Bridgmn growth of rystls Duits, H.A.E.; Molenr, J. Published: 01/01/1987 Doument Version Publisher s PDF, lso known s Version of Reord (inludes finl pge, issue nd volume numbers) Plese hek the doument version of this publition: A submitted mnusript is the uthor's version of the rtile upon submission nd before peer-review. There n be importnt differenes between the submitted version nd the offiil published version of reord. People interested in the reserh re dvised to ontt the uthor for the finl version of the publition, or visit the DOI to the publisher's website. The finl uthor version nd the glley proof re versions of the publition fter peer review. The finl published version fetures the finl lyout of the pper inluding the volume, issue nd pge numbers. Link to publition Cittion for published version (APA): Duits, H. A. E., & Molenr, J. (1987). Bridgmn growth of rystls. (WD report; Vol. 8709). Nijmegen: Rdboud Universiteit Nijmegen. Generl rights Copyright nd morl rights for the publitions mde essible in the publi portl re retined by the uthors nd/or other opyright owners nd it is ondition of essing publitions tht users reognise nd bide by the legl requirents ssoited with these rights. Users my downlod nd print one opy of ny publition from the publi portl for the purpose of privte study or reserh. You my not further distribute the mteril or use it for ny profit-mking tivity or ommeril gin You my freely distribute the URL identifying the publition in the publi portl? Tke down poliy If you believe tht this doument brehes opyright plese ontt us providing detils, nd we will rove ess to the work immeditely nd investigte your lim. Downlod dte: 14. Jn. 2019
WD 87-09 BRIDGMAN GROWTH OF CRYSTALS H.A.E. Duits J. Molenr ugustus 1987
BRIDGMAN' GROWTH OF CRYSTALS H.A.E. Duits J. Molenr Mthtil Consulting Deprtment Toernooiveld 6525 ED Nijmegen The Netherlnds
CONTENTS Introdution The model The pty Bridgmn-oven Results nd Disussion Conlusions Reommendtion Referenes Symbols Viewftors -- 1 --
Introdution Crystl growth by mens of the Bridgmn-oven tehnique is widely epted nd frequently pplied nowdys. However, the understnding of the physil proesses in the oven is fr from omplete yet. A lot of numeril models nd onsidertions hve been introdued but these pprohes nerly lwys tret idelised situtions, so tht the results re not diretly pplible in prtie. In this report the results of detiled lultions on the Bridgmn-oven geometry re presented. This work hs been rried out in support of rystl growth projet t the Fulty of Siene, University of Nijmegen. The lultions hve been performed in diret onnetion with the growth of Bismuth Gllium Oxide (BGO) rystls but re nerly independent of the rystl under onsidertion nd thus useful for ll kinds of Bridgmn-oven pplitions. The generl fetures of the Bridgmn-oven re s follows. An mpoule ontining BGO moves down through heter on top of ooler. Both heter nd ooler re ylindrilly shped. At the interfe diphrgm is sometimes mounted to redue the ylinder rdius lolly. The wll tperture in heter nd ooler re hosen to be higher respetively lower thn the melting tperture of BGO, so tht solidifition will tke ple somewhere t the interfe. Importnt prmeters during the soldifition proess re the vertil tperture grdient in the BGO nd the form of the liquid/solid interfe, determined by the isotherml ontours in the mpoule. The tehniques to lulte tperture profiles in relisti ovens re presented in the following 2 setions. The speed of the mpoule through the oven is suh low tht dynmi effets re negleted. Only sttionry sttes re studied. For this purpose use is mde of the pkge ESACAP, developed t E.S.A., Noordwijk, in whih the syst is treted s the nlogue of n eletril network. In the lst setion we present results nd nlyze in detil how tperture grdient nd form of the interfe re ffeted by the geometril prmeters of the oven nd the terml properties of the used mterils. From these results lot of insight is gined in the seletion of the relevnt prmeters in order to obtin speifi tperture distribution in the mpoule during the solidifition proes. Speifition of the most desirble onditions requires deep knowledge of metllurgy nd flls therefore outside the sope of this projet. -- 2 --
