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1 W a InthOpn, th wold s ladn publsh of Opn Ass books Bult by sntsts, fo sntsts,800 6,000 0M Opn ass books avalabl Intnatonal authos and dtos Downloads Ou authos a amon th 54 Counts dlvd to TOP % most td sntsts.% Contbutos fom top 500 unvsts Slton of ou books ndxd n th Book Ctaton Indx n Wb of Sn Co Collton (BKCI) Intstd n publshn wth us? Contat book.dpatmnt@nthopn.om Numbs dsplayd abov a basd on latst data olltd. Fo mo nfomaton vst

2 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 4 Lodana Tanas and Stfan Balnt Wst Unvsty of Tmsoaa Romana. Intoduton. Cystal owth fom th mlt by E.F.G. thnqu Modn nnn dos not only nd ystals of abtay shaps but also plat, od and tub-shapd ystals,.., ystals of shaps that allow th us as fnal poduts wthout addtonal mahnn. Thfo, th owth of ystals of spfd szs and shaps wth ontolld dft and mputy stutus a qud. In th as of ystals own fom th mlt, ths poblm appas to b solvd by pofld-ontan ystallzaton as n th as of astn. Howv, ths soluton s not always possbl, fo xampl own vy thn plat-shapd ystals fom th mlt (to say nothn of mo omplatd shaps), xluds ontan applaton ompltly.[ Tatahnko, 99] Th thnqus whh allow th shapn of th latal ystal sufa wthout ontat wth th ontan walls a appopat fo th abov pupos. In th as of ths thnqus th shaps and th dmnsons of th own ystals a ontolld by th ntfa and mnsus-shapn apllay fo and by th hat- and mass-xhan ondtons n th ystal-mlt systm. Th d-dfnd flm-fd owth (EFG) thnqu s of ths typ. Whnv th E.F.G. thnqu s mployd, a shapn dv s usd (F. ). In th dv a apllay hannl s manufatud (F. ) n whh th mlt ass and fds th owth poss. Fquntly, a wttabl sold body s usd to as th mlt olumn abov th shap, wh a thn flm s fomd. Whn a wttabl body s n ontat wth th mlt, an qulbum lqud olumn mban th sufa of th body s fomd. Th olumn fomaton s ausd by th apllay fos bn psnt. Suh lqud onfuaton s usually alld a mnsus (F. ) and n th E.F.G. thnqu, ts low bounday (F. pont C) s attahd to th shap d of th shap. Lt b th tmpatu of th mnsus upp hozontal ston (F. AB ) th tmpatu of th lqud ystallzaton. So, abov th plan of ths ston, th mlt tansfoms n sold phas. Now st th lqud phas nto upwad moton wth th onstant at, v, kpn th poston of th phas-tanston plan nvaabl by slton of th hat ondtons. Whn th moton stats, th ystallzd poston of th mnsus wll

3 9 Cystallzaton Sn and Thnoloy ontnuously fom a sold upwad o downwad tapn body. In th patula as whn th ln tannt at th tpl pont B to th lqud mnsus sufa maks a spf anl (anl of owth) wth th vtal, th latal wall of th ystal wll b vtal. Thus, th ntal body, alld th sd, svs to fom a mnsus whh lat on dtmns th fom of th ystallzd podut, th phas tanston poston bn fxd. z nn as flow out as flow mlt/sold ntfa nn f sufa mnsus mlt shap 0 z () A h h B z () tub out f sufa C H H ff apllay hannl R ubl mlt R F.. Pototyp tubula ystal owth by E.F.G. mthod Basd on ths dspton, a onluson an b dawn that th dmnsons and shaps of th spmns bn pulld by th E.F.G. thnqu dpnd upon th follown fatos: () th shap omty; () th pssu of fdn th mlt to th shap; () th ystallzaton font poston; (v) th sd s shap. Th sd s shap s only mpotant fo statonay pulln; n ths as ts oss-ston should ond wth th dsd podut s ossston. Fquntly, spally whn omplatd pofls a own, th pulln poss s ad out und unstatonay ondtons by lown th ystallzaton sufa, whh thn nhans th dpndn of th shaps on th ystal oss-ston. Wth suh an appoah appld to th pulln poss, th dmnsons and th shap of th own ystal a dtmnd by th abov-mntond fatos and by th pulln at-to-ystallzaton font dsplamnt ato.[tatahnko, 99].. Bakound hstoy of tub owth fom th mlt by E.F.G. mthod Th thnoloy of own tubs an hav a snfant mpat fo xampl on th sola ll thnoloy. Th owth of slon tubs by E.F.G. poss was fst potd by Es t al. 4 [Es t al.,980]. Tubs w own wth a damt of 95 0 [m], wall thknss n th

4 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 9 5 an of 50 0 [m] at ats up to 0 [m/s]. In [Es t al.,980] a thoy of tub owth by th E.F.G. poss s dvlopd to show th dpndn of th tub wall thknss on th owth vaabls. Th thoy onns th alulatons of th shap of th lqud-vapo ntfa (o mnsus) and of th hat flow n th systm. Th nn and out mnsus shaps, (F.), a both alulatd fom Lapla s apllay quaton, n whh th pssu dffn Δp aoss a pont on mnsus s onsdd to b Δp = ρ H ff = onstant, wh H ff psnts th fftv hht of th owth ntfa abov th hozontal lqud lvl n th ubl (F.). Aodn to [Suk t al.,977], [Swatz t al., 975], t nluds th ffts of th vsous flow of th mlt n th shap apllay and n th mnsus flm, as wll as that of th hydostat had. Th abov appoxmaton fo Δp s vald fo slon bbon owth [Suk t al.,977], [Kaljs t al., 990], whn H ff >> h, wh h s th hht of th owth ntfa abov th shap top (.. th mnsus hht). Anoth appoxmaton usd n [Es t al.,980], onnn th mnsus, s that th nn and out mnsus shaps a appoxmatd by ula smnts. Wth ths latvly tht tolans onnn th mns n onjunton wth th hat flow alulaton n th systm, th pdtv modl dvlopd n [Es t al.,980] has bn shown to b a usful tool n undstandn th fasbl lmts of wall thknss ontol. A mo ps pdtv modl would qu an nas of th aptabl tolan an ntodud by appoxmaton. Lat, ths poss was sald up by Kaljs t al. [Kaljs t al., 990] to ow 5 0 [m] damt slon tubs, and th stss bhavo n th own tub was nvstatd. It has bn alzd that numal nvstatons a nssay fo th mpovmnt of th thnoloy. Sn th owth systm onssts of a small d tp ( 0 m wdth) and a thn 6 tub (od of 00 0 [m] wall thknss) th wdth of th mlt/sold ntfa and mnsus a aodnly vy small. Thfo, t s ssntal to obtan an auat soluton fo th tmpatu and ntfa poston n ths tny on. In [Rajndan t al., 99] an axsymmt fnt lmnt modl of mant and thmal fld was psntd fo an ndutvly hatd funa. Lat th sam modl was usd to dtmn th tal paamts ontolln slon abd pptaton on th d wall [Rajndan t al., 994]. Rajndan t al. also dvlopd a th dmnsonal mant nduton modl fo an otaonal E.F.G, systm. Rntly, n [Roy t al., 000a], [Roy t al., 000b], a n numal modl fo an ndutvly hatd la damt S tub owth systm was potd. In [Sun t al., 004] a numal modl basd on mult-blok mthod and mult-d thnqu s dvlopd fo nduton hatn and thmal tanspot n an E.F.G. systm. Th modl s appld to nvstat th owth of la otaon slon tubs of up to 50 0 m damt. A D dynam stss modl fo th owth of hollow slon polyons s potd n [Bhnkn t al., 005]. In [Makntosh t al., 006] th hallns fxd n bnn E.F.G. thnoloy nto la-sal manufatun, and onon dvlopmnt of funa dsns fo owth of tubs fo la waf poduton usn hxaons wth 0 m fa wdths, and wall thknsss n h an m s dsbd. In [Kasjanow t al., 00] th authos psnt a D oupld ltomant and thmal modln of E.F.G. slon tub owth, sussfully valdatd by xpmntal tsts wth ndustal nstallatons.

