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MATERIALS SCIENCE RESEARCH • Volume 9
MASS TRANSPORT PHENOMENA IN CERAMICS
Edited by
A. R. Cooper and A. H. Heuer Department of Metallurgy and Materials Science
Case Western Reserve University Cleveland, Ohio
PLENUM PRESS • NEW YORK AND LONDON
Library of Congress Cataloging in Publication Data
University Conference on Ceramic Science, Case Western Reserve Uni­ versity, 11th, 1974. Mass transport phenomena in ceramics.
(Materials science research; v. 9) "Proceedings of the eleventh University Conference on Ceramic Science,
held at Case Western Reserve University from June 3-5, 1974." Includes bibliographical references and index. 1. Mass transfer-Congresses. 2. Ceramics-Congresses. I. Cooper,
Alfred R., 1924- II. Heuer, Arthur Harold, 1936- III. Title. IV. Series. TP156.M3U53 1974 666 75-20154 ISBN-13: 978-1-4684-3152-0 e-ISBN-13: 978-1-4684-3150-6 DOl: 10.1007/978-1-4684-3150-6
Proceedings of the Eleventh University Conference on Ceramic Science, held at Case Western Reserve University from June 3-5, 1974
©1975 Plenum Press, New York A Division of Plenum Publishing Corporation 227 West 17th Street, New York, N.Y. 10011 Softcover reprint of the hardcover 18t edition 1975
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PREFACE
The Eleventh University Conference on Ceramic Science held at Case Western Reserve University fran June 3 - 5, 1974 was devoted to the subject of M:l.ss Transport Phenanena in Ceramics. '!his book follows closely the fonn of the oonference. While the active participation at the meeting was not reoorded, it is clear that many of the contributors have benefited fran the ranarks, suggestions, and criticisms of the participants. Fur­ thennore, the session chainnen -- Delbert Day (Univ. of Missouri), WU. IaCourse {AlfrErl Univ.) , W. Richard ott (Rutgers Univ.) , A.L. FriErlberg (Univ. of Illinois), v. Stubican (Penn. State Univ.), and R. Loehman (Univ. of Florida) -- successfully kept the meeting to a reasonable schedule, but also stimulated the lively discussion.
The book divides naturally into four sections, focusing on correlation and ooup1ing effects in diffusion in ionic materials, understanding of fast ion transport, diffusion and electrical con­ ductivity in crystalline and glassy oxides and applications of diffusion to oxidation and other processes of current interest.
The editors have benefited fran the cheerful help and assis­ tance of many people. !-1rs. Karyn P1etka typed the entire manu­ script with unusual accuracy and tolerance. Mr. MakmJd E1Lei1, Ajit Sane, Leslie M:l.jor and Ms. Jenny Sang provided the subject index. The authors have been cooperative and understanding and we ack.now1Erlge our enjoyment in working with them.
The conference receivErl financial assistance fran U.S. Army Research Office, Durham, N.C., Air Force Office of Scientific Research, Arlington, Va., and Aerospace Research laboratory, Wright Patterson Air Force Base, Dayton, Ohio. This support made it possible to bring together the distinguished group of contri­ but.ors to this volt:roe.
Cleveland, Ohio March, 1975
CONI'ENTS
Non-Random Diffusion in Ionic Crystals • . . . . JOM R. Manning
Correlation Effects in Ionic Transport Processes • . A. D. LeClaire
Correlation and Isotope Effects for Cation Diffusion in Sirrple OXides . • • . • • • • • • • • • •
N. L. Peterson and W. K. Chen
Coupling, Cross Tenus, Correlation and Conduction A. R. Cooper
Irreversible Thermodynamics in Materials Problems F. A. Nichols, G. P. Marino and H. Ocken
OXygen Redistribution in U02 Due to a Temperature Gradient . . . . • . • . . . • . . • . • .
D. D. Marchant and H. K. Bcwen
The Thermodynamics of Diffusion Controlled Metamorphic Processes .•.•..
George W. Fisher
FAST ION TRANSPORI'
Willem Van Gaol
Hiroshi Sato and Ryoichi Kikuchi
Alkali Ion Transport in Materials of the Beta Alumina Family • • • • • • • • • • • •
Robert A. Huggins
Ionic Conductivity of Doped Cerium Dioxide H. L. Tuller and A. S. NCMick
lJ::M Temperature Oxygen Transport in
CONTENTS
149
155
177
Nonstoichiametric ce02 • • • • • • • • • •• 187 S. P. Ray and A. S. Nowick
Self Diffusion Constant Measurement by Continuous­ Wave Nuclear Magnetic Resonance • • • •
J. Stepisnik, J. Bjorkstam and C. H. Wei
DIFFUSION AND ELECTRICAL mNDUCTIVITY IN CRYSTALLINE AND GlASSY OXIDES
On the Interpretation of Lattice Diffusion in Magnesium Oxide • • • • • • • • • • • •
Bernhardt J. Wuensch
H. P. R. Frederikse and W. R. Hosler
Concentration Dependent Diffusion of W in Ti02: Analysis of Electronic Effects in Ion Diffusion
o. W. Johnson, J. W. DeFord, and S. -H. Paek
Measurement of Chemical Diffusion Coefficients in Non-Stoichiometric Oxides using Solid State Electrochemical Techniques • • . • • • • •
B. C. H. Steele
G. H. Frischat
Oxygen Diffusion in Liquid Silicates and Relation to Their Viscosity •
Y. Oishi, R. Terai, and H. Ueda
201
211
233
253
269
285
297
CONl'ENTS
Silicon and Oxygen Diffusion in Oxide Glasses Helmut A. Schaeffer
Alkali Ion Conductivity in Fused Silica Edward M. Clausen
Cation Migration in Electrcx:1e Glasses F. G. K. Baucke
A Structural Model of Ionic and Electronic Relaxation in Glass • • • • • • • •
J. Aitken and R. K. MacCrone
Molecular Diffusion in Glasses and Oxides J. E. Shelby
APPLICATIONS OF DIFFUSION TO OXIDATION AND Ol'HER PROCESSES OF CURRENT INTEREST
Mass Transport Phenomena in Oxidation of Metals Per Kofstad
The High-Temperature Oxidation of Hot-Pressed Silicon Carbide • • • • • • • • • • • •
J. W. Hinze, W. C. Tripp and H. C. Graham
Sintering of Silicon Carbide • • • • SVante Prochazka
Determination of Phase Diagrams Using DIffusion Techniques • • ••• • • • • • • • •
Joseph A. Pask and llhan A. Aksay
Arrbipolar Diffusion and Its Application to Diffusion Creep • • • • • • • • • •
R. S. Gordon
Crystal Growth and Glass Formation D. R. Uhlmann
The Effect of Diffusion and Shear on Concentration Correlations in Binary Systems • • • • • •
J. H. Heasley
The Dissolution Kinetics of Lithiated NiO in Aqueous Acid Solutions • • . . • • •
Chin-Ho Lee, Alan Riga and Ernest Yeager
Subject Index
John R. Manning
ABSTRACT
Non-random diffusion can result both from the presence of atonic driving forces and from the notion of defects in a crystal. Defect-related non-random effects appear in two different ways in the kinetic diffusion equations, as correlation effects and as defect-wind effects. In the present paper, the origin and magni­ tude of these effects during diffusion in a driving force are dis­ cussed. Kinetic expressions for the drift velocity <vF> are derived from expressions for the effective frequencies of independent atom j unps and are related to the tracer diffusion coefficient D*. For impurity diffusion in an electric field, deviations from the Nemst­ Einstein relation result from defect-wind effects. Recently developed equations for the ionic-irnpuri ty drift-rrobili ty when diffusion occurs via divacancies noving on one of the sub-lattices in the NaCl structure are summarized. Extensions of the sirople equations derived here to more complex situations are discussed.
INTRODUCTION
Diffusion in crystals occurs by atoms following more-or-less random walks. Each atom noves through the crystal by making a series of elanentary atom jumps from one lattice site to another. An atomic driving force will provide a bias to the directions of the individual junps so that they are no longer random in direc­ tion. Thus , driving forces, such as electric fields, are one
*Contribution of the National Bureau of Standards, not subj ect to copyright.
1
2 JOHN R. MANNING
source of non-random effects. If each jUIl"q? has the same bias, as for self-diffusion in a constant driving force, effects from driving forces are easily treated by making simple rrodifications in the random walk equations. Havever, when the biases are not constant or when other non-random effects occur, such as the defect-related non-random effects discussed later in this paper, the analysis is more corrplex.
A category of non-randan effect which can arise even in the absence of driving forces occurs when a given atom jUIl"q? is influenced by the directions of previous jurrps taken by atans and defects in the crystal. 'Ihe resulting "correlation effects" and "defect-wind effects" can occur either with or without the presence of atomic driving forces. 'Ihese non-random effects do not provide the same bias on each jump.
In the present paper, the origin and magnitude of correlation and defect wind effects will be discussed, particularly for the case where there is an atonic driving force. Electric field effects will be emphasized and explicitly discussed. Nevertheless, most of the equations will apply equally well to effects from other driving forces and other sources of defect fluxes, such as tem­ perature gradients, stress fields, or gradients in concentration or stoichiorretry.
Electric field effects are empahsized here since they provide a particularly straight-forward exarrple of a driving force. Also, electric fields have inportant effects on ions moving in ionic crystals. In the present discussion, correlation and wind effects resulting fran motion of individual defects will be emphasized. Non-stoichiometric crystals with large vacancy concentrations would require a more canplex treatment since interactions between large groups of defects then should be considered.
