Gupta D. (Ed.). Diffusion Processes in Advanced Technological Materials

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62 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

Figure 1.25 Diffusion of radioactive tracers in amorphous metallic alloys and a

bulk metallic glass. (See Table 1.6 for compositions and diffusion parameters.)

the unrelaxed state is likely to be seen in techniques such as the decay of

the satellite peaks in the compositionally modulated films and RBS,

requiring short anneals, rather than in the radioactive tracer sectioning

technique where annealing periods may extend to hundreds of hours. The

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DIFFUSION IN BULK SOLIDS AND THIN FILMS, GUPTA 63

activation volumes ∆VV

∼ 0.0 and ∆E ∼ 0 in the relaxed glasses, as

listed in Table 1.6.

There is, however, an ambiguity in the transition metal-metal glasses.

While diffusion of early transition metals (such as Co, Cu, and Fe) in the

metal-metal glasses is consistent with a cooperative mechanism, diffusion

changes to a vacancy mechanism for late-transition tracers (Hf and Zr), as

evidenced by sizable values of ∆E and ∆VV

(see Table 1.6). The differ-

ence is largely attributed to the larger atomic size prevalent in the late

transition metals.

Diffusion coefficients in the bulk metallic glasses are apparently non-

Arrhenius, with a knee present at about the glass transition temperature

) (see Figure 1.25). This effect has been explained by Faupel et al..

[106]

There are actually two atomic diffusion mechanisms operating in bulk

metallic glasses: Below T

, diffusion occurs by a cooperative mechanism

in the relaxed glasses or early transition metal-metal glasses that is sup-

ported by (a) isotope effect ∆E ∼ 0, (b) small lifetimes of positrons, indi-

cating little free volume, and (c) molecular dynamics simulations. Above

, the diffusion occurs by a vacancy mechanism as the frozen structure

attempts to revert to the undercooled liquid state. Each has its own

Arrhenius dependence, resulting in a curved plot.

1.7 Summary

In this chapter, we first provided important mathematical bases of diffu-

sion for a variety of experimental situations and diffusion mechanisms in

crystalline solids. The thermo-chemical effects encountered in homogeneous

and inhomogeneous alloys were then discussed within linear and nonlinear

diffusion regimes. Diffusion processes operating in common engineering

metals and alloys with extended defects such as the grain boundaries and

dislocations were discussed and illustrated by experimental results in some

model alloys. In thin films, which are of great importance to the information

technology discipline, grain boundary diffusion plays a vital role in reliabil-

ity at low temperatures. Therefore, the diffusion process in grain boundaries

was described in some detail and its interrelationship with the diffusion in

the lattice and the grain boundary energy was established. Finally, diffusion

in the recently discovered quasicrystalline and amorphous metallic alloys

was briefly discussed. Tabulated data of diffusion coefficients in a number

of materials under varying physical and chemical conditions were provided

for ready reference. This chapter was confined to diffusion in relatively

model materials. The following chapters discuss materials of technological

importance individually. It is hoped that the topics covered will be useful to

materials scientists in general without getting them deeply involved in the

mathematical and analytical complexities of the science of diffusion.

Ch_01.qxd 11/30/04 8:36 AM Page 63

Acknowledgment

The author is grateful to Dr. S. J. Rothman of Argonne National

Laboratory for critically reviewing this chapter and for many stimulating

discussions.

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2 Solid State Diffusion and

Bulk Properties

Gyanendra P.Tiwari and Radhey S. Mehrotra

Bhabha Atomic Research Centre, Trombay, India

Yoshiaki Iijima

Department of Materials Science, Tohoku University, Japan

2.1 Introduction

The basic equation for self-diffusion in cubic crystalline lattice via a

monovacancy mechanism

[1]

is as follows:

D  fa

u exp(∆S

R) exp(∆H

RT) (1)

where

D  diffusion coefficient

f  correlation factor

a  lattice parameter

u  jump frequency

∆S

 activation entropy

∆H

 activation enthalpy

R  universal gas constant

T  temperature, in degrees Kelvin

Furthermore, the enthalpy and free energy (∆G

)of activation for diffusion

are given by:

∆H

 ∆H

 ∆H

, (2.1)

 DH

 TDS

. (2.2)

The terms ∆H

and ∆H

are the activation energies for formation and migra-

tion of vacancies. Equation (1) was derived by Wert and Zener

[2]

on the basis

of absolute reaction rate theory. This equation is applicable for self-diffusion

in all crystalline lattices. For diffusion in complex lattices and changes in the

diffusion mode, introduction of a numerical factor is necessary. The present

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discussion is focused on the self-diffusion in metallic elements, where the

migration of a monovacancy is the dominant mode of diffusive motion.

