Modification of the Kolmogorov-Johnson-Mehl-Avrami rate equation for non-isothermal experiments and its analytical solution

Transcription

1 Modfco of he ologorov-johso-mehl-avr re eqo for o-soherl eeres d s lycl solo J.Frjs #, P.Ror RM, Dere of Physcs, Uversy of ro, Cs Molv, df. PII, 77 ro, Clo, S. Absrc Avr s odel descrbes he kecs of hse rsforo der he sso of slly rdo cleo. I hs er we rovde qs-ec lycl solo of Avr s odel whe he rsforo kes lce der coos heg. hs solo hs bee obed wh dffere cvo eerges for boh cleo d growh res. he relo obed s lso solo of he so-clled ologorov- Johso-Mehl-Avr rsforo re eqo. he corresodg o-soherl ologorov-johso-mehl-avr rsforo re eqo oly dffers fro he oe obed der soherl codos by cos reer, whch oly deeds o he ro bewee cleo d growh re cvo eerges. Coseqely, or correco llows s o eed he ologorov-johso-mehl-avr rsforo re eqo o coos heg codos. # Corresodg hor: jord.frjs@dg.es

2 . Irodco Phse rsos re og he os or ocs erls scece. Crysllzo of orhos erls d oher sold se rsforos slly volves cleo d growh. hese rsforos re geerlly descrbed by ologorov-johso-mehl-avr [-5] odel. A solo of ologorov-johso- Mehl-Avr s odel der soherl codos c be obed ssg h cleo d growh res re e deede [5] e[ k ] where s he rsfored hse frco, k s he overll re cos h geerlly deeds o eerre, whle s slly kow s Avr s eoe s e. For he res of he er eq. wll be referred o s he ologorov-johso- Mehl-Avr JMA relo. Dffereo of eq. resls he well kow JMA re eqo: d d [ l ] k Alhogh hs eqo s obed fro he soherl solo eq., coses he bss for lyzg o-soherl eeres [6-8]. hs s becse he rsforo re sees o deed oly o eerre hrogh k d o he rsfored frco. However, deedg o he herl hsory e.g. he heg re, gve vle of wll corresod o dffere se d coseqely wll evolve dffere re. Ideed, he JMA re eqo s vld for o-soherl rsforos oly whe very rclr codos re e see Seco 3.. Dese hese severe los, o-soherl eeres re cooly erreed wh he JMA re eqo. As oed o by severl hors [7-], lycl ehods bsed o he JMA re eqo hve bee develoed regrdless of s vldy. I rclr, he wdesred ssger ehod [] c be led o y rsforo descrbed by he JMA re eqo [6]. ve hogh oe wold eec erroeos coclsos fro hs correc se of he JMA re eqo, he fc s h good greee wh oher ec ehods s ofe obed. hs s srog dco h y roeres of he ec solo re shred by he JMA re eqo. I hs work qs-ec solo of Avr s odel for he coos heg cse s obed by osg oly Arrhe eerre deedece for boh cleo

3 d growh re. Or solo roves o be he ec solo of JMA re eqo whch he kec cos, k, s slghly odfed wh resec o he soherl cse. So he vldy of he JMA re eqo s eeded beyod he severe los of he soherl JMA re eqo.. he soherl JMA re eqo For he rsforos volvg cleo d growh d ssg h he cle of he ew hse re rdoly dsrbed, Avr [,3] obed he followg relo e[ ] 3 e where e s he eeded rsfored frco,.e. he reslg rsfored frco f cle grow hrogh ech oher d overl who l erferece e τ v τ, dτ 4 s he cleo re d vτ, s he vole rsfored e by sgle cles fored e τ z dz v τ, σ 5 τ σ s she fcor e.g. σ 4π/3 for shercl grs, s he growh re d deeds o he growh echs [7,9,] e.g. 3 for hree desol growh. qs. 3-5 show he kecs of he rsforo der very geerl ssos bo he re coss y e or eerre deedece d for y herl hsory. he JMA relo eq. s he rclr solo for soherl codos rovded h boh d do o deed o e. he overll re cos s gve by: σ k I os rccl sos s ossble o sse Arrhe eerre deedece for boh d [,3] e / d e / 7 where d re he cvo eerges for cleo d growh resecvely, d s he olz cos. Sbso of eq. 7 o eq. 6 gves 6 σ k e k e 8 3

