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Flow modelling with relevance to vertical gradient freeze crystal growth under the influence of a travelling magnetic field
Journal of Crystal Growth 318 (2011) 150–155
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Flow modelling with relevance to vertical gradient freeze crystal growth under the influence of a travelling magnetic field a ¨ K. Niemietz a,n, V. Galindo b, O. Patzold , G. Gerbeth b, M. Stelter a a b
TU Bergakademie Freiberg, Institut f¨ ur NE-Metallurgie und Reinststoffe, Leipziger Straße 34, 09599 Freiberg, Germany Forschungszentrum Dresden-Rossendorf, MHD Department, P.O. Box 510119, 01314 Dresden, Germany
a r t i c l e i n f o
a b s t r a c t
Available online 23 October 2010
Results on the experimental and numerical modelling of the melt flow typically observed in vertical gradient freeze (VGF) crystal growth with a travelling magnetic field (TMF) are presented. Particular attention is paid on the transition from a laminar to a time-dependent flow, which represents a crucial problem in VGF growth. Low-temperature model experiments at around 80 1C were performed using a GaInSn melt in a resistance furnace with concentric, separately adjustable heating zones. The TMF was created by an external coil system, and the flow velocity was measured by means of the ultrasonic Doppler velocimetry (UDV). The melt flow was simulated numerically using a finite volume code based on the open source code library OpenFOAM. As a criterion for the stability of the flow the turbulent kinetic energy was calculated under the influence of the TMF and thermal buoyancy. The results obtained are compared to isothermal TMF flow modelling at ambient temperature. The stability limit of the melt flow is found to be significantly influenced by the mutual interaction of buoyant and TMF-driven flows. Both experimental and numerical results show the stabilizing effect of natural, VGF-type buoyancy on the TMF-induced flow. & 2010 Elsevier B.V. All rights reserved.
Keywords: A1. Fluid flows A1. Magnetic fields A1. Numerical simulation A1. Stirring A2. Vertical gradient freeze technique
1. Introduction The vertical gradient freeze (VGF) method is an important technology for the melt growth of bulk compound semiconductors [1,2]. The application of a time-dependent magnetic field, such as a travelling magnetic field (TMF) allows the temperature and concentration fields to be tailored during growth by inducing an additional melt flow [3–5]. Detailed knowledge about the melt flow resulting from the superposition of buoyant and Lorentz forces is necessary to use the full potential of the VGF–TMF technology. Of particular interest is the transition to time-dependent flow patterns, which is a crucial problem in bulk crystal growth. However, direct measurements of the melt flow in an enclosed, high-temperature growth furnace are extremely difficult. Numerical and experimental modelling is widely used to study the flow phenomena associated to VGF–TMF growth in a lowtemperature liquid metal. In Refs. [6,7] the TMF-induced basic flow and its stability under isothermal conditions were investigated in detail. More recently, the effect of an axially stratified thermal field has been included [8] which can be considered as model for an idealized VGF–TMF growth under a radially uniform temperature field. In this case, natural buoyancy only appears as a secondary
n
Correspondig author: Tel.: +493731393915; fax: + 493731392268 E-mail address: [email protected] (K. Niemietz).
0022-0248/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2010.10.077
phenomenon due to the TMF-induced flow. Nevertheless, it affects the basic flow and the transition to time-dependent flow patterns significantly [8]. In this paper, modelling is extended to a cylindrical melt under a certain radial temperature gradient that corresponds to a more realistic situation in crystal growth. The outer temperature profiles are adjusted to give the natural double vortex flow which typically exists in a real VGF setup due to the interaction of radial heating, the release of latent heat on solidification, and the discontinuous heat conductivity at the solid–liquid interface [5,9]. Experimental and numerical results are presented and discussed with focus on the mutual interaction of buoyant and TMF-induced flows and the consequences for the stability of the flow.
2. Experimental setup The experimental setup is shown in Fig. 1. It consists of a model furnace for creating a VGF-type buoyant flow and an external TMF coil system. The TMF (flux density: 0–3.5 mT, frequency: 20–800 Hz) is described in detail in Ref. [6]. It is created by six circular coils fed by a three-phase current with the phase shift between adjacent coils being 601. The resulting travelling field in the inner region of the coils behaves like an axially propagating magnetic wave with a wavelength of lTMF ¼0.55 m. The direction of the coil currents can
K. Niemietz et al. / Journal of Crystal Growth 318 (2011) 150–155
151
Fig. 1. (a) Photograph of the TMF coil system and the model furnace. The DC field coils also shown are not relevant for the experiments presented in this work. (b) Scheme of the VGF-type model furnace.
