Concentration dependence, mode coupling

The concentration dependence of ionic mobility at high ion concentrations and also in the melt is still an unsolved problem. A mode coupling theory of ionic mobility has recently been derived which is applicable only to low concentrations [18]. In this latter theory, the solvent was replaced by a dielectric continuum and only the ions were explicitly considered. It was shown that one can describe ion atmosphere relaxation in terms of charge density relaxation and the elctrophoretic effect in terms of charge current density relaxation. This theory could explain not only the concentration dependence of ionic conductivity but also the frequency dependence of conductivity, such as the well-known Debye-Falkenhagen effect [18]. However, because the theory does not treat the solvent molecules explicitly, the detailed coupling between the ion and solvent molecules have not been taken into account. The limitation of this approach is most evident in the calculation of the viscosity. The MCT theory is found to be valid only to very low values of the concentration. 

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]. 

Thus, in the fluid state, there are two relaxation processes, the a and the with relaxation times that scale with proximity to the critical point with differing exponents, -y and — l/2fl, respectively. For spherical particles, y — 2.58 and l/2a = 1.66 thus the a process is predicted to slow more dramatically as the transition is approached than the process. Figure 4-22 shows the relaxation times t and extracted from the relaxation data of Fig. 4-20 for the colloidal fluids. The power laws given by Eqns. (4-33) and (4-34) fit these experimental concentration dependencies well, supporting the mode-coupling theory of this transition. 

The concentration dependence of the viscosity of polyelectrolyte solutions has been discussed by several authors [Cohen et al., 1988 Cohen and Priel, 1990 Borsali et al., 1992, 1994 Antonietti et al., 1997]. Several groups [Borsali et al 1992, 1994 Antonietti et al., 1997] have used the mode-mode coupling approximation of Hess and Klein [1983]. In the weakly charged polyelectrolyte limit, the latter formulation leads to an expression for the time-dependent viscosity of the form [Borsali et al., 1992]... 

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. 

We have inferred (2) that the intermediate time scale regime corresponds to probe motions that are heavily coupled to incompletely relaxed polymer modes. Mode properties supporting this inference include the hi ly non-exponential nature of the relaxations, the increase of t and tf with increasing c, and the increased concentration dependences of t and X( with increasing probe diameter, expected because larger probes can couple strongly to more polymer chains than small probes do. The detailed nature of the coupling or the identity of the polymer modes in question have not been completely clarified by the present analysis. 

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