According to Gibbs' phase rule, a section of the phase diagram of a three component system at P,T = const is a partition of the concentration triangle into.
ISSN 00125016, Doklady Physical Chemistry, 2012, Vol. 443, Part 1, pp. 53–56. © Pleiades Publishing, Ltd., 2012. Original Russian Text © V.I. Kosyakov, V.A. Shestakov, E.V. Grachev, 2012, published in Doklady Akademii Nauk, 2012, Vol. 443, No. 2, pp. 191–194.
PHYSICAL CHEMISTRY
Triangulation Schemes of ThreeComponent Systems V. I. Kosyakov, V. A. Shestakov, and E. V. Grachev Presented by Academician F.A. Kuznetsov September 6, 2011 Received September 21, 2011
DOI: 10.1134/S0012501612030025
According to Gibbs’ phase rule, a section of the phase diagram of a threecomponent system at P,T = const is a partition of the concentration triangle into the regions of the existence of phases, two, and three phase complexes. To describe the topology of such a section, it is convenient to denote one, two, and threephase regions by points, lines, and triangles, respectively. If the system at given Р and Т contains no phases with unlimited solubility of two or three com ponents, the topology of the section can be repre sented as a partition of the concentration triangle into elementary triangles with the vertices corresponding to components or compounds. The scheme of triangu lation of the concentration triangle uniquely repre sents the structure of the network of lines of monova riant equilibria in the melting diagram. Such schemes also describe the structure of the piecewise continuous solidus surface of the phase diagrams of ternary sys tems. The triangulation schemes of the solidus surface should be constructed for analyzing possible variants of the structure of the liquidus surface of ternary sys tems [1, 2]. Let us denote the solid phases by α, β, and γ and the liquid by L. Each threephase region αβγ of the solidus surface corresponds to the invariant point of the liquidus surface that characterizes the equilib rium Lαβγ between the liquid phase and the solid phases. The monovariant equilibrium lines Lαβ, Lαγ, and Lβγ issuing from this point correspond to the sides of the abc triangle. Therefore, the figure formed by monovariant equilibrium lines and the invariant points in the liquidus surface is dual to the triangulation scheme of the solidus surface. This fact was used for classifying the melting diagrams of ternary systems with solid phases of constant compositions [3]. Note also that the topology of the phase diagram of a ternary system can be regarded as the topology of the partition of the Р–Т space into regions each of which is represented by a certain triangulation scheme. The
union of such schemes with the identified correspond ing region of the Р–Т space allows one to describe the structure of the entire diagram [4]. The importance of studying triangulation schemes was noted as long ago as early in the investigation of phase diagrams [1, 5]. Triangulation schemes were described in textbooks (see, e.g., [6, 7]), however, without much attention. In some specialized mono graphs on the structure of the phase diagrams of ter nary systems, e.g., [8], the triangulation problem was not considered at all. In principle, this problem can be solved by thermodynamic methods [9, 10]. To con struct an isobaricisothermal section of a diagram, it is necessary to know the Gibbs free energy of each of the phases of the system and to construct the surface G(x, y) corresponding to the minimal value of the Gibbs free energy G of the heterogeneous sample at any point of the composition space (х, у). In the absence of information on the thermodynamic prop erties of the phases of the system, a possible triangula tion scheme can be constructed by geometric thermo dynamics. According to this approach, to the set of points corresponding to the compositions of the phases, arbitrary values are assigned, on the basis of which the minimal surface G(x, y) is constructed. Its projection on the concentration triangle is a triangula tion scheme. Exhaustive search of the assigned values generally enables one to construct all the possible tri angulation schemes for a given set of the phases of the system. This approach is convenient for visualizing the thermodynamic description, but poorly suitable for the exhaustive search of all the possible variants of tri angulation, especially for systems with a large number of compounds. Experimental data on phase diagrams show a wide variety of triangulation schemes for real physicochem ical systems. The amount of such information contin uously increases, because of which it is urgent to solve the problems of convolution and classification of information on triangulation schemes of ternary sys tems, and also the problem of predicting such schemes. However, at the present time, there are no reliable data and algorithms for a priori prediction of the composition and structure of possible crystalline
Nikolaev Institute of Inorganic Chemistry, Siberian Branch, Russian Academy of Sciences, pr. Akademika Lavrent’eva 3, Novosibirsk, 630090 Russia 53
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KOSYAKOV et al.
