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Solute effects in diffusion
Thin Solid Films, 25 (1975) 1-14 © ElsevierSequoia S.A., Lausanne---Printedin Switzerland
1
S O L U T E E F F E C T S IN D I F F U S I O N *
A. D. LE CLAIRE A.E.R.E., Harwell, Oxon. ( Gt. Britain)
(ReceivedSeptember27, 1974)
Small additions o f solutes to ionic substances can produce drastic changes in diffusion rates and ionic conductivities and these effects, dominated by the requirements of charge compensation, are very well understood and regularly employed in studies o f such materials. In metals and other substances the effects of solute additions on diffusion rates are usually very much smaller and rather more difficult to treat theoretically so that, until recently, they have been less widely studied. Nevertheless, knowledge of the effects, especially when combined with other data, is of considerable value in the identification and detailing of diffusion mechanisms. There is now for volume diffusion in metals an amount of reliable quantitative data and a review will be given of its essential features and of the theoretical treatments that have been developed for its interpretation and application. Some examples will be given of solute effects in grain boundary and surface diffusion processes but study o f these has been rather less quantitative and systematic and in the absence of any detailed theoretical understanding only general comment is at present possible.
1. INTRODUCTION Most detailed investigations of transport processes in ionic solids have generally included a study of the effects on conductivity tr and on diffusion rates D of the addition of small amounts of solute, a procedure known as "doping ''~. This is a valuable technique because when the " d o p e " is aliovalent to the host lattice ions it changes the defect concentrations by an extent readily calculable from considerations of charge neutrality and alters the diffusion rates and conductivity in a predictable way, determined by the nature and properties of the intrinsic defects present. Thus defect species can be identified or confirmed and many of their properties ascertained. Because the changes in defect concentrations are comparable with the dope concentration very small amounts of dope can drastically alter the defect levels and produce large effects * Paper presented at the International Conferenceon Low Temperature Diffusion and Applications to Thin Films, Yorktown Heights, New York, U.S.A., August 12-14, 1974.
2
A. D. LE CLAIRE
on D and a. Typical doping concentrations are of up to a few tenths of a molar percent. Similar techniques have been applied very much less frequently to other classes of solids because the effects of solute additions on the transport processes are more difficult to calculate and are usually very much smaller and less striking than in ionic solids. Because the effects are small careful measurements are necessary. This is especially so as these have to be confined to the low concentrations of solute, less than a few percent, for which theoretical descriptions of sufficient detail for useful comparison with experiment can be readily developed. Systematic studies of solute effects in suitably dilute solutions began several years ago with the advent of really precise diffusion measurement techniques; there is now a substantial and growing body of reliable solute-effect data, particularly extensive for metal systems. With the theoretical developments that this has stimulated it has been used, together with the results from other types of measurement, to provide a deeper and more quantitative understanding of diffusion processes. In a few cases the data have been instrumental in indicating previously unsuspected processes for atomic migration. The more quantitative aspects of these studies have been in relation to volume diffusion. For surface and grain boundary diffusion, observations of significant solute or impurity effects have also frequently been made but the interpretations of these have usually been of a more qualitative or semi-quantitative nature, contributing little to a better understanding of the processes themselves. This paper summarizes some of the more salient features of experimental and theoretical studies of solute effects. Because the data on metal systems are the most extensive, most attention will be given to them. Nothing more will be said about ionic crystals, because the effects there are very well known and understood, nor about semiconductors or other covalent systems because of the comparative dearth of data. 2.
SOLUTE ENHANCEMENT IN VOLUME DIFFUSION: PHENOMENOLOGY
We consider the addition to a pure solvent of a small fractional concentration c of solute. This is observed to change the self-diffusion coefficient of the solvent from D(0) to D(c) and we wish to relate these two coefficients. Around any isolated solute atom we expect there to be a "region of influence '" within which the solvent jump frequencies may differ from Wo and the defect concentrations from nv; w0 and nv are the values in pure solvent and, by definition, outside the regions of influence. At concentrations sufficiently low that these regions of influence rarely overlap, their effects will be additive and will provide a change in the overall solvent diffusion rate proportional to c. At rather higher concentrations we have to consider pairs of solute atoms whose regions of influence overlap. Such close pairs will also perturb the solvent jump frequencies and defect concentration, to an extent in general different from the added effect of two isolated solutes. They will contribute to a change in the solvent diffusion coefficient proportional to their concentration, i.e. to c 2. Similarly, groupings
SOLUTE EFFECTS IN DIFFUSION
3
of three, four etc. solute atoms with overlapping regions of influence will contribute terms proportional to c 3, c 4 etc. We therefore expect D(c) to have the form 2
D(c) = D(O) + k
Ic +
k2c 2 + k3 c3 + . . .
(1)
which is usually written as D(c) = D(0) (1 + b l c + b 2 c Z + b 3 c 3 + . . . )
(2)
where the bi are called "solute enhancement factors". The argument is clearly only appropriate for dilute solutions, of no more than 5-10 ~ say, when the independence of the effect of different solute groupings is a reasonable approximation, although such a power series is useful as an empirical representation over any range of concentration. Experimental results from dilute solutions usually fit eqn. (2) very well. Only rarely have the data been sufficiently precise and extensive to require the c 3 term to be retained and most frequently they have been adequately represented in terms of bl alone 3--17 A few measurements 6' 8, over rather large concentration ranges, have shown a better fit to the equation
D(c) = D(0) exp (bc)
(3)
than to eqn. (2). Equation (3) has also occasionally been used in preference to (2) for low concentration ranges 18. While at sufficiently low concentrations it may, of course, be difficult to distinguish experimentally between the two representations, there seem to be no simple theoretical arguments to justify the form of eqn. (3), other than for exceptional cases where the necessary relations obtain between the bi, i.e. b2 = ½bl z etc. The solute diffusion coefficient Di is also found at low concentrations to vary with c in a way similar to eqn. (2): Di(c ) = Di(0 ) (1 + B 1c + B 2c2 + . . . )
(4)
Di(0) is the limiting value of the solute " i m p u r i t y " diffusion coefficient at vanishingly small concentrations. In this case the linear term derives from close solute pairs, the c 2 term from threefold groupings, and so on. Di(c) is proportional to F~, the number of jumps per solute atom in unit time. Isolated solute atoms contribute to the total solute jump rate in proportion to c, solute pairs in proportion to c z, and so on. Dividing by c to give F i these contribute to eqn. (4) respectively the constant, the linear term and so on. Some typical values of bi and B~ are given in Table I to illustrate general properties, bl is the factor for which the most extensive data are available. No negative values greater than of the order of - 1 0 have been reported but the more prevalent positive values of b~ cover a wide range and may reach very large values. Values of b2 are very roughly an order of magnitude greater than b~. b 3 is only known for the two systems indicated in the table and is one to two orders of magnitude greater than b z. When, as is usual, b~, b2 and b3 are of the same sign, D(c) has an accelerating rate of increase, or decrease, with c.
4 TABLE
A. D. LE CLAIRE
I
bl
b2 ×10 -1
Ag-Cd
9.2
b3 ×10 -2
-
Temp.
B1
Bz
(°C)
727
-
-
3,4
-
-
5
-
-
4,5
Ag-Sb
65.7
-
-
54
95
-
Ag-Pb
87
-
-
85
125
-
-
-
727
-
727
-7.5
Pb-Cd
Pb-Hg
Pb-Au
-8.2 - 5
V Ti U-Co
727
3
-
-
6
50
-
4,6
-
-
7
8
-
-
-
1020
-
-
9
44
-
199
-
-
-
10
30.0
41
442
248
14
100
-
19.1
107
-
301
22
39
93
295
23
49
90
274
45.2
-
-
29
- 74
248
300
26
- 70
66
272
27
18
147
252
19
+37
39
249
28
12
226
226
18
+45
58
224
4300
-
215
-
1300
-
-
200
- 2100
0
5700
c~-Fe-Si • -Fe-Co
Ref
(°C)
-
-
- 1.2
Temp.
