Handbook of Solvents - George Wypych - ChemTech - Ventech!

192 Christian Wohlfarth
The resulting curve in the T vs. ϕ 2 diagram, see Figure 4.4.19a, is the binodal curve. Applying
Equations [4.4.3 to 4.4.5 and 4.4.13a], one gets two relations which have to be solved
simultaneously to fit an empirical χ(T)-function to experimental binodal (coexistence) data.
For the most simple case of χ being only a function of T (or P) and not of ϕ 2 (the so-called
one-parameter approach) these relations read:
and
where:
I ( −ϕ2)
II ( 1−ϕ2)
ln 1
I II I2 II2
( ϕ2 ϕ2) χ( ϕ ϕ 2 2 )
1 1 ⎛ ⎞
+ ⎜ − ⎟ − + − = 0
⎝ r ⎠
[4.4.58a]
1
1 ( ) ( )
1
I
2
I II I2 II2
ln 2 2 0
II
1 1
r r
ϕ ⎛ ⎞
−⎜ − ⎟ ϕ − ϕ + χ ϕ − ϕ = [4.4.58b]
ϕ ⎝ ⎠
2
I
ϕ 2
II
ϕ 2
volume fraction of the polymer in coexisting phase I
volume fraction of the polymer in coexisting phase II
r ratio of molar volumes V 2/V 1 what is the number of segments with V seg =V 1
χ Flory-Huggins interaction function of the solvent
and solvent activities result from Equation [4.4.13a]. However, this simple approach is of
limited quality. More sophisticated models have to be applied to improve calculation results.
A special curve is obtained with the border line to the instability region, i.e., the
spinodal curve, for which the second derivative in Equation [4.4.57] is equal to zero. If one
applies again the one-parameter approach with an empirical χ(T)-function, the following
simple result can be derived:
2χ
spinodal ( T )
1 1
− −
rϕ
1−ϕ
spinodal spinodal
2 2
= 0
[4.4.59]
where:
T spinodal
spinodal temperature
spinodal
ϕ 2 volume fraction of the polymer at the spinodal curve
which has to fit an empirical χ(T)-function. An example for the spinodal relation of a
polydisperse polymer was given above by Equations [4.4.44 and 4.4.45].
The common point of spinodal and binodal curve is the critical point. The critical point
conditions are:
2 2
3 3
(∂ Δ G / ∂ ϕ ) = 0 ( ∂ Δ G / ∂ ϕ ) = 0
mix 2 P, T mix 2 P, T
and [4.4.60]
4 4
( ∂ ΔmixG / ∂ ϕ2)
P, T > 0
If one applies again the one-parameter approach with an empirical χ(T)-function, two
simple results can be derived: