Mode coupling terms

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 ... 

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],... 

To obtain nonlinear coupling terms, we consider two linearly independent, not identical E modes, namely. 

The friction coefficient of a large B particle with radius ct in a fluid with viscosity r is well known and is given by the Stokes law, Q, = 67tT CT for stick boundary conditions or ( = 4jit ct for slip boundary conditions. For smaller particles, kinetic and mode coupling theories, as well as considerations based on microscopic boundary layers, show that the friction coefficient can be written approximately in terms of microscopic and hydrodynamic contributions as ( 1 = (,(H 1 + (,/( 1. The physical basis of this form can be understood as follows for a B particle with radius ct a hydrodynamic description of the solvent should... 

Cnm is the quantum average of proton coupling, and AG m is the activation barrier for n — m transition. With explicit computation of Cnm, and considering reaction symmetry within the Q mode, a new term, Ea = h2(y2/2inu, whose physical interpretation is the coupling term between Q mode and solvent polarity, is introduced, accompanied by removal of the summation term in (13) ... 

Often, to increase peak resolution and capacity for a complex sample mixture, the term hyphenation can be referred also to different chromatographic modes coupled together. 

The coupling to the soft mode introduces here oscillatory terms that can become important close to Tc. The fact that the amplitude, as well as the coherence length of the order parameter fluctuations, increase on approaching Tc from either side also probably brings higher order coupling terms into... 

Even with all of its sophistication, the mode coupling theory is still a perturbation theory where dynamics is described in terms of a subset of dynamical variables chosen from the products of hydrodynamic modes. It fails, for example, to describe rare events, such as the activated processes or stringlike cooperative motions often found to dictate dynamics in glassy liquids [13-15]. 

While the hydrodynamic theory always predicts this near equivalence of the friction and the viscosity, microscopic theories seem to provide a rather different picture. In the mode coupling theory (MCT), the friction on a tagged molecule is expressed in terms of contributions from the binary, density, and transverse current modes. The latter can of course be expressed in terms of viscosity. However, in a neat liquid the friction coefficient is primarily determined not by the transverse current mode but rather by the binary collision and the density fluctuation terms [59]. Thus for neat liquids there is no a priori reason for such an intimate relation between the friction and viscosity to hold. 

Since the goal is to study the friction on a solute which is different in size from the solvent, in the mode coupling expressions of friction the terms representing the coupling between the solute and the solvent are calculated using the solute-solvent interaction potential. Thus the binary terms cu q) and y dn are all calculated from V12 (r), and all the other solvent static and dynamical quantities are calculated from v(r). 

The isomerization rate is calculated using the Grote-Hynes formula, given by Eqs. (320) and (322). The frequency-dependent friction ( (z)) and viscosity (rj) has been obtained from the mode coupling theory presented in Section IX. For convenience the rate is expressed in terms of the dimensionless quantity k in the following form ... 

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