be reversed resulting in an axially upward or downward directed Lorentz force due to the interaction between the induced current density and the magnetic field. The Lorentz force drives a meridional, single vortex flow going up or down at the edge of the melt. The model furnace (Fig. 1(b)) consists of three concentrically arranged resistance heaters with separate temperature control. It allows for the flexible and accurate adjustment of the thermal field for T450 1C. Three S-type thermocouples fixed at the inner surfaces of the platinum heater supports are used for the temperature control. The maximum temperature fluctuations at the control thermocouples are found to be below 0.5 K. Cylindrical melt containers with a maximum diameter of 75 mm can be used. A set of Al2O3 pedestals of different heights is available to fix the position of the container relative to the heaters. The setup of the model experiments is sketched in Fig. 2(a). GaInSn (melting point: 10.5 1C) was used as the low-temperature liquid metal. The material parameters of GaInSn are given in Section 3. It was filled in a silica container with an inner diameter of D ¼73 mm to a height of H¼80 mm (melt aspect ratio H/D 1.1). The melt flow was characterized by means of ultrasonic Doppler velocimetry (UDV) using the commercial DOP 2000 measurement system from Signal Processing SA Lausanne. The system is equipped with an 8 MHz transducer (diameter: 5 mm) which is certified up to 250 1C. Touching the transducer onto the melt surface the measured UDV profile represents mean values of the velocity component being parallel to the propagation direction of the ultrasonic beam and averaged over its cross section [10]. The spatial resolution of the velocity profiles is about 0.7 mm. The experiments described below were performed with the probe in the central position, i.e., the flow velocity along the axis of the melt was measured. Axial temperature profiles within the melt were detected by means of thermocouples. Fig. 2(b) indicates the thermal field imposed to the melt. It is characterized by a radial temperature gradient which changes its sign in the axial direction being zero at about the mid-height of the melt. In this way, the VGF-type natural flow consisting of two counter-rotating vortices as indicated in Fig. 2(c) is well reproduced. Moreover, with the radial gradient being stronger in the upper part than in the lower part of the melt (see Fig. 2(b)), the upper flow vortex becomes stronger, too. This represents just the hydrodynamic situation usually found in VGF growth as known from global thermal simulations of the solidification process [5].
Finally, the VGF-type thermal and flow fields of Fig. 2(b) and (c) result in the axial UDV profile along the centreline of the melt plotted in Fig. 2(d). Negative velocities indicate melt flow towards the probe face and vice versa. Model experiments were carried out either at room temperature or under the thermal field mentioned above. The melt container was arranged in an axially centred position relative to the furnace heaters as well as to the external coil system. The TMF was used with a constant frequency of 50 Hz whereas the induction was systematically varied.
3. Numerical simulation The flow in the liquid metal GaInSn was simulated numerically using the incompressible Navier–Stokes equation in the Boussinesq approximation together with the heat transfer equation. The governing equations were solved with the help of the finite element code Fidap (Ansys Inc.) and the finite volume code OpenFOAM [11]. In a dimensionless form, with R, R2/n, r, (n/R)2, and DT being the distance, time, pressure, and temperature scale, respectively, the governing equations system is given by
d! u ! ! 2 þ ð! u Ur Þ ! u ¼ r þ r ! u þ GrðTT0 Þ eg þ f em dt
ð1Þ
and 1 2 r T ð! u UrÞT ¼ Pr
ð2Þ
Here the Grashof number Gr ¼gbDTR3/n2 describes the ratio of buoyant to viscous forces acting on the fluid. Pr ¼ n/a is the Prandtl number which expresses the ratio of the momentum diffusivity to thermal diffusivity a, g the gravity constant, b the thermal expansion coefficient, and DT a characteristic temperature differ! ence. The induced electromagnetic force density term f em ¼ ! ! / j B ST averaged over one period T¼2p/o and the incompressibility condition rU! u ¼ 0 were taken into account. In the low induction and low frequency approximation the force term becomes [7] ! 1 ! f em ¼ 7 Fr 2 ez 2
with
F¼
sokB20 R5 4mu
ð3Þ
152
K. Niemietz et al. / Journal of Crystal Growth 318 (2011) 150–155
The dimensionless magnetic force number F characterizes the relative impact of the travelling field on the melt. Experiments including buoyant flow were carried out between 70 and 90 1C. In the numerical simulation the material properties of GaInSn at 72 1C were used [12]: density r ¼6340 kg/m3, viscosity Z ¼1.67 mPas, thermal conductivity l ¼22.6 W/mK, specific heat
cp ¼361 J/kgK, thermal expansion coefficient b ¼0.00007 l/K, and electric conductivity s ¼2.89 106 S/m. The velocity of sound in GaInSn is c¼2860 m/s. No-slip and stress free conditions were set as the velocity boundary conditions at the inner container walls and the top surface of the melt, respectively. In order to define proper thermal
Fig. 2. (a) Scheme of the setup of the model experiments. (b) Axial temperature profiles measured along the axis and the edge of the melt. (c) Sketch of a VGF-type buoyant flow resulting from the radial temperature differences indicated in (b). (d) Axial UDV profile measured along the centreline of the melt under the thermal field shown in (b). The solid line represents a numerically calculated velocity profile (see Section 3).
