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THEORY OF CORRELATION EFFECTS FOR DIFFUSION IN A TWO SUBLATTICE STRUCTURE
J. PhysChem SolidsVol58, No.
@i!9
PII: S0022-3697(96)00126-6
Pergamon
2, pp. 287–294, 1997 G 1997 Elsevier Science Ltd Printed in Great Britain. All rkhts reserved 0022-3697/97 $i7J30 + 0.00
THEORY OF CORRELATION EFFECTS FOR DIFFUSION SUBLATTICE STRUCTURE
IN A TWO
Z. QIN and A. R. ALLNATT Department of Chemi8try,Universityof WesternOntario, London, Ontario, N6A 5B7,Canada (Received12February1996;accepted21May 1996) Abstract-The tracer diffusioncorrelation factor has been studied for a simplecubic lattice gas with two energeticallyinequivalentsublattiees and sites restricted to accommodate a maximumof one atom. The modelservesas a paradigmfor solid8withsiteinequivaleneebut no algebraicformulaehavepreviouslybeen
derivedand comparedwithavailableMonteCarlosimulationdata. An expressionbasedon a kinetic equationtheorywhichneglectsthehighestorderinconcentrationfluctuationsis shownto leadto a good descriptionof the simulationresults.The timedependenceof the associatedtimecorrelationfunction appearsto be unusuallycomplexfor diffusionin a two sublattice 8ystem.~ 1997Elsevier ScienceLtd.
All rights reserved
Keywmis:A. alloys,D.diffusion,D. transportproperties.
1. INTRODUCTION
There has been much activity in recent years in the simulation of the transport coefficientsof lattice gas models of concentrated alloys and highly defective solid materials [1–3]and the results have led to many insights into the factors controlling the nature and magnitude of the often large correlation effectswhich have been observed. On the other hand, progress in the interpretation of such results through detailed statistical mechanical calculationshas been relatively slow and incompletedespite steady advances by several different theoretical methods. The latter include the path probability method [4–6], the solution of time-dependentkineticequations [7–9],and the use of random walk methods which build on the successof such methods in dilute alloys [10–12]. Important among the models studied by both simulation and theory are those in which atomic transport occurs on two inequivalent interpenetrating sublattices, as for example in lattice gas models of ordered alloys, e.g. [13], or highly defective solids, e.g. [14], based on nearest neighbour interactions between the atoms. The complexityof suchmodelsled to the introduction by Murch and coworkers [15,12] of a simpler two sublattice model in which the jump frequency of an atom to a neighboring vacant site depends only on the chemical species of the atom and the sublattice from whichit jumps. In the limitsthat Cv+ Oand the jump frequency of an atom is independent of its sublattice this reduces to the well-known Manning
random alloy model [16,3]. Simulations have been made of the correlation function for the electrical conductivity[15,17]and of the tracer diffusioncorrelation factor [15,18]for the pure crystal limit of the Murch model and it was found that the results exhibited certain features characteristic of a two sublattioe highly defective solid. The sole analytical result for these pure crystal systems is a formula for the conductivity due to Richards [19]which also provides insight into the relaxation time associated with the frequencydependent conductivity [17].An extensive theoretical study of the incoherentscatteringfunction and the tracer diffusioncoefficienthas also beenmade [20]but the final stages were numerical and did not provideany algebraicformula for the diffusioncoefficient. The objective of the present paper is to show that the kinetic equation method successfully employed in earlier studies of the random alloy [8,9, 21]can be extended to provide a useful approximate expressionfor the tracer diffusioncoefficientin the pure crystallimitof the Murch model.The method also provides an interesting insight into the time dependenceof the associated relaxation process. 2. APPROXIMATEKINETICTHEORY
We consider a simplecubic lattice of N sites which is made up of two interpenetrating f.c.c. sublattices (a and b) so that the nearest-neighboursof an a site are all b sitesand viceversa.Each siteis occupiedbyeither a host atom, a tracer atom or a vacancy;the tracers are
287
Z. QIN and A. R ALLNATT
288
An atom can migrate by exchange with a nearest-neighbour vacancy with jump frequencies denoted by w“ and w*for host atoms and by W;and w! for tracer atoms, where the superscript a (b) signifiesthat the atom jumps from an a (b)sublatticesite.The concentrations of host atoms, tracer atoms and vacancies on the sublattices a and b, denoted (c”,Cb), (c+,c$-)and (c;, c$), respectively,are each definedas the fraction of the sites on the relevant sublattice occupiedby the species.The principle of detailed balance relates the rates of jumps in the two directions between neighboring sites and yieldsthe expression present in very small concentration.
which determine the time dependence of the time correlation functions is d~v~(l’, 1’ – r; t) dt = O(r)[w~@v~(lX – r. 1’;t) — w~f/&T(lx, lx – r; t)] + ~ 19(r’)[~’”y@VTH(lx – r’, lx – r, I“x;f) r’#r — W14’VTH (1”’,1“’– r, 1“’– r’; f)]
+~
~(i)
[@thTV
(1”’, lx – (r – +),
1“’ – r;
t)
r’#r — w;@vTv (1’, lx – r, 1’ – (r – r’); t)].
and a similar equation for tracer atoms. These equations, in combination with the condition that the total atom concentration c is constant and equal to (c”+ cb)/2, determine the equilibrium distributions of atoms and vacancies between the sublattices at equilibrium, see Ref. [15]for details. Linear responsetheory leads to exact time correlation function expressions for transport properties which can be simplified in structure for particular models [3].A minor extension of arguments detailed earlier for the Manning random alloymodel [3]shows that in the present case the expressionfor the tracer correlation factor, ,f_,reducesto
+ kt$+”(ja(o) + Cbcbb(o)]} (2) where, for later convenience,we have used the notation that ~(p) is the Laplacetransform withrespectto tof a function F(t). The time correlation functions CX}, (t) are definedby the equation Cxy(t)= E(s. So/S2)&,(s;I:
SO)
We have introduced the notation that x = a if x = b and vice versa, and have defined O(r) to be 1 if r is a nearest-neighbour vector and zero otherwise. Here and whereverconvenientall subscriptsand arguments of @functions referring to the initial state have been omitted where they are the same as in eqn (4). The subscript H denotes the host atom speciesand @my whereX, Y = T, H, is the conditional prob(1,m, n; t), ability of observingparticles of speciesV,X, Y at the different sites 1, m, n, respectively, at time t.The probability ?/~vTv can be eliminated by the identity r/&TV(l,
= ‘@VT(l, m; t)–‘@vTH(l, III, II; ~) –4JVTT(11 w u ~)j
f = I - {W;[Ca~a.(O) + Cbcab(())]
(X, Y= a, b).
