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ARTICLE IN PRESS

Journal of Crystal Growth 266 (2004) 239–245

In?uence of the crucible geometry on the shape of the melt–crystal interface during growth of sapphire crys

tal using a heat exchanger method

Jyh-Chen Chena,*, Chung-Wei Lub

b a Department of Mechanical Engineering, National Central University, Chung-Li 32054, Taiwan, ROC Project Quality Improvement Committee, Chung Shan Institute of Science and Technology, P.O. Box No. 90008-18 Lung-Tan, Tao-Yuan 32500, Taiwan, ROC

Abstract Computer simulations using the commercial code FIDAP, which is based on ?nite element techniques, were performed to investigate the effect of the shape of the crucible on the temperature distribution, velocity distribution and shape of the melt–crystal interface, during the application of the heat exchanger method (HEM) of growing sapphire crystals. Heat transfer from the furnace to the crucible and heat extraction from the heat exchanger can be modeled by the convection boundary conditions. Cylindrical crucibles with differently curved corners at their base are considered. The curved base of the crucible decreases the convexity of the melt–crystal interface and suppresses the appearance of ‘‘hot spots’’. A hemispherically shaped crucible base yields the lowest maximum convexity. The variation in convexity of the melt–crystal interface is less abrupt for a cylindrical crucible with curved corners at the base than one without curved corners. The effects of the thickness and the conductivity of the crucible are also addressed. The convexity of the melt–crystal interface decreases as the thickness of the crucible wall increases. The convexity also declines as the conductivity of the crucible increases. r 2004 Elsevier B.V. All rights reserved.

PACS: 02.70.Dh; 81.30.Fb; 95.30.Tg; 02.60.Cb Keywords: A1. Convexity; A1. Crucible geometry; A2. Heat exchanger method; A2. Single crystal growth; B1. Sapphire

1. Introduction Their excellent optical, mechanical, thermal and chemical characteristics make large sapphires the preferred dome material from which to make the outer windows of IR air-to-air missiles [1]. Large

*Corresponding author. Tel.: +886-3-426-7321; fax: +8863-425-4501. E-mail addresses: jcchen@cc.ncu.edu.tw (J.-C. Chen), lucw1@ms5.hinet.net (C.-W. Lu).

columnar sapphire crystals can be grown by various techniques. The cost of the fabrication of a hemispherical dome is high, because of the hardness of the sapphire. Growing single sapphire crystals with a dome-shaped geometry can markedly shorten the time required for the fabrication of the dome. Dome-shaped sapphires have been primarily grown by the heat exchanger (HEM) [2–4], gradient solidi?cation (GSM) [5,6] and edgede?ned ?lm-fed method (EFG) [7]. However, the

0022-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2004.02.051

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240 J.-C. Chen, C.-W. Lu / Journal of Crystal Growth 266 (2004) 239–245

application of these processes to the production of low-cost near-net-shaped sapphire domes remains immature, because both the inside and outside curvatures of the dome must be controlled with a high degree of accuracy [8]. The thermal and the velocity ?elds imposed on the crystal during these growth processes are very dif?cult to evaluate. Wang et al. [9] used a transient numerical method to determine the temperature and velocity ?elds of an ideal HEM system using a simple cylindrical crucible. The initial BGO melt temperature was maintained in excess of the melting temperature; the cylindrical crucible wall was adiabatic and the heat exchanger was kept at a very low temperature (from 0 C to 500 C). The authors found that the temperature of the heat exchanger insigni?cantly affected the maximum de?ection of the melt–solid interface before the interface crossed the corners of the cylindrical crucible. Brandon et al. [10] had examined the growth of cylindrical sapphire single crystals numerically, using a GSM system. They considered quasi-steady-state solutions for the melt convection and the shape of the melt–crystal interface. The sensitivity of the thermal ?elds to the parameter values was markedly dependent on the diameter of the crucible. They found that the effect of furnace gradients on the interface in an intermediate size crucible was larger than the interface in crucibles with different diameters. Lu and Chen [11,12] used FIDAP to simulate the HEM crystal growth process in a cylindrical crucible; both the energy input and the energy output were modeled under convection boundary conditions. They found that the contact angle was obtuse before the solid–melt interface touched the sidewall of the crucible, and that hot spots always appeared. The effects of various thermal ?elds on the shape of interface and the rate of increase of the size of the crystal were also investigated. The crucible shape for growing a near-netshaped sapphire dome is very complicated. Therefore, the velocity and temperature ?elds during the growth process are very dif?cult to simulate with a numerical method. Untill now, the effect of the shape and thickness of the crucible on the HEM crystal growth has not been considered. Hence, in the present study, we employ FIDAP to study the

correlation between the shape of the crucible and the melt–crystal convexity, as well as the temperature and velocity distributions during HEM sapphire crystal growth. The effects of the thickness and the conductivity of the crucible are also considered.