4 where we hve defed he overll cvo eergy s: 9 As eoed he rodco, dffereo of eq. leds o eq. whch wll be referred o s he soherl JMA re eqo for he res of he er. 3. he o-soherl cse Alhogh o-soherl eeres c se y rbrry herl hsory, he os sl eeres erfored herl lyss volve heg cos re, d/d. herefore, d for he res of he er, we wll del wh hs rclr osoherl codo. 3. he sokec cse. he JMA re eqo c be led o he o-soherl cse whe he rsforo re deeds eclsvely o eerre d o he degree of rsforo [6,8,9] d o o he herl hsory. hs codo s flflled rclr cses sch s se sro, where cleo s coleed ror o crysl growh [9,4], or he sglr sokec so where d hve he se cvo eergy []. For cos heg re, rodcg eq. 7 o eq. 5 gves he vole rsfored by sgle cles: v τ, σ ' where, τ, s he l eerre d he fco s defed s see Aed A: e d Accordgly, he eeded rsforo frco, e, c be dedced fer sbsg eq. o eq. 4, d ssg h he rsforo re s eglgble : 4

5 5 [ ] d e e σ where /. hs egrl c be solved lyclly whe sokec cse. Ideed, by sbsg eq. A.3 o eq. oe obs [ ] e k d d d σ 3 d, sbsg e o eq. 3 gves, flly, he rsfored frco: k 4 Dervo of eq. 4 wh resec o e shows h s ec solo of he soherl JMA re eqo. Coseqely, hs eqo s vld for o-soherl codos rovded h sokec cse. he lerre shows [,5] h he solo,, for he se sro cse lso obeys he soherl JMA re eqo eq.. 3.b he geerl cse. Whe, he egrl of eq. hs o lycl solo. For os eeres / >>, hs s slly roed by s frs er seres of / [7,,6- ] see Aed A: e 5 wh hs frs order roo, ber of hors [8-] obed decl solo of eq.. We wll follow dffere roch o solve he egrl of eq.. he fc h he rges of he eoel fcos d re dffere kes ossble o solve lyclly. We overcoe hs roble by relcg hese rges by coo verged rge see Aed. Wh hs roo, eq. c be solved d he corresodg rsfored frco s see Aed C: C k 6

6 where C s cos h deeds o, d :! C 7 Π I s worh og h or roe solo cocdes wh he ec sokec solo eq. 4, ece for he cos C. As coseqece, he roe solo for he geerl o-soherl cse s lso solo of he soherl JMA re eqo wh he overll re cos k lled by he cos C: d d [ l ] C k I he res of he er, we wll refer o eq. 8 s he o-soherl JMA re eqo. As eeced whe, C redces o y d or solo eq. 6 cocdes wh he ec solo for hs rclr l eq c Accrcy of he o-soherl JMA re eqo We wll lyze he ccrcy of or solo by corg o he ec solo h resls fro he ercl egro of Avr s odel eq We wll lso show h s ch ore ccre h: he solo of he soherl JMA re eqo eq. wh / d; b h of Vzqez e l. [8] d oher hors [9-] eq. D., whch cosss relcg by s frs order roo e-/ eq. 6 see Aed D. I fc, oe c esly se h he relve error clclg e wh resec o he ec solo for he foreeoed solos oly deeds o wo reers: he ro / d he vle of /. Whe hs error s sll sy <., he se coclso les o he relve error of. A ler eso of eq. 3 rovdes he relo bewee he relve error of Δ/ d h of e Δ e / e : [ l ] Δ Δ e 9 e I Fg., Δ/ hs bee loed verss / for dffere orlzed eerres /. Fro Fg. c be coclded h y codo or solo s he os ccre oe. For.5, Fg. ells s h he relve error of or solo s lower h.7 for / greer h. he bsole error s hs lower h.35. I 6

