  We consider in this review the statistical mechanical description of a very
general microscopic lattice model of a compressible and interacting
multi-component mixture of linear polymers of fixed lengths. The model contains
several microscopic, i.e. bare parameters determining the thermodynamic state
of the system. General arguments are given to show that these parameters must
be independent not only of the lattice properties but also of the thermodynamic
state, and that the voids representing free volume must be carefully treated,
if thermodynamics has to be properly obeyed. These facts have not always been
appreciated in the literature. We focus on mixing functions, some of which have
not been properly calculated in the literature. In general, mixing is
non-isometric (non-zero volume of mixing) and the entropy of mixing is
non-ideal. We have recently developed a lattice theory for the general model,
which goes beyond the random mixing approximation (RMA) limit and is
thermodynamically consistent in the entire parameter space. The theory contains
terms that do not have a continuum analog except in the RMA limit or for
point-like particles. Both the free volume and the total volume determine the
thermodynamics of the system. The RMA limit of our theory gives rise to a new
theory, which can be taken as the extension of the conventional incompressible
Flory-Huggins theory and is similar in simplicity. Using our complete theory,
we calculate the effects of size disparity and interactions on the
thermodynamics of the model. Cohesive energies are not constant in general.
Non-isometry can make the energy of mixing negative, even when all exchange
interactions are repulsive. Consequently, Scatchard-Hildebrand theory cannot be
substantiated in general. Various unusual features are noted and discussed.
