FOR. Journal. Stress Development and Fracture of Surface Nucleated Cristobalite on Silica Glass. Ryan C. Breneman,, * and John W.

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1 Journl J. Am. Cerm. Soc., 97 [11] (014) DOI: /jce The Americn Cermic Society Stress Development nd Frcture of Surfce Nucleted Cristoblite on Silic Glss Ryn C. Brenemn,, * nd John W. Hllorn** Deprtment of Mteril Science nd Engineering, University of Michign, Ann Arbor, MI Stress development nd frcture of isolted cristoblite spherulites in morphous silic mtrix were observed. High purity bulk silic ws nneled to produce prtil surfce crystlliztion consisting of isolted nd impinged spherulites in n morphous mtrix. The stress stte of the morphous silic surrounding cristoblite spherulites ws qulittively exmined using crossed-polrs microscopy. Frcture ws observed to occur with mny spherulites encircled by crcks in the mtrix nd other spherulites observed to self-frcture in mudcrcking pttern. The frcture ws found to be size dependent with encircling mtrix crcks occurring s minority phenomen in spherulites 0 70 microns in dimeter nd mud-crcking self-frcture to occur in ll spherulites over 70 microns in dimeter. The stresses develop s result of the strin ssocited with the 4.9% volume reduction in the cristoblite on trnsitioning from bet-to-lph phse t ~50 C. Observed frcture behviors were modeled. Mtrix crcks encircling spherulites were found to be consistent with Weibull filure model of the glss under stress field derived from the Eshelby inclusion model. Self-frctured spherulite filure ws found to be consistent with filure model bsed on thin films under bixil stress. I. Introduction FOR over century it hs been understood tht tht crystlliztion to cristoblite wekens glss. The erliest reports of the crystlliztion of morphous silic to cristoblite come from scientists frustrted by the clouding nd subsequent brekge of fused-qurtz lb-wre exposed to extreme tempertures, such s reported by Crookes in We know this wekening effect is result of extensive microfrcture upon the trnsition of cristoblite from the hightemperture bet-phse to the low-temperture lph-phse t ~50 C. 4 This micro-frcture is cused by the 4.9% volume reduction on trnsition from lph-to-bet. 5 The development of stress in glss-cermics hs been widely reported upon in the literture; however, the work tends to be directed towrd bulk-nucleted glss-cermics in which the multiphse glss-cermic cn be used s stiff lod bering mteril. 6 9 Less well studied is the development of stress nd filure in surfce nucleting systems. This is likely becuse crystlliztion in such systems is considered flw rther thn ttribute. To our knowledge, no one hs looked specificlly t the stress development in the morphous siliccristoblite system. Crystlliztion in cristoblite occurs through the formtion of surfce spherulites which then grow nd impinge to form L. Pinckney contributing editor Mnuscript No Received My 14, 014; pproved July 4, 014. *Member, The Americn Cermic Society. **Fellow, The Americn Cermic Society. Author to whom correspondence should be ddressed. e-mil: brenemn@umich.edu 48 fully crystlline surfce over n morphous core. The lterl growth cross the surfce hs been reported to occur fster thn growth in thickness. 10 During crystlliztion the glss mtrix is bove the nneling temperture so no stress develops. On cooling smll therml mismtch stresses cn develop between the bet-cristoblite nd morphous silic. Although bet-cristoblite hs very low therml expnsion, the therml expnsion of morphous silic is nerly zero. On cooling from the strin temperture of ~1100 C 00 C betcristoblite will experience volume shrinkge of ~0.58% reltive to the morphous silic. 11 Upon cooling to ~50 C bet-cristoblite trnsforms to lph-cristoblite, resulting in 4.9% volume reduction nd the development of stress between the cristoblite nd the morphous mtrix. 