6
6
727
×10 -2
-
7
Ag-Au Ag-Pd Cu-Fe
B3
×10 -1
20
-
1427
- 1.1 28.7 714
-
12.4
895
-
1100
-
822
-
175
11
12,13
150 -
14 15 16
-
17
B1 is generally of the same sign as and comparable in magnitude with bl but usually smaller. Thus D(c) and Di(c) generally change in the same direction with increasing c, but D(c) usually changes more rapidly than Di(c), as might be expected. An important exception is the Pb-Au system for which B 1 is negative while bl is positive, a significant result in elucidating the Au diffusion mechanism, as we shall see. What little data there are on B2 and B 3 suggest that these are respectively approximately one and two orders of magnitude greater than B a. There are other general features of solute effects that are of significance and interest. Solutes that are "fast diffusers" in the solvent generally enhance the solvent diffusion rates, i.e. bl is positive when Di(0)>D(0 ). Similarly, b 1 is negative when Di(0)
SOLUTE EFFECTS IN DIFFUSION
5
From time to time analytical expressions are proposed that incorporate this or other correlations with the melting point and describe quite well the variation of Di and D with c for particular systems, often over extensive ranges of concentration 8' 19. A notable example is the Ag-Pd system 4 for which both D and O i a r e found to be dependent exponentially only on -TM(c,p)/T, TMbeing the solidus temperature at composition c and pressure p. However, such simple relations are very restricted in their application. The effects on measured diffusion coefficients of unavoidable impurities in the samples used can be estimated when their b~ or B~ are known. For a measured D to be affected by less than about 1 ~o (the minimum error from other sources say) there must be a concentration of about 10-2/bl or less of the offending impurity. For example, samples of 99.99 ~ purity are adequate provided the weighted mean bl of the impurities present is less than 102. Sometimes measurements are made to establish that results are independent of purity, and occasionally it is found that they are not so. A striking example is the measurements of Irmer and Feller-Kniepmeier 2° of the self-diffusion coefficient of 0c-Fe of different purities. The least pure, with a total C + O major impurity content of about 70 ppm, showed D(0) values as much as 35 times greater than those from the purest samples, which contained about 8 ppm of impurity. There was no significant change in the activation energy. Most previous measurements have given values comparable with those from the least pure Used in these studies, suggesting that they were all seriously affected by impurities. If these strikingly large effects are at all representative of interstitial impurities in b.c.c, transition metals, then many measurements of diffusion rates in such metals may be suspect because they are often very difficult to free from such impurities. Furthermore, the results add considerable verisimilitude to Kidson's proposal 21'22 that the behaviour of the so-called "anomalous b.c.c, metals" may be due to the enhancing effect of impurities like O, almost inVariably present in the more strongly anomalous metals such as 13-Ti and 13-Zr. 3. CALCULATION OF
b~ AND
Bi
We have first to stipulate the "region of influence" within which the solvent properties are perturbed by the solute, In f.c.c, crystals with diffusion by vacancies it is usually assumed that this extends out to fourth nearest neighbours from a solute atom. We define w1 as the frequency for solvent-vacancy jumps between two first nearest neighbours of a solute and w3 as the frequency for vacancy jumps from first neighbour sites by exchange with solvent atoms on second, third or fourth neighbour sites, w4 is the frequency of the reverse (association) jumps of vacancies from these sites onto first neighbour sites. All other vacancysolvent jumps are assumed to occur at the same frequency wo as in the pure solvent. With the frequency w2 for vacancy-solute exchanges, we have a total of five jump frequencies in terms of which to discuss the solute and solvent diffusion properties in a very dilute alloy, where we assume for the moment that solute pairs can be ignored. If there is an interaction energy AE 1 between a solute and a vacancy on
6
A . I ) . LE C L A I R E
a first neighbour site, the probability of there being a vacancy on any particular such site is exp { - ( E + AE1)/RT }, compared with exp ( - E/RT) for all other sites. (E is the vacancy formation energy in the pure solvent.) To maintain dynamic equilibrium between these vacancy concentrations demands a relation 1/'4/11' 3 =
exp( - AE1/RT)
(5)
between w3 and w4. We cannot presume that there is any binding energy on second, third or fourth neighbour sites without introducing additional perturbed j u m p frequencies from these sites. We now calculate D(c) by counting all the solvent jumps of each frequency w 1, w a, w 4, w o that occur in unit time and dividing by N ( 1 - c ) , the number of solvent atoms, to give the average number of jumps F per solvent a t o m in unit time. More particularly, we must weight the contribution F, from each type of j u m p ~ by its partial correlation factor f~, remembering that there may often be two or more types of j u m p having the same frequency but differing in their f~. If x is the j u m p distance, D(c) is then given by
D(c) = 1 x 2 ~/-~f.. = D(0) (1 + b l c + . . . )
(6)
from which b t can be determined. The result for f.c.c, crystals, where there are altogether 13 types of j u m p 23 (~ = 13), is b~ = - 1 8 + 4
w~ (4 wl )~1+14 Z2'~
Wo
w3 fo
fo]
(7)
Z1 and Z2 are mean values o f the J~ and are known tabulated functions of the three frequency ratios w2/wl, wl/w 3 and w4/Wo. Note that bl depends therefore on w2, and that it can be written to contain the perturbed vacancy concentration by using eqn. (5). b~ has also been calculated for b.c.c, crystals. With the "region of influence "' extending, as for f.c.c, crystals, to just those sites that can be reached by one j u m p from a solute's first neighbour sites, i.e. the second, third and fifth sites in this case, bl is given by z~ w4 v2 . ~/~2 b, = - 2 0 + 14 ~ .~oo+O f 7
(8)
The v 2 and /~2 are mean values of J; again and functions of the ratios w2/w3 and w4/wo. There are no w~ type j u m p s in b.c.c, crystals. Equation (5) still applies here. An alternative b.c.c, model has been discussed in which vacancy-solute binding, with energy - A E 2, may occur at second as well as first neighbour sites 24. We then need to introduce an additional frequency ws for dissociative vacancy jumps from second neighbours (to fourth), and to distinguish dissociative jumps from first to second and from first to third or fifth neighbours by different frequencies w 3 and w~ respectively. The corresponding associative jumps are w 4 and w~,. In the case when w~, = wo,
SOLUTE EFFECTS IN DIFFUSION
b 1 = - 2 0 + 6 w3 vl #1_ w~ foo+ 14 fo
7
(9)
vl and/~1 are functions of the ratios w2/w'a and w3/w'3 and wo = w) e x p ( - A E I / R T ) = w 5 e x p ( - A E 2 / R T )
(10)
Similar equations can be derived for other structures and assumed mechanisms of diffusion once the perturbed frequencies and defect concentrations have been specified. A case recently discussed is that of diffusion in solutions where the solute atoms are distributed interstitially and substitutionally in dissociative equilibrium 2'25'26. A substitutional solute can jump to an interstitial site with frequency v1, creating a vacancy alongside, and can perform the reverse jump with frequency v2 say. Clearly, vz/v ~ = exp { - ( E + I + B ) / k T}
(11)
if I is the formation energy of a free interstitial and - B its binding energy to a vacancy. The vacancy-interstitial pair may dissociate by an interstitial jump away from the vacancy, with frequency kl, or by a jump of the vacancy by exchange with a solvent atom, not a neighbour of the interstitial, with frequency w 1. Particular interest has attached to the contribution to solute diffusion and solvent enhancement arising from the interstitial-vacancy pair. It can move by reorientation without dissociation, by migration of the interstitial to another interstitial site with frequency k z, or by a jump of a mutual nearest neighbour solvent atom into the vacancy, with frequency w2. In one calculation with this model, due to Miller 15, it is ~assumed that interstitial jump frequencies are much greater than solvent jump frequencies, i.e. kz, vz >> w~, w2, and that the fraction of free interstitials is very small. One then finds bl = { - 6 + ( 2 w2+4 Wl) fWofo ~ x exp (-k-T) } exp (k/---T)
