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

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1 Diﬀusion: Introduction and Case Studies in Metals and Binary Alloys 49

Fig. 1.25. Self-diﬀusion in three L1

structure intermetallics (Ni

Ge, Ni

Ga, Ni

Al)

and in Ni normalized to the melting temperatures (1405 K, 1383 K, 1635 K, and

1726 K).

1.11 Interdiﬀusion in Substitutional Binary Alloys

1.11.1 Boltzmann-Matano Method

A diﬀusion coeﬃcient in a binary alloy measured in a chemical composition

gradient is denoted as chemical or interdiﬀusion coeﬃcient

D (see also Sect.

1.3 and Chap. 5). The experimental procedure of an interdiﬀusion study is

schematically illustrated in Fig. 1.26. A diﬀusion couple consisting of two

homogeneous, homo-phase alloys with diﬀerent compositions (A

1−X

and

1−Y

), but within the same phase ﬁeld of the phase diagram, is formed.

Usually the thicknesses of the couple members are chosen very large as com-

pared to the average diﬀusion length. Then each couple member can be con-

sidered to be semi-inﬁnite. The interdiﬀusion proﬁle is measured after the

diﬀusion anneal, e.g., by electron microprobe analysis.

Boltzmann-Matano analysis, which is based on Boltzmann’s transforma-

tion of Fick’s second law [83] and a procedure suggested by Matano [84], is

employed to evaluate – in general concentration-dependent – interdiﬀusion

coeﬃcients from an experimental proﬁle according to

D(c

∗



2 t

∂c

∂x



∗



−1

∗



−

− x)dc. (1.82)

50 Helmut Mehrer

Fig. 1.26. Schematic illustration

of a single-phase interdiﬀusion

experiment. The original inter-

face (Kirkendall plane) is marked

by inert markers. The position of

the Matano plane (x

)isalsoin-

dicated.

Here t is the time of the diﬀusion anneal and x the position. In (1.82) x

denotes the position of the so-called Matano plane, which is deﬁned by



−

− x) dc =0 . (1.83)

−

and c

refer to the initial compositions of the two starting alloys (see

Fig. 1.26). To deduce

D one determines the Matano plane ﬁrst. Afterwards,

the area corresponding to the integral in (1.82) and the slope of the tangent

to the c(x) curve at the concentration c

∗

must be calculated (see Fig. 1.26).

This procedure is repeated for various choices of c

∗

.Inthisway

D values are

obtained for various compositions.

In (1.82) and in the rest of this chapter volume changes have been neglected

for reasons of simplicity. There are systems in which the volume will change

in the course of interdiﬀusion. The generalisation of (1.82) for non-constant

volume was given by Sauer and Freise [85] and can be found in textbooks [2,

3, 5].

1 Diﬀusion: Introduction and Case Studies in Metals and Binary Alloys 51

1.11.2 Darken’s Equations

As already mentioned in Sect. 1.3 the intrinsic diﬀusion coeﬃcients D

and

are related to the interdiﬀusion coeﬃcient

D and to the velocity of the

marker movement v

(Kirkendall velocity) in a Kirkendall experiment (see

Fig. 1.27). The relations deduced by Darken [14] are as follows:

D = X

+ X

, (1.84)

where X

and X

denote the mole fractions of the elements A and B respec-

tively. The Kirkendall velocity is given by

=(D

− D

)

∂X

∂x

(1.85)

with ∂X

/∂x denoting the concentration gradient at the Kirkendall plane.

Using (1.84) and (1.85) the intrinsic diﬀusion coeﬃcients D

and D

can be

deduced if the interdiﬀusion coeﬃcient and the Kirkendall velocity are known

from the experiment.