s
(3) Here s and and s. are nearest neighbour vectors of magnitudes and we define @.ry(r; t : I’0),for anYPair of lattice vectors r and ro, by the relation
(4) where, on the right hand side, $ is the conditional probability of observing a vacancy at the site 1’ on sublattice x and a tracer at the site l“’– r at time t if there was a vacancy at l; and the same tracer at l; – ro, respectively,at time zero. The kinetic equation for the probability functions
(5)
(6)
where, since we assume that cT+ O, rJw~ can be neglectedwhenm, and n, are differentsites. In order to introduce our basic approximation we note that the definitionof @vTH can be written as
7)VTH(IX, my,n’; f) = ~f)(1x)4T)(mJ’)4 H)(nz) 0.7 X
G7~(t)p~v) (lo)P~T) (10– ‘0)~6
(7)
in which Pfl is the probability of finding the systemin configuration @and G70(t) is the conditional probability that a system at thermodynamic equilibrium initially in configuration/3will be in configuration ~ after time t.The occupancyvariable P$x)(1)is unity if site 1is occupied by a particle of speciesX in system configuration~ and is zero otherwise.We can expand each occupancy variable of species V and H as its average value, which is the concentration of the species,plus a fluctuation, e.g.
p$)(l”’)= c; + Ap\v)(l-’),
(8)
Diffusionin a two sublattice structure
so that @VTH(lx,my, n’; t) = cfJc’@T(mY;t)+ Cz@dv’r(lx,My;t) + c~?#TAH(my,n’; t)
(9)
+ ‘@AVTAH(lx, my, n=;1).
The subscript AX denotes that the corresponding occupancyvariable in @is a fluctuation Ap(x)rather than p(x).In like manner we have ?/@(lx,my;t) = c~~(my; t) +@AvT(lx,my;t). (10) It is useful to note that, since we assume CT~ O,it my;t) is just the can readily be shown that @AHT(lx, negativeof @AvT(lx, my;t). Our basic approximation is to neglectall the terms of secondorder in fluctuations,suchas ~AwAH,in the kineticequation. The sametype of approximationhas beenusedfor the binary random alloymodel [9]under the name simple decoupling approximation (since it decouplesthe hierarchy of kineticequations [3]).By a summation over lx and some straightforward manipulations the kinetic equation is then transformed to
289
Weuse the notation that F(k) is the Fourier transform with respectto r of a function F(r). Equations (3)and (14)yield, by manipulations similar to those encountered in earlier work [3],expressionsfor the Laplacetransformed time correlation functions: C..(P) = [Qm(P) + (cbwf – @) Q(p)] /f2(p)
Cab(p) = [Q.b(p) – CaWfQ(P)]/~(P),
(17a) (17b)
plus equations obtained by interchanging a and b. Here we have defined Q(P) = Qaa(p)Qbb(p) Q(p) = 1 + ~
- Qab(p)Qba(p)
(18)
[(c%: – l’7’)Qxx(p)+ c’w&(p)]
x=a,b + (W%b – Caw!wa– cbw@~)Q(p)
(19)
and Qxy(P) = ~(%/~2)%4s s = Gxy(o;p) –
– %P)
G.y(–2so; p), (20)
d&(r; t)
+ zWxf#+(r;t) dt -~e(r’){cj~~xy(r -r’;t) 8 + (cxwx+ c~wf)q5zy(r– r’;t)}= Sxy(r;t) (11)
where z is the number of nearest neighbors per site (six for the simplecubic lattice) and Sxy(r;t) = f3(r)[c’”ti&y(-r;
f)
(12)
+ (Vi – cxw$)f#xy(r;t)]
Wx= Cxwx+ C+wx+ C;w+.
(13)
Terms independentof r whichmake no contributions to the subsequentexpressionsfor the time correlation functions CXY(t)have been omittedineqn(11). The pair of equations obtained from eqn (11) by putting x = a, b may be solvedafter Fourier–Laplace transformation; the result is ~xy(k~)
=
~
f-%z(k~)[~zy(b)
Z=.y,f
+ Ozy(kt
=
0)1
where the second equality followsfrom the structure of the simplecubic lattice. Before turning, in the next section, to the final step of evaluatingthe functions GXY appearing in eqn (20), we note that the results above are exact in the limit Cv-0 and that –w~~Xy(0)is then the averageof the cosine of the angle between an initial tracer-vacancy exchange,before which the tracer is on sublattice y, and a final exchangeof the same vacancy and tracer, after whichthe tracer is on sublatticex, whenthere are no restrictions on the number of tracer-vacancy exchanges between the initial and final exchanges. The above analysisthen relates these average cosines to quantities –WXQXY(0) whichare definedin the same way as the –W~Cxy(0) except that they refer to averages of cosines between successiveexchangesof a particular host atom and a vacancy in a tracer-free crystal. Furthermore GXYcan be identified as the Green function for a vacancy random walk which starts on they sublattice and ends on the x sublattice.
(14) 3. CALCULATION ANDRESULTSFORTHETRACER CORRELATION FACTOR
where GXX(k,p)= [p+ ZWX– c~w@(k)]/D(k,p)
(15a)
Equations (2)and (17)-(20)can be combinedto give a formalexpressionfor the tracer correlationfactor. In (15b) ~xx(k,p) = (Cxti + c@7)0(k)/D(k,p) the case of a finite concentration of vacancies the D(k,p) = [p+ Z@ – c$w@(k)] functions Gxy(p) in eqn (20) cannot be determined analytically(even for the limit of p = O required to x [P+ z~b – clwfe(k)] – w“wb[fl(k)]2. calculate the tracer correlation factor). We consider (16) two approximationsleadingto analyticalresults.
