Chapter 19 Radiation Effects in Metals:

Void Swclling and Irradiation Crecp

19.1 INTRODUCTION

L'n til ahou I ] 967 the most dr trimen tal radiation e ffe c't ex pected to be str ffe re d I»’ th t stainless-stet•l t Padding of the fuel eleme nts of the p roje ‹ ted liquid -me tal -i oole d fast breeder re acto r I Lfi'l f“B R J was emb rit tie me nt due I.o ex‹'es- sis'e hardening at losv tr mp‹'raturrs or helium agglome rati‹›n at grain boundaries at h igli temp‹'raturr s. These proble me, however, were at lt ast q u ali tati ve ly understood , anti su ffi- cient experimental data had been amassed to permit em brit tie mcii t to be circ uni vented bj care ful dPsign. Sin‹'e th at time a n timber of u ru'xpe‹ ted phen ome na have been un covere d by microseop ie e xamination of fuel ele mt n ts an d st rti ct u rat componr nts th at had been irradiated in a fast reactor t'nvir‹i nmen t f‹ir long pe riod s. I n aflditio n to the chemical at tae k of the inside of the cladding b \ the fuv 1 ( Chap. 1.2.1 , s feel.s i rrad rate d to large fast-iu'ti Iron fJ ti‹ ii ‹'‹ s exhibited d ram at ie de nsity decre ases. tJsin g transmission electro n microscop y. Cawt ho rne and F tilt on ' demo nstra t‹'d that th is swell ing was due to the forma tion of small cavi tit'.s within the grains of the me tal. These voids, wh ich did n ct co ntain sufficic nt gas ( i I an y ) to he classed as bubbles. ranged in size from the smallest obst'rs'ahle to greater than 1000 A. Further research has sh own that voids form in stainless steel on ly at te mpe ratu res between 350 to 600°C. U nfo rttinate 1 \ ’, th is range falls squarely with in the temperature xone in which the cladding of LMFBR fuel pins is designed to operate t Table 10.2 ) .

Void fo rmatio n is not unique to s tain less steel; in fact, steel is one of the allo y s most resist an t to this phe nome n on. Nearly all met aIs s well by this mech un /s m over a tempera - ture band from 0. 3 to 0.55 of the absolute mel ting temperature.

The seve rity of metal swelling under irradi atio n also depends o n the fast-n eutron exposure tand to a much smaller extent o n the fast-ne utron flu x ) . There appears to be an incubation period u p to a fast fl uen ce of 10' 2 ne utrons c m 2 in which no observable swell ing of steel occurs. Thereafter swell ing I measured , as in the case of fuel swelling by fission gases, as IV V ) increases as ( 'I›t ) ' , where the exponent n is greater th an unity'. Very' fe w data at fluences above 10 2 " neutrons / cm 2 exist, and , because of

tli‹* man j v=ari ablt's non trolli rig s z'e 11 i Hg, t'xt rapo I ation o f

th‹' duse depei d‹* c‹’ t‹› th‹‘ de sign flri‹’i ce of the L11FBR ( ” 't X T 02 ’ nerl tr‹›ns ‹'m' ) is \ ’r‘r \ ” ilJsec \ I rt›. ñ onse que n th ,

tll ‹' r‹' h as bet n in tense aeti v ity in dew lo p iiig theoretical models th at cats at'curatelj pre di ct swell ing at large fl ut'tice s and in de i'ising experimental techniq ues ot hP r than neutron irradiation to produce voids in me tals in short times. Of special it t‹ rest is the fl uen ct' to ›vh ich th e power law A V \ ' ( '1'I I e xte rids and i 1' an d at wli at fl uence the sz'ell i n g saturates. I.evt link off of th e swell in g vu inc has not i ct been obs‹ rved in read ter irradiated xtee 1, Eli t h igh - ent'rgy ion bombard me n t ( Sec. 1.7.9 ) h as sh own' that

smell i rig of stainlt'ss stet•l saturates at II tie rices ap prtiac hing

1.0 ru*titro us i,triz 'flit' li igh t qui \ 'ale tit nt•ut ron fl lien t'e

ion -i rrad iati‹in st tidies, tak t'n wit li t xtrapol ation of lo w- fl urncc' ne utron -i rradi ation data, suggest th at ty pe US sta init ss st.t*t 1, which is the most li k e lv I>âI FE 11 cladding, will 5 well h y 5 t o 10’ in a c om mt'r‹ ial rt iict‹›r. 'I'he rami fi ‹'ations of v‹i I u me in ere ase s of th is magn i ttide on ftit'l-‹'lt'men I design arr p ro found, ‹in d flu' n'medi es are c‹is t I y . SO bit of f h t ti ii dest rable .s id r e ffe‹'ts cif .s well i n g t'8Il be flllrYiflted b \ ’ tier related phenomenon uf irradiation

‹ ree p. The r ffects of swelling and irradiation crt'ep on ‹'ore desi gIi are discusse d in Cls ap. 21.

