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Factors affecting the isotherm shape of semi-transparent BaF2 crystals grown by Bridgman method
Journal of Crystal Growth 237–239 (2002) 1762–1768
Factors affecting the isotherm shape of semi-transparent BaF2 crystals grown by Bridgman method F. Barvinschia,*, O. Bunoiub, I. Nicoarab, D. Nicoarab, J.L. Santaillerc, T. Duffard a
Department of Physics, Politehnica University of Timisoara, P-ta Regina Maria, 1, Ro-1900 Timisoara, Romania b West University of Timisoara, Bv.V.Parvan, 4, Ro-1900 Timisoara, Romania c CEA/CEREM/DEM, 17, rue des Martyrs, F-38054 Grenoble, France d EPM-MADYLAM, ENSHMG, BP 95, F-38402 St Martin d’Heres, France
Abstract The effect of physical parameters and geometrical configurations on the interface shape in semi-transparent BaF2 crystals grown by the vertical Bridgman process are studied both experimentally and numerically. Values of thermophysical parameters are suggested in the case of BaF2. r 2002 Elsevier Science B.V. All rights reserved. PACS: 47.70; 61.50.C Keywords: A1. Computer simulation: A1. Radiation: A2. Bridgman technique: B1. Barium compounds
1. Introduction During crystal growth from the melt, thermal gradients are the origin of buoyancy convection, of solid–liquid (S–L) interface deflection and of stresses in the solid. Therefore, high-quality crystals can be grown when the temperature distribution in the crystal and the melt is accurately controlled. In the case of a Bridgman furnace, which consists in a heating system and a number of screens, it is difficult to define and then to get an optimised thermal environment during the crystal growth, if undertaken in a strictly experimental way. The numerical simulation of heat transfer in such a complex furnace is therefore *Corresponding author. Tel.: +40-56-20-3417; fax: +40-5620-4364. E-mail address: fl[email protected] (F. Barvinschi).
useful for this optimisation process [1–3]. While most of the analyses do not take into account the radiative heat transfer within the crystalline and molten phases, it has been pointed out [4–7] that the role of internal radiation transfer of energy during the crystal growth process of semi-transparent materials cannot be neglected. There are semi-transparent materials, such as LiF, CaF2, BaF2, whose optical absorption coefficient is not so high as to consider the molten phase opaque [8,9]. Section 2 of this study presents the experimental demarcation of the S–L interface during the growth of BaF2 crystals by the vertical Bridgman method. Experimental convex interface shapes, when seen from the melt, are quenched for different diameter crystals. In order to explain the experimental results, the numerical simulation of heat exchanges in the Bridgman system is
0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 2 3 2 8 - 4
F. Barvinschi et al. / Journal of Crystal Growth 237–239 (2002) 1762–1768
performed by using the FIDAP software [10]. A first steady-state heat transport model in the whole furnace is solved (Section 3). The optimum position of a molybdenum screen, positioned at several places between the graphite heater and the outer screen system, is numerically determined in order to obtain the reduction of the S–L interface deflection. In Section 4, the model is restricted to the central part of the Bridgman furnace, taking into account the semi-transparency of BaF2. By adjusting the physical properties of both phases, it has been possible to obtain results which are in qualitative concordance with the conclusions of [6,7,11] and with our experimental observations.
2. Experimental results BaF2 crystals were grown in a shaped graphite heater using the conventional Bridgman–Stockbarger method, the characteristics of the system being described in Ref. [12,13]. The heater is surrounded by a set of concentric screens made of graphite and stainless steel. Two crystals with different geometrical characteristics (indicated in Table 1) were grown in a crucible made of pure spectral graphite. The melt was doped with CoF2 (0.5% weight) in order to clearly visualise the striations associated with the S–L interface marking. To get the demarcation, the power of the
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furnace was interrupted for 5 min, when the grown crystal had a length of a few centimetres, and then the power and the standard translation rate were restored. On the photographs (Figs. 1a and b) the quenched interfaces are seen for both crystals. The interface is convex towards the melt, meaning that the thermal conductivity of the melt is lower than that of the crystal, in agreement with Refs. [8,9].
