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

7.3 Vacancy Mechanism of Self-diffusion 113 from giving a comprehensive review of all methods. Instead we rather strive for a physical understanding of the underlying ideas: we consider explicitely low 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 equiva- lent, we introduce the abbreviation cos θ j ≡cos θ i,j  and get: f = 1 + lim n→∞ 2 n n−1  j=1 (n − j)cos θ j . (7.21) Here cos θ 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 cos θ j  converge rapidly versus zero. Executing the limit n →∞, Eq. (7.21) can be written as: f =1+2 ∞  j=1 cos θ j  =1+2(cos θ 1 + cos θ 2  + ) . (7.22) 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 considered 2 , which are equal in length and opposite in sign. Since then cos θ j = ±1, we get from Eq. (7.22) f =1+2 ∞  j=1  p + j − p − j  , (7.23) 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 tracer jumps, say jumps 1 and 2, the probabilities fulfill the following equations: p + 2 = p + 1 p + 1 + p − 1 p − 1 , p − 2 = p + 1 p − 1 + p − 1 p + 1 . (7.24) Introducing the abbreviations t j ≡ p + j − p − j and t 1 ≡ t,weget t 2 = p + 1  p + 1 − p − 1     t −p − 1  p + 1 − p − 1     t = t 2 . (7.25) From this we obtain by induction the recursion formula t j = t j . (7.26) 2 Jumps with vanishing x-components can be omitted. 114 7 Correlation in Solid-State Diffusion The three-dimensional analogue of Eq. (7.26) was derived by Compaan and Haven [20] and can be written as cos θ j  =(cos θ) j , (7.27) where θ is the angle between two consecutive tracer jumps. With this recur- sion expression we get from Eq. (7.22) f =1+2cos θ  cos θ +(cos θ) 2 +  . (7.28) The expression in square brackets is a converging geometrical series with the sum 1/(1 −cosθ). As result for the correlation factor of vacancy-mediated diffusion in a cubic coordination lattice, we get f = 1+cos θ 1 −cosθ . (7.29) We note that Eq. (7.29) reduces correlation between non-consecutive pairs of tracer jumps within the same encounter to the correlation between two consecutive jumps. Equation (7.29) is valid for self- and solute diffusion via a vacancy mechanism. The remaining task is to calculate the average value cos θ.Atthispoint, it may suffice to make a few remarks: starting from Eq. (7.29), we consider the 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 cos θ = Z  l=1 P l cos δ l . (7.30) 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 l is the corresponding probability. A computation of P l must take into account all vacancy trajectories in the lattice 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 small number 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 the infinite number of all longer vacancy trajectories. Then, nothing else than the immediate back-jump of the tracer to site 1 in Fig. 7.3 can occur. Any other tracer jump requires vacancy trajectories with several vacancy jumps. For example, 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 cos θ≈P back · cos 180 ◦ = − 1 Z . (7.31) Inserting this estimate in Eq. (7.29), we get f ≈ Z − 1 Z +1 =1− 2 Z +1 . (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 the diffusion mechanism considered. The correlation factor decreases the tracer diffusion 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 correlations factors is of interest. The correlation factor is quite sensitive to the diffusion mechanism. 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 identifi- cation of the diffusion mechanism(s) is certainly of prime importance for the understanding of diffusion processes in solids. Unfortunately, a direct mea- surement of f is hardly possible. However, measurements of the isotope effect 116 7 Correlation in Solid-State Diffusion Table 7.2. Correlation factors of self-diffusion in several lattices Lattice Mechanism Correlation factor f Reference 1d chain vacancy 0 see text honeycomb vacancy 1/3 [20] 2d-square vacancy 0.467 [20, 18] 2d hexagonal vacancy 0.56006 [20] diamond vacancy 1/2 [20] simple cubic vacancy 0.6531 [18] bcc cubic vacancy 