Experimental study on the flow instability of a binary mixture driven by rotation and surface-tension gradient in a shallow Czochralski configuration

International Journal of Thermal Sciences 118 (2017) 236e246

Contents lists available at ScienceDirect

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

Experimental study on the flow instability of a binary mixture driven by rotation and surface-tension gradient in a shallow Czochralski configuration Jia-Jia Yu, You-Rong Li*, Lu Zhang, Shuang Ye, Chun-Mei Wu 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 21 August 2016 Received in revised form 30 April 2017 Accepted 1 May 2017

A series of experiments have been conducted on the flow instability of a binary mixture driven by the combined surface-tension gradient and rotations of the crystal and crucible in a shallow Czochralski configuration with the aspect ratio and radius ratio of 0.043 and 0.5 respectively. The working fluid was the n-decane/n-hexane mixture with an initial mass fraction of 50%. The results indicate that the hydrothermal waves (HTWs) instability appears when the thermal capillary Reynolds number exceeds a critical value. Furthermore, the critical thermal capillary Reynolds number increases at first, then decreases with the increase of the crystal or crucible rotation rate. At the combined crystal and crucible rotation, it decreases first, and then almost maintains a constant value with the increase of the crystal rotation rate. The Soret effect in the binary mixture makes the flow more unstable than that in the pure fluid. The propagating direction of the HTWs is the same as the crucible rotation direction at the beginning, and then influenced by the crystal rotation with the increase of the crystal rotation rate at a small thermal capillary Reynolds number. In addition, the propagation angle of the HTWs is greatly affected by rotations. © 2017 Elsevier Masson SAS. All rights reserved.

Keywords: Flow instability Binary mixture Rotation Czochralski configuration Experiment

1. Introduction When a tangential temperature gradient is imposed on a fluid layer with a free surface on the ground, the thermocapillarybuoyancy flow driven by the combination of the surface-tension gradient and buoyancy force occurs in the fluid layer. This classic flow frequently arises in many natural and industrial processes. For example, in the Czochralski (Cz) crystal growth, the thermocapillary-buoyancy flow has an important influence on the quality of the crystal materials [1e3]. Up to now, a body of researches have dedicated to the rich flow patterns of the thermocapillary-buoyancy flow of pure fluids in a non-rotation Cz configuration. Li et al. [4,5] conducted a series of three-dimensional (3D) numerical simulations to explore the flow patterns and transition in a Cz configuration with a small aspect ratio (H ¼ depth/radius). They found that the steady twodimensional (2D), steady 3D and oscillatory 3D flows orderly

* Corresponding author. E-mail address: [email protected] (Y.-R. Li). http://dx.doi.org/10.1016/j.ijthermalsci.2017.05.001 1290-0729/© 2017 Elsevier Masson SAS. All rights reserved.

appear with the increase of the radial temperature difference. 3D oscillatory flow is characterized by the traveling spoke pattern in either the clockwise or counterclockwise direction [4]. With a higher aspect ratio, Berdnikov et al. [6] observed the ‘cold pump’ under the crystal by experiments and simulations. Hintz et al. [7] found a surface roll driven by a surface tension which takes up a large part of the fluid volume directly under the free surface and is clearly separated from the rest of the fluid volume. As they reported, the ‘cold pump’ and the separation of the surface roll is mainly due to the buoyancy force. It is a common industrial practice to rotate the crystal or crucible in a Cz crystal growth process. Jing et al. [8,9] explored the thermocapillary-buoyancy flow of an oxide melt in the Cz configuration with the crystal rotation. It revealed that the crystal rotation can suppress the inward flow and result in a wave-spoke pattern. However, this pattern disappears when the crystal rotation rate exceeds a critical value. They believed that the disappearance of the wave-spoke pattern is dominated by rotation-driven convection, which totally agree with previous experimental results [10]. More than that, Lee and Chun [10] predicted three distinct flow regimes in a wide span of Prandtl (Pr) numbers from l02 to 103 which are

J.-J. Yu et al. / International Journal of Thermal Sciences 118 (2017) 236e246

Nomenclature C Bo d D DT g H m n Pr r R Rec ReC Res ReT Rs

mass fraction dynamic Bond number depth, m mass diffusivity of species, m2/s thermo-diffusion coefficient, m2/(K$s) gravitational acceleration, m/s2 aspect ratio azimuthal wave number rotation rate, rpm Prandtl number radius, m radius ratio crucible rotation Reynolds number solutal capillary Reynolds number crystal rotation Reynolds number thermal capillary Reynolds number capillary ratio

rotation-driven flow, baroclinic wave flow and buoyancy-driven flow. Hintz and Schwabe [11] observed a ‘cold jet’ under the crystal and a separated surface tension driven convective roll at the free surface at all fluid depths and all temperature differences between the crystal and crucible with medium Prandtl number fluid. The thermocapillary force strongly affects this flow. In addition, the critical crystal rotation rate increases linearly with the temperature difference, while is independent of the depth of the liquid layer. Nevertheless, despite the enormous progress in understanding the thermocapillary-buoyancy flow in the Cz configuration, most of studies in this field ignore the effects of the co- or counter-rotation of the crystal and crucible on the thermocapillary-buoyancy flow. Carruther and Nassau [12] performed some experiments on the thermocapillary-buoyancy flow at a low thermal capillary Reynolds number in the combined crystal and crucible rotation Cz configuration. They reported two distinct types of the flow, 2D flow in the outer region and 3D flow under the crystal. The distinct flow regime depends on whether the crystal or crucible rotation dominates this flow, and whether rotations of the crystal and crucible are in the same directions or counter directions. Recently, the flow stability diagrams with different rotation directions and rates of the crystal and crucible were shown by Wu et al. [13,14] based on a series of 3D numerical simulations. Furthermore, the transition from the HTWs to the rotating waves was observed by Shen et al. [15] with the increase of the crystal rotation rate and at a constant crucible rotation rate. The working fluids in those studies listed above are pure fluids, while few investigations on the thermocapillary-buoyancy flow of binary mixtures in the Cz configuration are carried out. The thermocapillary-buoyancy flow of binary mixtures in the Cz configuration becomes much more complex due to the solute concentration gradient generated by the Soret effect in binary mixtures. Abbasoglu and Sezai [16] pointed out that the solute concentration distribution due to the Soret effect significantly affects the quality of the crystal, which should not be ignored in the crystal growth process. In our previous studies, we conducted a serious of 3D simulations [17] and experiments [18] to understand the characteristics of the thermocapillary-buoyancy flow of binary mixtures with the Soret effect in a non-rotation annular pool. The experimental results on the flow instability and flow patterns in a rotation Cz configuration, considering the solutal

