The role of divacancies in void swelling

Scripta METALLURGICA

Vol. 13, pp. 635-639, 1979 Printed in the U.S.A.

Pergamon Press Ltd. All rights reserved.

THE ROLE OF DIVACANCIES IN VOID SWELLING*

M. H. Yoo

Metals and Ceramics Division, Oak Ridge National Laboratory Oak Ridge, Tennessee 37830

(Received May 2, 1979) (Revised May 14, 1979) In a recent paper [ l ] , the effect of divacancies on the growth kinetics of dislocation loops and voids has been investigated by incorporating into the rate equations the reactions of vacancy association, divacancy dissociation, divacancy-interstitial interaction, and divacancy loss to unsaturable sinks. Based on the available defect parameters for f.c.c, metals, i t was found that the general role of divacancies in diffusion processes is most important in Al and Ni, moderately important in Cu and Ag, and not important in Au. Since the previous analysis [ l ] was based on the parameters of monovacancies and divacancies from available self-diffusion data [2], i t is worthwhile to examine the s e n s i t i v i t y of the divacancy role in void swelling under irradiation by varying divacancy parameters, such as the binding energy of divacancy, E~v, and the preference factor of dislocation for divacancies, ~ v . The defect generation rate, G, was taken as a constant for both monovacancies and i n t e r s t i t i a l s , and zero for divacancies in the previous paper [ l ] , which is the case for an electron irradiation situation. During neutron or heavy-ion irradiation, displacement damage cascades may produce an appreciable population of divacancies. Therefore, the effect on void swelling of the divacancy source from depleted zones created by displacement cascades also w i l l be investigated in the present paper. The rate equations for mobile defect concentrations are: ~v = G v - RCvCi - KvCv + 2(KdC2v- KaC~) + R'C~vCi '

(I)

~i = Gi - RCvCi - KiCi - R'C2vCi '

(2)

~2v = G2v + KaCv - (Kd + R'Ci + K2v) C2v ' (3) where Ca is the average defect concentration (atom fraction) with ~ = v, 2v, or i for vacancies, divacancies, or s e l f - i n t e r s t i t i a l s . The total vacancy and i n t e r s t i t i a l generation rates are given by [3] Gv = Gf(l - Ev) + G~ ,

(4)

Gi : Gf(l - c i ) + G~ ,

(5)

where f is the f r a c t i o n o f p o i n t defects s u r v i v i n g recombination w i t h i n displacement cascades (0 < f < l ) . ~v and ~i are the numerical f a c t o r s (0 < Ev < I , 0 < Ei < I ) accounting f o r the reduction of f r e e vacancies and i n t e r s t i t i a l s , r e s p e c t i v e l y , due to athermal n u c l e a t i o n o f t h e i r r e s p e c t i v e d e f e c t c l u s t e r s in displacement cascades. GG is the emission r a t e o f thermal vacancies from a l l o f the i n t e r n a l s o u r c e s , a n d G ~ i s the a d d i t i o n a l generation r a t e of i n t e r s t i t i a l s due to the deposited s e l f - i n t e r s t i t i a l s during s e l f - i o n bombardment. Consider a case o f heavy-ion i r r a d i a t i o n where Gf : 10-3 dpa/s and Gi = O. I f the defect c l u s t e r s created from depleted zones are assumed to be a l l mobile divacancies, then ~i = 0 and the source term o f divacancies in Eq. (3) i s :

G2v = ~Gf~v . (6) The numerical value of ~v refers to the fraction of the total population of vacancies, created by da~ge, that are involved in forming divacancies. Research sponsored by the Materials Sciences Division, U.S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

635 0036-9748/79/070635-05502.00/0

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The reaction rate constants in Eqs. ( I ) , (2), and (3) are the same as in the previous papers [1,4]. The recombination rate constant is R = 4~r0(Di + Dv)/~, where r 0 is the radius of the recombination volume, DS is the diffusion coefficient of defect type s, and R is the atomic volume. To a f i r s t approximation, the loss rate constant for s-type defects to dislocations and voids may be given as K~ = Ds(Z~L + 4~RvNv), where L is the dislocation density, Nv the void density, and Rv is the void radius. ZS is the capture efficiency factor of dislocations for ~-type point defects, ZS = 2~/Zm(Ro/rS) and r~ = rc(l + as), where Ro and r c are the outer and inner cut-off radii of dislocations, respectively, r S is the effective capture radius of dislocations for ~-type defects, which is related to aS, the preference factor of dislocations for s-type defects as written above [4-~]. The reaction rate constants involving divacancies in an f.c.c, l a t t i c e may be given as [7] Ka = 84vexp(-~/kT) ,

