Diffusion in multicomponent solids. Anton Van der Ven Department of Materials Science and Engineering University of Michigan Ann Arbor, MI

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1 Diffusion in multicomponent solids nton Van der Ven Department of Materials Science and Engineering University of Michigan nn rbor, MI

2 Coarse graining time Diffusion in a crystal Two levels of time coarse graining

3 Δ Γ = ν * exp kt Coarse graining time Diffusion in a crystal Two levels of time coarse graining Short-time coarse graining: transition state theory E - MD simulations - Harmonic approximation Vineyard, J. Phys. Chem. Solids 3, 121 (1957).

4 Coarse graining time Diffusion in a crystal Two levels of time coarse graining Short-time coarse graining: transition state theory Δ Γ = ν * exp kt E - MD simulations - Harmonic approximation second level of coarse graining that leads to Fick s law J = D C Green-Kubo Kinetic coefficients derived from fluctuations at equilibrium Vineyard, J. Phys. Chem. Solids 3, 121 (1957). Zwanzig, nnu. Rev. Phys. Chem. 16, 67 (1965).

5 Interstitial diffusion C diffusion in bcc Iron (steel) Li diffusion in transition metal oxide host O diffusion on Pt-(111) surface In all examples, diffusion occurs on a rigid lattice which is externally imposed by a host or substrate

6 Example of interstitial diffusion

7 Irreversible thermodynamics: interstitial diffusion of one component J = L µ D = L dµ dc J = D C

8 Notation M = number of lattice sites N = number of diffusing atoms v s = volume per lattice site x = N/M C=x/v s

9 Interstitial diffusion: one component Kubo-Green relations (linear response statistical mechanics) D = L Θ Thermodynamic factor Kinetic coefficient L = Θ = 1 (2d)tMv s kt µ C N i=1 ( ) Δ r R i t 2 R. Gomer, Rep. Prog. Phys. 53, 917 (1990)/. Van der Ven, G. Ceder, Handbook of Materials Modeling, chapt. 1.17, Ed. S. Yip, Springer (2005).

10 Trajectories Δ r R j t ( ) Δ r R i t ( ) D J = 1 2d ( )t 1 N N i=1 ( ) Δ r R i t 2

11 More familiar form D = D J ~ Θ Thermodynamic factor ~ Θ = µ kt ln x Self diffusion coefficient D J = 1 2d ( )t 1 N N i=1 ( ) Δ r R i t 2 R. Gomer, Rep. Prog. Phys. 53, 917 (1990)/

12 Common approximation D Thermodynamic factor = ~ D * Θ ~ Θ = µ kt ln x Tracer diffusion coefficient D* = ( ΔR i ( t) ) 2 ( 2d)t R. Gomer, Rep. Prog. Phys. 53, 917 (1990)/

13 ~ D = D J Θ Diffusion coefficient at 300 K Thermodynamic factor Θ. Van der Ven, G. Ceder, M. sta, P.D. Tepesch, Phys Rev. 64 (2001)

14 Interstitial diffusion (two components) C & N diffusion in bcc Iron (steel) Li & Na diffusion in transition metal oxide host O & S diffusion on Pt-(111) surface In all examples, diffusion occurs on a rigid lattice which is externally imposed by a host or substrate

15 Diffusion of two species on a lattice J = L µ L µ J = L µ L µ D = L Θ D D D D = L L L L µ C µ C µ C µ C J = D C D C J = D C D C

16 lternative factorization = x kt x kt x kt x kt L L L L D D D D µ µ µ µ ~ ~ ~ ~ ( ) ( ) ( )tm d t R t R L j i ij 2 ~ Δ Δ = ξ ξ ς ς r r Kubo-Green. Van der Ven, G. Ceder, Handbook of Materials Modeling, chapt. 1.17, Ed. S. Yip, Springer (2005)..R. llnatt,.. Lidiard, tomic Transport in Solids (Cambridge Univ. Press, 1993).

17 Kinetic coefficients (fcc lattice in dilute vacancy limit, ideal solution) ( ) ( ) ( )tm d t R t R L j i ij 2 ~ Δ Δ = ξ ξ ς ς r r

18 Diffusion in an alloy: substitutional diffusion Not interstitial diffusion Instead, diffusing atoms form the lattice Dilute concentration of vacancies

19 Thermodynamic driving forces for substitutional diffusion J = L ~ µ L ~ µ J = L ~ µ L ~ µ ~ µ ~ µ = µ µ V = µ µ V. Van der Ven, G. Ceder, Handbook of Materials Modeling, chapt. 1.17, Ed. S. Yip, Springer (2005).