NON-RANOOM DIFEUSION VIA MONOVACANCY MEX.::HANISM
Correlation and defect wind effects depend strongly on the diffusion mechanism. When diffusion proceeds by the simple inter­ stitial mechanism with no atomic driving forces, the individual interstitial atans can follow truly random walks. 'Ihe kinetic dif­ fusion equations then may be obtained from the simple mathematical theory of randan walks. On the other hand, for most other diffu­ sion mechanisms, especially those which require a mobile defect at a neighboring site, the individual atans do not follow random walks. 'Ihe direction of an atan jUIl"q? I for exarrple, will depend on which particular neighboring site is occupied by a defect at the time of the jump. Consequently, when diffusion of atans occurs via motion of vacancies, di vacancies, intersti tialcies, crowdions I or even more-cOITlplex mobile defects, defect-dependent non-random diffusion
NON-RANDCM DIFFUSION IN IONIC CRYSTALS
will occur [1,2,3].
To illustrate the origin of these defect-dependent non­ random effects, let us consider diffusion by the rronovacancy mechanism. In this mechanism, the elementary atom jump involves the jumping of an atom into a neighboring vacant lattice site. '!hese vacant sites, or vacancies, are assu:rred to be present in thenncrlynamic equilibriun in concentrations which depend on exp (-Ef/kT) where Ef is the fonnation energy for vacancies, k is Boltzmann's constant and T is the absolute temperature. In discussing the monovacancy mechanism, one assu:rres that each vacancy rroves independently in that it is not bollild to other vacancies and does not interfere with diffusion processes involv­ ing oLher vacancies. By contrast, in the divacancy mechanism, it is assuned that vacancies move as part of vacancy pairs.
3
With respect to non-random diffusion, the important thing about the vacancy mechanism, and also the di vacancy mechanism, is that an atom cannot move lliltil a vacancy arrives at a site neighboring on it. '!hus, the independent diffusion process is not the single atom jurrp but instead requires a sequence of jumps. In this sequence, the vacancy is first created at a vacancy source. '!hen the vacancy rroves through the crystal by exchanging with the various atans in the crystal lliltil it arrives at a site neighbor­ ing on the diffusing atom i. After arrival, the vacancy can exchange with atan i one or rrore times. Finally the vacancy will move pennanently away and be destroyed at a vacancy sink.
'!his sequence of interrelated vacancy jumps may be contrasted to the situation in the simple interstitial mechanism, where the atcm jurrps directly fran one interstitial site to another, with th~ direction of each atom junp being independent of the direction of jump of any other neighboring atcm or defect. en the other hand, for diffusion via an interstitialcy mechanism, an intersti­ tialcy must approach the atom much as described above for diffusion via a vacancy mechanism.
In principle, one must follav the complete paths of the vacancies if one wishes to detennine the kinetic diffusion pro­ cesses affecting the particular atcm i. In simple situations, how­ ever, there are several ways to simplify the calculation.
TO present the defect-dependent non-randan diffusion equa­ tions in as simple a fonn as possible while still allowing discus­ sion of the influence of driving forces on these equations, atten­ tion will be restricted to high-synmetry crystals, such as cubic crystals, with mirror synmetry across all lattice planes normal to the diffusion direction. '!he diffusion direction x will be chosen nonnal to low-index planes separated by a regular inter­ planar spacing b, and it will be assumed that the x components of
4 JOHN R. MANNING
the possible atcm jumps are all +b, zero, or -b. Thus, an atom or vacancy cannot reach a lattice plane that is 2b away without stopping at a lattice site one interplanar distance <May in that direction. As examples, <lOO>-diffusion directions in a variety of cubic struc­ tures satisfy these rules provided that only jumps to nearest neigh­ bor sites are allcwed. Also, <lOO>-diffusion directions in tetra­ gonal or orthorhombic crystals would be sui table. In cubic crystals diffusion is isotropic, so this arbitrary choice of dif­ fusion direction does not make L~e result any less general.
Driving forces can be expected to create a net vacancy flux, which normally will be distorted in the vicinity of an impurity atom i. (A distortion is expected unless the diffusion properties of i are identically the same as those of all other atoms in the crystal. ) Nevertheless, if there is a mirror crystal symnetry plane passing through i nonnal 'co the flux, the average vacancy concentrations on this plane will not be altered from those in the absence of a flux. Thus, sites along this plane will maintain equilibrium vacancy concentrations and for present purposes can be treated as effective vacancy sources-and-sinks. Any sequence of vacancy jUllIJS thus is regarded as terminated when the vacancy reaches this syrrmetry plane (but not tenninated by exchange with atom i itself since after such an exchange the atan and vacancy still are not on the same plane). Further jurrps by the vacancy starting from the syrrmetry plane will begin a new independent sequence, uncorrelated to the previous sequence. Since the only way for a vacancy during a single jurrp sequence to move from one side of atom i to the other is by exchange with atom i itself, a single vacancy sequence cannot provide atom i with two consecutive jumps in the sane sense. Instead jurrps by atom i from a single vacancy sequence will cause alternate +b and -b displacements and the net displacement from any sequence will be either ±b or zero.
After one exchange with atom i in a given direction, the vacancy is in the proper position to cause a reverse jump, moving the atan in the opposite direction. Such a second jurrp would cancel the effect of the first jurrp. Similarly, the fourth exchange of the vacancy with i will cancel the third exchange, the sixth exchange will cancel the fifth exchange, and so on. After an odd nunber of exchanges the net displacerrEl1t of i equals b, the interplanar spacing; and after an even number of exchanges the net displacement is zero.
After the vacancy has arrived at a site a, neighboring on atom i on the +x side, it will have a probability P+ of eventually causing a +b jurrp of atan i. This jurrp may occur on the next jurrp of the vacancy or the vacancy may first wander away from atom i and only later return to exchange with it without arrival of the vacancy at any effective vacancy sink. After a P+ exchange the vacancy will be on the -x side of atom i in proper position
NON-RANDa-1 DIFFUSION IN IONIC CRYSTALS
to cause a -b jump. The probability of the vacancy causing such a -b jump (again on any of its subsequent jumps before arrival at an effective vacancy sink) is defined as P _. If there is a driving force, P _ will differ from P +; but, in the absence of a driving force, P_ equals P+ since the only distinction between P+ and P _ is the direction in which the exchange occurs.
The frequency '.1+ with which atom i undergoes a displace:rrent, always of magnitude b, in the +x direction therefore is:
where vn+ is the frequency of arrival of fresh vacancies at sites a neighboring on i on the plane on the +x side of i. (These vacancies then are in the proper position to allow a +b jump of atom i on the next jump.) The sun involving P+ and P_ tenus in Eqn. (1) gives the probability of the vacancy causing a net atom displace:rrent +b after arrival at site a. Thus, '.1+ is the effective frequency of independent displacements +b.
5
In Eqn. (1), the initial term P+ within the brackets represents the probability of an initial exchange of an a-site vacancy with atan i. The second term P~_ represents the probabil­ ity of a two-junp sequence where the second junp cancels the effect of the first jump. Hence, this term appears with a minus sign. The third term represents the probability of a three-junp sequence, again providing a net displace:rrent +b of atan i, and so on.
A fresh vacancy is defined as one which in its sequence of jumps has not previously arrived at a site from which it could cause a +b or -b j unp of the tracer. As noted previously, any vacancy which arrives at the mirror symmetry plane passing through atom i normal to the diffusion direction ends one sequence of jumps and in effect becomes a new fresh vacancy during its subsequent pa.th. Thus, in its subsequent path it again can have a first arrival at a site a.
In Eqn. (1), v + represents a sun over all possible paths from any vacancy source (or effective fresh vacancy source on the mirror symmetry plane) to any site a which will allow a +b junp of atom i. For simplicity, it is assuned in the present equation that all a sites are equivalent to each other. Then vn+ is a simple scalar quantity, while still representing a sum over all a sites. This condition will be satisfied in any crystal which has suf­ ficient rotational or mirror symmetry around the diffusion direc­ tion that rotations or reflections of the crystal can be applied to bring any site a into the original position of any other site a and these rotations or reflections also reproduce in the trans­ formed orientation all other lattice sites in the original orien­ tation. In such a situation, a vacancy at anyone a site has the
6 JOHN R. MANNING
same effect of diffusion along the x-axis as does a vacancy at any other a site. This condition will be satisfied by many crystals (cubic, tetragonal, orthorhanbic) for which the previously­ introduced mirror symmetry nor.mal to the diffusion direction applies.
P+ can be expressed in terms of the possible vacancy jump frequencies from sites a. In particular, P+ depends on the junp frequency w2+ for a vacancy on site a to cause a +b jump of atan i and on the carpeting jump frequencies w+rr for the vacancy to start a path which leads it away from site a to an effective vacancy sink without exchangin:j with atom i or returning to any site a. Here w+rr represents the sum over all jumps (to any sites) which actually start a vacancy on such a path of non-exchange and non-return. In w+rr ' one must exclude paths which eventually return the vacancy to a site a or which involve a direct jump from one site a to another site a. Detailed calculations sheM that these a to a transitions do not affect P+.
Physically the reason for omitting a to a transitions may be seen fran the definition of P +. In the calculation of P +, one must follow the vacancy until it either exchan:Jes with atan i or pemanently rroves away by reaching an effective vacancy sink. There is unit probability that one of these two events will even­ tuallyoccur and P+ is the probability of the former, Le., exchange with atan L The relative probabilities of an eventual w2+ junp or w+rr junp path are the same regardless of which a site contains the vacancy. Thus, transitions fran one a site to another do not affect P+ and these transitions may be anitted fran further consideration.
