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 9

Fig. 1.1. Initial conﬁgurations for diﬀusion experiments:

a) Thin layer of A

∗

on A: tracer diﬀusion in pure elements.

b) Thin layer of A

∗

or B

∗

on homogeneous alloy: tracer diﬀusion of alloy compo-

nents.

c) Thin layer of C

∗

on element A or homogeneous alloy: Impurity diﬀusion.

d) Diﬀusion couple between metal-hydrogen alloy and a pure metal.

e) Diﬀusion couple between pure end-members.

f) Diﬀusion couple between two homogeneous alloys.

depends on the alloy composition. For ideal solutions the chemical potentials

are

= µ

+ RT ln

+ n

, (1.17)

where µ

depend on T and p only. In this case the gradient of the chemical

potential is directly proportional to the logarithmic gradient of the concen-

tration. In non-ideal solutions the gradient of the chemical potential gives

rise to an ‘internal’ driving force. As a consequence the interdiﬀusion diﬀu-

sion coeﬃcient is concentration-dependent and Fick’s equation in the form

of (1.13) must be used.

Examples of diﬀusion couples which entail an interdiﬀusion coeﬃcient are

(see Fig. 1.1):

(i) Pure end-member diﬀusion couples consisting of two samples of pure ele-

ments joined together (e.g. Ni/Pd, Cu/Ag, ...).

(ii) Incremental diﬀusion couples consisting of two samples of homogenous

alloys joined together (e.g. Ni

/Ni

,Ni/Ni

, ...).

(iii) Diﬀusion couples which involve solutions of hydrogen in a metal (e.g. Pd-

H/Pd, Ag

1−X

/Ag

1−Y

, ...). Binary metal-hydrogen systems are often

special in the sense that hydrogen is the only mobile component.

Interdiﬀusion results in a composition gradient in the diﬀusion zone. Inter-

diﬀusion proﬁles are analysed by the Boltzmann-Matano method or related

procedures. This method will be described in Sect. 1.11. It permits to deduce

the concentration dependence of the interdiﬀusion coeﬃcient

10 Helmut Mehrer

D =

D(c) (1.18)

from the experimental diﬀusion proﬁle.

1.3.3 Intrinsic Diﬀusion Coeﬃcients

The intrinsic diﬀusion coeﬃcients (sometimes also component diﬀusion coeﬃ-

cients) D

and D

of a binary A-B alloy describe diﬀusion of the components

A and B relative to the lattice planes. The diﬀusion rates of A and B atoms

are usually not equal. Therefore, in an interdiﬀusion experiment a net ﬂux of

atoms across any lattice plane exists. The shift of lattice planes with respect

to a sample ﬁxed axis is denoted as Kirkendall eﬀect, which is illustrated

in Fig. 1.27 in Sect. 1.11. The Kirkendall shift can be observed by incorpo-

rating inert markers at the initial interface of a diﬀusion couple. This shift

was for the ﬁrst time observed for Cu/Cu-Zn diﬀusion couples by Kirkendall

and coworkers [13]. In the following decades work on many diﬀerent alloy

systems and a variety of markers demonstrated that the Kirkendall eﬀect is

a widespread phenomenon of interdiﬀusion.

The intrinsic diﬀusion coeﬃcients D

and D

of a substitutional binary

A-B alloy are related to the interdiﬀusion coeﬃcient

D and the marker veloc-

ity v

(Kirkendall velocity). These relations were deduced for the ﬁrst time

by Darken [14] and reﬁned later on by Manning [15]. They will be discussed

in Sect. 1.11. If the quantities

D and v

are known from experiment the

intrinsic diﬀusion coeﬃcients can be deduced.

