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Order-disorder transformations in a generalized Ising alloy
Physica 64 (1973) 571-586 0 North-Holland Publishing Co.
ORDER-DISORDER
TRANSFORMATIONS ISING
IN A GENERALIZED
ALLOY
C. M. VAN BAAL Laboratorium voor Metaalkpnde, Technische Hogeschool De&t, Delft, Nederland
Received 29 June 1972
Synopsis The Kikuchi approximation is applied to the order-disorder transformations in an fee binary alloy that is assumed to conform to a generalized Ising model. This model takes formally into account all interactions between 2, 3 and 4 nearest neighbours. The results show: (1) that the use of the Kikuchi approximation results in a more realistic phase diagram compared to previously used approximations; (2) that the inclusion of 3- and 4-particle interactions of a moderate strength compared to that of the usual pair interaction suffices to account for most of the variations shown by actua1 alloy systems. This result lends considerable support to the belief in the Ising model as a valid first approximation for such systems. 1. Introduction. It is well known that the Ising model if used to describe the thermodynamical behaviour of a binary alloy shows serious deficiencies. Disregarding the fact that the model is limited to a fixed lattice, the most important of these shortcomings is that only interactions between nearest-neighbour pairs
of atoms are taken into this limitation is, as the very well able to make pragmatic view, one can
account. It is, however, theory of concentrated reliable predictions in say that the Ising model
difficult to estimate how serious alloys at the present time is not this respect. But taking a more in so far has been successful that
it gives an essentially correct description of the two most prominent types of behaviour found in binary alloys: decomposition and ordering. Therefore one is intuitively inclined to regard the Ising model as a first approximation, to ascribe the many details of its predictions that do not agree with experiment to its sirnplicity, and to expect that the inclusion of higher-order interactions will give improved results. It is clear, however, that it must be pdssible to achieve this improvement by rather modest means or the idea of a first approximation will lose its sense. Another deficiency of the Ising model is that, at least in three dimensions, only approximate solutions are known. This makes it difficult to decide if the differences that are found between approximate solutions and experimental observations 571
512
C. M. VAN BAAL
must be ascribed to the poor quality of the approximations or to the limitations of the model. From the many investigations that have been made in this direction (and which have been reviewed by Dombl)) so much can be said, however, that the shortcomings of approximate solutions are serious only in the neighbourhood of the critical point. Large discrepancies between (even approximate) solutions of the Ising model and experimental facts that have no relation to critical phenomena therefore must be ascribed to the model. Although they do not seem to have received much attention, at least two important discrepancies of this kind are obvious: (1) the Ising model can, on a reduced temperature scale, only give one phase diagram, whereas actual phase diagrams show a marked individuality; (2) the model is strictly symmetric with respect to its two constituent elements and thus cannot account for the asymmetry that is apparent in all known phase diagrams. Therefore, if one still assumes that the Jsing model is a valid first approximation and if one wishes to improve the correspondence between theory and experiment, one will have to generalize the model by the inclusion ofhigherorder interactions. Several authors have already suggested such an extension of the Ising model, but it seems that only in Cowley’s theory2) this suggestion has been realized. In this theory interactions between pairs of more distant atoms than nearest neighbours are taken approximately into account. But although in several ‘cases, notably that of Cu3Au, this seems to give an improved agreement with experimental observations (for pertinent references see the review by Guttman3)) this extension by its nature will not be able to account for the asymmetry that is also present in the Cu-Au system. In fact no modei of a binary alloy that incorporates only pair interactions will ever result in an asymmetric phase diagram. To achieve this one has to take into account interactions between at least three particles. In recent times an approximation to the solution of the Ising problem has been proposed by Kikuchi4) and shown to lead to considerable improvement over previous approximations if applied to a ferromagnet. Besides this, Kikuchi’s method is almost naturally suited to incorporate higher-order interactions, both between more particles and between pairs of more distant particles. It is the purpose of this paper to show that this approximation, in conjunction with a rather small admixture of more-particle interactions, at least for order-disorder phenomena on an fee lattice leads to much more satisfactory phase diagrams of binary alloys than previous approximations. 2. General considerations. The Kikuchi approximation is a general method of finding solutions to the Ising problem. A very clear and detailed exposition of the method has been given by Hijmans and De Bee?) and we will follow as closely as is practical their nomenclature and notation. As far as is known, however, the Kikuchi method has not previously been applied to order-disorder problems. Now order-disorder problems (and antiferromagnetism) are in so far more
ORDER-DISORDER
complicated magnetism)
IN A GENERALIZED
ISING ALLOY
573
than the corresponding decomposition problems (and ferrothat in the former one has to subdivide the crystal lattice into two or
more sublattices. This subdivision the different configurations of the interrelations in some detail. This discuss a few choices that have to
entails a greater number of variables to describe crystal and it will be necessary to expose their will be done in the next section. Here we shall be made beforehand.
The kind of subdivision one chooses depends for a large part on the type of ordered structure one wishes to study. So in an fee lattice a subdivision in two sublattices, each consisting of the lattice points in alternate (100) planes, is necessary to describe the CuAu (E Ll,) structure, another subdivision in two sublattices, now consisting of the lattice sites in alternate (111) planes has to be used for the CuPt (Z Ll 1) structure, whereas a subdivision into four simple cubic sublattices (of which three are equivalent) must be used for the Cu,Au (G LlJ structure. All these subdivisions are specializations of a subdivision of the fee lattice into 8 fee sublattices and in this way could be treated together. But as this paper will present results only for the CuAu and Cu3Au types of order, we have chosen a subdivision into four sublattices to expose the general scheme. A second and more important choice one has to make from the start is that of the basic cluster of lattice sites that plays the principal part in the Kikuchi approximation. It has been shown by Kurata ef af.4) that the choice of a larger cluster in general (but not always) leads to a better approximation. On the other hand the labour involved in the calculations grows about exponentially with the TABLE I
Curie temperature of an fee Ising ferromagnet Size of cluster Mean field Quasi-chemical Kikuchi Domb
1 2 4 -
T, 1.ooo 0.914 0.835 0.816
cluster size. We have chosen a cluster of 4 lattice sites, together forming a regular tetrahedron, on the basis of the following considerations. It turns out that, depending on the type of order, a tetrahedral cluster will lead to a set of 6 or 7 equilibrium equations. One can estimate that a somewhat larger cluster, one consisting of 2 tetrahedra, e.g., would have> resulted in a set of about 25 more complicated equations. Although not impossible, the solution of such a set would seem to present a formidable task. On the other hand the resulting gain in accuracy probably would not have been very impressive, as might be judged from table I. This gives the Curie temperature T, (in reduced units and taken from Domb’))
574
C. M. VAN BAAL
of an fee lsing ferromagnet,
calculated
by means
of successively
higher
approxi-
mations. Now the Curie temperature can be taken as a fairly sensitive measure of the quality of an approximation and the Kikuchi approximation is seen to differ here only by about 2 % from the value found by Domb from a high-temperature series expansion which probably is exact within 0.1 ‘A. To this must be added the fact that in all cases we have calculated it has been found that the free energy of the disordered state becomes equal to that of the ordered state at a temperature below the critical temperature (here taken as the temperature at which the ordered state becomes inherently unstable). And it is well known that in the Ising problem approximations are better the farther away one is from the critical temperature. In a recent paper Kikuchi and Brush6) have shown that for a two-dimensional ferromagnet the choice of a “one-dimensional” basic cluster in general gives better results than that of a closed “two-dimensional” atoms. They expect that an analogous result will lattices. The use of such a cluster, however, would 4-particle interactions. Concluding, one can say that a tetrahedral basic compromise between accuracy and generality on possibility on the other.
one of the same number of hold for three-dimensional make it difficult to include cluster seems to form a fair the one hand and practical
3. Definitions. We assume that the fee iattice has N lattice sites of which cON are filled with A atoms and c,N with B atoms. The lattice is thought to be subdivided into 4 similar simple cubic sublattices, 1abelIed a, b, c, d. In an ordered state the numbers of A and B atoms on these sublattices in general will be different. We call the fraction of the sites on sublattice k which in a certain state are occupied by A atoms cOk, the fraction occupied by B atoms elk. Evidently one has: COk
+
Clk
1,
=
c,+c,
~&,,=4cO. ;
=I,
(1)
A bond between nearest-neighbour sites connecting a site on k with one on I is called a kl bond. The number of kl bonds is N and, as there are six sets kl, there are in total 6N nearest-neighbour bonds. A number of kl bonds in a given configuration of the lattice connects two A atoms (called an AA pair) and this number is given byp,,,N. In the same wayp,,, is the fraction of kl bonds connecting two a B atom on k and an A atom on I and B atoms, Plkt the fraction connecting p, ,k that of bonds between a B atom on I and an A atom on k. In generalp, k, # p1 Ik. As an A atom on k must belong to either an AA pair or an AB pair, we have for all sets kl: POkl
+
Pllk
=
COk>
Plkl
+
P2kl
=
Clke
(2)
ORDER-DISORDER
There
IN A GENERALIZED
are 2N sets of 4 lattice
that together
form a regular
sites, connected
tetrahedron.
