Phenomenological Coefficients in Solid State Diffusion (an introduction)

Transcription

1 Phenomenologial Coeffiients in Solid State iffusion (an introdution) Graeme E Murh and Irina V elova iffusion in Solids Group Shool of Engineering The University of Newastle Callaghan New South Wales ustralia Collaborators: idiard (Reading and Oxford), R llnatt (UWO), K Chaturvedi (Kurukshetra), M Martin (RWTH ahen). Researh supported by the ustralian Researh Counil G day!

2 Talk Outline: 1. Fik s First aw and the Onsager Flux Equations.. The meaning of the phenomenologial oeffiients. 3. llnatt s Equation for the phenomenologial oeffiients and the Einstein Equation. 4. Correlation effets in phenomenologial oeffiients and in traer diffusion oeffiients. 5. How to make use of phenomenologial oeffiients: The arken and Manning approahes. The Sum-Rule. 6. Some appliations. 7. Conlusions.

3 Fik s First aw (1855): J i i dc dx i eause it does not reognize all of the diret and indiret driving fores ating on speies i, Fik s First aw is frequently insuffiient as a ondition for desribing the flux. The atual driving fore for diffusion is not the onentration gradient but the hemial potential gradient.

4 The Onsager (1934) Flux Equations of irreversible proesses provide the general formalism through the postulate of linear relations between the fluxes and the driving fores: J i j ij X j ij ji (reiproity ondition) ij : the phenomenologial oeffiients (independent of driving fore) X j : the driving fores

5 Consider a binary system. The Onsager Flux Equations are: J X + X J X + X Consider a hypothetial situation where is harged and is not, and the system is plaed in an eletri field E. The driving fores are then: X -q E and X 0 The fluxes are then: J - q E and J - q E The atoms respond only to the diret fore q E. The atoms only respond to the indiret fore q E and are then dragged along by the atoms.

6 What are these phenomenologial oeffiients? ij < R R i j (llnatt 198) 6VkTt > R i : the olletive displaement or displaement of the enter-of-mass of speies i in time t. E.g. < R > 6VkTt < R R 6VkTt > If the moving speies does not interfere with the moving speies e.g. and do not ompete for the same defets or and do not interat (i.e. different sublatties) <R R > 0 and 0. However, in most ases in solid-state diffusion the off-diagonal oeffiients an be signifiant. They an be positive or negative.

7 ij < R i R j 6VkTt > llnatt s (198) equation for the ij is a generalization of the Einstein (1905) equation for the traer or self-diffusion oeffiient: < r > * 6t r displaement of a traer atom in time t The Einstein Equation is frequently used in Moleular ynamis simulations, see Poster 31: Zhao et al., Poster 37: eroy et al., Poster 38: eroy et al., Poster 54: Plant et al., Poster 4: Chihara et al., Poster 39: Habasaki et al.

8 The relationship between the Einstein Equation and the llnatt Equation an be appreiated if we onsider a binary system of * and in whih we allow the traer * onentration to be very low. Then we would have that: * * * VkT

9 In solid-state diffusion, the motion of the atoms normally takes plae by disrete jumps or hops (often alled the hopping model ). I an hop too! efets suh as vaanies provide the vehiles for atom motion. The hopping model is frequently used diretly or indiretly in the modelling of solid state diffusion, see Poster 18: Maas et al., Poster 8: Sholl, Poster 41: Kalnin et al., Poster 49: Radhenko et al.

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11 In solid-state diffusion, the motion of the atoms normally takes plae by disrete jumps or hops (often alled the hopping model ). I an hop too! It is usual then to partition diffusion oeffiients suh as the traer diffusion oeffiient in the following way: j * f j (Z v w j a ) orrelated part unorrelated part Z: oordination number v : vaany onentration w j : exhange frequeny of an atom of type j with a vaany a: jump distane f j : traer orrelation fator of atoms of type j. It is an expression of the orrelation between the diretions of the suessive jumps of a given atom of type j.

12 The traer orrelation fator an be expressed in terms of the osine of the angle between the first jump and all subsequent jumps of a given atom (the traer): 1.0 f 1+ m 1 < osθ (m) > 0.5 m Cos(θ ) Example of the onvergene of the osine between the first traer jump and the m th traer jump m

13 Phenomenologial oeffiients an be partitioned in a similar way to diffusion oeffiients: ij f (j) ij (Z v w j a N j / 6 V k T) orrelated part unorrelated part f ij (j) : olletive orrelation fator. It is an expression of the orrelation between the diretions of suessive jumps of the enters-of-mass of the speies present.

