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
255
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 ironbased 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:
256
xE i   C   E ixC 
yE i   O   E iyO 
zE i   N   E iz N  ,
(1)
nE i   B  E in B 
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:

 E   O   K
 E   N   K
 E   B   K
VC ,i  K Co ,i  E i   C   K C ,i .a EixC
x
VO ,i  K Oo ,i
V N ,i  K No ,i
VB ,i  K Bo ,i
y
i
O ,i
.a EiyO
N ,i
.a Eiz N
z
i
n
i


,

B ,i
.a EinB
(2)
Where K Jo ,i 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:
257
K C ,i 
K N ,i
[ Ei ] x [ C ]
[ Ei ] y [ O ]
; K O ,i 
;
a EixC
a EiyO
[ Ei ] z [ N ]
[ Ei ] n [ B ]

; K B ,i 
,
a Ezx N
a EinB
(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:
dm J ,i
dm Ei
dm EiJ
(4)

mM J d
mM Ei d
mM EiJ d
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 . Then for any
reactant, for example for elements J we can have, instead of (4):
V J ,i  

m Jk  1  m Jk m k  1 [ J ] k  1  m k [ J ] k
,
(5)

m k M J 
100 m k M J 
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:
V Jk,i  
i
m Jk  1  m Jk  m k M J   V Jk,i ,
(6)
i 1
We also express the concentrations of all the substances with the help of (4,5):
i
[ J ] k  1  ([ J ] k  100 M J   V Jk,i ) / S k  1 ,
(7)
i 1
i
i
i
i
i 1
i 1
i 1
i 1
[ Ei ] k  1  ([ Ei ] k  100 M Ei  (  xVCk,i   yVOk,i   zV Nk ,i   nV Bk,i )) / S k  1
(8)
[ EiJ ] k 1  [ EiJ ] k  S0k .100 M EiJ VJk,i ,
(9)
k 1
k 1
k
Where S  m / m is the relative change of alloy matrix mass during the
next period:
i
i
i
i
i 1
i 1
i 1
i 1
S k  1  1  (( M C VCk,i  M O VOk,i  M N V Nk ,i  M B VBk,i ) 
258
i
i
i
i
i 1
i 1
i 1
i 1
 M Ei (  xVCk,i   yVOk,i   zV Nk ,i   nV Bk,i )) ,
(10)
and S 0k  m k / m 0 are relative alloy matrix mass changes in a moment k
rather than initial mass, that equals
S 00  S 0 * S 1 * S 2 * ...* S k ,
(11)
0
0
0
For the starting conditions S = m /m = 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 ( Gio ) to be equal to standard
condition of all reactants - pure condensed substances:
aE C
aE B
(12)
K Cp ,i  x ix ; K Bp ,i  x in ,
a Ei .aC
a Ei .a B
G Eo
ix ,C
  RT ln K Cp ,i ; G Eo ix ,B   RT ln K Bp ,i ,
(13)
Where
a i   i .N i 
[i ] i [i ]
,
/
M i i 1 M i
(14)
ai , i , N i , M i are activities, activity factors, molar shares and molar masses
of components i. From equations (3,12,14) we obtain:
K C ,i 
M Exix ,C .M C .S 1 x
K Cp ,i . Eix . C
; K B ,i
n
M EiB
.M B .S 1 n
,

n
K Bp ,i . Ei
. B
Where
259
(15)
i
S
i 1
[ Ei ] [ C ] [ N ] [ O ] [ B ]
,




Mi
MC
MN
MO M B
(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:
a EiyO
yEi + 1/2 O2 = EiyO
K Op 1,i  y 0 ,5
a Ei .PO2
K Np 1,i 
zEi + 1/2 N2 = EizN
a EizN
z
Ei
a .PN0 2,5
,
(17)
Using the reaction equilibrium data:
1/2O2 = [O]
K Op 
1/2N2=[N]
K Op 
[ O ]. f [ O ]
PO02,5
[ N ]. f [ N ]
PN0 2,5
,
(18)
From (3,17,18) obtaining:
K Op .M Eiy .S y
K Np .M Eiz .S z
,
(19)
;
K