5 Therefore, the trnsformtion volumetric strin is ~8.4 times lrger thn the therml expnsion volumetric strin mismtch. In this work we mke the pproximtion of only considering the much lrger trnsformtion strin. In the cse of isolted spherulites, the bet-cristoblite develops stress free t high temperture, but on trnsformtion to lph-cristoblite the spherulite shrinks wy from the mtrix s shown schemticlly in Fig. 1. In the present study the stress stte of isolted spherulites nd resulting frcture re observed. The frcture is modeled to understnd the reltionship between spherulite size nd mode of frcture. II. Procedure (1) Mterils A high purity bulk silic glss ws used in this study. GE 14 (GE Momentive, Willoughby, OH) glss prepred s 75 mm 9 5 mm 9.5 mm slides were used s the strting mteril. The purity of this mteril is greter thn 99.9%. The primry contminnts re luminum with 14 ppm, titnium with 1.1 ppm, nd hydroxyl with <5 ppm. All other contminnts mount to <4.5 ppm in totl. It should be noted tht this is very low hydroxyl content tht reflects the mteril s-received mteril. Heting under mbient conditions will increse the hydroxyl content through bsorption of tmospheric H O. 1,1 This will result in higher hydroxyl content in the mteril t crystlliztion. () Anneling nd Observtion GE 14 slides were nneled t 100 C 1400 C with hold times rnging from 10 min to 6 h. Anneling ws conducted under mbient tmosphere with lbortory reltive humidity rnging from 15% to 60%. During nneling the slides were held on n lumin plte nd supported by lumin spcers t four points ner ech corner. The slide ws supported in this wy to minimize contct contmintion tht my impct crystlliztion. 1,14 The extent of bet-cristoblite formtion, including the size nd re frction of the spherulites, did not chnge systemticlly with the nneling time nd temperture. Insted the behvior ws quite vrible from one specimen to the next nd from one loction to nother on

2 484 Journl of the Americn Cermic Society Brenemn nd Hllorn Vol. 97, No. 11 Fig. 1. Illustrtion of the bet-to-lph trnsformtion for n isolted spherulite in glss mtrix. The 4.9% volume decrese on trnsformtion to lph from bet on cooling cuses the spherulite to try to pull wy from mtrix inducing stress nd/or filure. the sme specimen. As the pprent kinetics of cristoblite formtion ws so vrible, we focus insted on the regulr nd systemtic behvior of isolted spherulite fter trnsformtion to lph-cristoblite. Following nneling, smples were exmined by opticl microscopy. Polrized light ws used in conjunction with Nomrski prism to chieve contrst between the cristoblite nd the morphous silic mtrix. The stress fields surrounding spherulites were qulittively exmined using crossedpolrizers opticl microscopy (CPLOM). III. Results Spherulites were observed to surfce nuclete nd grow spheruliticlly s hs been reported by others.,15,16 Though often indistinct when spherulite symmetry cn be determined it ppers to be -fold bout the center of the spherulite. This indictes the cubic bet-cristoblite grew with <111> orienttion. This grees with the results of Presser et l. who found the orienttion to be <111> nd growth to occur on the octhedrl fces. 15 When observed t room temperture, spherulites re in the tetrgonl lph-cristoblite phse hving trnsformed from cubic bet-cristoblite. This lph-cristoblite is highly twinned s result of the loss of symmetry through the displcive bet-lph trnsformtion nd lso under stress due to the 4.9% volume reduction ssocited with the trnsformtion. While mny spherulites remin intct some were observed to frcture within the spherulite while others were observed with encircling crcks in the glss mtrix. A typicl imge with frctured nd unfrctured spherulites in the morphous mtrix cn be seen in Fig.. An exmple of n encircling glss mtrix crck nd close up of frctured spherulite cn be seen in Fig.. Amorphous silic is birefringent under stress. A spherulite surrounded by rdil stress field will polrize trnsmitted light to n orienttion prllel to the stress. In crossed-polrizers opticl microscopy (CPLOM), two polrizers oriented 90 degrees prt re utilized, one bove nd one below the smple. If the smple does not induce ny polriztion, the light tht psses the first polrizer will be extinguished upon reching the second due to the misoriention of 90 degrees. In the cse of rdil tensile field the light pssing through the smple will reorient to be prllel to the stress field. The reoriented light psses through the second polrizer resulting in n illumintion effect. In our cse of rdil stress field the light will lso be polrized. This rdil polriztion effect in the stressed morphous silic results in n illumintion effect with intensity tht scles with the degree of misorienttion reltive to the polrizers. In the cse of rdil stress field (such s tht surrounding n