How short-range attractions impact the structural order, self-diffusivity, and viscosity of a fluid
We present molecular simulation data for viscosity, self-diffusivity, and the local structural ordering of (i) a hard-sphere fluid and (ii) a square-well fluid with short-range attractions. The latter fluid exhibits a region of dynamic anomalies in its phase diagram, where its mobility increases upon isochoric cooling, which is found to be a subset of a larger region of structural anomalies, in which its pair correlations strengthen upon isochoric heating. This “cascade of anomalies” qualitatively resembles that found in recent simulations of liquid water. The results for the hard-sphere and square-well systems also show that the breakdown of the Stokes-Einstein relation upon supercooling occurs for conditions where viscosity and self-diffusivity develop different couplings to the degree of pairwise structural ordering of the liquid. We discuss how these couplings reflect dynamic heterogeneities. Finally, we note that the simulation data suggests how repulsive and attractive glasses may generally be characterized by two distinct levels of short-range structural order.
Colloidal fluids play a central role both in technology and in the study of condensed matter. Regarding the latter, suspensions of colloids are interesting model systems because they can behave collectively in ways that are similar to atomic and molecular liquids while simultaneously being both large and slow enough to allow experimental measurements of their real-space structure and dynamics. Despite the similarities between the behaviors of colloids, atoms, and molecules, there also are important, qualitative differences.Russel et al. (1989); Sciortino (2002) For example, the effective interparticle attractions between colloids can be “tuned” to be short-ranged relative to the particle diameter , unlike the dispersion attractions in atomic and molecular systems that decay more gradually with interparticle separation , i.e., as .
Short-ranged attractive (SRA) interactions have nontrivial implications for the static structural, equilibrium thermodynamic, and dynamic behaviors of colloidal suspensions. For instance, whereas cooling a simple atomic liquid generally slows down its dynamic processes, several recent studies have demonstrated that reducing the temperature (or increasing the attractive interactions) of an SRA fluid can have a non-monotonic effect on its mobility. In fact, concentrated SRA fluids can vitrify not only upon cooling, forming an “attractive” glass or gel, but also upon heating, forming a “repulsive” or hard-sphere (HS) glass.Eckert and Bartsch (2002); Pham et al. (2002); Bergenholtz and Fuchs (1999); Fabbian et al. (1999); Dawson et al. (2001); Zaccarelli et al. (2002); Sciortino (2002) Since the structural and mechanical properties of these glassy states can be quite different, there is a broad interest in understanding the properties of the precursor supercooled fluids from which they are formed.
In this article, we explore whether the unusual effect that temperature has on the dynamics of SRA fluids reflects a more general connection between the static structure and the dynamics of condensed phases. In doing so, we find it instructive to view the behavior of SRA fluids in the context of another well-studied system with anomalous dynamical trends, liquid water. Cold water behaves differently than simple fluids in that its mobility increases upon isothermal compression over a broad range of conditions. Interestingly, the state points where water shows this behavior is a subset of a larger region on its phase diagram where its local structural order anomalously decreases upon compression.Errington and Debenedetti (2001); Sastry (2001) In other words, there is a “cascade of anomalies”Errington and Debenedetti (2001) where the pressure-induced disordering of liquid water emerges at a lower density, and ultimately disappears at a higher density, when compared to the pressure-induced increase in its mobility. It has been recently arguedMittal et al. (2006); Errington et al. (2006); Sharma et al. (2006) that this behavior follows from the fact that water approximately obeys a scaling relationship between its self-diffusivity and the structural order parameter , where is the contribution to the fluid’s excess entropy due to its static oxygen-oxygen pair correlations, and is the Boltzmann constant.
Here, we use molecular dynamics simulations to systematically investigate the relationships between static structural order (), self-diffusivity (), and viscosity () for both a HS fluid and an SRA fluid. One of our main aims is to understand whether the latter shows a cascade of anomalies similar to that of liquid water. Specifically, we are interested in whether the region on the phase diagram of an SRA fluid where it becomes more mobile (higher , lower ) upon cooling is a subset of a larger set of conditions where the fluid becomes more ordered (higher ) upon heating. A second, and related, goal of this study is to explore whether the relationships between , , and can provide generic insights into the breakdown of the Stokes-Einstein (SE) relation () in deeply supercooled liquids and also into the structural properties of repulsive and attractive glasses.
