Irradiation creep by cascade-induced point defect fluctuations

1257

JoumalofNuclearMaterials 103& 104(1981)1257-1262 North-Holland Publishing Company

IR~IATION

CREEP BY ~SC~E-INURED

POINT DEFECT ~UCTUATIONS*

L. K. Mansur,t W. A. Coghlan,f T. C. Reiley,+ and W. G. Wolfer* TMetals and Ceramics Division, Oak Ridge National Laboratory Oak Ridge, Tennessee 37830 *Nuclear Engineering Department, University of Wisconsin Madison, Wisconsin 53706 A theory of irradiation creep caused by the cascade-inducedlocal fluctuations in point defect concentrationsis developed. The climb excursions of pinned dislocation segments caused by these fluctuationsare described by a climb frequency vs climb height spectrum. A fraction of such excursions enable dislocation segments to bs released from obstacles, leading to stress directed glide and resulting in irradiation creep.

1.

INTRODUCTION

A mechanism of irradiation creep by dislocation climb excursions caused by cascade-inducedpoint defect concentration fluctuationswas proposed in an earlier paper by Mansur, Coghlan, and Brailsford [l], and described in more detail in a later discussion [2]. In the present paper we summarize the methods developed to treat this mechanism. A more complete description of the theory undsrlying these results is available elsewhere [3].

i

produce creep rates at least one to two orders of magnitude lower than those observed experimentally. 2.

CASCADE DIFFUSION THEORY

The deposition of point defects by a cascade is spatially localized. For simplicity we model it as two spherical Gaussians, one for interstitials and one for vacancies. Point defect recombination in the bulk is neglected. For a cascade occurring at position pc and time tc the distribution is described by [l]

The quasi-chemicalrate theory of point defect PI, -312 C,(P - pc, t - tc) * [4aDa(t - t,)l reactions, widely used in the theory of radiation effects, excludes the possibility of cascadeinduced creep. Spatial and temporal averaging, x exp[ - (0 - o,)2i(4Do(t- tc))l whereby the discrete nature of point defect production is lost as a matter of definition, is invoked. The rate theory is, therefore, suitable x exp[-DuSo(t- t,)J . (1) for treating cumulative mechanisms of irradiation creep which depend on the net long term flow of where one type of point defect to a given dislocation. a = i or v for interstitialor vacancy, The cascade diffusion theory, by retaining the respectively, information on point defect concentration fluctu= the point defect diffusion Do ations, is suitable for treating both fluctuacoefficient, tion-dependentas well as cumulative proces= the strength of all sinks in the ses [I]. However, one element of the present So material for point defects of concept has been raised previously. Without type a, and considering cascades, Gittus raised the possi5,-l/2 = the absorption mean free path bility that climb enabled-glideof dislocations (diffusion length). leading to creep could occur even without a net cumulative flow of one type of defect to a disCascades throughout the material contribute to location. The absorption of interstitialsand the concentrationat an arbitrary reference vacancies was described as being unbalanced on a point, so that the concentrationthere is a local scale because of the statisticalnature of linear superposition. Nearly all the concentrapoint defect absorption [4]. A creep rate was tion at a given reference point results from calculated on this basis. Nichols and Dollins cascades within about seven diffusion lengths 151 reanalyzed the model proposed by Gittus and 111. A nearby cascade contributes greatly since presented an alternative random walk analysis there is less distance to spread the point defect for dislocation segments. They concluded that profile and less intervening sink-containing statistical absorption of point defects would ____ *Research sponsored by the Division of Materials Sciences, U. S. Department of Energy under contract W-7405-eng-26 with the Union Carbide Corporation.

