Experimental investigation on the effect of crystal and crucible rotation on thermocapillary convection in a Czochralski configuration

International Journal of Thermal Sciences 104 (2016) 20e28

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International Journal of Thermal Sciences journal homepage: www.elsevier.com/locate/ijts

Experimental investigation on the effect of crystal and crucible rotation on thermocapillary convection in a Czochralski configuration Ting Shen, Chun-Mei Wu, You-Rong Li* Key Laboratory of Low-grade Energy Utilization Technologies and Systems of Ministry of Education, College of Power Engineering, Chongqing University, Chongqing 400044, China

a r t i c l e i n f o

a b s t r a c t

Article history: Received 14 June 2015 Received in revised form 19 December 2015 Accepted 21 December 2015 Available online xxx

Experimental investigation is conducted to understand the effects of crystal and crucible rotations on the thermocapillary convection in a Czochralski configuration. A transparent fluid (0.65cSt silicone oil with Pr ¼ 6.7) is contained in a cylindrical model crucible with the inner diameter of 92 mm and the depth of 2 mm. A copper disc with the diameter of 46 mm is used to simulate the crystal. The rotation rates of the crystal and crucible are 0 ~ ± 60 rpm and 0 ~ ± 1 rpm, respectively. Results indicate that there is a transition from stable axisymmetric flow to a three-dimensional oscillatory flow with the increase of the thermocapillary Reynolds number. The critical thermocapillary Reynolds number for the flow pattern transition decreases with the increase of the crystal rotation rate, and it is greater for the crystalecrucible co-rotation, compared to the counter-rotation. When the crystal rotation rate is small, the thermocapillary force is dominant, and the oscillatory flow behaves as the hydrothermal waves. The crystal rotation has only a slight effect on the wave number and propagation angle of the hydrothermal waves. At a high crystal rotation rate, the hydrothermal waves will transit to rotating waves. The temperature fluctuation of the hydrothermal waves is suppressed by the crucible rotation. © 2016 Elsevier Masson SAS. All rights reserved.

Keywords: Thermocapillary convection Flow pattern Rotation Czochralski method

1. Introduction The Czochralski (Cz) method is the most frequently-employed crystal growth technique in the industry. In Cz crystal growth, the quality of a single crystal depends mainly on the flow of the melt [1,2], which is generally driven by the buoyancy and thermocapillary forces, centrifugal and Coriolis forces due to the crystal and crucible rotations. Therefore, many researches were devoted to the melt flow in the Cz system during the past few decades [3e7]. In the experimental research on model Czochralski-oxide melts with crystal and crucible rotation and differential heating, Jones [8] reported four types of flow patterns as the rotation rate was increased. Tanaka et al. [9] observed the similar flow patterns depending on the crucible rotation rate in the experiments of the silicon melt without crystal. Nakamura et al. [10] found that the number of thermal waves increases with the increase of crucible rotation rate in a Cz configuration with a carbon-dummy crystal. Haslavsky et al. [11,12] studied the effect of crystal rotation rate (0e5 rpm) on steady-oscillatory convection utilizing

* Corresponding author. Tel.: þ86 23 6511 2284; fax: þ86 23 6510 2473. E-mail address: [email protected] (Y.-R. Li). http://dx.doi.org/10.1016/j.ijthermalsci.2015.12.016 1290-0729/© 2016 Elsevier Masson SAS. All rights reserved.

thermocouples and interferometer. They found that the critical temperature difference significantly decreases even at a low crystal rotation rate. The oscillation frequency is power dependences of Grashof numbers. When the buoyancy is strong enough, the effect of crystal rotation can be neglected. Lee and Chun [13,14] performed a set of experiments on the effect of disk rotation on oscillatory buoyancy-driven convection in a Cz model melts. They found three distinct flow regimes for a wide range of Pr number (102e103), which are the buoyancy-driven flow, baroclinic-wave, and rotation-driven flow. The regular baroclinic thermal waves were also observed at a low thermal Rossby number. With the increase of the thermal Rossby number, these regular waves become irregular. Schwabe et al. [15,16] grew the NaNO3 crystals with large convex interface deflection from the melt in the model experiments. They showed that the critical rotation rate connected with a flow pattern transition increases as the convex interface deflection of crystal-melt or the strength of buoyant-thermocapillary flow increases. The crystal rotation rate for growing flat crystal-melt interface was also measured. It was concluded that the buoyantthermocapillary convection is stronger at a larger rotation rate [7]. Kanda et al. [17] investigated the effects of the crystal and crucible rotations on the instability behaviors and flow patterns. At

