Diffusion Solids Fundamentals Diffusion Controlled Solid State Episode 2 Part 4 docx

Springer, Diffusion Studies of Solids by Quasielastic Neutron Scattering, in:Diffusion in Condensed Matter, J.. 19 Diffusion in Dilute Substitutional AlloysDiffusion in alloys is more comple

Fig 18.2 Diffusion of H in Pd, Ni and Fe according to Alefeld and V¨olkl[13]

bility of H in Ni other methods such as Gorsky effect, NMR, and QENS havelimited applicability for the Ni-H system The Arrhenius parameters below

the magnetic transition (Curie temperature T C = 627 K) seem to be onlyslightly different than above (see Fig 18.2)

Diffusion of hydrogen in iron has considerable importance due to the

technical challenge of hydrogen embrittlement of iron and steels The absolutevalues for H-diffusion in Fe are higher than those for Pd and Ni (see Fig 18.2)

In spite of the technological interest and the large number of studies (forreferences see [19]), the scatter of the data is rather large especially belowroom temperature Several reasons for this scatter have been discussed in theliterature: surface effects, trapping of hydrogen by impurities, dislocations,grain boundaries, or precipitates, and the formation of molecular hydrogen

in micropores, either already existing in the material or produced by excessloading (for references see [18])

Hydrogen in niobium has been studied over a very wide temperature

region (see Fig 18.3) Because of the high diffusivities and its relatively smallchanges with temperature the Gorsky technique could be used for both H and

D in nearly the whole temperature range Tritium diffusion has been studied

in a more limited temperature range Isotope effects of hydrogen diffusion in

Nb are very evident from Fig 18.3 Similar isotope effect studies are availablefor V and Ta (see Table 18.3)

18.2 Hydrogen Diffusion in Metals 323

Fig 18.3 Diffusion of H, D, and T in Nb according to [13]

18.2.4 Non-Classical Isotope Effects

Classical rate theory for the isotope effect in diffusion predicts, if many-body

effects are neglected, for the ratio of the diffusivities [33, 34],

For several metal-hydrogen systems non-classical isotope effects have beenreported Figure 18.3 and Table 18.3 show that in group-V transition metalshydrogen (H) diffuses more rapidly than deuterium (D), and deuterium dif-fuses more rapidly than tritium (T) In addition, the activation enthalpies ofhydrogen isotopes are different and hence the ratio of the diffusivities varieswith temperature

Interestingly, the characteristic features of the dependence of the ity on the isotope mass are correlated with the structure of the host metal [13,14] In the bcc metals V, Nb and Ta, hydrogen diffuses faster than deuterium

diffusiv-Table 18.3 Activation parameters for diffusion of H, D, and T in Nb, Ta and V

according to Alefeld and V¨olkl[13]

Eq (18.8) within the experimental errors However, again the activation thalpies are mass-dependent, but in contrast to bcc metals one observes

This fact leads to an inverse isotope effect at lower temperatures For example,

in Pd below about 773 K deuterium diffuses faster than hydrogen

Because of the non-classical behaviour of hydrogen and the large ature region over which diffusion coefficients were measured, it is by no meansevident that the diffusion coefficient of hydrogen in metals should obey a lin-ear Arrhenius relation over the entire temperature range The break observed

temper-in the Arrhenius relation for H temper-in Nb (see Fig 18.3) was first observed withthe Gorsky effect method and confirmed independently by measurements ofresistivity changes [35] and by QENS [36] The deviation of hydrogen diffu-sion in Nb from an Arrhenius law at lower temperatures (Fig 18.3) has alsobeen observed in the tantalum-hydrogen system It has been attributed toincoherent tunneling

References

1 A.D Le Claire, Diffusion of C, N, and O in Metals Chap 8 in: Diffusion in

Solid Metals and Alloys, H Mehrer (Vol.Ed.), Landolt-B¨ornstein, NumericalData and Functional Relationships in Science and Technology, New Series,Group III: Crystal and Solid State Physics, Vol 26, Springer-Verlag, 1990, p.471