The origin.s of i'oid swe fling of me tals are q ual itati ve I y under.stuod. fiulIisi‹›n of fast neutrons with laftire atoms produces \ large numbers of Yacanc}’— in tersti tial pai re ( see Chap. 17 ) . host of th‹’se point de fec'ts evrntuall \ ' reccm- hine with each r›ther ur migrate to site kS il the* solid where the poin t defects lose the ir identity . The most e ffective sinks are d relocations, eith er th osc w ) i ie h are part of th e natural dis locatio n network of th e me tal or dislocation loops created h}' conde nsation of radiatio n-produced in ter- stit ials. Precipitates and grain boundaries dlso act to remove point de fects from the medi u m. The dynamic balance between the point-defe ct creation and removal proce sses during i rrad iat ion sustains concentrations of vacancies and in terstitials far in ‹'xcess of thermal equilih riu m ( see Fig. 13.1 7 ) .

Nucleation of segregated cluste re of inte rstitials and vacaH cies can take place provided ( hat the tempe ratu re i.s high enough so that both in terstitials and sae an cies are

463

464

mubil‹ in the sulid, but nut su high tlzaL Llzc* poin I de Fetus are re moved b;’ recuillbiila£iulJ ur migra£iun tu sin ks so quitk I ¿’ th at li igh su pers at utation cannot be main taiiied. 'f'he I}’pe cf cluste r l"ormed h \ ill te rstitials is in \ ’ariabl \ ’ a dj slocati on loop. Vacan cies, li owever, can agglo me rate either in to platelc* ts, win ie h roll apse in t‹› disluc at ion luups, or in to th re'+-di mention at clustc*r5, \ vh ich are term‹'d ›'ui0s. The atomic' structu re s of inte rsti t ial arid van an cy loops are shown in Fig. Id. I .

The collection of interstitial atoms as cxl ra planrs in the la ttice causes the solid to swell. If the vac an cies condensed in to analogous vacan cy loops, the lattice con- tractioiarou n d these loo ps would ca use shri n kage of the surrounding solid by an a moun t that just coun te rbalan ces the swell ing due to interstitial loops. loo we ver, whell the vacaii cies agglomerate iii to void s. n o lattic'e c'on t raction is available to cance 1 the dilatation due to the in te rstit ial loops, and a net volume in crrasr of the solid ensues. I n irradiated zirconiu ni, for example', large vacancy loops. but no voids and he n ce no swell in g, arr obse rvvd .

The relative stability of voids a lid wacan cy loo ps can be

assessr d by comparing the energy di t'ference between the part i cular cluster conta inilig m vac an cies and the perfect lattice. For th e void this di ffere nce is ju st the ‹• uergy required to form the surf ace of the void:

( 10. 1 )

w |iere ) is ( he su r fac't ( i* ns in ii of th e so 1 id ( ap pro x i mate 1} 500 dykes cm for stainless steel ) and It is the radius of the s'oid, w his li is rt• lated to the n uniber ot‘ va‹'an cies iH the cavity by

‹ t9.2 )

whert‘ \ / is the a to mic \ ol M me , cr tI \ t’ volume con trihu ted b}’ each \ acanc¿ tu th e ›'oid . ’I'he P ne rg}’ of Lh e void is thMs

Th e e ue rgy of a faul ted dislo c'atiOii loop composed of m

\ 'acanc ie s in a disk u I radius Rj is

( 10.1 )

where rg is the energy’ per unit length ( i.e., the line tension ) of the disloc‘atiun comprising the pe riphe n uf the loop. Accord ing to kq. 8. \ 0 d ”Gb* , where h is the Burgers

\ ’ector of the faulted loop.

The term q,t is the energy per tin it area of the stacking fault encl osed by the loop. As show n in Sec. 3.6, tht sequence of close-packe d ( 111 ) planes in the fee stru ctu re is ordered 123123... . \ Vhen part of one of these planes is removed or a section of an othe r ( 111 ) plane is inserted , the stacking sequen ce is disturbed, but the atoms surrounding the stacking fault are surrounded by the same number ( 12.1 of nearest neighbors as in the pe rfect lattice. However, the configuration of the next nearest neighbors is slightly al tered, and the stacking-fault con figuratio n is somewhat more energetic than the perfect lattice. This energy di f ference is manifest as the stacking- fault energy. Because the energy difference is due to second-order atomic

arrange me n ts, q’ t is snJal I, t \ ’ pic“al \ ’alutx he iM g “ \ 0 d \ ’ices,:’«m.