We emphasize that the intrinsic diﬀusion coeﬃcients and the tracer diﬀu-

sion coeﬃcients are diﬀerent. D

and D

pertain to diﬀusion in a composition

gradient whereas D

∗

and D

∗

are determined in a homogenous alloy. They

are approximately related (see Sect. 1.11 for details) via

= D

∗

Φ and D

= D

∗

Φ, (1.19)

where Φ denotes the so-called thermodynamic factor (see Sect. 1.11). In a

metal-hydrogen system ususally only the H atoms are mobile. Then the in-

trinsic diﬀusion coeﬃcient and the chemical diﬀusion coeﬃcient of hydrogen

are identical.

1.4 Experimental Methods

Methods for the measurement of diﬀusion coeﬃcients can be grouped into two

major categories: Direct methods are based on Fick’s laws and the phenom-

enological deﬁnition of the diﬀusion coeﬃcients given in Sect. 1.3. Indirect

methods are not based directly on Fick’s laws. Their interpretation requires a

microscopic model of the atomic jump processes and then uses the Einstein-

Smoluchowski relation to deduce a diﬀusion coeﬃcient.

1 Diﬀusion: Introduction and Case Studies in Metals and Binary Alloys 11

1.4.1 Direct Methods

Trace r Metho d

The tracer method is the most direct and most accurate technique for the

determination of diﬀusion coeﬃcients in solids. Since very tiny amounts of

trace isotopes can be applied in tracer experiments, the chemical composi-

tion of the sample is practically not inﬂuenced by the tracer. In this way

self-diﬀusion and impurity diﬀusion can be studied in a material which is

homogeneous – apart from the tracer gradient.

As indicated schematically in Fig. 1.2 the tracer is usually deposited onto

a polished, ﬂat surface of the diﬀusion sample. Normally a radioactive isotope

of the investigated atomic species is used as tracer. Enriched stable isotopes

have also been used in a few cases. Evaporation, dripping of a liquid solution,

and electrodeposition of the tracer onto the surface are common deposition

techniques. Sometimes the tracer is ion-implanted as a thin layer below the

sample surface in order to overcome disturbing surface oxide hold-up and sol-

ubility problems [16]. The sample is usually encapsulated in quartz ampoules

under vacuum or inert (e.g. Ar) atmosphere and an isothermal diﬀusion an-

neal is performed at temperature T for some diﬀusion time t. For tempera-

tures below 1500 K quartz ampoules and resistance furnaces can be used. For

higher temperatures more sophisticated heating techniques are necessary.

The best way to determine the resulting concentration-depth proﬁle is

serial sectioning of the sample and subsequent determination of the amount

of tracer per section.

For average diﬀusion lengths of at least several ten micrometers mechan-

ical sectioning techniques are applicable (for a review see, e.g., [17]). Lathes

and microtomes are appropriate for ductile, grinding devices for brittle sam-

ples. For extended diﬀusion anneals and diﬀusivities D>10

−15

−1

lathe

sectioning is suﬃcient whereas diﬀusivities D>10

−17

−1

are accessible

by microtome sectioning. In favourable cases, grinder sectioning can be used

for diﬀusivities D>10

−18

−1

- - -

ln c

slope:−

4 D

∗

Fig. 1.2. Schematic illustration of the tracer method: The major steps – deposition

of the tracer, diﬀusion anneal, serial sectioning, and evaluation of the penetration

proﬁle – are indicated.

12 Helmut Mehrer

Diﬀusion studies at lower temperatures require measurements of very

small diﬀusivities. Measurements of diﬀusion proﬁles with diﬀusion lengths

in the micrometer or nanometer range are possible using sputter section-

ing techniques. Devices for serial sectioning of radioactive diﬀusion samples

by ion-beam-sputtering are described in [18, 19]. Such devices permit serial

sectioning of shallow diﬀusion zones, which correspond to average diﬀusion

lengths between several nm and a few µm. This implies that for anneal-

ing times of about 10

s a diﬀusivity range between D ≈ 10

−24

−1

and

D ≈ 10

−16

−1

can be examined.