Fig. 3. Axial velocity profiles along the centreline of the GaInSn melt under VGF-type buoyancy and TMF-induced flow as a function of the magnetic force number F of a downward directed (a) and an upward directed TMF (b). (c), (d) Maximum flow velocity vs. measurement time at the transition from laminar to time-dependent flow under an upward directed TMF and VGF-type buoyancy (c) as well as isothermal conditions (d).
K. Niemietz et al. / Journal of Crystal Growth 318 (2011) 150–155
boundary conditions, a global thermal modelling of the furnace was used. The heat fluxes at the external container walls were approximated using polynomial expressions in the coordinates:
heat flux at the container bottom: q(r)¼q1 + (q2 q1)(r/R)3 heat flux at the container side walls: q(z) ¼q3 + (q4 q3)(z/H)3 heat flux at the melt surface: q(r) ¼q5 + (q6 q5)(r/R) The parameter q1,q2yq5 were employed to fit the heat flux distributions coming from the global modelling of the furnace. The heat flux density q6 was adjusted so that the total heat flux normal RR to the melt surfaces qdS becomes zero. As initial condition T¼72 1C was chosen.
Table 1 Critical TMF force number of the transition from a laminar to an oscillating flow under isothermal conditions and natural buoyancy. Buoyant flow
TMF-induced flow
Critical force number, Fc
(isothermal) VGF-type VGF-type
Down/up Down Up
1.5y2.0 105 4 105 3.25 105
153
The relevance of thermal boundary conditions was validated by calculating the resulting buoyant flow in the melt. In Fig. 2(d) calculated and measured velocity profiles are compared. In general, a good qualitative agreement between experimental and numerical results is found. The remaining discrepancies might arise for instance from the divergence of the ultrasound beam [6] or the impact of the horizontal melt boundaries on the velocity profile [13] which were not considered here. For high values of the magnetic force number the flow is no longer stationary and the use of a turbulence-model in the flow simulation is strictly necessary. The standard k o SST model is applied here. In addition to the Navier–Stokes equation, two additional differential equations have to be solved (one for the turbulent kinetic energy k and one for the isotropic dissipation rate o). The turbulent kinetic energy is 0 the average of the velocity fluctuations to the square: k ¼ u 2 =2, where 0 n is the deviation of n from its mean value u. In order to define initial conditions for k and o, a degree of turbulence ofE10% is assumed.
4. Results and discussion 4.1. Model experiments The results of the model experiments are shown in Fig. 3. Fig. 3(a) and (b) illustrate the evolution of the melt flow under the
F=0
F = 0.8 × 105
F = 1.2 × 105
F = 1.6 × 105
74
73.5
73.1
Fig. 4. Temperature distribution (left, isoline spacing 0.1 K) and stream function (right) for different values of the magnetic force number F of an upward directed TMF.
154
K. Niemietz et al. / Journal of Crystal Growth 318 (2011) 150–155
influence of an upward and downward directed TMF with increasing field strength. It can be clearly seen that the pure buoyant flow at F¼0 is superimposed by the TMF-induced flow. Finally, the induced flow dominates completely, i.e., a single vortex flow pattern is detected. This state is reached at about F ¼1.05 105 (upward directed TMF) and F ¼1.48 105 (downward directed TMF), respectively. The different values of the magnetic force parameter F for this transition can be explained by the different intensity of the natural flow vortices to be counterbalanced by the TMF. An upward directed TMF drives a flow against the relatively weak buoyancy in the lower part of the melt. Hence, the field strength necessary to dominate the flow ought to be lower in the case of a downward directed TMF, which has to counterbalance the relatively strong buoyancy in the upper part of the melt. From the view of a crystal grower the transition from a laminar to a time-dependent flow regime at a certain critical force number Fc is of great interest. In Fig. 3 the maximum velocities of the UDV profiles around the stability threshold of the melt flow are plotted. The transition to time-dependent flow conditions is characterized by the appearance of an oscillating flow velocity. Further increase of the field strength leads to an irregularly fluctuating flow velocity. It can be seen that the TMF-driven flow is stabilized by a laminar, VGF-type buoyant flow. At F ¼3.24 105, for instance, the
v
combined VGF–TMF flow just starts to oscillate (see Fig. 3(c)) whereas at the same force number, the TMF-induced flow under isothermal conditions (see Fig. 3(d)) is already characterized by strong velocity fluctuations. The results of the experimental stability analysis are summarized in Table 1 in terms of the critical force number of the travelling field which was calculated from the magnetic induction at the transition to an oscillating flow. Compared to isothermal conditions Fc is found to be significantly higher under the influence of natural buoyancy. The slightly different critical values observed under upward and downward directed TMF can be attributed to the asymmetric vorticity of the natural flow.
