Solid-fluid phase

Runge, K. J. Chester, G. V., Solid-fluid phase transition of quantum hard spheres at finite temperature, Phys. Rev. B 1988, 38, 135-162... 

Ghoussoub, J. and Leroy., Y. (2001) Solid-fluid phase transformation within grain boundaries during compaction by pressure solution, J. Mech. Phys. Solids 49, 2385-2430... 

Pure solid + fluid phase equilibrium calculations are challenging but can, in principle, be modeled if the triple point of the pure solid and the enthalpy of fusion are known, the physical state of the solid does not change with temperature and pressure, and a chemical potential model (or equivalent), with known coefficients, for solid constituents is available. These conditions are rarely met even for simple mixtures and it is difficult to generalize multiphase behavior prediction results involving even well-defined solids. The presence of polymorphs, solid-solid transitions, and solid compounds provide additional modeling challenges, for example, ice, gas hydrates, and solid hydrocarbons all have multiple forms. 

When examining the phase behavior of compounds in carbon dioxide, it is important to note that both solid-fluid phase behavior (solubility) and liquid-liquid phase behavior (miscibility) are reported in the literature. In general, the designation of one material as more or less C02-philic than another refers to whether solubility or miscibility pressures (for a given concentration) are relatively lower or higher than those of the comparison material. 

Ashour I, Almehaideb R, Fateen SE, Aly G. Representation of solid fluid phase equilibria using cubic equations of state. Fluid Phase Equilibria 2000 167 41-61. 

Figure 4 Binary solid-fluid phase behavior of (a) type I and (b) type II systems (13).

Computer simulations of molecular models by MC and MD techniques have revolutionized our understanding of the solid-fluid phase transition. 

Solid-fluid phase diagrams of binary hard sphere mixtures have been studied quite extensively using MC simulations. Kranendonk and Frenkel [202-205] and Kofke [206] have studied the solid-fluid equilibrium for binary hard sphere mixtures for the case of substitutionally disordered solid solutions. Several interesting features emerge from these studies. Azeotropy and solid-solid immiscibility appear very quickly in the phase diagram as the size ratio is changed from unity. This is primarily a consequence of the nonideality in the solid phase. Another aspect of these results concerns the empirical Hume-Rothery rule, developed in the context of metal alloy phase equilibrium, that mixtures of spherical molecules with diameter ratios below about 0.85 should exhibit only limited solubility in the solid phase [207]. The simulation results for hard sphere tend to be consistent with this rule. However, it should be noted that the Hume-Rothery rule was formulated in terms of the ratio of nearest neighbor distances in the pure metals rather than hard sphere diameters. Thus, this observation should be interpreted as an indication that molecular size effects are important in metal alloy equilibria rather than as a quantitative confirmation of the Hume-Rothery rule. 

The cell theory plus fluid phase equation of state has been extensively applied by Cottin and Monson [101,108] to all types of solid-fluid phase behavior in hard-sphere mixtures. This approach seems to give the best overall quantitative agreement with the computer simulation results. Cottin and Monson [225] have also used this approach to make an analysis of the relative importance of departures from ideal solution behavior in the solid and fluid phases of hard-sphere mixtures. They showed that for size ratios between 1.0 and 0.7 the solid phase nonideality is much more important and that using the ideal solution approximation in the fluid phase does not change the calculated phase diagrams significantly. 

In somewhat earlier work, Vlot et al. [229,230] made calculations of Lennard-Jones binary mixtures in which the pure components are identical but in which the unlike interactions have departures from the Lorentz-Berthelot combining rules. They use this as a model of mixtures of enantiomers. A variety of solid-fluid phase behavior can be obtained from the model. Both substitutionally ordered and substitutionally disordered solid solutions were found to occur. 

Although these simplified models of hydrogen-bonded systems give a far from complete picture of the solid-fluid phase behavior of water, this kind of approach to identifying the key features required in the molecular model is an instructive one. Indeed, the inability of the PMW to generate reentrant melting of the low-density solid at thermodynamically stable states is an important result. It shows us that more than just short-range directional forces are required for this to occur. 

Although most of the studies of this model have focused on the fluid phase in connection with the theory of electrolyte solutions, its solid-fluid phase behavior has been the subject of two recent computer simulation studies in addition to theoretical studies. Smit et al. [272] and Vega et al. [142] have made MC simulation studies to determine the solid-fluid and solid-solid equilibria in this model. Two solid phases are encountered. At low temperature the substitutionally ordered CsCl structure is stable due to the influence of the coulombic interactions under these conditions. At high temperatures where packing of equal-sized hard spheres determines the stability a substitutionally disordered fee structure is stable. There is a triple point where the fluid and two solid phases coexist in addition to a vapor-liquid-solid triple point. This behavior can be qualitatively described by using the cell theory for the solid phase and perturbation theory for the fluid phase [142]. Predictions from density functional theory [273] are less accurate for this system. 

Alder and Wainwright [5,6] published the first paper of a molecular dynamics simulation of a condensed phase fluid system and this paper began a trend that did have a strong impact on statistical mechanics. These authors tackled one of the open questions of the day, whether a solid-fluid phase transition existed in a system of hard spheres. This problem could not be solved by existing analytical methods and Alder and Wainwright s simulation demonstrated that such analytically intractable problems could be studied and solved by direct MD simulation of the equations of motion of a many-body system. Of course, the simulation was modest by today s standards and was carried out on systems containing 32 and 108 hard spheres. This research set the stage for the development of MD as a basic tool in statistical mechanics. 

Equilibrium Treatment of Solidification. As an example of liquid-solid phase change in solid-fluid flow systems with the assumption of local thermal equilibrium imposed, consider the formulation of solid-fluid phase change (solidification/melting or sublimation/frosting) of a binary mixture. For this problem, the equilibrium condition extends to the local thermodynamic equilibrium where the local phasic temperature (thermal equilibrium), pressure (mechanical equilibrium), and chemical potential (chemical equilibrium) are assumed to be equal between the solid and the fluid phases. This is stated as... 

Heidug, W. K. 1995. Intergranular solid-fluid phase transformations under stress The effect of surface forces. J. Geophys. Res., 100 pp. 5931-5940. 

Deiters, U.K. and Swaid, I. (1984) Calculation of fluid-fluid and solid-fluid phase equilibria in binary mixtures at high pressures, Ber. Bunsenges. Phys. Chem. 88, 791-796. 

If a is smaller than one mixed solid-fluid phase occurs, the edge free energies are zero and the crystal interface solid surface is in essence rough and the crystal can grow without a layer mechanism. At this stage the overall crystallographic orientation (hkl) will also be lost. 

In order to confirm the influence of A on the phase diagram and regions of anomalous behavior, we carried out several simulations restricted to the small region in the phase diagram between the densities p = 0.20 and p = 0.28 (which corresp>ond roughly to 2 < P < 4) and temperatures T = 0.60 and T = 0.94. Figure 6 shows the sohd-fluid phase boimdary for the systems with X/a = 0.10, 0.20, 0.40, 0.50 and 0.70. The increase of the interparticle separation shifts the solid-fluid phase boimdary to lower temperatures. 

GRE Gregorowicz, J., Solid-fluid phase behaviom of linear polyethylene solutions in propane, ethane and ethylene at high pressures, J. Supercrit Fluids, 43, 357, 2007. 

One fluid phase in contact with one solid phase liquid-solid gas/vapor-solid supercritical fluid-solid (solid-fluid phase membrane contactors) (Fig. 26.2). 

Figure 26.2 Membrane contactor for solid-fluid phase contacting.

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