lph-spherulite) the observed intensity re t mxim t 45 misorienttion to the polrizers. This results in Mltese Cross pttern centered on the middle of the Fig.. Isolted spherulites in morphous silic mtrix viewed by reflected polrized light. Spherulites cn be observed in both frctured nd unfrctured stte. Smple ws nneled t 175 C for h. (A) Isolted spherulite with encircling mtrix crck. (B) Isolted spherulite surrounded by un-crcked mtrix. (C) Impinged spherulites. rdil stress field. Conversely the misorienttion nd intensity re t minimum when ligned with one polrizer or the other. The end results for polrizers oriented verticlly nd horizontlly re intensity mxim t 45, 15, 5, nd 15 nd minim t 0, 90, 180, nd 70. The stress fields surrounding the lph-cristoblite spherulites were imged qulittively by CPLOM, s seen in Fig.. An imge with three neighboring spherulites imged both by reflected polrized light nd by cross-polrized light cn be seen in Fig. 4. As discussed in the previous section, the intensity of rdilly symmetric field hs mxim t 45, 15, 5, nd 15 nd minim t 0, 90, 180, nd 70. In Fig. 4 the spherulites re oriented digonlly reltive to the polrizers to mximize the intensity in between them. The stress concentrtion cn be seen to be very intense in between the spherulites. In Fig. 5 n S-crck cn be seen, likely inititing from the stress concentrtion between two djcent spherulites. At room-temperture spherulites were found to exist in one of three sttes: (1) Intct, with intct spherulites surrounded by intct but stressed glss mtrix; () Mtrix Crcked, with intct spherulites surrounded by n encircling crck pssing through the glss mtrix; () Self-Frcture, with mud-crck self-frcture of the cristoblite spherulite itself. These three sttes were found to coexist in the sme smple. This cn be seen in Fig. 6, in which exmples of the three sttes cn be seen in reflected polrized nd trnsmitted cross-polrized imging. A survey ws conducted on the filure mode for isolted spherulites to look t the incidence of intct spherulites, mtrix crcking, nd spherulite self-frcture s function of spherulite size. In totl 74 spherulites were observed from seven seprte nneled smples. The results re presented s cumultive plot s function of size in Fig. 7 or spherulites 0 microns nd lrger; the spherulite size ws recorded with n ccurcy of 5 microns. For spherulites smller thn 0 microns, the ccurcy is microns. It cn be seen tht no frcture of the cristoblite or encircling crcks in the mtrix occur for spherulites less thn 15 microns in dimeter. For intermedite sizes, some spherulites hve mtrix crcked with n encircling crck through the glss mtrix. Spherulites 70 microns nd greter in dimeter re lmost ll self-frctured through the cristoblite in dry mud mnner. This crcking pttern is similr to the pttern of frcture seen when spherulites impinge creting fully crystlline surfce s seen in Fig. 8. Unlike the encircling crcks in the mtrix seen in intermedite sized spherulites the mud-crcking self-frcture seems to occur in lmost ll spherulites bove certin size, this is illustrted in Fig. 9. The occurrence of both mtrix crcks in the glss

3 November 014 Frcture of Surfce Cristoblite 485 () (b) Fig.. Close up imges of frctured spherulite nd spherulite with surrounding mtrix crck viewed by reflected polrized light. Smple ws nneled t 175 C for h ) frctured spherulite illustrting mud crcking b) spherulite with encircling crck pssing through glss mtrix. () (b) Fig. 4. Spherulites imged under polrized reflected light () nd imged by cross-polrized trnsmitted light (b). Cross-polrized imging highlights the stress concentrtion between spherulites. Smple ws nneled t 140 C for 40 min. nd spherulite self-frcture of the cristoblite pper to be size dependent. However, mtrix crcking ws lwys minority of the spherulites in given smple; wheres bove 70 microns lmost ll spherulites were observed to be selffrctured. This suggests tht while both behviors re size dependent they my depend on size in different mnner. Fig. 5. Trnsmitted cross-polrized light imge illustrting stress fields nd mtrix crcking. Smple ws nneled t 140 C for 40 min. IV. Discussion (1) Observed Filure Modes As seen in Fig. 6, following the bet-lph trnsition, isolted spherulites were observed in one of three sttes: intct, mtrix crcked, or self-frctured. Intct here refers to intct spherulites in intct