4.2. Numerical results The experimental results are confirmed by numerical simulation. Temperature distributions and flow patterns under different inductions of an upward directed TMF are shown in Fig. 4. On increasing field strength the natural buoyant flow is replaced by a single vortex TMF-driven flow. In good agreement with the model experiments (see Fig. 3a and the related text) the induced flow completely dominates already at F¼ 1.2 105. The simulation with
TMF (F = 4 × 105) Ψ
k
v¯ max = 3.3 mm/s k max = 2.5 mm2/s2
0 TMF (F = 4 × 105) and thermal convection v
¯v max = 3.3 mm/s
R k
k max= 0.92 mm2/s2
Fig. 5. Velocity distribution (left), stream function (middle) and isolines of the turbulent kinetic energy (right) induced by an upward directed TMF with F¼ 4 105 in an isothermal melt (top) as well as under the influence of VGF-type thermal convection (bottom).
K. Niemietz et al. / Journal of Crystal Growth 318 (2011) 150–155
a downward directed TMF give analogous results (not shown in Fig. 4) with the buoyant flow becoming fully suppressed at about F ¼2 105. In Fig. 5 the results of the numerical stability analysis are presented. The mean flow velocity, stream function, and turbulent kinetic energy were calculated under the influence of an upward directed TMF with F ¼4 105 4Fc (see Table 1) with or without VGF-type buoyancy. From the contours of the mean velocity and stream function it can be concluded that the TMF-driven basic flow essentially remains unchanged, no matter whether buoyancy is taken into account or not. On the other hand, amplitude and distribution of the fluctuations of the melt flow are significantly influenced by natural buoyancy. As indicated in Fig. 5 the maximum value of the turbulent kinetic energy decreases by more than 60% which means almost a halving of the fluctuations of the pffiffiffiffiffiffiffiffiffiffiffi 0 measured velocity component nz ¼ 2k=3. So the stabilizing effect of VGF-type thermal convection on the TMF-induced flow turns out again in good agreement with the experimental observations.
5. Conclusion The melt flow induced by a TMF in combination with a VGF-type thermal field has been studied experimentally and numerically. With respect to VGF–TMF crystal growth the following results are worth pointing out: (i) The asymmetric vorticity of the buoyant flow leads to a slightly different impact of upward and downward travelling fields. (ii) The combined flow turns out to be more stable than the TMF-driven flow only. Laminar flow patterns are sustained up to higher field strength than indicated by isothermal flow modelling. (iii) Beyond the flow stability threshold VGF-type buoyancy results in an effective damping of the TMF-induced velocity fluctuations whereas the mean flow velocity is essentially preserved. Relative to the basic flow the velocity fluctuations in a VGF–TMF melt appear much less pronounced than in an isothermal melt under the influence of a travelling field.
155
Acknowledgment Financial support from Deutsche Forschungsgemeinschaft in frame of the Collaborative Research Centre SFB 609 is gratefully acknowledged.
