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Atomic distribution of alloying additions between sublattices in the intermetallic compounds Ni3Al and NiAl I: Phenomenological treatment
~ou~~a~ of the Less-Cordon
Metals,
273
139 (1988) 273 - 282
ATOMIC DISTRIBUTION OF ALLOYING ADDITIONS BETWEEN SUBLATTICES IN THE INTERMETALLIC COMPOUNDS NisAI AND NiAl I: PHENOMENOLOGICAL TREATMENT N. V. ALLAVERDOVA, V. I. BOGDANOV
V. K. PORTNOY,
M.V. Lomonosou Moscow State ~nive~ity, ~19~99, GSP, Moscow 1172312 (U.S.S.R.)
L. A. KUCHERENKO, Dependent
of e~e~~t~~
A. V. RUBAN, and Lenin Hills,
(Received April 21, 1987 ; in revised form October 5,1987)
Summary The atomic distribution of an alloying additive in an ordered phase with two types of site is treated using the statistical theory of ordering and the pairwise interaction approximation without correlations. The suggested method is based on the minimization of the ordering energy in terms of the long-range order parameters. The ~h~a~teristic vector diagram of motion for the system is plotted. This diagram permits a unique de~rmination of the required distribution.
1. Phenomenological aspects The atomic distribution of an alloying additive between sublattices in an ordered phase determines the superconducting transition temperature [ 13, the mechanical properties of intermetallic compounds [Z] and many other physical properties. The determination of the atomic distribution is p~icul~ly impo~nt for making the right choice of alloy composition in the physical metallurgy of in~~etallic compounds with a narrow homogeneity range. We consider here the ways of tackling this problem for highly ordered phases with just a small amount of alloying additive, i.e. phases with a high degree of long-range order and a composition approaching stoichiometry, In this case the atoms of the binary phase and the atoms of an alloying addition at the sites of every sublattice form the corresponding weak solution. Statistical and thermodynamic studies [3] have shown that in these highly ordered phases the correlation effects are negligible and the configuration-dependent parts of the the~odyn~ic quantities vary exponentially with temperature at sufficiently low temperatures. It follows that, at a sufficiently large ratio Vale, where V is an ordering energy, the free energy can be calculated using 002%5088:88,‘$3.50
@ Elsevier Sequoia/Printed in The Netherlands
274
the self-consistent field approximation. If the ordering temperature is high, the corresponding phase will be highly ordered even at relatively high temperatures (of the order of the Debye temperature or higher). All the information about the atomic distribution in an ordered phase is contained in the long-range order (LRO) parameters. Therefore, the problem is either to determine the values of these parameters in the three-component ordered phase or to determine the most advantageous set of values for the parameters. In order to determine the LRO parameters it is necessary to solve the set of self-consistent field equations (of Gorsky-Bragg-Williams type) which describe the temperature and concentration dependence of the parameters. However, this system is of the transcendental type and involves, therefore, computational difficulties. Also, the concentrations of all the components need to be determined. At the same time, it is not specified which lattice the alloying additions occupy; an arbitrary choice of composition for the phases with a narrow homogeneity region may correspond to the two-phase region in the phase diagram. That is, the phase with such a composition may not exist. Hence, the main obstacle to using the self-consistent field equations is, in this case, the concentration dependence of the entropy term in the expression for the free energy. There are, however, good reasons to believe that, for highly ordered phases, the configurational entropy does not determine the atomic distribution of the alloying additions in the sublattices. The contribution of correlations to the configurational entropy is proportional to exp(-V/M’) [ 31. If V/M’ > 1, this contribution to the free energy can be neglected. Furthermore, the entropy of an ideal solution determines only the statistical atomic redistribution of the alloying addition and the original binary phase. The second way of solving the problem of the atomic redistribution of an alloying additive between sublattices is to minimize the change in the free energy by reducing changes in the LRO parameters. Since the entropy contribution can be neglected in this case, the only quantity to be minimized is the ordering energy. The question naturally arises of the temperature and concentration dependence of the atomic distribution of the alloying additive and the effect of the alloying additive on the atomic redistribution of the original binary phase. In the self-consistent field approximation the change in temperature will cause no more than the statistical redistribution of all the atoms of ordered phase. The effect of concentration on the atomic distribution between sublattices is explained on the basis of the characteristic vector diagram in the space defined by the LRO parameters.
