The effects of helium on the evolution of voids under cascade damage

4X6

Journal

THE EFFECTS OF HELIUM ON THE EVOLUTION K. KITAJIMA. Research Imtrrure

Y. AKASHI for Applied

and

Mrchunm.

of Nuclear

Materials

OF VOIDS UNDER

133&l 34 (19X5) 4X6-490

CASCADE

DAMAGE

E. KURAMOTO Kyushu

Unicersrty

87. Kmugu

-shl.

Fuh uoktr 810, Jtrpm

Equilibrium conditions of voids are examined under irradiations when cascades are created together with the coproductton of helium. In this situation the transport equations which describe the evolution of the voids are derived in the form of the Fokker-Planck equation. Solutions of the equation are then surveyed subject to approprtate boundary conditions. Approxtmate expressions are obtained for the nucleation rates in the cases of homogeneous nucleation. direct void formatton from cascades. and heterogeneous nucleation associated with precipitates. The characteristics of these types of nucleation are compared with each other and are used to explain the available experimental data.

1. Introduction Production of helium atoms by transmutation is known to be an important factor for the irradiation response of materials under fusion reactor environments Considerable efforts have been devoted to examine these effects using various simulation environments such as thermal neutron reactors [l], e.g. HFIR. or dual ion beam accelerators [2]. Theoretical interest has been concentrated on the role of helium atoms in the formation of voids, particularly in the nucleation aspects [3.4]. Recent observations of bi-modal size distributions of voids in simulation irradiations provides a new challenge for such theoretical studies. An extensive review of this field has been presented by Mansur [5,6]. In the present paper we propose a method of analysis of the problem within the frame-work of the statistical treatment of cascade damage previously developed [7-l 11. Generally the nucleation is complicated because of its sensitivity to many metallurgical factors and particularly many uncertainties remained concerning the physical nature of small clusters. We shall discuss, however. the characteristic mechanisms of various types of nucleation, i.e. homogeneous. heterogeneous (associated with precipitates) one and that induced directly by cascades. abstracting the available experimental evidences

2. The equations describing the evolution of voids containing helium Consider a void containing n vacancies and x helium atoms. the evolution of the void on a microscopic scale is given by dn/dt

=

d.\-/dt

= I/,( M. x, r. w).

U,>(n,

x,

(2) where $I = +( n. x, t) is the probability density of the cluster (17, x), F, and F, are the coefficients of the drift terms. and D,,,, and D,, are those of the diffusion terms, defined as the first and the second moments of the increments An and AX during the time interval At. 20,,,,=

F,, = (G/At).

0022-3115/85/$03.30 Physics

[(3-(LG)2j/At.ctc.

(3)

The interval At is determined by the characteristic correlation time of each stage. Eq. (2). therefore, provides a hierarchy of equations corresponding to the processes cited above. We take here the equation with largest At, i.e. the process of successive cascade--void collisions. Since each of the stages are statistically nearly independent, D,,,, can be approximated by (4)

n,,,, = cQ,,,.i.

where Q,,,., indicates

the contribution of each component. Expressions for t;, and D,,,, due to cascade diffusion, and F, and D,, due to cascade collisions were presented in the previous papers [7.10]. When the size of the void is much smaller than that of the cascade. r,; and II,, are expressed by F,

=P,R(AHe).

I (AHe)=soJ/.

+L= [

R=(4?7/3)r3.

v=/(T)/2.

t. w).

1 &il(. 1. I.,,>,,, o==n(u,

+trz)/4.

(1)

where w is a parameter specifying the initial and boundary conditions of the void. U, and U, consist of several fluctuation components with different time scales, i.e. those associated with thermal activations, the decay of cascades by diffusion

(North-Holland

various point defects. and the sputtering out of helium atoms contained in a void by successive cascade-void collisions. For each stage. the evolution of micro-structure, the void in this case. can be represented hy the Fokker-Planck equation of

(J Elsevier Science Publishers Publishing

Division)

B.V

~‘p/E,n,n = i (4M,Mz),‘(

M, + Ml )‘.