References [1] J.B. Mullin, Progress in the melt growth of III–V compounds, J. Cryst. Growth 264 (2004) 578–592. ¨ ¨ ¨ [2] M. Jurisch, F. Borner, Th. Bunger, St. Eichler, T. Flade, U. Kretzer, A. Kohler, J. Stenzenberger, B. Weinert, LEC- and VGF-growth of SI GaAs single crystals—recent developments and current issues, J. Cryst. Growth 275 (2005) 283–291. [3] Ch. Frank-Rotsch, P. Rudolph, Vertical gradient freeze of 4 inch Ge crystals in a heater magnet module, J. Cryst. Growth 311 (2009) 2294–2299. ¨ [4] R. Lantzsch, I. Grants, O. Patzold, M. Stelter, G. Gerbeth, Vertical gradient freeze growth with external magnetic field, J. Cryst. Growth 310 (2008) 1518–1522. [5] P. Schwesig, M. Hainke, J. Friedrich, G. Mueller, Comparative numerical study of the effects of rotating and travelling magnetic fields on the interface shape and thermal stress in the VGF growth of InP crystals, J. Cryst. Growth 266 (2004) 224–228. ¨ [6] R. Lantzsch, V. Galindo, C. Zhang, O. Patzold, G. Gerbeth, M. Stelter, Experimental and numerical results on the fluid flow driven by a traveling magnetic field, J. Cryst. Growth 305 (2007) 249–256. [7] I. Grants, G. Gerbeth, Stability of melt flow due to a travelling magnetic field in a closed ampoule, J. Cryst. Growth 269 (2004) 630–638. [8] I. Grants, A. Klyukin, G. Gerbeth, Instability of the melt flow in VGF growth with a travelling magnetic field, J. Cryst. Growth 311 (2009) 4255–4264. ¨ [9] O. Patzold, I. Grants, U. Wunderwald, K. Jenkner, A. Cr\’’oll, G. Gerbeth, Vertical gradient freeze growth of GaAs with a rotating magnetic field, J. Cryst. Growth 245 (2002) 237–246. [10] Y. Takeda, Development of an ultrasound velocity profile monitor, Nucl. Eng. Design 126 (1991) 277–284. [11] OpenFOAMs: the open source CFD toolbox /http://www.openfoam.com/S. ¨ ¨ [12] U. Muller, L. Buhler, Magnetofluiddynamics in Channels and Containers, Springer Verlag, 2001. [13] A. Cramer, C. Zhang, S. Eckert., Local flow structures in liquid metals measured by ultrasonic Doppler velocimetry, Flow Meas. Instrum. 15 (2004) 145–153.
Contents lists available at ScienceDirect
Journal of Crystal Growth journal homepage: www.elsevier.com/locate/jcrysgro
Flow modelling with relevance to vertical gradient freeze crystal growth under the influence of a travelling magnetic field a ¨ K. Niemietz a,n, V. Galindo b, O. Patzold , G. Gerbeth b, M. Stelter a a b
TU Bergakademie Freiberg, Institut f¨ ur NE-Metallurgie und Reinststoffe, Leipziger Straße 34, 09599 Freiberg, Germany Forschungszentrum Dresden-Rossendorf, MHD Department, P.O. Box 510119, 01314 Dresden, Germany
a r t i c l e i n f o
a b s t r a c t
Available online 23 October 2010
Results on the experimental and numerical modelling of the melt flow typically observed in vertical gradient freeze (VGF) crystal growth with a travelling magnetic field (TMF) are presented. Particular attention is paid on the transition from a laminar to a time-dependent flow, which represents a crucial problem in VGF growth. Low-temperature model experiments at around 80 1C were performed using a GaInSn melt in a resistance furnace with concentric, separately adjustable heating zones. The TMF was created by an external coil system, and the flow velocity was measured by means of the ultrasonic Doppler velocimetry (UDV). The melt flow was simulated numerically using a finite volume code based on the open source code library OpenFOAM. As a criterion for the stability of the flow the turbulent kinetic energy was calculated under the influence of the TMF and thermal buoyancy. The results obtained are compared to isothermal TMF flow modelling at ambient temperature. The stability limit of the melt flow is found to be significantly influenced by the mutual interaction of buoyant and TMF-driven flows. Both experimental and numerical results show the stabilizing effect of natural, VGF-type buoyancy on the TMF-induced flow. & 2010 Elsevier B.V. All rights reserved.
Keywords: A1. Fluid flows A1. Magnetic fields A1. Numerical simulation A1. Stirring A2. Vertical gradient freeze technique
1. Introduction The vertical gradient freeze (VGF) method is an important technology for the melt growth of bulk compound semiconductors [1,2]. The application of a time-dependent magnetic field, such as a travelling magnetic field (TMF) allows the temperature and concentration fields to be tailored during growth by inducing an additional melt flow [3–5]. Detailed knowledge about the melt flow resulting from the superposition of buoyant and Lorentz forces is necessary to use the full potential of the VGF–TMF technology. Of particular interest is the transition to time-dependent flow patterns, which is a crucial problem in bulk crystal growth. However, direct measurements of the melt flow in an enclosed, high-temperature growth furnace are extremely difficult. Numerical and experimental modelling is widely used to study the flow phenomena associated to VGF–TMF growth in a lowtemperature liquid metal. In Refs. [6,7] the TMF-induced basic flow and its stability under isothermal conditions were investigated in detail. More recently, the effect of an axially stratified thermal field has been included [8] which can be considered as model for an idealized VGF–TMF growth under a radially uniform temperature field. In this case, natural buoyancy only appears as a secondary
n
Correspondig author: Tel.: +493731393915; fax: + 493731392268 E-mail address: [email protected] (K. Niemietz).
