Variation Principle Treatment of the Excluded-Volume Problem

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Abstract
A variation principle is used to develop a ``single‐segment'' self‐consistent‐field approach to the excluded‐volume problem. The zeroth‐order self‐consistent field (based on freely orienting chain statistics) is computed. A partial differential equation is developed for the description of the ``random flight in a field'' problem emerging from the self‐consistent‐field approach. The appropriate boundary value problem is solved in one limit and the mean‐square end‐to‐end distance for the polymer molecule is calculated. This is found to be proportional to L4/3, where L is the number of segments in the chain, a result which is in satisfactory agreement with earlier computer calculations. The end‐to‐end distribution going with the zeroth‐order self‐consistent field is found to be a displaced skew Gaussian.

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