Correlation effects and super-arrhenius diffusion in binary Lennard-Jones mixtures

Transcription

1 Correlation effects and super-arrhenius diffusion in binary Lennard-Jones mixtures Vanessa K. de Souza and David J. Wales University Chemical Laboratories, Lensfield Road, Cambridge CB2 1EW, United Kingdom Received 8 April 2006; revised manuscript received 26 June 2006; published 3 October 2006 An underlying Arrhenius temperature dependence of the diffusion constant can be extracted from the fragile, super-arrhenius diffusion of a binary Lennard-Jones mixture. The Arrhenius form is found by considering time intervals and temperatures where diffusive motion is present but the system is nonergodic, with averages taken over an ergodic trajectory. The Arrhenius diffusion can be related to the true super-arrhenius behavior by a factor that depends on the average angle between steps in successive time intervals. This correction factor accounts for the fact that on average, successive displacements are negatively correlated. This negative correlation can be linked directly with the higher apparent activation energy for diffusion in fragile glass formers at lower temperature. Arrhenius dependence of the diffusion constant for nonergodic systems is found over a range of time scales. As time intervals are increased, the time windows become long enough to register negative correlation and the correction decreases. The Arrhenius form changes with time as ergodicity is approached and this change can be characterized by the variation in gradient and intercept with time. We introduce a formalism to describe super-arrhenius behavior based on Arrhenius and non-arrhenius contributions to the diffusion constant. DOI: /PhysRevB PACS number s : Ja, Pf, Fs I. INTRODUCTION Supercooled liquids have failed to crystallize on cooling below their melting temperature and exist in a metastable state. Viscosity and relaxation times increase with further cooling, and eventually a glass, or non-crystalline solid, is formed at the glass transition temperature, T g. A glass has the microscopic disorder of a liquid, but is defined as a solid by its mechanical properties. The Arrhenius law defines a relationship where an observable property scales with temperature as exp ±B/T for a constant B. Such a functional form is expected when a transport or rate process encounters a potential or free energy barrier of magnitude B. Some supercooled liquids exhibit super-arrhenius behavior, for which the temperature dependence of relaxation times or transport properties is stronger than predicted by the Arrhenius law. The Vogel- Tammann-Fulcher VTF equation 1 3 is commonly used to fit such behavior. An alternative is provided by mode-coupling theory MCT 4 which predicts that the diffusion constant shows a power law behavior as a function of temperature. The degree of super-arrhenius behavior was used by Angell to classify supercooled liquids as strong or fragile. 5 Strong materials include many network glass formers, such as silica and germanium dioxide, which have tetrahedrally coordinated structures, while fragile materials, such as orthoterphenyl, are often bound by more isotropic forces. The strong or fragile character of a supercooled liquid appears to be correlated with various other properties, including the response to a mechanical or dielectric perturbation In simple liquids, relaxation functions usually exhibit exponential decay, i.e., the Debye form. Many supercooled liquids show departure from Debye relaxation with stretched exponential behavior, 10,11 which correlates somewhat with their fragility. 10 Experimentally, dielectric loss spectroscopy has been applied to study the dielectric relaxation time, a measure of the response of the polarization of a material to an applied electric field. The dielectric loss factor is proportional to the rate of energy dissipation and is measured as a function of frequency. Peaks in this spectrum are often assumed to correspond to distinct mechanisms with dielectric relaxation rates given by the peak positions. For most glass formers, Johari and Goldstein found that the temperature dependence of the frequency of the main peak in the dielectric relaxation spectrum exhibited super-arrhenius behavior. However, a second peak, identified with a faster relaxation time and Arrhenius temperature dependence, was also characterized. 12,13 The two peaks are usually identified as a slower non-arrhenius process and a faster process, respectively. The process responsible for the peak has been associated with structural relaxation, which is frozen out at the glass transition on the experimental time scale, and the process has been associated with some form of secondary relaxation, which is sometimes interpreted as a faster process corresponding to a smaller barrier. 12 The idea of separation into two components, one Arrhenius and the other super-arrhenius, may also be applicable to diffusion. In this contribution we show that for a binary Lennard-Jones BLJ glass former, it is possible to separate out an Arrhenius dependence from the ergodic super- Arrhenius behavior. A. Dynamic heterogeneity Heterogeneous dynamics has been suggested as an explanation for fragile, nonexponential relaxation in supercooled liquids. A large variation in the relaxation times of individual particles could result in nonexponential relaxation, even if the relaxation of each particle is exponential. An alternative possibility is homogeneous dynamics in which each particle undergoes nonexponential relaxation. Glasses are amorphous solids and therefore all the atoms are in slightly different environments for practically all in /2006/74 13 / The American Physical Society

