- Email: [email protected]
Three-dimensional numerical simulation of oxide melt flow in Czochralski configuration
Journal of Crystal Growth 216 (2000) 372}388
Three-dimensional numerical simulation of oxide melt #ow in Czochralski con"guration C.J. Jing , N. Imaishi *, T. Sato , Y. Miyazawa Institute of Advanced Material Study, Kyushu University, 6-1 Kasuga Kouen, Kasuga-shi, Fukuoka 816-8580, Japan National Institute for Research in Inorganic Materials, 1-1 Namiki, Tsukuba-shi, Ibaraki 305-0044, Japan Received 26 July 1999; accepted 20 March 2000 Communicated by J.J. Derby
Abstract In order to understand the well-known surface pattern of oxide melt #ow in a Czochralski crucible, a set of three-dimensional numerical simulations of LiNbO melt #ow in a crucible (47 mm ;46 mm) with a rotating disc (23.5 mm, rotating at X) was conducted using the "nite-di!erence method. The crucible bottom was assumed to be adiabatic and the side wall was heated by a constant heat #ux. Several case studies with a constant temperature of the side wall were considered for comparison. The heat loss from the melt surface caused by radiation to the ambient at a constant temperature was taken into account. The disc/melt interface was assumed to be #at and at melting- point temperature. To investigate the role of surface tension in the melt #ow, numerical simulations including and neglecting the Marangoni e!ect were conducted at each disc-rotation rate. When neglecting the Marangoni e!ect, the interaction between the buoyancy and the disc rotation produces a wavy ring-like pattern on the melt surface only at approximately X"40 rpm. However, if the Marangoni e!ect is taken into account, a spoke pattern appears on the surface over a wide range of disc-rotation rate (0 rpm)X)45 rpm). At higher disc-rotation rates, the spoke pattern disappears. The #ow in the bulk can be classi"ed into three di!erent types according to the disc-rotation rate, i.e., (1) thermally driven axisymmetric #ow with no reverse #ow at low rotation rates (0}20 rpm), (2) thermally driven axisymmetric #ow with reverse #ow beneath the disc at moderate rotation rates (20}30 rpm) and (3) nonaxisymmetric unsteady #ow at higher rotation rates (*40 rpm). The investigation of the e!ect of the thermal boundary condition indicated that these #ow patterns are sensitive to the thermal boundary condition of the crucible side wall. 2000 Elsevier Science B.V. All rights reserved. PACS: 02.60.Cb; 47.20.Dr; 81.10.Fq Keywords: Surface pattern; Oxide melt; Marangoni e!ect; Numerical simulation
1. Introduction During the past decade, a number of authors have reported surface patterns occurring in Czoch* Corresponding author. Tel.: #81-92-583-7793; fax: #8192-583-7796. E-mail address: [email protected] (N. Imaishi).
ralski (Cz) oxide melts. It should be noted that in oxide Cz furnaces, the crystal rotates whereas the crucible does not. Whi$n et al. [1] reported various surface patterns in molten bismuth}silicon oxide under various crystal rotation rates. A report of similar surface patterns was presented by Takagi et al. [2]. So far, numerous experimental investigations on oxide melt #ow have aimed at examining
0022-0248/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 4 2 7 - 9
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
the e!ect of crystal rotation on the melt #ow and #ow transition. Some simulation experiments with water or water/glycerine solutions, the so-called `cold-model experimentsa, have demonstrated that with an increase in crystal rotation rate, the melt #ow exhibits three mode types: (1) buoyancy-driven axisymmetric #ow at low rotation rates, (2) nonaxisymmetric #ow at moderate rotation rates, and (3) rotation-driven #ow at high rotation rates [3}5]. Miller and Pernell [4] reported on the melt #ow and surface patterns at several crystal rotation rates. Jones [5] observed a vortex pattern when the rate of crystal rotation was slightly higher than a certain threshold value. The majority of #ows observed in actual oxide melts or cold models are three-dimensional and time dependent. Leister and Peric [6] were the "rst to attempt fully three-dimensional, time-dependent simulations of the #ow for the model experimental system employed by Jones [5]. Later, Xiao and Derby [7], and Rojo and Derby [8] performed a three-dimensional, time-dependent numerical simulation for oxide melt #ow and identi"ed a fourfold wavy pattern in the bulk temperature (but not on the melt surface). In these three-dimensional calculations, only the interaction between buoyancy and rotation was considered. The Marangoni (thermocapillary and/or solutocapillary) e!ect was
373
assumed to be secondary and negligible. In our previous paper [9], a set of three-dimensional numerical simulations indicated that the spoke pattern on an open surface (without crystal) is caused by the Marangoni instability in the thermal boundary layer beneath the melt surface. In this paper, another set of numerical simulations was conducted to clarify the in#uences of the crystal rotation rate and the Marangoni e!ect on the oxide melt #ow and the surface patterns in the oxide melt in an RF-heated crucible.
2. Problem formulation 2.1. Basic assumptions and governing equations The schematic in Fig. 1a illustrates that the con"guration of the model considered in the present paper consists of a crucible of radius r and height h and a rotating disc (simulating a crystal) of radius r . The three-dimensional simulation of oxide #ow in the open crucible, i.e., without a rotating disc, was performed in our previous paper [9]. Thus, in the present paper we employed assumptions identical to those in Ref. [9] for both the oxide melt and the boundary in addition to assuming a #at and nondeformable disc/melt interface maintained at
Fig. 1. The con"guration of the system (a) and the grid used (b).
374
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
a constant temperature ¹ (the melt-point). Re garding the boundary conditions, the key points are repeated here. At the free surface, the thermal Marangoni e!ect is considered in some cases, and the crucible side wall is assumed to be heated with a constant heat #ux q. For the sake of investigating the e!ect of the thermal boundary condition, several case studies which used a "xed wall temperature were considered. The heat loss from the free surface is caused by radiation to the ambient at an e!ective temperature ¹ . The fundamental equations consist the continuity equation, the momentum equation and the energy equation, which are identical to those in Ref. [9]. The boundary conditions, described in Fig. 1a, are also identical to those in Ref. [9] with the exception of the following additional boundary condition at the disc/melt interface: < "< "0, < "RRe , H"1 P X F S (Z"H, 0)R)R , 0)h)2p), (1) where < , < and < are the dimensionless velocity P F X components in the R, h and Z direction, respectively, H is the dimensionless temperature, R, Z and h are the dimensionless cylindrical coordinates and R is the dimensionless radius of the disc. The disc-rotation Reynolds number, Re , is de"ned S as ru Re " , S l
(2)
where u is the disc-rotation angular velocity and l is the kinematic viscosity of the oxide melt.
