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Kinetic paths in two order parameters: Theory
OOOI-616Oj89 $3.00+0.00
Acta merall. Vol. 37, No. 3, pp. 823429, 1989 Printed in Great Britain. All rights reserved
KINETIC
Copyright e 1989Pergamon Press plc
PATHS IN TWO ORDER
PARAMETERS:
THEORY
B. FULTZ Department of Materials Science 138-78, California Institute of Technology, Pasadena, CA 91125, U.S.A. (Received 24 March 1988; in revised form 18 July 1988)
Abstract-It is shown that when a crystalline alloy is characterized by two order parameters, the alloy may develop various combinations of the two order parameters en route to a final equilibrium state. The kinetic evolution of the two order parameters in a ternary alloy with B2 order is analyzed with an activated state rate theory. Elementary vacancy jump processes are considered in a Master Equation formulation. A two dimensional transport equation describing the evolution of the alloy through the two order parameters is developed, and the chain of states in this two dimensional order parameter space is found by numerical solution of the transport equation. It is shown that either different interatomic interactions or different activation barrier heights for the diffusive jumps of the different atoms will cause the alloy to evolve through different nonequilibrium states, or along different “kinetic paths”. R&sum&-On montre que quand un alliage cristallin est caracterise par deux parametres d’ordre, il peut dbvelopper diverses combinaisons des deux parametres d’ordre en allant vers un Ctat d’equilibre final. L’ivolution cinetique des deux parametres d’ordre dans un alliage temaire B2 est analyde par une thtorie d&tat active. Les mecanismes elementaires de saut de lacunes sont formulis dans une equation directrice. Une equation de transport bi-dimensionnelle decrivant l’tvolution de l’alliage a travers ces duex parametres d’ordre est developpee, et on trouve la chaine d&tats dans cet espace bidimensionnel des parametres d’ordre en rtsolvant numtriquement l’equation de transport. On montre que tant les differentes interactions d’activation pour les sauts par diffusion des differents atomes font que l’alliage passe par differents Ctats hors d’equilibre, ou par differents “chemins cinetiques”. Zusammenfassung-Es wird gezeigt, dal3 eine kristalline Legierung, die durch zwei Ordnungsparameter charakterisiert ist, auf dem Wege zum endgiiltigen Gleichgewichtszustand verschiedene Kombinationen der beiden Ordnungsparameter aufweisen kann. Die kinetische Entwicklung der beiden Ordnungsparameter wird in einer temiiren Legierung mit Ordnung B2 mit Hilfe einer Ratentheorie aktivierter Zustlnde analysiert. Elementare Sprungprozesse der Leerstellen werden in der Formulierung von Mastergleichungen betrachtet. Es wird eine zweidimensionale Transportgleichung, welche die Entwicklung der Legierung mit den beiden Ordnungsparametern beschreibt, entwickelt; die Kette der Zustande in diesem zweidimensionalen Raum der Ordnungsparameter wird ilber eine numerische Losung der Transportgleichung erhalten. Es wird gezeigt, dat3 entweder interatomare Wechselwirkungen oder verschiedene Hohen der Aktivierungsbarriere bei den difusiven Spriingen der verschiedenen Atome dafiir sorgen, da8 die Legierung sich iiber verschiedene Nichtgleichgewichtszustlnde, oder fiber verschiedene “kinetische Wege” entwickelt.
1. INTRODUCTION
Recently a number of kinetic theories have been developed to predict the rates of order-disorder transformations on crystal lattices [l-5]. These theories are concerned with one physically measurable order parameter. Nevertheless, in two of these theories [14], as well as in studies of diffusion in ternary alloys [6, 71, several different relaxation processes were found. Furthermore, different nonequilibrium combinations of short and long range order parameters were obtained by varying the initial state of a binary alloy [3]. These different relaxation processes and different combinations of order parameters hint at a rich variety of observable nonequilibrium states for alloys with two physically measurable order parameters. Any alloy characterized by a single order parameter will necessarily move along the same chain of nonequilibrium states for all temperatures and initial
conditions. On the other hand, the kinetic chain of states need not remain fixed when two or more parameters characterize the state of order in the alloy. The kinetic process must still have a beginning and an end state, but it is possible in principle to connect these ends by different chains of states, here termed different “kinetic paths”. For a given value of one order parameter, states on different chains need not have equal values of the other order parameters. A rich variety in the nonequilibrium states appears possible, and this variety must be understood in terms of a kinetic theory. The present paper develops a kinetic theory for B2 ordering of a ternary alloy with vacancies. This problem has been treated for a binary alloy with vacancies in the Path Probability Method of Kikuchi and Sato [l-3], and for f.c.c. alloys by Radelaar [5]. The use of ternary alloys makes it easier to obtain two independent, physically measurable, order parameters. In principle, three independent pair cor-
823
824
FULTZ:
KINETIC PATH IN TWO ORDER PARAMETERS
relations exist for a ternary alloy. However, we constrain the B2 structure to be composed of two interpenetrating sublattices. The average occupations of these sublattices by the three species of atoms and vacancies provide three independent order parameters. Two of these parameters are the physically measurable pair correlations. The third order parameter characterizes the vacancy distribution, which while physically unobservable, is shown to have a major effect on kinetic processes. The present goal is to find the minimum requirements for obtaining variations in kinetic paths. An equation for the path of the B2 ternary alloy through the state space of its two physically measurable order parameters is found. The formulation is a point approximation, but it gives equilibrium states similar to those of the Bethe approximation. This equation is solved numerically for a variety of: (1) chemical interactions between the atoms, (2) activation barrier heights for atomic jumps, and (3) initial states of the alloy. A wide variety of different kinetic paths is shown possible within this model, and reasons for some of these paths are given.
2. B2 ORDERING OF TERNARY ALLOYS A convenient choice of a ternary alloy with B2 order has a concentration of B-atoms, cs=f, with the half of atoms as C-atoms and A-atoms: cc = c, and c,, = f - c. The B2 structure is represented as two interpenetrating simple cubic sublattices: CL, which is rich in A-atoms and C-atoms, and /I, which is rich in B-atoms. The order parameter “j” equals the number of A-atoms on the a sublattice. The parameter j can range from (N/2) - cN (perfect order), to (N/4 - cN/2) (zero order), to 0 (perfect order of opposite phase), where N is the total number of sites in the crystal. The order parameter “7 equals the number of C-atoms on the a sublattice, and : 0 < i < cN. The number of vacancies on the a-sublattice is “k”, where 0 < k < c,N, and c, is the concentration of vacancies in the alloy. There are 2 atoms in each first neighbor shell about a given atom (in B2, 2 = 8). Note that the first-neighbor sites of an atom on the a-sublattice are all on the /?-sublattice, and vice -versa. Consider the possible diffusive jumps out of the state Ai,, . For the 2 atoms that could jump into the neighboring vacant site, there is a Boltzmann probability, p, for the success of each candidate jump that is determined by the difference between the initial energy of the atom and its maximum potential energy during the diffusive jump, E P = exp( -BE)
where
(1)
and EA = E:, - NM V,
- N,, V,, - N,, V,,
Es = J% - NABVAB -
NB,~BB
-
%C~BC
E,=E$-N,,V,,-N,V,-N,V,,.