3. Global modelling of heat transfer The temperature distribution in growing crystals depends strongly upon the boundary conditions on their outer surface, which are a priori unknown. As a first approach, the global calculation of the steady-state heat transfer throughout the whole furnace is performed. The growth of a 24 mm diameter crystal is simulated by using (i) a reduced number of external parameters (power input, water temperature in the double jacket facility wall), (ii) the geometrical size of the crystal and or the set-up and (iii) the physical properties of the materials. A detail of the schematic diagram of the Bridgman furnace is shown in Fig. 2. The thermo-physical properties of interest are listed in Table 1. In order to obtain the temperature distribution, the whole system is divided into subdomains. Heat transfer in all the subdomains is described by the
Table 1 Operating parameters and material properties used Geometrical characteristics D ¼ 11 and 24 mm L ¼ 70 and 30 mm
Graphite Ka ¼ 45 W m1 K1 ea ¼ 0:8
BaF2 solid kC ¼ 2:4 W m1 K1 aC ¼ 0:120:3 cm1
Aspect ratio a ¼ D=L ¼ 0:16; 0.34 and 0.8
Molybdenum kMo ¼ 90 W m1 K1
BaF2 liquid kL ¼ 0:24 W m1 K1
Crucible wall thickness wa ¼ 1:5 mm
eMo ¼ 0:4 Stainless steel
aL ¼ 123 cm–1 Tm ¼ 1628 K
Pulling rate: v ¼ 3 mm h1
kst ¼ 20 W m1 K1
DH ¼ 3:8 105 J Kg1
Current intensity I ¼ 290 and 306 A
est ¼ 0:2
rL;C ¼ 4830 Kg m3 n ¼ 1:47
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Fig. 1. S–L interfaces revealed by quenching experiments on BaF2 crystals :(a) aspect ratio a ¼ 0:16; solidified fraction f ¼ 0:88; interface deflection dexp ¼ 170:05 mm; (b) aspect ratio a ¼ 0:8; solidified fraction f ¼ 0:64; interface deflection dexp ¼ 3:170:05 mm.
steady-state equation: rðkðT; r; zÞrTÞ þ Hðr; zÞ ¼ 0;
ð1Þ
where kðT; r; zÞ is the thermal conductivity of the considered domain and Hðr; zÞ is the heat source term that vanishes in all the subdomains, except in the heater. The following boundary conditions are used. The temperature and the heat flux between adjoining subdomains are continuous: Ti ¼ Tj
ð2Þ
and on the boundary between the subdomains i and j: ~ ij ¼ ðkj rTÞj N ~ ij ; ðki rTÞi N
ð3Þ
~ ij is the unitary normal to the boundary where N between subdomains. The release of the latent heat to be considered at the S–L boundary is about r DHv ¼ 1100 J m2
and the thermal flux in the sample is k rT ¼ 4000 J m2; this 25% ratio may have an effect on the interface curvature, but will be neglected in the present calculations because experimental curvatures are obtained by holding up the pulling rate. The Grashof number in the melt is about 104, which is a moderate value. The Prandtl number of molten BaF2 is 0.2 and convection in the melt will then be supposed to have no significant influence on the heat transfer in this system and will be neglected as well. The crystal growth process takes place in a vacuum of 103–104 Torr so that there is no gaseous convection in the system. Between the boundaries separated by vacuum, the radiative heat transfer is taken into account and the following boundary condition is used: qT ki ¼ qir ; ð4Þ qN i
F. Barvinschi et al. / Journal of Crystal Growth 237–239 (2002) 1762–1768
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intensity in the resistive heater, in which the Joule effect is calculated by taking into account the resistivity of the graphite and the actual geometry (section) of the heater. The steady-state temperature distribution for different growth conditions is calculated by finite element analysis. The computation domains are discretised with four nodes quadrilateral elements in a linear approximation. The number of elements has been varied from 7000 to 11,000 for the whole model in order to get the interface deflections practically unaffected by the mesh density. The Newton Raphson method with a relaxation factor of 0.6 has been used to solve the highly non-linear radiative problem. Convergence has been reached after about 130 iterations. 3.1. Influence of a molybdenum screen position on the interface shape
Fig. 2. Vertical section showing a central detail of the growth system.
