Identification of self consistent field interaction parameter from continuum Monte Carlo simulation of model polymer blends

Monte Carlo simulations of binary AB polymer blends have been performed to evaluate the effective interaction parameter $\chi_{e}$ of self consistent field theory, and to quantify corrections to RPA predictions for fluctuations. We consider a model with a non-bonded pair interaction $v_{ij}(r) = \epsilon_{ij}f(r)$ for which $f(r)$ is of the repulsive Lennard-Jones form, $\epsilon_{AA}=\epsilon_{BB}$, and $\epsilon_{AB} = \epsilon_{AA} + \Delta \epsilon$. Using thermodynamic perturbation theory, to first order in $\Delta\epsilon$, we obtain an interaction free energy with the composition dependence predicted by Flory-Huggins theory, with an effective interaction parameter $\chi_{e} = \Delta\epsilon z_{c}$. Here, $z_{c}$ is an effective coordination number given by the average of the sum of values of $f(r)$ for interactions between a test monomer and nearby monomers on other chains, in a reference system with $\Delta\epsilon=0$. Results for composition fluctuations in semigrand ensemble simulations of blends with a range of values of $\Delta\epsilon \neq 0$, for several chain lengths, are compared to RPA predictions calculated using this perturbatively defined $\chi_{e}$ parameter.