The model We shll use the following model to study Bridgmn growth. The oven nd the mpoule re ssumed to be onentri ylinders. Owing to the ylinder symmetry the probl is essentilly 2-dimensionl. The mpoule is 20 long with dimeter of 3. The thikness of the mpoule wll is 0.025. The mpoule is divided long its length into 2 setions nd long its dimeter into 0.5 setions. The BGO in the mpoule is lso divided long its length into 2 setions nd top nd bottom re rdilly divided into 0.5 setions. The het ondution xilly through eh setion is desribed by: q = k A DT L Here, q is the het flow, k the therml ondutivity, A the ross-setionl re, DT the tperture differene over one setion nd L the setion length. Repling q by urrent I nd T by voltge V for the eletril nlogue, this eqution beomes: I = k A DV L If we interprete this eqution s Ohm's lw ondution resistne R results, given by: R = L k A There is lso het ondution rdilly through the ylinder nd the rdil ondution resistne R is given by: [2] r R = r Here rl is the inner rdius nd r2 is the outer rdius of onentri ylinder. Free onvetion in the melt is negleted beuse it is smll for vertil growth. The movent of the mpoule tkes ple suh slow tht in the lultions the onsequene of the dynmi effets n be negleted. At eh moment the tperture distribution is ssumed to be sttionry. The oven wll is 60 long nd divided into 2 setions. It is ssumed tht rdition is only exhnged between oven wll nd mpoule wll. The tperture t the oven wll is suh high tht onvetion nd ondution in the ir gp re negleted. The rdition of one setion of the oven to setion of the mpoule is pproximtely given by the Stefn-Boltzmnn reltion: q = sea (T 4_T 4) o o -- 3 --
Here, s is the Stefn-Boltzmnn onstnt, E the viewftor, A 0 the surfe re of the oven setion, T the surfe tperture of the oven setion nd 0 T the surfe tperture of the mpoule setion. The viewftor will depend on the issivity of the oven wll nd the mpoule, on the surfe res of both setions nd on the lotion of both res. For the purpose of omputtion the surfes re idelised s being opque-grey. For one surfe totlly enlosed by the other the viewftor is: In ll other ses we hve: E = e e F o Ao-A Here, e is the issivity of the oven wll, e the issivity of the mpoule o nd FA o- A does not depend on e o nd e but only on the geometry. In the eletril nlogue the eqution beomes: I = sea {V 4_V 4) o o Using Ohm's lw, the rdition resistne Rr is found to be: R = [sea (V 2 +V 2 }(V +V }]-l r o o o In only one se refletion will be tken into ount. It is when het is itted from surfe to nother surfe totlly enlosed by the first one. Trnsmission of rdition does not tke ple in the model beuse the surfes re ssumed to be opque. The viewftors mentioned bove re lulted with these ssumptions. Using the formuls bove the rdition or ondution resistne between two nodes of the model n be lulted. Combining the nodes with eh other results in omplex eletril network. In this network the tpertures of ll the nodes nd the het flows between th n be lulted, if one speifies tperture profile t the oven wll. The vlues of the onstnts in the model re: Stefn-Boltzmnn onstnt : s = 5.6693 10-8 w m24 0 Therml ondutivity of the mpoule k = 80.0 :C. Emissivity of the oven : e = 0.8. 0 Length of the oven : 1 = 0.60 m. 0 Length of the mpoule : 1 = 0.20 m. Wll thikness of the mpoule : w = 0.25 lo-3 m. -- 4 --
