Scripta Materialia 49 (2003) 619–623

www.actamat-journals.com

Discussion concerning the rationalisation of diffusion

Andrew Howe *

Department of Engineering Materials, University of Sheffield, Sheffield S1 3JD, UK

Received 6 February 2003; received in revised form 6 February 2003; accepted 11 June 2003

Abstract

Further support is given for the proposed rationalisation of interstitial diffusion, which is also extended to the

substitutional case. A discussion is presented in order to answer various issues raised by Erdelyi and Beke in their

response to the original paper.

� 2003 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

Keywords: Alloys; Diffusion; Theory; Modeling

1. Introduction

Not surprisingly, daring to query the funda-

mental formulation of solute diffusion [1] has

prompted serious argument [2], and I welcomesuch debate. In brief, the proposal is that the un-

derlying form of Fick�s first law:

J ¼ �DdCdx

ð1Þ

for isothermal diffusion, where J is the solute flux,D is the diffusion coefficient, and C the concen-

tration at a position, x, is better written as:

J ¼ � dðDCÞdx

¼ �D � dCdx

� C � dDdx

ð2Þ

It is also proposed that for simple systems

D ¼ cDI, and thus alternatively:

*Address: Corus Group plc, Swinden Technology Centre,

Rotherham S60 3AR, UK. Tel.: +44-1709-825328; fax: +44-

1709-825337.

E-mail address: andy.howe@corusgroup.com (A. Howe).

1359-6462/03/$ - see front matter � 2003 Acta Materialia Inc. Publisdoi:10.1016/S1359-6462(03)00352-X

J ¼ �DIdðcCÞdx

¼ cDI 1�

þ Cd lncdC

�� dC=dx ð3Þ

where DI is the value which the diffusion coefficientwould have were it an ideal solution, and c is theactivity coefficient, as the departure from ideality.The term DI houses the �datum� value of the acti-vation energy, which is then altered by RT ln c fordepartures from ideality. Mathematically, it is

identical to invoking c simply as a factor within D[1].

Both these aspects can be found in the literature,

and the claim that their synthesis is wrong [2]

might stem partly from a matter of definition. Theproposal is a necessary result if the diffusion co-

efficient is defined in terms of the atomistic

mechanism of diffusion, as is commonly assumed,

but the issue is confused by the treatment of Eq.

(1) as being true by definition, at the expense of a

strict, mechanistic interpretation of D. Not sur-prisingly, this loss of physical meaning of D leadsto inconsistencies, complications and errors, whicha mechanistic definition would avoid.

hed by Elsevier Ltd. All rights reserved.

Fig. 1. The partial fluxes J12 and J21 differ as the product of theconcentrations C and jump frequencies C in their source planesand thus according to the perceived extra energy, G� l1 orG� l2, in a chemical potential gradient.

620 A. Howe / Scripta Materialia 49 (2003) 619–623

2. Basis of the argument

The argument is restricted to one-dimensional

diffusion and relatively dilute solutions to avoidunnecessary complications of concept. (My ter-

minology in the original paper appears to have

caused some confusion [2], so I will now adopt that

as employed by Philibert [3].) The diffusion coef-

ficient on this basis, is simply:

D ¼ k2C ð4Þ

where k is the jump distance and C is the jumpfrequency in that direction. The rate at which sol-

ute is jumping from a given plane must be given by

the product of the jump frequency and the con-centration. The net flux between two adjacent

planes must therefore derive from the difference

between these products, which necessarily leads to

Eq. (2) rather than Eq. (1). I had only noted this

stated explicitly in the text of Christian ([4, p. 349])

although it can be inferred from Philibert [3] also.

Philibert defines the flux equation as:

J ¼ �DdCdx

þ V � C ð5Þ

where V is the drift velocity, resulting from effectsother than the concentration gradient. When

providing an atomistic interpretation of this

equation ([3, pp. 33–34]), his D is related to the

average jump frequency from plane 1 to plane 2

and from plane 2 to plane 1, and V is defined interms of the difference between these frequencies.

However, on substituting his expansions for D andV into Eq. (5), it can be shown that this generatesEq. (2) (with D, like C, being defined in relation toeach plane rather than averaged) but, curiously,

this is not mentioned in the text.

In their criticism of my original paper, Erdelyi

and Beke cite the text of Manning [5] and indeedthis is a highly relevant text. Manning invokes Eq.

