Correlation between the combinatory entropy of polymer and ideal liquid solutions: 3. Derivation of equation for local concentration

Abstract

The local concentration in binary solutions of polymer in solvent and in simple liquid solution has been determined by an equation derived from the maximum condition of the number of configurations in the binary solution with two hypothetical regions, a small region and the rest. The equation for a solution of flexible polymer with r segments in solvent is given by: 1n( ф 2 1X 2 X 2 1ф 2 )+{ k−(1−k) r }1n( ф 2 X 2 )+k(1−r −1)(ф 2−X 2)+k 1n( N ∗ t n ∗ t )=0 where X i is the local volume fraction of polymer ( i = 2) and solvent ( i = 1), ф i is the mean or macroscopic volume fraction, k is a parameter defined by k = ± {1 + ( V 0 2/ V 0 1)( ∂n 2/ ∂n 1) v } and V 0 i is the molar volume of i and n i is the number of molecules of i in the small region, n∗ t = n 1 + rn 2 , and N∗ t is the total number of molecules in the solution. It is found that the maximum concentration fluctuation in the small region around the mean concentration ф 1 characterized by |X 1 − ф 1| is proportional to | k| and is zero for k = 0.