Liquid-Liquid Phase Equilibria
The phenomenon of liquid-liquid demixing is the basis of many experimental investigations to determine the thermodynamic properties of polymer-containing systems. Central items are the composition of coexisting phases (fixing the tie lines), conditions for incipient phase separations (cloud points) for spinodal decomposition. Another interesting aspect concerns the fractionation of polymers associated with phase separation.
Here we discuss a special case, namely the phase behavior of flowing, polymer containing mixtures which is of great practical importance during the processing of polymeric mixtures, for example the co-extrusion of polymer blends.
The calculation is based on a generalized Gibbs energy of mixing Gshear, which is the sum of Gz (zero shear) and Es the energy the mixture stores until it reaches the steady state for a given shear rate. Depending on the curvatures of Es and Gz one can distinguish two different principally different shear effects on the demixing behavior as shown in the depicted graph:
Shear induced mixing results if the
Es curve is always situated above its tangent and if there is
a "hump" in the plot of Gz versus concentration (in
the above graph the volume fraction of polymer). Under these conditions
the hump in the curve of Gz vanishes by addition of Es,
and Gshear exhibits a positive curvature within the whole composition
range. This situation - typical for molar masses of the polymer below a
certain characteristic M value and comparatively low shear rates - is exemplified
in the left part of the figure where the already demixed system becomes
homogeneous under shear.
The device presented in the scheme was assembled from commercially available optical pieces using components of a shear rate controlled rheometer.
Its central part consists of a rotor-stator system (Searle typ) which allows the simultaneous measurement of the viscosity of the liquid contained in the gap and of the ratio I/I0, the intensity I of the light having passed the solution twice, divided by I0, the intensity of the primary laser beam. For this purpose it was necessary to replace a part of the stator by a glass tube of 0.5 cm height where the laser beam can pass the sample. In its present configuration the apparatus can be operated in the T-interval from 10 to 100 °C, the usual heating rate is 0.1 K/min.
A typical example for experimental data (red
symbols) obtained with the above described rheo-optical device is shown
in this figure. This graph gives the shear effects on the phase separation
for a ternary system (C. Krause; R. Horst; B. A. Wolf, Macromolecules,
submitted) made up of two highly incompatible polymers (polystyrene (PS),
poly(n-butyl methacrylate) (PBMA)) and a solvent (cyclohexanone, CHO) which
is thermodynamically good for both components. The results are presented
for constant values of w*PBMA, the weight fraction of PBMA in
the blend PS/PBMA (w*PBMA = wPBMA /(wPS
+ wPBMA)) and of wpol, the over-all weight fraction
of the polymer (wpol = wPS + wPBMA). The
corresponding theoretical calculations (black curves) are also depicted
in this figure.
This investigation has demonstrated the usefulness of the theoretical concept developed for the description of shear effects also for the modeling of ternary system using the concentration dependent interaction parameters required for a realistic description of the actual behavior in the absence of shear. The theoretical predictions are in good qualitative agreement with the experimental findings, there is, however, still some quantitative discrepancy. Disregarding experimental inaccuracies, the explanation could lie in neglects on the theoretical side, in an improper modeling of the stored energy or in principal deficiencies of the present approach.
The next graph shows a phase diagram recalculated for the system trans-decaline/polystyrene (the parameters can be found in B.A. Wolf, Macromolecules 17, 615 (1984)) for the given shear rate of 1000 s-1and for the stagnant solution according to a new procedure ("Calculation of phase diagrams not requiring the derivatives of the Gibbs energy demonstrated for a mixture of two homopolymers with the corresponding copolymer" R. Horst Macromol. Theory Simul. 4, 449-458 (1995))
This phase diagram
exhibits many of the phenomena which can be observed in sheared systems.
The black lines give the spinodal and the binodal curve for the stagnant
UCST-system, all colored lines and symbols stand for the sheared solution: