## Polymer Phase Equilibria

Spinodal and Binodal

From the thermodynamic models for polymer blends and dissolved polymers, it is possible to calculate the thermodynamic conditions (temperature, pressure, and composition) under which phase separation occurs.

Whether two polymers are mutually miscible or a polymer
is soluble in a solvent, depends on the sign of the Gibbs
free energy, Δ*g _{m}*, which is related to the energy
and entropy of mixing:

Δ*g _{mix}* = Δ

*h*-

_{mix}*T*Δ

*s*= Δ

_{mix}*h*-

_{mix}*T*∂Δ

*g*/ ∂

_{mix}*T*|

_{p,φ}< 0

However, this criterion alone is not sufficient to determine if a
mixture is truly stable against phasing into two polymer mixtures of
different composition. It only states that the mixture will not
separate into two phases of pure components, like pure solvent and
polymer. As will be shown below, phasing can occur even for negative
Gibbs free energies of mixing, Δ*g _{mix}* < 0,
whereas a mixture is stable if the free energy curve does not show
local minima.

Δ*g _{mix}* < 0 and ∂

^{2}Δ

*g*/ ∂

_{mix}*φ*

_{p}

^{2}

The dependency of Δ*g _{mix}* on composition
(i.e. volume fraction) at constant temperature is illustrated in the
figure below.
If Δ

*g*is positive over the entire range of composition, as illustrated by curve I, then the two components are completely immiscible, or in other words, the solvent and the polymer form two separate phases. The other extreme, is total miscibiltiy, which is illustrated by curve III. This is the case when Δ

_{mix}*g*is positive and the second derivative of Δ

_{mix}*g*is negative over the entire range of composition:

_{mix} Δ*g _{mix}* < 0 and (∂

^{2}Δ

*g*/ ∂

_{mix}*φ*

_{P}

^{2})

_{p,T}> 0

### Gibbs Free Energy of Mixing

as a Function of Composition

The third case describes partial miscibility. This curve has two minima at *φ _{P,B}^{1}* and

*φ*. Any mixture with a composition between these two concentrations will spontaneously phase separate into a solvent-rich and polymer-rich phase of composition

_{P,B}^{2}*φ*and

_{P,B}^{1}*φ*which are the points of the common tangent as illustrated by curve II. For a given polymer fraction

_{P,B}^{2}*φ*, the volume fractions

_{P}*φ*and

_{P,L}*φ*of the two coexisting mixed phases can be calculated from

_{P,H} *φ _{P}* =

*φ*

_{P,L}*φ*+

_{P,B}^{1}*φ*

_{P,H}*φ*

_{P,B}^{2}=

*φ*

_{P,L}*φ*+ (1 -

_{P,B}^{1}*φ*)

_{P,L}*φ*

_{P,B}^{2}Solving this equation for *φ _{P,l}* gives

*φ _{P,L}* = (

*φ*-

_{P,B}^{2}*φ*) / (

_{P}*φ*-

_{P,B}^{2}*φ*)

_{P,B}^{1} *φ _{P,H}* = (

*φ*-

_{P}*φ*) / (

_{P,B}^{1}*φ*-

_{P,B}^{2}*φ*)

_{P,B}^{1}The two minima *φ _{P,B}^{1}* and

*φ*are the so called binodal points, which can be found by differentiating the Gibbs free energy of mixing, Δ

_{P,B}^{2}*g*, with respect to the composition,

_{mix}*φ*. At constant pressure (

_{P}*p*) and temperature (

*T*), the binodal condition reads

∂Δ*g _{mix}* / ∂

*φ*|

_{P}*φ*= ∂Δ

_{P,B}^{1}*g*/ ∂

_{mix}*φ*|

_{P}*φ*

_{P,B}^{2-}the condition for equilibrium between the two phases in a binary mixture can be also expressed by equality of the chemical potentials in the two phases:

*μ*_{P1}^{1} = *μ*_{P1}^{2}, and *μ*_{P2}^{1} = *μ*_{P2}^{2}

*μ*_{Pi} = ∂Δ*g _{mix}* / ∂

*n*|

_{i}_{p,T, nj}

where the subscripts *P*_{1} and
*P*_{2}
are the designations for the two compounds and the superscripts 1
and 2 for the two phases.

For a symmetrical polymer blend, i.e. for
a blend of two polymers with equal chain length, *r _{P1}* =

*r*=

_{P2}*r*, the expression for the binodal reads:

*χ _{1,2}* ·

*r*= ln[

*φ*/

_{P2}*φ*] / (1 - 2

_{P1 }*φ*)

_{P1}where *φ _{P1}* and

*φ*are the volume fractions of the two polymers and

_{P2 }*χ*is the polymer-polymer interaction parameter.

_{1,2}For the general case *r _{P1}* ≠

*r*, no simple analytical solution exists, thus, the

_{P2}*binodal*points are usually estimated using numerical methods. This is often done with the aid of a computer. The binodal points can be also predicted by constructing the common tangent, as indicated in the figure above.

A very different situation is encountered
when the composition lies between *φ _{P,B}^{1}* and

*φ*or between

_{P,S}^{1}*φ*and

_{P,B}^{2}*φ*(see curve II), where

_{P,S}^{2}*φ*and

_{P,S}^{1}*φ*are the points of inflection. We already stated that points on a concave curve, (∂

_{P,S}^{2}^{2}Δ

*g*/∂

_{mix}*φ*

_{P}

^{2})

_{p,T}> 0, are stable against phase separation, simply because on a concave curve, the total free energy of neighboring compositions is greater than that of the mixture in question, i.e., the free energy will always increase when the mixture separates into any two phases. Similarly, a mixture on a concave part of a curve with a convex portions is stable against phase separation into compositions on the concave part, that is, all points that lie between the minimum and the point of inflection of the concave portion are stable in regard to neighboring compositions. However, these compositions are not stable against separation into phases of composition

*φ*and

_{P,B}^{1}*φ*. Thus, the mixture is only stable against small concentration fluctuations, but not stable against large fluctuations. This is why this region is called meta-stable. The point that separates the unstable region from the meta-stable is called a

_{P,B}^{2}*spinodal*point. It is the point of inflection at constant

*T*and

*p*:

(∂^{2}Δ*g _{mix}* / ∂

*φ*

_{P}^{2})

_{p,T}= 0

Another important point is the point at which
the two coexisting phases coincide. This point is called the *
critical point*. In a binary phase diagram, these are the lower
(LCST) and upper critical solution temperature (UCST) of the phase
curves. It is also the point where the binodal and spinodal
coincide. These points are easy to calculate; if the temperature
increases, the local maxima diminish and eventually vanish, thus the
points of inflection and the contact points on the double tangent
approach each other until they coincide. At this particular point,
both the second and the third derivative of Δ*g _{mix}* vanish:

(∂^{2}Δ*g _{mix}* / ∂

*φ*

_{P}

^{2})

_{p,T}= (∂

^{3}Δ

*g*/ ∂

_{mix}*φ*

_{P}

^{3})

_{p,T}= 0

The critical value of *χ* where the miscibility gap begins is given by^{1,2}

*χ _{c}* = ½ · (

*r*

_{P1}^{-½}+

*r*

_{P2}^{-½})

^{2}

And the critical concentration is located at

*φ _{P1,c}* =

*r*

_{P1}^{½}/ (

*r*

_{P1}^{½}+

*r*

_{P2}^{½})

##### References

- Gert Strobl,
*The Physics of Polymers*, Berlin 2007 - Paul L. Flory,
*Principles of Polymer Chemistry*, Ithaca, New york, 1953