290
Z. QIN and A. R. ALLNATT
also known [23] that the IR result can be transformed to the exact one by taking MO = 2~0/(1–JO) where ,fi is the Bardeen–Herring correlation factor (~0= 0.6531for the simple cubic lattice), and we have shown (unpublished work) that this is also true for the two sublattice result, eqn (21), in the limit Cv~ O. As is shown in Fig. 1, the modified IR result, corresponding to taking MO= 2~0/(1–~0) = 3.765 in eqn (21) gives improved results at all concentrations as well as having the correct limiting value; however, it does not appear to be the result of a systematic approximation to the kinetic theory. A more satisfactory approximation appears in Section 3.2. The specialstructure of the two sublatticeequations isalsoapparent in the timecorrelation functionsin the IR approximation; these are
3.1. Immediate reversal (IR) approximation In the case of a uniform lattice (w”= Wb,W;= w!) with Cv+ O,it is known [23]that a useful short time approximation for the singletimecorrelationfunction Q(t) is obtained by assumingthat the dominant contribution comesfrom the finaltracerjump whichis an immediate reversal of the initial tracer jump, without any other vacancy jumps in between. In the present context this immediate reversal (IR) approximation correspondsto neglectingall terms containingO(k)in the Green functions. When this approximation is made in eqns (17) then the result for the tracer correlation factor is found to be
C..(t) = exp(–zwvt){[z(w$– wv)/A] where MO= 5. Predictions of the IR approximation for ~ as a function of total atom concentration c for differentvalues of the jump frequencyratio
(23)
Cab(t)= –[cbw~/A]exp(–zwvt)sinh(At),
(24)
plus two expressionswith the a, b labelsinterchanged, where
(22)
13= WafWb
x sinh(At) + cosh(At)}
are compared in Fig. 1with the Monte Carlo resultsof Murch [15],which take w; = w’ (x = a, b). A particularly unsatisfactory aspect of the result is that the low vacancyconcentration limit is incorrect although it would be exact for an exact solution of the simple decouplingapproximation. In the case of a uniform lattice with Cv-+ Oit is
W$= [(z– I)mf + c“’w;]/z
(25)
Wv= (w; + w$)/2
(26)
A* =
Z2(W$
–
W$W~)
+
CaCbWfW:.
(27)
It is clear that when the linear combination of four time correlation functions in eqn (2) for f is consid-
1
0.8 0.6 f
0,4 *-
0.2 1
0 0
0.2
I
1
0.6
0.4
I
0.8
1
c Fig. 1.Thetracercorrelationfactorf asafunctionofatomconcentration c forvahresofthejumpfrequency ratio@asmarked. —–0– – MonteCarlosimulation,[24]for ~ = 1 and [15]for others;- - - - - IR approximation;— modifiedIR approximation.
Diffusionin a two sublatticestructure
ered as a single overall time correlation function the result is not a single exponential decay in time, although it can be shown that the expression does have the correct zeroth and first moments. This is in contrast to the single exponential in time, which correctly reproduces the zeroth and first moments of the exact time correlation function, found for the IR approximation to the binary random alloy model [9], the uniform lattice limit of the present diffusion calculation [23],and evenfor the frequencydependent conductivity of a pure crystal in the present model (wherewefind the result is the sameas Richards [19]). Evidentlythe time dependenceof the relaxation associated with diffusionin two sublattice models is especially complex, and care needs to be made in proposing simple approximations by analogy with simplersystems.
3.2. Small-k
approximation
For p = O eqn (15) can be written in the compact form PO+
Pl~(k)
“’(k’ 0)= 6(a0 + al y(k) + &(k)]2)
(28)
291
? = r/a. We also defined h(F)= cos(AF)/7 = exp(–l JIF)/7
for J2 >0 for J2 <0
A*= 6(ao + al + a2)/(al + 2cx2).
and for the simplecubic lattice [3] ~(k) a @(k)/6 = [cos(kXa) + cos(lrya) + cos(kZa)]/3
(30) where a is the nearest neighbour distance. The dominant contributions in the Fourier inversion of eqn (28) come from small k because at large k the factor exp(ik.r) undergoes rapid oscillations. In this limit y(k) + 1 – (lCa)2/6. Substituting this and extending the range of integration over all k-space we obtain
where we omit the p = O argument in ~ and put
(33)
This small-k approximation is well-known [3]when applied to the Green function for a random walker on a perfect lattice where, for example, the approximate and exact values for the simplecubic lattice are (0.477,0.516),(0.338,0.331),(0.276,0.261)for 7 = 1, J2, ~3, respectively.In viewof the errors introduced by our basic assumption (neglect of second order fluctuations) the small-k approximation will be adequate, and has the advantage of allowing an analytical result. We also need an approximation for the Green functions when r = O.This can be obtained by convertingeqn (28)to an integralequation by multiplying through by the right hand denominator in eqn (28) and inverting the Fourier transform. After some straightforward calculation one finds GXY(0) = IPO–6a1~XY(a)–4@xy(fi~) – ~2~Xy(2a)]/(6ao+ CY2).
where
(32)
(34)
The Glyon the right hand sidecan be calculated from eqn (31).Equations (2),(17),(20),(31)and (34)can be combinedto givean expressionfor the tracer correlation factor, whichweneednot repeat herein viewofits length. As can be seenfrom Fig. 2, the small-kapproximation provides a good approximation to Murch’s Monte Carlo results [15]for f as a function of c for four values of ~ (= 1,0.1,0.02,0.0025).The discrepancies become more significant as the jump frequencyratio deviatesmore from one. Other Monte Carlo simulations have been made by Tahir-Kheli et al. [18]for ~ in the range 0.086-1 with Cvheld constant at 0.5, and with c“ (the concentration of atoms on the low energy lattice) held constant at 0.75 and 0.875.As shown in Table 1, our small-k approximation results are in comparable or better agreement with these simulation results as the numerical results from the approximate theory of Holdsworth er al. [20].For these results, as for the Monte Carlo results of Murch shownin Figs 1 and 2, the modified IR approximation gives results close to the small-kapproximation and the IR approximation itself is less satisfactory. There are no simulation results for the case when the jump frequencies of a tracer and a host atom are different. The behaviour of the small-k approximation for ~ as a function of c at @= 0.01
292
Z, QIN and A. R. ALLNATT
1
0.8
0.6 f 0.4
L
“v 002
●
0.0025
,“ ,*’
‘ -- “
4
0.2 0
0,2
0
I
I
0,4
1
0.6
0.8
1
c Fig.2. Thetmcercorrelation factor~ ma functionofatom concentrationcforvalues ofthejump frequencyratioflas marked. —-0– Monte Carlo simulation,[24]for ~?= 1and [15]for others; -––— small-kapproximation,
Table 1.Simulatedand theoreticalvaluesof the tracer correlation factor ,f’whentracer and host atoms have the samejump frequencies.The first sevenvaluescorrespondto a total vacancyconcentration of 0.5 c“
c’
/3=
0.5 0.5448 0.6017 0.6586 0.7155 0.75 0.7724 0.75 0.75 0.75 0.875 0.875 0.875 0.875 0,875
0.5 0.4552 0.3983 0.3414 0.2845 0.25 0.2276 0.75 0,5 0,475 0.875 0.75 0.625 0.5 0.375
1 0.69793 0.43808 0.26865 0,15807 0.11111 0.08681 1 0.33333 0,3016 1 0.42857 0.23810 0.14286 0.08571
w“/wlh
Simulation[18]
~ Small-kapprox.
Theory of [20]
0.846 0.842 0.830 0.806 0.767 0.733 0.707 0.751 0.774 0.774 0.703 0.719 0.724 0.712 0.682
0.8507 0.8474 0.8338 0,8092 0,7721 0.7422 0.7191 0.7579 0.7776 0.7766 0.7056 0,7235 0.7252 0.7131 0.6864
0.8494 0.8471 0.8363 0.8159 0.7838 0.7569 0.7354 0.7584 0.7841 0.7841 0.7077 0.7253 0.7345 0.7289 0.7075
for Q s w~/wa= w~/wbin the range O.1–10is shown in Fig. 3. At the limit c = 1 these results are in good agreement with the prediction of eqn (21) with MO= 2~0/(1–~0), which is exact in this limit; the results are 0.1, 0.4, 0.7, 1.0, 1.6°Alow for a = 0.1, 0.5, 1, 2, 10,respectively. 4. DISCUSSIONAND CONCLUSIONS
A formula for the tracer diffusioncorrelation factor has been derived for a two sublatticemodel of a pure crystal with an arbitrary vacancy content using a simple decoupling approximation similar to that employed in [20]for the same model in the context
of a theoryof the incoherentscatteringfunction. It can be shownthat in the limitingcase of a uniform lattice (w”= w~, w; = w:) the simple decoupling approximation expressionsof Section 2 lead directly to the result of Tahir-Kheli and Elliott [22]for the tracer correlation factor. This is encouragingsincethe result in [22]is remarkably accurate (better than 1/2°/0for the simple cubic lattice [3]).In this limiting case the only quantity requiring numerical calculation can be expressed in terms of the Bardeen–Herring correlation factor ~. for tracer diffusionin the limit Cv-+ O, which is of course a well-knownquantity but for the full two sublattice model such an identification in terms of known functions does not seem possible.
Diffusionina two sublattice structure
293
1
0.8 0.6 f
0.4 0.2 I
0 0
0.2
1
I
0.4
0.6
I
0.8
1
c Fig. 3. The tracer correlation factor _f in the small-k approximation as a function of atom concentration c for the jump frequencyratio ~ = 0.01and the ratio a as marked.
However, the additional use of the small-k approximation allows an algebraic result to be obtained which provides a useful description of the available simulation results and also givesresultscloseto those obtained numerically in [20].The calculations could be extended to other lattices in which the set of nearest-neighbour lattice vectors is the same for all sites independent of sublattice. Another analytical result was obtained by introducing the immediate reversal (IR) approximation into the simpledecouplingexpressions,but this givesless satisfactory results for ~ unless an empirical adjustment is made so as to reproduce the exact limiting behaviourfor smallvacancycontents. For the random alloy on a uniform lattice this adjustment actually converts the IR transport coefficient(but not its time correlation function)exactlyto the simpledecoupling result [9], but for the present two sublattice model this is not the case except for Cv~ O. A further indicator of the complexity of diffusion in the two sublattice model is provided by the time dependence of the associated time correlation function; the simplest viable approximation (the IR approximation) no longer corresponds to a single exponential in time, unlike other simple models enumerated in Section 3.1.
Acknowledgement—This researchreceivedfinancialsupport fromtheNaturalSciences andEngineering ResearchCouncil ofCanada.
REFERENCES 1. MurchG. E., Dz@rionin CrystallineSolids.(Eds G. E. Murch and A. S. Nowick), p. 379. Academic Press, Orlando (1984). 2. Murch G. E. and Zhang L., D@sion inMaterials.(Eds A. L. Laskar, J. L. Bocquet,G. Brebecand C. Monty), p. 251.KluwerAcademicPublishers,Dordrccht (1990). 3. Allnatt A. R. and Lidiard A. B., Atomic Transportin Solids.CambridgeUniversityPress, Cambridge(1993). 4. Sato H., Prog. Theoret. Phys., Supplement115, 317 (1994). 5. Sato H., NontraditionalMethodrinDz@sion.(EdsG. E. Murch, H. K. Bimbaum and J. R. Cost), p. 203. Met. Soc.AIME, New York (1984). 6. Akbar S. A., J. Mater. Sci. 27,3125(1992). 7. HoldsworthP. C. W. and Elliott R. J., Phil.Mag.A 54, 601(1986). 8. Moleko L. K., Allnatt A. R. and Allnatt E. L., Phil. Mag.A 59, 141(1989). 9. Qin Z., Allnatt A. R. and Allnatt E. L., Phil.Mag.A 71, 291(1995). 10.QinZ. andMurchG.E.,PhiLA4ag.A 66,957(1992);68, 819(1993);70,481 (1994). 11. Qin Z. and Allnatt A. R., 71,307 (1995). 12. Belova1.V.,IvoryM. E. and Murch G. E., Phil.Mag.A 72,671 (1995). 13. Zhang L., Oates W. A. and Murch G. E., Phil.Mug.A 58,937 (1988);60,277 (1989);Qin Z. and Murch G. E., Phil.Mag.A 67,757 (1993). 14. Murch G. E. and Thorn R. J., PhiLMug. A 36, 517 (1977);Murch G. E. and ZhangL., SolidStateIonics47, 125(1991). 15. Murch G. E., J. Phys. Chem.Solids43,243 (1982). 16. ManningJ. R., Dzj%sionKineticsfor Atomsin Crystals. Van Nostrand Reinhold,Princeton,New York (1968). 17. Allnatt A. R. and Allnatt E. L., Phil.Mug.Lett. 64,393 (1991). 18. Tahir-KheliR. A., ElliottR. J. and HoIdsworthP. C. W., Phys.Rev. B. 34,3233 (1986).
