Simplicity and scaling - size of a real polymer in three (or any) dimensions

TL;DR

This paper analyzes how the size of a real polymer scales in different dimensions, deriving a formula for the scaling exponent that aligns with known results in two dimensions and predicts a new value for three dimensions.

Contribution

It provides a theoretical derivation of the polymer size scaling exponent in various dimensions, including a novel prediction for three dimensions.

Findings

Scaling exponent in 2D is 3/4.

Scaling exponent in 4D is 1/2.

Predicted scaling exponent in 3D is 7/12.

Abstract

We examine the scaling of the linear dimension of the system size of a real polymer solution at constant excess free energy and in two different spacial dimensionalities, d=d0 and d=d1. Standard results for the functional form of the excess free energy lead to the conclusion that the scaling exponent nu(d) satisfies nu(d0) - nu(d1) = 1/d0 - 1/d1. Taking the critical dimensionality as a point of reference (nu(4)=1/2) gives a scaling exponent nu(d) = 1/4 +1/d, in agreement with the accepted result for two-dimensions (nu(2) = 3/4) and the first term in the epsilon (d-4) expansion. For the unsolved case of three dimensions it predicts nu(3)=7/12. Several simplifying features of this result are pointed out.