LETTERS TO

A discussion of “self-diffusion and inter- diffusion in gold-nickel allays”:(1)

A correlation with absolute rate theory*

THE EDITOR

one obtains

55

Fisher et .Z.c2) derived from absolute rate theory the following expression for the self-diffusion coefficient of a component in a homogeneous alloy:

+ %~,"Y,lY2")~ (3)

Di = cc ri*ai2{(yi/yi*)[1 + (d In yi/d In cJ])

where D1” and D,” are the diffusion coeflicients relating to standard state conditions and have the form

x exp (--AFdofEZ’), (1) .Dio -di2vi* exp (- AF,“/RT). (4)

In order to retain a general form, one may accept

where a is the number of paths of jump distance vi in equations (1) and (3). However, to facilitate com-

either the forward or reverse direction, vi* is the parison with experimental data, it is necessary to

frequency of passage of an activated complex across evaluate yi* and AFi” by independent means. Since

the free energy barrier, yi and yi* are the activity this is not possible at the present time, it is expedient

coefficients of the normal and activated species, ci is the to include both factors together. i.e., place ya* in the

concentration of the ith species, R is gas constant, T is exponential. yi * formally expresses the effect of

absolute temperature, and APi” is the free energy composition on the environment of the activated

difference between normal and activated species i in solute (including concentration and distribution of

the standard state of infinite dilution of activated vacancies in diffusion by the vacancy mechanism).

species in pure species i. At the time Fisher et aZ.t2) Yi * is not unity, even though the concentration of

developed equation (l), radioactive tracer techniques activated complexes is very low, because the standard

for determining self-diffusion coefficients in alloys were state of activated species refers to infinite dilution in

not available, so they made special assumptions about pure normal species. Inclusion of yi* in the ex-

yi* in order to compare equation (1) with experiment. ponential provides the means by which variation of Making such assumptions about yi* is now un- the free energy of activation AP,* with solute con-

necessary as will be shown in the following. centration may be expressed.

Combining equation (1) with the Darkenc3) equation for chemical diffusion in a binary system,

Thus,

I), = CW~*&~Y~[~ + (d In ~$2 In ci)]

x exp (-- AFi*/RZ’). (5)

-

TABLE 1. Compilation of activity coefficients of activated species,$ nickel-gold, 900°C.

-._-...-_ .-.... ___-.-.

Nau Diu ’ ( x 100) YAU YAU*

’ D&(cm=/ (cm~jsec x 109) se0 x 10”) YNi YNi*

__I_____._..~.-

0

1: 15 20 25

1: 40 45 50

xi 65 70 75 80 85 90 95

100

! 0.010 0.017

j 0.030 0.048 0.080 0.12

/ -11.0 9.2 5.9 4.2 3.2 2.6

990 486 177

;: 19 I

/ 0.19 0.25 0.42 0.58 0.75 0.90 1.06 1.1s 1.30 1.32 1.35 1.32 1.25 1.10 0.90

/ 2.2 / 10 !

/ 1.9, 6.2 /

1.4, 1.3, 1.3, 1.2, 1.1, 1.1, 1.0, 1.0, 1.0;

i :%I

I 1.8 1.4 :

::5 0.82 0.79 I 0.73 /

/ 0.72 0.74 ! 0.83 ! 1.00

0.06 0.4 1.2 2.5 4.8 8.2

13 19 28 36

:;

I; 105 114 120 120 115 105

90

i_.._+_. I-- _ _ 1.00 / 1 .oo 1.0, :fp

1:1”,

I / 0.15 0.05

0.02, 0.01, 1.2, 0.008,

! 1.2,

’ 0.006,

I 1.3, 0.004, 1.4, 1.5, /

1

;;;;;5

::2 0:0019,” 0.0017,

f.7 214

/ I 0.0015,

/ 0.0013, 0.0013,

2.7 0.0014, 3.2 0.0015 3.9 0.0019 4.8 j 0.0025 7.2

-8.0 : EE __..~_ --_-.-____

$ Pure solvent is taken as the standard state for normal species: iuflnite dilution of activated Au in pyre Au and of activated Ni & puce Ni are i;aken as standard states for activated species. Also DA, = 9 x 1O-1o cm2/sec, and I&, = 6 x lo-r5 cm*/sec. Diffusion data were taken from reference (1); activity data from reference (4).

