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Effect of thermo-solutal Marangoni convection on the azimuthal wave number in a liquid bridge
Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Effect of thermo-solutal Marangoni convection on the azimuthal wave number in a liquid bridge H. Minakuchi a,n, Y. Okano b, S. Dost c a
Department of Mechanical Systems Engineering, University of the Ryukyus, 1 Senbaru, Nishihara, Okinawa 903-0213, Japan Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, 1–3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan c Crystal Growth Laboratory, University of Victoria, Victoria, BC, Canada V8W 3P6 b
art ic l e i nf o
Keywords: A1. Computer simulation A2. Marangoni convection A2. Floating zone technique A1. Half-zone
a b s t r a c t A numerical simulation study was carried out to investigate the effect of thermo-solutal Marangoni convection on the flow patterns and the azimuthal wave number (m) in a liquid bridge under zerogravity. The liquid bridge in the model represents a three dimensional half-zone configuration of the Floating Zone (FZ) growth system. Three dimensional field equations of the liquid zone, i.e. continuity, momentum, energy, and diffusion equations, were solved by the PISO algorithm. The physical properties of the silicon-germanium melt were used (Pr ¼6.37 10 3 and Sc ¼14.0, where Pr and Sc stand for the Prandtl number and the Schmidt number). The aspect ratio Asp was set to 0.5 (Asp¼ L/a, where L and a stand for the length of free surface and the radius of liquid bridge). Computations were performed using the open source software OpenFOAM. The numerical simulation results show that the co-existence of thermal and solutal Marangoni convections significantly affects the azimuthal wave number m in the liquid bridge. & 2016 Elsevier B.V. All rights reserved.
1. Introduction The variation in the surface tension drives a convective flow along the free surface in a melt. This flow is known as Marangoni convection, and may be produced by temperature (thermal Marangoni convection) and/or concentration (solutal Marangoni convection) gradients in the melt. It is known that such Marangoni convective flows may become unstable much easier than the buoyancy driven natural convection. When a free surface exists in a crystal growth system such as the Floating-zone (FZ), unstable Marangoni flows that develop in the melt (liquid bridge) may induce undesirable growth striations in the grown crystals. Therefore, a large number of studies have been conducted for the thermal Marangoni convection in the FZ systems (see, for instance, Refs. [1–6]). For simplicity and to reduce computational cost, a half-zone (HZ) model as described in Fig. 1 is often used [7–11]. The HZ model is considered as a liquid bridge between two hot and cold discs by establishing a temperature gradient along the free surface of the liquid bridge. For small thermal Marangoni numbers, the Marangoni convection in the half-zone is steady and axisymmetric. However, at high thermal Marangoni numbers the n
Corresponding author. E-mail address: [email protected] (H. Minakuchi).
flow in the melt becomes 3-D and asymmetric, and its mode depends on the Prandtl number (Pr). For low-Pr fluids, the steady, 2-D, and axisymmetric flow becomes steady but 3-D and asymmetric in the first bifurcation, and becomes oscillatory in the second bifurcation. In high-Pr fluids, the steady, 2-D, and axisymmetric flow loses its stability and becomes directly 3-D and unsteady with the increasing thermal Marangoni number. The 3-D flows have m-fold symmetry of the flow pattern. The symmetry number m (azimuthal wave number) depends strongly on the aspect ratio, Asp(Asp ¼height/radius), of the half-zone. In addition, in the growth of alloys such as SixGe1 x, it was shown that it is necessary to consider not only the thermal Marangoni convection but also the solutal Marangoni convection due to surface tension differences of the components of the alloy [12– 15]. Therefore, in order to shed light on the relative contributions of thermal and solutal Marangoni convections occurring in the melt of a half-zone of the FZ system, we have carried out threedimensional numerical simulations in the present study.
2. Mathematical model In FZ growth of single crystals, various factors such as growth rate, deformation of free and liquid/solid interfaces, the concentration condition at the liquid-solid interface and the
http://dx.doi.org/10.1016/j.jcrysgro.2016.09.028 0022-0248/& 2016 Elsevier B.V. All rights reserved.
