On the scaling of the critical solution temperature of binary polymer blends with chain length

Monte Carlo simulations have been performed to examine the sealing with chain length of the upper critical solution temperature (T-c) of binary, symmetric blends of polymers. Critical parameters were obtained by histogram reweighting analysis of semigrand canonical ensemble simulations near the critical temperature. For several continuum space polymeric models, it is found that T-c/<(epsilon)over bar> scales approximately as either rho(c)(A + B N) or rho(c)N/(B-1 + A/root N), where N is chain length, rho(c) is the density at T-c, <(epsilon)over tilde> is a chemical mismatch parameter, and A and B are constants that depend on the specific characteristics of the model, Constant A provides a measure of the correction to the Flory-Huggins' prediction T-c similar to N. The effect of such correction term becomes unimportant for large N, <(epsilon)over tilde>, and temperature. Excellent agreement is found, however, with the mean-held prediction that chi(E) similar to 2/N (for all systems studied), where chi(E) is a continuum space, enthalpic "chi" parameter which takes into account the variations of local structure of the fluid for different chain lengths at the critical point.