T. Shen et al. / International Journal of Thermal Sciences 104 (2016) 20e28

relatively small difference between the crystal and crucible rotation rates, the barotropic instability generates multiple vertically coherent vortices, whose number decreases with the increase of the rotation rate difference. Spoke patterns on the free surface in different depths (3, 8, and 200 mm) of the silicon melt were observed by CCD camera [18]. It was found that the number of spokes depends on the depth of the silicon melt. Son et al. [19e21] investigated the thermal field of Wood's metal which had a similar Prandtl (Pr ¼ 0.019) number with the silicon melt for various crucible and crystal rotation rates in a Cz system. The temperature fluctuations and the flow velocities were measured using thermocouples and incorporated magnetic probe, respectively. The results revealed that the rotations of the crucible and crystal influence the frequency and amplitude of temperature fluctuation, and the flow fields. The maximum thermal fluctuation is located under the crystal, and migrates towards the crucible sidewall with the increase of the crystal rotation rate. Near the edge of the crystal the thermal fluctuation and the azimuthal velocity decrease as the rate of crystal counter-rotation with the crucible increases. However, with the increase of the co-rotation rate of the crystal, the thermal fluctuation remains strongly, while the azimuthal velocity decreases. After that, the corresponding numerical simulations were also performed in Refs. [22,23]. In the works mentioned above, the buoyancy plays an important role in the flow. However, under the microgravity condition, the buoyancy effect is excluded. Schwabe et al. [24,25] investigated the thermocapillary flow of the fluid with Pr ¼ 6.7 in an open annular gap with various aspect ratios and depths (d ¼ 2.5e20 mm). In a shallow pool, the concentric steady convection rolls embedded into the main thermocapillary roll were observed at a small temperature difference. At a higher temperature difference, the hydrothermal waves were experimentally identified and the number of m-fold temperature patterns decreases with the decrease of the aspect ratio of the annular pool. In the deep pool, the oscillation was found to be more complicated. Under normal gravity, the HTWs only exist in the shallow pool, and the longitudinal rolls will replace it in thicker layers [26]. Shi et al. [27] analyzed the thermocapillary flow in a rotating annular pool of 0.65cst silicone oil with the depth of 1 mm by numerical simulation and linear stability analysis. Recently, Wu et al. [28,29] performed a series of numerical simulations in a shallow cylindrical pool with a disc on the free surface. The stability diagrams with various crystal and crucible rotation rates were presented, which showed the critical conditions for the onset of flow instabilities. In this work, the effect of rotation on the thermocapillary convection was experimentally studied in a shallow Cz configuration with the depth of 2 mm. The flow patterns and temperature signals are presented at different temperature differences between the crystal and sidewall of the crucible, and rotation rates. The critical thermocapillary Reynolds numbers for the onset of flow instabilities are determined. 2. Experimental apparatus and methods The schematic diagram of the experimental apparatus is shown in Fig. 1. A copper disc with radius of rs ¼ 23 ± 0.1 mm is in contacted with the free surface of the test liquid that is contained in a cylindrical copper crucible with radius of rc ¼ 46 ± 0.1 mm. The working fluid is the transparent 0.65cSt silicone oil. The physical properties are listed in Table 1. In order to prevent the formation of meniscus, a step plane at the inner rim of the crucible is processed. The upper surface of the plane is painted with FC725 to avoid wetting by the silicone oil [25]. To form an adiabatic boundary condition, a high quality plexiglass (thickness 10 ± 0.1 mm) with small thermal conductivity is used as the

