Critical phase behavior theory

Much of the traditional theory behind critical phase behavior has been discussed in the context of phase... 

In view of the above developments, it is now possible to formulate theories of the complex phase behavior and critical phenomena that one observes in stractured continua. Furthermore, there is currently little data on the transport properties, rheological characteristics, and thermomechaiucal properties of such materials, but the thermodynamics and dynamics of these materials subject to long-range interparticle interactions (e.g., disjoiiung pressure effects, phase separation, and viscoelastic behavior) can now be approached systematically. Such studies will lead to sigiuficant intellectual and practical advances. 

Micellar aggregates are considered in chapter 3 and a critical concentration is defined on the basis of a change in the shape of the size distribution of aggregates. This is followed by the examination, via a second order perturbation theory, of the phase behavior of a sterically stabilized non-aqueous colloidal dispersion containing free polymer molecules. This chapter is also concerned with the thermodynamic stability of microemulsions, which is treated via a new thermodynamic formalism. In addition, a molecular thermodynamics approach is suggested, which can predict the structural and compositional characteristics of microemulsions. Thermodynamic approaches similar to that used for microemulsions are applied to the phase transition in monolayers of insoluble surfactants and to lamellar liquid crystals. 

The phase behavior that is exhibited by a critical or supercritical mixture of several components is usually not simple Street (jO reports six classes of phase behavior diagrams In the simplest classes of systems (classes 1 and 2), the critical lines are continuous between the critical points of pure components Study of reaction equilibrium at SCF conditions requires knowledge of critical properties of the reacting mixture at various levels of conversion Three different approaches to evaluate critical properties are available, viz, empirical correlations, rigorous thermodynamics criteria and the theory of conformal solutions (10) The thermodynamic method is more general and reliable because it is consistent with the calculation of other thermodynamic properties of the reacting mixture (11) ... 

We have also examined the effects of charge asymmetry on the phase behavior of primitive electrolytes. For 2-1 electrolytes (the cation has a charge of 2, while the anions have unit charge), if the sizes of cations and anions are the same, the MSA theory predicts that the critical point is identical to that of the RPM (same anion-cation charge and size), namely, T = 0.049 and p = 0.062. Our simulations predict that the critical temperature is reduced signihcantly, to 7 = 0.046, and the critical density increases to p = 0.105. 

Most of the studies reported in this chapter fail to include the phase behavior of the reacting mixture. Since multiple phases can occur in the mixture critical region, reaction studies need to be complemented with phase behavior studies so that we may gain an understanding of the fundamentals of the thermodynamics and kinetics of chemical reactions in solution. Chapter 5 describes how a simple cubic equation of state can be used to extend and complement the phase behavior studies. An equation of state can be used to determine the location of phase-border curves in P-T space and, with transition-state theory, to correlate the pressure dependence of the reaction rate constant when the pressure effect is large (i.e., at relatively high pressures). 

The lower critical solution temperature (LCST) phase behavior exhibited by the nanocrystals is often found for low molecular weight solutes in supercritical fluids (25,26) and also for polymers dissolved in SCFs, and results from compressibility differences between the polymer and the solvent (15). As the temperature increases or the pressure decreases, the solvent prefers to leave the solute to increase its volume and entropy. The same mechanism that governs phase separation in supercritical fluids also drives flocculation of two surfaces with steric stabilizers, as has been shown with theory (22) and simulation (23). 

Nies, E., Stroeks, A., Simha, R., and Jain, R. K., LCST [lower critical solution temperature] phase behavior according to the Simha-Somcynsky theory apphcation to the n-hexane/polyethylene system. Colloid Polym. Sci., 268, 731-743 (1990). 

NIE Nies, E., Li, T., Berghmans, H., Heenan, R.K., and King, S.M., Upper critical solution temperature phase behavior, composition fluctuations, and complex formation in poly(vinyl methyl ether)/D20 solutions Small-angle neutron-scattering experiments and Wertheim lattice thermodynamic perturbation theory predictions, J. 