0.7272, (0.72149) [18, 20] fcc cubic vacancy 0.7815 [20, 18] fcc cubic divacancy 0.4579 [19] bcc cubic divacancy 0.335 to 0.469 [17] fcc cubic 100 dumb-bell interstitial 0.4395 [25] any lattice direct interstital 1 diamond colinear interstitialcy 0.727 [21] CaF 2 (F ) non-colinear interstitialcy 0.9855 [20] CaF 2 (Ca) colinear interstitialcy 4/5 [20] CaF 2 (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 diffusion are listed in Table 7.3. These values are mostly within 10 % of the correct values, indicating that a large amount of correlation results from the first backward exchange of vacancy and tracer. For the diamond and honeycomb lattices and the 1d chain the ‘rule of thumb’ values coincide with the exact values. The exact values confirm a trend suggested already by the ‘rule of thumb’: 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 in dilute fcc alloys have been treated by Howard and Manning [12] on the basis of the ‘five-frequency-model’ (see below) proposed by Lidiard [13]. Here we concentrate on the more important case of solute diffusion. Results for 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 basedonEq.(7.12)andexactvalues Lattice Z 1 − 2/Z Correlation factor fcc 12 0.833 0.781 bcc 8 0.750 0.727 simple cubic 6 0.667 0.653 diamond 4 0.500 0.500 honeycomb 3 1/3 1/3 2d square 4 0.5 0.467 2d hexagonal 6 2/3 0.56006 1d chain 2 0 0 In a dilute alloy, isolated solute atoms are surrounded by atoms of the solvent on the host lattice. Accordingly, each solute atom can be considered 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: p = C eq 1V exp  G B k B T  . (7.33) The probability p to find a vacancy on a nearest-neighbour site of a solute atom is different from the probability C eq 1V to find a vacancy on an arbitrary site of the solvent. This difference is due to the Gibbs free energy of binding G B for the vacancy-solute pair. For G B > 0(G B < 0) the probability p is enhanced (reduced) with respect to C eq 1V . The presence of the solute also influences atom-vacancy exchange rates in its surroundings. The exchange rates between vacancy and solute and between vacancy and solvent atoms near a solute atom are different from those in the pure solvent. The exchange rates enter the expression for the correlation factor. Correlation factors of solute diffusion in the fcc, bcc, and diamond lattices are considered in what follows. Note that it is common practice in the diffusion literature to use the index ‘2’ to distinguish the solute correlation factor, f 2 , from the correlation factor of self-diffusion in the pure solvent, f, and the solute diffusivity, D 2 , 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, 118 7 Correlation in Solid-State Diffusion Table 7.4. Vacancy-atom exchange rates of the ‘five-frequency model’ ω 2 : solute-vacancy exchange rate ω 1 : rotation rate of the solute-vacancy pair ω 3 : dissociation rate of the solute-vacancy pair ω 4 : 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. 7.4 and 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 this model 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 ω 4 (ω 3 ) 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 balancing equation ω 3 ω 4 =exp  − G B kT  . (7.34) Before we discuss the correct expression for the correlation factor of solute diffusion, 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. Suppose that a first solute-vacancy exchange has occurred. The crudest approxima- tion for f 2 takes 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, the term ‘rate’ is more appropriate. 