Statistical theory of microstructural evolution under cascade damage

Statistical theory of microstructural evolution under cascade damage

A revjew cover\ is presented on the recent progress the general stnchastic theoretical in the statistical consequences of high energy cascades. ...

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A revjew cover\

is presented

on the recent progress

the general stnchastic

theoretical

in the statistical

consequences of high energy cascades. C‘haracterization of prtmary knock-on atom (PKA) rlmulations. Model account

in which

estimauons

are performed

the diffusional

of caxades

cascade defect

structures

and the collGnal

energy

are represented

on the component

have a peak at some PKA

energy

the

stages

of

nucleation

and

growth

using

the

Russell further developed the theories for the effects of helium production and the problem of radiation irlduced precipitations [13.14]. These theories. however, were rather confined to the homogeneous irradiations, and did not take into account the effect of various fluctuations associated with cascades. which is considered to play an important role particularly in the nucleation stages. The problem may generally

formulations.

be in the

frame-work

the sense that the statistical scopic state is to he described in

of

dependence

of cascades

as spatially

that increase5 generally

statistical

mechanics

behaviour of the macroby the microscopic dy-

0022-3115/X5/$03.30 1~ Elsevier Science Publishers (dearth-Holland Physics Publishing Division)

R.V

evolution

under

cascade damagr.

It

various approximations and the predictions for the

effects of cascades. The estimalmns

Problems of cascade damage are important not only in the research for correlations among the data of various simulation irradiations using electrons. ions or neutrons with various energies. but also in the prediction of radiation effects in fusion environments. Cascades, the localized production of displacements. influence the evolution of micro-structures through some characteristic processes, e.g. the absorption by pre-existing defect clusters of the fluctuating flux of defects diffused from cascades [l]. the production of various types of residual clusters directly from cascades [2,3], and the resolution of clusters, precipitates or bubbles, by atomic collisions inside the cascades 14.51. A number of investigations have been devoted to the analyses for each of these processes from the theoretical side including computer simulations [6.7]. On the other hand. the evolution of micro-structures composed of these processes has been pursued on the stages of nucleation and growth of clusters. Wiedersich [8] and Russell [9] proposed the theory of the nucleation of voids based on the classical theory of nucleation, but using the assun~ption of constrained thermal equilihrium. The formal rate theory, however. has proved to provide the same results without recourse to the artificial assumption [lo]. Wolfer [11.12] introduced the approximation for the treatment of discrete systems and Fokker-Planck

of microstructural

processes. e.g. evaluation

1. introduction

unified

theory

hasia. the methodological phases of

correlated

are carried aggregates

of voids, bubbles

out

baaed On computer

sub-cascades.

of unit-size

and precipitates.

made using these theories

predict

taking

Into

that the effect\

as the dose rate decreases.

namical equations. The present state of the theories [15]. however, is confined to those for the nearly thermal equilibrium state, and cannot be applied to systems much removed from the thermal equilibrium state as containing high energy atomic collisions. It is considered to be very important at the present stage. therefore, to establish the methodological basis for the analyses of cascade effects on micro-structural evolutions. In previous papers [16-191 the author proposed a formulation based on the stochastic theory. In this paper, a review is presented summarizing this work. This includes the general theoretical basis for the stochastic treatments of cascade damage. the methodological phases of various approximations and the theoretical predictions for the consequences of high-energy cascades.

2. General basis for the statistical theory of cascade damage Cascade damage is characterized by the highly athermal processes of atomic collisions produced by the energetic particles injected from an outside source of irradiation. This procces is followed by thermal diffusion of free defects which results in temporal and spatial fluctuations of fluxes of free defects. The ranges of temporal correlations of the fluctuations, moreover, spread widely in time with several orders of magnitude, depending on the mobilities of point defects, i.e. seif-interstitiais, vacancies and their complexes with solute atoms. -7.1.

