| dc.description.abstract | In this chapter, the existing mode coupling theory [14] has been extended to calculate the diffusion of a tagged molecule and to study the dynamics of a neat liquid.
It has been shown that in the earlier theory the definition of the memory function of the binary propagator for the Green’s function was erroneous. This led to a wrong definition of the short-time dynamics of the propagators in the ring collision terms. The proposed way to describe the memory function of the propagator is to consider the inertial motion of the solute and short-time collective dynamics of the solvent instead of the full dynamics of the solvent as has been done in the earlier formulation. A derivation of the frequency-dependent friction is presented with this new definition. It is found that although the expressions for the ring collision terms differ, the binary term remains unchanged. The short-time dynamics of the propagators in the ring collision terms are now found to be given by the product of the inertial motion of the solute and short-time collective dynamics of the solvent.
It has been stated that both formulations give similar results for solutes having the same mass as the solvent. A graphical analysis is presented which shows why these two formulations give similar results. It is found that for solutes having the same mass as the solvent, the product of the inertial dynamics of the solute and the short-time collective dynamics of the solvent is almost the same as the product of the inertial dynamics of the solute and the full dynamics of the solvent. The error shows up for solutes which are heavier than the solvent as will be shown in Chapter 6.
A novel scheme has been introduced to calculate the frequency-dependent friction self-consistently with the mean square displacement or the velocity autocorrelation function over the whole frequency and time plane. This scheme allows the calculation of the friction to be fully self-consistent where only the density and the temperature are required as input. No adjustable parameters have been used at any stage of the calculation.
The time dependence of the different contributions to the friction/diffusion has been studied. It is found that the binary term contributes at very short time, the contribution from the density fluctuation comes in the intermediate time whereas the current term contributes in the long time. The time scale of decay of the binary collision is found to be in the femtosecond regime. It is found to decrease at lower density and higher temperature due to decrease in the correlations.
The velocity autocorrelation function calculated at different densities and temperatures shows that at low density and high temperature the VACF decays monotonically to zero whereas at higher density it shows a negative dip and then approaches zero. The negative dip arises from the backscattering of the particle.
The calculated values of the diffusion coefficient are found to be in good agreement with the computer simulation studies [26, 27]. It is also found that the binary collision term and the coupling to the density fluctuation make major contributions to the diffusion in the intermediate and high-density regime. The current term contribution, although increases at lower density or higher temperature, remains quite small over the whole density and temperature range studied here.
A derivation of the viscosity is also presented in this chapter following the formulation of Balucani [28, 29] and Geszti [30]. The calculated values of the viscosity are found to be in good agreement with the computer simulation studies [26].
Thus, the study presented in this chapter suggests that the extended mode coupling theory (with the proper description and with the new self-consistent scheme) can describe the diffusion of a tagged particle in a Lennard-Jones neat liquid. Hence, the theory is expected to provide accurate description of the solvent dynamics which, as mentioned in the Introduction of this chapter, is required to describe many chemical and physical processes. It will be shown in Chapter 3 that the theory indeed provides a good description of the solvent dynamics. Since the theory is successful in describing the dynamics and diffusion in neat liquids, an extension of this theory for binary systems (where the solute has different size, interaction energy and mass) is also expected to be successful in describing the solute diffusion. It will be shown in the next few chapters (4, 5 and 6) that the size, interaction energy and mass dependence of the solute diffusion for a binary system at infinite solute dilution can be well described by this extended mode coupling theory.
In this chapter, the relation between friction and viscosity in both normal and supercooled liquids is studied. Let us first summarize the results obtained in the normal liquid regime. First, it is shown that the short-time viscous and frictional responses in a neat liquid occur essentially on the same time scale. The time dependence of this response is largely Gaussian which is followed by a slow long-time decay. This biphasic response is a hallmark of dynamical processes in dense liquid. This is clearly reflected in the imaginary part of the frequency-dependent friction and viscosity. The study of frequency-dependent friction at a constant temperature and different densities demonstrates the emergence of the solvent cage in dense liquids. The second important result is that the time scales of the initial decay are of the order of 100 fs, which is typically the time scale observed both in polar and non-polar solvation dynamics. In the present case, the ultrafast dynamics originates clearly from the nearest neighbor static correlation. The third important result is the demonstration that the apparent validity of a Stokes-like relation between friction and viscosity has its root both in statics and dynamics. While the initial values of these quantities primarily determine the ratio, the nearly identical dynamics sustains this ratio even in the frequency plane.The friction is determined by the microscopic terms and numerically the friction converges to a value close to (but less than) 6?R6\pi R6?R. Thus, this analysis seems to suggest that the origin behind the validity of the Stokes relation for the same size solute is that both the microscopic friction and the viscosity are determined essentially by the same dynamical variables. It has been emphasized that the occurrence of 6?6\pi6? is not to be taken as a signature of the stick boundary condition.
The above discussion leads to the following important point: For a neat liquid, the Navier–Stokes hydrodynamics cannot be used to justify apparent numerical validity of the Stokes relation. In this case, the validity of a Stokes-like relation between the viscosity and the friction can be explained only when the contributions from the bare (that is, the binary) and the density modes are both taken into account. Another point of interest is that while in hydrodynamics-based analysis it is believed that it is the viscosity which determines the friction, the present study suggests that perhaps it is more meaningful to think in terms of the reverse scenario. In the region where hydrodynamics is not valid but a Stokes-like relation is obeyed, it is the diffusion (or the friction) which determines the viscosity and not vice versa.
In the second part, it is shown that a phenomenological model of the heterogeneity in the liquid can explain the decoupling of the friction and viscosity in the supercooled liquids. The expressions for friction and viscosity remain the same as in the normal liquid regime, but the dynamical variables such as the dynamic structure factor develop a long-time tail within a very narrow density range. This long-time tail of the dynamic structure factor accounts for the rapid rise of the value of viscosity and friction over this narrow density range. The decoupling arises due to the existence of the slowly relaxing solid-like microdomains in the supercooled liquids. The presence of these solid-like domains gives rise to inhomogeneity in the liquid with some domains which are solid-like and others which are liquid-like. In each of these domains, the SE law will be valid, but the overall value of the diffusion/friction decouples from the measured viscosity.
Note that the study presented here can explain decoupling in neat liquids only in a semi-quantitative manner. To present a quantitative study of the decoupling, a detailed calculation which considers different domains in the solvent is to be performed. The crux of the problem is to find an accurate distribution function, like P[F(q,t)]P[F(q,t)]P[F(q,t)], of the domains. This remains an unsolved problem. | |