Pa) (W
(w
to p, and is determined by the bond energies: {V} (expressed in units of kT), and numbers of bonds between the three species of atoms: {N}. It is assumed that the activation barrier height, E+, depends only on the species of atom whose jump is being considered. Attempt frequencies (i.e. pre-exponential factors) for all diffusive jumps are assumed equal. A chain of kinetic steps cannot be constructed by having B-atoms jumping onto the /?-sublattice independently of the jumps of A- and C-atoms off the B-lattice; there are a tied number of sites on the /I-sublattice, and most of them are occupied. It is tempting to assume that atomic exchanges between the Z- and fi-sublattices occur in pairs. Such a treatment is provided in the Appendix, together with cautions for its use. Here we use vacancy populations on the a- and B-sublattices as agents for the migration of atoms from the /?- and I-sublattices, respectively. With a single order parameter the kinetic Master Equation assumes its familiar canonical form The rate of each candidate jump is proportional
where the state of the alloy is represented by the vector: A, whose components are the states of order. Except for the states of perfect order, a particular Aj is many-fold degenerate; a specific state of order may be realized with a variety of atomic arrangements. The W-matrix induces transitions of the alloy between different states of order. The present interest is in two measurable order parameters and a vacancy order parameter. When the state of the alloy is characterized by three independent order parameters, the Master Equation is $4,
= c
Wi,t.r,,A,m-
c
Wb,ijkA,k
(4)
I.m.n
1.m.n
where the state of the alloy is now a matrix of rank three, and transitions between these states are induced by a W-matrix of rank six. In a vacancy mechanism of atomic transport, the elementary kinetic step involves the jump of a single atom into a vacant neighboring lattice site. This process will cause the number of atoms properly placed on a sublattice will change by at most + 1 or - 1, and the vacancy concentration of the sublattice changes in the opposite direction. The W-matrix assumes the simple off-diagonal structure K/+Ik-l.ijk#O
Wr)-Ik+t.ijk#O
W r+l,k-l.i,kfO
FULTZ:
KINETIC
PATH
IN TWO ORDER
825
PARAMETERS
Wt-I,k+I.,]k#O wtjk-l.,,k#”
2
W ,]ktI.,,kf”
wit
I I’k’
- 1.il.k’
-
Wt.-I J k’ I.t'I'k' I
= J,(i’, j’, k’). w,k.,k
=
(9)
O,
The rate of change in the vacancy distribution also be considered
and all other W ,,k.lmn
+
(5a-h)
O.
must
dk
As an example of an elementary kinetic step, + wi,‘k’+ ,.,‘,‘k w,C, + IA 1.1, A consider the atomic jump rate of an A-atom from the /?-sublattice into a vacancy on the a-sublattice: ,.I, ‘k w, + 1,‘k’ ,.,‘,‘k’ w,,,k W ‘I +,k _ ,,,Jk.This rate is composed of three factors: = J,(i’, j’, k’). (10) (1) the probability of a vacancy on the cx-sublattice (2) the probability of an A-atom in one of the Equations (@-(IO) form a set of coupled differeight neighboring sites of the vacancy, and (3) the ential equations that give the absolute rates of change Boltzmann factor of equation (2). In the Boltzmann of the order parameters. For the present purpose, factor the numbers of A-, B-, and C-atoms on the however, it is interesting to follow the relative rates sevent a-sublattice sites are used to determine NAA, of change of the two observable order parameters, i NAe, and NAc. We assume that the atoms are placed and j. Representing these two order parameters as randomly over their sublattice sites. This averaged orthogonal axes in a two dimensional space. the treatment of neighboring sites is analogous to that trajectories of alloys through this space arc sought. used in the point approximation of equilibrium therAt each state of order (i’, j’, k’), the direction of the modynamics. The expression for W,, +,k_ I,yk is thus trajectory is a vector quantity J(i’, j’,k’)
W t,+Ik-I.l]k=
x{(z-cN-j)$z}exp(-BE,,)
(6)
(r-i-j)
E,,=EX-Y,,ji(Z-1)-Y,, x;(z-l)-v~,i;(z-l).
(7)
Similar expressions can be written for the other five nonzero elements of the W-matrix in equation S(a-f). Equations 5(e) and (f) are the rates of change of k due to the interchanges of vacancies with B-atoms, whose order is not followed explicitly. The six rates { W} from equation (6) and its five analogs depend only on the state of order in the alloy (i.e. on the set: {I, j, k}). For short time intervals these six rates may be assumed to occur independently and in parallel. The net flux of alloys between the states parameterized by j ’ and j ’ + 1, and between the states parameterized by i’ and i’ + 1, is
di -$
I,J,k, =
wi)'
+ I k’ - I.i]‘k’
g
W,,,.+,,_,J,‘,-
E
J,(i’, j’, k’)
-
wi,‘k:,~,~
+ 1k - I
wr~‘-,k’+l.i,‘k’
(8)
iOne of the eight neighboring sites is occupied by the vacancy.
= J,(i’, j’,k’)?+
J,(i’, j’, k’);.
