Transition flow modes in Czochralski convection

j. . . . . . . .

ELSEVIER

CRYSTAL QIROWTH

Journal of Crystal Growth 180 (1997) 305-314

Transition flow modes in Czochralski convection J u n g - I 1 C h o i a, S e u n g t a e K i m a, H y u n g

J i n S u n g a'*, A k i f u m i N a k a n o b, H i d e S. K o y a m a b

~Department of Mechanical Engineering, Korea Advanced Institute of Science and Technology, 373-1, Kusong-dong, Yusong-gu, Taejon 305-701, South Korea bDepartment of Mechanical Engineering, Tokyo Denki University, Chiyoda-ku, Tokyo 101, Japan

Received 2 December 1996; accepted 25 March 1997

Abstract A laboratory experiment of critical transition flow modes in Czochralski convection was performed. Numerical computation was also made to delineate the dynamic transition mechanism of flow modes. The period of temperature oscillation (tp) and the interval of temperature oscillation (A0) were scrutinized to capture the critical transition regime. The mixed convection parameter was varied in a range 0.134 ~< Ra/PrRe 2 ~< 3.804. The computed results were shown to be in good agreement with the experimental data. The influence of the Prandtl number on the transition was examined for Pr = 910, 4445 and 8889. To understand the transition mechanism, the detailed temperature oscillation modes, the isolines of meridional temperature and the axial velocity profiles were investigated. The effect of the rotations of crucible on the transient flow modes were also examined. Keywords." Czochralski convection; Crystal growth; Rotating flows

1. Introduction It is known that high quality of the crystals grown by a Czochralski process is affected by the fluid flow in the melt Eli. In the Czochralski process, convection is generally driven by buoyancy and rotations of the crystal and crucible. These mutual interactions cause extreme complexities in the problem. Inhomogeneity stemming from vary-

*Corresponding author. Fax: + 82 42 869 5027; e-mail: [email protected].

ing impurity concentration, which is customarily known as the growth striation, is generated in the crystal when the growth conditions are not timeinvariant. These striations affect significantly the quality of the crystal [2, 3]. The striation is undesirable in semiconductors because it gives rise to fluctuations in resistivity along the length of the crystal [4]. The present study aims to examine the phenomena of transient oscillating flows in the melt. A literature survey reveals that studies on the dynamic patterns of convection in a Czochralski melt are numerous. Whiffin et al. [-5] observed the rotational flow patterns on the surface of molten

0022-0248/97/$17.00 Copyright © 1997 Elsevier Science B.V. All rights reserved PII S0022-0248(97)002 1 2- 1

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Jung-II Choi et al. / Journal o f Crystal Growth 180 (1997) 305-314

bismuth silicon oxide and reported the transition of flow modes with a change in the rotation rate. Jones [6, 7] investigated the details of temperature oscillation in a model Czochralski melt. M u n a k a t a and Tanasawa [8] made experimental and numerical study on the oscillatory transient flow in a Czochralski model. Recently, Sung et al. [9] made a systematical study on the oscillatory transient flow in a Czochralski model. The temperature oscillations were computed over a broad range of the mixed convection parameter, 0.225 ~< Ra/PrRe 2 ~ 0.929. They clarified that the oscillatory flows are essential characteristics in Czochralski convection systems, and these bring forth the major cause of striations. The computed periods of oscillation were shown to be in excellent agreement with the experimental results of Ozoe et al. [-10]. It is noted that most earlier studies have dealt with the overall characteristics of temperature oscillation in Czochralski convection. Little attention has been given to the critical transition, where a sudden discontinuity takes place in the temperature oscillation of the melt. As the rotation rate of the crystal is increased, there is a transition for flow driven inwards under the crystal by buoyancy force to flow pushed outwards by rotation. This is generally termed Kobayashi's criterion [11]. A few researchers reported the transition phenomena I-7, 9-12], however, it is necessary to perform further systematic studies on the critical transition. The present paper addresses this point. To concentrate on the presence of the transition, the period of temperature oscillation (fp) and the interval of temperature oscillation (A0) are scrutinized in the vicinity of the transition regime, where the buoyancy effect is comparable to the rotation effect. It is noted that real semiconductor melts are low Prandtl number ones. Although the parametric values in this study are not directly relevant to a practical crystal growth system, it is important to find out the critical transition between the Prandtl number (Pr) and the critical Reynolds number (Rec). The influence of the Prandtl number (Pr = 910, 4445 and 8889) and the crucible rotation (Rec = - 12.85 and Rec = 12.85) on the transition is investigated as a function of the mixed convection parameter (Ra/PrRe2). Numerical analysis