The expression for P+ therefore is simply
Also
SummiIB" the series in Eqn. (1) yields
v+ = vrr + P+ (l_P_p+)-1 (l-P_)
(2)
(3)
(4)
This equation may be recast into a form which explicitly displays the correlation factor and a familiar expression for the jump fre­ quency. Upon application of Eqn. (3) relating P+ to I-P+, one obtains
vrr+ (l-P+) (l-P_)
v+ = w N Z (l-P_P+) w2+ Nva za +rr Va a
(5)
NON-RANDOH DIF.EUSION IN IONIC CRYSTALS 7
'Where NVo: is the equilibrium vacancy concentration at site 0: and Zo: is the number of equivalent sites 0:. According to Eqn. (5), the effective jurrp frequency v+ of jurrps +b can be expressed as the product of four factors,
where
(6)
(7)
(8)
(9)
Here rb+ is the basic jurrp frequency for an atom to jurrp to a given site in b1.e +x direction. 'Ihis basic frequency is the jurrp frequency which would apply if the only non-randan effects were those fran driving forces, i.e., in the absence of defect-related non-randan effects. These defect-related non-random effects appear in Eqn. (6) as the oorrelation factor f and the vacancy wind factor '4.
The basic jurrp frequency rb+ will be affected by driving forces along the x-axis since these driving forces influence the vacancy jurrp frequency w2+. By contrast, the correlation factor f is unaffected by driving forces to first order, since P+ and P_ appear symmetrically in Eqn. (8). The vacancy wind factor '4 given in Fqn. (9) is unity if there are no vacancy fluxes. A driving force nonnally will provide a vacancy flux and make ~ differ fran unity.
In Fqn. (9), v1l+ is the frequency with which vacancies leave vacancy sources-and- sinks at sites 11 and travel to sites 0: neigh­ boring on the atan, with the 0: sites here being those on the +x side of the atan. Similarly, w+1I NVo: Zo: is the frequency with which vacancies follow the reverse paths fran sites 0: to sites 11.
If there is a net vacancy flux, vacancies will travel more fre­ quently in one sense along these paths than the other. Thus, a net vacancy flux makes '4 differ from unity, whereas in the absence of such a flux ~ equals unity. This general oonclusion will apply even when there are interactions between defects.
'lb first order in the driving forces, Eqn. (8) reduces to
1 - P f = 1 + P (10)
where P is the average of P+ and P_. Physically the oorrelation factor occurs here because a vacancy can exchange with an atom more than once during a single independent sequence of jurrps.
8 JOHN" R. MANNING
Notian .of the atom itself changes L'le local probability .of a vacancy being on the +x or -x side .of the atan.
By contrast, the vacancy wind factar arises because .of motian .of ather atans. When there is a driving farce along the x-axis motion .of these ather atoms (i. e., ather than the atom i whase jump frequency v+ is being calculated) will bring vacancies up to sites ex by paths contributing to v'Tf+ more frequently than motion .of these ather atoms will carry vacancies in the apposite directian alang these paths.
Atonic driving farces which bias the directians .of individual atan jurrps are nat the .only influences which can yield a net vacancy flux and hence cause G.t ta differ fran unity. For exarrple, variaus types .of concentration gradients .or stoichianeb:y gradients can alsa produce net vacancy fluxes. In the present discussian, however, attentian will be restricted ta vacancy fluxes resulting from atomic driving farces, and particularly fran electric fields.
Equatian (6) yields the effective jurrp frequency v+ far an atan jurrp in the +x directian. A similar equatian is faund far the effective jurrp frequency v_ in the -x direction,
(11)
where f again is given by Eqn. (8), rb- is .obtained from Eqn. (7) merely by replacing all subscripts + with subscripts -, where the - subscript refers ta atOll jumps in the -x directian, and G is .obtained from Eqn. (9) by replacing subscripts + with subscripts - where the - subscript refers ta a site in the -x directian fram the atom.
CALCULATION OF DIFFUSION COEFFICIENT D* AND DRIFT VELCX:ITY <vF> FRCM EFFEX::TIVE JUMP FREQUENCIES
The basic kinetic diffusian equation far planar diffusian in the x-direction relates the atom flux J ta the concentratian c and cancentratian gradient ac/ax .of the diffusing species,
J = -D* (ac/ax) + <vF> c (12)
This equatian contains DNa measureable coefficients, the tracer diffusian coefficient D* and ti1e drift velacity from atonic driving farces <vp>' If atonic driving farces are the .only nan­ randan effects wh~ch .occur, as far diffusian by a simple intersti­ tial mechanism, D* and <vF> are related by the well-known Nernst­ Einstein equatian [4],
(13)
NON-RANDCM DIFFUSION IN IONIC CRYSTALS 9
where F is the atanic driving force. For diffusion in an electric field E
F = q E (14)
where q is the charge of the diffusing ion. The drift rnobili ty 11 is defined as the drift velocity in unit electric field. Thus,
(15)
and for the specific case of an electric field, the Nernst-Einstein equation becanes
..L=...3. D* kT
(l6)
~Vhen defect-related non-random effects occur, as for diffu­ sion by a vacancy mechanism, it is found that the Nernst-Einstein equation no longer applies. The deviations which are found can be described by considering the ways in which the effective junp frequencies v+ and v_ differ from the basic jump frequencies fb+ and fb-.
In tenns of effective jump frequencies, if v+ and v_ are independent of location in the crystal, as for diffusion in a han~eneous crystal with a constant electric field,
D* = ~ b 2 (v+ + v_) (17)
and
<v > = b(v - v ) F +- (18)
These equations are very similar in appearance, the only differences being the square and half which appear in Eqn. (17) and the fact that the v's are added in Eqn. (17) but subtracted in Eqn. (18). Because of this slinilari ty, it is not surprising that D* and <vF> can be related to one another.
When Eqns. (6) and (11) are inserted into Eqn. (17), one finds
D* = ~ Zab2f (G+fb+ + GJb-) (19)
'lb first order, G+ and G_ can be written as
G = 1 + ~G (20) ± -
where G is a small quantity calculated from Eqn. (9). Also
fb± = fbo exp(±bF/2kT) (21)
10 JOHN R. MANNING
where fbo is the value of fb+ or fb _ in the absence of atomic driving force F. Expanding the exponential to first order yields
fb± = fbo [1 ± (bF/2kT) + • . . . ] (22)
When Eqns. (20) and (22) for G± and fb± are substituted into B:Jn. (19), one finds
D* = Z b2 f f a bo (23)
When these same equations are substituted into Eqn. (18) for <vF>' one finds
<VF -> = za b f (G+ fb+ - G_ fb_)
and to first order
(24)
<vF> = za b 2 f fbo [(fb+ - fb_)f~ b-l + (G+ - G_)b-l ] (25)
(26)
where
(27)
The tenn B, which arises from the vacancy wind factors G+ and G_, is the tenn which gives rise to deviations from the Nernst-Einstein relation, Eqn. (l3) or:f.qn. (16). In the special case of self~iffusion, 1 + B equals f-l. This relationship, especially when derived by methods other than that given above, has inspired statements in the literature that <vF> /D* differs fran F/kT because of the correlation factor. It would appear more appropriate to say that the deviations are proportional to 1 + B, with B arising from a defect~ind effect.
'Ihe factor I'.G in B:Jn. (27) depends on the driving forces and jump frequencies influencing the motion of the vacancy as it approaches or leaves the vicinity of atom i. In the special case of self~iffusion, these forces and frequencies are the same as those for motion of atom i itself. As a result, I'.G for self­ diffusion is directly proportional to the force F, and B for self­ diffusion is independent of F. Here F is the force which acts specifically on atom i. For impurity diffusion by a vacancy mechanism, hONever, the forces contributing to the vacancy flux and to I'.G often differ from F. Then the expression for B can become rather complex, depending on the jump frequencies and driving forces for all atoms and ions in the crystal.
NON-RANDCl1 DIFEUSION IN IONIC CRYSTAlS
Detailed calculations show that the vacancy wind effect for .impurity diffusion usually differs from that for self-mffusion. In particular, 1 + B for .impurity diffusion can differ greatly from f-l and in some cases will even be negative [5].
For diffusion by a monovacancy mechanism, negative values
11
of 1 + B are obtained mainly when the irrpurity and vacancy are tightly bound together. In an ionic crystal, the force from an electric field will create a flux of solvent ions on the sub­ lattice containing the impurity. '!his ion flux will cause an equal and opposite flux of vacancies. For vacancies bound to impurity ions, the vacancy-solvent flux merely moves vacancies fran the dONn-stream side of irrpurities to the up-stream side. Because of this vacancy redistribution, the impurities are pro­ vided with a greater than randan opportunity of jurt"q?ing up-stream (in a direction opposite to the solvent ion flux) than of jumping
dONn-stream. '!hen, if the impurity charge is less than twice the solvent ion charge, the vacancy-irrpurity carplexes may actually move up-stream, opposite to the normal direction of flow of the charged solvent and impurity ions.
Even if the drift mobilities of particular irrpuri ty ions do becane negative, the vacancy wind effect will not lead to a negative ionic conductivity in the crystal. Physically, this result can be described in terms of the charge flows carried by the solvent ions. For example, one can regard the vacancy­ impurity canplex as occupying lattice sites that oth.el:wise would be occupied by two solvent ions. Movanent of this conplex up­ stream will be accorrpanied by a countennotion of solvent ions down-stream with the conplex in effect exchanging places with two solvent ions. Since the motion of the canplex up-stream occurs only if the .impurity charge is less than twice the solvent charge, this interchange of positions actually moves net ionic charge dONn-stream. A positive value of the ionic conducti vi ty thus is obtained, as expected.