D,, = (1/2)P,R(AHe)‘. where

(5)

P, is the production

correlation

range

of

the

rate of cascades. cascade

collision

R is the interacting

Table I Reolution

rate of helium atoms contained in a void by cascade

with varrous PKA energies PKA energ! (keV) Penetration depth (I/a)

40

20

50

23

Ill li

5 X.?

; I’( l/l1 ) 1. ‘.,,,,,I K, t,’ = (4~,‘3)(//2~)’

591

719

12x

60

65 000

6300

1100

?OO

with the voids. 4 is the integrated flux of colliding atoms in a cascade which has energy E large enough to transfer energy greater than E, to the helium atoms contained in a void by the collision. E, is the critical energy of a helium atom required for its ejection into the matrix and is nearly equal to 100 eV for Ni. (I,. M,. (I? and M, are the atomic size and the mass of helium and a matrix atom respectively. N and 1 are the mean nutnher and mean free path of a colliding atom which has energy E in a cascade with a PKA energy of T. !2’ is estimated using the Kinchin-Pease model with h’(E) = 2” and E = T/2”. I(E) is estimated from the penetration range calculated using the TRIM code. Results for the calculation of F, are shown tn table 1: we see in table I that I+Lincreases when Tdecreases.

3. Solution of the Fokker-Planck

equations

Using the expressions for the transfer terms stated above. the final form of the equation describing the evolution of voids is expressed by,

+ Z/p

1

20,,,,=4nrD,C,.

i~Z/p+H, 2 D, ,

= 4rrD,C,

.

i

The nucleation rate of voids may be calculated using the solution of eq. (6) under the assumption of a quasisteady state and the appropriate boundary conditions. In this calculation, however. the discrete character of the quantities must be taken into account when the values of II and .Y are small. For this purpose we propose here the following simple approximation for the boundary conditions

for n = 1

(

f;,-&,+=O.

and assume that the values of $( II = 1. .Y= 0) = C, and @(II = 1, x = l)= Cv_He are given by the condition of halance of C,. and CL_Hc%as is cited in the appendix. A general view of the evolution of void embryos can be obtained by nodal line analysis in (n, .Y) space, (see fig. 1 in section 3.1) [4]. The two crossing points of the curves F,, = 0. and F, = 0 represent the stable bubble and the unstable voids at the critical size.

In the previous paper, we introduced the flux band approximation [lo], where an approximate expression for the nucleation rate J, is obtained in the form of integral along the critical path parallel to flux vector of void embryo:

JT= b@(l)/iXdt

h@=i-G-J,. ac

exp(G),

(8)

i~=J[(F;,m+F,i)-i,n(D,,,,-D,.)(F;:i-F:m) F, = 47rrD,,C;

- Kzs

/(o,,,,.i+D,,~~)]~d~.

(l+.Z)+(l/g)exp-(p-F)fj/kT

1

b~=I[(D,,,,m+D,,i)-.im(D,,,-D,,)

.

+ K:.x.

/(Q,,.i+ (6)

where D,. C,., D,, Ck are the mobilities and the concentrations of vacancies and interstitial helium atoms respectively. Z is the flux bias factor, Z = 0.95, jI is the supersaturation of vacancies. K: is the resolution coefficient of helium calculated from eq. (5). I’ is the surface tension, r is the radius of the void. and the pressure p caused by helium in a void is given by Van der Waals equation of state. p = AT/( II - b Y )s2, where S2 is the atomic volume. H, is the factor ,.,tused by cascade diffusion calculated previously [7].

D,,n?)]Gdv.

m=cosl9.

,j = sin 0.

@=

J,=

I

+dn.