7 os rccl sos / s greer h 3, hs wold be eve lower <.. We coclde h wh he ccrcy of eerel d, or solo c brely be dsgshed fro he ec solo. I s worh eog h Fg. fls o gve he correc resl for he sokec cse /, eve hogh he roo of Vázqez e l. crve b s ore ccre h he solo of he soherl JMA re eqo crve. I he l where >>, cleo kes lce before growh [3] d he se sro roo s he os rore descro. Coversely, whe >> eher heerogeeos cleo does d he se sro roo s lso he rore descro, or crysllzo s drve by el growh. Aoher wy o es he ccrcy s by log he crysllzo re s fco of eerre. hs hs bee doe for he rclr d vles of orhos slco crysllzo []. he resl hs bee loed Fg.. I s cler h or solo d he ec solo re rcclly dsgshble he shf of he ek eerre s less h. ºC. Moreover, lhogh he soherl JMA re eqo s o ccre solo, redcs he correc ek she sce he oly dfferece wh resec o he o-soherl JMA re eqo s cos fcor. 4. ec lyss of hse rsforos der codos of cos heg re Mos of he lyses of herolycl eeres re bsed o he soherl JMA eqo [6-8,7] eq.. Sce or solo obeys forlly decl eqo eq. 8 for he o-soherl cse, hs es h hese lyses c lso be led o o-soherl eeres rovded h he overll re cos k of he soherl JMA re eqo s odfed by he cos fcor C. A coo fere of os herolycl echqes s he defco of wo reers whch rodce srgh le whe loed gs ech oher. he cvo eergy d he rsforo order re obed fro hs rereseo. I geerl, os ehods re socoversol,.e. hese wo reers re reled o deerg he eerres whch eqvle se of coverso s reched. 7

8 he os wdesred ehod s he ssger ehod [] where l / s loed P verss d P s he ek eerre,.e.. he ssger lo reles o d / P P he ler relosh; A l P P he cvo eergy c be obed fro A, wheres cos foro bo he re-eoel fcor of he kec cos. For he o-soherl JMA re eqo c be show h Aed F: where g' s: d A d l k Cg' P [ ] l [ l ] g' P P. he vle of P c be obed for he o-soherl JMA re eqo eq. 8 fer clclg he secod dervve of s ec solo eq. 6 d eqg o zero: P P k C e e P P P by sbsg by s frs order roo he ls er, oe ges: C P k 3 whch ells s h he ek eerre, e, core eq. 3 wh eq. 6 d coseqely: e.63 4 P he revos resl does o deed o y reer d ws roosed by Hederso [6] s es for he lcbly of he JMA odel. Moreover, g' P, so he ssger cos er s redced o l kc. Devos of P fro.63 re eglgble rel sos / P > [8]. he redco of he ek eerre for he o-soherl JMA re eqo s loed 8

9 Fg. 3 d cored wh he ec solo. For wde rge of heg res << /, dscreces vry fro. o.4. Coseqely, wh eerel ccrcy, eq. 8 c be cosdered ec. he Cos-Ms ehod [3,4] c be worked o fro eq. 6 by kg wce he logrh: l[ l ] l l kc 5 P y sbsg by s frs order roo, we ob l[ l ] l e l kc 6 I c be esly verfed erclly h he lo of l y ehbs cler ler red: l y y be redced o: verss y for <y<6 [7]. herefore, eq. 6 c l[ l ].5 l kc l l s fco of recrocl eerre s ler wh hs, he lo of [ ] sloe of -.5 /. Followg slr rocedre, we c verfy he vldy of he ehod develoed by Ploy [5] whch he cvo eergy s obed fro he sloe of he ler relosh bewee d / d l d /. Mos of he boveeoed socoversol ehods rely o relcg gve sge by he frs er of s seres eq. 5. However, ore ccre socoversol ehods do o se y hecl roo [6] or re bsed o ore recse roo [8,7,7,8]. oeheless, sce or solo s ec solo of he JMA re eqo, her lcbly s oclly eeded o he geerl o-soherl cse. Ozw s ehod [9] s wdely sed ec ehod, d c be ferred by kg wce he logrh of eq. 6 Log[ l ] Log kc Log 8 9

