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

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7 Fluctuations and Growth Phenomena in Surface Diﬀusion 333

w ﬂuctuating step width

z tip-surface separation

α atom polarizability

β step stiﬀness

Γ step mobility

γ 1D “surface” tension

∆ amplitude of step ﬂuctuations

ζ instrumental correlation length

θ coverage

κ atom detachment rate in the steady state model

Fermi wavevector

µ chemical potential

prefactor at step edge

prefactor at terraces

ξ step correlation length

Σ area of elementary hops along step

EC evaporation and condensation mechanism

FIM ﬁeld ion microscopy

LEED low energy electron diﬀraction

LEEM low energy electron microscopy

ML monolayer

QSE quantum size eﬀects

RHEED reﬂection high energy electron diﬀraction

STM scanning tunneling microscopy

SD step diﬀusion

TD terrace diﬀusion

References

1. Surface Diﬀusion: Atomistic and Collective Processes, ed by M.C. Tringides

(Plenum, New York 1997)

2. Collective Diﬀusion on Surfaces: Correlation Eﬀects and Adatom Interactions,

ed by M.C. Tringides, Z. Chvoj (Kluwer, Dordrecht 2001)

3. W.N. Unertl. In: Handbook of Surface Science Vol. 1, Physical Structure,edby

W.N. Unertl (Elsevier, Amsterdam 1996)

4. C. Ratsch, P. Ruggerone, M. Scheﬄer. In: Surface Diﬀusion: Atomistic and

Collective Processes, ed by M.C. Tringides (Plenum, New York 1997)

5. Gun-Do Lee, C.Z. Wang, Z.Y. Lu, K.M. Ho: Surf. Sci. 426, L427 (1999)

6. H. Brune: Surf. Sci. Rep. 31, 121 (1998)

7. M.C. Tringides, R. Gomer: Surf. Sci. 265, 283 (1992)

8. T.L. Einstein. In: Handbook of Surface Science Vol. 1, Physical Structure,ed

by W.N. Unertl (Elsevier, Amsterdam 1996)

9. D. Meade, D. Vanderbilt: Phys. Rev. Lett. 63, 1404 (1989)

10. Ordering in Two Dimensions, ed by S.K. Sinha (North Holland, New York

1980)

334 Michael C. Tringides and Myron Hupalo

11. R. Gomer: Surf. Sci. 38, 373 (1973)

12. A. Sadiq, K. Binder: Surf. Sci. 35, 617 (1984)

13. X. Xiao, X.D. Zhu, W. Daum, Y.R. Shen: Phys. Rev. Lett. 61, 2883 (1988)

14. J.W. Frenken, B.J. Hinch, J.P. Toennies, Ch. W¨oll: Phys. Rev. B 41, 938 (1990)

15. J.R. Sanchez, D. Quinn, M.C. Tringides: Surf. Sci. 391, 101 (1997)

16. R. Gomer: Rep. Prog. Phys. 53, 917 (1990)

17. P. Nikunen, I. Vattulainen, T. Ala-Nissila: Surf. Sci. 447, L162 (2000)

18. D. Foster: Hydrodynamic Fluctuations, Broken Symmetry and Correlation

Functions (Addison Wesley, Reading MA 1990)

19. C. Uebing. In: Surface Diﬀusion: Atomistic and Collective Processes,edby

M.C. Tringides (Plenum, New York 1997)

20. M.R. Dudek, M.C. Tringides. In: Collective Diﬀusion on Surfaces: Correlation

Eﬀects and Adatom Interactions, ed by M.C. Tringides, Z. Chvoj (Kluwer,

Dordrecht 2001)

21. A.T. Loburets, A.G. Naumovets, Y.S. Vedula. In: Surface Diﬀusion: Atomistic

and Collective Processes, ed by M. Tringides (Plenum Press, New York 1997)

pp 509–528

22. M.C. Tringides, M. Gupalo, Q. Li, X. Wang. In: Proceedings of XI Max Born

Symposium, ed by R. Kutner, A. Pelkalski, K. Sznajd-Weron (Springer, Berlin

Heidelberg New York 1999)

23. M.L.Lozano,M.C.Tringides:Europhys.Lett.30, 537 (1995)

24. Z.Chvoj,E.H.Conrad,M.C.Tringides:Phys.Rev.B62, 4672 (2000)

25. M.C. Tringides, M.I. Larson, H. Pfnuer, H. Frischart, K. Budde, M. Henzler:

J. Appl. Phys. 82, 1507 (1997)

26. E.H. Conrad, A. Menzel, S. Kiriukhin, M.C. Tringides: Phys. Rev. Lett. 81,

3175 (1998)