References 325

2 A.S Nowick, B.S Berry, Anelastic Relaxation in Crystalline Solids, Academic

Press, New York, (1972)

3 R.D Batist, Mechanical Spectroscopy, in: Materials Science and Technology,

Vol 2B: Characterisation of Materials, R.W Cahn, P Haasen, E.J Cramer(Eds.), VCH, Weinheim, 1994, p 159

4 H Kronm¨uller, Nachwirkung in Ferromagnetika, Springer Tracts in Natural

Philosophy, Springer-Verlag (1968)

5 J Philibert, Atom Movements – Diffusion and Mass Transport in Solids, Les

Editions de Physique, Les Ulis, 1991

6 Th Heumann, Diffusion in Metallen, Springer-Verlag, Berlin, 1992

7 A.J Bosman, PhD Thesis, University of Amsterdam (1960)

8 D.C Parris, R.B McLellan, Acta Metall 24, 523 (1976)

9 R.P Smith, Acta Metall 1, 576 (1953)

10 I.V Belova, G.E Murch, Philos Mag 36, 4515 (2006)

11 T Graham, Phil Trans Roy Soc (London) 156, 399 (1866)

12 A Sieverts, Z Phys Chem 88, 451 (1914)

13 G Alefeld, J V¨olkl (Eds.), Hydrogen in Metals I – Application-oriented

Prop-erties, Topics in Applied Physics, Vol 28, Springer-Verlag, 1978

14 G Alefeld, J V¨olkl (Eds.), Hydrogen in Metals II – Basic Properties, Topics

in Applied Physics, Vol 29, Springer-Verlag, 1979

15 H Wipf (Ed.), Hydrogen in Metals III – Properties and Applications, Topics in

Applied Physics Vol 73, Springer-Verlag, 1997

16 H.K Birnbaum, C.A Wert, Ber Bunsenges Phys Chem 76, 806 (1972)

17 R Hempelmann, J Less Common Metals 101, 69 (1984)

18 J V¨olkl, G Alefeld, in Diffusion in Solids – Recent Developments, A.S Nowick,

J.J Burton (Eds.) Academic Press, 1975, p 232

19 G.V Kidson, The Diffusion of H, D, and T in Solid Metals, Chap 9 in:

Dif-fusion in Solid Metals and Alloys, H Mehrer (Vol Ed.), Landolt-B¨ornstein,Numerical Data and Functional Relationships in Science and Technology, NewSeries, Group III: Crystal and Solid State Physics, Vol 26, Springer-Verlag,

1990, p 504

20 H Matzke, V.V Rondinella, Diffusion in Carbides, Nitrides, Hydrides, and

Borides, in: Diffusion in Semiconductors and Non-metallic Solids, D.L Beke

(Vol.Ed), Landolt-B¨ornstein, Numerical Data and Functional Relationships inScience and Technology, Vol 33, Subvolume B1, Springer-Verlag, 1999

21 H Z¨uchner, N Boes, Ber Bunsenges Phys Chem 76, 783 (1972)

22 H Kronm¨uller, Magnetic After-Effects of Hydrogen Isotopes in Ferromagnetic

Metals and Alloys, Ch 11 in [13]

23 R.B McLellan, W.A Oates, Acta Metall 21, 181 (1973)

24 R.M Cotts, Ber Bunsenges Phys Chem 76, 760 (1972)

25 G Majer, Die Methoden der Kernspinresonanz zum Studium der Diffusion von

Wasserstoff in Metallen und intermetallischen Verbindungen, Cuvillier Verlag,

G¨ottingen, 2000

26 P Heitjans, S Indris, M Wilkening, Solid-State Diffusion and NMR, in:

Diffu-sion Fundamentals – Leipzig 2005, J K¨arger, F Grinberg, P Heitjans (Eds.),Leipziger Universit¨atsverlag 2005

27 R Hempelmann, Quasielastic Neutron Scattering and Solid-State Diffusion,

Oxford Science Publication, 2000

28 T Springer, Z Phys Chem NF 115, 317 (1979)

29 T Springer, Diffusion Studies of Solids by Quasielastic Neutron Scattering, in:

Diffusion in Condensed Matter, J K¨arger, P Heitjans, R Haberlandt (Eds.),Vieweg-Verlag, 1998, p 59

30 T Springer, R.E Lechner, Diffusion Studies of Solids by Quasielastic Neutron

Scattering, in: Diffusion in Condensed Matter – Methods, Materials, Models, P.