5 94 Cystallzaton Sn and Thnoloy Th stat of th at at onnn th alulaton of th mnsus shap n nal n th as of th owth by E.F.G. mthod s summazd n [Tatahnko, 99]. Aodn to [Tatahnko, 99], fo th nal quaton dsbn th sufa of a lqud mnsus possssn axal symmty, th s no omplt analyss and soluton. Fo th nal quaton only numal ntaton was ad out fo a numb of poss paamt valus that a of patal ntst at th momnt. Th authos of paps [Boodn&Boodn&Sdoov&Ptkov, 999],[Boodn&Boodn&Zhdanov, 999] onsd automatd ystal owth posss basd on wht snsos and omputs. Thy v an xpsson fo th wht of th mnsus, ontatd wth a ystal and shap of abtay shap, n whh th a two tms latd to th hydodynam fato. In [Rosolnko t al., 00] t s shown that th hydodynam fato s too small to b onsdd n th automatd ystal owth and t s not la what quaton (of non Lapla typ) was onsdd fo th mnsus sufa. Fnally, n [Yan t al., 006] th authos psnt thotal and numal study of mnsus dynams und symmt and asymmt onfuatons. A mnsus dynams modl s dvlopd to onsd mnsus shap and ts dynams, hat and mass tansf aound th d-top and mnsus. Analyss vals th olatons btwn tub thknss, fftv mlt hht, pull-at, d-top tmpatu and ystal nvonmntal tmpatu. Th pupos of ths hapt s th mathmatal dspton of th owth poss of a snl ystal ylndal tub own by th d-dfnd flm-fd owth (EFG) thnqu. Th mathmatal modl dfnd by a st of th dffntal quatons ovnn th voluton of th out adus and th nn adus of th tub and of th ystallzaton font lvl s th on onsdd n [Tatahnko, 99]. Ths systm ontans two funtons whh psnt th anl mad by th tannt ln to th out (nn) mnsus sufa at th th-phas pont wth th hozontal. Th mnsus sufa s dsbd mathmatally by th soluton of th ax-symmt Youn-Lapla dffntal quaton. Th analyss of th dpndn of solutons of th Youn-Lapla dffntal quaton on th pssu dffn aoss th f sufa, vals nssay o suffnt ondtons fo th xstn of solutons whh psnt onvx o onav out o nn f sufas of a mnsus. Ths ondtons a xpssd n tms of nqualts whh a usd fo th ho of th pssu dffn, n od to obtan a snl-ystal ylndal tub wth spfd szs. A numal podu fo dtmnn th funtons appan n th systm of dffntal quatons ovnn th voluton s psntd. Fnally, a podu s psntd fo sttn th pulln at, apllay and thmal ondtons to ow a ylndal tub wth po stablshd nn and out adus. Th ht hand tms of th systm of dffntal quatons sv as tools fo sttn th abov paamts. At th nd a numal smulaton of th owth poss s psntd. Th sults psntd n ths hapt w obtand by th authos and hav nv bn nludd n a book onnn ths top. Sn th alulus and smulaton n ths modl an b mad by a P.C., th nfomaton obtand n ths way s lss xpssv than an xpmnt and an b usful fo xpmnt plann.

6 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 95. Th systm of dffntal quatons whh ovns th voluton of th tub s nn adus, out adus and th lvl of th ystallzaton font h Aodn to [Tatahnko, 99] th systm of dffntal quatons whh ovns th voluton of th tub s nn adus, th out adus and th lvl of th ystallzaton font h s: d = vtan (, h, p) dt d = vtan (, h, p) dt dh = v G(,, h) G(,, h) dt () In quatons () and () : v s th pulln at, (, h, p ) ( (, h, p ) ) s th anl btwn th tannt ln to th out (nn) mnsus at th th phas pont of oodnats (, h ) ( (, h ) ) and th hozontal O axs (F. b), s th owth anl (F. ), p ( p ) s th ontollabl pat of th pssu dffn aoss th f sufa vn by: m m p = p p H ( p = p p H ) () wh p m s th hydodynam pssu n th mlt und th f sufa, whh an b nltd n nal, wth spt to th hydostat pssu H ( H ); p ( p ) s th pssu of th as flow, ntodud n od to las th hat fom th out (nn) wall of th tub; H ( H ) s th mlt olumn hht btwn th hozontal ubl mlt lvl and th shap out (nn) top lvl (F. a); s th mlt dnsty; s th avty alaton. Th anl (, h, p ) ( (, h, p ) ) flutuats du to th flutuatons of: th out (nn) adus ( ), th lvl h of th ystallzaton font and th out (nn) pssu p ( p ) F.. Flutuatons at th tpl pont