ST t T

237

Soret coefficient, 1/K time, s temperature,  C

Greek symbols a propagation angle,  bC solutal expansion coefficient bT thermal expansion coefficient, 1/K gC solutal coefficient of surface tension, N/m gT temperature coefficient of surface tension, N/(m$K) n kinematic viscosity, m2/s m dynamic viscosity, kg/(m$s) r density, kg/m3 Subscripts c crucible cri critical s crystal 0 initial

capillary and buoyancy forces, are rarely reported in literatures. However, in Cz crystal growth, the quality of a crystal strongly depends on the flow of the melt, which is driven by the capillary and buoyancy forces, centrifugal and Coriolis forces due to the rotations of the crystal and crucible. Herein, we will report a series of experiments on the thermocapillary-buoyancy flow of a binary mixture with the Soret effect in a rotation shallow Cz configuration. A direct comparison with the results in pure fluids will be carried out to reveal the effects of the rotation and the Soret effect on the flow instability. Therefore, understanding the complex flow patterns and instabilities of binary mixtures in a rotation Cz configuration will help to deal with the radial segregation of one component in binary melts [16] and improve the crystal qualities during the Cz crystal growth.

2. Experiments 2.1. Experimental apparatus Fig. 1 shows the schematic diagram of the experimental setup, which is an improved version of the experimental apparatus in Ref. [18]. Especially, the shallow annular pool is replaced by a copper cylindrical pool and a copper disk on the free surface of working fluids. This experimental setup is applied to imitate the Cz configuration. In the following sections, the cylindrical pool and disk are called as the crucible and crystal, respectively. The radius of the crucible and crystal are rc ¼ 46 ± 0.1 mm and rs ¼ 23 ± 0.1 mm, respectively. The depth of the fluid layer in the Cz configuration is 2 mm. The rim-edge of the crucible is finished sharply to avoid the effect of meniscus to obtain a relatively flat liquid layer. A step plane at the inner rim of the crucible [19] is processed to prevent the formation of meniscus, as shown in the inset to Fig. 1. The bottom of the crucible is manufactured by plexiglass with thickness 10 ± 0.1 mm for its good heat insulation property. Ref. [4] reported that the flow instabilities are not significantly affected by the small vertical heat flux. Therefore, in the present work, the bottom is considered to be adiabatic. The position of the disk is adjusted by an automatic lifting platform with a precision of ±3 mm. Temperatures Tc and Ts (Ts < Tc) of the crucible sidewall and crystal are controlled and adjusted by circulating hot and cold water from two thermostatic baths with different constant

238

J.-J. Yu et al. / International Journal of Thermal Sciences 118 (2017) 236e246

Fig. 1. The experimental apparatus.

temperatures. The temperature fluctuation of the thermostatic baths is ±0.1 K. Eight T-type thermocouples with a precision of ±0.1 K are applied to measure the temperature difference. Two typical thermocouples are displayed in Fig. 1. All temperature data are obtained through the conversion of HP data acquisition (HP 34970A). An infrared thermal imager (FLIR SC325 320  240 pixels) with an accuracy of ±0.05 K is placed right above the Cz configuration, which is applied to record the snapshots of temperature on the free surface in real time. Two servo motors (Panasonic, MSMD100W and MSMD-400W) whose resolution ratio of rotation is 0.036 are available to well control the rotations of the crucible and crystal. A vibration isolation platform is used to weaken the influence of vibration as much as possible. N-hexane (pure fluid), n-decane/n-hexane (binary mixture with an initial mass fraction of 50%) are chosen as working fluids. Their detailed physical properties are referred to Refs. [17,18] and listed in Table 1. The n-decane and n-hexane (analytical pure) were purchased from Chongqing Chuandong Chemical (Group) Co., Ltd. The ambient temperature is maintained at 25  C. The temperature difference between the crucible and crystal is limited to 12  C to keep the working fluids cold and relieve the rate of evaporation. Furthermore, the evaporation fluid volume of n-hexane and ndecane/n-hexane is tiny enough to be ignored in the experiments

for the small evaporation rate, short experimental time and relative low temperature [18,20]. But, the heat loss at the free surface which is due to the evaporation heat transfer to the environment slightly stabilizes the flow. The procedure of experiments is shown as follows. Firstly, the fluid layer is established by the automatic lifting platform. Secondly, a stable rotation is controlled by two servo motors, which is necessary before observing flow patterns. Thirdly, the copper walls of the crystal and crucible are cooled and heated by the circulating temperature-controlled water. In the meantime, temperature difference is increased. Lastly, the infrared thermal imager and thermocouples record the temperature fluctuation on the free surface at 60 Hz and temperature difference between the crystal and crucible at 1.3 Hz, respectively. 2.2. Data reduction The surface tension s and density r are allowed to vary linearly according to the temperature and n-decane concentration, which can be expressed as:

sðT; CÞ ¼ s0  gT ðT  T0 Þ  gC ðC  C0 Þ;

(1)

Table 1 Physical properties of n-hexane and n-decane/n-hexane mixture at 25  C. Property

Unit

Value n-hexane

n-decane/n-hexane

Density, r Thermal diffusivity, a Mass diffusivity of species, D Soret coefficient, ST Dynamic viscosity, m Thermal expansion coefficient, bT Solutal expansion coefficient, bC Temperature coefficient of surface tension, gT Solutal coefficient of surface tension, gC Prandtl number, Pr Lewis number, Le

kg/m3 m2/s m2/s 1/K kg/(m$s) 1/K e N/(m$K) N/m e e

654.79 8.56  108 e e 2.96  104 1.36  103 e 1.02  104 e 5.29 e

689.82 7.50  108 2.69  109 2.42  103 4.70  104 1.20  103 0.103 6.98  105 5.19  103 9.08 27.89

J.-J. Yu et al. / International Journal of Thermal Sciences 118 (2017) 236e246

rðT; CÞ ¼ r0 ½1  bT ðT  T0 Þ  bC ðC  C0 Þ;

(2)

where s0 ¼ s(T0, C0), r0 ¼ r(T0, C0), gT ¼ (vs/vT)C, gC ¼ (vs/vC)T, bT ¼ (vr/vT)C/r0 and bC ¼ (vr/vC)T/r0. T0 and C0 are the reference temperature and solute concentration, which are 25  C and 50% respectively. For a binary mixture, the solute concentration difference DC arises for the Soret effect, which is calculated by the following equation [21,22].