where qir is the radiative flux at the boundary of the ith subdomain. The radiative heat fluxes are determined on the basis of the view factor calculation [14]. The axis of the system is an axis of reflective symmetry in the temperature profile, thus: qT k ¼ 0: ð5Þ qr r¼0
The thermal conductivity of BaF2 changes from its value in the molten state to its value in the solid state in an interval of 0.5 K around the melting temperature. In order to check the role of internal radiation in the charge, both opaque and fully transparent BaF2 samples are considered. The global model of the furnace is submitted to external boundary conditions, i.e. (i) a constant temperature on the wall, taken as the temperature of the cooling water, and (ii) a constant current
It has been already noticed that, in the case of the Bridgman furnace, the decrease of the radial dissipation of heat prevents high values of the S–L interface curvature. The addition of a thermal screen of molybdenum between the heater and the graphite screens should minimise the radial transport of heat. The heat transfer equation has been solved for three molybdenum screen positions in the furnace, noted as positions 1, 2 and 3 in Fig. 3. For all cases, the crucible was considered in the same position, its bottom at z ¼ 80 mm representing 40 mm of pulling. The results suggest that the lowest screen position leads to a 37% reduction of the S–L interface deflection. The thinner part of the heater represents the region with the highest level of heat generation by the Joule effect. As the results show, the molybdenum screen placed in front of this region prevents the radial dissipation of the heat, thus the radial thermal gradient in the sample decreases and produces a lower S–L interface deflection. 3.2. Influence of the thermal conductivity and transparency on the interface shape The S–L interface curvature depends on the thermal conductivity of the melt and of the solid. In the case of fluorides, the S–L interface is convex
F. Barvinschi et al. / Journal of Crystal Growth 237–239 (2002) 1762–1768
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4. Local model of heat transfer in the central part of the Bridgman furnace 4.1. Semi-transparent model
Fig. 3. The effect of the molybdenum screen position on the S–L interface position and shape. f indicates the solidified fraction and the interface deflection d is in mm.
In order to look more deeply into the coupled conduction–radiation heat transfer, the model is now restricted to the central part of the Bridgman furnace. The cylindrical crucible containing the BaF2 sample exchanges heat by radiation with an infinite surrounding medium whose axial temperature distribution, TA ¼ f ðZÞ; is chosen in agreement with the numerical results obtained in the global modelling. A steady-state solution of the heat transport equation is computed for several positions of the crucible in the furnace, considering crystals with different geometrical characteristics, i.e. a ¼ D=L ¼ 0:16 and 0.34. The semi-transparent properties of BaF2 are considered by using the diffusive approximation of Rosseland [14], i.e. the optically thick material has a thermal conductivity dependent on temperature (for both phases) given by ktot ¼ k þ
Table 2 Influence of some thermo-physical properties on the S–L interface deflection at a solidified fraction f ¼ 0:4 k ¼ kL =kC
Opaque d (mm)
Transparent d (mm)
0.1 1 10
4.7 1.4 4
3.2 1.2 3.5
if seen from the melt, thus k ¼ kL =kC o1: In order to study the effect of the heat transfer by internal radiation on the interface deflection, d; the BaF2 sample is considered either as opaque or fully transparent material. The numerical results for a 24 mm diameter charge are summarised in Table 2, showing that an opaque sample displays the most deflected S–L interfaces, for the considered values of k: In the case of a fully transparent charge, the axial heat transport is increased by radiation, leading to a higher axial heat flux than that in an opaque sample, thus preventing the increase of the S–L interface curvature.