2. Characteristic
vector diagram
In order to describe the ordering in the three-component phase (AkB, is the main phase and C is the alloying additive) with two types of sites (A
275
sites and B sites), it is necessary to introduce two LRO parameters. These parameters, nA and qz may be chosen such that in the completely ordered state when the element C is absent, 7A = qa = 1. In this case nA(%-CA.
h&--B qB=
qA= (11
'
P
where 01 = m/k + m, fl= k/h + m, nA(A) is a probability of finding atoms A at sites A, CA and CB are the concentrations of components. In the self-consistent field approximation the ordering energy AU is a quadratic expression of the LRO parameters [ 41. For the three-component phase with two types of site, it can be demonstrated (see Appendix A) that the expression for AU takes the form
(2) are the Fourier transforms of the corresponding VAAC, VBBC, VABC mixing energies. We shall minimize the change in the free energy (the ordering energy in our approximation) by reducing the changes in the LRO parameters arising with the introduction of an alloying additive. Consider the limiting transition from a binary to a three-component ordered phase. The region of existence of the binary ordered phase is shown in terms of the LRO parameters by the segment DO (Fig. l), where r)A = vz = r). Starting with an arbitrary initial point on this segment, we can trace the evolution of the system with the introduction of a third element. It is clear that the element replacing A will decrease VA, the element replacing B will decrease r)z and the element replacing both A and B will change both the LRO parameters equally. where
DcO,O, Fig. 1. The region of existence of the binary ordered phase (segment DO). The direction of the vector R = -grad(AU) shows the most advantageous change in the LRO parameters with the introduction of an alloying addition.
276
When minimizing the change in the ordering energy, we consider the ordering energy as a potential acting in the generalized space defined by the LRO parameters. The most advantageous direction of motion of the system in this space will then be determined, as in the method of steepest descent, by the vector: R = -grad(AU). In fact, R is a generalized strength describing the response of the system to the introduction of a third element. Thus, all the information about the atomic distribution of an alloying addition between sublattices in a highly ordered phase is determined by the direction of the vector (3) Here fiA and 7jB are the corresponding unit vectors. It follows from eqn. (3) that the direction of R is independent of the position of the point S on the segment DO. The point S will determine only IR I. In the general case, therefore, it is useless to determine the initial point of the binary system more accurately, but it is clear that for highly ordered phases it will be near the point 0. On the basis of a simple analysis we choose five typical directions which determine the atomic distribution of an alloying addition in the ordered phase (Fig. 2). The direction a(P) corresponds to the motion of the system with increasing qB(TA), in which case qA(vB) may vary arbitrarily. This situation can take place in cases where the atoms of the alloying additive replace atoms A(B). The direction e corresponds to the equally advantageous increase in 7A and 7)B. In this case the alloying additive will replace atoms A and atoms B with equal probability. We can also distinguish two limiting directions: oiim and Piim. If R is between oii, and a@, pi,,) (from here on we mean the region obtained by the clockwise motion from the first to the second vector), the alloying
Fig. 2. Characteristic alloying addition.
directions
of the vector
R specifying
the atomic
distribution
of an
277
additive will replace only atoms A(B), because in this case the increase in ?~(TJ) is advantageous and the decrease in qA(qB) is disadvantageous. Now let us show that the solution of the Gorsky-Bragg-Williams type equations does not exist within the statistical theory of ordering, if for some element the vector R is in the region (a lim, &im)e To this end, we consider the sets of transcendental self-consistent field equations for the threecomponent phase using the method of statistical concentration waves (Appendix B) ln
(1 - CA- CB - a(qA - ??B))tc - b?A)_ VAACks KT t1 --CA --B +@(?A--?~))(c +oo)A) -
In(l-CA-CB t1
--CA
-
a(??A
CB + @(?A
-
‘?d)(C
+ b/B)
-
%d)(c
+ qA)
--
qA
VABCks
KT VABcks KT
Vmcks = -qB+ KT
‘?’ qA
Here k, is the superstructure vector, K is the Boltzmann constant. Subtracting in the second equation from the first in eqns. obtain In (‘A - PrlA)(cB - anlB) = f(CA
+ aqA)(CB
KT
+ hB)
=
{(V AAC k s - V,,k,)TjA
Xf??A,
VB)
+ (VBB, -
(4)
(4), we
V AB ~)rf~) (5)
With the numerator being the probability of finding one type of atom at a site of the other type and the denominator being the probability of finding one type of atom at a site of the same type (see eqn. (l)), we obtain an obvious condition x(?A,
