Handbook of Solvents - George Wypych - ChemTech - Ventech!

4.4 Measurement of solvent activity 191
equilibrium is reached after both phases are sharply separated and clear, Rehage et al. 193 After
separating both phases, concentrations and distribution functions were measured. Acceptable
results can be obtained for low polymer concentrations (up to 20 wt%). Scholte and
Koningsveld 194 developed a method for highly viscous polymer solutions at higher concentrations
by constructing a modified ultracentrifuge where the equilibrium is quickly established
during cooling by action of gravitational forces. After some hours, concentrations,
phase volume ratios and concentration differences can be determined. Rietfeld with his
low-speed centrifuge 195 and Gordon with a centrifugal homogenizer 196 improved this technique
and expanded its applicability up to polymer melts, e.g., Koningsveld et al. 197
The methods for obtaining spinodal data have already been discussed above with the
light scattering technique, please see Subchapter 4.4.3.2.2.
Special methods are necessary to measure the critical point. For solutions of
monodisperse polymers, it is the maximum of the binodal. Binodals of polymer solutions
can be rather broad and flat. The exact position of the critical point can be obtained by the
method of the rectilinear diameter. Due to universality of critical behavior, a relation like
Equation [4.4.56] is valid, deGennes: 198
where:
I II crit crit
( ϕ2 −ϕ2) 2−ϕ2 ∝( 1−T
T )
1−α
/ / [4.4.56]
volume fraction of the polymer in coexisting phase I
I
ϕ 2
II
ϕ 2 volume fraction of the polymer in coexisting phase II
crit
ϕ 2 volume fraction of the polymer at the critical point
T crit
critical temperature
α critical exponent
and critical points can be obtained by using regression methods to fit LLE-data to Equation
[4.4.56].
For solutions of polydisperse polymers, such a procedure cannot be used because the
critical concentration must be known in advance to measure its corresponding coexistence
curve. Additionally, the critical point is not the maximum in this case but a point at the
right-hand side shoulder of the cloud-point curve. Two different methods were developed to
solve this problem, the phase-volume-ratio method, e.g., Koningsveld, 199 where one uses
the fact that this ratio is exactly equal to one only at the critical point, and the coexistence
concentration plot, e.g. Wolf, 200 I II
where an isoplethal diagram of values of ϕ 2 and ϕ2 vs. ϕ02
gives the critical point as the intersection point of cloud-point and shadow curves.
Since LLE-measurements do not provide a direct result with respect to solvent activities,
Equation [4.4.8] and the stability conditions are the starting points of data reduction. As
pointed out above, the following explanations are reduced to the strictly binary solution of a
monodisperse polymer. The thermodynamic stability condition with respect to demixing is
given for this case by (see Prausnitz et al. 49 ):
2 2 ( ∂ ΔmixG/ ∂ ϕ2)
PT
, > [4.4.57]
0
I II
If this condition is not fulfilled between some concentrations ϕ2 and ϕ 2 , demixing is
obtained and the minimum of the Gibbs free energy of mixing between both concentrations
is given by the double tangent at the corresponding curve of ΔmixG vs. ϕ 2 , Equation [4.4.8].