(II)
With only one order parameter, J would lx a scalar, and the alloy would necessarily pass through the same states of order, albeit with rates and directions that are temperature-dependent. This is expected if, for instance, the A-atoms and C-atoms are chemically identical, and the alloy begins in an initial state of disorder. In this case the alloy is expected to take a kinetic path that is a straight line from disorder to order in its normalized two dimensional i-j state space:
J2 = I J, c 5-l (initially disordered, no chemical differentiation of ternaries). However at low temperatures when the C-atoms are chemically distinct, the next section shows that J(i’, j’, k’) varies with temperature, and the kinetic path is generally nonlinear. To compare the present end states with well-known equilibrium results from binary alloys, the case when c = 0 was considered. Critical energies were sought that would let equations (8) and (IO) equal zero for nontrivial j (i.e. j #(N/2) - cN). To do this, small departures of i and j from the disordered state were examined so that the Boltzmann factors had large common factors times second factors, nearly unity, that could be accurately linearized in the departures from disorder. Nontrivial solutions were found in cases when: 4V, = VAA + V,, - 2 VA, = (4/Z - I) = (417)= 0.5714. Some critical energies, 4 V,, from thermodynamic theories are; mean field: 0.500, Bethe pair approximation: 0.575, best known: 0.6294 [8]. Evidentlv the Ioresent kinetic theory_ .provides end
826
FULTZ:
KINETIC
PATH
IN TWO ORDER
PARAMETERS
(the vacancy distribution relaxes quickly), equations (8x10) form a “stiff’ set of differential equations, whose integration can be troublesome. Succeeding values of i and j were obtained by the forward Euler method, the fourth-order Runge-Kutta method with and without adaptive step size control, and the Bulirsch-Stoer method. All methods gave essentially the same results when they were stable. Graphs were made of the points in the i-j state space. Examples of such graphs are shown in Figs l-5. (Here all energies, { V}, are in units of kT. Higher values of { V} correspond to lower temperatures with respect to the critical temperature of the disorder-order transformation.) +N-cN Figure 1 was generated by starting the alloys at CN tcN different initial states of order, and allowing them to 1 - porometer evolve at the same low temperature. Notice that all Fig. 1. Kinetic paths of alloys that started at various initial alloys reached the same end states, which are the states. Initial and final states are indicated by squares and equilibrium states for that temperature. These end circles, respectively. For the light lines: VA, = V,, = V, states therefore move to a higher degree of order at = V,,=O.8; VA,= V,=O; EX=E;=E:; c=O.25; lower temperatures (larger V). The temperaturec, = 0.003. For the dark lines: VA, = V,, = V,, = VA, = 0.3; all other energies and concentrations are the same. dependence of these kinetic paths can be roughly understood in terms of the temperature-dependence of the equilibrium end states; the larger the difference in value of an order parameter in the present and final states that are in reasonable agreement with the Bethe states of the alloy, the more rapid will be the change pair approximation for B2 ordering of an equiatomic of this order parameter. However, there are instances alloy. These critical energies, 4VC= (4/Z - I), are in Fig. 1 where one of the order parameters changed also reasonably close to results of the thermodynamic in the “wrong” direction with respect to the direct pair approximation for other equiatomic alloys. For the 2-dimensional square lattice the present path from initial to final state. Figure 2 shows different kinetic paths at various critical energy, 4V, = (4/Z - I) = (4/3), whereas the thermodynamic pair approximation gives: I .387; temperatures from alloys that started near full disorfor the simple cubic lattice: 4VC= (4/Z - 1) der (k = c, (N/2), j = (N/4) - (civ!Z) + 6,, and i = c(N/2) + &, where 6, and 6, are small positive = (4/5) whereas the pair approximation gives: quantities needed to force the system off the saddle 0.8108. Mean field theories (point approximations) point at zero order.) These same initial conditions provide the critical energy: 4V, = (4/Z). However were used in obtaining Figs 3-5. In Fig. 2 the A- and due to the use of vacancies, our point approximation gives results closer to those of the pair approximation. There is a feature of kinetic paths in the i-j state space that facilitates interpretations of kinetics experiments. While the choice of path is determined by kinetics, the choice of path is independent of the vacancy concentration. An increase in vacancy concentration will cause a proportional increase in the k c rates of equation (6) and its five analogs, so the E *N-+34magnitudes of changes in order parameters of equations (8x10) will increase proportionally. The same kinetic path will therefore be taken by alloys with .7 .3 different vacancy concentrations, or with varying .5 vacancy concentrations, although the alloy will proceed down the path at different rates. D
+N-cNi
3. APPLICATIONS
AND DISCUSSION
Using equations (6) and (7) and their five analogs, equation (11) was integrated by numerical procedures. Initial values of i and’j were chosen, and the relevant W-matrix elements were obtained. Because of the much shorter time scale of the /c-parameter
0
-“B
%cN
Y
1 - parameter
Fig. 2. Kinetic paths of alloys that started at the same initial state, but at different temperatures. The energies: VA,, = V,, = V,-- = VA, = X (where X is marked on the figure): V,,, = V, = 0; EX = I?,+= E; + 8X. c = 0.25; c, = 0.003.
FULTZ:
+N-CNJ
0
KINETIC
+CN
827
PATH IN TWO ORDER PARAMETERS
1
CN
I - porometer Fig. 3. Kinetic paths of alloys that started at the same initial state, but at different temperatures. The energies:
(where X is marked on the figure); v,,. = v,, = 2X; v,, = v,,. = 0; E: = EB = EF. c = 0.25; c, = 0.003.
VAA = V,, = X
C-atoms are chemically identical, and without a difference in activation barrier heights for their diffusive jumps, the kinetic paths would be straight lines from the center of the figure towards the right corner. The equilibrium end states in Fig. 2 are in fact identical to those obtained with: EX = Ei = E,+. However, with a greater diffusive mobility of the C-atoms, E,* < Ez = E,f, there is a temperaturedependent variation from the linear kinetic path. In essence, the lower activation barrier height for the jumps of C-atoms allows the i order parameter to relax relatively more quickly than the j order parameter, especially at low temperatures. Differences in the activation barrier heights {E*} can therefore lead to different kinetic paths, whose choice is temperature-dependent. If the atomic species are differentiated chemically by giving the C-atoms chemical bonds that are twice as strong as those of the A-atoms ( Vcc = V,c = 2 VA,, = 2V,, and VAB= Vsc = 0), the kinetic paths shown in Fig. 3 are obtained. Although all alloys started at the same nearly disordered initial state, there is a marked temperature-dependence of the kinetic paths. The larger chemical bond strengths of the C-atoms cause the i order parameter to relax relatively more quickly, and the kinetic path deviates from the straight line that is taken without chemical differentiation. Differences in chemical bond strengths (V} can therefore lead to differentiate kinetic paths, whose choice is temperature-dependent. As shown in Fig. 4, a peculiar reversal of the i parameter occurs when the B-atoms are chemically very strong, and the C-atoms are chemically weak but the very mobile (Ei z Ei > E$). By monitoring distribution of the various species, it was found that a rapid buildup of the chemically strong B-atoms on the /?-sublattice causes a depletion of vacancies on the
I - porometer Fig. 4. A kinetic path that shows overshoot. The energies: V,, = 0.4; I’,, = 1; VA, = I’,,. = V,., = I’,,- = 0: E: = E; = Ef + 6. c = 0.02; c, = 0.003.
/I-sublattice. The C-atoms prefer the z-sublattice, and with an excess of vacancies on this sublattice, J, becomes quite positive. The i parameter increases rapidly, and over-shoots its equilibrium value. Later as the slower moving, but chemically stronger, Aatoms settle onto the a-sublattice, the A-atoms force the C-atoms off the z-sublattice, and the i parameter decreases to its equilibrium value. As shown in Fig. 5, chemical differentiation of the atomic species can cause an individual order parameter to start to change in the “wrong” direction. With the interaction parameters of Fig. 5, both the Aand C-atoms prefer the z-sublattice, but the A-atoms prefer it quite strongly, and force the vacancies off the a-sublattice. The C-atoms therefore have a preponderance of jumps from the x-sublattice to the fi-sublattice, and the i parameter starts to decrease.
-SCN
CN
I -parameter Fig. 5. A kinetic path that initially goes backwards. The energies: VAA = 1 3; I’,, = I; I’,, = VA, = 0.8; VAs = V,, = 0: E: = E; = E: c = 0.02; C, = 0.003.
