Free energy functional expansion for inhomogeneous polymer blends

Abstract

Density functional theory for inhomogeneous polymer systems is reformulated using the new ideal system of noninteracting Gaussian chains to replace the Flory–Huggins-like formulation of McMullen and Freed in which the polymer chains have unspecified connectivity. The price paid for introducing this more realistic ideal system is the fact that the density-field relation may only be inverted in powers of the density gradients, so the ideal free energy functional is obtained as a density gradient expansion. The relevant expansion parameter involves the radius of gyration of the polymer, as expected. However, the coefficient of the square gradient term (and those of higher gradients) involves the spatially varying density in the interface as postulated by de Gennes and first derived rigorously here. The nonideal free energy functional is treated by expansions about a homogeneous reference system, and the correlation functions are evaluated in the random phase approximation (RPA). Although truncations are made at second order, there are no difficulties in including higher order terms provided the RPA approximation is retained. The theory is formulated in general for compressible polymer systems, and the incompressible case follows as a special limiting situation. We also analyze the contribution from higher order terms in a traditional Landau-type free energy functional expansion for inhomogeneous polymer systems in which coefficients are evaluated in a homogeneous reference system. Despite the difference of the former coefficients from the de Gennes postulate, it is shown that this Landau expansion may be resummed to produce the identical functional that we obtain by rigorous density functional methods.