Section 2.2 surveys the interrelationships between diffusion parameters

and some bulk properties of the solids. These relationships, which can be

qualified as the “enlightened empiricism,” have contributed significantly to the

growth in the field of diffusion in solids by serving as the reference point for

the rationalization of the diffusion data. The most important developments in

this area have been (a) the establishment of a firm basis for the correlation

between diffusion and melting parameters, (b) the Zener’s hypothesis

relating the activation entropy of diffusion (∆S

) with the temperature

dependence of elastic constants, and (c) a formal theory relating the vacancy

formation energy with the Debye temperature. The main feature of such rela-

tionships is a constant, which directly relates the concerned diffusion param-

eter, most often the activation energy for self-diffusion, with a bulk property.

For any particular class of solids, the magnitude of the constant depends on

the crystal structure and the mode of diffusion. Starting with Van Leimpt,

[3]

there exists a fairly long history of empirical relationships between the dif-

fusion parameters and bulk properties of the solids.

[2–13]

The formation of a

vacancy results in the breaking of bonds between the atoms in the matrix.

Similarly, the migration of an atom from its equilibrium position to the next

site causes distortions in the bonds between the diffusing atom and its near-

est neighbors. The kinetics of both of these steps in diffusion are governed

by the nature and strength of the cohesion in the matrix. Therefore, in trying

to relate the diffusion parameters with the bulk properties, we are, in fact,

seeking to correlate them, albeit indirectly, with the cohesive energy. This

explains, to some extent, why the correlations of the diffusion parameters

with melting point and elastic constants have been sought extensively in the lit-

70 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS

Figure 2.1 Plots of activation energy for self-diffusion in metals against their

cohesive energies; slopes 0.57 (FCC), 0.65 (HCP), and 0.60 (BCC).

Notes: 1. Activation energies were obtained from Smithells’Metal Reference Book, vol. VII

(E. A. Brandes and G. B. Brook, eds.), Butterworths Pub. (1992).

2. Cohesive energies were obtained from C. Kittel, Introduction to Solid State Physics, VII

ed., Wiley (1986).

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SOLID STATE DIFFUSION AND BULK PROPERTIES, TIWARI ET AL.71

erature. Figure 2.1 shows plots of ∆H

for metals against the cohesive energy,

E. The slopes vary only marginally between FCC, HCP, and BCC structures.

Figure 2.1 demonstrates a general rule that activation energy for self-

diffusion in any matrix depends basically on its cohesive strength.

From cohesive energy considerations, diffusion is easier in metals and

polymers because the breaking and bending of bonds in them requires

proportionately lesser energy as compared to that in ionic and covalent

solids. In other words, energy fluctuations needed for diffusion to occur

are smaller in metals and organic solids than in ionic and covalent solids.

Following this, a thermodynamic equation relating the activation vol-

ume to the activation entropy is derived in Sec. 2.3 for pressure-dependent

diffusion measurements. This equation is further used to obtain estimates

of the activation entropy of diffusion, the frequency of diffusion, and the

specific heat of the activated complex. It is shown that the equation

derived here gives reliable values of all associated parameters.

Disregarding the chronological sequence, the important relationships

between diffusion parameters and bulk properties discussed here have

helped (a) in the growth of the subject, through being a reference point for

the rationalization of diffusion data in the absence of a formal theory, and

(b) in the estimation of diffusion rates when the experimental data are lack-

ing. Indeed, when diffusion measurements are attempted in a new solid,

these empirical rules greatly help in choosing the experimental technique

and the subsequent thermal treatments.

2.2 Correlations with Bulk Properties

2.2.1 The Melting Parameters

Melting of a solid leads to the destruction of its long-range periodicity.

Therefore, melting point can be considered, on a relative scale, to be a

measure of the pair-wise binding energy of the matrix.

[14, 15]

Around a vacancy,

pair-wise bonds are broken and, at the same time, there is an increase in the

overall volume of the matrix. Hence, vacancy formation energy has been

linked intuitively to the melting point.

[16, 17]

Figure 2.2 shows plots between the

vacancy formation energy and the melting point. The data on the activation

enthalpy for the formation of monovacancies in various metals are satisfacto-

rily represented by a straight line passing from the origin and are given by:

 80.2 T

(3)

for FCC and HCP metals, and

 97 T

(4)

for BCC metals.

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