But the results with "ner grids showed very similar spoke patterns and the local velocity and temperature converged within few percents. Then, in the present work, a non-uniform staggered grid of 40P;40X;60F, as shown in Fig. 1b, was used. The numerical simulations were performed for the LiNbO melt under the following conditions. The radius of the crucible is r "23.5 mm, the radius of the disc is r "0.5r and the depth of the melt h"46 mm. The thermophysical properties of the LiNiO melt at 1573 K are taken from Shigematsu et al. [10] and listed in Ref. [9]. Thus, the nondimensional parameters are R "1.0, R "0.5, H"1.945, Gr"2.62;10, Re "2.32; c 10, Pr"13.6 and R "0.34, respectively, where Gr is the Grashof number Gr"gb¹ r/l; Re is
c the capillary Reynolds number Re "c ¹ r /kl; c 2 Pr is the Prandtl number Pr"l/a and R is the radiation number R "pe¹ r /j. g denotes
the acceleration due to gravity, b the thermal expansion coe$cient of the melt, ¹ the melting
point, c "!Rc/R¹ the temperature coe$cient 2 of surface tension, k the viscosity, a the thermal di!usivity, p the Stefan}Boltzmann constant, e the emissivity, and j the thermal conductivity. According to Jing et al. [9], the e!ective ambient temperature was set at H "0.8. The heat #ux Q on the crucible side wall was estimated to be Q"0.05 in order to keep the calculated melt temperature in a reasonable region. The results based on this set of parameters will be presented below.
3. Results and discussion 2.2. Numerical method and calculation conditions The fundamental equations were solved by the "nite-di!erence method. The details of the numerical method are described in Ref. [9]. A set of simulations had been conducted with di!erent grid resolutions with 35P;40X;60F, 40P;35X;60F 40P;40X;56F and coarser grids for X"10 rpm. The results with grids of 30P;30X;40F or less showed quite di!erent #ow patterns when the Marangoni e!ect was taken into account, e.g. a surface pattern with hot spots rather than spokes.
A number of computer simulations were performed to understand the in#uence of the disc rotation as well as the Marangoni e!ect on the oxide melt #ow. The rotation rate of the disc X (X"60u/2p) varied from 0 to 80 rpm, which corresponds to the disc-rotation Reynolds number, Re , from 0 to 412. In order to highlight the signi"S cance of the Marangoni e!ect on the oxide melt #ow, for each rotation rate, calculations were conducted with and without considering the Marangoni e!ect.
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
3.1. Results neglecting the Marangoni ewect When the Marangoni e!ect is neglected, i.e., the melt #ow is driven by buoyancy force and disc rotation, the numerical results indicated several distinct #ow features. At the disc-rotation rate X"0 and 10 rpm, the melt #ow is axisymmetric and steady. Fig. 2 shows the #ow and temperature "elds for the case of X"10 rpm. With X"20 and 30 rpm, the #ow becomes periodically oscillatory but remains axisymmetric. The oscillation is quite
375
similar to that of the 2D simulation performed by Munakada and Tanasawa [11]. When the discrotation rate is slightly higher than 21 rpm, reverse #ow occurs below the disc. At about 3 rpm more than the critical value, a disc-rotation-driven reverse-#ow cell appears. Fig. 3 shows a snapshot of the top and meriodial views of the velocity vectors and isotherms for the case of X"30 rpm. When the disc-rotation rate is increased to X"40 rpm, the melt #ow becomes nonaxisymmetric and time dependent. A gray-scale expression of the surface
Fig. 2. The top views, (a)}(c), and the meridional views, (c)}(f), of the velocity vectors and isotherms of the axisymmetric, steady #ow solution at X"10 rpm when neglecting the Marangoni e!ect. The isotherms, (b) and (e), and the contours of the azimuthal velocity, (f), are evenly spaced. (c) is the visualized gray-scale surface temperature. The other nondimensional parameters are Gr"2.62;10, Re "0, Pr"13.6, R "0.34, Q"0.05, H "0.8. c
376
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 3. The snapshots of the top and meridional views of the velocity vectors, (a) and (c), the isotherms, (b) and (d), and the contours of the azimuthal velocity, (e), of the periodically oscillatory #ow and temperature "elds with reverse #ow at X"30 rpm and the other conditions identical to those in Fig. 2.
temperature "eld shows a distorted surface ringlike pattern. This pattern travels in the azimuthal direction very quickly (about 20 rpm), as shown in Figs. 4a}f. Figs. 4g}l show the corresponding isothermal surface at ¹"1548 K. Fig. 5 shows the snapshot of the #ow and temperature "elds at several cross sections at the time corresponding to Fig. 4f. The melt from the disc is forced downwards when it meets the inward #ow from the wall and forms a vertical free shear layer at approximately 1.5 cm beneath the surface. The unsteady and asymmetric #ow could be caused by the shear instability and/or the baroclinic instability. Due to further mixing of the reverse #ow and the thermally driven #ow, the shear layer disappears in the lower melt zone. The numerical results indicate that the dark line on the surface corresponds to the meeting line
of the #ow from the disc (reverse #ow) and that from the wall (thermally driven #ow) and the melt temperature has a locally low value. This can be explained by the fact that the cold melt from the disc is further cooled by radiation. With an increase of the disc-rotation rate, centrifugal force pushes the shear layer outwards. At X'53 rpm the reverse #ow prevails in the upper melt zone and the surface pattern disappears. The #ow is developed in the #ow region dominated by the e!ect of disc rotation. 3.2. Results considering the Marangoni ewect When the Marangoni e!ect is considered, the calculated oxide melt #ow is quite di!erent from that when the Marangoni e!ect is neglected. The
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
377
Fig. 4. The time evolution of the surface pattern (a) and an isothermal surface (b) of the nonaxisymmetric and unsteady #ow under the condition of X"40 rpm and Marangoni e!ect neglected. The other nondimensional parameters are identical to those in Fig. 2.
main feature is the appearance of the surface spoke pattern (refer to Jing et al. [9] for the mechanism of the spoke pattern). Fig. 6 shows the surface patterns at several crystal rotation rates. When the disc is
stationary (X"0 rpm), the surface pattern is regular and the spokes are straight. Once the disc begins rotation, the spokes become skewed in the discrotation direction and the surface pattern starts to
378
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 5. The snapshot of the nonaxisymmetric and unsteady #ow and temperature "elds of the free surface (a}c), the horizontal cut of Z"0.95H (d}f ), and the meridional cuts of h"0 (g}i) and h"90 ( j}l). The calculation conditions are identical to those in Fig. 4 and the time corresponds to that in Fig. 4f.