2. Modeling Fig. 1a schematically depicts the physical system used in this study. The crystal is grown by reducing the environmental temperature of the furnace and increasing the ability of the heat exchanger to remove heat. Molybdenum crucibles with variously curved bases are considered. The transfer of heat from the furnace to the crucible and the extraction of heat from the heat exchanger are modeled by the convection boundary conditions. Neither the effects of the surface tension at the gas–melt interface nor the gap resistance between the crucible base and the support plate are taken into account. The computation domain is shown in Fig. 1b. A quasi-steady-state [12] is assumed for the governing equations and the energy balance condition at the melt–solid interface. The steady-state, axisymmetric energy equations for the problem may be expressed in cylindrical coordinates in the following form: (A) Energy equation in the melt: qT qT 1q qT q2 T rCp u ?v r ?k ? 2 : ?1? qr qz r qr qr qz (B) Energy equation in the solid: 1q qT q2 T r ? 2 ? r ? q ? 0: k r qr qr qz

?2?

Here q represents the radiant heat ?ux. The divergence of the radiative heat ?ux denotes the net rate at which thermal energy radiates from this point. Radiative heat transfer enhances heat transport within the crystalline phase of optically thick media [13]. Brandon and his co-workers [14,15] used the arti?cially enhanced thermal conductivity of the solid to include the radiative effect. In our study [16], the maximum convexity is slightly decreased by an increase of the assumed conductivity value of the sapphire crystal. We

ARTICLE IN PRESS

J.-C. Chen, C.-W. Lu / Journal of Crystal Growth 266 (2004) 239–245 241

melt and crystal insulation heater crucible plat heat exchanger

(A) the temperature is equal to the melting point at the solidi?cation interface T ? Tm ; ?4? (B) energy balance at the melt–solid interface boundary ?kni ?rT ?l ? kni ?rT ?s ? 0; ?5? (C) convection heat transfer in the heat exchanger qT ? hc ?T ? Tref ? at 0prpRc ; z ? 0; ?k ?6? qz (D) symmetry about the centerline qT ? 0 at r ? 0; 0pzpH ; ?7? qr (E) convection heat transferred by the heating element: qT ?i? ? k ? hI ?T ? Tref ? at r ? R; qr ?8? Ra pzpH ; qT ?ii? ? k ? hI ?T ? Tiref ? at z ? 0; qz Rc orpR ? Ra ; qT qT ? ?iii? ? k ? hI ?T ? Tiref ?; qz qr at 0ozoRa ; R ? Ra orpR; ?9?

gas inlet tube

(a)

Z

melt

H

n

Ra

solid

r

Crucible

Rc R

radiation + convection convection

?10?

(b)

Fig. 1. (a) Schematic diagram of an HEM system; (b) the computation domain.

?iv?

qT ? hI ?T ? Tiref ? qr 0prpR: ?k

at z ? H ; ?11?

focus on the in?uence of the crucible geometry on the shape of the melt–crystal interface. Therefore, we neglect the effect of the internal radiant heat ?ux of the crystal. (C) Energy equation for the crucible wall (where relevant): kc 1q qT q2 T r ? 2 ? 0: r qr qr qz ? 3?

