Heitjans P., Karger J. (Eds.). Diffusion in Condensed Matter: Methods, Materials, Models

Подождите немного. Документ загружается.

782 Klaus W. Kehr et al.

0.2

0.4

0.6

0 0.5 1

coll

Fig. 18.22. Collective diﬀusion coeﬃcient in the RT model in d =3withexpo-

nential distribution of site energies. Continuous lines: EMA; symbols: MC results

for α =1.43 (), 1.0(+),0.5(2), and 0.333 (×). Limit c → 1(*).

Fig. 18.22 shows numerical results for the RT model in d = 3 together

with EMA results. It is evident that D

coll

(c → 0) −→ D

s.p.

,alsoforα<1,

where D

s.p.

= 0. There is qualitative agreement between the simulations and

the EMA, particularly at lower particle concentrations.

The behaviour of D

coll

for c  1 can be understood from the following

qualitative consideration: Since only one particle can occupy a given site, the

equilibrium occupancy of a site is given by the Fermi-Dirac distribution. In

the limit of low temperatures, all levels are occupied up to a pseudo Fermi

level E

∗

(see Fig. 18.23). For the exponential distribution



∗

−∞

exp





= c, (18.114)

from which follows

∗

= E

ln(c) . (18.115)

At small concentrations and low T , diﬀusion is mainly carried by parti-

cles at or above the Fermi level, and the coeﬃcient of collective diﬀusion is

approximately given by the single-particle diﬀusion expression,

−1

coll

≈



∗

dEν

(E)

Γ (E)

. (18.116)

The result for small c is (note that α<1)

coll

≈ Γ



− 1



(

−1)

. (18.117)

That is, D

coll

exhibits a power-law dependence on the concentration.

18 Diﬀusion of Particles on Lattices 783

(E)

-E

Fig. 18.23. Exponential density of site

energies; the levels are ﬁlled up to the

“Fermi” level E

∗

Power-law dependence of the conductivity on the concentration has been

observed in ionic conductors, see, e. g., [95]. Of course, the actual systems

are much more complex than described by the site-exclusion lattice gas in

the RT model. Indeed, power-law dependence of the mobility was obtained

in numerical simulations of models that were designed to describe superionic

conductors [96, 97]. Nonetheless, it is interesting to note that already rather

simple models exhibit these eﬀects.

18.6 Conclusion

This chapter was concerned with the theoretical description of diﬀusion of

particles on lattices. Two diﬀerent dimensions of diﬃculties were encountered:

Diﬀusion of single versus many particles and diﬀusion in ordered versus dis-

ordered lattices. The diﬀusion of a single particle on a uniform lattice is a

well-understood textbook problem. Of practical interest are extensions of this

problem, for example two-state models for diﬀusion and trapping of single

particles. The problem of diﬀusion of a single particle in disordered lattices is

now well understood with respect to the asymptotic properties, if local (site

and/or bond) disorder is present. One remaining topic for future work is dif-

fusion in the presence of extended defects. Also the frequency dependence

seems to be not completely understood.

Various aspects of the diﬀusion of many particles on uniform lattices have

been clariﬁed in the past, in particular the diﬀerences between collective and

tracer diﬀusion, the inherent correlation eﬀects in tracer diﬀusion, and the

anomalous diﬀusion of tagged particles on linear chains (single-ﬁle diﬀusion).

Open problems can be found in the consideration of various forms of the

interactions of the particles.

784 Klaus W. Kehr et al.

Least well understood is, of course, the problem of diﬀusion of many par-

ticles in lattices with quenched, disordered transition rates. Since this is a

problem of current interest, the corresponding remarks will be more detailed

than those concerning the previous sections. In the preceding section diﬀu-

sion of site-exclusion lattice gases in disordered lattices was considered. There

exist now reasonable appoximate descriptions for the coeﬃcient of collective

diﬀusion in form of the phenomenological theory and the EMA. The approxi-

mations are rather accurate for two-level models of site energies if the lattice-

gas concentration c is smaller than the trap concentration c

. However, one

should be aware of the limitations of the approximations. Simulation results

in d = 3 show that there are large discrepancies between the approximations

and the numerical results when c ≈ c

. Hence an exact theory for collective

diﬀusion for the site-exclusion model in disordered linear chains is desired.