828
FULTZ:
KINETIC PATH IN TWO ORDER PARAMETERS
However, all of the A-atoms can be accommodated on the a-sublattice by displacing only the B-atoms. This occurs later and the i parameter then increases as expected. Variations in kinetic paths are a means of varying the microstructure of nonequilibrium materials. An experimental study of variations in kinetic paths must be able either to prepare alloys with various degrees of initial order, or must employ alloys that have chemically distinct elements. Some variations in initial state may be possible through an initial quench from a temperature between the critical temperatures of the two order parameters. For experiments starting with an initial state of disorder, it behooves the experimentalist to find, for example, a ternary alloy with pairs of components that differ in their heats of formation of intermetallic compounds (implying different {V}), or components that differ in their diffusive mobilities (implying different {E*}). With such an alloy system a’ variety of kinetic paths, and therefore novel microstructures, can be obtained by thermal processing. 4. CONCLUSION When the state of an alloy is characterized by two or more order parameters, it is possible for the alloy to traverse a rich variety of kinetic paths in order parameter space. In other words, the Markov chain of nonequilibrium alloy states can be temperaturedependent not only in its absolute rate, but also in its choice of states in the chain. In the present formulation for a ternary alloy with two order parameters, the alloy generally follows different chains of states at different temperatures. Variations in either the chemical bond energies or the activation barrier heights will cause the kinetic path to vary with temperature. The agents of kinetic change are vacancies, and their distribution between the two sublattices is important in determining the kinetic path. When the vacancy concentrations on the two sublattices become substantially unbalanced, as they will when one species of atom prefers a sublattice so strongly that the vacancies are forced off, atoms may jump onto the sublattice with the surplus of vacancies even if this sublattice is energetically unfavorable for them. Order parameters can hence change backwards, and this leads to a wide variety of kinetic paths. Acknow\e&ements-In the process of implementing a similar theory for B2 and DO, ordering, Mr L. Anthony
3. K. Gschwend, H. Sato and R. Kikuchi, J. them. Phys. 69, 5006 (1978). 4. Y. Saito and R. Kubo, J. Stat. Phys. 15, 233 (1976). 5. S. Radelaar, J. Phys. Chem. Svlidy 31, 219 (1970). 6. D. de Fontaine, J. Phys. Chem. Solids 34, 1285 (1973).
7. M. Murakami, D. de Fontaine, J. M. Sanchez and J. Fodor, Acra metall. 22, 709 (1974).
8. D. de Fontaine, Solid St. Phys. 34, 73 (1979). APPENDIX The numerical inconvenience of the “stiff’
equations
(8x10) motivated an alternate formulation of kinetic paths. The formulation presented in this Appendix makes nominal use of vacancies but maintains a constant vacancy distribution between the two sublattices of the 82 structure. It becomes equivalent to a mechanism of interchanging first neighbor pairs of atoms. Kinetic paths predicted with this mechanism depend on chemical interaction energies and activation barrier heights, but the paths are shown lo disagree with those of the vacancy mechanism of the main text when unsymmetrical interatomic interactions or activation barrier heights lead to unequal vacancy populations on the a- and /J-sublattices. A chain of kinetic steps cannot be constructed by having B-atoms jumping onto the /? sublattice independently of th; iumos of the A and C-atoms off of the B sublattice. To con&uct a chain of states, as elementary iinks six pairs of jumps are used. These links are depicted schematically in ’ Fig. Al. Processes that involve jumps of two atoms of the same type, since they contribute nothing to the changes in order parameters, are ignored. Unlike the two elementary vacancy jumps of which they are composed, a kinetic path can be constructed by connecting these six links together in any order. These six links can therefore be considered to operate independently of each other and in parallel. Here the rate: Wan, for example, designates the jump of an A-atom lo the a sublattice followed by the jump of a B-atom to the /l sublattice. During this process the vacancy is initially on the z sublattice, jumps to the /? sublattice, and ends on the z sublattice. None of the six elementary links change the vacancy distribution. After this process any of the six elementary links can then follow. The average rates of the six links, {wAs. wl\c, wBA,wBc,wc-.. wCB},involve two processes in series, and have the form
;r\ B
c
B
A yr\
wBC
wBA
independently checked many of the numerical integrations perfbrmed in-the present woik. This work was supported by the U.S. Department of Energy under contract DE-FGO386ER45270.
;A C
REFERENCES I. H. Sate and R. Kikuchi, Ada mekall. 24, 797 (1976). 2. H. Sate, K. Gschwend and R. Kikuchi. J. Physique CI, 357
(I 977).
vc%A
B
C
A ;/\
wCB
Fig. Al. The six independent transition processes in B2 ordering of the ternary alloy.
FULTZ:
KINETIC
PATH
IN TWO ORDER
\
PARAMETERS
829
There are six types of jumps of atoms into vacancies: {B’,(A). B’,,(B), B’&). B,,(A), B,,(B). W,,(C)}, which when combined in pairs, are used to generate the six links H.‘c*.rice}.. {w*rl. WAC,H’sA.H’BC.9 Because the atomic jump processes are grouped in pairs (Fig. Al), these elementary processes can occur in any order. These processes arc assumed to occur in parallel, so the average rate of flow of alloys between states A,, and Ai,, , is obtained by recognizing that the links ~‘~sand H’~< serve to increase j. and ~‘s,, and w‘(~decrease j. (The links wcs and rvs(- have no effect on j.) The rates of change of j and i are Zj r
=
was(j',j')+WA(.(i'.i')
1,
- [waA(i’, j ‘) + wcA(i’, j ‘)]. and
1 -parameter
Fig. A2. Discrepancy in kinetic paths predicted by interchanging pairs of atoms (formulation in Appendix) and by using vacancies as the agents of change (main text). Both cases started at the same nearly disordered state, and used the energies: V,, = 0.5; V,, =0.5; V,, = V,, = 2; V,, = VW = 0; E: = Et = E:. c = 0.25. For the vacancy mechanism: c, = 0.003.
I H’
(Al)
-_-__-
AB .__.
W&V
‘_
+
w,(B)
where, for example
(A21
W,I(A)=(:-cN-j)iZexp(-/?E,& with EAa= E: - V,, j 5 (Z - I) - v,,
- V,,i$
N _ -i-j ( 2
i(Z-1) >
(Z - 1).
(A4a)
(A3)
di (2r I,,.=
H’(.l\(i’,j’)+W,(.B(i’,j’) - [w,Ji’,
j’) + wBJi’. j’)].
(A4b)
The numerical integration of these equations is straightforward, and can be accurately and quickly performed with the forward Euler method. An example of a significant discrepancy between the kinetic path obtained with this present “interchange” mechanism and the “vacancy” mechanism of the main text is shown in Fig. A2. The path labeled “vacancy” is similar to the kinetic paths of Fig. 3. However, the curve labeled “interchange” is substantially different, and examining the individual rates (w} revealed the cause. Although the Catoms are capable of quick jumps from either the a- or fi-sublattices, the rates: n’xc, H’~. ~1~~.wcs that involve the motion of the C-atoms are dominated by the slower-moving A- and B-atoms because of the harmonic mean in equation (Al). Of the motions of the A- and B-atoms, the rate: IV&A) is large because of the large VAc. Consequently the rate wAcis large, and the i parameter changes slowly at first. On the other hand, when the vacancies are allowed to change their distribution between the two sublattices, they will be depleted from the a-sublattice by the energetic C-atoms. This retards the increase of the j parameter, and the kinetic path marked “vacancies” in Fig. A2 is obtained.