ν=
B
∑u
m
and
∑ mu
m
= 2 е. Using the above rela
tions, we obtain
∑ (6 − m)u
D1
m
= 2(M + 6)
(4)
or D2
D4 T1
D3 T2 A
C
Fig. 1. Example of triangulation of a diagram with four binary compounds and two ternary compounds.
phases in a given chemical system and also for progno sis of their thermodynamic properties, although this field is actively being developed [11]. It can be stated that, although the knowledge of the rules of triangula tion is important for understanding the properties of the set of the phase diagrams of ternary systems and studying complex diagrams with a large number of phases, an exhaustive analysis of this problem is still unavailable in the literature. Let us consider an isobaricisothermal section of the diagram of a system comprising М binary com pounds and N ternary compounds. The triangulation scheme of this section can be regarded as a graph with ν vertices, f faces, and е edges. Of them, v1 = e1 = М + 3 vertices and edges are external because they belong to the boundary face of the graph, and the other v2 = N vertices and е2 edges are within the boundary face. The graph consists of the boundary face and f2 = f – 1 tri angular faces. Since each edge belongs to two neigh boring faces, we have М + 3 + 3(f – 1) = 2e. (1) Using relation (1) and Euler’s theorem f = e – v + 2, (2) we find
f = M + 2N + 2, e = 2M + 3N + 3, (3) e2 = M + 3N , ν = M + N + 3. Figure 1 gives an example of triangulation for the system at M = 4 and N = 2. For this diagram, f = 10, v = 9, v1 = 7, е = 17, and е2 = 10. In calculating v1, all the vertices of the triangle were taken into account. In this case, one of the vertices is bivalent. Let us denote the number of such vertices by u2; this number should meet the inequality 0 ≤ u2 ≤ 3. As an additional characteristic of the graph, it is convenient to use the enumeration of the numbers of vertices of various degrees. For the graph in Fig. 1, such a vertex degree code is written as 2134415261. Let uт be the number of vertices of degree m. Then
4u2 + 3u3 + 2u4 + u5 (5) + 0u6 − u7 − 2u8 − … = 2(M + 6). Possible formulas for the vertices of the triangulation graph should meet relation (5). Using this relation, the complete list of vertex degree codes for a given number of binary compounds can be made. Note that the degree of a graph vertex should not exceed N + M + 2. Possible formulas for the graph vertices are also restricted in terms of the number of ternary com pounds:
∑ (4 − m)u
m
= 6 − 2N
(6)
or
2u2 + u3 + 0u4 − u5 − 2u6 (7) − 3u7 − 4u8 − … = 6 − 2N . Summation of relations (5) and (7) gives one more equation relating the numbers of various vertices of the graph to the numbers of compounds: 3 u2 + 2 u3 + u4 + 0u5 − u6 (8) − 2 u7 − 3 u8 − … = М − N + 9. Relations (5), (7), and (8) can be used for deriving possible formulas of triangulation graphs. They have an important specific feature facilitating this proce dure, namely, zero factors at u6, u4, and u5 in rela tions (5), (7), and (8), respectively. Let us consider as an example the problem of enu merating triangulation schemes at М = 2 and N =1. In this case, f = 6, v = 6, v2 = 1, е = 10, and е2 = 5. Pos sible vertex degree codes are conveniently derived while varying u2. At u2 = 0, a single code, 3551, is pos sible. At u2 = 1, there are two codes, 213243 and 21334151. At u2 = 2, there is a single code, 22314251. Fig ure 2 shows ten possible triangulation schemes for the considered case of diagrams with four vertex degree codes: two variants for each of the codes 3551 and 22314251, and three variants for each of the codes 21334151 and 213243. With increasing number of binary and ternary compounds in a system, the number R of possible variants of triangulation rapidly increases. The table illustrates the dependence of R on M and N as determined using a triangulation graph generation program of our own design. This program enables one to construct the complete set of nonisomorphic triangulations for given M and N or determine their number. A wide diversity of possible triangulation schemes and their realizations in real chemical systems requires one to develop approaches to the systematization of information in this field and to the topological classi
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TRIANGULATION SCHEMES OF THREECOMPONENT SYSTEMS C
55
C
5 1
35
D2 D1
D2 D1 T
T
A
B
A
B
C
C
C
1 2 3
234
D2
D1
D2
D2
D1
T
D1
T
T A
B
C
A
B A
C
C
B
21334151 D1
D2
D2 D2 D1
T D1
T
T A
B
A
B A
C
B
C
22314251 D1
D2
T A
D2
D1
B
A
T B
Fig. 2. Triangulation schemes at M = 2 and N = 1. The vertex degree codes are presented.
fication of isobaricisothermal sections of phase dia grams. Such a classification can be based on partition ing the set of triangulation graphs into subsets of graphs with certain properties (e.g., graphs with M = 0, 1, 2, …). Let A be the set of graphs each of which the boundary face is a polygon and the other faces are tri angles. Let a, b, c… be subsets of this set with given attributes (e.g., with different values of M = 0, 1, 2, …). The introduction of additional attributes (e.g., the number of bivalent vertices u2 = 0, 1, 2, …) makes pos sible the partition of the set а into subsets ab, ac, ad, …. Extension of the list of attributes leads to the parti tion of ab into ab1, ab2, ab3, etc. The totality of possible partitions of the set А with respect to given sets of attributes allows one to con struct various hierarchical or network classification DOKLADY PHYSICAL CHEMISTRY
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schemes reflecting the topological properties of iso baricisothermal sections of the phase diagrams of ter nary systems. In constructing a hierarchical scheme of classification of triangulation graphs, the first main attribute is conveniently chosen to be the number М of binary compounds. Then, it is reasonable to specify the number of binary compounds in each of the boundary binary systems (code of boundary systems [3]) because their distribution over the edges of the concentration triangle affects the topology of triangu lation schemes. The second main attribute is the num ber N of ternary compounds. Necessary attributes are the numbers of vertices of different degrees, which determine the vertex degree code. The example in Fig. 2 demonstrates the existence of isomeric graphs with identical sets of vertices but different packings of 2012
KOSYAKOV et al.
