Ternary-Phase Diagrams

A typical ternary phase diagram of soybean oil (triglyceride), sunflower oil monoglyceride and water at 25 °C [8] is shown in Fig. 5.7. It clearly shows the LC phase and the inverse micellar (L2) phase. This inverse micellar phase is relevant to the formation of water-in-oil emulsions. The interfacial tension between the micellar L2 phase and water is about 1-2 mN m and that between the L2 and water is even lower. It is proposed that the L2 phase forms an interfacial film during emulsification, and the droplet size distribution should then be expected to be related mainly to the interfacial and rheological properties of the L2 phase. 

2 Multi-component systems Ternary phase diagrams 

For three-component (C = 3) or ternary systems the Gibbs phase rule reads Ph + F = C + 2 = 5. In the simplest case the components of the system are three elements, but a ternary system may for example also have three oxides or fluorides as components. As a rule of thumb the number of independent components in a system can be determined by the number of elements in the system. If the oxidation state of all elements are equal in all phases, the number of components is reduced by 1. The Gibbs phase rule implies that five phases will coexist in invariant phase equilibria, four in univariant and three in divariant phase equilibria. With only a single phase present F = 4, and the equilibrium state of a ternary system can only be represented graphically by reducing the number of intensive variables. 

It is sometimes convenient to fix the pressure and decrease the degrees of freedom by one in dealing with condensed phases such as substances with low vapour pressure. The Gibbs phase rule for a ternary system at isobaric conditions is Ph + F = C + 1=4, and there are four phases present in an invariant equilibrium, three in univariant equilibria and two in divariant phase fields. Finally, three dimensions are needed to describe the stability field for the single phases e.g. temperature and two compositional terms. It is most convenient to measure composition in terms of mole fractions also for ternary systems. The sum of the mole fractions is unity thus, in a ternary system A-B-C  

In the present case there are no ternary invariant equilibria in the system, partly due to the complete solid solubility of the A-B system. In a ternary system composed from three binary eutectic sub-systems, three univariant lines would meet in a ternary eutectic equilibrium  

A sample in the primary crystallization field of phase C will behave differently during crystallization. Here phase C precipitates with composition identical to C (no solid solubility) during cooling keeping the A B ratio in the melt constant until the melt hits the intersection of the two primary crystallization fields. At this temperature a will start to precipitate together with further C and from this point on the cooling process corresponds to that observed for the sample with overall composition P after this sample reaches the same stage of the crystallization path. 

If we consider three-component (ternary) systems, for instance, we have C = 3 and, according to the phase rule, the variance is given byV=C — P + 2 = 5 Porin 

In the case of a ternary system, the formation of several, binary and ternary, stoichiometric phases, and different types of variable composition phases can be observed. One may differentiate between these phases by using terms such as point compounds (or point phases), that is, phases represented in the composition field by points, line phases , field phases , etc. 

A simple example of a real ternary diagram is shown in Fig. 2.26, where the isothermal section, determined at 200°C, of the Al-Bi-Sb system is shown together with the relevant binary diagrams Al-Bi showing a miscibility gap in the liquid state and complete insolubility in the solid state, Bi-Sb with complete mutual 

The Al-Zn-Si is another example of simple ternary system its isothermal section at 307°C is shown in Fig. 2.27 together with its boundary binaries. Si, in the solid state, is practically insoluble in A1 or in Zn and in their binary solutions. In the 

For two-dimensional representation, the temperature can be projected on an equilateral triangle, with liquidus temperatures represented by isotherms. The diagram is divided into areas representing the equilibrium between a liquid and a solid phase. The boundary curves represent the equilibrium between two solids and a liquid, and the intersections of the three boundary curves represent the points of four phases in equilibrium. 

In another method of two-dimensional representation, a constant-temperature plane is cut through the diagram. This plane indicates the phases at equilibrium at that temperature. The interpretation of ternary diagrams is not different from that of binary diagrams. The composition of each 

Representation of ternary diagram (a) system in which ternary eutectic is present, (b) system forming a complete series of solid solutions. 

These lines form composition triangles in which three phases are present at equilibrium. 

Pressure-temperature phase diagram for carbon, Am. Sci., 52 395,1964. 