20 Textbook treatment of substitional diffusion Not Rigorous J = L ~ µ L ~ µ J = L ~ µ L ~ µ

21 Textbook treatment of substitional diffusion µ V Traditional = 0 dµ V Not Rigorous = 0 Gibbs-Duhem x dµ x dµ = 0 J = L ~ µ L ~ µ J = L ~ µ L ~ µ ssume vacancy concentration in equilibrium everywhere + J J = D = D C C

22 Textbook treatment of substitional diffusion µ V J J Traditional = 0 = D = D dµ V C C Not Rigorous = 0 Gibbs-Duhem x dµ x dµ = 0 J = L ~ µ L ~ µ J = L ~ µ L ~ µ + ssume vacancy concentration in equilibrium everywhere J J = D = D Rigorous C C D D C C

23 Lattice frame and laboratory frame of reference lattice m V m ( J J ) v = V J = V + Fluxes in the laboratory frame ~ J = J + xjv ~ J = J + x J V

24 Lattice frame and laboratory frame of reference J + x J J V = W ~ ~ = D C V C lattice m V m ( J J ) v = V J = V + ~ W Drift = D D Fluxes in the laboratory frame ~ J = J + xjv ~ J = J + x J V Interdiffusion ~ D = x D + x D

25 Rigorous treatment J J = D = D C C D D C C

26 Rigorous treatment J J = D = D C C D D C C Diagonalize the D-matrix Yields a mode corresponding to (a) density relaxation (b) interdiffusion D D D D = E λ E λ K. W. Kehr, et al, Phys. Rev. 39, 4891 (1989)

27 Physical meaning of modes λ+ and λ-

28 Physical meaning of modes λ + and λ - Density fluctuations relax with a time constant of λ + K. W. Kehr, et al, Phys. Rev. 39, 4891 (1989)

29 Physical meaning of modes λ + and λ - Density fluctuations relax with a time constant of λ + Compositional inhomogeneities decay with a time constant of λ K. W. Kehr, et al, Phys. Rev. 39, 4891 (1989)

30 Comparisons of different treatments Random alloy Traditional and rigorous treatment are equivalent only when Γ =Γ Γ =10xΓ Γ =100xΓ

31 Example: Li x CoO 2

32 Intercalation Oxide as Cathode in Rechargeable Lithium attery LixCoO2

33 First-Principles (Density Functional Theory) Cluster Expansions Fit V α, V β, V γ, to first-principles energies ( ) F σ = V + V σ + V σ σ + V σ σ σ +... o α i β j k γ l m n i j, k l, m, n Monte Carlo Z = exp σ F ( σ ) k T

34

35 First principles energies (LD) of different lithium-vacancy configurations. Van der Ven, et al, Phys. Rev. 58 (6), p (1998).

36 Cluster expansion for Li x CoO 2 ( ),,, E σ = V + V σ + V σ σ + V σ σ σ +... o i i i j i j i j k i j k i i, j i, j, k. Van der Ven, et al, Phys. Rev. 58 (6), p (1998).

37 Calculated Li x CoO 2 phase diagram. Van der Ven, et al, Phys. Rev. 58 (6), p (1998).

38 Calculated phase diagram Experimental phase diagram Reimers, Dahn, J.Electrochem. Soc, (1992) Ohzuku, Ueda, J. Electrochem. Soc. (1994) matucci et al, J. Electrochem. Soc. (1996)

39 Predicted phases confirmed experimentally Confirmed experimentally with TEM Y. Shao-Horn, S. Levasseur, F. Weill, C. Delmas, J. Electrochem. Soc. 150 (2003), 366

40 Predicted phases confirmed experimentally Confirmed experimentally by Z. Chen, Z. Lu, J.R. Dahn J. Electrochem. Soc. 149, 1604 (2002) Confirmed experimentally with TEM Y. Shao-Horn, S. Levasseur, F. Weill, C. Delmas, J. Electrochem. Soc. 150 (2003), 366