ISING ALLOY
by 6 nearest-neighbour
Every tetrahedron
575 bonds,
has one of its sites
on each of the 4 sublattices. The fractions of the 2N tetrahedra occupied by 4 A atoms and 4 B atoms are called t,, and t,, respectively. The tetrahedra with 3 A atoms or 3 B atoms and 1 atom of the other kind each can be of 4 different types. They are labelled according to the sublattice on which the minority atom is situated and their fractions are given by t Ik and tJk, respectively. Finally there are 6 types of tetrahedra with 2 A atoms and 2 B atoms. These are labelled after the sublattices occupied by the two B atoms and their fractions are given by the 6 numbers tzkl. An AA pair with atoms on sublattices a and b is a part of 2 spatially different tetrahedra. Fixing attention on one of these, this must be either a tetrahedron with 4 A atoms or one with 3 A atoms and its B atom on c or d, or one with 2 A atoms and 2 B atoms with the latter on c and d. From this we find: POab
=
tO
and 5 similar we find:
+
tl,
+
tld
equations
Plab
=
tl,
P2ab
=
t2ab
+
+
+
(3)
t2cd,
for the other possible
t2ac
+
t2ad
t3c
+
t3d
+
+
f3b,
f4,
P]ba
combinations
=
tlb
+
t2bc
kl. In the same way
+
t2bd + t&, (4)
together with 15 other equations. With the help of (2), (3) and (4) we can express all pair and atom fractions in the 16 tetrahedron fractions and, as in an alloy the overall composition is fixed, we can with the help of (1) again express 2 of these 16 fractions in terms of the 14 remaining ones. It seems natural to take for the 2 dependent fractions to and t4. The 14 independent fractions then are the variables in which we shall try to express the (nonequilibrium) free energy of the alloy. 4. Free energy. To find the free energy of a certain state of the alloy, specified by given values of the 14 independent variables, we must calculate the energy and the entropy. For the energy we assume that we can ascribe to each tetrahedron, pair or atom in the lattice a certain energy. This energy will be assumed to depend on the kind and number of the atoms that make up the cluster, but not on the sublattices on which the atoms happen to be situated. Thus there are 5 different tetrahedron energies e: (i = 0, 1, 2, 3, 4 indicates the number of B atoms of the tetrahedron), 3 different pair energies eP (i = 0, 1, 2) and 2 different atom energies ef (i = 0, 1). We now form the sum: U, = 2N c t,,e:, i.m
C. M. VAN BAAL
576
(where here and in the following summation over m means a summation over all different values the lattice labels may assume). U, is not the correct value of the total energy. Each pair in the lattice is a part of two tetrahedra and so in (5a) the energy content of the pairs is counted twice. Also each atom belongs to 8 different tetrahedra and thus the energy of the atoms is contained eightfold in U, . To correct for this we must subtract
from U, the pair energy
U,:
UP = Nxp,,e;, i,m
(5b)
and 7 times the atom energy
U,:
Ua = $N 1 ci,el.
(5c)
i.m
But even then the correct energy has not been found because each atom is a part of 12 pairs and thus, in correcting for the pair energy, 12 times the atom energy has been incorrectly subtracted. So finally we have for the total energy U: u = u, -
UP + 5u,.
(6)
with U,, UP, U, given by (5a), (5b) and (5~). Now this seems to be a rather roundabout way to calculate a quantity that could have been found in a much more direct way. But Hijmans and De Boer’) have shown that the same “overlap correction numbers” (- 1 and + 5) that we have found here, appear in the expression for the entropy of the Kikuchi approximation. But before calculating the entropy we wish to simplify (6). Writing: ,m = 4t, ,
P
and making u =
1 t2nz = et, 3 m
use of (l)-(4),
240
-
we directly
c ts,, = 4t, 2 m find for the energy
per atom u = U/N:
12 [(l + a) t, + 2t, + (1 + /?) t3] E,
(7)
in which: E = J(-2ei
+ 4ey - 2e; + e\ - 2ei + ek),
(1 + LX)E =6(-3eE
+ 6eg - 3ei + 3eb - 4ei + ek), (8)
(1 +/3)E=t(-3eg+6ey-3e;+eb-4e\+3ek); UO
=
5e”, - 6e: + 2ek + co (se”, - 5e”, - 6eP, + 6e’: + 2eb
2ek).
ORDER-DISORDER
For an alloy of constant determination
IN A GENERALIZED
composition
of the equilibrium.
ISING ALLOY
u0 is a constant If one assumes
577
and plays no part in the
that
only
nearest-neighbour
pair interactions are important (we shall call this the pure Ising 01 = /j = 0, whereas E reduces to the usual expression:
case) one has
in which VA, = eg - 2et, is the excess energy of an AA pair over that of 2 A atoms, with analogous definitions for V,, and VA,. For ordering alloys E > 0. In LYE.BE are comprised all the 3- and 4-particle interactions between nearest neighbours that could possibly be present in an fee binary alloy. It has already been pointed out in the introduction that their influence on the resulting orderdisorder transformations will be a good measure of the validity of the Isingmodel as a model for such an alloy. Considering now the entropy, one can repeat step for step the reasoning of Hijmans and De Boer for this case, coming to the same result: in the Kikuchi approximation the entropy S can be written, just like the energy, as the sum of a “tetrahedron entropy” S, , a “pair entropy” S, and an “atom entropy” S,, with the same overlap correction numbers : s = s, - s, + 5s,. S,, S, to the by the entities comply
(9)
and S, are found by applying the well-known Boltzmann entropy formula tetrahedra, pairs and atoms as independententities. So they are determined number of ways in which given numbers of the different types of these can be distributed over the available places. Of course these numbers must with the relations (l)-(4). We thus find:
S, = -2Nk
C ti,,,In
tirn,
i.m
SD= -Nk i,m CpirnInpi,,
(10)
sa= - Wki,m C cjmIn tin,. From (7), (9) and (10) we finally find for the free energy per atomf dropping the constant u0 :
= (U - 2X)/N,
.f = - 12 [(1 + ol) tl -t 2t, + (1 + p) t3] E + kT C (2ti, In ti, - pim lnp,, i,m
+ icim In cim).
We have already shown that (11) contains 14 independent with respect to these will give us 14 equilibrium equations
(11) variables. Minimizingf and from their solutions
578
C. M. VAN BAAL
we could find the most general phase diagram that would approximation. We shall not execute this large programme,
be possible in this however, but limit
ourselves to a few structures that appear to be the most common. In a disordered alloy, including alloys with only short-range order, there is no distinction between the sublattices and so we have: tl, = tlb = t,, = tld, r2ab = f2ac =
t2ad
=
f2bc
=
t2bd
=
f2cd,
(12)
t& = t3b = tsc = t3d. If, on the other hand, one or more of these equalities are violated we must have an ordered state. Now, instead of analyzing the numerous possibilities there obviously are, we will follow Shockley7) and classify the ordered states according to the way in which the different sublattices are occupied. Shockley finds that there are in this way 4 possible types of ordered structure: (1) the cubic Cu,Au structure in which 3 sublattices are equally and preferentially occupied by A atoms and the fourth sublattice contains an excess of B atoms; (2) the tetragonal CuAu structure in which 2 sublattices are equally and preferentially occupied by A atoms and the other 2 also are equally occupied but now preferentially by B atoms; (3) a second tetragonal structure in which again 2 sublattices are equally occupied, but the remaining two each have a different occupation; (4) an orthorhombic structure in which all 4 sublattices have a different occupation. The last 2 of these possible types of order do not seem to have been observed experimentally. We therefore have limited our calculations to the first named two structures, although it would have been interesting to investigate if the fact that the other two have not been found could perhaps be explained by the fact that they are less stable also in our model. The specialization of eq. (11) to the Cu,Au, the CuAu and the disordered states is easy and we shall not give it in detail. One finds that for the Cu,Au structure there remain 6 independent variables, for the CuAu structure 7 and for the disordered state 3. Minimizing the resultant expressions for the free energy thus leads to 6, 7 and 3 equilibrium equations that have to be solved to find the equilibrium values of the independent variables and hence the energy, the entropy and the free energy. The numerical methods by which we have solved the three sets of equilibrium equations and some of the difficulties we have met are described in the appendix. 5. Results. By the methods described in the appendix the three sets of equilibrium equations have been solved in the appropriate concentration and temperature regions for three sets of values of the energy parameters (a, b), namely (0, 0), (0, 0.2) and (0, -0.2). These sets also include the sets (0.2, 0) and (-0.2, 0) as the latter correspond to a simple interchange of A and B atoms.