7 96 Cystallzaton Sn and Thnoloy In th quaton () : s th latnt mltn hat;, a th thmal ondutvty j j offnts n th mlt and th ystal sptvly; G, G a th tmpatu adnts at th ntfa n th mlt (=) and n th ystal (=) sptvly, vn by th fomulas: j v k δ h G(,,h)= T0 T n( 0) β j SINH(β j h) (F j ) B χ v k T m T n( 0) k h δ SINH(β j h) β j COSH(β j h) k (F j ) B χ j v k G (,, h)= Tm Tn(0) k h SINH( j L) ( F ) j B v k L SINH( j L) j COSH( j L) j k ( Fj ) B () (4) wh - th thmal dffusvty offnt qual to apaty, B - th Bot numb qual to, - th dnsty, - th hat k ( = - th mlt, = - th ystal), - th offnt of th hat-xhan wth nvonmnt ( k - th onvtv hat-xhan offnt and - th lnazd adaton hat-xhan offnt), F ( j,, ) th ystal (mnsus) oss ston pmt to ts aa ato: j and F =, fo a thk-walld tub wth small nn adus, fo whh hat s movd fom th xtnal sufa only, j and F = fo a tub of not to la nn adus fo whh hat s movd fom th xtnal sufa only, j and F = j fo a tub fo whh hat s movd fom both th out and nn sufas([tatahnko, 99], pp. 9-40, 46). T 0 - th mlt tmpatu at th mnsus bass, T m - mltn tmpatu, T n(0) - th nvonmnt tmpatu at z = 0, k - th vtal tmpatu adnt n th funa, - th out adus of th tub qual to th upp adus of th out mnsus, - th nn adus of th tub qual to th upp adus of th nn mnsus, L - th tub lnth and v v =, j = ( F ), =,, =,, j B j, SINH and COSH a th hypbol sn 4 and hypbol osn funtons. Du to th supooln n ths adnts t s assumd that: T0 T m, k 0, Tn(0) T m.

8 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 97 In th follown stons w wll show n whh way (, h, p ) and (, h, p ) an b found statn fom th Youn-Lapla quaton of a apllay sufa n qulbum.. Th ho of th pssu of th as flow and th mlt lvl n slon tub owth In a snl ystal tub owth by d-dfnd flm-fd owth (E.F.G.) thnqu, n hydostat appoxmaton, th f sufa of a stat mnsus s dsbd by th Youn- Lapla apllay quaton [Fnn, 986]: z p R R (5) H γ s th mlt sufa tnson, ρ dnots th mlt dnsty, s th avty alaton, / R,/ R dnot th man nomal uvatus of th f sufa at a pont M of th f sufa, z s th oodnat of M wth spt to th Oz axs, dtd vtally upwads, p s th pssu dffn aoss th f sufa. To alulat th out and nn f sufa shap of th stat mnsus t s onvnnt to mploy th Youn-Lapla q.(5) n ts dffntal fom. Ths fom of th q.(5) an b obtand as a nssay ondton fo th mnmum of th f ny of th mlt olumn [Fnn, 986].Fo a tub of out adus R R R R, R and nn adus R,, th ax-symmt dffntal quaton of th out f sufa s vn by: zp z" z' z' z' fo [, R ] (6) whh s th Eul quaton fo th ny funtonal R I z z' z p,, z d z z z R z ( R ) 0 (7) Th ax-symmt dffntal quaton of th nn f sufa s vn by: zp z" z' z' z' fo R, (8) whh s th Eul quaton fo th ny funtonal: I z z' z p zd, zr ( ) 0, z R z z (9) R

9 p p p p 98 Cystallzaton Sn and Thnoloy In paps [Balnt & Balnt, 009b], [Balnt&Balnt&Tanas, 008], [Balnt & Tanas, 008], Balnt, Tanas, 0] som mathmatal thoms and oollas hav bn oously povn adn th xstn of an appopat mnsus. Ths sults a psntd n Appndxs. In th follown w wll shown n whh way th nqualts an b usd fo aton of th appopat mnsus.. Convx f sufa aton In ths ston, t wll b shown n whh way th nqualts psntd n Appndx an b usd fo th aton of an appopat stat onvx mnsus by th ho of p and p [Balnt, Tanas, 0]. Inqualts (A..) stablsh th an wh th pssu dffn p has to b hosn n od to obtan a stat mnsus wth onvx out f sufa, appopat fo th owth of a tub of out adus qual to R n. If th pssu dffn satsfs (A..), thn a stat mnsus wth onvx out f sufa s obtand whh s appopat fo th owth of a tub of out adus R n, R. If th pssu dffn satsfs nqualty (A..4) and th valu of p s los to th valu of th ht hand mmb of th nqualty (A..4) thn a stat mnsus wth onvx out f sufa s obtand whh s appopat fo th owth of a tub of out adus R R qual to. If th pssu dffn satsfs nqualty (A..5), thn a stat mnsus wth onvx out f sufa s obtand whh s appopat fo th owth of a tub of out adus n th an R n R,. n Thom 5 (Appndx ) shows that a stat mnsus havn a onvx out f sufa, appopat fo th owth of a tub of out adus stuatd n th an R n stabl. R,, s n Inqualts (A..6) stablsh th an wh th pssu dffn p has to b hosn n od to obtan a stat mnsus wth onvx nn f sufa appopat fo th owth of a tub of nn adus qual to m R. If th pssu dffn p satsfs (A..7) thn a stat mnsus wth onvx nn f sufa s obtand whh s appopat fo th owth of a tub of nn adus m R.

10 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 99 If th pssu dffn p satsfs th nqualty (A..9) and th valu of p s los to th valu of th ht hand tm of th nqualty (A..9) thn a stat mnsus wth onvx nn f sufa s obtand whh s appopat fo th owth of a tub of nn adus R R qual to. If th pssu dffn p satsfs nqualty (A..0) thn a stat mnsus wth onvx nn f sufa s obtand whh s appopat fo th owth of a tub of nn adus whh s n th an m' R, m R. Thom 0 (Appndx ) shows that a stat mnsus havn a ovx nn f sufa appopat fo th owth of a tub of nn adus stuatd n th an s stabl. Fo numal llustatons, th nn adus of th shap was takn out adus of th shap was hosn R R R R, 4. 0 R [m] and [m] [Ess, 980]. Computatons w pfomd n MathCAD 4. and fo S th follown numal valus w onsdd: =.5 0 [k/m ]; = 7. 0 [N/m]; =0 o ; = o ; =9.8[m/s ]. To at a onvx mnsus appopat fo th owth of a tub havn th out adus qual to [m] ( n.06 ), aodn to th Thom (Appndx ), p has to b hosn n th an: , 6.5 [Pa]. Aodn to th Coollay (Appndx ), fom ths an fo th valus of p small than 70.5 [Pa] th pont wh ' ( ) tan z s los to [m]. Hn, w hav to fnd fo p 70.5 [Pa] th pont fo whh th abov ondton s satsfd. Ths an b mad by ntatn numally th follown systm fo zr ( ) 0, ( R ) and p 70.5 [Pa] (s F. ): dz tan d d p z tan d os (0) Sn th obtand s [m], and t s small than th dsd valu [m], th valu of p has to b hosn n th an , 70.5 [Pa]. Th sults of th ntatons of th systm (0) fo zr ( ) 0, ( R ) and dffnt valus of p n ths an, a psntd n F. 4. Ths fu shows that th out adus [m] s obtand fo p 98 [Pa].