21

bottom of the crucible. The temperature difference DT between the crucible sidewall (Tc) and the model crystal (Ts) is measured using T-type thermocouples with precision of ±0.1 K. As shown in Fig. 1, four standing thermocouples are located near the corner of the model crystal and other four thermocouples in inner sidewall of the crucible uniformly. The wall temperatures Ts and Tc (Tc > Ts) are controlled by two thermostatic baths, respectively. The precision of the thermostatic baths is ±0.1 K. To measure the temperature under the free surface, a thermocouple denoted as P is located at 1 mm below the free surface and 5 mm from the crystal. All temperature data are converted into a computer through a data acquisition system (HP 34970A). The rotations of model crystal and crucible are controlled by Panasonic servo motors (MSMD-100W and MSMD-400W, respectively). The resolution ratio of rotation for servo motors is 0.036 . The position of crystal is adjusted by using an automatic lifting platform with the precision of ±3 mm. Furthermore, to avoid the influence of vibration as much as possible, the whole experiment system is installed on a vibration isolation platform. Temperature fluctuation pattern on the free surface is observed by the schlieren method. As shown in Fig. 1, an optical fiber (diameter 1.5 mm) to transfer the light of a medical cold light source (150 W) is located below the pool as a point source. The light passes through the plexiglass and fluid layer, and then is projected on the screen. It is noted that the refractive index of liquid varies with the fluid density, which depends on the temperature. After the flow destabilization, the temperature of the liquid varies with time and space. As the light passes through the unstable convective liquid layer, the image projected on the screen can reflect the temperature fluctuation on the free surface. Then, the schlieren images on the screen are recorded using a digital camera. In order to avoid the influence of environmental light, the experiments are operated in a dark room. The ambient temperature is controlled at 293.15 K to weaken the volatile of the silicone oil. Based on the experimental results, when the crucible rotation rate exceeds 1 rpm, the temperature fluctuation induced by the thermocapillary convection is very small so that the observation on the flow pattern by employing the Schlieren method becomes very difficult. However, this kind of fluctuation still exists even at a very high crystal rotation rate. Therefore, the ranges of the crucible and crystal rotation rates are 0 ~ ± 1 rpm and 0 ~ ± 60 rpm, respectively. The minus rotation rate means the counter-clockwise direction. For the purpose of examining the effect of crystal rotation rate on the thermocapillary convection, the silicone oil with a depth of 2 mm is adopted in the experiments. It should be pointed out that there are some differences between the present experimental model and the industrial Cz progress [30], including shallow pool, isothermal cold model crystal and thermally isolated bottom et al. In general, the dynamic bond (Bo) number is used to evaluate the ratio of the buoyancy convection to thermocapillary convection, which is defined as follows [26]:

Bo ¼

rgrT d2 ; rT

(1)

where r is density, g is the gravitational acceleration (g ¼ 9.81 ms2), rT is the thermal expansion coefficient, rT is the temperature coefficient of surface tension. The value of Bo is equal to 0.5 at d ¼ 2 mm. Therefore, the thermocapillary force is dominant, as reported by Peng et al. [26]. Without rotation, it takes approximately 6 h to completely volatilize the working fluid with 2 mm in depth when DT is 8 K, which corresponds to the evaporating Biot number of 2.3. In our experiment, the experiment for

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T. Shen et al. / International Journal of Thermal Sciences 104 (2016) 20e28

Fig. 1. The experimental apparatus.

3.1. Critical conditions for the incipience of thermocapillary instability

Table 1 Physical properties of the 0.65cSt silicone oil at 293.15 K [26]. Item

Value

Thermal diffusivity, a Kinematic viscosity, n Density, r Temperature coefficient of surface tension, gT Thermal expansion coefficient, rT Prandtl number, Pr

Unit 7

0.97  10 0.65  106 760 8.0  105 1.34  103 6.7

m2 s1 m2 s1 kg m3 N m1 K1 K1 e

each case is controlled within 20e30 min. If the depth of silicone oil is thinner, the error caused by the volatile of fluid will become greater. The thermocapillary Reynolds number (ReT) is a dominant parameter for the thermocapillary convection, which is defined as:

ReT ¼

gT rc ðTc  Ts Þ ; mn

Fig. 2 gives the variation of the critical thermocapillary Reynolds number ReT,cri with the crystal rotation rate ns at different crucible rotation rates. It is found that ReT,cri decreases gradually with the increase of the crystal rotation rate, which indicates that the crystal rotation can destabilize the thermocapillary convection in the shallow pool. It agrees with the numerical results reported in Ref. [27] for the same fluid (Pr ¼ 6.7) in a shallow annular pool, as added in Fig. 2. However, the values of ReT,cri in the present experiments are higher than those in Ref. [27], this discrepancy is from different geometry that the simulation was conducted for an annular pool with a depth of 1 mm. Furthermore, Wu et al. [29] numerically predicted the ReT,cri for the flow destabilization when Pr ¼ 0.011, d ¼ 3 mm, and nc ¼ 0 rpm. They found the similar results that ReT,cri decreases with increasing ns at low rotation rate.