In l"tc m‹’ tale ctislucat iuiJ l‹Jups furm un th e clc c*- packed ( 111 ) planes in which the area per atom is 3* a 2 , .*4, where a„ is the lattice constant. The radius uf a vacanc}’ I oop created by rem oval of m atoms from ( o r the condensation o f m \ aca iicies on ) a ( 111 ) plane is

"*‹›

where I/ = aq ' 1 is the ato wit rolume iH the fev stru c't u re . Th e em'rgy of the faul ted vacate t j loop is tin're fore

I I the loop is min faulte d ( i.‹'., the stac‘king faul I is removed ) , the sccund Le rm on the right uf Eq. 1.9.6 is ahsenL, but The' resulting rrd uctio n iti env rgy is partially compensated by the larger Burgers vt cto r txt th e pc rfect loop compared to that uf the fariltt‘d luup.

Th‹ uuid and luup t‘nt'r¿ic's given b}’ I'Jqs. 19..J and \ 9.b arc‘ rather close tu ‹’acid uther, and conclusions concern ing the relati v+ stabiIit}’ of the two t \ ’pes uf ›’acanc} clusters arc' uncc'rtain because' in purtaI1 I para me term, such as II e di.slocatiun line te usion, are Mot accurately known. I£ appears that the void is thr stable form for small clusters ( small m ) . but, as m incrc'ases, the loop becomes the energetit-all y favored con figuration. I f the presence of the stacking-fa ult term in Eq. 19.5 is ig no re d temporaril y, the

tel the loop , and the energy bal an ee tips its favor of the loop at void rad ii of st i'eral te ns of angstroms. However, collapse of the emf›ryo void into a vacanv y loop is probably

impeded by the prese nce of small quan ti ties of he liu m gas in the i'oid , a nd th us voids may su n'ive and gro iv. Equation

19. fi also indicates th at loops rather than voids are favored in metals in which th e stack ing-fault energy is lo w. Gold , for example, has a ve n low stac ki ng-fault energy, and irradiation -produ ced voids have not he en observed in th is metal. O n the other hand, ve ids are easil y produced iH

nickt•l, for vfhich i.f is large. The sta cking-fault energy iii stainless steel lies between th csv two e x I remes, and i oid s

c'an be prod need iH th is all oy but on ly at mu ch higher fl uences than that required for void formation in nickel. This observation is cons iste nt with the pre crding discussion of the ef fe ct of stacking-f“ault energy on the relative stability of voids and vacancy loops, but man y other factors influence the relative resis lances to void formation of a complex alloy, such as steel, and of pure metals, such as gold or n i ek el.

Gran ted that, given a choice between forming loo ps or

voids, vacancies will condense as the latter, there remains the question of w hy the irrad iation-prod uce d point defects form separate interstitial loops and voids in the first place. Since vacancies and inte rstitials are formed in eq ua1 n u mbe re by fast-neu tron bombardment, one would expect that poin t defects of both tv- pcs would diffuse to voids at equal rates and hence produce no net growth of the voids.

Inasmu ch as the v ords represent accumulated excess vacan- cies, the in terstitials must be pre fe re ntiall y absorbed else- where in the solid. The pre fere ntial in ie rstitial sink is

VOI D S! V’ELLIL“G AUD IR RA DII TION C R EEP 465

undoub tedl y the d is locations, e it he r th ose bet on gin g to the original network in tht• metal or the in te rstitial loops. I t was noted in Sec. 1.3.9 th at dislocations exhibit a sligh tly larger capture rad ius for in terstitials than for vacancies, and it is this fact which fun damentall \ provides the me chan is m for i'oid formation. The pre fere n ce of dislocations i'o r intersti- tials is due to the inte ractio n of the strain field arotin d the dislocation with the strain field established b ) ’ the misfit of an in ie r› titi al atom in the latti cc' ( the stra in field around a vacancy is much smaller than that around an interstitial ) . This strain-field interaction cause.s an attraction of in ter slit ials for dislocations iv he n the two are in prox imi ty . Ham" has sh own th at the di recte d due it r in te rsti tials toward dislocations can be in cor porated into a di f fusional model ‹if the transport process if the dislocation line is assi gn ed a somewhat larger capture radi u.s for in ie rsti tial s

to voids' ' 7 con tain mu ch detailed in to rmation pertinent to the experimental and theoretical status of the subject. Emphasis here is placed on vo ids fo rmed in neutron-irradi - ated stainless steel. Void formation in the potential cladding materials nic kel and its allo y s vanadiu m and mol \ 'bde nu m will not be conside re d in detail. I n addition to fast-neutron irradiation, i'oids ma y be formed by bo mbard- ing metals with lieav y ions ( e.g., protons, carbon, and sel f-ions ) or iv it h electrons. The results of these investiga- tion s are su mmari zed in Refs. 4 and 5.