ORDER-DISORDER
It has already of an ordered
IN A GENERALIZED
been mentioned
state becomes
ISING
that we always have found
579
ALLOY
that the free energy
equal to that of the corresponding
disordered
state
at a temperature below the critical temperature (taken here as the temperature above which the ordered state ceases to be one of minimum free energy). From general thermodynamics we know that in such a case the temperature T, at which both free energies are equal (for brevity we will call this temperature the equality temperature) for only one composition coincides with the actual transformation temperature T,. For all other compositions one has a temperature region in which the homogeneous alloy is not stable and separates in 2 phases of different compositions. The boundaries of this 2-phase region can be found from the points of tangency of the common tangent to the free-energy curves of both phases plotted as functions of composition at a fixed temperature. For the pure Ising case (a = /3 = 0) we have determined these boundaries betwRen each two of the three phases for which the free energy has been calculated. The result is shown in fig. 1 (where only the left half of the complete phase diagram has been given because of the symmetry between A and B atoms that is inherent in the pure Ising model). The most significant and new feature of this diagram is the three-phase equilibrium found at kT/E M 0.81 and the consequent two-phase equilibrium between the Cu,Au and CuAu ordered phases at lower temperatures. It must be mentioned that although the three-phase equilibrium in fig. 1 has been drawn as if it were of the eutectoid kind, our calculations have not been I
I
I
/
I
L
,
0.35
0.40
0.45
a=@=0
100
0.90 -
0.80 -
0.70 -
t W”
-
kT/E
010
0.15
a20
0.25
0.30
0.50
Fig. 1. The phase diagram of an ordering pure Ising fee alloy in the Kikuchi approximation (CC)indicates the disordered state.
580
C. M. VAN BAAL
detailed enough to decide that this actually was the case or whether it was rather a peritectoid. A distinction between the two, however, did not seem important enough to warrant a more thorough investigation. Fig. 2 shows the results for the generalized Ising model for o( = 0, p = 0.2 and DL= 0, p = -0.2 (for comparison those for the pure Ising case have also been indicated). Here we only give the equality temperature T, as a function of composition because it is felt that this would make the phase relationships sufficiently 1
1.6Oi-
I
a=0
I -
1.40
p z-o.2 .., p.0
----
/ //
p-0.2
/
fl\
\
\ \ \
Cl I 0.10
0.20
I 0.30
I 0.40
I 0.50
I 0.60
I 0.70
I 0.80
090
Fig. 2. The equality-temperature diagram (see text) of a generalized king alloy for two values of the higher-order energy parameter B (the dotted line indicates the pure Ising case).
Most prominent here is the sensitivity of the equality temperatures to variations in the strength of the higher-order interactions. For although the direction of the influence of these interactions could have easily been predicted from their definition in (8), it is somewhat surprising that a relative strength of these interactions of 0.2 produces the large changes depicted in fig. 2. Another interesting point is that for B = -0.2 the AB3 phase does not have a maximum transformation temperature in the neighbourhood of the stoichiometric composition. Experimentally this behaviour has been found in CuAu3 for compositions between 25 and 32 % Cu*). clear.
ORDER-DISORDER
Finally
IN A GENERALIZED
we give in table II the values
total heat of ordering AU,jNkTe for stoichiometric compositions.
ISING
of the latent
[in which
581
ALLOY
heat AU,/NkT,
AU, = U (T = a)
and of the -
U (T = 0)]
TABLE II Values of latent heat and total heat of ordering
A lJ,/NkTe
A U,/NkT,
A3
AB
Ah
A3
AB
ABA
p = --0.2
0.2253
0.3691
0.1925
0.8015
0.9896
1.0134
p=o
0.2371
0.2656
0.2371
0.7794
1.0563
0.7794
B = 0.2
0.2490
0.2984
0.2637
0.7590
1.2658
0.7302
6. Discussion and comparisons. It will have been noted that we have not given results for low temperatures. We have refrained from doing this mainly for the following reason. As a model of an alloy the Ising model (and also our generalized model) is a rather crude one. Even if theory in the future will provide a sound basis for the model, one has to expect that many refinements will have to be incorporated before a sizeable portion of all binary-alloy systems can be described adequately. Now at lower temperatures the free energy is to a larger extent determined by the energy, and deficiencies in the description of the energy wilI become more and more pronounced. This means that the model at low temperatures will lose the contact with reality that it might have at higher temperatures. It seems therefore pointless to perform calculations of a detailed nature in this low-temperature region before a better founded and more sophisticated model is available. The most complete results which were obtained earlier and can be compared to the present ones are due to Shockley7). Shockley made use of the BraggWilliams approximation and he found that in this approximation both the CuAu and CuBAu structures formed stable phases. It was disappointing, however, that both phases had their (coinciding) maximum transformation temperatures at c = 0.50, a result which for the Cu,Au structure is at variance with all experimental findings. This situation was not improved by the work of Peierls and its extension by Easthopeg), who applied the method of Bethe to the C+Au case. Easthope could only show that for compositions with c > 0.25 the critical temperature rose with the concentration, thus confirming partially the previous result of Shockley. Yang and Li”), using an approximation that seems to be intermediate between the Bethe approximation (or its equivalent, the quasichemical approximation) and the Kikuchi approximationl), were the first to find 3 distinct maximum transformation temperatures at c = 0.255, c = 0.50 and c = 0.745, respectively.
582
C. M. VAN BAAL
Li, however, did not find a 2-phase region between the CuAu and Cu,Au ordered phases. A more quantitative comparison is given in table III. This gives, for alloys of stoichiometric compositions, the critical temperature T, or the transformation temperature T, calculated by means of the various approximations mentioned. Also included is a value found by Fosdick as the result of a Monte Carlo calculation performed with a lattice of 500 sites. In a comparison with the exact results available for a 2-dimensional square ferromagnet this method has been shown to be accurate to within a few percent13). The agreement between Fosdick’s value and that of the Kikuchi approximation presents a new argument, next to that given in section 2, in favour of this approximation. It must be mentioned, however, that in Fosdick’s results the transition from the ordered to the disordered state seems to be more nearly continuous than in all other approximations. Probably this is due to the fact that the Monte-Carlo method, being essentially a kinetic method, in the neighbourhood of a critical point has the same difficulty in reaching equilibrium as actual thermodynamic systems experience in this region. TABLE III
Critical (T,) or transformation
Shockley’) (Bragg-Williams) Easthopeg) (Bethe-Peierls) Schapink’l) (Quasichemical) Yang, Li’O) Fosdickl’) (Monte Carlo) Present work (Kikuchi) Present work (Kikuchi)
(Tt) temperatures
T. T, T, Tt Tt T, T,
in units of kT/E
Cu3Au
CuAu
1.642 0.8911
2.000 _ 1.7869
0.8228 Z1.0 0.9967 0.9623
0.7306 _ 0.9718 0.9467
Several authors have estimated that the influence of three-body interactions on the energy of the fluid or solid state of the rare-gas elements is appreciable (see the review by Scott14)), but as far as is known more-particle interactions have not been considered in the theory of alloys. Also no previous Ising-model calculations exist with which we could compare our results. Fosdick, however, also applied the Monte-Carlo method to a 25 % alloy in which interactions between pairs of next-nearest neighbours were included. His results are so far in good agreement with those reported here, that he also finds that the influence of higher-order interactions on the transformation temperature is large : if the ratio of the strength of the next-nearest neighbour interaction to that of the nearest neighbours is -0.25, the reduced transformation temperature is raised from 1.0 to 1.6. This is quite comparable to our result, where a change from /3 = 0 to /3 = 0.2 raises this temperature from 0.96 to 1.50.
ORDER-DISORDER
IN A GENERALIZED
ISING
ALLOY
583
Many authors have compared their results on the Tsing model others with the CuAu system. So in a recent review of order-disorder
or those of phenomena
in metals3), Guttman compared the phase diagram calculated by Li’O) with this alloy system, noting the following discrepancies : 1) The maximum transformation temperatures of the 3 ordered phases Cu3Au, CuAu and CuAu, were calculated to be about equal, whereas actually that of CuAu, is more than 20% lower than those of the other two. 2) .Li finds that at no temperature the phases Cu,Au and CuAu are in equilibrium, whereas in fact such an equilibrium has been observedl’). 3) An orthorhombic phase, called CuAu-II (= Llo_J exists that has not been found in the calculations. To this could be added the already mentioned, more recent result of Battermans:), who found that for &Au, the transformation temperature for Cu concentrations above 25 % is higher than that for the 25 % alloy. One can understand a priori that phases with large unit cells like CuAu-II can never be found in a model with only nearest-neighbour interactions. With respect to the other discrepancies, however, it follows from figs. 1 and 2 that with a judicious choice of the energy parameters 01 and p they can be made to disappear. A survey of the other alloy systems listed by Guttmann, in which phases of the CuAu or Cu,Au type have been observed and for which details are given in the reference works of Hansen-Anderko, Elliott and ShunkId) shows that these, with a few exceptions, present no essentially new features compared with the Cu-Au system. The variations in transformation temperature that have been found all fall within the limits shown in fig. 2 and so a range of about -0.2 to 0.2 for both 01 and p is capable to account for these variations. Exceptions are Cu,Pt, FePd and FePd3, for which maximum transformation temperatures of the CuAu or Cu,Au structures have been reported at 17 % Pt, 58 % Pd and 66 % Pd, respectively. Such large deviations from stoichiometry do not appear to be covered by our model. In spite of the sketched general agreement, however, it follows from what has been said in the first part of this section that we do not believe that further work in this direction is urgently required. We would rather wish to conclude from this agreement that the Ising model appears to be a rather good first approximation as a model for a binary alloy, and that a relatively moderate inclusion of moreparticle interactions seems to be capable to account for many of the discrepancies between predictions of the model and the behaviour of actual alloys. Theories of alloys ought to reflect these facts. The values given in table II for the latent-heat and for the total-energy change of the transitions are comparable with those of other approximations. Like the latter they are too high by a factor 2 to 4 compared to the few experimental values available3). The cause of this discrepancy is not clear. Experimental values have
584
a tendency differences.