The following important aspects of diﬀusion of lattice gases in disordered

lattices were omitted in this chapter:

– Tracer diﬀusion

The theory of tracer diﬀusion is more complicated than that of collective

diﬀusion, in view of the inherent backward correlations for tracer diﬀu-

sion. The inclusion of backward correlations in the presence of quenched

disorder appears to be very diﬃcult.

– Eﬀects of interactions

Even for lattices with uniform transition rates, interactions can only be

treated by approximations like mean ﬁeld. If disorder is present, and the

particles have interactions, one has to resort to numerical simulations,

see [57–59, 96, 97] and Chap. 20.

– Frequency dependence

Although frequency dependence was investigated in numerical works (see

the references given in the preceeding item), many of the observed features

are not yet understood in a qualitative way (see however Chap. 21).

One could easily add more points to this list of partially understood problems.

The main message perhaps should be that the ﬁeld of diﬀusion of particles in

condensed matter systems is still an open ﬁeld with many unsolved problems.

18.7 Appendix

18.7.1 Derivation of the Result for the Diﬀusion Coeﬃcient for

Arbitrarily Disordered Transition Rates

Consider a segment of a linear chain with general (nonsymmetric) transition

rates, as indicated in Fig. 18.3 of Sect. 18.3.2. A constant current I is fed into

site 0 and extracted at site N. Kirchhoﬀ’s node equation for site 0 is

− Γ

= I. (18.118)

18 Diﬀusion of Particles on Lattices 785

It expresses the fact that the sum of all currents into site 0 is zero. The master

equation for site 1 is

(t)=Γ

(t)+Γ

(t) − (Γ

+ Γ

(t) . (18.119)

One has in the stationary situation

−Γ

+(Γ

+ Γ

− Γ

=0. (18.120)

Add this equation to the node equation (18.118) to obtain

− Γ

= I. (18.121)

This procedure is continued and the equation for site i is

i+1,i

− Γ

i,i+1

i+1

= I. (18.122)

The last equation is Kirchhoﬀ’s node equation at site N ,

N,N−1

N−1

− Γ

N−1,N

= I. (18.123)

The aim is to relate I to the diﬀerence P

−P

. This can be achieved by

ﬁrst expressing P

N−1

through P

, I, using (18.123),

N−1

N−1,N

N,N−1

. (18.124)

The quantity P

N−2

can be expressed through P

N−1

and I,

N−2

N−2,N−1

N−1,N−2

N−1

N−1,N−2

. (18.125)

Insert P

N−1

to obtain a relation between P

N−2

, P

,andI,

N−2

N−2,N−1

N−1,N−2

N−1,N

N,N−1



N−2,N−1

N−1,N−2

N,N−1

N−1,N−2



I. (18.126)

Continue this procedure with P

N−3

,etc.Theﬁnalresultis

×···×Γ

N−1,N

×···×Γ

N,N−1



×···×Γ

N−2,N−1

×···×Γ

N−1,N−2

N,N−1

×···×Γ

N−3,N−2

×···×Γ

N−2,N−3

N−1,N−2

+ ...



I. (18.127)

786 Klaus W. Kehr et al.

The expression (18.127) can be further simpliﬁed by introducing the re-

lation of detailed balance which holds in thermal equilibrium. It is clear that

a non-equilibrium situation is maintained in the segment of the chain. The

relation of detailed balance will be used to express the ratio of two rates by

equilibrium occupation factors. An implicit assumption is the validity of the

Arrhenius law. The equilibrium occupation probabilities are proportional to

∼ exp(−βE

) ; (18.128)

note that E

≤ 0. The condition of detailed balance is

= Γ

(18.129)

=exp[− β(E

− E

)] . (18.130)

The product of two factors is

=exp[−β(E

−E

)] exp[−β(E

−E

)] = exp[−β(E

−E

)] (18.131)

and the product

N−1

i=0

i,i+1

i+1,i

=exp[−β(E

− E

)] . (18.132)

Equation (18.127) can then be written as

− exp[−β(E

− E

)]P



exp[−β(E

N−1

− E

)]Γ

N,N−1

exp[−β(E

N−2

− E

)]Γ

N−1,N−2

+···+

exp[−β(E

− E

)]Γ



I. (18.133)

It will be required that P

= P

if I =0,andP

= P

only through

maintaining a current through the chain. This requirement will be satisﬁed if

= E

, i. e., if the initial and ﬁnal site energies are equal. It is henceforth

assumed that always E

= E

, which is a reasonable assumption.