r)B) 4
0
(6)
The condition in eqn. (6) determines the feasible range of the LRO parameters in the three-component phase with two types of site. It is easy to see that X(qA, qB) = 0 defines a straight line passing through the origin and perpendicular to R (Fig. 3). In this case R is always directed towards the
Fig. 3. Feasible area (shading) of the LRO parameters. (cylimv Plim)*
The vector R is in the region
278
feasible range of the LRO parameters. If R is in the region (p,i,, Ollim) (Fig. 3), the region of the binary phase (the segment DO) and that of the threecomponent phase (the shaded area) are separated. This indicates that the limiting transition from binary to three-component ordered phase is not feasible (the common point D corresponds to the disordered state). It is of great interest that in the approximations we employ any element may be dissolved in a given binary phase provided such a phase exists. To illustrate this, we rewrite the expression for R in a different form R=-09~(6~
VBCA
+~?B~AcB)
The above equation can be obtained A. However, it is clear that V BCA
+ VACB=
VA,,%
using the formalism
VAB
of Appendix
(7)
where VA, is the Fourier transform of the mixing energy of the binary phase and VA, < 0 is the condition of existence of the phase. The relation in eqn. (7) essentially restricts the direction and length of the possible vectors R. It follows from the analytical geometry that the ends of the vectors will always be on the straight line r (Fig. 4) perpendicular to DO and, so, parallel to the directions oii, and &im. Hence, the vector is always in the region (oii, &im) except for VA, > 0. The result obtained has a simple physical meaning. An infinitely small amount of an alloying addition will lead to a certain loss in stability of the binary phase (the ratio of the number of atoms of the binary phase to that of the alloying additive Ngh/Nadd + m), only if the atomic interaction potential between the alloying additive and the phase is infinitely large.
/
Blim
Fig. 4. Possible directions of the vector R in the three-component of the vector must be on straight line r.
ordered phase. The end
219
Thus, the five characteristic directions ((Y, (Xiim,/3,Piim, E) divide a set of all possible directions into four subsets: (aiim, a) is the region of the atomic distribution of an alloying addition in sublattice Ai; (a, E) is the region of the atomic distribution of an alloying addition between sublattice A and B with the preferential filling of sublattice a as R gets closer to cx; (E, 0) is the analogous region but with the preferential filling of sublattice B; (p, /3iim)is the region of the atomic distribution of an alloying addition in sublattice B. Up to this moment we have discussed the atomic distribution of an alloying additive between sublattices. It is to be noted, however, that the introduction of an alloying additive can lead to the atomic redistribution of the main components of the original phase. For example, if atoms C can fill only sites A and the total amount of elements A and C is greater than the stoichiometric concentration, a deficiency in atoms B at sites B will be filled by atoms A. Here again the vector diagram enables us to draw a definite conclusion about the atomic distribution of all the elements. For example, if the response vector of the system is in the region (O(lim,a) and is not coincident with CY,the alloying additions can fill only sites A and any deficiency of atoms in sublattice B will be filled by element A, but if the response vector is in the region (a, E), C atoms will primarily fill sublattice B in the case of a deficiency in atoms B. The choice of the space of LRO parameters with two dimensions T)A, 7)B is not indispensable. It is clear that the result will be the same if we choose two other LRO parameters with a different normalization. As an example, Fig. 5 presents the diagram for a space with dimensions qA and qc. The parameter qc is normalized such that qc = 1 when atoms of element C replace all atoms of element A. In this case, however, the analysis is no
Fig. 5. Characteristic and 7)A.
directions
of the vector
R in the space of the LRO parameters
qc
280
longer straightforward and it is more convenient to work with the space of LRO parameters corresponding to the main phase elements. The expression for the response vector in the space (nA, Q) will take the form
R S(q.0)
=
-apr7(%iV*m+ iiCVACl3)
References S. V. Vinsovsky, Yu. A. Izyumov and E. Z. Kurmaev, Superconductivity of Tradition deter and their Alloys, Nauka, Moscow, 1977, p. 387. R. S. Rawlings, A. E. Staton-Bevan, The alloying behaviour and mechanical properties of polycrystalline Ni+.l (y-phase) with ternary additions. J. Mater. Sci., 10 (3) (1975) 505 - 514. M. A. Krivoglaz, To the theory of highly ordered and weak solid solutions, SOU. J. Phys. Chem., 31 (9) (1957) 1930 - 1942. A. G. Khachaturyan, Theory of Phase Trcmsformations and the Solid Solution Structure, Nauka, Moscow, 1974, p. 384.
