Thermal Physics · Problem 10

Boltzmann Distribution and Helmholtz Free Energy — Charles Kittel & Herbert Kroemer

Physics — Statistical Mechanics · elasticity-of-polymers

Problem

Elasticity of polymers. The thermodynamic identity for a one-dimensional system is τdσ=dU−f dl(95)\tau d\sigma = dU - f\,dl \tag{95} when ff is the external force exerted on the line and dldl is the extension of the line. By analogy with (32) we form the derivative to find −fτ=(∂σ∂l)U.(96)-\frac{f}{\tau} = \left(\frac{\partial\sigma}{\partial l}\right)_U. \tag{96} The direction of the force is opposite to the conventional direction of the pressure. We consider a polymeric chain of NN links each of length ρ\rho, with each link equally likely to be directed to the right and to the left. (a) Show that the number of arrangements that give a head-to-tail length of l=2∣s∣ρl = 2|s|\rho is g(N,−s)+g(N,s)=2N!(12N+s)!(12N−s)!.(97)g(N,-s) + g(N,s) = \frac{2N!}{(\tfrac{1}{2}N+s)!(\tfrac{1}{2}N-s)!}. \tag{97} (b) For ∣s∣≪N|s| \ll N show that σ(l)=log⁡[2g(N,0)]−l2/2Nρ2.(98)\sigma(l) = \log[2g(N,0)] - l^2/2N\rho^2. \tag{98} (c) Show that the force at extension ll is f=lτ/Nρ2.(99)f = l\tau/N\rho^2. \tag{99} The force is proportional to the temperature. The force arises because the polymer wants to curl up: the entropy is higher in a random coil than in an uncoiled configuration. Warming a rubber band makes it contract; warming a steel wire makes it expand. The theory of rubber elasticity is discussed by H. M. James and E. Guth, Journal of Chemical Physics 11, 455 (1943); Journal of Polymer Science 4, 153 (1949); see also L. R. G. Treloar, Physics of rubber elasticity, Oxford, 1958.

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