The reason why diﬀusion occurs is always a decrease in the Gibbs free

energy of the system. This implies that the atoms are diﬀusing from regions

where their chemical potential is high to regions where it is low, i.e., the

driving force for diﬀusion is the gradient of the chemical potential (see also

Sect. 1.3.2). As a consequence the interdiﬀusion coeﬃcient

D is related to the

tracer self-diﬀusion coeﬃcients of the components in a homogeneous alloy via

D =(X

∗

+ X

∗

) Φ. (1.86)

Here X

and X

denote the mole fractions of the components and Φ is the

thermodynamic factor of the alloy. Using a classical result from thermody-

namics of binary systems, the thermodynamic factor can be written as [2]

Fig. 1.27. Schematic illustration of the Kirkendall eﬀect in a diﬀusion couple

composed of two pure metals A and B. The circles represent inert markers inserted

at the original interface. The line at time t represent the position of the markers

after the diﬀusion anneal. The solid line is a graph of the concentration of A atoms

in the couple after diﬀusion time t. As drawn, the B atoms diﬀuse into A faster than

those of A diﬀuse into B. As a result the interface moves to the right (D

52 Helmut Mehrer

Φ =

∂ln a

∂ln X

=1+

∂ln γ

∂ln X

, (1.87)

where G denotes Gibbs free energy, a

the (chemical) activity, and γ

the

coeﬃcient of activity of species i ( = A or B). According to the Gibbs-Duhem

relation for binary systems, the thermodynamic factor is identical for both

species. Equations (1.84), (1.85), and (1.86) are denoted as the equations of

Darken.

The thermodynamic factor is unity for ideal solutions. It is larger than

unity for phases with negative deviations from ideality and smaller than unity

in the opposite case. Negative deviations are expected for systems with or-

der. Therefore thermodynamic factors of intermetallic compounds are often

larger, sometimes even considerably larger than unity due to the attractive

interaction between the constituents of the intermetallic phase. As a con-

sequence interdiﬀusion coeﬃcients are often larger than the bracket term in

(1.86). The latter corresponds to a weighted average of the tracer diﬀusivities.

Activation enthalpies of interdiﬀusion are often smaller than those for tracer

diﬀusion due to the temperature variation of the thermodynamic factor.

1.11.3 Darken-Manning Relations

The Darken relation between tracer diﬀusivities and the chemical diﬀusion

coeﬃcient (1.86) is widely used in practice. Although it remains a very useful

equation it was recognized soon that from a theoretical viewpoint (1.86) is

incomplete. Manning [15, 86] developed equations for vacancy-mediated dif-

fusion in a so-called ’random alloy’ (vacancies and A and B atoms distributed

at random on the same lattice). In general, the results are like those obtained

by Darken but with ’vacancy-wind’ corrections

. The intrinsic diﬀusion co-

eﬃcients and the interdiﬀusion coeﬃcient, respectively, are given by

= D

∗



1 − f

∗

− D

∗

)

∗

+ X

∗

)



(1.88)

and an analogous equation for D

,and

D =(X

∗

+ X

∗

) Φ



1 − f

∗

− D

∗

)

∗

+ X

∗

)(X

∗

+ X

∗

)



. (1.89)

The factors in square brackets are Manning’s vacancy-wind corrections to

Darken’s equations. One also ﬁnds that the Kirkendall velocity is increased

by the addition of a factor f

−1

to (1.85). As usual f denotes the correla-

tion factor of self-diﬀusion. In substitutional alloys the Manning corrections,

A transparent derivation can be found in [4].

1 Diﬀusion: Introduction and Case Studies in Metals and Binary Alloys 53

while signiﬁcant, will never be very large even when one of the tracer self-

diﬀusivities is much greater than the other. For instance the Kirkendall ve-

locity is increased by about 30 % in an fcc alloy.

It is a debated question whether the Darken-Manning equations can be

applied to ordered intermetallics. The answer depends on the structure, the

type of disorder, and the diﬀusion mechanism. For example, according to

Belova and Murch [87] (1.89) can be used for B2 compounds with antisite

disorder.