27. M. Kammler, M. Horn von Hoegen, N. Voss, M.C. Tringides, A. Menzel, E.H.

Conrad:PhysRev.B65, 075312 (2002)

28. A. Menzel, K. Wiesenfeld, E.H. Conrad, M.C. Tringides: Phys. Rev. B 61, 2997

(2000)

29. S. Brauer, G. B. Stephenson, M. Sutton, R. Bruning, E. Dufresne, S.G.J.

Mochrie, G. Grubel, J. Als-Nilesen, D.L. Abernathy: Phys. Rev. Lett. 74, 2010

(1995)

30. B. Poelsema, G. Comsa: Scattering of Thermal Energy Atoms from Disordered

Surfaces (Springer, Berlin Heidelberg New York 1989)

31. N.C. Bartelt, R.M. Tromp: Phys. Rev B 54, 11731 (1996)

32. H.P. Bonzel, W.W. Mullins: Surf. Sci. 350, 285 (1996)

33. T. Zambelli, J. Trost, J. Wintterlin, G. Ertl: Phys. Rev. Lett. 76, 795 (1996)

34. S. Renisch, R. Schuster, J. Wintterlin, G. Ertl: Phys. Rev. Lett. 82, 3839 (1999)

35. N.C. Bartelt, W. Theis, R.M. Tromp: Phys. Rev. B 54 11741 (1996)

36. Q. Li, M.C. Tringides: Surf. Sci. 365, 495 (1996)

37. F.Bozso,I.W.Lyo,P.Avouris:J.Vac.Sci.Techn.B9, 428 (1991)

38. U. Hofer, T.F. Heinz: Phys. Rev. Lett. 66, 1994 (1991)

39. A. Memsouk, Y. Boughaleb, R. Nassif, K.Saadouni: Phys. Lett. A 248 (2001)

40. X.Wang,Q.Li,M.C.Tringides:Phys.Rev.B57, 7275 (1998)

41. M. Horn von Hoegen, F.K. LeGoues, M. Copel, M.C. Reuter, R.M. Tromp:

Phys. Rev Lett. 67, 1130 (1991)

42. J. Tersoﬀ, C. Teichert, M.G. Lagally: Phys. Rev. Lett. 76, 1675 (1996)

43. K. Budde, E. Abram, V. Yeh, M.C. Tringides: Phys. Rev. B 61, 10602 (2000)

7 Fluctuations and Growth Phenomena in Surface Diﬀusion 335

44. M. Hupalo, S. Kremmer, V. Yeh, L. Berbil-Bautista, E. Abram, M.C. Tringides:

Surf. Sci. 493, 526 (2001)

45. W.B. Su, S.H. Chang, W.B. Jian, C.S. Chang, L.J. Chen, T.T. Tsong: Phys.

Rev. Lett. 64, 1589 (2001)

46. F.K. Schulte: Surf. Sci. 55, 427 (1976)

47. N. Trivedi, N. Ashcroft: Phys. Rev. B 38, 12298 (1988)

48. M. Jalochowski, E. Bauer: Phys. Rev. B 38, 5272 (1988)

49. C.M. Wei, M.Y. Chou: Phys. Rev. B 66, 233408 (2002)

50. B.J. Hinch, C. Koziol, J.P. Toennies, G. Zhang: Europhys. Lett. 10(4), 341

(1989)

51. D. Schmicker, T. Hibna, K.A. Edwards, P.B. Howest, J.E. MacDonald, M.A.

James, M. Breeman, G.T. Barkema: J. Phys.: Cond. Matt. 9, 969 (1997)

52. A. Grottini, D. Cvetko, L. Floreano, R. Gotter, A. Morgante, F. Tommasini:

Phys.Rev.Lett.79, 1527 (1997)

53. J. Braun, J.P. Toennies, Ch. W¨oll: Phys. Rev. B 60, 11707 (1999)

54. A.R. Smith, K.J. Chao, Q. Niu, C.K. Shih: Science 273, 226 (1996)

55. Z.Y. Zhang, Q. Niu, C.K. Shih: Phys. Rev. Lett. 80, 5381 (1998)

56. V. Yeh, L. Berbil-Bautista, C.Z. Wang, K.M. Ho, M.C. Tringides: Phys. Rev.

Lett. 85, 5158 (2000)