Therml ondutivity of fluid BOO : k 1 = 0.45 ~ Therml ondutivity of solid BOO : k s Emissivity of the.poule : e = 0.8 Dimeter of the oven : d = 0. 06. 0 Dimeter of the.poule : d = 0.03. = 0.45 J!_ llc The isotherm interfe hs teperture of 1055 C. To indite the position of the.poule with respet to the oven we need only one oordinte xis. nmely the ylinder xis, with the origin t the top of the oven nd pointing downwrds. The bottom of the oven is t the position of 60. "1be boundry between the heter nd the ooler is then t 30. "1be interfe shll be BOlle where ner the boundry of the heter nd the ooler. 1be tperture profile t the oven wll is g.lven in figure 1. Ner the boundry between heter nd ooler the te.perture grdient is steepest. nmely 30 -. 'nle peks ner the boundry inrese the tellperture grdient. Without those peks the teperture grdient t the wll would be only 25 ;.................... -- 5 --
The pty Bridgmn-oven The sitution in the Bridgmn-oven is s follows: Definition: (1) The wll tperture of the oven is kept onstnt. {2) There is no rdition of het to the enviroment. The tperture t point on the xis of the oven is lulted by introduing n infinitesimlly smll test volume t tht point. The het flow from smll ylindril surfe A on the xis of the oven v to ylindril setion Ai of the oven wll is given by: qi =sa E.(T 4_Ti4) v 1 v Subsript v refers to the test volume nd subsript i refers to setion i on the oven wll. Divide the oven wll into n setions of equl length. This results in n rdition equtions. The totl sum of the het flow hs to be zero in the sttionry se: n I q i=1 i = o Combining these equtions yields: or E (T 4_T 4) = 0 i=1 i v i T 4 = v n I E. i=l 1 The sttionry tperture Tv is, of ourse, independent of the issivity e This n be shown s follows. v By negleting non-horizontl refletion, one of the Ei' sy Ej' is given by: 1 Av 1-1 Ej = [---+---(----1)] e Ai e v 0 nd fori E {1,,n}, i # j -- 6 --
Beuse A «1, this yields: v Ej = ev Define j = 1 nd i = e 0 FAv-Aj fori {l,.,n},i; j, this results in: T 4 = v n I i=l i It is obvious now tht T is independent of e. v v Given the oven wll tperture, it is reltively esy to lulte the tperture on the xis of the oven. Compring the tperture of the oven wll nd the tperture on the xis it is seen in figure 2 tht the tperture profile on the xis tends to beome more sooth. -- 7 --
The vertil tperture profile in the mpoule will be even more smoothed out then the tperture profile on the xis in the pty oven. When the mpoule is in the oven there is through ondution het flow from the top to the bottom of the mpoule. This mens tht t the top of the mpoule the tperture is lower thn the orresponding tperture in the pty oven nd t the bottom of the mpoule the tperture is higher thn the orresponding tperture in the pty oven. When the mpoule is in the middle of the oven the tperture grdient t the interfe will be smller thn the tperture grdient in the pty oven. The tperture grdient in the pty oven ppers to be 24 _Q_. -- 8 --
Results nd Disussion The model bove is used to lulte the tperture distribution in the oven s funtion of severl prmeters. To study the behviour of the tperture profile in time we ple the mpoule t five different positions nmely when the mpoule is 6, 8, 10, 12, 14 in the ooler. When the mpoule is 4 in the ooler ll BOO is still melt. All BOO hs beome solid when the mpoule is 16 in the ooler. The quntities of min interest re the shpe or the onvexity of the interfe nd the vlue of the vertil tperture grdient t the interfe. We define the onvexity to be negtive if the shpe of the interfe is onve, seen from the solid. Let L be the position of BOO with tperture s 1055 C t the xis nd L the position of