1.12 Multiphase Diﬀusion in Binary Systems

The experimental procedure in a multiphase diﬀusion experiment is schemat-

ically illustrated in the upper part of Fig. 1.28. A diﬀusion couple is formed

from the two elements A and B or from two compounds A

1−X

and

1−Y

. Let us suppose that the phase diagram of the system contains inter-

metallics. Then interdiﬀusion will give rise to the formation of intermetallic

layers. A hypothetical phase diagram (containing only one intermetallic com-

pound) and a concentration proﬁle resulting from interdiﬀusion is illustrated

in the lower part of Fig. 1.28.

Fig. 1.28. Schematic illustration of a multi-phase diﬀusion experiment. Upper

part: Diﬀusion couple before and after a diﬀusion anneal. Lower part: Interrelation

between a hypothetical phase diagram and an interdiﬀusion proﬁle after a diﬀusion

anneal at temperature T

54 Helmut Mehrer

The name multi-phase diﬀusion emphasizes the diﬀusional aspect of the

whole process. However, there is also the aspect of a chemical reaction be-

tween the atomic species at the phase boundaries. The overall kinetics of

phase formation and growth can be either governed by diﬀusion across the

growing product phase(s) or by reaction(s) occurring at the interfaces. In the

ﬁrst case the process is denoted to be diﬀusion controlled; in the second case

it is called to be reaction controlled. In general the kinetics of formation and

growth of intermetallic layers will be the result of both processes. For this

phenomenon the name reaction (reactive) diﬀusion has been suggested. In

very thin layers gradients are very large and diﬀusion processes proceed very

fast. Then the process is in the interface-reaction controlled regime. However,

during progress of reaction diﬀusion the layer thickness increases. Then dif-

fusion of the reacting species needs more and more time. Finally, diﬀusion

processes determine the overall kinetics [88,89].

When the whole process is diﬀusion controlled, the growth of the inter-

metallic layers in a binary system obeys a parabolic growth law

∆x

=2k

t, (1.90)

where ∆x

denotes the thickness of the layer. Kidson has shown that growth

constants have the following meaning [90]:



γβ

−

βγ

− C

γβ



−



βα

−

αβ

− C

βα



. (1.91)

αβ

denotes the equilibrium composition (in atomic fractions) on the α-side

of an α/β-interface,

αβ

the interdiﬀusion coeﬃcient in the α-phase near the

α/β -interface, and K

αβ

is the composition gradient in the α-phase near the

α/β-interface in a diagram of concentration versus x/

√

t. As already men-

tioned, in the derivation of (1.91) any explicit inﬂuence of interface processes

like phase nucleation, atomic transfer across the interface, and the creation

and/or annihilation of point defects at the interfaces has been disregarded.

This is only justiﬁed for long diﬀusion times. Then the growth process is

diﬀusion controlled [91, 92].

We consider as an example multiphase diﬀusion in the system Ni-Al [93].

Apart from the primary and terminal solid solutions ﬁve intermetallic com-

pounds are present in the phase diagram of Ni–Al: Al

Ni, Al

, AlNi,

and Ni

Al [94]. Fig. 1.29 shows an optical micrograph of a diﬀusion

couple of pure Al and Ni after a diﬀusion anneal of 48 h at 823 K. Layers

of the Al-rich intermetallic compounds Al

Ni and Al

are clearly visible.

Their compositions can be identiﬁed in the proﬁle across the diﬀusion zone

measured by electron microprobe analysis. The layer thicknesses of Ni-rich

compounds – at the relatively low temperature of the diﬀusion anneal – are

too small to be detectable in Fig. 1.29. As can be seen from Fig. 1.30 the

1 Diﬀusion: Introduction and Case Studies in Metals and Binary Alloys 55

50 m

(Al)

(Ni)

Al Ni

Fig. 1.29. Optical micrograph of Al–Ni diﬀusion couple after a diﬀusion anneal of

48 h at 823 K (left). Composition proﬁle obtained by electron microprobe analysis

(right).