0022-0248/$ - see front matter & 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jcrysgro.2010.10.077
phenomenon due to the TMF-induced flow. Nevertheless, it affects the basic flow and the transition to time-dependent flow patterns significantly [8]. In this paper, modelling is extended to a cylindrical melt under a certain radial temperature gradient that corresponds to a more realistic situation in crystal growth. The outer temperature profiles are adjusted to give the natural double vortex flow which typically exists in a real VGF setup due to the interaction of radial heating, the release of latent heat on solidification, and the discontinuous heat conductivity at the solid–liquid interface [5,9]. Experimental and numerical results are presented and discussed with focus on the mutual interaction of buoyant and TMF-induced flows and the consequences for the stability of the flow.
2. Experimental setup The experimental setup is shown in Fig. 1. It consists of a model furnace for creating a VGF-type buoyant flow and an external TMF coil system. The TMF (flux density: 0–3.5 mT, frequency: 20–800 Hz) is described in detail in Ref. [6]. It is created by six circular coils fed by a three-phase current with the phase shift between adjacent coils being 601. The resulting travelling field in the inner region of the coils behaves like an axially propagating magnetic wave with a wavelength of lTMF ¼0.55 m. The direction of the coil currents can
K. Niemietz et al. / Journal of Crystal Growth 318 (2011) 150–155
151
Fig. 1. (a) Photograph of the TMF coil system and the model furnace. The DC field coils also shown are not relevant for the experiments presented in this work. (b) Scheme of the VGF-type model furnace.
be reversed resulting in an axially upward or downward directed Lorentz force due to the interaction between the induced current density and the magnetic field. The Lorentz force drives a meridional, single vortex flow going up or down at the edge of the melt. The model furnace (Fig. 1(b)) consists of three concentrically arranged resistance heaters with separate temperature control. It allows for the flexible and accurate adjustment of the thermal field for T450 1C. Three S-type thermocouples fixed at the inner surfaces of the platinum heater supports are used for the temperature control. The maximum temperature fluctuations at the control thermocouples are found to be below 0.5 K. Cylindrical melt containers with a maximum diameter of 75 mm can be used. A set of Al2O3 pedestals of different heights is available to fix the position of the container relative to the heaters. The setup of the model experiments is sketched in Fig. 2(a). GaInSn (melting point: 10.5 1C) was used as the low-temperature liquid metal. The material parameters of GaInSn are given in Section 3. It was filled in a silica container with an inner diameter of D ¼73 mm to a height of H¼80 mm (melt aspect ratio H/D 1.1). The melt flow was characterized by means of ultrasonic Doppler velocimetry (UDV) using the commercial DOP 2000 measurement system from Signal Processing SA Lausanne. The system is equipped with an 8 MHz transducer (diameter: 5 mm) which is certified up to 250 1C. Touching the transducer onto the melt surface the measured UDV profile represents mean values of the velocity component being parallel to the propagation direction of the ultrasonic beam and averaged over its cross section [10]. The spatial resolution of the velocity profiles is about 0.7 mm. The experiments described below were performed with the probe in the central position, i.e., the flow velocity along the axis of the melt was measured. Axial temperature profiles within the melt were detected by means of thermocouples. Fig. 2(b) indicates the thermal field imposed to the melt. It is characterized by a radial temperature gradient which changes its sign in the axial direction being zero at about the mid-height of the melt. In this way, the VGF-type natural flow consisting of two counter-rotating vortices as indicated in Fig. 2(c) is well reproduced. Moreover, with the radial gradient being stronger in the upper part than in the lower part of the melt (see Fig. 2(b)), the upper flow vortex becomes stronger, too. This represents just the hydrodynamic situation usually found in VGF growth as known from global thermal simulations of the solidification process [5].
Finally, the VGF-type thermal and flow fields of Fig. 2(b) and (c) result in the axial UDV profile along the centreline of the melt plotted in Fig. 2(d). Negative velocities indicate melt flow towards the probe face and vice versa. Model experiments were carried out either at room temperature or under the thermal field mentioned above. The melt container was arranged in an axially centred position relative to the furnace heaters as well as to the external coil system. The TMF was used with a constant frequency of 50 Hz whereas the induction was systematically varied.