mtrix, while mtrix crcked refers to intct spherulites encircled by crck through the surrounding glss mtrix. Self-frctured refers to spherulites in which the cristoblite hs frctured. No spherulites were observed to hve mtrix crcks nd lso be self-frctured. These two different filure modes both pper to relieve stress, but lso pper to be seprte phenomen. Both filure modes seem to be size dependent with the popultion occurring t different spherulite dimeters s seen in Fig. 7. Smll spherulites with dimeters 15 microns nd smller re ll intct, displying no crcks in the mtrix or frcture of the spherulite. Spherulites between 15 nd 70 microns in dimeter sometimes re encircled by crck through the mtrix, with frequency of mtrix crcking incresing with spherulite dimeter. Spherulites bove 70 microns in dimeter self-frcture

4 486 Journl of the Americn Cermic Society Brenemn nd Hllorn Vol. 97, No. 11 () (b) Fig. 6. Reflected polrized nd crossed-polrized light imges showing three behviors of spherulites: (1) Stressed by no frcture; () Mtrix crcking round spherulite; nd () Frcture of the spherulite. Smple ws nneled t 140 C for 40 min. Fig. 8. Frctured surfce of smple in which spherulites hve begun impinging () nd in which spherulites hve ll impinged (b) producing fully crystlline surfce over n morphous bulk. Smple ws nneled t 15 C for 40 min. Fig. 7. Cumultive distribution by size of isolted spherulites for ech filure behvior. through the cristoblite, with very few exceptions. There were mny more smll intct spherulites thn lrge spherulites with crcks. Figure 7 hs the size distribution for 479 intct spherulites, 19 lrger spherulites with crcks in the glss mtrix, nd 116 lrge spherulites which hd self-frctured. In the following sections, the stress distribution nd the mechnics of the mtrix crcking nd spherulite self-frcture re modeled. () Stress Distribution Surrounding Alph-Cristoblite Spherulites To understnd why the behvior depends on the size of the spherulite, we model the stress distribution. The spherulite cn be considered hemisphericl inclusion on the surfce Fig. 9. A field of view with mny spherulites of ~90 microns in dimeter, ll of which hve self-frctured. The blurry drk bckground shpes re crystlliztion on the fr surfce. Smple ws nneled t 15 C for 40 min. of the glss. The well-known Eshelby model for sphericl inclusion cn then be used to pproximte the stress stte in the surrounding mtrix. In the clssic Eshelby solution sphericl prticle becomes strined reltive to the mtrix due to therml expnsion mismtch. This strin leds to the development of hydrosttic internl pressure nd stress in the surrounding mtrix. In the cse of the spherulite in the mtrix trnsformtion strin cuses the inclusion (spherulite) to shrink reltive to the mtrix. This is nlogous to the Eshelby tretment of stress development on cooling for n inclusion with greter therml expnsion thn the mtrix. From the Eshelby solution in this cse, the stress inside the

5 November 014 Frcture of Surfce Cristoblite 487 inclusion is uniform tension, which is expressed s pressure p, where: 17 p ¼ E m p ðdtþ ð tþ ¼ r ðpþ r ¼ r ðpþ h ¼ r ðpþ / (1) Here p is the internl Eshelby pressure, E is the elstic modulus, refers to the coefficient of therml expnsion of the mtrix nd inclusion respectively, nd DT is the chnge in temperture. The three stresses on the right-hnd side re the rdil, ngulr, nd zimuthl stress components in sphericl coordintes. These stress components re ll equl since inside the prticle there is hydrosttic tension. Notice tht the stress inside the prticle is independent of size of the inclusion in this model. For the cse of the bet-to-lph phse trnsformtion of cristoblite spherulites in glss mtrix, the therml expnsion mismtch strin in Eq. (1) for the Eshelby form is replced by the trnsformtion strin, e trns : p ¼ Ee trns ð tþ ¼ rðpþ r ¼ r ðpþ h ¼ r ðpþ / () Similrly the mtrix stresses cn be modeled. The stress in the glss mtrix surrounding the spherulite hs rdil tension, nd ngulr nd zimuthl compression. Rdil tension in the surrounding glss mtrix dmps with distnce s 1/r nd increses with the cube of the prticle size : r ðmþ r ¼ Ee trns ð tþ r ¼ p r () This rdil tension hs the potentil to cuse the glss mtrix to crck. The plne of the crck will be ny concentric shell in the glss mtrix round the spherulite. There lso re compressive stresses