compound Al

indeed shows a parabolic growth. The growth constant

obtained from these experiments is thermally activated with an enthalpy of

126 kJ mol

−1

We emphasize that according to (1.91) growth constants have a complex

meaning. They depend on diﬀusivities in the bordering layers as well as on

the diﬀusivity of the growing phase, on the concentration gradients on both

sides of the interfaces, and on the (in general temperature dependent) solubil-

ity limits of the phases. If one deduces an activation enthalpy for the growth

process it has a complex meaning as well. Its value is usually not identical

with the activation enthalpy of interdiﬀusion in the growing layer. The ac-

tivation enthalpy assigned to the growth of an intermetallic layer is at most

only an average of several more fundamental activation enthalpies. If a linear

Arrhenius plot is observed for the growth constant, this is only empirically

useful. It indicates either that one of the terms dominates over the others or

that the activation enthalpies are similar to each other. In addition, growth

constants may be inﬂuenced by mass transport along diﬀusion short circuits

like grain boundaries in the growing layer.

56 Helmut Mehrer

Fig. 1.30. Parabolic

growth of the inter-

metallic compound

in Al–Ni diﬀu-

sion couples.

1.13 Conclusion

Diﬀusion is an important topic of materials science since the phenomenon of

diﬀusion is of both practical importance and of fundamental interest. In this

chapter we have outlined the continuum description and the basic atomic

mechanisms of bulk diﬀusion in solids. The various diﬀusion coeﬃcients,

which are relevant for diﬀusion in solid elements and binary alloys, have been

introduced: the tracer diﬀusion coeﬃcient(s) of self-diﬀusion, tracer diﬀusion

coeﬃcients of foreign elements, the chemical (or interdiﬀusion) coeﬃcient in

a composition gradient, and the intrinsic diﬀusion coeﬃcients. The most im-

portant experimental method for diﬀusion studies is the tracer technique in

combination with mechanical and/or sputter sectioning for depth proﬁling.

Various other techniques for the measurement of diﬀusion proﬁles have been

also considered. Indirect methods including mechanical and magnetic relax-

ation and nuclear methods such as nuclear magnetic relaxation, M¨oßbauer

spectroscopy and quasielastic neutron scattering have been mentioned only

brieﬂy since they are the main topics of Chaps. 2, 3 and 9.

Self-diﬀusion is the most basic diﬀusion process in a solid. It is well studied

for most metallic elements. In close-packed metals self-diﬀusion occurs pre-

dominantly by a vacancy mechanism with some small contributions of diva-

cancies at temperatures above 2/3 of the melting temperature. Self-diﬀusion

in bcc metals is also vacancy-mediated. In a homologous temperature scale

1 Diﬀusion: Introduction and Case Studies in Metals and Binary Alloys 57

it is faster and the range of diﬀusivities is much wider than for close-packed

metals.

Diﬀusion of small foreign atoms proceeds much faster than self-diﬀusion

of the host metal. Small foreign atoms such as H, C, N, and O are usually

incorporated in octahedral or tetrahedral interstitial sites of the host lattice

and diﬀuse via a direct interstitial mechanism. Diﬀusion of hydrogen is a

particularly interesting case for scientiﬁc and application-oriented reasons. Its

diﬀusion is extremely rapid, non-classical isotope eﬀects occur, and quantum

eﬀects may cause strong deviations from a linear Arrhenius-like temperature

dependence of H diﬀusion.

Diﬀusion of solutes in substitutional alloys is vacancy-mediated. Solute

diﬀusion in very dilute alloys (impurity diﬀusion) is normally described by

diﬀusion coeﬃcients which lie in a relatively narrow band around those of

self-diﬀusion of the host (solvent). This rather small diﬀusivity dispersion

is understandable since solute and solvent atoms are located on sites of the

same lattice and both use vacancies as diﬀusion vehicles. It reﬂects the high

eﬃciency for screening of electrical point charges of some metallic hosts such

as the noble metals and zinc. This eﬃcient screening limits the solute-vacancy

interaction to relatively small values.