10 hs, he sloe of he lo [ l ] yelds he vle of. Log gs Log gve eerre 5. Coclso We hve obed he ore ccre roe solo of Avr s odel der codos of cos eerre sc re. hs solo s lso solo of he JMA re eqo. Ideed, s he solo of eqo h dffers fro he soherl JMA re eqo by cos fcor. As coseqece, os of he clsscl kec lyss echqes, whch re bsed o he JMA re eqo, re vld for he geerl o-soherl cse. I vew of hs resl, oe c dersd he oeworhy sccess d ssfcory resls obed fro ehods bsed o JMA regrdless of he fc h hey rely o he correc sso of he vldy of he soherl JMA eqo for he o-soherl cse. Wh eerel ccrcy, he o-soherl JMA re eqo d s lycl solo c be cosdered s qs-ec descro of rsforo kecs. ble I srzes he resls obed hs er.

11 Aed A: he fco he fco e d A. s reled o he eoel egrl [3] ccordg o: A. Is frs dervve s d e A.3 d d, fro he syoc eso of [3], c be develoed s e! ~ A.4 e Aed : roe solo for [ b ] d I hs ed we wll se h relo. s ec for he frs order syoc eso of, e b b I [ b ] d. b b Wh he ew vrble b /, he egrl I c be wre s b I b b b e d b b. e, we erfor he followg roo whch s ec for he frs er he seres eso of eq. A.4 he he egrl I s roely b b b b b b b e b.3 I b e b [ ] d b.4

12 berg d relo A.3, he ler egrl c be solved d s he rgh hd sde er of.. Aed C: Aroe o-soherl solo of ologorov-johso-mehl- Avr s odel If we egre by rs, eq. becoes: [ ] [ ] d d dz z z e e e e σ σ C. Where /. Afer secod egro by rs, e s eressed s: [ ] d dz z z z e e e σ y sbsg he roo. oe obs [ ] d e 4 e σ Afer es of frs egro by rs d f we sbse. g, e redces o:! Π e σ If s relced by / he ler eresso c be rewre s: e C k C. where! Π C. Aed D. Aroe solo of Vázqez e l. he solo of Vázqez e l. [8]:

13 3 e e σ D. c be dedced fro or solo eq. 6 sly by sbsg by s frs order roo: e d rewrg C s see Aed : / C Oe c esly coe o he se coclso for he solos obed by oher hors [9-]. Aed. Proof of he dey Π! he lef hd sde er c be develoed sle frcos Π!! D. he dey s he esblshed f we c rove h:!!! D. so of he rgh hd sde er of D. resls!! j j Π Π Π D.3 whch les j j Π D.4 Le s evle he revos relo for -,,,. For ech of hese vles ll he ers of he s re eql o zero ece oe, he oly oe h cos he fcor,.e.:

14 Π j L L!! j hs relo D. d coseqely he dey re roved. Aed F. ssger lo. Cosder re eqo wh he geerl for: d g wh e d F. where g s rbrry fco. For cos heg re eere he ek eerre P s deered by he codo:, d d P F. whch leds o he relosh: e P P g' P F.3 where g P s he frs dervve of g wh resec o evled he of d d. qo F.3 c be esly rsfored o for h s sble for ssger lo: l l g' P F.4 P P Ackowledgees We whsh o hk Dr. Frcesc-Xver Mssed for develog ed. hs work ws sored by he Ssh Progr col de Merles der corc MA--436-C4-. 4