16n2 sT 3 ; 3a
ð6Þ
where n is the refractive index, s is the Stefan– Boltzmann constant and a is the Rosseland mean absorption coefficient. In the case of BaF2, the melt absorption coefficient is about 10 times higher than that of the crystal [8,9]. Since the Rosseland model does not solve an extra transport equation for the incident radiation (as the P-1 model does, [15]), it is faster and requires less memory. 4.2. Influence of the semi-transparency of the sample on the interface shape It is common knowledge that the interface convexity changes when growth proceeds, due to the modification of the boundary conditions seen by the crucible. S–L interface deflections associated with various geometrical characteristics of the charge are given in Table 3, both for opaque and semi-transparent models. It can be observed that the most deflected interfaces are obtained when the sample is supposed to be opaque, for
F. Barvinschi et al. / Journal of Crystal Growth 237–239 (2002) 1762–1768 Table 3 Influence of the semi-transparency on the interface deflection. The solidified fraction is f ¼ 0:3 a ¼ D=L
Opaque d (mm)
Semi-transparent a ¼ aL =aC ¼ 10 d (mm)
1.37 3.48
1.02 2.94
Fig. 4. Isotherms in opaque (a) and semi-transparent (b) BaF2 sample (a ¼ 0:16), for the same position of the crucible in the furnace. Isotherm increment is 5 K.
both aspect ratio values. This behaviour, as it is shown in the Fig. 4 as well, it is in concordance with the reported results of [6,7,11]. We conclude this study by choosing the ratio between the absorption coefficient of the melt and of the crystal a ¼ aL =aC ¼ 10: With this ratio, the computed interface deflection values obtained by the local model are in good agreement with the experimental results (compare the S–L interface deflections in Fig. 1a. and Fig. 4b), suggesting that
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the diffusive Rosseland model of radiation is accurate enough for this material.
5. Conclusions Control strategies for generating a desired temperature distribution during a crystal growth process are difficult to obtain if only experimental attempts are used. The modelling of the heat transfer in the system may be useful in the design of the real growth configuration. However, it is important to take into account all the relevant physical phenomena. The importance of semitransparency of both crystal and melt is studied by comparing the experimental and the computed results corresponding to the growth of BaF2 crystals. By using an appropriate experimental procedure, the solid–liquid interface shape during the Bridgman growth of BaF2 crystals is revealed by quenching the solid–liquid interface during a few minutes. A steady-state simulation of the global Bridgman furnace was performed in order to calculate the temperature distribution in the equipment. The effect of a molybdenum screen on the S–L interface shape and position is determined, resulting in the optimisation of the furnace design. A melt thermal conductivity 10 times smaller than that of the solid is validated for BaF2. The coupled radiative–conductive heat transfer in crystal and melt has been studied in a local model on the basis of the diffusive radiation model. The good agreement between calculated and experimental solid–liquid interface deflections shows that our models work can be used for the design of a furnace and for clarifying some aspects of the growth process.
Acknowledgements This work was supported in part by the Ministry of Education, Romania, through Grant No. 5032/ 17.1/1996 and in part by the CE, through a TEMPUS grant IMG-RO-2095.
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References [1] J.J. Derby, R.A. Brown, J. Crystal Growth 87 (1988) 251. [2] M.J. Crochet, D. Dupret, Y. Ryckmans, J. Crystal Growth 97 (1989) 173. [3] S. Brandon, J.J. Derby, et al., J. Crystal Growth 132 (1989) 261. [4] T.I. Butaeva, A.V. Gevorkyan, et al., Docl. Acad. Nauk Arm. SSR 84 (1987) 38. [5] V.S. Yuferev, M.G. Vasiliev, J. Crystal Growth 82 (1987) 31. [6] S. Brandon, J.J. Derby, J. Crystal Growth 110 (1991) 481. [7] S. Brandon, J.J. Derby, J. Crystal Growth 121 (1992) 473.