Appendix A The ordering energy of a ~~lti~orn~o~e~t phase wits two types of site Let us generalize the expression for the ordering energy of the binary solid solution obtained in ref. Al to the case of a multicomponent phase. We assume that n kinds of atoms are distributed somehow in the Ising lattice {R} with the corresponding occupation numbers C,(R). In the absence of vacancies one of these become dependent on the other numbers
A-1.
i
C,(R) = 1
(Al)
i=l
We average the occupation numbers over the Gibbs ensemble and introduce the single-particle functions
which represent the probability of finding the ith kind of atom at site R. Naturally, the condition in eqn. (Al) holds true for these functions as well. In the self-consistent field approximation the hamiltonian of the multicomponent system can be recorded in the following form H’“’ = ~ I: R,R’
~
Vfj(R, R’) Ci(R)Cj(R’)
= Hlo(“’ + H,“’
i, j=l
where Ha(“) is the confi~ration~ly independent part of the hamiltonian and Hi(“) is the config~ation~y dependent part. Using the condition (A) Hi(n) can be rewritten in the form i
28;
$ 2 “ilVijn(R,
Hl(“) =
R')Ci(R)Cj(R')
R,R'i, j=l
where V,j”(R,
R’) =
R’) + V,n(R’,R’) -
Gij(R,
Vi”(R,
R’) - Vj,(R,
R’)
Averaging over the Gibbs ensemble gives the expression for the ordering energy Au
=
(HI’“‘)
ni1Vij”(R,
2
= i
R’)ni(R)nj(R’)
(AZ)
R,R' i,j=l
The single-particle functions will be defined as n@)
= ci + ‘I)&?)
(A3)
where Ci is the ith element concentration and vi is the LRO parameter. The coefficients E@) are normalized such that in the highly ordered state vi = 1 for the element filling sites A (the original phase A,B,) and qj = -1 for the element filling sites B. ei(R = ii> =
_!?_.=
k = B> = - _ k+m
E#
a;
k+m
=P
(A4)
However, the coefficients
E(R) (they are chosen equal for all elements: can be represented as a superposition of static concentration waves
eI(R)
= e,(R)
E(R) =
= . . . c,_,(R)
= E(R))
exp(ik,R) $~Cr, m
+ 7:
exp(--ik,R)}
(-45)
where k, is the superstructure vector and the index m takes on all values throughout the rotation of this vector (12,); yrn are the normalization factors. Taking into account eqn. (A5) and also the fact that in the pairing interaction approximation V(R, R’) = V(R - R’), we obtain zV(R,
R’)@)
= V(k,)e(R)
(A6)
Using this result and substituting eqn. (A5) into eqn. (A2) we arrive at the following expression for the ordering energy AU=
1
n-l
2Nk + m)
Z C vijn(ks)E2(RhiTlj R i,j=l
Or, taking account of the definition in eqn. (A4) AU = io$
nzl Vijn(ks)qiqj
(A7)
i,j=l
Reference Al
A.
G.
for Appendix
A
Khachatwyan, Theory Structure, Nauka, Moscow, 1974,
of Phase
p. 384.
Transformations
and
the Solid
Solution
282
Appendix B
B.1. The set of tmnsce~denta~ equations for LRU parameters in the multicomponent phase on the basis of AkB, with two types of site The sought equation is obtained in the statistical theory of ordering by minimizing the free energy. In the seIf-consistent field approx~ation the LRO parameters affect only the ordering energy AU and the entropy S AF=AU-TS
W)
Further, we use the definitions (Al), (A3), (A4) and the expression for the ordering energy (A7). First we take into account the definitions (A3) and (A4) and obt&n the entropy c
nj(R) Inni
+
in the form s= k* +
C&(G
+ a~?~)InG + Wi) + mWi -P77i)
{I-m(Ci
- WG
-@%I
+
ocrli))
In{1
WG
-mm(Ci
-Prli)
-Prli) -k(Ci
-+
an7i)Il 032)
Then, substituting eqns. (B2) and (A7) into eqn. (Bl) and minimizing the resulting equation with respect to each qi, we eventually obtain the set of n - 1 equations
1 l-
In
n-1
C
(Cj
-W?ji)
(Ci
n .-.1 1 - 2 (Cj 1 j= 1
-P?li)
i
j=l
+
&j)
t
= K$ ‘zi Vijn tk8 hj (Ci
+
@Vi)
j-l
033)
Metals,
273
139 (1988) 273 - 282
ATOMIC DISTRIBUTION OF ALLOYING ADDITIONS BETWEEN SUBLATTICES IN THE INTERMETALLIC COMPOUNDS NisAI AND NiAl I: PHENOMENOLOGICAL TREATMENT N. V. ALLAVERDOVA, V. I. BOGDANOV
V. K. PORTNOY,
M.V. Lomonosou Moscow State ~nive~ity, ~19~99, GSP, Moscow 1172312 (U.S.S.R.)