Derivutron

of Fokkor-

Plunck

eyuution

The microstructure of an irradiated system at the time t is st~tisticaIly represented by the number density. or the probability density, of various defect clusters Q( s,, I) in the space ‘c,. where .Y,. I = 1 q. are the variables defining the clusters. The time change of the

K. Kitujima

/

Statisticaltheory

statistical state, on the other hand, is generally described by the stochastic process [20], x,(r), P( x,(t, o)}; the function of time x,(t) are the stochastic variables representing the time history of the cluster, and P{ x,(t, 0)) represent the probability density of the variable X,(I) to take on the value of x,(t, w), where w is a parameter describing a particular microscopic state. Since P{ x,(t, w)} include all joint probabilities of the time sequence of x,(t,), it is equivalent to the probability density in the phase space in the statistical mechanics. The Chapman-Smoluchowsky equation is in the usual notation, P(x,.

2) =/P(x,,,

0)+(x,>

and the Langevin dx,,‘dr=u,(x,, Defining

1; x,0,0>

dx,,,

(1)

equation, t, 0).

correlation

(3)

(4) transition where $=+(5,. T; x,, r) is the backward probability density. Using eq. (4) the first and second moments of Ax, = (x, - x,~) can be derived in the abridged form

=-& j%, dt 0

{Ax,Ax,-AX,

D,, = (l/Ar)iJrj

Ax,}/AI

jR,,+

c D,.

D(j)=

,=I

Where D, represents the separate contribution of the i th component having a smaller correlation time than that of the J th process, since fluctuations of each component are considered to be approximately independent of each other. 2.2. Method of calculution of the momenfs

we can derive the time change of the transition probability density in the form of an integro-differential equation [5],

= F,

formulation of highly athermal processes [16]. Corresponding to the physical nature of cascade damage, as stated previously, a correlation function R consists of several component processes with correlation times of different orders of magnitude. Then the whole process may be viewed as a hierarchy of these processes, each of which is described by the Fokker-Planck equations. The second moment term in the equation of the jth process, D(j), is expressed by

(2)

functions,

R,,=(u,-u,),,,(u,)*,~and(i=(u,-u,),,,,~,,.

Ax/At

65

euolution

of microstructural

D,,

D,,

i =j,

=2D,,

i=j,

=

d.$ dr dr’.

The problem is thus reduced to the calculation of the transfer terms F, and D,, in eq. (6) by means of the dynamical equations describing the microscopic processes. Firstly consider the diffusional effects of cascades. The underlying physical assumptions are: (1) mobile elements are confined to point defects and limited sorts of small clusters. and the clusters of defects are generally immobile, (2) the dominant cause of fluctuations is associated with cascades. and the direct contribution of the neighbouring clusters in the fluctuation is very small. Consider a cluster placed at the origin, as shown in fig. 1. The spatial and temporal region of the production of cascades, which have any effects on the cluster. is limited to the correlation range R(r, (t). At] in fig. 1. where the diffusion range rl.( t) is determined by the mobility of mobile defects 9 by rl_( /) = (pQ/)“‘. Since the production of a cascade is random and homogeneous both in space and time. the probability of finding

!

(5)

rL,I

When Ar is sufficiently larger than the correlation time of AZ. and in this case we of R,,, D,, is independent can derive the Fokker-Planck equation from eq. (1)

A similar expression to eq. (5) is obtained for the transfer coefficients in non-equilibrium thermodynamics [15], but it may be noted that the derivation of eq. (4) using eq. (3) is a formal one free from any mathematical approximations or physical limitations so that it is considered to be appropriate in the present

0

Ati

Correlation

tI

range

t2

Ate

Atv

time Fig. 1. The ranges of production spatial

and temporal

correlation

of cascades which have strong with

the evolution

placed at the origin through the diffusion vacancies. and atomic collisions,

of interstitial

respectively.

of volda atoms.

66

X: Kifajima

\ Statrsticd

rheory of microstructural

N cascades in the specified N regions (dr dt),. i = 1. N, and no other cascades in the correlation range, d P( N f. is provided by d~(N)=exp~~~~~(A~)~~~(d~df),

‘..

xp,(drdr),.,,

(8)

where p, is the production rate of cascades, and 52(A t) is the volume of the correlation range defined by Q(At)

=~~‘(4/3)*~~(~)

dt.

(9)

Using eq. (8) the mean value of f(x) i(.x)=c/f(x) .?G

dP(N),

C/dP(N)= v

is defined

by

1.