294
Z. QIN and A. R. ALLNATT
19. Richards P. M., J. Chem. Ph~s. 68,2125 (1978); Fasl Ion Transport in Solid.~(Eds P. Vashista, J. N. Mundy and G. K. Shenoy), pp. 349. North-Holland, Amsterdam (1979), 20. Holdsworth P. C. W., Elliott, R. J. and Tahir-Kheli R. A., Phys. Rev. B. 34,3221 (1986).
21. Allnatt A. R,, Phi[. A4ag. A 64, 709 (1991). 22. Tahir-KheliR. A, and Elliott R, J., Phys, Rev. B. 27,844 (1983). 23. Qin Z. and Alhratt A, R., Phil. Mag. A 70,677 (1994). 24. Murch G. E., Phil. Mug. .4, 49,21 (1984).
@i!9
PII: S0022-3697(96)00126-6
Pergamon
2, pp. 287–294, 1997 G 1997 Elsevier Science Ltd Printed in Great Britain. All rkhts reserved 0022-3697/97 $i7J30 + 0.00
THEORY OF CORRELATION EFFECTS FOR DIFFUSION SUBLATTICE STRUCTURE
IN A TWO
Z. QIN and A. R. ALLNATT Department of Chemi8try,Universityof WesternOntario, London, Ontario, N6A 5B7,Canada (Received12February1996;accepted21May 1996) Abstract-The tracer diffusioncorrelation factor has been studied for a simplecubic lattice gas with two energeticallyinequivalentsublattiees and sites restricted to accommodate a maximumof one atom. The modelservesas a paradigmfor solid8withsiteinequivaleneebut no algebraicformulaehavepreviouslybeen
derivedand comparedwithavailableMonteCarlosimulationdata. An expressionbasedon a kinetic equationtheorywhichneglectsthehighestorderinconcentrationfluctuationsis shownto leadto a good descriptionof the simulationresults.The timedependenceof the associatedtimecorrelationfunction appearsto be unusuallycomplexfor diffusionin a two sublattice 8ystem.~ 1997Elsevier ScienceLtd.
All rights reserved
Keywmis:A. alloys,D.diffusion,D. transportproperties.
1. INTRODUCTION
There has been much activity in recent years in the simulation of the transport coefficientsof lattice gas models of concentrated alloys and highly defective solid materials [1–3]and the results have led to many insights into the factors controlling the nature and magnitude of the often large correlation effectswhich have been observed. On the other hand, progress in the interpretation of such results through detailed statistical mechanical calculationshas been relatively slow and incompletedespite steady advances by several different theoretical methods. The latter include the path probability method [4–6], the solution of time-dependentkineticequations [7–9],and the use of random walk methods which build on the successof such methods in dilute alloys [10–12]. Important among the models studied by both simulation and theory are those in which atomic transport occurs on two inequivalent interpenetrating sublattices, as for example in lattice gas models of ordered alloys, e.g. [13], or highly defective solids, e.g. [14], based on nearest neighbour interactions between the atoms. The complexityof suchmodelsled to the introduction by Murch and coworkers [15,12] of a simpler two sublattice model in which the jump frequency of an atom to a neighboring vacant site depends only on the chemical species of the atom and the sublattice from whichit jumps. In the limitsthat Cv+ Oand the jump frequency of an atom is independent of its sublattice this reduces to the well-known Manning
random alloy model [16,3]. Simulations have been made of the correlation function for the electrical conductivity[15,17]and of the tracer diffusioncorrelation factor [15,18]for the pure crystal limit of the Murch model and it was found that the results exhibited certain features characteristic of a two sublattioe highly defective solid. The sole analytical result for these pure crystal systems is a formula for the conductivity due to Richards [19]which also provides insight into the relaxation time associated with the frequencydependent conductivity [17].An extensive theoretical study of the incoherentscatteringfunction and the tracer diffusioncoefficienthas also beenmade [20]but the final stages were numerical and did not provideany algebraicformula for the diffusioncoefficient. The objective of the present paper is to show that the kinetic equation method successfully employed in earlier studies of the random alloy [8,9, 21]can be extended to provide a useful approximate expressionfor the tracer diffusioncoefficientin the pure crystallimitof the Murch model.The method also provides an interesting insight into the time dependenceof the associated relaxation process. 2. APPROXIMATEKINETICTHEORY
We consider a simplecubic lattice of N sites which is made up of two interpenetrating f.c.c. sublattices (a and b) so that the nearest-neighboursof an a site are all b sitesand viceversa.Each siteis occupiedbyeither a host atom, a tracer atom or a vacancy;the tracers are
287
Z. QIN and A. R ALLNATT
288
An atom can migrate by exchange with a nearest-neighbour vacancy with jump frequencies denoted by w“ and w*for host atoms and by W;and w! for tracer atoms, where the superscript a (b) signifiesthat the atom jumps from an a (b)sublatticesite.The concentrations of host atoms, tracer atoms and vacancies on the sublattices a and b, denoted (c”,Cb), (c+,c$-)and (c;, c$), respectively,are each definedas the fraction of the sites on the relevant sublattice occupiedby the species.The principle of detailed balance relates the rates of jumps in the two directions between neighboring sites and yieldsthe expression present in very small concentration.
which determine the time dependence of the time correlation functions is d~v~(l’, 1’ – r; t) dt = O(r)[w~@v~(lX – r. 1’;t) — w~f/&T(lx, lx – r; t)] + ~ 19(r’)[~’”y@VTH(lx – r’, lx – r, I“x;f) r’#r — W14’VTH (1”’,1“’– r, 1“’– r’; f)]
+~
~(i)
[@thTV
(1”’, lx – (r – +),
1“’ – r;
t)
r’#r — w;@vTv (1’, lx – r, 1’ – (r – r’); t)].
