Statistical thermodynamics of polymer solutions. VI. Comblike branched molecules with branches placed randomly along the backbone

Abstract

Abstract Thermodynamic interactions between arbitrarily branched flexible “comb” molecules (with branches of uniform length affixed to backbones of uniform length but with the number of branches on a molecule and their placement arbitrary) are treated by Zimm's perturbation method to obtain results valid to the double contact approximation. Thus, bimolecular cluster configurations with one and two intermolecular contacts, but not more, are correctly accounted for. This general result is applied to “random” combs (with the number of branches per comb uniform, but the placement random) and to “heterogeneous” combs (with both the number of branches on a molecule and their placement random). Results for the random combs are very similar to those reported earlier for “symmetrical” combs (with f uniform branches disposed at points along the backbone chain so as to subdivide it into f + 1 equal sections), but the second virial coefficient for the random model is slightly the larger when the number of branches per chain is small and the fraction of the molecule in the backbone is at least a few per cent. Since random combs are uniform in mass, the osmotic pressure and light‐scattering second virial coefficients are alike; but for a system of heterogeneous combs the virial coefficients differ from one another, and from the random comb result, when the mean number of branches f̄ per chain is small and they are of appreciable length in comparison to the backbone. This behavior reflects the heterogeneity in both molecular mass and dimensions existing under these circumstances. As f̄ increases and/or the mean fraction of segments in the chain backbone becomes large, coincidence with the random comb result is approached. The double‐contact second virial coefficients for both random and heterogeneous combs are obtained in closed analytical expression that are much more easily reduced to numerical results than the expression for symmetrical combs given earlier. These two models also correspond better than the symmetrical combs to materials actually obtainable.