Crystals, liquid equilibria

Since an isotope ratio cannot be fractionated by crystal-liquid equilibria and therefore is indicative of the magmatic source, the correlation between an isotope ratio and a major or trace element can be used as a guide to the major element or trace element composition of the source. Typically, ratios of highly incompatible trace elemente (Section 4.7.1 and Table 6.7) are most usehil in characterizing the... 

Their remarkable properties mean that the lanthanides, as trace elements, are sensitive indexes of crystal-liquid equilibria in typical geological systems, whose mineralogy is dominated by major elements such as Si, Al, Fe, Mg, Ca, Na and K. Under reducing conditions, as in the terrestrial mantle or in the moon, Eu is reduced from the trivalent to the divalent state. This leads to an increase in ionic radius of... 

Following the development of more precise analytical methods for the lanthanides about 20 years ago, a quantitative approach to modelling trace element distribution during crystal-liquid equilibria was developed (see Haskin 1984 and DePaolo 1981b for relevant equations). The distribution coefficient, K, is a measure of the partitioning behaviour of an element between a crystal and a co-existing melt ... 

Geometrical analysis of the state equation surfaces of liquid-vapour and crystal-liquid equilibria (Equation 1.2-33, Figures 1.20 and 1.21), analysis of expcrimotilal data and computer simulation results lead to the conclusion of that there is no spinodal of the liquid crystal phase transition while the spinodal of crystal —> liquid transition does exist (Skripov, 1975 Skripov and Koverda, 1984). Consequently, the liquid crystdiization occurs only through the formation of critical nuclei (through the metastable state) and no barrierless transition is possible. 

A. Thermodynamic Model for Crystal-Liquid Equilibria, and Supporting Data... 

B. Influence of Third Components on Crystal-Liquid Equilibria... 

Kimberlites and carbonatites rise to the crustal surface from great depths, often with unusually explosive violence. It has been suggested that alkali basalts come from depths intermediate to those of kimberlites and continental or island tholeiites. Thus, there appears to be a trend of increasing light lanthanide enrichment of lavas with increasing depth of source. This apparent trend cannot yet be interpreted in terms of mantle composition in any general way. The relative importance of hypothesized gas extraction of lanthanides and crystal-liquid equilibria for producing severely fractionated lanthanide distributions is unknown. 

In general, one can expect to observe the types of phase diagrams that are found with low molecular weight systems in crystal-liquid equilibrium. For polymeric systems the liquid composition can usually be determined in a straightforward manner. However, establishing the composition in the solid state is quite difficult and presents a major problem in properly analyzing phase diagrams. 

Since the crystal-liquid equilibrium can also be regulated by chemical processes, the force-length-temperature relations of axially oriented crystalline systems will be influenced accordingly. The formal analysis of the problem is similar to that for a nonreacting system with... 

Once the equilibrium phases have been determined, a parametrization of the Interfacial p(r) is constructed by allowing the order parameters used in the equilibrium calculation to vary with z, the coordinate perpendicular to the interface. The shape of these z-dependent order parameter profiles, as well as the interfacial excess free energy, can then be determined from the minimization condition, together with the boundary conditions [the order parameters must tend toward their crystal (liquid) equilibrium values as z goes to -foo (—oo)]. Many authors also assume that the Fourier components vary slowly across the interface, allowing a square gradient approximation to be used. 

The solubility of a solid in the liquid phase of a mixture depends on the properties of the two phases for the components that crystallize, the equilibrium is governed by the following equation [ XI... 

Whether a task can be performed concurrently with other tasks depends on two factors. One is whether the input information for the activity under consideration depends on the output from other activities. The other is the availability of manpower and equipment. Consider a team that has only one chemical engineer to design both the reactor and the crystallizer. Even though reaction kinetics, solid-liquid equilibrium data and crystallization kinetics can be measured in parallel, the total time for these activities is determined by what the single individual can achieve. 

During Rayleigh s crystallization process (formation of crystals with equilibrium limited to the solid/liquid interface), the relative concentration of a trace element in liquid Q with respect to the initial concentration in system Cq is described by the Doerner-Hoskins equation (Doerner and Hoskins, 1925 we will omit hereafter subscripts in the indicization of coefficients, for the sake of simplicity) ... 

Figure 6 shows the phase diagrams plotting temperature T vs c for PHIC-toluene systems with different Mw or N [64], indicating c( and cA to be insensitive to T, as is generally the case with lyotropic polymer liquid crystal systems. This feature reflects that the phase equilibrium behavior in such systems is mainly governed by the hard-core repulsion of the polymers. The weak temperature dependence in Fig. 6 may be associated with the temperature variation of chain stiffness [64]. We assume in the following theoretical treatment that liquid crystalline polymer chains in solution interact only by hardcore repulsion. The isotropic-liquid crystal phase equilibrium in such a solution is then the balance between S and Sor, as explained in the last part of Sect. 2.2. 

It is proposed that in mixed organic base-alkali systems, the presence of the organic base changes the solid-liquid equilibrium and stabilizes larger sol-like aluminosilicate species ( 25 m/ ). The alkali ion affects agglomeration of the sol particles to larger amorphous precipitate particles from 100 to 500 min size which subsequently crystallize to zeolite. 

Stability Limit 1, With the exception of helium and certain apparent exceptions discussed below. Fig. I gives a universal phase diagram liir all pure compounds The triple point of one P and one T is the single point at which all three phases, crystal, liquid, and gas. are in equilibrium. The triple point pressure is normally below atmospheric. Those substances, c.g.. CO . / - 3H85 mm. 7, = -5ft.fi C. for which it lies above, sublime without melting ai atmospheric pressure. 

Partial Miscibility in the Solid State So far, we have described (solid + liquid) phase equilibrium systems in which the solid phase that crystallizes is a pure compound, either as one of the original components or as a molecular addition compound. Sometimes solid solutions crystallize from solution instead of pure substances, and, depending on the system, the solubility can vary from small to complete miscibility over the entire range of concentration. Figure 14.26 shows the phase diagram for the (silver + copper) system.22 It is one in which limited solubility occurs in the solid state. Line AE is the (solid -I- liquid) equilibrium line for Ag, but the solid that crystallizes from solution is not pure Ag. Instead it is a solid solution with composition given by line AC. If a liquid with composition and temperature given by point a is... 

For an uncharged lyotropic lamellar liquid crystal, the equilibrium concentrations are given by the expressions26... 

Equilibrium form of crystals and droplets — Solid crystals — Fig-Equilibrium form of crystals and droplets — Liquid droplets — Fig- ure. Equilibrium form of a cubic crystal formed on a foreign substrate ure. Cross section of a liquid droplet formed on a flat foreign substrate (cross section)... 

Of interest in crystallization calculations is solid-liquid equilibrium. When the solid phase is a pure component, the following thermodynamic relationship holds ... 

Slaughter, D. W., and Doherty, M. E, Calculation of solid-liquid equilibrium and crystallization paths for melt crystallization processes. Submitted for publication (1994). 

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