and a similar equation for tracer atoms. These equations, in combination with the condition that the total atom concentration c is constant and equal to (c”+ cb)/2, determine the equilibrium distributions of atoms and vacancies between the sublattices at equilibrium, see Ref. [15]for details. Linear responsetheory leads to exact time correlation function expressions for transport properties which can be simplified in structure for particular models [3].A minor extension of arguments detailed earlier for the Manning random alloymodel [3]shows that in the present case the expressionfor the tracer correlation factor, ,f_,reducesto
+ kt$+”(ja(o) + Cbcbb(o)]} (2) where, for later convenience,we have used the notation that ~(p) is the Laplacetransform withrespectto tof a function F(t). The time correlation functions CX}, (t) are definedby the equation Cxy(t)= E(s. So/S2)&,(s;I:
SO)
We have introduced the notation that x = a if x = b and vice versa, and have defined O(r) to be 1 if r is a nearest-neighbour vector and zero otherwise. Here and whereverconvenientall subscriptsand arguments of @functions referring to the initial state have been omitted where they are the same as in eqn (4). The subscript H denotes the host atom speciesand @my whereX, Y = T, H, is the conditional prob(1,m, n; t), ability of observingparticles of speciesV,X, Y at the different sites 1, m, n, respectively, at time t.The probability ?/~vTv can be eliminated by the identity r/&TV(l,
= ‘@VT(l, m; t)–‘@vTH(l, III, II; ~) –4JVTT(11 w u ~)j
f = I - {W;[Ca~a.(O) + Cbcab(())]
(X, Y= a, b).
s
(3) Here s and and s. are nearest neighbour vectors of magnitudes and we define @.ry(r; t : I’0),for anYPair of lattice vectors r and ro, by the relation
(4) where, on the right hand side, $ is the conditional probability of observing a vacancy at the site 1’ on sublattice x and a tracer at the site l“’– r at time t if there was a vacancy at l; and the same tracer at l; – ro, respectively,at time zero. The kinetic equation for the probability functions
(5)
(6)
where, since we assume that cT+ O, rJw~ can be neglectedwhenm, and n, are differentsites. In order to introduce our basic approximation we note that the definitionof @vTH can be written as
7)VTH(IX, my,n’; f) = ~f)(1x)4T)(mJ’)4 H)(nz) 0.7 X
G7~(t)p~v) (lo)P~T) (10– ‘0)~6
(7)
in which Pfl is the probability of finding the systemin configuration @and G70(t) is the conditional probability that a system at thermodynamic equilibrium initially in configuration/3will be in configuration ~ after time t.The occupancyvariable P$x)(1)is unity if site 1is occupied by a particle of speciesX in system configuration~ and is zero otherwise.We can expand each occupancy variable of species V and H as its average value, which is the concentration of the species,plus a fluctuation, e.g.
p$)(l”’)= c; + Ap\v)(l-’),
(8)
Diffusionin a two sublattice structure
so that @VTH(lx,my, n’; t) = cfJc’@T(mY;t)+ Cz@dv’r(lx,My;t) + c~?#TAH(my,n’; t)
(9)
+ ‘@AVTAH(lx, my, n=;1).
The subscript AX denotes that the corresponding occupancyvariable in @is a fluctuation Ap(x)rather than p(x).In like manner we have ?/@(lx,my;t) = c~~(my; t) +@AvT(lx,my;t). (10) It is useful to note that, since we assume CT~ O,it my;t) is just the can readily be shown that @AHT(lx, negativeof @AvT(lx, my;t). Our basic approximation is to neglectall the terms of secondorder in fluctuations,suchas ~AwAH,in the kineticequation. The sametype of approximationhas beenusedfor the binary random alloymodel [9]under the name simple decoupling approximation (since it decouplesthe hierarchy of kineticequations [3]).By a summation over lx and some straightforward manipulations the kinetic equation is then transformed to
289
Weuse the notation that F(k) is the Fourier transform with respectto r of a function F(r). Equations (3)and (14)yield, by manipulations similar to those encountered in earlier work [3],expressionsfor the Laplacetransformed time correlation functions: C..(P) = [Qm(P) + (cbwf – @) Q(p)] /f2(p)
Cab(p) = [Q.b(p) – CaWfQ(P)]/~(P),
(17a) (17b)
plus equations obtained by interchanging a and b. Here we have defined Q(P) = Qaa(p)Qbb(p) Q(p) = 1 + ~
- Qab(p)Qba(p)
(18)
[(c%: – l’7’)Qxx(p)+ c’w&(p)]
x=a,b + (W%b – Caw!wa– cbw@~)Q(p)
(19)
and Qxy(P) = ~(%/~2)%4s s = Gxy(o;p) –
– %P)
G.y(–2so; p), (20)
d&(r; t)
+ zWxf#+(r;t) dt -~e(r’){cj~~xy(r -r’;t) 8 + (cxwx+ c~wf)q5zy(r– r’;t)}= Sxy(r;t) (11)
where z is the number of nearest neighbors per site (six for the simplecubic lattice) and Sxy(r;t) = f3(r)[c’”ti&y(-r;
f)
(12)
+ (Vi – cxw$)f#xy(r;t)]
Wx= Cxwx+ C+wx+ C;w+.
(13)
Terms independentof r whichmake no contributions to the subsequentexpressionsfor the time correlation functions CXY(t)have been omittedineqn(11). The pair of equations obtained from eqn (11) by putting x = a, b may be solvedafter Fourier–Laplace transformation; the result is ~xy(k~)
=
~
f-%z(k~)[~zy(b)
Z=.y,f
+ Ozy(kt
=
0)1
where the second equality followsfrom the structure of the simplecubic lattice. Before turning, in the next section, to the final step of evaluatingthe functions GXY appearing in eqn (20), we note that the results above are exact in the limit Cv-0 and that –w~~Xy(0)is then the averageof the cosine of the angle between an initial tracer-vacancy exchange,before which the tracer is on sublattice y, and a final exchangeof the same vacancy and tracer, after whichthe tracer is on sublatticex, whenthere are no restrictions on the number of tracer-vacancy exchanges between the initial and final exchanges. The above analysisthen relates these average cosines to quantities –WXQXY(0) whichare definedin the same way as the –W~Cxy(0) except that they refer to averages of cosines between successiveexchangesof a particular host atom and a vacancy in a tracer-free crystal. Furthermore GXYcan be identified as the Green function for a vacancy random walk which starts on they sublattice and ends on the x sublattice.