DEVIATIONS FRCM NERNST-EINSTEIN REIATION FOR DIFFUSION CN FCC IA'ITICE VIA DIVACANCY MEOfANISM
Recently, results have been obtained [6] for impurity diffu­ sion via divacancies on a face-centered cubic lattice, which is the same lattice as each sub-lattice in the NaCl structure. In this calculation, it was assumed that the two vacancies in each divacancy are tightly bound to one another so that the two vacancies in the divacancy are always nearest neighbors of each other. '!he di vacancy migrates by means of individual vacancy jumps which move the divacancy through the close-packed fcc lat­ tice without dissociation.
12 JOHN R. MANNING
The simple treatment of Eqn. (1) which was applied to the monovacancy mechanism must be modified here, since in the ill vacancy mechanism not all configurations ex which would allow an atom jump in the +x direction are equivalent. Instead there are two types of oonfigurations from which at least one of the vacancies in the divacancy can cause a +x atom jump [7]. In configuration 1, only one of the vacancies in the divacancy has a different x coordinate than the diffusing atom whereas in oonfiguration 2 beth vacancies are on the plane in the +x direction fran the atan. Then \!+ and \!rr+ in Eqn. (1) becane two-oomponent row vectors. The carponents of \!+ are the effective jump frequencies for jumps caused by divacancies in oonfigurations 1 and 2, and the cc:mponents of \!rr+ are the arrival frequency of fresh divacancies at configuration of type 1 am 2. The summation involving the P+ and P_ quantities in Eqn. (1) also must be rrodified, with the quantities in the summation becc:ming 2 x 2 matrices and including possible transi­ tions between configurations 1 and configurations 2.
Wi th this approach, matrix equations similar, but not identi­ cal, to Eqns. (6) and (10) can be developed; and it is found that the diffusion flux J of a dilute impurity diffusing via divacancies in a oonstant electric field E in an otherwise pure fcc crystal is given by [6]
cD* q. E J = - -D* ~~ + k~
(3.89wl + 13.26 w3 ) qs
[1 +(7.75Wl + 6.75 w3 ) qi] (28)
where qi is the charge of the impuri ty ion, qs the charge of the solvent ions on that sub-lattice, WI is the jump frequency for a jump of a solvent ion where roth before and after the jump both members of the di vacancy pair neighOOr on the impurity, and w3 is the jump frequency for a vacancy jump with a solvent ion where before the jump roth members of the divacancy neighOOr on the impurity but after the jump only one neighOOrs on the impurity. 'IWo other jump frequencies also are allowed in this model: Wo for other allowed vacancy-solvent exchanges (with dissociation of the divacancy not allowed) and w2 for exchange of a vacancy with the impuri ty . These frequencies w and w do not appear in the expres­ sion, equal to 1 + B, inside tRe bra~ets in Eqn. (28), but w2 does enter into the expression for D*.
Comparison of Eqn. (28) with Eqns. (12), (14), and (15) yields
y _ qi (3.89 WI + 13.26 w3) qs
D* - kT [1 + (7.75 WI + 6.75 w) qi] (29)
which shows a significant deviation fran the Nernst-Einstein equa­ tion.
NON-RANDOM DIFFUSION IN IONIC CRYSTAIS
Fbr self-diffusion, where wI = w3 and qi = qs' this expres­ sion reduces to
)1 _ 2.183q D* - kT (30)
13
The techniques used to calculate 6G in the derivation of Egn. (28) also have been applied to calculate the correlation factor f for self-diffusion on an fcc lattice by a divacancy mechanism [8], yielding the result
where
8 WI + 9.15 w3
(31)
(32)
This expression for the correlation factor is similar in form to that obtained by Mehrer [9], though the numerical coefficients in this expression for <p are smaller than Mehrer's. Fbr self-diffusion, where WI = w2 = w3 ' Egn. (31) becomes
-1 f = 0.4581 = (2.183) (33)
Thus, again for self-diffusion,
~ = --.SL D* kTf (34 )
Nevertheless, for impurity diffusion, )1/D* can differ greatly from q/kTf, as can be seen by canparing Egns. (29) and (31).
An important part of the derivation of both Et]:ns. (28) and (31) was the establishment of an accurate value for w+7To = w7T+o' the effective escape or approach frequency of a vacancy from or to the impurity in the absence of a driving force, equal to the average of W+7T and W7T • This quantity appears both in the equa­ tion for P, which yields f by Egn. (10), and in the equation for 6G, obtained from Egns. (9) and (20). Thus, although the defect­ wind effect and the correlation factor effect are distinct effects, one depending solely on solvent ion jumps and the other depending to a large extent on the frequency of impurity ion jumps, the common term w+7TO enters into both effects.
14 JOHN R. MANNING
MORE GENERAL TRFATMENTS
In the discussion above, several simplifying ass1.lItptions were made to provide ease of treatment. For exanple, in Eqns. (17) and (18), it was assumed that v+ and v_ would not vary with position (diffusion coefficients and driving forces are constant); and throughout the discussion simple crystal structures and diffusion directions were assumed. In treating actual problems, these assumptions are usually not necessary.
It can be shown [10) that diffusion coefficient gradients and driving force gradients will not affect the kinetic expres­ sions for D* and <vF> as measured at a particular plane. Thus , even t.hough the average drift of a layer of tracer atans can be sanewhat affected by such gradients after the tracer atans move away fran the central plane, the diffusion flux at this central plane is not affected by these gradients. :fur driving forces resulting from electric fields, an essentially constant force will often be present throughout a particular diffusion zone. Then, the kinetic equation above can be applied directly. Dis­ cussions of modifications which are required when there are other driving forces or effects from diffusion coefficient gradients can be found, for example, in references 1 and 3.
In more canplex crystals and for diffusion by more canplex defects, the syrranetry planes used to simplify the above equations often will not be present. Nevertheless, it is possible in these cases to use canplete-path equations [11] which when reduced to a matrix fonn yield equations very similar to Eqn. (6)- (10). These matrix equations will allow treatment of non-rarn.cm effects by individual defects in any crystal.
REFERENCES
1. J. R. Manning, Diffusion Kinetics for AtOllS in Crystals (D. Van tbstrand, Princeton, N. J., 1968).
2 • A. D. IeClaire, in Physical Chenistry, An Advanced Treatise Vol. X, Solid State, edited by H. Eyring, D. Henderson, and W. Jost (AcadEmic Press, New York, 1970).
3. J. R. Mmning, in Diffusion (Arrerican Society for Metals, Metals Park, Ohio, 1973) pp. 1-24.
4. See e.g., A.B. Lidiard in Handbuch der Physik, edited by S. Flugge (Springer-Verlag, Berlin, 1957), Vol. 20, p. 324.
5. J.R. Manning, Phys. Rev. 139, A2027 (1965); also in Mass Transport in Oxides, National Bureau of Standards Special Publication 296, edited by J.B. Wachtman, Jr. and A.D. Franklin (U.S. (bvernment Printing Office, Washington, D.C., 1968), pp. 53-63.
NON-RANDClJ! DIFFUSION IN IONIC CRYSTALS
6. J. R. Manning, Abstract Bulletin of 'IMS!AIME, Fifth Annual Sprin;J Meetin;J, Philadelphia (1973) p. 148.
7. R.E. Howard, Phys. Rev. 144, 650 (1966). 8. J.R. Manning, Bull. Am. Phys. Soc., Ser. II, 18, No.3,
426 (1973). 9. H. M8hrer, J. Phys. F: IVEtal Phys. 2, Lll (1972).
10. J.R. Hanning, Phys. Rev. 139 A 126 (1965). 11. J.P. Stark and J.R. Manning, Phys. Rev. ~, 425 (1974).
15
A. D. Le Claire
1. INTRODUcrION
"Correlation Effects" in diffusion and other rrass transport phenomena were first discussed for solids by Bardeen and Herring [1] in 1951, although similar phenomena had been recognized as occurring in neutron diffusion as previously as 1936 [2] and in gases even earlier [3]. The effects derive from the fact that, in the nature of the defect processes by which atoms migrate in solids, the direction of any atom displacement is not at randan but is influenced in part by the directions in which the previous and earlier j1.lITpS of that atom occurred. That is to say, successive jump directions are correlated with one another [4].
The commonest situation is that of vacancy diffusion. SUp­ pose an atom has just made one j1.lITp by exchange with a vacancy. Because the vacancy is adjacent on the original site of the atom, the atom is rrost likely to make its next j1.lITp back to this site and least likely to rrake it in the same direction as the first j1.lITp. Consecutive pairs of jumps with zero net displacement occur rrore frequently, and those with maximum net displacement less frequently, than randani the effect is to reduce the diffusion coefficient, by a factor f, below the value calculated assuming displacements in any direction occurred with equal probability. f is called the "correlation factor".
Similar effects occur with other types of defect, as we shall see.
Correlation effects are closely related to the occurrence of non-zero cross-terms in the irreversible thermodynamic for.mulation
17
18 A. D. LE ClAIRE
of nass transport [5,7]: these aspects of the subject are dis­ cussed in the following paper by Cooper. In this paper we shall be concerned with the random walk. fonnulation for this provides the expressions from which correlation factors can be calculated.
2. RANOOM WALK EQUATIONS [6-8J
It is a straightforward matter to derive a quite general expression for the net flux of matter J, in a concentration gradient. This is
2 2 <X> dC <x > dC d <X >
J = c (Xo)-t- - dX 2t" - c (Xo ) dX dC 2t" (1)
+ higher order tenns
c(Xo ) is the concentration of the diffusing species at x = Xo' where J is evaluated. <X> and <X2> are respectively the rrean dis­ plaCEment and mean square displacement of a diffusing atom after a time t.
