Scaling in Good Solvents

The mobility ficiMn of one chain is measured in experiments (such as sedimentation, electrophoresis, etc.) where a given external force fior is applied to the chain, and the resulting drift velocity V is measured. At low forces (such that the chain shapes are unperturbed) 

Another, related, parameter, is the diffusion constant D of the chain it is measured by direct monitoring of concentration profiles or more conveniently by inelastic scattering of laser light. Diffusion is related to mobility through the Einstein formula 

A good, rigorous, starting point for the theoretical discussion of D is the formula relating D to the spontaneous fluctuations of the velocity V (due to Brownian motion). In three dimensions 

At this point we introduce the fvst approximation. We split the correlation function cc tm) into a concentration part and a velocity part. In modem jargon this corresponds to a mode-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. 

Here g(r) is the static pair correlation discussed in Chapter I and normalized always by fg(r)dr = N. fl is the total volume allowed for the chain, and the factor N/il ensures that if we integrate eq. (VI.22) over ri and rg, we get as desired. 

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