Atomic diffusion in alloys exposed to liquid sodium

Journal of Nuclear Materials 62 (1976) 247-256 0 North-Holland Publishing Company

ATOMIC DIFFUSION IN ALLOYS EXPOSED TO LIQUID SODIUM S.B. FISHER Central Electricity Generfft~ Board, Berkeley Nuclear Laboratories, Bffke~ey, GIor, UK Received I May 1976 Reasons for the observed enhancement of atomic diffusion coefficient in alloys exposed to sodium are explored. The diffusion coefficient in the presence of a chemical potential gradient is derived from a simple atom&tic model. It is concluded that two mechanisms could contribute to the enhancement. (a) A non-zero gradient of partial molar enthalpy. (b) An increase in the atomic mobility due to the presence of high diffusivity paths. It is shown that the effect of (a) is not of sufficient magnitude to account for the increase in coefficient. However the results can be analysed successfully in terms of grain boundary diffusion and the coefficients are determined for iron in NijCr (1.2 X 10s9 cm*/s) at 65O’C and nickel in iron (5 X lo-* cm*/s) at 730°C. The total absorbed diffusant is also calculated for the case of iron into NijCr, and a general expression for the absorbed diffusant is given. Les raisons de ~augmentation du coefficient de diffusion atomique observt?e dans des a&ages exposes au sodium sent explorees. Le coefficient de diffusion en presence d’un gradient de potentiel chimique est diduit d’un mod&le atomistique simple. On en conclut que deux m&mismes pourraient contribuer g l’augmentation des coefficients de diffusion: (a) un gradient d’enthalpie molaire partielle, (b) un accroissement de la mobiliti atomique due ii la presence de chemins 1 diffusivit6 prdf&entielle. I1 est montrd que l’effet de (a) n’est pas dune importance suffisante pour rendre compte de Paugmentation des coefficients de diffusion. Cependant les r&Rats peuvent Etre analysis avec succes en termes de diffusion intergranulaire. Les coefficients de diffusion du Fe dans Ni/Cr $65O”C (I,2 X 10m9 cm*/s) et du Ni dans Fe a 730°C (5 X 10” cm*/@ ont 6th d6terminis. L’element diffusant absorb6 a 8ti calcule aussi dans le cas du Fe diffusant dans NifCr. Une expression generate donnant les quantites d’element diffusant absorbi est proposde. Es werden Grtinde fiir die beobachtete Erhijhung des Diffusionskoeffizienten von Atomen in Legierungen angegeben, die Nat&m ausgesetzt sind. Der Diffusionskoeffiiient im Gradienten eines chemischen Potentials wird aus einem einfachen atomistischen Model1 abgeleitet. Daraus folgt, dass zwei Mechanismen zur Erhiihung beitragen konnten: (a) ein von null verschiedener Gradient der partiellen molaren Enthalpie, (b) eine Zunahme der Atom~we~chkeit durch vorhandene Wege mit hohem D~f~ionskoeffizienten. Es wird gezeigt, dass der Einfluss (a) nicht gross genug ist, urn die Erhijhung des Diffusionskoeffizienten zu erkiiiren. Die Ergebnisse konnen jedoch erfoigreich durch Korngrenzendiffusion erkliirt werden. Der Diffusionskoeffizient von Fe in Ni/Cr betrsgt 1,2* 10m9 cm2/s bei 650°C und von Ni in Fe 5 * lOwe cm2/s bei 730°C. Ferner wird die insgesamt aufgenommene diffundierende Komponente fii Fe in Ni/Cr berechnet, ein allgemeiner Ausdruck fiir die aufgenommene diffundierende Komponente wird angegeben.