DC ¼ Cc  Cs ¼ ST C0 ð1  C0 ÞðTc  Ts Þ

(3)

where ST ¼ DT/D is the Soret coefficient. D and DT are the mass diffusivity and thermo-diffusion coefficient respectively. The direction of separation due to the Soret effect was studied by Ecenarro et al. [23] based on twelve binary mixtures, whose components had molecules with different structures: rings, chains, tetrahedral. They concluded that the component with higher density in the binary mixture gathers at the cold area. Li et al. [24] and Platten [25] also pointed out that DC is the solute concentration of the heavier component in Eq. (3). Yu et al. [17] observed that the concentration of n-decane near the cold wall is higher than that near the hot wall in a shallow annular pool (see Figs. 2 and 3 in Ref. [17]). Several dimensionless parameters, ReT, ReC, Res, Rec, H, R and Rs, are defined as the following equations

ReT ¼ gT rc ðTc  Ts Þ=ðvmÞ; Res ¼ 2prs ns rc =60v; H ¼ d=rc ;

ReC ¼ gC rc DC=ðvmÞ;

. Rec ¼ 2pnc rc2 60v;

R ¼ rs =rc ;

Rs ¼ ReC =ReT ¼ gC DC=ðgT DTÞ:

(4) (5) (6) (7)

Bo ¼

rgbT d2 ; gT

239

(9)

where g is the gravitational acceleration (g ¼ 9.81 m$s2). It has been successfully predicted that the different regimes of the thermocapillary-buoyancy flow of pure fluids in an annular pool [27] and a rotating Cz configuration [15]. The dynamic Bond numbers of the binary mixture and pure fluid are respectively 0.47 and 0.34 at d ¼ 2 mm. Therefore, the thermal capillary force is dominant [15,27]. 3. Experimental uncertainty analysis The experimental uncertainty analysis method refers to Refs. [15,18]. The combined standard uncertainty of an experimental measurement variable R can be expressed as:

1=2  UR ¼ S2R þ B2R

(10)

where SR and BR represent the random and systematic standard uncertainties of the experimental measurement variable R respectively. The details of this method are described in Refs. [15,18]. Hugh et al. [28] reported that the 95% of the random standard uncertainty associated with the readability of an analog instrument can be taken as one-half of the least scale division. Therefore, the random standard uncertainties in temperatures of thermocouples and the infrared thermal imager are 0.05 K and 0.025 K, respectively. In addition, the random standard uncertainties of the radius, time and angle are 0.05 mm, 0.008s and 0.5 . According to Eqs. (4), (5), (7) and (10) in Ref. [18], the maximum relative combined standard uncertainties of the thermal capillary Reynolds number, temperature and propagation angle are 14.3%, 23.5% and 2.2%, respectively.

Substituting Eq. (3) into Eq. (7), we have

Rs ¼ gC ST C0 ð1  C0 Þ=gT :

4. Results and discussion

(8) 4.1. Only crystal or crucible rotation

Here, ReT and ReC are the thermal and solutal capillary Reynolds numbers. Res and Rec are the rotation Reynolds numbers of the crystal and crucible respectively. H, and R are the aspect ratio and radius ratio of the Cz configuration. Rs is the capillary ratio, which is used to measure the relationship between the solutal and thermal capillary effects [17,26]. For n-decane/n-hexane mixture with the initial mass fraction of 50%, the capillary ratio Rs is 0.045. The maximum temperature difference between the crystal and crucible is 12 K. The ranges of the crystal and crucible rotation rates are 0 ~ ±24.55 rpm and 0 ~ 7.67 rpm respectively. The depth of the fluid layer in crucible is 2 mm. Therefore, the maximum thermal capillary Reynolds number is 1.2  105. The rotation Reynolds numbers for the crystal and crucible, Res and Rec, range from 0 to ±4000 and 0 to 2500 respectively. The aspect ratio H and radius ratio R of the Cz configuration are respectively 0.043 and 0.5. When a horizontal temperature gradient is imposed on pure fluid layers without any rotations, the buoyancy flow is weak for a shallow pool, but for a large aspect ratio, the effect of buoyancy force is found to be important [27]. In addition, the buoyancy force is enhanced in the binary mixture due to the solute concentration difference generated by the Soret effect [17,18]. The dynamic Bond number is widely used to evaluate the relative strength between the capillary and buoyancy effects, which is defined as follows [15,27]:

4.1.1. Critical condition for the incipience of 3D oscillatory flow When the thermal capillary Reynolds number ReT is small, the thermocapillary-buoyancy flow is axisymmetric and steady, which is similar to that in pure fluids in Cz configurations [13,14] and binary mixtures with Soret effect in a shallow annular pool [17,18]. This typical flow is named as “basic flow”. When ReT exceeds a certain threshold value, the basic flow destabilizes to a 3D oscillatory flow. The amplitudes of the incubated 3D disturbances are amplified gradually [17]. The 3D oscillatory flow can be characterized by traveling fluctuations in the azimuthal direction with a constant angular velocity. This threshold value is defined as the critical thermal capillary Reynolds number (ReT, cri), which can be calculated by the following equation:

 ReT; cri ¼ gT rc ðTc  Ts Þcri ðvmÞ:

(11)

Here, (TcTs) cri is measured by increasing the temperature difference between the crystal and crucible slowly until the 3D oscillatory flow is observed by the infrared thermal imager [18]. Crystal or crucible rotation plays an important role on the critical thermal capillary Reynolds number for the transition from the basic flow to a 3D oscillatory flow, as shown in Fig. 2. The critical thermal capillary Reynolds numbers of the binary mixture and pure fluid are 2.7  104 and 3.7  104 in a non-rotation Cz configuration,