Dependence of the number of nonisomorphic triangula tions of phase diagrams on the number of binary (M) and ternary (N) compounds in a ternary system M 0 1 2 3 4 5
N 0
1
2
3
4
5
1 1 2 6 12 31
1 2 10 37 121 382
1 8 50 244 1003 3738
4 38 293 1682 7981 33398
16 211 1865 11992 62922 287654
78 1299 12611 87868 497160 –
triangular faces. Each of the packings can be charac terized by a graph automorphism group [12], which is equivalent to the indication of the maximum point symmetry group of the triangulation scheme. For more detailed classification, it is necessary to intro duce additional attributes, e.g., degrees of connected vertices. If the set of attributes is sufficient for uniquely describing the triangulation graph, this set can be used as the code of this graph. It is clear that the develop ment of the rules of coding and selection of codes cor responding to possible triangulation schemes from the set of codes can also be used for enumerating triangu lation schemes. The vertices of the triangulation graph of a real phase diagram correspond to real components and compounds. In addition, in this case, the coordinates of vertices in the concentration triangle are known, which imposes restrictions on possible triangulation schemes. In the experimental construction of an iso baricisothermal section of the phase diagram of a spe cific system, the phase composition of a set of samples after long annealing at a given temperature is found. The compositions of the samples are most often cho sen at nodes of a regular grid constructed on the con centration triangle or its fragment. An example of the optimization of choosing the compositions of the samples was given previously [13]. It was proposed to take the compositions of the sam ples at the intersection of two edges of possible trian gulation graphs. The phase analysis of a sample deter mines which of the edges is present in a real triangula tion scheme. In this case, the optimization problem reduces to the construction of the complete graph at vertices with given coordinates in the concentration triangle, the determination of the coordinates of the points of intersection of various edges, and the selec
tion of the compositions of recommended samples among these points. The performed software realization of the problems of enumeration of graphs and design of experiments in constructing diagram sections at Р, T = const and also the development of principles of classification of tri angulation schemes are among the main tasks of the triangulation problem. Note also that the use of the results of this work promotes the formation of a sys tems approach to the set of phase diagrams as opposed to the fragmentary presentation of information on the basis of describing individual examples of diagrams in most of the textbooks. REFERENCES 1. Kurnakov, N.S., Vvedenie v fizikokhimicheskii analiz (Introduction to Physicochemical Analysis), Lenin grad: ONTI, 1936. 2. Kosyakov, V.I. and Shestakov, V.A., Dokl. Phys. Chem., 2008, vol. 421, part 2, pp. 220–222. 3. Kosyakov, V.I., Shestakov, V.A., and Grachev, E.V., Zh. Neorg. Khim., 2010, vol. 55, no. 4, pp. 662–670. 4. Kosyakov, V.I., Zh. Neorg. Khim., 2010, vol. 55, no. 11, pp. 1894–1902. 5. Guertler, W., Metall Erz, 1920, vol. 8, pp. 192–195. 6. Vögel, R., Die heterogenen Gliechgewichte, Leipzig: Acad. Verlagsgesellshaft, 1959. 7. Anosov, V.Ya., Ozerova, N.I., and Fialkov, Yu.Ya., Osnovy fizikokhimicheskogo analiza (Fundamentals of Physicochemical Analysis), Moscow: Nauka, 1976. 8. Petrov, D.A., Troinye sistemy (Ternary Systems), Mos cow: Izd. AN SSSR, 1953. 9. Glazov, V.M. and Pavlova, L.M., Khimicheskaya termo dinamika i fazovye ravnovesiya (Chemical Thermody namics and Phase Equilibria), Moscow: Metallurgiya, 1988. 10. Hillert, M., Phase Equilibria, Phase Diagrams, and Phase Transformations. Their Thermodynamic Basis, Cambridge: Cambridge Univ. Press, 1998. 11. Kiseleva, N.N., Komp’yuternoe konstruirovanie neor ganicheskikh soedinenii. Ispol’zovanie baz dannykh i metodov iskusstvennogo intellekta (Computer Design of Inorganic Compounds: The Use of Artificial Intelli gence Databases and Methods), Moscow: Nauka, 2005. 12. Ore, O., Theory of Graphs, Providence, Rhode Island: American Mathematical Society, 1962. Translated under the title Teoriya grafov, Moscow: Nauka, 1968. 13. Niepel, L. and Malinovsky, M., Chem. Zv e sti, 1978, vol. 32, no. 6, pp. 810–820.
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