The values of f(x) do not depend on the choice of correlation range when the size of the range is taken to be sufficiently large as shown by simple calculations. The condition of p&At) = 1 was proposed in the previous paper [ 161 to determine the correlation time At and to perform various approximate calculations. This is based on the consideration that the diffusion range from each of the cascades produced in the time interval At cover the whole space on average as shown by the condition. then the statistical effects of the fluctuations may reach a stationary value in that time, i.e. At may give an approximation for the correlation time. In the case of a collisional process, on the other hand, the spatial range of cascade production. which induces any effect on the cluster, is limited geometrically to the region of overlapping of the cascade with the cluster, 52. Then the condition p$Ar = 1 proposed in the previous paper means one cascade hits the cluster on average in a time At, and the statistical effects of the collision may reach a stationary value in that time. Though the validity of the At must be checked by model calculations, it may provide a good measure of the correlation time from the physical considerations stated above. Since the spatial correlation range of the diffusional process is much larger than that of the collisional one. the relations between the temporal correlation ranges are expected to be inverse, i.e. Arc ,,,, B At,,,,, fig. 1. From these considerations, the diffusion term in the final equation (6). DC.,,, is expressed by DC.,, = D,, + Dcil + 0,. .

3. Characterization ous PICA energies

between the injected particle and the primary knock-on atom, PKA. Particularly, cascades are split into subcascades as the energies of PKA exceed several tens of keV. In order to evaluate the effects of these cascades on the processes discussed in the previous section. we introduce here the concept of unit-size sub-cascades, i.e. the cascades with high PKA energies are treated as a distribution of the unit-size sub-cascades. The calculations are performed on the iron lattice using the MARLOWE code for 30 cases of the cascades with PKA energies of 200, 500 and 1000 keV, respectively. In this calculation the positions of unit sub-cascades are assigned to those of the colliding atoms whose energies are reduced to the prescribed value of 20 keV [21]. fig. 2. Ten examples of the distributions of sub-cascades are illustrated in fig. 2, and in table 1 are listed the calculated values of the average number of sub-cascades contained in a cascade, N,, together with the average which is number, Na, of the group of sub-cascades, defined by the number of sub-cascades, N,a, contained in a region with a prescribed diameter of 2 R, = 5Ou, lOOa or 200~. where u is the lattice constant [22]. Since the average number of sub-cascades thus calculated is roughly proportional to the total displacement energy of these cascades, we assumed in the model that about 60 keV of displacement energy are deposited around a unit sub-cascade abstracting detailed varieties in the microstructure of cascades. In table 1 are also listed the average sizes of cascades. 2R,, estimated using the mean half-breadth of the spatial distribution

Table

and modeling of cascades with vari-

The microstructure of cascades is known to show great variety depending on the conditions of collision

1

Characteristics of distribution of

sub-cascades

sub-cascade

200 x,

2R,

keV

N, = 2.4 = 200n

1221

Average number of

PKA energy

N,g = 2

0.37

3

= 500

group iVp --__I__ IOOU 2oou 0.54

0.45

0.16

0.45

500 keV 4.X

2

0.56

0.60

0.53

4500

3

0.06

0.33

0.30

0.03

0.16

4

(10)

the contributions where D,,,. Dcd and D,.., represent from the thermal process. the process of cascade diffusion and that of the cascade collision, respectively.

evoluiion

5

0.03

6

0.03

7 0.03

8 1 MeV 6.4

2

0.44

0.79

0.51

soon

3

0.20

0.27

0.44

4

0.03

0.24

5

0.03

0.03

6

0.03

0.06

7

0.03

X

0.03

K. Kitajima

200

67

/ S~aiistical theory of microstruetu~a~ eoolution

500

KeV

KeV

I 1 IOOa

x

I MeV *A

93 0

A

+

&

.