The disorder average of (18.133) will now be taken. Each term on the

right-hand side of this equation can be averaged independently and the per-

tinent quantities are taken from common distributions. The result is then

− P

= N



i+1,i

exp[−β(E

− E

)]



I, (18.134)

where the dummy indices have been kept to indicate the direction of the

transition rate. The curly brackets represent the average over the diﬀerent

realizations of the disorder.

18 Diﬀusion of Particles on Lattices 787

The further requirement will be made that the initial site is a typical site

with respect to the disorder average. Hence it is assumed that

{exp(−βE)} =exp(−βE

) . (18.135)

This means that the thermal occupation of the initial (and ﬁnal) site is equal

to the average occupation. Large segments will be considered, where the

disorder average is approximated by the sum over all sites,

{exp(−βE)} =



j=0

exp(−βE

) . (18.136)

Thermal occupation factors will be deﬁned by

exp(−βE

)

{exp(−βE)}

exp(−βE

)



N−1

j=0

exp(−βE

)

. (18.137)

Relation (18.134) between P

−P

and the current I canthenbewrittenas

− P

= IN



i+1,i



. (18.138)

This relation represents Fick’s ﬁrst law. The following identiﬁcation can be

made (lattice constant a =1)

− P

∼

−∇nd=1. (18.139)

Fick’s ﬁrst law reads

I = −D∇n. (18.140)

Hence one ﬁnds

D =



i+1,i



−1

. (18.141)

18.7.2 Derivation of the Self-Consistency Condition for the

Eﬀective-Medium Approximation

The Fourier-Laplace transformation of the conditional probability P

(t)is

deﬁned by

P (k, s)=



∞

dt exp(−st)



exp(−ika)P

(t) . (18.142)

The equation for

P (k, s) is obtained by multiplying (18.51) and (18.52) with

exp(−ikal), summing over l, and integrating:

788 Klaus W. Kehr et al.

[ s +2

eﬀ

(s)(1 − cos ka)]

P (ks,)

=1+(e

−ika

− 1)[

eﬀ

(s) − Γ

][

(s) −

(s)] . (18.143)

Combining this equation with the master equation (18.56) for the eﬀective

medium one ﬁnds

P (k, s)=

E(k, s)+

E(k, s)(e

−ika

− 1)

×[

eﬀ

(s) − Γ

][

(s) −

(s)] . (18.144)

The inverse Fourier transform of

E(k, s)is

(s)=



−π

2π

exp(ikal)

s +2

eﬀ

(s)(1 − cos ka)

. (18.145)

Note that

(s)=

−l

(s). Multiplying (18.144) with 1/2π and integrating

over k one obtains

(s)=

(s)+[

(s) −

(s)][

eﬀ

(s) − Γ

][

(s) −

(s)] . (18.146)

Multiplying (18.144) with exp(ika)/2π and integrating one obtains

(s)=

(s)+[

(s) −

(s)][

eﬀ

(s) − Γ

][

(s) −

(s)] . (18.147)

The diﬀerence of the two equations is

(s) −

(s)=

(s) −

(s)

1+2[

(s) −

(s)][

eﬀ

(s) − Γ

]

(18.148)

and thus

(s)=

(s)+

[

(s) −

(s)]

[

eﬀ

(s) − Γ

]

1+2[

(s) −

(s)][

eﬀ

(s) − Γ

]

. (18.149)

The self-consistency requirement {

(s)} =

(s) (cf. (18.57)) leads to



[

(s) −

(s)]

[

eﬀ

(s) − Γ

]

1+2[

(s) −

(s)][

eﬀ

(s) − Γ

]



=0. (18.150)

The diﬀerence squared in the numerator is systematic and can be factored

out. From the master equation for the eﬀective medium in d = 1 follows

(s) −

(s)=

(s) − 1

2Γ

eﬀ

(s)

. (18.151)

The self-consistency condition can now be brought into the form that has

been given in the main text.

