The chain length dependence of the chemical potentials of macromolecular systems at zero density: Exact calculations and Monte Carlo simulations

In a previous paper we illustrated a Monte Carlo method to calculate the incremental chemical potential between polymer chains of length ν and ν+1 at all densities. In this document we present a statistical mechanics model that allows for its quantitative evaluation as a function of chain length and temperature for macromolecular systems at zero density. Through this approach the incremental chemical potential at zero density is shown to be a strong function of chain length at low temperatures, but becomes chain length independent even for short chains (ν≥3) at higher temperatures. The temperature where the chemical potential becomes essentially chain length independent is a definition of the Boyle transition for finite chain length polymers, and we show that this transition temperature has a well defined chain length dependence. Chain dimensions and incremental entropies associated with chain segments are also found to decrease significantly when the temperature is reduced below this θ temperature thus proving that chain segments are localized in the vicinity of other segments in the collapsed state. The consequences of these results on the modeling of the phase equilibria of polymer solutions are also examined.