MATHEMATICAL MODEL FOR FORECASTING
CHEMICAL AND PHASE COMPOSITION OF MULTICOMPONENT METAL ALLOYS AT EQULIBRIUM
V.N.Boronenkov, Ural State Technical University, Ekaterinburg, Russia
ABSTRACT
The mathematical model for calculation of composition of matrix, type and number of phases in metal alloys containing any number of components at equilibrium was built. The computer program for the prognosis of formation of up to 60 carbides, nitrides, oxides and borides for alloys of iron containing up to 15 elements and the solutions of these compounds was developed.
INTRODUCTION
Nowadays the mathematical modeling with use of the computers allows solving problems of forecasting of metal composition for various metal production and processing technologies. The implementation of computer models reduces labour and research time and also allows performing virtual experiments in the case when the realization of real experiments is problematic. One of the important problems is the prognosis of the number and composition of phases formed during interaction of metal components with one another in a liquid phase and in the process of cooling of a hardening alloy. Carbides, oxides, nitrides etc. can form such phases. Solution of a similar problem becomes rather complex in the case of the multi-component alloys. This is connected, foremost, to the attempts to account for the formation of all the phases, which existence is possible at the given conditions. Secondly, it is necessary to take into account kinetic braking that really determines the formation of the given set of phases. The number, morphology and composition of phases under specific crystallization conditions and cooling rates also has to be taken into consideration.
As the first step in the solution of this problem it is reasonable to use thermodynamic approach that allows for prediction of composition and number of phases formed at equilibrium. Among the merits of this method are relatively small amounts of the data necessary for calculations and high reliability of results, the latter depending solely on the precision of thermodynamic characteristics used. Besides, in the number of cases when the phase formation rates are rather high, the thermodynamic calculation method can provide the results that are close to experimental. Such conditions could be, for example, expected when the primary highly disperse phases are formed in molten metal in conditions substantially favouring the interactions: high temperatures, large specific particles surface and intense diffusion. For similar reasons the actual phase composition of an alloy could be close to equilibrium in some temperature range below the solidus point when the cooling rates are not too high. The presented method opens also the phenomenological accountability of the kinetic conditions of formation and growth of new phases when the kinetic parameters for the considered reactions are known.
By the present time a large number of studies on the calculation of phase equilibrium in multi-component and multiphase systems, mostly involving gases and pure condensed phases has been published (Ref.1, 2). In recent years’ research the techniques and computer programs for prediction of the appearance of the dozens of phases in systems, containing condensed solutions (ideal, as a rule) have been described (Ref.2). For the metal systems that usually form imperfect solutions, the similar methods were presented in works (Reg.3-9). Particularly, those dealt with the calculation of the phase diagrams. However, the employed techniques, for example the method of the Newton-Raffson (Ref.3-8), required a satisfactory initial approximation and proved to be excessively labour-consuming even when considering the formation of 3-4 phases. Nevertheless, it is worth mentioning the impressive results when analyzing the metal phase diagrams (Ref.5-9), describing the formation of carbides and nitrides in Fe-Ti-V-Cr alloys (Ref.3) and the calculations of number and composition of oxide inclusions in iron alloys (Ref.10).
The offered method allows solving the given problem for the considerably higher, basically arbitrary, number of phases with the help of rather simple computer program. This method is the further development of the technique presented in (Ref.11).
The method is based on the sequential calculations of the systems’ equilibrium approach. It makes use of the equations for the conditional reaction rates based on the action mass law. Therefore the method can be called kinetic or relaxational (Ref.1). Its important advantages are the absolute convergence and the existence of the unique single solution (Ref.12). Thus, the requirements for the solution of the system of equations and for the “good” initial approximation no longer hold in the case of the multi-component solutions. The problem of determination of a list of phases, which formation should be taken into account, that is complex in other methods, is automatically resolved. For considered multiphase systems the law of the action masses is expressed through the mass concentrations. The refinement of activity factors for the solution components, the determination of mass, composition of all phases, the correction of the list of all possible reactions etc. are done at every calculation cycle.
COMPUTATIONAL MODEL
The general approach will be presented based on the example of the iron-based multi-component alloys.
The method of full material balance on all 16 elements in the system (Fe, C, Si, Mn, Cr, Ti, V, Mo, W, Ni, Nb, Zr, Al, B, N, O) is used for the calculation of the chemical and phase composition of such an alloy at equilibrium. The possibility of formation of the following 60 individual phases of carbides, nitrides, borides and oxides is taken into account: Fe3C, SiC, Mn3C, Cr23C6, Cr7C3, TiC, VC, V2C, Mo2C, MoC, W2C, WC, Nb2C, NbC, ZrC, Zr2C, Al4C3 B4C, FeO, SiO2, MnO, Cr2O3, TiO2, V2O3, MoO2, WO2, NiO, Nb2O5, ZrO2, Al2O3, B2O3, Fe4N, Si3N4, Cr2N, TiN, VN, Mo2N, W2N, Nb3N, ZrN, AlN, BN, Fe2B, FeB, MnB, CrB2, CrB, TiB2, TiB, VB2, VB, MoB, WB, NiB, Ni4B3, NbB2, NbB, ZrB2, ZrB, AlB12. Besides, the formation of solutions (for example, mixed carbides, oxide inclusions) is taken into account as well. The alloy composition and temperature are the input parameters.
From the mathematics point of view the problem consists of finding 76 variables: concentrations of 16 elements in metal matrix and contents (mass%) of all 60 phases. To solve the problem it is necessary to solve the system of 76 equations minimum: 16 equations of mass balance for all elements of an alloy and 60 equations for the constants of equilibrium of all reactions of the phase formation. The solution compositions also have to be determined when the formed compounds are reciprocally soluble.