18 Diﬀusion of Particles on Lattices 789

18.7.3 Relation Between the Relative Displacement and the

Density Change

The displacements of two tagged particles from their equilibrium positions x

and x+ δx are considered. The notation is explained in Fig. 18.16. The initial

distance changes into the actual distance

δx −→ δx + u(x + δx) − u(x) . (18.152)

Because the number of particles in the interval between the two tagged par-

ticles is conserved, the local density change induced by the diﬀerence in the

displacements of the two particles is

δn =

nδx

δx + u(x + δx) − u(x)

− n. (18.153)

Here n is the (spatially constant) equilibrium particle density. A Taylor ex-

pansion for small δx is made, together with the assumption of small gradients,

∂u/∂x  1. From (18.153) follows in linear approximation

δn = −

∂u

∂x

. (18.154)

The inverse Fourier transformation is deﬁned by

f(x, t)=



2π

ikx

f(k, t) . (18.155)

The relation between density change and displacement gradient reads in

Fourier space

δn(k, t)=iku(k,t) . (18.156)

This is the relation used in the main text.

Notation

D = D

s.p.

Diﬀusion coeﬃcient of single, independent particles

coll

Collective (Fick’s) diﬀusion coeﬃcient

Tagged particle (tracer) diﬀusion coeﬃcient

Energy level of site i

Activation energy of barrier between sites i and j

f(c) Correlation factor for tagged-particle diﬀusion in lattice gas

f = f (c → 1) Correlation factor for metal physics

P (lt|00) Conditional probability of ﬁnding a particle on site l at time t

when it was at site 0 at t =0.

(t) Abbreviated form of P (lt|00)

790 Klaus W. Kehr et al.

P (l,t) Probability of ﬁnding a particle at site l at time t (applies to

lattice gases)

P (k,t) Fourier transform of P

(t)

P (l,s) Fourier-Laplace transform of P

(t)

Γ Transition rate between two neighbour sites in uniform lattice

Transition rate from site i to neighbour site j in disordered

lattice

eﬀ

(t) Time-dependent transition rate in eﬀective medium

l,l



Transition-rate matrix between two neighbour sites

Λ(k) Fourier transform of transition rate matrix

(E) Density of activation energies of random barriers

(E) Density of random site energies

ρ(Γ ) Probability density of random transition rates

 Random walk and/or thermal average

{} Average over (quenched) disordered rates.

References

1. A. Einstein: Ann. Phys. (Leipzig) 17, 549 (1905)

2. C.P. Flynn: Point Defects and Diﬀusion (Clarendon, Oxford 1972)

3. P. H¨anggi, P. Talkner, M. Borkovec: Rev. Mod. Phys. 62, 251 (1990)

4. P.W. Anderson. In: Ill-Condensed Matter, Les Houches, Session XXXI, ed by

R. Balian, R. Maynard, G. Toulouse (North Holland, Amsterdam 1979) p 159

5. K. Mussawisade. PhD thesis, Univ. K¨oln (2000)

6. N.M. Barber, B.W. Ninham: Random and Restricted Walks (Gordon and

Breach, New York 1970)

7. G.H. Weiss: Aspects and Applications of the Random Walk (Clarendon Press,

Oxford 1996)

8. N.G. van Kampen: Stochastic Processes in Physics and Chemistry (North Hol-

land, Amsterdam 1987)

9. F. Spitzer: Principles of Random Walks (Van Nostrand, Princeton NJ 1964)

10. H. Spohn: Large Scale Dynamics of Interacting Particles (Springer, Berlin Hei-

delberg New York 1991)

11. T.M. Liggett: Stochastic Interacting Systems: Contact, Voter and Exclusion

Processes (Springer, Berlin Heidelberg New York 1999)

12. Fractals and Disordered Systems, ed by A. Bunde, S. Havlin (Springer, Berlin

Heidelberg New York 1996)

13. S. Alexander, J. Bernasconi, W.R. Schneider, R. Orbach: Rev. Mod. Phys. 53,

175 (1981)

14. G.H. Weiss, R.J. Rubin: Adv. Chem. Phys. 52 363 (1983)

15. J.W.Haus,K.W.Kehr:Phys.Rep.150, 263 (1987)

16. S. Havlin, D. ben-Avraham: Adv.Phys. 36, 695 (1987)

17. J.-P. Bouchaud, A. Georges: Phys. Rep. 195, 127 (1990)

18. G.M. Sch¨utz. In: Phase Transitions and Critical Phenomena, vol 19, ed by C.

Domb, J. Lebowitz (Academic, London 2000)