Provided that the experimental conditions were chosen in such a way

that the deposited layer is thin compared with the mean diﬀusion length,

the distribution after the diﬀusion anneal is described by (1.9). If radioactive

tracers are used, the speciﬁc activity per section (count rate divided by the

section mass) is proportional to the tracer concentration. The count rate

is conveniently determined by nuclear counting facilities (γ-orβ-counting,

depending on the isotope). According to (1.9) a plot of the logarithm of

the speciﬁc activity versus the penetration distance squared is linear, if bulk

diﬀusion is the dominating diﬀusion process. Its slope equals −(4D

∗

−1

From the slope and the diﬀusion time the tracer diﬀusivity D

∗

is obtained.

An obvious advantage of the tracer method is that a determination of the

absolute tracer concentration is not necessary.

Fig. 1.3 shows a penetration proﬁle of the radioisotope

Fe in the inter-

metallic phase Fe

Si obtained by grinder sectioning [20]. Gaussian behaviour

as stated by (1.9) is observed over several orders of magnitude in concentra-

tion. An example for a penetration proﬁle of

Fe in the intermetallic phase

Al obtained with the sputtering device described in [18] is displayed in

Fig. 1.4 according to [21]. From diﬀusion proﬁles of the quality of Figs. 1.3

and 1.4 diﬀusion coeﬃcients can be determined with an accuracy of a few

percent.

Deviations from the Gaussian behaviour in experimentally determined

penetration proﬁles may occur for many reasons. We mention two frequent

ones:

– Grain boundaries in a polycrystal often act as diﬀusion short-circuits with

enhanced mobility of atoms. Grain boundaries usually cause a ‘grain-

boundary tail’ in the deeper penetrating part of the proﬁle. In this ‘tail’

region the concentration of the diﬀuser is enhanced with respect to mere

bulk diﬀusion.

– Evaporation losses of the tracer itself and/or of the diﬀusion sample will

cause deviations from Gaussian behaviour in the near-surface region.

For a more detailed discussion of implications and pitfalls of the tracer

method the reader is referred to [17]. The grain-boundary tails mentioned

above can be used for a systematic study of grain-boundary diﬀusion in bi-

or polycrystals as described in Chap. 8 and in [10].

1 Diﬀusion: Introduction and Case Studies in Metals and Binary Alloys 13



Fig. 1.3. Penetration proﬁle of the ra-

dioisotope

Fe in Fe

Si obtained by

grinder sectioning. The solid line rep-

resents a ﬁt of the thin-ﬁlm solution of

Fick’s second law to the data points.



Fig. 1.4. Penetration proﬁle of the ra-

dioisotope

Fe in Fe

Al obtained by

sputter sectioning. The solid line rep-

resents a ﬁt of the thin-ﬁlm solution of

Fick’s second law to the data points.

In some cases several tracer isotopes of the same element are available.

Diﬀerences between the isotopic masses lead to isotope eﬀects in diﬀusion.

Isotope eﬀects are interesting phenomena although the diﬀerences between

diﬀusivities of two isotopes of the same element are usually a few percent

only. An exception is hydrogen with its three isotopes H, D, and T, which

have signiﬁcantly diﬀerent masses (see Sect. 1.7.2). Isotope eﬀects of self- and

solute-diﬀusion in metals can contribute useful information about the atomic

mechanisms of diﬀusion. For a detailed discussion the reader is referred to [22–

24] (see also Chap. 6, Sect. 6.5.3).

Other Proﬁling and Detection Methods

Several other proﬁling and detection methods can be used to measure

concentration-depth proﬁles. We mention the more important ones:

1. Secondary Ion Mass Spectrometry (SIMS)

As already mentioned foreign elements or stable isotopes of the matrix

can be used as tracers in combination with SIMS for depth proﬁling. SIMS

is mainly appropriate for the diﬀusion of foreign elements. Contrary to

self-diﬀusion studies by radiotracer experiments, in the case of stable

tracers the natural abundance of the stable isotope in the matrix limits

the concentration range of the diﬀusion proﬁle. Highly enriched isotopes

should be used. An example of this technique can be found in a recent

14 Helmut Mehrer

Fig. 1.5. Interdif-

fusion proﬁle of a

–Fe

couple measured by

EMPA.

study of Ni self-diﬀusion in the intermetallic compound Ni

Al in which

the highly enriched stable

Ni isotope was used [25]. Average diﬀusion

lengths between several nm and several µm are accessible.