[8] P.I. Antonov, S.I. Baholdin, M.G. Vasiliev, V.M. Krimov, V.S. Yufereev, Izv. Akad. Nauk SSSR Ser. Fiz. 52 (1988) 1997. [9] V.D. Golyshev, M.A. Gonik, Izv. Akad. Nauk. SSSR Ser. Fiz. 52 (1988) 1896. [10] FIDAP Users Manual (Revision 7. 52), Evanston, USA. [11] D. Vizman, I. Nicoara, D. Nicoara, J. Crystal Growth 169 (1996) 161. [12] I. Nicoara, D. Nicoara, O. Aczel, Cryst. Res. Technol. 22 (1987) 1139. [13] D. Nicoara, I. Nicoara, Mater. Sci. Eng. A 102 (1988) L1. [14] R. Siegel, J.R. Howell, Thermal Heat Transfer, McGrawHill, New York, 1972. [15] F. Barvinschi, T. Duffar, J.L. Santailler, J. Optoel. Adv. Mater. 2 (4) (2000) 327.
Factors affecting the isotherm shape of semi-transparent BaF2 crystals grown by Bridgman method F. Barvinschia,*, O. Bunoiub, I. Nicoarab, D. Nicoarab, J.L. Santaillerc, T. Duffard a
Department of Physics, Politehnica University of Timisoara, P-ta Regina Maria, 1, Ro-1900 Timisoara, Romania b West University of Timisoara, Bv.V.Parvan, 4, Ro-1900 Timisoara, Romania c CEA/CEREM/DEM, 17, rue des Martyrs, F-38054 Grenoble, France d EPM-MADYLAM, ENSHMG, BP 95, F-38402 St Martin d’Heres, France
Abstract The effect of physical parameters and geometrical configurations on the interface shape in semi-transparent BaF2 crystals grown by the vertical Bridgman process are studied both experimentally and numerically. Values of thermophysical parameters are suggested in the case of BaF2. r 2002 Elsevier Science B.V. All rights reserved. PACS: 47.70; 61.50.C Keywords: A1. Computer simulation: A1. Radiation: A2. Bridgman technique: B1. Barium compounds
1. Introduction During crystal growth from the melt, thermal gradients are the origin of buoyancy convection, of solid–liquid (S–L) interface deflection and of stresses in the solid. Therefore, high-quality crystals can be grown when the temperature distribution in the crystal and the melt is accurately controlled. In the case of a Bridgman furnace, which consists in a heating system and a number of screens, it is difficult to define and then to get an optimised thermal environment during the crystal growth, if undertaken in a strictly experimental way. The numerical simulation of heat transfer in such a complex furnace is therefore *Corresponding author. Tel.: +40-56-20-3417; fax: +40-5620-4364. E-mail address: fl[email protected] (F. Barvinschi).
useful for this optimisation process [1–3]. While most of the analyses do not take into account the radiative heat transfer within the crystalline and molten phases, it has been pointed out [4–7] that the role of internal radiation transfer of energy during the crystal growth process of semi-transparent materials cannot be neglected. There are semi-transparent materials, such as LiF, CaF2, BaF2, whose optical absorption coefficient is not so high as to consider the molten phase opaque [8,9]. Section 2 of this study presents the experimental demarcation of the S–L interface during the growth of BaF2 crystals by the vertical Bridgman method. Experimental convex interface shapes, when seen from the melt, are quenched for different diameter crystals. In order to explain the experimental results, the numerical simulation of heat exchanges in the Bridgman system is
0022-0248/02/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 2 2 - 0 2 4 8 ( 0 1 ) 0 2 3 2 8 - 4
F. Barvinschi et al. / Journal of Crystal Growth 237–239 (2002) 1762–1768
performed by using the FIDAP software [10]. A first steady-state heat transport model in the whole furnace is solved (Section 3). The optimum position of a molybdenum screen, positioned at several places between the graphite heater and the outer screen system, is numerically determined in order to obtain the reduction of the S–L interface deflection. In Section 4, the model is restricted to the central part of the Bridgman furnace, taking into account the semi-transparency of BaF2. By adjusting the physical properties of both phases, it has been possible to obtain results which are in qualitative concordance with the conclusions of [6,7,11] and with our experimental observations.