7.5 Vacancy-mediated Solute Diffusion 119 immediate reversal of the first solute atom jump. The pertinent probability expressed in terms of jump rates is P back ≈ ω 2 ω 2 +4ω 1 +7ω 3 , (7.35) since the vacancy from its site next to the tracer can perform one ω 2 jump, four ω 1 jumps, and seven ω 3 jumps. Herewith, we get from Eqs. (7.29) and (7.35) for the solute correlation factor f 2 ≈ 2ω 1 +7ω 3 /2 ω 2 +2ω 1 +7ω 3 /2 . (7.36) This expression illustrates that a combination of vacancy jump rates enters the correlation factor. An exact expression for f 2 was derived by Manning [15]: f 2 = ω 1 +7F 3 ω 3 /2 ω 2 + ω 1 +7F 3 ω 3 /2 . (7.37) It has some similarity with the approximate relation. In Eq. (7.37) F 3 is the probability that, after a dissociation jump ω 3 , the vacancy will not return to a neighbour site of the solute. F 3 is sometimes called the es- cape probability. It is illustrated in Fig. 7.5 as a function of the ratio α = ω 4 /ω. Manning derived the following numerical expression for the escape probability: 7F 3 (α)=7− 10α 4 + 180.5α 3 + 927α 2 + 1341α 2α 4 +40.2α 3 + 254α 2 + 597α + 436 . (7.38) When α goes to zero, no association of the vacancy-solute complex occurs and we have F 3 =1.Forω = ω 4 the escape probability is F 3 =0.7357. When α goes to infinity, we have F =2/7 (see Fig. 7.5). The correlation factor f 2 is a function of all vacancy-atom exchange rates. We mention several special cases: 1. For self-diffusion all jump rates are equal (when isotope effects are ne- glected): ω = ω 1 = ω 2 = ω 3 = ω 4 and we get f 2 =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 ω 2  ω 1 ,ω 3 , , the correlation factor tends towards unity. Frequent vacancy-solvent exchanges randomise the vacancy posi- tion before the next solute jump. Then, solute diffusion is practically uncorrelated. 3. If vacancy-solute exchanges occur much faster than vacancy-solvent ex- changes, i.e. for ω 2  ω 1 ,ω 3 , , 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. 120 7 Correlation in Solid-State Diffusion Fig. 7.5. Escape probabilities F 3 for the fcc, bcc, and diamond structure. For fcc and bcc lattices F 3 is displayed as function of ω 4 /ω. For the diamond structure F 3 is a function of ω 4 /ω 5 .AfterManning [15] 4. Dissociation jumps ω 3 are very unlikely for a tightly bound solute-vacancy pair. Then, we get from Eq. (7.37) f 2 ≈ ω 1 ω 2 + ω 1 . (7.39) In this case the vacancy-solute pair can migrate as an entity via ω 1 and ω 2 -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 bcc lattice is illustrated in Fig. 7.6. It distinguishes four jump rates: the solute- vacancy exchange rate ω 2 , the dissociation rate ω 3 , the association rate ω 4 , and the vacancy-solvent exchange rate ω. As in the case of the fcc lattice, the rates ω 3 and ω 4 are related via the detailed balancing relation Eq. (7.34). In contrast to the fcc lattice, the bcc structure has no lattice sites which are simultaneously 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 correlation factor for solute diffusion can be written as f 2 = 7ω 3 F 3 2ω 2 +7ω 3 F 3 , (7.40) where the escape probability F 3 (α)isgivenby 7F 3 (α)= 528.4 + 779.03α + 267.5α 2 +24α 3 75.50 + 146.83α +69.46α 2 +8α 3 (7.41) illustrated in Fig. 7.5. The ratio α = ω 4 /ω has the same meaning as for the fcc case. 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 and next-nearest solute-vacancy pairs and then also several additional dissociation and 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): ω 2 for exchange with the solute, ω 3 for vacancy jumps from first to second-nearest neighbours of the solute, ω 4 for the reverse jump of ω 3 , ω 5 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 the correlation factor f 2 = 3ω 3 F 3 2ω 2 +3ω 3 F 3 (7.42) 122 7 Correlation in Solid-State Diffusion Fig. 7.7. ‘Five frequency model’ for jump rates in the diamond structure. ω 2 : jump rate of vacancy-tracer exchange, ω 3 (ω 4 ): jump rate of vacancy jump from first (second) to second (first) nearest neighbour of solute atom, ω 5 :jumprateof vacancy jump from second to third nearest neighbour of tracer where the escape probability F 3 (α)= 2.76 + 4.93α +2.05α 2 2.76 + 6.33α +4.52α 2 + α 3 (7.43) is a function of the ratio α = ω 4 /ω 5 . For self-diffusion, all frequencies are identical, F 3 =2/3andf 2 agrees with the value 1/2 of Table 7.2. When α goes to zero, F 3 goes to unity. In the other limit where α goes to infinity, F 3 goes to zero (see Fig.7.5). Additional complexity can arise in semiconducting solvents, when solute and vacancy carry electrical charges. In contrast to metallic solvents, where the solute-vacancy interaction is short-ranged, the Coulomb interaction can modify the vacancy behaviour over larger distances in the surroundings of achargedsolute. 