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Reports of the Academy of Sciences of the USSR
- Volume 136, No. 6
PHYSICAL CHEMISTRY
L. S. PALATNIK
GENERALIZED LEVER RULE
(Presented by Academician P. A. Rebinder on 27 IX 1960)
As is known, to determine the mass ratio of the phases of an equilibrium two-phase system, the lever rule is used. Its generalization to the case of equilibrium of a three-phase system is the rule of the center of gravity; for the case of a multicomponent multiphase equilibrium system containing an arbitrary number of components, we have proposed a generalized rule of the center of gravity \((^1)\). This rule is applied to the quantitative study of equilibrium states of complex multicomponent heterogeneous systems and has been used to derive certain general regularities and rules relating to the structure of equilibrium diagrams of such systems \((^2,\,^3)\).
In this article a generalized lever rule is proposed, representing a certain new generalization of the rules indicated above. It consists in the following: in studying various processes in equilibrium multicomponent systems, instead of the figurative points of each of the phases separately, we shall consider the figurative points of certain (two) conglomerates of phases (see below) that are in equilibrium. Such a generalization will make it possible to study more easily (stepwise or continuously) the most complex processes in multiphase systems, which appear intricate when considered by the usual methods.
Let \(M\) be the figurative point (or the corresponding composition vector inside the concentration simplex \((^3)\)) representing as a whole an equilibrium multicomponent heterogeneous system of unit mass. Then
\[ M=\sum_i X_i A_i=\sum_j m_j B_j=\sum_k v_k C_k, \tag{1} \]
where \(A_i\) is the vertex corresponding to the \(i\)-th component of the concentration simplex (or the vector of the \(i\)-th component); \(X_i\) is the concentration of the \(i\)-th component in the system; \(B_j\) is the \(j\)-th vertex in the phase simplex (the vector of the \(j\)-th phase); \(m_j\) is the mass of the \(j\)-th phase; \(C_k\) is the vector of the \(k\)-th conglomerate; \(v_k\) is the mass of the \(k\)-th conglomerate; \(i=1,2,\ldots,n\) is the number of the component; \(j=1,2,\ldots,r\) is the number of the phase; \(k=1,2,\ldots,s\) is the number of the conglomerate of phases.
Formula (1) describes the indicated system \(M\) through its components \(A_i\), through the coexisting phases \(B_j\) entering into it, or through the conglomerates of phases \(C_k\) into which the system \(M\) decomposes. Equations (1) simultaneously provide a transition from one barycentric coordinate system (the components of the system) to another (its phases or conglomerates of phases). In this case the geometrical dimensions \(R_A\), \(R_B\), and \(R_C\) of the concentration, phase, or conglomerate simplexes may in general be different. Thus, for example, for a three-component two-phase system the concentration simplex is the Gibbs triangle \((R_A=2)\), and the phase simplex is a line segment—the tie-line \((R_B=1)\); for a four-component three-phase system the concentration simplex is the Roozeboom–Fedorov tetrahedron \((R_A=3)\), while the phase simplex is a triangle \((R_B=2)\), etc. The dimensions \(R_A\) and \(R_B\) may also be identical—
for example, for a three-component three-phase system \((R_A = R_B = 2)\), for a four-component four-phase system \((R_A = R_B = 3)\), etc.
In the case \(s = 2\) \((R_C = 1)\), i.e., when the system decomposes into two conglomerates of phases, equations (1) become the equation of the generalized lever rule
\[ M = \sum_i X_i A_i = \sum_i m_j B_j = \nu L + (1 - \nu)D, \tag{2} \]
where \(L\) and \(D\) are the vectors of two conglomerates of phases with masses \(\nu\) and \((1-\nu)\), respectively. For example, in the case of crystallization of a multicomponent alloy, \(L\) corresponds to the liquid phase (or conglomerate of liquid phases), and \(D\) to the conglomerate of all solid phases that have separated out before the present equilibrium moment.
Proceeding from the law of conservation of mass for each of the \(n\) components of the system, we have
\[ \nu x_{iL} + (1-\nu)x_{iD} = X_i \quad (i = 1, 2, 3, \ldots, n), \tag{3} \]
where \(x_{iL}, x_{iD}, X_i\) are the concentrations of the \(i\)-th component, respectively, in the liquid phase, in the conglomerate of solid phases, and in the entire system \(M\) as a whole; \(\nu\) and \((1-\nu)\) are the masses of the liquid phase and of the conglomerate of solid phases.
The solution of equations (3) is
\[ \nu = \frac{X_i - x_{iD}}{x_{iL} - x_{iD}}, \qquad 1-\nu = \frac{x_{iL} - X_i}{x_{iL} - x_{iD}}, \tag{4} \]
whence follow the equations of the generalized lever rule
\[ \frac{\nu}{1-\nu} = \frac{X_i - x_{iD}}{x_{iL} - X_i}. \tag{5} \]
With the aid of equations (5) one can calculate the ratios of the masses \(\nu\) and \((1-\nu)\) of the coexisting liquid phase and conglomerate of solid phases, if the concentrations \(x_{iL}, x_{iD}\), and \(X_i\) are specified; or else calculate (analytically) the overall composition of the conglomerate \(x_{iD}\) for given \(\nu, x_{iL}\), and \(X_i\) in an equilibrium system containing an arbitrary number of components, etc.
