## Flory-Huggins Lattice Theory

of Polymer Solutions,
Part 1

The thermodynamics of (binary) regular polymer solutions^{1} were
first investigated by Paul Flory^{2} and Maurice Huggins^{3} 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
*N*_{0} lattice sites, as illustrated in the figure below. The
total number of lattice sites, *N*_{0}, is assumed to be equal to
the number of solvent molecules, *N*_{s}, and
polymer repeat units,* N _{p}r*:

*N*_{0} = *N*_{s} + *N _{p}r*

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 ln*M*! = *M* ln*M* - *M*, Flory and Huggins obtained following expression for the entropy of mixing:

Δ*S _{mix}* ≈ -

*k*(

*N*ln

_{p}*φ*+

_{p}*N*ln

_{s}*φ*)

_{s}Alternately, the entropy of mixing can be written as

Δ*s _{mix}* = Δ

*S*/

_{mix}*N*≈ -

_{0}*k*{

*φ*/ (

_{p }*r·v*) · ln

_{r}*φ*+

_{p}*φ*/

_{s}*v*· ln

_{s}*φ*}

_{s}where *φ _{s}* and

*φ*are the volume fractions of the solvent and polymer,

_{p} *φ _{s}* =

*N*/ (

_{s}v_{s}*N*+

_{s}v_{s}*N*),

_{p}rv_{r}*φ*=

_{p}*N*/ (

_{p}rv_{r}*N*v

_{s}_{s}+

*N*

_{p}

*rv*)

_{r}and *v _{s}* and

*v*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.

_{r}The Gibbs free energy of mixing, Δ*G _{mix}*, often includes an enthalpy part, that is, mixing can be an
endothermic or an exothermic process

Δ*G _{mix}* = Δ

*H*-

_{mix}*T*Δ

*S*

_{mix}where Δ*H _{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}* = Δ

*H*/ (

_{mix}*kT N*)

_{s}φ_{p}which combined with the entropy term leads to the free energy of mixing:

Δ*G _{mix}* /

*kT*=

*N*ln

_{p}*φ*+

_{p}*N*ln

_{s}*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

Δ*g _{mix}* /

*kT*≈

*φ*/

_{p}*r*· ln

*φ*+

_{p}*φ*ln

_{s}*φ*+

_{s}*χ*

_{ps}φ_{s}φ_{p}A more general expression for the free energy of mixing is

Δ*g' _{mix}* /

*kT*=

*φ*/ (

_{p}*r·v*) · ln

_{r}*φ*+

_{p}*φ*/

_{s}*v*· ln

_{s}*φ*+

_{s}*χ*/ √(

_{ps}φ_{s}φ_{p}*v*)

_{r}v_{s}where Δ*g' _{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*. This gives

_{s} Δ*μ _{s}*/

*RT*= ln [1-

*φ*] + (1 - 1/r)

_{p}*φ*+

_{p}*χ*

_{ps }φ_{p}^{2}

where *μ _{s}* is the
chemical potential of the solvent per mole.
Substitution of this expression into the osmotic pressure relation

*Π*= - Δ

*μ*/

_{s}*V*

_{s}gives

*Π* ≈ *RT* *V*_{s}^{-1} · {ln [1- *φ _{p}*] +

*φ*+

_{p}*χ*

_{ps}*φ*

_{p}^{2}}

where *V _{s}* is the molar volume of the solvent.

##### References

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.

- P. J. Flory,
*J. Chem. Phys.*9, 660 (1941); 10, 51 (1942) - M. L. Huggins,
*J. Phys*.*Chem.*46, 151 (1942); J. Am. Chem. Soc. 64, 1712 (1942) - Paul L. Flory,
*Principles of Polymer Chemistry*, Ithaca, New york, 1953 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*_{p}φ_{s}. If*ε*_{ps}is the change in energy for the formation of an unlike pair contact, Δ*ε*_{ps}=*ε*_{ps }- ½_{pp}+*ε*_{ss}), then the total enthalpy of mixing is simply Δ*H*= Δ_{mix}*ε*_{ps}(*z*-2)*rN*_{p}φ_{s}= Δ*ε*_{ps}*z N*_{s}φ_{p}. This is the well-known*van Laar equation*for mixing for 2-component mixtures, which can be recast to Δ*H*=_{mix}*kT χ**N*_{s}φ_{p}, with*χ**=*z*Δε*_{ps}/*kT*.