N ,i
K Op ,i . Eiy . f [ O ]
K Np ,i . Eiz . f [ N ]
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 K Op and K Np for liquid iron can be calculated from known
K O ,i 
equations (Ref. 13). For calculation of sizes K O ,i is possible to use also immediately
expressions for constants responses of deleting of the dissolved oxygen:
K O ,i  K O0 ,i . f [ yE ,i ] . f [ O ] ,
(20)
The temperature correlation's K O0 ,i for the majority of elements are known
(Ref. 13). The results are greatly influenced by the precision of activity factors i 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:
j
ln  i  ln  i 0    ij  N j
j2
j
lg f i   e ij .[ j ] ,
(21)
j2
The values of interaction parameters  ij , e ij and their temperature relations are
known (Ref. 13). The values of activity factors for the diluted solutions (i0) for
1873К (Ref. 13) and for other temperatures is recalculated based on the equation:
(22)
ln T   T ln  i0 ,
i
1873
260
In the case of concentrated alloys it is preferable to apply certain solution
models (regular, subregular etc.), supplemented with the empirically chosen
parameters. Usually for the iron-based alloys it is possible to limit the choice with the
regular solutions model. For an alloy with k components for a component i we have:
RT ln i 
k

j  1 ,i  j
k 1
X j Qij  
k
X
l 1 i  j 1
i
X j Ql , j ,
(23)
Where Ql , j is exchange energy, reviewed, for example in (Ref.15). If the
resulting compounds can form solutions, then aEiJ  1 . In this case information on
mutual solubility of compounds and type of these solutions is necessary. Current
activity values could be directly found from (13) in the most simple case, when the
ideal solution with unlimited component solubility is being formed. This is done
after the content of the dissolving compounds is found from (9). It is advisable to
apply equation (21,23) in the case of non-ideal solutions.
RESULTS
The computational program is realized in Excel 97 environment in language
VBA.
To illustrate a typical calculation procedure the changes phases content with
time for iron alloy with 3% C, 10% Cr, 1% Ti, V, Al, Ni, Mo, Nb, B, 0,1% N and
0,03% O2 at 15000C are presented in Figure 1.
1,4
0,035
TiC
NbC
1
EJ,i, %
0,03
0,025
NiB
0,8
0,02
BN
0,6
0,015
0,4
0,01
AlN
0,2
B 4C
TiN
0
0
50
100
150
200
250
300
BN, %
1,2
0,005
0
350
Time, s
Fig.1. Change of the contents of phases in time
As discussed above, the initially formed less stable phase (BN) gradually
dissolves, as boron and nitrogen are being depleted from the alloy to form more stable
compounds ( TiC, NbC, NiB, B4C, AlN, TiN).
The computational as well as experimental data on metal composition and
primary carbide content during abrasive-resistant alloy cladding (Ref.16) are
presented in Tables 1 and 2. The close correlation of experimental and computational
values testifies for the applicability of the thermodynamic method for the primary
carbide formation prediction in steels.
The calculation results for the content and composition of oxide inclusions
during welding with the rutile-type coated electrode are presented in Table 3.
The data about the seam metal structure were used for activity calculations
(Ref.17). The computational results are comparable to the experimental values
(Ref.18). The formation of pure oxides of iron and manganese is impossible in the
261
temperature range of metal in a welding bath: from 2000 K down to the metal
solidification temperature (1773 К). When the ideal solutions approximation is
applied the calculation produces the overestimated FeO contents and underestimated
SiO2 contents. The calculation for high temperatures results in the contents of an iron
oxide that hardly exceeds experimental. When the theory of regular solutions was
applied to the slag components’ activites calculations, close to experimental values for
the steel solidification temperatures were obtained.
To summarize some additional capabilities and perspectives of use of an
offered technique and computational programs:
1. In the computational program the assumptions of equilibrium state and the
availability of the set of specific initial parameters [EiJ] 0are followed. The scheme
of calculation completely is saved, but the S0 values in this case are obtained from
equation (11):
16
4
S 0  1   [ EiJ ] 0 / 100 ,
(24)
i 1 j 1
2. Simple additions to the program allow solving a practically important
problem - calculation of equilibrium compositions and masses of metal and slag at a
0
specific initial ratio of their masses K 0f  m0f / mme
. In this case it is more convenient
to express the amounts of oxides EiO through concentration in slag (EiO), instead as
in (9). From material balance follows:
[ Ei yO ] k
m
,
(25)
( Ei yO )k  k [ Ei yO ] k 
i [ Ei O ] k
mf
y
Sum  
100
i 1
0
i
Kf
(26)
S 0  1  Sum 
( EiyO )0 ,

