UNIT 3. Elements of applied physical metallurgy. Unit 3c Chapter 4: Crystal shape from growth instabilities

Transcription

1 UNI 3 Elemens of alie hysical meallgy Uni 3c Chae 4: Cysal shae fom gowh insabiliies

2 3/A.3.3 CYA HAPE FOM GOWH INABIIIE: ENIE he Mllins-eea moel ing soliificaion an iniially lana gowh fon can become nsable an yiel eniic sces. e s efe o he following case fo a e meal (gowh conolle eniely by hea flow): sable gowh nsable gowh laen hea of soliificaion (fo incemen of soli facion i.e. cysal gowh) exace fom walls Gowh fon moes ino sable hase: lii a > M laen hea of soliificaion injece ino lii Gowh fon moes ino measable hase: lii a < M

3 NB In he case of alloys gowh is also ien by gaiens of chemical foce (no only by emeae gaiens) an also he case wih exacion of laen hea of soliificaion fom he wall is nsable (see yical eniic sce of cas ingo alloys). he foma eamen of he chemical case is analogos o ha of he hemal case. NB If ohe foces ie soliificaion (e.g. elecoeosiion) he sabiliy coniions change an of cose also incooae infomaion egaing he new genealise foces.

4 ) MOE EQUAION We shall ea cysal gowh conolle by hea flow in n ien by a emeae gaien. his is eesenaie case of soliificaion a e meal fom he mel. he eamen can be exene o alloys in a saighfowa way by consieing he Chemical oenials in aiion o no essenial changes in fomalism an message. he fiel can be obaine fom he hea eaion wiho a soce. (n.b.: he laen hea of soliificaion will be accone fo in he BCs) Of cose: lii soli an y x y x y

5 .) BOUNAY CONIION A INEFACE (FUX) Physical bacgon BC geomey y ineface (x) x BC hysics y Coniniy eaion x Hea (laen hea of soliificaion) exace fom he (elemenal laye of newly-fome) soli as a esl of cysallisaion iffses ino he lii

6 BOUNAY CONIION A INEFACE (FUX) Fomalism (i) Coniniy of flx a ineface (ii) wih soce a ineface an (iii) moing ineface he soce flx is lie a ho slab moing wih elociy n. I.e. a [hea e ni olme ()] [ineface elociy]. sface ha moing caies wih iself a hea flx soli n lii n c c ( x ) n

7 n c n c ( x ) ( x ) n maeial soliifie in ime ni soli lii

8 n c c ( x ) n whee: (J m -3 ) is he laen hea e ni olme of soli n (m s - ) is he scala elociy of he -ineface an he ems c ae he aes of hea injecion ino he wo hases c ae he hea concances of he wo hases n.b.: Q la (J m -3 ) (m s - )Q la (J m - s - ) i.e. a hea (hemal enegy) flx n.b.: (m s - ) c (J K - m -3 ) (K m - ) c (J m - s - ): i.e. a hea flx aci hyoheses: (i) soliifying laye e ni ime is ey hin (incemenal) (ii) no conecion ( caie away js by hea iffsion)

9 .) BOUNAY CONIION A y ± ( y ) ( y ) c necooling ( y ) whence : ( y ) ( y ) M se o M c M necooling

10 Changing o cooinae sysem moing wih ineface y y x ansfomaion of BCs a y ± lim y ± ( y ) ±

11 3) Physical coniions on inefacial emeae Aoms a a ce ineface ae in a iffeen enegeic coniion w aoms a a fla ineface.

12 A simle moel fo aoms a a sheical ineface Consie a sysem consisiing of a e sbsance iie ino wo hases α an by a sheical sface. As oe aboe (ininsic cysal shae wih isooic sface ension σ) he eilibim coniions ae: µ α µ α α σ µ α α µ α σ µ Combining he s n Po wih he efiniion of he Gibbs fee enegy e ni mass: whence: µ a a a Va a µ V α σ µ α α Vα α