Please cite this article as: H. Minakuchi, et al., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.09.028i
H. Minakuchi et al. / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
Fig. 1. Schematics of the half-zone domain. Fig. 2. Initial conditions.
temperature condition at the free interface may play roles as described in [16]. However, since the objective of the present study is to shed light on the relative contributions of thermal and solutal Marangoni convections on flow structures of the melt under zero gravity, we have simplified the model as much as possible by not taking into account some of these factors in order to eliminate their possible influences. With this in mind, we have made the following assumptions in the model: (i) the Si-Ge melt is an incompressible, Newtonian fluid mixture, (ii) the solid/liquid interfaces are flat, and (iii) the system is under zero gravity. Since the gravity is taken zero, the liquid bridge remains cylindrical (and thus the liquid/gas interface is flat), and natural convection does not develop in the melt (due to the absence of the gravitational body force). The three-dimensional half-zone of the FZ liquid bridge is considered as the liquid phase of the model domain (its schematic description is given in Fig. 1). Three dimensional field equations of the liquid zone, i.e. continuity, momentum, energy, and diffusion equations, were solved by the PISO algorithm. The associated dimensionless thermal and solutal Marangoni numbers are defined as
MaT = −
MaC =
⎛ ∂σ ⎞ ΔTL ⎜ ⎟ ⎝ ∂T ⎠ μν
⎛ ∂σ ⎞ ΔCL ⎜ ⎟ ⎝ ∂C ⎠ μν
(1)
(2)
The thermal Marangoni number values were used from 0 to 3500 and the solutal Marangoni number was set to 1786. The computation was carried out using the open source software OpenFOAM. Details, such as the model description, numerical scheme, and validity of simulation, can be found in [13–15]. Physical properties of silicon–germanium melt were used (Prandtl number (Pr ¼6.37 10 3) and Schmidt number (Sc ¼14.0)). The length of free surface and aspect ratio were set to L¼5.0 10 3 m, Asp( ¼L/a) ¼0.5. The results of the conductive heat and mass transfer analysis were used as the initial conditions of computations as shown Fig. 2. Along the free surface, the thermal and solutal Marangoni flows are in the vertical direction.
Fig. 3. Melt concentration distribution at MaT ¼(a) 0, (b) 893, (c) 2000 and (d) 3036, MaC ¼ 1786 and 1000 s.
Please cite this article as: H. Minakuchi, et al., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.09.028i
H. Minakuchi et al. / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3
Fig. 4. Time dependency of the flow velocity at MaT ¼ (a) 0, (b) 893, (c) 2000 and (d) 3036 and MaC ¼1786.
Fig. 5. Dependence of the azimuthal wave number on the thermal Marangoni number at MaC ¼1786.
3. Results and discussion Figs. 3 and 4 present the computed melt concentration distribution in the horizontal (z ¼L/2) and vertical planes and the time dependency of the flow strength at a point of (R, y, Z) ¼ (0.999a, 0, 0.5L). When only the solutal Marangoni convection is considered (at MaT ¼ 0 and MaC ¼1786), the flow becomes unstable, and a 3-D oscillation occurs with the azimuthal wave number of m ¼ 4, as shown in Fig. 3(a). When the co-existence of thermal and solutal Marangoni convections are considered, the flows become unstable and the azimuthal wave numbers become 5, 6 and 7 at MaT ¼893, 2000 and 3036, respectively. The velocity and the azimuthal number increase when the MaT number increases. Fig. 5 shows the dependence of the azimuthal wave number on the thermal Marangoni number. The overlap region of the
Fig. 6. Snap shots of melt concentration at MaT ¼ 1000 and MaC ¼1786.
