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FORMATION l.

IN SOLIDS

INTRODUCTION

The final phase transformation involving the vapor that shall be considered in this monograph is the process of void information in solids. A special instance of this type, void formation at a free surface, has already been discussed in Section D-3-b-2. The growth aspect of this transition is thought to be simply a boundary value problem involving the diffusion of vacancies and hence will not be treated further. Also, data on the nucleation kinetics involved in void formation is sparse, and therefore only a brief account of these processes will be undertaken here. Void formation has been observed in diffusion couples(S1°,5m and in creep tsx2} at elevated temperatures where, because the specific interracial free energy is more isotropic at higher temperatures, the voids tend to form in the shape of segments of spheres. Also, vacancies quenched-in from elevated temperatures have been observed to lead to the formation of dislocation loops within monocrystalline grains.(S13,sx4~ According to a mechanism first proposed by Seitz {Sxs~and Frank t516~(see also refs. 517, 518, 519) these dislocation loops form by the nucleation of vacancy discs,* which subsequently collapse to form the loops.

2.

NUCLEATIONOF VOIDS AT ELEVATED TEMPERATURE

At elevated temperatures where vacancy mobility as determined from the diffusion coefficient is high, metastable equilibrium between single vacancies and embryos may be assumed. The nucleation kinetics are then given by equation (F-28) for homogeneous nucleation and (F-29) for heterogeneous nucleation except that in this case

AG~ = (--kT/f2) In (n,.ac/nv~e,)

(G-l)

where nv, c and nvae, represent, respectively, the actual and equilibrium concentration of vacancies.t An equation similar to (F-29) but differing in the pre-exponential factor has been derived by Resnick and Seigle. c52°~ Resniek and Seigle studied pore formation during the interdiffusion of copper and zinc. They found a critical vacancy supersaturation ratio (r/vac/r/vao)crit of about 1.5 and a contact angle 0 = 150° for the nucleation of * Bounded by low-index planes. T Note that n,~ exp (--A6T]kT) (equation F-28), the frequency with which an atom at the void surface diffuses into the crystal, equals nv,c~, exp (--AG~JkT) in which AG~a is the activational free energy for vacancy diffusion. 162

G.3]

NUCLEATION OF VACANCY DISCS

163

pores on a heterogeneity which was presumed to be zinc oxide. However, their correlation was only approximate, and thus it appears that further work is required for a definitive test of equation (F-29) as modified for this case.

3.

NUCLEATION OF VACANCY DISCS

The rate equation for vacancy-disc nucleation should differ from equation (F-28) only in the geometry of the critical nucleus. The free energy of forma= tion of the critical disc is AG* = 2rrr*Ztr + 2~r*e + 7rr*2hAG,, ---- --rr*~'/(2e + h A G , )

(G-2)

where e is the edge energy of the disc, tr is the interracial free energy of the low-index circular surface and h is the height of the disc. This expression is valid only when IhAGol>12al. Although numerous experiments have been performed in which vacancies have been quenched-in from elevated temperatures (see ref. 521), the vacancies may also diffuse to sinks such as the free surface, dislocations and grain boundaries. This consideration enormously complicates the analysis of vacancy removal.t~a, 5z2) Thus even though vacancy-disc dislocation loops have been observed, tSxz,514) the kinetics of disc nucleation have not been quantitatively studied. Nevertheless data have been obtained ~Sm)on the quenching-in of vacancies which may pertain to the interesting postulation of athermal nucleation by Fisher, Hollomon and Turnbull39~) They developed the kinetics of athermal nucleation for precipitation in condensed phases but their treatment can easily be extended to vacancy-disc nucleation. Let a crystal be equilibrated at temperature To. where the equilibrium concentration of vacancies is nvae and then quenched to a temperature 7". The metastable-equilibrium concentration of critical nuclei at T is n* ---- n o exp (--AG*/kT)

(G-3)

where n o is the concentration of lattice sites and AG* is given by equation (G-2). Now at the temperature T o the concentration of embryos (subcritical at T o of course) of the same size is n~ = n o exp (--AG°/kTo).

(G-4)

If the quench from TO. to T is very rapid so that embryos existing at To. do not dissociate during quenching, embryos which are critical at T will already exist in appreciable number if n~ (equation G-4) is greater than n* (equation G-3). Since the nuclei nl do not require thermal fluctuation to form at T, this process is called athermal nucleation. (99) The condition that ni ~ n* leads to the requirement that

(AG°/To) < lAG*IT)

(G-5)

164

v o l e FORMATION IN SOLIDS

[G.4

for athermal nucleation. Now AG O is given by equation (G-2) which becomes AG O ---- 2zrr~a + 2~r*e

(G-6)

in this case because AG~ = 0 at T o. Also r* ---- -- e/(2a + hAG~)

(G-7)

and hence upon substituting (G-2) and (G-6) into (G-5) one obtains the conditions for appreciable athermal nucleation ( r o / r ) > 2 -- [2~/(2a + h ± a 0 ] or

(To/T) ~ 2 -- (2a/hAG~).

(G-8)

The term (2a/hAGO is negative and varies in magnitude from a large number when T---~ T o to a value of about unity when T , ~ T O. Baurle and Koehler ~s~ performed careful experiments in which they quenched gold wires to room temperature and studied the kinetics of the decrease in vacancy concentration. They found anomalous behavior which indicated the presence of vacancy clusters in the as-quenched wires for T o ~ 700°C. They and Seitz ~524Janalyzed this anomaly on the supposition that the clusters formed during quenching. However, noting that vacancy clusters appeared when (To[T) ~ 3.3 it would appear in view of equation (G-8) that athermal nucleation, as proposed by Fisher et al., ~99~could account for these results.

4.

SUMMARYOF VOID FORMATION IN SOLIDS

Void nucleation and growth in solids should follow essentially the same kinetics as in the liquid. Again, there are no new statistical mechanical contributions to the free energy of formation of the critical nuclei. The kinetics of void nucleation in solids have not been studied quantitatively. However, this appears to be a promising area for study because it is one of the few situations in nature where homogeneous nucleation may occur. Finally, the possibility exists that athermal nucleation of vacancy clusters may occur during quenching experiments.