Microscopic theory of polymer internal viscosity: Mode coupling approximation for the Rouse model

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Abstract
The internal viscosity or friction coefficient η̂κ for normal mode κ of an N‐monomer Rouse‐type ringpolymer is defined microscopically using the Mori projection technique and is then computed within the simplest (trilinear) mode coupling approximation. One finds that η̂κ=ζγy[(κ/N)2], where y (x) is a nonanalytic function of x, ζ is the monomerfriction coefficient, and γ is proportional to the square of an equilibrium four point polymercorrelation function. This correlation function is nonvanishing for non‐Gaussian chains; e.g., γ=18/25 for freely jointed rigid rods. For κ/N→0, y (x) is found explicitly to yield lim (κ/N) →0η̂κ = (ζγ/2π√3)(πκ/N)2 ln(N/πκ). For 0.45≲κ/N≲0.50 numerical evaluation of y (x) shows η̂κ is approximately independent of κ. For the bulk of the κ range, 0.10≲κ/N≲0.40, η̂κ≃ζΦ [(κ/N)−0.05], where Φ is an order unity constant proportional to γ. For the freely jointed chain Φ≃1.7. This last form for η̂κ agrees well with the Cerf–Peterlin empirical form η̂κ=ζ (κ/N) φ. Experimental data are typically fit with φ∼1−3.

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