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Liquid-Liquid Phase Equilibria

Introduction

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.

 

Theoretical approach.

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:

Gibbs energy without shear, stored energy and the resulting energy of the flowing system
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Fig. 1. Gibbs energy without shear, stored energy and the resulting energy of the flowing system

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.
Shear induced demixing, given in the right part of the figure, happens more frequently on the high molecular side of molar masses and larger shear rates. Here the stored energy Es changes from a curve which is in the entire composition range located above its tangent to one exhibiting a hump. That means that a homogeneous stagnant mixture becomes phase-separated as it flows.

Rheo-optical apparatus.

The device presented in the scheme was assembled from commercially available optical pieces using components of a shear rate controlled rheometer.

Rheo-optical apparatus
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Fig. 2. Rheo-optical apparatus

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.

Experimental Data

Experimental data
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Fig. 3. Experimental data

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.
In the case of PBMA-rich solutions, a augmentation of the shear rate by 100 s-1 increases the cloud point temperatures by approx. 20 °C (extension of homogeneous region / shear induced mixing) in an approximately linear manner. The shear effects depend pronouncedly on the composition of the blend and are very prominent for mixtures that contain more of the higher molecular weight PBMA and less of the lower molecular weight PS.

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 Calculations

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))

A phase diagramm and the effects of shear on the phase behaviour
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Fig. 4. Phase diagramm and the effects of shear on the phase behaviour

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:
spinodal (yellow)
(meta)stable tie lines (blue)
two types of unstable tie lines (red and green)
The blue line and open triangles (in the top third of the diagramm) indicate a three phase equilibrium which only occurs in the sheared system. Up to a volume fractions of 0.082 the homogeneous region is increased, one observes the phenomenon of shear induced mixing (magenta shading). From 0.082 to approx. 0.17 the heterogeneous area becomes larger as the system flows (shear induced demixing, cyan shading).


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