Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 1 Part 6 potx

A crude estimate for the correlation factor can be obtained as follows: we consider the shortest vacancy trajectory, which consists of only one further vacancy jump after the first displa

7.3 Vacancy Mechanism of Self-diffusion 113from giving a comprehensive review of all methods Instead we rather strivefor a physical understanding of the underlying ideas: we consider explicitelylow vacancy concentrations and cubic coordination lattices Then, the aver-ages in Eq (7.7) refer to one complete encounter Since for a given value of

n there are (n − j) pairs of jump vectors separated by j jumps, and since all

vacancy-tracer pairs immediately after their exchange are physically lent, we introduce the abbreviation j i,j and get:

Here j is the average of the cosines of the angles between all pairs of vectors separated by j jumps in the same encounter With increasing j the

averages j converge rapidly versus zero Executing the limit n → ∞,

Eq (7.21) can be written as:

To get further insight, we consider – for simplicity reasons – the

x-dis-placements of a series of vacancy-tracer exchanges For a suitable choice of

the x-axis only two x-components of the jump vector need to be considered2,

which are equal in length and opposite in sign Since then cos θj = ±1, we

get from Eq (7.22)

#

where p+j (p −

j ) denote the probabilities that tracer jump j occurs in the same

(opposite) direction as the first jump If we consider two consecutive tracerjumps, say jumps 1 and 2, the probabilities fulfill the following equations:

The three-dimensional analogue of Eq (7.26) was derived by Compaanand Haven[20] and can be written as

where θ is the angle between two consecutive tracer jumps With this

recur-sion expresrecur-sion we get from Eq (7.22)

We note that Eq (7.29) reduces correlation between non-consecutive pairs

of tracer jumps within the same encounter to the correlation between twoconsecutive jumps Equation (7.29) is valid for self- and solute diffusion via

a vacancy mechanism

The remaining task is to calculate the average value

it may suffice to make a few remarks: starting from Eq (7.29), we considerthe situation immediately after a first vacancy-tracer exchange (Fig 7.3)

The next jump of the tracer atom will lead to one of its Z neighbouring sites

l in the lattice Therefore, we have

Z



l=1

In Eq (7.30) δ l denotes the angle between the first and the second tracer

jump, which displaces the tracer to site l P lis the corresponding probability

A computation of P l must take into account all vacancy trajectories in thelattice which start at site 1 and promote the tracer in its next jump to

site l An infinite number of such vacancy trajectories exist in the lattice.

One example for a vacancy trajectory, which starts at site 1 and ends at site

4, is illustrated in Fig 7.3 Some trajectories are short and consist of a smallnumber of vacancy jumps, others comprise many jumps

A crude estimate for the correlation factor can be obtained as follows: we

consider the shortest vacancy trajectory, which consists of only one further

vacancy jump after the first displacement of the tracer, i.e we disregard theinfinite number of all longer vacancy trajectories Then, nothing else than theimmediate back-jump of the tracer to site 1 in Fig 7.3 can occur Any othertracer jump requires vacancy trajectories with several vacancy jumps Forexample, for a tracer jump to site 4 the vacancy needs at least 4 jumps (e.g.,

7.4 Correlation Factors of Self-diffusion 115

Fig 7.3 Example of a vacancy trajectory immediately after vacancy-tracer

ex-change in a two-dimensional lattice

1 → 2 → 3 → 4 → tracer) The probability for an immediate back-jump is

P back = 1/Z and we have

back · cos 180 ◦=−1

Inserting this estimate in Eq (7.29), we get

f ≈ Z Z + 1 − 1 = 1− Z + 12 . (7.32)This equation is similar to the ‘rule of thumb’, Eq (7.12), discussed at the

beginning of this section Exact values of f are presented in the next section.

7.4 Correlation Factors of Self-diffusion

There are a number of publications devoted to calculations of correlation

factors for defect-mediated self-diffusion Values of f are collected in Table 7.2

for various lattices and for several diffusion mechanisms (see also the reviews

by Le Claire [27], Allnatt and Lidiard [30], and Murch [31])

The correlation factor depends on the type of the lattice and on thediffusion mechanism considered The correlation factor decreases the tracerdiffusion coefficient with respect to its (hypothetical) ‘random-walk value’.For self-diffusion this effect is often less than a factor of two Nevertheless,for a complete description of the atomic diffusion process it is necessary to

include f

There are, however, additional good reasons why a study of correlationsfactors is of interest The correlation factor is quite sensitive to the diffusionmechanism For example, the correlation factor for diffusion via divacancies

is smaller than that for monovacancies An experimental determination of f

could throw considerable light on the mechanism(s) of diffusion The cation of the diffusion mechanism(s) is certainly of prime importance for theunderstanding of diffusion processes in solids Unfortunately, a direct mea-

identifi-surement of f is hardly possible However, meaidentifi-surements of the isotope effect