56 AC!i’.4 METSLLURGICA, VOL. 8, 1960

Considering equation (3) it is apparent that the self (radioactive) diffusion coefficient in a homogeneous alloy,

includes the activity coefficient terms. Thus, since Bi = DCRIRT, the mobility terms in the Darken equations are functions of the activity coefficients, It is evident that, accepting equation (6), the phenom~no- logical treatment’3) of diffusion is consistent with the a,bsolute rate theory.

Equation (6) may be used to determine yi* from experimental data on DiR and y( as a function of Ni. The results of such a computation are presented in Table 1. The diffusion data for the system niekel- gold were taken from Reynolds et aE.tn, and the activity coefficients were taken from the work of Seigle et al. (4). It is felt that the resulting values for ptd* are reasonable in view of the following qualitative arguments. As the concentration of nickel is in- creased, the activated gold atom LLsees” a more compact lattice, and also the equilibrium conoen- tration of vacancies should decrease. Accordingly, it is reasonable to expect yAu* to become large in the nickel-rich regions as is observed in Table 1. By similar reasoning one would expect yxi* to become small in the gold-rich solutions.~

Hence in this ease, absolute rate theory gives a plausible rationalization of the results of diff~lsion measurements in the system gold-nickel.

References I. J. E. REYNOLDS, B. L. AVERBACK and M. COHEN, A&cl,

Met., 5, 2Q (1957).

4. L, L. SEIatE, M. COHEN and B. L. AVERBACH, Tmns. Rmer. Inst. Min. (Metall.) hgrs. 194, 1320 (1952).

* This work was supported by the U.S. OfEee of Naval Resemch. Received July 31, 1959; revised version October 22, lS5S.

I_ III this quditative argument non-ideal in%e~~&ons between solute atoms and vwcaneies are iwored.

Interstitial atoms in cold worked aluminum*

This question arose when it was found here that cold drawn Al wires (0.3 mm in diameter, above 90 per cent cold reduction) have a slightly higher density than the same wires recrystallized at 250°C for 15 min in vacuum (Al from the Aluminum Company of America, -+99.99% pure). Although there is some

evidence that the density of cold worked metals is lower than that of recrystallized ones,fl*s) there are also measurements showing no effect or the opposi~ behaviour. Such metals are, according to Tam- mann’l), Sn, Cu, Cd, Ag, Au, and there is co~rmation by others (Cu, f3) Ag’J) and other metaM”,@)). Micro- crack formation may be responsible for the lower density of cold worked metals.t’-Ii)

As density meas~ments(12} alone do not reveal the actual conditions in a crystalline lattice, precision lattice constant determinationsoa) of cold drawn and recrystallized Al wires were also made. In such a case, nr, the actual number of atoms per unit cell can be calculated from

where v is the volume of the unit (tell, d the density, N, Avogadro’s number(i*) and A the atomic weight of Al. If ‘it’ > n, n being the ideal number of atoms per unit ceft, excess atoms are present in the meta& if n’ = ‘L, the lattice is sound, and if n’ < n‘, vacant sites are predominant. For comparison, ?z’ of re- crystallized zone-refined Al (AIAG, Switzerland) was also determined. The results are summarized in Table 1.

The first line of the table shows that the lattice of the zone-refined Al is sound because the deviation n’ - P, = 0.00041 is within the limits of error (~0.00044, calculated from error propagation). On the other hand the 99.990/, Al contains excess atoms, as n’ - n is completely outside the error span. How- ever, the increased density can be explained by higher density segregations, observed e.g. by Cuff and Grant in such a metal.(r5j There is no such expla,nation for the cold worked metal (3rd line of the table) and, at present, we have to assume, there are interstitial atoms at a ~0nGent~tion of at least 4 atoms per 1000 unit cells. These atoms are forced into ~n~rstitial positions during coid work and may be calted arresting or blocking dislocations, because they resist the pro- cess of slipping, causing the strain hardening of the metal. At low temperatures these interstitial atoms do not move but leave their positions for the regular ones upon reception of some energy, e.g. in the form of heat (recovery of Al), decreasing the density of the metal.

Thus, according to this concept the cold worked metal consists of fibers or tractions of higher density strain-hardened metal (due to presence of interstitial atoms) and of micro cracks or fissures, which may or may not, depending on various obstacles, outbalance the density increase induced by cold work. The fissures close at higher temperatures causing a density