| dc.description.abstract | Isothermal–Isobaric (NPT) Ensemble Molecular Dynamics Simulations of Composition Fluctuations and Diffusion in Binary Mixtures
In this chapter, isothermal–isobaric (NPT) ensemble molecular dynamics simulations have been performed to study the composition fluctuations in non-phase-separating dense Lennard-Jones binary mixtures of two types, in both of which the interaction between the dissimilar species is more favored. One of the model binary mixtures (known as the Kob–Andersen model) is a well-known glass former.
It is apparent from these simulations that irrespective of the size of the spherical volumes considered, the composition fluctuations and their intercorrelations are well described by Gaussian statistics. The standard deviation of the distributions is found to be quite large, indicating broad distributions. The equilibrium spontaneous fluctuations in the two components are shown to be negatively correlated, indicating that the occurrence of total density fluctuations in these local volumes is very small.
The relative abundance of the composition fluctuation indicates the presence of many different compositions within small volumes. Thus, the system is indeed locally heterogeneous. It will be a worthwhile exercise to find the distribution of relaxation times of these regions. However, this is meaningful only when there is a broad spread in the relaxation times, expected in supercooled liquid.
It is found that the time correlations of fluctuations in N? and N? show non-exponential dynamics. As expected, the cross-correlation decay is slowest because in the binary mixtures of composition N?/N = 0.2, most of the B molecules are surrounded by A molecules. This result clearly suggests that the most slowly relaxing local configurations should involve B molecules surrounded by A molecules - that is, maximize the number of A–B bonds. Recently, Glotzer et al. [4] have explored the dynamics in the KA model along this line.
Diffusion and Mode-Coupling Analysis
The main results presented in this chapter are now summarized. Molecular dynamics simulations have been carried out to study the diffusion of small light particles in a solvent composed of larger massive particles for a fixed solvent-to-solute size ratio (S? = 5) but with a large variation in mass ratio (where the mass of the solute is kept constant). In addition, a mode-coupling theory (MCT) analysis of diffusion is also presented.
It is found that the solute dynamics remain surprisingly coupled to the solvent dynamics even in the limit of highly massive solvent. Most interestingly, with increase in mass ratio, the self-intermediate scattering function of the solute develops a stretching at long time which, for intermediate values of mass ratio, could be fitted to a single stretched exponential function with the stretching exponent ? 0.6. In the limit of very large mass ratio, the existence of two stretched exponentials separated by a power-law type plateau is observed.
This behavior is found to arise from increasingly heterogeneous environment probed by the solute particle as one increases the mass of the solvent particles. The MCT calculation of self-diffusion is found to agree qualitatively with the simulation results for small mass ratio. However, it fails to describe the simulated prediction at large mass ratios. The velocity correlation function of the solute shows interesting oscillatory structure.
Connection to Glassy Dynamics
Several of the results observed here are reminiscent of the relaxation of the self-intermediate scattering function, F?(k,t) observed in deeply supercooled liquids near their glass transition temperature. In that case also, one often observes a combination of power-law and stretched exponential in the decay of the intermediate scattering function.
It has been shown that even the breakdown of MCT at large mass ratio could be connected to its breakdown near the glass transition temperature because it is the neglect of the spatial hopping mode of particles which is responsible for the breakdown of MCT. It is to be noted that those hoppings which are mostly ballistic in nature (after a binary collision) have already been incorporated in MCT. However, MCT does not include the hoppings which involve collective displacement involving several molecules [15].
It should be pointed out that in the MCT calculation, the contribution of the current term has been neglected. While the current contribution may improve the agreement between the simulation and MCT result for small mass ratio (M? = 5), its contribution at larger mass ratio is not expected to change the results significantly because the discrepancy is very large.
Origin of Power-Law and Stretching
The origin of the power law remains to be investigated in more detail. Preliminary analysis shows that this may be due to the separation of the time scale between the first weakly stretched exponential (due to the dispersion in the binary-type interaction term) and the second, later more strongly stretched exponential (which is due to the coupling of the solute’s motion to the density mode of the slow solvent). This separation arises because these two motions are very different in nature. However, a quantitative theory of this stretching and power-law is not available at present.
While the origin of the stretching of F?(k,t) can be at least qualitatively understood in terms of the inhomogeneity experienced by the solute, the origin of hopping is less clear. In the supercooled liquid, hopping is found to be correlated with an anisotropic local stress [15], which is unlikely in the present system which is at lower density and pressure.
Finally, it is to be noted that the system investigated here is a good candidate to understand qualitative features of relaxation in a large variety of systems, such as concentrated solutions of polysaccharide in water and also motion of water in clay. Molecular dynamics simulation results for the time-dependent pair distribution functions in a strongly non-ideal glass-forming binary Lennard-Jones mixture are presented. In addition, a mean-field description of the pair dynamics is considered, and the comparison is made with the simulated distributions.
The main goal of this investigation was to explore the dynamics of supercooled liquids in a more collective way by following the relative motion of the atoms rather than their absolute motion. It is shown that the three pairs (AA, BB, and AB) behave differently. The analysis of the trajectory shows clear evidence of jump motions for all three pairs.
The relative diffusion constant of the BB pair (D??) is almost twice the value for the AA pair (D??). This clearly suggests the importance of jump dynamics for the B particles, and indeed, the motion of the B particles is found to be mostly discontinuous in nature, while the A particles show occasional hopping.