(12)
f l is the solvent correlation factor in the alloy and fo that in the pure solvent. Miller assumes that f l = fo. Warburton 26 has modified Miller's model by supposing that isolated interstitials are so rare that k~, wl can be assumed to be zero. He also assumes that k 2 type jumps are unlikely and puts k 2 = 0. bl is the same as eqn. (12), with w~ = 0. Warburton evaluates f l but does not give explicit values. The only calculation of the non-linear factors in eqn. (2) appears to be that of Bocquet 4 for b2. This derives from solvent-enhanced jumps in the neighbourhood of pairs of solute atoms. Such pairs are all those close enough together for there to be solvent-vacancy jumps that can no longer be unambiguously described as wl, w3 or w4. With f.c.c, crystals these are of eighth nearest neighbour separation or closer. Bocquet found it necessary to introduce a further six perturbed solvent jump frequencies and three more solute jump frequencies to describe the vacancy diffusion associated with such pairs. The resulting
8
A . D . LE CLAIRE
expression for b 2 is necessarily rather lengthy and we shall not reproduce it here. Bocquet also calculated the linear term B~ in eqn. (4) from his model in terms of the now four solute jump frequencies and the association energy Eb2 of a nearest neighbour pair of solute atoms. A more approximate calculation has been reported by Miller 2 giving B~ in terms of Eb2 alone: B~ = l l { e x p ( - E b 2 / k T ) -
1}
(13)
This follows from Bocquet's equation with the assumptions of a unique solute jump frequency w2 and a negligible vacancy-solute binding, i.e. w3 = w,,. 4. COMPARISON WITH EXPERIMENT All the equations for bi and Bicontain only frequency ratios and/or association energies. Calculations have occasionally been made of these quantities, using the electrostatic model for vacancy diffusion, to provide values of b i for comparison with experiment 4' 27. Results for bl have always produced the correct sign and, at least for weakly perturbing solutes, a fair agreement in magnitude. However, for b 2 some calculations for solutes in Ag gave signs opposite to those of b 1 and thus in marked disagreement with experiment 4. What calculations there are for B~, for a few solutes in Ag, are more encouraging in giving the correct sign, but strong attractive interactions between solute atoms (Eb2 negative) seem necessary for the results to approach the observed numerical values 4. This is evident too from eqn. (13). Studies of enhancement effects have been of most value and interest when the results have been considered in association with other diffusion properties. The more important of these have been the following: (i) the ratio of the impurity diffusion coefficient D2 (---Di(0)) to the solvent self-diffusion coefficient D o (=D(0)) given, for vacancy diffusion for example, by D:
WE f2 exp (
Do - Wo fo
AEI~ - -~-]
(14)
For other mechanisms wz and wo may be replaced by sums of terms, the exponential may contain other defect energies and there may be a numerical factor. Working with this ratio defect energies common to D 2 and Do (in this case E) are eliminated. (ii) the mass or isotope effect in impurity diffusion, defined as ( D ' - D~)/D ~
E(1) - (m~/m,)X/2 _ 1 This is intimately related to the impurity correlation factor f2 that appears in eqn. (14). The relation is particularly simple when f2 has the form f2 = u/(w2 +u)
with u depending only on solvent jump frequencies, for then
(15)
SOLUTEEFFECTSIN DIFFUSION E(1) = f 2 A K
9 (16)
allowing f2 to be determined when the kinetic energy factor AK can be estimated. In other cases E(1) may be quite a different function of the solvent and solute frequencies than that represented by f2, but readily calculable as
e(1) = E
81n D2
the wj being the solute or impurity jump frequencies 2s. We illustrate application first to cases of vacancy diffusion. For these, eqn. (15) is valid and, for f.c.c, crystals for example, u = w1 +3.5 w3 F(w4/wo)
(17)
with F a known function of w,,/Wo that varies slowlyTM 29 from 2/7 (w,,/Wo = 0o) to 1 (w4/wo = 0). Using eqn. (5) we note that D2/D o can be written D 2 _ f2 w2 wl w4 Do fo W1 W3 W0
(18)
Thus, each of the quantities f2, D2/D o and bl (eqn. (7)) is a function of the same three frequency ratios w2/w 1, wl/w 3 and w,/w o. It follows that when all three quantities have been measured we can solve eqns. (7), (15) with (17), and (18) to yield numerical values for these frequency ratios. This clearly provides very detailed information concerning diffusion processes in dilute solution and a most satisfactory form of data for testing theoretical calculations of jump frequencies. The necessary measurements have been carried out for such an analysis to be possible only on three systems, namely for Zn in Cu and Ag 3° and for Cu in Fe 9. Obviously any such analysis must yield positive values for the frequency ratios. If one or more of them is negative one must conclude that either (i) the jump frequency model is inadequate or (ii) diffusion is not occurring by the vacancy mechanism that is assumed in writing eqns. (7), (17) and (18). However, such incompatibilities can often be identified more simply by making use of other relations derived from the equations. Also it is not always necessary that the complete set of experimental data be available. As an example of (i), an earlier four-frequency model, used to interpret data on Fe diffusion in Cu and Ag, yielded a negative value of wl/w 3 and first indicated 3~the need to introduce the frequency w4 as distinct from wo. A relationship of particular value is the equation for bl in terms of D2/D o and f2, namely
~4~l(_WI/W3)+
4 f 0 02 14Z2~ b~ = - 18 -~ 1 - f z Do [ f o (4 wffwa + 14F )J
(19)
If the solute does not have a very strong perturbing effect it is a reasonable approximation to put ZI = )~2 = f o . With this and the approximation F = 1, eqn. (19) becomes the Lidiard equation32:
l0
A . D . LE CLAIRE
bl = - 18-~
4fo D2 1 -f: D O
(20)
Both eqns. (19) and (20) express the general observation that fast diffusers (D 2 > Do) tend to enhance solvent diffusion rates (bl positive) and slow diffusers to reduce them ( D 2 < D 0, bl negative). They also indicate why large negative values of b I are not observed, however small D2/D o. Equation (19) can be used to determine the minimum value of bl consistent with an experimental value of DE/D o. It is easily seen that bl is a minimum in the limit wl/w3--,o~ and {2--*0, implying that wz/wl--*~: brain = - 18+4 )~1 (D2/Do) = - 18+ 1.945 (D2/Do) (21) where the value for 7.1 at wl/w 3 and w 2 / w o ~ has been inserted ~°. This limit corresponds to the case of a very tight vacancy-impurity binding and very rapid impurity jumps. Such extreme conditions are unlikely to be common so we expect measured values of bl to be appreciably greater than bmi.. Most of them are. However, over the last three or four years some systems have been discovered for which the experimental bl is well below the bmi. given by eqn. (21). For Au and Ag in Pb 12 the observed h 1 is only approximately one-fiftieth of hmi"--so very much smaller that no elaboration of the frequency model could conceivably alter the situation and one must conclude that some mechanism other than that of vacancy diffusion is operating here. There is a similar but much less extreme situation for Cd in Pb l°, where bi ~ D 2 / D o. For Hg in Pb ~ b 1 ~ 1.5 D2/D o and slightly above bm~,, but so close to it that the extreme conditions necessary for a vacancy mechanism make such a mechanism questionable in this case too. Very similar analyses have been made of the data on f2, D2/Do and bl for dilute b.c.c, alloys, using eqns. (8) or (9) and the appropriate expressions for u in eqn. (15). The only difference is that, since there are no w~ jumps, the three experimental quantities are determined by just two frequency ratios. These can therefore be estimated from any two of the experimental quantities. For many systems positive values are found and b~ >b,,i,, consistent with vacancy diffusion. However, there are a few--Ti-Co, Zr-Co and U - C o - - f o r which b~ is an order of magnitude below b,,i,, indicating again some other mechanism than the vacancy one 24. Since, for these cases in which bl