[Z]. More recently Hofer and Wieling [3], have reported the measurement of nickel and chromium diffusion coefficients from concentration profiles observed in Armco iron exposed to flowing sodium at ~730’C. They determined coefficients of 3.2 X lo-l2 cm2/s and 2.9 X lo-l2 cm2js respective. ly by fitting the concentration profiles to simple error functions. Chemical diffusion coefficients of similar magnitudes have been found in experiments on interdiffusion in binary diffusion couples at these temperatures. Faulkner and Bridges [4] examined a wide range of

1. lnt~duction In 1972, Fisher, Hooper and Swallow [l] observed and measured the diffusion of iron into an 80/20 nickelchromium alloy exposed to static liquid sodium at 650°C within an iron vessel. They were able to fit a simple error function to the concentration ~st~bution observed, and concluded that the effective iron diffusion coefficient in the alloy at this temperature was =10-r3 cm2/s, approximately two orders of magnitude higher than typical values obtained from conventional tracer diffusion experiments in similar systems 241

248

S.B. Fisher /Atomic diffusion in alloys exposed to liquid sodium

couples between commercial alloys at 700°C and Ustand and Sorum [5] interdiffusion in the Fe-Ni, Ni-Co, and Te-Co systems at temperatures between 700-1400°C. In each case at =7OO”C the diffusion coefficient was found to be =2 X lo-l2 cm2/s. In none of the above experiments was any definite conclusion drawn as to the reason for the enhancement of the coefficient above the tracer value extrapolated from higher temperature measurements. It is the purpose of this paper to identify the causes for this apparent enhancement, particularly for the sodium experiments. To begin with we examine the experiments in more detail. 2. Experimental 2.1. Fisher, Hooper and Swallow [ I/ The concentration profile for the iron diffusion into the Ni/Cr alloy is shown in fig. 1. The surface concentration of iron is-=40%. There is some evidence for non-uniform distribution of iron parallel to the surface [I]. We can see that (fig. 2) diffusion of iron into an 80/20 Ni/Cr alloy does not lead to any phase change. The composition moves deeper into the austenite region in some direction between the limits indicated in the figure. We may summarize the findings as follows (a) The concentration of diffusant is such that we may expect additional thermodynamic driving forces on the atom other than from the entropy of mixing. (b) The concentration profile fits an error function solution of the form

C/C, = erfc (x/2*) which suggests matrix diffusion with a coefficient D. (c) There is some evidence from X-ray microscan photographs of a non-uniform distribution of diffusant parallel to the surface which suggests the presence of high diffusivity paths. 2.2. Hofer and Wieling (3/ The observations of Hofer and Wieling are similar to those of the previous authors (a) There is a relatively high concentration of diffusant compared to say conventional tracer experiments.

h S

1

0.2

I

0.4

5Opm

I

1 0.6

‘lh

Fig. 1. The concentration profile for the diffusion of iron into nickel/chromium.

(b) There is no phase change in the diffusion zone; the material remains in the o-phase. (c) The diffusion coefficient is obtained from an error function solution to the concentration profile and suggests matrix diffusion with D = 2 X lo-l2 cm2/s at 730°C. 2.3. Ustad and Sorum /5] Ustad and Sorum determined the interdiffusion coefficient 5 in binary couples in the temperature range 700-14OO’C. For Fe-Ni, for example, they found the following (i) The matrix diffusion coefficient 5 at 700°C was effectively =2 X lo-l2 cm2/s and was independent of concentration. (ii) At temperatures below lOOO”C,0” showed less

S.B. Fisher /Atomic diffusion in alloys exposed to liquid sodium

Isothermal

249

Section

Ni wt%

Fig. 2. The Fe/Ni/Cr phase diagram at 650°C showing the composition limits within which the alloy must lie. on temperature than at temperatures above 1000°C indicating a smaller activation energy (and different diffusion mechanism) at the lower temperature. (iii) D varied by approximately an order of magnitude with composition, at temperatures above ~lOOO°C. (iv) Below lOOO”C,X-ray photographs of the diffusion zone showed the presence of high diffusivity paths. Ustad and Sorum concluded that grain boundary diffusion was the controlling mechanism at these temperatures. No attempt was made to analyse the results on this basis. In addition to the experiments outlined above, Hooper et al. [6] examined mass transport in 304 stainless steel exposed to sodium. The element concentration profiles obtained were complicated by the phase changes and surface compound formation taking place during the exposures. However it is possible to obtain some estimate of the diffusion coefficients for loss of manganese and chromium from their specimens using, agains, error function solutions. If we assume dependence

that the surface is held at a concentration C’ where CO is the initial concentration in the material, then (C - C’)/(C, - C’) = erf (x/2fi) The values of D are again found to be e10-13 cm2/s. In a more recent report [7] on the behaviour of 304 steels in sodium the authors concluded that deformation twins acted as channels for element diffusion in their experiments, and that in the absence of twins, in solution-treated unstable steels, ferrite formation resulting from element depletion occurred only adjacent to grain boundaries. This indicates the importance of the latter with regard to diffusion at these temperatures. There are clearly two possible reasons for diffusion enhancement in these experiments. (1) Chemical diffusion - can the thermodynamic ‘forces’ additional to the entropy of mixing account for the two orders of magnitude increase in diffusion coefficient? (2) Grain boundary diffusion - can we analyse the concentration profdes, in the experiments where there

250

S.B. Fisher /Atomic diffusion in alloys exposed to liquid sodium

was no phase change, in terms of grain boundary diffusion? If grain boundary diffusion is the dominant mechanism, why do the profiles fit error function solutions of the type C/C, = erfc(x/2fi), where D is an ‘enhanced’ matrix diffusion coefficient? We shall explore these two areas in the subsequent sections.

3. Chemical diffusion 3.1. Darken (81

Darken’s analysis of diffusion in a chemical potential gradient is phenomenological and assumes no mechanism or model. He relates the diffusion coefficient D, for a component, a, in a non-ideal solution to the value Da*in a self-diffusion experiment by the relation D,=D:(l+$$),

(1)

where ya is the chemical activity of component a, and Na is the fractional concentration of a. This relation follows from the assumption of an additional thermodynamic ‘force’ acting on an atom 6F = a(kT In ra)/ijx ,

(2)

where the chemical potential may be written I-(,=kTln~,tkTlnNa,

(3)

so that the total ‘force’ on the atom is ap,/ax, and in an ideal solution this reduces to a(kT In NJ/ax, i.e. that ‘force’ which arises solely from the entropy of mixing. Subsequent treatments of this effect follow the same lines (for example see Manning, [9] and Shewmon, [lo]). The physical meaning of relation (2) is not obvious. Accordingly we shall derive an expression for the diffusion coefficient in a chemical gradient from a simple atomic model. 3.2. The atom&tic model for chemical diffusion

For ease of reference we shall begin by reproducing some standard results in order to illustrate the development of the model. Consider the diffusion of atoms across the plane Y from position A (fig. 3(a)) to posi-

lb1

Fig. 3. (a) Schematic energy diagram with no enthalpy gadient. Cb) Schematic energy diagram with a gradient of partial molar enthalpy.

tion B and vice versa through the activation energy barrier Ag. First of all let us assume that there is purely a concentration gradient of atoms of type (i) across this barrier. Then the flux of atoms A + B may be written o ~0 ci(X>exp(-4rlkT)

,

where Q is the atomic spacing, v. is the Debye frequency and ci(X) is the concentration of i-type atoms on the A plane. Similarly the flux of atoms from B + A may be written &i , a ~0 ci(x) + z * a ew(-&t/W 1 ( so that the net flux J, from A to B is given by J =

-Vo

‘2

a2

exp(-&/kT)

= Di 2

.

This of course corresponds to Fick’s first law and this derivation may be found in a number of texts [lo] . We now ask how this net flux A + B may be increased to give an apparent enhancement to the diffusion coefficient. For a fixed (a&x) there are two ways this may be achieved. 1) We may increase Di by reducing the activation energy barrier &. This of course can be achieved in practice along certain restricted high diffusivity paths

S.3. Fisher f Atomic diffusion in alloys exposed to liquid sodium

e.g. surface diffusion, grain boundary diffusion, dislocation diffusion. 2) We may reduce the flux in the reverse B + A direction. This would be achieved in practice if the free energy of the atom at site B is less than at site A, (see fig. 3(b)) i.e. the atom preferred to sit surrounded by a lower concentration of its own species. This additional energy term is then the enthalpy of mixing one atom of type i to a solution of a certain composition of i and a, say. We have already mentioned these two factors in the previous section, from a consideration of the experimental evidence for diffusion enhancement. Grain boundary diffusion will be discussed in the next section. An enhanced diffusion coefficient will now be derived from the mechanism described under 2). From fig. 3(b), the net flux AB, in the presence of an enthalpy gradient, may be expressed

JAB= ~0~@)a! ew I(-& + &42)/~~~ -voczc(x) +g a: exp({-&

- A~/2)/kT}

i 1 = v. c(x)oc exp(-Ag/kT)

ex~-A~l2kT)

= 2~0 c(x)o( exp(-Ag/kT)

-vuo2 $ exp(-Ag/kT)

sinh(A@/2kT) exp(--A$/2kT).