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
379
Fig. 6. The calculated surface patterns at several disc-rotation rates when considering the Marangoni e!ect.
migrate slowly in the same azimuthal direction as the disc rotation. As shown in Fig. 7a, the rotation rate of the spoke pattern increases with the discrotation rate but the number of spokes remains constant over the range of 0)X)30 rpm (Fig. 7b). With the increase of the disc-rotation rate, the skewed parts extend outward. At X)30 rpm, the surface spoke pattern remains regular but at X*40 rpm the surface pattern becomes irregular. Further increase in the disc-rotation rate bring about the disappearance of the surface spoke pattern. When considering the Marangoni e!ect, the bulk #ow shows two distinct #ow zones. The #ow and temperature "eld near the free surface exhibit
three-dimensional characteristics. In the lower melt zone, however, the #ow and temperature distribution become almost axisymmetric. The #ow and temperature "elds at the disc-rotation rate of X"0 rpm are shown in Fig. 8. The #ow characteristics at X"10 and 20 rpm are axisymmetric, unsteady and there is no reverse #ow beneath the disc. The #ow and temperature "elds at X"30 rpm are still regular but have a reverse #ow, as shown in Fig. 9. From Figs. 8 and 9, it is evident that the three-dimensional structure appearing under the disc penetrates about 1 cm from the surface. The penetration depth decreases with the disc-rotation rate, as shown in Fig. 7c. Even though the melt #ow near the surface exhibits three dimensional
380
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 7. The dependence of the rotation rate of the surface patterns (a), number of the spokes (b) and penetration depth of the surface patterns (c) on the disc-rotation rate.
and oscillatory behavior (the rotation of the surface pattern is the cause of the oscillation), the #ow in the lower melt zone remains approximately axisymmetric and quasisteady. When the disc-rotation rate exceeds 40 rpm, as shown in Fig. 10 (X"40 rpm) and Fig. 11 (X"50 rpm), the bulk #ow becomes highly nonaxisymmetric and time dependent. At X"50 rpm, the spoke pattern disappears and a highly asymmetric #ow pattern is formed on the surface as well as in the bulk, as shown in Fig. 11. With a further increase of the crystal rotation rate, the centrifugal e!ect plays a dominant role and the Marangoni e!ect becomes secondary. The #ow becomes similar to that when the Marangoni e!ect is neglected. These results coincide with the experimental observation performed by Miller and Penell [4]. The critical disc-rotation rate for the onset of the reverse #ow in the presence of the Marangoni e!ect is estimated to be about 24 rpm, i.e., 3 rpm higher than that in the absence of the Marangoni e!ect. From the standpoint of the crystal grower, the Marangoni e!ect could a!ect the quality of the crystal. Fig. 12 shows the temperature distribution on the horizontal cross section 0.8 mm beneath the disc/melt interface at three disc-rotation rates which represent the three typical #ow regions. As can be seen in these "gures, at low rotation rate
(X)30 rpm), the Marangoni e!ect could a!ect the quality of the crystal edge and at high rotation rate it could improve the quality of the crystal because it suppresses the nonaxisymmetric #ow beneath the disc. Furthermore, the Marangoni e!ect a!ects the circular motion of the melt. Figs. 13a and b show the in#uence of the Marangoni e!ect on the maximum radial velocity, generally occurring at the free surface, and the circular #ow rate, de"ned by the rising #ow (or descending #ow) across the horizontal plane of Z"0.5H, at various disc-rotation rates, respectively. The Marangoni e!ect accelerates the maximum radial velocity by about three times in the low rotation rate range (X)30 rpm) and the acceleration e!ect is weakened by the increase in the disc-rotation rate. The circular #ow is enhanced in the low rotation rate range (X)20 rpm) and in contrast is lower down by the Marangoni e!ect in the high rotation rate range (X*30 rpm). This could be attributed to the reverse #ow generated in the high rotation rate range. As seen in Figs. 13c}e, the Marangoni e!ect decreases the average temperature of the bulk melt (Fig. 13c) and crucible side wall (Fig. 13d) and increases the average temperature of the free surface (Fig. 13e). 3.3. Ewect of disc-rotation rate As shown in Figs. 13c}e, when the Marangoni e!ect is neglected, the e!ect of the rotation rate on the temperature "eld is signi"cant. The average temperatures of the melt (Fig. 13c), the crucible side wall (Fig. 13d) and the free surface (Fig. 13e) increase with the rotation rate and reach maximum values at approximately X"30 rpm. Further increase of the rotation rate decreases these characteristic values. Because the crucible wall is set at a constant heat #ux, the abrupt increase in the wall temperature means that the heat transfer in the radial direction becomes abruptly weak. The decrease in radial heat transfer could correspond to a transition of #ow structure in the melt, as was pointed out by Kakimoto et al. [12]. When the Marangoni e!ect is considered, the in#uence of the rotation rate on the temperature "eld is monotonic and becomes relatively weak.
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
381
Fig. 8. The velocity vectors and isotherms of the surface, (a) and (b), the meridional cut of h"0, (c) and (d), the circumferential extension plane of R"0.8, (e) and (f), and the bird's eye view of an isothermal surface of ¹"1540 K (g), of the axisymmetric and steady #ow at disc-rotation rate X"0 rpm when considering the Marangoni e!ect. The isotherms are spaced evenly. The other nondimensional parameters are Gr"2.62;10, Re "2.42;10, Pr"13.6, R "0.34, Q"0.05 and H "0.8. c
382
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 9. The velocity vectors and isotherms of the surface, (a) and (b), the meridional cut of h"0, (d) and (e), the circumferential extension plane of R"0.8, (g) and (h), the visualized gray-scale surface spoke pattern (c), the contours of the azimuthal velocity (f ) and the bird's eye view of the isothermal surface of ¹"1539 K (i), of the axisymmetric and periodically oscillatory #ow at discrotation rate X"30 rpm when considering the Marangoni e!ect. The other nondimensional parameters are identical to those shown in Fig. 8.