is also made to delineate the dynamic transition mechanism of flow modes in Czochralski convection. This paper reports observation of the flow transition experimentally and numerically. After describing the experimental apparatus (Section 2), the numerical method is described (Section 3). Results are discussed in Section 4 and some conclusions are drawn in Section 5.

2. Experimental apparatus A high-quality precision-controlled turntable apparatus has been fabricated. An overview of this turntable is sketched in Fig. 1. The major components are two sets of turntable gears and controllers. Cylindrical containers correspond to a single crystal and a crucible in a Czochralski model. The rotation rate was controlled to an accuracy of 0.7%. As angular-roll bearing unit was employed for the turntable axis. This has the advantage of suppressing eccentric motions and vertical vibrations. In addition to the selection of a precision bearing set (JIS P4-Class), a forced-circulation lubrication method was adopted to withstand highspeed rotations. An AC-servo m o t o r (Nikki Denso Co., 7.5 kW, NA 20-370F) was used for the power drive. This is suitable for large-torque device with easy speed control. A rotary encoder was built into the servo motor, which permitted a precision digital-control of the rotation rate. As sketched in Fig. 1, cylindrical containers for the crucible and crystal were mounted to the flange of the lower and upper turntable, respectively. The material of the model crucible was pyrex-glass. The inner radius was Rc = 71.5 mm, and the thickness of the container wall was 3.5 mm. The model crystal was made of aluminum and its outer radius Rs = 35.8 mm. This model was rotated by a separable drive attached to the lower frame. An outer tank surrounded the model crucible was installed, and a precision-temperature control was achieved by circulating a constant temperature water (accurate + 0.1°C) between the outer surface of model crucible and the inner surface of the outer tank. In the series of experiments, high-precision temperature measurements of the working fluid inside the

Jung-ll Choi et al. ,/Journal o f C~stal Growth 180 (1997) 305-314

307

®i ®---x Constant temperature bath

(~Thermocouple

®-~,

.. - . . ~ pump J

-® ®

,

u

Constant temperature bath

Fig. I. Schematic diagram of the apparatus: (l) silicon oil, (2) heated water, (3) cooled water, (4) thermocouple, (5) rotating frame of model crystal, (6) rotating frame of crucible, (7) constant temperature bath (cold), (8) constant temperature bath (hot), (9) crucible, (10) a l u m i n u m disk plate.

Table 1 Properties of the fluid Properties

Pr = 910

4445

8889

K i n e m a t i c viscosity (v, m2/s) Volumetric coefficient ([], K - 1) T h e r m a l diffusivity (:~,m2/s)

1.0 x 10 4 9.5 x 10 -4 1.1 x 10 7

5.0 × 10 4 9.5 × 1 0 - 4 1.1 x 10 7

1.0 × 10 -3 9.4 x 10 4 1.1 x 10 -7

model crucible were of primal importance. Three kinds of silicone oil were used for experiments. Their physical properties at 25°C are listed in Table 1. A constant temperature water at 20.0 _+0.1°C was forced to be circulated into the model crystal and a water at 30.0_+ 0.1°C was circulated inside of the outer tank so as to attain a thermal equilibrium between the interior working fluid and the outer circulating water. Measurements of the fluid temperature were performed at the location of 42.9 m m below the center of the model crystal by a L-type thermocouple.

3. Numerical model

A model of the numerical analysis of the Czochralski growth is shown in Fig. 2. A viscous incompressible flow in a cylindrical crucible of height H and radius Rc is considered. The radius of the crystal rod is Rs. It is known that the combined buoyancy and rotation driven flow for low Prandtl number melts is characterized by precessing waves, i.e., it is non-axisymmetric [5 7]. Only three-dimensional calculation can produce relevant results [13]. However, in this present study for high Prandtl number ones, it is found that the flow

Jung-ll Choi et al. /Journal of Co,stal Growth 180 (1997) 305 314

308

which are defined as co r

5w 8r

8u 5£

1 80 r 5z'

u-

1 8@ w - r 5r .