(14) 3. CALCULATION ANDRESULTSFORTHETRACER CORRELATION FACTOR
where GXX(k,p)= [p+ ZWX– c~w@(k)]/D(k,p)
(15a)
Equations (2)and (17)-(20)can be combinedto give a formalexpressionfor the tracer correlationfactor. In (15b) ~xx(k,p) = (Cxti + c@7)0(k)/D(k,p) the case of a finite concentration of vacancies the D(k,p) = [p+ Z@ – c$w@(k)] functions Gxy(p) in eqn (20) cannot be determined analytically(even for the limit of p = O required to x [P+ z~b – clwfe(k)] – w“wb[fl(k)]2. calculate the tracer correlation factor). We consider (16) two approximationsleadingto analyticalresults.
290
Z. QIN and A. R. ALLNATT
also known [23] that the IR result can be transformed to the exact one by taking MO = 2~0/(1–JO) where ,fi is the Bardeen–Herring correlation factor (~0= 0.6531for the simple cubic lattice), and we have shown (unpublished work) that this is also true for the two sublattice result, eqn (21), in the limit Cv~ O. As is shown in Fig. 1, the modified IR result, corresponding to taking MO= 2~0/(1–~0) = 3.765 in eqn (21) gives improved results at all concentrations as well as having the correct limiting value; however, it does not appear to be the result of a systematic approximation to the kinetic theory. A more satisfactory approximation appears in Section 3.2. The specialstructure of the two sublatticeequations isalsoapparent in the timecorrelation functionsin the IR approximation; these are
3.1. Immediate reversal (IR) approximation In the case of a uniform lattice (w”= Wb,W;= w!) with Cv+ O,it is known [23]that a useful short time approximation for the singletimecorrelationfunction Q(t) is obtained by assumingthat the dominant contribution comesfrom the finaltracerjump whichis an immediate reversal of the initial tracer jump, without any other vacancy jumps in between. In the present context this immediate reversal (IR) approximation correspondsto neglectingall terms containingO(k)in the Green functions. When this approximation is made in eqns (17) then the result for the tracer correlation factor is found to be
C..(t) = exp(–zwvt){[z(w$– wv)/A] where MO= 5. Predictions of the IR approximation for ~ as a function of total atom concentration c for differentvalues of the jump frequencyratio
(23)
Cab(t)= –[cbw~/A]exp(–zwvt)sinh(At),
(24)
plus two expressionswith the a, b labelsinterchanged, where
(22)
13= WafWb
x sinh(At) + cosh(At)}
are compared in Fig. 1with the Monte Carlo resultsof Murch [15],which take w; = w’ (x = a, b). A particularly unsatisfactory aspect of the result is that the low vacancyconcentration limit is incorrect although it would be exact for an exact solution of the simple decouplingapproximation. In the case of a uniform lattice with Cv-+ Oit is
W$= [(z– I)mf + c“’w;]/z
(25)
Wv= (w; + w$)/2
(26)
A* =
Z2(W$
–
W$W~)
+
CaCbWfW:.
(27)
It is clear that when the linear combination of four time correlation functions in eqn (2) for f is consid-
1
0.8 0.6 f
0,4 *-
0.2 1
0 0
0.2
I
1
0.6
0.4
I
0.8
1
c Fig. 1.Thetracercorrelationfactorf asafunctionofatomconcentration c forvahresofthejumpfrequency ratio@asmarked. —–0– – MonteCarlosimulation,[24]for ~ = 1 and [15]for others;- - - - - IR approximation;— modifiedIR approximation.
Diffusionin a two sublatticestructure
ered as a single overall time correlation function the result is not a single exponential decay in time, although it can be shown that the expression does have the correct zeroth and first moments. This is in contrast to the single exponential in time, which correctly reproduces the zeroth and first moments of the exact time correlation function, found for the IR approximation to the binary random alloy model [9], the uniform lattice limit of the present diffusion calculation [23],and evenfor the frequencydependent conductivity of a pure crystal in the present model (wherewefind the result is the sameas Richards [19]). Evidentlythe time dependenceof the relaxation associated with diffusionin two sublattice models is especially complex, and care needs to be made in proposing simple approximations by analogy with simplersystems.
3.2. Small-k
approximation
For p = O eqn (15) can be written in the compact form PO+
Pl~(k)
“’(k’ 0)= 6(a0 + al y(k) + &(k)]2)
(28)
291
? = r/a. We also defined h(F)= cos(AF)/7 = exp(–l JIF)/7
for J2 >0 for J2 <0
A*= 6(ao + al + a2)/(al + 2cx2).
and for the simplecubic lattice [3] ~(k) a @(k)/6 = [cos(kXa) + cos(lrya) + cos(kZa)]/3
(30) where a is the nearest neighbour distance. The dominant contributions in the Fourier inversion of eqn (28) come from small k because at large k the factor exp(ik.r) undergoes rapid oscillations. In this limit y(k) + 1 – (lCa)2/6. Substituting this and extending the range of integration over all k-space we obtain
where we omit the p = O argument in ~ and put
(33)
This small-k approximation is well-known [3]when applied to the Green function for a random walker on a perfect lattice where, for example, the approximate and exact values for the simplecubic lattice are (0.477,0.516),(0.338,0.331),(0.276,0.261)for 7 = 1, J2, ~3, respectively.In viewof the errors introduced by our basic assumption (neglect of second order fluctuations) the small-k approximation will be adequate, and has the advantage of allowing an analytical result. We also need an approximation for the Green functions when r = O.This can be obtained by convertingeqn (28)to an integralequation by multiplying through by the right hand denominator in eqn (28) and inverting the Fourier transform. After some straightforward calculation one finds GXY(0) = IPO–6a1~XY(a)–4@xy(fi~) – ~2~Xy(2a)]/(6ao+ CY2).