Acta merall. Vol. 37, No. 3, pp. 823429, 1989 Printed in Great Britain. All rights reserved
KINETIC
Copyright e 1989Pergamon Press plc
PATHS IN TWO ORDER
PARAMETERS:
THEORY
B. FULTZ Department of Materials Science 138-78, California Institute of Technology, Pasadena, CA 91125, U.S.A. (Received 24 March 1988; in revised form 18 July 1988)
Abstract-It is shown that when a crystalline alloy is characterized by two order parameters, the alloy may develop various combinations of the two order parameters en route to a final equilibrium state. The kinetic evolution of the two order parameters in a ternary alloy with B2 order is analyzed with an activated state rate theory. Elementary vacancy jump processes are considered in a Master Equation formulation. A two dimensional transport equation describing the evolution of the alloy through the two order parameters is developed, and the chain of states in this two dimensional order parameter space is found by numerical solution of the transport equation. It is shown that either different interatomic interactions or different activation barrier heights for the diffusive jumps of the different atoms will cause the alloy to evolve through different nonequilibrium states, or along different “kinetic paths”. R&sum&-On montre que quand un alliage cristallin est caracterise par deux parametres d’ordre, il peut dbvelopper diverses combinaisons des deux parametres d’ordre en allant vers un Ctat d’equilibre final. L’ivolution cinetique des deux parametres d’ordre dans un alliage temaire B2 est analyde par une thtorie d&tat active. Les mecanismes elementaires de saut de lacunes sont formulis dans une equation directrice. Une equation de transport bi-dimensionnelle decrivant l’tvolution de l’alliage a travers ces duex parametres d’ordre est developpee, et on trouve la chaine d&tats dans cet espace bidimensionnel des parametres d’ordre en rtsolvant numtriquement l’equation de transport. On montre que tant les differentes interactions d’activation pour les sauts par diffusion des differents atomes font que l’alliage passe par differents Ctats hors d’equilibre, ou par differents “chemins cinetiques”. Zusammenfassung-Es wird gezeigt, dal3 eine kristalline Legierung, die durch zwei Ordnungsparameter charakterisiert ist, auf dem Wege zum endgiiltigen Gleichgewichtszustand verschiedene Kombinationen der beiden Ordnungsparameter aufweisen kann. Die kinetische Entwicklung der beiden Ordnungsparameter wird in einer temiiren Legierung mit Ordnung B2 mit Hilfe einer Ratentheorie aktivierter Zustlnde analysiert. Elementare Sprungprozesse der Leerstellen werden in der Formulierung von Mastergleichungen betrachtet. Es wird eine zweidimensionale Transportgleichung, welche die Entwicklung der Legierung mit den beiden Ordnungsparametern beschreibt, entwickelt; die Kette der Zustande in diesem zweidimensionalen Raum der Ordnungsparameter wird ilber eine numerische Losung der Transportgleichung erhalten. Es wird gezeigt, dat3 entweder interatomare Wechselwirkungen oder verschiedene Hohen der Aktivierungsbarriere bei den difusiven Spriingen der verschiedenen Atome dafiir sorgen, da8 die Legierung sich iiber verschiedene Nichtgleichgewichtszustlnde, oder fiber verschiedene “kinetische Wege” entwickelt.
1. INTRODUCTION
Recently a number of kinetic theories have been developed to predict the rates of order-disorder transformations on crystal lattices [l-5]. These theories are concerned with one physically measurable order parameter. Nevertheless, in two of these theories [14], as well as in studies of diffusion in ternary alloys [6, 71, several different relaxation processes were found. Furthermore, different nonequilibrium combinations of short and long range order parameters were obtained by varying the initial state of a binary alloy [3]. These different relaxation processes and different combinations of order parameters hint at a rich variety of observable nonequilibrium states for alloys with two physically measurable order parameters. Any alloy characterized by a single order parameter will necessarily move along the same chain of nonequilibrium states for all temperatures and initial
conditions. On the other hand, the kinetic chain of states need not remain fixed when two or more parameters characterize the state of order in the alloy. The kinetic process must still have a beginning and an end state, but it is possible in principle to connect these ends by different chains of states, here termed different “kinetic paths”. For a given value of one order parameter, states on different chains need not have equal values of the other order parameters. A rich variety in the nonequilibrium states appears possible, and this variety must be understood in terms of a kinetic theory. The present paper develops a kinetic theory for B2 ordering of a ternary alloy with vacancies. This problem has been treated for a binary alloy with vacancies in the Path Probability Method of Kikuchi and Sato [l-3], and for f.c.c. alloys by Radelaar [5]. The use of ternary alloys makes it easier to obtain two independent, physically measurable, order parameters. In principle, three independent pair cor-
823
824
FULTZ:
KINETIC PATH IN TWO ORDER PARAMETERS
relations exist for a ternary alloy. However, we constrain the B2 structure to be composed of two interpenetrating sublattices. The average occupations of these sublattices by the three species of atoms and vacancies provide three independent order parameters. Two of these parameters are the physically measurable pair correlations. The third order parameter characterizes the vacancy distribution, which while physically unobservable, is shown to have a major effect on kinetic processes. The present goal is to find the minimum requirements for obtaining variations in kinetic paths. An equation for the path of the B2 ternary alloy through the state space of its two physically measurable order parameters is found. The formulation is a point approximation, but it gives equilibrium states similar to those of the Bethe approximation. This equation is solved numerically for a variety of: (1) chemical interactions between the atoms, (2) activation barrier heights for atomic jumps, and (3) initial states of the alloy. A wide variety of different kinetic paths is shown possible within this model, and reasons for some of these paths are given.
2. B2 ORDERING OF TERNARY ALLOYS A convenient choice of a ternary alloy with B2 order has a concentration of B-atoms, cs=f, with the half of atoms as C-atoms and A-atoms: cc = c, and c,, = f - c. The B2 structure is represented as two interpenetrating simple cubic sublattices: CL, which is rich in A-atoms and C-atoms, and /I, which is rich in B-atoms. The order parameter “j” equals the number of A-atoms on the a sublattice. The parameter j can range from (N/2) - cN (perfect order), to (N/4 - cN/2) (zero order), to 0 (perfect order of opposite phase), where N is the total number of sites in the crystal. The order parameter “7 equals the number of C-atoms on the a sublattice, and : 0 < i < cN. The number of vacancies on the a-sublattice is “k”, where 0 < k < c,N, and c, is the concentration of vacancies in the alloy. There are 2 atoms in each first neighbor shell about a given atom (in B2, 2 = 8). Note that the first-neighbor sites of an atom on the a-sublattice are all on the /?-sublattice, and vice -versa. Consider the possible diffusive jumps out of the state Ai,, . For the 2 atoms that could jump into the neighboring vacant site, there is a Boltzmann probability, p, for the success of each candidate jump that is determined by the difference between the initial energy of the atom and its maximum potential energy during the diffusive jump, E P = exp( -BE)
where
(1)
and EA = E:, - NM V,
- N,, V,, - N,, V,,
Es = J% - NABVAB -
NB,~BB
-
%C~BC
E,=E$-N,,V,,-N,V,-N,V,,.
