Near critical point

Near critical points, special care must be taken, because the inequality L will almost certainly not be satisfied also, cridcal slowing down will be observed. In these circumstances a quantitative investigation of finite size effects and correlation times, with some consideration of the appropriate scaling laws, must be undertaken. Examples of this will be seen later one of the most encouraging developments of recent years has been the establishment of reliable and systematic methods of studying critical phenomena by simulation. 

A second observation relates to calculations near critical points. The coexistence lines in Fig. 10.7 do not extend above a temperature of T = 11.6 because above that temperature significant overlap exists between the liquid and vapor peaks of the histograms. This overlap renders calculations of the liquid and gas densities imprecise. 

Larger system sizes suffer less from this effect and can be used to obtain coexistence densities near critical points. 

The last several decades have seen the growing scientific importance of phenomena near critical points, those conditions of system properties where two coexisting phases, such as liquid and vapor, become identical. This is exemplified by the 1982 Nobel Prize awarded to Kenneth Wilson for his theoretical studies of... 

Figure 8.5 P-V projection of state diagram for H2O near critical point. and are critical pressure and critical volume of compound.

This chapter deals with critical phenomena in simple ionic fluids. Prototypical ionic fluids, in the sense considered here, are molten salts and electrolyte solutions. Ionic states occur, however, in many other systems as well we quote, for example, metallic fluids or solutions of complex particles such as charged macromolecules, colloids, or micelles. Although for simple atomic and molecular fluids thermodynamic anomalies near critical points have been extensively studied for a century now [1], for a long time the work on ionic fluids remained scarce [2, 3]. Reviewing the rudimentary information available in 1990, Pitzer [4] noted fundamental differences in critical behavior between ionic and nonionic fluids. 

Physical properties. Gas-oil ratios for the Pre-Cretaceous fluids vary from fairly low values of 1224 and 1691 scf/bbl for wells 30/7a-6 and 30/ 7a-3 to a maximum of 6287 scf/bbl in 30/7a-l Iz (Table 1). Most of the petroleums are near-critical point fluids, being either single-phase... 

The origin of the Ginzburg-Landau approach lies in the study of the thermal behavior near critical points, which is characterized by a set of universal critical exponents. One of the advantages of this approach is that many techniques that have been developed in this context can be applied to Ginzburg-Landau models of ternary amphiphilic systems as well. 

The description of thermodynamic anomalies observed near critical points has been presented in many books and reviews [124-135]. Sufficiently close to a critical point, thermodynamic properties A vary as simple power laws of the distance e from the critical point. 

It is now established both theoretically and experimentally that many thermodynamic variables assume a simple power-law behaviour at or near critical points in both pure and mixed fluids. The actual functional dependence of one variable on another can be characterized by the so-called critical indices a, 5, etc. The critical index j8, for example, defines both the shape of the gas-liquid coexistence curve for a pure fluid and the liquid-liquid coexistence curve of a binary mixture in the vicinity of either an upper or a lower critical solution temperature. The correspondence between critical phenomena in one-, two-,... 

Fleming, RD. and Vinatieri, J.E., Quantitative interpretation of phase volume behavior of multicomponent systems near critical points, AIChE J., 25, 493, 1979. 

Fredrickson, G.H. and Larson, R.G. (1987) Viscoelasticity of homogeneous polymer melts near critical point J. Chem. Phys., 86 (3), 1553-1560. 

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... 

Anisimov, M.A., Gorodetskii, E.E., Kulikov, V.D., and Sengers, J.V. (1995) A gen-eral isomorphism approach to thermodynamic and transport properties of bin2u y fluid mixtures near critical points, Physica A 220, 277-324. 

Principle of critical simplification. In accordance with this principle (Yablonsky et al., 2003), the behavior near critical points, for instance ignition or extinction points in catalytic combustion reactions, is governed by the kinetic parameters of only one reaction—adsorption for ignition and desorption for extinction— which is not necessarily the rate-limiting one. 

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