Polydisperse systems coexistence

The semigrand ensemble method can be implemented in different forms for calculation of equilibrium properties, and phase equilibria for inert or reacting mixtures. Recently, it has been applied to simulate phase coexistence for binary polymer blends [85], where advantage was taken of the fact that identity exchanges are employed in lieu of insertions or deletions of full molecules. The semigrand ensemble also provides a convenient framework to treat polydisperse systems (see Section III.F). 

Inspection of Figs 5 and 6 shows that at high probe concentration, limiting values of D tend toward Di. Thus, a reasonable estimate of D for monodisperse systems can be obtained by measuring D at relatively large probe concentrations, for example, 2 to 5 mM. For polydisperse systems, additional equilibria must be considered. A three-state model was used to develop a version of Eq. (12) for two micellar size distributions, and applied to systems with coexisting globular and rodshaped micelles [4]. 

The spinodal curve and the critical points (including multiple critical points) only depend on few moments of the molar-mass distribution of the polydisperse system while the cloud-point curve the shadow curve and the coexistence curves are strongly influenced by the whole curvature of the distribution function. The methods used that include the real molar-mass distribution or an assumed analytical distribution replaced by several hundred discrete components have been reviewed by Kamide. In the 1980s continuous thermodynamics was applied, for example, by Ratzsch and Kehlen to calculate the phase equilibrium of a solution of polyethene in supercritical ethene as a function of pressures at T= 403.15 K. The Flory s model was used with an equation of state to describe the poly-dispersity of polyethene with a a Wesslau distribution. Ratzsch and Wohlfarth applied continuous thermodynamics to the high-pressure phase equilibrium of ethene [ethylene]-I-poly(but-3-enoic acid ethene) [poly(ethylene-co-vinylace-tate)] and to the corresponding quasiternary system including ethenyl ethanoate [vinylacetate]. In addition to Flory s equation of state Ratzsch and Wohlfarth also tested the Schotte model as a suitable means to describe the phase equilibrium neglecting the polydispersity with respect to chemical composition of the... 

For the understanding of these processes and for the design and evaluation of new separation processes, it is crucial to get a better insight in the underlying principles and phase equilibria. Especially in the investigation of complex systems such as polydisperse polymers with additives or drugs with impurities, it is necessary to get information on the composition of the coexisting phases. Up to now, there is almost no such information available. 

The classic biological example of these systems is bile salt (BS)-lecithin (L)-cholesterol (Ch) micelles which have been studied in detail by QLS [239], In TC-L-Ch systems, particle size and polydispersity were studied as functions of Ch mole fraction (= 0-15%), L/TC molar ratio (0-1.6), temperature (5-85°C), and total lipid concentration (3 and 10 g/dl) in 0.15 M NaCl. For values below the established solubilization limits (A )> added Ch has little influence on the size of simple TC micelles, on the coexistence of simple and mixed TC-L micelles, or on the growth of mixed disk TC-L micelles. For supersaturated systems >1), 10 g/dl simple micellar systems (L/TC = 0) exist as metastable micellar solutions even at = 5.3. Metastability is decreased in coexisting systems... 

Fig. 21. Conversion and composition of phases at equilibrium (coexistence curves) calculated for an initial volume fraction of modifier, 4>mo = 0.177 in a castor oil (CO)-modified DGEBA-EDA system (Reprinted from Polymer, 35, CC. Riccardi, J. Borrajo, RJJ. Williams, Thermodynamic analysjs of phase separatimt in rubber-modilied thermosetting polymers influence of the reactive polj ner polydispersity, 5541-5550 Copyright (1994), with laid permission from Butterworth-Heinemiann journals, Elsevier Science Ltd, The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK)...

The occurrence of a secondary phase separation inside dispersed phase particles, associated with the low conversion level of the p-phase when compared to the overall conversion, explains the experimental observation that phase separation is still going on in the system even after gelation or vitrification of the a-phase [26-31]. A similar thermodynamic analysis was performed by Clarke et al. [105], who analyzed the phase behaviour of a linear monodisperse polymer with a branched polydisperse polymer, within the framework of the Flory-Huggins lattice model. The polydispersity of the branched polymer was treated with a power law statistics, cut off at some upper degree of polymerization dependent on conversion and functionality of the starting monomer. Cloud-point and coexistence curves were calculated numerically for various conversions. Spinodal curves were calculated analytically up to the gel point. It was shown that secondary phase separation was not only possible but highly probable, as previously discussed. 

Later, Burning and Lekkerkerker [37] observed isotropic—nematic phase separation in a dispersion of sterically stabilized boehmite rods, which approximate hard rods, in cyclohexane. Buitenhuis et al. [43] studied the effect of added 35 kDa polystyrene (/ g = 5.9nm) on the hquid crystal phase behaviour of sterically stabilized boehmite rods with average length L = 1.1 nm and average diameter D = ll.lnm in ortho-dichlorobenzene. Different phase equihbria were observed. Two biphasic equilibria dilute isotropic phase Ij + nematic N, concentrated isotropic phase I2 + nematic N and a triphasic equilibrium 1 -F I2 + N (see photo. Fig. 6.20). In this system the boehmite rods are quite polydisperse. Therefore comparison with theory should be done with an approach including polydisperse rods. We further note no li +12 coexistence was observed experimentally but... 

Quaternary and higher systems have been investigated. In such cases, many phases may coexist. Indeed, since the phase behaviour is dependent on the molecular weight of the polymer, polydispersity itself results in a multicomponent system. 

Some differences between BCP/surfactant systems and surfactant/co-surfactant mixtures arise from the large difference in size and from the polymer-specific chain entropy contributions to the free energy of the systems. Moreover, in contrast to simple surfactant-based samples, the polydispersity of the polymer chains is also an important issue for structure formation. Polydispersity might lead to coexistence of different structures (e.g., spheres and worm-like structures) in the same polymer solution. This is due to the fact that small differences in the degree of polymerization of one of the blocks might lead to different packing parameters for a part of the size distribution. 

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