Pa) (W
(w
to p, and is determined by the bond energies: {V} (expressed in units of kT), and numbers of bonds between the three species of atoms: {N}. It is assumed that the activation barrier height, E+, depends only on the species of atom whose jump is being considered. Attempt frequencies (i.e. pre-exponential factors) for all diffusive jumps are assumed equal. A chain of kinetic steps cannot be constructed by having B-atoms jumping onto the /?-sublattice independently of the jumps of A- and C-atoms off the B-lattice; there are a tied number of sites on the /I-sublattice, and most of them are occupied. It is tempting to assume that atomic exchanges between the Z- and fi-sublattices occur in pairs. Such a treatment is provided in the Appendix, together with cautions for its use. Here we use vacancy populations on the a- and B-sublattices as agents for the migration of atoms from the /?- and I-sublattices, respectively. With a single order parameter the kinetic Master Equation assumes its familiar canonical form The rate of each candidate jump is proportional
where the state of the alloy is represented by the vector: A, whose components are the states of order. Except for the states of perfect order, a particular Aj is many-fold degenerate; a specific state of order may be realized with a variety of atomic arrangements. The W-matrix induces transitions of the alloy between different states of order. The present interest is in two measurable order parameters and a vacancy order parameter. When the state of the alloy is characterized by three independent order parameters, the Master Equation is $4,
= c
Wi,t.r,,A,m-
c
Wb,ijkA,k
(4)
I.m.n
1.m.n
where the state of the alloy is now a matrix of rank three, and transitions between these states are induced by a W-matrix of rank six. In a vacancy mechanism of atomic transport, the elementary kinetic step involves the jump of a single atom into a vacant neighboring lattice site. This process will cause the number of atoms properly placed on a sublattice will change by at most + 1 or - 1, and the vacancy concentration of the sublattice changes in the opposite direction. The W-matrix assumes the simple off-diagonal structure K/+Ik-l.ijk#O
Wr)-Ik+t.ijk#O
W r+l,k-l.i,kfO
FULTZ:
KINETIC
PATH
IN TWO ORDER
825
PARAMETERS
Wt-I,k+I.,]k#O wtjk-l.,,k#”
2
W ,]ktI.,,kf”
wit
I I’k’
- 1.il.k’
-
Wt.-I J k’ I.t'I'k' I
= J,(i’, j’, k’). w,k.,k
=
(9)
O,
The rate of change in the vacancy distribution also be considered
and all other W ,,k.lmn
+
(5a-h)
O.
must
dk
As an example of an elementary kinetic step, + wi,‘k’+ ,.,‘,‘k w,C, + IA 1.1, A consider the atomic jump rate of an A-atom from the /?-sublattice into a vacancy on the a-sublattice: ,.I, ‘k w, + 1,‘k’ ,.,‘,‘k’ w,,,k W ‘I +,k _ ,,,Jk.This rate is composed of three factors: = J,(i’, j’, k’). (10) (1) the probability of a vacancy on the cx-sublattice (2) the probability of an A-atom in one of the Equations (@-(IO) form a set of coupled differeight neighboring sites of the vacancy, and (3) the ential equations that give the absolute rates of change Boltzmann factor of equation (2). In the Boltzmann of the order parameters. For the present purpose, factor the numbers of A-, B-, and C-atoms on the however, it is interesting to follow the relative rates sevent a-sublattice sites are used to determine NAA, of change of the two observable order parameters, i NAe, and NAc. We assume that the atoms are placed and j. Representing these two order parameters as randomly over their sublattice sites. This averaged orthogonal axes in a two dimensional space. the treatment of neighboring sites is analogous to that trajectories of alloys through this space arc sought. used in the point approximation of equilibrium therAt each state of order (i’, j’, k’), the direction of the modynamics. The expression for W,, +,k_ I,yk is thus trajectory is a vector quantity J(i’, j’,k’)
W t,+Ik-I.l]k=
x{(z-cN-j)$z}exp(-BE,,)
(6)
(r-i-j)
E,,=EX-Y,,ji(Z-1)-Y,, x;(z-l)-v~,i;(z-l).
(7)
Similar expressions can be written for the other five nonzero elements of the W-matrix in equation S(a-f). Equations 5(e) and (f) are the rates of change of k due to the interchanges of vacancies with B-atoms, whose order is not followed explicitly. The six rates { W} from equation (6) and its five analogs depend only on the state of order in the alloy (i.e. on the set: {I, j, k}). For short time intervals these six rates may be assumed to occur independently and in parallel. The net flux of alloys between the states parameterized by j ’ and j ’ + 1, and between the states parameterized by i’ and i’ + 1, is
di -$
I,J,k, =
wi)'
+ I k’ - I.i]‘k’
g
W,,,.+,,_,J,‘,-
E
J,(i’, j’, k’)
-
wi,‘k:,~,~
+ 1k - I
wr~‘-,k’+l.i,‘k’
(8)
iOne of the eight neighboring sites is occupied by the vacancy.
= J,(i’, j’,k’)?+
J,(i’, j’, k’);.