Table 7.2 Correlation factors of self-diffusion in several lattices

Lattice Mechanism Correlation factor f Reference

CaF2(F ) non-colinear interstitialcy 0.9855 [20]

CaF2(Ca) non-colinear interstitialcy 1 [20]

of diffusion, which is closely related to f (see Chap 9), and in some cases

also measurements of the Haven ratio (see Chap 11) can throw some light

on the diffusion mechanism

The ‘rule of thumb’ values from Eq (7.12) for vacancy-mediated diffusionare listed in Table 7.3 These values are mostly within 10 % of the correctvalues, indicating that a large amount of correlation results from the firstbackward exchange of vacancy and tracer For the diamond and honeycomblattices and the 1d chain the ‘rule of thumb’ values coincide with the exactvalues The exact values confirm a trend suggested already by the ‘rule ofthumb’: correlation becomes more important, when the coordination number

Z decreases.

7.5 Vacancy-mediated Solute Diffusion

Diffusion in binary alloys is more complex than self-diffusion in pure crystals

For dilute alloys there are two major aspects of diffusion: solute diffusion and solvent diffusion In this section, we confine ourselves to correlations effects

of solute diffusion in dilute alloys Correlation effects of solvent diffusion indilute fcc alloys have been treated by Howard and Manning [12] on thebasis of the ‘five-frequency-model’ (see below) proposed by Lidiard [13].Here we concentrate on the more important case of solute diffusion Resultsfor correlation effects of solvent diffusion can be found in Chap 19

7.5 Vacancy-mediated Solute Diffusion 117

Table 7.3 Comparison between ‘rule of thumb’ estimates of correlation factors

based on Eq (7.12) and exact values

Lattice Z 1− 2/Z Correlation factor

to diffuse in a pure solvent In Chap 5 we have already discussed vacancies

in dilute substitutional alloys We remind the reader to the Lomer expression(5.31), which we repeat for convenience:

G B for the vacancy-solute pair For G B > 0 (G B < 0) the probability p is enhanced (reduced) with respect to C 1V eq

The presence of the solute also influences atom-vacancy exchange rates

in its surroundings The exchange rates between vacancy and solute andbetween vacancy and solvent atoms near a solute atom are different fromthose in the pure solvent The exchange rates enter the expression for thecorrelation factor Correlation factors of solute diffusion in the fcc, bcc, anddiamond lattices are considered in what follows Note that it is commonpractice in the diffusion literature to use the index ‘2’ to distinguish the

solute correlation factor, f2, from the correlation factor of self-diffusion in the pure solvent, f , and the solute diffusivity, D2, from the self-diffusivity in

the pure solvent, D

7.5.1 Face-Centered Cubic Solvents

The influence of a solute atom on the vacancy-atom exchange rates is often

described by the so-called ‘five-frequency model’ proposed by Lidiard [13,

Table 7.4 Vacancy-atom exchange rates of the ‘five-frequency model’

ω2: solute-vacancy exchange rate

ω1: rotation rate of the solute-vacancy pair

ω3: dissociation rate of the solute-vacancy pair

ω4: association rate of the solute-vacancy pair

ω: vacancy-atom exchange rate in the solvent

Fig 7.4 Left: ‘Five-frequency model’ for diffusion in dilute fcc alloys Right:

‘En-ergy landscape’ for vacancy jumps in the neighbourhood of a solute atom

14]3 The five types of vacancy-atom exchange rates are illustrated in Fig 7.4and listed in Table 7.4 Each exchange leads to a different local configura-tion of solute atom, vacancy, and solvent atoms In the framework of thismodel two categories of vacancies can be distinguished: vacancies located

in the first coordination shell of the solute and vacancies located on lattice

sites beyond this shell The vacancy jump with rate ω4 (ω3) forms

(dissoci-ates) the vacancy-solute pair Association and dissociation rate are related

to the Gibbs free energy of binding of the pair via the detailed balancingequation



Before we discuss the correct expression for the correlation factor of solutediffusion, let us consider – as we did in the case of self-diffusion – an esti-mate, which may provide a better understanding of the final result Supposethat a first solute-vacancy exchange has occurred The crudest approxima-

tion for f2takes into account only that vacancy trajectory which leads to an

3 Lidiard uses the word ‘frequency’ instead of rate Frequency and rate have the

same dimension, but they are physically different In the authors opinion, theterm ‘rate’ is more appropriate

7.5 Vacancy-mediated Solute Diffusion 119immediate reversal of the first solute atom jump The pertinent probabilityexpressed in terms of jump rates is

P back ≈ ω2

since the vacancy from its site next to the tracer can perform one ω2 jump,

four ω1 jumps, and seven ω3 jumps Herewith, we get from Eqs (7.29) and(7.35) for the solute correlation factor

