Reduced phase diagrams

In a reduced-phase diagram for a two-component system, T = 2 for a single phase and an area is the appropriate representation. F = 1 for two phases in equilibrium, and a curve that relates the two variables is the appropriate representation. As the composition of the two phases generally is different, two conjugate curves are required. 

Other two-component systems may exhibit either limited solubility or complete insolubility in the solid state. An example with limited solubUity is the silver-copper system, of which the reduced-phase diagram is shown in Figure 13.5. Region L represents a liquid phase, with F = 2, and S and 5s represent solid-solution phases rich in Ag and Cu, respectively, so they are properly called one-phase areas. S2 is a two-phase region, with F= 1, and the curves AB and DF represent the compositions of the two solid-solution phases that are in equilibrium at any... 

Figure 14.3. The reduced phase diagram at constant pressure for the two-component system diopside-anoithite, in which the pure sohds are completely insoluble in each other. Data from N. L. Bowen, Am. J. Sci. Ser. 4, 40, 161 (1915).

The interpretation of Equations (14.63) and (14.64) can be illustrated graphically by the reduced-phase diagram for a two-component system at constant pressure, as shown for the system diopside-anorthite in Figure 14.3. 

Fig. 9.7 Reduced phase diagram for the diamond crystal/liquid coexistence curves determined for carbon (using a Tersoff-II potential [65]—black line) [98] and silicon (using a StiUinger-Weber potential [62]—red line) [95]. The pressure and temperature axes are scaled by the respective values at which dT/dp = 0 for the coexistence curve as described in the text. The magenta lines shows the LDA/HDA coexistence curve for Si (thick line) determined from a two state model and the associated spinodals (thin lines). The light blue lines show the liquid/amorphous coexistence curve determined for carbon (thick line) and the associated spinodals (thin lines). Both sets of curves are scaled as for the Uquid/diamond coexistence curves...

Figure A2.5.10. Phase diagram for the van der Waals fluid, shown as reduced temperature versus reduced density p. . The region under the smooth coexistence curve is a two-phase liquid-gas region as indicated by the horizontal tie-lines. The critical point at the top of the curve has the coordinates (1,1). The dashed line is the diameter, and the dotted curve is the spinodal curve.

Flalf a century later Van Konynenburg and Scott (1970, 1980) [3] used the van der Waals equation to derive detailed phase diagrams for two-component systems with various parameters. Unlike van Laar they did not restrict their treatment to the geometric mean for a g, and for the special case of b = hgg = h g (equalsized molecules), they defined two reduced variables. 

While the phase rule requires tliree components for an unsymmetrical tricritical point, theory can reduce this requirement to two components with a continuous variation of the interaction parameters. Lindli et al (1984) calculated a phase diagram from the van der Waals equation for binary mixtures and found (in accord with figure A2.5.13 that a tricritical point occurred at sufficiently large values of the parameter (a measure of the difference between the two components). 

One can effectively reduce the tliree components to two with quasibinary mixtures in which the second component is a mixture of very similar higher hydrocarbons. Figure A2.5.31 shows a phase diagram [40] calculated from a generalized van der Waals equation for mixtures of ethane n = 2) with nomial hydrocarbons of different carbon number n.2 (treated as continuous). It is evident that, for some values of the parameter n, those to the left of the tricritical point at = 16.48, all that will be observed with increasing... 

The lead—copper phase diagram (1) is shown in Figure 9. Copper is an alloying element as well as an impurity in lead. The lead—copper system has a eutectic point at 0.06% copper and 326°C. In lead refining, the copper content can thus be reduced to about 0.08% merely by cooling. Further refining requites chemical treatment. The solubiUty of copper in lead decreases to about 0.005% at 0°C. 

In conclusion, we have presented a new formulation of the CVM which allows continuous atomic displacement from lattice point and applied the scheme to the calculations of the phase diagrams of binary alloy systems. For treating 3D systems, the memory space can be reduced by storing only point distribution function f(r), but not the pair distribution function g(r,r ). Therefore, continuous CVM scheme can be applicable for the calculations of phase diagrams of 3D alloy systems [6,7], with the use of the standard type of computers. 

The changes in osmotic coefficients with temperature and concentration make it difficult to solve the above equations accurately, but accurate determinations of the composition and relative amounts of the concentrated liquid and ice can be made from phase diagrams which are plots of the freezing points of solutions versus their concentration. From these, it is possible to determine the exact NaCl concentration at any temperature. Examples are shown in Figure 9 for solutions of 0 to 2.0 M glycerol in 0.15 M NaCl. This figure nicely illustrates how the presence of glycerol reduces the concentration of NaCl in the residual unfrozen solution. 

The hydrophobic interaction results in the existence of a lower critical solution temperature and in the striking result that raising the temperature reduces the solubility, as can be seen in liquid-liquid phase diagrams (see Figure 5.2a). In general, the solution behaviour of water-soluble polymers... 

Based on the analogy between polymer solutions and magnetic systems [4,101], static scaling considerations were also applied to develop a phase diagram, where the reduced temperature x = (T — 0)/0 (0 0-temperature) and the monomer concentration c enter as variables [102,103]. This phase diagram covers 0- and good solvent conditions for dilute and semi-dilute solutions. The latter will be treated in detail below. 

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