2. Experimental results BaF2 crystals were grown in a shaped graphite heater using the conventional Bridgman–Stockbarger method, the characteristics of the system being described in Ref. [12,13]. The heater is surrounded by a set of concentric screens made of graphite and stainless steel. Two crystals with different geometrical characteristics (indicated in Table 1) were grown in a crucible made of pure spectral graphite. The melt was doped with CoF2 (0.5% weight) in order to clearly visualise the striations associated with the S–L interface marking. To get the demarcation, the power of the
1763
furnace was interrupted for 5 min, when the grown crystal had a length of a few centimetres, and then the power and the standard translation rate were restored. On the photographs (Figs. 1a and b) the quenched interfaces are seen for both crystals. The interface is convex towards the melt, meaning that the thermal conductivity of the melt is lower than that of the crystal, in agreement with Refs. [8,9].
3. Global modelling of heat transfer The temperature distribution in growing crystals depends strongly upon the boundary conditions on their outer surface, which are a priori unknown. As a first approach, the global calculation of the steady-state heat transfer throughout the whole furnace is performed. The growth of a 24 mm diameter crystal is simulated by using (i) a reduced number of external parameters (power input, water temperature in the double jacket facility wall), (ii) the geometrical size of the crystal and or the set-up and (iii) the physical properties of the materials. A detail of the schematic diagram of the Bridgman furnace is shown in Fig. 2. The thermo-physical properties of interest are listed in Table 1. In order to obtain the temperature distribution, the whole system is divided into subdomains. Heat transfer in all the subdomains is described by the
Table 1 Operating parameters and material properties used Geometrical characteristics D ¼ 11 and 24 mm L ¼ 70 and 30 mm
Graphite Ka ¼ 45 W m1 K1 ea ¼ 0:8
BaF2 solid kC ¼ 2:4 W m1 K1 aC ¼ 0:120:3 cm1
Aspect ratio a ¼ D=L ¼ 0:16; 0.34 and 0.8
Molybdenum kMo ¼ 90 W m1 K1
BaF2 liquid kL ¼ 0:24 W m1 K1
Crucible wall thickness wa ¼ 1:5 mm
eMo ¼ 0:4 Stainless steel
aL ¼ 123 cm–1 Tm ¼ 1628 K
Pulling rate: v ¼ 3 mm h1
kst ¼ 20 W m1 K1
DH ¼ 3:8 105 J Kg1
Current intensity I ¼ 290 and 306 A
est ¼ 0:2
rL;C ¼ 4830 Kg m3 n ¼ 1:47
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Fig. 1. S–L interfaces revealed by quenching experiments on BaF2 crystals :(a) aspect ratio a ¼ 0:16; solidified fraction f ¼ 0:88; interface deflection dexp ¼ 170:05 mm; (b) aspect ratio a ¼ 0:8; solidified fraction f ¼ 0:64; interface deflection dexp ¼ 3:170:05 mm.
