Flory-Huggins Lattice Theory
of Polymer Solutions, Part 1

The thermodynamics of (binary) regular polymer solutions¹ were first investigated by Paul Flory² and Maurice Huggins³ independently in the early 1940s. They assumed a rigid lattice frame, that is, the molecules in the pure liquids and in their solution / mixture are considered to be distributed over N₀ lattice sites, as illustrated in the figure below. The total number of lattice sites, N₀, is assumed to be equal to the number of solvent molecules, N_s, and polymer repeat units, N_pr:

N₀ = N_s + N_pr

where N_p is the number of polymer molecules each consisting of r repeat units.

     Two-dimensional Lattice

The model has been described in great detail by Flory in his famous book "Principles of Polymer Chemistry" (1953).⁴ Following the standard theory of mixing for small molecules of similar size and using Stirling's approximation lnM! = M lnM - M, Flory and Huggins obtained following expression for the entropy of mixing:

ΔS_mix ≈ -k (N_p ln φ_p + N_s ln φ_s )

Alternately, the entropy of mixing can be written as

Δs_mix = ΔS_mix / N₀ ≈ -k {φ_p / (r·v_r) · ln φ_p + φ_s / v_s · ln φ_s}

where φ_s and φ_p are the volume fractions of the solvent and polymer,

φ_s = N_sv_s / (N_sv_s + N_prv_r),   φ_p = N_prv_r / (N_sv_s + N_prv_r)

and v_s and v_r are the volumes of a solvent molecule and of a polymer repeat unit, respectively. Obviously, the solvent needs not necessarily to be made of a single unit. The solvent may, in fact, consist of several repeat units or of another polymer.

The Gibbs free energy of mixing⁶, ΔG_mix, includes an enthalpy part, that is, mixing can be an endothermic or an exothermic process

ΔG_mix ≈ ΔF_mix = ΔU_mix - T ΔS_mix

where ΔU_mix is the heat of mixing. Flory and Huggins introduced a new parameter, the so called Flory-Huggins interaction parameter to describe the the polymer-solvent interaction:⁵

χ_ps = ΔU_mix / (kT N_s φ_p)

which combined with the entropy term leads to the free energy of mixing:

ΔF_mix / kT = N_p ln φ_p + N_s ln φ_s + χ_ps N_s φ_p

Assuming equal lattice volumes for both repeat units, the free energy per lattice site can be written

Δf_mix / kT ≈ φ_p / r · ln φ_p + φ_s ln φ_s + χ_ps φ_s φ_p

A more general expression for the free energy of mixing is

Δf_mix / kT = φ_p / (r·v_r) · ln φ_p + φ_s / v_s · ln φ_s + χ_ps φ_s φ_p / √(v_rv_s)

where Δf'_mix is the free energy of mixing per unit volume. These equations are the starting point for many other important equations. For example, the partial molar free energy of mixing (chemical potential) can be obtained by differentiation of the expression above with respect to N_s. This gives

 Δμ_s/RT =  ln [1- φ_p] + (1 - 1/r) φ_p + χ_ps φ_p²

where μ_s is the chemical potential of the solvent per mole. Substitution of this expression into the osmotic pressure relation Π = - Δμ_s / V_s gives

Π ≈ RT V_s^-1 · {ln [1- φ_p] + φ_p + χ_psφ_p²}

where V_s is the molar volume of the solvent.

References & Notes

1. The word regular implies that the molecules mix in a totally random manner. Thus, there is no segregation or preference of any molecule for another molecule.

2. P. J. Flory, J. Chem. Phys. 9, 660 (1941); 10, 51 (1942)
3. M. L. Huggins, J. Phys. Chem. 46, 151 (1942); J. Am. Chem. Soc. 64, 1712 (1942)
4. Paul L. Flory, Principles of Polymer Chemistry, Ithaca, New york, 1953

5. This equation can be easily deduced; The total number of contacts between a polymer and its neighbors is z-2 per chain. The probability that a cell is occupied by a solvent molecule is equal to its volume fraction φ_s. Then the total number of solvent-monomer contacts is simply (z-2) rN_pφ_s. If ε_ps is the change in energy for the formation of an unlike pair contact, Δε_ps =  ε_ps - ½ (ε_pp + ε_ss), then the total enthalpy of mixing is simply ΔH_mix = Δε_ps (z-2) rN_pφ_s = Δε_ps z N_sφ_p. This is the well-known van Laar equation for mixing for 2-component mixtures, which can be recast to ΔH_mix = kT χ N_sφ_p, with χ = z Δε_ps / kT.

6. This quantity is the Helmholz energy when the change in volume upon mixing polymer and solvent molecules is negligible. Then ΔG = ΔH - TΔS = ΔU + pΔV - TΔS reduces to ΔF = ΔU - TΔS.