Phase Separation and Equilibrium

There is a long history of observations of phase equilibrium in polymer solutions. Phase separation in polymer solutions was first considered in detail from a statistical thermodynamic view by Flory [27,40,41). In his original papers on the statistical thermodynamics of these systems, the conditions for equilibrium between two separated phases are the classical conditions of when the partial molar Gibbs free energies are the same for each phase. The partial molar free energies are 

For a monodisperse flexible polymer chain solution, we have for the solvent (1) and polymer 

When AHi = 0 (andB = 0), AGj decreases continuously as 2 increases from zero to unity. When AH 0, AGj decreases more slowly, and conditions are eventually reached where phase separation can occur. This is observed to take place first for large x (i.e., the highest degree of polymerization). 

Flory s analysis for heterogeneous solutions [41] shows that this phase separation occurs first for the highest molecular weight species. This explains the well-known observation that poor solutions can fractionate polymers according to molecular weight. The critical volume fraction for polymer miscibility is 

This is discussed by them and by later authors including Huggins and Okamato [42] among others. 

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Obviously, construction of a mathematical model of this process, with our present limited knowledge about some of the critical details of the process, requires good insight and many qualitative judgments to pose a solvable mathematical problem with some claim to realism. For example what dictates the point of phase separation does equilibrium or rate of diffusion govern the monomer partitioning between phase if it is the former what are the partition coefficients for each monomer which polymeric species go to each phase and so on. 

It is important to note that catalysts for alkoxysilane hydrolysis are usually catalysts for condensation. In typical silane surface treatment applications, alkoxysilane reaction products are removed from equilibrium by phase separation and deposition of condensation products. The overall complexity of hydrolysis and condensation has not allowed simultaneous determination of the kinetics of silanol formation and reaction. Equilibrium data for silanol formation and condensation, until now, have not been reported. 

Phase separation and immiscibility is a well-known phenomenon for almost all solid solutions [3], It arises from a lower free energy of the separated components due to a large strain in the mixed crystal. It is therefore expected to occur primarily under close-to-equilibrium growth conditions. 

The second is to examine the dynamics of phase separation and phase dissolution which can be pursued by scattering techniques. This topic involves the fundamental problem of self-organization in polymer systems under non-equilibrium conditions. 

It is quite remarkable how many firm deductions are based on a single hypothesis. Beginning with the Margules formulation (3.13.12) for r , thermodynamics leads directly to the specification of AGm, as shown in (3.13.13). All other mixing functions are then found from (3.13.3)—(3.13.7). When phase separation does occur the composition of the two phases in equilibrium is specified by Eq. (3.13.14). The critical value of B required for incipient phase separation and the critical composition of the mixture are specified by Eq. (3.13.15c). Finally, one may construct diagrams such as shown in Fig. 3.13.3 by which deviations from Raoult s Law are predicted. The foregoing is a beautiful illustration of the power of thermodynamic methodology. 

In contrast to natural structures the morphological features of structures in fabricated foods are in principle within our control. The source of the many structures of foods, even those made from a single raw material (e.g., wheat flour), lies in the ingredient mix and the fact that thermodynamic equilibrium is practically never required or achieved during processing. These metastable structures can be attained because they are favored kinetically, that is, the approach to equilibrium is slow. At any point during the development of a particular structure a process of shape stabilization sets in, usually by vitrification, partial crystallization, phase separation and/or formation of a network (Figure 12.5). 

It is the authors experience that, when phase separation takes place in such systems, for example with PVC/chlorinated polyethylene, it is a rather indeterminate process. It appears to take place at different temperatures using different techniques for establishing phase separation and for different heating rates. The process may more resemble a gradual increase in equilibrium phase separation extent over a wide temperature range and time. 

Frequently, vapor-liquid phase separators follow and are combined with the con onent separators, and equilibrium is assume between the exit streams of this combination. Here, the phase separators are omitted as shown in Figure 3.3.1 to keep the two kinds of separators divided according to their major function - one where essentially component separation occurs and the other where essentially phase separation occurs. 

This method starts off by fixing the temperature and pressure and iterating around the vapor fraction to calculate the equilibrium phase separation and compositions. The first step is an isothermal Hash calculation. If T and P are in fact the independent variables, the solution obtained in the first step is the desired solution. If either Tori and one more variable are specified, then another, outer iterative loop is required. The outer loop iterates around P or T (whichever is not fixed) until the other specified variable is satisfied. 