BOO with tperture 1055 C ner the wll of the mpoule. The onvexity is then given by: onvexity = Another point of interest is the vertil position of the interfe. In the following we denote therefore where the tperture t the xis equls 1055 C. Note tht the top of the oven is t position of 0 nd fter 30 there is the top edge of the ooler. The defult oven: position of the bottom of the mpoule in onvexity grdient position interfe in 36 38 40 42 44 2.5 0.7 0.7 0.3-0.6 20.2 18.3 18.2 18.3 20.4 33.8 31.8 29.8 27.8 25.9 Tble 1. Results for the defult oven, desribed in the previous setion. Comments on tble 1: Note tht the movents of the mpoule nd the interfe re in opposite diretions. This rrkble result n be explined s follows. The top nd the bottom of the mpoule hve lrge surfe res. Let us ll the rdition from the oven wll to the top of the mpoule end-gin nd from the bottom of the mpoule to the oven wll end-loss. When the mpoule is 6 in the ooler, the lower prt reeives lot of het from the upper prt in the heter. Tht is why the position of the interfe is in the ooler. In this position the end-loss is lrge. Thus the tperture rpidly dereses ner the bottom of the mpoule nd the tperture grdient t the interfe is lrge. The -- 9 --
onvexity of the interfe profile n be explined s follows. If the mpoule is prtly in the ooler the lower prt of the vertil surfe exhnges het both with ooler nd heter. The bottom surfe, however, merely looses het towrds the ooler. Thus the tperture in the middle of the bottom is ooler thn the tperture on the outside of the bottom. When the mpoule goes down the prt of the mpoule in the ooler inreses nd the position of the interfe moves towrds the heter. When the mpoule hs been lowered 10 in the ooler the position of the interfe is in the heter. When the mpoule is lowered further the position of the interfe shifts towrds the top of the mpoule. Then the onvexity will hve opposite sign beuse of symmetry rguments. Influene of issivity: position of the bottom onvexity grdient position interfe of the mpoule in in 36 2.3 17.9 33.9 38 0.7 17.0 31.8 40 0.7 17.0 29.8 42 0.7 17.0 27.8 44-1.5 18.1 25.8 Tble 2. Results for the defult oven with e vlue e = 0.8. = 0.35 insted of the defult Comments on tble 2: Comprison tbles 1 nd 2 mkes ler tht hnging the issivity only effets the grdient nd the onvexity, wheres the position of the interfe hrdly hnges. The effets re reltively smll. When the issivity of the mpoule beomes smller, there is less het bsorption. The tperture devition in the mpoule will smooth out little beuse the ondutivity in the mpoule nd BOO rins the sme. Thus the tperture grdient t the interfe is smller ompred with the defult oven. -- 10 --
Influene of ondutivity of BOO: position of the bottom of the mpoule in onvexity grdient position interfe in 36 38 40 42 44 1.8 0.7 0.7 0.7-0.8 19.2 17.9 17.9 17.9 19.3 33.8 31.8 29.8 27.8 25.9 Tble 3. Results for the defult oven with k 1 = the defult vlues k 1 = 0.45 nd ks = 0.45. 1.35 nd k = 1.35 insted of s Comments on tble 3: When the ondutivities of BGO beomes lrger. the tperture profile in the mpoule smoothes out nd the tperture grdient t the interfe is smller ompred with the defult oven. The position of the interfe nd the onvexity do not hnge muh. position of the bottom of the mpoule in onvexity grdient position interfe in 36 38 40 42 44 2.9 0.7 0.7 0.7-2.0 20.7 18.4 18.4 18.4 20.6 33.8 31.8 29.8 27.8 25.9 Tble 4. Results for the defult oven with k 1 = 0.15 nd ks = 0.15 insted of the defult vlues k 1 = 0.45 nd ks = 0.45. Comments on tble 4: When the ondutivities of BGO beomes smller. The tperture profile in the mpoule will beome steeper so the tperture grdient t the interfe is little higher. -- 11 --
