Systems with negligible mutual solubility

Often the liquid and solid are not the same metal and sometimes they are mutually insoluble. When the equilibrium value of the molar fraction of solid component A in a liquid B X 1, by equalling the chemical potentials of A in the solid and liquid phases, X is related to the regular solution parameter X given by equation (4.3) by  

Typical examples are the Pb / Fe and Cu / W couples for which the solubility of Fe in molten Pb and W in molten Cu at temperatures close to the melting point of the liquid is as low as a few ppm. In this type of system for which spreading times are of the order of 10-2 second for millimetre-size droplets (see Section 2.1.1), the final contact angles can be as low as 10° (Table 5.1). This indicates that strong A-B interactions can be established at the interface, even for nearly insoluble metallic couples, i.e., for systems having weak A-B interactions in the bulk liquid(X 0). 

The first term of the right hand side of equation (5.8) is the interfacial energy of pure solid A in contact with pure liquid A, while the second term takes into account the fact that the liquid is not pure A but pure B. 

Assuming perfect wetting of pure molten A on pure solid A at T = Tp is also valid for supercooled liquid A, i.e., at T Tp, then 

T aking into account the fact that deoxidation of solid metals is very difficult at low temperatures, it is usual to observe non-wetting contact angles for insoluble metal/metal systems. 

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Figure 2.11. The Au-Si diagram is an example of a simple eutectic system with complete mutual solubility in the liquid state and no (or negligible) solubility in the solid state at a temperature of 363°C the liquid having the composition of 18.6 at.% Si solidifies with the simultaneous crystallization of the practically pure gold and silicon mechanically mixed. In the Cr-U system a slightly more complex situation due to the solid-state transformations of uranium is shown.

Among the first ones to present a complete theoretical approach are Burton, Prim, and Slitcher (1953). The authors developed a mathematical formulation of the problem for metallic systems with partial solid solubility crystallized from the melt in a suspension process. Their theory is based on a boundary layer model and does not account for the mutual dependence of heat and mass transfer but regards the influence of heat transfer as negligible. 

The main problem with modeling the MMA-H2O system is the absence of reliable phase equilibrium data. The only data available in hterature on the MMA-water system consisted of mutual solubility data of water in MMA at standard pressure and different temperatures [45]. In this study, the interaction parameters are fitted to isobaric data. Because of this restriction, the parameters are fitted to one single set of data points, i.e. the mutual solubility of water and MMA at a certain temperature. This procedure provides interaction parameters that are temperature and pressure dependent. However the effect of pressure on the mixing of MMA has been neglected, as the compressibility of this liquid-liquid system is generally assumed to be negligible. A similar temperature dependency can be observed for the Stryjek-Vera parameters. 

Given a nonionic solute that has a relatively low solubility in each of the two liquids, and given equations that permit estimates of its solubility in each liquid to be made, the distribution ratio would be approximately the ratio of these solubilities. The approximation arises from several sources. One is that, in the ternary (solvent extraction) system, the two liquid phases are not the pure liquid solvents where the solubilities have been measured or estimated, but rather, their mutually saturated solutions. The lower the mutual solubility of the two solvents, the better can the approximation be made. Even at low concentrations, however, the solute may not obey Henry s law in one or both of the solvents (i.e., not form a dilute ideal solution with it). It may, for instance, dimerize or form a regular solution with an appreciable value of b(J) (see section 2.2). Such complications become negligible at very low concentrations, but not necessarily in the saturated solutions. 

Mikhalenko and Kuz ma (1977) presented an isothermal section of the Sm-Cr-B system at 800°C as based on X-ray and metallographic data of 30 alloys. Sample preparation as described for (La, Ce)-Cr-B. Mutual solid solubilities of binary compounds were observed to be negligible. Two ternary compounds have been characterized SmCrB4 with YCrB4-type, Pbam, a = 5.993(5), h = 11.61(1), c = 3.511(4) (Mikhalenko and Kuz ma, 1977), and SmCrjB with CeCrjB -type, Immm, a =6.571(4), 6 = 8.306(5), c = 3.102(3) (Mikhalenko and Kuz ma, 1975). Phase equilibria after Mikhalenko and Kuz ma (1977) did not include the well established SmjBj binary for which a structure proposal exists (La Placa, unpublished). From this a two-phase equilibrium Sm2B5-SmCrB4 is likely (fig. 43). For the Cr-B binary system, see Y-Cr-B. 

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