Fig. 8 Left The phase behavior of amphiphiles as observed with the model of [114,115], is shown in the main panel, plotted as a function of rescaled temperature kgT/e and attraction width w,. ja at zero lateral tension. Each symbol corresponds to one simulation and identifies different bilayer phases. Crosses denote the gel phase, solid circles mark fluid bilayers, and vertical crosses indicate the region where bilayers are unstable. The dashed lines are merely guides to the eye. The inset shows the pair potential between tail beads (solid line) and the purely repulsive head-head and head-tail interaction (dashed line). Reprinted with permission from Ref. 114. Copyright (2005) by the American Physical Society. Right Phase separation and budding sequence for a vesicle containing a 50 50 mixture of two lipids. The vesicle is in equilibrium with a very dilute vapor of amphiphiles (i.e., the lipids seen floating in the exterior volume). From [114]...

Using Flory-Huggins theory it is possible to account for the equilibrium thermodynamic properties of polymer solutions, particularly the fact that polymer solutions show major deviations from ideal solution behavior, as for example, the vapor pressure of solvent above a polymer solution invariably is very much lower than predicted from Raoult s law. The theory also accounts for the phase separation and fractionation behavior of polymer solutions, melting point depressions in crystalline polymers, and swelling of polymer networks. However, the theory is only able to predict general trends and fails to achieve precise agreement with experimental data. 

Presence of Complex Molecules.— The Phase Rule, we have seen, takes no account of molecular complexity, and so it is found that the system water— vapour or the system acetic acid— vapour behaves as a univariant system of one component, although in the liquid and sometimes also in the vapour different molecular species (simple and associated molecules) are present. Such systems, however, it should be pointed out, can behave as one-component systems only if at each tmperature there exists an equilibrium between the different molecular species [pseudo-components) in each phase separately and as between the two phases and only if these equilibria are established sufficiently rapidly. By this is meant that the time required for establishing equilibrium is short compared with that required for determining the vapour pressure. When these conditions are satisfied, the system will behave as a univariant system of one component. 

Exciplex states only exist at the interface between the two dissimilar polymers in the blend. Reducing the density of these interfaces in the polymer blend is expected to reduce the amount of exciplex observed. For example, annealing mobilizes the polymers and causes the film to move closer to thermodynamic equilibrium, i.e. the two polymers phase separate and the density of heterojunction sites decreases. Indeed, we observe that the amount of exciplex emission is reduced by the annealing treatment [30]. 

Although by starting with Eq. 11.2-2 one can proceed directly to the calculation of the liquid-liquid phase equilibrium state, this equation does not provide in.sight into the reason that phase separation and critical solution temperature behavior occur. To obtain this insight it is necessary to study the Gibbs energy versus composition diagram for various mixtures. For an ideal binary mixture, we have (Table 9.3-1)... 

Although any of the designs mentioned above will provide the location of phase boundaries (versus temperature and pressure), it is also important to know the compositions of the two phases in equilibrium. Note that while tie lines (lines connecting phases in equilibrium on T-x or p-x diagrams) are horizontal for simple binary mixtures, this is not true for phase separation in multicomponent systems (most notably polymer-fluid systems where the polymer sample contains chains of various lengths). Consequently, ports which allow withdrawal of samples following phase separation and equilibration are an important feature of view cells. Such ports also allow for the measurement of partition coefficients of solutes between, for example, aqueous and CO2 phases. 

Unit operation model (black box models such as mixers, separators, component splitters, etc. models of phase separation and relaxation, heat-transfer model, multistage models, pumps and compressors, reactor models such as equilibrium reactor, stoichiometric reactor, tubular reactor, etc. see Chapter 2). 

The relation between phase separation and viscosity has been studied as a function of time during which the scale of phase separation increases. Under these circumstances, the viscosity has been observed to increase by as much as five orders of magnitude during an isothermal heat treatment for times of several hundred hours. The viscosity initially changes rapidly as the connectivity of the structure and the compositions of the equilibrium phases approach their final values, and then more slowly as coarsening, or growth in the scale of the microstructure, occurs. 

It becomes apparent that under pol)merization conditions a single type of equilibrium morphology may not always be likely. The competition between phase separation and the second-stage pol)merization kinetics can generate nonequilibrium morphologies due to the diffrisional resistance provided by the high local viscosity. In such cases, particles with occlusions or multiple surface domains (lumps) may coexist. 

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