Remarkable exceptions from ‘normal’ behaviour of solute diﬀusion are

observed for polyvalent metals: Transition-metal solutes in aluminium are

extremely slow diﬀusers with high activation energies, high pre-exponential

factors and high activation volumes. Very likely this indicates a strong repul-

sion between vacancy and solute somewhere on the vacancy-solute exchange

path. Fast solute diﬀusion is observed, e.g., for noble metal solutes in lead

and tin and for late-transition-element solutes in group IVA solvents (zirco-

nium, titanium and hafnium). The fast diﬀusion has been attributed to the

dissociative mechanism. The latter operates for solutes which are dissolved

on substitutional and interstitial sites. Even when its interstitial fraction is

very small it can be responsible for the rapid diﬀusion. Fast solute diﬀusion

is also observed in the semiconductors silicon and germanium (see Chap. 4).

Systematic studies of diﬀusion in intermetallics (intermetallic compounds

and ordered alloys) have become available only recently. An understanding

of diﬀusion in terms of atomic mechanisms is more complex than for metal-

lic elements. The current knowledge about self-diﬀusion has been illustrated

for binary intermetallics with B2-, L1

-, and D0

-structures. Important fac-

tors which inﬂuence diﬀusion such as the crystal structure, the state of order

and disorder, the temperature and the composition have been illustrated. In

a broad sense diﬀusion is mediated by vacancy-type defects, which include

triple-defects and antistructure-bridge mechanisms. Relevant atomic mech-

anism must take into account that the degree of order in the material is

maintained during diﬀusion. Despite of the progress made in recent years,

diﬀusion in intermetallics is a ﬁeld that deserves further attention.

58 Helmut Mehrer

The ﬁeld of interdiﬀusion in binary alloys has been only touched in this

chapter. The Boltzmann-Matano method (or a related method) is necessary

to deduce interdiﬀusion coeﬃcients from composition proﬁles. A detailed in-

terpretation of the interdiﬀusion coeﬃcient requires also measurements of

the Kirkendall eﬀect. The analysis given by Darken and its reﬁnement by

Manning permits to deduce the intrinsic diﬀusion coeﬃcients. The latter are

related to the tracer diﬀusion coeﬃcients of the components in homogeneous

alloys via the thermodynamic factor of the alloy and the so-called vacancy-

wind corrections.

Multiphase diﬀusion or reaction diﬀusion occurs in a binary alloy system

with one or several intermediate phases. In an appropriate diﬀusion couple

layers of the intermediate phase(s) form and grow. Their growth is often diﬀu-

sion controlled. Then their growth constants are related to the interdiﬀusion

coeﬃcients.

Notation

a cubic lattice parameter

activity (chemical) of species i

c volume concentration of diﬀusing species

) mole fraction of vacancies (in thermal equilibrium)

) mole fraction of divacancies (in thermal equilibrium)

) mole fraction of interstitial solutes (in equilibrium)

) mole fraction of substitutional solutes (in equilibrium)

D diﬀusion coeﬃcient (second rank tensor)

III

diﬀusion coeﬃcients in principal diﬀusion directions

D diﬀusion coeﬃcient, also self-diﬀusion coeﬃcient in a cubic

crystal

D interdiﬀusion coeﬃcient (also chemical diﬀusion coeﬃcient)

diﬀusion coeﬃcient of solute (impurity) in dilute alloy (also

∗

)

diﬀusion coeﬃcient of a solute in interstitial sites

∗

, tracer self-diﬀusion coeﬃcients of in pure metal A

∗

, D

∗

tracer diﬀusion coeﬃcients of component A or B in a binary

material

∗

, D

∗

, tracer diﬀusion coeﬃcients of impurity C in pure metal A

or in a binary A-B material

pre-exponential factor of diﬀusion (including diﬀusion en-

tropy)



pre-exponential factor of diﬀusion (without diﬀusion en-

tropy)

f correlation factor, correlation factor of self-diﬀusion

correlation factor of solute diﬀusion

g geometry factor