2 VANESSA K. DE SOUZA AND DAVID J. WALES stantaneous configurations. Hence we expect the dynamics of different atoms to be distinct on time scales that are too short to produce identical averages. 14 This dynamic heterogeneity, or heterogeneous relaxation, has been inferred from experiments and computer simulations. 14,19 22 It is generally studied by monitoring the most mobile and least mobile particles ,22 In fact, heterogeneity should be the usual outcome of experiments on solids that probe local short-time transport properties. For example, when vacancies migrate in a face-centred-cubic solid only the atoms surrounding a vacancy will be mobile until the vacancy moves. 23 In isotropic diffusion, atomic displacements show a Gaussian distribution. 24 However, in supercooled liquids and glasses, this distribution is distorted even for relatively long time scales, and the change can be quantified by a non- Gaussian parameter, 25 2 t = 3 r4 t 5 r 2 t 2 1, where denotes an average over all atoms of a particular type and all time origins. 2 t is zero when the atomic motion is homogeneous, which occurs during ballistic motion at t 0 and ergodic diffusion at t. For intermediate times, corresponding to the relaxation of short-range caged motion, 2 t increases, adhering to a master curve at different temperatures. 26 A common dependence of the form 2 t t has been found in this regime for different classes of materials. This relationship extends for longer times at lower temperature where there is more pronounced heterogeneity. 27 At the temperatures studied in this contribution, which are well above T g, heterogeneity is evident for time regimes that correspond to nonergodic behavior. However, we are able to describe super-arrhenius behavior without any reference to heterogeneity, suggesting that this property is not the fundamental cause of super-arrhenius diffusion. B. Ergodicity The ergodic hypothesis states that for a system in thermodynamic equilibrium, the time average of a dynamical variable must converge to its ensemble average. In general, a system is ergodic when averaging times are infinite. On finite time scales, a system that samples all regions of phase space with equal likelihood can be considered effectively ergodic. For a metastable state, such as a supercooled liquid, the equilibrium averages on a given time scale will not agree with an average taken over the whole of phase space. However, agreement is restored if the phase space average is restricted to the relevant region of phase space. Hence it is necessary to associate equilibrium with the observation time scale. 28 A supercooled liquid can be considered as a locally ergodic equilibrium for a certain time scale and region of phase space, which excludes the crystal. In order to study the properties of supercooled liquids we must be able to diagnose when ergodicity is attained within this restricted region of phase space. 1 In order to identify ergodicity we use a measure introduced by Thirumalai et al. The original method was based on the time-averaged energies of individual particles for simulations starting from two different initial states of the same system, which they called the energy metric. 29 This measure was used to show that effective ergodicity is broken in the transition from supercooled liquids to glasses. 29 Subsequently, Mountain and Thirumalai suggested that in the absence of a multivalley structure in the relevant region of phase space, it is sufficient to examine the properties of a single trajectory using an energy fluctuation metric to diagnose ergodicity. 30 For a fluid in equilibrium, all particles of a given type have identical average characteristics over a long enough time scale. If a system consists of N particles of each type, such that N =N, where N is the total number of particles, and the time-averaged energy of the jth particle of type, is j t;, the total fluctuation metric is defined as where t = N 1 j t; t; 2, N j=1 t; = 1 N N j t;. j=1 The time-averaged energy of a particle is given by j t; =t 1 0 t E j t ; dt, where E j t ; is the value of the energy of the jth particle at time t, which, if the potential is pairwise additive, is the sum of the kinetic energy and half of the potential energy involving j. If the system is ergodic within a well-defined region of configuration space, t should vanish for long times, as the average energy of each individual particle reaches the ensemble average for that species. In computer simulations of a glass, the long-time limit of the metric is not zero, but instead reaches an almost constant value, indicating that the energy per particle is not self-averaging on the molecular dynamics time scale. 29,31 For a particular trajectory, the form of t plotted against 1/t can be used to determine ergodic and nonergodic time scales. This approach is particularly effective when comparisons are made among results for different total energies. 32 In this work we expand on the suggestion 33 that super- Arrhenius diffusive behavior can be separated into Arrhenius behavior and a quantitative correction factor, which takes into account the correlation between displacements in successive time windows. This separation is identified on nonergodic time scales, when dynamic heterogeneity is evident. We show how the energy fluctuation metric, t, and the non-gaussian parameter, t, can be used to identify these time scales, consider the nature of the correction factor, particularly in relation to the heterogeneity of the system, and extend these ideas to produce a new formalism to describe super-arrhenius behavior

3 CORRELATION EFFECTS AND SUPER-ARRHENIUS TABLE I. Summary of the total system energies studied in MD simulations of 60-, 256-, and 320-atom binary Lennard-Jones mixtures. The cutoff for the potential is given in units of where, =A or B. This cutoff is just under half the shortest box length for the AA interaction. Atoms Density Energy Cutoff , , , , II. SIMULATION DETAILS The trajectories considered in this study were produced by molecular dynamics MD simulations of 60-, 256-, and 320- atom binary mixtures of 48 type A and 12 type B particles, 204 type A and 52 type B particles, and 256 type A and 64 type B particles, respectively, interacting via a Lennard- Jones potential in a periodically repeated cell. 26,34 44 For the 60- and 256-atom systems this cell was cubic, and for the 320-atom system, a tetragonal cell found to describe the lowest energy crystal structure for this system was used. 45 Interaction parameters 26,46 AA =1.0, AB =0.8, BB =0.88, AA =1.0, AB =1.5, and BB =0.5 were considered along with the Stoddard-Ford quadratic cutoff, 47 which ensures that both the potential energy and its first derivative are continuous. Cutoffs for the AA interaction were just under half the shortest box length, as specified in Table I. All particles have the same mass, m. A velocity Verlet 48 algorithm was employed to propagate the trajectory with a time step of in reduced units of m AA / AA 1/2 used throughout. Energy 2 fluctuation metrics for the total energy and potential energy were calculated. The contributions to the total fluctuation metric from type A, A t, and type B, B t, particles are 2 examined separately and are given in reduced units of AA used throughout, where t = A t + B t. However, the results for each type of particle are qualitatively the same. 3 Number densities of 1.1 and 1.3 in reduced units of AA used throughout were examined. For the 256- and 320- atom systems, each microcanonical run at constant energy included 10 5 initial steps followed by 10 6 steps of data collection. For the 60-atom system, 10 7 initial steps were followed by 10 7 steps of data collection in each run. The final configuration at a particular energy was used as a starting configuration for the subsequent simulation, with the total energy decreased each time as shown in Table I. Self-diffusion coefficients, presented previously, 32 were obtained using Einstein s proportionality relation for the mean square displacement of a particle as a function of observation time in the limit that t. The ensemble average taken for the mean square displacement is an average over all atoms of type A and for many time origins. The gradient was calculated by linear regression. All results are presented for the majority A species unless stated otherwise. However, results for the B species are qualitatively the same. Further simulation details for the trajectories used to obtain the ergodic diffusion constants can be found in Ref. 32. FIG. 1. Potential energy fluctuation metric, A t, for the majority A species in a 256-atom BLJ mixture of number density 1.3. The total length of the trajectory is t total and many different time origins are used to describe the average behavior for half of the total trajectory length. A series of lines is shown with the total system energy, and hence temperature, decreasing towards the top of the figure. Results are shown for temperatures between 1.66 and AA /k B. As the temperature is lowered, longer simulation times are required for ergodic behavior. For the lowest temperatures shown, glassy behavior is observed, the potential energy per particle is not self-averaging on the molecular dynamics time scale, and the system is nonergodic. At the highest temperatures, A t approaches the long-time limit of zero, which is a characteristic of ergodic trajectories. Trajectories at temperatures down to AA /k B are found to be ergodic. The trajectories that we consider to be ergodic for this simulation length are indicated by dashed lines and the inset includes their short time behavior. III. RESULTS The energy fluctuation metric enables us to diagnose effective ergodicity for a particular trajectory over a range of different time scales. We have previously used the metric to diagnose ergodicity for a particular trajectory from one time origin. 32 However this measure can also be calculated for many time origins, as shown for a range of different energies in Fig. 1. Ergodic behavior, for which we require smooth monotonic decay of t towards zero, can be deduced from this figure. By definition, over an ergodic trajectory the average energy of each particle reaches the ensemble average for that species and we would not expect to observe dynamic heterogeneity in the mobilities of the particles. This expectation is supported by examination of the non-gaussian parameter in Eq. 1, which decays close to zero for all ergodic trajectories. However, this observation may not preclude spatial heterogeneity in the form of clusters of immobile and mobile particles. In order to characterize differences between the most mobile and least mobile particles, and to identify their dynamics as homogeneous or heterogeneous, we considered time scales over which the energy is not self-averaging, where t indicates nonergodicity, and where the non-gaussian parameter does not decay to zero. If the entire trajectory is ergodic, we can divide it up into shorter nonergodic segments and average over the corresponding short time dynam

4 VANESSA K. DE SOUZA AND DAVID J. WALES FIG. 2. Color online Variation of the logarithm of D with 1/T for the most mobile and least mobile A atoms, as well as the average, in a 60-atom BLJ mixture of number density 1.3. The D were calculated over time intervals of 25, 250, and 2500, represented by crosses, squares, and circles, respectively. The true diffusion constant obtained by averaging over all A atoms on the longest time scale is shown by filled diamonds. ics. Further insight is obtained by attempting to recover longtime behavior from these short segments of trajectory, which requires an examination of correlation effects. A. Mobile and immobile atoms Diffusion constants are a measure of mobility, and Einstein s relation should be used in the limit that t. We will study mobility using an incorrect diffusion constant, calculated from the average mean square displacement over different time scales. The mean square displacement after a period of time, t, is given by the following formula: r 2 t N 1 N i=1 r i t r i 0 2 N 1 N r i t 2, i=1 4 where denotes an average over many time origins, and r i t is the position vector of atom i at time t. The mean square displacement used to calculate D is given by r 2 t, N 1 m N j=1 r i j 2, i=1 5 FIG. 3. Minority B-atom contribution to the total energy potential and kinetic energy fluctuation metric for a 60-atom BLJ mixture at a number density of 1.3, plotted against t total /t, where t total =50 000, the total length of the trajectories. Overall nonergodic runs are shown by solid lines, and these are excluded from the D analysis. The ergodic runs from which D has been calculated