in the ngulr nd zimuthl directions. These re: r ðmþ h ¼ r ðmþ / ¼ Ee trns ð tþ ¼ p r r (4) These compressive forces re constnt with the ngles h, /, nd dmp out in mgnitude with the squre of the rdil distnce from the spherulite. They will not cuse frcture, but they will suppress ny ngulr or zimuthl component to the crck pth. Therefore, mtrix crck will tend to circle round the spherulite. This model could explin why crcks initited some distnce from the spherulite encircle but do not pproch the spherulite. () Mechnics of Glss Mtrix Frcture Mny spherulites hve encircling crcks through the mtrix. Such crcks could be cused by the tensile field surrounding the spherulite inititing crcking in the glss. The encircling crcks cn occur some distnce from the edge of the spherulite. This my be due to the crcks inititing from flws in the glss, s often observed for the frcture of glss. This would imply tht n interction between the stress field intensity nd the flw distribution of the morphous silic leds to the observed spcing between the spherulite nd the encircling crck. The highest mgnitude of the mtrix tension occurs t r =, the boundry between the spherulite nd the mtrix. However, for n incrementl sphericl shell outside the spherulite, while the stress decreses s 1/r, the volume under stress increses s 4p/(r ). So in the region where the stress is highest, the stressed volume is smllest. Frcture in glss is often determined by the lrgest flw found in the stressed volume, so the problem cn be considered in the context of Weibull sttistics. The probbility, F, of inititing n encircling mtrix crck cn be described s: F ¼ 1 exp4 Z 1 rðþ r m VðrÞ dr5 (5) r o where r o nd m re the chrcteristic strength nd Weibull modulus for two-prmeter Weibull, nd V o is the reference volume ssocited with the chrcteristic strength. Since the stress nd stressed volume chnge with distnce r, we ssess the filure probbility integrted for ll r greter thn the spherulite rdius. Eqution (5) models how the probbility of filure vries with stressed volume V(r). Substituting vlues for the stress nd volume terms llows the integrtion s follows: F ¼ 1 exp4 Followed by: Z 1 V o m p pr r o r dr5 (6) V o F ¼ 1 exp p p m Z 1 4 r m dr5 (7) V o And finlly: r o F ¼ 1 exp p p m V o r o m So from Eq. (8), the probbility of crck initition in the mtrix surrounding single spherulite is simply function of the Weibull prmeters, the internl pressure p derived from the Eshelby pproximtion, nd the spherulite size,. If we consider the size when the probbility of mtrix crcking is prticulr vlue, sy 7% (i.e., 1/e) to simplify the exponentil, we hve criticl spherulite size is F=0.7 where: F¼0:7 ¼ ½m ŠV 1 m o r o (9) p p Thus, the likelihood of mtrix crcking should increse strongly with the spherulite size, but remins sttisticl phenomenon. This seems to mtch the behvior seen in Fig. 7. Glss mtrix crcking cused by spherulites is clerly relted to size with lrger sizes being ssocited with higher rtes of mtrix crcking; however, mtrix crcking is lwys observed to coexist with unfrctured spherulites. The Weibull model suggests tht s spherulites get lrger the probbility of ctivting flw nd triggering mtrix crck increses, but it remins sttisticl phenomenon dependent on the Weibull prmeters of the glss. Thus, the observed mtrix crcks behve in the mnner of clssic brittle filure induced by stress field thn cn be pproximted by Eshelby methods. (4) Mechnics of Spherulite Self-Frcture of Alph- Cristoblite The self-frctured spherulites cn be considered seprtely from glss mtrix crcking, becuse the frcture occurs within the cristoblite nd the crck morphology differs. The geometry of the spherulites my lso chnge s they grow s others hve reported fster growth lterlly thn in depth; 10 however, we were unble to ccurtely mesure depth of the spherulites nd so cnnot confirm this. Self-frctured spherulites exhibit crcking pttern reminiscent of dried mud. This pttern of crcking is not uncommon in thin-film pplictions. In mny (8)