where
(32)
(34)
The Glyon the right hand sidecan be calculated from eqn (31).Equations (2),(17),(20),(31)and (34)can be combinedto givean expressionfor the tracer correlation factor, whichweneednot repeat herein viewofits length. As can be seenfrom Fig. 2, the small-kapproximation provides a good approximation to Murch’s Monte Carlo results [15]for f as a function of c for four values of ~ (= 1,0.1,0.02,0.0025).The discrepancies become more significant as the jump frequencyratio deviatesmore from one. Other Monte Carlo simulations have been made by Tahir-Kheli et al. [18]for ~ in the range 0.086-1 with Cvheld constant at 0.5, and with c“ (the concentration of atoms on the low energy lattice) held constant at 0.75 and 0.875.As shown in Table 1, our small-k approximation results are in comparable or better agreement with these simulation results as the numerical results from the approximate theory of Holdsworth er al. [20].For these results, as for the Monte Carlo results of Murch shownin Figs 1 and 2, the modified IR approximation gives results close to the small-kapproximation and the IR approximation itself is less satisfactory. There are no simulation results for the case when the jump frequencies of a tracer and a host atom are different. The behaviour of the small-k approximation for ~ as a function of c at @= 0.01
292
Z, QIN and A. R. ALLNATT
1
0.8
0.6 f 0.4
L
“v 002
●
0.0025
,“ ,*’
‘ -- “
4
0.2 0
0,2
0
I
I
0,4
1
0.6
0.8
1
c Fig.2. Thetmcercorrelation factor~ ma functionofatom concentrationcforvalues ofthejump frequencyratioflas marked. —-0– Monte Carlo simulation,[24]for ~?= 1and [15]for others; -––— small-kapproximation,
Table 1.Simulatedand theoreticalvaluesof the tracer correlation factor ,f’whentracer and host atoms have the samejump frequencies.The first sevenvaluescorrespondto a total vacancyconcentration of 0.5 c“
c’
/3=
0.5 0.5448 0.6017 0.6586 0.7155 0.75 0.7724 0.75 0.75 0.75 0.875 0.875 0.875 0.875 0,875
0.5 0.4552 0.3983 0.3414 0.2845 0.25 0.2276 0.75 0,5 0,475 0.875 0.75 0.625 0.5 0.375
1 0.69793 0.43808 0.26865 0,15807 0.11111 0.08681 1 0.33333 0,3016 1 0.42857 0.23810 0.14286 0.08571
w“/wlh
Simulation[18]
~ Small-kapprox.
Theory of [20]
0.846 0.842 0.830 0.806 0.767 0.733 0.707 0.751 0.774 0.774 0.703 0.719 0.724 0.712 0.682
0.8507 0.8474 0.8338 0,8092 0,7721 0.7422 0.7191 0.7579 0.7776 0.7766 0.7056 0,7235 0.7252 0.7131 0.6864
0.8494 0.8471 0.8363 0.8159 0.7838 0.7569 0.7354 0.7584 0.7841 0.7841 0.7077 0.7253 0.7345 0.7289 0.7075
for Q s w~/wa= w~/wbin the range O.1–10is shown in Fig. 3. At the limit c = 1 these results are in good agreement with the prediction of eqn (21) with MO= 2~0/(1–~0), which is exact in this limit; the results are 0.1, 0.4, 0.7, 1.0, 1.6°Alow for a = 0.1, 0.5, 1, 2, 10,respectively. 4. DISCUSSIONAND CONCLUSIONS
A formula for the tracer diffusioncorrelation factor has been derived for a two sublatticemodel of a pure crystal with an arbitrary vacancy content using a simple decoupling approximation similar to that employed in [20]for the same model in the context
of a theoryof the incoherentscatteringfunction. It can be shownthat in the limitingcase of a uniform lattice (w”= w~, w; = w:) the simple decoupling approximation expressionsof Section 2 lead directly to the result of Tahir-Kheli and Elliott [22]for the tracer correlation factor. This is encouragingsincethe result in [22]is remarkably accurate (better than 1/2°/0for the simple cubic lattice [3]).In this limiting case the only quantity requiring numerical calculation can be expressed in terms of the Bardeen–Herring correlation factor ~. for tracer diffusionin the limit Cv-+ O, which is of course a well-knownquantity but for the full two sublattice model such an identification in terms of known functions does not seem possible.
Diffusionina two sublattice structure
293
1
0.8 0.6 f
0.4 0.2 I
0 0
0.2
1
I
0.4
0.6
I
0.8
1
c Fig. 3. The tracer correlation factor _f in the small-k approximation as a function of atom concentration c for the jump frequencyratio ~ = 0.01and the ratio a as marked.
However, the additional use of the small-k approximation allows an algebraic result to be obtained which provides a useful description of the available simulation results and also givesresultscloseto those obtained numerically in [20].The calculations could be extended to other lattices in which the set of nearest-neighbour lattice vectors is the same for all sites independent of sublattice. Another analytical result was obtained by introducing the immediate reversal (IR) approximation into the simpledecouplingexpressions,but this givesless satisfactory results for ~ unless an empirical adjustment is made so as to reproduce the exact limiting behaviourfor smallvacancycontents. For the random alloy on a uniform lattice this adjustment actually converts the IR transport coefficient(but not its time correlation function)exactlyto the simpledecoupling result [9], but for the present two sublattice model this is not the case except for Cv~ O. A further indicator of the complexity of diffusion in the two sublattice model is provided by the time dependence of the associated time correlation function; the simplest viable approximation (the IR approximation) no longer corresponds to a single exponential in time, unlike other simple models enumerated in Section 3.1.
Acknowledgement—This researchreceivedfinancialsupport fromtheNaturalSciences andEngineering ResearchCouncil ofCanada.
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19. Richards P. M., J. Chem. Ph~s. 68,2125 (1978); Fasl Ion Transport in Solid.~(Eds P. Vashista, J. N. Mundy and G. K. Shenoy), pp. 349. North-Holland, Amsterdam (1979), 20. Holdsworth P. C. W., Elliott, R. J. and Tahir-Kheli R. A., Phys. Rev. B. 34,3221 (1986).
21. Allnatt A. R,, Phi[. A4ag. A 64, 709 (1991). 22. Tahir-KheliR. A, and Elliott R, J., Phys, Rev. B. 27,844 (1983). 23. Qin Z. and Alhratt A, R., Phil. Mag. A 70,677 (1994). 24. Murch G. E., Phil. Mug. .4, 49,21 (1984).