41 Calculated phase diagram? Experimental phase diagram Reimers, Dahn, J.Electrochem. Soc, (1992) Ohzuku, Ueda, J. Electrochem. Soc. (1994) matucci et al, J. Electrochem. Soc. (1996) M. Menetrier et al J. Mater Chem. 9, 1135 (1999)

42 Effect of metal insulator transition Holes in the valence band localize in space C.. Marianetti et al, Nature Materials, 3, 627 (2004). LD & GG fails to accurately describe localized electronic states

43 Diffusion Fick s Law J = D C

44 Interstitial diffusion and configurational disorder Kubo-Green relations D = D J ~ Θ Thermodynamic factor ~ Θ = µ kt ln x Self diffusion coefficient D J = 1 2d ( )t 1 N N i=1 ( ) Δ r R i t 2

45 Individual hops: Transition state theory ΔE Γ = ν * exp Δ kt E

46 Kinetically resolved activation barrier ΔE kra = E activated state 1 ( 2 E 1 + E ) 2 ΔE barrier = ΔE kra + 1 ( 2 E final E initial ). Van der Ven, G. Ceder, M. sta, P.D. Tepesch, Phys Rev. 64 (2001)

47 Migration mechanism in Li x CoO 2 Lithium Cobalt Oxygen Single vacancy hop mechanism Divacancy hop Mechanism

48 Lithium Single vacancy hop Divacancy hop Cobalt Oxygen

49 Many types of hop possibilities in the lithium plane

50 Migration barriers depend configuration and concentration Single-vacancy mechanism Divacancy mechanism

51 Local Cluster expansion for divacancy migration barrier

52 Calculated diffusion coefficient (First Principles cluster expansion + kinetic Monte Carlo) D = Θ D J Diffusion coefficient at 300 K Thermodynamic factor Θ. Van der Ven, G. Ceder, M. sta, P.D. Tepesch, Phys Rev. 64 (2001)

53 vailable migration mechanisms for each lithium ion Channels into a divacancy Number of vacancies around lithium Channels into isolated vacancies

54 Diffusion occurs with a divacancy mechanism

55 Diffusion and phase transformations in l-li alloys Dark field TEM. Kalogeridis, J. Pesieka, E. Nembach, cta Mater 47 (1999) 1953 Dark field in situ TEM, peak aged l-li specimen under full load H. Rosner, W. Liu, E. Nembach, Phil Mag, 79 (1999) 2935

56 fcc l-li alloy inary cluster expansion σ i σ i = +1 = 1 Li at site i l at site i Fit to LD energies of 70 different l-li arrangements on fcc ( ),,, E σ = V + V σ + V σ σ + V σ σ σ +... o i i i j i j i j k i j k i i, j i, j, k. Van der Ven, G. Ceder, Phys. Rev. 71, (2005)

57 Calculated thermodynamic and kinetic properties of l-li alloy First principles cluster expansion + Monte Carlo. Van der Ven, G. Ceder, Phys. Rev. 71, (2005)

58 Expand environment dependence of vacancy formation energy Local cluster expansion* (perturbation to binary cluster expansion) Fit to 23 vacancy LD formation energies in different l-li arrangements (107 atom supercells). σ i σ i = +1 = 1 Li at site i l at site i. Van der Ven, G. Ceder, Phys. Rev. 71, (2005)

59 Equilibrium vacancy concentration (Monte Carlo applied to cluster expansion). Van der Ven, G. Ceder, Phys. Rev. 71, (2005)

60 Vacancy surrounds itself by l Short range order around a vacancy 750 Kelvin. Van der Ven, G. Ceder, Phys. Rev. 71, (2005)

61 Vacancies reside on lithium sublattice in L Kelvin Li l

62 Migration barriers for lithium and aluminum differ by ~150 mev l barriers Li barriers Calculated (LD) in 107 atom supercells v * l Hz v * Li Hz

63 Calculated interdiffusion coefficient Li l. Van der Ven, G. Ceder, Phys. Rev. Lett. 94, (2005).

64 Hop mechanisms

65 Frequency of hop angles between successive hops. Van der Ven, G. Ceder, Phys. Rev. Lett. 94, (2005).

66 Conclusion Green-Kubo formalism yields rigorous expressions for diffusion coefficients Discussed diffusion formalism for both interstitial and substitutional diffusion Intriguing hop mechanisms in multicomponent solids that can depend on ordering Thermodynamics plays a crucial role!