C. M. VAN BAAL
to be too low, but it seems doubtful
that this could explain
7. Conclusions. 1) The generalized Ising model, in which actions between nearest-neighbour atoms on a face-centered
the large
all possible intercubic lattice are
included, appears to be capable to account for nearly all the experimentally known details with respect to the phase diagram of ordering fee binary alloys. 2) As the strength of the 3- and 4-particle interactions in this model that are needed to achieve this agreement, appears to be moderately small compared to that of the nearest-neighbour pair interactions, it follows that the Ising model in all probability is a valid first approximation as a model for such alloys. 3) The Kikuchi approximation that was used to calculate several phase diagrams for the generalized Ising model, here as in the ferromagnetic case, appears to offer a substantial improvement compared to other approximations. Acknowledgement. It is a pleasure to thank R.H.Duyts for the enthousiasm and ingenuity with which he carried through a part of the numerical labour.
APPENDIX
DetaiZs of the calculations.
The three sets of equilibrium equations that result from the minimization of the free energies can be written in the form of simultaneous aIgebraic equations of degrees up to 12. Most of the enormous number of possible solutions of these equations will not be acceptable, however, because the definitions of the variables tin,, pinl , cim imply that they are all real and restricted to the interval: (13) These constraints,
however,
offer an extra difficulty.
A general
and elegant
way
to solve minimization problems in the presence of constraints is that in which one makes use of so-called penalty functionsi7). In this method one adds to the function to be minimized (heref) a penalty function g of all the variables ,yi that are to obey a constraint of the type (13). g has the following properties: (I) it is positive everywhere; (2) it is small for values of si in the allowed interval and large elsewhere; (3) it contains a parameter q such that for q + 0, g --f 0. A possible function g for our case would be: g = q xi (1 - 2xi)‘“, with n a positive integer. For a suitably chosen initial value of q one now minimizes f + g without constraints, solves the resulting equations and uses the solution as the starting point for the solution of the same equations, but with a lower value of q. Iteration of this process with successively lower values of q will generally converge to the described solution. We have applied this method with the penalty function given
ORDER-DISORDER
above and with one suggested to find a suitable
IN A GENERALIZED
by Bracken
wa) to decrease
ISING ALLOY
and McGormickL7).
the penalty
parameter
585
After some trials
q, the method
was found
to be successful. In the mean time, however, another procedure was developed that took less computing time, So for the greatest part of the calculations the latter procedure was followed. Our real problem was, here as well as with the penalty-function method, to find a starting point. Once this was found, it was possible, by slowly varying the parameters in the equilibrium equations (temperature, 01, ,4), to obtain the other desired results. To find this starting point a Monte-Carlo method was used. Within rather wide but plausible ranges a set of random variables (6 or 7; for the disordered state the solution at very high temperature is known and can be started from there) was generated. It was then checked if all variables ti,, Qirn, cim calculated with this set met the constraint (13). If not, a partly new set was generated and again checked. When a succesful set was found the free energy was calculated. Then the process was repeated, the second free energy was compared to the first and if lower, kept, else discarded. After having done this a number of times (usually 50), the allowed ranges of the variables were centered around the best point found and reduced by a factor 1.2. The whole procedure then was repeated some 50 times. The final set of variables in all but a few cases proved to be accurate enough (deviating by less then 0.1 ‘A from the exact values) to serve as a starting point for a solution of the equilibrium equations. The solution itself in all cases was found by a straightforward extension of the NewtonRaphson tangent method’*). Quite another and initially very disturbing difficulty was encountered when we tried to solve the equilibrium equations for the CuAu structure. Our first trials for co = 0.5 all gave a solution that, if visualized, corresponded to a state of the crystal that is called a “domain structure”: different parts of the lattice are ordered in the same way, but with respect to different sublattices. So here in half the crystal the A atoms were situated on sublattice a, in the other half on sublattice c. Such structures are well known to occur in real crystals, they are in fact the only ones known. Evidently they are highly ordered, but in our case the Kikuchi approximation ascribes to this ordered state the very high entropy of nearly k In 2 per atom, making it a stable state up to very high temperatures. A careful examination of the derivation of the equilibrium equations failed to give a clue as to the cause of this evident anomaly. Turning to the next-lower approximation, the quasichemical approximation, we found to our surprise that the same state there has a large negative entropy. From this we must conclude that although approximations like the quasichemical or the Kikuchi approximation appear to give in general acceptable results, there are “corners” in which they are unreliable. The large and negative entropy found in the quasichemical approximation led us, however, to the idea of a stratagem by which we could hope to avoid the sharp
586 but undesirable
C. M. VAN BAAL
minimum.
As both
the entropy
in the quasichemical
approxi-
mation, which we shall call So, and that of the Kikuchi approximation (to be called Sk) in general are reasonable approximations to the true entropy, evidently a linear combination of the form (1 - w) So + IV&, 0 < w < 1, also will be a reasonable approximation. Now, as this “mixed” entropy up to MI = 0.8 is negative for the “domain state” we wish to avoid, this state in all probability will not correspond to a solution of the equilibrium equations derived from a free energy with this mixed entropy. So a solution of these equations could be in the neighbourhood of the true solution. We then could hope that by slowly increasing w up to the value 1, where we would be back at the Kikuchi entropy, this solution could be attained. This stratagem was found to work and even led to two ordered states that afterwards appeared to belong to two branches of the same solution. order paraAlong them c,,~ (which for c0 = 0.5 is identical with the long-range meter S as defined by Bragg and Williams) shows about the same behaviour as in Fowler and Guggenheim’s solution of the Bragg-Williams approximation for the Cu,Au structurelg). A difference is, however, that in our case also the lower branch corresponds to a local minimum in the free energy and so represents a possible (although metastable) equilibrium state of the model.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) JO) 11) 12) 13) 14) 15) 16) 17) 18)
Domb, C., Advances in Physics 9 (1960) 149. Cowley, J.W., Phys. Rev. 77 (1950) 669; 120 (1960) 1648; A138 (1965) 1384. Guttman, L., in: Solid State Physics (F. Seitz and D. Turnbull, eds.) 3 (1956) 146. Kikuchi, R., Phys. Rev. 81 (1951) 988; Kurata, M., Kikuchi, R. and Watari, T., J. them. Phys. 21 (1953) 434. Hijmans, J. and De Boer, J., Physica 21 (1955) 471, 485, 499. Kikuchi, R. and Brush, S.G., J. them. Phys. 47 (1967) 195. Shockley, W., J. them. Phys. 6 (1938) 130. Batterman, B.W., J. appl. Phys. 28 (1957) 556. Peierls, R., Proc. Cambr. Phil. Sot. 32 (1936) 477; Easthope, C.A., Proc. Cambr. Phil. Sot. 33 (1937) 502. Yang, C.N., J. them. Phys. 13 (1945) 66; Yang, C.N. and Li, Y. Y., Chin. J. Phys. 7 (1947) 59; Li, Y.Y., J. them. Phys. 17 (1948) 447. Schapink, F. W., Thesis Delft (1969). Fosdick, L.D., Phys. Rev. 116 (1959) 565. Fosdick, L.D., in: Alder, B., Fernbach, S. and Rotenberg, M., Methods in Computational Physics, Vol. 1, Academic Press (New York, 1963). Scott, R.L., in: Eyring, H., Henderson, D. and Jost, W., Physical Chemistry, Vol. VIIJA, Academic Press (New York, 1971). Rhines, F.N., Bond, W.E. and Rummel, R.A., Trans. A.S.M. 47 (1955) 578. Hansen, M. and Anderko, K., Constitution of Binary Alloys (1958), McGraw-Hill; Elliott, R.P., jr]., 1st Supplement (1965) id.; Shunk, F. A., id., 2nd Suppl. (1969) id. Bracken, J. and McCormick, G. P., Selected Applications of Nonlinear Programming, Wiley (New York, 1968). Whittaker, E. and Robinson, G., Calculus of Observations, Blackie (London, 1948).