+

e

u A

A A

Q

0

Fig. 2. Distribution of unit-size sub-cascades in a cascade with PICA energies of 200. 500 keV and 1 MeV. 10 examples. calculated using the MARLOWE code. the projection of the co-ordinates of the sub-cascades to X-Y plane, [22].

of displacement energy deposition, calculated using the TRIM code. It can be seen from table 1 and fig. 2 that the spatial density of the sub-cascade decreases with the increase of PK.4 energy. Based on this model and. further, assuming equal spacing among unit sub-cascades, the PKA energy dependence of the diffusiona effects of cascades on the nucleation rate of voids are calculated [18,22] and is shown in fig. 3. The calculation shows that the factor On,,/<,. which is the controlling factor in the nucleation rate, increases in proportion to the number of subcascades when the spatial size of the diffusional correlation range. R,,,,, is much larger than that of the cascade,

but decreases to the value in the R,,,. i.e. R,,,,% R,,. case of the production of a unit size single cascade when the former is smaller than the latter, Rdirr < R,,\. Since R,,,,[ increases with the decrease of pC by RJ,II= constant X9J,/spc- I/‘. . 9,. the mobility of vacancies, as described from the condition p,J;?(Al)= 1, [16], the tendency of the curve in fig. 3 can be easily understood, i.e. (I&,/F, ) increases, comes to a peak then decreases with the increase of PKA energy for a given value of p,, and the peak energy becomes higher, the lower the production rate pc. For the collisional effects of cascades, it can be predicted that there should be no effects of sub-cascades

/

60

100

/

/

/

/

/

6

200 PKA

-1

1000

500

energy

kV

Fig. 3. PKA energy dependence of the nucleation rate of voids. Full line. taking into account the sub-cascade structure: dashed line. aasummg a concentrated the number normalued

of sub-cascades:

single cascade hy eq. b/k,

by that of 60 keV. p/D

the value of = 3.5 x

(1I ). .V\ is D,,,,,/&

lO’~~~p,,U,.

when the size of the spatial correlation range of collision. I?.,,,, . is smaller than that of mean spacing among sub-cascades R ,uhc. In the case of collisional effects, it may be more appropriate to take the energy of the unit sub-cascade smaller than 20 keV, since collisional effects like the resolution of clusters is most effective in small but highly condensed areas such as 5 keV subcascades. Highly concentrated aggregates of sub-cascades may result in the large residual defect, e.g. the direct formation of voids [18]. Table 1 suggests the possibility that a highly concentrated region of sub-cascades increases with increasing PKA energy. though its probability is quite small.

cascade fluctuations reduces the nucleation potentlul mainly through the diffusion term D,,,, bince the effect5 of cascades is small in f;, [l]. The estimation for D,,,, in case of single PKA energ? cascade is obtained in the form. 1161.

evolution

Consider now the simple case of the nucleation of a void where the cluster is represented by one variable x = II. the number of vacancies contained in a void. The one dimensional equation (6) predicts that the critical size for the nucleation of the void is determined by the condition. c;, = 0. The clusters smaller than this size can grow only by statistical fluctuations. as described by the diffusion term D,,,(a/an)p. The nucleation rate. J. of the clusters is derived under the assumption of a quasiequilibrium state by, [ll].

J=

X ,Y\ 11’ ‘9,

’ ip,’

‘.

(17)

where ?v, IS the total number of free vacancies hhed from a cascade. D ,,,, , and D ,,,,, the vacancv abaorptlc>n terms in D,,,, and D,,,. Fig. 4 shows the calculations for the nucleation rate of voids in pure iron [23.24]. It can be seen that the effects of cascade fluctuations I large when the value of J is small as in the case of small IT,. as expected from the above estimation. Then consider the effects of the production of hellurn on the nucleation of voids. The void containing helium is represented by the two variables. ( )I. _\-). in rq. (6). v.here .Y is the number of helium atoms. The criticat point of nucleation is represented hy the nodal point determined by the cross point of the two nadal line\ $ = 0. where the nucleation potential show> ;I haddIe point. fig. 5. Applying Gauss’s formula to eq. (6) which represents the continuity equation of the flux of embryos. i.e. div J = 0.

then assuming that the flux is concentrated withln the range of the narrow band where the nucleation potential shows a groove from physical considerations. we obtain the approximate expression for the nucleation rate of the voids in the form of line integral [18]. J, = ~,,(l)@(l)//rd5 G=lCI J, =

4. Prediction of microstructural

/ = constant

Dnn.\/Dth

/

j;/i&

Jdq.

exp(C;).