azimuthal wave number exists, however, the azimuthal wave number increases almost only with the increase of the thermal Marangoni number. Fig. 6 shows the snap shots of melt concentration at m¼ 5.5 (MaT ¼1000 and MaC ¼1786). When the thermal Marangoni numbers are about 1000, the azimuthal wave number goes back and forth (with 5 and 6 appearance) as shown in Fig. 6. Because the flow pattern was unstable, the azimuthal wave number was assumed 5.5. Fig. 7 shows the dependency of the average velocity and a cycle
Please cite this article as: H. Minakuchi, et al., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.09.028i
H. Minakuchi et al. / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
dimensional conclusions:
half-zone
liquid
bridge
led
to
the
following
1. The azimuthal wave number m depends not only on the aspect ratio but also on the thermal Marangoni number due to the effect of thermo-solutal Marangoni convection on flow patterns. 2. The azimuthal wave number increases almost only with the increase of the thermal Marangoni number. 3. These results indicate the possibility of controlling the azimuthal wave number by controlling the Marangoni number.
References
Fig. 7. Dependence of the average velocity and a cycle of velocity oscillation on the thermal Marangoni number at a point (R, y, Z) ¼ (0.999a, 0, 0.5L).
of velocity oscillation on the thermal Marangoni number at the point (R, y, Z) ¼ (0.999a, 0, 0.5L). Since the average velocity increases when the thermal Marangoni number increases, the flow instability increases. Furthermore, from the destabilization of the flow in the melt, it is thought that the azimuthal wave number increases by the cycle of velocity oscillation being decreased.
4. Conclusions The present numerical simulation study carried out to examine the role of thermo-solutal Marangoni convection in a three-
[1] H. Minakuchi, Y. Okano, S. Dost, J. Cryst. Growth 266 (2004) 140. [2] T.P. Lyubimova, R.V. Skuridin, I.S. Faizrakhmanova, Eur. Phys. J. Spec. Top. 192 (2011) 41. [3] C.W. Lan, J.H. Chian, J. Cryst. Growth 230 (2001) 172. [4] Z. Zeng, H. Mizuseki, K. Simamura, T. Fukuda, K. Higashino, Y. Kawazoe, Int. J. Heat Mass Transf. 44 (2001) 3765. [5] M. Lappa, R. Savino, R. Monti, Int. J. Heat Fluid Flow 44 (2001) 1983. [6] H. Minakuchi, Y. Okano, S. Dost, Int. J. Mater. Prod. Technol. 22 (2005) 151. [7] N. Imaishi, S. Yasuhiro, Y. Akiyama, S. Yoda, J. Cryst. Growth 230 (2001) 164. [8] Z. Zeng, H. Mizuseki, K. Simamura, T. Fukuda, K. Higashino, Y. Kawazoe, Int. J. Heat Mass Transf. 44 (2001) 3765. [9] M. Lappa, R. Savino, R. Monti, Int. J. Heat Transf. 44 (2001) 1983. [10] M. Sakurai, N. Ohishi, A. Hirata, J. Cryst. Growth 308 (2007) 352. [11] I. Ueno, Y. Abe, K. Noguchi, H. Kawamura, Adv. Space Res. 41 (2008) 2145. [12] T.A. Campbell, M. Schweizer, P. Dold, A. Crӧll, K.W. Benz, J. Cryst. Growth 226 (2001) 231. [13] H. Minakuchi, Y. Takagi, Y. Okano, K. Mizoguchi, S. Gima, S. Dost, J. Adv. Res. Phys. 3 (2012) 011201. [14] H. Minakuchi, Y. Takagi, Y. Okano, T. Nosoko, S. Gima, S. Dost, Trans. JSASS Aerosp. Technol. Jpn. 10 (2012), Ph_15. [15] H. Minakuchi, Y. Takagi, Y. Okano, S. Gima, S. Dost, J. Cryst. Growth 385 (2014) 61. [16] T.P. Lyubimova, R.V. Skuridin, I.S. Faizrakhmanova, J. Cryst. Growth 303 (2007) 274.