I

(JnM+Jxj)dq.

where 5 and n are the co-ordinates parallel thogonal to the flux vector of the void embryo

J,,=

i

F,r-$%t,)G.

and orJ:

J,=(kj$+-

The approximation (8) is based on the assumption that the main part of the flux is confined in a rather narrow

488 Table 2 The calculated values of the integral /d[ exp(G) in the expression of J, in eq. (8). Parameters are /3. K. Cas = H,. the effects of cascade diffusion in eq. (6). and /I = n r,/ nc. the heterogeneous nucleation rn eq (11). (a) D = 0.5. Cas = 0 K

p =I650

850

450

250

0

4.8 El0 6.3 E2 1.9 E2

1.9 El5 1.0 ES 6.3 E2

1.3 El7 4.0 E3 6.9 E2

7.0 1.5 2.2 4.3 1.1

1.2 1.6 2.0 2.4 3 5

150 E22 El0 E5 E3 E3

1.2 1.6 1.9 1.2 1.1

50

El6 E8 ES E4 E3

2.2 1.4 2.5 5.1 5.9

._.~__

E26 El7 El2 F.7 E3

-_. (b) fi = 250, K = 1.6. D = 0.5

K

Cas = 0.25

0.5

1.2 1.6 2.0 2.4

1.5 4.X 3.2 1 I)

7.4 2.1 2.1 9.8

region

formed

using

EX E4 E3 E3

around the

..-_-___

p = 250, K = 1.6, Cas = 0 (c)

band

,/,,c,.

D-D,,(,;I)

the path.

3.7 8.4 2.2 9.x

Calculations

approximate

D

1.0 E6 E4 E3 E2

path

going through the two critical points *. This

ES E3 E3 E2

= i;;/F;,

is based

-

2 1 0.5 0.25

were per-

dx/dn

(d)

on

4.2 4.1 2.2 8.5

E4 E4 E5 E5

the considerations field of (F,. tial

D = 0.5 2.0 2.5 1.8 3.4

for the mapping

ES IF-4 El 1’2

of the drift

F, ) and the form of the nucleation

G derived

approximation

h 0.1 0.2 0.4 0.x

from

it. The

error

introduced

was checked by numerical

flo\\ poten-

h!

thih

integration

01

eys. (6) and (7) for some special caseb. the details

of

which will be presented in another paper. The rest&r of the calculations

using

ey.

(8) are

illustrated

in frgs.

I

and 2 and table 2. in which the values of the denominu-

01 I

I

II

I

15

n ‘k

II 2

II

I

I 25

Fig. 1. Example of nodal lines calculated for the case of type 316 stainless steels irradiated at a temperature of T= 773 K with a dose rate of P = IO-” and a Constant hehum production rate. Material constants cited in ref. [6] are used. Z = 0.95, I = (2I‘O)/( akT) = 14.94. Curves correspond to the values of B and I(’ = DxC, /K; as indicated in the figure.

* Graham

[IZ] presented a path integral representation of Green’s function solution of the general Fokker-Planck equations. which may be usefull to provide a more reasnnahle approximation for the critical path [13].

I

1.5

“‘16

,

d

I

2

-

25

Fig. 2. The characteristic lines and critical path defined h? du/‘drr = F,,‘F,,. & = 250. K = 1.6: full cur\e: ii = ( D, ,C, )/( D,,,,C, ) = 1 ; dashed curves: D = 025 and 4. we table 2c: h. the effects of heterogeneou\ nucleation. see table 2d: d. direct f<)rmatron of vord from ca\cnde\

K. Kitajima

et nl. /

Evolution of vo&

under

489

cascade damage

tor in eq. (8) are evaluated for various values of supersaturation ,B, the parameters showing the effects of helium (0&)/(&C,) = D, ( DxCx)/Kz = K',and that of cascade diffusion H,. Values of the numerator must be determined by the approximate solution of eq. (6) near the origin under the boundary condition of eq. (7).