If A#/2kT is small but non-zero, then JAB = 2~~0C(X)CXexp(-Ag/kT) -v. cz2 exp(-AgfkT)

fx2

* A@/2kT

.$

arp ac

=-c(x)k7j;‘v~~-pw?&

between this quantity and the other partial molal quantities for free energy (chemical potential) and entropy. FT =fiim-TSim=RTlnai,

(5)

where ai is the chemical activity. F~=RTln~i+RTlnNi,

(6)

where yi is the activity coefficient. In ideal solutions SF (ideal) = R In Nr Hr=O,

F~=RTlnNi.

(7)

For reguhzr non-ideal solutions (i.e. when SF (ideal) = -R In Ni but Hr f 0), then Ff = RT h Ni, SO that HP %RTln Tie

03)

The term (l/kT) (%#$I In c) therefore reduces to a In rifa In c and eq. (4) is identical to that of Darken

181. Not all non-ideal solutions adhere to the relation

X {exp(A@/2kT) - exp(-A$/2kT))

-v0cx2 $ ex~-~/kT)

251

ac

Now $Jas defined above is simply the partial molal enthalpy (q), and we can write down relationships

SF = Sr (ideal) . Then ST (excess) = ,Sy - .S‘r (ideal) F?

=H/” - 331 (excess) + Tin Ni

+“’

- 7Sp (excess) + Ff’ (ideal) .

I%

In this case I#has enthalpy and entropy components and Q,z Fr (excess). This is the excess partial molal free energy and it is tabulated for a number of binary alloys as a function of composition [ 11J. One can crudely estimate the e~ancement of diffusion coefficient due to this enthalpy term in the iron/nickel system from values-of ,FFe (excess) as a function of composition. This is only possible for the liquid alloys in the Fe-Ni system, as there are insufficient data for the solid phase. We find (1 /kT) @$/a In c) to be =2, so that only a threefold increase in diffusion coefficient with increasing nickel content may be expected. In practice one finds an order of magnitude increase [S], and there may well be a con-

252

S.B. Fisher /Atomic diffusion in alloys exposed to liquid sodium

tribution here from an increase in Di with composition i.e. a diffusion coefficient gradient. Clearly then we must investigate the alternative explanation for the hundredfold increase in coefficient observed in the sodium experiments over the values for volume diffusion obtained from tracer measurements [2].

the grain is described by an empirical sink function appropriate to a real polycrystalline solid. The numerical results of the model are expressed as average concentration of diffusant as a function of an effective penetration o, a reduced time 7, and effective exposure e2r, where these parameters are defined as follows:

4. Grain boundary diffusion

w=

(A6)‘i2 (D2t)l14 ’

The experimental conditions for diffusion in the sodium experiments were identical to those for standard tracer diffusion experiments (with the exception of the high concentration of diffusant). The surface concentration of diffusant was assumed constant at a level, in the sodium experiments, dictated by the activity of the sodium for the diffusant. The concentration profile of diffusant perpendicular to the specimen surface was measured. There are a number of theories which may be applied for the interpretation of the results of the above type of measurement when grain boundary diffusion is important. That described by Fisher [ 121 predicts simply that log C varies linearly with penetration depth, however his analytical result contained many assumptions (ses for example Shewmon’s discussion [lo]). Subsequent treatments [ 13-151, clarified certain aspects of the problem, but the effects of grain size, volume diffusion from the surface, and exposure time, were not fully identified until the work of Levine and MacCallum [ 161. It is with the results of their work that we choose to analyse the sodium ex; periments.