I d.$.

@=

J

p dv.

&=/I&p

Fe.

dq,(13)

where .$ and 1) are the new coordinates parallel and orthogonal to the flux vector J. and I$. D,, are those in the expression of eq. (6) obtained by changing variables. Then assuming that the critical path. i.e. the flux line 7 = constant. is approximated by the curve d.r,/d.v? = F,/F, in eq. (6) *. we obtained the nucleation rate of a void J, in the case of the continuous production of helium. which provides the explanations for the bi-modal distribution of voids [ 19.261. fig. 5. In commercial alloys. however, nucleation of voids IS considered to be heterogeneous. since small voids are

4,(l)p(l)/l]rdnew(G).

G = -

1‘I(I;;,,'Dnn)du.

G‘ may be called the nucleation

(11)

* Graham’s

path Integral

representation

solutmn of the general Fokker-Planck

potential.

The effect of

a more

rraaonablr

approximation

for

of Green’s

function

equation may prow& the critical

path

[25].

K, K~tujrma

/ Statrsticai

theory of microstruciura/

69

evolution

r I

P = lx ld6dpo/sec

I

E=

! i0 5 iL-L__, 0

o-4

550°C

%

1

0.2

04

06

08

JO

I

8

1.2

!,4

NV.COS

35o*c ) 1

,E=O% ,,E:O05% ‘,’ E=006%

,E : 0 96 ,E = 0 I 96 F:o17%:

I 45oOc

15

“‘/6

2

-

/i

25

Fig. 5. Nucleation of voids under irradiation with co-production of helium. 316 stainless steel. irradiation temperature r= 773 K. dose rate p =lOeh dpa/s, supersaturation of vacancies p = 250, K2 = (D,,C,)/K:. I = (2~u)/‘(~~~~). Thick lines. nodal lines; thin lines, characteristic lines and critical path: full curve, D = (D, ,C, )/( D,,C,) = 1; dotted curve. D = 0.25. and 4. K and D indicate the effects of helium, h. miniCas = D nn,v/D,h,v the effects of cascade diffusion: mum size of the void embryo produced attached to precipitates; d. the size of the void embryo produced directly from cascades. The effects of these parameters on the nucleation rate of voids were calculated in table 2 in ref. [19].

i?--

associated with precipitates. For this case too, the analysis presented above can be applied. and the approximate expressions for the nucleation rate J,, is obtained by shortening the integral path in eq. (13) under the assumption that the total number of sites of nucleation N, is fixed,

preferentiaI1~ //

E =0 %

1

_:=-::--:

E=0246

J 55o’c

/

E =037%j

P = 5 x id’ &o/see

/$

L-i_..___i-I-.L-.i_~L.-ii 0

02

04

0.6 N”

08

IO

12

14

.CDS

Fig. 4. PKA energy dependence of the nucleation rate of voids in pure iron. N,.,,, is the total number of vacancies shed from a cascade normalized by the value of 100 keV cascade cited in ref. [16]. Measured values of dislocation density were used, [23,24]. E is the factor of excessive loss of vacancies due to cascade collapse [2].

where h is the minimum size of the void embryo associated with the presipitates and pH is the total number of the embryos. A similar expression can be obtained for the effects of the direct formation of void embryos produced as the residual clusters of the cascades [19]. Similar considerations can be applied to the evolution of precipitates under cascade damage. Many variables may in general be required to describe precipitates, i.e. the numbers of impurity atoms, vacancies. interstitials and ordering parameters even for the simplest case’ of non-coherent precipitates. Evolution of the precipitates is a competition between the two factors. i.e. the growth of the precipitates caused by the segregation of