The organization of the paper is as follows. In Section II, we describe the two model systems examined in this study and also the simulation methods used to carry out the investigation. Then, in Section III, we present our simulation results and discuss their relevance for understanding the connection between structural order and mobility in HS and SRA fluids. Finally, in Section IV, we present some concluding remarks.
Ii Modeling and Simulation
We focus on two model systems: a fluid of HS particles and a fluid of square-well particles with short-range attractions, the latter of which we denote the SW-SRA fluid. The interparticle potential between particles 1 and 2 for these systems is given by
where is the distance between the particle centers, is the interparticle diameter, and the parameters and set the magnitude and range of the interparticle attraction, respectively. For the HS fluid, one has , and thus there are no interparticle attractions. We choose for the SW-SRA fluid, a range similar to that of other SW-SRA fluids known to exhibit anomalous dynamical behaviorFoffi et al. (2002); Zaccarelli et al. (2002). In order to avoid crystallization in our study, and thus allow study of the supercooled fluid states, we have drawn the individual particle diameters of both systems from a Gaussian distribution with an average of and standard deviation of . For practical reasons, we truncated this distribution so that all particle diameters lie in the range . We have implicitly non-dimensionalized all reported quantities in this investigation by appropriate combinations of the characteristic length scale, and a model-dependent time scale . For the HS fluid, the characteristic time scale is cast in terms of the temperature , where is the particle mass, while for the SW-SRA fluid it is defined in terms of the attractive strength of the potential, .
To explore the behavior of these model systems, molecular dynamics simulations were performed using an event-driven algorithm Rapaport (2004) in the microcanonical ensemble. For all runs, particles were simulated in a cubic simulation cell of volume with periodic boundary conditions. Particle packing fractions in the range were investigated for both model systems, and temperatures in the range were examined for the SW-SRA fluid. For each thermodynamic state point, between three and ten independent simulations were performed in order to estimate the errors in the transport coefficients. Self-diffusivities were calculated by fitting the long time () mean-squared displacements to the Einstein relation for self-diffusivity . Zero-shear viscosities were calculated using the impulse limit of the Einstein formula.Alder et al. (1970) The structural order parameter was computed using the expression
where is the number density, and is the average pair correlation function. Note that for an ideal gas, and for a perfect crystalline lattice. Thus, as has been discussed at length elsewhere Truskett et al. (2000); Mittal et al. (2006); Errington et al. (2006), one can view as a scalar measure of the amount of pair-wise translational order of the system.
Iii Results and Discussion
iii.1 Transport and Structural Properties
We now examine some of the state-dependent transport and structural properties of the HS and SW-SRA fluids. For the HS fluid, the quantities , , and (non-dimensionalized as outlined in Section II) depend only on the packing fraction , whereas the corresponding dimensionless properties for the SW-SRA fluid depend on both packing fraction and temperature .
The transport properties of the HS fluid are displayed in Fig. 1. As expected, with increasing packing fraction , the self-diffusivity monotonically decreases, while the viscosity monotonically increases. The changes in both of these properties become more pronounced for , conditions for which can be accurately fitted using either Vogel-Fulcher or power-law functional forms Kumar et al. (2006) with a predicted divergence near .
Figure 1(c) illustrates that the SE relation in the slip limit, , is approximately obeyed by this HS fluid for . However, for higher packing fractions, large positive deviations from the slip limit of the SE relation become apparent. Kumar et al. (2006) This type of SE “breakdown” has been observed in studies of several glass-forming fluids in their deeply supercooled states.Rössler (1990); Cicerone et al. (1995); Cicerone and Ediger (1993); Fujara et al. (1992)
We note that, as observed by previous investigators,Kumar et al. (2006) the SE relation is not strictly obeyed by the HS fluid even at lower packing fractions. For example, the product varies by approximately as the packing fraction is increased from to . However, the breakdown of the SE relation is typically understood to occur when the product begins to exhibit a pronounced positive deviation from the slip value. In this work, we identify the breakdown of the SE relation by the heuristic criterion, . As can be seen from Figure 1(c), this threshold is crossed at for the HS fluid investigated here.