Th e bulk of the in formatio n o n void formation in metals has bee n obta in ed by transmission electron mic ros- copy i Sec. 18.1 ) . 2’liis tcp chn ique permits the no/d dis li‘ibu - lion ( ’unc I ion, N ( R ) d R - n umbe r of voids/ cm w it li radii between R arid R * d R, to be measure d. Often, on lj the total void n limb er de nsity,

than for vacancies ( see problem 13.7 ) . The preferred migration of in terstitials to dislocation s leaves th e matrix of the metal sligh I ly de pleted in in terstitials relative to vacancies: so nonpreferential sinks, such as voids. absorb

the aide rage void si ze ,

N J N ( R ) d R

( 19.7 )

vacancies a t a somewhat greater rate than in terstitials and growth results.

I n su mmary , the conditions neces.sary for void swell in g

1. Both in ie rstitials and i'acan cies must be mobile in tta e sol id. ’£his require me n I is easil j me t by in te rstitials, wli ich

R ) " R N ( R ) d R

or the void say elli n g,

V

H N ( il ) d R

( 19.8 )

( 19.9 )

can migrate in mrtals at s'e ry low te mperatures. I f the vac an cies are not mo bile as we 11 , ther Will simpl y be an n ih ilated by the cl oud of mcv in g in ters titials.

2. Point defects must be ca pable of being removed at sinks prov ided by structural defe cts in the solid in additio n to being destro \ ’vd h \ ’ recumb in at iun. ñ1oreove r, une of the will ks must ha•'e a prt•ference for the’ illtersti tials ill orcler to pt'rmit establish ment of the ‹'xc'c'ss \ acancj population necessary fOr voids to form.

3. Th e su prrsatu ration o f vacan cies must be large en ough to pc rmit voids and dislocation loops to be n u clt'ated . e ith e r ho mog ene on sl y or hete roge neon slj , and to grow. At temperatures su fficie ntl y li igh th at the thermal eqtiil ibri u m conce nt ration of vacancies at the \ ’oirl surface is comparable to that sustained in the mat rix by i rradiation, void nucleation an d growth erase. A I la igh tempe rattt res voids thermally emit vacate cies as fast as the irradiatio n - prod freed vacancies arrive from thr bul k of the solid.

4. Tra‹'e quantities of insoluble gases must br p mse n t to stabili ze the e mbryo void s and pre ve n I collapse to wac'ancy loo ps. T ransm utation lu'liu m provides the ne ce seat gas con ie nt in n e u tron - irradiate d mc tal s, al th oug h othe r gas- eous i mpu ritit's ( oxjge n , nil roge n, and li y d roge n ) pre se n t in mos I metals can perform the same fun ction. Altli o ugh some he liu m gas is undoub tedly prvse nt in voids, there i.s de finitel y n of en‹›ugh to ‹'lass th we cav ities a.s eq uil i briu m b ub bles.

19.2 OBSERVED CHARACTERISTICS

OF VOIDS

Ex‹ r'He la t sum marie s of tht' expc'rime ntal obse matrons of voids in metals h ai'e bee n pre sr n ted by be me n t and by iNorris. I n acld itio n , the papers in two con ferenrt s de i ot ed

ar‹ re ported. If the ve id d is trib ut ion is n arrow, th e en elf in g may be ex pressed by

( 19.1U )

Most tht'oretiral treat me rite are c'ontt'n t to pred ict th e as erage ve id si ze , assunii rig th at the void de n sitj is a spv‹ i fied n timber rather th an th e complete void distribution fun ction.

Sz'ell ing can also be e x pe ri me n tall y dete rmin ed by inn me rsi n g a samp 1 e of kn own we igh I in a fluid to rneasu re the .solid vol time. I to we ve r, on lj the electron microscope c an gru \ ide d aha utJ \ 'oicl size ancl dt‘tJ sit}’. In arlrl itiun, H› is tool can provide in formation on the evolution of the dislocation structure of the irradiated metal. 'I'h is in forma- tion consists of:

I . 1'h e de nsi £} of we £wurk d islocation s ( i.e., dislocati uns other than those comprising the loop.s ) .

