A practical density functional for polydisperse polymers

– The Flory Huggins equation of state for monodisperse polymers can be turned into a density functional by adding a square gradient term, with a coefficient fixed by appeal to RPA (random phase approximation). We present instead a model nonlocal functional in which each polymer is replaced by a deterministic, penetrable particle of known shape. This reproduces the RPA and square gradient theories in the small deviation and/or weak gradient limits, and can readily be extended to polydisperse chains. The utility of the new functional is shown for the case of a polydisperse polymer solution at coexistence in a poor solvent. Introduction. – Recent work on polydisperse thermodynamics has shown the value of using, as thermodynamic coordinates, whatever linear combinations of densities actually occur in the excess free energy of the system. These linear combinations are ‘generalized moments’ (moments for short) of the complete set of densities, which are in principle infinite in number; however, in most models, particularly in polymeric systems, they form a much reduced set. For example, the Flory Huggins model of length-polydisperse homopolymers and solvent on a unit lattice has (setting kT = 1) Fex = (1− φ) ln(1− φ) + χφ(1− φ), where φ = ∫ dNNρ(N) is the monomer density, or first moment of the distribution ρ(N) that specifies the density of chains of length N . Thus there is only one moment in Fex. It was shown [1–3] how to construct a ‘projected’ free energy, expressed as a function of the required moments, whose predictions for cloud curves, shadow curves, and critical points coincide with those of the full model. It was also shown [2, 3] how, by systematically adding additional moments, the full phase behaviour (describing states of coexistence midway along a tie-ine as well as at its endpoints) could be recovered to any desired accuracy. An important extension of this approach is to address interface thermodynamic behaviour, for example to predict the interfacial tensions between coexisting phases. (In doing this, we can suppose that the composition of such phases was calculated already, e.g., by the methods just outlined.) This requires a model, not merely of the equation of state, but of the free energy functional. In some cases (such as polydisperse hard spheres) accepted functionals include ones where the excess free energy depend only on a few nonlocal ‘weighted densities’. The