L.K. Mansur et al. J Cascade-induced point defect fluctuations

1258

material to absorb point defects. However, there are relatively few cascades per unit radial distance for a unit of time because the shell volume corresponding to a radial increment is small. For large distances the situation is reversed. Between these extremes, at intermediate distances, each cascade makes a significant contribution and yet there are a reasonable number of cascades per unit time. Figure 1 is a plot of the importance of each increment of radial distance. 0.4 0

0

1

2

5

6

Figure 1 : Average Concentration of Point Defects at Reference Point Contributed by Cascades in the Shell of Material at Each Radial Distance, Normalized to the Total Concentration Contributed by all Cascades in the Infinite Medium. Not all cascades that have occurred contribute The superposition discussed above has equally. a finite, species and distance dependent, "memory" for cascades that have occurred earlier in For the interstitial component, this time. memory is typically orders of magnitude shorter than the mean time between cascades. At any point, the interstitial concentration is nearly always zero. This is punctuated rarely for short time intervals by large spikes of interstitial concentration. For the vacancy concentration, on the other hand, the memory of previous cascades is typically much longer than the time Figure 2 shows the calculated between cascades. vacancy concentration for 250 keV cascades at a dose rate of 10-e dpa/s and fractional survival rate of 20%. The interstitial profile is a series of large spikes of duration
DISLOCATION CLIMB EXCURSIONS PINNED SEGMENTS

AND RELEASE OF

The concentration profiles apply to an arbitrary point on a dislocation segment pinned at an obstacle to glide, such as a precipitate parIf the obstacle ticle or a dislocation loop.

7

a 400

TIME

% (50

200

($1

Figure 2: Vacancy Concentration Behavior with Time for a Dose Rate of 10-6 dpa/s, Temperature of 500°C and Cascades Producing u = 500 Frenkel Pairs. dimension is small with res ect to the point defect diffusion length s-P 12, these Profiles also apply over the entire length of a dislocation segment of the order of the glide obstacle size. Every point on the segment then experiences similar and correlated concentration fluctuations. Nearby cascades produce larger and less frequent excursions than those produced by distant cascades. This leads to the concept of the dislocation climb frequency vs climb height spectrum. The concept is direct for interstitial climb because, as shown in the previous section, the interstitials from one cascade arrive at the reference point highly correlated in time and vastly separated in time from the interstitials of another cascade. We describe methods used to obtain the spectrum. Three methods of increasing refinement have been developed. The first is a geometric method that approximates a cascade as a spreading spherical continuous fluid traveling outward from a center. The diffusional effects of random walk and absorption at distributed sinks are ignored. The dislocation is modeled as a cylinder whose radius equals the point defect capture radius of a dislocation. The dislocation segment simply

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L. K. Mansur et al. / Cascade-induced point defect fluctuations

absorbs all defects interceptedby the projected area of the cylinder. The method is described more fully in Ref. [3]. The spectrum resulting from cascades of size v = 1000 using this method is plotted in Figure 3. Here v denotes the number of vacancies or interstitial6available for diffusional processes after in-cascade recombination.

where thermal emission is ignored and h is given in units of the lattice dimension b. Here, n is the climb height from a unit cascade and v is the number of defects in the cascade. Ed is the capture efficiency of dislocations for point defects of the type under consideration. The symbol denotes that this is the result for mean continuum climb. The integral can be represented in closed form.

p(p,t)dt

AVERAGE

=”

exp(~f~p, .

Cascades occurring at distances equal to or less than P give climb height equal to or larger than the value in Eq. (2). The frequency of climb excursions of height or greater is then given by

CL I ME

Ffz= 4rZ$p3/3

I”

0

2

4

6

6

h, CLIMB

10

12

HEIGHT

14

16

16

20

(b)