L. A. KUCHERENKO, Dependent
of e~e~~t~~
A. V. RUBAN, and Lenin Hills,
(Received April 21, 1987 ; in revised form October 5,1987)
Summary The atomic distribution of an alloying additive in an ordered phase with two types of site is treated using the statistical theory of ordering and the pairwise interaction approximation without correlations. The suggested method is based on the minimization of the ordering energy in terms of the long-range order parameters. The ~h~a~teristic vector diagram of motion for the system is plotted. This diagram permits a unique de~rmination of the required distribution.
1. Phenomenological aspects The atomic distribution of an alloying additive between sublattices in an ordered phase determines the superconducting transition temperature [ 13, the mechanical properties of intermetallic compounds [Z] and many other physical properties. The determination of the atomic distribution is p~icul~ly impo~nt for making the right choice of alloy composition in the physical metallurgy of in~~etallic compounds with a narrow homogeneity range. We consider here the ways of tackling this problem for highly ordered phases with just a small amount of alloying additive, i.e. phases with a high degree of long-range order and a composition approaching stoichiometry, In this case the atoms of the binary phase and the atoms of an alloying addition at the sites of every sublattice form the corresponding weak solution. Statistical and thermodynamic studies [3] have shown that in these highly ordered phases the correlation effects are negligible and the configuration-dependent parts of the the~odyn~ic quantities vary exponentially with temperature at sufficiently low temperatures. It follows that, at a sufficiently large ratio Vale, where V is an ordering energy, the free energy can be calculated using 002%5088:88,‘$3.50
@ Elsevier Sequoia/Printed in The Netherlands
274
the self-consistent field approximation. If the ordering temperature is high, the corresponding phase will be highly ordered even at relatively high temperatures (of the order of the Debye temperature or higher). All the information about the atomic distribution in an ordered phase is contained in the long-range order (LRO) parameters. Therefore, the problem is either to determine the values of these parameters in the three-component ordered phase or to determine the most advantageous set of values for the parameters. In order to determine the LRO parameters it is necessary to solve the set of self-consistent field equations (of Gorsky-Bragg-Williams type) which describe the temperature and concentration dependence of the parameters. However, this system is of the transcendental type and involves, therefore, computational difficulties. Also, the concentrations of all the components need to be determined. At the same time, it is not specified which lattice the alloying additions occupy; an arbitrary choice of composition for the phases with a narrow homogeneity region may correspond to the two-phase region in the phase diagram. That is, the phase with such a composition may not exist. Hence, the main obstacle to using the self-consistent field equations is, in this case, the concentration dependence of the entropy term in the expression for the free energy. There are, however, good reasons to believe that, for highly ordered phases, the configurational entropy does not determine the atomic distribution of the alloying additions in the sublattices. The contribution of correlations to the configurational entropy is proportional to exp(-V/M’) [ 31. If V/M’ > 1, this contribution to the free energy can be neglected. Furthermore, the entropy of an ideal solution determines only the statistical atomic redistribution of the alloying addition and the original binary phase. The second way of solving the problem of the atomic redistribution of an alloying additive between sublattices is to minimize the change in the free energy by reducing changes in the LRO parameters. Since the entropy contribution can be neglected in this case, the only quantity to be minimized is the ordering energy. The question naturally arises of the temperature and concentration dependence of the atomic distribution of the alloying additive and the effect of the alloying additive on the atomic redistribution of the original binary phase. In the self-consistent field approximation the change in temperature will cause no more than the statistical redistribution of all the atoms of ordered phase. The effect of concentration on the atomic distribution between sublattices is explained on the basis of the characteristic vector diagram in the space defined by the LRO parameters.
