The procedure to study the phase equilibrium lines “solid - liquid” was developed for the binary systems with using a coefficient of Bjerrum Guggenheim. The universality of this procedure to solve a mathematical problem of phase diagrams and its efficiency to find the behavioristic characteristics of a solvent and the dissolved components in the equilibrium phases were demonstrated. Two types of the generalized mathematical expression as the modified Le Chatelier-Shreder equation were proposed to describe analytically the liquidus and solidus lines of phase diagrams of the whole class of systems

G. Narikbayeva

I. Korsukova

S. Baisanov

V. Tolokonnikova

Ye. Mukhambetgaliyev

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97METALURGIJA 59 (2020) 1, 97-100S. BAISANOV, V. TOLOKONNIKOVA, G. NARIKBAYEVA, I. KORSUKOVA, YE. MUKHAMBETGALIYEVMATHEMATICAL METHOD OF PHASE EQUILIBRIUM OF BINARY SYSTEM Cr-Si BASED ON BJERRUM GUGGENHEIM CONCEPT Received – Primljeno: 2019-06-06Accepted – Prihvaćeno: 2019-09-11Preliminary Note – Prethodno priopćenjeS. Baisanov, V. Tolokonnikova, G. Narikbayeva, I. Korsukova, Ye. Mukhambetgaliyev, Chemical-Metallurgical Institute, Karaganda, Ka-zakhstan, e-mail: met.rasplav@mail.ruThe procedure to study the phase equilibrium lines “solid - liquid” was developed for the binary systems with using a coefficient of Bjerrum Guggenheim. The universality of this procedure to solve a mathematical problem of phase diagrams and its efficiency to find the behavioristic characteristics of a solvent and the dissolved components in the equilibrium phases were demonstrated. Two types of the generalized mathematical expression as the modified Le Chatelier-Shreder equation were proposed to describe analytically the liquidus and solidus lines of phase diagrams of the whole class of systems. Key words: Cr-Si system, mathematical method, phase diagrams, Le Chatelier-Shreder equation, coefficient of Bjer-rum Guggenheim. INTRODUCTIONNo universal solution have yet been found for a problem of a heterogeneous phase equilibrium which results might be suitable to calculate and construct the phase diagrams and phase constitution of various sys-tem types.Thereby each system requires the individual solu-tion that is accompanied by considerable labor inputs especially in case of studying the multicomponent sys-tems.The heart of the theory of regular and subregular so-lutions, the surrounded atom theory [1] and the cluster theory of solutions suppose that properties and structure of melts are defined generally with sizes of particles and forces of interparticle interaction.All listed directions have one common fault, name-ly, require preliminary determination of the conditional interaction parameters and enthalpies of mixture of melts that provide the special researches [1]. Besides the mathematical apparatus of each direc-tion and their versions is only used for the certain groups of systems [2-4], and for others is needed the develop-ment of aspecial mathematical apparatus or introduc-tion of complex correction factors.WAYS OF STUDYSolution to inverse problems under Gibbs is connect-ed with decoding the phase diagrams with the obtaining of its thermodynamic information, namely relative and excess functions of mixture of components in melt.ISSN 0543-5846METABK 59(1) 97-100 (2020)UDC – UDK 669.26.782.001.5: 519.216: 541.121.084.2: 512.3: 164.3=111In fact the problem consists of two parts: develop-ment of the correct mathematical models of phase dia-grams in an analytical form, and further, based on the fundamental laws of thermodynamics, their transforma-tion to the thermodynamic functions of equilibrium phases [5,6]. The analysis of some phase diagrams of binary met-al and oxide systems found the general regularity con-sisting in a deviation of energy parameters of compo-nents of a real system from ideal one.For the real system the monovariant phase equilib-rium lines are calculated with Le Chatelier-Shreder equation:   (1)where, Hm,i, Tm,i – enthalpy and temperature of melting i – components; R – universal gas constant; Cp– change of heat capacity during melting of components; T – the melting point of the components and the crystallization of melts. A measure of deviation was used as coefficient of Bjerrum Guggenheim known in paper [7], (Фi), which for “solid-liquid” equilibrium is written in the following form of:    (2)where,  – Gibbs energy of melting compo-nent to its ideal component .98S. BAISANOV et al.: MATHEMATICAL METHOD OF PHASE EQUILIBRIUM OF BINARY SYSTEM Cr-Si BASED... METALURGIJA 59 (2020) 1, 97-100The liquidus and solidus lines on the phase diagrams in generalized form are represented as: The coefficient of Bjerrum Guggenheim (2) charac-terizes a range of deviation of the energy condition of this component from ideality in equilibrium conditions. Thus when this deviation is more might be observed a strong interparticle interaction of this component with other components. This interaction might be divided into two constituents: the Van der Waals (interatomic or intermolecular interaction in the melt of any system) and directed (connected with directed hetero- and ho-meopolar bonds between components in the melt). Moreover, in both cases the interacting force and range of deviation of component property from ideality will depend on the particle quantity which is in other states.In the theory of the statistical thermodynamics the first is defined as , and the second as difference of quantity of the particles which are com-pletely in the associated and free states. Thus, Фi change along the phase equilibrium line might be written as:  (3)оr   ,   (4)where Gas,i- Gibbs free energy of melting on reaching which transition of this component completely in the associated state is possible; Ai, Bi are empirical coeffi-cients. Therefore, the equations (3 and 4) permit under the statistical thermodynamics to explain the physical meaning of dependence of Фi on the crystallizing com-ponent activities.It should be pointed out that in the depending on a character of interpartial interaction in melts for various systems will be observed a different type of Фi change along liquidus and solidus lines. If in the melt along a liquidus line the Van der Waals interaction forces (equa-tion 3) are prevail between the elementary structural units of the studied component with the second, then the rectilinear dependence will be observed (the elementary structural units are meant as element atoms or mole-cules of compounds). At formation of associates from initial elements or dissociation of the studied compound in melt, the true amount of atoms or molecules of this phase will depend stronger (on very complex depend-ences including dissociation rate of compounds) on temperature and composition of melt that must inevita-bly lead to a nonlinear dependence (equation 4). Equations (1, 3, 4) permit to show out the semi-em-pirical dependences as the generalized Le Chatelier - Shreder equations to calculate liquidus and solidus lines for any real system in crystallization of a relevant phase on the following dependences:  exp  Mr,  exp  Nr. (5)The solution of system of the equations (5) gives the mathematical expressions of solidus and liquidus lines for the real systems:  . (6)Based on equations (4-6) the processing procedure of experimental data was developed. The open experi-mental results in binary phase diagrams contain two co-ordinates for each point of liquidus and solidus lines, namely: concentration of a crystallizing component XiL(S) and temperature K. A key feature of various crys-tallization regions of binary systems is an existence of close correlations between the thermodynamic proper-ties along curves of monovariant phase equilibrium. This correlation for the simple systems is observed as rectilinear dependence of osmotic coefficient of Bjer-rum Guggenheim of the crystallizing component Фi on its activity aiL / aiS, in liquid and solid phases as:    (7)Where coefficients A and В are constants deter-mined with the nature of components and the interpar-tial interactions. The experimental data of various bi-nary metal systems were used to receive these values at a concrete temperature Tm under equations: Ф′1 =  and Ф′2 = , (8)where xiL(S) is concentration of a crystallizing com-ponent at this temperature; aiL(S) – activity of a crystalliz-ing component for ideal solution calculated under Le Chatelier - Shreder equation (1).Processing of the received results with the least-squares method permits to find values of constants and coefficients of correlation for various systems under equations 3 and 4, and to show the analytical expres-sions of surfaces of fields of crystallization phases in any quasi-system found at setup of the phase diagram with the TDA method (equations 5 and 6). However the simple analysis of crystallization fields in various systems [8] on a relative position of liquidus 99S. BAISANOV et al.: MATHEMATICAL METHOD OF PHASE EQUILIBRIUM OF BINARY SYSTEM Cr-Si BASED...METALURGIJA 59 (2020) 1, 97-100lines does not practically present a structure of melt and authenticity of the experimental data except weak or strong decrease in melting temperature of the studied phase with the second components. The studying of phase diagrams of systems based on iron under Bjerrum Guggenheim concept was made according to the well-known classification [5] to show analytically the phase equilibrium lines “solid - liquid” and define behavior of components in melts of metal systems [9,10,11]. Cr – Si system. As demonstration material of the developed proce-dure of the mathematical description of the phase equi-librium lines based on Bjerrum Guggenheim concept, the formation of crystallization fields of phases in Cr–Si system was studied. For this system the following compounds: Cr3Si, Cr5Si3, CrSi and CrSi2 were determined.In order to calculate the mathematical expressions of the liquidus and solidus lines under specially made pro-gram, values of enthalpy and temperature of melting of silicium and chrome were used. ∆ Hm,Si = 50 210J / mol, Tm,Si = 1 703 K,  ∆Hm,Cr = 19 246,4 J / mol, Tm,Cr = 2 130 K  [9,12]. The basic data for crystallization of silicium and chrome were taken from the phase diagram as data ar-ray [13], each temperature (from chrome melting tem-perature) which relevant a structure to the eutectic tem-perature. The equations of dependence of Bjerrum Guggen-heim’s coefficient from activities in liquid and solid phases with correlation coefficient were received. Ф”Si = 3,6254 – 2,7841· aLSi / aSSi Rxy = – 0,9986 (9) Ф”Cr = 1,5593 – 1,9582· aLCr / aSCr Rxy = – 0,9969 (10)Table 1 demonstrates the comparative analysis of the experimental and calculated data for crystallization of silicium and chrome from melting temperature 1 420 K to 1 335 K which shows good repeatability of experi-mental and calculated data, thus the received mathemat-ical expressions of liquidus and solidus lines describe ideally a line of crystallization of phases.  Figure 1 illustrates the diagrammatic curves for the first and second component, and they are rectilinear. It suggests that presence of particles of two types in melt that is Si and Cr, and of the Van der Waals interaction forces between them.  CONCLUSIONSBased on the above it can be concluded that for the analytical description of the liquidus and solidus lines of the phase diagrams of the whole class of systems were proposed two types of the generalized mathemati-cal expression as modified Le Chatelier-Shreder equa-tion (equation 6). The main correlations of thermody-namic properties and structures of phases along phase equilibrium lines “solid - liquid” permitting to set up the melt diagrams of the unstudied systems based on limit-ed quantity of experimental data were also determined. This study was made under the project of Commit-tee of Science of the Ministry of Education and Science of the Republic of Kazakhstan for 2 018 - 2 020,  IRN AP05130225 / SPh. REFERENCES [1]  Sryvalin I.T., Yessin O.A.// News of Higher Educational Institutions. Ferrous metallurgy (1961) 9, 13-17. [2]  Lukas H.L., Fries S.G., Sundman B. Computational Ther-modynamics, the Calphad Method. - Cambridge: Universi-ty Press, 2007. - 324 p.[3]  Zakharov M.A. Calculation of the main types of the phase diagrams of binary solutions within the generalized lattice model // The Bulletin of the Novgorod State University 98 (2016) 7, 22-26.[4]  Voronin G.F. New opportunities of thermodynamic calcu-lation and set-up of the phase diagrams of heterogeneous systems JPhCh 77 (2003) 10, 1874-1883. [5]  Glazov V.M., Pavlova L.M. Chemical thermodynamics and phase equilibriums. M.: Metallurgy, 1988. 559 p. [6]  Baisanov S.O., Tolokonnikova V.V. Development of fun-damental theory bases of liquid phase for binary systems Table 1 Comparative experimental and calculated data for Cr – Si system  Ф”Si,cal. Ф”Si,exp. XLSi, cal. XLSi,exp. Ф”Cr,cal. Ф”Cr,exp. XLCr,cal. XLCr, exp.0,8990 0,9002 0,9770 0,9770 0,0801 0,0744 0,0301 0,02300,9562 0,9588 0,9569 0,9570 0,0921 0,0917 0,0435 0,04301,0128 1,0277 0,9391 0,9400 0,1041 0,1055 0,0577 0,06001,0689 1,0645 0,9233 0,9230 0,1161 0,1190 0,0722 0,07701,1245 1,1114 0,9090 0,9080 0,1282 0,1314 0,0867 0,09201,1795 1,1674 0,8960 0,8950 0,1404 0,1429 0,1008 0,10501,2339 1,2328 0,8841 0,8840 0,1526 0,1535 0,1144 0,11601,2878 1,2869 0,8731 0,8730 0,1648 0,1645 0,1275 0,12701,3411 1,3429 0,8628 0,8630 0,1771 0,1752 0,1399 0,13701,3675 1,3784 0,8580 0,8590 0,1832 0,1801 0,1458 0,1410Figure 1  Dependences Ф”Cr and Ф’Cr of Bjerrum Guggenheim’s coefficients from activities for system Cr–Si, where a) –Ф”Cr = f(aLCr / aSCr);  b) – Ф’Si = f(aLSi / aSSi) 100S. BAISANOV et al.: MATHEMATICAL METHOD OF PHASE EQUILIBRIUM OF BINARY SYSTEM Cr-Si BASED... METALURGIJA 59 (2020) 1, 97-100from thermodynamics of heterogeneous phase equili-briums. – Karaganda: Glassir, 2017. - 129 p. [7]   Guggenheim E.A. 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Analysis of processes with statistical me-thods, M.: Mir, 1973. 959 p.Note:  The responsible translator for English language is Nataliya. Drag, Karaganda, Kazakhstan 

Mathematical method of phase equilibrium of binary system Cr-Si based on Bjerrum Guggenheim concept

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