3. Numerical simulation The flow in the liquid metal GaInSn was simulated numerically using the incompressible Navier–Stokes equation in the Boussinesq approximation together with the heat transfer equation. The governing equations were solved with the help of the finite element code Fidap (Ansys Inc.) and the finite volume code OpenFOAM [11]. In a dimensionless form, with R, R2/n, r, (n/R)2, and DT being the distance, time, pressure, and temperature scale, respectively, the governing equations system is given by
d! u ! ! 2 þ ð! u Ur Þ ! u ¼ r þ r ! u þ GrðTT0 Þ eg þ f em dt
ð1Þ
and 1 2 r T ð! u UrÞT ¼ Pr
ð2Þ
Here the Grashof number Gr ¼gbDTR3/n2 describes the ratio of buoyant to viscous forces acting on the fluid. Pr ¼ n/a is the Prandtl number which expresses the ratio of the momentum diffusivity to thermal diffusivity a, g the gravity constant, b the thermal expansion coefficient, and DT a characteristic temperature differ! ence. The induced electromagnetic force density term f em ¼ ! ! / j B ST averaged over one period T¼2p/o and the incompressibility condition rU! u ¼ 0 were taken into account. In the low induction and low frequency approximation the force term becomes [7] ! 1 ! f em ¼ 7 Fr 2 ez 2
with
F¼
sokB20 R5 4mu
ð3Þ
152
K. Niemietz et al. / Journal of Crystal Growth 318 (2011) 150–155
The dimensionless magnetic force number F characterizes the relative impact of the travelling field on the melt. Experiments including buoyant flow were carried out between 70 and 90 1C. In the numerical simulation the material properties of GaInSn at 72 1C were used [12]: density r ¼6340 kg/m3, viscosity Z ¼1.67 mPas, thermal conductivity l ¼22.6 W/mK, specific heat
cp ¼361 J/kgK, thermal expansion coefficient b ¼0.00007 l/K, and electric conductivity s ¼2.89 106 S/m. The velocity of sound in GaInSn is c¼2860 m/s. No-slip and stress free conditions were set as the velocity boundary conditions at the inner container walls and the top surface of the melt, respectively. In order to define proper thermal
Fig. 2. (a) Scheme of the setup of the model experiments. (b) Axial temperature profiles measured along the axis and the edge of the melt. (c) Sketch of a VGF-type buoyant flow resulting from the radial temperature differences indicated in (b). (d) Axial UDV profile measured along the centreline of the melt under the thermal field shown in (b). The solid line represents a numerically calculated velocity profile (see Section 3).
Fig. 3. Axial velocity profiles along the centreline of the GaInSn melt under VGF-type buoyancy and TMF-induced flow as a function of the magnetic force number F of a downward directed (a) and an upward directed TMF (b). (c), (d) Maximum flow velocity vs. measurement time at the transition from laminar to time-dependent flow under an upward directed TMF and VGF-type buoyancy (c) as well as isothermal conditions (d).
K. Niemietz et al. / Journal of Crystal Growth 318 (2011) 150–155
boundary conditions, a global thermal modelling of the furnace was used. The heat fluxes at the external container walls were approximated using polynomial expressions in the coordinates:
heat flux at the container bottom: q(r)¼q1 + (q2 q1)(r/R)3 heat flux at the container side walls: q(z) ¼q3 + (q4 q3)(z/H)3 heat flux at the melt surface: q(r) ¼q5 + (q6 q5)(r/R) The parameter q1,q2yq5 were employed to fit the heat flux distributions coming from the global modelling of the furnace. The heat flux density q6 was adjusted so that the total heat flux normal RR to the melt surfaces qdS becomes zero. As initial condition T¼72 1C was chosen.
Table 1 Critical TMF force number of the transition from a laminar to an oscillating flow under isothermal conditions and natural buoyancy. Buoyant flow
TMF-induced flow
Critical force number, Fc
(isothermal) VGF-type VGF-type
Down/up Down Up
1.5y2.0 105 4 105 3.25 105
153
The relevance of thermal boundary conditions was validated by calculating the resulting buoyant flow in the melt. In Fig. 2(d) calculated and measured velocity profiles are compared. In general, a good qualitative agreement between experimental and numerical results is found. The remaining discrepancies might arise for instance from the divergence of the ultrasound beam [6] or the impact of the horizontal melt boundaries on the velocity profile [13] which were not considered here. For high values of the magnetic force number the flow is no longer stationary and the use of a turbulence-model in the flow simulation is strictly necessary. The standard k o SST model is applied here. In addition to the Navier–Stokes equation, two additional differential equations have to be solved (one for the turbulent kinetic energy k and one for the isotropic dissipation rate o). The turbulent kinetic energy is 0 the average of the velocity fluctuations to the square: k ¼ u 2 =2, where 0 n is the deviation of n from its mean value u. In order to define initial conditions for k and o, a degree of turbulence ofE10% is assumed.
4. Results and discussion 4.1. Model experiments The results of the model experiments are shown in Fig. 3. Fig. 3(a) and (b) illustrate the evolution of the melt flow under the
F=0
F = 0.8 × 105
F = 1.2 × 105
F = 1.6 × 105
74
73.5
73.1
Fig. 4. Temperature distribution (left, isoline spacing 0.1 K) and stream function (right) for different values of the magnetic force number F of an upward directed TMF.