Fbr the simple but important cases of tracer diffusion in effectively chemically homogeneous systEmS, diffusion properties do not vary with posi tion in the sample and the third tenn in (1) is zero. Also, <X> and other odd moments are zero. Provided we can ignore the higher order terms, eqn. (1) reduces to Fick' s law with
(2)
which is Einstein's equation. The asterisk indicates the restric­ tion to tracer type diffusion in homogeneous SYStEmS. There are similar equations for Dy* and Dz*. In isotropic crystals
-+2 D* = D* = D* = D* = <R > / 6t (3 ) x Y z
~ -+ <R > is the mean square of the total vector displacement R of an atom in time t.
Equations (2) and (3) apply for any tracer self-diffusion co­ efficient in a chEmically homogeneous crystal. In the general case of diffusion in a chEmical concentration gradient all three tenus of equation (1) have to be considered in developing expressions for the intrinsic diffusion coefficients and the chemical interdif­ fusion coefficient ~ [7]. These also of course contain correlation effects, but the correlation factors involved are, to a very good approximation, numerically the same as those that occur with self diffusion, so we confine our considerations to this simple case.
CORRELATION EFFECTS IN IONIC TRANSPORT PROCESSES 19
3. GENERAL EQUATICNS FOR CORREIATIQN FACroRS [7,8J
Each R in eqn. (3) is the vector sum of individual displace­ ments h of an atom. Let there be on average N displacem:mts in time t, or r in unit tline (N = rt). 'Ihen
<R2> = < (l:r. )2> ~
~ ~ ~ ~ ~
If all the ;i have the same magnitude r and if <;i·;i .> (for a given j) has the same value whichever junp is represented b;?;i, then all jumps are said to be of the same type. 'Ihis is the case, for example, for self or impurity diffusion by the vacancy mechanism in a f .c.c. or b.c.c. crystal or sub-lattice. It is then convenient to write
+ + 2 <r .• r. . > = r <Cos8 . > (5) ~ ~+J J
where <CoS8j> is the average value of the rosine of the aIBle between anyone jump am the jth next junp. Equations (3), (4) and (5) then yield
D* = ! rr2f (6) 6
(7)
f is obviously the "correlation factor", as defined in the Introduc­ tion, because if successive displacements take place at random all the <Cos8y will be zero and f = l.
Generally, for diffusion by a defect mechanism f is not unity. We have imicated already in the Introduction how correlation effects cane about with diffusion by vacancies. It will be evident fran what vas said that <Cos8F will be negative, that <Cos82> will be positive but snaUer than <CoS8l>' and so on. We expect the various <CoS8j> to be related and it is easily swwn that if each junp vector is an axis of at least two-fold symmetry
<CoS8.> = <CoS8 >j J 1
Equation (7) row becanes
for vacancy diffusion. Clearly fv~l.
More generally, when atans are diffusing by two or more types of jump it turns out to be more convenient to work with the com­ ponents of -;: i along the diffusion direction. If this is the-X=­ axis these are xi arrl we put X =2l:Xi in eqn. (2). In place of (4) we then have <x2> in terms of <Xi> and the <Xi Xi+j>. Jumps are now of the same type, or equivalent, if for each of then <xf> arrl <Xi xi+j> have the same values. Let n be the number of different types and r a the number of jumps of type a made by an a tom in unit time. We now easily find for D.x the result
where
D X
f aX
1 n 2 - l: r x f 2 a=l a a aX
= 1 + 2 l: <x X . > Ix2 j a aJ . a
(10)
(11)
Xa is the camnon value of Xi for jumps of type a and Xaj is the x­ cc:mponent of the jth jump following an a type jump. fax is the "partial correlation factor" for diffusion along X associated with jumps of type a. Generally there will also be a set of fay and f aZ for diffusion along the y and z direction respectively ..
In terms of an overall correlation factor for diffusion along x, fx' eqns. (12) may be rewritten
Dx <} l: r a x~) f X (12) a
where
(13)
and cq, = r a/ l: r is the fraction of all jumps that are of type a. Equation (ll~ ai\:i (13) reduce to eqn. (7) when n = 1.
Obvious applications of these equations are to self diffusion by vacancies in anisotropic crystals and to solvent diffusion, for example in a dilute alloy, or doped ionic crystal. In the latter cases there are different types of jumps because the <Xi xi+j> will generally depend on the proximity to an impurity atan ot the par­ ticular solvent jump represented by xi. c a and n can generally be readily determined by inspection.
There are similar correlation effects with intersti tialcy dif­ fusion, as occurs for example for the cations in A:J halides and the anions in many fluorite type structures. Figure 1 sho.vs three
CORRELATION EFFEx:::TS IN IONIC TRAN'SPORT PROCESSES 21
x X X X X X
)~X X
"'- X
(a) (b) ( c )
Figure 1 -- Interstitialcy Diffusion.
consecutive jUIrg?S of a tracer atom. As with vacancy diffusion, the second junp is more likely than rand.an to be, as shONl1, in the reverse direction to the first because, in this case, an intersti­ tial atom is immaiiately available to induce such a jUIrg? A jurtp of the tracer in a different direction would first require inter­ stitialcy defect migration out of configuration (b), through the surrounding lattice, so as to bring an atan into one of the other interstitial sites adjacent to the tracer. lbwever, there is a difference row fran vacancy diffusion: the third junp of the tracer out of v.7hichever i-site it occupies after its second junp (Fig. lc) will occur with equal probability to any of the neighbor­ ing and. equivalent lattice (1) sites. Thus it is not correlated with the secorrl nor, therefore, with the first jump. It follows that <Cosej> = 0 for all j ~ 2. Also <Cosel> is non-zero only for correlated consecutive p::tirs, i-+l-+i, of jurtps. These are half the total number of p::tirs so equation (7) becanes
fI = 1 + <Cosel >· 1 . ~-+ -+~
(14)
Equation (9) and (14) contain only <Cosel >. In other words a calculation of f requires consideration of the correlations only between consecutive junps. The same result is obtained in the
22 A. D. LE CLAIRE
further develop:nent of the more general equations (11) or (13): a relation analogous to (8) allows the surrmation in eqn. (11) to be carried out to give a result in terms related only to the <xa Xal>'
Another carmon rrechanism is that of silrq;lle interstitial dif­ fusion, where atans diffuse by junps fran one interstitial site to another. This occurs in metals for such solutes as H, C, N, 0 etc. , and is a possible mechanism by which ions in excess of stoichio­ metric and accommodated interstitialy, may diffuse. Provided the concentration of interstitials is low enough all junp directions are equally probable at every jump so that f = 1. However, at higher concentrations some jump directions of an atan will be blocked by adjacent interstitial sites being occupied. SUch blocking will have the effect of giving <OOS81> a negative value, because at least the site fran which an atan last junped must be vacant, for a time anyway. f will therefore decrease below unity as the concentration increases. In the limit of near full occupancy of interstitial sites the process becomes the vacancy mechanism and the correlation factor beaames that for diffusion by vacancies on the lattice of interstitial sites.
With regard to diffusion by vacancies, as their concentration increases vacancies additional to the one that effected a jump of an atom will, with increasiI1<J frEqUency, be found adjacent to that atan: this can only serve to make more randan the direction of the atan' s next jump and so increase f y • The lowest value of fv will occur when the vacancy concentration is low enough for t.lEre to be negligibly few encounters of a tracer atom with rrore than one vacancy at a time. This will usually be the case in stroI1<Jly ionic crystals and in metals, where intrinsic vacancy concentra­ tions are :;; 'VIO-3. However, rrany non-stoichianetric catlpOunds may have vacancy concentrations very much in excess of intrinsic and these changes in fv may be linportant.
Similar consideration apply to diffusion by interstitialcy jumps. As the concentration of defects increases we expect suc­ cessive jumps of a particular atom to be decorrelated by the prox­ imity of other interstitials and fI (eqn. 14) to begin to rise above the value for isolated defects. As interstitial sites becane progressively filled, either interstitialcy junps begin to take on the nature of single vacancy junps that move two atoms at a time, or the migration process goes over into single atom junps. In either event the limitiI1<J correlation factor is one for diffusion by vacancies, and may be greater or less than the original fi for isolated interstitialcy jumps.
Let us rr:JW consider the correlation factors for the diffusion of the defects thEmSelves. This is important because in many
CORREIATION EFFECTS IN IONIC TRANSPORT PROCESSES 23
cases ionic conductivity is most sinply thought of in terms of the migration of the defects rather than the ions.
When defects are present in low enough concentration that they rarely encounter one another, and provided they are in a pure lattice or sub-lattice, (i.e. only one species of atom or ion present), they can obviously jurrp with equal probability in any of the crystallographically allowed directions and their correla­ tion factors will be unity.
If these conditions are not satisfied defect migration will generally be non-random and the diffusion coefficient for defects will contain a correlation factor, fda
When there are tw::> or rrore species of atom present a vacancy, say, will generally exchange with one species of atom or ion more readily than with the other. This makes the vacancy migration non-random and can be shown always to give an overall correlation factor for diffusion of vacancies less than unity. Ibwever, the partial correlation factors associated with vacancy jumps by -­ exchange with the slower moving species may be greater than unity [9] •
Strong interactions between defects may also serve to induce additional non-randanness into their migration and further contri­ bute to changes in fd fran unity. Such interactions may induce clustering, or may impose a minimum distance of approach for two defects, preventing them for example from occupying adjacent sites, or again, in extrerre cases generate an ordered or near ordered distribution of defects. There is a good deal of evidence that such situations often arise with the high defect concentra­ tions associated with non stoichiometry in many canpounds [10]. When they do, defect migration may be very non-random and correla­ tion factors accordingly low. With perfect order, to take an extreme case, any defect jump will put it on a ''wrong'' site and so will be followed with high probability by a reverse jump. fd will then be very srrall.