11 00 Cystallzaton Sn and Thnoloy F.. Th sults of th ntaton of systms (0) and () fo p 70.5 p [Pa] [Pa] and Takn p m 0 [Ess t al., 980], [Rossolnko t al., 00], [Yan t al., 006], th mlt olumn hht n ths as s H 98 p, wh p 0 s th pssu of th as flow (ntodud n th funa fo las, th hat fom th out wall of th tub). Whn p H 0, thn H [m].. th shap s out top lvl has to b wth [m] abov th ubl mlt lvl. F. 4. Th tub out adus and nn adus vsus p and p To at a onvx mnsus appopat fo th owth of a tub havn th nn adus 4.50 [m] ( m.057 ), aodn to th Thom 6 (Appndx ), p has to b hosn n th an: 7., [Pa]. Aodn to th Coollay 8 (Appndx ), fom ths an fo th valus of p small than [Pa], th pont wh th

12 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 0 ondton ( ) tan z s satsfd s los to [m]. Thfo, w hav to fnd now fo p [Pa] th pont wh th abov ondton s ahvd. Ths an b mad by ntatn numally th systm: dz tan d d z p tan d os () fo zr ( ) 0, ( R ) and p [Pa]. (s F. ). Sn th obtand s [m] and t s hh than th dsd valu [m], w hav to hoos th valu of p n th an [Pa]. Th sults of th ntatons of th systm () fo valus of p n ths an a psntd n F. 4. zr ( ) 0, ( R ) and fo dffnt Ths fu shows that th nn adus 4.50 [m] s obtand fo p 44 [Pa]. Takn p m 0 [Ess t al., 980], [Rossolnko t al., 00], [Yan t al., 006], th mlt olumn hht s H 44 p, wh p 0 s th pssu of th as flow (ntodud n th funa fo lasn th hat fom th nn wall of th tub). Whn p 0, thn H [m],.. th shap s nn top lvl has to b wth H [m] abov th ubl mlt lvl. Whn p 44, thn th ubl mlt lvl has to b abov th shap s nn top lvl. H s natv,.. To at a onvx mnsus appopat fo th owth of a tub wth th out adus [m] and nn adus 4.50 [m], whn th shap s nn top s at th sam lvl as th shap s out top, w hav to tak: 98 p 44 p It follows that th pssu of th as flow, ntodud n th funa fo lasn th hat fom th nn wall of th tub has to b hh than th pssu of th as flow, ntodud n th funa fo lasn th hat fom th out wall of th tub and w hav to tak: p p 6 [Pa]... Conav f sufa aton In ths ston, t wll b shown n whh way th nqualts psntd n Appndx an b usd fo th aton of an appopat stat onav mnsus by th ho of p and p [Balnt&Balnt, 009a].

13 p 0 Cystallzaton Sn and Thnoloy Inqualts (A..) stablsh th an n whh th pssu dffn p has to b hosn n od to obtan a stat mnsus wth onav out f sufa appopat fo th owth of a tub of out adus qual to R n. If th pssu dffn satsfs nqualty (A..) thn a stat mnsus wth onav out f sufa s obtand whh s appopat fo th owth of a tub of out adus n th an R n R,. n Thom (Appndx ) shows that a stat mnsus havn a onav out f sufa R R appopat fo th owth of a tub of out adus, R s stabl. Inqualts (A..) stablsh th an n whh th pssu dffn p has to b hosn n od to obtan a stat mnsus wth onvx nn f sufa appopat fo th owth of a tub of nn adus qual to m R. If th pssu dffn p satsfs nqualty (A..4) thn a stat mnsus wth onav nn f sufa s obtand whh s appopat fo th owth of a tub of nn adus n th an m' R, m R. Thom 6 (Appndx ) shows that a stat mnsus havn a onav nn f sufa appopat fo th owth of a tub of nn adus R R R, s stabl. Computatons w pfomd fo an InSb tub owth: 658 / k m ; 4. 0 / N m ; 8.9 ; If th xsts a onav out f sufa, appopat fo th owth of a tub of out adus [ m ] ( n.06 ), thn aodn to th Thom (Appndx ) ths an b obtand fo a valu of p n th an (4.85; 64.49)[ Pa ]. Takn nto aount th abov fat, n od to at a onav out f sufa, appopat fo th owth of a tub of whh out adus s qual to [ m ] w hav solvd th.v.p. (A..) fo dffnt valus of p n th an (4.85; 64.49)[ Pa ]. Mo psly, w hav ntatd th systm (0) fo dffnt z ( R ) 0, z '( R ) tan and p. Th obtand out ad vsus p a psntd n F.5, whh shows that th dsd out adus [ m] s obtand fo p ' 49.7[ Pa ].

14 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 0 F. 5. Out ad vsus p n th an (4.85; 64.49)[ Pa ]. Atually, as t an b sn n th sam fu, fo p ' 49.7[ Pa ] w an also obtan a R R sond out adus.8 0 [ m], whh s not n th dsd an, R. Moov, th out f sufa of ths mnsus s not lobally onav; t s a onvxonav mnsus (F.6). Takn nto aount p m 0 [Ess, 980], [Rossolnko, 00], [Yan, 006], th mlt olumn hht n ths as s H' [ p' p ], wh p 0 s th pssu of th as flow (ntodud n th funa fo lasn th hat fom th out sd of th tub wall). Whn p 0, thn H ' s natv, H '. 0 [ m],.. th ubl mlt lvl has to b wth H '.0 [ m ] abov th shap top lvl. F. 6. Non lobally onav out f sufa obtand fo p ' 49.7[ Pa ].

15 04 Cystallzaton Sn and Thnoloy If th xsts a onav nn f sufa fo th owth of a tub of nn adus ' 4.50 [ m ]( m.057 ), thn aodn to th Thom 4 (Appndx ), ths an b obtand fo a valu of p whh s n th an.46,.07 [ Pa ]. Takn nto aount th abov fat, n od to at a onav nn f sufa, appopat fo th owth of a tub whos nn adus s qual to ' 4.50 [ m ], w hav solvd th.v.p. (A..8) fo dffnt valus of p n th an.46,.07 [ Pa ]. Mo psly, w hav ntatd th systm () fo dffnt z ( R ) 0, z '( R ) tan and p. Th obtand nn ad vsus p a psntd n F.7 whh shows that th dsd nn adus [ m ] s obtand fo p ' 6.[ Pa ]. F. 7. Inn ad vsus p n th an.46,.07 [ Pa ]. Takn p m 0 olumn hht n ths as s [Ess t al., 980], [Rossolnko t al., 00], [Yan t al. 006], th mlt H' [ p' p ], wh p 0 s th pssu of th as flow (ntodud n th funa fo lasn th hat fom th nn sd of th tub wall). Whn p 0, thn H ' s postv, wth H H ' 0.50 [ m ],.. th ubl mlt lvl has to b ' 0.50 [ m ] und th shap top lvl. To at a onav mnsus, appopat fo th owth of a tub wth out adus b ' 4.50 H ' [m] and nn adus [m] th mlt olumn hhts (wth spt to th ubl mlt lvl) hav to 6. p and H 49.7 p. Whn th shap s out top s at th sam lvl as th shap s nn top, wth spt to th ubl mlt lvl, thn th laton H H holds. It follows that th pssu of th as flow, ntodud n th

