Melt and Solution Equilibria

At this point, for clarification purposes, some remarks should be given to the terms melt and solution equilibria. Particularly driven from the application, that is, whether one refers to melt or to solution crystallization (the term solution used here to characterize a liquid homogeneous mixture with a classical solvent as one component), it is frequently distinguished between both, what is not mandatory from thermodynamic viewpoint. Generally, a solution is a homogeneous 

In the following sections, with regard to their application and the differences in determination, melt and solution equilibria will be discussed separately. 

Thermodynamic Description of SLE Liquidus Curve in the Phase Diagram 

The thermodynamic description of the liquidus curve in the phase diagram is based on classical equilibrium considerations, which will not be elaborated in detail here. Only the frequently used simplified final relationship should be briefly derived. 

The system to be considered is an arbitrary substance A, where pure A in the solid phase, A], is in equilibrium with A in the liquid phase, [A] . 

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Figure 3.3 Bina phase diagrams of (a) two compounds A and B and (b) compound A and a solvent demonstrating the formal analogy between melt and solution equilibria.

Temperature, pressure, and concentration can affect phase equilibria in a two-component or binary system, although the effect of pressure is usually negligible and data can be shown on a two-dimensional temperature-concentration plot. Three basic types of binary system — eutectics, solid solutions, and systems with compound formation—are considered and, although the terminology used is specific to melt systems, the types of behaviour described may also be exhibited by aqueous solutions of salts, since, as Mullin 3-1 points out, there is no fundamental difference in behaviour between a melt and a solution. 

Phase equilibria of vaporization, sublimation, melting, extraction, adsorption, etc. can also be represented by the methods of this section within the accuracy of the expressions for the chemical potentials. One simply treats the phase transition as if it were an equilibrium reaction step and enlarges the list of species so that each member has a designated phase. Thus, if Ai and A2 denote liquid and gaseous species i, respectively, the vaporization of Ai can be represented stoichiometrically as —Aj + A2 = 0 then Eq. (2.3-17) provides a vapor pressure equation for species i. The same can be done for fusion and sublimation equilibria and for solubilities in ideal solutions. 

The aforementioned equilibria may be attained only in the liquid state, i.e., in the melt or solution. When the temperature is low enough to allow crystallization of the polymer, then the fraction of crystallized polymer does not take part in the equilibrium. Only the amorphous fraction of the polymer is involved in the monomer—polymer equilibrium and as a result, the monomer content decreases with increasing crystallinity [21]. In this way, the monomer content may be lowered substantially. The effect of crystallization or phase separation is very important in the polymerization of five- and six-membered lactams which can thus be forced to polymerize to higher yields. Dilution and lowering of the melting temperature, on the other hand, increases the equilibrium monomer content. 

Many properties of pure polymers (and of polymer solutions) can be estimated with group contributions (GC). Examples of properties for which (GC) methods have been developed are the density, the solubility parameter, the melting and glass transition temperatures, as well as the surface tension. Phase equilibria for polymer solutions and blends can also be estimated with GC methods, as we discuss in Section 16.4 and 16.5. Here we review the GC principle, and in the following sections we discuss estimation methods for the density and the solubility parameter. These two properties are relevant for many thermodynamic models used for polymers, e.g., the Hansen and Flory-Hug-gins models discussed in Section 16.3 and the free-volume activity coefficient models discussed in Section 16.4. 

Many databases (some available in computer form) and reliable GC methods are available for estimating many pure polymer properties and phase equilibria of polymer solutions such as densities, solubility parameters, glass and melting temperatures, and solvent activity coefficients. 

On following the solubility curve of the hexahydrate from the ordinary temperature upwards, it is seen that at a temperature of 29 8° represented by the point H, it cuts the solubility curve of the a-tetrahydrate. This point, therefore, represents an invariant system in which the three phases hexahydrate, a-tetrahydrate, and solution can coexist under constant pressure. It is also the transition point for these two hydrates. Since, at temperatures above 29 8°, the a-tetrahydrate is the stable form, it is evident from the data given before (p. 184), as also from Fig. 79, that the portion of the solubility curve of the hexahydrate lying above this temperature represents metastable equilibria. The realisation of the metastable melting-point of the hexahydrate is, therefore, due to suspended transformation. At the transition point, 29 8°, the solubility of the hexahydrate and a-tetrahydrate is 100 6 parts of CaClg in 100 parts of water. 