Naturally erough, we expect any non-randanness in the defect migration to contribute to the non-random migration of the atoms or ions themselves -- the tv.o must be intimately related. Any effect that reduces fd from unity will reduce fI or fv and we might expect the ratio fdlfv I to be roughly constant. There are one or tw::> cases where this sort of proportionality has been found to hold, but its general validity is by no means established. (See later.)
These general remarks have been with regard to the romronly discussed vacancy, inter$ti tial and intersti tialcy :rrechanisrns of diffusion. There is evidence that other and rrore carrplex defects
24 A. D. LE CIAIRE
may need to be invoked in discussing atom migration in many types of non-stoichianetric carpoW1ds [10], but little study has so far been made of their modes of migration or of the associated correla­ tion effects.
4. THE CAICULATION OF CORRElATION FACI'ORS
Space forbids anything IIDre than a brief :rrention of the three general methods that have been used to calculate correlation fac­ tors. These are (a) random walk methods (b) Monte-Carlo methods and (c) nett flux methods.
Random Walk Methods [7,8,12-15]
These start from equations like (9), (11) or (14) and calcu­ late <Cosel>' <xa xal> etc. directly.
If it is <Cosel> we are dealing with then given an initial tracer jump from site i to site k
z <Cosel > = E PI Cos (i~k k~l)
1=1 (15)
where z is the coordination nunber and PI the relative probability that the second junp of the tracer will be from k to 1. Fbr such a j unp to occur the vacancy, if we assume a vacancy mechanism, will have had to migrate from i to I, so that to calculate the Pl we need to know the relative probabilities of the vacancy occurring at each of the z nearest neighbor sites of the tracer. To find these we need to consider all possible sequences of junps, or tra­ j ectories, the vacancy can pursue through the lattice that start at i and finish at each of the 1. Systematic means have been devised for doing this and are described in the references.
Alth:>ugh the random walk has been used in the very large majority of calculations of f so far rrade, it has two :i.rrportant limi ta tions .
(i) It must be valid to assume that the defect concentra­ tions are sufficiently snall, .::;0.1% say, that all possibilities for exchanges between a particular atom and a particular defect are fully exploiterl before another defect approaches that atom.
This limits application of the method in its present fonn to diffusion in pure substances and stoichiometric catl?Ounds where only intrinsic defect concentrations prevail. It cannot be readily
CORRELATION EFFECTS IN IONIC TRANSPORT PROCESSES 25
adapted to calculate f in crystals where defect concentrations are high.
(ii) It must be possible to specify adequately and simply all possible vacancy trajectories bebNeen successive junps of a tracer. This requires that the region of crystal arcund the tracer, through which the trajectories are bein:] considered, be c::a:tpJsed of identi­ cal atoms. If different species of atoms are present the atomic configuration around the tracer nay be changed by the passage of a defect. The random walk method, as at presented fo:r:mulated, has no way of siIrply taking into account such changes.
The method is therefore restricted also to self diffusion in pure substances and to diffusion in very dilute solutions. M::>st estimates of f have been for such cases. It cannot be used for concentrated alloys or solutions, unless the approximatiIlJ assunp­ tions is made, as has been [9], that defects have an average migra­ tion rate duriIlJ their trajectories between tracer jurrps.
These inherent limitations of the random walk method are, in principle, absent from the M:mte-Carlo and the M9an Flux method. Although these are rot capable of the same precision as the random walk method, they are becaning important as interest grows in systems to which the randan walk method is inapplicable.
Monte-Carlo M3thods [11,17,18]
These detenuine f through direct estimates of <R2> or <x2> and the use of the general definitions of f, for example
f = <R2> /N<r~> 1
appropriate for isotropic crystals.
(16)
The configuration of an adequately sized crystal, with its defects, is represented digitally in a canputer array and the actual paths of a number of atons are simulated, subject to what­ ever restriction there nay be on atan jumps arising from the nature of the defect beiIlJ studied, from interactions between the defects or between like and unlike atans (ordering effects), and so forth. with increasing number of individual paths rronitored and/or of steps within each path, the mean <R2> converges to a limiting value that is used in egn. (16) to give f.
Nett Flux M3thods [19,20]
These first deter.mine the diffusion coefficient D through a calculation of the nett flux J in a ooncentration gradient dC/dx --
26 A. D. LE CIAIRE
D = -J (dC/dX) -- using a rrodel sufficiently detailed to contain the essential non-randan aspects of atomic migration. The factor f can usually be identified in the resulting expression.
One starts with sane suitably detailed specifications of the atomic and defect configurations that occur in the crystal. In a condition of steady state diffusion under a concentration gradient the distribution of configurations rEmains in dynamic equilibrium, constant in time. But there is a net flux across any plane that can be calculated in tenus of the rates at which atc:ms by jumping move into and out of the various configurations. Provided the configurations specified include all those that would be explicitly or implicitly taken into account in a calculation of, say, the PI in equation (15), this method yields a D that autanatically con­ tains the same factor the randan walk method calculates and iden­ tifies as due to correlations between successive atom jumps. The difference between the two methods is essentially in the averaging methods they employ, but the way in which configurations can be specified is particularly convenient for including such effects as ordering and high defect concentrations.
Diffusion by round vacancy-impurity pairs was first treated this way [21], the configurations considered being the various orientations of pairs and their distribution in the steady state. fure recently Kikuchi and Sato [19,20] have elal::orated the method, fonnulating it in tenus of the path probability theory of irre­ versible phenomena.
5. RESULTS OF CORRELATIO'J FACIDR CAICUlATICNS
Self Diffusion in Pure SUbstances
In very many of these cases there is no more than one defect frequency involved so that f is just a numerical factor. Sane values for such cases are shown in Table I.
Results for many more cases of interstitialcy diffusion are reported in reference 14. In practice, two or more ~s of interstitialcy jump may be operative, when fI will be a function of the ratio of their jump frequencies. In the H.C.P. case, fre­ quencies for jumps within a basal plane, wA' and between planes, WE' are assumed equal. For the general case see reference 13.
f is also known for self-diffusion by bound divacancy pairs in ionic crystals (NaCl and CsCl) as a function of the ratio of cation- and anion-vacancy jump frequencies [12].
CORRELATION EFFECTS IN IONIC TRANSPORI' PROCESSES 27
TABLE I
Vacancy Diffusion
Dianond Simple Cubic B.C.C. F.C.C. H.C.P.
Hexag. Net (2 Dim) Square Net (2 pim) Trian:]. Net (2 Dim)
Interstitialcy Diffusion
F.C.C. and Cation Diffusion in 'Fluorite' B. C. C. and Cation Diffusion in 'CsCl'
(110) jumps
= 0.7812 = 0.7815
0.3333 0.4669 0.5601
fI 0.80 1.0 0.666 0.832 1.0 0.912 2/3 32/33 0.9643 0.90
Divacangy Diffusion ~v
F.C.C. [43] 0.458
Exchange, Interstitial Diff.
~
0.40 1.0 0.333 0.622 1.0 1.82 0.333 0.727 1.446 0.6
All these values of ·f are calculated by the randcm walk method and so are appropriate only for crystals with low defect concen­ trations. For crystals with high defect concentrations there have been very few calculations, but enou:Jh to illustrate many of the points made in Section 3.
Fig. 2 shows sane results of calculation by de Bruin and Murch rll], usin:] the M:mte Carlo method, for diffusion by vacancies in a simple cubic lattice. Curve 1 gives fv for tracer self-diffusion by non-interacting vacancies and shows the rise fran fva = 0.653 (see Table I) to, ultirrately, unity. Curves (2) and (3) relate to vacancies between which there are interactions that prohibit any two vacancies occupying adjacent sites. Curve (3) shows fd for the vacancy diffusion and Curve (2) shows fv for tracer diffusion. Note that the ratio fd/fv is rOu:Jhly constant. It is of interest too that the effects of concentration increase and of interactions do not become very :important until beyond
28 A. D. LE CLAIRE
atout 5-10%, so the results shown in Table I IrI3.y, in some cases anyway, be useful beyond the intrinsic range of defect concentra­ tions.
de Bruin and :M..lrch have also used the M::>nte-carlo method to calculate f for diffusion by interstitialcy j'l.lllpS at high defect concentration, but only for a planar square lattice. f changes fran 2/3 for isolated interstitialcy j'l.lllps to 0.467 for vacancy diffusion (Table I) when nearly all interstitial sites are filled.
These same authors have also calculated f for anion self dif­ fusion in U0:2+x crystals by migration of the Willis 2: 2: 2 clustered defect, the most corrplex defect for which correlation data are available. They find f = 1.587, greater, surprisinJly, than unity. This is attributed to the unusually canplex se::J:uence of atom dis­ placements involved in a Willis cluster junp, but a convincing description of this sequence has not yet been published. f was shawn to decrease with increasing density of defects, due to steric effects assumed in the mcxiel calculation.
The results of both these calculations again show, surprisingly, that the correlation factors sean to rerrain fairly constant up to at least a 5-10% concentration of defects.
12
10
~
10 20 30 40 50 60 70 80 90 100
Fract ion of lattice - sites occupied by defects 96
Cbrrelation factors for vacancy diffusion mechanism in a simple cubic array. [11]
CORRELATION EFFECTS IN IONIC TRANSPORT PROCESSES 29
The effects of interactions that order the atoms and defects are vividly shown in the results of nett flux calculation by Sato and Kikuchi [20], using the path probability method. These are reviewed elsewhere in this volume.
Diffusion of Impurities
By this we mean diffusion of a solute present in law enough concentrations for each atom to diffuse effectively through pure solvent. If w2 is the jump frequency for the solute impurity atoms, the correlation factor f2 for their diffusion turns out, in a large number of cornrronl y occurring cases, to have the form
(17)
where u contains the frequencies for jumps only of rearest neighbor and more distant solvent atoms. f2 is small when w2 » u because one impurity jump is then likely to be followed by second in the opp0site direction, and <Cos8p -+ -1. When w2 « u, f2 is near unity -- the many exchanges a vacancy, for example, will make with solvent atoms between two exchanges with the impurity tend to randomise the position of the vacancy with respect to the impurity and make <CoS81> -+ O.