57. M. Hupalo, V. Yeh, L. Berbil-Bautista, S. Kremmer, E. Abram, M.C. Tringides:

Phys Rev. B 64, 155307 (2001)

58. H.H. Weitering, A.R.H.F. Ettema, T. Hibma: Phys. Rev. B 45, 9126 (1992)

59. M.C. Tringides. In: Morphological Organizations in Epitaxial Growth and Re-

newal, ed by Z. Zhang, M.G. Lagally (Word Scientiﬁc, Singapore 1998)

60. M. Jalochowski, M. Hoﬀmann, E. Bauer: Phys. Rev. B 51, 7231 (1995)

61. R. Biswas, K. Roos, M.C. Tringides: Phys. Rev. B 50, 10932 (1994)

62. S. Stepanovskyy, V. Yeh, M. Hupalo, M.C. Tringides: Surf. Sci. 515, 187 (2002)

63. G. Rosenfeld, R. Servanty, C. Teichert, B. Polsema, G. Comsa: Phys. Rev. Lett.

72, 895 (1993)

64. J. Vrijmoeth, H.A. van der Vegt, J.A. Meyer, E. Vlieg, R.J. Behm: Phys. Rev.

Lett. 72, 383 (1994)

65. M. Stanley, C. Papageorgopolos, K.R. Roos, M.C. Tringides: Surf. Sci. 355,

L264 (1996)

66. Z. Chvoj, C. Ghosh, T.S. Rahman, M.C. Tringides: J. Phys. C 15, 5223 (2003)

67. K.R.Roos,M.C.Tringides:Phys.Rev.Lett.87 149604 (2001)

68. K. Bromann, H. Brune, H. Roder, K. Kern: Phys. Rev. Lett. 75, 677 (1995)

69. K.R.Roos,M.C.Tringides:Phys.Rev.Lett.85, 1480 (2000)

70. J. Tersoﬀ, A.W. Denier van Der Gon, R.M. Tromp: Phys. Rev. Lett. 72, 266

(1994)

71. J.A. Venables, G.D.T. Spiller, M. Hanbucken: Rep. Prog. Phys. 47, 399 (1984)

72. J. Krug, P. Politi, T. Michely: Phys. Rev. B 61, 14037 (2000)

73. K.R. Roos, M.C. Tringides: Surf. Sci. Rev. and Lett.

5, 3 (1998)

74. M.C. Tringides, R. Gomer: Surf. Sci. 145, 1212 (1984)

75. K. Morgenstern, G. Rosenfeld, Laegsgaard, F. Besenbacher, G. Comsa: Phys.

Rev. Lett. 80, 556 (1998)

76. Z. Chvoj, M.C. Tringides: Phys. Rev. B 66, 035418 (2002)

77. M. Giesen, C. Steimer, H. Ibach: Surf. Sci. 471, 80 (2001)

8 Grain Boundary Diﬀusion in Metals

Christian Herzig and Yuri Mishin

8.1 Introduction

Grain boundary (GB) diﬀusion plays an important role in many processes

taking place in engineering materials at elevated temperatures. Such processes

include Coble creep, sintering, diﬀusion-induced grain boundary migration,

discontinuous reactions (such as discontinuous precipitation, discontinuous

coarsening, etc.), recrystallization, and grain growth. GB diﬀusion is impor-

tant not only at high temperatures but sometimes also at relatively low, even

ambient temperatures. In particular, the service life of many microelectronic

elements and devices is limited by room temperature diﬀusion or electromi-

gration of detrimental impurities along GBs resulting in the degradation of

service characteristics.