2. The total d islocation line length of the loops. which is determined bj thy ai'erage diamc• ter of the loops and th e nu mber density of the loops.

19.2.1 The Void Distribution Function

F igu e 19.1 shows the void-size distribu tions for stain - less steel irradiated at di ffere li t temperatures b ut to the same fl uc nce. The distributions at low temperatures are appro ximatel y fiaussian, z'itli the peak ,sh i1“ted to the large r

\ 'oid siz‹‘s as thc’ te mperat \ mm* is int reased. The \ ’e ’ narrow cl istrihutiuns at luw tt‘mp‹’ra£rir‹ s indic'ate th at. alth‹›ugh void n uc leati on li as o r!cur red , th e I ow growth ratt prt•ve li ts

› id rrom at tai n iris large sizes in the allot ted irrad iatioii time. A I h igli temperatures the d is trib utio n fun ction is i e o’ broad and con tains some i'ery large i'oids and a small proportion of litt ie one s. Th is ty pe of distribti tion suggests

466 F L’L'DAME S'TAL AS PEC US GF N UCLE A R REAC TOR F UF. L ELFz MEN US

indicates a rapid in crease in void size at low temperatures and a smaller rate of increase at high tern peratures. Similar nonlinear behavior is seen along the fl uence ax is. Figure

19. 2 ( b } shows that the void n u mber density decreases with increasing temperature and increases with fluence. Observed vo id densities range from 10 1 to 10' voids cm .

VOID DIAM E TE R. : \

Fig. 19.1 Void-size distribution N ( R ) in type 316 stainless steel irradiated to a fluence o f 6 x 10 2.2 neu tro ns/cm° at various temperatures. ( A fter d. I. B ramman et al., p. 1.25, Ref. 6. )

that n ucleation has ceased and a constant densi tj of v'oids is in the proce ss of grow ing.

19.2.2 Void Size and Density

The ze roth and fi rst monie n Is o f th r• i'oi d d ist ri b u ti on fut ctiun, which represen t the \ ’oid nrlmbe r dei \ sit \ ’ and avtrnge \ 'uid size, respecti \ ’eI}’, are shown its the three - dimens in na1 re prest•ti latin n s of Fig. 19 . 2. Pigti re IS. 2 ( a )

19.2.3 Void Swelling

According to Eq. 19.10, the de pendence of volume s welling on temperature and fl uence could be constructed by multiplying the cube of the surface heights of Fig. 19.2{a ) by the surface heights in Fig. 19.2 ( b ) . Cuts th rough this three-dimensional representation of volume ssvelliti g are show n in Figs. 19. 3 and 19.4. Figure 19.3 shows the restriction of swelling to the temperat ure band

350 tu 600”C with peak swelling occurring at 500”C. Figure 19.4 indi cafes a po wer-law increase o f void swelling with neutron fl ue nee. 'I'he fluence dependen ce is of the form

A V

V

where the expone nt n js about un its' at 400‘C and in creases to about 2 at high te mperattires. Other functional forms have been suggested fo r the fl uen ce de pe nden ce of swell ing. Because of the scatter ct the data, swelling can eq hall y well be fitted to a linear eq nation w ith an in c ubat ion period du ring which voids are absen I:

( IS. 1 2 )

l b I

Fig. 19. 2 Void size ( a ) a nd n u mber de nsity ( b ) in fast reactor irradiated a us ten it ie sta i nless steel as a l'u nc- tion of fast -rie u tron fl ue nce and irradiation tern per at ure. [ After T. T. C la nd so n , R . W. Ba rk er, a nd R. L. Fish, V xc f. .ñ p pl. Tr e 11 ii o 1. , 9 : 1 0 ( 19 7 0 ) . ]

467

—

are fret from disl ocatit›ns. V oids easil y fo rm in tliest xont s and are res possible fri r the second h ump in tlit' sivelli rig t‘urve t'or t \ ’ pe 30 i st a in lt ss steel. ’I’lit• clislocatiu n st ru cture int roduc‹•d by c olcl work ing of ty pe 3.16 stainless stee 1 appt ars to be nior‹* stable. The majo r di fferr ii ce betwet n tlu'se t wo steels is the 2 to 3"zr molybdenum additioti to t \ ’pe ,31fi stai n less stet'l. This alloying e Ie me n t can en ffi - eien t I j rt d tir'e th e inn b ili ty of dislucations ( by pi nn ing ) tt› cli rn ini str tlit• rt'co ierj pr‹ ) t ess.