Because of the local influence of the solute atans, solvent jumps in their neighborhood will differ in frequency fran the value Wo in pure solvent. Thus a vacancy on a 1st nearest neigh­ bor (n.n.) site to a solute moves to another 1st n.n. site with frequency wI (in f .c.c.) or may dissociate fran the impurity with frequency w3' w4 denotes the frequency for the reverse association jumps and w5 that for vacancy jumps fran 2nd n.n. to more distant sites. All more distant jumps are assumed of frequency wo ' as is also w5 in many cases.
u is known in terms of these frequencies for vacancy and interstitialcy diffusion in several cammon types of lattice [8, 22,23]. Fbr example, for f.c.c. lattices
(18)
where F is a function of w4/wo' slowly varying fran 2 at w4/wo = 00 to 7 at w4/wo = O. F(w4/wo = 1) = 5.15.
There rray be a binding energy between impurities and vacancies, at least at 1st n.n. separations. If this is Bl it will be related to w4 and w3 by
(19)
Solvent Diffusion in Very Dilute Solution
Because of the changed vacancy concentration (due to Bl) and altered jump fre::ruencies near an impurity, the solvent diffusion coefficient D (c) in a solution of concentration c will differ fran its value D (0) in pure sol vent. At low concentrations we anticipate a relation
D (c) = D (0) (1 + be) (20)
and this is found experimentally. Calculating the correlation effects entails consideration of a large munber n of jump' types' corresponding to the many configurations of vacancy, tracer (sol­ vent) atan and impurity from and into which tracer jumps can occur. Assuming the same range of influence of a solute atom as in the expressions for f2' calculations have been made for f.c.c. [24] (n = 13) and b.c.c. [25] (n = 15) crystals. Results are expressed in terms of the 'enhancement factor' b. For f.c.c. crystals, for example,
(21)
Xl and X2 are mean partial correlation factors associated with WI and with w3 and w4 jumps respectively and are functions of w3/wl' w2/wl, and w 4/wo.
Diffusion in Cbncentrated Solutions
The calculation of correlation factors for diffusion in con­ centrated solutions is a more difficult problem because of the large nunber of different atomic configurations and of jump types that would need to be specified in any accurate trea-t::rrent. Approx­ imate values of f have been obtained by assuming an averag.e jump frequency for a vacancy in its trajectories between two consecutive exchanges with a tracer [9]. The problem is then reduced to that of impurity diffusion, with just the additional feature of corre­ lation effects in the trajectory motion of the vacancy, and can be solved with the random walk e::ruations.
The path probability nethod is particularly suited to this more complex type of problem and more detailed though still approxi­ mate calculations, incorporating ordering effects, have been made using it [19]. There are also sone very recent calculations by the fbnte-Carlo nethod [42] for random alloys.
6. THE MEASUREMENT AND USE OF CORRELATIQ"J FACTORS [8]
The correlation factor is just a fraction and so is not a very important contribution to the magnitude of D itself, except
CORRELATION EFFEX::TS IN IONIC TRANSPORl' PROCESSES
in extreme cases. lbwever, it is of course always essential for a proper analysis of diffusion measurerrents.
31
If f is independent of T, as when only one junp fra;ruency is involved, it is just an additional factor in A in the Arrhenius equation D = A exp (-Q/Rl'). However, when more than one fre:JUency is involved f varies with T and partitions between the A and the Q in a way that can be significant in theoretical discussion of their rmgnitudes. Fran the definition of Q as Q = -RdlnD/d(l/T) , there must always in the equation for Q be a tenn
-C = Riln.f/d (l/T) (22)
'!hen, A must contain the factor (f exp - C/Rl'), a constant if C is constant.
Theoretical and experimental evidence indicates that C can be an :important contribution to Q2 for :impurity diffusion, being carparable with, although less than, the difference between Q2 and the solvent self-diffusion Q,.
Aside fran such consequences of the variations of f with T, its rmin importance derives fran its dependence, apart from crystal structure, on the mechanisn by which atom jurrq;>s occur and, When there is more than one, on the relative values of all the junp frequencies concerned. It follows that by experimentally isolating f we can in principle derive inforrmtion about the junp mechanism operatin:r in a particular case and, when the llEChanism is known, inforrmtion on the relative junp frequencies.
'!here are two methods by which a knowledge of correlation factors can be exploited in this way.
(1) By canparin:J tracer diffusion coefficients with ionic conductivities. '!his is the rrore pcM'erful and unambiguous method, but applicable only to good ionic conductors.
(2) By comparin:J the diffusion coefficients of two different isotopes of the diffusing species -- the Mass Effect method. '!his is of general application but results are not always canpletely unambiguous •
Ccroparison of Diffusion and Ionic Conductivity. We can measure for each sublattice of an ionic crystal the tracer self diffusion coefficient Dr of the ions and their contribution cr to the total conductivity. Dr is
1 2 DT = 6" r T r T fT (23)
32 A. D. LE CIAIRE
fT is the number of j unps of a tracer per ooi t time and rT the dis­ plaCEment. The conductivity in a pure ionic crystal is due to the net drift of the lattice defects in the electric field -- each defect behaves as an effective carrier of charge equal to the ionic charge q, am
a = nd q ~ (24)
nd is the density of charge carriers and \1 their nobility. We can also write
1 2 D = -6 f r f (25) q q q q
for the diffusion coefficient Dq of the charge carriers. fq is the nunber of junps per ooit time of a carrier (defect) and r the charge displacEment at a jump. fq is the correlation fador for defect migration.
we now make use of the Nernst-Einstein equation relating Dq and )J for any type of charge carrier
Dq = kT )J/q
a
r (~)2 r
q
(26)
(27)
The large maJority of practical applications of this equation have been to D and a in sub-lattices of one atomic species with intrinsic defect concentrations. In such cases fq = 1 and equa­ tion (27) contains only fT' Now consider same special cases.
When the defects are vacancies rT = rq and Nr T = nd r q where N = number of ions on the sub-lattice per unit volume. Equation (27) then gives
where
(28)
(29 )
If the defects are intersti tials moving by simple interstitial j unps, then clearly D'J""Da = 1.
CORRELATION EFFECTS IN IONIC TRANSPORT PROCESSES 33
If the interstitials nove by interstitialcy jumps, two ions move for each defect junp and NT T - 2 r gl'la • If they are co-linear junps charge q is displaced rq = 2rT wliile a tracer moves rT. Then
I D /D = - f (30) T a 2 I
If, hCMeVer, the jumps are non-colinear, rT/rq depends upon their geanetry. For the possibilities in 1lg Br for example, rTirg = 1378 arrl 13/4. The last column of 'Iable I lists the resultmg values of Dr/Da (= HR), along with values for other cases of inter­ sti tialcy j unps.
The ratio DrVDa is sometimes called 'the correlation factor' in the present context, but clearly it contains nore than f and this tenninology is misleading. It derives fran eqn. (28) having been the first relation to be derived and discussed on this topic, by Haven [26]. Accordingly the tenn "Haven Ratio", HR, has been suggested for it.
When there are two or more types of defect present, and mechanisms for their migration, these contribute additively to Dr, Dq and (J and the above equations are easily generalized. In all cases, for a given mechanisn or combination of mechanisms in specified proportions, there is a unique theoretical value of HR. Thus, experimental measurements of this ratio are of considerable diagnostic value in establishing or identifying the operative mechanism in any particular case. It is to be ranembered that neutral defects (e.g. anion-cation vacancy pairs) contribute to Dr but not to a, so HR is larger than in their absence. Similarly, any electronic conductivity can make HR anomalously small.
fq will generally differ fran unity when there are interactions between defects or when there is more than one species of ion on the sub-lattice. In these cases equation (27) is of less general Vcilue because comparatively little is so far knCMn about fq •
We have havever quoted a few studies of the way f is changed fran unity by defect interactions. These of course ~e fT as well, and it a~s that ~n the ratio fr/fq at least the dras~c effects of defect lllteract~Ons and order~ng are to some apprec~able extent canceled out.
Ebr exan:ple, a cancellation is evident in de Bruin and Murch's calculations (Fig. 2), fran which fylfq '" 0.68. This is close to the limiting value fv = 0.653 for free "'Vacancies. Similarly, in Sate arrl Kikuchi's S-Alumina calculations the influence of ordering on fT arrl on fq is very rruch less evident in their ratio.
These indications suggest that equation (35) with fg = I may be still valuable as a help in identifying migration mecfianisms
34 A. D. LE CLAIRE
even when infonnation is lacking on the possible existence and details of defect interactions, ordering effects etc. in the system being sttrlied. However, this is a very tentative conclu­ sion based on sparse evidence and further study is needed on the extent of its validity.
Equation (27) is appropriate for pure sub-lattices. When there are two or more ionic species present, one works with the ratio of Dr to the mobility \l of each ion. Thus for ions A of charge qA diffusing by the vacancy mechanism e.g.
(31)
where G contains tracer cnrrelation effects and the effects of non­ random defect migration. It is the analogue of frr/f and equals fV when all ions are identical. Manning [7] has d.ev~lOped expres­ S1-ons for G, for the case that A is a dilute i.rrg?urity, in tenus of the frequencies wO' WI, w3 etc. already defined.
We now mention a few examples of the use of eqn. (27).
