in the lattice space. It reflects the fact of volume exclusion between molecules. Such
a model facilitates the calculation of the combinatorial entropy between molecules
of various species, i.e. the number of ways to arrange the placements of the
molecules in the solution space (Guggenheim 1952 ; Flory 1953 ; Prigogine
1957a). Therefore, the lattice model has become the basic tool in classic statistical
thermodynamic theories of solutions. In 1939, Chang first investigated the combi-
national entropy of dimers in a lattice solution (Chang 1939 ). Later on, Flory 1942
and Huggins 1942 separately derived the currently well-accepted lattice theory for
long-chain polymers. On the other hand, the lattice model also facilitates the
calculation of molecular attractions that often drive phase transitions. For instance,
the Ising model has become the classic lattice model in the condensed matter
physics for a statistical mechanic analysis of phase transitions.
8.2.2 Basic Assumptions of Flory-Huggins Lattice Theory
8.2.2.1 Incompressible Mixture
Assuming polymer solutions are located in the lattice space, with each lattice site
containing the coordination numberq, one can assign the solvent molecules to those
single sites with the numberN 1 , and each polymer chain to occupy consecutive
connected sites with the number ofr, as illustrated in Fig.8.3. The number of polymer
chains isN 2. Supposing that the size of each chain unit is comparable with one solvent
molecule, as given by
vmonomer¼vsolvent (8.1)Fig. 8.3 Illustration of the
lattice model of a polymer
chain consecutively
occupying the sites in the
lattice space
150 8 Statistical Thermodynamics of Polymer Solutions