240

J.-J. Yu et al. / International Journal of Thermal Sciences 118 (2017) 236e246

Fig. 2. Variations of the critical thermal capillary Reynolds numbers of the binary mixture and pure fluid with the crystal (a) or crucible rotation Reynolds number (b).

respectively. These values are about two times as large as the critical results in an annular pool filled with the same working fluids (see Fig. 8 in Ref. [17] and Fig. 5 in Ref. [18]). This discrepancy is due to the different geometry that the characteristic length in this work is much larger than that in the annular pool in Refs. [17,18] and the flow under the crystal is very complex [6,12e14,29]. For both the binary mixture and pure fluid, the critical thermal capillary Reynolds number increases at first, then decreases with the increase of the crystal or crucible rotation, as shown in Fig. 2. For the Cz configuration with the crystal rotation, when the crystal rotation Reynolds number is small, the centrifugal force generated by the crystal rotation suppresses the radial inward flow driven by the thermal capillary force in pure fluids [4,10,11,13,14] or thermal and solutal capillary forces in binary mixtures [17,18]. Therefore, the flow stability is improved by the centrifugal force. However, when the crystal rotation Reynolds number is greater than 2000 in the binary mixture and 1000 in the pure fluid, the crystal rotation destabilizes the flow. Ozoe et al. [29] found that the cold plume from the periphery of the rotation crystal merges to the central cold plume. The cold plume descends from the bottom center of the crystal to the bottom of the crucible. The cold plume is enhanced with the increase of the crystal rotation rate, which is an essential element for the flow destabilization [6,29]. It is speculated that the role of the cold plume is more dominant than the centrifugal force, when the crystal rotation Reynolds number is over 2000 in the binary mixture and 1000 in the pure fluid. It is noted that the variation rate of the critical thermal capillary Reynolds number with the variation of the crystal rotation Reynolds number in the pure fluid is larger than that in the binary mixture. It means that the

pure fluid is more sensitive to the crystal rotation than the binary mixture. The solute concentration near the crystal is higher than that near the crucible for the Soret effect in binary mixtures [17,18,23,25]. The solutal capillary force has the same direction with thermal capillary force, which enhances the inward capillary flow [17,18], compared with the pure fluid. However, dot dashed line in Fig. 2(a) from Ref. [15] presents a decreasing trend of the critical thermal capillary Reynolds number with the crystal rotation rate, which is partly similar to the results of the binary mixture. When the crucible rotation Reynolds number is relatively small, the radial inward flow is also suppressed by the centrifugal force due to the crucible rotation, as shown in Fig. 2(b). It's a common conclusion in pure fluids reported in Refs. [15,30] at a low crucible rotation Reynolds number. When the crucible rotation Reynolds number is over 1500 in both the binary mixture and pure fluid, the critical thermal capillary Reynolds number decreases with the future increase of the crucible rotation Reynolds number. The direction of HTWs propagation is against the surface flow, which is resulting from the combination of the radial flow and the azimuthal flow generated by the Coriolis force. The HTWs in the Cz configuration with the crucible rotation can receive the energy more effectively [31,32]. The critical thermal capillary Reynolds number decreases with the increase of the crucible rotation Reynolds number. Based on the variations of the critical thermal capillary Reynolds number shown in Fig. 2, it's reasonable to speculate that the flow instability also can be driven by the crystal or crucible rotation in the binary mixture even though the temperature gradient is not considered. It has been well predicted in pure fluids [33,34], while it has not been reported in binary mixtures. In Fig. 2, the critical thermal capillary Reynolds number in the binary mixture is always lower than that in the pure fluid, because the n-decane gathers near the crystal for the Soret effect in the binary mixture. The capillary flow is enhanced by the coupling thermal and solutal capillary forces in the same direction [17,18]. 4.1.2. Characteristics of 3D oscillatory flow Fig. 3 shows the snapshots of the temperature fluctuation on the free surface of the binary mixture at ReT ¼ 6.0  104 and Rec ¼ 0. When the thermal capillary Reynolds number exceeds the critical value, moving spoke patterns are observed on the free surface in the binary mixture. This flow pattern is similar to the result in pure fluids in a Cz configuration [4,5,30,35] and binary mixtures with Soret effect in a shallow annular pool [17,18]. Smith and Davis [35] first predicted the HTWs instability in pure fluids by the linear stability analysis. The variation of the propagation angle a of the HTWs with the Prandtl number was exhibited in Fig. 14 of Ref. [35]. From Fig. 3, it can be found that the propagation angle of the moving spoke patterns ranges from 35 to 48 , which is close to 35 predicted for the pure fluid of Pr z 9 in Ref. [35]. The difference could be attributed to the absence of the solutal capillary force and effect of the geometry of the pool. Therefore, the flow instability should be referred to the HTWs instability which was well defined in Ref. [35], even though the Soret effect, crystal and crucible rotations are taken into consideration in a Cz configuration filled with a binary mixture in the experimental range. Since the thermocapillary-buoyancy flow is dominant in the experimental range, the HTWs is the only flow pattern for the medium Prandtl fluids. Nevertheless, the HTWs being invaded by the rotating waves reported in Ref. [15] was not observed for two reasons. Primarily, Res in this work is much smaller than that in Ref. [15], which is one-third of Res used in Ref. [15]. Although the rotation in the present work does not shake the dominance of the HTWs instability, it affects the critical thermal capillary Reynolds number and characteristic parameters of the HTWs. The high