are shown by dashed lines. The vertical line corresponds to t=2500. Four ergodic trajectories are separated from the other ergodic runs at this point; they are nonergodic on shorter time scales, but are approaching ergodicity. The same trend is shown by A t, the majority A-atom contribution. where t=m for integer m and r i j =r i j r i j. In each time interval of length, we can identify a range of different mobilities. Here we define mobility in terms of the displacement between two end points. Figure 2 includes such short-time-averaged diffusion constants, D, for the most mobile A atom, least mobile A atom and the average over all A atoms for time intervals of =25, 250, and 2500 in a 60-atom BLJ mixture of number density 1.3. The true diffusion constants calculated from the full ergodic trajectories and averaged over all A atoms are also shown. On the shortest time scale all the atoms, whether more or less mobile, show diffusion that appears to be Arrhenius, without the distinctive super-arrhenius curvature of the true diffusion constant. Arrhenius-type behavior is also found when we average over all the atoms in this time interval. The results for individual atoms in Fig. 2 run parallel to the true diffusion constant at high temperature, but deviate at low temperature. The deviation occurs where curvature in the true diffusion constant becomes apparent, around T = 1.0 in units of AA /k B used throughout. As is increased, we would expect D to approach the true long-term diffusion constant, such that eventually ergodicity is reached and D no longer changes. Figure 2 shows that super-arrhenius curvature does appear with increasing and we see that it is only the lower temperature results that change. At high temperature it appears that ergodicity was already reached on the shortest time scale considered. Arrhenius behavior is observed on time scales for which the low temperature results, where the true super-arrhenius behavior is most pronounced, are nonergodic. On averaging over longer time windows, super-arrhenius behavior is progressively recovered. The behavior of the energy fluctuation metric, as plotted in Fig. 3, shows that at the four lowest temperatures, ergodicity is approached at a time of around 2500, as D D. Breaking down the contributions to D from atoms of differing mobility reveals essentially the same increasing super-arrhenius behavior with increasing, independent of the atomic mobility Fig. 2. A difference in the diffusion constants calculated for the most mobile and least mobile atoms remains, and when ergodicity is approached the difference in ln D appears to become constant. This difference reflects the shape of the distribution of square displacements,

5 CORRELATION EFFECTS AND SUPER-ARRHENIUS and we would expect such a distribution of mobilities even on an ergodic time scale. 24 The slight difference in the relative positions of the curves for increasing reflects the narrowing of the distribution with time, which results from an averaging effect as heterogeneity disappears from the system. If barriers of different heights can be associated with atoms of greater and lesser mobility, this would be reflected in differing gradients. However, we find that there is little difference even on the shortest time scales. Hence we find no evidence that super-arrhenius diffusion results from averaging over a distribution of particles that each behave in Arrhenius fashion with different barriers. Instead, the analysis below shows that super-arrhenius behavior results from a negative correlation between the atomic displacements in successive time windows. If the windows are too short then there is insufficient time for reversals in direction to be registered in lower temperature trajectories, and the calculated D is too large. The magnitude of the discrepancy between D and D for a particular value of increases with decreasing temperature, in agreement with the suggestion that non-arrhenius transport properties are linked to the increasing time scale required to achieve effective ergodicity at low temperature. 41 B. Correlation and super-arrhenius behavior We can write an expression for the mean square displacement after a period of time, t, in terms of atomic displacements in m time intervals of length, m r i t 2 = j=1 m r i j 2 +2 r i j r i k j k = r i j 2 +2 r i j r i k cos jk. 6 j=1 j k When determining the mean square displacement for a time interval, we include the first term in Eq. 6 but not the second term. As we see from the behavior of D, this second term is nonzero at low temperature, and by effectively averaging over the atoms too early, we miscalculate the diffusion constant and find Arrhenius temperature dependence. The second term is therefore responsible for the super- Arrhenius behavior. The cos jk term between displacements in different time intervals can be calculated. 33 For =25, a distribution of cos jk for k= j+1, describing the angle between displacements in adjacent time intervals, shows a significant bias towards negative values, which is largest for lower temperatures. For k= j+2, describing the angle between displacements in time intervals separated by one intervening time interval, there is little difference between the probability distributions at different temperatures and no bias towards positive or negative values of cos jk, leading to an average value of zero. 33 Figure 4 shows the average value of cos jk for a larger 256-atom system and time intervals of length =2.5. We see from Fig. 4 that, except at the lowest temperatures, where there is a small net negative value for k = j +2, FIG. 4. Color online The average value of cos j,j+n between atomic displacements in different time intervals of length 2.5 is shown for a 256-atom system of number density 1.3. The time intervals are separated by n 1 intervening time intervals, so for adjacent steps n=1. Results are shown for k B T/ AA between and 1.66, with temperature decreasing towards the bottom of the figure. cos jk 0 for adjacent time intervals only. For k j+2, where the cos jk distribution is flat, resembling a random walk, it is likely that the angle and magnitude