The simulated transport properties of the SW-SRA fluid are displayed in Fig. 2 as a function of reciprocal temperature along isochores. The most striking feature of this plot is that exhibits a maxima with inverse temperature, a behavior that has also been observed in both experiments and computer simulations of other SRA fluids. Eckert and Bartsch (2002); Pham et al. (2002); Bergenholtz and Fuchs (1999); Fabbian et al. (1999); Dawson et al. (2001); Zaccarelli et al. (2002); Puertas et al. (2003, 2002, 2004) As discussed in Section I, this trend becomes pronounced at high , where the system can ultimately form either a repulsive glass by isochoric heating or an attractive glass or gel via isochoric cooling.
The behavior of the zero-shear viscosity of the SW-SRA fluid, displayed in Fig. 2(b), qualitatively mirrors that of its self-diffusivity. For a more quantitative comparison, the SE relationship is plotted in Figure 2(c). Note that the slip limit of the SE relation is again approximately obeyed for the SW-SRA fluid over a broad range of and . However, for all values of investigated here, the SE relation breaks down at sufficiently low , as the attractive glass transition is approached. At high , on the other hand, only the highest isochore studied showed a significant breakdown of the SE relation. This asymmetry between high and low behaviors has also been observed in other model SRA fluids.Puertas et al. (2003, 2002); Foffi et al. (2002); Zaccarelli et al. (2002) It simply reflects the fact that one must reach a relatively high value of in these systems to achieve the level of frustration required to form a repulsive glass, whereas interparticle attractions can induce formation of the attractive glass at much lower particle concentrations.
It is natural to wonder whether the decrease in mobility that the HS and SW-SRA fluids experience upon approach to the glass transition is generally correlated with an increase in the amount of local structural order that they exhibit. To explore this issue, we first examine the behavior of the structural order parameter for the HS fluid. As can be seen in Fig. 3(a), for this system monotonically increases with packing fraction, indicating a strengthening of the pair-wise interparticle correlations. From the radial distribution functions shown in Fig. 3(b), it is also evident that these correlations correspond to the progressive development of well-defined coordination shells around the particles. The contributions of these shells to the translational order parameter become readily apparent when we investigate the following cumulative order integral,
Note that one recovers from this integral in the large limit, and thus quantifies the average amount of translational ordering on length scales smaller than surrounding a particle. In Fig. 3(c), we observe nearly step-wise increases in at the locations of the various coordination shells. It is also clear that increasing of the HS fluid has two main effects on its structural order. It strengthens the ordering within the individual coordination shells, and it uniformly increases the number of coordination shells (i.e., the range of order).
The behavior of the structural order parameter for the SW-SRA fluid is displayed in Fig. 4(a) as a function of reciprocal temperature along isochores. In accord with what might be expected based on the behavior of this fluid’s transport coefficients, displays broad minima. At low , heating the fluid decreases its structural order (similar to what happens in normal molecular fluids), but, at high , heating anomalously increases its structural order.
To determine the origins of this trend, we focus on the behaviors of the radial distribution function and the cumulative order integral for the SW-SRA fluid along the isochore. As can be seen in Fig. 4(b)-(e), heating induces subtle changes in that result in nontrivial cumulative changes in the structural order. In particular, heating the cold fluid () results in a barely detectable decrease in the height of the first peak of , and it has little effect on the other coordination shells. However, as is shown in Fig. 4(c), even these subtle modifications to result in significant changes to the cumulative order integral of the fluid. In particular, heating the cold fluid decreases the total amount of pair-wise structural order, even though the range of the order remains essentially the same. This change is qualitatively consistent with the thermal weakening of transient clustered configurations of physically “bonded” particles observed in SRA fluids at lower temperatures.Krekelberg et al. (2006a, b); Puertas et al. (2004); Sciortino (2002)
The effect of increasing temperature on the structure of the warm SW-SRA fluid () is different [see Fig. 4(d) and (e)]. In particular, heating continues to decrease the height of the first peak in , but it also broadens the peak and enhances the interparticle correlations associated with the other, more distant, coordination shells. The net effect is an increase in both the total amount of translational structural order and its range. This structural change is due to the fact that heating the warm fluid collapses the open channels of free volume that form at intermediate due to weak interparticle clustering,Krekelberg et al. (2006b) essentially jamming the particles into a less efficient, and more correlated, packing arrangement.Pham et al. (2002); Sciortino (2002); Zaccarelli et al. (2002); Krekelberg et al. (2006b)
It is clear from the results presented in this section that there is a qualitative (negative) correlation between pair-wise structural order and mobility in both the HS and SW-SRA fluids. In the following two sections, we explore the extent to which this connection can be made quantitative.