2. Characteristic
vector diagram
In order to describe the ordering in the three-component phase (AkB, is the main phase and C is the alloying additive) with two types of sites (A
275
sites and B sites), it is necessary to introduce two LRO parameters. These parameters, nA and qz may be chosen such that in the completely ordered state when the element C is absent, 7A = qa = 1. In this case nA(%-CA.
h&--B qB=
qA= (11
'
P
where 01 = m/k + m, fl= k/h + m, nA(A) is a probability of finding atoms A at sites A, CA and CB are the concentrations of components. In the self-consistent field approximation the ordering energy AU is a quadratic expression of the LRO parameters [ 41. For the three-component phase with two types of site, it can be demonstrated (see Appendix A) that the expression for AU takes the form
(2) are the Fourier transforms of the corresponding VAAC, VBBC, VABC mixing energies. We shall minimize the change in the free energy (the ordering energy in our approximation) by reducing the changes in the LRO parameters arising with the introduction of an alloying additive. Consider the limiting transition from a binary to a three-component ordered phase. The region of existence of the binary ordered phase is shown in terms of the LRO parameters by the segment DO (Fig. l), where r)A = vz = r). Starting with an arbitrary initial point on this segment, we can trace the evolution of the system with the introduction of a third element. It is clear that the element replacing A will decrease VA, the element replacing B will decrease r)z and the element replacing both A and B will change both the LRO parameters equally. where
DcO,O, Fig. 1. The region of existence of the binary ordered phase (segment DO). The direction of the vector R = -grad(AU) shows the most advantageous change in the LRO parameters with the introduction of an alloying addition.
276
When minimizing the change in the ordering energy, we consider the ordering energy as a potential acting in the generalized space defined by the LRO parameters. The most advantageous direction of motion of the system in this space will then be determined, as in the method of steepest descent, by the vector: R = -grad(AU). In fact, R is a generalized strength describing the response of the system to the introduction of a third element. Thus, all the information about the atomic distribution of an alloying addition between sublattices in a highly ordered phase is determined by the direction of the vector (3) Here fiA and 7jB are the corresponding unit vectors. It follows from eqn. (3) that the direction of R is independent of the position of the point S on the segment DO. The point S will determine only IR I. In the general case, therefore, it is useless to determine the initial point of the binary system more accurately, but it is clear that for highly ordered phases it will be near the point 0. On the basis of a simple analysis we choose five typical directions which determine the atomic distribution of an alloying addition in the ordered phase (Fig. 2). The direction a(P) corresponds to the motion of the system with increasing qB(TA), in which case qA(vB) may vary arbitrarily. This situation can take place in cases where the atoms of the alloying additive replace atoms A(B). The direction e corresponds to the equally advantageous increase in 7A and 7)B. In this case the alloying additive will replace atoms A and atoms B with equal probability. We can also distinguish two limiting directions: oiim and Piim. If R is between oii, and a@, pi,,) (from here on we mean the region obtained by the clockwise motion from the first to the second vector), the alloying
Fig. 2. Characteristic alloying addition.
directions
of the vector
R specifying
the atomic
distribution
of an
277
additive will replace only atoms A(B), because in this case the increase in ?~(TJ) is advantageous and the decrease in qA(qB) is disadvantageous. Now let us show that the solution of the Gorsky-Bragg-Williams type equations does not exist within the statistical theory of ordering, if for some element the vector R is in the region (a lim, &im)e To this end, we consider the sets of transcendental self-consistent field equations for the threecomponent phase using the method of statistical concentration waves (Appendix B) ln
(1 - CA- CB - a(qA - ??B))tc - b?A)_ VAACks KT t1 --CA --B +@(?A--?~))(c +oo)A) -
In(l-CA-CB t1
--CA
-
a(??A
CB + @(?A
-
‘?d)(C
+ b/B)
-
%d)(c
+ qA)
--
qA
VABCks
KT VABcks KT
Vmcks = -qB+ KT
‘?’ qA
Here k, is the superstructure vector, K is the Boltzmann constant. Subtracting in the second equation from the first in eqns. obtain In (‘A - PrlA)(cB - anlB) = f(CA
+ aqA)(CB
KT
+ hB)
=
{(V AAC k s - V,,k,)TjA
Xf??A,
VB)
+ (VBB, -
(4)
(4), we
V AB ~)rf~) (5)
With the numerator being the probability of finding one type of atom at a site of the other type and the denominator being the probability of finding one type of atom at a site of the same type (see eqn. (l)), we obtain an obvious condition x(?A,
