Semidilute Solutions and Hydrodynamic Screening

As the polymer concentration is increased, the chains begin to overlap at about the overlap concentration c, which depends (Section 2.7) on the chain length according to 

The main results of the theory of collective dynamics of chains in semidilute solutions with hydrodynamic screening are the following. The mean square displacement of a labeled chain follows the diffusion law  

This diffusion constant describing the diffusion of a labeled chain is called the tracer diffusion constant. It decreases with N and the monomer concentration c. The sharpness of the concentration dependence of Dt depends on the size exponent. For example, in good solutions (v = 3/5), Dt and in theta solutions (v = 1/2), Dj The longest relaxation time t is 

One of the convenient ways of measuring the diffusion coefficient is by monitoring the dynamical correlations of the local density of the scattering entities in experiments such as the dynamic light scattering. In dilute solutions, the number density of polymer chains can be monitored. In semidilute solutions or within polymer chains in dilute solutions, monomer density can be monitored. As an example, let us consider the time-dependent correlation of fluctuations in local concentration. Let the local monomer concentration, in number of monomers per unit volume, at the spatial location r and time t be 

Dc is called the cooperative or mutual diffusion coefficient, as thermodynamic correlations among all chains are used in writing the above equations. In contrast, the tracer diffusion coefficient Dt measures the diffusion of a tagged polymer. Defining the fluctuation in the local monomer density 8cp(r, f) as the deviation of Cp (r, t) from its average value Cp = c, combining Equations 7.59 and 7.60, and taking the Eourier transforms yield 

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