It has some similarity with the approximate relation In Eq (7.37) F3 is

the probability that, after a dissociation jump ω3, the vacancy will not return to a neighbour site of the solute F3 is sometimes called the es- cape probability It is illustrated in Fig 7.5 as a function of the ratio

α = ω4/ω Manning derived the following numerical expression for the escape

probability:

7F3(α) = 7 − 10α4+ 180.5α3+ 927α2+ 1341α

2α4+ 40.2α3+ 254α2+ 597α + 436 . (7.38)When α goes to zero, no association of the vacancy-solute complex occurs and we have F3 = 1 For ω = ω4 the escape probability is F3 = 0.7357 When

α goes to infinity, we have F = 2/7 (see Fig 7.5) The correlation factor f2

is a function of all vacancy-atom exchange rates We mention several specialcases:

1 For self-diffusion all jump rates are equal (when isotope effects are

ne-glected): ω = ω1 = ω2 = ω3 = ω4 and we get f2 = 0.7814 This value

agrees (as it should) with the value listed in Table 7.2

2 If the vacancy-solute exchange is much slower than vacancy-solvent

ex-changes, i.e for ω2  ω1, ω3, , the correlation factor tends towards

unity Frequent vacancy-solvent exchanges randomise the vacancy tion before the next solute jump Then, solute diffusion is practicallyuncorrelated

posi-3 If vacancy-solute exchanges occur much faster than vacancy-solvent

ex-changes, i.e for ω2  ω1, ω3, , the correlation factor tends to zero

and the motion of the solute atom is highly correlated The solute atom

‘rattles’ frequently back and forth between two adjacent lattice sites

Fig 7.5 Escape probabilities F3 for the fcc, bcc, and diamond structure For fcc

and bcc lattices F3 is displayed as function of ω4/ω For the diamond structure F3

is a function of ω4/ω5 After Manning [15]

4 Dissociation jumps ω3are very unlikely for a tightly bound solute-vacancypair Then, we get from Eq (7.37)

f2≈ ω1

In this case the vacancy-solute pair can migrate as an entity via ω1 and

ω2-jumps Such pairs are sometimes called ‘Johnson molecules’.

7.5.2 Body-Centered Cubic Solvents

The simplest model for vacancy-exchange rates of solute diffusion in a bcclattice is illustrated in Fig 7.6 It distinguishes four jump rates: the solute-

vacancy exchange rate ω2, the dissociation rate ω3, the association rate ω4, and the vacancy-solvent exchange rate ω As in the case of the fcc lattice, the rates ω3and ω4are related via the detailed balancing relation Eq (7.34)

In contrast to the fcc lattice, the bcc structure has no lattice sites which aresimultaneously nearest neighbours to both the solute atom and the vacancy of

a solute-vacancy pair Hence an analogue to the rotation rate in the fcc lattice

7.5 Vacancy-mediated Solute Diffusion 121

Fig 7.6 ‘Four-frequency model’ of solute diffusion in the bcc lattice

does not exist in the bcc lattice According to Manning [15] the correlationfactor for solute diffusion can be written as

More sophisticated models for solute diffusion in bcc metals are available

in the literature (see, e.g., [16]), which we will, however, not describe here.They take into account different solute-vacancy interactions for nearest andnext-nearest solute-vacancy pairs and then also several additional dissociationand association jump rates

7.5.3 Diamond Structure Solvents

In the diamond structure one usually considers the following vacancy-jump

rates (see Fig 7.7): ω2 for exchange with the solute, ω3 for vacancy jumps

from first to second-nearest neighbours of the solute, ω4for the reverse jump

of ω3, ω5 for jumps from second- to third- or fifth-nearest neighbours of the

solute, and ω for jumps originating at third-nearest neighbours or further

apart from the solute Manning [15] derived the following expression for thecorrelation factor

f2= 3ω3 F3

Fig 7.7 ‘Five frequency model’ for jump rates in the diamond structure ω2:

jump rate of vacancy-tracer exchange, ω3 (ω4): jump rate of vacancy jump from

first (second) to second (first) nearest neighbour of solute atom, ω5: jump rate ofvacancy jump from second to third nearest neighbour of tracer

where the escape probability

F3(α) = 2.76 + 4.93α + 2.05α

2

2.76 + 6.33α + 4.52α2+ α3 (7.43)

is a function of the ratio α = ω4 /ω5 For self-diffusion, all frequencies are

identical, F3 = 2/3 and f2 agrees with the value 1/2 of Table 7.2 When α goes to zero, F3 goes to unity In the other limit where α goes to infinity, F3

goes to zero (see Fig.7.5)

Additional complexity can arise in semiconducting solvents, when soluteand vacancy carry electrical charges In contrast to metallic solvents, wherethe solute-vacancy interaction is short-ranged, the Coulomb interaction canmodify the vacancy behaviour over larger distances in the surroundings of