r)B) 4
0
(6)
The condition in eqn. (6) determines the feasible range of the LRO parameters in the three-component phase with two types of site. It is easy to see that X(qA, qB) = 0 defines a straight line passing through the origin and perpendicular to R (Fig. 3). In this case R is always directed towards the
Fig. 3. Feasible area (shading) of the LRO parameters. (cylimv Plim)*
The vector R is in the region
278
feasible range of the LRO parameters. If R is in the region (p,i,, Ollim) (Fig. 3), the region of the binary phase (the segment DO) and that of the threecomponent phase (the shaded area) are separated. This indicates that the limiting transition from binary to three-component ordered phase is not feasible (the common point D corresponds to the disordered state). It is of great interest that in the approximations we employ any element may be dissolved in a given binary phase provided such a phase exists. To illustrate this, we rewrite the expression for R in a different form R=-09~(6~
VBCA
+~?B~AcB)
The above equation can be obtained A. However, it is clear that V BCA
+ VACB=
VA,,%
using the formalism
VAB
of Appendix
(7)
where VA, is the Fourier transform of the mixing energy of the binary phase and VA, < 0 is the condition of existence of the phase. The relation in eqn. (7) essentially restricts the direction and length of the possible vectors R. It follows from the analytical geometry that the ends of the vectors will always be on the straight line r (Fig. 4) perpendicular to DO and, so, parallel to the directions oii, and &im. Hence, the vector is always in the region (oii, &im) except for VA, > 0. The result obtained has a simple physical meaning. An infinitely small amount of an alloying addition will lead to a certain loss in stability of the binary phase (the ratio of the number of atoms of the binary phase to that of the alloying additive Ngh/Nadd + m), only if the atomic interaction potential between the alloying additive and the phase is infinitely large.
/
Blim
Fig. 4. Possible directions of the vector R in the three-component of the vector must be on straight line r.
ordered phase. The end
219
Thus, the five characteristic directions ((Y, (Xiim,/3,Piim, E) divide a set of all possible directions into four subsets: (aiim, a) is the region of the atomic distribution of an alloying addition in sublattice Ai; (a, E) is the region of the atomic distribution of an alloying addition between sublattice A and B with the preferential filling of sublattice a as R gets closer to cx; (E, 0) is the analogous region but with the preferential filling of sublattice B; (p, /3iim)is the region of the atomic distribution of an alloying addition in sublattice B. Up to this moment we have discussed the atomic distribution of an alloying additive between sublattices. It is to be noted, however, that the introduction of an alloying additive can lead to the atomic redistribution of the main components of the original phase. For example, if atoms C can fill only sites A and the total amount of elements A and C is greater than the stoichiometric concentration, a deficiency in atoms B at sites B will be filled by atoms A. Here again the vector diagram enables us to draw a definite conclusion about the atomic distribution of all the elements. For example, if the response vector of the system is in the region (O(lim,a) and is not coincident with CY,the alloying additions can fill only sites A and any deficiency of atoms in sublattice B will be filled by element A, but if the response vector is in the region (a, E), C atoms will primarily fill sublattice B in the case of a deficiency in atoms B. The choice of the space of LRO parameters with two dimensions T)A, 7)B is not indispensable. It is clear that the result will be the same if we choose two other LRO parameters with a different normalization. As an example, Fig. 5 presents the diagram for a space with dimensions qA and qc. The parameter qc is normalized such that qc = 1 when atoms of element C replace all atoms of element A. In this case, however, the analysis is no
Fig. 5. Characteristic and 7)A.
directions
of the vector
R in the space of the LRO parameters
qc
280
longer straightforward and it is more convenient to work with the space of LRO parameters corresponding to the main phase elements. The expression for the response vector in the space (nA, Q) will take the form
R S(q.0)
=
-apr7(%iV*m+ iiCVACl3)
References S. V. Vinsovsky, Yu. A. Izyumov and E. Z. Kurmaev, Superconductivity of Tradition deter and their Alloys, Nauka, Moscow, 1977, p. 387. R. S. Rawlings, A. E. Staton-Bevan, The alloying behaviour and mechanical properties of polycrystalline Ni+.l (y-phase) with ternary additions. J. Mater. Sci., 10 (3) (1975) 505 - 514. M. A. Krivoglaz, To the theory of highly ordered and weak solid solutions, SOU. J. Phys. Chem., 31 (9) (1957) 1930 - 1942. A. G. Khachaturyan, Theory of Phase Trcmsformations and the Solid Solution Structure, Nauka, Moscow, 1974, p. 384.