of the displacements are independent and separable so that taking an average in this way is justified. However, for k= j+1, the angle and magnitude of the displacements may not be separable. This effect would result in a loss of information when an average value of cos jk is used. However, we find this makes little difference to the correction, as discussed in Sec. III C. If we assume that m is large and the magnitudes of the displacements in adjacent steps are similar, such that only the average behavior of cos jk is important, the following formula for the mean square displacement is obtained: r 2 t 1 N m N i=1 r i j cos j,j+1. j=1 The mean square displacement is calculated for integer values of m and averaged over many time origins. This formula would be exact if the displacements for every atom in every time interval had the same magnitude. Using this formula, which adjusts the mean square displacement, a new diffusion coefficient, D *, is obtained and we can recover the correct super-arrhenius behavior for different values of. 33 This analysis was repeated for larger BLJ systems of 256 and 320 atoms. Figure 5 shows D calculated for several different values of, with and without the cos jk correction for the 256-atom system at a density of 1.1. Figure 6 shows D for 256- and 320-atom systems at a higher density of 1.3. This figure shows that increasing the system size from 256 to 320 atoms results in no further change in D. The ergodic diffusion constants also remain unchanged. This convergence of the diffusion constants with increasing system size shows that for a system of at least 256 atoms, finitesize effects are small

6 VANESSA K. DE SOUZA AND DAVID J. WALES FIG. 5. Color online D for the majority A species in a BLJ mixture of 256 atoms at a number density of 1.1 calculated for different time intervals,, as shown in the legend. Filled diamonds show the true diffusion constants for an ergodic trajectory and the solid line is a VTF fit. a D calculated over a range of ; b D corrected by a factor containing the average angle between successive displacements for the same values of. The correction recovers the correct diffusion constants for all but the smallest values of. For both densities, without the correction, the smallest considered, corresponding to 500 MD steps, results in diffusion that appears to be sub- Arrhenius. This result is easily understood, since at low temperatures for the shortest times, a plateau in the mean square displacement shows that the particles are still caged. This caging explains why the cos j,j+1 correction is not sufficient at such short times to recover the true super- Arrhenius behavior. For these low temperatures, as shown in Fig. 4 for the higher density, some negative correlation is evident beyond the correlation of adjacent steps. Departure of the non-gaussian parameter from a master curve of the approximate form 2 t provides a convenient cutoff for the end of the plateau region of the mean square displacement. After this time we observe recovery towards long-time diffusive behavior. Previously, 49 a cage rearrangement time scale has been defined by the time taken for a maximum in the non-gaussian parameter to be reached. However, in this instance, there is no need to delay the cutoff until the maximum. Using the master curve criterion, we can rationalize the results at the two lowest temperatures for the 256-atom system in Fig. 6. The vertical lines in Fig. 7 show the five lowest values of for which D has been calculated. For D calculated before departure from the master FIG. 6. Color online D for the majority A species in BLJ mixtures of 256 atoms open symbols and 320 atoms filled symbols at a number density of 1.3 calculated for different time intervals,, as shown in the legend. Filled diamonds show the true diffusion constants for ergodic trajectories of the 256-atom system and the solid line is a VTF fit. a D calculated over a range of ; b D corrected by a factor containing the average angle between successive displacements for the same values of. curve, i.e., during caging, the cos j,j+1 correction for correlation between adjacent steps is not sufficient to regain the true diffusion constants. C. The correction factor The behavior of cos j,j+1 for a particular temperature changes with. Initially, for the values of considered, cos j,j+1 is negative, and an atom is most likely to move backwards relative to the displacement vector in the previous time window. If all directions were equally likely, cos j,j+1 would be zero. We expect it to become zero when increases sufficiently for ergodicity to be achieved. On this time scale any backward motion is relatively fast and has already made a direct contribution to the calculated diffusion constant. The short time behavior of cos j,j+1 for three different temperatures is shown in Fig. 8. cos j,j+1 is positive during ballistic motion and reaches a negative minimum before a slow recovery. Figure 8 also shows the behavior of r i j r i j+1 / r i j 2, which gives the exact correction for the correlation between adjacent steps from Eq

7 CORRELATION EFFECTS AND SUPER-ARRHENIUS FIG. 7. Color online Non-Gaussian parameter, 2, vs time for the majority A species of a 256-atom BLJ mixture with number density 1.3. Results are shown for k B T/ AA in the range to 1.66, illustrating how 2 t generally increases as T decreases. The dashed vertical lines show the five lowest values of for which D has been calculated. Departures from a master curve bold indicating the end of caging are marked for the two lowest temperatures. 