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Single Chain Statistics

Consider the probability distribution function tex2html_wrap_inline711 of a chain consisting of N freely jointed units of length b, with the end-to-end vector tex2html_wrap_inline717 . By introducing the auxiliary field tex2html_wrap_inline719 , it is rather trivial [21] to show:


Here tex2html_wrap_inline721 is the characteristic function for the freely jointed chain. Its successive derivatives with respect to tex2html_wrap_inline719 give the chain moments, from which the tex2html_wrap_inline711 can be inferred very accurately [5, 6]. However, we shall present a perturbative method here for simplicity.

Approximating the characteristic function to tex2html_wrap_inline727 gives rise to the usual Gaussian distribution function, but more accurately we can write it as a power series expansion in tex2html_wrap_inline729 :


which gives the probability distribution function in the form of:



This integration can be performed with the saddle point approximation to give the leading correction to the Gaussian probability. The saddle point is found to be at:


To evaluate the full correction, we write the standard Taylor expansion for tex2html_wrap_inline731 about the saddle point tex2html_wrap_inline733 :


where tex2html_wrap_inline735 measures the distance away from the saddle point, and tex2html_wrap_inline737 denotes that all derivatives are to be evaluated at the same saddle point tex2html_wrap_inline733 . The leading Gaussian behaviour [1] dictates the likely scaling relation: tex2html_wrap_inline741 ; and this guides our expansion in orders of 1/N throughout this paper. Some details of this expansion are provided in the appendix. The Eq.(3) can be rewritten as:


with summations over repeated indices implied here. The first exponential in the integrand describes a Gaussian average which, on its own, gives the prefactor in the saddle point approximation; the second exponential can be expanded into a power series in k, and their average with respect to the Gaussian distribution is straightforward thanks to Wick's theoremgif. The resulting probability distribution can be written in the following exponential form:


Note that this probability distribution only depends on the magnitude of tex2html_wrap_inline717 , as the system is isotropic. The exponent is a power series in tex2html_wrap_inline751 , and for large N, should be well approximated by the first few terms. In principle, this probability distribution function can be written to a higer accuracy by the inclusion of higher order terms; but in practice the expansion becomes difficult for short chains.

The probability distribution function given by the inverse Langevin approximation [3] is equivalent to taking the saddle point value tex2html_wrap_inline755 :


the exponent of which agrees with that of Eq.(8) only partially. In comparison, the multiplicative factors in the exponent, such as tex2html_wrap_inline757 , are new and necessary for a self-consistent approximation in orders of 1/N. Therefore, expanding the inverse Langevin function in a power series, e.g. in Refs.[20, 22], will give incomplete correction to Gaussian result at order 1/N. The error in the inverse Langevin approximation comes mainly from the use of Stirling's approximation, which can be viewed as a saddle point approximation of a Gamma function integral.

In Eq.(8,9) we have non-Gaussian probability distributions, which contains the leading order (in 1/N) corrections due to the finite extensibility. However, the exact expectation values are available for a 3-dimensional freely jointed chain: tex2html_wrap_inline767 and tex2html_wrap_inline769 , which account for the finite extensibility of the chains. These expectation values can be readily verified with the probability distribution of Eq.(8), but not with that of Eq.(9).

next up previous
Next: Network Average Up: Rubber Elasticity with Finite Previous: Introduction

Yong Mao
Wed Apr 30 14:46:27 BST 1997