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
383
Fig. 10. The snapshot of vectors and isotherms of the free surface, (a) and (b), the horizontal cut of Z"0.5H, (d) and (e), the meridional cuts of h"0, (g) and (h), and h"90, ( j) and (k), of the nonaxisymmetric and unsteady #ow and temperature "elds at X"40 rpm when considering the Marangoni e!ect. (c) is the gray-scale surface temperature "eld, (f ) the contours of the axial velocity of the horizontal cut of Z"0.5H, and (i) and (l) are the contours of the azimuthal velocity of the meridional cut of h"0 and 90, respectively. The other parameters are identical to those shown in Fig. 8.
384
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 11. The snapshot of an unusual #ow pattern obtained at X"50 rpm. Vectors and isotherms of the free surface, (a) and (b), the horizontal cut of Z"0.5H, (d) and (e), the meridional cuts of h"0, (g) and (h), and h"90, ( j) and (k), of the nonaxisymmetric and unsteady #ow and temperature "elds at X"50 rpm when considering the Marangoni e!ect. (c) is the gray-scale surface temperature "eld, (f ) the contours of the axial velocity of the horizontal cut of Z"0.5H, and (i) and (l) are the contours of the azimuthal velocity of the meridional cut of h"0 and 90, respectively. The other parameters are identical to those shown in Fig. 8.
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
385
Fig. 12. The temperature distribution on the horizontal plane 0.8 mm beneath the disc/melt interface at the disc-rotation rate X"10, 30 and 40 rpm when the Marangoni e!ect is neglected (a) and when the Marangoni e!ect is considered (b).
3.4. The ewect of the thermal boundary condition at the crucible side wall The e!ect of the thermal boundary condition at the crucible side wall was investigated for several cases. When the Marangoni e!ect is neglected, the crucible side wall is "xed at a constant temperature of 1561 K, which corresponds to the average temperature of the crucible wall at X"40 rpm in the constant heat #ux case (Fig. 4 or Fig. 5), the melt #ow exhibits a wavy ring-like pattern of mode m"2 on the surface, as shown in Fig. 14b. The wavy ring-like pattern and the entire body of the #ow and temperature "elds rotate in the same di-
rection as that of the disc rotation but at approximately 2.4 rpm. This pattern was observed in the range of disc rotation of 33(X(43 rpm (see Figs. 14a and c). On the other hand, when the Marangoni e!ect is considered, with a "xed crucible wall temperature at 1550 K, which corresponds to the average crucible wall temperature of Fig. 10, the irregular surface spoke pattern disappears and no pattern is generated on the surface (Fig. 14d) and the melt #ow becomes approximately axisymmetric. The #ow at X"50 rpm with a constant wall temperature, 1545 K (the average wall temperature of case Fig. 6f ), becomes approximately axisymmetric and
386
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 13. The in#uence of the disc-rotation rate and Marangoni e!ect on the maximum radial velocity (a), the circular #ow rate (b), the averaged melt temperature (c), the averaged crucible wall temperature (d) and the average surface temperature (e).
a regular ring-like pattern, as shown in Fig. 14e, is generated. Further increase of the rotation rate pushes the ring toward the crucible wall until it disappears. The present numerical simulations indicate the importance of the thermal boundary condition at the crucible side wall. Under a constant heat #ux thermal boundary condition on the crucible side
wall, the calculated #ow at X'50 rpm only has numerical signi"cance because the wall temperature becomes lower than the disc temperature (the melting point in the case of crystal growth). The realistic thermal boundary condition of the crucible side wall during growth of the oxide crystal should be neither the constant heat #ux nor the constant temperature but between them. To obtain realistic
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
387
Fig. 14. The surface patterns obtained at various disc-rotation rates when the constant temperature thermal boundary condition is used for the crucible side wall.
numerical results, both the RF heat generation and the thermal conduction in the crucible wall must be included.
4. Conclusions The three-dimensional numerical simulations of the oxide melt #ow in a crucible with a rotating disc were performed by the "nite-di!erence method with the HS-MAC algorithm. The signi"cance of the Marangoni e!ect in the oxide melt #ow was
investigated at various disc-rotation rates. It was determined that the Marangoni e!ect is an essential factor for forming the surface spoke pattern. The surface pattern is con"ned to a thin layer close to the surface. The Marangoni e!ect greatly accelerates the circulation rate. These conclusions are identical to those achieved in our previous study [9] for the open crucible. On the other hand, even if the Marangoni e!ect is neglected, the crystal rotation, in cooperation with the buoyancy, can also produce a surface pattern, although it di!ers from the spoke pattern. The rotation shows a tendency to
388
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
weaken the Marangoni e!ect, at lower rotation rates, it prevents the surface spoke pattern structure from penetrating into the melt and at higher rotation rates, it prevents the generation of the spoke pattern.
Acknowledgements This work was partly supported by JSPS Research for Future Program; `Atomic-scale surface and interface dynamicsa.
References [1] P.A.C. Whi$n, T.M. Bruton, J.C. Brice, J. Crystal Growth 32 (1976) 205.
[2] K. Takagi, T. Fukazawa, M. Ishii, J. Crystal Growth 32 (1976) 89. [3] C.D. Brandle, J. Crystal Growth 42 (1977) 400. [4] D.C. Miller, T.L. Pernell, J. Crystal Growth 57 (1982) 253. [5] A.D.W. Jones, J. Crystal Growth 94 (1989) 421. [6] H.-J. Leister, M. Peric, J. Crystal Growth 123 (1992) 567. [7] Q. Xiao, J.J. Derby, J. Crystal Growth 152 (1995) 169. [8] J.C. Rojo, J.J. Derby, J. Crystal Growth 198/199 (1999) 154. [9] C.J. Jing, N. Imaishi, Y. Yasuhiro, Y. Miyazawa, J. Crystal Growth 200 (1999) 204. [10] K. Shigematsu, Y. Anzai, S. Morita, M. Yamada, H. Yokoyama, Jpn. J. Appl. Phys. 26 (1987) 1988. [11] T. Munakada, I. Tanasawa, J. Crystal Growth 106 (1990) 566. [12] K. Kakimoto, M. Watanabe, M. Eguchi, T. Hibiya, J. Crystal Growth 126 (1993) 435.