2 =0.1 is also marked by a dashed line. This expression is exact because it does not assume that the magnitude and direction of the displacements are separable. The most significant difference between the two corrections occurs at the minima in Fig. 8. However, for these small values of, neither correction is sufficient to regain the true diffusion, as negative correlation extends beyond adjacent steps. For longer times there is little difference between these two corrections. cos j,j+1 appears to overestimate slightly the magnitude of the exact correction. This effect increases with decreasing temperature and the maximum difference appears to correlate with the maximum in the non-gaussian parameter. However, for the temperature range considered, the difference is relatively small and it appears to be sufficient to correct the whole distribution by the same constant factor, cos j,j+1. A correction of this form does not alter the value of the non-gaussian parameter. Hence, super- Arrhenius behavior is recovered without removing heterogeneity from the system. Previous work has highlighted the importance of correlation effects. Reversible and irreversible cage-breaking processes or jumps in a BLJ system have been studied, and the ratio of reversible to irreversible jumps was found to increase with decreasing temperature. 50 Jumps which become increasingly reversible with decreasing temperature have also been found for Cu 33 Zr ,52 For dynamics at short times during caging, computer simulations of hard spheres 53,54 and experiments on colloidal suspensions 55 suggest that a backdragging effect determines the local slope of the meansquare displacement. Our observation that super-arrhenius behavior can be recovered from the negative correlation between displacements leads us naturally to consider correlated transitions between local minima on the potential energy surface. Such a correlation could be a direct result of a reduced number of connections, because if the number of connections is limited then return to a previous minimum becomes more likely. In FIG. 8. Color online Solid lines show the changing value of cos j,j+1 between adjacent steps with increasing step size for temperatures of 0.93, 1.16, and 1.66 in the 256-atom BLJ system of number density 1.3. Dashed lines show r i j r i j+1 / r i j 2, the exact correction for correlation between adjacent steps. The data are taken from a trajectory of length t total =500. The systems shown are determined by the energy fluctuation metric to be ergodic over this simulation length. an amorphous region of configuration space, connectivity probably decreases at low energy, and this effect has been suggested previously as an explanation for the increasing free energy barrier to relaxation, which produces super- Arrhenius diffusion. 56,57 Correlation in the direction of successive transitions between minima has been considered previously. Middleton and Wales found that no positive correlation occurred between successive transitions for a 60-atom BLJ system, and negative correlation is visible in Fig. 6 of their report. 58 Studies by Keyes and Chowdhary for a 32-atom Lennard-Jones system 59 and Doliwa and Heuer for a 65-atom BLJ system 39,60 also suggest that negative correlation exists, in agreement with the present work. An increased apparent activation energy as a result of an increased probability of return to the original ground state has also been found in a model system with a simple hierarchical landscape. 61 We can directly link the increased negative correlation between displacements in successive time windows at lower temperatures with the higher apparent activation energy for diffusion in fragile glasses. If D is Arrhenius, we can also deduce the form needed for the correction to give super-arrhenius behavior. One possibility is that ln 1+2 cos j,j+1 behaves like a/t b, where b 1. In the next section, we will attempt to characterize this behavior. D. Nonergodic Arrhenius diffusion It is not only at short times that an Arrhenius component of diffusion is recovered from the super-arrhenius behavior. If only nonergodic results are considered, the Arrhenius dependence is seen for the whole range of, as shown in Fig. 9. As increases, time windows become long enough to register negative correlation, the Arrhenius form changes and the correction decreases. In order to study the Arrhenius dependence we must determine the time when systems at different temperatures have

8 VANESSA K. DE SOUZA AND DAVID J. WALES FIG. 9. Color online D for the majority A species in a BLJ mixture of 256 atoms at a number density of 1.3 calculated for different time intervals,, as shown in the legend. The diffusion constants shown are for systems where particles have escaped their cages but the non-gaussian parameter is above a certain cutoff value, which is 0.05 for the upper panel and 0.2 for the lower panel. Filled diamonds show the true diffusion constants for the ergodic trajectory and the solid line is a VTF fit. FIG. 10. The decay of the non-gaussian parameter, 2 top panel, is compared to the decay of the potential energy fluctuation metric for the majority A species, A t bottom panel. Ergodic trajectories as described by the energy fluctuation metric for a trajectory of length t total are indicated by dashed lines. Decay of 2 to below 0.1 is seen to correspond to an ergodic trajectory as determined by the energy fluctuation metric. become ergodic. For time intervals longer than this ergodic time, t erg, the diffusion constant should be excluded from fits to the Arrhenius form, because the system begins to show super-arrhenius curvature. In order to determine the ergodicity or nonergodicity of simulation trajectories at low temperature we have used the energy fluctuation metric, t. 32 However, this determination did not require a precise cutoff between ergodic and nonergodic times, and in fact the transition occurs slowly over a period of time. We now need to determine a consistent cutoff between ergodicity and nonergodicity. Decay of t or t / 0 to a specific value can be used to provide the cutoff or a cutoff for decay of the non-gaussian parameter, 2 t, can be used. 2 t describes the heterogeneity of the system, giving a measure of the increasing time scales required for ergodicity at lower temperatures, as heterogeneity no longer exists once the trajectory is ergodic. Once we have an ergodic trajectory, we have found that using a cutoff for the decay of 2 t is a convenient measure of an ergodic time. Figure 10 shows that for low temperatures, decay of 2 t to below 0.1 coincides with a system that t would describe as approaching ergodicity. We shall use decay of 2 t to below 0.05, 