iii.2 Connection between structure and mobility anomalies
As discussed in Section III.1, at high and , both the self-diffusivity and the structural order of the SW-SRA fluid behave in a manner that is anomalous when compared to simple molecular fluids [see Figs. 2(a) and 4(a)]. Whereas increasing of a simple fluid generally increases its mobility, isochorically heating the warm SW-SRA fluid can result in slower single-particle dynamics. This latter behavior is characterized by the following inequality:
The structural order of a simple molecular fluid, on the other hand, normally decreases when it is isochorically heated. Therefore, we denote conditions for which the following inequality holds (i.e., order increases upon heating),
as “structurally anomalous”.
The locations of the regions for self-diffusivity and structural anomalies of the SW-SRA fluid in the - plane, as determined by numerical differentiation of the data in Figs. 2(a) and 4(a), respectively, are displayed in Fig. 5(a). A schematic based on the data is provided in Fig. 5(b). The most striking point is that the region of structural anomalies appears to completely envelop the region of self-diffusivity anomalies. In other words, the SW-SRA fluid exhibits unusual -dependencies for its single-particle dynamics only for those state points where it also exibits unusual -dependencies for its structural order.
This “cascade” of structural and dynamic anomalies shown by the SW-SRA fluid is very similar to that observed originally in simulations of the SPC/E model of water Errington and Debenedetti (2001) and later in simulations of other simpler models that also show waterlike behavior. Yan et al. (2006); de Oliveira et al. (2006a, b); Errington et al. (2006); Shell et al. (2002) Recall that cold water behaves differently from simple fluids over a wide range of conditions in that its mobility increases, while its structural order decreases, when it is isothermally compressed. Thus, the generic similarity between the behavior of water and the SW-SRA fluid shown in Fig. 5 is that, in both cases, the region on the phase diagram where mobility anomalies occur is a subset of the region where structural anomalies are found. It has been recently arguedMittal et al. (2006); Errington et al. (2006); Sharma et al. (2006) that this type of behavior for water follows directly from the fact that liquid water approximately obeys a scaling relationship between its self-diffusivity and its translational structural order parameter over a wide range of temperature and density. In the next section, we test whether there exist similar scaling relations between and the transport coefficients of the HS and SW-SRA fluids. We also discuss how such relations might provide insights into the breakdown of the SE relation for these systems.
iii.3 Structure-property relations and the breakdown of Stokes-Einstein
As alluded to above, recent molecular dynamics simulationsMittal et al. (2006), motivated by other earlier observations of RosenfeldRosenfeld (1977, 1999) and DzugutovDzugutov (1996), have demonstrated that the following simple relation is approximately obeyed by various model fluids in their equilibrium liquid states:
|where and are parameters which may depend on packing fraction (i.e., density), but not on . Results from other earlier theoretical studies,Rosenfeld (1977) and considerations based on the Stokes-Einstein relation, suggest that a similar relationship should approximately hold for the zero-shear viscosity of these equilibrium fluids:|
where, again, and may depend on packing fraction (density) only.
However, given that the SE relation breaks down as a liquid is supercooled, it is apparent that Eq. (6a) and (6b), with coefficients fit to higher temperature equilibrium fluid data, cannot also describe the transport coefficients in deeply supercooled liquid states. Nonetheless, it has recently been shown that the functional form of Eq. (6a) can in fact approximately describe the isochoric self-diffusivity data of model supercooled liquids over a broad range of temperatures if a different pair of parameters and are adopted.Mittal et al. (2006) In other words, there appears to be a crossover upon cooling where the self-diffusivities of fluids transition from being approximately described by for equilibrium states to being approximately described by for supercooled conditions. Of course, even this latter relation must eventually fail for conditions very near the glass transition where both and rapidly vanish, while remains finite. Truskett et al. (2000) We will return to this last point at the end of the section.