(7)

Kd = 14vexp[-(Em + Eb )/kT] (8) v 2v R" = 4~r6(Di + D2v)/n , (9) where v is the attempt frequency of vacancy jump (v ~ 1013s-i), E~ the defect migration energy, and r 6 is the effective capture radius of divacancies for i n t e r s t i t i a l s . The rate equations (1), (2), and (3), which u t i l i z e the reaction rate constants discussed above, must be solved together with the rate equation of void growth* ~V = [Dv(Cv - C~) - DiCi + 2D2vC2v]/RV , where C~ is the local equilibrium concentration of vacancies at a void [4,6].

(lO)

In this paper, examplecalculations were made for the case of Ni irradiated by Gf = lO-3 dpa/s up to lO dpa at temperatures 200-800°C. The coupled rate equations (1), (2), (3), and (lO) were solved by the usual numerical integration method [4]. The input parameters for the present analysis are listed in Tables l a~d 2, which are consistent with those for Ni in other calculations [1,8]. The vacancy parameters are from Seeger and Mehrer [2], and the i n t e r s t i t i a l parameters are from Lampert and Schaefer [9]. The temperature dependence of the radius of recombination volume is approximated based on the calculated [ l O , l l ] and experimentally v e r i f i e d [12] result. The temperature dependenceof sink densities is approximated based on the experimental result by Packan et al. [13]. Table 2. TemperatureDependentSink Densities in Nickel TableI. DefectParametersfor Nickel h

h

£,NV =£~0A£I "~ , t.m L = ~oAil-~ Sv f : 1.5 k

Di : 1.8 x lO "2 cm2/s

Efv = 1.39eV

Em i = 0.15eV

D~ = 1 . 6 x lO -2 cm2/s

r o = 8.38 aoT't/3

Evm = 1 . 3 8 eV

a o = 0 . 3 5 nm

AS = 1.8 k 2v Eb = 0.3 eV ~v

r

= 1J/m2

6i = l.O

NV Ao A1 A2

D~v = 1.3 x lO-3 cm2/s 62v = 0.5

A3

Em = 0.82 eV

A~

2V

6v = 0

31.43 2 . 4 7 x 10-2 -6.78

x lO-S

6 . 9 4 x 10 -e -2.60

L 22.81

x 10-11

1 . 4 6 x 10-3 -6.66

x lO-S

9 . 3 6 x 10"9 -6.61

x 10 - 1 2

Figure 1 shows the s e n s i t i v i t y of the temperature dependent void swelling curve to the specific value of divacancy binding energy. The melting point of Ni in the absolute temperature is Tm: 1728 K. As was discussed in the previous paper [ l ] , there arise two d i s t i n c t temperature reglmes of void swelling insofar as the effect of divacancies on void swelling is concerned. Void swelling is enhanced in the recombination dominant regime at low temperatures (<590°C) and is suppressed in the sink dominant regime at higher temperatures. Whenthe defect annealing during irradiation is dictated by the mutual recombination, the net f l u x of surviving vacancies to voids increases because of the r e l a t i v e l y high mobility of divacancies (see Table l ) . Whereas, in the temperature regime (590-710°C) where the defect annealing is dictated by the defect loss to dislocations void swelling is suppressed because the preferential absorption of divacancies to dislocations occurs due to the non-zero preference factor used for the present calculation, 62v =0.5. The increase in E~v decreases Kd in Eq. (8), which in turn increases the divacancy Here, for simplicity, voids are considered as neutral sinks.

Vol.

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VOID SWELLING

637

r/Tm 03

s

0.4

I

0.5

I

7"/Tm

06

I

07

I

6

J

05

0.4

05

0.6

07

I

I

I

I

I

8Zv/Bi

EbvleV) 5 --

0.4

,,~i~l Ill

~4

_

.o

!i

l

M

0.VACA.,ES.//.

A:

_ I,// /ii ,

,,,oo,

-

¢n a

,_

200

1<'//

500

400 500 600 TEMPERATURE (=C)

FIG.

700

_

1 --

800

200

1

.,,/.. 300

400 500 600 TEMPERATURE ('C)

700

800

FIG. 2

Effect of the binding energy of divacancies on the temperature dependent void swelling.