2. Electron Microprobe Analysis (EMPA)

In EMPA an electron beam of several tens of keV with a diameter of

about one micrometer stimulates X-ray emission in the diﬀusion zone

of the sample. The diﬀusion proﬁle can be obtained by analysing the

intensity of the characteristic radiation of the elements in a line scan along

the diﬀusion direction. The detection limit is about 10

−3

to 10

−4

mole

fractions depending on the element. Light elements cannot be analysed.

Because of the ﬁnite size of the excited volume (several µm

)onlyfairly

large diﬀusion coeﬃcients ≥ 10

−15

−1

can be measured. EMPA is

mainly appropriate for interdiﬀusion studies. An example of a single-

phase interdiﬀusion proﬁle resulting from a Fe

–Fe

couple is

shown in Fig. 1.5 according to [26]. The analysis of interdiﬀusion proﬁles

is discussed in Sect. 1.11.

3. Auger Electron Spectroscopy (AES)

AES in combination with sputter proﬁling can be used to measure dif-

fusion proﬁles in the range of several nm to several µm. It is, however,

only applicable to diﬀusion of foreign atoms since AES only discriminates

between diﬀerent elements.

4. Rutherford Backscattering Spectrometry (RBS)

In RBS experiments a high-energy beam of monoenergetic α-particles is

used. These particles are preferentially scattered by heavy nuclei in the

sample and the energy spectrum of scattered α-particles can be used to

determine the concentration-depth distribution of scattering nuclei. This

technique is mainly suitable for detecting heavy elements in a matrix of

1 Diﬀusion: Introduction and Case Studies in Metals and Binary Alloys 15

substantially lower atomic weight. Due to the energy straggling of the

incident beam the proﬁle depth is limited to less than a few µm.

5. Nuclear Reaction Analysis (NRA)

High energy particles can also be used to study diﬀusion of light elements,

if the nuclei undergo a suitable resonant nuclear reaction. An example is

diﬀusion of boron in an alloy. During irradiation with high energy pro-

tons α-particles are emitted from the nuclear reaction

B+p→

B+α.

The concentration proﬁle of

B can be determined from the number and

energy of emitted α-particles as a function of the incident proton energy.

Like in RBS energy straggling limits the depth resolution of NRA.

RBS and NRA methods need a depth calibration which is based on not

always very accurate data for the stopping power in the matrix for those

particles emitted by the nuclear reaction. Also the depth resolution is usu-

ally inferior to that achievable in careful radiotracer and SIMS proﬁling

studies.

6. Field Gradient Nuclear Magnetic Resonance (FG NMR, PFG NMR)

Nuclear magnetic resonance (NMR) measurements in a magnetic ﬁeld

gradient (FG) or in a pulsed ﬁeld gradient (PFG, see Chap. 10) provide

a direct macroscopic method for diﬀusion studies. In a magnetic ﬁeld

gradient the Larmor frequency of a nuclear moment depends on its po-

sition. FG NMR and PFG NMR utilize the fact that nuclear spins that

diﬀuse in a magnetic ﬁeld gradient experience an irreversible phase shift,

which leads to a decrease in transversal magnetization. This decrease can

be observed in so-called spin-echo experiments [27, 28]. A measurement

of the diﬀusion-related part of the spin echo provides the diﬀusion co-

eﬃcient without any further hypothesis. In contrast to tracer diﬀusion,

FG NMR and PFG NMR techniques permit diﬀusion measurements in

isotopically pure systems. These techniques are applicable for relatively

large diﬀusion coeﬃcients D  10

−13

−1

[29].