The structure-property scalings discussed above suggest several interesting questions concerning the liquid state. For example, does the aforementioned crossover between scaling relations occur near the breakdown of the SE relation? Furthermore, does a similar crossover for the dependence of occur upon supercooling? If so, does it coincide with the crossover point for ? Below, we use our molecular dynamics simulation results to investigate these questions for the HS and SW-SRA fluids. The idea is that the new information that we gain about how and couple to pair structure should give insights into the SE breakdown and the general effects that supercooling has on liquids.
First, we consider the HS fluid. Figure 6 displays the transport coefficients and the SE relationship for this system as a function of the structural order order parameter, , which itself depends only on packing fraction . Although the basic scaling relations outlined above describe the behavior of attractive fluids along isochores, they apply equally well for athermal fluids (such as this) along isotherms, keeping in mind that the glass transition in the latter is approached by compression and not by cooling. One important aspect to note in Fig. 6 is that there is, in fact, a crossover between “equilibrium” (low ) and “supercooled” (high ) states. In fact, Eq. (6) provides a quantitative fit to the data for packing fractions in the range (). Interestingly, the point at which both transport coefficients diverge from this equilibrium scaling relationship closely coincides with the packing fraction () where , i.e., the breakdown of the SE relation.
It is also evident from Fig. 6 that one can use the scaling form of Eq. (6) with different pairs of coefficients to approximately describe the dependencies of the transport coefficients for the supercooled HS fluid. The fits of Eq. (6) for these supercooled states, however, are not as accurate as those for the equilibrium fluid data below the crossover. In fact, our only goal in fitting the supercooled liquid data to this exponential form is that it allows us to extract simple quantitative measures [ and ] of the couplings that exist between the transport coefficients and the static structure of the fluid. It can be seen both from the raw data in Fig. 6 and from the values of these coefficients for the “equilibrium” and “supercooled” HS fluid [(, ) and (, ), respectively] that the breakdown of the SE relation coincides with qualitative change in how static structure correlates to and . In the equilibrium fluid, both transport coefficients show weak, and roughly equivalent, couplings to . However, after the SE breakdown, the transport coefficients develop much stronger couplings to structural order. This is presumably due to the integral role that cooperative structural rearrangements play in the relaxation of deeply supercooled liquids.Debenedetti (1996)
It is clear from Fig. 6 that the reason that the SE relation breaks down in this system is because the viscosity of the deeply supercooled HS fluid becomes much more sensitive to changes in static structural order than the self-diffusivity. This is consistent with the observations of previous studies that have correlated the breakdown of the SE relationship to the onset of heterogeneous dynamics.Kumar et al. (2006); Yamamoto and Onuki (1998); Ediger (2000); Stillinger and Hodgdon (1994); Tarjus and Kivelson (1995) Heterogeneous dynamics is typically characterized by the presence of many particles that transiently exhibit exceedingly high or low values mobility relative to the mean. The highly mobile particles have been shown to readily diffuse distances on the order of a particle diameter by so-called “hopping” motions. It is presumably the presence of these highly mobile particles that allows the self-diffusivity to maintain a weaker coupling to static structure than the viscosity.
Displayed in Fig. 7 are the scaled transport coefficients, and , of the SW-SRA fluid as a function of the structural order parameter . The factor is included here to remove the trivial “thermal velocity” contribution to these transport coefficients, which then allows them to be directly compared to the dimensionless values of and of the HS fluid. Along these lines, it should be noted that the high temperature data points for , , and of the SW-SRA fluid (, denoted in Fig. 7 by filled symbols) are indeed approximately described by the corresponding data for , and of the HS fluid. One consequence of this is that the breakdown of the SE relation for the SW-SRA fluid upon approaching the repulsive glass transition by heating occurs at approximately the same value of as the breakdown of the SE relation for the HS fluid upon compression. In contrast, the value of of the SW-SRA fluid at the breakdown of the SE relation upon cooling toward the attractive glass is different for each packing fraction studied. As might be expected, fluids with lower packing fractions show departures from the slip limit of the SE relation for conditions where they exhibit lower amounts of translational structural order.