J.-J. Yu et al. / International Journal of Thermal Sciences 118 (2017) 236e246

rotation rate in Ref. [15] is not available in the real Cz crystal growth industry [36,37]. Secondly, the solutal capillary effect plays an important role in enhancing the capillary flow in the binary mixture, which is absent in the pure fluid in Ref. [15]. Comparing the moving spokes marked by the dashed arrow in Fig. 3(a) and (b), we find that the curve fringes of the HTWs are stretched by the increasing crystal rotation, especially near the edge of the crystal. The effect of the crystal rotation on the flow becomes more important with the decrease of the distance from the crystal. Shen et al. [15] and Son et al. [38] also reported this phenomenon. Fig. 4 shows the time dependencies of the temperature at the monitoring point P (in the center of the free surface and shown in Fig. 3(a)) at ReT ¼ 5.0  104 and Rec ¼ 0 on different crystal rotation Reynolds numbers. The temperature oscillation at the monitoring point P is obtained by the infrared camera FLIR SC325. When the crystal rotation Reynolds numbers is small, the temperature fluctuation is also small and relatively periodic, as shown in Fig. 4(a). The capillary flow is much stronger than the flow generated by the centrifugal force due to the crystal rotation. Fig. 4(b) exhibits an irregular and small-amplitude temperature oscillation signal at Res ¼ 1500. However, a reverse transition that the temperature fluctuation becomes regular and sharp is observed at Res ¼ 4000, as shown in Fig. 4(c). In this case, the cold plume from the periphery of the crystal is further strengthened by the crystal rotation and merges to the central cold plume generated by the buoyancy [39]. The strong cold plume descends from the bottom of the crystal to the bottom of the crucible, which enhances the thermocapillarybuoyancy flow and promotes the flow destabilization [29,39]. Therefore, there is a reverse transition of the temperature fluctuation with the increase of the crystal rotation rate. As reported by Shen in Ref. [15], when the crystal rotation rate is large enough, the regular rotating waves appear in the pool and the temperature fluctuations become very sharp, as shown in Fig. 4 in Ref. [15]. Although the thermocapillary-buoyancy flow is dominant, as shown in Fig. 3, it is reasonable to speculate that the centrifugal force affects the thermocapillary-buoyancy flow. Fig. 5 displays the effect of the crucible rotation on the temperature fluctuation on the free surface at a relatively high thermal capillary Reynolds number. When the crucible rotation Reynolds number is small, two groups of the HTWs which are propagating in opposite direction always coexist, as shown in Fig. 5(a). It is similar to the results of binary mixtures in an annular pool [17,18] and the results of pure fluids in a non-rotation Cz configuration [4]. At Rec ¼ 2500, the HTWs become irregular and rough, which is pointed out by the dashed arrow in Fig. 5(b). It's very interesting

241

that some cells near the crucible can be found at a high crucible rotation Reynolds number, as illustrated by the dot dashed arrow in Fig. 5(b). These cells propagate in the same direction with the crucible rotation. As shown in Refs. [17,18], when the aspect ratio is larger than 0.11, some cells near the outer cylinder are generated by the buoyancy force. The aspect ratio, H ¼ 0.043, is much smaller than 0.11 for the same working fluid. Thus, these weak moving cells near the crucible are possibly due to the crucible rotation at a large crucible rotation Reynolds number. These propagating cells near the crucible become stronger and more obvious with the further increase of the crucible rotation Reynolds number. Similar evolution of the flow pattern with the increase of the crystal rotation rate or the crucible rotation rate can be observed in the pure fluid. The variation of the wave number m and propagation angle a of the HTWs with the crystal or crucible rotation is shown in Fig. 6. Obviously, the propagation angle of the HTWs in the binary mixture and pure fluid increases first, and then decreases with the increase of the crystal rotation rate. However, the crystal rotation has a slight influence on the wave number of the HTWs for both the binary mixture and pure fluid. Shen et al. [15] also reported that the wave number is independent of the crystal rotation. Furthermore, the propagation angle and wave number of the HTWs in the binary mixture or pure fluid show a diminishing tendency with the increase of the crucible rotation. Comparing the binary mixture and the pure fluid, it can be found that this tendency of the pure fluid is more obvious than that of the binary mixture, especially in Fig. 6(a) and (c). It is speculated that the capillary effect in the binary mixture is enhanced by the solute concentration gradient. The thermocapillary-buoyancy flow in the pure fluid is much easier to be affected by the crystal or crucible rotation than that in the binary mixture. The propagation angle and the wave number of the pure fluid are lager and smaller than those of the binary mixture respectively, which agrees with the conclusion in Refs. [17,18]. When the thermal capillary Reynolds number increases, the propagation angle and wave number decreases, which is also reported in an annular pool in Ref. [18]. Furthermore, these two characteristic parameters of the binary mixture are much more sensitive to the temperature variation than those of the pure fluid for the coupling effect between the temperature and solute concentration gradients [17,18]. 4.2. Combined crystal and crucible rotation 4.2.1. Critical condition for the incipience of 3D oscillatory flow Fig. 7 shows the influence of co- and counter-rotation of the

Fig. 3. Snapshots of temperature fluctuation on the free surface of the binary mixture at Rec ¼ 0 and ReT ¼ 6.0  104. (a) Res ¼ 1000; (b) Res ¼ 4000.

242

J.-J. Yu et al. / International Journal of Thermal Sciences 118 (2017) 236e246

as displayed in Fig. 7. It is different from that of the binary mixture at the combined crystal and crucible rotation, while it is similar to that at the only crystal or crucible rotation, comparing Fig. 2 with Fig. 7. Here, the crucible rotation Reynolds number is 1000, which is close to the locations of the peaks shown in Fig. 2(b). In addition, the Soret effect in the binary mixture facilitates the flow destabilization [17,18]. The flow in the binary mixture at Rec ¼ 1000 is very sensitive to any unstable element. Once the crystal rotates, the critical thermal capillary Reynolds number decreases. However, the flow in the pure fluid is more stable than that in the binary mixture. The cold plume from the periphery of the rotation crystal in the pure fluid is weaker than that in the binary mixture at the same crystal rotation Reynolds number for the Soret effect. Therefore, the role of the crystal rotation on the flow destabilization in the pure fluid is different from that in the binary mixture at the combined crystal and crucible rotation, while it is similar to that at the only crystal or crucible rotation. The critical thermal capillary Reynolds number in the binary mixture is smaller than that in the pure fluid, as depicted in Fig. 7. The solute concentration difference generated by the Soret effect in the binary mixture is responsible for this difference. Therefore, the Soret effect in the binary mixture has an impact on accelerating the flow transition from the basic flow to 3D oscillatory flow in a non-rotation annular pool [17,18], crystal or crucible rotation Cz configuration (see Fig. 2) and combined crystal and crucible rotation Cz configuration (see Fig. 7).