16 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 05 funa fo lasn th hat fom th nn wall of th tub, p has to b hh than th pssu of th as flow ntodud n th funa fo lasn th hat fom th out wall of th tub, p ; p p [ Pa ]. 4. Th anls (, h, p) and (, h, p ) whh appa n systm () dsbn th dynams of th out and nn adus of a tub, own by th E.F.G. thnqu 4. Th podu fo th dtmnaton of th anls (, h, p) and (, h, p ) fo a onvx f sufa Th anls (, h, p ) ( (, h, p ) ) psnt th anls btwn th tannt ln to th out f sufa (nn f sufa) of th mnsus at th th phas pont, of oodnats, h, h and th hozontal axs O. Ths anls an flutuat dun th owth. Th dvaton of th tannt to th ystal out (nn) f sufa at th tpl pont fom th vtal s th dffn (, h, p ), (, h, p ) (F. ), wh s th owth anl. Th dvaton an flutuat also and th out (nn) adus ( ) s onstant whn th dvaton s onstant qual to zo Th anls (, h, p ) and (, h, p ) annot b obtand dtly fom th Youn- Lapla quaton. Fo ths ason fo ths quaton th follown staty s adoptd: two ondtons a mposd at th out adus R of th shap (nn adus R of th shap) ( z ( R ) 0; ' z ( R ) tan( ) ( z ( R ) 0, ' z ( R ) tan( )). In th last ondton ) s a paamt whh an flutuat n a tan an, dun th owth. Fo dffnt valus of ( ) n a vn an th soluton Youn-Lapla quaton whh satsfs th ondtons R (( z( R ) 0, ' z ( R ) tan( )) at R ) s found. Wth z ( ;, p ) ( z ( ;, p )) of th z ( R ) 0; ' z ( R ) tan( ) at ' z ( ;, p ) ( z ( ;, p )) th funton ( ;, p ) atan z ( ;, p ) ' ( ( ;, p ) atan z ( ;, p ) ) s onstutd. Aft that, fom h z ( ;, p ) ( h z ( ;, p )) ( ) s xpssd as funton of, h and p (, h and p ). ( ; h, p ) ( ( ; h, p )) s ntodud n ( ;, p ) ( ( ;, p )) obtann th funton (, h, p ) ( ; ( ; h, p ), p ) ( (, h, p ) ( ; ( ; h, p ), p )). To th bst of ou knowld, th s no alothm n th ltatu onnn th onstuton of (, h, p ) ( (, h, p ) ) at th lvl of nalty psntd h.

17 06 Cystallzaton Sn and Thnoloy Du to th nonlnaty, th abov dsbd podu an t b alzd analytally. Ths s th ason why fo th onstuton of th funton ( ;, p ) n [Balnt&Tanas, 00] th follown numal podu was onvd: Stp. Fo a vn 0 ; 0 0, 0 0 that,, and E n E n wh: R n an n, n s found suh R R 0 0 n n E n, sn R tan R n n n os R 0 n E ( n, ) os sn R n R () Stp. Fo a an 0, s dtmnd suh that 0 and fo Stp. Fo s onsdd. vy, th nqualty,, p th an p, p dfnd by: E n E n holds. sup, nf,,, p E n p E n () Stp 4. In th an, a st of l dffnt valus of s hosn. Stp 5. In th an p, p a st of m dffnt valus of p s hosn. Stp 6. In a vn an, possssn th popty j st of j valus of s hosn:. Stp 7. Fo a vn k a p, k, m and q, q, l th soluton of th systm (0) q ospondn to th ondtons: z ( R ) 0, ( R ) s dtmnd numally q k obtann th funtons (pofls uvs Rfs [Tatahnko, 99]): z z ( ;, p ) q k and ( ;, p ). Stp 8. Th valus q k s k q s fo whh ( ;, ) k q s p,, k, m, q, l a dtmnd.

18 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 07 Stp 9. Th valus Stp 0. Fttn th data q k k q s kq s h z ( ;, p ) a found. k q s h k q s k p and s, th funton,, h p s found. Fo th sam ason as n th as of (, h, p ) fo th onstuton of (, h, p ) th follown numal podu was onvd: 0 0 Stp. Fo and 0 0,, R R m, R E m E m wh: an m, m s dtmnd suh that 0 0 E m, sn R m tan m R m R sn E ( m, ) os os m R R 0 (4) Stp. Fo a an, s dtmnd suh that 0 and fo vy,, th nqualty,, Stp. Fo p th an p, p dfnd by: s onsdd. E m E m holds. sup, nf,,, p E m p E m (5) Stp 4. In th an, a st of l dffnt valus of a hosn. Stp 5. In th an p, p a st of n dffnt valus of p a hosn. Stp 6. In a vn an,, possssn th popty, a st j of j valus of a hosn:. Stp 7. Fo a vn p k, k, n and q, q, l th soluton of th systm () whh satsfs th ondtons: z ( R ) 0, ( R ) s found numally obtann th q q k funtons (pofls uvs Rf. [Tatahnko, 99]): z z ( ;, p ) and q k ( ;, p ).

19 08 Cystallzaton Sn and Thnoloy Stp 8. Th valus q k s k q s fo whh ( ;, ) k qs p,, k, n, q, l a dtmnd. Stp 9. Th valus q k k q s kq s Stp 0. Th funton,, h z ( ;, p ) a found. h p s found by fttn th data k q s Fo th as of a slon tub and th out f sufa th funton:, h, p h k q s k p and s. a p a p h a p h a p h a( p ) a p a p a4 p h a5 p h a6 p h wth: a ( p ) p a ( p ) p a ( p ) p a ( p ) p 4 a ( p ) p a ( p ) p 5 6 a ( p ) p a ( p ) p 7 8 a ( p ) p a ( p ) p 9 was obtand and fo th nn f sufa th funton: ; h, p 0 b p b p h b p h b p h b( p ) b p b p b4 p h b5 p h b6 p h wth b ( p ) p b ( p ) p b ( p ) p b ( p ) p 4 b ( p ) p b ( p ) p 5 6 b ( p ) p b ( p ) p 7 8 b ( p ) p b ( p ) p 9 was obtand. Fo p 000Pa and 4 (, h, p ) a psntd n F. 8.: 0 p Pa th funtons (, h, p ) and F. 8. Th aphs of (, h, p ) ( p 000 [ Pa ] ) and (, h, p ) ( p 4 [ Pa ] )

