Tie-triangle

For a ternai y system, the phase diagram appears much like that in conventional liquid-liquid equilibrium. However, because a SCF solvent is compressible, the slopes of the tie lines (distribution coefficients) and the size of the two-phase region can vary significantly with pressure as well as temperature. Furthermore, at lower pressures, LLV tie-triangles appear upon the ternary diagrams and can become quite large. 

The composition of lithium-manga-nese-oxide spinel electrodes that are of interest for lithium battery applications fall within the Li[Mn2]04 - Li4Mn5Ot2 -Li2[Mn4]0() tie-triangle of the Li-Mn-0... 

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. 

An additional requirement is that the reactant material must have two phases present in the tie-triangle, but the matrix phase only one. This is another way of saying that the stability window of the matrix phase must span the reaction potential, but that the binary titration curve of the reactant material must have a plateau at the tie-triangle potential. It has been shown that one can evaluate the possibility that these conditions are met from knowledge of the binary titration curves, without having to perform a large number of ternary experiments. 

Yamaguchi, S. and Kunieda, H. (1997) Determination of a three-phase tie triangle (the hydrophile-lipophile balance plane) in a composition tetrahedron Evaluation of the composition of adsorbed mixed-surfactant and the monomeric solubilities of short-chain surfactant. Langmuir, 13, 6995-7002. 

Within a tie triangle on a ternary phase diagram, a point will correspond to ... 

At fixed T and P, a three-phase ternary has P = 0, which defines a point. On a triangular diagram, a three-phase situation produces three points, each giving the composition of one of the phases. The three points can be connected to form a triangle, and the relative amounts in the three phases can be found by a tie-triangle rule (see Appendix H). 

Analogous results are obtained for N /N and N /N. The result (B.6.17) is a form of the tie-triangle rule for three components in three-phase equilibrium (see H.2). 

When ternary mixtures exhibit three-phase equilibria, a tie-triangle rule can be used to obtain the relative amounts in the three phases. This is analogous to the lever rule for binaries in two-phase equilibria. Consider a ternary mixture of components 1, 2, and 3 in three-phase equilibrium at the overall composition represented by a point P, as in Figure H.2. In the figure, the tie-triangle ABC bounds the three-phase region, the... 

This tie-triangle rule is a consequence of material balances on the system (see (B.6.17) in Appendix B) and therefore it applies to any three-phase equilibrium situation involving ternary mixtures. 

The poinl that represents the microemulsion middle phase in central type 111 diagram moves from one side of the diagram to the other, as indicated in Fig. 7. It is seen that at both extremes of type HI diagram occurrence, the three-phase tie triangle collapses when an upper lobe and bottom tie lines merge together (penultimate diagram on each side) (28,29). 

Elder data on phase relations were reviewed and critically assessed by Raynor and Rivlin [1988Ray], who in turn refer to several older reviews and assessments. They suggested best choice versions of liquidus projection, including position of tie-triangles for some solid-liquid monovariant equilibria, Scheil scheme and a number of isothermal sections for 1100 to 600°C. This assessment was updated by [1994Rag], who included literature up to 1991. 

The lines subdividing phase fields (aFe) and a in Figs. 2 and 3 are taken from [1976Hag]. As a transition between those is of a second order, no two-phase region should exist. Position of the tie lines between the fields ((aFe)+7) and (a +y) (which indeed are degenerated three-phase tie triangles ((aFe)+a +y)) are drawn rather arbitrarily, which is indicated by question marks. 

When the system undergoes the transition from a triphasic to a biphasic case, the tie-triangle flattens and the microemulsion becomes a micellar phase and merges together with an excess phase. The flattened triangle thus vanishes into a tie-line, the so-called bottom tie-line in the seventies, recently called the CMC tie-line by some authors [51]. These authors studied the same kind of phase transition due to a temperatiue change in a nonionic system and probably wanted to point out that this line joins the representative point of the excess phases that are believed to be at the CMC in their respective solvents. 

Provided that optimization criteria are selected, it is possible to find the best set of pseudocomponents, so that the plane of the cut coincides with one of the tie-triangle planes, according to the desired application. Such a description has been proposed for enhanced oil-recovery simulation [102,103]. 

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