
LiquidLiquid Phase EquilibriaIntroductionThe phenomenon of liquidliquid demixing is the basis of many experimental investigations to determine the thermodynamic properties of polymercontaining 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 coextrusion of polymer blends.
Theoretical approach.The calculation is based on a generalized Gibbs energy of mixing G_{shear}, which is the sum of G_{z} (zero shear) and E_{s} the energy the mixture stores until it reaches the steady state for a given shear rate. Depending on the curvatures of E_{s} and G_{z} 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
E_{s} curve is always situated above its tangent and if there is
a "hump" in the plot of G_{z} versus concentration (in
the above graph the volume fraction of polymer). Under these conditions
the hump in the curve of G_{z} vanishes by addition of E_{s},
and G_{shear} 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. Rheooptical apparatus.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 rotorstator system (Searle typ) which allows the simultaneous measurement of the viscosity of the liquid contained in the gap and of the ratio I/I_{0}, the intensity I of the light having passed the solution twice, divided by I_{0}, 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 Tinterval from 10 to 100 °C, the usual heating rate is 0.1 K/min. Experimental Data
A typical example for experimental data (red
symbols) obtained with the above described rheooptical 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(nbutyl 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} = w_{PBMA }/(w_{PS}
+ w_{PBMA})) and of w_{pol}, the overall weight fraction
of the polymer (w_{pol} = w_{PS} + w_{PBMA}). 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. Detailed CalculationsThe next graph shows a phase diagram recalculated for the system transdecaline/polystyrene (the parameters can be found in B.A. Wolf, Macromolecules 17, 615 (1984)) for the given shear rate of 1000 s^{1}and 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, 449458 (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
UCSTsystem, all colored lines and symbols stand for the sheared solution: 