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influence of an upward and downward directed TMF with increasing field strength. It can be clearly seen that the pure buoyant flow at F¼0 is superimposed by the TMF-induced flow. Finally, the induced flow dominates completely, i.e., a single vortex flow pattern is detected. This state is reached at about F ¼1.05 105 (upward directed TMF) and F ¼1.48 105 (downward directed TMF), respectively. The different values of the magnetic force parameter F for this transition can be explained by the different intensity of the natural flow vortices to be counterbalanced by the TMF. An upward directed TMF drives a flow against the relatively weak buoyancy in the lower part of the melt. Hence, the field strength necessary to dominate the flow ought to be lower in the case of a downward directed TMF, which has to counterbalance the relatively strong buoyancy in the upper part of the melt. From the view of a crystal grower the transition from a laminar to a time-dependent flow regime at a certain critical force number Fc is of great interest. In Fig. 3 the maximum velocities of the UDV profiles around the stability threshold of the melt flow are plotted. The transition to time-dependent flow conditions is characterized by the appearance of an oscillating flow velocity. Further increase of the field strength leads to an irregularly fluctuating flow velocity. It can be seen that the TMF-driven flow is stabilized by a laminar, VGF-type buoyant flow. At F ¼3.24 105, for instance, the
v
combined VGF–TMF flow just starts to oscillate (see Fig. 3(c)) whereas at the same force number, the TMF-induced flow under isothermal conditions (see Fig. 3(d)) is already characterized by strong velocity fluctuations. The results of the experimental stability analysis are summarized in Table 1 in terms of the critical force number of the travelling field which was calculated from the magnetic induction at the transition to an oscillating flow. Compared to isothermal conditions Fc is found to be significantly higher under the influence of natural buoyancy. The slightly different critical values observed under upward and downward directed TMF can be attributed to the asymmetric vorticity of the natural flow.
4.2. Numerical results The experimental results are confirmed by numerical simulation. Temperature distributions and flow patterns under different inductions of an upward directed TMF are shown in Fig. 4. On increasing field strength the natural buoyant flow is replaced by a single vortex TMF-driven flow. In good agreement with the model experiments (see Fig. 3a and the related text) the induced flow completely dominates already at F¼ 1.2 105. The simulation with
TMF (F = 4 × 105) Ψ
k
v¯ max = 3.3 mm/s k max = 2.5 mm2/s2
0 TMF (F = 4 × 105) and thermal convection v
¯v max = 3.3 mm/s
R k
k max= 0.92 mm2/s2
Fig. 5. Velocity distribution (left), stream function (middle) and isolines of the turbulent kinetic energy (right) induced by an upward directed TMF with F¼ 4 105 in an isothermal melt (top) as well as under the influence of VGF-type thermal convection (bottom).
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a downward directed TMF give analogous results (not shown in Fig. 4) with the buoyant flow becoming fully suppressed at about F ¼2 105. In Fig. 5 the results of the numerical stability analysis are presented. The mean flow velocity, stream function, and turbulent kinetic energy were calculated under the influence of an upward directed TMF with F ¼4 105 4Fc (see Table 1) with or without VGF-type buoyancy. From the contours of the mean velocity and stream function it can be concluded that the TMF-driven basic flow essentially remains unchanged, no matter whether buoyancy is taken into account or not. On the other hand, amplitude and distribution of the fluctuations of the melt flow are significantly influenced by natural buoyancy. As indicated in Fig. 5 the maximum value of the turbulent kinetic energy decreases by more than 60% which means almost a halving of the fluctuations of the pffiffiffiffiffiffiffiffiffiffiffi 0 measured velocity component nz ¼ 2k=3. So the stabilizing effect of VGF-type thermal convection on the TMF-induced flow turns out again in good agreement with the experimental observations.
5. Conclusion The melt flow induced by a TMF in combination with a VGF-type thermal field has been studied experimentally and numerically. With respect to VGF–TMF crystal growth the following results are worth pointing out: (i) The asymmetric vorticity of the buoyant flow leads to a slightly different impact of upward and downward travelling fields. (ii) The combined flow turns out to be more stable than the TMF-driven flow only. Laminar flow patterns are sustained up to higher field strength than indicated by isothermal flow modelling. (iii) Beyond the flow stability threshold VGF-type buoyancy results in an effective damping of the TMF-induced velocity fluctuations whereas the mean flow velocity is essentially preserved. Relative to the basic flow the velocity fluctuations in a VGF–TMF melt appear much less pronounced than in an isothermal melt under the influence of a travelling field.
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Acknowledgment Financial support from Deutsche Forschungsgemeinschaft in frame of the Collaborative Research Centre SFB 609 is gratefully acknowledged.
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