SOLUTE EFFECTS IN DIFFUSION
11
Miller 2'25 suggested that the diffusion mechanism is dominated by tightly bound vacancy-interstitial pairs; the values of bl for this are given in eqn. (12) and D2/D o is
D_~z=4k2+v z Do
fz
exp ( - ~ )
(22)
2Wo fo
f2 was estimated by Miller and, with his approximations (k2, v2:~k ~, w 1, w2),
fz ~ = (4 w2+8 w1 +20 ka)/(4 kz+v2) With the approximation Warburton introduced into his treatment 26 of the model
fz w = 4 w2/(4 w2 + v2) Thus, from eqn. (12) for b ~, ignoring the term 6 exp ( - / / k T) which is negligible compared with b~,
D2 ( 1 btM=ft Doo
5 kl
)-'
(23)
q W z q - 2 Wl
and D2 4 w2+v 2
blW= fl Do
v2
(24)
Miller estimated that f~ ~fo, so if k~ is very small, i.e. the pairs are very stable,
bl u = )Co (D2/Do)
(25)
in near agreement with the results for Cd in Pb. However, bl u cannot be greater than D2/D o, as found for Hg in Pb. Warburton's assumptions demand a more careful consideration of f l ; this was evaluated as a function of w2/v 2 to give the result from eqn. (24) that 3.80 (D2/Do) < bl w < 0.83 (D2/Do)
(26)
the upper limit being for w2/v 2 = ~ , the lower for w2/v2 = 0. Both the Cd in Pb and Hg in Pb data can now be accommodated and w2/v 2 can be evaluated: about ½ for Hg, about ~ for Cd. These systems have provided excellent examples of the value of solute enhancement effects in studying diffusion migration mechanisms. 5. SOLUTE EFFECTS IN GRAIN BOUNDARY AND SURFACE DIFFUSION
Any measurement of a grain boundary diffusion coefficient DQb in a solid solution is usually complicated by the lack of adequate knowledge on the extent of solute segregation at the boundary. The fractional concentration cb of solute in equilibrium in the boundary may differ very appreciably from the fractional concentration c e in the lattice. The ratio Cb/Ct is the "segregation coefficient" k. If Fb is the free energy of binding of a solute atom in the boundary 34
12
A . D . LE CLAIRE
k
=--Cb =
exp
l+ct
{
exp
-1
-1
ct
~ exp ( ~ - ~ ) = exp ( - - ~ )
exp ( ~ T )
the latter equality obtaining when ce exp (F/kT)<< 1. Values of k as large as 103 or 104 have been reported, corresponding to UZ +7 k T t o U,,~ + 9 kT. We need to know k for two reasons: first, to establish the boundary concentration at which Dg u is being measured; second, to be able to evaluate Dgb itself, because the standard methods for studying grain boundary diffusion rates measure only the product P = D0b fi k, 3 being the effective width of the boundary. (k arises because in solving the grain boundary diffusion equations the relation cb = k ce is introduced as a boundary condition linking the solutions within and without the boundary at the grain boundary surfaces.) Only with measurements of grain boundary self-diffusion rates in pure solvents is there no problem, for then k = 1 and, given ~, D0b itself can be evaluated. In other cases very little is known about k for the systems that have been studied and most investigators have either ignored this factor altogether or implicitly assumed k = 1. Their reported values of "Dg b'' are then actually of Dgbk so that their activation energies "Qgb" are Q0b--U, plus any contribution there may be from a temperature dependence of 6. "In view of the possible magnitudes of k and U it does not seem possible, therefore, to draw any reliably firm conclusions from the results about the relative magnitudes of the solvent self and solute impurity D0b values or of the effects of solute concentration on the solvent and solute D~b values, in other words to extract enhancement factors bi and Bi for grain boundary diffusion. Variations of D~b between systems and with concentration may be masked by or confounded with corresponding variations in k. One can only discuss the variation of P and its activation energy. It is certainly well established that quite small concentrations of solute, even residual impurities, can in many cases have a marked effect on grain boundary self-diffusion rates. Thus the value of Dgb around 600~C for self-diffusion in an a-Fe of 99.7 ~o purity was reported to be more than four times that measured on a purer sample of 99.96~o purity 35. The difference decreased slightly with increase in temperature. In the y-phase it was only some 50~o. Impurities may also decrease Dgb, as has been found 36 for self-diffusion in Pb with additions of T1, In or Sn, where reductions by a factor of approximately one-third arise at concentrations of about 1½~o. Effects of similar magnitude have been found in many other systems: when they are large in relation to the total amount of impurity they are usually attributed to the concentrating effect of segregation but there is little understanding of the mechanism by which Dgb is changed. There are a number of measurements of the grain boundary diffusion of solutes but here, as we have seen, we are very much involved with k. It would be desirable to establish the relation between the impurity Dgb for various impurities in a solvent and the D0b for self-diffusion in that solvent, for the corresponding relationship in volume diffusion has proved so useful in the
SOLUTE EFFECTS IN DIFFUSION
13
understanding of volume migration processes. Without knowing k this is not possible. The tendency is rather to assume that these relationships are similar in volume and in grain boundary diffusion and that any unexpected magnitude or type of behaviour in a Dg b, or P, is due to segregation. Thus, for example, P for Ru in the substructure boundaries of Cu is very much larger than P for Ni in Cu, yet the impurity volume D i for Ru in this solvent is much less than that for Ni 37. Attributing this behaviour to a large k value for Ru, and assuming k ~ 1 for Ni on the grounds of its complete solubility, k was calculated to be of the order of 100. The ratio P / D i for Ru is about 100 times its value for Ni and it was assumed that these ratios would be much the same in the absence of any segregation. When k is large we expect larger than normal dislocation or grain boundary enhancements of volume diffusion rates to be revealed in departures from linearity of the Arrhenius plots. For Ru in Cu such departures set in at the unusually high temperature of about 0.9 TM, even in single crystals. Similar behaviour was found 38 for S diffusion in Cu and attributed again to segregation, with k ~ 100. Other measurements that have been made include studies of the effect of one solute on the grain boundary diffusion of another and, in a few cases, of "Dg b'' over ranges of temperature and composition for solute and solvent, but little by way of general principles of behaviour has emerged. Studies of solute effects in surface diffusion mostly concern the influence of residual impurities or of foreign atmospheres on surface self-diffusion rates. This can be very severe and there are many examples of D s being changed by several orders of magnitude, with corresponding large changes in activation energy. These are attributed, naturally, to adsorbed layers of foreign atoms. The effects can be to increase or to decrease D s. D s for W surface self-diffusion is much reduced by oxygen or water vapour39; there are large increases in Ds following S adsorption on Ag 40 or C1 adsorption on Cu 41 An interesting feature of tracer surface self-diffusion measurements is that they often show volume-like behaviour in that In c is proportional to x 2, rather than to x or x 6/5 as might be expected (c is the concentration of diffusant at a distance diffused x) 42. This is Harrison's type C condition of diffusion and means that the diffusant is wholly confined to the surface layers with no loss of diffusant into the substrate. This often occurs though, even at temperatures where appreciable loss into the substrate might be expected. It has been suggested 42 that this is due to impurities segregated or adsorbed at the surface to form a barrier layer that inhibits inward diffusion. If this is so, the D~ measured may refer to diffusion over the contaminant layer and not represent a true surface self-diffusion coefficient. In the absence o f this effect and with true Harrison type C conditions obtaining, measurements would yield directly a Ds for solute or impurity surface diffusion without the need to know k or 6. This requires that k be so large that k 2 62 >> Dtt, D t being the lattice diffusion coefficient.