20 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu Th podu fo th dtmnaton of th anls (, h, p) and (, h, p ) fo a onav f sufa Th numal podu fo th onstuton of th funton (, h, p) fo a onav f sufa (whn ) s smla to thos appld fo a onvx f sufa. Only Stp - Stp. psnt th som dffns. Fo th out f sufa w hav to onsd: Stp. Fo a vn 0 ; 0, that 0 0,, and E n E n wh: R n an n, n s found suh R R 0 n n E n, sn R tan nsn R n n R n 0 E ( n, ) os os R n R (6) Stp. Fo a an, s dtmnd suh that 0 and sup E n, nf E n, holds. Stp. Fo s onsdd. th nqualty,, p th an p, p dfnd by: sup, nf,,, p E n p E n (7) Fo th nn f sufa w hav to mak: Stp. Fo 0 0 R R and m, R 0 E m, E m ', 0 wh: an m, m s dtmnd suh that E m, sn R mtan os ( m) R mr 0 E ( m', ) os sn m' R R Stp. Fo a an, s dtmnd suh that 0 sup E m, nf E m ', holds. th nqualty,, (8) and

21 0 Cystallzaton Sn and Thnoloy Stp. Fo p th an p, p dfnd by: s onsdd. sup, nf ',,, p E m p E m (9) In th as of of th InSb tub onsdd n ston., fo th out f sufa th funton:, h, p a p a p h a p h a p h a( p ) a p a p a4 p h a5 p h a6 p h wth a ( p ) = p a ( p ) = p a ( p ) = p a ( p ) = p a ( p ) = p a ( p ) = p 5 a ( p ) = p a ( p ) = p 7 a ( p ) = p a ( p ) = p 9 was obtand. Fo th nn f sufa th funton: ; h, p b( p ) b p b p h b4 p h b5 p h b p b p b p h b p h wth b ( p ) p b ( p ) p b ( p ) p b ( p ) p 4 b ( p ) p b ( p ) p 5 6 b ( p ) p b ( p ) p 7 b ( p ) p 9 was obtand. Fo p 80Pa and 90 h p a psntd n F. 9: (,, ) 8 p Pa th funtons (, h, p ) and F. 9. Th aphs of (,, ) h p ( 90 p Pa ) and (,, ) h p ( 80 p Pa )

22 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 5. Sttn th pulln at, th thmal and apllay ondtons In ths ston t wll b shown that th sults psntd n th abov stons an b usd fo sttn th pulln at, th thmal and apllay ondtons n vw of an xpmnt [Tanas&Balnt, 00]. Aodn to [Tatahnko, 99] at th lvl of th ystallzaton font h th ystallzaton at v s vn by: j j j j v = G(,, h) G(,, h), j =,,. (0) Th dffn btwn th pulln at v and th ystallzaton at v j s qual to th ystallzaton font dsplamnt at j dh dt, j =,,. j In od to kp th ystallzaton font lvl h onstant, th pulln at and th thmal ondtons hav to satsfy th follown ondtons: j j v G(,, h) G(,, h) = 0, j =,, () Whn th ad, and th lnth L of th tub, whh has to b own, a po vn, and h s known, thn th ondton () an b add as an quaton n whh th pulln at v s unknown. If ths quaton has a postv soluton v, t dpnds on th follown paamts: h, T n(0), T 0, and k. Th sttn of th pulln at, thmal ondtons mans th ho of v, T n(0), T 0 and k suh that th follown ondtons b satsfd: 00 < n(0) < m < 0 T T T ; T 0< < n(0) 00 k ; L quaton () has a postv soluton v n an aptabl an. v s patally th sam fo vy L : L0 L L ( L 0 = th sd lnth). Th sttn of th apllay ondton mans to tak th tub ad, (po vn) and th ystallzaton font lvl h, dtmnd fom () (fo th abov hosn v, T n(0), T 0, k ) and fnd th pssus p, p solvn th followns quatons: If th solutons p, (, h, p )= and (, h, p )= () p of ths quatons a n th an fo whh, was buld up, thn th valus p, p wll b usd to st p, p, H, H usn () wth p =0 o m H p p =, p p H = ()

23 Cystallzaton Sn and Thnoloy Fo th owth of a slon tub wth onvx pofl uvs th follown numal data wll b usd: =.5 0 [k/m ]; =. 0 [k/m ]; T m = 68 [K]; =60[W/m K]; =.6 6 [W/m K]; =.80 [J/k]; = ; = ; = 9 [J/k K]; = 70 [J/k K]; = [K]; = 8.58 [K]; R =4.0 [m]; R =4.80 [m]; R = [m]; R =4.660 [m]; L =0.4[m]; L =0.[m]; L =0.[m]. Stp. A stabl stat out mnsus s hosn, whos haatst paamts, h, a n th an wh (, h, p ) s vald and fo whh s los to. In th as onsdd h suh a stat mnsus s obtand fo p = 980 [Pa] and ts haatst paamts a: 4 p = [m] and h = [m]. Stp. An ntal nput fo T 0, T n(0) and k has to b hosn. Fo T 0, th stat an b T0 = T. Connn T (0) and k th stat an b T (0) = T and m T = n(0) 00 k. L Usn ths nput and th valus, h, n th valu of th pulln ats v, v,, v40 vn by th quaton () hav to b found. If all ths valus a postv, thn: f th ava v and standad dvaton of th st of valus of v, a aptabl, thn th ava pulln at v and th ntal nput thmal ondtons an b st, ls th ntal nput thmal ondtons hav to b st lown n nal T n(0) and/o nasn T 0. Stp. Consd v, T n(0), T 0, k obtand abov and solv quaton () fo ths valus hoosn = and = (th dsd ad) and h unknown. Dnot by h th obtand soluton. Rpla, fndn Stp 4. Usn p, p. p, p fnd, h n quaton () and solv ths quatons H H, fo p = p (n th as of an opn ubl) o fnd p p fo H H =0 (n th as of a losd ubl). Follown th abov stps fo th onsdd slon tub owth, som of th omputd possbl sttns, a psntd n Tabl. Fo th abov sttns th owth poss stablty analyss s mad thouh th systm of nonlna odnay dffntal quatons () whh ovns th voluton of,, h fo th stablshd sttns. It mans to vfy fst of all that th dsd, and th obtand h s a stady stat of (). Futhmo to vfy f at th stat,, h a ptubd (.. th sd szs a dffnt fom, ) aft a pod of tanston th valus,, h a ovd. In oth wods, n m

24 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu to vfy f th stady stat (,, h ) s asymptotally stabl. Ths last qumnt s satsfd f th Huwtz ondtons a satsfd [Tatahnko, 99]..: a a a >0, a a a a a a >0 ( a a a )( a a a a a a a a ) ( a a a a a a )>0 (, h, p) (, h, p) (, h, p) a = v a = v 0 a = v R R h (, h, p) (, h, p) (, h, p) a = v =0 a = v a = v R R h S (,, h) S (,, h) S (,, h ) a = a = a = R R h S(,, h)= v G(,, h) G(,, h) (4) (5) v F F F F F F H H T p p T n (0) h k p p Tabl. Possbl sttns of v, T 0, T n(0), p, p, p p a a a a a a a F F F Tabl. Th offnts aj of th lnazd systm n th stady stats Th valus of th numbs a j n th onsdd ass a vn n Tabl. It s asy to vfy that n all ass th Huwtz ondton a satsfd. In F. 0 smulatons of th slon tub owth s psntd whn th sd lnth s 0 and th ad of th sd a R = [m]; R = [m]. Th mnsus 4 hht at th stat s h = [m].