(5)

Here, the velocity components (u, v, w) in the cylindrical coordinate system represent the radial, azimuthal and axial velocities, respectively. The swirl velocity F in Eq. (2) is F = rv. The equations have been made dimensionless by adopting the following non-dimensional quantities: (r, z) = (r*, z*)/H, t = t*(Qs), (u, v, w) = (u*, v*, w*)/f~sH, 0 = (T - rc)/ (TH - Tc), in which the asterisk denotes the dimensional counterparts. The following three non-dimensional parameters emerge in the governing equations, Re, Pr and Ra. Re represents the rotational Reynolds number of the crystal Re = (2sH2/v, Pr is the Prandtl number Pr = v/:~ and Ra is the Rayleigh number Ra = f i g ( T n - T c )

C

H

z

I F

f~c

H3/~v.

The boundary and the growth interface conditions are written as

Fig. 2. Model of Czochralski growth.

z = O, 0 < r < Rc/H:

can be assumed to be axisymmetric [-8 10]. We assumed that both the flow and temperature fields are axisymmetric with respect to the z (vertical) axis. The governing equations, in dimensionless form, are as follows [-9, 14]: 8~o 8(uco) 5(woo) 1 F, 0 ( l & o ~ _ -~ ~ 5t + ~ + 5z Re L \ r ]

_

Ra

50

-}- P~TRje2/'Sr

+

8209q

@ = o,

aO

~=0,

5z2j

O~--O~--O, 51" ~z

(1)

80

~,=0,

8t+r

5r

+

5~-PrR~L

[1 a f 80"~

8207

r~rOrr)

+Sz2J '

(3) 8(l&p)

. 8f18@~

co

F = r 2,

@=0, .,=0, ~5F= 0 , r=0,0
1

0 = 1,

(6)

0=0,

(7)

z = 1, R s / H < r < Rc/H:

2F 8F uco t.2 8z -~ r '

8(wO)

~c

F = .Q~ r 2,

z = 1, O < r < Rs/H:

5C 5(u/')+a(wr) 1 [ 5(!5V'/ 82rq u v 5 7 + 7 F 8~---Re r~kr 5,'J+~-d] r' (2) 1 8(ruO)

80

8r - 0Z -- 0,

(4)

For a rotationally symmetric flow, computation time can be reduced if the problem is formulated by using the vorticity (05) and stream function (@),

~,o=0,

50 ~=0,

(8)

80 v-=0,

V=0,

(9)

(Jr

r = Rc/H, O < z < 1:

0 = 0,

80 80 5r-0z-0'

F

= Oc(Rc) 2 OskH]'

0 = 1.

(101 The above equations were solved by adopting a numerical scheme based on a finite-difference

Jung-ll Choi et al. / Journal of Crystal Growth 180 (1997) 305-314

scheme. All the computations were conducted on the (41 x 81) stretched grid network. The initial conditions were the solutions for purely natural convection, i.e., the case where the crucible and crystal were stationary. Details of the present numerical procedure were available in Sung et al. [9]. The SIP (Strongly Implicit Procedure) solver was employed in numerical computation [15]. The computations were implemented on an HP-715 workstation, and a typical computer CPU time was approximately more than 20 h for one set of calculations in transition regime. Convergence was declared when the maximum changes in dimensionless values between two successive iterations were less than 10 4. Several trial calculations were repeated to monitor the sensitivity of the results to the grid size, and the outcome of these tests was satisfactory.

4. R e s u l t s

and discussion

As mentioned earlier, the oscillatory flow in the melt is inherent. The oscillatory flow modes are characterized by the dimensionless mixed convection parameter Ra/PrRe 2 [-9, 10, 14]. The experimental conditions are listed in Table 1. The reliability and accuracy of the present numerical simulation were ascertained in the preceding study of Sung et al. [9]. It was seen that the numerical

3oo! 500

400

t P

'

'

I

.

.

.

.

.......

I

.

.

.

.

,,

I

•

~ : Pr--4445 + ~ . ,,

.

.

.

experiment Computation _ experiment Computation experiment

Pr=88~

• l

_

.

