Ternary and Higher Systems

The occurrence of irreversible fatigue phenomena and grain-size effects indicates that important features of the SMA phenomenon lie outside the domain of equilibrium thermodynamics. Nevertheless, details of the SMA T-x (and T-P-x) phase diagram are clearly important for the understanding and engineering of this curious thermal effect. 

Ternary A/B/C systems (c = 3) present further challenges to thermodynamic description. According to the phase rule, the number of degrees of freedom 

The marked values in (7.94) illustrate how every interior point of the Gibbs-Roozeboom triangle can be uniquely associated with the composition variables xA, xB, xc of a given ternary A/B/C system (in this case, for vA = 0.2, xB = 0.3, xq = 0.5). 

Why does the Gibbs-Roozeboom triangle work The answer becomes apparent from the following elementary geometrical theorem  

Theorem For any interior point of an equilateral triangle, the sum of perpendiculars to the three edges is a constant. 

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The thermodynamic consistency test for binary systems described above can be extended to ternary (and higher) systems with techniques similar to those described by Herington (H3). The necessary calculations become quite tedious, and unless extensive multicomponent data are available, they are usually not worthwhile. 

The work of Vera and co-workers nasHed to a semi-empirical expression for the excess Gibbs energy which is consistent with our choice of the saturated solution as the standard state for the electrolyte. Vera has, however, shown that pure water is a more convenient standard state for hLO in place of the saturated solution used by Vega and Funk (19). This is particularly convenient for ternary and higher systems since it avoids the complication of having a composition-dependent standard state. 

Chapters 6 and 7 dealt with solid state reactions in which the product separates the reactants spatially. For binary (or quasi-binary) systems, reactive growth is the only mode possible for an isothermal heterogeneous solid state reaction if local equilibrium prevails and phase transitions are disregarded. In ternary (and higher) systems, another reactive growth mode can occur. This is the internal reaction mode. The reaction product does not form at the contacting surfaces of the two reactants as discussed in Chapters 6 and 7, but instead forms within the interior of one of the reactants or within a solvent crystal. 

This is the main reason why up to now no reliable prediction of the LLE behavior is possible. Even the calculation of the LLE behavior of ternary systems using binary parameters can lead to poor results for the distribution coefficients and the binodal curve. Fortunately, it is quite easy to measure LLE data of ternary and higher systems up to atmospheric pressure. 

While the calculation of binary LLE can be performed graphically, the calculation of LLE for ternary and higher systems has to be performed iteratively. One possible procedure for a multicomponent system is shown in Figure 5.73 in the form of a flow diagram. The method takes into account the isoactivity conditions (Eq. (5.73)) and the material balance. 

In this chapter, we will discuss the generation modes of thermodynamically stable point defects and the defect-chemical logic to calculate the equilibrium defect structure of a given system. As a stereotype of systems, we will consider only a binary oxide MO, but the idea and logic can be readily extended to other binary, ternary and higher systems with minor modifications [2-6]. 

Sch] Schiirmann, E., Schweinichen, J.V., Vblker R., Fischer, H., Calculation of the a/6 resp. y Liquidus Surfaces of Iron and the Liquidus Surface of Carbon as Well as the Lines of Double Saturation in Iron Rich Carbon Containing Ternary and Higher Systems Fe-C-Xi-X2 (in German), Giessereiforsch., 39, 104-113 (1987) (Review, Phase Relations, Phase Diagram, Thermodyn., 19)... 

The crystal structures of the borides of the rare earth metals (M g) are describedand phase equilibria in ternary and higher order systems containing rare earths and B, including information on structures, magnetic and electrical properties as well as low-T phase equilibria, are available. Phase equilibria and crystal structure in binary and ternary systems containing an actinide metal and B are... 

If the desired catalyst is to consist of two or more catalytic metals after leaching or if a promoter metal is to be included, the precursor alloy becomes even more complicated with respect to phase diagrams. The approximate proportion of reactive metal (aluminum) in these ternary and higher alloys usually remains the same as for the binary metal system for the best results, although the different catalytic activities, leaching behavior and strengths of the various intermetallic phases need to be considered for each alloy system. 

Rogl, P. (1984) Phase equilibria in ternary and higher order systems with rare earth elements and boron. In Handbook on the Physics and Chemistry of Rare Earths, eds. Gschneidner Jr., K.A. and Eyring, L. (North-Holland, Amsterdam), Vol. 6, p. 335. 

They can also be made identical in the general case if the conditions t = —vB/vA and j — 1. The equivalences break down in ternary and higher-order systems as there is the introduction of more compositional variables in the associate model than for the two-sublattice case. This was considered (Hillert et al. 1985) to demonstrate the advantages of the sub-lattice model, but as mentioned previously it turns out that the number of excess terms to describe Fe-Mn-S is very similar. 

In vertical sections through ternary and higher-order systems or isothermal sections for quaternary and higher systems, the position becomes more complex as tie-lines do not lie in the plane of the diagram. They therefore cannot be used to define the positions of phase boundaries and the procedures described above become inoperative. New concepts are required, such as viewing the diagram in a... 

In Section 4.3.3, it was explained how to construct the reaction (diffusion) path for ternary and higher solid solution systems. In practice, one plots, for example, in a ternary system, the composition variables (measured along the pertinent space coordinate of the reacting solid) into a Gibbs phase triangle, noting that the spatial information is thereby lost. For certain boundary conditions, such a reaction path is independent of reaction time and therefore characterizes the diffusion process. For a one dimensional ternary system with stable interfaces, these boundary conditions are c,-( = oo,f) = c°( oo) q( <0,0) = c (-oo) c,(f>0,0) = c (+oo). 

P. Rogl, Phase equilibria in ternary and higher order systems with rare earth elements and boron 335... 

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