(2) in combination with variously defined drift

velocities. However, on p. 232 he states that: ‘‘by

definition of the diffusion coefficient, when there

are no driving forces, J ¼ �D � dC=dx.’’ D is thusdefined in terms of the average energy barrier as

perceived by adjacent planes (p. 230) rather than

the actual ones, and is no longer strictly related tothe jump frequency as in Eq. (4). Manning ac-

knowledges that only his tracer diffusion coeffi-

cient is thus directly related (p. 23). After tortuous

argument, his result is the same as Philibert�s, i.e.Eq. (5), with identical definition of V :

V ¼ DFRT

where F ¼ �RT � d lncdx

ð6Þ

for this isothermal case and in the absence of other

potential driving forces, F , such as electrostaticfields or stress gradients. This V term in Eq. (5)

represents the dD=dx term in Eq. (2) as describedabove, and therefore:

dDdx

¼ Dd lncdx

ð7Þ

This can only be true if D is the product of some�ideal� value, DI, and the activity coefficient, c,which is consistent with Fig. 1 [1], although this

point does not appear in either text:

dDdx

¼ dðcDIÞdx

¼ DIdcdx

¼ cDI1

c� dcdx

¼ cDId lncdx

¼ Dd lncdx

ð8Þ

Manning describes the perceived energy barrier

from adjacent planes 1 and 2 as being G� E andGþ E respectively, such that the total energy dif-ference is 2E which is identified as kF . By simplesubstitution for F from Eq. (6), this energy dif-

ference is RT � d ln c, where d ln c is the difference inthis term evident between plane 1 and plane 2. This

0

2

4

6

8

10

12

0 1 2 3 4 5 6 7At % C

D (E

-11

m2/

s)

ig. 2. The effect of carbon concentration on its diffusivity in

cc-iron at 1000 �C, Experimental data (12); N Predictions

rom current work; j Predictions from traditional formulation.

A. Howe / Scripta Materialia 49 (2003) 619–623 621

is fully consistent with the analysis from [1], where

the resultant G�s in planes 1 and 2 are modified byRT ln c for D, and by RT ln cC for the flux, J . ThusC and hence D varies according to the gradient ofthe non-ideal part of the chemical potential, and

the flux is driven by the full chemical potential

gradient. Conventionally, no such c factor is

adopted and the flux equation is translated as:

J ¼ �DdCdx

þ VC ¼ �DdCdx

� CD � d lncdx

¼ �DhdCdx

ð9Þ

where h ¼ ð1þ C � ðd lnc=dCÞÞ as opposed to Eq.(3), which acknowledges the presence of c. Thusboth aspects of the proposed rationalisation have

been implied already in the standard literature,

though neither is now generally accepted. How-ever, it must be straightforward to test whether

the diffusivity (or mobility) houses such a factor

and whether Eqs. (3) or (9) agrees better with

experiment. Indeed, several authors have found

empirically that Eq. (3) (whether invoked in

terms of the activity gradient, dðcCÞ=dx, or themobility and thus diffusivity being proportional

to c) is successful for the bulk diffusion of simplesystems ([4, p. 366], [6, p. 463], [7–10]). Krishtal

[11] even considered Eq. (3) to be standard

theory for the �true� diffusion coefficients, corre-sponding to the DI values in this work. Fur-thermore, some proponents of this alternative

view appear to have suffered severely in discus-

sions of their work, even when clearly demon-

strating much better agreement with experiment[10].

Comparison with experiment can be aided with

Eq. (3) re-written in terms of the value of D at

infinite dilution, D0, rather than the value which

the system would have were it ideal. Thus:

D ¼ D0cHh ð10Þ

where cH is the �Henrian� activity coefficient, i.e.relative to that at infinite dilution. One of the most

thoroughly characterised interstitial systems is

probably that of carbon in iron. It is relevant

therefore to perform a direct comparison to seewhether Eqs. (9) or (10) is closer to reality, based

F

f

f

on the same value for D0 and the same, source

thermodynamic data, for this system. For this

purpose, experimental values for the diffusion co-

efficient were taken from Wells et al. [12], averag-ing their results for 999, 1000 and 1005 �C forgreater confidence in the nominal data at 1000 �C.The required thermodynamic data were taken

from the JMatPro commercial software package

[13]. In Fig. 2, the traditional expansion of the

value for infinite dilution by the thermodynamic

factor, h, can be seen to be significantly in error,whereas the results from the present analysis areseen to be substantially better, and very close to

the experimental results up to 5 at.% (�1.1 wt.%).Perhaps more complicated formulations can ex-

hibit even greater accuracy, but clearly much more

can be achieved with a simple analysis than is

commonly assumed.