7 = (4D#i2)t,

E = 26/n,

where y is the perpendicular penetration, A = D, ID,, D1 diffusivity in boundary, D2 diffusivity in grain, 6 boundary thickness, and (0 is the mean linear dimension of a grain. Fig. 4 shows the concentration profiles (as a function of e2r) resulting from the numerical evaluation of the complex integral relating average concentration with penetration depth when grain boundary diffusion is dominant. Also on the graph are values of the con-

4. I. The Levine-MacGdium model The salient features of the LM model may be summarised as follows. (1) Diffusion from the specimen surface through the grains, and diffusion around the grains through the boundaries are treated separately. This is valid in certain regions of interest. (2) Diffusion in the boundaries is assumed to be much more rapid than diffusion into the lattice. There is a good deal of experimental proof of this, especially from chemical diffusion experiments [ 51. (3) m, r) = 1 wheny = 0. (4) The model assumes a finite grain size. Flux into

Fig. 4. The results of the Levine-MacCallum model for grain boundary diffusion.

253

S.B. Fisher /Atomic diffusion in alloys exposed to liquid sodium

centration

evaluated from the error function Iron

m

Nickel

/Chromr

650%

C= erfc(j-&)

= erfc (($)l”+]

(10)

for various values of (A2/r). This of course represents bulk diffusion through the grains from the surface. The general features of their results are as follows: (i) For e2r 5 1, d(log C)/d(c#) is constant and we may use the numerical value to determine D, if we know the lattice diffusion coefficient D2. This is only true if over the experimental range chosen for measurement of the concentration profile the contribution from bulk diffusion << contribution from boundary diffusion, i.e. A2/r 2 lo3 and e2r = 1. (ii) If e2r 2 1 the concentration profile from GB diffusion passes through unit concentration at the surface and therefore could be fitted approximately to an error function profile with A2/r 2 1.O over the concentration range loo + 10-l) i.e. it would have the appearance of bulk diffusion with an enhanced lattice diffusion coefficient. This offers us a possible explanation of our ability to fit the measured profile to an error function. We shall now proceed to investigate in detail whether we can analyse the results of the sodium experiments successfully in terms of GB diffusion. 4.2. The LM model applied to the sodium diffusion experiments

4.2. I. Iron diffusion into nickel/chrome [ 1J We shall begin with the approximate calculation of A2/r and e2r to determine’if the condition GB diffusion >> lattice diffusion is fulfilled. We take, at 65O”C,D, = 2 X 10eg cm2/s [2], and D2 - lo-l5 cm2/s, but this value may be enhanced through chemical diffusion. We may anticipate a value lo- l5 + lo-l4 depending on concentration. For simplicity we choose an intermediate value 5 X 1C115 cm2/s and assume there is no concentration dependence. Then A2/r 2 lo3 and e2r = 1. These are ideal conditions for the analysis. We note d(log C)/do6/5 * -0.33, and we expect the concentration profile to pass through unit concentration at the surface. In fig. 5 the results of this experiment are plotted as log C against y 6/5. The graph is linear and passes through unit concentration at y = 0.

I 10

20

30

50

40

y6’5

(xl05

60

x)

I

Fig. 5. Log (concentration) versus (penetration depth)6’5 for iron into Ni/Cr.

From the slope -d(log Q/d(#), and knowing -d(log C’)/do6/5 w 0.33 we can determine Dl , the boundary diffusivity. d(log C)/du6/5 = d(log C)/dc#

- d(c.#)/dy(+

= -0.33 k615 , where k = &/(A8)l12 (D2t)114. We find for D2 a 5 X lo-l5 cm2/s and 6 a S X lo-* cm, that 4

=

1.2 X 10Bg cm2/s.

Perkins et al. [2] give for GB diffusion of iron in an Fe/Cr( 17)/Ni( 12) alloy in the temperature range (600-1030°C) SD,cB = 5.3 X lo-’ cm3/s and $ = 42.4 kcal/mole. At 650°C therefore Dl = 1.1 X lob9 cm2/s in excellent agreement with the value determined from the sodium experiment. 4.2.2. Nickel into iron /3J Krishtal et al. [ 171, have examined volume and grain boundary diffusion of nickel into Armco iron between lOOO-1200°C and find for D2, Do = 1 with 9 * 64.7 kcal/mole and for D1, Do - 40 with @= 41.5 k&/mole. At 730°C this gives D, = 6 X lo-l5

cm2/s,

D, = 4 X lo-* cm2/s.