Thus, when a multicomponent system \(M\) decomposes into two subsystems—the liquid phase and the conglomerate of solid phases, represented by the figurative points \(L\) and \(D\)—the “conglomerate simplex” degenerates into an ordinary (one-dimensional) tie-line, to which the ordinary lever rule is applicable.
In the course of a reversible process, at any of its stages the points \(M, L\), and \(D\) lie on one common straight line \(LMD\), situated in the \((n-1)\)-dimensional space of the concentration simplex (and at the same time in the \((r-1)\)-dimensional space of the phase simplex). Throughout the entire process the point \(M\), obviously, always remains fixed, while the points \(L\) and \(D\) may move simultaneously within the concentration simplex (sliding along the boundary between regions of phase states) or may alternately remain fixed (in which case one of the points will move along a straight line within a single-phase region). The tie-line (straight-line segment) \(LMD\), and together with it the segments \(LM\) and \(MD\), may rotate in the multidimensional concentration simplex about the point \(M\), or may retain their direction, lengthening or shortening (but only not bending). In this case the vector \(D\) at any moment of the reversible process can be decomposed into its constituent vectors, representing individual phases or other, simpler conglomerates of phases (eutectics, etc.), which makes it possible to record easily any state of the system throughout the entire process under study.
With the aid of the generalized lever rule it is possible to describe very complex crystallization processes. As an example, let us consider the process
crystallization of the three-component system A—B—C, in which a ternary compound S is formed with an incongruent melting point (in the liquid state there is complete mutual solubility, and in the solid state complete insolubility of the components). Let us follow the crystallization process for some composition from the very beginning to its complete completion.
In Fig. 1 the projection of the liquidus surfaces onto the concentration triangle \(ABC\) is shown. The region \(EP_2P_1E\) of primary crystallization of S
Fig. 1
lies outside the point S, which corresponds to the composition of the solid ternary compound. The ternary point \(P_1\) corresponds to a nonvariant, singly incongruent crystallization process (\(-L - A + B + S\)). The plus and minus signs were determined on the basis of the center-of-gravity rule: the plus sign corresponds to a congruent process, the minus sign to an incongruent process (for example, at point \(P_1\) crystals of S and B separate out, while A disappears together with the liquid). Point \(P_2\) corresponds to a nonvariant, doubly incongruent process (\(-L - A - C + S\)); point \(E\) to a nonvariant congruent process (\(-L + B + C + S\)), in which the ternary eutectic \((B + C + S)\) crystallizes.
The line \(P_1E\) represents a monovariant congruent process (\(-L + B + S\)), while the lines \(P_2P_1\) and \(P_2E\) represent monovariant incongruent crystallization processes (\(-L - A + S\) and \(-L - C + S\)), etc. The arrows indicate the direction of decreasing temperature during the crystallization process.
In Fig. 1 the stages of crystallization from a liquid whose initial composition is indicated by point 19 are shown by heavy lines. As is evident from Fig. 1, on the basis of the generalized lever rule it is easy to establish the trajectories of the figurative points of the liquid phase and, correspondingly, of the conglomerate of solid phases coexisting at any moment of the crystallization process. For point 19 there arise 7 stages of crystallization, designated by Roman numerals.
with the sectors: \(I—VII\) for the liquid phase and \(I'—VII'\) for the conglomerate of solid phases. At the various stages of crystallization, the following solid phases coexist in the conglomerates:
\[ I'(A);\quad II'(A+C);\quad III'(A+C+S);\quad IV'(C+S); \]
\[ V'(S);\quad VI'(B+S);\quad VII'(B+C+S). \]
During these 7 stages the variance of the system changes as follows: \(y=2, 1, 0, 1, 2, 1, 0\). The mass ratio \(\dfrac{v}{1-v}\) is readily calculated for any moment of crystallization. With the aid of the proposed rule it is easy to establish all stages of crystallization for any other compositions of the \(A—B—C\) system.
We have applied the proposed rule to the construction of various polythermal sections (for example, \(A\alpha\), \(\gamma_1\alpha_1\), \(\gamma_2\alpha_2\), \(\beta\alpha_3\), etc.), on the basis of which the entire spatial equilibrium-state diagram of the indicated complex ternary system can be represented.
In the case of multicomponent heterogeneous systems \((n \geq 4)\), the generalized lever rule loses its visual clarity, but it can be used in analytical form (equations (4) and (5)).
Thus, from equations (5), when experimental values of \(X_x\), \(x_{iD}\), and \(\dfrac{v}{1-v}\) are available, one can find analytically the equation of the trajectory of the figurative point \(x_L\) in the multidimensional concentration simplex. From such data, important information can be extracted about the structure of the equilibrium diagram of a multicomponent system.
Kharkov State University
named after A. M. Gorky
Received
26 IX 1960.
CITED LITERATURE
¹ L. S. Palatnik, DAN, 95, 1227 (1954).
² L. S. Palatnik, V. M. Kontorovich, ZhFKh, 28, 1599 (1954).
³ L. S. Palatnik, A. I. Landau, DAN, 102, 125 (1955); 109, 954 (1956); 114, 837 (1957); ZhFKh, 30, 2399 (1956); 31, 304 (1957); 31, 2739 (1957); ZhNKh, 3, 637 (1958); ZhTF, 28, 2340 (1958).