Appendix A The ordering energy of a ~~lti~orn~o~e~t phase wits two types of site Let us generalize the expression for the ordering energy of the binary solid solution obtained in ref. Al to the case of a multicomponent phase. We assume that n kinds of atoms are distributed somehow in the Ising lattice {R} with the corresponding occupation numbers C,(R). In the absence of vacancies one of these become dependent on the other numbers
A-1.
i
C,(R) = 1
(Al)
i=l
We average the occupation numbers over the Gibbs ensemble and introduce the single-particle functions
which represent the probability of finding the ith kind of atom at site R. Naturally, the condition in eqn. (Al) holds true for these functions as well. In the self-consistent field approximation the hamiltonian of the multicomponent system can be recorded in the following form H’“’ = ~ I: R,R’
~
Vfj(R, R’) Ci(R)Cj(R’)
= Hlo(“’ + H,“’
i, j=l
where Ha(“) is the confi~ration~ly independent part of the hamiltonian and Hi(“) is the config~ation~y dependent part. Using the condition (A) Hi(n) can be rewritten in the form i
28;
$ 2 “ilVijn(R,
Hl(“) =
R')Ci(R)Cj(R')
R,R'i, j=l
where V,j”(R,
R’) =
R’) + V,n(R’,R’) -
Gij(R,
Vi”(R,
R’) - Vj,(R,
R’)
Averaging over the Gibbs ensemble gives the expression for the ordering energy Au
=
(HI’“‘)
ni1Vij”(R,
2
= i
R’)ni(R)nj(R’)
(AZ)
R,R' i,j=l
The single-particle functions will be defined as n@)
= ci + ‘I)&?)
(A3)
where Ci is the ith element concentration and vi is the LRO parameter. The coefficients E@) are normalized such that in the highly ordered state vi = 1 for the element filling sites A (the original phase A,B,) and qj = -1 for the element filling sites B. ei(R = ii> =
_!?_.=
k = B> = - _ k+m
E#
a;
k+m
=P
(A4)
However, the coefficients
E(R) (they are chosen equal for all elements: can be represented as a superposition of static concentration waves
eI(R)
= e,(R)
E(R) =
= . . . c,_,(R)
= E(R))
exp(ik,R) $~Cr, m
+ 7:
exp(--ik,R)}
(-45)
where k, is the superstructure vector and the index m takes on all values throughout the rotation of this vector (12,); yrn are the normalization factors. Taking into account eqn. (A5) and also the fact that in the pairing interaction approximation V(R, R’) = V(R - R’), we obtain zV(R,
R’)@)
= V(k,)e(R)
(A6)
Using this result and substituting eqn. (A5) into eqn. (A2) we arrive at the following expression for the ordering energy AU=
1
n-l
2Nk + m)
Z C vijn(ks)E2(RhiTlj R i,j=l
Or, taking account of the definition in eqn. (A4) AU = io$
nzl Vijn(ks)qiqj
(A7)
i,j=l
Reference Al
A.
G.
for Appendix
A
Khachatwyan, Theory Structure, Nauka, Moscow, 1974,
of Phase
p. 384.
Transformations
and
the Solid
Solution
282
Appendix B
B.1. The set of tmnsce~denta~ equations for LRU parameters in the multicomponent phase on the basis of AkB, with two types of site The sought equation is obtained in the statistical theory of ordering by minimizing the free energy. In the seIf-consistent field approx~ation the LRO parameters affect only the ordering energy AU and the entropy S AF=AU-TS
W)
Further, we use the definitions (Al), (A3), (A4) and the expression for the ordering energy (A7). First we take into account the definitions (A3) and (A4) and obt&n the entropy c
nj(R) Inni
+
in the form s= k* +
C&(G
+ a~?~)InG + Wi) + mWi -P77i)
{I-m(Ci
- WG
-@%I
+
ocrli))
In{1
WG
-mm(Ci
-Prli)
-Prli) -k(Ci
-+
an7i)Il 032)
Then, substituting eqns. (B2) and (A7) into eqn. (Bl) and minimizing the resulting equation with respect to each qi, we eventually obtain the set of n - 1 equations
1 l-
In
n-1
C
(Cj
-W?ji)
(Ci
n .-.1 1 - 2 (Cj 1 j= 1
-P?li)
i
j=l
+
&j)
t
= K$ ‘zi Vijn tk8 hj (Ci
+
@Vi)
j-l
033)