Mode coupling theories functions

Rostov, K.S. Freed, K.F., Mode coupling theory for calculating the memory functions of flexible chain molecules influence on the long time dynamics of oligoglycines, J. Chem. Phys. 1997,106, 771-783... 

Scattering Function in the Framework of the Idealized Mode-Coupling Theory A Monte Carlo Study for Polymer Melts. 

The relaxation equations for the time correlation functions are derived formally by using the projection operator technique [12]. This relaxation equation has the same structure as a generalized Langevin equation. The mode coupling theory provides microscopic, albeit approximate, expressions for the wavevector- and frequency-dependent memory functions. One important aspect of the mode coupling theory is the intimate relation between the static microscopic structure of the liquid and the transport properties. In fact, even now, realistic calculations using MCT is often not possible because of the nonavailability of the static pair correlation functions for complex inter-molecular potential. 

There have been several treatments to calculate correlation functions and the transport coefficients near the critical point (Fixman [35], Kawasaki [36] and Kadanoff and Swift [37]). All these treatments embody essentially the same physical ideas and contains the genesis of the modem mode coupling theory. Here we discuss the treatment of Kadanoff and Swift [37] because this is physically the most transparent one and seems to have influenced the latter development of the mode coupling theory in a more significant manner. 

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...

It should be noted that although in Eq. (90) only the connected motion of the solute and the solvent is retained, in the argument presented on the time scale it is the disconnected parts which have been considered. This is because in the latter part, for the derivation of the expression of Ci. the solute and the solvent motions are assumed to be disconnected. This assumption is the same as those made in the density functional theory and also in mode coupling theories where a four-point correlation function is approximated as the product of two two-point correlation functions. This approximation when incorporated in Ci. means that after the binary collision takes place, the disturbances in the medium will propagate independently. A more exact calculation would be to consider the whole four-point correlation function, thus considering the dynamics of the solute and the solvent to be correlated even after the binary collision is over. Such a calculation is quite cumbersome and has not been performed yet. 

Mode coupling theory provides the following rationale for the known validity of the Stokes relation between the zero frequency friction and the viscosity. According to MCT, both these quantities are primarily determined by the static and dynamic structure factors of the solvent. Hence both vary similarly with density and temperature. This calls into question the justification of the use of the generalized hydrodynamics for molecular processes. The question gathers further relevance from the fact that the time (t) correlation function determining friction (the force-force) and that determining viscosity (the stress-stress) are microscopically different. 

Here we present a different prescription to calculate the dynamic structure factor or the intermediate scattering function in the supercooled regime. This is a quantitative approach based on the basic result of the mode coupling theory. The effect of the mode coupling term in the intermediate scattering function is written in a simpler way by the following expression ... 

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 (

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. 

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. 

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. 

The mode coupling theory of molecular liquids could be a rich area of research because there are a large number of experimental results that are still unexplained. For example, there is still no fully self-consistent theory of orientational relaxation in dense dipolar liquids. Preliminary work in this area indicated that the long-time dynamics of the orientational time correlation functions can show highly non-exponential dynamics as a result of strong in-termolecular correlations [189, 190]. The formulation of this problem, however, poses formidable difficulties. First, we need to derive an expression for the wavevector-dependent orientational correlation functions C >m(k, t), which are defined as... 

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],... 

A mode coupling theory is recently developed [135] which goes beyond the time-dependent density functional theory method. In this theory a projection operator formalism is used to derive an expression for the coupling vertex projecting the fluctuating transition frequency onto the subspace spanned by the product of the solvent self-density and solvent collective density modes. The theory has been applied to the case of nonpolar solvation dynamics of dense Lennard-Jones fluid. Also it has been extended to the case of solvation dynamics of the LJ fluid in the supercritical state [135],... 

Figure 4.7 Cole-Davidson exponent as a function of temperature for glycerol (circles), propylene glycol (squares), and propylene carbonate (triangles). is the temperature at which the crossover from Arrhenius to VFTH behavior is observed, while is the best fit to the critical temperature of the mode-coupling theory (see Section 4.6). (From Schonhals et al. 1993, reprinted with permission from the American Physical Society.)...

Figure 4.24 Normalized intermediate scattering function versus time for NiZr from molecular d)uiamics simulations at wavevectors q = (a) 21.6 nm- and (b) 10.8 nm. The lines are fits using a modified mode-coupling theory. (From Teichler 1996, reprinted with permission from the American Physi-cal Society.)...

Slow relaxation of water was further discussed in more broad contexts. In particular, since the IS picture is expected to become appropriate as the temperature is decreased, special attention has been paid to water in supercooled states, and dynamics in supercooled states has been investigated in relation to applicability of the mode-coupling theory [51]. It was found that the bond lifetime of individual molecules obeys the thermal process, whereas the bond correlation function shows power-law behavior [52,53], The behavior below or above the temperature at which the mode-coupling theory can be applied was also studied and the transition between IS structures, which is just the network rearrangement dynamics just mentioned above, has clearly been identified in supercooled regions. 

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