Heitjans, J K¨arger (Eds.), Springer-Verlag, 2005

31 H Zabel, Quasielastic Neutron Scattering: a Powerful Tool for Investigating

Diffusion in Solids, in: Nontraditional Methods in Diffusion, G.E Murch, H.K.

Birnbaum, J.R Cost (Eds.), The Metallurgical Society of AIME, Warrendale,

1984, p 1

32 K Sk¨old, Quasielastic Neutron Scattering Studies of Metal Hydrides, Ch 11

in [13]

33 C Wert, C Zener, Phys Rev 76, 1169 (1949)

34 C H Vineyard, J Phys Chem Sol 3 121 (1957)

35 H Wipf, G Alefeld, Phys Stat Sol (a) 23, 175 (1974)

36 D Richter, B Alefeld, A Heidemann, N Wakabayashi, J Phys F: Metal Phys

19 Diffusion in Dilute Substitutional Alloys

Diffusion in alloys is more complex than self-diffusion in pure metals In thischapter, we consider dilute substitutional binary alloys of metals A and Bwith the mole fraction of B atoms much smaller than that of A atoms Then

A is denoted as the solvent (or matrix ) and B is denoted as the solute

Dif-fusion in a dilute alloy has two aspects: solute diffusion and solvent diffusion

We consider at first solute diffusion at infinite dilution This is often called

impurity diffusion Impurity diffusion implies concentrations of the solute less

than 1 % In practice, the sensitivity for detection of radioactive solutes ables one to study diffusion of impurities at very high dilution of less than 1ppm In very dilute substitutional alloys solute and solvent diffusion can beanalysed in terms of vacancy-atom exchange rates

en-19.1 Diffusion of Impurities

Impurity diffusion is a topic of diffusion research to which much scientific workhas been devoted We consider at first ‘normal’ behaviour of substitutionalimpurities, which is illustrated for the solvent silver Similar behaviour isobserved for the other noble metals, for hexagonal Zn and Cd, and for Ni.There are exceptions from ‘normal’ impurity diffusion A prominent ex-ample is the slow diffusion of transition elements in the trivalent solventaluminium Another example is the very fast diffusion of impurities in so-called ‘open’ metals Lead is the most famous open metal, for which very fastimpurity diffusion has been observed The rapid diffusion of Au in Pb wasdiscovered by the diffusion pioneer Roberts-Austen in 1896 (see Chap 1)

It is beyond the scope of this book to give a comprehensive overview ofimpurity diffusion Instead, we refer the reader to the chapter of Le Claireand Neumannin the data collection edited by the present author [1] and tothe review by Neumann and Tuijn [2]

19.1.1 ‘Normal’ Impurity Diffusion

Figure 19.1 shows an Arrhenius diagram for diffusion of many substitutionalforeign atoms in a Ag matrix together with self-diffusion of Ag A comparison

Fig 19.1 Diffusion of substitutional impurities in Ag and self-diffusion of Ag

(dashed line) Diffusion parameters from [1, 2]

of impurity diffusion and self-diffusion coefficients, D2 and D, reveals the

following features of ‘normal’ diffusion of substitutional impurities:

– The diffusivities of impurities, D2, lie in a relatively narrow band aroundself-diffusivities, D, of the solvent atoms The following limits apply in

the temperature range between the melting temperature T m and about

– The activation enthalpies of impurity and self-diffusion, ∆H2 and ∆H,

are not much different:

0.75 < ∆H2

19.1 Diffusion of Impurities 329Substitutional impurities in other fcc metals (Cu, Au, Ni) and in the hcp met-als (Zn, Cd) behave similarly as in Ag (for references see the chapter of LeClaire and Neumann in the data collection [1]) Like self-diffusion of thesolvent, diffusion of substitutional impurities occurs via the vacancy mecha-nism For impurity diffusion in a very dilute alloy it is justified to assume thatsolute atoms are isolated, i.e interaction with other solute atoms (formation

of solute pairs, triplets, etc.) is negligible The theory of vacancy-mediateddiffusion of substitutional impurities takes into account three aspects:

1 The formation of solute-vacancy pairs: we remind the reader of theLomer equation introduced in Chap 5, which shows that the proba-bility of a vacancy occupying a nearest-neighbour site of a substitutionalimpurity is given by



− H 1V F − H B

kBT

(19.4)

where G B = H B − T S B is the Gibbs free energy of solute-vacancy

inter-action, C 1V eq the atomic fraction of vacancies in the pure matrix in thermal

equilibrium H F

1V and S F

1V denote the formation enthalpy and entropy of

a monovacancy For GB > 0 the interaction is attractive, for G B < 0 it

is repulsive

2 In contrast to the case of self-diffusion in the pure matrix, for impuritydiffusion it is necessary to consider several atom-vacancy exchange rates.Five types of exchanges, between vacancy, impurity and host atoms havebeen introduced in Lidiards ‘five-frequency model’ (see Chap 7)

3 The impurity correlation factor is then no longer a constant depending

on the lattice geometry as in the case of self-diffusion It depends on thevarious jump rates of the vacancy (see Chap 7)

From the atomistic description developed in Chap 8, we get for the impuritydiffusion coefficient of vacancy-mediated diffusion in cubic Bravais lattices

where a is the lattice parameter, f2 the impurity correlation factor, and ω2

the vacancy-impurity exchange rate

To be specific, we consider in the following fcc solvents As discussed inChap 7, within the framework of the ‘five-frequency model’ the correlationfactor can be written as [3]

f2= ω1+ (7/2)F3ω3

ω2+ ω1+ (7/2)F3ω3

The various jumps rates in an fcc lattice are illustrated in Fig 7.4 For

con-venience we remind the reader of their meaning: ω is the rotation rate of

the solute-vacancy complex, ω3and ω4 denote rates of its dissociation or

as-sociation The escape probability F3 is a function of the ratio ω/ω4, where

ω denotes the vacancy jump rate in the pure matrix It is also useful to

remember that in detailed thermal equilibrium according to

the dissociation and association rates are related to the Gibbs free energy of

binding of the vacancy-impurity complex, G B = H B − T S B Equation (19.5)can then be recast to give

where H B and S B denote the binding enthalpy and entropy of the

vacancy-impurity complex and H M

2 and S M

2 the enthalpy and entropy of the

vacancy-impurity exchange jump ω2 Thus, the activation enthalpy of impurity sion is given by

diffu-∆H2= H 1V F − H B + H2M + C , (19.9)where

im-reasons, namely: (i) correlation effects, because f2 = f, (ii) differences in

the atom-vacancy exchange rates between impurity and solvent atoms,

be-cause ω2 = ω, and (iii) interaction between impurity and vacancy, because

ω4/ω3= 1 or G B = 0.

Since solute and solvent atoms are located on the same lattice and sincethe diffusion of both is mediated by vacancies, the rather small diffusivity dis-persion (see Fig 19.1) is not too surprising It reflects the high efficiency ofscreening of point charges in some metals, which normally limits the vacancy-

impurity interaction enthalpy H B to values between 0.1 and 0.3 eV Such ues are small relative to the vacancy formation enthalpies (see Chap 5) Using

val-1 Sometimes in the literature C is defined with the opposite sign.

charge ∆Ze (e = electron charge) is responsible for ∆Q Vacancy and

im-purity are considered to behave as point charges−Ze and Z2e, respectively.