Figure 3 : The Spectrum of Frequencies of Climb Heights Equal to h or Greater as a Function of h, Using Three Methods of Increasing Refinement. The top curve includes the important physical processes, while the middle curve illustrates the effect of not accounting for discrete point defect absorption, and the bottom shows the effect of neglecting diffusive processes. In these calculationsv = 1000, dose rate = 1 x 10-6 dpa/s, and dislocation density = 5 x 1014 m-2. To account more accurately for diffusional behavior, the cascade diffusion theory has been applied to obtain the spectrum. Two levels of refinement have been developed. Cascades are inserted into the material discretely. In the subsequent diffusion and absorption at distributed sinks of the point defects, continuum diffusion theory is used for convenience since this does not obscure the direct effect of discrete point defect production. However, the absorption of point defects at the dislocation segment in question must be modeled. This absorption may be taken as if from a continuous point defect fluid or as discrete absorption of point defects in atomic units. Both methods have been developed; the contrast in results is significant and shows that it is important to account for the discrete nature of point defect absorption as well as the discrete nature of point defect production. The climb height from the total flux of point defects of one type from a cascade that occurred at time t = 0 and distance p from the reference point is given by [3]

“ZiDa 3 nv = V bL

(3)

m

/ &o,t) dt 0'

(2)

.

(4)

Here Z is the macroscopic cross section for cascade-producingcollisions and $ is the particle flux. Figure 3 shows the distributionFft vs h. However, absorption of defects takes place in atomic units. The above results can be generalized to account for this. Equations (2) and (3) give the mean continuum climb height corresponding to a unit cascade where v = 1. A mean continuum climb height p from a unit cascade can also be reinterpreted,in a discrete picture, as equivalent to the probability of absorbing one defect. For v trials of this type, each with probability of success n, the binomial distribution gives the probabilityof exactly h out of u successes,

0’

ph = h n

h

(1 - !l)V-h)

where 0;: are the binomial coefficientsand h is now an integral multiple of b. We require the probability of climbs of h or greater which we denote as Ph, where PO = 1 'h = 'h-1 -ph-1

(6a)

’

(6b)

The frequency of dislocation climb excursions of height h or greater is again proportional to the cascade frequency per unit volume and to the geometrically weighted value of Ph. It is given by the expression m2 F; = 4nE$/o Phdp . 0

(7)

The superscriptD denotes that the results are calculated accounting for discrete point defect

absorption. The function I is plotted in Figure 3. FE is shown as a curve but actually only exists as discrete points at each integral value of h. It can be seen that the frequency of large climb excursions is

nonphysical. dislocation rate calculations

segment below.

release and creep

Next, we derive the dislocation segment frequency of release from obstacles and calculate for comparison the release frequencies characteristic of two other proposed mechanisms of irradiation creep. In an ensemble of obstacles, each pinning a dislocation segment, dislocations will occupy various positions along the obstacle. We assume that these positions are at various integral multiples of b away from the release position, so that

y&D=

JD’

J

where Rj is the probability of finding a dislocation at position jb away from the release position. Rj = Lj/L where Lj is the dislocation density represented by segments at position j and L = iLj is the total dislocation density. J In Eq. (8) hB denotes the size of the obstacle. If the distribution were uniform then Lj/L = (hB)-1 for all j. However, the distribution is not uniform because dislocations near the unpinning position are released more frequently than dislocations far from it. The distribution is expected to be depleted near the end of the obstacle. In Ref. [3] a method is given for estimating the shape of the discrete distribution Lj/L. Net dislocation climb in one direction also may occur, driven by the cumulative processes of swelling, where all dislocations climb by net interstitial absorption, and preferred point defect absorption, where stress aligned dislocations climb by net interstitial absorption while nonaligned dislocations climb by net vacancy absorption. The release rates resulting from these net climb velocities, for each mechanism in isolation, are =

of aligned and

The release frequencies, as calculated by Eqs. (8), (9), and (10) are given in Table 1. Several obstacle heights and cascade sizes are shown for comparison. The results shown for the cascade theory are the contribution from climb excursions in the interstitial climb direction. The climb excursions in the vacancy climb direction can be shown to add another contribution of the same order.

Table

1.