(2)

where n is the kinematic viscosity and m is the dynamic viscosity. The critical thermocapillary Reynolds number (ReT,cri) for the flow destabilization is measured by increasing the temperature difference DT slowly until the instability flow pattern is observed on the screen. 3. Results and discussion When the radial temperature difference DT between the crucible wall and the model crystal is small, the thermocapillary convection driven by surface tension gradient is stable and axisymmetric. However, when DT exceeds a threshold value, the stable and axisymmetric flow transits to a three-dimensional (3-D) oscillatory flow. This flow transition was also reported in Refs. [24,26]. Rotations of the crystal and crucible influence the critical thermocapillary Reynolds number ReT,cri and the flow pattern of the thermocapillary convection considerably.

Fig. 2. Variations of the critical thermocapillary Reynolds number ReT,cri with the crystal rotation rate, and the numerical results reported in Ref. [27] at d ¼ 1 mm in an annular pool.

T. Shen et al. / International Journal of Thermal Sciences 104 (2016) 20e28

For the co-rotation of the crystal and crucible (nc ¼ 1 rpm), the ReT,cri is slightly higher than that for the counter-rotation (nc ¼ 1 rpm), which hints that the flow for the co-rotation of the crystal and crucible is more stable than that for the counterrotation. The ReT,cri at ns > 25 rpm is not listed herein. Since when ns is above 25 rpm, the oscillatory flow driven by rotation appears and the hydrothermal waves (HTWs) related to the thermocapillary instability are fuzzy and hard to be identified. 3.2. The effect of crystal rotation Fig. 3 shows the snapshots of temperature fluctuations on the free surface at nc ¼ 0 rpm and ReT ¼ 91,685. Once the thermocapillary Reynolds number ReT exceeds the critical value, a moving spoke pattern, which corresponds to the hydrothermal waves (HTWs) [27,31,32], appears on the free surface. When ns < 20 rpm, the thermocapillary force is dominant, the wave number m of the HTWs is 27, as shown in Fig. 3(aec). The hydrothermal wave instability is responsible for this flow pattern [26,32]. In this case, the curve fringes of the HTWs are stretched by the crystal rotation, especially near the edge of the crystal. This phenomenon was clarified by Son and Yi [21]. It was concluded that the effect of crystal rotation becomes important with the decreasing distance from the crystal. With the increase of the crystal rotation rate, its effect is enhanced gradually, the temperature fluctuation pattern on the free surface becomes irregular and fuzzy, as shown in Fig. 3(d). With the further increase of the crystal rotation rate, the rotation effects on the velocity and temperature fluctuations near the crystal become important. The centrifugal force near the crystal

23

surpasses the thermocapillary force, and then the outward flow driven by crystal rotation encounters the inward flow induced by the surface tension gradient. Hence, the HTWs are extruded toward crucible sidewall, and another wave pattern appears near the crystal. For example, when ns ¼ 45 rpm, the new wave pattern which rotates along the same direction with the crystal rotation is observed near the crystal. As shown in Fig. 3(e), its wave number m is 5, which is far less than that of the HTWs. However, the azimuthal propagation velocity is very high. This new flow pattern is attributed to a rotating wave driven by the crystal rotation, which was also observed by Hirsa et al. [33] in a lid-driven cylinder with a free surface at DT ¼ 0 K. In their experiment, the rotating waves with azimuthal wave number m ¼ 4 was spontaneously transited from the axisymmetry flow. In Fig. 3(e), the rotating waves near the crystal coexist with the HTWs that are warped near the crucible sidewall. For the rotatingthermocapillary flow, Wu et al. [29] reported that the rotating crystal induces a clockwise cold flow cell underneath the crystal and the thermocapillary force drives a counter-clockwise flow cell. The circular shear layer appears between the two opposite cells. With the increase of the crystal rotation rate, the cold flow cell moves outwardly. In the shallow pool, the shear layer divides the flow field into two regimes, thus, two types of waves could coexist in the pool. At ns ¼ 60 rpm, the rotating waves (m ¼ 4) occupy the entire liquid pool and the HTWs disappear, as shown in Fig. 3(f). When the driven force induced by crystal rotation is dominant, the rotating waves occupy the entire liquid pool. The inward flow induced by thermocapillary force on the free surface is suppressed by the outward flow. Kanda [17] investigated the circular shear