0.1, and 0.2 as cutoffs for the maximum value of considered for a particular temperature. D values calculated before departure from the master curve in 2 t are also excluded, as they correspond to caging. For these small values of, negative correlation extends beyond adjacent steps. Linear regression is then used to fit to the remaining nonergodic results at time intervals between 2.5 and 250, as shown in Fig. 9 for cutoffs in 2 t of 0.05 and 0.2. For each value of the linear regression fit of ln D to A/T+B yields a gradient A and an intercept, B. Figure 11 reveals that both A and B can be fitted quite well to the form a ln / 0 +1 b +c where 0 =1. For time intervals in the Arrhenius regime, the variation of the diffusion constant with temperature and is then the following for 0 t erg : ln D T, = a ln / 0 +1 b c T + d ln / 0 +1 e +lnd 0, with fitted parameters a,b,c,d,e, and ln D 0, and 0 =1. At =t erg, the function in Eq. 8 reaches a minimum and then begins to increase, while the real diffusion constant becomes constant. This is why we have to be careful about excluding ergodic results from the Arrhenius fits

9 CORRELATION EFFECTS AND SUPER-ARRHENIUS FIG. 11. Color online Gradient A upper panel and intercept B lower panel for Arrhenius fits to diffusion constants calculated for nonergodic time intervals as a function of, for a BLJ mixture of 256 atoms at a number density of 1.3. Three different cutoffs for the non-gaussian parameter are used as the criterion for ergodicity, as shown in the legend. Functions of the form a ln + 0 / 0 b +c are fitted for both the gradient and the intercept for each cutoff. Minimization of Eq. 8 with respect to gives an ergodic time, and hence an ergodic diffusion constant as a function of temperature, with the following forms: t erg = 0 exp j h 1 and ln D erg T = m T T n c T +lnd 0, 9 as shown in Fig. 12 for all three cutoffs, where h, j, m, and n are combinations of the fitted parameters in Eq. 8. The diffusion coefficient consists of an Arrhenius dependence hypothetically seen at =0 and a correction of the form we anticipated at the end of the previous section. If the Arrhenius process describes a single or multiple barrier event with a constant activation energy, the non-arrhenius behavior could arise from an entropic contribution. 56 If the particles are restricted in their choice of direction, negatively correlated motion is likely and a higher apparent activation energy results. We find that the diffusion constant is predicted well by Eq. 9 for temperatures where ergodic times remain within the range of time intervals studied, with deviation occurring only at high and low temperature. The choice of cutoff for 2 t appears to make little difference to the results. D T, is linked to the local behavior of the mean square displacement after time and at a particular temperature. A mean square displacement can be recovered using the following expression: r 2 T, =6 exp ln D T,. 10 Figure 13 shows both the mean square displacement obtained using this form and the mean square displacement obtained directly from the MD trajectory. The fitting parameters used in D T, are taken for a cutoff of 0.2, although the choice of cutoff makes little difference, as above. The arrows on the figure mark t erg, found by minimization of D T,, for the different temperatures shown. The mean square displacement obtained from D T, generally agrees FIG. 12. Color online An ergodic time upper panel and an ergodic diffusion constant lower panel as a function of temperature are estimated using the variation of the gradient and intercept of Arrhenius fits to nonergodic diffusion constants, D, as a function of. Filled black diamonds show the true diffusion constants for the ergodic trajectory. very well below t erg and above times for which caging is evident, i.e., in the range where the expression for D T, is valid. FIG. 13. Color online The mean square displacement, r 2 t, of a 256-atom BLJ mixture at a number density of 1.3 is shown by solid lines for temperatures, from top to bottom, T=1.656, 1.410, 1.161, 1.044, 0.991, and These results can be compared with the mean square displacement predicted using our equation for D T, dashed lines at equivalent temperatures. Arrows mark the ergodic time obtained by minimization of D T,

10 VANESSA K. DE SOUZA AND DAVID J. WALES FIG. 14. Color online Black squares show the true diffusion constants for ergodic trajectories of a 256-atom BLJ system of number density 1.2. The solid line shows a fit of the form, m/t n c/t+lnd 0, and dashed lines show the two components of this fit: an Arrhenius term, c/t+lnd 0, and a correction factor, m/t n. E. Fitting to super-arrhenius behavior The form of the diffusion constant given in Eq. 9 can be used to fit the ergodic diffusion constants. Figure 14 shows the new fit for a 256-atom system at a number density of 1.2. The form of the mean square displacement and diffusion constants for this system agree very well with those found for 320 atoms and also for a larger system of 1000 particles with a shifted potential. 26,62 Figure 14 also shows the two components of the fit, the Arrhenius part, c/t+lnd 0, and the correction, m/t n, separately. The Arrhenius form follows the high temperature results and deviates at T 1, where this system is expected to enter a landscapeinfluenced regime. 34 The correction is close to zero until T 1 and then its magnitude increases dramatically. An estimated variance or residual mean square for the fit is calculated by dividing the residual sum of squares by the number of degrees of freedom, given by subtracting the number of fitted parameters from the number of data FIG. 15. Color online Black squares show the true diffusion constants for ergodic trajectories of a 256-atom BLJ system of number density 1.2. The solid line shows a fit of the form, m/t n c/t+lnd 0, and the dot-dash line is an MCT fit, both for T 1. A VTF fit over the full temperature range of the data is also shown by a dotted line. points. 