Fig. 4. The time dependencies of the temperature at the monitoring point P in the binary mixture at Rec ¼ 0 and ReT ¼ 5.0  104.

crystal and crucible on the critical thermal capillary Reynolds number for the incipience of 3D oscillatory flow in the binary mixture and pure fluid at Rec ¼ 1000. The critical thermal capillary Reynolds number of the binary mixture decreases at the beginning, and then keeps constant with the increase of the crystal rotation Reynolds number at the combined crystal and crucible rotation. It agrees with the results in Ref. [40] (Fig. 3 in Ref. [40]). However, the critical thermal capillary Reynolds number of the pure fluid increases at first, then decreases, as the crystal Reynolds number is increased. Shen et al. [15] observed that the critical thermal capillary Reynolds number of a pure fluid presents a diminishing trend,

4.2.2. Characteristics of 3D oscillatory flow Fig. 8 shows the effects of the thermal capillary Reynolds number, co- and counter-rotation rates of the crystal and crucible on the temperature fluctuation patterns on the free surface. At a small thermal capillary Reynolds number, the HTWs only propagate along the anticlockwise direction in the inner four-fifths of the radial length of the free surface, which fade in the outer region, as shown in Fig. 8(a). With the increase of the thermal capillary Reynolds number, the HTWs become much clearer and stronger because of the enhanced capillary force, as illustrated in Fig. 8(b). In addition, two groups of the HTWs in opposite directions with the wave source and sink were observed, which are also reported in Refs. [17,18] and clearly explained by Garnier et al. [41]. The propagation direction of the HTWs is mainly influenced by the crucible rotation at a low thermal capillary Reynolds number. However, this role of the rotation crucible on the propagation direction is weakened by the enhanced capillary force with the increase of the thermal capillary Reynolds number. Of course, this phenomenon can also be observed with only crucible rotating. The typical example of the pure fluid is shown in Fig. 8(c) and (d). In addition, Shen et al. [15] also obtained similar results by experiments (see Fig. 7 in Ref. [15]). Comparing Fig. 8 (a) and (e, f), it is found that two groups of the HTWs in opposite directions coexist in the case of the counter-rotation of the crystal and crucible at a high crystal rotation Reynolds number. Within the experimental range, two groups of the HTWs in opposite directions are only observed in the cases where the thermal capillary Reynolds numbers are high, as exhibited in Fig. 8(b) and (d), or the crystal and crucible rotate in the contrary directions at a high crystal rotation rate, as shown in Fig. 8(f). Otherwise, the propagation direction of the HTWs is same as the crucible rotation, as displayed in Fig. 8 (a), (c) and (e). Fig. 9 shows the time dependencies of the temperature at the monitoring point P at the combined crystal and crucible rotation. Whatever the rotation direction of the crystal is, the temperature fluctuation becomes sharper and the amplitude of the temperature fluctuation becomes larger with the increase of the crystal rotation Reynolds number. This variation of the temperature fluctuation with the crystal rotation Reynolds number is similar to the results when the crystal only rotates. Shen et al. [15] and Schwabe et al.

J.-J. Yu et al. / International Journal of Thermal Sciences 118 (2017) 236e246

243

Fig. 5. Snapshots of temperature fluctuation on the free surface in the binary mixture at Res ¼ 0 and ReT ¼ 6.0  104. (a) Rec ¼ 500; (b) Rec ¼ 2500.

[42] also reported similar evolution of the temperature oscillation signal when the crystal rotation plays an important role on the flow instability (see Fig. 4 in Ref. [15] and Fig. 1 in Ref. [42]). The wave propagation angle a and wave number m of the HTWs at the combined crystal and crucible rotation are revealed in Fig. 10. The wave propagation angle and wave number of the HTWs in the binary mixture present a decreasing tendency with the increase of the crystal rotation Reynolds number, which range from 45 to 35 and from 51 to 29 in the co- and counter-rotation system at different thermal capillary Reynolds numbers, respectively. In addition, these tendencies of the wave propagation angle and wave number in the binary mixture are becoming more and more

obvious with the increase of the thermal capillary Reynolds number. However, they slightly decrease with the increase of the thermal capillary Reynolds number, which is similar to the results in shallow annular pool in Ref. [18] and the results at crystal or crucible rotation (see Fig. 6). These tendencies of the propagation angle and wave number are different from those reported by Shen et al. [15] that the influence of the crystal rotation on the propagation angle and wave number are slight in a co- or counter Cz configuration. Nevertheless, a higher crystal rotation Reynolds number results in a lower wave number at the combined crystal and crucible rotation, which agrees with the simulation results in Ref. [34]. In the pure fluid, the wave propagation angle decreases at first, and

Fig. 6. Variations of the propagation angle a (a and c) and wave number m (b and d) of the HTWs of the binary mixture and pure fluid with crystal (a and b) or crucible rotation Reynolds number (c and d). ,: pure fluid, Rec ¼ 0, ReT ¼ 8.1  104; -: binary mixture, Rec ¼ 0, ReT ¼ 7.0  104; B: pure fluid, Res ¼ 0, ReT ¼ 8.1  104; C: binary mixture, Res ¼ 0, ReT ¼ 7.0  104; △: pure fluid, Res ¼ 0, ReT ¼ 1.4  105; :: binary mixture, Res ¼ 0, ReT ¼ 9.0  104.

244

J.-J. Yu et al. / International Journal of Thermal Sciences 118 (2017) 236e246

mixture with an initial mass fraction of 50% as the working fluid. The basic characteristics of the thermocapillary-buoyancy flow have been discussed and compared with these in pure fluids at various rotation rates of crucible and crystal. Based on previous works in an annular pool [17,18], the Prandtl number (5.52  Pr  9.08) of working fluids does not strongly affect the features of the thermocapillary-buoyancy flow of binary mixtures with the Soret effect. This work is, by no means, complete since some questions remain unanswered. Therefore, we reach the following conclusions of the binary mixture with the medium Prandtl number.