Do the SW-SRA transport coefficients also follow “equilibrium” and “supercooled” exponential scaling branches when plotted versus ? For and intermediate temperatures, the data approximately collapse onto the same exponential scaling relation obtained from the fit of the equilibrium HS fluid data (shown in all panels of Fig. 7 as a blue line). At sufficiently high or low temperatures, however, the SW-SRA data transitions to “supercooled” exponential scalings with different coupling coefficients and . Interestingly, these transitions closely coincide with the breakdown of the SE relation. In all cases, similar to the HS fluid, exhibits a considerably weaker dependence on than does (i.e., , see Fig. 8) after the SE breakdown. Although the difference between the coupling coefficients increases with , the relative magnitude remains approximately constant () and very similar to that of the HS fluid. This latter structure-property connection is general in the sense that it approximately holds for these model liquids as they become “supercooled” via heating, cooling, or compression.
We would like to conclude this section with a speculation. Although it is not yet possible to equilibrate molecular dynamics simulations very close to the repulsive or attractive glass transitions in the SW-SRA fluid, the data shown in Fig. 7(a) and (b) are suggestive of what may happen to the structural order of the fluid in those limits. As discussed earlier, the high temperature behaviors of the SW-SRA transport coefficients, when plotted versus , approximately follow the trends of the HS fluid. Therefore, it is reasonable to suspect that the structural order parameter of the high temperature (repulsive) glass will be close to that of the HS glass [Fig. 9(a)]. At low temperatures, the transport coefficients of the SW-SRA fluid as a function of also appear as if they may asymptote to a different, lower, limiting value [Fig. 9(a)], i.e., a value of the structural order parameter characteristic of the attractive glass. The resulting picture is that the repulsive and attractive glass lines closely follow two “iso-” curves, with a sharp transition between the two in a narrow temperature range, shown schematically in Fig. 9(b)). Since can be readily obtained from static pair correlations, this is a speculation that could be tested via experiments of SRA colloidal fluids. Weeks et al. (2000) It would also be interesting to explore in future studies the extent to which liquid-state (e.g., mode-coupling) theories for SRA fluids are able to reproduce the empirical connection between and dynamics found in our simulations and to test the predictions that they make about how the structural order varies along the ideal repulsive and attractive glass lines.
We have presented new molecular simulation data for viscosity, self-diffusivity, and the local structural ordering of both a hard-sphere fluid and a square-well fluid with short-range attractions (relative to the particle diameter). We found that the latter system has a region of mobility anomalies in the temperature-packing fraction plane, where its self-diffusivity increases upon isochoric cooling. This region is entirely enclosed within a wider set of state points where the fluid’s pair correlations strengthen upon isochoric heating. This type of “cascade of anomalies” is very similar to that found in recent simulations of liquid water, and it follows from a broader connection between static structure and dynamics in condensed phase systems.
Both the hard-sphere and square-well fluids show that the breakdown of the Stokes-Einstein relation upon supercooling occurs for conditions where viscosity and self-diffusivity develop different couplings to the degree of pairwise structural ordering of the liquid. We discussed how these couplings reflect dynamic heterogeneities. Finally, we provided an experimentally testable hypothesis about how repulsive and attractive glasses may be generally characterized by two distinct levels of short-range structural order. In future work, we will investigate whether there are similar connections between non-equilibrium dynamics (e.g. shear-dependent viscosity) and structural order in these systems.
WPK acknowledges financial support of the National Science Foundation for a Graduate Research Fellowship. TMT acknowledges financial support of the National Science Foundation (CTS 0448721), the David and Lucile Packard Foundation, and the Alfred P. Sloan Foundation. VG acknowledges financial support of the Robert A. Welch Foundation and the Alfred P. Sloan Foundation. Computer simulations for this study were performed at the Texas Advanced Computing Center (TACC).
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