14
A. D. LE CLAIRE
REFERENCES
1 See, for example, L. W. Barr and A. B. Lidiard, Defects in ionic solids, in Physical Chemistry. Academic Press, New York, 1970, Chap. 3. 2 J.W. Miller, Diffusion Processes, Vol. 1, Gordon and Breach, London, 1971. 3 A.H. Schoen, Ph.D. Thesis, Univ. of Illinois, 1958. 4 J.-L. Bocquet, C.E.N. Saclay Rep. CEA-R-4292, 1972. 5 E. Sonder, Phys. Rev., 100 (1955) 1662. 6 R.E. Hoffman, D. Turnbull and E. W. Hart, Acta Met., 3 (1955) 417; R. E. Hoffman, Acta Met., 6 (1958) 95. 7 W.C. Mallard, A. B. Gardner, R. F. Bass and L. M. Slifkin, Phys. Ret'., 129 (1963) 617. 8 N.H. Nachtrieb, J. Petit and J. Wehrenberg, J. Chem. Phys., 26 (1957) 106: R. L. Rowland and N. H. Nachtrieb, J. Phys, Chem,, 67 (1963) 2817. 9 J.-L. Bocquet, Acta Met., 20 (1972) 1347. 10 J.W. Miller, Phys. Re~:., 181 (1969) 1095; Ph.D. Thesis, Harvard Univ.. 1969. 11 W.K. Warburton, Phys, Ret'., B7 (1973) 1330. 12 J.W. Miller, Phys. Re~:., B2 (1970) 1624. 13 W.K. Warburton, Scripta Met., 7 (1973) 105. 14 R.J. BorgandD. Y . F . Lai, J. Appl. Phys.,41(1970) 5193. 15 C.M. Walter and N. L. Peterson, Phys. Retd., 178 (1969) 922. 16 J.F. Murdock and C. J. McHargue, Acta Met., 16 (1968) 493. 17 N.L. Peterson and S. J. Rothman, Phys. Ret'., 136 (1964) A842. 18 C.J. Santoro, Phys. Retd., 179 (1969) 593. 19 J. Kucera, B. Million and J. Plskova, Phys. Status Solidi (a), 11 (1972) 361: J. Kucera, B. Million, J. Ruzickova, V. Foldyna and A. Jakobova, Acta Met., 22 (1974): H. A. Resing and N. H. Nachtrieb, J. Phys. Chem. Solids, 21 (1961)40. 20 V. lrmer and M. Feller-Kniepmeier, Phil. Mag., 25 (1972) 1345. 21 G.V. Kidson, Can. J. Phys., 41 (1963) 1563. 22 A.D. Le Claire, G. V. Kidson and others, in D(ffusion in B.C.C. Metals, Am. Soc. Metals, Metals Park, Ohio, 1965. 23 R.E. Howard and J. R. Manning, Phys. Ret'., 154 (1967) 561. 24 M.J. Jones and A. D. Le Claire, Phil. Mag., 26 (1972) 1191. 25 J.W. Miller, Phys. Ret'., 188 (1969) 1074. 26 W.K. Warburton, Phys. Ret,., B7(1973) 1341. 27 A.D. Le Claire, Phil. Mag., 7 (1962) 141. 28 A.D. Le Claire, Correlation effects in diffusion in solids. In Physical Chemistry, Academic Press, New York, 1970, Chap. 5. 29 J.R. Manning, Phys. Ret'., 128 (1962) 2169. 30 N.L. Peterson and S. J. Rothman, Phys. Retd., B2 (1970) 1540; S. J. Rothman and N. L. Peterson, Phys. Retd., 154 (1967) 1952. 31 J.G. Mullen, Phys. Ret:., 121 (1961) 1649. 32 A.B. Lidiard, Phil. Mag., 5 (1960) 1171. 33 T.J. Turner, S. Painter and C. H. Nielsenn, Solid State Commun., 11 (1972) 577. 34 D. McLean, Grain Boundaries in Metals, Clarendon Press, Oxford, 1957. 35 P. Guiraldenq and P. Lacombe, Acta Met., 13 (1965) 51. 36 J.P. Stark and W. R. Upthegrove, Trans. ASM, 59 (1966) 486. 37 J. Bernardini and J. Cabane, Acta Met., 21 (1973) 1571. 38 F. Moya, G. E. Moya and F. Cabane-Brouty, Phys. Status Solidi, 35 (1969) 893. 39 E.W. Miiller, Z. Phys., 126 (1949) 642. 40 J. Perdereau and G. E. Rhead, Surface Sci., 7 (1967) 175. 41 F. Delamare and G. E. Rhead, C.R. Acad. Sci. (Paris), C270 (1970) 249. 42 T. Suzuoka, J. Phys. Soe. Japan, 20 (1965) 1259.
1
S O L U T E E F F E C T S IN D I F F U S I O N *
A. D. LE CLAIRE A.E.R.E., Harwell, Oxon. ( Gt. Britain)
(ReceivedSeptember27, 1974)
Small additions o f solutes to ionic substances can produce drastic changes in diffusion rates and ionic conductivities and these effects, dominated by the requirements of charge compensation, are very well understood and regularly employed in studies o f such materials. In metals and other substances the effects of solute additions on diffusion rates are usually very much smaller and rather more difficult to treat theoretically so that, until recently, they have been less widely studied. Nevertheless, knowledge of the effects, especially when combined with other data, is of considerable value in the identification and detailing of diffusion mechanisms. There is now for volume diffusion in metals an amount of reliable quantitative data and a review will be given of its essential features and of the theoretical treatments that have been developed for its interpretation and application. Some examples will be given of solute effects in grain boundary and surface diffusion processes but study o f these has been rather less quantitative and systematic and in the absence of any detailed theoretical understanding only general comment is at present possible.
1. INTRODUCTION Most detailed investigations of transport processes in ionic solids have generally included a study of the effects on conductivity tr and on diffusion rates D of the addition of small amounts of solute, a procedure known as "doping ''~. This is a valuable technique because when the " d o p e " is aliovalent to the host lattice ions it changes the defect concentrations by an extent readily calculable from considerations of charge neutrality and alters the diffusion rates and conductivity in a predictable way, determined by the nature and properties of the intrinsic defects present. Thus defect species can be identified or confirmed and many of their properties ascertained. Because the changes in defect concentrations are comparable with the dope concentration very small amounts of dope can drastically alter the defect levels and produce large effects * Paper presented at the International Conferenceon Low Temperature Diffusion and Applications to Thin Films, Yorktown Heights, New York, U.S.A., August 12-14, 1974.
2
A. D. LE CLAIRE
on D and a. Typical doping concentrations are of up to a few tenths of a molar percent. Similar techniques have been applied very much less frequently to other classes of solids because the effects of solute additions on the transport processes are more difficult to calculate and are usually very much smaller and less striking than in ionic solids. Because the effects are small careful measurements are necessary. This is especially so as these have to be confined to the low concentrations of solute, less than a few percent, for which theoretical descriptions of sufficient detail for useful comparison with experiment can be readily developed. Systematic studies of solute effects in suitably dilute solutions began several years ago with the advent of really precise diffusion measurement techniques; there is now a substantial and growing body of reliable solute-effect data, particularly extensive for metal systems. With the theoretical developments that this has stimulated it has been used, together with the results from other types of measurement, to provide a deeper and more quantitative understanding of diffusion processes. In a few cases the data have been instrumental in indicating previously unsuspected processes for atomic migration. The more quantitative aspects of these studies have been in relation to volume diffusion. For surface and grain boundary diffusion, observations of significant solute or impurity effects have also frequently been made but the interpretations of these have usually been of a more qualitative or semi-quantitative nature, contributing little to a better understanding of the processes themselves. This paper summarizes some of the more salient features of experimental and theoretical studies of solute effects. Because the data on metal systems are the most extensive, most attention will be given to them. Nothing more will be said about ionic crystals, because the effects there are very well known and understood, nor about semiconductors or other covalent systems because of the comparative dearth of data. 2.
SOLUTE ENHANCEMENT IN VOLUME DIFFUSION: PHENOMENOLOGY
We consider the addition to a pure solvent of a small fractional concentration c of solute. This is observed to change the self-diffusion coefficient of the solvent from D(0) to D(c) and we wish to relate these two coefficients. Around any isolated solute atom we expect there to be a "region of influence '" within which the solvent jump frequencies may differ from Wo and the defect concentrations from nv; w0 and nv are the values in pure solvent and, by definition, outside the regions of influence. At concentrations sufficiently low that these regions of influence rarely overlap, their effects will be additive and will provide a change in the overall solvent diffusion rate proportional to c. At rather higher concentrations we have to consider pairs of solute atoms whose regions of influence overlap. Such close pairs will also perturb the solvent jump frequencies and defect concentration, to an extent in general different from the added effect of two isolated solutes. They will contribute to a change in the solvent diffusion coefficient proportional to their concentration, i.e. to c 2. Similarly, groupings
SOLUTE EFFECTS IN DIFFUSION
3
of three, four etc. solute atoms with overlapping regions of influence will contribute terms proportional to c 3, c 4 etc. We therefore expect D(c) to have the form 2
D(c) = D(O) + k
Ic +
k2c 2 + k3 c3 + . . .
(1)
which is usually written as D(c) = D(0) (1 + b l c + b 2 c Z + b 3 c 3 + . . . )
(2)
where the bi are called "solute enhancement factors". The argument is clearly only appropriate for dilute solutions, of no more than 5-10 ~ say, when the independence of the effect of different solute groupings is a reasonable approximation, although such a power series is useful as an empirical representation over any range of concentration. Experimental results from dilute solutions usually fit eqn. (2) very well. Only rarely have the data been sufficiently precise and extensive to require the c 3 term to be retained and most frequently they have been adequately represented in terms of bl alone 3--17 A few measurements 6' 8, over rather large concentration ranges, have shown a better fit to the equation
D(c) = D(0) exp (bc)
(3)
than to eqn. (2). Equation (3) has also occasionally been used in preference to (2) for low concentration ranges 18. While at sufficiently low concentrations it may, of course, be difficult to distinguish experimentally between the two representations, there seem to be no simple theoretical arguments to justify the form of eqn. (3), other than for exceptional cases where the necessary relations obtain between the bi, i.e. b2 = ½bl z etc. The solute diffusion coefficient Di is also found at low concentrations to vary with c in a way similar to eqn. (2): Di(c ) = Di(0 ) (1 + B 1c + B 2c2 + . . . )
(4)
Di(0) is the limiting value of the solute " i m p u r i t y " diffusion coefficient at vanishingly small concentrations. In this case the linear term derives from close solute pairs, the c 2 term from threefold groupings, and so on. Di(c) is proportional to F~, the number of jumps per solute atom in unit time. Isolated solute atoms contribute to the total solute jump rate in proportion to c, solute pairs in proportion to c z, and so on. Dividing by c to give F i these contribute to eqn. (4) respectively the constant, the linear term and so on. Some typical values of bi and B~ are given in Table I to illustrate general properties, bl is the factor for which the most extensive data are available. No negative values greater than of the order of - 1 0 have been reported but the more prevalent positive values of b~ cover a wide range and may reach very large values. Values of b2 are very roughly an order of magnitude greater than b~. b 3 is only known for the two systems indicated in the table and is one to two orders of magnitude greater than b z. When, as is usual, b~, b2 and b3 are of the same sign, D(c) has an accelerating rate of increase, or decrease, with c.
