On the growth of inert gas bubbles subjected to stress

JOURNAL

OF NUCLEAR

ON THE

MATERIALS

GROWTH

42 (1972) 235-236. 0

OF INERT

NORTH-HOLLAND

GAS BUBBLES

PUBLISHINQ

SUBJECTED

CO., AMSTERDAM

TO STRESS

G. W. LEWTHWAITE

Received 1 November 1971

where yi, is the grain boundary tension, ‘ys the surface tension and r the radius of ourvature of the spherical caps. The equilibrium conditions are :

Bubbles of inert gases lying on grain boundaries are believed to be the cause of the hightemperature embrittlement of many irradiated metals. Hyam and Sumner 1) showed that under the action of stress bubbles can become unstable and enlarge continuously and Barnes 2) subsequently invoked this effect to explain the embrittlement of stainless steels and nickel alloys which have been exposed in reactors to fluxes of neutrons. Apart from a remark by Barnes that bubbles on grain boundaries will be subject to peripheral forces due to the grain there is ~pp~~ntly no boundary tension discussion of the influence of this tension on the stress required to promote continuous growth of bubbles. Nelson et a1.3) have discussed the shape adopted by a grain boundary bubble which has achieved thermod~amic equilibrium. They show that in general such bubbles will adopt irregular, facetted shapes determined by the relative orientation of the grains and the variation of surface energy with orientation ;

yb = zy, co8 8,

Pf

P = 2y&,

(2)

where P is the pressure of the gas in t,he bubble. On applying a hy~ost&tic tension PB the radius will change to satisfy the equation: P&-t Pr = 2ys/rl

(3)

and by assuming that the perfect gas laws are obeyed, the critical stress for continuous growth is: PC = 4y,/3@. (4) Por a similar bubble (that is, one contail~ing the same number of gas atoms) isolated in the lattice, the equivalent expression is P’c = 4ys/3p%o.

such complications probably exclude a simple analysis of the effect of grain boundary tension on the critical stress. By adopting the usual simplification that grain boundary bubbles are lenticular, however, an expression for the critical stress can easily be obtained. This expression resembles that of Hyam and Sumner but with the bubble radius replaced by the radius of the spherical caps comprising the lentioular shape. Because this radius is larger than that of a similar bubble isolated in the grain, the stress required for continuous growth is lower. Suppose, then, that the shape adopted by a grain boundary bubble is as shown in fig. 1, 235

r and ro are related through

f5l the equation

ro2 = @2( 1 - cos f!J)z( 2 + cos 0). Therefore,

(6)

using eq. (1) we have:

PC/PC= (I

--yb/+s)(2

x yb,&s)*/v’2.

(7)

groin boundary

Fig. 1. Bubble on grain boundary.

236 The

G.

table

Values Of

below

yb/ys

for

LEWTHWAITE

various

yb[ys. TABLE

P,/P’,

P,jPfc

gives

W.

From the data quoted

stress required precipitate.

I

0.1 ’ 0.2 0.3 / 0.4 / 0.96 / 0.92 1 0.88 ! 0.84

so that, for, say, O= $z, PC/PC N 0.7 where 0 is t.he contact angle and P, is now t,he crit,ica,l

0.5 j 0.6 0.79 / 0.75

by McLean 4), values

for yb/ys approaching 0.4 can be seen to be achieved by several metals and alloys and from table 1 we see that bubbles lying on such grain boundaries will become unstable at a stress approximately 16% lower than similar bubbles within the grains. Similar reasoning can be applied to other, analogous situations. The binding of inert gas bubbles to precipitates has been considered by Nelson 5) and, as an example, we can use his results for a bubble attached to a large, flat,, rigid pre~ipita~. For this situation: r(Js= $994 - (1 - 00s 0)2(2 + cos 6))

for the bubble

attached

to t’hc

It has been shown that the expression for the stress required for continuous growth of bubbles lying on gram boundaries can be written in a manner which explicitly contains the grain-boundary energy. Examination of this expression shows that suoh bubbles are more unstable with respect to stress than similar bubbles within the grains. A similar effect, is found for bubbles attached to precipitates.

References 1) E. D. Hyam and G. Sumner, Proc. IAEA Symposium, Venice (1962) 323 “) R. S. Barnes, Nature 206 (1965) 1307 3) R. S. Nelson, D. .I. Mazey and R. S. Barnes, Phil. Mag. 11 (1965) 91 4) D. McLean, Grain boundariesin metals (Clarendon Press, Oxford, 1957) 5) R. S. Nelson, J. Nucl. Mater. 19 (1960) 149