Fig. 3. Snapshots of temperature fluctuations on the free surface at nc ¼ 0 rpm and ReT ¼ 91,685 (DT ¼ 8.0 K).

T. Shen et al. / International Journal of Thermal Sciences 104 (2016) 20e28

Fig. 5 gives the variations of the fundamental oscillatory frequency (f) obtained from the FFT spectrum and amplitude (A) of the temperature fluctuation at the monitoring point P. With the crystal rotation rate increasing from 0 to 20 rpm, the frequency decreases slightly, and then increases. Hintz and Schwabe [3] reported a similar variation trend of the frequency versus the crystal rotation rate in the range of ns ¼ 7e15 rpm, using 0.65cSt silicone oil even though in the deep pool of d ¼ 30 mm. When ns < 10 rpm, the azimuthal velocity of the fluid near the crystal edge increases with

0.25

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293.0

293.0

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292.8

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292.4 0

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292.5

292.8 292.0

292.6 292.4

0

20

40

60

80

100

t-t0 (s)

(c) ns=30 rpm

120

-0.2

Fig. 5. Variations of the fundamental oscillatory frequency (circle) obtained from the FFT spectrum and amplitude (square) of the temperature fluctuation at the monitoring point P with the crystal rotation rate at nc ¼ 0 rpm and ReT ¼ 91,685 (DT ¼ 8 K). Hollow symbol: this work; solid symbol: frequencies in Ref. [3] at DT ¼ 4 K and d ¼ 30 mm.

293.2

40

0.0

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A (K)

0.15

293.4

0

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T (K)

T (K)

instability and the elliptic instability at different crystal rotation rate without temperature gradient. It was reported that multiple (2e4) coherent vortices have common characteristics with the circular shear instability, and the number of vortices decreases with the increase of the crystal rotation rate. In this experiment, the coherent rotating waves (m ¼ 4e5) are observed in the pool, and the wave number decreases with the increase of the crystal rotation rate. Therefore, the oscillatory flow at high rotation rate is generated by the circular shear instability. Moreover, when the effects of rotation and surface tension gradient are comparable, both the HTWs and rotating waves are observed on the free surface (as shown in Fig. 3(e)). It indicates that, the flow instability for this case is due to the mixed circular shear and hydrothermal wave instabilities [29]. In order to examine the temperature fluctuation characteristics, Fig. 4 shows the time dependencies of the temperature measured at the monitoring point P for different crystal rotation rate. In the thermocapillary-flow-dominated regime, the 3-D oscillation is incubated when the thermocapillary Reynolds number exceeds the certain threshold value, and the temperature fluctuation is small, as shown in Fig. 4(a). At ns ¼ 20 rpm, being consistent with the surface fluctuation pattern in Fig. 3(d), the signal is chaotic for the irregular fluctuation, as shown in Fig. 4(b). Schwabe and Sumathi [16] also reported that the signal became chaotic at about ns ¼ 17.7 rpm for the crucible with the diameter of 75 mm and the depth of 48 mm. When the regular rotating waves appear in the liquid pool, the temperature fluctuations become very sharp, since the cold fluid underneath the crystal is driven outward and the hot fluid near the sidewall of crucible flows back periodically, as shown in Fig. 4(c,d).

f (Hz)

24

291.5 0

20

40

60

80

t-t0 (s)

(d) ns=50 rpm

Fig. 4. The time dependencies of the temperature at the monitoring point P at nc ¼ 0 rpm and ReT ¼ 91,685.