4 TABLE
A. D. LE CLAIRE
I
bl
b2 ×10 -1
Ag-Cd
9.2
b3 ×10 -2
-
Temp.
B1
Bz
(°C)
727
-
-
3,4
-
-
5
-
-
4,5
Ag-Sb
65.7
-
-
54
95
-
Ag-Pb
87
-
-
85
125
-
-
-
727
-
727
-7.5
Pb-Cd
Pb-Hg
Pb-Au
-8.2 - 5
V Ti U-Co
727
3
-
-
6
50
-
4,6
-
-
7
8
-
-
-
1020
-
-
9
44
-
199
-
-
-
10
30.0
41
442
248
14
100
-
19.1
107
-
301
22
39
93
295
23
49
90
274
45.2
-
-
29
- 74
248
300
26
- 70
66
272
27
18
147
252
19
+37
39
249
28
12
226
226
18
+45
58
224
4300
-
215
-
1300
-
-
200
- 2100
0
5700
c~-Fe-Si • -Fe-Co
Ref
(°C)
-
-
- 1.2
Temp.
6
6
727
×10 -2
-
7
Ag-Au Ag-Pd Cu-Fe
B3
×10 -1
20
-
1427
- 1.1 28.7 714
-
12.4
895
-
1100
-
822
-
175
11
12,13
150 -
14 15 16
-
17
B1 is generally of the same sign as and comparable in magnitude with bl but usually smaller. Thus D(c) and Di(c) generally change in the same direction with increasing c, but D(c) usually changes more rapidly than Di(c), as might be expected. An important exception is the Pb-Au system for which B 1 is negative while bl is positive, a significant result in elucidating the Au diffusion mechanism, as we shall see. What little data there are on B2 and B 3 suggest that these are respectively approximately one and two orders of magnitude greater than B a. There are other general features of solute effects that are of significance and interest. Solutes that are "fast diffusers" in the solvent generally enhance the solvent diffusion rates, i.e. bl is positive when Di(0)>D(0 ). Similarly, b 1 is negative when Di(0)
SOLUTE EFFECTS IN DIFFUSION
5
From time to time analytical expressions are proposed that incorporate this or other correlations with the melting point and describe quite well the variation of Di and D with c for particular systems, often over extensive ranges of concentration 8' 19. A notable example is the Ag-Pd system 4 for which both D and O i a r e found to be dependent exponentially only on -TM(c,p)/T, TMbeing the solidus temperature at composition c and pressure p. However, such simple relations are very restricted in their application. The effects on measured diffusion coefficients of unavoidable impurities in the samples used can be estimated when their b~ or B~ are known. For a measured D to be affected by less than about 1 ~o (the minimum error from other sources say) there must be a concentration of about 10-2/bl or less of the offending impurity. For example, samples of 99.99 ~ purity are adequate provided the weighted mean bl of the impurities present is less than 102. Sometimes measurements are made to establish that results are independent of purity, and occasionally it is found that they are not so. A striking example is the measurements of Irmer and Feller-Kniepmeier 2° of the self-diffusion coefficient of 0c-Fe of different purities. The least pure, with a total C + O major impurity content of about 70 ppm, showed D(0) values as much as 35 times greater than those from the purest samples, which contained about 8 ppm of impurity. There was no significant change in the activation energy. Most previous measurements have given values comparable with those from the least pure Used in these studies, suggesting that they were all seriously affected by impurities. If these strikingly large effects are at all representative of interstitial impurities in b.c.c, transition metals, then many measurements of diffusion rates in such metals may be suspect because they are often very difficult to free from such impurities. Furthermore, the results add considerable verisimilitude to Kidson's proposal 21'22 that the behaviour of the so-called "anomalous b.c.c, metals" may be due to the enhancing effect of impurities like O, almost inVariably present in the more strongly anomalous metals such as 13-Ti and 13-Zr. 3. CALCULATION OF
b~ AND
Bi
We have first to stipulate the "region of influence" within which the solvent properties are perturbed by the solute, In f.c.c, crystals with diffusion by vacancies it is usually assumed that this extends out to fourth nearest neighbours from a solute atom. We define w1 as the frequency for solvent-vacancy jumps between two first nearest neighbours of a solute and w3 as the frequency for vacancy jumps from first neighbour sites by exchange with solvent atoms on second, third or fourth neighbour sites, w4 is the frequency of the reverse (association) jumps of vacancies from these sites onto first neighbour sites. All other vacancysolvent jumps are assumed to occur at the same frequency wo as in the pure solvent. With the frequency w2 for vacancy-solute exchanges, we have a total of five jump frequencies in terms of which to discuss the solute and solvent diffusion properties in a very dilute alloy, where we assume for the moment that solute pairs can be ignored. If there is an interaction energy AE 1 between a solute and a vacancy on
6
A . I ) . LE C L A I R E
a first neighbour site, the probability of there being a vacancy on any particular such site is exp { - ( E + AE1)/RT }, compared with exp ( - E/RT) for all other sites. (E is the vacancy formation energy in the pure solvent.) To maintain dynamic equilibrium between these vacancy concentrations demands a relation 1/'4/11' 3 =
exp( - AE1/RT)
(5)
between w3 and w4. We cannot presume that there is any binding energy on second, third or fourth neighbour sites without introducing additional perturbed j u m p frequencies from these sites. We now calculate D(c) by counting all the solvent jumps of each frequency w 1, w a, w 4, w o that occur in unit time and dividing by N ( 1 - c ) , the number of solvent atoms, to give the average number of jumps F per solvent a t o m in unit time. More particularly, we must weight the contribution F, from each type of j u m p ~ by its partial correlation factor f~, remembering that there may often be two or more types of j u m p having the same frequency but differing in their f~. If x is the j u m p distance, D(c) is then given by
D(c) = 1 x 2 ~/-~f.. = D(0) (1 + b l c + . . . )
(6)
from which b t can be determined. The result for f.c.c, crystals, where there are altogether 13 types of j u m p 23 (~ = 13), is b~ = - 1 8 + 4
w~ (4 wl )~1+14 Z2'~
Wo
w3 fo
fo]
(7)
Z1 and Z2 are mean values o f the J~ and are known tabulated functions of the three frequency ratios w2/wl, wl/w 3 and w4/Wo. Note that bl depends therefore on w2, and that it can be written to contain the perturbed vacancy concentration by using eqn. (5). b~ has also been calculated for b.c.c, crystals. With the "region of influence "' extending, as for f.c.c, crystals, to just those sites that can be reached by one j u m p from a solute's first neighbour sites, i.e. the second, third and fifth sites in this case, bl is given by z~ w4 v2 . ~/~2 b, = - 2 0 + 14 ~ .~oo+O f 7
(8)
The v 2 and /~2 are mean values of J; again and functions of the ratios w2/w3 and w4/wo. There are no w~ type j u m p s in b.c.c, crystals. Equation (5) still applies here. An alternative b.c.c, model has been discussed in which vacancy-solute binding, with energy - A E 2, may occur at second as well as first neighbour sites 24. We then need to introduce an additional frequency ws for dissociative vacancy jumps from second neighbours (to fourth), and to distinguish dissociative jumps from first to second and from first to third or fifth neighbours by different frequencies w 3 and w~ respectively. The corresponding associative jumps are w 4 and w~,. In the case when w~, = wo,
SOLUTE EFFECTS IN DIFFUSION
b 1 = - 2 0 + 6 w3 vl #1_ w~ foo+ 14 fo
7
(9)
vl and/~1 are functions of the ratios w2/w'a and w3/w'3 and wo = w) e x p ( - A E I / R T ) = w 5 e x p ( - A E 2 / R T )
(10)
Similar equations can be derived for other structures and assumed mechanisms of diffusion once the perturbed frequencies and defect concentrations have been specified. A case recently discussed is that of diffusion in solutions where the solute atoms are distributed interstitially and substitutionally in dissociative equilibrium 2'25'26. A substitutional solute can jump to an interstitial site with frequency v1, creating a vacancy alongside, and can perform the reverse jump with frequency v2 say. Clearly, vz/v ~ = exp { - ( E + I + B ) / k T}
(11)
