Description of Phase Separation

Mathematical Description of Phase Separation. The thermodynamic state of a system of two or more components with limited miscibility can be described in terms of the free energy of mixing.40 At constant pressure and temperature, three different states can be distinguished  

In this state, the free energy of mixing is positive 

The free energy of mixing of a system describes the thermodynamic state of the system and thus provides information about the system stability. If a system is unstable and separates in two coexisting phases, transport of individual components has to take place. The transport processes are determined by thermodynamic parameters, which are expressed by driving forces, and by kinetic parameters, which are determined by diffusivities, i.e., the diffusion coefficient. Fick s law relates the diffusion coefficient to concentration gradients. However, the actual driving forces for any mass transport are gradients in the chemical po- 

D] is the diffusion coefficient of component i, /X is its chemical potential and Xj its mol fraction. Bj is a mobility term, which is always positive. 

The stable, unstable and equilibrium states of the system are not only characterized by positive, negative or disappearing values of the free energy of mixing but, since the diffusion coefficient is directly related to the chemical potential of the individual components, which again is a function of the Gibb s free energy, the state of a system is also characterized by  

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For the description of phase separation we choose again the generic Cahn-Hilliard model in one spatial dimension [124, 125]... 

Phenomenological Description of Phase Separation. Phase separation due to thermal gelation, evaporation of solvent and addition of nonsolvent can be illustrated with the aid of the phase diagram of a polymer solution. 

We have shown how models for volumetric equations of state can be used with stability criteria to predict vapor-liquid phase separations. However, not all phase equilibria are conveniently described by volumetric equations of state for example, liquid-liquid, solid-solid, and solid-fluid equilibria are usually correlated using models for the excess Gibbs energy g. When solid phases are present, one motivation for not using a PvT equation is to avoid the introduction of spurious fluid-solid critical points, as discussed in 8.2.5. A second motivation is that properties of liquids and solids are little affected by moderate changes in pressure, so PvT equations can be unnecessarily complicated when applied to condensed phases. In contrast, g -models often do not contain pressure or density instead, they attempt to account only for the effects of temperature and composition. Such models are thereby limited to descriptions of phase separations that are driven by diffusional instabilities, and the stability behavior must be of class I (see 8.4.2). In this section we show how a g -model can describe liquid-liquid and solid-solid equilibria. 

The above references list experimental critical parameters. In particular we may compare the compressibility factor Eq.(4.62), because it is a pure number. It turns out that the above model does not make quantitative predictions—except in selected cases. But for us it provides a valuable exercise. In fact the model is not wrong—it is incomplete. It turns out that association of ions into aggregates is the most important ingredient for a more accurate description of phase separation in molten salts and... 

This is illustrated in Figure 17.1. The energies of the van der Waals complexes are a better description of the separated species for describing liquid-phase reactions. The energies of the products separated by large distances are generally more relevant to gas-phase reactions. 

A detailed description of AA, BB, CC step-growth copolymerization with phase separation is an involved task. Generally, the system we are attempting to model is a polymerization which proceeds homogeneously until some critical point when phase separation occurs into what we will call hard and soft domains. Each chemical species present is assumed to distribute itself between the two phases at the instant of phase separation as dictated by equilibrium thermodynamics. The polymerization proceeds now in the separate domains, perhaps at differen-rates. The monomers continue to distribute themselves between the phases, according to thermodynamic dictates, insofar as the time scales of diffusion and reaction will allow. Newly-formed polymer goes to one or the other phase, also dictated by the thermodynamic preference of its built-in chain micro — architecture. 

Equations of state relate the phase properties to one another and are an essential part of the full, quantitative description of phase transition phenomena. They are expressions that find their ultimate justification in experimental validation rather than in mathematical rigor. Multiparameter equations of state continue to be developed with parameters tuned for particular applications. This type of applied research has been essential to effective design of many reaction and separation processes. 

This document is organized into three sections. The first defines terms basic to the description of polymer mixtures. The second defines terms commonly encountered in descriptions of phase-domain behaviour of polymer mixtures. The third defines terms commonly encountered in the descriptions of the morphologies of phase-separated polymer mixtures. 

The following table lists the liquid crystalline materials that are useful as gas chromatographic stationary phases in both packed and open tubular column applications. In each case, the name, structure, and transition temperatures are provided (where available), along with a description of the separations that have been done using these materials. The table has been divided into two sections. The first section contains information on phases that have either smectic or nematic phases or both, while the second section contains mesogens that have a cholesteric phase. It should be noted that each material may be used for separations other than those listed, but the listing contains the applications reported in the literature. 

The mathematical model comprises a set of partial differential equations of convective diffusion and heat conduction as well as the Navier-Stokes equations written for each phase separately. For the description of reactive separation processes (e.g. reactive absorption, reactive distillation), the reaction terms are introduced either as source terms in the convective diffusion and heat conduction equations or in the boundary condition at the channel wall, depending on whether the reaction is homogeneous or heterogeneous. The solution yields local concentration and temperature fields, which are used for calculation of the concentration and temperature profiles along the column. 

The formation of the sponge-structured membranes can be easily rationalized utilizing the description of the precipitation process given above. With finger-structured membranes the formation process is more complex and cannot entirely be described by the thermodynamic and kinetic arguments of phase separation processes. 

As mentioned earlier, necessary for the description of the fluid flow is the knowledge of the material properties density and viscosity. Thus for the complex rheology of the ceramic material additionally the preparatory work of designation of the parameters rjB and To (7) is required. If in addition the formation of phase separation at the wall by a wall slip boundary condition is to be included, the rheological measurement of the parameters Ts and k is indispensable. 

The ceU, which is provided with sapphire windows and magnetic stirring, is a modification of the one described by Van Hest and Diepen [4]. A detailed description of this apparatus and the experimental techniques used is given by De Loos et al. [5]. The cloud-point pressures of mixtures of known composition have been measured as a function of temperature by visual observation of the onset of phase separation of the homogeneous phase by lowering the pressure (cloud-point isopleths). The cloud-points have been determined with an absolute error of 0.03 K in temperature and 0.1 MPa in pressure. 

The column is the core of the LC separation technique. To expand the field of MLC, a rapid description of two separation techniques that do not use a column, but can use a micellar phase is presented. The two techniques are field flow fractionation (FFF) and capillary electrophoresis (CE). 

Studying the relation between critical concentration and chemical polydispersity by model calculations yield results similar to those outlined in Sect. 3.3.1 [42]. Generally, chemical polydispersity increases the critical segment-mole fraction. The additional term in facilitates a description of more complicated types of phase diagrams, particularly with parameters exceeding a certain positive value. This leads to the prediction of phase separation of the pure copolymer ensemble B [42]. 

Drexler and Ballschmiter [714] separated pheophytin a and b, chlorophyll a, a, and b, and xanthophyll in <10min using a diol coliunn (A = 425 nm) and a 100/2 hexane/ethanol mobile phase. In the same work, a description of the separation of... 

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