1.4.2 Indirect Methods

Indirect methods are based on phenomena which are inﬂuenced by the dif-

fusive jumps of atoms. Some of these methods are often sensitive to one or

a few atomic jumps only. Quantities like relaxation times, relaxation rates

or linewidths are measured. Using a microscopic model of the jump process

the mean residence time of the diﬀusing species is determined and then via

the Einstein-Smoluchowski relation (see Sect. 1.6) the diﬀusivity is deduced.

Indirect methods can be grouped into two categories – relaxation methods

(mechanical and magnetic) and nuclear methods.

16 Helmut Mehrer

Relaxation Methods (Mechanical and Magnetic)

Mechanical relaxation methods make use of the fact that atomic motion in

a material can be induced by external inﬂuences such as the application

of constant or oscillating mechanical stress. In ferromagnetic materials the

interaction between the magnetic moments and local order can give rise to

various relaxation phenomena similar to those observed in anelasticity. A

great variety of experimental devices have been used for such studies. Their

description is, however, beyond the scope of this chapter.

Some of the more important relaxation phenomena related to diﬀusion

are the following [30–32]:

The Snoek eﬀect is observed in bcc metals which contain interstitial

solutes such as C, N, or O. These solutes occupy octahedral or tetrahedral

interstitial sites. These sites have tetragonal symmetry, which is lower than

the cubic symmetry of the matrix. Therefore the lattice distortions caused by

interstitial solutes give rise to elastic dipoles. Under the inﬂuence of external

stress these dipoles can reorient (para-elasticity). The reorientation of solutes

gives rise to a strain relaxation or an internal friction peak. The relaxation

time or the (frequency or temperature) position of the internal friction peak

can be used to deduce information about the mean residence time of a solute.

A Snoek eﬀect of interstitial solutes in fcc metals cannot be observed, because

the interstitial sites have cubic symmetry.

The Gorski eﬀect is due to solutes in a solvent which produce a lattice

dilatation. In a macroscopic strain gradient solutes redistribute by diﬀusion.

This redistribution gives rise to an anelastic relaxation. The Gorski eﬀect

is detectable if the diﬀusion coeﬃcient of the solute is high enough. Gorski

eﬀect measurements have been widely used for studies of hydrogen diﬀusion

in metals [30].

In substitutional A-B alloys the reorientation of solute-solvent pairs under

the inﬂuence of stress can give rise to an anelastic relaxation called Zener

eﬀect.

Nuclear Methods

Examples of nuclear methods are NMR, M¨oßbauer spectroscopy (MBS), and

quasielastic neutron scattering (QENS). Since MBS, QENS, NMR and PFG

NMR are the subjects of the Chaps. 2, 3, 9 and 10 and QENS also of a recent

textbook [33] we conﬁne ourselves here to a few remarks:

The width of the resonance line and the spin-lattice relaxation rate T

−1

in NMR have contributions which are due to the thermally activated jumps

of atoms. Measurements of the ’diﬀusional narrowing’ of the linewidth or

of T

−1

as a function of temperature permit a determination of the mean

residence time τ of the atoms. NMR methods are mainly appropriate for

self-diﬀusion measurements on solid or liquid metals. In favourable cases (e.g.

Li and Na) self-diﬀusion coeﬃcients between 10

−18

and 10

−10

−1

are

1 Diﬀusion: Introduction and Case Studies in Metals and Binary Alloys 17

accessible (see [6]). In the case of foreign atom diﬀusion, NMR studies suﬀer

from the fact that a signal from nuclear spins of the minority component must

be detected. Nevertheless, detailed studies were conducted, e. g., in the case

of Li-based Li-Mg and Li-Ag alloys via the spin-lattice relaxation of polarized

radioactive

Li nuclei [34].

The linewidth in MBS and QENS both have a contribution ∆Γ which is

due to the diﬀusion jumps of atoms. This diﬀusion broadening can be ob-

served only in systems with fairly high diﬀusivities since ∆Γ must at least

be comparable with the natural linewidth in MBS experiments or with the

energy resolution of the neutron spectrometer in QENS experiments. Appro-

priate probes for MBS must be available. The usual working horse in MBS

is the isotope

Fe although there are a few other M¨oßbauer isotopes avail-

able (e.g.