Calculate the saturated vapor pressure and fugacity along the vapor-liquid coexistence curve for benzene, water and acetic acid between 25 "C and the critical temperature of the components. For the real vapor phase behavior, use the virial equation truncated after the second virial coefficient in case of water and benzene and the chemical theory in case of acetic acid. Discuss the results. Inside which temperature range are the results reliable Do the calculations lead to under- or overprediction of the fugacity outside the reliable temperature range ... 

Freed et al. developed the lattice cluster theory (LCT) specifically to account for diversity of segmental structures, affecting blend miscibility, viz., the critical point [ c, cj)c, chain swelling [T ], as well as the scale and intensity of composition fluctuations (Freed and Bawendi 1989 Foreman and Freed 1997 Freed and Dudowicz 1998, 2005 Dudowicz et al. 2002). As an example, several monomeric and polymeric stmctures are shown in Fig. 18.15, and LCT predictions of the phase behavior the authors are discussed in details. 

The thermodynamic behavior of fluids near critical points is drastically different from the critical behavior implied by classical equations of state. This difference is caused by long-range fluctuations of the order parameter associated with the critical phase transition. In one-component fluids near the vapor-liquid critical point the order parameter may be identified with the density or in incompressible liquid mixtures near the consolute point with the concentration. To account for the effects of the critical fluctuations in practice, a crossover theory has been developed to bridge the gap between nonclassical critical behavior asymptotically close to the critical point and classical behavior further away from the critical point. We shall demonstrate how this theory can be used to incorporate the effects of critical fluctuations into classical cubic equations of state like the van der Waals equation. Furthermore, we shall show how the crossover theory can be applied to represent the thermodynamic properties of one-component fluids as well as phase-equilibria properties of liquid mixtures including closed solubility loops. We shall also consider crossover critical phenomena in complex fluids, such as solutions of electrolytes and polymer solutions. When the structure of a complex fluid is characterized by a nanoscopic or mesoscopic length scale which is comparable to the size of the critical fluctuations, a specific sharp and even nonmonotonic crossover from classical behavior to asymptotic critical behavior is observed. In polymer solutions the crossover temperature corresponds to a state where the correlation length is equal to the radius of gyration of the polymer molecules. A... 

This result emphasizes that mean-field theory is a very accurate model to describe the behavior of polsrmer systems for most of the phase diagram, except in a narrow window near the critical temperature and concentration. This implies that concentration fluctuations do not significantly influence the phase behavior... 

This is a simple model and cannot account for all the issues of mixture thermodynamics. Interaction parameters deduced from various phase behavior information are often believed to include other effects than purely enthalpic ones. This way, the LCST (lower critical solution temperature) behavior observed in polymer blends can be explained and accounted for quantitatively. These theories refine the binary interaction parameter by removing extraneous effects. EOS effects do not favor phase... 

We stress that design for controllability can either aim at reducing control bandwidth limitations, imposed by fundamental process properties, or at reducing the control requirements imposed by disturbance sensitivities. Based on results from linear systems theory we have presented simple model based tools, based on the decomposed models above, which can be used to improve stability, non-minimum phase behavior and disturbance sensitivities in plants with recycle. One important conclusion of the presented results is that the phase-lag properties of the individual process units play a crucial role for the disturbance sensitivity of an integrated plant. In particular, by a careful design of the recycle loop phase lag, it is possible to tailor the effect of process interactions such that they serve to effectively dampen the effect of disturbances in the most critical frequency region, that is, around the bandwidth of the control system. 

There are several different theoretical approaches to the problem. The Landau molecular field theory was applied by de Gennes to liquid-crystal phase transitions. (89) The Maier-Saupe theory focuses attention on the role of intermolecular attractive forces.(90) Onsager s classical theory is based on the analysis of the second virial coefficient of very long rodlike particles.(91) This theory was the first to show that a solution of rigid, asymmetric molecules should separate into two phases above a critical concentration that depends on the axial ratio of the solute. One of these phases is isotropic, the other anisotropic. The phase separation is, according to this theory, solely a consequence of shape asymmetry. There is no need to involve the intervention of intermolecular attractive forces. Lattice methods are also well suited for treating solutions, and phase behavior, of asymmetric shaped molecules.(80,92,93)... 

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