The boundary conditions are as follows:

The symbol hc represents the convection coef?cient of the heat exchanger, hI is the heat transfer coef?cient inside the furnace, k is the thermal conductivity of the melt and the solid sapphire, kc is the thermal conductivity of the molybdenum crucible, Tref is the temperature of the working ?uid inside the heat exchanger, Tiref is the environmental temperature outside the crucible, and Rc is the radius of the cooling-zone. Subscripts l and s denote the liquid and the solid phases, respectively. The steady, axisymmetric governing equations in cylindrical coordinates of the ?ow ?eld are as follows:

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242 J.-C. Chen, C.-W. Lu / Journal of Crystal Growth 266 (2004) 239–245

Table 1 Physical properties of the sapphire Physical properties Density (r) (kg/m ) Dynamic viscosity (m)(Pa s) Melting point (Tm ) (K) Speci?c heat (CP ) (kJ/kg K) Thermal conductivity of the melt or solid (k) (W/m K) Thermal conductivity of the molybdenum crucible (kc ) (W/mK) Thermal expansion coef?cient (b) (K?1)

3

3. Results and discussion

Values 3500 0.0475–5E9 2323 1.56 3.5 10–50 1.8E?5

As in previous studies [11,12] the convexity is de?ned by D ? max Zh ? min Zh ; ?16?

(A) continuity equation in the melt 1q qv ?ru? ? ? 0; r qr qz (B) in the r-direction qu qu r u ?v qr qz qP q 1q q2 u ?ru? ? 2 ; ?? ?m qr qr r qr qz (C) in the z-direction qv qv 1q qv q2 v r ?m ? 2 r u ?v qr qz r qr qr qz ? rbg?T ? Tm ?: ?14? ?12?

?13?

where Zh is the height of the interface in the z direction. The convexity of the interface is an important growth parameter that affects the quality of the crystal. The formation of crystal facets can be prevented by lower convexity at the interface, except for a certain orientation of the crystal seed [18]. Higher quality crystals may be grown when there is less convexity at the melt– solid interface. The shape of the melt–solid interface is nearly hemispherical during the initial HEM growth [11]. Fig. 2 illustrates the isothermals and streamlines in the melt, for cylindrical crucibles with or without curved bases. The radius of curvature of the corners of the base is de?ned by Ra : Clearly under speci?c heat transport conditions the convexity is lower when the corner of the crucible base is curved than when it is not, because the curve increases the magnitude of min Zh when the solidi?cation front touches the curved wall of the crucible. The authors’ earlier results [11,12] indicate that a cylindrical crucible without a curved base will have ‘‘hot spots’’ in the corners during

The velocity ?eld is symmetrical at the centerline qv ?0 qr and u ? 0 at r ? 0; 0pzpH : ?15?

A no-slip boundary condition is applied at the crucible surface. These moving boundary problems were solved using the ?nite element package FIDAP 8.0 [17]. The mesh used in these calculations consisted of a bilinear four-node quadrilateral element. A discontinuous piecewise approximation of pressure was made using the mixed method. The nonlinear equations were solved using the Newton–Raphson method. The total number of nodes was dependent on the shape of crucible and is approximately 10,000. The convergence criterion for the solution is that the relative error should be less than 0.0001. Table 1 presents the growth parameters used in the computations.

Fig. 2. Isothermals and streamlines in the melt for hI ? 100 W/ m2 K, Tref ? 1573 K, Tiref ? 2330 K, hc ? 200 W/m2 K and Rc ? 0:01 m with two different cylindrical crucible geometries: (a) crucible without a curved base, and (b) crucible with a curved base. (Temperature ?elds in the crystal region is Tmax ? 2323 K, DT ? ?10 K, and in the melt region is Tmin ? 2323 K, DT ? 1 K. The increment of the stream function is Dc ? ?2:5E?6).

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J.-C. Chen, C.-W. Lu / Journal of Crystal Growth 266 (2004) 239–245 243

the growth of the crystal. According to experimental observations [19], when ‘‘hot spots’’ exist in the corners of the crucible, the crystal will not grow to the corner until the amount of heat output reaches a certain value. This phenomenon may increase the dislocation and reduce the local quality of the crystal. The presence of a curved crucible base suppresses the appearance of corner ‘‘hot spots’’. Only a cellular ?ow is generated by the buoyancy force in the melt region, which is established by the heating of the melt in the radial direction of the crucible wall. The radial temperature gradient is smaller for a crucible with a curved base than for one without such a base. Therefore, the strength of the ?ow in a curved corner crucible