25 4 Cystallzaton Sn and Thnoloy F. 0. Th voluton of th out adus, th nn adus of th tub and th mnsus hht obtand ntatn numally th systm () fo F =/ R, 4 v [m/s], T 0 = 74.8 [K] and T (0) = 400[ K ] n 6. Conlusons Known th matal onstants (dnsty, hat ondutvty, t), th sz of th snl ystal tub whh wll b own fom that matal, th sz of th shap whh wll b usd and th ooln as tmpatu at th ntan, t s possbl to pdt valus of pulln at, tmpatu at th mnsus bass, ooln as tmpatu at th xt, vtal tmpatu adnt n th funa, nn and out walls ooln as pssu dffns, mlt olumn hht dffns, ystallzaton font lvl, whh an b usd fo a stabl owth. Aodn to th modl th pdtd valus a not unqu.. th a sval possblty to obtan a tub wth po vn sz fom a vn matal usn th sam shap. So, vn f n ou omputaton th matal and th sz of th shap and tub s th sam as n [Ess] xpmnt, th omputd data vn n Tabl an b dffnt fom that usd n th al xpmnt. Fo ths ason ou pupos s not ly to ompa th omputd sults wth th xpmntal data. Moov w want to val that a tub of po vn sz an b obtand by dffnt sttns and th modl pmt to omput suh sttns. Th ho of a spf sttn s th patal ystal ow dson. Th modl povd possbl sttns and an b hlpful n a nw xpmnt plannn. Connn th lmts of th modl t s la that t s lmtd n applablty, as all modls. Th man lmts a thos ntodud by appoxmatons mad n quatons dfnn th modl. 7. Appndx. Inqualts fo snl ystal tub owth by E.F.G. thnqu - Convx out and nn f sufa Consd th dffntal quaton (6) fo 0. R R R,, suh that 0,,

26 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 5 Dfnton. A soluton z z x of th q. (6) dsbs th out f sufa of a stat mnsus on th ntval, R f t posssss th follown popts: zr 0, z' tan, z' R tan and z s sttly dasn on [, R ]. Th dsbd out f sufa s onvx on [, ] R f z" 0 [, R ]. Thom. If th xsts a soluton of th q. (6), whh dsbs a onvx out f R sufa of a stat mnsus on th losd ntval[, R ], thn fo n, p satsfy: n os sn p R n R n R n sn tan R n n n os R (A..) Thom. Lt b n suh that p R n. If p satsfs th nqualty: R R n os sn R n R (A..) thn th xsts R n, R suh that th soluton of th ntal valu poblm: zp R R z" z' ( z') z' fo R zr 0, ' tan z R (A..) on th ntval [, R ] dsbs th onvx out f sufa of a stat mnsus. Coollay. If fo p th follown nqualty holds: p ( ) os sn R R R (A..4) R R R R thn th xsts, R (los to ) suh that th soluton of th.v.p. () on th ntval [, R ] dsbs a onvx out f sufa of a stat mnsus.

27 6 Coollay 4. If fo R n' n th follown nqualts holds: R R Cystallzaton Sn and Thnoloy ( ) n' n' sn R tan 'os ' ' n R n n R ( ) n p os sn R n R (A..5) R R thn th xsts, suh that th soluton of th.v.p. (A..) on th ntval n n [, R ] dsbs a onvx out f sufa of a stat mnsus. Thom 5. If a soluton z z of th q. (6) dsbs a onvx out f sufa of a stat mnsus on th ntval [, R ], thn t s a wak mnmum fo th ny funtonal of th mlt olumn (7). Dfnton. A soluton z z x of th q.(8) dsbs th nn f sufa of a stat mnsus on th ntval R,, popts: z' R tan, z' tan, 0 R R R f t posssss th follown z R and on [ R, ]. Th dsbd nn f sufa s onvx on [, ] z s sttly nasn R f z" 0, [R, ]. Thom 6. If th xsts a soluton of th q. (8), whh dsbs a onvx nn f sufa of a stat mnsus on th losd ntval [ R, ] and m R wth R R m R, thn th follown nqualts hold: os os p m R R sn R m tan sn m R mr (A..6) Thom 7. Lt m b suh that R R m R. If p satsfs th nqualty: p os os m R R (A..7)

28 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 7 thn th xsts [ R, m R ], suh that th soluton of th ntal valu poblm: zp R z" z' ( z') z' fo R zr 0, ' tan z R R (A..8) on th ntval [ R, ] dsbs th onvx nn f sufa of a stat mnsus. Coollay 8. If fo p th follown nqualty holds, ( ) p os os R R R (A..9) R R R R thn th xsts R, (los to ) suh that th soluton of th.v.p. (A..8) on th ntval [ R, ] dsbs a onvx nn f sufa of a stat mnsus. Coollay 9. If fo R R m' m R th follown nqualts hold ( ) sn R m' tan sn ' m R m' R (A..0) ( ) p os os m R R thn th xsts n th ntval [ m' R, m R ] suh that th soluton of th.v.p. (A..8) on th ntval [ R, ] dsbs a onvx nn f sufa of a stat mnsus. Thom 0. If a soluton z z of th q. (8) dsbs a onvx nn f sufa of a stat mnsus on th ntval[ R, ], thn t s a wak mnmum fo th ny funtonal of th mlt olumn (9). 8. Appndx. Inqualts fo snl ystal tub owth by E.F.G. thnqu - Conav out and nn f sufa Consd th quaton (6) fo 0. R R 0< R < < R,, suh that 0,,

29 8 Cystallzaton Sn and Thnoloy Dfnton. Th out f sufa s onav on [, ] R f z" 0, [, R ]. R Thom. If th xsts a onav soluton z = z( ) of th quaton (6) thn n = and p satsfy th follown nqualts: n n os os p sn n R R n R n n tan sn n R (A..) Thom. If fo R < n < n< and p th nqualts hold: R R n n n sn R tan sn < p < n R n R n os os. n R R (A..) thn th xsts R n R, and a onav soluton of th quaton (6). n Thom. A onav soluton z( ) of th quaton (6) s a wak mnmum of th f ny funtonal of th mlt olumn (7). Consd now th dffntal quaton (8) fo that 0, 0,. R R 0< R < < < R and, suh Thom 4. If th xsts a onav soluton z = z( ) of th quaton (8) thn m = R and p satsfs th follown nqualts: os sn p sn m R R m R ( m) R tan os m R (A..) Thom 5. If fo R < m < m< and fo p th follown nqualts hold: R R