ORDER-DISORDER
TRANSFORMATIONS ISING
IN A GENERALIZED
ALLOY
C. M. VAN BAAL Laboratorium voor Metaalkpnde, Technische Hogeschool De&t, Delft, Nederland
Received 29 June 1972
Synopsis The Kikuchi approximation is applied to the order-disorder transformations in an fee binary alloy that is assumed to conform to a generalized Ising model. This model takes formally into account all interactions between 2, 3 and 4 nearest neighbours. The results show: (1) that the use of the Kikuchi approximation results in a more realistic phase diagram compared to previously used approximations; (2) that the inclusion of 3- and 4-particle interactions of a moderate strength compared to that of the usual pair interaction suffices to account for most of the variations shown by actua1 alloy systems. This result lends considerable support to the belief in the Ising model as a valid first approximation for such systems. 1. Introduction. It is well known that the Ising model if used to describe the thermodynamical behaviour of a binary alloy shows serious deficiencies. Disregarding the fact that the model is limited to a fixed lattice, the most important of these shortcomings is that only interactions between nearest-neighbour pairs
of atoms are taken into this limitation is, as the very well able to make pragmatic view, one can
account. It is, however, theory of concentrated reliable predictions in say that the Ising model
difficult to estimate how serious alloys at the present time is not this respect. But taking a more in so far has been successful that
it gives an essentially correct description of the two most prominent types of behaviour found in binary alloys: decomposition and ordering. Therefore one is intuitively inclined to regard the Ising model as a first approximation, to ascribe the many details of its predictions that do not agree with experiment to its sirnplicity, and to expect that the inclusion of higher-order interactions will give improved results. It is clear, however, that it must be pdssible to achieve this improvement by rather modest means or the idea of a first approximation will lose its sense. Another deficiency of the Ising model is that, at least in three dimensions, only approximate solutions are known. This makes it difficult to decide if the differences that are found between approximate solutions and experimental observations 571
512
C. M. VAN BAAL
must be ascribed to the poor quality of the approximations or to the limitations of the model. From the many investigations that have been made in this direction (and which have been reviewed by Dombl)) so much can be said, however, that the shortcomings of approximate solutions are serious only in the neighbourhood of the critical point. Large discrepancies between (even approximate) solutions of the Ising model and experimental facts that have no relation to critical phenomena therefore must be ascribed to the model. Although they do not seem to have received much attention, at least two important discrepancies of this kind are obvious: (1) the Ising model can, on a reduced temperature scale, only give one phase diagram, whereas actual phase diagrams show a marked individuality; (2) the model is strictly symmetric with respect to its two constituent elements and thus cannot account for the asymmetry that is apparent in all known phase diagrams. Therefore, if one still assumes that the Jsing model is a valid first approximation and if one wishes to improve the correspondence between theory and experiment, one will have to generalize the model by the inclusion ofhigherorder interactions. Several authors have already suggested such an extension of the Ising model, but it seems that only in Cowley’s theory2) this suggestion has been realized. In this theory interactions between pairs of more distant atoms than nearest neighbours are taken approximately into account. But although in several ‘cases, notably that of Cu3Au, this seems to give an improved agreement with experimental observations (for pertinent references see the review by Guttman3)) this extension by its nature will not be able to account for the asymmetry that is also present in the Cu-Au system. In fact no modei of a binary alloy that incorporates only pair interactions will ever result in an asymmetric phase diagram. To achieve this one has to take into account interactions between at least three particles. In recent times an approximation to the solution of the Ising problem has been proposed by Kikuchi4) and shown to lead to considerable improvement over previous approximations if applied to a ferromagnet. Besides this, Kikuchi’s method is almost naturally suited to incorporate higher-order interactions, both between more particles and between pairs of more distant particles. It is the purpose of this paper to show that this approximation, in conjunction with a rather small admixture of more-particle interactions, at least for order-disorder phenomena on an fee lattice leads to much more satisfactory phase diagrams of binary alloys than previous approximations. 2. General considerations. The Kikuchi approximation is a general method of finding solutions to the Ising problem. A very clear and detailed exposition of the method has been given by Hijmans and De Bee?) and we will follow as closely as is practical their nomenclature and notation. As far as is known, however, the Kikuchi method has not previously been applied to order-disorder problems. Now order-disorder problems (and antiferromagnetism) are in so far more
ORDER-DISORDER
complicated magnetism)
IN A GENERALIZED
ISING ALLOY
573
than the corresponding decomposition problems (and ferrothat in the former one has to subdivide the crystal lattice into two or
more sublattices. This subdivision the different configurations of the interrelations in some detail. This discuss a few choices that have to
entails a greater number of variables to describe crystal and it will be necessary to expose their will be done in the next section. Here we shall be made beforehand.
The kind of subdivision one chooses depends for a large part on the type of ordered structure one wishes to study. So in an fee lattice a subdivision in two sublattices, each consisting of the lattice points in alternate (100) planes, is necessary to describe the CuAu (E Ll,) structure, another subdivision in two sublattices, now consisting of the lattice sites in alternate (111) planes has to be used for the CuPt (Z Ll 1) structure, whereas a subdivision into four simple cubic sublattices (of which three are equivalent) must be used for the Cu,Au (G LlJ structure. All these subdivisions are specializations of a subdivision of the fee lattice into 8 fee sublattices and in this way could be treated together. But as this paper will present results only for the CuAu and Cu3Au types of order, we have chosen a subdivision into four sublattices to expose the general scheme. A second and more important choice one has to make from the start is that of the basic cluster of lattice sites that plays the principal part in the Kikuchi approximation. It has been shown by Kurata ef af.4) that the choice of a larger cluster in general (but not always) leads to a better approximation. On the other hand the labour involved in the calculations grows about exponentially with the TABLE I
Curie temperature of an fee Ising ferromagnet Size of cluster Mean field Quasi-chemical Kikuchi Domb
1 2 4 -
T, 1.ooo 0.914 0.835 0.816
cluster size. We have chosen a cluster of 4 lattice sites, together forming a regular tetrahedron, on the basis of the following considerations. It turns out that, depending on the type of order, a tetrahedral cluster will lead to a set of 6 or 7 equilibrium equations. One can estimate that a somewhat larger cluster, one consisting of 2 tetrahedra, e.g., would have> resulted in a set of about 25 more complicated equations. Although not impossible, the solution of such a set would seem to present a formidable task. On the other hand the resulting gain in accuracy probably would not have been very impressive, as might be judged from table I. This gives the Curie temperature T, (in reduced units and taken from Domb’))
574
C. M. VAN BAAL
of an fee lsing ferromagnet,
calculated
by means
of successively
higher
approxi-
mations. Now the Curie temperature can be taken as a fairly sensitive measure of the quality of an approximation and the Kikuchi approximation is seen to differ here only by about 2 % from the value found by Domb from a high-temperature series expansion which probably is exact within 0.1 ‘A. To this must be added the fact that in all cases we have calculated it has been found that the free energy of the disordered state becomes equal to that of the ordered state at a temperature below the critical temperature (here taken as the temperature at which the ordered state becomes inherently unstable). And it is well known that in the Ising problem approximations are better the farther away one is from the critical temperature. In a recent paper Kikuchi and Brush6) have shown that for a two-dimensional ferromagnet the choice of a “one-dimensional” basic cluster in general gives better results than that of a closed “two-dimensional” atoms. They expect that an analogous result will lattices. The use of such a cluster, however, would 4-particle interactions. Concluding, one can say that a tetrahedral basic compromise between accuracy and generality on possibility on the other.
one of the same number of hold for three-dimensional make it difficult to include cluster seems to form a fair the one hand and practical
3. Definitions. We assume that the fee iattice has N lattice sites of which cON are filled with A atoms and c,N with B atoms. The lattice is thought to be subdivided into 4 similar simple cubic sublattices, 1abelIed a, b, c, d. In an ordered state the numbers of A and B atoms on these sublattices in general will be different. We call the fraction of the sites on sublattice k which in a certain state are occupied by A atoms cOk, the fraction occupied by B atoms elk. Evidently one has: COk
+
Clk
1,
=
c,+c,
~&,,=4cO. ;
=I,
(1)
A bond between nearest-neighbour sites connecting a site on k with one on I is called a kl bond. The number of kl bonds is N and, as there are six sets kl, there are in total 6N nearest-neighbour bonds. A number of kl bonds in a given configuration of the lattice connects two A atoms (called an AA pair) and this number is given byp,,,N. In the same wayp,,, is the fraction of kl bonds connecting two a B atom on k and an A atom on I and B atoms, Plkt the fraction connecting p, ,k that of bonds between a B atom on I and an A atom on k. In generalp, k, # p1 Ik. As an A atom on k must belong to either an AA pair or an AB pair, we have for all sets kl: POkl
+
Pllk
=
COk>
Plkl
+
P2kl
=
Clke
(2)
ORDER-DISORDER
There
IN A GENERALIZED
are 2N sets of 4 lattice
that together
form a regular
sites, connected
tetrahedron.