T. Shen et al. / International Journal of Thermal Sciences 104 (2016) 20e28

the crystal rotation rate. Therefore, the temperature in this region is more uniform in the azimuthal direction, which suppresses the temperature fluctuation. Subsequently, the temperature fluctuation amplitude A and frequency f at the monitoring point P decreases slightly with the increase of the crystal rotation rate. Once ns > 10 rpm, the HTWs (as shown in Fig. 3(c)) propagate in the azimuthal direction and move outwards, simultaneously, due to the stronger effects of the centrifugal force. Thus, the temperature oscillation frequency increases. Because the temperature signals transit to irregular, the data at ns > 20 rpm are not listed in Fig. 5. Fig. 6 gives the Fourier spectra of the temperature oscillation at the monitoring point P at ReT ¼ 36,674. When ReT ¼ 36,674 and ns < 15 rpm, the flow is stable. The temperature oscillation is considered to be the “background noise” [16], which is caused by the noise in the thermocouple reading. In this case, no fundamental frequency is detected, as shown in Fig. 6(a). At ns ¼ 20 rpm, the temperature periodically oscillates with small amplitude. As shown in Fig. 6(b), two main oscillation frequencies are detected and f1 ¼ 0.6245 Hz is found to have the largest power in the Fourier spectra. This phenomenon was also as reported by Hintz and Schwabe [3]. In this case, the temperature fluctuation is generated by the interaction of the crystal rotation and surface tension gradient. When the crystal rotation rate exceeds 30 rpm, large temperature fluctuation amplitudes are observed. The corresponding spectrum has multiple frequencies (f1 ¼ f2/2 ¼ f3/3 ¼ …), as illustrated in Fig. 6(c). This type of flow is considered as the rotation-driven flow, the corresponding surface pattern is similar to that in Fig. 3(f). The phenomena of multiple frequencies was also reported in the thermocapillary convection of pure fluid [34] and binary mixture by Yu et al. [35]. 3.3. The effects of combined crystal and crucible rotations Fig. 7 shows the effects of co- and counter-rotation of the crystal and crucible on the thermocapillary convection at nc ¼ 1 rpm and ReT ¼ 91,685. When jns j  20 rpm; the oscillatory flow behaves as the HTWs, which is driven by the surface tension gradient, as shown in Fig. 7(a,b,d,e). In this case, the propagation direction of the HTWs is always along counterclockwise, which is the same direction of the crucible rotation. It is indicated that azimuthal propagation direction of the HTWs is dominated by the crucible rotation. Comparing Fig. 7 with Fig. 3, it can be observed that the shadowgraph of the HTWs with the rotating crucible was dimmer than that in a standing crucible, which hints that the temperature oscillation of the HTWs is suppressed by the crucible rotation. As shown in Ref. [36], the crucible rotation can suppress the

25

thermocapillary flow at small crucible rotation rate. Specifically, at nc ¼ 0 rpm and ns ¼ 45 rpm, the HTWs and the rotating waves coexist in the pool. However, the HTWs at nc ¼ 1 rpm and ns ¼ ±45 rpm are completely suppressed, as shown in Fig. 7(c,f). At a high crystal rotation rate, the HTWs transit to the rotating waves. These rotating waves always travel in the same direction with the crystal rotation. Therefore, the propagation direction of the rotating waves is dominated by the crystal rotation (see Fig. 7(c,f)). The propagation angle of the HTWs is an important characteristic parameter, which is defined as the angle between the temperature gradient direction and the wave propagating normal direction. Fig. 8 shows the variations of the wave number m, the propagation angle q of the HTWs with the crystal rotation rate at different crucible rotation rates. It can be found that the wave number and the propagation angle of the HTWs vary from 25 to 27 and from 41 to 47 approximately at jns j  20 rpm; respectively. Therefore, the effects of the crystal rotation rate on the wave number and the propagation angle are slight. Meanwhile, numerical results in Ref. [26] at Pr ¼ 6.7, d ¼ 2 mm and DT ¼ 10 K in an annular pool are also presented in Fig. 8(a). Obviously, when the crystal is standing, the values of the wave number and the propagation angle of the HTWs in this work is close to these in Ref. [26]. Furthermore, the azimuthal propagation velocity u of the HTWs with the crystal rotation rate at different crucible rotation rates is also given in Fig. 9. Obviously, the azimuthal propagation velocity u of the HTWs is independent of the crystal rotation rate. However, with the increase of the crucible rotation rate, the azimuthal propagation velocity of the HTWs increases, which indicates that it is mainly dominated by the crucible rotation. 4. Experimental uncertainty analysis The combined standard uncertainty of an experimental measurement parameter R is defined as [37]:

1=2
UR ¼ S2R þ B2R

(3)

where SR and BR are the random and systematic standard uncertainties of the parameter R, respectively. SR and BR can be expressed as [35]:

S2R ¼

n  X vR i¼1

vXi

2 sXi

Fig. 6. The Fourier spectra of the temperature oscillation at the monitoring point P at ReT ¼ 36,674 (DT ¼ 3.2 K).