The composition of an internal point such as E is also found from the composition axes. The point 

The amount of a phase present at a vertex of a triangle is 100 %. The amounts of the phases 

The amounts of the three phases present at point J can be determined by an extension of the lever rule. The phase triangle made up of Wig049-W02-W03 

Amount of W18O49 = Amount of Z1O2 = Amount of WO2 present = 

This method is called the triangle rule. Note that, just as in the lever rule, we assume that the composition scales are linear. If they are not, actual compositions must be used, not distances. However, this is rarely the case in ternary diagrams, which always use a linear scale for the composition axes. 

The presence of a solvent in the crystallization process alleviates these problems, mainly by lowering the activation energy barrier for the rearrangement of the solute molecules into a new crystalline structure. Thus, the solvent has the classical function of a catalyst and will only affect the kinetics but not the thermodynamics of the transformation. This is valid as long as the solvent does not become part of the crystal structure, that is, is not forming a solvate with the solute. With respect to co-crystals this signifies that the existence of a thermodynamically stable binary compound is not a function of the solvent. 

The phase boundaries between the two-phase regions and the liquid phase at a given temperature ean be approximated by the following polynomial equations (see Rager and Hilfiker (2009), Further Reading, ref. 13)  

In these equations, and are again the mole fractions of components A and B, 5a and 5b are their solubilities in the solvent S, ATab is the solubility product of the compound AB, and a, and bj are virial coefficients describing deviations from ideality in the ternary mixture. The equilibrium constants 5a, 5b, and ATab are correlated with the Gibbs free energy of each crystalline 

It follows from this equation that the solubility product of a thermodynamically stable co-crystal is lower than the product of the solubilities of the individual components. 

The stability domains of the co-crystals overlap near the crossing point of the solubility curves of A and B. 

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In the absence of special syimnetry, the phase mle requires a minimum of tliree components for a tricritical point to occur. Synnnetrical tricritical points do have such syimnetry, but it is easiest to illustrate such phenomena with a tme ternary system with the necessary syimnetry. A ternary system comprised of a pair of enantiomers (optically active d- and /-isomers) together with a third optically inert substance could satisfy this condition. While liquid-liquid phase separation between enantiomers has not yet been found, ternary phase diagrams like those shown in figure A2.5.30 can be imagined in these diagrams there is a necessary syimnetry around a horizontal axis that represents equal amounts of the two enantiomers. 

Figure C2.3.10. Ternary phase diagram of surfactant, oil and water illustrating tire (regular) and (reverse) L2 microemulsion domains.

Ternary Alloys. Almost ah commercial ahoys are of ternary or higher complexity. Ahoy type is defined by the nature of the principal ahoying additions, and phase reactions in several classes of ahoys can be described by reference to ternary phase diagrams. Minor ahoying additions may have a powerflil influence on properties of the product because of the influence on the morphology and distribution of constituents, dispersoids, and precipitates. Phase diagrams, which represent equhibrium, may not be indicative of these effects. 

Fig. 26. Ternary-phase diagram for the system styrene—PS—polybutadiene mbber.

We have recently shown that the hydrophobic hexafluorophosphate ILs can in fact be made totally miscible with water by addition of alcohols [47, 48] the ternary phase diagram for [BMIM][PF(3]/water/ethanol (left part of Figure 3.3-7) shows the... 

Figure 3.3-7 Ethanol/water/[BMIM][PFg] ternary phase diagram (a, left) and solute distribution...

It was shown some time ago that one can also use a similar thermodynamic approach to explain and/or predict the composition dependence of the potential of electrodes in ternary systems [22-25], This followed from the development of the analysis methodology for the determination of the stability windows of electrolyte phases in ternary systems [26]. In these cases, one uses isothermal sections of ternary phase diagrams, the so-called Gibbs triangles, upon which to plot compositions. In ternary systems, the Gibbs Phase Rule tells us... 

In order for this concept to be applicable, the matrix and the reactant phase must be thermodynamically stable in contact with each other. One can evaluate this possibility if one has information about the relevant phase diagram — which typically involves a ternary system — as well as the titration curves of the component binary systems. In a ternary system, the two materials must lie at comers of the same constant-potential tie-triangle in the relevant isothermal ternary phase diagram in order to not interact. The potential of the tie-triangle determines the electrode reaction potential, of course. 

B. Ternary Phase Diagram from Binary Data. 196... 

Once the solubility and coexistence curves have been determined, the complete ternary phase diagram may be constructed. This procedure is illustrated by the example shown in Fig. 28. The following parameters have been used ... 