200

100

%

i

i

i

1.0

i

i

i

I

i

i

2.0

i

i

I

3.0

i

i

i

i

4.0

Ra/PrRe2 Fig. 3. E x p e r i m e n t a l a n d n u m e r i c a l results of the time period of t e m p e r a t u r e oscillation (tp) for P r = 910, 4445 and 8889.

309

predictions are in broad agreement with the experimental results. Fig. 3 shows experimental and numerical results of the time period (tp) a t the position (r, z ) = (0.0,0.7) for three Prandtl numbers (Pr = 910, 4445 and 8889). For relatively higher Prandtl number flows (Pr = 4445 and 8889), as Ra/PrRe 2 increases (Ra/PrRe 2 > 0.6), i.e., the buoyancy effect is dominant, tp gradually decreases. When Ra/PrRe 2 becomes larger than the onset value of temperature oscillation, the temperature oscillation is gradually attenuated and it eventually vanishes. Although the onset values of experiment and computation are not coincident, the numerical prediction of the period is globally consistent with the experiment. When the forced-convection due to rotation dominates the buoyancy effect i.e., R a / P r R e 2 < 0.47, as Ra/PrRe 2 increases, t v still decreases. It is noted that, when the effect of the forced convection is comparable to the effect of the buoyancy force, a discontinuity of tp is clearly displayed in the vicinity of the region 0.4 ~< Ra/PrRe2 ~< 0.6 for both experiment and computation. For lower Prandtl number (Pr = 910), the transition is clearly captured in the numerical result (0.62 ~< Ra/PrRe 2 ~< 0.84). However, the transition is not observed in the present experiment. It was not feasible to detect the distinctive period of temperature oscillation (tv) in the forced-convection dominant regime (Ra/PrRe 2 ~< 1.15). This may be caused by the fact that, due to the lower kinematic viscosity (Pr = 910), the melt flow is reheated before the cold plume descends fully to the crucible bottom wall. Experimental results of the interval of temperature oscillation (A0) are also displayed in Fig. 4 under the same Prandtl number conditions. The afore-said transition phenomena is evident near the transition regime 0.4 ~< Ra/PrRe 2 ~< 0.6 for Pr = 4445 and 8889. In the forced-convection dominant regime, the absolute value of A0 is very small. This is caused by the higher rotation speed of the top rod. However, the value of A0 in the buoyancy-convection dominant regime is very large. It is known that this big temperature oscillation is associated with a quick descent of cold plume in the melt, i.e., the temperature rises gradually and afterwards it drops rapidly [-9].

310

Jung-Il Choi et al. /Journal of C~stal Growth 180 (]997) 305-314 0.5

. . . .

i

. . . .

i

. . . .

i

• 0.4

. . . . . . . .

. . . .

Pr=8889 Pr-~445 Pr=910

*

• 0.3

A0 0.2

~

~,

~

/

-

~

0.1

210 . . . . 310 . . . . 4.0 Ra/PrRe 2 Fig. 4. Experimentalresults of the interval of temperature oscillation (A0) for Pr = 910, 4445 and 8889. 0'%10 . . . .

110 . . . .

1.0

Pr=4445 0.8

0.6

0.4

~orced c o n v e c t i o n .dominantregime

Intermediate regime Buoyancy convection dominant regime

0.2 Ra/PrRe2= 0.195 0.469 0.0

0

1000

t 0.488 1000

0.508 I000

1.223 1000

1000

t Fig. 5. Features Ra/PrRe 2.

of

temperature

oscillation

for

several

To look into the detailed transient flow modes in the transition regime, several temperature oscillations for Ra/PrRe 2 = 0.195, 0.469, 0.488, 0.508 and 1.223 are exhibited numerically in Fig. 5. The case of Pr = 4445 is selected. As mentioned earlier, the transition takes place in the regime 0.47 ~< Ra/PrRe 2 ~<0.51. As seen in Fig. 5, depending on the flow regime (Ra/PrRe2), the patterns of temperature oscillation are entirely different. The interval of temperature oscillation (A0) in the forced-convection dominant regime (Ra/PrRe 2 = 0.195) is small, while A0 in

the buoyancy-convection dominant regime (Ra/ Pr R e 2 = 1.223) is large. However, the average temperature for Ra/PrRe2 = 0.195 is higher than that for Ra/PrRe 2 = 1.223. In the intermediate regime, Ra/PrRe 2 = 0.469, 0.488 and 0.508, the flow oscillation patterns are definitely transient. For R a / P r R e 2 = 0.469, the temperature oscillations are slightly irregular due to the higher rotation speed of the crystal rod. When Ra/PrRe 2 is increased further, i.e., Ra/PrRe2 = 0.488, the flow becomes suddenly reversed from inward to outward. This sudden change is caused by the quickly descending cold plume and subsequent gradually relating by the circulating fluid. However, closer to the buoyancy-convection dominant regime (Ra/PrRe2 = 1.223), the temperature oscillations return to a more organized pattern. Contour plots of the isotherms (0) in the meridional plane are illustrated in Fig. 6 for three cases; (a) Ra/PrRe 2 = 0.195, (b) Ra/PrRe 2 = 0.488 and (c) Ra/PrRe 2 = 1.223. For Ra/PrRe 2 = 0.195, as time elapses, a cold plume starts to descend from the periphery of the rotation rod, but does not penetrate to the central axis. Since the buoyancy force is not so strong, the forced-convection due to rotation dominates the transient flow mode. However, for Ra/PrRe 2 = 1.223, it is seen that a cold plume from the edge of the rotating rod descends periodically to the bottom center of the rotating rod. In the intermediate regime, i.e., Ra/PrRe 2 = 0.488, the above-stated two distinctive features coexist: a cold plume descends along the centre and from the periphery of the crystal. The flow moves inward under the crystal by buoyancy force and then is pushed outwards by rotation. In an effort to discuss the mechanism of transition in the melt, the axial velocity (w) profiles at r = 0 are illustrated in Fig. 7. Three flow regimes are also selected, i.e., Ra/PrRe 2 = 0.195, 0.488 and 1.223. A closer inspection of Fig. 7a for the forcedconvection dominant regime (Ra/PrRe2 = 0.195) reveals that, as time proceeds, a clockwise recirculating cell is detected near the upper region (z > 0.6). However, only large counterclock-wise flow recirculations are present at Ra/PrRe 2 = 1.223. As pointed out by Kobayashi [11], the interface inversion beneath the crystal rod is closely connected to transient melt flow pattern,

311

Jung-ll Choi et al. / Journal of Crystal Growth 180 (1997) 305-314

~b~ " ~

~

~

~

r~L--

~

¢,

! ill

t=0

t p/6

2t f,/6

3t v/6

4t p/6

5t v/6

t [,

o

Fig. 6. Contour plots of isotherms (0) in the meridional plane for Pr = 4445: (a) Ra/PrRe 2 = 0.195, (b) Ra/PrRe 2 = 0.488, (c) Ra/PrRe 2 = 1.223.

1.o(a! R,a(prRe2=0:l,95 (b) Ra/PrRe2--0.488 (c) Ra/PrRe2=l.223

0.8

~

"

o.~

f

Z

t---o 0.4

Z

---2tJ6 I! /f V'~

4

...........4tp/6 [t I ~I!

]

3q6rl I t!

1

............ 5tJ6 [ \ i~:~!

1 0.0 ....

I ........ -0.01

W

0.00

t 0.01

-0.01

0.00

W

0.01

-0.02-0.01 0.00 0.01

W

Fig. 7. Axial velocity profiles w at r = 0 for Pr = 4445: (a) Ra/PrRe z = 0 195 (b) Ra/PrRe 2 = 0.488, (c) Ra/PrRe 2 = 1.223.

Jung-II Choi et al. / Journal of Crystal Growth 180 (1997) 305 314

312

600

(a) Rec= 12.85 ~ experiment ~ computation i Rec=-12"85 • experiment

500 400

f~ t ~

~

tp 300 200

100 ,

0 0.5

(b) 0.4 . . . . . . . .

,

,

i

. . . .

,

t

i

.... ! . . . . . .

!

.

.

.

.

.

.

.