if I is the formation energy of a free interstitial and - B its binding energy to a vacancy. The vacancy-interstitial pair may dissociate by an interstitial jump away from the vacancy, with frequency kl, or by a jump of the vacancy by exchange with a solvent atom, not a neighbour of the interstitial, with frequency w 1. Particular interest has attached to the contribution to solute diffusion and solvent enhancement arising from the interstitial-vacancy pair. It can move by reorientation without dissociation, by migration of the interstitial to another interstitial site with frequency k z, or by a jump of a mutual nearest neighbour solvent atom into the vacancy, with frequency w2. In one calculation with this model, due to Miller 15, it is ~assumed that interstitial jump frequencies are much greater than solvent jump frequencies, i.e. kz, vz >> w~, w2, and that the fraction of free interstitials is very small. One then finds bl = { - 6 + ( 2 w2+4 Wl) fWofo ~ x exp (-k-T) } exp (k/---T)
(12)
f l is the solvent correlation factor in the alloy and fo that in the pure solvent. Miller assumes that f l = fo. Warburton 26 has modified Miller's model by supposing that isolated interstitials are so rare that k~, wl can be assumed to be zero. He also assumes that k 2 type jumps are unlikely and puts k 2 = 0. bl is the same as eqn. (12), with w~ = 0. Warburton evaluates f l but does not give explicit values. The only calculation of the non-linear factors in eqn. (2) appears to be that of Bocquet 4 for b2. This derives from solvent-enhanced jumps in the neighbourhood of pairs of solute atoms. Such pairs are all those close enough together for there to be solvent-vacancy jumps that can no longer be unambiguously described as wl, w3 or w4. With f.c.c, crystals these are of eighth nearest neighbour separation or closer. Bocquet found it necessary to introduce a further six perturbed solvent jump frequencies and three more solute jump frequencies to describe the vacancy diffusion associated with such pairs. The resulting
8
A . D . LE CLAIRE
expression for b 2 is necessarily rather lengthy and we shall not reproduce it here. Bocquet also calculated the linear term B~ in eqn. (4) from his model in terms of the now four solute jump frequencies and the association energy Eb2 of a nearest neighbour pair of solute atoms. A more approximate calculation has been reported by Miller 2 giving B~ in terms of Eb2 alone: B~ = l l { e x p ( - E b 2 / k T ) -
1}
(13)
This follows from Bocquet's equation with the assumptions of a unique solute jump frequency w2 and a negligible vacancy-solute binding, i.e. w3 = w,,. 4. COMPARISON WITH EXPERIMENT All the equations for bi and Bicontain only frequency ratios and/or association energies. Calculations have occasionally been made of these quantities, using the electrostatic model for vacancy diffusion, to provide values of b i for comparison with experiment 4' 27. Results for bl have always produced the correct sign and, at least for weakly perturbing solutes, a fair agreement in magnitude. However, for b 2 some calculations for solutes in Ag gave signs opposite to those of b 1 and thus in marked disagreement with experiment 4. What calculations there are for B~, for a few solutes in Ag, are more encouraging in giving the correct sign, but strong attractive interactions between solute atoms (Eb2 negative) seem necessary for the results to approach the observed numerical values 4. This is evident too from eqn. (13). Studies of enhancement effects have been of most value and interest when the results have been considered in association with other diffusion properties. The more important of these have been the following: (i) the ratio of the impurity diffusion coefficient D2 (---Di(0)) to the solvent self-diffusion coefficient D o (=D(0)) given, for vacancy diffusion for example, by D:
WE f2 exp (
Do - Wo fo
AEI~ - -~-]
(14)
For other mechanisms wz and wo may be replaced by sums of terms, the exponential may contain other defect energies and there may be a numerical factor. Working with this ratio defect energies common to D 2 and Do (in this case E) are eliminated. (ii) the mass or isotope effect in impurity diffusion, defined as ( D ' - D~)/D ~
E(1) - (m~/m,)X/2 _ 1 This is intimately related to the impurity correlation factor f2 that appears in eqn. (14). The relation is particularly simple when f2 has the form f2 = u/(w2 +u)
with u depending only on solvent jump frequencies, for then
(15)
SOLUTEEFFECTSIN DIFFUSION E(1) = f 2 A K
9 (16)
allowing f2 to be determined when the kinetic energy factor AK can be estimated. In other cases E(1) may be quite a different function of the solvent and solute frequencies than that represented by f2, but readily calculable as
e(1) = E
81n D2
the wj being the solute or impurity jump frequencies 2s. We illustrate application first to cases of vacancy diffusion. For these, eqn. (15) is valid and, for f.c.c, crystals for example, u = w1 +3.5 w3 F(w4/wo)
(17)
with F a known function of w,,/Wo that varies slowlyTM 29 from 2/7 (w,,/Wo = 0o) to 1 (w4/wo = 0). Using eqn. (5) we note that D2/D o can be written D 2 _ f2 w2 wl w4 Do fo W1 W3 W0
(18)
Thus, each of the quantities f2, D2/D o and bl (eqn. (7)) is a function of the same three frequency ratios w2/w 1, wl/w 3 and w,/w o. It follows that when all three quantities have been measured we can solve eqns. (7), (15) with (17), and (18) to yield numerical values for these frequency ratios. This clearly provides very detailed information concerning diffusion processes in dilute solution and a most satisfactory form of data for testing theoretical calculations of jump frequencies. The necessary measurements have been carried out for such an analysis to be possible only on three systems, namely for Zn in Cu and Ag 3° and for Cu in Fe 9. Obviously any such analysis must yield positive values for the frequency ratios. If one or more of them is negative one must conclude that either (i) the jump frequency model is inadequate or (ii) diffusion is not occurring by the vacancy mechanism that is assumed in writing eqns. (7), (17) and (18). However, such incompatibilities can often be identified more simply by making use of other relations derived from the equations. Also it is not always necessary that the complete set of experimental data be available. As an example of (i), an earlier four-frequency model, used to interpret data on Fe diffusion in Cu and Ag, yielded a negative value of wl/w 3 and first indicated 3~the need to introduce the frequency w4 as distinct from wo. A relationship of particular value is the equation for bl in terms of D2/D o and f2, namely
~4~l(_WI/W3)+
4 f 0 02 14Z2~ b~ = - 18 -~ 1 - f z Do [ f o (4 wffwa + 14F )J
(19)
If the solute does not have a very strong perturbing effect it is a reasonable approximation to put ZI = )~2 = f o . With this and the approximation F = 1, eqn. (19) becomes the Lidiard equation32:
l0
A . D . LE CLAIRE
bl = - 18-~
4fo D2 1 -f: D O
(20)
Both eqns. (19) and (20) express the general observation that fast diffusers (D 2 > Do) tend to enhance solvent diffusion rates (bl positive) and slow diffusers to reduce them ( D 2 < D 0, bl negative). They also indicate why large negative values of b I are not observed, however small D2/D o. Equation (19) can be used to determine the minimum value of bl consistent with an experimental value of DE/D o. It is easily seen that bl is a minimum in the limit wl/w3--,o~ and {2--*0, implying that wz/wl--*~: brain = - 18+4 )~1 (D2/Do) = - 18+ 1.945 (D2/Do) (21) where the value for 7.1 at wl/w 3 and w 2 / w o ~ has been inserted ~°. This limit corresponds to the case of a very tight vacancy-impurity binding and very rapid impurity jumps. Such extreme conditions are unlikely to be common so we expect measured values of bl to be appreciably greater than bmi.. Most of them are. However, over the last three or four years some systems have been discovered for which the experimental bl is well below the bmi. given by eqn. (21). For Au and Ag in Pb 12 the observed h 1 is only approximately one-fiftieth of hmi"--so very much smaller that no elaboration of the frequency model could conceivably alter the situation and one must conclude that some mechanism other than that of vacancy diffusion is operating here. There is a similar but much less extreme situation for Cd in Pb l°, where bi ~ D 2 / D o. For Hg in Pb ~ b 1 ~ 1.5 D2/D o and slightly above bm~,, but so close to it that the extreme conditions necessary for a vacancy mechanism make such a mechanism questionable in this case too. Very similar analyses have been made of the data on f2, D2/Do and bl for dilute b.c.c, alloys, using eqns. (8) or (9) and the appropriate expressions for u in eqn. (15). The only difference is that, since there are no w~ jumps, the three experimental quantities are determined by just two frequency ratios. These can therefore be estimated from any two of the experimental quantities. For many systems positive values are found and b~ >b,,i,, consistent with vacancy diffusion. However, there are a few--Ti-Co, Zr-Co and U - C o - - f o r which b~ is an order of magnitude below b,,i,, indicating again some other mechanism than the vacancy one 24. Since, for these cases in which bl