6 488 Journl of the Americn Cermic Society Brenemn nd Hllorn Vol. 97, No. 11 thin-film pplictions stress develops between the film nd the substrte, such s tht due to mismtch in therml expnsion. Often such films develop chnnel crcks to llevite this stress, resulting in dried-mud morphology. This my suggest the lrgest spherulites re behving similr to thin films under stress. In fct, s cn be seen in Fig. 8 when the spherulites impinge nd crete fully crystlline surfce the frcture pttern remins similr to tht of self-frctured spherulites. This supports the hypothesis tht self-frctured spherulites re behving like thin films. A cristoblite lyer on n morphous silic mtrix will ttempt to shrink on pssing the bet-tolph trnsition, but it will be constrined by the underlying mtrix. The constrint from the underlying glss results in bixil stresses. This sitution is directly nlogous to thin film on thick substrte experiencing bixil tension. A thin film under bixil stress on n infinite mtrix is well-defined system. In such systems the stress often rises from strin resulting from the therml expnsion mismtch between the film nd substrte. In the cse of the spherulites the strin is the strin from the bet-to-lph trnsition. Cristoblite nd morphous silic both hve n elstic modulus of 7 GP which further simplifies the reltionship. The frcture of thin films on infinite substrtes with identicl elstic moduli is governed by: ðe=ð1 t ÞÞG ss r o h ¼ 1:98 (10) in which G ss is the stedy-stte energy relese due to frcture. 18 Frcture is expected if G ss г, where г is the cohesive energy of the mteril. This provides criticl thickness, h c, for filure given trnsformtion strin, e trns s follows: C ðe=ð1 t ÞÞe trns1:98 ¼ h c (11) For given strin energy, elstic modulus, Poisson s rtio, nd cohesive energy there is criticl film thickness t which the film crcks spontneously. One cn suppose tht for spherulites the thickness, h, is function of the dimeter,. If we mke n pproximtion tht = b*h we cn rewrite Eq. (11) in terms of spherulite dimeter: b C ðe=ð1 t ÞÞe trns1:98 ¼ c (1) This suggests tht bove criticl dimeter the spherulite is expected to be ble to lower its energy by spontneously frcturing. This mtches the behvior seen for spherulite frcture. Above certin size the spherulites re observed to nerly universlly self-frcture. A field of spherulites bove this criticl size cn be seen in Fig. 9. The criticl vlue for spherulite dimeter, c, ppers to be ~70 microns s very few spherulites re unfrctured bove tht size. It seems tht self-frctured spherulites re exhibiting frcture behvior similr to thin films. Upon reching certin size the strin energy ssocited with the bixil constrint of the trnsformtion strin induces frcture of the spherulite. V. Conclusions Cristoblite spherulites were formed on the surfce of fully dense morphous silic nd llowed to pss the bet-to-lph trnsition upon cooling. Stress fields were observed to surround unfrctured smples by crossed-polr microscopy. Frctured spherulites were found to hve filed by one of two mechnisms, either by encircling crcks pssing through the mtrix or mud-crcking of the spherulite itself. Mtrix crcks were observed to lwys coexist with unfrctured spherulites, nd the frequency of crcking ws observed to increse with spherulite size. Spherulite frcture on the other hnd, ws observed in lrger spherulites, nd nerly ll spherulites bove 70 microns in dimeter were self-frctured. Applying Weibull sttistics to glss crcks induced by n Eshelby stress field produced model for mtrix crcking. This model predicts probbilistic mtrix hoop crcking with dependence on spherulite size, which ligns with observtion. The occurrence of mtrix crcks is size dependent but sttisticl phenomenon. A seprte model bsed on thin-film mechnics ws formulted for spherulite frcture. The spherulite behves s thin film under bixil stress over n infinite mtrix. This model predicts criticl size t which the spherulite will frcture. This criticl size mtches the observtion tht bove certin size, nerly ll spherulites hve self-frctured. Bsed on the models nd observtions, complete picture of filure cn be developed. Upon trnsitioning from bet-lph stress is induced by the ~5% volume reduction. In spherulites bove criticl size this results in spontneous self-frcture. In spherulites below the criticl size, the stress field my induce crcking if the stress field ctivtes sufficiently lrge flw. Lrger spherulites induce lrger mtrix stresses over lrger res incresing their probbility of ctivting flw, but mtrix crcking remins sttisticl phenomen. Acknowledgments This reserch ws supported by the Defense Advnce Reserch Projects Agency (DARPA) under grnt no. 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