Three-dimensional numerical simulation of oxide melt #ow in Czochralski con"guration C.J. Jing , N. Imaishi *, T. Sato , Y. Miyazawa Institute of Advanced Material Study, Kyushu University, 6-1 Kasuga Kouen, Kasuga-shi, Fukuoka 816-8580, Japan National Institute for Research in Inorganic Materials, 1-1 Namiki, Tsukuba-shi, Ibaraki 305-0044, Japan Received 26 July 1999; accepted 20 March 2000 Communicated by J.J. Derby
Abstract In order to understand the well-known surface pattern of oxide melt #ow in a Czochralski crucible, a set of three-dimensional numerical simulations of LiNbO melt #ow in a crucible (47 mm ;46 mm) with a rotating disc (23.5 mm, rotating at X) was conducted using the "nite-di!erence method. The crucible bottom was assumed to be adiabatic and the side wall was heated by a constant heat #ux. Several case studies with a constant temperature of the side wall were considered for comparison. The heat loss from the melt surface caused by radiation to the ambient at a constant temperature was taken into account. The disc/melt interface was assumed to be #at and at melting- point temperature. To investigate the role of surface tension in the melt #ow, numerical simulations including and neglecting the Marangoni e!ect were conducted at each disc-rotation rate. When neglecting the Marangoni e!ect, the interaction between the buoyancy and the disc rotation produces a wavy ring-like pattern on the melt surface only at approximately X"40 rpm. However, if the Marangoni e!ect is taken into account, a spoke pattern appears on the surface over a wide range of disc-rotation rate (0 rpm)X)45 rpm). At higher disc-rotation rates, the spoke pattern disappears. The #ow in the bulk can be classi"ed into three di!erent types according to the disc-rotation rate, i.e., (1) thermally driven axisymmetric #ow with no reverse #ow at low rotation rates (0}20 rpm), (2) thermally driven axisymmetric #ow with reverse #ow beneath the disc at moderate rotation rates (20}30 rpm) and (3) nonaxisymmetric unsteady #ow at higher rotation rates (*40 rpm). The investigation of the e!ect of the thermal boundary condition indicated that these #ow patterns are sensitive to the thermal boundary condition of the crucible side wall. 2000 Elsevier Science B.V. All rights reserved. PACS: 02.60.Cb; 47.20.Dr; 81.10.Fq Keywords: Surface pattern; Oxide melt; Marangoni e!ect; Numerical simulation
1. Introduction During the past decade, a number of authors have reported surface patterns occurring in Czoch* Corresponding author. Tel.: #81-92-583-7793; fax: #8192-583-7796. E-mail address: [email protected] (N. Imaishi).
ralski (Cz) oxide melts. It should be noted that in oxide Cz furnaces, the crystal rotates whereas the crucible does not. Whi$n et al. [1] reported various surface patterns in molten bismuth}silicon oxide under various crystal rotation rates. A report of similar surface patterns was presented by Takagi et al. [2]. So far, numerous experimental investigations on oxide melt #ow have aimed at examining
0022-0248/00/$ - see front matter 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 0 ) 0 0 4 2 7 - 9
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
the e!ect of crystal rotation on the melt #ow and #ow transition. Some simulation experiments with water or water/glycerine solutions, the so-called `cold-model experimentsa, have demonstrated that with an increase in crystal rotation rate, the melt #ow exhibits three mode types: (1) buoyancy-driven axisymmetric #ow at low rotation rates, (2) nonaxisymmetric #ow at moderate rotation rates, and (3) rotation-driven #ow at high rotation rates [3}5]. Miller and Pernell [4] reported on the melt #ow and surface patterns at several crystal rotation rates. Jones [5] observed a vortex pattern when the rate of crystal rotation was slightly higher than a certain threshold value. The majority of #ows observed in actual oxide melts or cold models are three-dimensional and time dependent. Leister and Peric [6] were the "rst to attempt fully three-dimensional, time-dependent simulations of the #ow for the model experimental system employed by Jones [5]. Later, Xiao and Derby [7], and Rojo and Derby [8] performed a three-dimensional, time-dependent numerical simulation for oxide melt #ow and identi"ed a fourfold wavy pattern in the bulk temperature (but not on the melt surface). In these three-dimensional calculations, only the interaction between buoyancy and rotation was considered. The Marangoni (thermocapillary and/or solutocapillary) e!ect was
373
assumed to be secondary and negligible. In our previous paper [9], a set of three-dimensional numerical simulations indicated that the spoke pattern on an open surface (without crystal) is caused by the Marangoni instability in the thermal boundary layer beneath the melt surface. In this paper, another set of numerical simulations was conducted to clarify the in#uences of the crystal rotation rate and the Marangoni e!ect on the oxide melt #ow and the surface patterns in the oxide melt in an RF-heated crucible.
2. Problem formulation 2.1. Basic assumptions and governing equations The schematic in Fig. 1a illustrates that the con"guration of the model considered in the present paper consists of a crucible of radius r and height h and a rotating disc (simulating a crystal) of radius r . The three-dimensional simulation of oxide #ow in the open crucible, i.e., without a rotating disc, was performed in our previous paper [9]. Thus, in the present paper we employed assumptions identical to those in Ref. [9] for both the oxide melt and the boundary in addition to assuming a #at and nondeformable disc/melt interface maintained at
Fig. 1. The con"guration of the system (a) and the grid used (b).
374
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
a constant temperature ¹ (the melt-point). Re garding the boundary conditions, the key points are repeated here. At the free surface, the thermal Marangoni e!ect is considered in some cases, and the crucible side wall is assumed to be heated with a constant heat #ux q. For the sake of investigating the e!ect of the thermal boundary condition, several case studies which used a "xed wall temperature were considered. The heat loss from the free surface is caused by radiation to the ambient at an e!ective temperature ¹ . The fundamental equations consist the continuity equation, the momentum equation and the energy equation, which are identical to those in Ref. [9]. The boundary conditions, described in Fig. 1a, are also identical to those in Ref. [9] with the exception of the following additional boundary condition at the disc/melt interface: < "< "0, < "RRe , H"1 P X F S (Z"H, 0)R)R , 0)h)2p), (1) where < , < and < are the dimensionless velocity P F X components in the R, h and Z direction, respectively, H is the dimensionless temperature, R, Z and h are the dimensionless cylindrical coordinates and R is the dimensionless radius of the disc. The disc-rotation Reynolds number, Re , is de"ned S as ru Re " , S l
(2)
where u is the disc-rotation angular velocity and l is the kinematic viscosity of the oxide melt.