Case I. At sufficiently low pressures, the solubility curve does not intersect the coexistence curve. In this case, the gas solubility is too low for liquid-liquid immiscibility, since the coexistence curve describes only liquid-phase behavior. Stated in another way, the points on the coexistence curve are not allowed because the fugacity f2L on this curve exceeds the prescribed vapor-phase value f2v. The ternary phase diagram therefore consists of only the solubility curve, as shown in Fig. 28a where V stands for vapor phase. 

Figure 3. Partial Ternary Phase Diagram for the CaC -rich Side of the CaCl2-CaF2-CaO System.

Present production processes use two stage counter-current extraction to remove americium from molten plutonium with magnesium chloride based salts. Both 35 mole % NaCl - 35 mole % KC1 - 3D mole % MgCl2 and 50 mole % NaCl-26 mole % CaCl2 - 24 mole % Mg Cl 2 are used for americium extraction. Figures 4 and 5 show the ternary phase diagrams for these salt systemsU0). 

Figure 4. Ternary Phase Diagram for the NaCI-CaCi2-MgCl2 System.

An example for a partially known ternary phase diagram is the sodium octane 1 -sulfonate/ 1-decanol/water system [61]. Figure 34 shows the isotropic areas L, and L2 for the water-rich surfactant phase with solubilized alcohol and for the solvent-rich surfactant phase with solubilized water, respectively. Furthermore, the lamellar neat phase D and the anisotropic hexagonal middle phase E are indicated (for systematics, cf. Ref. 62). For the quaternary sodium octane 1-sulfonate (A)/l-butanol (B)/n-tetradecane (0)/water (W) system, the tricritical point which characterizes the transition of three coexisting phases into one liquid phase is at 40.1°C A, 0.042 (mass parts) B, 0.958 (A + B = 56 wt %) O, 0.54 W, 0.46 [63]. For both the binary phase equilibrium dodecane... 

As with the pnictates, a quasi-ternary phase diagram can be developed to map out possible compounds in this composition phase space using key chalcotetre-late building blocks. We have begun to make use of the peritectic nature of the starting materials, as this has facilitated reactions between phases. 

The National Materials Science Film series includes an excellent treatment of ternary phase diagrams and of mechanical and electrical properties. 

Figure 24 shows the ternary phase diagram (solubility isotherm) of an unsolvated conglomerate that consists of physical mixtures of the two enantiomers that are capable of forming a racemic eutectic mixture. It corresponds to an isothermal (horizontal) cross section of the three-dimensional diagram shown in Fig. 21. Examples include A-acetyl-leucine in acetone, adrenaline in water, and methadone in water (each at 25°C) [141].

Figure 25 shows the ternary phase diagram (solubility isotherm) for an unsolvated racemic compound. Examples of this type include benzylidenecamphor in methanol, or /V-acetylvaline in acetone [141]. In Fig. 25, A and A represent the...

Figure 26 shows the ternary phase diagrams (solubility isotherms) for three types of solid solution. The solubilities of the pure enantiomers are equal to SA, and the solid-liquid equilibria are represented by the curves ArA. The point r represents the equilibrium for the pseudoracemate, R, whose solubility is equal to 2Sd. In Fig. 26a the pseudoracemate has the same solubility as the enantiomers, that is, 2Sd = SA, and the solubility curve AA is a straight line parallel to the base of the triangle. In Figs. 26b and c, the solid solutions including the pseudoracemate are, respectively, more and less soluble than the enantiomers. 

Figure 4.17 Isothermal sections of the ternary phase diagram A-B-C shown in Figure 4.16 at (a) 650 °C and (b) 450 °C [17]. Here L denotes liq. Reprinted with permission of The American Ceramic Society, www.ceramics.org. Copyright [1984], All rights reserved.

Let us now consider two real ternary systems to illustrate the complexity of ternary phase diagrams in some detail. While the first is a system in which the solid state situation is rather simple and attention is primarily given to the liquidus surfaces, the solid state is the focus of the second example. 

FIGURE 3.14 Ternary phase diagram of (Zr,Y)02-La203-Mn304 system at 1400°C in air. (Zr,Y)02 denotes 3 mol% Y203-Zr02. Symbols + are the experimental data. (From Jiang, S.P. et al., J. Euro. Ceram. Soc., 23 1865-1873, 2003. With permission.)... 

Solubilities of Trichlorides in Water as Taken from Ternary Phase Diagrams... 

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