Fig. 7. Variations of the critical thermal capillary Reynolds number in the binary mixture and pure fluid with the crystal rotation Reynolds number at the combined crystal and crucible rotation. >: Rec ¼ 340, Res > 0, Ref. [15]; *: Rec ¼ 340, Res > 0, Ref. [15]; ,: pure fluid, Rec ¼ 1000, Res > 0; B: pure fluid, Rec ¼ 1000, Res < 0; -: binary mixture, Rec ¼ 1000, Res > 0; C: binary mixture, Rec ¼ 1000, Res < 0.

then increases with the increase of the crystal rotation Reynolds number at the co- or counter-rotation, while the wave number is about 35 and independent on the crystal rotation. Compared with the results in the binary mixture, the wave propagation angle and wave number in the pure fluid are bigger and smaller respectively than those in the binary mixture, which is also reported in Fig. 4 of Ref. [18]. 5. Conclusions In order to assess the influence of the crystal and crucible rotations on the flow instability in a shallow Cz configuration, a series of experiments have been performed using the n-decane/n-hexane

(1) For the binary mixture, the critical thermal capillary Reynolds number for the incipience of the 3D oscillatory flow increases at first, then decreases with the increase of the crystal or crucible rotation Reynolds number. At the combined crystal and crucible rotation, it decreases at first, then maintains a constant value with the increase of the crystal rotation Reynolds number. There is no obvious difference for the critical thermal capillary Reynolds number between the co- and counter-rotation. (2) Due to the Soret effect in the binary mixture, the critical thermal capillary Reynolds number in the binary mixture is always smaller than that in the pure fluid. At the crystal or crucible rotation only, the variation tendencies of the critical thermal capillary Reynolds number in the binary mixture are the same as those in the pure fluid. At the combined crystal and crucible rotation, the critical thermal capillary Reynolds number of the pure fluid increases at first, then decreases with the increase of the crystal rotation Reynolds number, which is different with that of the binary mixture. (3) For the binary mixture, two groups of the HTWs propagating along opposite directions can be observed in the cases that

Fig. 8. Snapshots of temperature fluctuation on the free surface in the binary mixture and pure fluid at the combined crystal and crucible rotation. (a) binary mixture, Rec ¼ 1000, Res ¼ 500, ReT ¼ 4.0  104; (b) binary mixture, Rec ¼ 1000, Res ¼ 500, ReT ¼ 1.1  105; (c) pure fluid, Rec ¼ 1000, Res ¼ 0, ReT ¼ 5.4  104; (d) pure fluid, Rec ¼ 1000, Res ¼ 0, ReT ¼ 1.1  105; (e) binary mixture, Rec ¼ 1000, Res ¼ 2000, ReT ¼ 4.0  104; (f) binary mixture, Rec ¼ 1000, Res ¼ 4000, ReT ¼ 4.0  104.

J.-J. Yu et al. / International Journal of Thermal Sciences 118 (2017) 236e246

245

Fig. 10. Variations of the propagation angle a (a) and wave number m (b) of the HTWs in the binary mixture (solid symbols) and pure fluid (hollow symbols) at the combined crystal and crucible rotation at Rec ¼ 1000.

fluid, while there are some differences in the characteristic parameters of the HTWs. When the crystal or crucible rotates only, the propagation angle of the HTWs of the binary mixture is smaller than that of the pure fluid, while the wave number is in contras. At the combined crystal and crucible rotation, the difference in the wave number of the HTWs between the binary mixture and pure fluid is related to the crystal rotation Reynolds number. Fig. 9. The time dependencies of the temperature at the monitoring point P in the binary mixture at Rec ¼ 1000 and ReT ¼ 4.0  104.

the thermal capillary Reynolds number is high or the crystal and crucible rotate in the contrary directions at a high crystal rotation rate. Otherwise, the propagation direction of the HTWs is the same as that of the crucible rotation. Some moving cells near the crucible are attributed to the crucible rotation at a relatively high crucible rotation Reynolds number. The propagation angle of the HTWs increases at first, then decreases with the increase of the crystal rotation Reynolds number, while the wave number is independent of the crystal rotation. At the combined crystal and crucible rotation, the propagation angle and wave number present a decreasing tendency with the increase of the crystal rotation Reynolds number. (4) The effects of the crystal and crucible rotations on the flow pattern in the binary mixture are similar to those in the pure

Acknowledgements This work is supported by National Natural Science Foundation of China (Grant No. 51406019) and Chongqing University Postgraduates' Innovation Project (Grant No: CYB15015). We thank Dr. Yimin Ding from The Pennsylvania State University for helpful grammatical edits. References [1] Chang CE, Wilcox WR. Analysis of surface tension driven flow in floating zone melting. Int J Heat Mass Transf 1976;19:355e66. [2] Azami T, Nakamura S, Eguchi M, Hibiya T. The role of surface-tension-driven flow in the formation of a surface pattern on a Czochralski silicon melt. J Cryst Growth 2001;233:99e107. [3] Uecker R, Wilke H, Schlom DG, Velickov B, Reiche P, Polity A, et al. Spiral formation during Czochralski growth of rare-earth scandates. J Cryst Growth 2006;295:84e91. [4] Li YR, Quan XJ, Peng L, Imaishi N, Wu SY, Zeng DL. Three-dimensional thermocapillary-buoyancy flow in a shallow molten silicon pool with Cz configuration. Int J Heat Mass Transf 2005;48:1952e60. [5] Peng L, Li YR, Liu YJ, Imaishi N, Jen TC, Chen QH. Bifurcation and hysteresis of flow pattern transition in a shallow molten silicon pool with Cz configuration.