S.B. Fisher /Atomic

254

diffusion in alloys exposed to liquid sodium

log c

Nickel

and Chromium Into

Armo Iron. (73O’C)

OS’\. ,“.:>*>.. 2-

0

*A.>.

01O-0

‘\.,>.\*

I -

-0-2 -0

-

.

3 -

-0.4

\

-

.

Chromium

-0 5-

.0.7 _

-0-6

\ I 2

I

.

I 3

4

y6’s(x IO31 Fig. 6. Log (concentration)

versus (penetration depth)6’5 for Ni, Cr into Armco iron.

We can therefore calculate A2/re lo5 and e2r = 1, and proceed with a legitimate analysis of the results in terms of GB diffusion. Fig. 6 shows the graph of log C agains@. In the same way as in the previous section we find that for D2 x 6 X lo-l5 cm2/s 4 = 5 X lo-* cm2/s which compares favourably with the figure of 4 X lo-* cm2/s [17]. 4.2.3. Chromium diffusion in Armco iron The volume diffusion of chromium in o-iron is much greater than the volume diffusion of nickel in&iron. Huntz et al. [ 181 find D, = 2.5 cm2/s with # = 57.5 kcal/mole. At 730°C this gives D = 4 X lo-l3 cm2/s. Bowen and Leak [19] find D0 * 8 cm2/s with $I=S59 kcal/mole. For GB diffusion Huntz [ 181 finds D,, (730°C) = 1.5 X lo-* cm2/s. This gives A2/r = 0.1 and e2r = 40. The effects of lattice and grain boundary diffusion are therefore of comparable magnitude and we cannot analyse these results from the LM model.

6oCo. We shall show how the total mass of absorbed diffusant may be calculated when grain boundary diffusion is dominant and t, D1, D2, (I), and the activity of the activated species in the sodium are known. We take as an example the diffusion of iron into the nickel/chromium specimen. The absorbed diffusant over a surface area A cm2 at time t is given by, y(whenC=O)

T=A

s y=o

Cti> t)dy

= AC0 Jexp(-2.3

m v6j5)dy

(11)

(12)

where -m = d(log C)/dy6i5, and Co is the surface concentration. This relation (12) is quite general if e2r 2 1 and A2/r ?. 103. We do not need to know m if both D, and D2 are known since, -d(log C)/d(w6j5) = 0.33 . In general then

5. Evaluation of the absorbed diffusant

The total number of atoms of diffusant entering a component immersed in liquid sodium is of particular importance with regard to activated species i.e. 54Mn,

T=AC’Jexp(-gv1.2)dy

(13)

where g = 2.3 X 0.33 X k615, and C’ is not necessarily CO but can be determined from the value of e27 using the LM numerical solutions. The integral (13) has no

S.B. Fisher /Atomic

diffusion in alloys exposed to liquid sodium

analytic solution. However the concentration converges rapidly withy and we wish to consider diffusion into specimens which are thick compared to the diffusion length. For example, in the iron into Ni/Cr experiment there is a 10% reduction in C from the surface value in only 0.001” at 650°C and at 15 weeks exposure. In practice we are concerned with component thicknesses in excess of 0.010”. We may therefore take the limits of the integral to be 0 + 00, so that T=AC,,

7

exp(-2.3

my6i5)dy

6. Discussion and conclusions A simple atomistic model has been used to derive an expression for the diffusion coefficient in a chemical potential gradient. Da

for the particular case under consideration. This may be transformed as follows, 00 s

x-li6

exp(-2.3

m x)dx

(15)

0

and this integral (15) has a gamma function T = 0.83 Co

of 6oCo is ~1% then after 15 weeks exposure the component would have absorbed an activity equivalent to 1 millicurie for every 3 cm2 of surface area.