But the results with "ner grids showed very similar spoke patterns and the local velocity and temperature converged within few percents. Then, in the present work, a non-uniform staggered grid of 40P;40X;60F, as shown in Fig. 1b, was used. The numerical simulations were performed for the LiNbO melt under the following conditions. The radius of the crucible is r "23.5 mm, the radius of the disc is r "0.5r and the depth of the melt h"46 mm. The thermophysical properties of the LiNiO melt at 1573 K are taken from Shigematsu et al. [10] and listed in Ref. [9]. Thus, the nondimensional parameters are R "1.0, R "0.5, H"1.945, Gr"2.62;10, Re "2.32; c 10, Pr"13.6 and R "0.34, respectively, where Gr is the Grashof number Gr"gb¹ r/l; Re is
c the capillary Reynolds number Re "c ¹ r /kl; c 2 Pr is the Prandtl number Pr"l/a and R is the radiation number R "pe¹ r /j. g denotes
the acceleration due to gravity, b the thermal expansion coe$cient of the melt, ¹ the melting
point, c "!Rc/R¹ the temperature coe$cient 2 of surface tension, k the viscosity, a the thermal di!usivity, p the Stefan}Boltzmann constant, e the emissivity, and j the thermal conductivity. According to Jing et al. [9], the e!ective ambient temperature was set at H "0.8. The heat #ux Q on the crucible side wall was estimated to be Q"0.05 in order to keep the calculated melt temperature in a reasonable region. The results based on this set of parameters will be presented below.
3. Results and discussion 2.2. Numerical method and calculation conditions The fundamental equations were solved by the "nite-di!erence method. The details of the numerical method are described in Ref. [9]. A set of simulations had been conducted with di!erent grid resolutions with 35P;40X;60F, 40P;35X;60F 40P;40X;56F and coarser grids for X"10 rpm. The results with grids of 30P;30X;40F or less showed quite di!erent #ow patterns when the Marangoni e!ect was taken into account, e.g. a surface pattern with hot spots rather than spokes.
A number of computer simulations were performed to understand the in#uence of the disc rotation as well as the Marangoni e!ect on the oxide melt #ow. The rotation rate of the disc X (X"60u/2p) varied from 0 to 80 rpm, which corresponds to the disc-rotation Reynolds number, Re , from 0 to 412. In order to highlight the signi"S cance of the Marangoni e!ect on the oxide melt #ow, for each rotation rate, calculations were conducted with and without considering the Marangoni e!ect.
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
3.1. Results neglecting the Marangoni ewect When the Marangoni e!ect is neglected, i.e., the melt #ow is driven by buoyancy force and disc rotation, the numerical results indicated several distinct #ow features. At the disc-rotation rate X"0 and 10 rpm, the melt #ow is axisymmetric and steady. Fig. 2 shows the #ow and temperature "elds for the case of X"10 rpm. With X"20 and 30 rpm, the #ow becomes periodically oscillatory but remains axisymmetric. The oscillation is quite
375
similar to that of the 2D simulation performed by Munakada and Tanasawa [11]. When the discrotation rate is slightly higher than 21 rpm, reverse #ow occurs below the disc. At about 3 rpm more than the critical value, a disc-rotation-driven reverse-#ow cell appears. Fig. 3 shows a snapshot of the top and meriodial views of the velocity vectors and isotherms for the case of X"30 rpm. When the disc-rotation rate is increased to X"40 rpm, the melt #ow becomes nonaxisymmetric and time dependent. A gray-scale expression of the surface
Fig. 2. The top views, (a)}(c), and the meridional views, (c)}(f), of the velocity vectors and isotherms of the axisymmetric, steady #ow solution at X"10 rpm when neglecting the Marangoni e!ect. The isotherms, (b) and (e), and the contours of the azimuthal velocity, (f), are evenly spaced. (c) is the visualized gray-scale surface temperature. The other nondimensional parameters are Gr"2.62;10, Re "0, Pr"13.6, R "0.34, Q"0.05, H "0.8. c
376
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 3. The snapshots of the top and meridional views of the velocity vectors, (a) and (c), the isotherms, (b) and (d), and the contours of the azimuthal velocity, (e), of the periodically oscillatory #ow and temperature "elds with reverse #ow at X"30 rpm and the other conditions identical to those in Fig. 2.
temperature "eld shows a distorted surface ringlike pattern. This pattern travels in the azimuthal direction very quickly (about 20 rpm), as shown in Figs. 4a}f. Figs. 4g}l show the corresponding isothermal surface at ¹"1548 K. Fig. 5 shows the snapshot of the #ow and temperature "elds at several cross sections at the time corresponding to Fig. 4f. The melt from the disc is forced downwards when it meets the inward #ow from the wall and forms a vertical free shear layer at approximately 1.5 cm beneath the surface. The unsteady and asymmetric #ow could be caused by the shear instability and/or the baroclinic instability. Due to further mixing of the reverse #ow and the thermally driven #ow, the shear layer disappears in the lower melt zone. The numerical results indicate that the dark line on the surface corresponds to the meeting line
of the #ow from the disc (reverse #ow) and that from the wall (thermally driven #ow) and the melt temperature has a locally low value. This can be explained by the fact that the cold melt from the disc is further cooled by radiation. With an increase of the disc-rotation rate, centrifugal force pushes the shear layer outwards. At X'53 rpm the reverse #ow prevails in the upper melt zone and the surface pattern disappears. The #ow is developed in the #ow region dominated by the e!ect of disc rotation. 3.2. Results considering the Marangoni ewect When the Marangoni e!ect is considered, the calculated oxide melt #ow is quite di!erent from that when the Marangoni e!ect is neglected. The
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
377
Fig. 4. The time evolution of the surface pattern (a) and an isothermal surface (b) of the nonaxisymmetric and unsteady #ow under the condition of X"40 rpm and Marangoni e!ect neglected. The other nondimensional parameters are identical to those in Fig. 2.
main feature is the appearance of the surface spoke pattern (refer to Jing et al. [9] for the mechanism of the spoke pattern). Fig. 6 shows the surface patterns at several crystal rotation rates. When the disc is
stationary (X"0 rpm), the surface pattern is regular and the spokes are straight. Once the disc begins rotation, the spokes become skewed in the discrotation direction and the surface pattern starts to
378
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 5. The snapshot of the nonaxisymmetric and unsteady #ow and temperature "elds of the free surface (a}c), the horizontal cut of Z"0.95H (d}f ), and the meridional cuts of h"0 (g}i) and h"90 ( j}l). The calculation conditions are identical to those in Fig. 4 and the time corresponds to that in Fig. 4f.