(4)

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T. Shen et al. / International Journal of Thermal Sciences 104 (2016) 20e28

Fig. 7. Snapshots of temperature fluctuations on free surface at nc ¼ 1 rpm and ReT ¼ 91,685 (DT ¼ 8 K).

B2R ¼

n  X vR i¼1

vXi

2 bXi

(5)

respectively. Additionally, the maximum relative error of the pool depth d during the experiment time is 8.3%. 5. Conclusions

where sXi and bXi stand for the random and systematic standard uncertainties of the parameter Xi (i ¼ 1, 2,…, n). Therefore, the relative combined standard uncertainty of the thermocapillary Reynolds number can be expressed as [35]:

UReT ¼ ReT

"

sTs Tc  Ts

2

 þ

sTc Tc  Ts

2

 þ

src rc

2

 þ

bTs Tc  Ts

2

 þ

bTc Tc  Ts

It was reported by Coleman and Steele [38] that 95% of sXi associated with the readability can be taken as half of the minimum scale division. Therefore, the random standard uncertainties in temperature, radius, time, angle in the present experiments are 0.05 K, 0.05 mm, 0.1 s, and 0.25 , respectively. According to above analysis, the relative combined standard uncertainties of oscillation frequency ranges from 5.8% to 15.0%. The maximum relative combined standard uncertainty of the propagation angle of the HTWs and the thermocapillary Reynolds number is 4.9% and 1.3%,

In the present work, the effects of the crystal and crucible rotations on thermocapillary convection in a shallow Cz configuration with silicone oil (Pr ¼ 6.7) are experimentally investigated. The

2

 þ

brc rc

2 #1=2 (6)

basic characteristics of the rotating-thermocapillary convection at various crucible and crystal rotation rates are discussed. From the experimental results, the following conclusions can be drawn: (1) With or without the crucible rotation, the critical thermocapillary Reynolds number decreases with the increase of the crystal rotation rate. Therefore, the crystal rotation can destabilize the thermocapillary convection. Additionally, the thermocapillary flow for the crystalecrucible counter-

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References

θ°

m

T. Shen et al. / International Journal of Thermal Sciences 104 (2016) 20e28

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Fig. 8. Variations of the wave number m (hollow symbols) and propagation angle q (solid symbols) of the HTWs with the crystal rotation rate at ReT ¼ 91,685. Triangle: nc ¼ 0 rpm; circle: nc ¼ 0.5 rpm; square: nc ¼ 1 rpm; star: results in Ref. [26].

Fig. 9. Variations of the azimuthal propagation velocity u of the HTWs with the crystal rotation rate at ReT ¼ 91,685. Triangle: nc ¼ 0 rpm; circle: nc ¼ 0.5 rpm; square: nc ¼ 1 rpm.

rotation is prone to become unstable, as compared to the crystalecrucible co-rotation. (2) When the crystal rotation rate is below 20 rpm, the crystal rotation has slight influence on the wave number, propagation angle and fundamental frequency of the HTWs. As the crystal rotation rate is gradually increased, the rotating waves and the HTWs will coexist in the fluid pool. Finally, the rotating waves occupy the entire pool. The crucible rotation can suppress the temperature fluctuation of the HTWs, and dominates the azimuthal propagation direction of the HTWs. (4) When the thermocapillary force is dominant, the hydrothermal wave instability is responsible for the oscillatory flow. When the effects of the rotation and surface tension gradient are comparable, the oscillatory flow is induced by the mixed circular shear and hydrothermal wave instabilities. Furthermore, the circular shear instability is taken as the mechanism of the oscillatory flow as the rotationdriven oscillatory flow is predominant. Acknowledgments This work is supported by National Natural Science Foundation of China (Grant No. 51176209, 51406019).

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