300 300 500 600

1RRAD|AT1ON TEMPERATURE, C

700

19. 2.5 Effect of Precipitates

’l'li t t t‘fect of all oj compt›sitio n is eve n more d ramati-

‹ all \ ’ t x li ibited in the s welling be havior of nicke 1 and the li igh li ickel-co n ten t alloy lncoiit*l ( Fig. 19.6 ) . N it kel with

0. t"‹ i mpurities swel 1s confide rabl y less than li igh -purity' nir-kel , and 1 ncont 1 act uall y de nsi fies during irradiatio li. 'l'h t' ext:t'llt'nt swelli rig rt'sistanct• tif I n ct›n el is probably due to tht flue Ni , Nb prer'ipitatt• that is present in this matt rial. '1’his p rt cip itatt particle' is cohvrt'ii t, wh icli means that its lattice ci›iistaiit is ‹ 1 use to th at of tht' inatri x, and the precipitate mat rix in tt rf‹ict• is con tin uousl y loo nded . 11 will be sh own latt'r th at culieren t precipitate's ac't as

Fig. 19. 3 Effect of irradiatio n teiriperature on swelling of type 30.4 stainless steel at a fluence of 5 x 10 2.2 neutrons/ cm°. , transmission electron microscopy. , immersio n density l After S. D. Hark news and Che-Yu Li, Met. "Ei ans.,

2: 14.57 ( 197 1 ) . ]

"the i ncubation p‹ riod , ( 'l›t ) „ , is of th I Ord er of 10' ' neu fret us ‹'m' an c1 is bt'l it's ed to re p rt sen I the nt ti mo n dt›se lit't'de d to p rOd ttce t'nO fig h Int I iu ni to pt riiii I voi c1 titit'l c'a- tion to p ro cc ed . The in ductio n pe rind maj al so be req u i red to build u p a su fficient de nsity o f interstitial loo ps to allo w the pre fe rential absorption of inte rstitials by dislocations to su ffi ci en II j bia.s t he poin t de fe ct po pul ati on i n the metal i n favor of vacancies so a.s tO pe rmi I va can cy a Sql orne rat itin into voids.

Neithe r of the above e m pt rical formu 1 ation s o f the fl ue nce de pendP nce of void .swell ing indi e ates satu rati on ( i .e., le ve 1 ing off ) Of this phen omeno n.

19.2.4 The Effect of Cold Work

€iold work , w hich increases the de usr ty of network d isl ocati ons, h as a significant e ffect on the swell in g cha rac- teristics of austenitic steels. L'p to a point, cold work ing inn pro ves the resist a net o f steel to swell i ng, as is shown by the smaller swelling of 20”‹ cold worked ty pe 3 lG stainless steel compared w ith the solu tion-treated ( i .e., an nealed ) material ( Fig. 19.5 ) . Excessive cold work may not be bene ficial, as indi cated b y the curve for ty pe 304 stai n less steel in Fig. 19.5. For th is steel, two swe fling peaks are observed. The low -tempe ratu re h u mp is associated w ith normal void formation in a iTie tal of const an t microstru c- ture . The high -temperature peak is pro bab ly due to the instability of the disl ovation network in trod u ced by cold work . Above 600‘C e xte nsi ve recovery an d recy stall i zation occur in the steel, and large segments of the microstructure

FLUENCE ( E > 0. 1 MeV 1 , neutrons/cm*

Fig. 19.4 Effect of fast-neutron fluence on swelling in type 316 ( ‹ ) and in type 34.7 ( ) stainless steels. Irradiation temperatures were between 4.70 and 540°C. ( After W. K. Appleby et a1., p. 156, Ref. 6. )

468

316 ( Soluhon t eu«*ll

3 1 G 120“ CW )

l

400

500 600 700

I R RA DI AT ION TEMPE R ATU R E, C

800

Fig. 19.5 Effect of cold work ( CW ) on the swelling behavior of austenitic stainless steels. The curves for ty pe 316 stainless steel are plots of Eqs. 19.12a and 19. 12b for a f luence of 5 X 10 2 ° neu trons/cm 2 . The curve for 50*i CW t ype 304 stain less steel is for a fluence of J4 X 10° 2 neutrons cm 2 . ( ‹ \ fter S traalsu nd et al., p. 142, Ref. 6. )