0
100( 1  K f ) i 1
K kf 
Sum
1

k
S0
100S0k
Sum 
K 0f
i
 [ Ei O ]
i 1
y
k
,
(27)
j
 ( Mj
)
O )0 ,
(28)
100( 1  K j 1
In equation (28) (MjmO)0 are initial concentration of oxides and salts in slag
which are not included in number of considered connections.
3. The method can be used for construction of phase diagrams of metal alloys
and other systems. It should be noted that no new data is required for calculation
compared to the known methods (Ref.6-10). However, as the thermodynamic data
accuracy requirements are substantially higher here, care must be taken when
applying equations (20, 22) for the component activity factors calculations.
Nevertheless, the use of equations (20, 22) is justified in the case of high-resistance
compounds that form even at low reagent concentrations. This allows expecting
sufficiently accurate iron-based multi-component alloys phase composition
predictions with the described method.
4. The use of the presented technique is promising when analyzing the
homogeneous equilibrium in polymeric oxide melts containing large number of
complex ions. For example, for systems MeO - SiO2 the fact of formation of all
polymeric anions can be represented by one equation (29):
iSiO4  Sii O32i(i11cc )  ( i  1  c )O 2 ,
(29)
0
f
m
262
Where i and c are integers. It allows to express all reaction rates as one equation and
to make the very simple program for calculation of concentration of all compounds.
5. It is expected that the offered approach would be effective when studying
equilibrium and in other cases, when the analyzed system includes a large number of
ongoing processes and multiple phases.
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1.
263
Table 1 - Composition of alloy matrix, %mass.
№
Element
C
Cr
Mo
of alloy
Exper.
1,9
6,9
2,78
1
Calc.
1,33
7,26
2,92
Exper.
Calc.
2
1,9
0,73
5,5
6,04
3,1
3,4
Ti
Nb
Ni
1,55
0,88
3,9
0,35
0,5
0,52
3,9
2,06
6,0
0,39
-
Table 2 – Carbide contents, %mass.
№ of alloy
Parameter
Carbides
1
experimental 8 % (TiC + NbC)
calculated
2,88 % TiC + 3,75 % NbC
2
experimental 15 % (TiC + NbC)
calculated
2,53 % TiC + 6,37 % NbC
Table 3. Composition of oxide inclusions, %mass.
№
T, K
FeO
SiO2
MnO
TiO2
Calc.
1
1773
0
82,14
0
11,67
6,19
Total
inclus.,
%mass.
0,149
2
1773
17,97
46,17
9,28
14,72
5,22
0,181
3
1773
3,27
57,7
13,8
15,96
5,65
0,167
4
5
2000
2000
0
43,2
84,9
28,9
0
6,1
8,9
12,2
6,1
4,5
0,140
0,210
6
2000
15,22
49,9
11,78
13,00
4,66
0,205
Exp.
---
0,95,3
55,573,1
6,619,0
5,311,5
7,213,1
0,109-0,131
264
Al2O3
Oxide
condition
Pure
phases
Ideal
solution
Regular
solution
Pure phases
Ideal
solution
Regular
solution
Ref.18