Please cite this article as: H. Minakuchi, et al., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.09.028i
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Effect of thermo-solutal Marangoni convection on the azimuthal wave number in a liquid bridge H. Minakuchi a,n, Y. Okano b, S. Dost c a
Department of Mechanical Systems Engineering, University of the Ryukyus, 1 Senbaru, Nishihara, Okinawa 903-0213, Japan Department of Materials Engineering Science, Graduate School of Engineering Science, Osaka University, 1–3 Machikaneyama, Toyonaka, Osaka 560-8531, Japan c Crystal Growth Laboratory, University of Victoria, Victoria, BC, Canada V8W 3P6 b
art ic l e i nf o
Keywords: A1. Computer simulation A2. Marangoni convection A2. Floating zone technique A1. Half-zone
a b s t r a c t A numerical simulation study was carried out to investigate the effect of thermo-solutal Marangoni convection on the flow patterns and the azimuthal wave number (m) in a liquid bridge under zerogravity. The liquid bridge in the model represents a three dimensional half-zone configuration of the Floating Zone (FZ) growth system. Three dimensional field equations of the liquid zone, i.e. continuity, momentum, energy, and diffusion equations, were solved by the PISO algorithm. The physical properties of the silicon-germanium melt were used (Pr ¼6.37 10 3 and Sc ¼14.0, where Pr and Sc stand for the Prandtl number and the Schmidt number). The aspect ratio Asp was set to 0.5 (Asp¼ L/a, where L and a stand for the length of free surface and the radius of liquid bridge). Computations were performed using the open source software OpenFOAM. The numerical simulation results show that the co-existence of thermal and solutal Marangoni convections significantly affects the azimuthal wave number m in the liquid bridge. & 2016 Elsevier B.V. All rights reserved.
1. Introduction The variation in the surface tension drives a convective flow along the free surface in a melt. This flow is known as Marangoni convection, and may be produced by temperature (thermal Marangoni convection) and/or concentration (solutal Marangoni convection) gradients in the melt. It is known that such Marangoni convective flows may become unstable much easier than the buoyancy driven natural convection. When a free surface exists in a crystal growth system such as the Floating-zone (FZ), unstable Marangoni flows that develop in the melt (liquid bridge) may induce undesirable growth striations in the grown crystals. Therefore, a large number of studies have been conducted for the thermal Marangoni convection in the FZ systems (see, for instance, Refs. [1–6]). For simplicity and to reduce computational cost, a half-zone (HZ) model as described in Fig. 1 is often used [7–11]. The HZ model is considered as a liquid bridge between two hot and cold discs by establishing a temperature gradient along the free surface of the liquid bridge. For small thermal Marangoni numbers, the Marangoni convection in the half-zone is steady and axisymmetric. However, at high thermal Marangoni numbers the n
Corresponding author. E-mail address: [email protected] (H. Minakuchi).
flow in the melt becomes 3-D and asymmetric, and its mode depends on the Prandtl number (Pr). For low-Pr fluids, the steady, 2-D, and axisymmetric flow becomes steady but 3-D and asymmetric in the first bifurcation, and becomes oscillatory in the second bifurcation. In high-Pr fluids, the steady, 2-D, and axisymmetric flow loses its stability and becomes directly 3-D and unsteady with the increasing thermal Marangoni number. The 3-D flows have m-fold symmetry of the flow pattern. The symmetry number m (azimuthal wave number) depends strongly on the aspect ratio, Asp(Asp ¼height/radius), of the half-zone. In addition, in the growth of alloys such as SixGe1 x, it was shown that it is necessary to consider not only the thermal Marangoni convection but also the solutal Marangoni convection due to surface tension differences of the components of the alloy [12– 15]. Therefore, in order to shed light on the relative contributions of thermal and solutal Marangoni convections occurring in the melt of a half-zone of the FZ system, we have carried out threedimensional numerical simulations in the present study.
2. Mathematical model In FZ growth of single crystals, various factors such as growth rate, deformation of free and liquid/solid interfaces, the concentration condition at the liquid-solid interface and the
http://dx.doi.org/10.1016/j.jcrysgro.2016.09.028 0022-0248/& 2016 Elsevier B.V. All rights reserved.