ISING ALLOY
by 6 nearest-neighbour
Every tetrahedron
575 bonds,
has one of its sites
on each of the 4 sublattices. The fractions of the 2N tetrahedra occupied by 4 A atoms and 4 B atoms are called t,, and t,, respectively. The tetrahedra with 3 A atoms or 3 B atoms and 1 atom of the other kind each can be of 4 different types. They are labelled according to the sublattice on which the minority atom is situated and their fractions are given by t Ik and tJk, respectively. Finally there are 6 types of tetrahedra with 2 A atoms and 2 B atoms. These are labelled after the sublattices occupied by the two B atoms and their fractions are given by the 6 numbers tzkl. An AA pair with atoms on sublattices a and b is a part of 2 spatially different tetrahedra. Fixing attention on one of these, this must be either a tetrahedron with 4 A atoms or one with 3 A atoms and its B atom on c or d, or one with 2 A atoms and 2 B atoms with the latter on c and d. From this we find: POab
=
tO
and 5 similar we find:
+
tl,
+
tld
equations
Plab
=
tl,
P2ab
=
t2ab
+
+
+
(3)
t2cd,
for the other possible
t2ac
+
t2ad
t3c
+
t3d
+
+
f3b,
f4,
P]ba
combinations
=
tlb
+
t2bc
kl. In the same way
+
t2bd + t&, (4)
together with 15 other equations. With the help of (2), (3) and (4) we can express all pair and atom fractions in the 16 tetrahedron fractions and, as in an alloy the overall composition is fixed, we can with the help of (1) again express 2 of these 16 fractions in terms of the 14 remaining ones. It seems natural to take for the 2 dependent fractions to and t4. The 14 independent fractions then are the variables in which we shall try to express the (nonequilibrium) free energy of the alloy. 4. Free energy. To find the free energy of a certain state of the alloy, specified by given values of the 14 independent variables, we must calculate the energy and the entropy. For the energy we assume that we can ascribe to each tetrahedron, pair or atom in the lattice a certain energy. This energy will be assumed to depend on the kind and number of the atoms that make up the cluster, but not on the sublattices on which the atoms happen to be situated. Thus there are 5 different tetrahedron energies e: (i = 0, 1, 2, 3, 4 indicates the number of B atoms of the tetrahedron), 3 different pair energies eP (i = 0, 1, 2) and 2 different atom energies ef (i = 0, 1). We now form the sum: U, = 2N c t,,e:, i.m
C. M. VAN BAAL
576
(where here and in the following summation over m means a summation over all different values the lattice labels may assume). U, is not the correct value of the total energy. Each pair in the lattice is a part of two tetrahedra and so in (5a) the energy content of the pairs is counted twice. Also each atom belongs to 8 different tetrahedra and thus the energy of the atoms is contained eightfold in U, . To correct for this we must subtract
from U, the pair energy
U,:
UP = Nxp,,e;, i,m
(5b)
and 7 times the atom energy
U,:
Ua = $N 1 ci,el.
(5c)
i.m
But even then the correct energy has not been found because each atom is a part of 12 pairs and thus, in correcting for the pair energy, 12 times the atom energy has been incorrectly subtracted. So finally we have for the total energy U: u = u, -
UP + 5u,.
(6)
with U,, UP, U, given by (5a), (5b) and (5~). Now this seems to be a rather roundabout way to calculate a quantity that could have been found in a much more direct way. But Hijmans and De Boer’) have shown that the same “overlap correction numbers” (- 1 and + 5) that we have found here, appear in the expression for the entropy of the Kikuchi approximation. But before calculating the entropy we wish to simplify (6). Writing: ,m = 4t, ,
P
and making u =
1 t2nz = et, 3 m
use of (l)-(4),
240
-
we directly
c ts,, = 4t, 2 m find for the energy
per atom u = U/N:
12 [(l + a) t, + 2t, + (1 + /?) t3] E,
(7)
in which: E = J(-2ei
+ 4ey - 2e; + e\ - 2ei + ek),
(1 + LX)E =6(-3eE
+ 6eg - 3ei + 3eb - 4ei + ek), (8)
(1 +/3)E=t(-3eg+6ey-3e;+eb-4e\+3ek); UO
=
5e”, - 6e: + 2ek + co (se”, - 5e”, - 6eP, + 6e’: + 2eb
2ek).
ORDER-DISORDER
For an alloy of constant determination
IN A GENERALIZED
composition
of the equilibrium.
ISING ALLOY
u0 is a constant If one assumes
577
and plays no part in the
that
only
nearest-neighbour
pair interactions are important (we shall call this the pure Ising 01 = /j = 0, whereas E reduces to the usual expression:
case) one has
in which VA, = eg - 2et, is the excess energy of an AA pair over that of 2 A atoms, with analogous definitions for V,, and VA,. For ordering alloys E > 0. In LYE.BE are comprised all the 3- and 4-particle interactions between nearest neighbours that could possibly be present in an fee binary alloy. It has already been pointed out in the introduction that their influence on the resulting orderdisorder transformations will be a good measure of the validity of the Isingmodel as a model for such an alloy. Considering now the entropy, one can repeat step for step the reasoning of Hijmans and De Boer for this case, coming to the same result: in the Kikuchi approximation the entropy S can be written, just like the energy, as the sum of a “tetrahedron entropy” S, , a “pair entropy” S, and an “atom entropy” S,, with the same overlap correction numbers : s = s, - s, + 5s,. S,, S, to the by the entities comply
(9)
and S, are found by applying the well-known Boltzmann entropy formula tetrahedra, pairs and atoms as independententities. So they are determined number of ways in which given numbers of the different types of these can be distributed over the available places. Of course these numbers must with the relations (l)-(4). We thus find:
S, = -2Nk
C ti,,,In
tirn,
i.m
SD= -Nk i,m CpirnInpi,,
(10)
sa= - Wki,m C cjmIn tin,. From (7), (9) and (10) we finally find for the free energy per atomf dropping the constant u0 :
= (U - 2X)/N,
.f = - 12 [(1 + ol) tl -t 2t, + (1 + p) t3] E + kT C (2ti, In ti, - pim lnp,, i,m
+ icim In cim).
We have already shown that (11) contains 14 independent with respect to these will give us 14 equilibrium equations
(11) variables. Minimizingf and from their solutions
578
C. M. VAN BAAL
we could find the most general phase diagram that would approximation. We shall not execute this large programme,
be possible in this however, but limit
ourselves to a few structures that appear to be the most common. In a disordered alloy, including alloys with only short-range order, there is no distinction between the sublattices and so we have: tl, = tlb = t,, = tld, r2ab = f2ac =
t2ad
=
f2bc
=
t2bd
=
f2cd,
(12)
t& = t3b = tsc = t3d. If, on the other hand, one or more of these equalities are violated we must have an ordered state. Now, instead of analyzing the numerous possibilities there obviously are, we will follow Shockley7) and classify the ordered states according to the way in which the different sublattices are occupied. Shockley finds that there are in this way 4 possible types of ordered structure: (1) the cubic Cu,Au structure in which 3 sublattices are equally and preferentially occupied by A atoms and the fourth sublattice contains an excess of B atoms; (2) the tetragonal CuAu structure in which 2 sublattices are equally and preferentially occupied by A atoms and the other 2 also are equally occupied but now preferentially by B atoms; (3) a second tetragonal structure in which again 2 sublattices are equally occupied, but the remaining two each have a different occupation; (4) an orthorhombic structure in which all 4 sublattices have a different occupation. The last 2 of these possible types of order do not seem to have been observed experimentally. We therefore have limited our calculations to the first named two structures, although it would have been interesting to investigate if the fact that the other two have not been found could perhaps be explained by the fact that they are less stable also in our model. The specialization of eq. (11) to the Cu,Au, the CuAu and the disordered states is easy and we shall not give it in detail. One finds that for the Cu,Au structure there remain 6 independent variables, for the CuAu structure 7 and for the disordered state 3. Minimizing the resultant expressions for the free energy thus leads to 6, 7 and 3 equilibrium equations that have to be solved to find the equilibrium values of the independent variables and hence the energy, the entropy and the free energy. The numerical methods by which we have solved the three sets of equilibrium equations and some of the difficulties we have met are described in the appendix. 5. Results. By the methods described in the appendix the three sets of equilibrium equations have been solved in the appropriate concentration and temperature regions for three sets of values of the energy parameters (a, b), namely (0, 0), (0, 0.2) and (0, -0.2). These sets also include the sets (0.2, 0) and (-0.2, 0) as the latter correspond to a simple interchange of A and B atoms.