In the Thomas-Fermi approximation, the excess charge of the impurity givesrise to a perturbing potential

V (r) = α ∆Ze

This equation describes a screened Coulomb potential with a screening radius

1/q, which is independent of ∆Z and can be calculated from the Fermi energy

of the host α is a dimensionless screening parameter, which depends on

∆Z [5] In the electrostatic model, the interaction enthalpy H B is equal to

the electrostatic energy V (d) of the vacancy located at a nearest-neighbour distance, d, of the impurity For the calculation of the differences in the

migration enthalpies in the second and third term of Eq (19.12), Le Clairedescribes the saddle-point configuration by two ‘half-vacancies’ located on

two neighbouring sites Each half-vacancy carries a charge Ze/2 at a distance

observed for the following solvents: noble metals and group-IIB hexagonal

metals Zn and Cd (for references see [1]) Calculations of ∆Q based on the

electrostatic model yield good agreement with experiments for impurity fusion in these metals

dif-For transition metal solutes in noble metals and for other solvents such asthe alkali metals, the divalent magnesium, and the trivalent aluminium the

values of ∆Q calculated on the basis of a screened Coulomb potential are at

variance with the experimentel values [5, 6] There are several reasons for thefailure of the electrostatic model:

(i) The choice of a screened Coulomb potential may not be appropriate forcertain solute-solvent combinations A self-consistent potential has anoscillatory form with so-called Friedel oscillations The vacancy-solute

interaction can then, for example, be attractive at the nearest neighbourpositions but repulsive at the saddle-point position.

(ii) A model based only on the difference in valence between solute andsolvent atoms disregards the effects of atomic size

(iii) Finally, the electrostatic model can only predict ∆Q It says nothing

about pre-exponential factors

Various attempts have been made in the literature to improve the theory ofimpurity diffusion For details the reader may consult older textbooks [6, 8]and the review of Neumann and Tuijn [2]

to those of Al self-diffusion, almost independent of their valence

The pressure dependence for diffusion of several impurities in aluminiumhas been studied by Rummel et al [10] and Th¨urer et al.[11] (see alsoChap 8) The activation volumes of non-transition element diffusers (Zn, Ge)are close to one atomic volume (Ω) and not much different from the activa-tion volume of self-diffusion reported by Beyeler and Adda [12] However,the transition elements (Co, Mn) are diffusers with high activation volumesbetween 2.7 and 1.67 Ω These findings have been attributed to differences

in vacancy-impurity interaction between transition and non-transition ments [10]

ele-The large ∆H2values of transition element solutes according to Eq (19.9)

indicate a strong repulsion between solute and vacancy (H B < 0) and/or

a large activation enthalpy H M

2 for solute-vacancy exchange jumps Ab tio calculations, based on the local density functional theory, of the solute-vacancy interaction energy for 3d and 4sp and for 4d and 5sp solutes inaluminium have been performed by Hoshino et al [13] They have demon-strated that for 3d and 4d impurities this interaction is indeed strongly repul-sive, whereas for 4sp and 5sp impurities it is weakly attractive Unfortunately,

ini-according to the author’s knowledge ab initio calculations of H M

2 and of theactivation volumes are not available

19.2 Impurity Diffusion in ‘Open’ Metals – Dissociative Mechanism 333

Fig 19.2 Diffusion of several impurities in Al and self-diffusion of Al (dashed line)

Roberts-Aus-Figure 19.3 shows an Arrhenius diagram of impurities in Pb together withself-diffusion (for references see the chapter of Le Claire and Neumann

in [1] and the review of Neumann and Tuijn [2]) Some solutes (e.g., Tl,Sn) show ‘normal’ behaviour However, noble metals, Ni-group elements, and

Zn have diffusivities which are three or more orders of magnitude faster thanself-diffusion Fast diffusion of 3d transition metals is also observed in someother polyvalent metals (In, Sn, Sb, Ti, Zr, Hf), and for noble metal diffusion