The dynamic inhomogeneity present in a supercooled liquid is generally characterized by the well-known non-Gaussian parameter ??(t), which describes deviations from Gaussian behavior in the motion of a single atom. In this chapter, this concept has been generalized, and a non-Gaussian parameter for the pair dynamics [???(t)] is introduced to measure deviations from Gaussian behavior in the relative motion of atoms. At intermediate times, all three pair distribution functions show significant deviations from Gaussian behavior, with different degrees.
It is found that for the nearest-neighbor AA and AB pairs, which are confined to a strong effective potential and merely make anharmonic motions in it, the dynamics can be treated at the mean-field level. However, as the motion of a nearest-neighbor BB pair is highly anharmonic, one must include the effects of fluctuations about the mean-force field to get a correct description of the dynamics.
While the mean-field treatment provides a reasonably accurate description of pair dynamics (at least for AA and AB pairs), it must be supplemented with the time-dependent diffusion coefficient D(t). This is a limitation of the mean-field approach because at present we do not have any theoretical means to calculate D(t) from first principles.
The mode-coupling theory (MCT) does not work because it neglects hopping, which is the dominant mode of mass transport in deeply supercooled liquids, even when the system is quite far from the glass transition. As discussed recently, hopping can be coupled to anisotropy in the local stress tensor [26]. The calculation of the latter is also nontrivial.Large-scale computer simulations of a supercooled Lennard-Jones (LJ) polydisperse system have been carried out with large variations in temperature at a fixed high pressure. Characteristic of a fragile glass former, the super-Arrhenius temperature dependence is observed for the viscosity and the self-diffusion coefficients of different size particles.
Interestingly, it is shown that the critical glass transition temperature (from VFT) for diffusion (T???) increases with the size of the particles, and the critical temperature for viscosity (T??) is larger than that for diffusion. Furthermore, a marked deviation from Stokesian diffusion is observed, where the dependence on size of the particles is highly nonlinear.
At low temperatures, it is found that hopping is the dominant mode for mass transport for both the smallest and largest size particles. However, the crossover from continuous Brownian to hopping motion takes place at shorter time scales for the smaller size particles.
In the present system, the size of all the particles is different. It would be interesting to see whether the jump motion executed by the individual particles occurs over a single energy barrier or via a number of “intermediate” inherent structures in the potential energy landscape. Recent molecular dynamics simulations on LJ binary mixtures [4] have shown that such a transition does not correspond to transitions of the system over single energy barriers.
In addition, in a deeply supercooled liquid, the jump motions are associated with strong nearest-neighbor correlations, in which several neighboring atoms jump at successive close times [7, 8, 28]. Similar correlations have been observed here also. Recently, a computer simulation study of a deeply supercooled binary mixture [9] has shown that local anisotropy in the stress is responsible (at least partly) for particle hopping. However, the molecular origin of the jump motions observed here (highly disordered system) is not clear.
Density Functional Theory (DFT) Analysis
In this chapter, the standard form of density functional theory (DFT) has been used to calculate the free energy penalty to create soft localized density fluctuations in a hard-sphere liquid. The scaled density has been varied continuously from 0.99 to 1.09, where the uniform liquid of density 1.04 is used as a reference system.
It is found that the free energy required is much less to create a local inhomogeneity of small size compared to that for a large size. This is attributed to the sharp maximum of the static structure factor S(k) at intermediate wave numbers (ka ? 2?) and also to the very low compressibility of supercooled liquid at low wave numbers.
It is shown that the liquid almost behaves like a homogeneous liquid on the length scale larger than about 5.0a (where a is the molecular diameter), which agrees qualitatively with recent experimental results [8, 9]. In addition, it is shown that the inclusion of the surface energy effect will more likely reduce the probability of small-size “density droplets” than droplets of large size. However, it is suggested that the surface effect has very small contribution to the total free energy cost for forming these droplets.
Comparison of Approximations
The results obtained for hard-sphere liquid using the Percus–Yevick (PY) approximation for the direct pair correlation function [36] have been compared with the soft mean spherical approximation (SMSA) [37] applied to the dense Lennard-Jones liquid. The fluctuations are found to be somewhat larger in size for the continuous potentials due to the increase in spatial correlation length.
Orientational Correlation Functions
Theoretically obtained inhomogeneous probability distributions have been used to calculate both the equilibrium and nonequilibrium orientational correlation functions. It is found that the spatially heterogeneous distribution of the density is responsible for the non-exponential nature of the rotational relaxation.
For nonequilibrium distribution, the average rotational correlation time of molecules relative to that of equilibrium distribution is found to increase with the average density of the liquid. The theoretical results have been compared with the experimental results of Ediger and co-workers [2, 3, 4] obtained recently by photobleaching technique. Good qualitative agreement is found between the theoretical results presented here and the experimental results.
Relaxation of Nonequilibrium Distribution
Relaxation of the nonequilibrium distribution has been studied qualitatively using the simple non-interacting two-state exchange model, where only two different domains with different densities were considered. The results obtained in this model show that the extremely slow relaxation process observed near the glass transition point [2, 3, 4] cannot be explained by the translational diffusion of a molecule between regions of different dynamics.
It is suggested that the transitions between different local minima of the free energy near the glass transition could be responsible for this very slow relaxation. | |
