A Unified Framework for Polymer Density-Functional Theories and Some Numerical Issues with Their Applications to Tangent Hard-Sphere Chains

Polymer density-functional theories (PDFTs) are able to predict microscopic details that are often neglected by the more widely used polymer self-consistent field theory (PSCFT), and provide more accurate equations of state due to the incorporation of compressibility and the correlation effects. PDFTs are, however, numerically more complicated and less developed than PSCFT, particularly for systems with multi-dimensional inhomogeneity. Here we show that two popular PDFTs, one proposed by Yu and Wu (J. Chem. Phys. 117, 2368, 2002) and the other by Chapman and co-workers (J. Chem. Phys. 127, 244904, 2007), can be put into the same unified framework as the PSCFT. Later-versions (J. Chem. Phys. 118, 3835, 2003; J. Chem. Theory Comput. 8, 1393, 2012) are based on the modified fundamental measure theory to account for the hard-sphere excluded-volume effects, and the only difference between them lies in the excess Helmholtz free-energy density for intra-chain correlations. The unified framework will help promote the application of PDFTs by the broader polymer community. We also present improved numerical methods for faster and more accurate evaluation of the integrals in PDFTs for tangent hard-sphere chains which have discontinuous or non-differentiable integrands.