18 Diﬀusion of Particles on Lattices 791

19. J.R. Manning: Diﬀusion Kinetics for Atoms in Crystals (Van Nostrand, Prince-

ton 1968)

20. J. Philibert: Atom Movements. Diﬀusion and Mass Transport in Solids (Les

Editions de Physique, Les Ulis 1991)

21. S. Chandrasekhar: Rev. Mod. Phys. 15, 1 (1943)

22. R. Kutner, I. Sosnowska: J.Phys.Chem. Solids 38, 741 (1977)

23. K.W.Kehr,D.Richter,R.H.Swendsen:J.Phys.F.8, 433 (1978)

24. I.S. Anderson, A. Heidemann, J.E. Bonnet, D.K. Ross, S.K.P. Wilson, M.W.

McKergow:J.LessCommonMet.101, 405 (1984)

25. K.W. Schroeder: Z. Physik B25, 91 (1976)

26. D. Richter, T. Springer: Phys. Rev. B 18, 126 (1978)

27. W. Dieterich (unpublished)

28. R. Kutner: Physica A 224, 558 (1996)

29. T. Wichmann. PhD thesis, Univ. K¨oln (1996)

30. J.W. Haus, K.W. Kehr, J.W. Lyklema: Phys.Rev.B 25, 2905 (1982)

31. K. Mussawisade Diploma thesis, Univ. K¨oln (1996)

32. R. Zwanzig: J. Stat. Phys. 28,127 (1982)

33. P.J.H.Denteneer,M.H.Ernst:J.Phys.C16, L961 (1983); Phys.Rev.B 29, 1755

(1984)

34. S. Alexander, R. Orbach: Physica 107B, 675 (1981)

35. S. Summerﬁeld: Solid State Commun. 39, 401 (1981)

36. T. Odagaki, M. Lax: Phys. Rev. B 24 5284 (1981)

37. I. Webman: Phys. Rev. Lett. 47, 1496 (1981)

38. J.W. Haus, K.W. Kehr, K. Kitahara: Phys. Rev. B 25, 4918 (1982); Z.Physik

B 50, 161 (1983)

39. S. Kirkpatrick: Rev. Mod. Phys. 45, 574 (1973)

40. J. Bernasconi: Phys. Rev. B 7, 2252 (1973)

41. H. Ambaye, K.W. Kehr: Phys. Rev. E 51, 5101 (1995)

42. P. Argyrakis, A. Milchev, V. Pereyra, K.W. Kehr: Phys. Rev. E 52, 3623 (1995)

43. B.I. Shklovskii, A.L. Efros: Soviet Physics JETP 33, 468 (1971)

44. V. Ambegaokar, B.I. Halperin, J.S. Langer: Phys. Rev. B 4, 2612 (1971)

45. D. Stauﬀer, A. Aharony: Introduction to Percolation Theory,2ndedn(Taylor

& Francis, London, 1994)

46. J. Bernasconi, W.R. Schneider, H.J. Wiesmann: Phys. Rev. B 16, 5250 (1977)

47. S. Tyc, B.I. Halperin: Phys. Rev. B 39, 877 (1989)

48. P. Le Doussal: Phys. Rev. B 39, 881 (1989)

49. R.A. Tahir-Kheli: Phys. Rev. B 28, 3049 (1983)

50. M.H. Ernst, T.M. Nieuwenhuizen, P.F.J. van Velthoven, J. Phys. A

20, 5335

(1987)

51. M. Braun, K.W. Kehr: Phil.Mag.A 61, 855 (1990)

52. S. Havlin, B.L. Trus, G.H. Weiss: J. Phys. A 19, L817 (1986)

53. Y. Limoge, J.L. Bocquet: Phys. Rev. Lett. 65, 60 (1990)

54. K. Mussawisade, T. Wichmann, K.W. Kehr: J. Phys.: Condens. Matter 9, 1181

(1997)

55. M. Wagener, W. Schirmacher: Ber. Bunsenges. Phys. Chem. 95, 983 (1991)

56. A. Hoerner, A. Milchev, P. Argyrakis: Phys. Rev. E 52, 3570 (1995)

57. W. Dieterich, D. Knoedler, P. Pendzig: J. Non-Cryst. Solids 172-174 1237

(1994)

58. P. Maass, M. Meyer, A. Bunde: Phys. Rev. B 51, 8164 (1995)