(II)
With only one order parameter, J would lx a scalar, and the alloy would necessarily pass through the same states of order, albeit with rates and directions that are temperature-dependent. This is expected if, for instance, the A-atoms and C-atoms are chemically identical, and the alloy begins in an initial state of disorder. In this case the alloy is expected to take a kinetic path that is a straight line from disorder to order in its normalized two dimensional i-j state space:
J2 = I J, c 5-l (initially disordered, no chemical differentiation of ternaries). However at low temperatures when the C-atoms are chemically distinct, the next section shows that J(i’, j’, k’) varies with temperature, and the kinetic path is generally nonlinear. To compare the present end states with well-known equilibrium results from binary alloys, the case when c = 0 was considered. Critical energies were sought that would let equations (8) and (IO) equal zero for nontrivial j (i.e. j #(N/2) - cN). To do this, small departures of i and j from the disordered state were examined so that the Boltzmann factors had large common factors times second factors, nearly unity, that could be accurately linearized in the departures from disorder. Nontrivial solutions were found in cases when: 4V, = VAA + V,, - 2 VA, = (4/Z - I) = (417)= 0.5714. Some critical energies, 4 V,, from thermodynamic theories are; mean field: 0.500, Bethe pair approximation: 0.575, best known: 0.6294 [8]. Evidentlv the Ioresent kinetic theory_ .provides end
826
FULTZ:
KINETIC
PATH
IN TWO ORDER
PARAMETERS
(the vacancy distribution relaxes quickly), equations (8x10) form a “stiff’ set of differential equations, whose integration can be troublesome. Succeeding values of i and j were obtained by the forward Euler method, the fourth-order Runge-Kutta method with and without adaptive step size control, and the Bulirsch-Stoer method. All methods gave essentially the same results when they were stable. Graphs were made of the points in the i-j state space. Examples of such graphs are shown in Figs l-5. (Here all energies, { V}, are in units of kT. Higher values of { V} correspond to lower temperatures with respect to the critical temperature of the disorder-order transformation.) +N-cN Figure 1 was generated by starting the alloys at CN tcN different initial states of order, and allowing them to 1 - porometer evolve at the same low temperature. Notice that all Fig. 1. Kinetic paths of alloys that started at various initial alloys reached the same end states, which are the states. Initial and final states are indicated by squares and equilibrium states for that temperature. These end circles, respectively. For the light lines: VA, = V,, = V, states therefore move to a higher degree of order at = V,,=O.8; VA,= V,=O; EX=E;=E:; c=O.25; lower temperatures (larger V). The temperaturec, = 0.003. For the dark lines: VA, = V,, = V,, = VA, = 0.3; all other energies and concentrations are the same. dependence of these kinetic paths can be roughly understood in terms of the temperature-dependence of the equilibrium end states; the larger the difference in value of an order parameter in the present and final states that are in reasonable agreement with the Bethe states of the alloy, the more rapid will be the change pair approximation for B2 ordering of an equiatomic of this order parameter. However, there are instances alloy. These critical energies, 4VC= (4/Z - I), are in Fig. 1 where one of the order parameters changed also reasonably close to results of the thermodynamic in the “wrong” direction with respect to the direct pair approximation for other equiatomic alloys. For the 2-dimensional square lattice the present path from initial to final state. Figure 2 shows different kinetic paths at various critical energy, 4V, = (4/Z - I) = (4/3), whereas the thermodynamic pair approximation gives: I .387; temperatures from alloys that started near full disorfor the simple cubic lattice: 4VC= (4/Z - 1) der (k = c, (N/2), j = (N/4) - (civ!Z) + 6,, and i = c(N/2) + &, where 6, and 6, are small positive = (4/5) whereas the pair approximation gives: quantities needed to force the system off the saddle 0.8108. Mean field theories (point approximations) point at zero order.) These same initial conditions provide the critical energy: 4V, = (4/Z). However were used in obtaining Figs 3-5. In Fig. 2 the A- and due to the use of vacancies, our point approximation gives results closer to those of the pair approximation. There is a feature of kinetic paths in the i-j state space that facilitates interpretations of kinetics experiments. While the choice of path is determined by kinetics, the choice of path is independent of the vacancy concentration. An increase in vacancy concentration will cause a proportional increase in the k c rates of equation (6) and its five analogs, so the E *N-+34magnitudes of changes in order parameters of equations (8x10) will increase proportionally. The same kinetic path will therefore be taken by alloys with .7 .3 different vacancy concentrations, or with varying .5 vacancy concentrations, although the alloy will proceed down the path at different rates. D
+N-cNi
3. APPLICATIONS
AND DISCUSSION
Using equations (6) and (7) and their five analogs, equation (11) was integrated by numerical procedures. Initial values of i and’j were chosen, and the relevant W-matrix elements were obtained. Because of the much shorter time scale of the /c-parameter
0
-“B
%cN
Y
1 - parameter
Fig. 2. Kinetic paths of alloys that started at the same initial state, but at different temperatures. The energies: VA,, = V,, = V,-- = VA, = X (where X is marked on the figure): V,,, = V, = 0; EX = I?,+= E; + 8X. c = 0.25; c, = 0.003.
FULTZ:
+N-CNJ
0
KINETIC
+CN
827
PATH IN TWO ORDER PARAMETERS
1
CN
I - porometer Fig. 3. Kinetic paths of alloys that started at the same initial state, but at different temperatures. The energies:
(where X is marked on the figure); v,,. = v,, = 2X; v,, = v,,. = 0; E: = EB = EF. c = 0.25; c, = 0.003.
VAA = V,, = X
C-atoms are chemically identical, and without a difference in activation barrier heights for their diffusive jumps, the kinetic paths would be straight lines from the center of the figure towards the right corner. The equilibrium end states in Fig. 2 are in fact identical to those obtained with: EX = Ei = E,+. However, with a greater diffusive mobility of the C-atoms, E,* < Ez = E,f, there is a temperaturedependent variation from the linear kinetic path. In essence, the lower activation barrier height for the jumps of C-atoms allows the i order parameter to relax relatively more quickly than the j order parameter, especially at low temperatures. Differences in the activation barrier heights {E*} can therefore lead to different kinetic paths, whose choice is temperature-dependent. If the atomic species are differentiated chemically by giving the C-atoms chemical bonds that are twice as strong as those of the A-atoms ( Vcc = V,c = 2 VA,, = 2V,, and VAB= Vsc = 0), the kinetic paths shown in Fig. 3 are obtained. Although all alloys started at the same nearly disordered initial state, there is a marked temperature-dependence of the kinetic paths. The larger chemical bond strengths of the C-atoms cause the i order parameter to relax relatively more quickly, and the kinetic path deviates from the straight line that is taken without chemical differentiation. Differences in chemical bond strengths (V} can therefore lead to differentiate kinetic paths, whose choice is temperature-dependent. As shown in Fig. 4, a peculiar reversal of the i parameter occurs when the B-atoms are chemically very strong, and the C-atoms are chemically weak but the very mobile (Ei z Ei > E$). By monitoring distribution of the various species, it was found that a rapid buildup of the chemically strong B-atoms on the /?-sublattice causes a depletion of vacancies on the
I - porometer Fig. 4. A kinetic path that shows overshoot. The energies: V,, = 0.4; I’,, = 1; VA, = I’,,. = V,., = I’,,- = 0: E: = E; = Ef + 6. c = 0.02; c, = 0.003.
/I-sublattice. The C-atoms prefer the z-sublattice, and with an excess of vacancies on this sublattice, J, becomes quite positive. The i parameter increases rapidly, and over-shoots its equilibrium value. Later as the slower moving, but chemically stronger, Aatoms settle onto the a-sublattice, the A-atoms force the C-atoms off the z-sublattice, and the i parameter decreases to its equilibrium value. As shown in Fig. 5, chemical differentiation of the atomic species can cause an individual order parameter to start to change in the “wrong” direction. With the interaction parameters of Fig. 5, both the Aand C-atoms prefer the z-sublattice, but the A-atoms prefer it quite strongly, and force the vacancies off the a-sublattice. The C-atoms therefore have a preponderance of jumps from the x-sublattice to the fi-sublattice, and the i parameter starts to decrease.
-SCN
CN
I -parameter Fig. 5. A kinetic path that initially goes backwards. The energies: VAA = 1 3; I’,, = I; I’,, = VA, = 0.8; VAs = V,, = 0: E: = E; = E: c = 0.02; C, = 0.003.
