . τ p /τ x , where τ p ≡ h t p i is the average persistence time and τ x ≡ h t x i isthe average exchange time. At low temperatures the av-erage persistence time becomes much larger than the ex-change time. This de-coupling mirrors that of the break-down of the Stokes-Einstein relation of Fig. 2.The de-coupling of exchange and persistence timesin the WCA mixture is similar to that observed inKCMs.[6, 13] These distributions reflecting the time se-ries of individual particle displacement events are non-Poissonian. Event times cluster with periods of quies-cence followed by periods of high activity. This clusteringof event times is one way to view dynamic heterogeneity,or more precisely, a coexistence in space-time betweenactive and inactive dynamical phases.[27] Phase separa-tion in trajectory space is a distinguishing prediction ofdynamic facilitation.[27] De-coupling of persistence andexchange processes is a consequence of this phase sepa-ration.By describing dynamics in terms of exchange and per-sistence, one is able to picture particle motion as acontinuous-time random walk.[28] An individual parti-cle makes random walk steps at random times drawnfrom the persistence and exchange time distributions.[5]Along with accounting for the breakdown of Stokes- Einstein relations,[5] this simple picture also accounts forthe non-Fickian to Fickian crossover[7] and the shape[29]of the van Hove correlation function in a wide range ofsystems.[11]In carrying out this work JPG was supported by EP-SRC grant GR/S54074/01, LM was supported by DOEgrant under Contract No. DE-AC02-05CH11231, andDC was initially supported by NSF grant No. CHE-0543158 and currently supported by ONR grant N00014-07-1-0689. Calculations were made possible through ac-cess to the NERSC super-computer facility at LawrenceBerkeley National Laboratory. [1] G. H. Fredrickson and H. C. Andersen, Phys. Rev. Lett. , 1244 (1984).[2] F. Ritort and P. Sollich, Adv. Phys. , 219 (2003).[3] J.P. Garrahan and D. Chandler, Phys. Rev. Lett. ,035704 (2002).[4] J.P. Garrahan and D. Chandler, Proc. Natl. Acad. Sci.USA , 9710 (2003).[5] Y. Jung, J.P. Garrahan and D. Chandler, Phys. Rev. E , 061205 (2004).[6] Y. Jung, J.P. Garrahan and D. Chandler, J. Chem. Phys. , 084509 (2005).[7] L. Berthier, D. Chandler and J.P. Garrahan, Europhys.Lett. 69, 320 (2005).[8] See e.g., I. Chang and H. Sillescu, J. Phys. Chem. B101, 8794 (1997); S.F. Swallen, P.A. Bonvallet, R.J.McMahon, and M.D. Ediger, Phys. Rev. Lett. 90, 015901(2003); L. Berthier, Phys. Rev. E , 020201 (2004); S.K.Kumar, G. Szamel, J.F. Douglas, J. Chem. Phys. ,214501 (2006).[9] For reviews on dynamic heterogeneity see: H. Sillescu, J.Non-Cryst. Solids , 81 (1999); M.D. Ediger, Annu.Rev. Phys. Chem. , 99 (2000); S.C. Glotzer, J. Non-Cryst. Solids, , 342 (2000); R. Richert, J. Phys. Con-dens. Matter , R703 (2002); H. C. Andersen, Proc.Natl. Acad. Sci. U. S. A. , 6686 (2005).[10] D. Chandler, J.P. Garrahan, R.L. Jack, L. Maibaum andA.C. Pan, Phys. Rev. E , 051501 (2006).[11] P. Chaudhuri, L. Berthier and W. Kob, Phys. Rev. Lett. , 060604 (2007).[12] See for example, E.R. Weeks, J.C. Crocker, A.C. Levitt,A. Schofield and D.A. Weitz, Science , 627 (2000);V. Prasad, D. Semwogerere and E.R. Weeks, J. Phys.Condens. Matter , 113102 (2007); R. Besseling, E.R.Weeks, A.B. Schofield, W.C.K. Poon, Phys. Rev. Lett. , 028301 (2007); Y. Gao and M.L. Kilfoil, Phys. Rev.Lett. , 078301 (2007).