63 By this criterion the fit is better than for the VTF form, which has one less parameter, as shown in Table II for 256-atom systems of varying number density. The VTF form for a 256-atom system at a number density of 1.2 is shown in Fig. 15. An MCT fit is also shown in Fig. 15 and can be compared withafittoeq. 9. For these fits, data at temperatures above T=1 have been excluded, as for this density it has been found previously that power law behavior is only followed at lower temperatures. 26 Our new form has a slightly smaller estimated variance Table III. It is also worth noting that high temperature results outside the fitting region are still predicted extremely well by our fit, and no kinetic arrest is predicted by our model. Various other functional forms with varying numbers of parameters have been used to fit super- Arrhenius behavior, 64,65 including the Ferry 66 or Bässler form, 67 which, in common with Eq. 9, has no kinetic arrest. TABLE II. Fitted parameters for the ergodic diffusion constants of a 256-atom BLJ system at number densities of 1.1, 1.2, and 1.3. Fitting parameters for the VTF form D=D 0 exp A/ T T 0 and the proposed form D=D 0 exp c/t exp m/t n are also shown. The estimated variance, 2 est, is calculated by dividing the residual sum of squares by the number of degrees of freedom Ref. 63. Parameter AA AA AA A 1.108± ± ±0.066 T ± ± ±0.008 ln D ± ± ± est n 9.99± ± ±0.54 m 0.306± ± ±0.028 c 1.836± ± ±0.32 ln D ± ± ± est

11 CORRELATION EFFECTS AND SUPER-ARRHENIUS TABLE III. Fitted parameters for the ergodic diffusion constants of a 256-atom BLJ system of number density 1.2. Data for temperatures below T=1 were fitted. Fitting parameters for the MCT form D=D 0 T T C and the proposed form D=D 0 exp c/t exp m/t n are shown. The estimated variance, 2 est, is calculated by dividing the residual sum of squares by its degrees of freedom Ref. 63. MCT IV. CONCLUSIONS New form 1.766±0.078 m 0.528±0.034 T C 0.448±0.006 n 6.48±2.09 c 2.74±0.44 ln D ±0.064 ln D ± est est Heterogeneous dynamics has been suggested as an explanation for fragile, super-arrhenius diffusion in supercooled liquids. Using an energy fluctuation metric 30,32 to determine ergodicity and the non-gaussian parameter 25 to determine heterogeneity, we have identified heterogeneity with nonergodic time scales. Over an ergodic trajectory, heterogeneity is no longer evident, and the non-gaussian parameter decays approximately to zero. In order to identify a connection between heterogeneity and super-arrhenius diffusion, we have examined the dynamics of 60-, 256-, and 320-atom binary Lennard-Jones mixtures on nonergodic time scales. For temperatures where super-arrhenius diffusion is evident, we have been able to separate an Arrhenius-like temperature dependence from the true diffusion constant. We considered time intervals where diffusive motion occurs, but measurements of the total particle energy show nonergodicity, averaged over an ergodic trajectory. Arrhenius behavior is then seen for all atoms as well as for the average. As the time intervals for averaging are increased, and ergodicity is approached, the correct super-arrhenius behavior is recovered. The short-time Arrhenius diffusion can be corrected to give the true super- Arrhenius behavior by including a factor containing the average angle between steps in successive time intervals but without including an explicit description of the heterogeneity of the system. We find that both the short-time Arrhenius diffusion constants and the ergodic diffusion constants appear to have converged by a system size of 256 atoms. Therefore we are confident that any finite size effects are small. The correction factor shows that displacements are negatively correlated on average. The negative correlation is therefore directly linked to the increase in effective activation energy at low temperature. For shorter times, where a plateau in the mean square displacement is evident, the correction factor is not sufficient to regain super-arrhenius curvature. We can identify the point at which diffusive motion begins with departure from a master curve in the non- Gaussian parameter. We are able to describe the nonergodic diffusion constants by an Arrhenius form that changes with time, and using this form we can accurately reproduce the mean square displacement within our region of interest. An alternative equation to describe super-arrhenius behavior based on Arrhenius and non-arrhenius contributions is introduced, and appears to fit the data at least as well as the VTF or MCT forms. ACKNOWLEDGMENT V.K.D. is grateful to the EPSRC for financial support. 1 H. Vogel, Z. Phys. 22, G. S. Fulcher, J. Am. Ceram. Soc. 8, G. Tammann and W. Z. Hesse, Z. Anorg. Allg. Chem. 156, W. Götze, in Liquids, Freezing and the Glass Transition, Proceedings of Les Houches, Session LI, 1989, edited by J.-P. Hansen, D. Levesque, and J. Zinn-Justin North-Holland, Amsterdam, C. A. Angell, J. Non-Cryst. Solids , G. P. Johari, Philos. Mag. B 46, G. P. Johari, G. E. Johnson, and J. W. Goodby, Nature London 297, K. L. Ngai, R. W. Rendell, and D. J. Plazek, J. Chem. Phys. 94, R. Böhmer and C. A. Angell, Phys. Rev. B 45, R. Böhmer, K. L. Ngai, C. A. Angell, and D. J. Plazek, J. Chem. Phys. 99, R. Kohlrausch, Ann. Phys. 12, G. P. Johari and M. Goldstein, J. Chem. Phys. 53, G. P. Johari and M. Goldstein, J. Chem. Phys. 55, K. Vollmayr-Lee, W. Kob, K. Binder, and A. Zippelius, J. Chem. Phys. 116, W. K. Kegel and A. van Blaaderen, Science 287, E. R. Weeks, J. C. Crocker, A. C. Lewitt, A. Schofield, and D. A. Weitz, Science 287, M. Y. Cicerone, F. R. Blackburn, and M. D. Ediger, J. Chem. Phys. 102, M. D. Ediger, Annu. Rev. Phys. Chem. 51, H. R. Schober, Phys. Chem. Chem. Phys. 6, C. Oligschleger and H. R. Schober, Phys. Rev. B 59, A. I. Mel cuk, R. A. Ramos, H. Gould, W. Klein, and R. D. Mountain, Phys. Rev. Lett. 75, W. Kob, C. Donati, S. J. Plimpton, P. H. Poole, and S. C. Glotzer, Phys. Rev. Lett. 79, D. J. Wales and J. Uppenbrink, Phys. Rev. B 50, W. Jost, Diffusion in Solids, Liquids and Gases, 3rd ed. Academic Press, New York, A. Rahman, Phys. Rev. 136, A W. Kob and H. C. Andersen, Phys. Rev. E 51, D. Caprion, J. Matsui, and H. R. Schober, Phys. Rev. Lett. 85, S.-K. Ma, Statistical Mechanics World Scientific, Singapore,