30 A Mathmatal Modl fo Snl Cystal Cylndal Tub Gowth by th Ed-Dfnd Flm-Fd Gowth (EFG) Thnqu 9 sn ( m ) R tan m R os < p < os sn mr m R R (A..4) thn th xsts n th ntval m R, m R and a onav soluton of th q. (8). Thom 6. A onav soluton z( ) of th quaton (8) s a wak mnmum of th f ny funtonal of th mlt olumn (9). 9. Rfns St. Balnt, A.M. Balnt (009), On th aton of th stabl dop-lk stat mnsus, appopat fo th owth of a snl ystal tub wth po spfd nn and out ad, Mathmatal Poblms n Ennn, vol. 009, Atl ID:4858 (009), pp - St. Balnt, A.M. Balnt (009), Inqualts fo snl ystal tub owth by d-dfnd flm-fd (E.F.G.) thnqu, Jounal of Inqualts and Applatons, vol.009, Atl ID: 706, pp.-8 St.Balnt, A.M.Balnt, L.Tanas (008) - Th fft of th pssu on th stat mnsus shap n th as of tub owth by d-dfnd flm-fd owth (E.F.G.) mthod, Jounal of Cystal Gowth, Vol. 0, pp.8-90 St. Balnt, L.Tanas (008), Nonlna bounday valu poblms fo sond od dffntal quatons dsbn onav qulbum apllay sufas, Nonlna Studs 5, Vol. 4, pp St.Balnt, L.Tanas(0), Som poblms onnn th valuaton of th shap and sz of th mnsus oun n slon tub owth - Mathmats n Ennn, Sn and Aospa Vol., pp St.Balnt, L.Tanas (00), A podu fo th dtmnaton of th anls 0 (, h; p) and 0 (, h; p) whh appas n th nonlna systm of dffntal quatons dsbn th dynams of th out and nn adus of a tub, own by th d-dfnd flm-fd owth (EFG) thnqu, Nonlna Analyss: Ral Wold Applatons, Vol. (Issu 5), pp St.Balnt, L.Tanas (0), Th ho of th pssu of th as flow and th mlt lvl n slon tub owth, Mathmats n Ennn, Sn and Aospa, Vol. 4, pp. H.Bhnkn, A.Sdl and D.Fank (005), A D dynam stss modl fo th owth of hollow slon polyons, Jounal of.cystal Gowth, Vol 75, pp A.V.Boodn, V.A.Boodn, V.V.Sdoov and I.S.Ptkov (999), Influn of owth poss paamts on wht snso adns n th Stpanov (EFG) thnqu, Jounal of.cystal Gowth, Vol. 98/99, pp.5-9. A.V.Boodn, V.A.Boodn and A.V.Zhdanov (999), Smulaton of th pssu dstbuton n th mlt fo sapph bbon owth by th Stpanov (EFG) thnqu, Jounal of.cystal Gowth, Vol. 98/99, pp.0-6. L.Es, R.W.Stomont, T.Suk, A.S.Taylo (980), Th owth of slon tubs by th EFG poss, Jounal of.cystal Gowth, Vol. 50, pp.00-.

31 0 Cystallzaton Sn and Thnoloy R. Fnn, Equlbum apllay sufas (986), Vol. 84, Gundlhn d mathmatshn Wssnshaftn, Spn, Nw Yok, NY, USA. I.P.Kaljs, A.A.Mnna, R.W.Stomont and I.W.Hudnson (990), Stss n thn hollow slon ylnds own by th d-dfnd flm-fd owth thnqu, Jounal of.cystal Gowth, Vol 04, pp.4-9. H.Kasjanow, A.Nkanoov, B.Nak, H.Bhnkn, D.Fank and A.Sdl (007), Doupld ltomant and thmal modln of EFG slon tub owth, Jounal of.cystal Gowth, Vol. 0, pp B.Makntosh, A.Sdl, M.Qulltt, B.Bathy, D.Yats and J.Kaljs (006) La slon ystal hollow-tub owth by th d-dfnd flm-fd owth (EFG) mthod, Jounal of.cystal Gowth, Vol 87, pp S.Rajndam, M.Laouss, B.R. Bathy and J.P.Kaljs (99), Slon abd ontol n th EFG systm, Jounal of.cystal Gowth, Vol. 8, pp.8-4. S.Rajndam, K.Holms and A.Mnna (994), Th-dmnsonal mant nduton modl of an otaonal d-dfnd flm-fd owth systm, Jounal of.cystal Gowth, Vol. 7(No -), pp S.N.Rossolnko (00), Mns masss and whts n Stpanov (EFG) thnqu: bbon, od, tub, Jounal of.cystal Gowth, vol., pp A.Roy, B.Makntosh, J.P.Kaljs, Q.S.Chn, H.Zhan and V.Pasad(000), A numal modl fo ndutvly hatd ylndal slon tub owth systm, Jounal of Cystal Gowth, Vol, pp A.Roy, H.Zhan, V.Pasad, B.Makntosh, M.Qulltt and J.Kaljs (000), Gowth of la damt slon tub by EFG thnqu: modln and xpmnt, Jounal of.cystal Gowth, Vol 0, pp.4-. D.Sun, Ch.Wan, H.Zhan, B.Makntosh, D.Yats and J.Kaljs (004), A mult-blok mthod and mult-d thnqu fo la damt EFG slon tub owth, Jounal of.cystal Gowth, Vol 66, pp T.Suk, B.Chalms and A.I.Mlavsky (977), Th d flm-fd owth of ontolld shap ystals, Jounal of.cystal Gowth, Vol. 4, pp J.C.Swatz, T.Suk and B.Chalms (975), Th EFG poss appld to th owth of slon bbons, J.Elton.Mat, Vol. 4, pp L.Tanas, St.Balnt (00), Modl basd, pulln at, thmal and apllay ondtons sttn fo slon tub owth, Jounal of Cystal Gowth, vol., pp V.A.Tatahnko (99), Shapd ystal owth, Kluw Aadm Publshs, Dodht. B.Yan, L.L.Zhn, B.MaKntosh, D.Yats and J.Kaljs (006), Mnsus dynams and mlt soldfaton n th EFG slon tub owth poss, Jounal of.cystal Gowth, Vol. 9, pp

32 0 Th Autho(s). Lns InthOpn. Ths s an opn ass atl dstbutd und th tms of th Catv Commons Attbuton.0 Lns, whh pmts unsttd us, dstbuton, and poduton n any mdum, povdd th onal wok s poply td.