The fact that GBs provide high diﬀusivity (“short circuit”) paths in met-

als has been known since a few decades. First indications of enhanced atomic

mobility at GBs were obtained as early as in the 1920–1930s, for example

from grain size dependence of creep and sintering rates in polycrystalline

materials. However, the ﬁrst direct proof of GB diﬀusion was obtained in the

early 1950s using autoradiography [1]: the additional blackening of autoradi-

ographic images along GBs indicated that the radiotracer atoms penetrated

into GBs much faster than in the regular lattice [2]. These observations were

immediately followed by two important events: the appearance of the nowa-

days classic Fisher model of boundary diﬀusion [3], on one hand, and the

development and extensive use of the radiotracer serial sectioning technique,

on the other hand. It was largely due to these events that GB diﬀusion studies

were put on a quantitative basis and GB diﬀusion measurements became the

subject of many investigations and publications. Over the ﬁve decades that

followed, the experimental techniques for GB diﬀusion measurements have

been drastically improved and extended to a wide range of temperatures and

a broad spectrum of metallic, semiconductor, and ceramic materials. On the

theoretical side, the Fisher model, being still the footing stone of GB diﬀu-

sion theory, has been subject to careful mathematical analysis and extended

to various situations encountered in diﬀusion experiments and technological

processes. For an overview of both fundamentals and recent achievements in

this area see the book of Kaur et al. [4] and review articles [5–7]. A com-

338 Christian Herzig and Yuri Mishin

surface

grain

boundary

Fig. 8.1. Schematic geometry of the Fisher model

of grain boundary diﬀusion.

plete collection of experimental data published before the end of 1980s can

be found in the handbook of Kaur et al. [8].

This chapter presents a brief review of fundamental aspects of GB dif-

fusion with emphasis on metals and metallic alloys. In Sects. 8.2 and 8.3

we introduce classiﬁcations of GB diﬀusion kinetics and mathematical mod-

els applied for the analysis of experimental concentration proﬁles. We also

give a brief summary of the current knowledge of GB diﬀusion in metals. In

Sect. 8.4 we discuss in more detail the problem of GB diﬀusion accompanied

by GB segregation and discuss the recent progress in this area. In Sect. 8.5

we summarize. The idea which we wish to emphasize in this review is that

many problems in this area could only be solved by combining new theoretical

models with precise measurements using novel experimental techniques.

8.2 Fundamentals of Grain Boundary Diﬀusion

8.2.1 Basic Equations of Grain Boundary Diﬀusion

Most mathematical treatments of GB diﬀusion are based on the Fisher model

[3] describing diﬀusion along a single GB. In the Fisher model, a GB is

represented as a high-diﬀusivity uniform and isotropic slab embedded in a

low-diﬀusivity isotropic crystal perpendicular to its surface (Fig. 8.1). The

GB is thus described by two physical parameters: the GB width δ and the

GB diﬀusion coeﬃcient D

such that D

 D, D being the volume diﬀusion

coeﬃcient. In a typical diﬀusion experiment, a layer of foreign atoms or tracer

atoms of the same material is created at the surface and the specimen is

annealed at a constant temperature T for a time t. During the anneal the

atoms diﬀuse from the surface into the specimen in two ways: directly into

the grains and, much faster, along the GB. In turn, the atoms which diﬀuse

along the GB eventually leave it and continue to diﬀuse in the lattice regions

adjacent to the GB, thus giving rise to a volume diﬀusion ﬁeld around the

GB.

Mathematically, the diﬀusion process is described by two coupled equa-

tions:

8 Grain Boundary Diﬀusion 339

∂c

∂t

= D



∂

∂x

∂

∂y



, where |x| >δ/2, (8.1)

∂c

∂t

= D

∂

∂y



∂c

∂x



x=δ/2

. (8.2)

These equations represent diﬀusion in the volume and along the GB, respec-

tively. c(x, y, t) is the volume concentration of the diﬀusing atoms and c

(y,t)

is their concentration in the GB. The second term in the right-hand side of

(8.2) takes into account the leakage of the diﬀusing atoms from the GB to the

volume. Any solution of (8.1) and (8.2) should meet the surface condition,

which can be diﬀerent in diﬀerent experiments (see below), as well as the

natural initial and boundary conditions at x →±∞and y →∞. The join-

ing condition between functions c(x, y, t)andc

(y,t) depends on whether

we study self-diﬀusion or impurity diﬀusion. For self-diﬀusion, the joining

condition simply reﬂects the continuity of the concentration across the GB,

(y,t)=c(± δ/2,y,t). (8.3)

For impurity diﬀusion, the joining condition involves the equilibrium segre-

gation factor s and reads

(y,t)=sc(±δ/2,y,t). (8.4)

The physical meaning of this relation will be discussed later.

8.2.2 Surface Conditions

Fisher [3] postulated a constant source condition at the surface,

c(x, 0,t)=c

=const. (8.5)

This condition is also called a thick-layer condition because it can be estab-

lished by depositing a thick layer of diﬀusant, h  (Dt)

1/2

. The constant

source condition also applies when diﬀusion takes place from a gas phase.