Release

---___________

Frequencies for Climb-Enabled Glide Creep --_-_

Climb Mechanism

Obstacle Height hB

---__----------

Release Frequency WCS-1)

Cascade

(v = 1000)

10 40

6.3 x 10-4 4.4 x 10-5

Cascade

(v = 100)

10 40

5.4 x 10-4 3.6 x 10-5

10 40

4.2 x 1O-7 1.1 x 10-7

10 40

1.4 x 10-5 3.5 x 10-6 -~--.--

R.F. j!:

WS

efficiencies for interstitials nonaligned dislocations.

(Ac/V)/(LbhB)

(9)

4$1AZ.

DiGi. wp= ._.--% QbhB

(10)

The superscript S denotes climb by swelling and the superscript P denotes climb by preferred Here A$/V is the swelling rate, absorption. Ci is the average (i.e., equivalent to rate theory) interstitial concentration per unit volume and AZi is the difference in capture

Preferred

Absorption

Swelling __---~

4.

DISCUSSION

AND SUMMARY

The creep rate resulting from the calculated release frequencies of Eqs. (8), (9), and (10) depends in detail on further microstructural modeling. To make an estimate we adopt, as an example, a simple model [[i9]. The creep rate is taken as proportional to the effective glide velocity, dw, where d is the obstacle spacing, and to L, the density of participating dislocations, so that . E = abLdw

,

(11)

where a is a constant z unity [9]. The comparison in Table 1 shows that the release frequencies by cascade-induced dislocation climb excursions are larger than or, for large obstacles, comparable to those by swelling-driven climb or preferred absorption-driven climb. These models give creep rates of the order of those observed experimentally. We, therefore, conclude that cascade-induced creep is a viable mechanism of irradiation creep. A number of refinements on the basic structure of the theory developed in this paper are pasSome of these considerations are sible. discussed in Ref. [3].

L. K. Mansur et al. / Cascade-induced

We summarize the present work as follows. Following the earlier proposal, we have developed a theory of cascade-induced creep. The central result of this work is the calculation of the spectrum of climb frequency versus climb height for a dislocation segment residing on an obstacle and subject to cascade-induced fluctuations in the local concentrations of point defects. The recently developed cascade diffusion theory is used to evaluate these fluctuations. Release of dislocation segments from obstacles is calculated by combining the above spectrum with the probability distribution describing dislocation segment residence positions. The results are compared with release frequencies for swellingdriven and preferred absorption glide mechanisms of irradiation creep. It is found for typical conditions that release frequencies of the cascade-induced mechanism are larger than the release frequencies of the other mechanisms.

point defect fluctuations

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REFERENCES

Ill Mansur, L. K., Coghlan, W. A., and Brailsford, A. D., J. Nucl. Mater. 85 and 86 (1979) 591. 121 Mansur, L. K., "Cascade-Induced Creep," presentation at Conf. on Fundamental Mechanisms of Radiation-Induced Creep and Growth, May 1979, Chalk River, Ontario, Canada. 131 Mansur, L. K., Coghlan, W. A., Reiley, T. C., and Wolfer, W. G., "A Theory of Irradiation Creep by Cascade-Induced Point Defect Fluctuations," to be published. [41 Gittus, J. H., Phil. Mag. 25 (1971) 345. [51 Nichols, F. A., and Dollins, C. C., Rad. Eff. 27 (1975) 23. [61 Michel, D. J., Hendrick, P. L., and Pieper. A. G., J. NUC~. Mater. 75 (1978) 1. _ [71 Simonen, E. P., J. Nucl. Mater. 90 (1980) 282. [al Gurol, H., Ghoniem, N. M., and Wolfer, W. G., "The Role of Dispersed Barriers in the Pulsed Irradiation Creep of Magnetic Fusion Reactor Materials," to be published in Journal of Nuclear Materials. 191 Wolfer, W. G., "Multi-Axial Creep Law According to the Climb-Controlled Glide Mechanism," to be published.