## Flory-Huggins Lattice Theory

of Polymer Solutions, Part 2

The classical thermodynamics of (binary) polymer solutions was first developed by Paul Flory^{1} and Maurice Huggins^{2} independently in the early 1940s.
They assumed a rigid lattice framework and a regular solution (random mixing)
and obtained for the Gibbs free energy of mixing per unit volume following expression:

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*

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}The two parameters *v _{s}* and

*v*are the volumes of a solvent molecule and of a polymer repeat unit with one and

_{r}*r*repeat units respectively. The parameter

*v*is the average segmental volume, (

*v*)

_{r}v_{s}^{1/2}.

Assuming equal lattice volumes for both repeat units, as
was assumed by Flory^{1}, 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}At first it was assumed that the interaction parameter* χ* is independent of the concentration *φ _{p}*.
But it became soon evident that

*χ*is a function of both the temperature

*T*and concentration of the polymer. Furthermore,

*χ*is also a function of the molecular weight of the polymer, not only at low but also at high polymer concentrations. However, there are quite a number of cases where

*χ*seems to be independent of composition, i.e. good solvents, as was proposed by the original Flory-Huggins theory. Some examples are polyisobutylene (PIB) in cyclohexane and poly(dimethyl siloxane) (PDMS) in n-octane. For this situation, the prediction of the free energy of mixing and of the phase curves (binodal and spinodal) is least difficult.

### Temperature Dependence of Interaction Parameter

The temperature dependence of the interaction parameter is quite obvious from the definition of the *χ* parameter:

*χ*_{}(*T*) = *-z*/2 (*ε*_{pp} + *ε*_{ss} - 2 *ε*_{ps}) / *kT* =
*B* / *T*

where *B* describes the temperature dependence of *χ*. According to this equation, *χ* is a linear function of
the inverse temperature. For the case B > 0 and (very) low
temperatures, the free energy curve is concave and homogenous
mixtures are unstable because the second derivative of the free
energy of mixing is negative and the contribution of the entropy is
small.

In most cases, mixing of a polymer with a solvent results in a volume change, that is, local packing effects lead to a temperature-independent additive constant in the
Flory expression for the interaction parameter. Empirically, the temperature dependence of
*χ* is often written as a sum of two terms:

*χ*(*T*) = *A* + *B */* T*

The temperature independent term *A* is the so-called "entropic part" of of *χ*, while *B* / *T* is called the "enthalpic part".
The parameter *A* and *B* have been tabulated for many polymer-solvent
and polymer-polymer
systems.

### Binodal Phase curve

The phase boundary between the single- and two-phase region, the so-called binodal, can be determined from the common tangent of the free
energy of mixing at the compositions *φ _{P,B}^{1}* and

*φ*, corresponding to the equilibrium composition of the two phases:

_{P,B}^{2} ∂Δ*g _{mix}* / ∂

*φ*|

_{P}*φ*= ∂Δ

_{P,B}^{1}*g*/ ∂

_{mix}*φ*|

_{P}*φ*

_{P,B}^{2-}The derivative of Δ*g _{mix}* is

∂Δ*g _{mix}* / ∂

*φ*=

_{P}*kT*· [

*r*

^{-1 }ln

*φ*+

_{p}*r*

^{-1}- ln

*φ*- 1 +

_{s}*χ (1 - 2φ*)]

_{p}And for the general case of two polymers with
*r* and *s* repeat units per polymer:

∂Δ*g _{mix}* / ∂

*φ*=

_{p1}*kT*· [

*r*

^{-1 }ln

*φ*+

_{p1}*r*

^{ -1}- s

^{-1}ln (1-

*φ*) - s

_{p1}^{-1}+

*χ (1 - 2φ*)]

_{p1}For the simple case of polymers of equal length, *r* = *s* =
*n*, the tangent is horizontal:

∂Δ*g _{mix}* / ∂

*φ*=

_{p1}*kT*· [

*n*

^{-1 }ln

*φ*- n

_{p1}^{-1}ln (1-

*φ*) +

_{p1}*χ (1 - 2φ*)] = 0

_{p1}This equation can be solved for the Flory-Huggins parameter corresponding to the binodal (phase boundary):

*n* · * χ _{B}* = ln

*φ*- ln (1-

_{p1}*φ*) / (1 - 2

_{p1}*φ*) = ln [

_{p1}*φ*/ (1-

_{p1}*φ*)] / (1 - 2

_{p1}*φ*)

_{p1}And with the temperature dependence of the interaction parameter, *χ* = *A* + *B */* T*, this relation can be solved for the binodal of the phase diagram, i.e.
for the binodal temperature as a function of composition:

*T _{B}* =

*B*/ {ln[

*φ*/ (1 -

_{p1 }*φ*)] / (2

_{p1}*n φ*-

_{p1}*n*) -

*A*}

### Spinodal Phase Curve

The boundary between the unstable and metastable region, the so-called spinodal, can be calculated from the inflection points of the free energy curve:

(∂^{2}Δ*g _{mix}* / ∂

*φ*

_{P}

^{2})

_{p,T}=

*kT*{

*(φ*

_{p}*r*)

^{-1}+ [(1 -

*φ*)

_{p}*s*]

^{-1}- 2

*χ*} = 0

This equation can be solved for the Flory-Huggins interaction parameter corresponding to the spinodal phase boundary:

*χ _{S}* = ½ {(

*φ*

_{p}*r*)

^{-1}+ [(1 -

*φ*)

_{p}*s*]

^{-1}}

The spinodal phase boundary can also be written as a temperature-composition relation:

*T _{S}* =

*B*/ {(2

*φ*

_{p}*r*)

^{-1}+ [2(1 -

*φ*)

_{p}*s*]

^{-1}-

*A*}

In a binary blend, the lowest point of the spinodal corresponds to the critical point^{4}:

*χ _{c}* = ½ · (

*r*

^{ -½}+

*s*

^{-½})

^{2}

or

*T _{c}* =

*B*/ (χ

_{c}-

*A*) =

*B*/ {½ · (

*r*

^{ -½}+

*s*

^{-½})

^{2}-

*A*}

For a *symmetrical polymer mixture*, *r* = *s* = *n*, the whole phase curve becomes symmetric with the critical point:

*χ _{c}* = 2 /

*n*and

*φ*= 1/2-

_{c}whereas in polymer solutions, *r* = *n*
and *s* = 1, the whole phase curve becomes
very asymmetrical with the critical point:

*χ _{c}* ≈ ½ +
1 /

*n*

^{½}and

*φ*= 1 /

_{c}*n*

^{½}

### Spinodal and Binodal Phase Diagram

The figure above shows the product *n χ* as
a function of polymer concentration *φ _{p}* for a symmetrical
blend of polypropylene (PP) and head-to-head polypropylene (hhPP).
The binodal (solid curve) and spinodal (dashed curve) have been calculated
with the equations above and the parameters

*A*= -0.0036 and

*B*= 1.84. The two curves coincide at the critical point.

##### References and Notes

- 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 *Note, the spinodal and binodal of any binary mixture meet at the critical point.*