that the total number

3.2. Dtrect formation

~PH=JTH,

of embryos

,eH is given by

Since the total number of embryos is reduced by the growth of those voids that have exceeded the critical size at the rate JTH, the nucleation rate, we have d

of a cavity from u cascude

JT,,= Experimental evidence was obtained recently by the D-T neutron irradiation of pure Ni and Cu using RTNS-II that indicates voids were formed even at very low fluences of 1 X lOI nvt during irradiations at 400°C 1141. Similar evidence was also reported after fission neutron irradiation of Cu [IS]. All this evidence suggests the possibility of the formation of a void as a residual cluster produced directly by a cascade. By assuming the initial size of the void to be )I~, and its production rate Pd(n,), we can impose the production term in eq. (6). The following approximate solutions of eq. (6) are derived near n = nd Q(d)=

[~~/2(~,.~)“‘2)exp(--iL/~--ndl).

p = ( DYX)‘?

(9)

Using eq. (9) we can obtain an expression for the nucleation rate Jr, by considerations analogous to eq. (8)

J

cp(d)bdn

= Pdb(d),‘D,,,

(10)

where the integration path must be chosen taking into account the drift field in the (I?, x) plane as illustrated in fig. 2.

~(h)~(h)/~~d~ exp(G),

and we obtain

from eq. (8)

@h=exp(-G)(@(h)b(h)-J,,,~id~exp(G)). ~,,=Q)(hjO(h)~~~‘dSiliji)exp(-G)

x (1-l’dCexp(G)/~XdEex~(G)~~ =dP(h)l>(h)H(h). J,,

= ~~/~(~)~,~d~

exp(G).

The values of the integral in the denominator calculated in table 2.

(11) of Jr,, are

3.4. Futher growth of ooids Consider now irradiation with a constant helium production rate. As the fluence increases the nucleation rate changes following the decrease of C, and the increase of total helium as described by the eqs. (Al) and (A2). These processes can be calculated using the values of the nucleation rate. i.e. as the number of voids exceeding the critical size increases with the rate shown by eqs. (9). (10) or (ll), the sink strength presented by these voids and the associated void embryos increases, thus C, and consequently the nucleation rate decreases.

3.3. Heterogeneous nucleation In commercial alloys most voids are nucleated heterogeneously, i.e. associated with precipitates, solute atoms [16] or dislocations. Precipitates may be particularly favourable sites for nucleation since they can have a large cross section for gathering vacancies and helium atoms and thus assist the initial growth of voids by the low surface tension and the high diffusivity of the vacancy along the phase boundary [17]. Consider a simple model. i.e. precipitates of a fixed size and density, which make voids easy to grow up to a size n,, but have no influence on their further growth. When nh is smaller than n,, the critical size of void growth, the embryos evolve by nucleation processes similar to the homogeneous situation as shown in fig. 2. In this case the form of the distribution function + is nearly equal to that for homogeneous nucleation for n > n,, except

4. Discussion When we compare the theoretical calculations with the experimental results, problems remain on both sides. On the experimental side, in most cases where commercial alloys were studied, the nucleation of voids is considered to be heterogeneous. but for wchich the effects of various factors are not yet separated. It is interesting, however, that the specimens of solution annealed type 316 steel, when irradiated in IIFIR, show larger swelling rates than when irradiated in EBR-II. in contrast to the smaller swelling of cold worked 316 [IS]. This evidence can be explained qualitatively by the present model, i.e. helium enhances the nucleation of a small number of voids attached to imhomogeneous precipitates, and these voids can continue to grow after

exceeding the critical size in the former. where initial critical size is small because of low sink strength. On the other hand in the latter. where initial critical size is large because of large sink strength due to high dislocation density, helium enhances the homogeneous nucleation of a high densitv of bubbles. and the resulting increase in sinh strength due to these bubbles then suppresses further growth of voids. On the theoretical side, many unknown factors remain concerning the physical nature of small clusters [19]. e.g. vacancy clusters do not always have the form of voids but would rather become dislocation loops in most cases. or stacking fault tetrahedra in some FCC metals and alloys. Progress has been made recently on the nature of vacancyyhelium complexes [N]. The effects of the cascade are very important in the nucleation of voids as shown in the present calculations. detailed conclusions must. however. await further rele\ ant data.