14 The olletive orrelation fators an be expressed in terms of the osine of the angle between the first jump and all subsequent jumps of the same speies (diagonal fator) or another speies (off-diagonal fator) iagonal olletive orrelation fator: f ii 1 + m 1 < osθ (m) ii > Off-diagonal olletive orrelation fator (binary ase only): f () m 1 < osθ (m) > + C C n n m 1 < osθ (m) >

15 m ) Cos(θ m Example of the onvergene of the osine between the first olletive jump and the m th olletive jump (of the same speies ).

16 The phenomenologial oeffiients are extremely diffiult to measure diretly in the solid state. If we want them, how do we proeed? First Strategy: Find relations between the phenomenologial oeffiients and the (measurable) traer diffusion oeffiients:

17 Example 1: The arken Relations (1948) : C * / kt ii i i ij 0

18 Example : The Manning Relations (1971) for the random alloy: ii C i kt * i 1 + M C 0 k i C * i k * k ij 0 * i C C i kt(m k j C k * j * k ) The Manning Relations an also be derived on the basis of two intuitive assumptions without reourse to the random alloy model (idiard 1986). They have also been derived for binary ordered strutures (elova and Murh 1997)

19 Seond Strategy: Find relations between the phenomenologial oeffiients themselves in order to redue their number. Sum-Rules

20 Possible vaany jumps after time t Initial vaany-atom jump Vetor summation of 0 Shemati illustration for the origin of the Sum-Rule.

21 The Sum-Rule for the phenomenologial oeffiients in a multiomponent random system is (Moleko and llnatt 1986): M w / w i 1 ij j i w j j V For the binary system, the Sum-Rule is: N V w kt a e.g. For the random binary system : w w, N V w kt a w w In the binary system there is then only one independent phenomenologial oeffiient, not three. (In the ternary random system there are three independent phenomenologial oeffiients, not six.)

22 nalogous Sum-Rule expressions have sine been derived for: iffusion via divaanies in the random alloy (elova and Murh 005). iffusion via dumb-bell interstitials in the random alloy (Sharma, Chaturvedi, elova and Murh 000). iffusion via vaany-pairs in strongly ioni ompounds (elova and Murh 004). iffusion via vaanies in substitutional intermetallis (elova and Murh 001, llnatt, elova and Murh 005). iffusion via vaanies in the five-frequeny impurity diffusion model (elova and Murh 005).

23 ppliation of the Onsager flux equations and the Sum-Rule (binary alloy): The Onsager flux equations are: J X + J X + X X X i μ x i kt C i C x i 1 + ln γ ln The intrinsi diffusion oeffiients (found from the Kirkendall shift and the interdiffusion oeffiient) are: I kt φ C C I Ci Ji i x I kt C C φ oal lattie referene frame φ is the thermodynami fator : ln γ φ 1+ ln

24 The ratio of the intrinsi diffusion oeffiients is: I I ppliation of the Sum-Rule then gives (elova and Murh 1997): I I w w Measurement of the ratio of the intrinsi diffusivities diretly thus gives the ratio of the exhange frequenies. There are no ompliating orrelation fators.

25 If we had simply used the arken relations (where all off-diagonal phenomenologial oeffiients are put equal to zero) we would then have obtained: I I * * (This is a very rough approximation)

26 The interdiffusion oeffiient: I ~ + I J 0 C x 0 J ab. referene ~ frame fter appliation of the llnatt Equation for the ij : ij < R i R j 6VkTt > (R R) ~ < φ elova and Murh Nt > This general equation for the interdiffusion oeffiient in a binary alloy is, in effet, an extension of the Einstein Equation (1905).

27 With diret aess to the ratio of the atom-vaany exhange frequenies, one an also use a diffusion kinetis theory to gain aess to the traer orrelation fators: The g-cd system: a) The ratio I g /I Cd ( w g /w Cd ) as a funtion of Cd at 873K. (Iorio et al. 1973); b) Corresponding traer orrelation fators using the Moleko et al. (1986) Self onsistent diffusion kinetis formalism. Cd is more orrelated (more jump reversals) in its motion than g.

28 emixing of and ations in (,)O in an oxygen potential gradient (gives a gradient of ation vaanies): P O J V Initially P O Cation Vaanies P O < P O v P O J +J P O J fter V Segregation w > w ( is blue and is pink) Veloity v ~ ms -1

29 (+) P O Initially emixing of and ations in (,)O in an eletri field. ( ) (pplied Eletri Field) Cation Vaanies P O w > w J ( is blue and is pink) J (+) ( ) P O J V fter Segregation P O Veloity v

30 nalysis of demixing of and ations in a mixed oxide (,)O in an oxygen potential gradient and eletri field: Onsager Flux Equations: J ( μ μ q E) ( μ μ V V q E), J ( μ μ q E) ( μ μ V V q E), μ i : hemial potential of omponent i (, or V (vaanies) E: Eletri field q i : harge on omponent i Steady-State Condition: J i v i N 0, i, ( μ ( μ μ μ V V q q E) E) 1 vn vn

31 ppliation of the Sum-Rule gives simply that: vnkt ( μ μ V ) + qe w a V vnkt ( μ μ V ) + q E w a V We now assume random mixing of the two ations. The demixed steady-state omposition profile of, say,, is given by: d dξ V vn a 1 w + w + (q kt q )E

32 Steady-state demixed profile of Co in (Co,Mg)O in an oxygen potential gradient (Experimental data: Shmalzried et al. 1979). Co omposition w Co /w Mg 5 V / V Experimental ata nalytial Result ξ (redued distane) Initial omposition: Co 0.51 The fitting parameter is w Co /w Mg.

