Phase separation, mathematical

Remember that the hump which causes the instability with respect to phase separation arises from an unfavorable AH considerations of configurational entropy alone favor mixing. Since AS is multiplied by T in the evaluation of AGj, we anticipate that as the temperature increases, curves like that shown in Fig. 8.2b will gradually smooth out and eventually pass over to the form shown in Fig. 8.2a. The temperature at which the wiggles in the curve finally vanish will be a critical temperature for this particular phase separation. We shall presently turn to the Flory-Huggins theory for some mathematical descriptions of this critical point. The following example reminds us of a similar problem encountered elsewhere in physical chemistry. 

For the phase separation problem, the maximum and minima in Fig. 8.2b and the inflection points between them must also merge into a common point at the critical temperature for the two-phase region. This is the mathematical criterion for the smoothing out of wiggles, as the critical point was described above. 

We have put this model into mathematical form. Although we have yet no quantitative predictions, a very general model has been formulated and is described in more detail in Appendix A. We have learned and applied here some lessons from Kilkson s work (17) on interfacial polycondensation although our problem is considerably more difficult, since phase separation occurs during the polymerization at some critical value of a sequence distribution parameter, and not at the start of the reaction. Quantitative results will be presented in a forthcoming pub1ication. 

A mathematical model for this polymerization reaction based on homogeneous, isothermal reaction is inadequate to predict all of these effects, particularly the breadth of the MWD. For this reason a model taking explicit account of the phase separation has been formulated and is currently under investigation. 

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. 

We have seen above in two instances, those of liquid-liquid phase separation and polymer devolatilization that computation of the phase equilibria involved is essentially a problem of mathematical formulation of the chemical potential (or activity) of each component in the solution. 

The growth rate, characterized by the change of the radius with time, is proportional to the driving force for the phase separation, given by the differences between 2 > the chemical composition of the second phase in the continuous phase at any time, and, its equihbrium composition given by the binodal line. The proportionahty factor, given by the quotient of the diffusion constant, D, and the radius, r, is called mass transfer coefficient. Furthermore the difference between the initial amount of solvent, (])o, and c]) must be considered. The growth rate is mathematically expressed by [101]... 

As just mentioned, there are a large number of unsolved problems in membrane biophysics, including the questions of local anisotropic diffusion, hysteresis, protein-lipid phase separations, the role of fluctuations in membrane fusion, and the mathematical problems of diffusion in two dimensions Stokes paradox). 

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. 

FJ and rj. denote, as in Chapter 111, the amount of the charged component i or uncharged component k present in the actual system, per unit area of interface, in excess of that which would be present in an idealized pair of phases separated by a mathematical plane (the Gibbs surface) up to which the phases continue without any change in composition. 

Most RP-HPLC separations are done in the iso-cratic mode (i.e., where the composition of the mobile phase is held constant during the analysis). This approach is suitable when the sample consists of analytes having similar properties or where their hydrophobic-ities encompass a small or moderate range. Under these conditions, all solutes in the sample will be eluted over a reasonable time span (i.e., not too short to prevent resolution of individual analytes and not too long to result in an inconvenient analysis period). Therefore, proper selection of the mobile-phase composition is essential in the development of any re-versed-phase separation method. Fortunately, due to the decades of long practice of RP-HPLC, there exists in the literature and from commercial sources, a wealth of information on suitable mobile-phase compositions for particular types of sample, especially for the Cig stationary phase. In addition, the retention of solutes on hydrophobic phases has been modeled mathematically and there exist computer programs for assisting in the optimization of mobile-phase composition in the solution of various separation problems. 

Gas chromatography is based on the distribution of a compound between two phases. In gas-solid chromatography (GSC) the phases are gas and solid, the injected compound is carried by the gas through a column filled with solid phase, and pmrtitioning occurs via the sorption-desorption of the compound (probe) as it travels past the solid. Superimposed upon the forward velocity is radial notion of the probe molecules caused by random diffusion through the stationary phase. Separation of two or more components injected simultaneously occurs as a result of differing affinities for the stationary phase. In gas-liquid chromatography (GLC), the stationary phase is a liquid coated onto a solid suppx>rt. The mathematical treatment is equivalent for GLC and CSC. 

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 ... 

Many aspects of the formation of symmetric or asymmetric membranes can be rationalized by applying the basic thermodynamic and kinetic relations of phase separation. There are, however, other parameters-such as surface tension, polymer relaxation, sol and gel structures-which are not directly related to the thermodynamics of phase separation but which will have a strong effect on membrane structures and properties. A mathematical treatment of the formation of porous structures is difficult. But many aspects of membrane structures and the effect of various preparation parameters Can be qualitatively interpreted. 

In this chapter, we discuss the thermodynamic properties of surface phases. In the previous chapters we have assumed that heterogeneous systems consist of a number of completely homogeneous phases separated by sharply defined mathematical surfaces. It is clear from either molecular or macroscopic considerations that this assumption cannot rigorously apply. Molecules in the vicinity of the interface between any two phases experience a different environment from molecules in the bulk of the phases. Thus, the densities of the various components and the densities of energy and entropy in the vicinity of the interface will be different from the corresponding densities in the bulk phases. However, the influence of the interface does not extend for more than a few molecular dimensions (about 10 cm) into the phases, and the phases may therefore be assumed to be uniform except in the immediate neighborhood of the interfaces. The interface between two phases is in reality a thin region in which the physical properties vary continuously from the bulk properties of one phase to the bulk properties of the other phase. 