Appendi\

The equations of equilibrium for its vacancy and interstitial concentration are the ones ordinary used [6]: (d/d/)(;

=/‘(I

-e)+G,

-R,.,D,C\C\ (d/dr)C

= P-R,,.,

R,,,D,C,C,

-D&CA-;,,i + R,,,,D,C\,C,.

D,C,~k;,,/

R,,,D,C;C,

D,C,,C, .

(Al 1

where j indicates a type of sink. i.e. dislocation. void embryo and growing void etc.. and R ,,,. R ,,,~. R ~,,, are binding coefficients among vacancy. interstitial atom, interstitial and substitutional helium. respectively. P is the effective dose rate. E the factor of excessive vacancy loss bv cascade collapsin,. 0 and ti,. the thermal vacancy generation rate from \,arious clusters. For

the equilibrium

ML’ assume

of interstitial

helium

atoms.

C;.

that trapping and detrapping are balanced for each component of the traps of helium, i.e. dislocatmn. void. and vacancy [4]. For instance. the absorption of helium by the dislocation D,C, Xt,d is in balance with

resolution of helium trapped to the dislocation KtC,. i.e. C, = D,C,(kt,,/Kt). Using similar relations for other traps. we have the

D,C, = C,?“/(

kf,,,/K;

+ X;,,/K:

+ R,.,,/R,,.,

D, B). (Al)

where CJ”“l’ IS the concentration of total helium Introduced by irradiation. B = (D,C,)/( DvC, ). Kt can be calculated

by a similar

method

for the case

of voids.

References [II G.R. Odette. P.J. h4aziasz and J.A. Spitznagel. J. Nucl. Mater. 103-104 (1981) 1289. A. Ayrault and A.P.L. -Turner. J. Nucl. PI A. Kohyama, Mater. 117 (1983) 151. 131 H. Wiedersich. Radiat. Effects 2 (1972) 111. 141 K.C. Russell. Acta Met. 26 (197X) 1615. 151 Hishinuma and L.K. Mansur. J. Nucl. Mat. 11X (lYX3) 01. [61 L.K. Mansur and W.A. Coghlan. J. Nucl. Mat. 119 (1983): 122-123 (1984) 459. N. Yoshida and K. Kuramoto. J. Nucl. 171 K. Kitajima. Mater. 104-10.5 (1981) 1355. K. Kuramoto and N. Yoshida. J. Nucl. [Xl K. KitaJima. Mater. 10X-109 (19X2) 267. [91 K. Kitajima, Int. Conf. Point Defects and Defect Intcrachens in Metals, Kyoto (19X2) X47. [lOI K. Kitajima. J. Nut!. Mater. 122-123 (1984) 60X. [Ill K. Kitajima. these Proceedings. [I21 R. Graham. 2. Phvs. 826 (1977) 281: 397. 1131 I.W. Chen and A. Tatao. Proc. 12th ASTM Int. Symp. Effects of Radiation of Materials. Williamsburg (19X4). J. Nucl. Mater. [I41 N. Yoshida. K. KitaJima. E. Kuramoto. 122.123 (1984) 664. Radiation Effect\. J. [I51 C.A. English. Int. Conf. Neutron Nucl. Mater. 108X109 (1982) 104. 1161 J. Takamura. Int. Conf. Point Defects and Defect Interactmns in Metals, Kyoto (1982) p. 431. 1171 L.K. Mansur. M.R. Havns and E.H. Lee. Proc. Symp. on Irradiation Phase Stability of the AIME Meeting, Pittsburgh. (1980). 1181 P.J. Maziasz. J. Nucl. Mater. 122 -123 (1984) 472. 1191 W.G. Wolfer. J. Nucl. Mater. 1222123 (1984) 367. WI H. Ullmaier. Symp. on Helium in Metals, Jtilich. Radtat. Effects 7X. Ne. 14 (1983).