The equilibrium condition of a system is completely described by a set of independent reactions:
, (1)
Where Ei, C, O, N, B are elements dissolved of metal. It is clear from the below explanations that the number of considered elements and formed phases can be easily increased without altering the basic calculation technique. The formation of sulphides and silicides etc., for example, could be of practical interest.
Formation of several kinds of phases of one element (for example, carbides V2C and VC) is taken into account by varying the appropriate stoichiometric factor values x, y, z, n, which can be assigned fractional values. The mutual influence of reactions is automatically taken into account through the common reactants.
We are going to present the equations for the formation rates for carbides, oxides, nitrides and borides in a mass unit of an alloy (mol.J/kg.sec) as follows:
, (2)
Where are the formal rate factors. Their values re chosen based on the solution ease and stability considerations;
KJ,i are equilibrium concentration multiples for reactions (1). They depend on matrix composition and are updated in each cycle of calculation;
J - common label for C, O, N, B;
[Ei] and [J] - concentration of elements dissolved in matrix at the given time point (mass%);
aEiJ - activities of compounds EiJ in formed phases at the same time point. During formation of pure phases or the solutions saturated with the given compound we consider aEiJ = 1.
Let's underline, that the equations (2) are conditional and do not reflect the true relations between the reaction rates (1) and the reactants’ concentrations. However, they do reflect correctly the tendency of reaction rates (1) to drop to zero when approaching equilibrium. It allows employing these equations when analyzing the changes in a system, relaxing in the direction of equilibrium.
The values KJ,i in (2) equal:
, (3)
Unlike the corresponding values in equations (2) that pertain to the given moments in time, the concentrations and activities in (3) describe the equilibrium. Thus, they represent the solutions sought.
During system relaxation the concentrations of the dissolved elements decrease according to equations (2) and reach their equilibrium values as described by (3). The concentration differences of no more than 0.1 relative % for all reactions are accepted as the equilibrium criterion. The accuracy of the solution depends first of all on the accuracy of determination of KJ,i values at equilibrium compositions condition. Because of that this question will be discussed in more detail below.
The rates in equations (2) are:
(4)
Where mi and m are masses of component i and of alloy as a whole, kg; Mi - molar mass, kg/mol.
Let's split the process time into k intervals with duration Dt. Then for any reactant, for example for elements J we can have, instead of (4):
, (5)
Equations (4,5) allow finding the mass of each reactant at every subsequent time interval. For example, for each of the substances J, taking into account all the reactions it is involved in, we obtain:
, (6)
We also express the concentrations of all the substances with the help of (4,5):
, (7)
(8)
, (9)
Where is the relative change of alloy matrix mass during the next period:
, (10)
and are relative alloy matrix mass changes in a moment rather than initial mass, that equals
, (11)
For the starting conditions S0 = m0/m0 = 1. Therefore, equations (7,8) determine concentration of all elements in matrix, i.e. its composition. Equation (9) presents the contents of each of the compounds EiJ in convenient for the analysis form: as concentrations relative to the initial alloy mass.
It follows from equations (7,11) that solution of the presented problem does not require the solution of set of equations, as the calculation of compositions at every consequent moment is being based on data from the preceding step. This approach considerably simplifies the programming and the numerical solution realization.
As follows from equations (2, 7-9), the thermodynamic possibility of the formation of each of the compounds EiJ at the initial phase composition and temperature is defined at the first calculation step. The VJ,i > 0 condition agrees with that. Therefore, the contents of all the compounds that can be formed at these conditions will increase. As the concentrations of C, O, N and B decrease because of the compound formation with the elements with the higher chemical affinity, the conditions of thermodynamic instability prevail for the compounds of other elements. Those unstable compounds will start to dissolve up to their complete vanishing (correlates with VJ,i < 0). On the closing stages of calculation near equilibrium only the thermodynamically stable at given conditions compounds remain. Thus, in an offered method it is not required to determine a list of phases as before. It can be derived from the given system.
Let's look more closely at the methods of KJ,i values determination. In order to do that, we are going to use the equations for the reaction equilibrium constants (1). For the processes of formation of carbides and borides it is possible to assume the Gibbs energy of formation of these compounds () to be equal to standard condition of all reactants - pure condensed substances:
, (12)
, (13)
Where
, (14)
are activities, activity factors, molar shares and molar masses of components i. From equations (3,12,14) we obtain:
, (15)
Where
, (16)
As only the Gibbs energy values for the formation of the gaseous O and N are known (Ref. 13,14), it is impossible to directly use the equations (13, 15) in the case of oxides and nitrides formation:
yEi + 1/2 O2 = EiyO
zEi + 1/2 N2 = EizN , (17)
Using the reaction equilibrium data:
1/2O2 = [O]
1/2N2=[N] , (18)
From (3,17,18) obtaining:
, (19)
Here f[O] and f[N] are the activity factors for the dissolved oxygen and nitrogen normalized on other standard condition - an indefinitely diluted solution. The temperature correlation's and for liquid iron can be calculated from known equations (Ref. 13). For calculation of sizes is possible to use also immediately expressions for constants responses of deleting of the dissolved oxygen:
, (20)
The temperature correlation's for the majority of elements are known (Ref. 13). The results are greatly influenced by the precision of activity factors gi and fi determination in the multi-component systems. The computational methods were used because of the scarcity of the appropriate experimental data. The method of Wagner serves as a good approximation in the low concentrations range:
, (21)
The values of interaction parameters , and their temperature relations are known (Ref. 13). The values of activity factors for the diluted solutions (gi0) for 1873К (Ref. 13) and for other temperatures is recalculated based on the equation: