Nonlocal contributions to the thermodynamics of inhomogeneous homopolymer-solvent systems

Abstract

We derive several recent approximation schemes for the thermodynamics of inhomogeneous homopolymer solutions by functionally expanding the nonideal part of the free energy about the local density. The functional expansion contains all orders in the gradients of the monomer density even when truncated at low order. When applied to systems with slowly varying densities, it reduces to a widely used square-gradient theory of homopolymers and blends. We also recover the so-called Roe functional by considering infinitely long Gaussian chains. However, this result only applies to a small region near the center of the interface. We employ a one-parameter, exponential decay trial function and the variational theorem to obtain estimates of the surface tension between two coexisting homopolymer-solvent phases. This particular trial function facilitates a comparison of the numerically evaluated nonlocal free energy functional, the analytic summation of the full gradient expansion, the square-gradient approximation, and the Roe functional. Within the scope of a Gaussian-chain model and a Flory–Huggins local free energy, we compare the various approximations with regards to their predictions for the surface tension, the nonlocal free energy density, and the interfacial profiles. The full gradient expansion does not converge to a useful result near the center of the interface of moderately segregated phases or when the interface becomes narrow as for strongly segregated homopolymer solutions. The square-gradient theory overestimates the free energy density in the wings of the interface but underestimates this quantity near the interface’s center. These two errors partially cancel, and the square-gradient theory provides an accurate estimate of the surface tension of moderately segregated homopolymer solutions. Unlike many other theories of interfaces which derive from considerations of two-body correlations in a homogeneous reference phase, the lowest nontrivial contribution of the functional expansion for Gaussian chains derives from three-body correlations. This feature is in qualitative accord with recent lattice theories of interfaces that treat chain biases in the interfacial region and thus implicitly contain three-body correlations.