Later Suzouka [9, 10] introduced an instantaneous source, or thin-layer

condition,

c(x, y, 0) = Mδ(y), (∂c/∂y)

y=0

=0, (8.6)

where M is the amount of diﬀusant deposited per unit area of the surface.

This surface condition assumes that the initial layer is completely consumed

by the specimen during the diﬀusion experiment, i.e. h  (Dt)

1/2

.Notethat

(8.6) unrealistically assumes that the rate of surface diﬀusion is the same as

the rate of volume diﬀusion, which contradicts existing experimental data.

A more realistic thin-layer condition, called fast surface diﬀusion condi-

tion, was proposed in [4,11]. That condition postulates that surface diﬀusion

is much faster than volume and even GB diﬀusion, which results in a uniform

340 Christian Herzig and Yuri Mishin

surface concentration near the GB-surface intersect. The respective surface

condition is

c(x, y, 0) = Mδ(y),c(x, 0,t)=M/(πDt)

1/2

. (8.7)

This condition also assumes that h  (Dt)

1/2

In modern experiments thin-layer conditions are established more often

than the constant source condition. This is due to the use of extremely thin

radiotracer layers which do not disturb the structural or chemical state of

GBs in the course of the diﬀusion experiment.

8.2.3 Methods of Proﬁle Analysis

The basic equations of the Fisher model, (8.1) and (8.2), can be solved analyt-

ically. The integral representations of the exact solutions for the constant and

instantaneous sources were derived by Whipple [12] and Suzuoka [9, 10], re-

spectively. Using the same Fourier-Laplace transformation method, the exact

analytical solution was also obtained for the fast surface diﬀusion condition,

(8.7) [4]. Unfortunately, the practical signiﬁcance of exact solutions is very

limited because of their very complicated mathematical form. In practical

terms, any mathematical solution is only useful if it oﬀers a way to process

the experimental concentration proﬁle and determine GB diﬀusion charac-

teristics.

GB diﬀusion measurements typically employ the radiotracer serial section-

ing method. After the diﬀusion anneal, thin-layers of the material parallel to

the source surface are removed from the specimen (either mechanically or by

ion beam sputtering) and the radioactivity of each section is measured with

a crystalline γ-detector or a liquid scintillation counter. Experimental details

of this method can be found in the literature [13, 14], see also Chap. 1. The

quantity measured by this method is the average layered concentration of the

diﬀusant,

c, as a function of the penetration depth y. This function, called

a concentration (or penetration) proﬁle, bears information about GB diﬀu-

sion parameters and is therefore subject to a mathematical treatment with

the purpose of extracting that information. Much work has been devoted to

the development of approximate analytical solutions of the Fisher model and

simple recipes of proﬁle analysis.

Fisher [3] derived the following approximate solution for self-diﬀusion

from a constant source:

c ∝ c

exp(−π

−1/4

w), (8.8)

where the precise value of the proportionality constant is not important. Here

w is the reduced penetration depth deﬁned by

w =

(δD

)

1/2





1/4

. (8.9)

8 Grain Boundary Diﬀusion 341

Equation (8.8) suggests that the penetration proﬁle c(y)plottedinthecoor-

dinates log

c versus y should yield a straight line. Then, knowing the slope

∂ ln

c/∂y of that line we can determine the product δD

δD

=1.128(D/t)

1/2

(−∂ ln c/∂y)

−2

. (8.10)

The volume diﬀusion coeﬃcient D is assumed to be known from independent

measurements.

Fisher’s exponential solution, (8.8), is not very accurate. If a proﬁle calcu-

lated using Whipple’s [12] exact analytical solution is plotted in the coordi-

nates log

c versus y

with various powers n, the best straight line is obtained

with the power of n =6/5 [15, 16]. Moreover, the linear part of the proﬁle

scales with the reduced depth w deﬁned by (8.9) with a constant slope of

about −0.78:

∂ ln

∂w

6/5

≈−0.78. (8.11)

It immediately follows that, knowing the slope of the experimental proﬁle in

the coordinates log

c versus y

6/5

, we can calculate the product δD

δD

=1.322(D/t)

1/2



−∂ ln

c/∂y

6/5



−5/3

. (8.12)

Equations (8.11) and (8.12) are only valid under two conditions:

– The so-called LeClaire’s parameter β deﬁned by

β =

δD

2D(Dt)

1/2

(8.13)

must be large enough, in practical conditions β>10.

– Parameter

α =

2(Dt)

1/2

(8.14)

must be small enough, in practical conditions α<0.1.