Influene of hnging both dimeters: position of the bottom of the mpoule in onvexity grdient position interfe in 36 38 40 42 44 4.0 0.9 0.8 0.8-3.8 17.3 14.2 10.3 14.3 17.3 33.5 31.7 29.6 27.7 26.0 Tble 5. Results for the defult oven with d = 6 nd d 0 = 12 insted of the defult vlues d = 3 nd d = 6. o Comments on tble 5: The tperture grdient t the interfe is drmtilly smoothed out when the mpoule is ner the middle of the oven. Here, the het flow n go esily down thnks to the lrge dimeter of the mpoule. The onvexities t the strt nd t the end of the proess re lrger beuse of the enhned end-gin nd end-loss. Influene of dimeter of the mpoule: position of the bottom of the mpoule in onvexity grdient I position interfe I in 36 38 40 42 44 1.0 0.2 0.2 0.2-0.9 19.1 18.7 18.7 18.7 19.2 33.9 31.9 29.9 27.8 25.8 Tble 6. Results for the defult oven with d vlues d = 3. = 1 insted of the defult Comments on tble 6: The smll res t the top nd t the bottom of the mpoule n not reeive or loss muh het. So the onvexities t the top nd bottom re smller. The sme rgument holds for the smller tperture grdients there. -- 12 --
Influene of oven wll tperture profile: position of the bottom onvexity grdient position interfe of the mpoule in in 36 2.2 16.6 33.8 38 0.5 15.6 31.8 40 0.5 15.6 29.7 42 0.5 15.6 27.7 44-2.1 16.8 25.7 Tble 7. Results for the defult oven with oven wll tperture profile without the peks. The oven wll tperture hnges from 1100 C to 1000 C in 4, so the tperture grdient t the wll is 25 _Q_. The oven wll tperture is symmetri round the boundry between the heter nd the ooler. Comments on tble 7: The tperture grdient t the interfe is smller beuse to the reltively smll tperture grdient t the wll. Compred with the defult oven the onvexity nd the position of the interfe re nerly the sme. position of the bottom onvexity grdient I position interfe of the mpoule in I in 36 1.8 8.2 33.0 38 5.1 8.2 30.9 40 3.8 8.2 28.9 42 3.7 8.3 26.8 44-2.1 14.57 25.1 Tble 8. Results for the defult oven with n oven wll tperture profile similr to the previous one exept of heter tperture of 1080 C. The tperture grdient t the wll is 20 _Q_. Comments on tble 8: The position of the interfe shifts upwrds beuse the tperture of the heter is lower. The tperture grdient t the interfe is smller ording to the deresed tperture grdient t the wll. -- 13 --
position of the bottom onvexity grdient position interfe of the mpoule in in 36 2.7 14.6 34.3 38-0.7 12.9 32.5 40-0.7 12.9 30.5 42-0.7 12.9 28.4 44-2.4 14.6 26.5 Tble 9. Results for the defult oven with heter tperture of 1100 C nd ooler tperture of 1020 C. Agin there re no peks. The tperture grdient t the wll is 20_Q_. Comments on tble 9: In this se the effets re similr to the results in the preeding tble. As expeted, the interfe shifts downwrds now. In prtie the onfigurtion of the Bridgmn-oven n be hnged, by mounting diphrgm between heter nd ooler. The totl het flow from heter to ooler tkes then ple through the mpoule. Clultion of the tperture distribution for this onfigurtion of the Bridgmn-oven yields the following results. Influene of diphrg: position of the bottom onvexity grdient position interfe of the mpoule in in 36 2.0 20.8 33 7 38 0.3 20.0 31.6 40 0.6 24.4 29.8 42 0.8 19.9 27.9 44-1.5 20.9 25.9 Tble 10. Results for the defult oven with diphrgm. Comments on tble 10: The presene of diphrgm leds to enhnent of the vertil tperture grdient in the mpoule. This effet is of ourse most ppreible when the interfe is in the viinity of the diphrgm. Beuse the interfe moves onsiderbly during the solidifition proess, the influene of the diphrgm on the tperture grdient vries drstilly in time. It my be questioned whether this is desirble. -- 14 --