Muratova et al. (1975) determined the phase equilibria in an isothermal section of the system Gd-Ge-Si at 600 C. 134 alloys, prepared by arc melting and annealed at 600 °C for 250 h in evacuated silica tubes, were analyzed by X-ray powder diffraction techniques and metallography (fig. 26). Starting materials were 99.98% pure Gd and 99.99% pure Ge,Si. Phase equilibria are characterized by the formation of three complete solid solutions Gd5(Ge,Si)j with the MnjSij-type (Pbj/mcm), Gd2(Ge,Si)j with a defect AlB2 derivative type, and Gd(Ge,Si)2 t- Gd-Ge-Si alloys with a Gd content between 30 to 40 a/o Gd have been investigated earlier by Muratova and Bodak (1974). 

Electroanalytical techniques, essentially similar to those employed in aqueous solutions, can be adapted for use in melts to provide data on solution equilibria by way of stability constant determinations, ion transport through diffusion coefficient measurements, as well as mechanistic analysis and product identification from mathematical data treatment. Indeed, techniques such as linear sweep voltammetry and chronopotentiometry may often be applied rapidly to assess or confirm general characteristics or overall stoichiometry of electrode processes in melts, prior to more detailed kinetic or mechanistic investigations requiring more elaborate instrumentation and equipment, e.g., as demanded by impedance studies. Thus, answers to such preliminary questions as... 

Information about the phase equilibria in the Ce-Ni-Ge system is due to the work of Salamakha et al. (1996b) who employed X-ray phase analysis of 227 alloys which were arc melted and aimealed at 870 K (0-50 at.% Ce) and 670 K (50-100 at.% Ce) in evacuated quartz tubes for two weeks (fig. 45). Twenty ternary compounds were found and characterized. A large homogeneity range was observed for the CeNis compound by the same authors the maximum concentration of Ge in Ce(Ni,Ge)5 solid solution was indicated as 15 at.%. 

The phase equilibria in the Ce-Rh-Ge system have been investigated by Shapiev (1993) by means of X-ray and metallographic analyses of 219 ternary alloys, which were arc melted and subsequently annealed in evacuated silica tubes for 1000 hours at 870 K and finally quenched in water. Starting materials were Ce 99.0mass%, Rh 99.99 mass%, Ge 99.99 mass%. According to the phase diagram, twenty ternary compounds exist within the Ce-Rh-Ge system at 870 K (fig. 49). The limits of solid solutions originating at binary Ce-Rh and Rh-Ge compounds are not indicated by Shapiev (1993). 

The phase equilibria in the Sc-Y-Ge system (fig. 117) were investigated by Shpyrka and Mokra (1991) by means of X-ray phase and microstructural analyses of alloys which were arc melted and subsequently annealed in evacuated silica tubes for 350 h at 870 K and finally quenched in water. The starting materials were Sc 99.92 mass%, Y 99.90 mass%, and Ge 99.99 mass%. The phase relations are characterized by the existence of three ternary compoimds and the formation of a continuous solid solution (Sc, Y)5Gc3 originating at the isotypic binary compounds. The binary compounds ScGc2, Y2Ges and ScnGeio dissolve 18, 10 and 8at.% of third component, respectively. 

However, without dissociation, in an ideal solution at infinite dilution the contributions of the enthalpy of mixing and solvation to the dissolution process can considered to be negligible, thus giving ApH AsH. Therefore, the Schroder-van Laar equation using the melting enthalpy also for solution equilibria is often a good approximation to estimate the temperature dependence of the solubility of organic substances. 

Gaussian thread limit for N—For many physical problems (e.g., polymer solutions and melts, liquid-vapor equilibria, and thermal polymer blends and block copolymers), the Gaussian thread model has been shown to be reliable in the sense that it is qualitatively consistent with many aspects of the behavior predicted by numerical PRISM for more realistic semiflexible, nonzero thickness chain models. However, there are classes of physical problems where this is not the case. The athermal stiffness blend in certain regions of parameter space is one case, both in... 

FIGURE 11.2 Illustrations of other physical equilibria, (a) The melting and freezing of water in a slushy ice-water mixture when the surroundings are at 0.0 °C. (b) The movement of a solute back and forth between immiscible layers of water and an organic solvent in an extraction experiment. 

Some information about phase equilibria in the monoboride region of the Sc- Ru-B system is found from an investigation of superconductivity by Ku et al. (1979). Samples were arc melted and annealed below 1300°C, wrapped in Ta foils and then sealed under Ar in quartz tubes or sealed under Ar in Ta tubes. One sample SCRU4B4 has been annealed at 1400° in a Ta tube furnace under Ar and quenched to room temperature in a Ga-In eutectic solution. Compositions were checked by electronmicroprobe analysis. From this SCRUB4 and SCRU2B2 were reported as two nonsuperconducting single-phase compounds with unknown structure type. 

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