  We present a lattice model for polymer solutions, explicitly incorporating
interactions with solvent molecules and the contribution of vacancies. By
exploiting the well-known analogy between polymer systems and the $O(n)$-vector
spin model in the limit $n \to 0$, we derive an exact field-theoretic
expression for the partition function of the system. The latter is then
evaluated at the saddle-point, providing a mean-field estimate of the free
energy. The resulting expression, which conforms to the Flory-Huggins type, is
then used to analyze the system's stability with respect to phase separation,
complemented by a numerical approach based on convex hull evaluation. We
demonstrate that this simple lattice model can effectively explain the behavior
of polymer systems in explicit solvent, which has been predominantly
investigated through numerical simulations. This includes both single-chain
systems and polymer solutions. Our findings emphasize the fundamental role of
vacancies whose presence, rendering the system a ternary mixture, is a crucial
factor in the observed phase behavior.