3. Substitutional diffusion

Erdelyi and Beke�s paper [2] emphasises thesubstitutional case. The rationalisation is believed

relevant to substitutional diffusion also, although

the argument [1] was restricted to the interstitial,

or �independent� diffusion case, for the sake ofconceptual and mathematical simplicity, but brief

comments will be offered on it here. Of note,

Erdelyi and Beke described the case of substitu-tional diffusion by direct atom exchange, whereas I

622 A. Howe / Scripta Materialia 49 (2003) 619–623

believe it more useful to consider the usual case of

diffusion by the vacancy mechanism.

Considering the intrinsic diffusion with respect

to the lattice (i.e. specifically not the interdiffusionwith respect to the Boltzmann–Matano interface

which involves contributions to the �flux� fromatoms which are not actually moving), the picture

is still one of planes of solute atoms and their jump

frequencies. The jump frequencies are no longer

simply the product of the atomic vibration fre-

quency and the probability of a given atom having

sufficient energy to pass over an energy saddle-point on its jump path. This would apply only to

the proportion of solute atoms that happen to be

adjacent to a vacancy. This in turn will depend on

the vacancy concentration, for which it is usual to

assume that the small, equilibrium concentration

applies. This is given by an expression of the form

expð�Gv=RT Þ where Gv is the energy for vacancyformation, and thus, rather conveniently, thisprobability can be accommodated within the ac-

tivation energy expression for diffusion. A nice

account of this is presented by Shewmon [14].

Thus, although C is differently configured, the

point remains that there are planes of atoms with

associated jump frequencies. As before, the energy

difference controlling the jump probability will

depend on the solute�s chemical potentials in thepair of planes. Thus C cannot be assumed to be thesame, and Eq. (2) would result.

The extension to interdiffusion with respect to

the equal flux, Boltzmann–Matano interface, is

relatively straightforward. Traditionally, this �flux�,JAB, for a simple binary of A and B atoms with asmall, equilibrium concentration of vacancies, is

given by:

JAB ¼ �ðCADBh þ CBDAhÞ � dCAdx

ð11Þ

According to the revised analysis, the result is:

JAB ¼ �ðCADB þ CBDAÞ �dCAdx

� CACB �dDAdx

�� dDBdx

�ð12Þ

but as before, they are exactly equivalent if the

intrinsic diffusion coefficients should also house a cfactor as in Eq. (3).

4. Discussion

The proposed revision addresses the simpler,

typical cases of solute diffusion in the bulk matrixand does not purport to cover every eventuality.

As such, it should still have a very wide use, and

should be helpful in the consideration of the more

complicated cases. Certainly for the test case of

carbon steel, it appears to be sufficient for the

whole range of concentration of usual industrial

relevance, Fig. 2. Darken and Gurry [6] acknowl-

edged that the thermodynamic factor, h, would notbe sufficient for the general case, and that the

mobility (and thus diffusivity) should also include

some function of composition which would need

to be determined empirically for each case. The

present work revives earlier ideas that suggest one

can achieve considerable improvement from a

simple extension to the basic theory, which auto-

matically arises just from consideration of theatom jump process.

Erdelyi and Beke�s argument [2] centres on thenature of the jump frequencies, C, and the order ofexpansion of variables considered as a Taylor se-

ries. The jump frequencies employed here and in

standard texts (e.g. [3–5,14]) refer to the frequency

at which solute leaves the source plane. This is

conceptually simple and intuitive. Erdelyi andBeke�s frequencies are different in nature, being afrequency of atom exchange. However, for either

interstitial or vacancy diffusion the conceptual

model that I employ should be fundamentally ro-

bust. Regarding the influence of neighbouring

planes, only adjacent planes are interacting within

the timescale of the jump process: the probability

of a solute atom jumping two or more planes inthat timeframe is minute. Certainly, the sur-

rounding region exerts an influence, but that

should be adequately accounted for by the chem-

ical potential and activation energy. One would

expect that a solute atom is only �aware� of thechemical potential, l, where it is, and of the saddlepoint of its possible jump path (as is accommo-

dated by incorporating the c factor within C andthus D [1]). Provided that there is a jump path

available, the partial flux is controlled by the en-

ergy of the saddle point, and it does not matter

how much lower is the energy of the receiving site.