Similarly, for diﬀusion from an instantaneous source it was found that

∂ ln

∂w

6/5

≈−0.775 (8.15)

provided that α is small enough and β>10

[9, 10]. Therefore, the product

δD

can be determined from the linear part of the proﬁle log c versus y

6/5

δD

=1.308(D/t)

1/2



−∂ ln

c/∂y

6/5



−5/3

. (8.16)

If β<10

, the right-hand side of (8.15) is no longer constant; instead, it

becomes a function of β (and thus time). Then, (8.16) should be slightly

modiﬁed to become

δD

=1.206(D

0.585

0.605

)

1/1.19



−∂ ln

c/∂y

6/5



−5/2.975

(8.17)

342 Christian Herzig and Yuri Mishin

log c

surface

6/5

grain

boundary

volume

diffusion

GB diffusion

Fig. 8.2. Schematic shape of a typical pen-

etration proﬁle of grain boundary diﬀusion.

If α  1andβ  1, where α and β are

deﬁned by (8.13) and (8.14), then the tail

of the proﬁle is a straight line in the coordi-

nates log

c versus y

6/5

for 10

<β<10

,and

δD

=1.084(D

0.91

1.03

)

1/1.94



−∂ ln

c/∂y

6/5



−5/2.91

(8.18)

for 10 <β<10

The preceding relations describe only one part of the proﬁle, namely, the

part which is dominated by GB diﬀusion. The overall penetration proﬁle

typically consists of two parts as shown schematically in Fig. 8.2:

– A near-surface region caused by direct volume diﬀusion from the surface.

The concentration in this region follows a Gaussian function or an error

function, depending on the surface condition. If measured accurately, this

part can be used to determine the volume diﬀusion coeﬃcient.

– A long-penetration tail caused by the simultaneous GB diﬀusion and lat-

eral volume diﬀusion from the GB to the adjacent grains. It is this tail

that should become a straight line when plotted as log

c versus y

6/5

.The

slope of this line is used to determine the GB diﬀusivity.

It should be pointed out that the power n =6/5 has no physical meaning

and cannot be derived analytically. It simply provides a good numerical ap-

proximation of the exact proﬁle in a convenient concentration range and is

therefore widely used for the analysis of experimental proﬁles. A slightly more

accurate value of n can be obtained for each particular surface condition or

GB geometry in the sample [17]. The obtained values range from 1.1 to 1.2,

i.e. are rather close to 6/5. Given the scatter of data points on experimental

proﬁles, the use of one universal power n =6/5 and thus (8.12), (8.16)-(8.18)

is a well-justiﬁed procedure for all practical purposes. Carefully measured

experimental proﬁles do follow the y

6/5

-rule in a wide concentration range,

see examples in Fig. 8.3. Moreover, the linearity of log

c versus y

6/5

is often

used as a proof of predominant GB diﬀusion in the sample and a measure of

the proﬁle quality.

8 Grain Boundary Diﬀusion 343

Fig. 8.3. Penetration proﬁles for

self-diﬀusion in polycrystalline sil-

ver [23]. This diagram demon-

strates that the GB-related tail

of high quality proﬁles becomes a

straight line when plotted as log

versus y

6/5

Equations (8.12), (8.16)-(8.18) are only valid for self-diﬀusion in metals.

The case of impurity diﬀusion was ﬁrst analyzed by Bokshtein et al. [18] and

Gibbs [19] and later revisited by other authors (see [4] for references). It has

been shown that, provided the segregation factor s is a constant, all solutions

of the Fisher model remain the same as for self-diﬀusion except that δ should

be replaced by the product sδ. In particular, (8.12), (8.16)-(8.18) should be

modiﬁed by replacing their left-hand side by sδD

. After this modiﬁcation

they can be applied for the determination of the triple product sδD

impurity diﬀusion.

8.2.4 What Do We Know About Grain Boundary Diﬀusion?

It is not our intention to give a complete answer to this question, especially

because many topics in this area are still the subject of debates in the lit-

erature. We will focus on facts that appear to be well established and will

restrict our discussion to self-diﬀusion in metals.

The Grain Boundary Width

In the case of self-diﬀusion we can only determine the product δD

.We

thus need to know δ if we want to determine the GB diﬀusion coeﬃcient

. The assumption δ =0.5 nm introduced by Fisher [3] seems to be a good

approximation. This value of δ is well-consistent with evaluations of GB width