7.6 Concluding Remarks 123Bakker[24] A rather elegant method based on Laplace and Fourier trans-forms was developed by Benoist et al [25] Koiwa [26] designed a ma-trix method for return probabilities Reviews among others were provided

by Le Claire [27], Mehrer [28], and in appropriate chapters of textbooks

by Manning [29], Allnatt and Lidiard [30], and by Murch [31] MonteCarlo simulations were introduced by Murch and coworkers [32, 33] Mean-while Monte Carlo methods and combinations between analytical schemesand Monte Carlo simulations play an important rˆole in the calculations ofcorrelation factors For a recent example see, e.g., [34]

Correlation factors of self-diffusion in elements and in the sublattice ofcompounds are often well-defined numbers For a given lattice they are char-acterstic of a certain diffusion mechanism as indicated in Table 7.2 Oftenthe ‘rule of thumb’ permits already a good guess

Correlation factors for substitutional solutes are temperature-dependent.The reason is that each vacancy-atom exchange rate is temperature depen-dent according to

In Eq (7.44) the quantity G M

i denotes the Gibbs free energy of activation

for jump ωi and ν0

i is the pertaining attempt frequency At a first glance,

the temperature dependence of f2 looks rather complex However, in a tain temperature interval it can always be approximated by an Arrheniusexpression

cer-f2≈ f0

2exp



− C kT

The correlation factors of solute diffusion in fcc, bcc, and diamond latticeshave the following mathematical form, called ‘impurity form’, in common:

f2 = u

The quantity u depends on vacancy-solvent exchange rates only and not on the vacancy-solute exchange rate ω2 This should be remembered, when wediscuss isotope effects of diffusion in Chap 9

We stress once more that Eq (7.29) has been derived for a vacancy anism in a cubic coordination lattice with one type of jumps Equation (7.29)

mech-is valid, when the defect-tracer complex contains at least a twofold rotationaxis [20] This is indeed the case for vacancy-tracer pairs in cubic coordi-nation lattices These condition are, however, violated for divacancy-tracer

complexes in cubic crystals and for monovacancy-tracer complexes in cubic lattices For generalisations of Eq (7.29) to non-cubic lattices and tomore complex diffusion mechanisms the reader should consult the literaturecited at the entrance to this section.

non-We also remind the reader that our discussion of correlation effects hasbeen limited to pure solids, very dilute alloys and to low defect concentrations.There are a number of solid compounds with high disorder in (at least)one of their sublattices Examples are non-stoichiometric compounds withstructural vacancies, certain concentrated interstitial alloys, and fast ion-conductors such as silver iodide Such compounds can be viewed as solidswith high apparent defect concentrations Then the encounter model is nolonger useful In addition ‘defects’ may interact and correlation effects tend

to be magnified and become highly temperature dependent

In the last two decades or so intermetallic compounds have attractedconsiderable interest because of the substantial technological importance ofsome intermetallics such as aluminides and silicides Correlation effects inintermetallic compounds are complicated by the fact that at least two sub-lattices are involved In addition, several diffusion mechanisms have beensuggested Examples are the six-jump cycles of vacancies, the triple defectmechanism in B2 lattices, migration of minority atoms as antisite atoms in themajority sublattice of L12 and D03 structures, diffusion by intra-sublatticejumps of vacancies (see Chap 20) Along with the growth in computationalmaterials science, a number of computer simulations of atomic transport inintermetallics have been performed The progress with analytical and ran-dom walk calculations of correlation effects for diffusion in intermetallics isreviewed by Belova and Murch [35] Atomistic computer simulations ofdiffusion mechanisms using molecular dynamics methods are reviewed byMishin[36]

References

1 J Bardeen, C Herring, in: Atom Movements A.S.M Cleveland, 1951, p 87 –

288

2 J Bardeen, C Herring, in: Imperfections in Nearly Perfect Solids, W Shockley

(Ed.), Wiley, New York, 1952, p 262

3 R Friauf, Phys Rev 105, 843 (1957)

4 R Friauf, J Appl Phys Supp 33, 494 (1962)

5 E.G Montroll, Symp Appl Math 16, 193 (1964)

6 I Stewart, Sci Am 280, 111 (1999)

7 M Koiwa, J Phys Soc Japan 45, 781 (1978)

8 M Koiwa, S Ishioka, Philos Mag A40, 625 (1979); J Stat Phys 30, 477 (1983); Philos Mag A47, 927 (1983)

9 D Wolf, Mass Transport in Solids, Ch 7, (1983) p 149

10 M Eisenstadt, A.G Redfield, Phys Rev 132, 635 (1963)

11 G.E Murch, Solid State Ionics, 7, 177 (1982)