15 5 ble. Sry of he resls o-soherl JMA re eqo [ ] l k C d d Is solo C k e deede re cos k σ Correco fcor! Π C Acvo eergy ssger cos er C k l

16 Fgre cos Fgre. Sold crves: relve error clclg he rsfored frco fro or solo o-soherl JMA eqo, eq. 6 wh resec o he ec solo eq. for hree dffere vles of /. Dshed crves: relve error clclg he rsfored frco sg he soherl JMA re solo eq. d b he frs order solo of Vzqez e l. [8] eq. D. for /. All crves hve bee clcled for rsfored frco of.5. Fgre. hree desol crysllzo re of orhos slco s fco of eerre clcled fro: he ec solo sold crve, or solo eq. 8 doed crve, he soherl JMA re eqo crve d fro eq. D. crve b for heg re of 4 /. erel reers []: 3. ev, 5.3 ev,.6 7 /s,.5 44 s - -3, 3 d 4 /. Fgre 3. ssger lo corresodg o he crysllzo of orhos slco clcled fro: he ec solo sold crve, or solo doed crve, he soherl JMA re eqo crve d fro eq. D. crve b for heg re rgg for o /. erel reers []: 3. ev, 5.3 ev,.6 7 /s,.5 44 s - -3 d 3. 6

17 Refereces [] Avr M. J Che Phys 939;7:3. [] Avr M. J Che Phys 94;8:. [3] Avr M. J Che Phys 94;9:77. [4] Johso WA, Mehl RF. rs Aer Is M Me g 939;35:46. [5] ologorov A. Izv Akd k SSSR, Ser Fz 937;:355. [6] Hederso DW. J o-crys Solds 979;3:3. [7] Yo H, Uhl DR. J o-crys Solds 983;54:53. [8] Málek J. heroch Ac 995;67:6. [9] Hederso DW. J herl Al 979;5:35. [] rj JW, Jog WA, erg PJ. heroch Ac 98;45:35. [] ssger H. Al Che 957;9:7. [] Chrs JW. he heory of rsforo els d lloys, r I. Oford U: lsever Scece Ld.;. [3] Rbero-Frde J. J A Cer Soc 998; 8:654. [4] Ch JW. Ac Mell 956;4:449. [5] Vázqez J, Wger C, Vllres P, Jeez-ry R. J o-crys Solds 998;35-37:548. [6] Doyle CD. re 965;7:9. [7] Srk MJ. heroch Ac 3;44:63. [8] Vázqez J, Wger C, Vllres P,Jeez-ry R. Ac Mer 996;44:487; Vázqez J, rcí-rred D, Lóez-Aley PL, Vllres P, Jeez-ry R. Mer Che Phys 6;96:7. [9] Wold. J Phys Che Solds 99;53:5. [] Reberg, Wold, Peford-Log A. heroch Ac ;378:97. [] ee AW, Soer F, Meejer J. J M Sc ;37:3. [] Sell C, Lobrdo S, Prolo F. J Al Phys 998;84:5383. [3] Cos AW, Redfer JP. re 964;:68. [4] Ms, os, Yoko R. J M Sc 984;8:9. [5] Ploy O, Rybchkov ID, ovkov OS. re 966;:9. [6] Fred HL. J Poly Sc 964;6PC:83. [7] Málek J. heroch Ac 99;:57. [8] Málek J. heroch Ac ;355:39. 7

18 [9] Ozw. Polyer. 97;:5. [3] Wler, Chll WF. Hdbook of hecl fcos. I: Abrowz M, Seg IA, edors. ew York Y: Dover Pblcos Ic.;

19 Δ/. b.. o-soherl JMA / / Fg. 7

20 rsforo re /s / / ev c o-soherl JMA b eerre ºC Fg.. 8

21 -3.5 c o-soherl JMA l / / s P b / P - Fg. 3. 9