Predicting $\chi$ for polymers with stiffness mismatch from simulations

The Flory-Huggins $\chi$ parameter describes the excess free energy of mixing and governs phase behavior for polymer blends and block copolymers. For chemically distinct polymers, the value of $\chi$ is dominated by the mismatch in cohesive energy densities of the monomers. For blends of chemically similar polymers, the entropic portion of $\chi$, arising from non-ideal local packing, becomes more significant. Using polymer field theory, Fredrickson, Liu, and Bates predict that a difference in backbone stiffness can result in a positive $\chi$ for chains consisting of chemically identical monomers. To quantitatively investigate this phenomenon, we perform molecular dynamic (MD) simulations for bead-spring chains which differ only in stiffness. From the simulations, we apply a novel thermodynamic integration to extract $\chi$ as low as $10^{-3}$ per monomer for blends with mild stiffness mismatch. By introducing a standardized effective monomer, we map real polymers to our bead-spring chains and show that the predicted entropic portion of $\chi$ are consistent with experimental data.