246

J.-J. Yu et al. / International Journal of Thermal Sciences 118 (2017) 236e246

Numer Heat Tr A-Appl 2007;51:211e23. [6] Berdnikov VS, Prostomolotov AI, Verezub NA. The phenomenon of “cold plume” instability in Czochralski hydrodynamic model: physical and numerical simulation. J Cryst Growth 2014;401:106e10. [7] Hintz P, Schwabe D, Wilke H. Convection in a Czochralski crucible - Part 1: non-rotating crystal. J Cryst Growth 2001;222:343e55. [8] Jing CJ, Tsukada T, Hozawa M, Shimamura K, Ichinose N, Shishido T. Numerical studies of wave pattern in an oxide melt in the Czochralski crystal growth. J Cryst Growth 2004;265:505e17. [9] Jing CJ, Imaishi N, Sato T, Miyazawa Y. Three-dimensional numerical simulation of oxide melt flow in Czochralski configuration. J Cryst Growth 2000;216: 372e88. [10] Lee YS, Chun CH. Prandtl number effect on traveling thermal waves occurring in Czochralski crystal growth. Adv Spac Res 1999;24:1403e7. [11] Hintz P, Schwabe D. Convection in a Czochralski crucible - Part 2: rotating crystal. J Cryst Growth 2001;222:356e64. [12] Carruthers JR, Nassau K. Nonmixing cells due to crucible rotation during Czochralski crystal growth. J Appl Phys 1968;39:5205e14. [13] Wu CM, Li YR, Liao RJ. Rotating and thermocapillary-buoyancy-driven flow in a cylindrical enclosure with a partly free surface. Phys Fluids 2014;26:104105. [14] Wu CM, Ruan DF, Li YR, Liao RJ. Flow pattern transition driven by the combined Marangoni effect and rotation of crucible and crystal in a Czochralski configuration. Int J Therm Sci 2014;86:394e407. [15] Shen T, Wu CM, Li YR. Experimental investigation on the effect of crystal and crucible rotation on thermocapillary convection in a Czochralski configuration. Int J Therm Sci 2016;104:20e8. [16] Abbasoglu S, Sezai I. Three-dimensional modelling of melt flow and segregation during Czochralski growth of Ge x Si1x single crystals. Int J Therm Sci 2007;46:561e72. [17] Yu JJ, Li YR, Wu CM, Chen JC. Three-dimensional thermocapillary-buoyancy flow of a binary mixture with Soret effect in a shallow annular pool. Int J Heat Mass Transf 2015;90:1071e81. [18] Yu JJ, Ruan DF, Li YR, Chen JC. Experimental study on thermocapillary convection of binary mixture in a shallow annular pool with radial temperature gradient. Exp Therm Fluid Sci 2015;65:79e86. [19] Schwabe D, Zebib A, Sim BC. Oscillatory thermocapillary convection in open cylindrical annuli. Part 1. Experiments under microgravity. J Fluid Mech 2003;491:239e58. [20] Xu JJ, Davis SH. Convective thermocapillary instabilities in liquid bridges. Phys Fluids 1984;27:1102e7. [21] Blanco P, Polyakov P, Bou-Ali MM, Wiegand S. Thermal diffusion and molecular diffusion values for some alkane mixtures: a comparison between thermogravitational column and thermal diffusion forced Rayleigh scattering. J Phys Chem B 2008;112:8340e5. [22] Charrier-Mojtabi MC, Elhajjar B, Mojtabi A. Analytical and numerical stability analysis of Soret-driven flow in a horizontal porous layer. Phys Fluids 2007;19:124104. [23] Ecenarro O, Madariaga JA, Navarro J, Santamaria CM, Carrion JA, Saviron JM. Direction of separation and dependence of feed concentration in liquid thermogravitational columns. Sep Sci Technol 1991;26:1065e76.

[24] Li WB, Segre PN, Sengers JV, Gammon RW. Non-equilibrium fluctuations in liquid and liquid mixtures subjected to a stationary temperature gradient. J Phys Condens Matter 1996;6:A119e24. [25] Platten JK. The Soret effect: a review of recent experimental results. J Appl Mech-Trans ASME 2006;73:5e15. [26] Zhan JM, Chen ZW, Li YS, Nie YH. Three-dimensional double-diffusive Marangoni convection in a cubic cavity with horizontal temperature and concentration gradients. Phys Rev E 2010;82:066305. [27] Peng L, Li YR, Shi WY, Imaishi N. Three-dimensional thermocapillarybuoyancy flow of silicone oil in a differentially heated annular pool. Int J Heat Mass Transf 2007;50:872e80. [28] Coleman HW, Steele WG. Experimentation, alidation, and Uncertainty Analysis for Engineers. third ed. John Wiley & Sons; 2009. [29] Ozoe H, Toh K, Inoue T. Transition mechanism of flow modes in Czochralski convection. J Cryst Growth 1991;110:472e80. [30] Imaishi N, Kakimoto K. Convective instability in crystal growth system. Annu Rev Heat Transf 2012;12:187e221. http://dx.doi.org/10.1615/ AnnualRevHeatTransfer.v12.70. [31] Shi WY, Ermakov MK, Imaishi N. Effect of pool rotation on thermocapillary convection in shallow annular pool of silicone oil. J Cryst Growth 2006;294: 474e85. [32] Lappa M. Rotating thermal flows in natural and industrial processes. first ed. West Sussex: John wiley & Sons; 2012. [33] Kanda I. A laboratory study of two-dimensional and three-dimensional instabilities in a quasi-two-dimensional flow driven by differential rotation of a cylindrical tank and a disc on the free surface. Phys Fluids 2004;16:3325e40. [34] Wu CM, Li YR, Liao RJ. Instability of three-dimensional flow due to rotation and surface-tension driven effects in a shallow pool with partly free surface. J Heat Mass Transf 2014;79:968e80. [35] Smith MK, Davis SH. Instabilities of dynamic thermocapillary liquid layers. 1. Convective instabilities. J Fluid Mech 1983;132:119e44. [36] Basu B, Enger S, Breuer M, Durst F. Effect of crystal rotation on the threedimensional mixed convection in the oxide melt for Czochralski growth. J Cryst Growth 2001;230:148e54. [37] Carruthers JR, Benson KE. Solute striations in Czochralski grown silicon crystals: effect of crystal rotation and growth rates. Appl Phys Lett 1963;3: 100e2. [38] Son S, Yi K. Experimental study on the effect of crystal and crucible rotations on the thermal and velocity field in a low Prandtl number melt in a large crucible. J Cryst Growth 2005;275. e249e57. [39] Teitel M, Schwabe D, Gelfgat AY. Experimental and computational study of flow instabilities in a model of Czochralski growth. J Cryst Growth 2008;310: 1343e8. [40] Gelfgat AY. Three-dimensional stability calculations for hydrodynamic model of Czochralski growth. J Cryst Growth 2007;303:226e30. [41] Garnier N, Chiffaudel A, Daviaud F. Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. II. Convective/absolute transitions. Phys D Nonlinear Phenom 2003;174:30e55. [42] Schwabe D, Sumathi RR. An intermittent transition to chaotic flow by crystal rotation during Czochralski growth. J Cryst Growth 2005;275. e15e9.