ORDER-DISORDER
It has already of an ordered
IN A GENERALIZED
been mentioned
state becomes
ISING
that we always have found
579
ALLOY
that the free energy
equal to that of the corresponding
disordered
state
at a temperature below the critical temperature (taken here as the temperature above which the ordered state ceases to be one of minimum free energy). From general thermodynamics we know that in such a case the temperature T, at which both free energies are equal (for brevity we will call this temperature the equality temperature) for only one composition coincides with the actual transformation temperature T,. For all other compositions one has a temperature region in which the homogeneous alloy is not stable and separates in 2 phases of different compositions. The boundaries of this 2-phase region can be found from the points of tangency of the common tangent to the free-energy curves of both phases plotted as functions of composition at a fixed temperature. For the pure Ising case (a = /3 = 0) we have determined these boundaries betwRen each two of the three phases for which the free energy has been calculated. The result is shown in fig. 1 (where only the left half of the complete phase diagram has been given because of the symmetry between A and B atoms that is inherent in the pure Ising model). The most significant and new feature of this diagram is the three-phase equilibrium found at kT/E M 0.81 and the consequent two-phase equilibrium between the Cu,Au and CuAu ordered phases at lower temperatures. It must be mentioned that although the three-phase equilibrium in fig. 1 has been drawn as if it were of the eutectoid kind, our calculations have not been I
I
I
/
I
L
,
0.35
0.40
0.45
a=@=0
100
0.90 -
0.80 -
0.70 -
t W”
-
kT/E
010
0.15
a20
0.25
0.30
0.50
Fig. 1. The phase diagram of an ordering pure Ising fee alloy in the Kikuchi approximation (CC)indicates the disordered state.
580
C. M. VAN BAAL
detailed enough to decide that this actually was the case or whether it was rather a peritectoid. A distinction between the two, however, did not seem important enough to warrant a more thorough investigation. Fig. 2 shows the results for the generalized Ising model for o( = 0, p = 0.2 and DL= 0, p = -0.2 (for comparison those for the pure Ising case have also been indicated). Here we only give the equality temperature T, as a function of composition because it is felt that this would make the phase relationships sufficiently 1
1.6Oi-
I
a=0
I -
1.40
p z-o.2 .., p.0
----
/ //
p-0.2
/
fl\
\
\ \ \
Cl I 0.10
0.20
I 0.30
I 0.40
I 0.50
I 0.60
I 0.70
I 0.80
090
Fig. 2. The equality-temperature diagram (see text) of a generalized king alloy for two values of the higher-order energy parameter B (the dotted line indicates the pure Ising case).
Most prominent here is the sensitivity of the equality temperatures to variations in the strength of the higher-order interactions. For although the direction of the influence of these interactions could have easily been predicted from their definition in (8), it is somewhat surprising that a relative strength of these interactions of 0.2 produces the large changes depicted in fig. 2. Another interesting point is that for B = -0.2 the AB3 phase does not have a maximum transformation temperature in the neighbourhood of the stoichiometric composition. Experimentally this behaviour has been found in CuAu3 for compositions between 25 and 32 % Cu*). clear.
ORDER-DISORDER
Finally
IN A GENERALIZED
we give in table II the values
total heat of ordering AU,jNkTe for stoichiometric compositions.
ISING
of the latent
[in which
581
ALLOY
heat AU,/NkT,
AU, = U (T = a)
and of the -
U (T = 0)]
TABLE II Values of latent heat and total heat of ordering
A lJ,/NkTe
A U,/NkT,
A3
AB
Ah
A3
AB
ABA
p = --0.2
0.2253
0.3691
0.1925
0.8015
0.9896
1.0134
p=o
0.2371
0.2656
0.2371
0.7794
1.0563
0.7794
B = 0.2
0.2490
0.2984
0.2637
0.7590
1.2658
0.7302
6. Discussion and comparisons. It will have been noted that we have not given results for low temperatures. We have refrained from doing this mainly for the following reason. As a model of an alloy the Ising model (and also our generalized model) is a rather crude one. Even if theory in the future will provide a sound basis for the model, one has to expect that many refinements will have to be incorporated before a sizeable portion of all binary-alloy systems can be described adequately. Now at lower temperatures the free energy is to a larger extent determined by the energy, and deficiencies in the description of the energy wilI become more and more pronounced. This means that the model at low temperatures will lose the contact with reality that it might have at higher temperatures. It seems therefore pointless to perform calculations of a detailed nature in this low-temperature region before a better founded and more sophisticated model is available. The most complete results which were obtained earlier and can be compared to the present ones are due to Shockley7). Shockley made use of the BraggWilliams approximation and he found that in this approximation both the CuAu and CuBAu structures formed stable phases. It was disappointing, however, that both phases had their (coinciding) maximum transformation temperatures at c = 0.50, a result which for the Cu,Au structure is at variance with all experimental findings. This situation was not improved by the work of Peierls and its extension by Easthopeg), who applied the method of Bethe to the C+Au case. Easthope could only show that for compositions with c > 0.25 the critical temperature rose with the concentration, thus confirming partially the previous result of Shockley. Yang and Li”), using an approximation that seems to be intermediate between the Bethe approximation (or its equivalent, the quasichemical approximation) and the Kikuchi approximationl), were the first to find 3 distinct maximum transformation temperatures at c = 0.255, c = 0.50 and c = 0.745, respectively.
582
C. M. VAN BAAL
Li, however, did not find a 2-phase region between the CuAu and Cu,Au ordered phases. A more quantitative comparison is given in table III. This gives, for alloys of stoichiometric compositions, the critical temperature T, or the transformation temperature T, calculated by means of the various approximations mentioned. Also included is a value found by Fosdick as the result of a Monte Carlo calculation performed with a lattice of 500 sites. In a comparison with the exact results available for a 2-dimensional square ferromagnet this method has been shown to be accurate to within a few percent13). The agreement between Fosdick’s value and that of the Kikuchi approximation presents a new argument, next to that given in section 2, in favour of this approximation. It must be mentioned, however, that in Fosdick’s results the transition from the ordered to the disordered state seems to be more nearly continuous than in all other approximations. Probably this is due to the fact that the Monte-Carlo method, being essentially a kinetic method, in the neighbourhood of a critical point has the same difficulty in reaching equilibrium as actual thermodynamic systems experience in this region. TABLE III
Critical (T,) or transformation
Shockley’) (Bragg-Williams) Easthopeg) (Bethe-Peierls) Schapink’l) (Quasichemical) Yang, Li’O) Fosdickl’) (Monte Carlo) Present work (Kikuchi) Present work (Kikuchi)
(Tt) temperatures
T. T, T, Tt Tt T, T,
in units of kT/E
Cu3Au
CuAu
1.642 0.8911
2.000 _ 1.7869
0.8228 Z1.0 0.9967 0.9623
0.7306 _ 0.9718 0.9467
Several authors have estimated that the influence of three-body interactions on the energy of the fluid or solid state of the rare-gas elements is appreciable (see the review by Scott14)), but as far as is known more-particle interactions have not been considered in the theory of alloys. Also no previous Ising-model calculations exist with which we could compare our results. Fosdick, however, also applied the Monte-Carlo method to a 25 % alloy in which interactions between pairs of next-nearest neighbours were included. His results are so far in good agreement with those reported here, that he also finds that the influence of higher-order interactions on the transformation temperature is large : if the ratio of the strength of the next-nearest neighbour interaction to that of the nearest neighbours is -0.25, the reduced transformation temperature is raised from 1.0 to 1.6. This is quite comparable to our result, where a change from /3 = 0 to /3 = 0.2 raises this temperature from 0.96 to 1.50.
ORDER-DISORDER
IN A GENERALIZED
ISING
ALLOY
583
Many authors have compared their results on the Tsing model others with the CuAu system. So in a recent review of order-disorder
or those of phenomena
in metals3), Guttman compared the phase diagram calculated by Li’O) with this alloy system, noting the following discrepancies : 1) The maximum transformation temperatures of the 3 ordered phases Cu3Au, CuAu and CuAu, were calculated to be about equal, whereas actually that of CuAu, is more than 20% lower than those of the other two. 2) .Li finds that at no temperature the phases Cu,Au and CuAu are in equilibrium, whereas in fact such an equilibrium has been observedl’). 3) An orthorhombic phase, called CuAu-II (= Llo_J exists that has not been found in the calculations. To this could be added the already mentioned, more recent result of Battermans:), who found that for &Au, the transformation temperature for Cu concentrations above 25 % is higher than that for the 25 % alloy. One can understand a priori that phases with large unit cells like CuAu-II can never be found in a model with only nearest-neighbour interactions. With respect to the other discrepancies, however, it follows from figs. 1 and 2 that with a judicious choice of the energy parameters 01 and p they can be made to disappear. A survey of the other alloy systems listed by Guttmann, in which phases of the CuAu or Cu,Au type have been observed and for which details are given in the reference works of Hansen-Anderko, Elliott and ShunkId) shows that these, with a few exceptions, present no essentially new features compared with the Cu-Au system. The variations in transformation temperature that have been found all fall within the limits shown in fig. 2 and so a range of about -0.2 to 0.2 for both 01 and p is capable to account for these variations. Exceptions are Cu,Pt, FePd and FePd3, for which maximum transformation temperatures of the CuAu or Cu,Au structures have been reported at 17 % Pt, 58 % Pd and 66 % Pd, respectively. Such large deviations from stoichiometry do not appear to be covered by our model. In spite of the sketched general agreement, however, it follows from what has been said in the first part of this section that we do not believe that further work in this direction is urgently required. We would rather wish to conclude from this agreement that the Ising model appears to be a rather good first approximation as a model for a binary alloy, and that a relatively moderate inclusion of moreparticle interactions seems to be capable to account for many of the discrepancies between predictions of the model and the behaviour of actual alloys. Theories of alloys ought to reflect these facts. The values given in table II for the latent-heat and for the total-energy change of the transitions are comparable with those of other approximations. Like the latter they are too high by a factor 2 to 4 compared to the few experimental values available3). The cause of this discrepancy is not clear. Experimental values have
584
a tendency differences.