—

-

1.0

0.8

0.6

0.4

0.2

0

0. 0 i 0 z. 0 s 0 n. 0 . 0 x 4 0 2 I

NE U TRON F LUE NC E E I MeV l. neu tr cns ‘cm*

Fig. 19.6 Swelling of high-purity nickel, nickel of 99.67c purity, and I nconel ( 739c N i—17?â Cw87o Fe ) at 4 25° C. [After J.J. Holmes, Trans. Amer. N ucl. Soc., 12: 11 7 ( 1969 ) .]

recombination sites for vacancies and in te rstitials and th us contribute to reducing swelling. Another way that a dispersion of fine precipitate particles in an alloy reduces swelling is b \ ’ impeding dislocation climb ( i.e., the}’ act in a man ner similar to molybde num in ty pe 31 ñ stain less steel } . Another nickel -based alloy wh ose mi crostructu re contains a fi ne dispersion Of coherent t precipitates is Nimonic PE lfi. Titanium and al U minu m are added in equal amounts to this all oy, and thy precipitates have the compositio n N i , ( Ti Al ) . This alloy .shows less swell ing than does ty pe 316 stainless steel at high fl ue n ces.

19.2.6 Empirical Void Swelling Formulas

I n view’ of the rudi me ntar y state of the theory of void formation in alloy s, empirical eq uations are used to account for the effects of void swelling in fuel-ele ment performance estimates. The equations used in curre nt core -design studies reflect the in fluence of the primar y variables of tempera- ture and fluen ce and the degree of cold work of the alloy. For ty pe 316 stainless steel, the swelling equation for solution -treated steel is

( ”1 ) - ( ‹I›t x 10— 2 * ) ** 0 S ! 0 * !’ l I T — 40 ) 10—’ 0 j exp ( 32.6 — 5100/T 0.015T ) ( 19.12a )

470 F L7NDAMEN TAL ASPECT!S GF N UCLE A R H EA C TO R F L'EL ELE MW L'T8

s

700

450

400

400

450

4.000

100

10 22

N E UTRON F LU EN CE , neu trons/c 2

10*^

N E UTRON F LU E NC E , neutrons/ 2

Fig. 19.8 G rap hs of equations correlating the size and density of the f aulted interstitial loops in type 3.16 stainless steel for various temperatures and neu tron fluences. ( A fter Ref. 10. )

and fuel-ele me nt perform an ce pre dirti one will be forced to rel y on empi rical correlations for L UP Blt design. Howe s e r, theoretical models r the process are val uab ie because the y offer guidan ce for experiments atid elucidate the factors that may preve n t or at least retard x'oid groivt li in el add i rig materials.

Void -for mation tlieorie s us vi all y div ide tht iiverall pro cess in to distinct nucle ation and growth phases, mod els for w hich are presented in the foll€i wi ng two sections. Pre di c- tioiof the e solution of the loop and void distribu I ion functions with irradiation time requires coupling thr hasic nucleation and growth laws in to poin I defe ct, loo p, an d void conservation state me nts. Progress in th is aspect of void s we hing th con’ is reviewed in Sec. 19.6.

19.3 NUCLEATION OF VOIDS

N ucleatioii of voids refers IO the rate at wit ich t iiij embryos of these de feet cluste rs appear in the solid . Once nucleated , the P mbrv os te nd to grow and are safe froin destruction . N ucle ation and growth are often treated as seque n tial ste ps in the overall proct'ss of void formation. Supersaturation of the solid with point defects is a prereq uisite to both nucleation an d growth , bu t a higlie r

.su pe maturation is required to force n ti cle ation than to con tinue gro z'th of e x is ting ernbr \ ’os.

’The’ mu.st cc›mn1on example uf nucleation is the tond‹‘ilsatiun u I wale* r \ apur ii air. I f the partial pres u re uf a ter in dust-free air is sIu wl \ ’ increased be \ ’und the* equilibr ium apur prt'ssrtre, not h ing happc'ns until the' supersaturatiun I i.e., th ‹’ ratiu uf the partial pre ssure tu the

\ ’apor pres.sure ) attains a ›’alue uf about 5 tu 6. Ac this puii t a l”ug, whic'h corrr'sponds tc Mucl‹‘atiun uf small liq slid drople to, appears. 1'h+' su pt'rsa turation uf the gas pIi ase falls abr \ iptl at the unsef ‹›f rl u ale a tiun, and the ne wl1’ born droplets c't›nsumt' the re raining e xcess water i-a por at a purt°1y di ffusion-limited rate un til the t•qu ilibrium vapor pre ssurt° in ( he gas phast• i.s aha int d. £“ormation tif voids and loo ps i n solid s maj not be as c'learly divisible in to n ucleation and growth phases because in th is case genera- tion of point defects acts to main tain the su pe rsatu ration. Nucleation of new v'oids and lo ops may proceed in parallel with the grow th of e.x is ting ones.