33 Steady-state demixed profile of Co in (Co,Ni)O in an eletri field (Experimental data: Martin 000) Co omposition qea/kt0.676 w Co /w Ni ξ (redued distane) Initial omposition: Co 0.93 The fitting parameter is w Co /w Ni. (Voltage 50 mv, sample thikness 600 μm) (qea/kt 0.676)

34 nalysis of interdiffusion in a strongly ioni diffusion ouple Z-Z: Z Z The ations and diffuse via vaanies on the ation sublattie. The anions Z diffuse via vaanies on the anion sublattie. The Onsager Flux Equations are (we assume that q q -qz): J J J Z ZZ X X X Z. + + X X where for the internal fores we have: X X X Z μ μ q Z E + q + q E E

35 ppliation of the eletro-neutrality onditions and the Gibbs-uhem relation gives for the intrinsi diffusion oeffiients (e.g. elova and Murh 004): J where I I N J N I φkt N Z + + ZZ ( + + ZZ )/ Z ) and I φkt N Z + + ZZ ( + + ZZ )/ Z ) φ: thermodynami fator

36 ppliation of the Sum-Rule to the ratio of the intrinsi diffusion oeffiients gives: I I w w (w (w + + w w Z Z ) ) where w Z is the anion vaany exhange frequeny. 1. For the limiting ase w Z >> w (w ), (anion mobility is relatively high): I /I w /w. (this is the same result as for the binary alloy) The mobility on the anion sublattie no longer determines the rate of ation interdiffusion).. For the limiting ase w Z << w (w ), (anion mobility is relatively low): I I. The almost immobile anion sublattie requires that the fluxes of the ations and are equal and opposite. no net ation vaany flux and no marker shift in interdiffusion.

37 The interdiffusion oeffiient: The general expression for ouple (Z-Z): ~ in a strongly ioni interdiffusion ~ (q + q )q Z ktφn ( + + ) I I ( ) ( d d ) Z Z κ ZZ where d I d I κ q + q + q q + q Z ZZ

38 Using llnatt s (198) equation for the phenomenologial oeffiients: 6VkTt R R j i ij > < ( )( ) ( ) > > + < + < > < φ Z Z ZZ R R R R p R R R kt ~ i ZZ i ) ( ii Z i p + 0 There are two limiting ases to onsider:

39 I. When the anions are muh more mobile than the ations (elova and Murh 005) : ( R R ) < ~ 6Nt Z > φ II. When the ations are muh more mobile than the anions (elova and Murh 005) : ~ φn Z a Va < ( R R)( wr wr) t < ( R + R ) > > These equations for the interdiffusion oeffiients in a strongly ioni ompound are, in effet, extensions of the Einstein Equation (1905).

40 We onsider further the ase where the anions are muh slower than the ations, e.g. in siliates, glasses, transition metal oxides. We apply the Sum-Rule and make use of the aurate self onsistent diffusion kinetis theory of Moleko et al. (1989): kt ( ) f ~ φ + 0 N (1 f0 ) elova and Murh 004: f 0 : the geometri traer orrelation fator (depends on lattie only) If were to be negleted (this is the arken approximation) it would be equivalent to implying that interdiffusion is impossible.

41 iret aess is now possible to in transition metal oxides, oxides, in siliates, in glasses et, i.e. whenever the anion mobility is low ompared with the ations. Example: Extration of from the interdiffusion oeffiient in (CoO-NiO). Expt. : J.J. Stiglih, Jr. et al. (1973).

42 Some other results that an be obtained at the same approximation level for strongly ioni ompounds when the ations are muh more mobile than the anions: ~ * * + * * φ (This is the Nernst-Plank Equation) and ~ * f 0 σ * d q C kt ion φ The testing of these equations in, say, a siliate, would require measurements of the traer diffusion oeffiients, the interdiffusion oeffiient, the ioni ondutivity and the thermodynami fator.

43 Conlusions: The Onsager flux expressions and llnatt s equation for the phenomenologial oeffiients an rightly be onsidered generalizations of Fik s First aw and the Einstein Equation The Onsager flux expressions and llnatt s equation, together with the Sum-rule, bring substantial simplifiations to many hemial diffusion problems. Vielen ank!