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
379
Fig. 6. The calculated surface patterns at several disc-rotation rates when considering the Marangoni e!ect.
migrate slowly in the same azimuthal direction as the disc rotation. As shown in Fig. 7a, the rotation rate of the spoke pattern increases with the discrotation rate but the number of spokes remains constant over the range of 0)X)30 rpm (Fig. 7b). With the increase of the disc-rotation rate, the skewed parts extend outward. At X)30 rpm, the surface spoke pattern remains regular but at X*40 rpm the surface pattern becomes irregular. Further increase in the disc-rotation rate bring about the disappearance of the surface spoke pattern. When considering the Marangoni e!ect, the bulk #ow shows two distinct #ow zones. The #ow and temperature "eld near the free surface exhibit
three-dimensional characteristics. In the lower melt zone, however, the #ow and temperature distribution become almost axisymmetric. The #ow and temperature "elds at the disc-rotation rate of X"0 rpm are shown in Fig. 8. The #ow characteristics at X"10 and 20 rpm are axisymmetric, unsteady and there is no reverse #ow beneath the disc. The #ow and temperature "elds at X"30 rpm are still regular but have a reverse #ow, as shown in Fig. 9. From Figs. 8 and 9, it is evident that the three-dimensional structure appearing under the disc penetrates about 1 cm from the surface. The penetration depth decreases with the disc-rotation rate, as shown in Fig. 7c. Even though the melt #ow near the surface exhibits three dimensional
380
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 7. The dependence of the rotation rate of the surface patterns (a), number of the spokes (b) and penetration depth of the surface patterns (c) on the disc-rotation rate.
and oscillatory behavior (the rotation of the surface pattern is the cause of the oscillation), the #ow in the lower melt zone remains approximately axisymmetric and quasisteady. When the disc-rotation rate exceeds 40 rpm, as shown in Fig. 10 (X"40 rpm) and Fig. 11 (X"50 rpm), the bulk #ow becomes highly nonaxisymmetric and time dependent. At X"50 rpm, the spoke pattern disappears and a highly asymmetric #ow pattern is formed on the surface as well as in the bulk, as shown in Fig. 11. With a further increase of the crystal rotation rate, the centrifugal e!ect plays a dominant role and the Marangoni e!ect becomes secondary. The #ow becomes similar to that when the Marangoni e!ect is neglected. These results coincide with the experimental observation performed by Miller and Penell [4]. The critical disc-rotation rate for the onset of the reverse #ow in the presence of the Marangoni e!ect is estimated to be about 24 rpm, i.e., 3 rpm higher than that in the absence of the Marangoni e!ect. From the standpoint of the crystal grower, the Marangoni e!ect could a!ect the quality of the crystal. Fig. 12 shows the temperature distribution on the horizontal cross section 0.8 mm beneath the disc/melt interface at three disc-rotation rates which represent the three typical #ow regions. As can be seen in these "gures, at low rotation rate
(X)30 rpm), the Marangoni e!ect could a!ect the quality of the crystal edge and at high rotation rate it could improve the quality of the crystal because it suppresses the nonaxisymmetric #ow beneath the disc. Furthermore, the Marangoni e!ect a!ects the circular motion of the melt. Figs. 13a and b show the in#uence of the Marangoni e!ect on the maximum radial velocity, generally occurring at the free surface, and the circular #ow rate, de"ned by the rising #ow (or descending #ow) across the horizontal plane of Z"0.5H, at various disc-rotation rates, respectively. The Marangoni e!ect accelerates the maximum radial velocity by about three times in the low rotation rate range (X)30 rpm) and the acceleration e!ect is weakened by the increase in the disc-rotation rate. The circular #ow is enhanced in the low rotation rate range (X)20 rpm) and in contrast is lower down by the Marangoni e!ect in the high rotation rate range (X*30 rpm). This could be attributed to the reverse #ow generated in the high rotation rate range. As seen in Figs. 13c}e, the Marangoni e!ect decreases the average temperature of the bulk melt (Fig. 13c) and crucible side wall (Fig. 13d) and increases the average temperature of the free surface (Fig. 13e). 3.3. Ewect of disc-rotation rate As shown in Figs. 13c}e, when the Marangoni e!ect is neglected, the e!ect of the rotation rate on the temperature "eld is signi"cant. The average temperatures of the melt (Fig. 13c), the crucible side wall (Fig. 13d) and the free surface (Fig. 13e) increase with the rotation rate and reach maximum values at approximately X"30 rpm. Further increase of the rotation rate decreases these characteristic values. Because the crucible wall is set at a constant heat #ux, the abrupt increase in the wall temperature means that the heat transfer in the radial direction becomes abruptly weak. The decrease in radial heat transfer could correspond to a transition of #ow structure in the melt, as was pointed out by Kakimoto et al. [12]. When the Marangoni e!ect is considered, the in#uence of the rotation rate on the temperature "eld is monotonic and becomes relatively weak.
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
381
Fig. 8. The velocity vectors and isotherms of the surface, (a) and (b), the meridional cut of h"0, (c) and (d), the circumferential extension plane of R"0.8, (e) and (f), and the bird's eye view of an isothermal surface of ¹"1540 K (g), of the axisymmetric and steady #ow at disc-rotation rate X"0 rpm when considering the Marangoni e!ect. The isotherms are spaced evenly. The other nondimensional parameters are Gr"2.62;10, Re "2.42;10, Pr"13.6, R "0.34, Q"0.05 and H "0.8. c
382
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 9. The velocity vectors and isotherms of the surface, (a) and (b), the meridional cut of h"0, (d) and (e), the circumferential extension plane of R"0.8, (g) and (h), the visualized gray-scale surface spoke pattern (c), the contours of the azimuthal velocity (f ) and the bird's eye view of the isothermal surface of ¹"1539 K (i), of the axisymmetric and periodically oscillatory #ow at discrotation rate X"30 rpm when considering the Marangoni e!ect. The other nondimensional parameters are identical to those shown in Fig. 8.