7 6

is less than in a crucible without such a curved corner. Fig. 3 plots D versus Tiref as a function of the two crucible geometries under particular heat transport conditions. The increase in the convexity that occurs when Tiref declines is smaller when the cylindrical crucible has a curved base (case (b)) than when the cylindrical crucible does not have a curved base (case (a)), before maximum convexity is attained. This means that, when the environmental temperature is reduced, the variation in the convexity of the melt–crystal interface is smoother for the cylindrical curved base crucible, before maximum convexity is reached. However, the results also show that case (b) requires a lower Tiref to reach maximum convexity. Fig. 3 also

7 6.5 6

Dmax (cm)

case (a) case (b)

5

5.5 5 4.5 4 3.5 3 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07

D (cm)

4 3 2 1 2320

2325

2330

2335

2340

2345

Ra (m)

Fig. 5. Maximum convexity (Dmax ) versus Ra for hI ? 100 W/ m2 K, Tref ? 1573 K, hc ? 200 W/m2 K and Rc ? 0:01 m.

Tiref (K)

Fig. 3. Convexity (D) versus Tiref as a function of the crucible geometry for hI ? 100 W/m2 K, Tref ? 1573 K, hc ? 200 W/ m2 K and Rc ? 0:01 m.

Table 2 Various growth parameters used in the computations Processing parameters Crucible radius (R) (m) Cooling zone radius (Rc ) (m) Crucible thickness (tw ) (m) Crucible height (H ) (m) Radius of curvature of the corner of the base (Ra ) (m) Environmental temperature outside the crucible (Tiref ) (K) Temperature of the working ?uid inside the heat exchanger (Tref ) (K) Heat transfer coef?cient inside the furnace (hI ) (W/m2 K) Heat exchanger convection coef?cient (hc ) (W/m2 K) Values 0.065 0.01 0.001–0.007 0.13 0.01–0.065 2325–2340 1573 100 200

7 6 5

Ra=0.065 m Ra=0.03 m Ra=0.01 m

D (cm)

4 3 2 1 2320

2325

2330

2335

2340

2345

Tiref (K)

Fig. 4. Convexity (D) versus Tiref for hI ? 100 W/m K, Tref ? 1573 K, hc ? 200 W/m2 K and Rc ? 0:01 m with Ra ? 0:01 m, Ra ? 0:03 m and Ra ? 0:065 m.

2

Note: hI includes the combined effect of radiation and convection.

ARTICLE IN PRESS

244 J.-C. Chen, C.-W. Lu / Journal of Crystal Growth 266 (2004) 239–245

indicates that the variation of D with Tiref in case (b) is less signi?cant around Dmax than elsewhere. The convexity is at its maximum in the region where the melt–solid interface touches the curved corners of the base of the crucible. The curved surface of the crucible causes the convexity to be insensitive to a change in Tiref ; until the melt–solid interface departs from the curved corner. Fig. 4 plots D versus Tiref for various values of Ra : The results show that the maximum convexity Dmax increases as Ra decreases. Fig. 4 clearly shows that Ra must be increased at the base of the crucible during HEM crystal growth to reduce Dmax : Fig. 5 plots Dmax versus Ra : The results indicate that a decrease in Dmax is directly

proportional to an increase in the radius of the curvature of the base (Ra ). In this study, when Ra ? 0:065 m, the base becomes hemispherical and has the lowest possible maximum convexity (Table 2). The effects of the thickness (tw ) and the conductivity (kc ) of the crucible are also considered herein. Fig. 6 plots the isothermals and streamlines in the melt for crucibles of: (a) 0.002, (b) 0.004 and (c) 0.006 m thick, when the convexity reaches its maximum value. When the thickness of the crucible is considered, some of the heat input from the furnace is removed by the heat exchanger through the crucible wall. The amount of heat removed through the crucible wall is proportional

Fig. 6. Isothermals and streamlines in the melt where the melt–solid interface touches the crucible corners, with walls of various thicknesses (a) tw ? 0:002 m, (b) tw ? 0:004 m and (c) tw ? 0:006 m for hI ? 100 W/m2 K, Tref ? 1573 K, Tiref ? 2330 K and Rc ? 0:01 m. (The temperature ?elds in the crystal region is Tmax ? 2323 K, DT ? ?10 K and in the melt region is Tmin ? 2323 K, DT ? 1 K. The increment of the stream function is Dc ? ?2:5E?6.)