SOLUTE EFFECTS IN DIFFUSION
11
Miller 2'25 suggested that the diffusion mechanism is dominated by tightly bound vacancy-interstitial pairs; the values of bl for this are given in eqn. (12) and D2/D o is
D_~z=4k2+v z Do
fz
exp ( - ~ )
(22)
2Wo fo
f2 was estimated by Miller and, with his approximations (k2, v2:~k ~, w 1, w2),
fz ~ = (4 w2+8 w1 +20 ka)/(4 kz+v2) With the approximation Warburton introduced into his treatment 26 of the model
fz w = 4 w2/(4 w2 + v2) Thus, from eqn. (12) for b ~, ignoring the term 6 exp ( - / / k T) which is negligible compared with b~,
D2 ( 1 btM=ft Doo
5 kl
)-'
(23)
q W z q - 2 Wl
and D2 4 w2+v 2
blW= fl Do
v2
(24)
Miller estimated that f~ ~fo, so if k~ is very small, i.e. the pairs are very stable,
bl u = )Co (D2/Do)
(25)
in near agreement with the results for Cd in Pb. However, bl u cannot be greater than D2/D o, as found for Hg in Pb. Warburton's assumptions demand a more careful consideration of f l ; this was evaluated as a function of w2/v 2 to give the result from eqn. (24) that 3.80 (D2/Do) < bl w < 0.83 (D2/Do)
(26)
the upper limit being for w2/v 2 = ~ , the lower for w2/v2 = 0. Both the Cd in Pb and Hg in Pb data can now be accommodated and w2/v 2 can be evaluated: about ½ for Hg, about ~ for Cd. These systems have provided excellent examples of the value of solute enhancement effects in studying diffusion migration mechanisms. 5. SOLUTE EFFECTS IN GRAIN BOUNDARY AND SURFACE DIFFUSION
Any measurement of a grain boundary diffusion coefficient DQb in a solid solution is usually complicated by the lack of adequate knowledge on the extent of solute segregation at the boundary. The fractional concentration cb of solute in equilibrium in the boundary may differ very appreciably from the fractional concentration c e in the lattice. The ratio Cb/Ct is the "segregation coefficient" k. If Fb is the free energy of binding of a solute atom in the boundary 34
12
A . D . LE CLAIRE
k
=--Cb =
exp
l+ct
{
exp
-1
-1
ct
~ exp ( ~ - ~ ) = exp ( - - ~ )
exp ( ~ T )
the latter equality obtaining when ce exp (F/kT)<< 1. Values of k as large as 103 or 104 have been reported, corresponding to UZ +7 k T t o U,,~ + 9 kT. We need to know k for two reasons: first, to establish the boundary concentration at which Dg u is being measured; second, to be able to evaluate Dgb itself, because the standard methods for studying grain boundary diffusion rates measure only the product P = D0b fi k, 3 being the effective width of the boundary. (k arises because in solving the grain boundary diffusion equations the relation cb = k ce is introduced as a boundary condition linking the solutions within and without the boundary at the grain boundary surfaces.) Only with measurements of grain boundary self-diffusion rates in pure solvents is there no problem, for then k = 1 and, given ~, D0b itself can be evaluated. In other cases very little is known about k for the systems that have been studied and most investigators have either ignored this factor altogether or implicitly assumed k = 1. Their reported values of "Dg b'' are then actually of Dgbk so that their activation energies "Qgb" are Q0b--U, plus any contribution there may be from a temperature dependence of 6. "In view of the possible magnitudes of k and U it does not seem possible, therefore, to draw any reliably firm conclusions from the results about the relative magnitudes of the solvent self and solute impurity D0b values or of the effects of solute concentration on the solvent and solute D~b values, in other words to extract enhancement factors bi and Bi for grain boundary diffusion. Variations of D~b between systems and with concentration may be masked by or confounded with corresponding variations in k. One can only discuss the variation of P and its activation energy. It is certainly well established that quite small concentrations of solute, even residual impurities, can in many cases have a marked effect on grain boundary self-diffusion rates. Thus the value of Dgb around 600~C for self-diffusion in an a-Fe of 99.7 ~o purity was reported to be more than four times that measured on a purer sample of 99.96~o purity 35. The difference decreased slightly with increase in temperature. In the y-phase it was only some 50~o. Impurities may also decrease Dgb, as has been found 36 for self-diffusion in Pb with additions of T1, In or Sn, where reductions by a factor of approximately one-third arise at concentrations of about 1½~o. Effects of similar magnitude have been found in many other systems: when they are large in relation to the total amount of impurity they are usually attributed to the concentrating effect of segregation but there is little understanding of the mechanism by which Dgb is changed. There are a number of measurements of the grain boundary diffusion of solutes but here, as we have seen, we are very much involved with k. It would be desirable to establish the relation between the impurity Dgb for various impurities in a solvent and the D0b for self-diffusion in that solvent, for the corresponding relationship in volume diffusion has proved so useful in the
SOLUTE EFFECTS IN DIFFUSION
13
understanding of volume migration processes. Without knowing k this is not possible. The tendency is rather to assume that these relationships are similar in volume and in grain boundary diffusion and that any unexpected magnitude or type of behaviour in a Dg b, or P, is due to segregation. Thus, for example, P for Ru in the substructure boundaries of Cu is very much larger than P for Ni in Cu, yet the impurity volume D i for Ru in this solvent is much less than that for Ni 37. Attributing this behaviour to a large k value for Ru, and assuming k ~ 1 for Ni on the grounds of its complete solubility, k was calculated to be of the order of 100. The ratio P / D i for Ru is about 100 times its value for Ni and it was assumed that these ratios would be much the same in the absence of any segregation. When k is large we expect larger than normal dislocation or grain boundary enhancements of volume diffusion rates to be revealed in departures from linearity of the Arrhenius plots. For Ru in Cu such departures set in at the unusually high temperature of about 0.9 TM, even in single crystals. Similar behaviour was found 38 for S diffusion in Cu and attributed again to segregation, with k ~ 100. Other measurements that have been made include studies of the effect of one solute on the grain boundary diffusion of another and, in a few cases, of "Dg b'' over ranges of temperature and composition for solute and solvent, but little by way of general principles of behaviour has emerged. Studies of solute effects in surface diffusion mostly concern the influence of residual impurities or of foreign atmospheres on surface self-diffusion rates. This can be very severe and there are many examples of D s being changed by several orders of magnitude, with corresponding large changes in activation energy. These are attributed, naturally, to adsorbed layers of foreign atoms. The effects can be to increase or to decrease D s. D s for W surface self-diffusion is much reduced by oxygen or water vapour39; there are large increases in Ds following S adsorption on Ag 40 or C1 adsorption on Cu 41 An interesting feature of tracer surface self-diffusion measurements is that they often show volume-like behaviour in that In c is proportional to x 2, rather than to x or x 6/5 as might be expected (c is the concentration of diffusant at a distance diffused x) 42. This is Harrison's type C condition of diffusion and means that the diffusant is wholly confined to the surface layers with no loss of diffusant into the substrate. This often occurs though, even at temperatures where appreciable loss into the substrate might be expected. It has been suggested 42 that this is due to impurities segregated or adsorbed at the surface to form a barrier layer that inhibits inward diffusion. If this is so, the D~ measured may refer to diffusion over the contaminant layer and not represent a true surface self-diffusion coefficient. In the absence o f this effect and with true Harrison type C conditions obtaining, measurements would yield directly a Ds for solute or impurity surface diffusion without the need to know k or 6. This requires that k be so large that k 2 62 >> Dtt, D t being the lattice diffusion coefficient.
14
A. D. LE CLAIRE
REFERENCES
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