Structure factor, mode coupling theory

From the above discussion, it is obvious that the mode coupling theory calculations are quite involved and numerically formidable. Balucani et al. [16] have made some simple approximations to incorporate the self-consistency between the self-dynamic structure factor and the friction. This required the knowledge of only the zero frequency friction. The full self-consistent calculation is more elaborate and will be discussed later in this chapter. 

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 ... 

It should be stressed at this point that the construction of the dynamic structure factor outlined above is based on the basic results of the mode coupling theory. 

When the solvent molecules are explicitly included, one needs to treat a ternary system (two ions and the dipolar solvent molecules). The additional slow variables to be included in the mode coupling theory are the products of the ion charge and solvent densities. This will explicitly introduce terms like Fis(k,t), which is the partial dynamic structure factor involving the ion and the solvent molecules. The calculation of the microscocpic terms of the friction, containing the density terms, does not appears to be difficult, but calculation of the current terms now appears to be formidable. 

A simple theory of the concentration dependence of viscosity has recently been developed by using the mode coupling theory expression of viscosity [197]. The slow variables chosen are the center of mass density and the charge density. The final expressions have essentially the same form as discussed in Section X the structure factors now involve the intermolecular correlations among the polyelectrolyte rods. Numerical calculation shows that the theory can explain the plateau in the concentration dependence of the viscosity, if one takes into account the anisotropy in the motion of the rod-like polymers. The problem, however, is far from complete. We are also not aware of any study of the frequency-dependent properties. Work on this problem is under progress [198]. 

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. 

In a mode coupling approach, a microscopic theory describing the polymer motion in entangled melts has recently been developed. While these theories describe well the different time regimes for segmental motion, unfortunately as a consequence of the necessary approximations a dynamic structure factor has not yet been derived [67,68]. 

Figure 2.24 shows the temporal evolution of the compositional structure factor S<(, (a) and the orientational structure factor Ss (b) for the temperature quench into the lu region (T/Tn, = 0.6, (f)Q = 0.55) in Figure 2.22. The structure factor for concentration has a maximum at q, which corresponds to the peak wavenumber of coi(q). With time the corresponding mode grows exponentially and the peak position qm is invariant. Then the time evolution of the structure faaor S<, is lhe same as that of the Cahn-Hilliard theory for isotropic SD [102]. The amplitude of the peak at q = 0 decreases with time because s > 0 and another peak appears at q. In this quench, the concentration fluctuation initially induces the SD and the orientational ordering within the domain subsequently takes place due to the coupling between the two order parameters concentration-induced SD.

The mode coupling approximation for m (0 yields a set of equations that needs to be solved self-consistently. Hereby the only input to the theory is the static equilibrium structure factor 5, that enters the memory kernel directly and via the direct correlation function that is given by the Ornstein-Zernicke expression = (1 - l/5,)/p, with p being the average density. In MCT, the dynamics of a fluid close to the glass transition is therefore completely determined by equilibrium quantities plus one time scale, here given by the short-time diffusion coefficient. The theory can thus make rather strong predictions as the only input, namely, the equilibrium structure factor, can often be calculated from the particle interactions, or even more directly can be taken from the simulations of the system whose dynamics is studied. 

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