Correlation functions, mode coupling theory

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. 

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],... 

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. 

Gottke et al. [5] offered a theoretical treatment of collective motions of mesogens in the isotropic phase at short to intermediate time scales within the framework of the Mode coupling theory (MCT). The wavenumber-dependent collective orientational time correlation function C/m(, t) is defined as... 

These are, of course, just the standard arguments used to motivate the mode coupling theories, which have proved useful in the description of critical phenomena and the asymptotic decay of correlation functions. ... 

As discussed in Chapter 6, molecular modeling and neutron scattering experiments greatly improved understanding of the liquid structures and their variability with T. The mode-coupling theory (MCT) originates from analysis of the flow behavior near a critical point involving nonlinear correlation functions [Kawasaki, 1966]. 

The decay rate (first cumulant) T of the time-dependent correlation function associated with the order parameter fluctuations can be measured by dynamic light scattering. Mode-coupling theories of critical phenomena including dynamical background contributions predict that is the sum of a critical part Tf and a background part Tb, i.e.. 

So far we have employed the exponential model for the memory kernel appearing in the GLE for the density correlation functions. In [60] the model based on the mode-coupling theory described in Sec. 5.2.4is applied to the calculation of the longitudinal current spectra of the same diatomic liquid as discussed here. It is found that the essential features of the results remained the same as far as the collective dynamics is concerned. It is also demonstrated that the results are in fair agreement with those determined from the molecular dynamics simulation. 

Another strong point of the simulation approach is its ability to selectively change parts of the model Hamiltonian. In this way one can compare a chemically realistic model of PB with a freely rotating chain version of the same polymer and does not have to switch to a completely different polymer with some of the same properties like is unavoidable in experiments [33]. With this approach we could establish that identical structure on the two-body correlation function level (single chain and liquid structure factors) does not imply identical dynamics which raises questions on the applicability of the mode-coupling theory of the glass transition to polymer melts. 

At this point we introduce the fvst approximation. We split the correlation functionmode-mode coupling theory, which has been applied with great success to binary mixtures near their critical point by Kawasaki S and Ferrell. However, for the present problem, the idea (in a different language) goes back to Kirkwood and Risemarm. s We make a further (minor) simplification. We assume that the essential time dependence is contained in the (tm) part of the correlation, while the (cc) part may be taken at equal times. This may be justified by a detailed study of the (cc) dynamics, along the lines of Section VI.2.2. 

Pomeau, Y. R6sibois, P. (1975). Time dependent correlation functions and mode-coupling theories. Phys. Rep., 19, 63-139. 

To what extent the schematic model systems A and B for a polymer melt show this typical relaxation behavior will be addressed in this subsection, by calculating various structural correlation functions that probe the dynamical changes of the melt on different length scales (Section 6.3.2.1). From these correlation functions it is possible to extract relaxation times the temperature dependence of which can be studied and compared to that of transport coefficients, such as the diffusion coefficient. This will be done in Section 6.3.2.2. The final paragraph of this subsection then deals with the calculation of the incoherent intermediate scattering function and its quantitative interpretation in the framework of the idealized mode coupling theory (MCT). " ... 

Kob, W. and Andersen H. C. 1995a. Testing mode-coupling theory for a supercooled binary Lennard-Jones mixture I The van Hove correlation function. Phys. Rev. E 51 4626-4641. 

When an atomic system is cooled below its glass temperature, it vitrifies, that is, it forms an amorphous solid [1]. Upon decreasing the temperature, the viscosity of the fluid increases dramatically, as well as the time scale for structural relaxation, until the solid forms concomitantly, the diffusion coefficient vanishes. This process is observed in atomic or molecular systems and is widely used in material processing. Several theories have been developed to rationalize this behavior, in particular, the mode coupling theory (MCT) that describes the fluid-to-glass transition kinetically, as the arrest of the local dynamics of particles. This becomes manifest in (metastable) nondecaying amplitudes in the correlation functions of density fluctuations, which are due to a feedback mechanism that has been called cage effect [2],... 

Alternatively the hydrodynamic equations of motion can determine directly the time evolution of the correlation functions instead of the mesoscopic variables, according to the Mori-Zwanzig model [42,52]. These equations are the starting point of the Mode-Coupling Theories (MCT) [60-62]. 

In the case that hydrodynamic interactions arising from the backflow by movements of segments (or particles) are dominant over friction of each segment with a viscous fluid medium, the diffusion coefficient is often evaluated theoretically as follows. According to the mode-mode coupling theory for fluids with the Oseen tensor of hydrodynamic interactions,the diffusion coefficient is expressed in terms of the static correlation function S([Pg.307]

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