C. M. VAN BAAL
to be too low, but it seems doubtful
that this could explain
7. Conclusions. 1) The generalized Ising model, in which actions between nearest-neighbour atoms on a face-centered
the large
all possible intercubic lattice are
included, appears to be capable to account for nearly all the experimentally known details with respect to the phase diagram of ordering fee binary alloys. 2) As the strength of the 3- and 4-particle interactions in this model that are needed to achieve this agreement, appears to be moderately small compared to that of the nearest-neighbour pair interactions, it follows that the Ising model in all probability is a valid first approximation as a model for such alloys. 3) The Kikuchi approximation that was used to calculate several phase diagrams for the generalized Ising model, here as in the ferromagnetic case, appears to offer a substantial improvement compared to other approximations. Acknowledgement. It is a pleasure to thank R.H.Duyts for the enthousiasm and ingenuity with which he carried through a part of the numerical labour.
APPENDIX
DetaiZs of the calculations.
The three sets of equilibrium equations that result from the minimization of the free energies can be written in the form of simultaneous aIgebraic equations of degrees up to 12. Most of the enormous number of possible solutions of these equations will not be acceptable, however, because the definitions of the variables tin,, pinl , cim imply that they are all real and restricted to the interval: (13) These constraints,
however,
offer an extra difficulty.
A general
and elegant
way
to solve minimization problems in the presence of constraints is that in which one makes use of so-called penalty functionsi7). In this method one adds to the function to be minimized (heref) a penalty function g of all the variables ,yi that are to obey a constraint of the type (13). g has the following properties: (I) it is positive everywhere; (2) it is small for values of si in the allowed interval and large elsewhere; (3) it contains a parameter q such that for q + 0, g --f 0. A possible function g for our case would be: g = q xi (1 - 2xi)‘“, with n a positive integer. For a suitably chosen initial value of q one now minimizes f + g without constraints, solves the resulting equations and uses the solution as the starting point for the solution of the same equations, but with a lower value of q. Iteration of this process with successively lower values of q will generally converge to the described solution. We have applied this method with the penalty function given
ORDER-DISORDER
above and with one suggested to find a suitable
IN A GENERALIZED
by Bracken
wa) to decrease
ISING ALLOY
and McGormickL7).
the penalty
parameter
585
After some trials
q, the method
was found
to be successful. In the mean time, however, another procedure was developed that took less computing time, So for the greatest part of the calculations the latter procedure was followed. Our real problem was, here as well as with the penalty-function method, to find a starting point. Once this was found, it was possible, by slowly varying the parameters in the equilibrium equations (temperature, 01, ,4), to obtain the other desired results. To find this starting point a Monte-Carlo method was used. Within rather wide but plausible ranges a set of random variables (6 or 7; for the disordered state the solution at very high temperature is known and can be started from there) was generated. It was then checked if all variables ti,, Qirn, cim calculated with this set met the constraint (13). If not, a partly new set was generated and again checked. When a succesful set was found the free energy was calculated. Then the process was repeated, the second free energy was compared to the first and if lower, kept, else discarded. After having done this a number of times (usually 50), the allowed ranges of the variables were centered around the best point found and reduced by a factor 1.2. The whole procedure then was repeated some 50 times. The final set of variables in all but a few cases proved to be accurate enough (deviating by less then 0.1 ‘A from the exact values) to serve as a starting point for a solution of the equilibrium equations. The solution itself in all cases was found by a straightforward extension of the NewtonRaphson tangent method’*). Quite another and initially very disturbing difficulty was encountered when we tried to solve the equilibrium equations for the CuAu structure. Our first trials for co = 0.5 all gave a solution that, if visualized, corresponded to a state of the crystal that is called a “domain structure”: different parts of the lattice are ordered in the same way, but with respect to different sublattices. So here in half the crystal the A atoms were situated on sublattice a, in the other half on sublattice c. Such structures are well known to occur in real crystals, they are in fact the only ones known. Evidently they are highly ordered, but in our case the Kikuchi approximation ascribes to this ordered state the very high entropy of nearly k In 2 per atom, making it a stable state up to very high temperatures. A careful examination of the derivation of the equilibrium equations failed to give a clue as to the cause of this evident anomaly. Turning to the next-lower approximation, the quasichemical approximation, we found to our surprise that the same state there has a large negative entropy. From this we must conclude that although approximations like the quasichemical or the Kikuchi approximation appear to give in general acceptable results, there are “corners” in which they are unreliable. The large and negative entropy found in the quasichemical approximation led us, however, to the idea of a stratagem by which we could hope to avoid the sharp
586 but undesirable
C. M. VAN BAAL
minimum.
As both
the entropy
in the quasichemical
approxi-
mation, which we shall call So, and that of the Kikuchi approximation (to be called Sk) in general are reasonable approximations to the true entropy, evidently a linear combination of the form (1 - w) So + IV&, 0 < w < 1, also will be a reasonable approximation. Now, as this “mixed” entropy up to MI = 0.8 is negative for the “domain state” we wish to avoid, this state in all probability will not correspond to a solution of the equilibrium equations derived from a free energy with this mixed entropy. So a solution of these equations could be in the neighbourhood of the true solution. We then could hope that by slowly increasing w up to the value 1, where we would be back at the Kikuchi entropy, this solution could be attained. This stratagem was found to work and even led to two ordered states that afterwards appeared to belong to two branches of the same solution. order paraAlong them c,,~ (which for c0 = 0.5 is identical with the long-range meter S as defined by Bragg and Williams) shows about the same behaviour as in Fowler and Guggenheim’s solution of the Bragg-Williams approximation for the Cu,Au structurelg). A difference is, however, that in our case also the lower branch corresponds to a local minimum in the free energy and so represents a possible (although metastable) equilibrium state of the model.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) JO) 11) 12) 13) 14) 15) 16) 17) 18)
Domb, C., Advances in Physics 9 (1960) 149. Cowley, J.W., Phys. Rev. 77 (1950) 669; 120 (1960) 1648; A138 (1965) 1384. Guttman, L., in: Solid State Physics (F. Seitz and D. Turnbull, eds.) 3 (1956) 146. Kikuchi, R., Phys. Rev. 81 (1951) 988; Kurata, M., Kikuchi, R. and Watari, T., J. them. Phys. 21 (1953) 434. Hijmans, J. and De Boer, J., Physica 21 (1955) 471, 485, 499. Kikuchi, R. and Brush, S.G., J. them. Phys. 47 (1967) 195. Shockley, W., J. them. Phys. 6 (1938) 130. Batterman, B.W., J. appl. Phys. 28 (1957) 556. Peierls, R., Proc. Cambr. Phil. Sot. 32 (1936) 477; Easthope, C.A., Proc. Cambr. Phil. Sot. 33 (1937) 502. Yang, C.N., J. them. Phys. 13 (1945) 66; Yang, C.N. and Li, Y. Y., Chin. J. Phys. 7 (1947) 59; Li, Y.Y., J. them. Phys. 17 (1948) 447. Schapink, F. W., Thesis Delft (1969). Fosdick, L.D., Phys. Rev. 116 (1959) 565. Fosdick, L.D., in: Alder, B., Fernbach, S. and Rotenberg, M., Methods in Computational Physics, Vol. 1, Academic Press (New York, 1963). Scott, R.L., in: Eyring, H., Henderson, D. and Jost, W., Physical Chemistry, Vol. VIIJA, Academic Press (New York, 1971). Rhines, F.N., Bond, W.E. and Rummel, R.A., Trans. A.S.M. 47 (1955) 578. Hansen, M. and Anderko, K., Constitution of Binary Alloys (1958), McGraw-Hill; Elliott, R.P., jr]., 1st Supplement (1965) id.; Shunk, F. A., id., 2nd Suppl. (1969) id. Bracken, J. and McCormick, G. P., Selected Applications of Nonlinear Programming, Wiley (New York, 1968). Whittaker, E. and Robinson, G., Calculus of Observations, Blackie (London, 1948).