The classical illustration of its value is Friauf's analysis of HR for hJ migration in AgBr [27J. HR ranges from 0.5 at low to 0.65 at high temperatures. The defects are known to be Frenkel pairs, so a temperature variation is expected, but HR is incnnsis­ tent with any combination of vacancy and interstitial mobility if the interstitials move only by simple interstitial jurt"ps: HR v..ould then have to be between O. 78 and 1. 'Ib explain the lower HR one must conclude there is appreciable migration by intersti tialcy jumps (see Table I). Detailed analysis showed that HR could be satisfactorily accounted for only on the assunption of both colinear and non-cnlinear jumps and the results gave the relative jump frequencies of these. No other type of experiment could so readily provide such detailed information on the nature of jump mechanisms .
Another example of the method is to de:monstra te the suppres­ sion of interstitials and augmenting of vacancies in hJBr and Agel when doped with CdBr2 i this changes HR towards the vacancy value of fv = 0.78 [28].
In the alkali and Tl-halides, SChottky defects prevail but one always finds HR > fv' This has been shown to be due to an appreciable cnntribution to Dr, but not of course to (J, from bound vacancy pairs [29 ,30J .
An example of current interest is diffusion and conductivity in the S-Aluninas. For Na and for hJ in single crystal of Na. S
CORRELATION EFFECl'S IN IONIC TRANSPORT PROCESSES 35
Al203 and hJ 13 Al203 respectively, with about 15% excess cation over stoichianetry, HR has been reported as '" 0.61. This is very close to the value of 0.6 for interstitialcy diffusion in a two dimensional trianJUlar lattice and provides evidence for this type of migration in the high rrobili ty planes of the 13 structure [31].
In the 13" structure, Na sites lie on a hexagonal lattice with packing such that interstitialcy diffusion seans unlikely. Fbr vacancy diffusion HR = 0.33, but this will be a lower limit in practice because of the high vacancy concentration due to 13" being always cation deficient.
~asurements of HR have been made for Na migration in poly­ crystalline 13" and mixed phase 13 13" sanples [32,33]. At sufficiently high temperatures (> 'V300°C) all sanples shCM camon values of HR that are indeed a little above 0.3 and slowly increasing with tem­ perature, reaching 0.4 at 'V600°C. Such a variation is consistent with there being a Na-Na nearest neighbor interaction energy E of 'V - 0.04 eV, as roughly estimated fran the Sato and Kikuchi theory. At temperatures below ~ 300°C HR falls below 0.3 and varies con­ siderably fran sample to sanple. The reasons for this are at present not understood.
The Mass Effect or Isotope Effect ~thod. Consider for simpli­ city cases of diffusion entailing a unique jump frequency w for the diffusing species. Fbr two isotopes of this species having dif­ ferent masses rna, m13 , junp frequencies will be slightly different -- wet and wl3 -- and therefore also the correlation factors fa and f 13 • The ratio of their diffusion coefficients is then, fran eg.n. (6) for example,
(32)
The f's have the fom of impurity correlation factors, even for self-diffusion, because the tracer is always an I :impurity I with jump rate different fran that of the host lattice atans. Thus fran eg.n. (17)
fa = u/(2wa + u) and fl3 = u/(2w13 + u) (33)
where u is the same in both because it contains only non-tracer jump frequencies. El:i.mina ting u and one of the f I S between (32) and (33) gives
(34)
The distinction between fa and f 13 is 'fJ.ON dropped because their difference is insignificant in this equation. We can 'fJ.ON forget aJ::x:mt the mass dependence of f and concentrate attention on its dependence on mechanisms and jump frequency ratios.
36 A. D. LE CIAIRE
Classical statistical mechanical calculations of w lead to the result [13,35] that
(wa/wS) - 1 = lIK( (mS /ma) 1/2 - 1) (35)
When a migrating atom jurrps, any coupling with the lattice may induce rrotion also in neighboring lattice atoms, and the lIK fac­ tor takes acoount of this. It is the fraction of the total kinetic energy, associatect with the whole rrotion at the saddle point of a jump, that actually resides in the migrating atom. Elementary treatments have often assumed lIK = 1, (i. e. wa /wS = (mS/mct) 1/2) but there is no a priori reason why a jurrp should be such an effectively one-atom process with no concomitant lattice atom rrotion.
From (34) and (35) we obtain
(36a)
n - 1/2 mS + (n-l) m
( rna + (n-l) m)
- 1
for processes where n atoms migrate simultaneously, m bein:;r the average mass of non-tracer atoms. (e.g. n = 2 for interstitialcy jurrps). E(n) is 'the mass effect for an n atom jurrp process' and can be calculated for any nonce Da/DS is known.
Equation slinilar to (36a) and (36b) can be derived for diffu­ sion where there are several jurrp frequencies or mechanisms and/or where f is not of the slinple form of eqn. (17) (e.g. Ref. 36).
MeasurEments of DO. /DS are best, and usually, made by diffusing two radioactive isotopes slinultaneously into the host crystal from a very thin layer of them deposited on its surface [34]. After diffusion, the sarrple is sliced and the activities co. and cS of the t:IM::> isotopes determined in each slice. A plot of ln ca/cS vs ln co. can then be shown to give a straight line with slope (00. /DS - J,.). Distance and tline measurements are not required, as in nonnal diffusion experlinents, and for this reason E (n) can be determined with sane precision, to within a few %.
We now oonsider the uses to which the basic equations (36a) and (36b) can be put.
CORREIATION EFF".EX:TS IN IONIC TRANSPORT PROCESSES 37
Obviously E (n) increases with n, but, since both f and ilK are fractional, only those E(n) '::1 are pe.nnissible, i.e. there is an upper limit no on n and this imnediately allows certain mechanisms to be rejected as incanpatible with the measurements. The values both of f am of ilK must then lie in the range E (no) to 1 and refer only to mechanisns with n.:>l1o. Unlike the method of the last sec­ tion, the mass effect method cannot give unique values of f, to aid in unambiguous identification of the mechanism, because theore­ tical values of ilK are not yet sufficiently well established. Nevertheless, it can be used to indicate the possibility of, or to confi:on, a mechanism in many cases. When this is well estab­ lished, so that f is known, it gives a value for ilK, a quantity oontaining significant infonnation on the dynamics of jurp pro­ cesses not easily obtained on other ways. W9 give a few examples.
For simple interstitial diffusion in many systems (e.g. C in a-Fe, Li in W, He in Si02 etc.), E(n ~ 2) > 1 arrl within experi­ mental error E (1) = 1. Thus, f = 1, as expected of interstitial diffusion am, more interestingly, ilK = 1. For Na in Soda glass E(2) < 1, indicating the possibility of interstitialcy diffusion [37] •
For self-diffusion in close packed structures of metals and ionic substances of the NaCl type, where there is substantial evi­ dence for a vacancy mechanism, it is found that no = 1 and E (1) < fv, consistent with this mechanism [8]. Using the known value of fv, ilK is found to be around O. 9 in metals, a little less, 'V(). 76, in the few ionic substances that have been studied, (e.g. cation migration in CoO, NiO) [38] imicating sane lattice atan partici­ pation in the jumps.
In sane cases E varies significantly with terrperature, a valuable indication that more than one mechanism of diffusion is operative [36,39].
The other main use of mq.ss effect measurements is to obtain, by determining f, infonnation on relative jump frequencies when f is a fl.IDction of these. For this we need to krPw ilK. In dilute solutions it is currently assuned that ilK, even for solute diffu­ sion, is the same as that dete:onined from E for the pure solvent self-diffusion, where f is known.
Values of f determined in this way are best made use of when canbined with a knavledge of other quantities that depend on the sarre jump frequencies. For exanple, in dilute f.c.c. solutions b (eqn. (21» arrl f2 (eqn. (17) am (18» are functions of w3/Wl,
w2/wl arrl w4/wo. So also is the ratio of the .impurity D2 and the solvent self-diffusion Do for, using eqn. (19).
38 A. D. LE CLAIRE
D2 w2f 2 exp B~kT
Do = Wo fO
W2 WI w4 = _._._. (37)
Thus if b and D2iDo are known, a measurEment of f2 allows numerical values of these frequency ratios to be obtained.
This procedure has been applied to a few metallic systems (Zn in Ag arrl in Cu, Fe in Cu) [40] but so far only to one Wnl.C
systEm (Co in NiO) [41]. When measurEments of f are made as a function of temperature the quantity C (eqn. 22) can also be determined [40]. SUch results are of central importance in an understanding of diffusion processes in dilute solutions.
The same can be done for b.c.c. solutions, althoUJh only two frequency ratios are involved here (because there are no WI jl.lll'ps) and only two of the experimental quantities are needed to determine thEm [25].
For impurity diffusion in ionic crystals a useful quantity to consider alon] with f2 is the ratio of the impurity nobility 112 to D2, given by equation (31) [7].
When complete data are not available useful but less quanti­ tive conclusions can still be drawn.
When such procedures yield positive values of the frequency ratios they provide confinnation of the vacancy mechanism on which the equations being used are based. O::casionally, experimental data are found to be quite incompatible with the equations, for positive frequency ratios, so the vacancy mechanism alone in such cases is not tenable. Such is the situation for diffusion of the noble metals and of Cd in Pb and the current hypothesis is that interstitial diffusion is responsible. Similar incompatibilities are found for Co diffusion in a-Zr, a-Ti and y-U [25]. None have so far been reported for ionic systems.
REFERENCES
1. J. Bardeen and C. Herrin3", In "Aton ~bvements" p. 87 A.S.M. Cleveland (1951).
2. E. Amaldi and E. Fermi, Ric. Sci. I 13 (1936). 3. In gas e sit is known as the "persistence of velocities effect". 4 • Extensive treatments of correlation effects are to be found
in the followin] texts and reviews, references 5 to 8. 5. R.E. lbv..ard and A.B. Lidiard, Reports on