Patterns usually appear due to the instability of a uniform state. However, such an instability does not necessarily lead to pattern formation. Let us consider, e.g., phase separation of a van-der-Waals fluid near the critical point Tc. For T > Tc, there exists only one phase, while for T < Tc, there exist two stable phases, corresponding to gas and liquid, and an unstable phase whose density is intermediate between those of the gas and the liquid. When an initially uniform fluid is cooled below Tc, the unstable phase is destroyed, and in the beginning one observes a mixture of stable-phase domains, i.e. hquid droplets and gas bubbles, which can be considered as a disordered pattern. However, the domain size of each phase grows with time (this phenomenon is called Ostwald ripening or coarsening). Finally, one observes a full separation of phases a liquid layer is formed in the bottom part of the cavity, and a gas layer at the top. Thus, the instabihty of a certain uniform state is not sufficient for getting stable patterns. Below we formulate some mathematical models that describe both phenomena, domain coarsening and pattern formation. 

To analyse the phenomenon of domain size growth in a quantitative way, let us consider a simpler physical system, a metallic alloy. There are two kinds of atoms, A and B, with volume fractions (j)A and 4>b, respectively. For the sake of simplicity, assume that the averaged volume fractions (pA) and 4>b) are equal. There exists a temperature Tc such that for T > Tc the fractions are mixed, i.e. the order parameter (p = (pA — thermodynamically stable phases, one with (p > 0 ( A-rich phase") and the other with (p < 0 ( B-rich phase"). A mathematical model of this phenomenon has been suggested by Cahn and Hilliard [25]. From the point of view of thermodynamics, phase separation can be described by means of the Ginzburg-Landau free energy functional... 

In an attempt to clarify the mechanism whereby electrolytes influence the solvency of water for a polymer, Garvey and Robb (1979) measured the heats of dilution of poly(vinyl pyrrolidone) in aqueous electrolyte solutions. The heat of dilution was found to be negative (i.e. mixing is favoured by the enthalpy changes) so that phase separation must inevitably be controlled by entropy effects. Thermodynamics demands that in the salting-out of a polymer by a second component, this component must be negatively adsorbed from the polymer. This can be expressed mathematically by... 

If all of the Jij are positive, the system will tend to phase separate and one can consider the energy of a lattice site surrounded by its average interaction with its surroundings. Mathematically, this allows a decoupling of the interaction term involving only single lattice sites. We thus approximate JijSiSj JijSi where 0 = (5,) is the average volume fraction of B particles. Now the sum in Eq. (1.63) can be evaluated since the sites are decoupled and where we approximate by... 

The phenomenon of phase separation naturally gives rise to the presence of interfaces whose study is the focus of the chapters that follow. In this section, we discuss the mathematical definition of an interface or surface (of zero thickness) and show how one can calculate the area and curvature of a general surface. These concepts are particularly useful in the statistical physics of surfaces, interfaces, and membranes since one often has terms in the energy that depend on the area e.g., surface tension) and/or curvature e.g., bending energy). Of course the physical problem is more complicated since often the interface shape and size is not known but is determined self-consistently by the system. Thus one often asks which surfaces have a given area or curvature. An additional complication is that the surfaces of interest are often not deterministic but rather stochastic thermal fluctuations of the surface... 

Khare V, Greenberg A, Krantz W. Vapor-induced phase separation—Effect of the humid air exposure step on membrane morphology Part I. Insights from mathematical modeling. J Membr Sci. 2005 258(1-2) 140-156. 

Times of gelation and point of phase separation can be described mathematically by the relation (23) t = fcV" (10)... 

The early theories of phase separation are of the mean-field, cell, or cell-hole lattice (statistical thermodynamics) type. The theory takes into account the configurational entropic and enthalpic contributions, but since these are weak, the effects on miscibility are not as predictable as that for other PO blends. Nevertheless, the enthalpy as a difference of the solubility parameters well correlate with the experimental data being independent on SCBD and SCB if SCB < 5/100 C. This observation is unexpected, since the miscibility was reportedly controlled by entropy, e.g., chain stiffening led to phase separation. The newer theoretical models attempt incorporating the model macromolecular chain stmctures using either mathematical modeling via MC, molecular... 

Despite the fact that, in some cases, small differences in AH and A5 can be observed (for various solutes, but on the same column with the same mobile phase), these differences could be found to be essentially insignificant when compared to a change in stationary-phase bonding density using enthalpy-entropy compensation. Enthalpy-entropy compensation is a term used to describe a compensation temperature, which is system independent for a class of similar experimental systems.Melander et have used the enthalpy-entropy compensation method in studies of hydrophobic interactions and separation mechanisms in reversed phase HPLC. Mathematically, enthalpy-entropy compensation can be expressed by the formula 9 ... 

---