5olute atoms, where diffusion of solute atoms associated with vacancies or self-interstitials plays an important role. and the resolution of them by collisions with cascades followed by diffusion of solute atoms. Eatimates made for these processes lead to the deduction of the presence of the two critical sizea. One is the limit size of growth a5 discussed by Nelson [4] and Frost and Russell [14] and the other i5 the critical size of the nucleation of precipitates (141. Growth of voids or dislocation loops combined with the precipitate5 can also be discussed thus extending the above model. Prediction5 for fusion reactor environments can be made using the estimations for the effects of high PKA energy cascades. presented in the previous section. Diffuaional effect of cascades on the nucleation of voids may be enhanced with the increase of PKA energy, and is expected to show a peak near the fusion condition. where the PKA energy is about 200 keV and the damage rate is as low, as IO- ’ dpa/s. The effect5 of the direct formation of voids from highly concentrated aggregates of sub-cascades. even if its probability is very small. may be large in the void nucleation processes. In commercial alloys, however, these effects may not be a5 large since the mobilities of mobile defect5 are expected to be reduced considerably by the presence of impurities. As for the effect5 of helium. two cases may be emphasized. Assuming heterogeneous void nucleation, in one ca5e. the formation of a high density of bubbles. produced attached to fine precipitates and whose size is smaller than the critical size of nucleation. may suppress the void nucleation by large sink effects caused by the bubbles themselves and results in a small rate of 5welling wtth large incubation dose. In the other ca5e. the low-density voids attached to large precipitates. and assisted by helium, will grow easily over the critical size of nucleation and reach the stage of a high rate of swelling with small incubation dose [27].

References

L.K. Mansur. W.A. Coglan and W.A. Brailsfold. .I. Nucl. Mater. X5-86 (1979) 591. 121 K. Bullough, B.K. Eyre and K. Kriahan. Proc. Kov. Sot. Land. A346 (1975) Xl. 111

[31 [41 [51 161 I71 PI [91 [lo]

R. Bullough. SM. Murphy and M.H. Wood. J. Nucl. Mater. 1222123 (1984) 4X9. R.A. Nelson. J.A. Hudson and D.J. Maze?. J. Nucl Mater. 44 (1972) 318. H.J. Frost and K.C. Russell. Acta Met. 30 (19X2) 953. H.L. Heinisch. J. Nucl. Mater. 1033104 (1981) 1325. P. Chou and N.M. Ghoniem. J. Nucl. Mater. 117 (19X3) 55. H. Wiederstch. Radiat. Effects 2 (1972) 111. K.C. Russell. Acta Met. 19 (1971) 753. C.F. Clement and M.H. Wood. J. Nucl. Mater. X9 (1980)

[I l] W.G. Wolfer, Breeder Reactor Structural Materials. Scottsdale. (AIME. New York. 1977) p. X41. 1121 W.G. Wolfer. J. Nucl. Mater. 122-123 (19X4) 367. [13] K.C. Russell. Acta Met. 26 (1978) 1615. [14] H.J. Frost and K.C. Russell. Phase Transformatton During Irradiation. ed.. F.V. Nolfi Jr (Appl. Set. Publ.. Neu York. 1983) p. 75. [ 15) I. Prigogme, introduction to Thermodynamics of Irreversible Processes. (Interscience. New York. 1967). [lh] K. Kitajima. N. Yoshida and E. Kuramoto, J. Nucl. Mater. 103-104 (1981) 1355. [17] K. Kitajima, E. Kuramoto and N. Yoahida. J. Nucl. Mater. 10X-109 (19X2) 267. [1X] K. Kttajima. J. Nucl. Mater. 122-123 (19X4) 60X. [19] K. Kttajima. Y. Akashi and E. Kuramoto. these Proceedtrigs. 1201 J.L. Do&. Stochastic Processes (J. Wiley and Sons. New York. 1953). [21] T. Muroga. K. KttaJtma and S. Ishino. these Proceedtngs. [22] K. Kitajima and T. Muroga. to he published. [23] E. Kuramoto. N. Yoshida. N. Taukuda, K. KitaJtma. N.H. Packan. M.B. Lewis and L.K. Manaur. J. Nucl. Mater. 1033104 (1981) 1091. [24] E. Kuramoto. N. Yoshida and K. Kltqima. Potnt Defects and Defect Interactions in Metals (1982) p. X99. (25) R. Graham. Z. Phys. 826 (1977) 281; 397. [26] L.K. Mansur and W.A. Coglan. J. Nucl. Mater. 64 (19X3)

[27] G.R. Odette (1984) 514.

and

R.E. Staller.

J. Nucl.

Mater.

1222123