A. Howe / Scripta Materialia 49 (2003) 619–623 623

As put by Borg and Dienes [15], the physics of the

jump process is contained in C. For the C as em-ployed in this work to be affected by other, nearby

planes, but not to be accommodated already bythe resultant l in the plane in question, one wouldassume it must therefore be affecting the value of G(or in extreme cases, perhaps the attempt fre-

quency � atom vibration frequency, m). It wouldbe relatively straightforward to apply the proposed

rationalisation to cases where G is dependent onthe local concentration.

Erdelyi and Beke [2] also criticise the originalpaper for its use of linear expansion of variables.

This might be a significant issue if using their

atom-exchange C�s as functions of concentrationin several planes, but I would argue that it is

actually incorrect to use higher order expansions in

my case. I invoke the simple but robust conceptual

model of planes of concentration and their asso-

ciated jump frequencies. The local gradients inthese variables are simply the difference between

their values in neighbouring planes divided by the

jump distance, e.g. dC=dx ¼ dC=k. Terms such asC and C do not have any meaning other than inthe atom planes (although the partial molal energy

does, e.g. to the saddle point at energy G); they arenot continuous functions. These gradient terms

will exhibit step changes either side of a given atomplane (giving rise to Fick�s second law of how theseconcentrations change with time). No function of

whatever order is being imposed on them; they are

free to be what they are. This becomes a somewhat

philosophical point, but diffusion is an inherently

discretised phenomenon. It can be envisaged as a

finite difference process operating at the inherent

grid spacing of the jump distance.

5. Conclusions

The recently proposed rationalisation of inter-

stitial diffusion combines aspects of theory and

experimental evidence already present in the liter-

ature. However, key aspects are commonly over-looked, and their synthesis into a coherent whole

appears to be new and controversial. It should

prove superior to standard formulations well

beyond the trivial case of infinite dilution as stated

by Erdelyi and Beke, and this has been demon-

strated for the case of carbon in fcc-iron. The

conceptual model upon which it is based is be-lieved to be simple but robust, and indeed it has a

strong �pedigree�, being invoked by the authors ofstandard texts. Cases of boundary diffusion, high

concentrations and very steep concentration gra-

dients are less straightforward and would require

further complications in the analysis, just as they

do for the standard formulation of Fick�s first law.However, employing the same principles as in thisanalysis that diffusion occurs by the mechanism of

atoms jumping between sites and must therefore

be fully describable in these terms, such extra

complications should also be addressable: the

complicated physics is �wrapped up� in the jumpfrequency, C, and thus in the diffusion coefficient,D. This of course assumes one uses a definition ofD based on the mechanism of diffusion, ratherthan defining it as the coefficient in the traditional

formulation of Fick�s first law.

References

[1] Howe AA. Scripta Mater 2002;47(10):663.

[2] Erdelyi Z, Beke DL. Scripta Mater, this issue.

[3] Philibert J. Atom movements: diffusion and mass transport

in solids. Les Ulis: Les Editions de Physique; 1991.

[4] Christian J. The theory of transformations in metals and

alloys. Oxford: Pergamon Press; 1965.

[5] Manning JR. Diffusion kinetics for atoms in crystals.

Princeton: Van Nostrand; 1968.

[6] Darken L, Gurry R. Physical chemistry of metals. Mc-

Graw-Hill; 1953.

[7] Birchenall CE, Mehl RF. Trans AIME 1947;171:143.

[8] LeClaire AD. Prog Met Phys 1949;1:306.

[9] Smith RP. Acta Met 1953;1:578.

[10] Guy AG. ASM 1952;44:382 (Part I) and 397 (Part II).

[11] Krishtal MA. Diffusion processes in iron alloys. Jerusalem,

Israel Program for Scientific Translations; 1970. p. 97.

[12] Wells C, Batz W, Mehl R. Trans AIME 1950;188:553.

[13] Saunders N, Li X, Miodownik AP, Schille J-P. In: Shao

J-C et al., editors. Materials design approaches and experi-

ences. Warrendale, PA: TMS; 2001. p. 185.

[14] Shewmon P. Diffusion in solids. Warrendale, PA: TMS;

1989.

[15] Borg RJ, Dienes GJ. An introduction to solid state

diffusion. London: Academic Press; 1988.