(14)

0

T= 0.83 AC0

solution,

I-V/6) (2.3 m)5/6 ’

T = A 0.94 Co (2.3

(16)

m)-".83.

we note this is dimensionally

(17)

correct

T [mass] =

= A [cm21 .0.94 . Co [mass/cm31 (2.3 m)-“.83

[cm]

For the iron diffusion case, . Co = 4 X 1O22 atoms/cm3

255

,

so that T * 4 X 10lg cm-2 atoms at 15 weeks exposure. In summary, we expect the diffusion of Fe, Ni, Co, Mn etc. at 650°C to be dominated by grain boundaries. Provided that this is true we can calculate approximately the mass of diffusant absorbed into a component at any time t if D, ,D2 and (1) are known from the integral. exp(-gy 6/5)dy . s Let us gain some idea for the magnitude lem for, say, the activated species 6oCo the same rate as iron [5] in the example sodium activity is such that the surface T = AC’

of the probdiffusing at above. If the concentration

=D;

where I$ is the partial molar enthalpy. This expression is equivalent to that derived phenomenologically by Darken [8]. There are two mechanisms through which we can obtain an enhancement of this diffusion coefticient. (1) Through increasing Da*by decreasing Ag; this is the high diffusivity path mechanism. (2) Through having a gradient of partial molar enthalpy such that the term a#/a In c is >O. We have shown that the second mechanism may be operative but that the magnitude cannot account for the increase in diffusion coefficient observed experimentally. In contrast it has been possible to analyse the experimental results successfully in terms of a grain boundary diffusion rate which is much greater than the rate of lattice diffusion. The values determined from the sodium experiments are: for iron into nickel-chrome, D,, = 1.2 X 10eg cm2/s, and for nickel into Armco iron, D,, a 5 X lo-* cm2/s. These values compare favourably with previous determinations. We have also shown that when grain boundary diffusion is dominant we are able to calculate the magnitude of the absorbed diffusant from the integral T = AC’ s

exp(-gy6/5)dy

0

which has a gamma function solution. A simple example is given for the activated species 6oCo. Assuming the same values of D, and D2 as-for iron in nickel/chrome, with a 1% concentration at the surface, the absorbed diffusant after 15 weeks exposure would be equivalent to 1 millicurie for every 3 cm2 of component surface.

256

S.3. Fisher /Atomic diffusion in attoys exposed to liquid sodium

Acknowledgements This paper is published with the permission of the Central Electricity Generating Board. References [l] S.B. Fisher, A. Hooper and G.A. Swallow, CEGB report RD/B/N2497,1972. [2] R.A. Perkins, R.A. Padgett and N.K. Tunati, Met. Trans. 4 (1973) 2535. f3] G. Hofer and N. Wieling, paper presented at Imperial College London on Chemical interactions of materials with their environments at high temperatures, 1975. [4] R.G. Faulkner and P.J. Bridges, International Nickel Tech. pub. P-Bl. 146 (1972). [S] T. Ustad and Ii. Sorum, Phys. Stat. Sol. 20 (1973) 285. [6] A. Hooper, G.A. Swallow, R.P. Sparry and S.B. Fisher, CEGB RDfB/N2687,1973.

[ 71 S.B. Fisher, G.A. Swallow and A.J. Hooper, CEGB report RD/B/N3201,1975. [8] L. Darken, Trans. AIME 1974 (1948) 194. (91 J.R. Manning, Diffusion kinetics for atoms in crystals (Van Nostrand, 1968). [lOI P.G. Shewmon, Diffusion in solids (M~raw-Hi, 1963). [ 111 Hultgren, Orr, Anderson and Kelly, Selected values of thermodynamic properties of metals and alloys (Wiley, 1963). [12] J.C. Fisher, J. Appl. Phys. 22 (1951) 74. 1131 D. Turnbull, Atom movements (ASM Cleveland, Ohio, 1951) p. 129. [1]4 G.M. Roe, Phys. Rev. 83 (1951) 871. [ 151 A.D. Le Claire, Phil. Mag. 42 (1951) 468. [16] H.S. Levine and C.J. MacCallum, Jour. Appl. Phys. 31 (1960) 595. [ 171 M.A. Krishtal, A.P. Mokrov, O.V. Stepanova and LA. Goncharenko, Zashch-Pokryt, Metal 2 (1968) 209. [ 181 A.M. Huntz, M. Anconturier and P. Lacombe, CR. Acad. Sci. Paris, Ser. C. 265 (1967) 554. [ 191 A.W. Bowen and G.M. Leak, Met. Trans. 1 (L970) 1965.