828
FULTZ:
KINETIC PATH IN TWO ORDER PARAMETERS
However, all of the A-atoms can be accommodated on the a-sublattice by displacing only the B-atoms. This occurs later and the i parameter then increases as expected. Variations in kinetic paths are a means of varying the microstructure of nonequilibrium materials. An experimental study of variations in kinetic paths must be able either to prepare alloys with various degrees of initial order, or must employ alloys that have chemically distinct elements. Some variations in initial state may be possible through an initial quench from a temperature between the critical temperatures of the two order parameters. For experiments starting with an initial state of disorder, it behooves the experimentalist to find, for example, a ternary alloy with pairs of components that differ in their heats of formation of intermetallic compounds (implying different {V}), or components that differ in their diffusive mobilities (implying different {E*}). With such an alloy system a’ variety of kinetic paths, and therefore novel microstructures, can be obtained by thermal processing. 4. CONCLUSION When the state of an alloy is characterized by two or more order parameters, it is possible for the alloy to traverse a rich variety of kinetic paths in order parameter space. In other words, the Markov chain of nonequilibrium alloy states can be temperaturedependent not only in its absolute rate, but also in its choice of states in the chain. In the present formulation for a ternary alloy with two order parameters, the alloy generally follows different chains of states at different temperatures. Variations in either the chemical bond energies or the activation barrier heights will cause the kinetic path to vary with temperature. The agents of kinetic change are vacancies, and their distribution between the two sublattices is important in determining the kinetic path. When the vacancy concentrations on the two sublattices become substantially unbalanced, as they will when one species of atom prefers a sublattice so strongly that the vacancies are forced off, atoms may jump onto the sublattice with the surplus of vacancies even if this sublattice is energetically unfavorable for them. Order parameters can hence change backwards, and this leads to a wide variety of kinetic paths. Acknow\e&ements-In the process of implementing a similar theory for B2 and DO, ordering, Mr L. Anthony
3. K. Gschwend, H. Sato and R. Kikuchi, J. them. Phys. 69, 5006 (1978). 4. Y. Saito and R. Kubo, J. Stat. Phys. 15, 233 (1976). 5. S. Radelaar, J. Phys. Chem. Svlidy 31, 219 (1970). 6. D. de Fontaine, J. Phys. Chem. Solids 34, 1285 (1973).
7. M. Murakami, D. de Fontaine, J. M. Sanchez and J. Fodor, Acra metall. 22, 709 (1974).
8. D. de Fontaine, Solid St. Phys. 34, 73 (1979). APPENDIX The numerical inconvenience of the “stiff’
equations
(8x10) motivated an alternate formulation of kinetic paths. The formulation presented in this Appendix makes nominal use of vacancies but maintains a constant vacancy distribution between the two sublattices of the 82 structure. It becomes equivalent to a mechanism of interchanging first neighbor pairs of atoms. Kinetic paths predicted with this mechanism depend on chemical interaction energies and activation barrier heights, but the paths are shown lo disagree with those of the vacancy mechanism of the main text when unsymmetrical interatomic interactions or activation barrier heights lead to unequal vacancy populations on the a- and /J-sublattices. A chain of kinetic steps cannot be constructed by having B-atoms jumping onto the /? sublattice independently of th; iumos of the A and C-atoms off of the B sublattice. To con&uct a chain of states, as elementary iinks six pairs of jumps are used. These links are depicted schematically in ’ Fig. Al. Processes that involve jumps of two atoms of the same type, since they contribute nothing to the changes in order parameters, are ignored. Unlike the two elementary vacancy jumps of which they are composed, a kinetic path can be constructed by connecting these six links together in any order. These six links can therefore be considered to operate independently of each other and in parallel. Here the rate: Wan, for example, designates the jump of an A-atom lo the a sublattice followed by the jump of a B-atom to the /l sublattice. During this process the vacancy is initially on the z sublattice, jumps to the /? sublattice, and ends on the z sublattice. None of the six elementary links change the vacancy distribution. After this process any of the six elementary links can then follow. The average rates of the six links, {wAs. wl\c, wBA,wBc,wc-.. wCB},involve two processes in series, and have the form
;r\ B
c
B
A yr\
wBC
wBA
independently checked many of the numerical integrations perfbrmed in-the present woik. This work was supported by the U.S. Department of Energy under contract DE-FGO386ER45270.
;A C
REFERENCES I. H. Sate and R. Kikuchi, Ada mekall. 24, 797 (1976). 2. H. Sate, K. Gschwend and R. Kikuchi. J. Physique CI, 357
(I 977).
vc%A
B
C
A ;/\
wCB
Fig. Al. The six independent transition processes in B2 ordering of the ternary alloy.
FULTZ:
KINETIC
PATH
IN TWO ORDER
\
PARAMETERS
829
There are six types of jumps of atoms into vacancies: {B’,(A). B’,,(B), B’&). B,,(A), B,,(B). W,,(C)}, which when combined in pairs, are used to generate the six links H.‘c*.rice}.. {w*rl. WAC,H’sA.H’BC.9 Because the atomic jump processes are grouped in pairs (Fig. Al), these elementary processes can occur in any order. These processes arc assumed to occur in parallel, so the average rate of flow of alloys between states A,, and Ai,, , is obtained by recognizing that the links ~‘~sand H’~< serve to increase j. and ~‘s,, and w‘(~decrease j. (The links wcs and rvs(- have no effect on j.) The rates of change of j and i are Zj r
=
was(j',j')+WA(.(i'.i')
1,
- [waA(i’, j ‘) + wcA(i’, j ‘)]. and
1 -parameter
Fig. A2. Discrepancy in kinetic paths predicted by interchanging pairs of atoms (formulation in Appendix) and by using vacancies as the agents of change (main text). Both cases started at the same nearly disordered state, and used the energies: V,, = 0.5; V,, =0.5; V,, = V,, = 2; V,, = VW = 0; E: = Et = E:. c = 0.25. For the vacancy mechanism: c, = 0.003.
I H’
(Al)
-_-__-
AB .__.
W&V
‘_
+
w,(B)
where, for example
(A21
W,I(A)=(:-cN-j)iZexp(-/?E,& with EAa= E: - V,, j 5 (Z - I) - v,,
- V,,i$
N _ -i-j ( 2
i(Z-1) >
(Z - 1).
(A4a)
(A3)
di (2r I,,.=
H’(.l\(i’,j’)+W,(.B(i’,j’) - [w,Ji’,
j’) + wBJi’. j’)].
(A4b)
The numerical integration of these equations is straightforward, and can be accurately and quickly performed with the forward Euler method. An example of a significant discrepancy between the kinetic path obtained with this present “interchange” mechanism and the “vacancy” mechanism of the main text is shown in Fig. A2. The path labeled “vacancy” is similar to the kinetic paths of Fig. 3. However, the curve labeled “interchange” is substantially different, and examining the individual rates (w} revealed the cause. Although the Catoms are capable of quick jumps from either the a- or fi-sublattices, the rates: n’xc, H’~. ~1~~.wcs that involve the motion of the C-atoms are dominated by the slower-moving A- and B-atoms because of the harmonic mean in equation (Al). Of the motions of the A- and B-atoms, the rate: IV&A) is large because of the large VAc. Consequently the rate wAcis large, and the i parameter changes slowly at first. On the other hand, when the vacancies are allowed to change their distribution between the two sublattices, they will be depleted from the a-sublattice by the energetic C-atoms. This retards the increase of the j parameter, and the kinetic path marked “vacancies” in Fig. A2 is obtained.