position of the bottom onvexity grdient position interfe of the mpoule in in 36 3.7 17.5 33.4 38 0.5 15.5 31.5 40 1.1 18.0 29.7 42 1.4 15.4 27.9 44-3.0 17.5 26.1 Tble 11. Results for the defult oven with diphrgm nd d 0 = 0.12 nd d = 0.06 insted of the defult vlues d = 0.06 nd d = 0.03. o Comments on tble 11: Also in this se the diphrgm hs n influene on the tperture grdient only when the interfe nd diphrgm re t the sme height. -- 15 --
Conlusions For the growth of high qulity rystls the shpe of the interfe nd the tperture grdient t the interfe re of importne. The results presented in this rport give insight whih ftors effet these quntities. It hs been found tht: 1 : e, k nd k only influene the tperture grdient t the interfe, 1 s but hrdly the onvex! ty. The influene on the tperture grdient is very smll. 2 : Pling diphrgm in the Bridgmn-oven hs only influene on the tperture grdient t the interfe. The vlue of the tperture grdient vries lot when the mpoule moves into the ooler. For stedy rystl growth diphrgm might be undesirble. 3 : Chnging the dimeter of the oven nd/or of the mpoule hs gret effet on the tperture grdient t the interfe nd on the onvexity. A smll dimeter of the mpoule, ompred with the dimeter of the oven, results in smll end effets. 4 : The hoie of the tperture profile t the oven wll is of gret importne for ll prmeters. In the defult oven this profile is symmetri with respet to the heter/ooler boundry. The interfe moves then during the proess in nerly symmetri wy from its lowest position in the ooler to its highest position in the heter. Introdution of n symmetri tperture profile t the wll shifts this trjetory downwrds or upwrds. The onsequene for the grdient nd onvexity n be esily dedued from the tbles in the preeding setion. ReOIIIIlelldtion It is quite generlly believed tht for the growth of high qulity rystls high tperture grdient nd onvex shpe of the interfe is fvourble. The following onfigurtion of the Bridgmn-oven will mth these requirents. The tperture of the heter should be muh higher then the tperture of the ooler, thus the tperture profile would be very symmetri. For exmple the tperture of the heter ould be hosen to be 1100 C nd the tperture of the ooler 800 C. With these prmeters vlues the interfe stys during the whole solidifition proess in the ooler with nerly onstnt grdient, prt, from end effets, whih depend only on the mpoule dimeter. -- 16 --
Referenes (1} C.E. Chng nd W.R. Wilox, Journl of Crystl Growth 21 (1974) 135. (2) C.L. Jones, P. Cpper nd J.J. Gosney, Journl of Crystl Growth 56 {1982) 581. {3) P.S. Rvishnkr nd T-Wei Fu, Journl of Crystl Growth 62 (1983) 425. (4} H.Y. Wong, Het Trnsfer for Engineers (Longmn London, 1977) {5) R. Siegel nd J.R. Howell, Therml Rdition Het Trnsfer {MGrw-Hill, In. 1972} (6) P.Stngerup, User's Mnul Esp (Horsholm Denmrk 1985) -- 17 --
Symbols s Stefn-Boltzmnn onstnt kl therml ondutivity of fluid BOO k s k e 0 therml ondutivity of solid BOO therml ondutivity of the mpoule issivity of the oven e issivity of the mpoule d 0 d dimeter of the oven dimeter of the mpoule 1 length of the oven 0 1 lenght of the mpoule w T 0 T A 0 A wll thikness of the wll tperture of the oven tperture of the mpoule surfe re of the oven surfe re of the mpoule FA -A o viewftor independent of e 0 nd e E viewftor -- 18 --
Viewftors Two onentri irulr ylinders T 1. -.., Y=~ Jr+1 b 8=-. I F A,-A, - B FA A..!... (X -.Ji!l - 4iiCI),- I 28 --. 6 --. Inner IW'f:e of irulr ylinder to bue T 1 FA,-A, ~ [,J't +2(.4(1 +R 2 )+(1-Ri)i -(I- R 2 +CI)] hmer surfe or ylidder betwee C 1 d C2tobueril between '2 nd ' FA A _I [Vi't +2d0 +RfJ+(t-.Rf)',- 4((2 - C&) - V1~--.-xt:-=r(:":",-.-=.T.I,""':'+-:':o=---.. ~wl3 e C -. + VCf + 2Cf0 + Rl) +(I - Rfi' -"'1 +2CJ(I +Rfl+U-ib'].,--. ' 1'1 R - --19--
Between tliru::, wll of. tube A, d dila,.az nd... retubd res --20--