13 ha ewies o: ( ) ( V V ) σv 0 α α his geneal elaionshi can be emloye fo he ose of esimaing he effec of cae on he meling oin. α he hs be: αlii soli. Woing a cons i esls: σvsoli 0 Of cose: V A e s consie a ansfomaion fom a / eilibim wih a fla ineface o a / eilibim wih a sheical sface of cae / an le s loo fo he iffeence in eilibim beween he wo cases s s s s s s

14 soli V 0 ) ( 0) ( σ V soli 0) ( ) ( σ soli V 0 0 σ Assming: 0 i esls: whence:

15 ha ewies o: V soli σ ( 0) σκ ( 0) / whence: ineface κσ M ineface genealising see nex slie wih geneal inefacial cae: κ ( x) x x ( x) being: ineface ( x) n.b.: κ>0 if he soli oes ino he lii ( x) a c x ( a 5 c.5) c κ c

16 4) OUION OF HE MOE EQUAION 4.) esaemen of Moel Eaions in he moing omain x ( x ) x x x x y since: 0 y

17 4.) Non-imensionalisaion of wih infomaion egaing BCs a ± ecalling ha: we efine: ( ( ) ) M M c ef / c M whence: ( ) 0 ( )

18 4..) Non-imensionalisaion of he flx BC n c c x n whence: M c c ; ineface / n c c c c x n x n n wih :

19 4..) Non-imensionalisaion of he hysical coniions of he inefacial ineface ineface ineface / c M M κσ ineface ef M c σκ o κ

20 4.) Nomalise Eaions in (x) omain We ao he asi-seay sae aoximaion (seay-sae eaion & ime-eenen ebaion of he solion): x We assme enaie solions of he ye comaible wih he BC a ± : 0 ex lii: 0 soli: 0 4.3) olion of he Eaions

21 An we a small ebaions o hese solions of he ye: ) aniies (i.e. being small an ex ex ex << ix x ix x Inseing hese enaie solions ino he seay-sae eaions one fins ha hese ae saisfie fo (see nex slies fo comaional eails): 0 0 oscillaes in sace ies o

22 Iem fo self-sy ix x ix i x ix ix ex ex ex ex ex ex ix ix ix ex ex ex ex ex 0 Inseing he aboe elaionshis ino he seay-sae eaion:.e..

23 Iem fo self-sy ix x ix i x ix ix ex ex ex ex Inseing he aboe elaionshis ino he seay-sae eaion: ix ix ix ex ex ex 0.e..

24 4.4) olion a he Ineface We assme he following fom fo he ineface: ( x ) ex( ix ) ) wih ) << an we ealae he soliions a he ineface: ( x ) ex ex( ix ) ( x ) ex( ix ) ince ) is small we can mae he following aoximaions: () ex an

25 () [ ] [ ] ix ix ix ix ix ix ix e ix since ex ex ex ex ex ex ex ex << ) ) ) an fo he same eason: ix ix ex ex

26 4.5) Alicaion of hysical coniion on inefacial We ecall he non-imensional fom of sch hysical coniion: ineface ef M c σκ o κ o which we aly he assme fom of he ineface. an he aboe secifie Aoximaions esling fom <<. Being: ex ex( ix ) ( x) κ x ( x) an: ( x) x i esls: κ ex( ix ) ( ix ) ex( ix ) x ex( ix ) <<

27 whence: κ o ewies o: ix ix ix o ex ex ex i.e. o

28 4.6) Alicaion of flx BCs (a he ineface) n x n We ecall he non-imensional fom of he flx BC: o which we aly he assme fom of he ineface. ix n ex ix ix ix ix n ex ex ex ex ex ex see aboe aoximaions

29 [ ] ix ix ix n ex ex ex see aboe aoximaions ix ix ix e e e hs: whence:

30 4.7) Comaing (4.5) an (4.6) o o o o an we obain he isesion eaion () o o

31 wih he aoximaion: we ge: since: assming also ha: l 3 ( ) he iffsion lengh >> l o we obain: o

32 / : sabilises ineface ~σ : sabilises ineface : amlifies insabiliy o ( ) : insabilises ineface : < 0 ex( ) 0 sable ineface gowh λ ci ci σ : > 0 ex ( ) nsable ineface gowh

33 In geneal: isance fom eilibim leas o enie fomaion