to nvt he less, nucle ation and gro w th prticcsse.s can be anal›-zt•d as iildivid ual phen ome na, the rates t›f wli ich are ftinc'tions of tht• poin I-de feet su pe rsa turatio us, the lit I itim con te n t of the solid, and the ie mpt rature . Afte r the basi c rate equatit›ns have bee n deri›’‹ d , simultaneous ope rat ion of

471

‹›r ‹he appr‹›pridL‹‘ cuns‹' ”avio n statement.s l”or \ ’oid s, loops,

fi‹‘d , and most the‹›ri‹’s ‹›f › ‹›id furmati‹›n in me Fall ha '‹' adopted the n ucle ation follou e d -bi’ -Pro ivth approar'h.

As with th e ‹ on de neat ion of u.'ate r. n ti cleation of voids and ltiops i n mt' ( all can be cl asst'rI e it h e r ‹is h ‹›inoge neou s or

poi n t de fects e xecu tin g ra ndo m wal ks in the' solid . 'I'he stab ilitj of th Use el usters re lativr to the iii di vid ual point defe‹'ts of whic h th ey are ‹ o inpt›srd ( i.e., voids coii tain vacancies and pe rli aps gas atoms wht're as lt›ops con ta in intrrstit i als ) is th e clri x ink f‹›rc e t'ur nti ‹ leatiu n. No ne of th e struct ural featu res of the sol id are nc'c'ded to vausr• atigl omt ration of th e poi n I de fe cm.

I lete roge neo us nucl eat ion re t'e re to flu' appear an t ' txt’ voids on distinc I strur'tural feat urt s t›f the solid. In iv-ute r conde utation , for e xa mple . d net part i‹‘1es provide he te roge- neous n ticleatioli sites. Hi metals th e In'tc'ruge neitic's th at i T1 a \ ' acc‘e le rate roid nucle a tit› n in chide p reex isti rig gas bubble s I ‹'on Gaining ei the r inipti ritj gast's in the as-fabri- cated rru t al or irradi at ion produ‹ r d hel itim or lij d role n which has pre‹ ipitate d int o equil ib ritim hub bles prior to iiu t'lt'd tion of i'oids ) , i licohe re ii t prt'‹'ip it ate parti‹'1c's, and dislo‹'ations. 'l'he de plc ted zones created in tlu ‹'ull isit›n

Therr is no ge n eral e‹xise us us on the pre do mi n an t void nticleatio n mechan is m. 3’he irn portaH ce of homogeneous nuch'atioli vis a-vis heir rogeneotis n tic'leation has bven dc‘bated si n ce irrad iatioiJ - pr‹›du ced ›’uirl s in n t tall ver‹ first discov‹' red ( Refs. 4, 5, 1.1 , an d 1.2 ) .

N u‹ I cation of \ ords in th e de plc' ted zoHes fo rmed in tlit' collisioH case ade is Int li ke I j because o I t1u• rapid th e rnaal

at the peak swell in g temperatu res in stai Hless s feel ( see Sec. 15.5 ) . Fu rt he rmore , irradi ati on of metals by etc ctro us results in copious void formation, even th ough de plc ted zones are not formed by th is ty pe of bombarding particle ( Sec. 18.5 ) .

I I has not been possible to unequivocall j dete rna inc the cond it ions under which hetc‘roget eo us iJ ucl‹'atiulJ r›f \ uid s on second -phase parti clv s is import an t. Bloo m' ' has foun d that whe ii the vo id con ce ntration is how t either by combination of low hue n ce at low teinprratu n' or high fluence at h igh tempr rature ) the voids are ofteil associated with dislocations or precipitates. A I constant flue n ce Brager and Straalsund ' ' observed what appears to be homo- geneou s nucleation at low temperatu res, whereas at high temperatures the voids were fewer in n umber and attached to pre ci pitates. Ne \ ’ertheless, the idea th at a fixed li umb er of heteroge neo us sites is re sponsible for all void nucleation is unacceptable on two coun Is. The con cept p redi cts th at the void co nce ntratio n should be ( 1 ) limited by the nu mbe r density of nucleation sites in the me tal and ( 2 ) rude pendent of irradiation ie mpe ratu re . Neither of these ex pectations is satis fled by void formation in stainless steel.