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Determination of the void nucleation rate from void size distributions
Journal of Nuclear Materials 68 (1977) 338-339 o North-Holland Publishing Company
DETERMINATIONOFTHEVOIDNUCLEATIONRATEFROMVOIDSIZEDISTRIBUTIONS A.D. BRAILSFORD Research StafJ
Ford Motor Company, Dearborn, Michigan 48121, USA
Received 12 April 1977
where IN(?) is the nucleation rate at time r and 6 [x] is the Dirac delta function. The solution (3) has the obvious property that
Recent advances in the theory of void nucleation [l] raise the question of how best to compare their prediction with the results of experimental observation. Straightforward integration of the void size distribution after successive time intervals to obtain the total void number is time-consuming and subject to error because the newly nucleated voids are below the experimental size resolution. Moreover, a major contribution to the change in the distribution may arise from void growth alone. The purpose of this note is to suggest an alternative approach to determining the nucleation rate as a function of time. Specifically, it is proposed that if the maximum void size, r,(r), is found as a function of time, 7, and the distribution&, t) of void radii, rv , is measured at time I, where t > T, then
(4) (where I, is the radius of a void at nucleation) since r(t, 7) spans the range from r(t, 7) E I, to r(t, 0) E r,,, as r varies over the domain of integration. Accordingly, from eq. (4), we find the standard result (neglecting void coalescence) that
IN(f)
N
$7f(rv, f) hv
.
(5)
rc As entails revert to r(t,
where p
=
[rm(t)2- r,(T)21u2 .
An analysis of an experiment based upon this result requires therefore only the determination of one complete size distribution coupled with prior observations of the largest voids. For these reasons it is proposed as an attractive alternative to the more obvious method of attack. The derivation of eq. (1) starts with Sears’ [2] solution of the continuity equation for f(rv, t), neglecting void coalescence events. This may be written down by inspection, for if a void nucleated at time T grows to a size r(t, T) at time I, then [2]
stated earlier, experimental evaluation of eq. (5) considerable labor. To seek an alternative, we to eq. (3) and change the variable of integration r), whence
L or
dr
1
[31
IN(T) = -
Msj-(?(t, T), t) .
(7)
since rv is an arbitrary radius in the distribution. Thus, in contrast with the form (S), eq. (7) depends only upon the complete distribution at one time, though now a knowledge of the void growth law is required to evaluate r(f, T). Fortunately this does not entail much detail, however, as we now indicate. 338
A.D. Brailsford /Determination of the void nucleation rate from void size distributions
Providing that higher order sink corrections [4] are negligible, the void growth rate is given by [4] dTV dt
1 = 6
[Dv{Cv
-
-GGl
W(N))
*
where the subscripts i and v refer to interstitials and vacancies respectively, the D’s are diffusivities and the ci and c, are far-field fractional concentrations of either defect. The quantity Cv(rv) accounts for vacancy thermal emission processes from the void and is dependent upon the void size, the surface free energy, temperature and pressure of any enclosed gas. Thus, in general, eq. (8) can only be integrated numerically. However, if the temperature is such that emission effects do not substantially effect the growth (i.e. roughly, at or below the peak swelling temperature), eq. (8) becomes 5 dr
ccI(&Cv rv
QCi)-_ A(t) ’
rv
where the final form follows because c, and Ci, being dependent upon properties of the entire void distribution, [and of course the dislocation density, pD(t)] are functions off only l. Basically, this is all that is required, for by integration of eq. (9) one finds
tit, 7)2 = rz t 2 ‘A(t’)dr
,
s 7
= rm(t)’
- r,(7)2
+ r: ,
For example, @
References
(11)
[l] D.R. Olander, (ERDA Report TID-26711-Pl, 1976) p. 470, provides a convenient review. [2] V.F. Sears, J. Nucl. Mat. 39 (1971) 18. [3] D.R. OIander, ref. 1, p.498. [4] A.D. Bra&ford, R. Bullough and M.R. Hayns, J. Nucl. Mat. 60 (1976) 246.
the totaI void sink strength (see ref. [4]) is 4nrvf (rv. t) drv, to the accuracy of eq. (8), and hence
manifestly only
a function
of
t.
We have been unable to locate sufficient data to explore the full implication of the above result adequately. However it is clear that it effectively provides a mapping of the void distribution at successively smaller sizes into the nucleation rate at successively larger times. This inversion is made particularly evident by the example f(rv, t) = CG(rv - rm(t)), for which it is verified readily that IN(~) = C’S(r), as one also expects on physical grounds. In summary, a method of estimating the void nucleation rate from one void size distribution and from observation of the maximum void radius at prior times has been proposed. Implicit in the method are the assumptions that both variations in the critical radius with dose and vacancy thermal emission processes during post-nucleation quasi-steady-state growth may be neglected. These hypotheses could be tested indirectly from additional experimental information. For example, repetition of the entire analysis on complete size distributions at two times (instead of just at one time) would serve to check on the necessary invariance of the derived nucleation rate over the time domain common to both. I am indebted to Dr. R. Bullough for this suggestion.