Please cite this article as: H. Minakuchi, et al., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.09.028i
H. Minakuchi et al. / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
Fig. 1. Schematics of the half-zone domain. Fig. 2. Initial conditions.
temperature condition at the free interface may play roles as described in [16]. However, since the objective of the present study is to shed light on the relative contributions of thermal and solutal Marangoni convections on flow structures of the melt under zero gravity, we have simplified the model as much as possible by not taking into account some of these factors in order to eliminate their possible influences. With this in mind, we have made the following assumptions in the model: (i) the Si-Ge melt is an incompressible, Newtonian fluid mixture, (ii) the solid/liquid interfaces are flat, and (iii) the system is under zero gravity. Since the gravity is taken zero, the liquid bridge remains cylindrical (and thus the liquid/gas interface is flat), and natural convection does not develop in the melt (due to the absence of the gravitational body force). The three-dimensional half-zone of the FZ liquid bridge is considered as the liquid phase of the model domain (its schematic description is given in Fig. 1). Three dimensional field equations of the liquid zone, i.e. continuity, momentum, energy, and diffusion equations, were solved by the PISO algorithm. The associated dimensionless thermal and solutal Marangoni numbers are defined as
MaT = −
MaC =
⎛ ∂σ ⎞ ΔTL ⎜ ⎟ ⎝ ∂T ⎠ μν
⎛ ∂σ ⎞ ΔCL ⎜ ⎟ ⎝ ∂C ⎠ μν
(1)
(2)
The thermal Marangoni number values were used from 0 to 3500 and the solutal Marangoni number was set to 1786. The computation was carried out using the open source software OpenFOAM. Details, such as the model description, numerical scheme, and validity of simulation, can be found in [13–15]. Physical properties of silicon–germanium melt were used (Prandtl number (Pr ¼6.37 10 3) and Schmidt number (Sc ¼14.0)). The length of free surface and aspect ratio were set to L¼5.0 10 3 m, Asp( ¼L/a) ¼0.5. The results of the conductive heat and mass transfer analysis were used as the initial conditions of computations as shown Fig. 2. Along the free surface, the thermal and solutal Marangoni flows are in the vertical direction.
Fig. 3. Melt concentration distribution at MaT ¼(a) 0, (b) 893, (c) 2000 and (d) 3036, MaC ¼ 1786 and 1000 s.
Please cite this article as: H. Minakuchi, et al., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.09.028i
H. Minakuchi et al. / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
3
Fig. 4. Time dependency of the flow velocity at MaT ¼ (a) 0, (b) 893, (c) 2000 and (d) 3036 and MaC ¼1786.
Fig. 5. Dependence of the azimuthal wave number on the thermal Marangoni number at MaC ¼1786.
3. Results and discussion Figs. 3 and 4 present the computed melt concentration distribution in the horizontal (z ¼L/2) and vertical planes and the time dependency of the flow strength at a point of (R, y, Z) ¼ (0.999a, 0, 0.5L). When only the solutal Marangoni convection is considered (at MaT ¼ 0 and MaC ¼1786), the flow becomes unstable, and a 3-D oscillation occurs with the azimuthal wave number of m ¼ 4, as shown in Fig. 3(a). When the co-existence of thermal and solutal Marangoni convections are considered, the flows become unstable and the azimuthal wave numbers become 5, 6 and 7 at MaT ¼893, 2000 and 3036, respectively. The velocity and the azimuthal number increase when the MaT number increases. Fig. 5 shows the dependence of the azimuthal wave number on the thermal Marangoni number. The overlap region of the
Fig. 6. Snap shots of melt concentration at MaT ¼ 1000 and MaC ¼1786.
