Size of a polymer molecule in solution. Part 1.—Excluded volume problem

Abstract

A study is made of the probability distribution of the end to end distance R of a polymer of N segments, length Nl=L, and of self repulsion ω. A simple method, capable of adoption in more complicated problems, is developed, using the idea of an effective step length.

The mean square value of R² is developed as a series which for large L is R²=ω^2/5L^6/5l^2/5(1.12 + 1.05 + 1.03 +…).

The probability distribution is developed in terms of the dimensionless parameter x=R²/l^2/5ω^2/5L^6/5, and for small x, logp(x)–x([graphic omitted]+[graphic omitted]+[graphic omitted]+…) but for large x a definite asymptotic form is derived log p(x)=–(⅗)π^1/2/3 x^{[graphic omitted]}.