A0 0.3 .... 0.2

Pr--4445 Ra=4.83x106

0.1 0oo20 '

i

o.s

t

,

1.0

,

,

' 1'.5 '

'

2.0

Ra/PrRe: Fig. 8. (a) T i m e p e r i o d of t e m p e r a t u r e o s c i l l a t i o n (tp) for Rec = 12.85 a n d Rec = 12.85; (b) I n t e r v a l of t e m p e r a t u r e o s c i l l a t i o n (A0) for Rec = 12.85 a n d Rec = - 12.85. P r = 4445.

e.g., a large recirculating cell [-9]. In the transition regime (Ra/PrRe 2 = 0.488), these two features coexist. Accordingly, the shape of the growth interface becomes convex for the buoyancy-convection dominant regime (Ra/PrRe 2 = 1.223), while the shape is concave for Ra/PrRe 2 = 0.195. These interface inversions are taken place at the transition regime. The influence of the crucible rotation (Rec) on the transition is examined. It is noted that the crucible was stationary in the afore-discussions (Rec = 0). The dimensionless time period (tp) and the interval of temperature oscillations (A0) are displayed in Fig. 8 for both Rec = 12.85 and Rec = - 12.85. The negative value of Rec indicates the case of counter-rotation and the positive represents the case of co-rotation between the crystal rod and the crucible. In the behavior of tp for Rec = 12.85, the agreement between experiment and computation is excellent in the buoyancy-convection dominant regime (Ra/PrRe 2 >i 0.5), while it is less satisfactory in the forced-convection dominant regime (Ra/PrRe2 ~< 0.35). For Rec =

- 1 2 . 8 5 , the agreement is also good between experiment and computation. It is seen in Fig. 8 that the transition for Rec = 12.85 is more distinctive than that for Rec = - 12.85. The magnitude of t o for R e c = 12.85 is larger than that for Rec = - 12.85, which is consistent with the numerical findings of Sung et al. [9]. When Rec is positive, the co-rotation of crucible aids the buoyancy force to be strengthened in the melt. This leads to the enlargement of the buoyancy-convection dominant regime. As a result, a sharp discontinuity is taken place in Fig. 8. If Rec is negative (Rec = - 12.85), the transition region is seen to be less distinctive. An examination of the distribution of A0 in Fig. 8b discloses that, as Ra/PrRe 2 increases in the buoyancy-convection dominant regime (Ra/PrRe2 ~> 0.5), A0 gradually decreases from the peak value of A0 for Rec = 12.85, while it still increases for Rec = - 12.85. This makes the transition regime to be wider when Rec is negative. The transient flow modes in the transition regime are illustrated numerically in Figs. 9 and 10 for Rec = 12.85 and Rec = - 12.85, respectively. The selected values of Ra/PrRe 2 for two cases represent the upper and lower limits of the transition, i.e., 0.343 ~< Ra/PrRe 2 ~< 0.476 for Rec = 12.85 and 0.359 ~< Ra/PrRe 2 ~< 0.586 for Rec = - 12.85. It is obvious that the transition regime for Rec = - 12.85 is wider than that for Rec = 12.85. A closer

1.0

Pr--4445 0.8

Re c = 12.85 0.6

0.4 Forced convection dominant regime

0.2

regime Buoyancy dominant

convection regime

Ra/PrRe2= 0.314

0.0

Intermediate

0.343 1000

0.933

0.476 t

1000

i

r

J

I

1000

J

i

i

i

1000

Fig. 9. F e a t u r e s of t e m p e r a t u r e o s c i l l a t i o n for Rec = 12.85 a n d P r = 4445.

Jung-ll Choi et al. /Journal of Crystal Growth 180 (1997) 305 314 1.0

. . . .

i

. . . .

Pr---4445 0.8

i

Re c = -12.85 0.6

0.4

Forced convection dominant regime

Buoyancy convection dominant regime

0.2

Ra/PrRe2= 0.140

0.0

Intermediate regime

0.359 ] i l l

0

500

500

0.586 0.933 l l r l l l l 500

500

t F i g . 10. F e a t u r e s o f t e m p e r a t u r e

oscillation for Rec =--

12.85

a n d P r = 4445.