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
383
Fig. 10. The snapshot of vectors and isotherms of the free surface, (a) and (b), the horizontal cut of Z"0.5H, (d) and (e), the meridional cuts of h"0, (g) and (h), and h"90, ( j) and (k), of the nonaxisymmetric and unsteady #ow and temperature "elds at X"40 rpm when considering the Marangoni e!ect. (c) is the gray-scale surface temperature "eld, (f ) the contours of the axial velocity of the horizontal cut of Z"0.5H, and (i) and (l) are the contours of the azimuthal velocity of the meridional cut of h"0 and 90, respectively. The other parameters are identical to those shown in Fig. 8.
384
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 11. The snapshot of an unusual #ow pattern obtained at X"50 rpm. Vectors and isotherms of the free surface, (a) and (b), the horizontal cut of Z"0.5H, (d) and (e), the meridional cuts of h"0, (g) and (h), and h"90, ( j) and (k), of the nonaxisymmetric and unsteady #ow and temperature "elds at X"50 rpm when considering the Marangoni e!ect. (c) is the gray-scale surface temperature "eld, (f ) the contours of the axial velocity of the horizontal cut of Z"0.5H, and (i) and (l) are the contours of the azimuthal velocity of the meridional cut of h"0 and 90, respectively. The other parameters are identical to those shown in Fig. 8.
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
385
Fig. 12. The temperature distribution on the horizontal plane 0.8 mm beneath the disc/melt interface at the disc-rotation rate X"10, 30 and 40 rpm when the Marangoni e!ect is neglected (a) and when the Marangoni e!ect is considered (b).
3.4. The ewect of the thermal boundary condition at the crucible side wall The e!ect of the thermal boundary condition at the crucible side wall was investigated for several cases. When the Marangoni e!ect is neglected, the crucible side wall is "xed at a constant temperature of 1561 K, which corresponds to the average temperature of the crucible wall at X"40 rpm in the constant heat #ux case (Fig. 4 or Fig. 5), the melt #ow exhibits a wavy ring-like pattern of mode m"2 on the surface, as shown in Fig. 14b. The wavy ring-like pattern and the entire body of the #ow and temperature "elds rotate in the same di-
rection as that of the disc rotation but at approximately 2.4 rpm. This pattern was observed in the range of disc rotation of 33(X(43 rpm (see Figs. 14a and c). On the other hand, when the Marangoni e!ect is considered, with a "xed crucible wall temperature at 1550 K, which corresponds to the average crucible wall temperature of Fig. 10, the irregular surface spoke pattern disappears and no pattern is generated on the surface (Fig. 14d) and the melt #ow becomes approximately axisymmetric. The #ow at X"50 rpm with a constant wall temperature, 1545 K (the average wall temperature of case Fig. 6f ), becomes approximately axisymmetric and
386
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
Fig. 13. The in#uence of the disc-rotation rate and Marangoni e!ect on the maximum radial velocity (a), the circular #ow rate (b), the averaged melt temperature (c), the averaged crucible wall temperature (d) and the average surface temperature (e).
a regular ring-like pattern, as shown in Fig. 14e, is generated. Further increase of the rotation rate pushes the ring toward the crucible wall until it disappears. The present numerical simulations indicate the importance of the thermal boundary condition at the crucible side wall. Under a constant heat #ux thermal boundary condition on the crucible side
wall, the calculated #ow at X'50 rpm only has numerical signi"cance because the wall temperature becomes lower than the disc temperature (the melting point in the case of crystal growth). The realistic thermal boundary condition of the crucible side wall during growth of the oxide crystal should be neither the constant heat #ux nor the constant temperature but between them. To obtain realistic
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
387
Fig. 14. The surface patterns obtained at various disc-rotation rates when the constant temperature thermal boundary condition is used for the crucible side wall.
numerical results, both the RF heat generation and the thermal conduction in the crucible wall must be included.
4. Conclusions The three-dimensional numerical simulations of the oxide melt #ow in a crucible with a rotating disc were performed by the "nite-di!erence method with the HS-MAC algorithm. The signi"cance of the Marangoni e!ect in the oxide melt #ow was
investigated at various disc-rotation rates. It was determined that the Marangoni e!ect is an essential factor for forming the surface spoke pattern. The surface pattern is con"ned to a thin layer close to the surface. The Marangoni e!ect greatly accelerates the circulation rate. These conclusions are identical to those achieved in our previous study [9] for the open crucible. On the other hand, even if the Marangoni e!ect is neglected, the crystal rotation, in cooperation with the buoyancy, can also produce a surface pattern, although it di!ers from the spoke pattern. The rotation shows a tendency to
388
C.J. Jing et al. / Journal of Crystal Growth 216 (2000) 372}388
weaken the Marangoni e!ect, at lower rotation rates, it prevents the surface spoke pattern structure from penetrating into the melt and at higher rotation rates, it prevents the generation of the spoke pattern.
Acknowledgements This work was partly supported by JSPS Research for Future Program; `Atomic-scale surface and interface dynamicsa.
References [1] P.A.C. Whi$n, T.M. Bruton, J.C. Brice, J. Crystal Growth 32 (1976) 205.
[2] K. Takagi, T. Fukazawa, M. Ishii, J. Crystal Growth 32 (1976) 89. [3] C.D. Brandle, J. Crystal Growth 42 (1977) 400. [4] D.C. Miller, T.L. Pernell, J. Crystal Growth 57 (1982) 253. [5] A.D.W. Jones, J. Crystal Growth 94 (1989) 421. [6] H.-J. Leister, M. Peric, J. Crystal Growth 123 (1992) 567. [7] Q. Xiao, J.J. Derby, J. Crystal Growth 152 (1995) 169. [8] J.C. Rojo, J.J. Derby, J. Crystal Growth 198/199 (1999) 154. [9] C.J. Jing, N. Imaishi, Y. Yasuhiro, Y. Miyazawa, J. Crystal Growth 200 (1999) 204. [10] K. Shigematsu, Y. Anzai, S. Morita, M. Yamada, H. Yokoyama, Jpn. J. Appl. Phys. 26 (1987) 1988. [11] T. Munakada, I. Tanasawa, J. Crystal Growth 106 (1990) 566. [12] K. Kakimoto, M. Watanabe, M. Eguchi, T. Hibiya, J. Crystal Growth 126 (1993) 435.