Mode coupling theory diffusion

XX. Mode Coupling Theory of Diffusion in One-Dimensional Lennard-Jones Rods... 

Figure 2. A pictorial representation of the mode coupling theory scheme for the calculation of the time-dependent friction (f) on a tagged molecule at time t. The rest of the notation is as follows Fs(q,t), self-scattering function F(q,t), intermediate scattering function D, self-diffusion coefficient t]s(t), time-dependnet shear viscosity Cu(q,t), longitudinal current correlation function C q,t), longitudinal current correlation functioa...

The most important prediction of the mode coupling theory is the temperature or the density dependence of the relaxation time, tmc(< )- MCT predicts that this relaxation time grows as a power law as the glass transition is approached (from the supercooled liquid side). This is because the diffusion coefficient Do of the liquid goes to zero in the following fashion ... 

Figure 8. The ratio of the self-diffusion coefficient of the solute (Di) to that of the solvent molecules (D ) plotted as a function of the solvent-solute size ratio (

Among the earlier theoretical formulations of the diffusion in 2-D systems, certainly the kinetic theory and the mode-mode coupling theory... 

There have been various approaches in the mode coupling theory [9, 37, 57, 176]. All these theories have exhibited the presence of t 3/2 of the velocity autocorrelation function in the asymptotic limit in three dimensions. Extending each of these theories for studies in two dimensions we can show that the velocity autocorrelation function has r1 tail in the asymptotic limit. Since the diffusion coefficient is related to Cv(t) through Eq. (337), it can be shown that D diverges in the long time due to the presence of this t l tail in the VACF. 

Figure 19. Time-dependent diffusion D i) of a two-dimensional system plotted against reduced time. The solid line represents the D t) obtained from the mode coupling theory (MCT) calculation, and the short-dashed line and the long-dashed line represent the D(t) obtained from simulated VACF and MSD, respectively. In the inset, fits to long-time D(t) to Eq. (351) are also shown. The plots are at p = 0.7932 and T = 0.7. The time is scaled by TJC = Jma2/c. D(t) is scaled by o2/. This figure has been taken from Ref. 175.

Note that the above study is performed for a simple system. There exists a large body of literature on the study of diffusion in complex quasi-two-dimensional systems—for example, a collodial suspension. In these systems the diffusion can have a finite value even at long time. Schofield, Marcus, and Rice [17] have recently carried out a mode coupling theory analysis of a quasi-two-dimensional colloids. In this work, equations for the dynamics of the memory functions were derived and solved self-consistently. An important aspect of this work is a detailed calculation of wavenumber- and frequency-dependent viscosity. It was found that the functional form of the dynamics of the suspension is determined principally by the binary collisions, although the mode coupling part has significant effect on the longtime diffusion. 

XX. MODE COUPLING THEORY OF DIFFUSION IN ONE-DIMENSIONAL LENNARD-JONES RODS... 

Recently a mode coupling theory study of diffusion and velocity correlation function of a one-dimensional LJ system was carried out [186]. This study reveals that the 1/f3 decay of the velocity correlation function could arise from the coupling of the tagged particle motion to the longitudinal current mode of the surrounding fluid. In this section a brief account of this study is presented. 

Mode coupling theory of binary mixtures where the constituents are of rather different sizes is a challenging task, as we have already discussed while addressing the mass depenence of diffusion. In addition to the problem with proper formulation of mode coupling terms, there is an additional difficulty of the nonavailability of the equilibrium two-particle correlation functions The existing integral equation theories become unstable when the size ratio exceeds a certain (low) value, like 1.5 or so [195],... 

Figure 47. Diffusion coefficient D as obtained from a molecular dynamics simulation study of a binary Lennard-Jones system reaching temperatures below the crossover temperature of mode coupling theory (MCT). Solid line represents interpolation by MCT power law note the large temperature range covered by the power law. (From Ref. 371.)...

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