azimuthal wave number exists, however, the azimuthal wave number increases almost only with the increase of the thermal Marangoni number. Fig. 6 shows the snap shots of melt concentration at m¼ 5.5 (MaT ¼1000 and MaC ¼1786). When the thermal Marangoni numbers are about 1000, the azimuthal wave number goes back and forth (with 5 and 6 appearance) as shown in Fig. 6. Because the flow pattern was unstable, the azimuthal wave number was assumed 5.5. Fig. 7 shows the dependency of the average velocity and a cycle
Please cite this article as: H. Minakuchi, et al., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.09.028i
H. Minakuchi et al. / Journal of Crystal Growth ∎ (∎∎∎∎) ∎∎∎–∎∎∎
4
dimensional conclusions:
half-zone
liquid
bridge
led
to
the
following
1. The azimuthal wave number m depends not only on the aspect ratio but also on the thermal Marangoni number due to the effect of thermo-solutal Marangoni convection on flow patterns. 2. The azimuthal wave number increases almost only with the increase of the thermal Marangoni number. 3. These results indicate the possibility of controlling the azimuthal wave number by controlling the Marangoni number.
References
Fig. 7. Dependence of the average velocity and a cycle of velocity oscillation on the thermal Marangoni number at a point (R, y, Z) ¼ (0.999a, 0, 0.5L).
of velocity oscillation on the thermal Marangoni number at the point (R, y, Z) ¼ (0.999a, 0, 0.5L). Since the average velocity increases when the thermal Marangoni number increases, the flow instability increases. Furthermore, from the destabilization of the flow in the melt, it is thought that the azimuthal wave number increases by the cycle of velocity oscillation being decreased.
4. Conclusions The present numerical simulation study carried out to examine the role of thermo-solutal Marangoni convection in a three-
[1] H. Minakuchi, Y. Okano, S. Dost, J. Cryst. Growth 266 (2004) 140. [2] T.P. Lyubimova, R.V. Skuridin, I.S. Faizrakhmanova, Eur. Phys. J. Spec. Top. 192 (2011) 41. [3] C.W. Lan, J.H. Chian, J. Cryst. Growth 230 (2001) 172. [4] Z. Zeng, H. Mizuseki, K. Simamura, T. Fukuda, K. Higashino, Y. Kawazoe, Int. J. Heat Mass Transf. 44 (2001) 3765. [5] M. Lappa, R. Savino, R. Monti, Int. J. Heat Fluid Flow 44 (2001) 1983. [6] H. Minakuchi, Y. Okano, S. Dost, Int. J. Mater. Prod. Technol. 22 (2005) 151. [7] N. Imaishi, S. Yasuhiro, Y. Akiyama, S. Yoda, J. Cryst. Growth 230 (2001) 164. [8] Z. Zeng, H. Mizuseki, K. Simamura, T. Fukuda, K. Higashino, Y. Kawazoe, Int. J. Heat Mass Transf. 44 (2001) 3765. [9] M. Lappa, R. Savino, R. Monti, Int. J. Heat Transf. 44 (2001) 1983. [10] M. Sakurai, N. Ohishi, A. Hirata, J. Cryst. Growth 308 (2007) 352. [11] I. Ueno, Y. Abe, K. Noguchi, H. Kawamura, Adv. Space Res. 41 (2008) 2145. [12] T.A. Campbell, M. Schweizer, P. Dold, A. Crӧll, K.W. Benz, J. Cryst. Growth 226 (2001) 231. [13] H. Minakuchi, Y. Takagi, Y. Okano, K. Mizoguchi, S. Gima, S. Dost, J. Adv. Res. Phys. 3 (2012) 011201. [14] H. Minakuchi, Y. Takagi, Y. Okano, T. Nosoko, S. Gima, S. Dost, Trans. JSASS Aerosp. Technol. Jpn. 10 (2012), Ph_15. [15] H. Minakuchi, Y. Takagi, Y. Okano, S. Gima, S. Dost, J. Cryst. Growth 385 (2014) 61. [16] T.P. Lyubimova, R.V. Skuridin, I.S. Faizrakhmanova, J. Cryst. Growth 303 (2007) 274.
Please cite this article as: H. Minakuchi, et al., Journal of Crystal Growth (2016), http://dx.doi.org/10.1016/j.jcrysgro.2016.09.028i