(10)
the last step following from the definition of rm(t). Result (1) then follows from eq. (7) upon substituting for r(t, T) and assuming r, < r,(t).
l
339
DETERMINATIONOFTHEVOIDNUCLEATIONRATEFROMVOIDSIZEDISTRIBUTIONS A.D. BRAILSFORD Research StafJ
Ford Motor Company, Dearborn, Michigan 48121, USA
Received 12 April 1977
where IN(?) is the nucleation rate at time r and 6 [x] is the Dirac delta function. The solution (3) has the obvious property that
Recent advances in the theory of void nucleation [l] raise the question of how best to compare their prediction with the results of experimental observation. Straightforward integration of the void size distribution after successive time intervals to obtain the total void number is time-consuming and subject to error because the newly nucleated voids are below the experimental size resolution. Moreover, a major contribution to the change in the distribution may arise from void growth alone. The purpose of this note is to suggest an alternative approach to determining the nucleation rate as a function of time. Specifically, it is proposed that if the maximum void size, r,(r), is found as a function of time, 7, and the distribution&, t) of void radii, rv , is measured at time I, where t > T, then
(4) (where I, is the radius of a void at nucleation) since r(t, 7) spans the range from r(t, 7) E I, to r(t, 0) E r,,, as r varies over the domain of integration. Accordingly, from eq. (4), we find the standard result (neglecting void coalescence) that
IN(f)
N
$7f(rv, f) hv
.
(5)
rc As entails revert to r(t,
where p
=
[rm(t)2- r,(T)21u2 .
An analysis of an experiment based upon this result requires therefore only the determination of one complete size distribution coupled with prior observations of the largest voids. For these reasons it is proposed as an attractive alternative to the more obvious method of attack. The derivation of eq. (1) starts with Sears’ [2] solution of the continuity equation for f(rv, t), neglecting void coalescence events. This may be written down by inspection, for if a void nucleated at time T grows to a size r(t, T) at time I, then [2]
stated earlier, experimental evaluation of eq. (5) considerable labor. To seek an alternative, we to eq. (3) and change the variable of integration r), whence
L or
dr
1
[31
IN(T) = -
Msj-(?(t, T), t) .
(7)
since rv is an arbitrary radius in the distribution. Thus, in contrast with the form (S), eq. (7) depends only upon the complete distribution at one time, though now a knowledge of the void growth law is required to evaluate r(f, T). Fortunately this does not entail much detail, however, as we now indicate. 338
A.D. Brailsford /Determination of the void nucleation rate from void size distributions
Providing that higher order sink corrections [4] are negligible, the void growth rate is given by [4] dTV dt
1 = 6
[Dv{Cv
-
-GGl
W(N))
*
where the subscripts i and v refer to interstitials and vacancies respectively, the D’s are diffusivities and the ci and c, are far-field fractional concentrations of either defect. The quantity Cv(rv) accounts for vacancy thermal emission processes from the void and is dependent upon the void size, the surface free energy, temperature and pressure of any enclosed gas. Thus, in general, eq. (8) can only be integrated numerically. However, if the temperature is such that emission effects do not substantially effect the growth (i.e. roughly, at or below the peak swelling temperature), eq. (8) becomes 5 dr
ccI(&Cv rv
QCi)-_ A(t) ’
rv
where the final form follows because c, and Ci, being dependent upon properties of the entire void distribution, [and of course the dislocation density, pD(t)] are functions off only l. Basically, this is all that is required, for by integration of eq. (9) one finds
tit, 7)2 = rz t 2 ‘A(t’)dr
,
s 7
= rm(t)’
- r,(7)2
+ r: ,
For example, @
References
(11)
[l] D.R. Olander, (ERDA Report TID-26711-Pl, 1976) p. 470, provides a convenient review. [2] V.F. Sears, J. Nucl. Mat. 39 (1971) 18. [3] D.R. OIander, ref. 1, p.498. [4] A.D. Bra&ford, R. Bullough and M.R. Hayns, J. Nucl. Mat. 60 (1976) 246.
the totaI void sink strength (see ref. [4]) is 4nrvf (rv. t) drv, to the accuracy of eq. (8), and hence
manifestly only
a function
of
t.
We have been unable to locate sufficient data to explore the full implication of the above result adequately. However it is clear that it effectively provides a mapping of the void distribution at successively smaller sizes into the nucleation rate at successively larger times. This inversion is made particularly evident by the example f(rv, t) = CG(rv - rm(t)), for which it is verified readily that IN(~) = C’S(r), as one also expects on physical grounds. In summary, a method of estimating the void nucleation rate from one void size distribution and from observation of the maximum void radius at prior times has been proposed. Implicit in the method are the assumptions that both variations in the critical radius with dose and vacancy thermal emission processes during post-nucleation quasi-steady-state growth may be neglected. These hypotheses could be tested indirectly from additional experimental information. For example, repetition of the entire analysis on complete size distributions at two times (instead of just at one time) would serve to check on the necessary invariance of the derived nucleation rate over the time domain common to both. I am indebted to Dr. R. Bullough for this suggestion.
(10)
the last step following from the definition of rm(t). Result (1) then follows from eq. (7) upon substituting for r(t, T) and assuming r, < r,(t).
l
339