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J.-C. Chen, C.-W. Lu / Journal of Crystal Growth 266 (2004) 239–245

9 8.5 8 7.5

— — — —

245

kc=10 W/mK kc=50 W/mK

7 6.5 6 5.5 5 4.5 4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

tw (cm)

Fig. 7. Maximum convexity (Dmax ) versus Ra for hI ? 100 W/ m2 K, Tref ? 1573 K, Tiref ? 2330 K with kc ? 10 W/m K and kc ? 50 W/m K.

crucible base decreases the convexity of the melt– crystal interface and suppresses the appearance of ‘‘hot spots’’. As the environmental temperature falls, the variation in the convexity of the melt– crystal interface is less abrupt in cylindrical crucibles with the curved bases. The decrease of the Dmax is directly proportional to the increase in the radius of the curvature of the base (Ra ); a hemispherical crucible base yields the lowest possible maximum convexity. The effects of the thickness and the conductivity of the crucible are also considered. The maximum convexity Dmax and the strength of the ?ow decrease as the thickness of the crucible wall increases. The maximum convexity also decreases as the conductivity of the crucible increases.

to its thickness, and to its thermal conductivity. Therefore, increasing the thickness of the crucible wall reduces the amount of heat added to the sapphire. Hence, when tw is higher, a smaller hc is required to enable the melt–solid interface to touch the corners of the crucible. Lu et al.’s simulation results [12] showed that a smaller hc corresponds to a lower Dmax : Fig. 6 indicates that as the thickness of the wall increases, the maximum convexity Dmax decreases because hc declines. The temperature gradient in the sapphire crystal decreases as tw increases. Hence, the strength of the ?ow degenerates as tw increases. Degeneration of the temperature gradients in the crystal region can reduce the thermal stress in the crystal. Fig. 7 plots Dmax versus tw for kc ?10 and 50 W/m K. The shapes that correspond to the two different values of kc seem to be similar to each other. The maximum convexity decreases markedly as the conductivity of the crucible increases.

Dmax (cm)

References

[1] J.A. Savage, J. Crystal Growth 113 (1991) 698. [2] C.P. Khattak, A.N. Scoville, F. Schmid, SPIE Proc. 683 (1986) 32. [3] F. Schmid, C.P. Khattak, SPIE Proc. 1112 (1989) 25. [4] C.P. Khattak, F. Schmid, SPIE Proc. 1760 (1992) 41. [5] S. Biderman, A. Horowitz, Y. Einav, G. Ben-Amar, D. Gazit, A. Stern, M. Weiss, SPIE Proc. 1535 (1991) 27. [6] A. Horowitz, S. Biderman, Y. Einav, G. Ben-Amar, D. Gazit, M. Weiss, J. Crystal Growth 128 (1993) 824. [7] V.N. Kurlov, B.M. Epelbaum, J. Crystal Growth 179 (1997) 175. [8] C.P. Khattak, F. Schmid, SPIE 505 Adv. Opt. Mater. (1984) 4. [9] J.H. Wang, D.H. Kim, J.S. Huh, J. Crystal Growth 174 (1997) 13. [10] S. Brandon, D. Gazit, A. Horowitz, J. Crystal Growth 167 (1996) 190. [11] C.W. Lu, J.C. Chen, J. Crystal Growth 225 (2001) 274. [12] C.W. Lu, J.C. Chen, Modelling Simul. Mater. Sci. Eng. 10 (2002) 147. [13] A.G. Petrosyan, J. Crystal Growth 139 (1994) 372. [14] Yongcai Liu, A. Virozub, S. Brandon, J. Crystal Growth 205 (1999) 333. [15] A. Virozub, S. Brandon, Modelling Simul. Mater. Sci. Eng. 10 (2002) 57. [16] C.W. Lu, Numerical simulation of sapphire crystal growth using HEM, D.Sc. Thesis, National Central University of Taiwan, 2002. [17] FIDAP User’s Manual, V 8.0, 1998. [18] B. Cockayne, J. Crystal Growth 3 (1968) 60. [19] D. Viechnicki, F. Schmid, J. Crystal Growth 26 (1974) 162.

4. Conclusion The effects of the crucible geometry on the shape of the melt–crystal interface during the application of the HEM of growing sapphire crystals has been investigated numerically using the FIDAP software. The presence of a curved

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