7.6 Concluding Remarks Nowadays, a number of methods for calculating correlation factors are avail- able: Compaan and Haven developed an analogue simulation method based on the similarity between Fick’s and Ohm’s law [20, 21]. 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Compaan, Y Haven, Trans Faraday Soc 52, 7 86 (19 56) K Compaan, Y Haven, Trans Faraday Soc 54, 14 98 (19 58) J.G Mullen, Phys Rev 12 4, 17 23 (19 61 ) R.E Howard, Phys Rev 14 4 65 0 (19 66 ) H Bakker, Phys Stat Sol 38, 16 7 (19 70) P Benoist, J.-L Bocquet, P Lafore, Acta Metall 25, 265 (19 77) M Koiwa, Philos Mag 36, 893 (19 77) A.D Le Claire, Correlation Effects in Diffusion in Solids, in: Physical Chemistry – an Advanced... -0.08 to -0.02 Bosman et al [13 ] Bosman et al [14 ] Al self-diffusion +1. 29 Beyeler and Adda [8] Ge in Al Zn in Al Mn in Al Co in Al +1. 16 to + 1. 24 +0.74 to +1. 09 +1. 67 +1. 64 to 1. 93 Th¨rer et al [17 ] u Erdelyi et al [18 ] Rummel et al [19 ] Rummel et al [19 ] Schottky pair formation: F VSP 1. 63 2.04 1. 23 1. 37 0. 21 0. 26 0.25 0.25 1. 03 1. 28 0.87 0.93 Yoon and Lazarus [22] Cation migration: M VVC Intrinsic... reviewed by Mishin [ 36] References 1 J Bardeen, C Herring, in: Atom Movements A.S.M Cleveland, 19 51, p 87 – 288 2 J Bardeen, C Herring, in: Imperfections in Nearly Perfect Solids, W Shockley (Ed.), Wiley, New York, 19 52, p 262 3 R Friauf, Phys Rev 10 5, 843 (19 57) 4 R Friauf, J Appl Phys Supp 33, 494 (19 62 ) 5 E.G Montroll, Symp Appl Math 16 , 19 3 (19 64 ) 6 I Stewart, Sci Am 280, 11 1 (19 99) 7 M Koiwa, J... +1. 09 +0 .66 to +0.88 Au self-diffusion +0.72 to +0.75 Beyeler and Adda [8] Beyeler and Adda [8], Rein and Mehrer [9] Dickerson et al [10 ], Beyeler and Adda [8] Werner and Mehrer [11 ], Rein and Mehrer [9] Mundy [12 ] Na self-diffusion +0.4 to +0.75 Ge diffusion in silicon -0 .68 to -0.28 S¨dervall et al [15 ], o Aziz et al [ 16 ] N in α-iron C in α-iron +0.05 -0.08 to -0.02 Bosman et al [13 ] Bosman et al [14 ]... Academic Press, New York and London, 19 70 H Mehrer, Korrelation bei der Diffusion in kubischen Metallen, Habilitation thesis, Universit¨t Stuttgart, 19 73 a J.R Manning, Diffusion Kinetics for Atoms in Crystals, van Norstrand Comp., Princeton 19 68 A.R Allnatt, A.B Lidiard, Atomic Transport in Solids, Cambridge University Press, 19 93 G.E Murch, Diffusion Kinetics in Solids, Ch.3 in: Phase Transformations... units kJ mol 1 K 1 or in eV per atom Note that 1 eV per atom = 96. 472 kJ mol 1 8 .1 Temperature Dependence 12 9 for T 1 ⇒ 0 yields the pre-exponential factor D0 It can usually be written as ∆S D0 = gf ν 0 a2 exp , (8.5) kB where ∆S is called the diffusion entropy, g is a geometrical factor, f is the correlation factor, ν 0 is the attempt frequency, and a some lattice parameter Combining Eqs (8 .1) and (8.5)... (entropies) of the diffusion-mediating F M defect For a monovacancy we have ∆H = H1V + H1V For solute diffusion in a dilute substitutional alloy the activation enthalpy is a more slightly complex quantity Combining Eqs (8.4), (8 .14 ), and (8 . 16 ) we get F M (8. 21) ∆H2 ⇒ HD − H B + H2 + C The correlation term C = −kB ∂ ln f2 ∂ (1/ T ) (8.22) arises from the temperature dependence of the correlation factor... exp − ∆H kB T (8 .1) In Eq (8 .1) D0 denotes the pre-exponential factor also called the frequency factor, ∆H the activation enthalpy of diffusion1 , T the absolute temperature, 1 In the literature the symbol Q is also used instead of ∆H 12 8 8 Dependence of Diffusion on Temperature and Pressure Fig 8 .1 Arrhenius plot of diffusion for various elements in Pb; activation parameters from [1] and kB the Boltzmann . Rev. 13 2, 63 5 (19 63 ) 11 . G.E. Murch, Solid State Ionics, 7, 17 7 (19 82) References 12 5 12 . R.E. Howard, J.R. Manning, Phys. Rev. 15 4, 5 61 (19 67 ) 13 . A.B. 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