Effect of the relative preference factor of dislocations for divacancies on the temperature dependent void swelling.

concentration according to Eqs. ( l ) , (2), and (3). Therefore, both the enhanced swelling at low temperatures (<5~0°C) and the suppressed swelling at high temperatures (590-710°C) should be strengthened as E~v is raised. This is evidenced in Fig. I . Another important divacancy parameter of interest is the preference factor of dislocations for divacancies, 62v. The void swelling shown as the solid curve in Fig. 2 is the same as that in Fig. l , which is the result based on the input parameters l i s t e d in Table I . As the ratio of the preference factor for divacancies to that for i n t e r s t i t i a l s , ~2v/6i, is raised from 0.5 to 0.7, the transition temperature above which void swelling is suppressed by divacancies is reduced from 590 to 470°C. On the other hand, when the ratio is lowered to 0.3, the enhanced void swelling occurs over the entire temperature range of void swelling. At a given temperature, therefore, whether the role of divacancies is to enhance or to suppress void growth depends c r u c i a l l y on the relative strength of the preference factor 62v. The void swelling with ~v = 0 in Fig. 3(a) is identical to the solid curve in Figs. l and 2. Figure 3(a) shows that an increase in ~v causes an increase in the enhanced void swelling, and when cv ~ 0.3 a peak on the swelling curve is developed at about 430°C. Since the increase in E. means a d i r e c t l y proportional increase in the source of divacancies by Eq. (6), the enhanced swelling in the recombination dominant temperature regime increases with increasing cv [14]. Figure 3(b) shows how the relative concentration of divacancies varies with temperature. The ratio of the thermal equilibrium concentration of divacancies to that of monovacancies increases monotonically with temperature reaching C2v/Cv ~ lO-4 at 610°C and lO-3 at 800°C. Under the irradiation condition by heavy-ion damage with Gf = lO-3 dpa/s at lO dpa, the ratio of the supersaturated concentration of divacancies to that of monovacancies is C2v/Cv > 3 x lO-2 at <300°C, and decreases with increasing temperature. The concentration ratio Increases as Ev increases, and this relation is p a r t i c u l a r l y prominent at low temperatures as shown in Fig. 3(b). Figure 3(c) shows the temperatur dependence of vacancy diffusion coefficient as the dashed curve and that of the effective d i f f u s i o r coefficient for vacancies under the i r r a d i a t i o n condition as a group of solid curves. The effective diffusion coefficient is defined as Deff DvCv + 2D2vC2v v : Cv + 2C2v

(ll)

The d i f f e r e n c e between D~f f and Dv is the measure o f enhanced d i f f u s i o n due to the presence of d i v a c a n c i e s , which is q u i t e s u b s t a n t i a l as depicted in Fig. 3(c). For instance, the r a t i o between the two, D~ff/D v is ~lO 3 a t 250°C and ~I0 at 470°C. The r e l a t i v e importance o f divacancies in void growth at temperatures where C~ < Cv can be measured bY the r e l a t i v e values o f d i f f u s i o n p o t e n t i a l [ I ] , DvCv and 2D~vC~v, the two . terms in . _ Eq. ( I 0 ) . An a l t e r n a t i v e way o f measuring the importance o f divacancies is to deal w l t h

638

VOID SWELLING

Vol.

T/Tin 0.3

0.4 I

I

0.6 I

0.7 I --/.//~(_~l

//

~

7

(a)

0.5 I

NO DIVACANCJES

5

13, No.

--

t

~/

4

-~ 3

= .

¢0 Q

>52I

0

I°'1--'] 2

I ~

\\

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I

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UNDER IRRADIATION

(v

o. 2--~

(b) =

----

~"/\\X

10-z -

10-3.

s

/ THERMAL EQUIL.--~ /

2

lO-4

I

1°-9 _

1

I

~

I

I/

/

I

/

I

(c)

10-1o

/ / . / . IC."~, = o8 /

,.."

-,o-,,

E Io-,, " / / ~ - o 2

/

/\

/Lo u

/

~__ 10 -13 ~ F, 10-14

,o-15

I

200

k__Dv

//

/ /

/

I 300

/

~

I

I

I

400 500 600 TEMPERATURE (°C)

Deff

I 700

800

FIG. 3 (a) The development of a low temperature swelling peak with increasing fraction of the total vacancies in depleted zones retained as divacancies, and the temperature dependence of (b) the relative concentration of divacancies and (c) the effective diffusion coefficient of vacancies.

Vol.

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VOID SWELLING

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Eq. (I0) without the t h i r d term, 2D2vC2v, and to substitute DvCv with D~ffc~, where C~ = Cv + C2v. In any case, the source for the swelling peak at ~430°C for ~v ~ 0.3 shown in Fig. 3(A) is the combined e f f e c t of the increasing d i f f u s i o n c o e f f i c i e n t and the decreasing defect concentrationwith increasing temperature. A so-called "second peak" of void swelling has been observed experimentally by several invest i g a t o r s [15], f o r example, in solution treated austenitic stainless steels a f t e r neutron i r r a d i a tion [16] and ion bombardment [17]. The development of the second peak at temperatures below the peak swelling temperature is generally a t t r i b u t e d to the solute trapping and segregation and the evolution of second-phase p a r t i c l e s . I t is i n t e r e s t i n g to note that the swelling data (Fig. 6) of Makin et a l . [17] shows q u a l i t a t i v e resemblance to the calculated r e s u l t shown in Fig. 3(a). The e f f e c t of displacement cascades on void swelling has been analyzed so f a r by considering athermal nucleation of vacancy type dislocation loops out of the depleted zones [18-2}]. In t h i s case, the role of continuously formed vacancy loops reduces the low temperature t a i l of void swelling d r a s t i c a l l y . The new approach adopted in the present paper is to consider only the mobile divacancies created from the displacement cascades. The low temperature swelling peak is also present in a neutron i r r a d i a t i o n with Gf : 10-6 dpa/s. The height of the low temperature swelling peak dependsLon the magnitude of Ev, and the temperature at which the peak develops depends not only on E~v and 62v but also on the r e l a t i v e m o b i l i t y of divacancies. A systematic analysis of the low temperature swelling peak as a function of the relevant parameters such as discussed above and the r e l a t i v e strengths of the low and high temperature swelling peaks as a function of i r r a d i a t i o n dose is needed. A specific experiment to v e r i f y theoretical predictions could be proposed following such an analysis. References I.

2. 3. 4. 5. 6. 7. 8. 9. I0. II. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

M. H. Yoo, Philos. Mag. (in press). A. Seeger and H. Mehrer, Vacancies and I n t e r s t i t i a l s in Metals, eds. A. Seeger, D. Schumacher, W. S c h i l l i n g , and J. Diehl, (North-Holland, Amsterdam, 1970) p. I . M. H. Yoo, J. Nucl. Mater. 79, 135 (1979). M. H. Yoo, J. Nucl. Mater. 68, 193 (1977). M. H. Yoo and L. K. Mansur, J. Nucl. Mater. 62, 282 (1976). L. K. Mansur, Nucl. Technol. 40, 5 (1978). A. C. Damask and G. J. Dienes, Point Defects in Metals (Gordon and Breach, New York, 1973) p. I I 0 . M. H. Yoo and L. K. Mansur, Fusion Reactor Materials ( I s t Topical Mtg. Miami Beach, Fla. Jan. 28-31, 1979); J. Nucl. Mater. (to be published). G. Lampert and H. E. Schaefer, Phys. Stat. Sol. B52, 475 (1972). K. Schroeder and K. Dettmann, Z. Phys. B22, 343 (1975). M. H. Yoo and W. H. Butler, Phys. Stat. Sol. B77, 181 (1976). R. Lennartz, F. Dworschak, and H. Wollenberger, J. de Phys. F7, 2011 (1977). N. H. Packan, K. F a r r e l l , and J. O. S t i e g l e r , J. Nucl. Mater. 78, 143 (1978). The range of Ev chosen for the present calculation is reasonable since the current computer calculation by M. T. Robinson gives Ev : 0.2 ~ 0.3 t y p i c a l l y for a cascade created by the primary recoil energy of 30 keV ~ 1MeV. Radiation Effects in Breeder Reactor Structural Materials, (Proc. Int. Conf., Scottsdale, A r i z . , June 19-23, 1977). J. S. Watkin, J. H. Gittus, and J. Standring, p. 467 of Reference 15. M. J. Makin, J. A. Hudson, D. J. Mazey, R. S. Nelson, G. P. Walters, and T. M. Williams, p. 645 of Reference 15. J. L. Straalsund, J. Nucl. Mater. 51, 302 (1974). R. Bullough, B. L. Eyre, and K. Krishan, Proc. Roy. Soc. (London) A346, 81 (1975). A. D. B r a i l s f o r d , J. Nucl. Mater. 71, 227 (1978). A.J.E. Foreman and M. J. Makin, J. Nucl. Mater. 7g, 43 (1979).