[13] A.C. Pan, J.P. Garrahan and D. Chandler, Phys. Rev.E. , 041106 (2005); A.C. Pan, J. Chem. Phys. ,164501 (2005).[14] The decoupling of the different relevant relaxation ratesis also evident from the behaviour of four-point dynamicsusceptibilities at different lengthscales, see for exampleRefs. 10, 15, 16.[15] S.C. Glotzer, V.N. Novikov and T.B. Schroeder, J. Chem.Phys. , 509 (2000). [16] L. Berthier, G. Biroli, J.P. Bouchaud, W. Kob, K.Miyazaki and D. Reichman, J. Chem. Phys. , 184504(2007).[17] L. Maibaum, PhD thesis, University of California Berke-ley (2005).[18] G. Wahnstrom, Phys. Rev. A , 3752 (1991).[19] See e.g., N. Lacevic, F.W. Starr, T.B. Schrøder and S.C.Glotzer, J. Chem. Phys. , 7372 (2003); N. Lacevicand S.C. Glotzer, J. Chem. Phys. , 4415 (2004).[20] J.D. Weeks, D. Chandler, H.C. Andersen, J. Chem. Phys. , 5237 (1971).[21] Molecular dynamics simulations were performed us-ing a microcanonical Verlet algorithm for Fig. 1 andLAMMPS[22] for Fig. 2, at temperatures T = 5 . . N A = N B = 500 and L = 9 .
172 for Fig. 1,and N A = N B = 4000 and L = 18 .
344 for Fig. 2, werefixed in each run. The system was prepared at each ther-modynamic state point by coupling to a heat bath fortime t cool using velocity rescaling or velocity reassign-ing every 50 time steps. No drift or aging was observedover the equilibration time t equil . in thermodynamic ordynamic observables before performing production runs.Both t cool and t equil . were significantly longer than thestructural relaxation time.[22] S. J. Plimpton, Large–scale Atomic / Molecular Mas-sively Parallel Simulator, http://lammps.sandia.gov; S.J. Plimpton, J. Comp. Phys. , 1 (1995).[23] See e.g., W. Kob and H.C. Andersen Phys. Rev. E ,4134 (1995); D.N. Perera and P. Harrowell, J. Chem.Phys. , 5441 (1999); E. Flenner and G. Szamel, Phys. Rev. E , 031508 (2005); L. Berthier and W. Kob, J.Phys. Condens. Matter , 205130 (2007).[24] In the context of renewal processes[25] exchange and per-sistence times are called total and excess lifetimes, re-spectively. If events occur over a very short time scaleas compared to waiting times the distributions of per-sistence and exchange times are related by p ( t p ) = h t x i − R ∞ t p p ( t x ) dt x .[6, 25] In our case, where an eventmay be built up over time (a displacement of distance d does not occur instantaneously), this relation worksonly as an approximation. Nevertheless, application ofthis formula gives reasonable estimates of the behaviour.[25] G.Grimmett and D. Stirzaker, Probability and RandomProcesses (Oxford University Press, 2001).[26] The distributions of persistence (or first-passage) timesfor smaller cut-off distances behave differently, see P. Al-legrini, J.F. Douglas and S.C. Glotzer, Phys. Rev. E , 5714 (1999). These authors also find that the typ-ical first-passage time for distances comparable to a par-ticle diameter scales with temperature roughly as τ α ,which is consistent with what is expected from dynamicfacilitation.[5, 6][27] M. Merolle, J.P. Garrahan and D. Chandler, Proc. Natl.Acad. Sci. USA , 10837 (2005); J.P. Garrahan, R.L.Jack, V. Lecomte, E. Pitard, K. van Duijvendijk, F. vanWijland, Phys. Rev. Lett. , 195702 (2007).[28] E.W. Montroll and G.H. Weiss, J. Math. Phys. , 167(1965).[29] D.A. Stariolo, G. Fabricius. J. Chem. Phys.125