119

Sn,

115

Eu,

161

Dy). MBS has been mainly used to study fast Fe

diﬀusion. QENS experiments are suitable for fast diﬀusing elements with a

sizable incoherent scattering cross section for neutrons. Examples are hydro-

gen diﬀusion in metals or hydrides and Na self-diﬀusion (see Chap. 3).

Neither MBS nor QENS are routine methods for diﬀusion measurements.

The most interesting aspect is that these methods can provide microscopic

information about the elementary jump process of atoms. For single crystals

∆Γ depends on the crystal orientation. This orientation dependence can be

used to deduce information about the jump direction and about the jump

length (see Chaps. 2 and 3), which is not accessible by conventional diﬀusion

studies.

For a more comprehensive discussion of experimental methods for the

determination of diﬀusion coeﬃcients we refer the reader to the already men-

tioned textbooks on diﬀusion [1–3] and to Chap. 1 in [6] as well as to a recent

article [29] where also an overview of the accessible windows for the mean

residence time τ are given.

1.5 Dependence of Diﬀusion on Thermodynamic

Variables

So far we have said nothing about the dependence of diﬀusion processes upon

thermodynamic variables, i.e. upon temperature T and hydrostatic pressure

p. In binary systems also variations of the diﬀusivity with the variable ’compo-

sition’ are of interest. These variations can range from very slight to striking.

They will be not considered in this section since they depend very much on

the system. Examples can be found in Sect. 1.10.

1.5.1 Temperature Dependence

It is well known that diﬀusion coeﬃcients in solids generally depend rather

strongly on temperature, being low at low temperatures but appreciable at

18 Helmut Mehrer

high temperatures. Empirically, measurements of diﬀusion coeﬃcients over a

certain temperature range may be often, but by no means always, described

by an Arrhenius relation

D = D

exp



−

∆H



. (1.20)

In (1.20) D

is denoted as pre-exponential factor and ∆H (or Q)

as activa-

tion enthalpy of diﬀusion. The pre-exponential factor can be written as

= D



exp

∆S

, (1.21)

where ∆S is the diﬀusion entropy and D



contains geometric factors, the

correlation factor, the lattice parameter squared, and an attempt frequency

of the order of the Debye frequency.

In an Arrhenius diagram the logarithm of the diﬀusivity is plotted versus

the reciprocal temperature T

−1

. For a diﬀusion process with a temperature-

independent activation enthalpy ∆H the Arrhenius diagram is a straight line

with slope −∆H/R. From its intercept for T

−1

→ 0 the pre-exponential

factor D

can be deduced. Such simple Arrhenius behaviour should, how-

ever, not be considered to be universal. Departures from it may arise for

many reasons, ranging from fundamental aspects of the atomic mechanism,

temperature dependent activation parameters

, eﬀects associated with im-

purities or microstructural features such as grain boundaries. Nevertheless,

(1.20) provides a very useful standard.

1.5.2 Pressure Dependence

The variation of diﬀusivity with hydrostatic pressure p is far less pronounced

than with temperature. Usually the diﬀusivity decreases as the pressure is

Equation (1.20) is often also written as

D = D

exp −

∆H

If the ﬁrst version of the Arrhenius equation is used the unit of ∆H is kJ mol

−1

If the second version is used the appropriate unit of ∆H is eV per atom. Note

that 1 eV per atom = 96.472 kJ mol

−1

. The gas constant R and the Boltzmann

constant k

are related via R = N

=8.314 ×10

−3

kJ mol

−1

,whereN

denotes the Avogadro constant.

The symbol Q for the activation enthalpy is also widely used in the literature.

Thermodynamics tells us that a temperature-dependent enthalpy according to

∂∆H

∂T

= T

∂∆S

∂T

(1.22)

automatically entails a temperature-dependent entropy and vice versa. If en-

thalpy and entropy increase with temperature the Arrhenius diagram will show

some upward curvature.