inspection of the patterns of temperature oscillation indicates that the pattern of temperature oscillation for Ra/PrRe 2 =0.933 at Rec = 12.85 is shifted to that for Ra/PrReZ = 0.586 at Rec = -- 12.85. This may be attributed to the fact that the swirl velocity becomes weak in the central axis ( r ~ 0 ) due to the counter-rotation (Rec = - 12.85). As mentioned earlier, the interval of temperature oscillation (A0) in the buoyancy-convection dominant regime is larger than that in the forced-convection dominant regime. Obviously, the interval of temperature oscillation for Ra/PrRe 2 = 0.476, Rec = 12.85 is larger than that for Ra/PrRe 2 = 0.586, Rec = - 12.85. It is also seen that the temperature for Ra/PrRe 2 = 0.314, Rec = 12.85 is higher than that for Ra/PrRe 2 = 0.140, Rec = - 12.85.

regime, where the effect of the forced convection is comparable to the effect of the buoyancy force. The influence of the Prandtl number on the transition is not substantial for higher Prandtl number flows (Pr = 910, 4445 and 8889). However, for a low Prandtl number flow (Pr = 910), the transition is not clearly captured. The detailed temperature oscillation and the isolines of meridional temperature revealed that two flow characters coexist: i.e., a cold plume descends along the centre and from the periphery of the crystal. The flow moves inward under the crystal by buoyancy force and then is pushed outwards by rotation. The axial velocity profiles along the central axis were investigated to understand the transition mechanism. The effect of the rotation of crucible on the transition flow modes were examined. The transition regime for Rec = - 12.85 is wider than that for Rec = 12.85.

6. N o m e n c l a t u r e

g H Pr Rc Rs Ra Re

gravitational acceleration melt height Prandtl number, v/~ crucible radius crystal radius Rayleigh number, figATH3/~v rotational Reynolds number of crystal rod, (2sHZ/v

Rec

rotational Reynolds number of crucible, (2cHZ/v

F

T t /;

5. Conclusions W

The experimental and numerical results disclosed the prominent features of transition flow mode in Czochralski convection. The period of temperature oscillation (tp) and the interval of temperature oscillation (A0) were scrutinized to capture the critical transition regime as a function of Ra/PrRe 2. It was found that the experimental results are in good agreement with the numerical predictions. The transition takes place near the

313

Z

dimensionless temperature dimensionless dimensionless dimensionless ent dimensionless dimensionless

radial space coordinate time radial velocity component azimuthal velocity componaxial velocity component axial space coordinate

6.1. Greek symbols

F 0 P

thermal diffusivity volumetric thermal expansion coefficient swirl velocity kinematic viscosity dimensionless temperature density

314 ~'~c

f2s (2)

4,

Jung-ll Choi et al. /Journal o['Cr),stal Growth 180 (1997) 305 314

rotation rate of crucible rotation rate of crystal rod vorticity, o~/r = ~ w / O r - ~u/Oz meridional stream fuction

6.2. Superscript dimensional variable References [1] W.E. Langlois, A. Rev. Fluid Mech. 17 (1985) 191. [2] E. Kuroda, H. Kozuka, Y. Takano, J. Crystal Growth 68 (1978) 613. [3] S. Kishino, M. Kamamori, N. Yoshihiro, M. Tajima, T. Lizuka, J. Appl. Phys. 50 (1979) 8240.

[4] H.P. Utech, M.C. Flemings, J. Appl. Phys. 37 (1966) 2021. [5] P.A.C. Whiffin, T.M. Burton, J.C. Brice, J. Crystal Growth 32 (1976) 205. [6] A.D.W. Jones, J. Crystal Growth 69 (1984) 165. [7] A.D.W. Jones, J. Crystal Growth 94 (1989) 421. [8] T. Munakata, I. Tanasawa, J. Crystal Growth 106 (1990) 566. [9] H.J. Sung, Y.J. Jung, H. Ozoe, Int. J. Heat Mass Transfer 38 (1995) 1627. [10] H. Ozoe, K. Toh, T. lnoue, J. Crystal Growth 110 (19911 472. [1l] N. Kobayashi, J. Crystal Growth 52 (1981) 425. [12] N. Kobayashi, J. Crystal Growth 147 (1995) 382. [13] H.-J. Eeister, M. Peric, J. Crystal Growth 123 (1992) 567. [14] J.-I. Choi, H.J. Sung, Int. J. Heat Mass Transfer 40 (1997) 1667. [15] H.L. Stone, SIAM J. Numer. Anal. 5 (1968) 530.