steady-state equation: rðkðT; r; zÞrTÞ þ Hðr; zÞ ¼ 0;
ð1Þ
where kðT; r; zÞ is the thermal conductivity of the considered domain and Hðr; zÞ is the heat source term that vanishes in all the subdomains, except in the heater. The following boundary conditions are used. The temperature and the heat flux between adjoining subdomains are continuous: Ti ¼ Tj
ð2Þ
and on the boundary between the subdomains i and j: ~ ij ¼ ðkj rTÞj N ~ ij ; ðki rTÞi N
ð3Þ
~ ij is the unitary normal to the boundary where N between subdomains. The release of the latent heat to be considered at the S–L boundary is about r DHv ¼ 1100 J m2
and the thermal flux in the sample is k rT ¼ 4000 J m2; this 25% ratio may have an effect on the interface curvature, but will be neglected in the present calculations because experimental curvatures are obtained by holding up the pulling rate. The Grashof number in the melt is about 104, which is a moderate value. The Prandtl number of molten BaF2 is 0.2 and convection in the melt will then be supposed to have no significant influence on the heat transfer in this system and will be neglected as well. The crystal growth process takes place in a vacuum of 103–104 Torr so that there is no gaseous convection in the system. Between the boundaries separated by vacuum, the radiative heat transfer is taken into account and the following boundary condition is used: qT ki ¼ qir ; ð4Þ qN i
F. Barvinschi et al. / Journal of Crystal Growth 237–239 (2002) 1762–1768
1765
intensity in the resistive heater, in which the Joule effect is calculated by taking into account the resistivity of the graphite and the actual geometry (section) of the heater. The steady-state temperature distribution for different growth conditions is calculated by finite element analysis. The computation domains are discretised with four nodes quadrilateral elements in a linear approximation. The number of elements has been varied from 7000 to 11,000 for the whole model in order to get the interface deflections practically unaffected by the mesh density. The Newton Raphson method with a relaxation factor of 0.6 has been used to solve the highly non-linear radiative problem. Convergence has been reached after about 130 iterations. 3.1. Influence of a molybdenum screen position on the interface shape
Fig. 2. Vertical section showing a central detail of the growth system.
where qir is the radiative flux at the boundary of the ith subdomain. The radiative heat fluxes are determined on the basis of the view factor calculation [14]. The axis of the system is an axis of reflective symmetry in the temperature profile, thus: qT k ¼ 0: ð5Þ qr r¼0
The thermal conductivity of BaF2 changes from its value in the molten state to its value in the solid state in an interval of 0.5 K around the melting temperature. In order to check the role of internal radiation in the charge, both opaque and fully transparent BaF2 samples are considered. The global model of the furnace is submitted to external boundary conditions, i.e. (i) a constant temperature on the wall, taken as the temperature of the cooling water, and (ii) a constant current
It has been already noticed that, in the case of the Bridgman furnace, the decrease of the radial dissipation of heat prevents high values of the S–L interface curvature. The addition of a thermal screen of molybdenum between the heater and the graphite screens should minimise the radial transport of heat. The heat transfer equation has been solved for three molybdenum screen positions in the furnace, noted as positions 1, 2 and 3 in Fig. 3. For all cases, the crucible was considered in the same position, its bottom at z ¼ 80 mm representing 40 mm of pulling. The results suggest that the lowest screen position leads to a 37% reduction of the S–L interface deflection. The thinner part of the heater represents the region with the highest level of heat generation by the Joule effect. As the results show, the molybdenum screen placed in front of this region prevents the radial dissipation of the heat, thus the radial thermal gradient in the sample decreases and produces a lower S–L interface deflection. 3.2. Influence of the thermal conductivity and transparency on the interface shape The S–L interface curvature depends on the thermal conductivity of the melt and of the solid. In the case of fluorides, the S–L interface is convex
F. Barvinschi et al. / Journal of Crystal Growth 237–239 (2002) 1762–1768
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4. Local model of heat transfer in the central part of the Bridgman furnace 4.1. Semi-transparent model
Fig. 3. The effect of the molybdenum screen position on the S–L interface position and shape. f indicates the solidified fraction and the interface deflection d is in mm.
In order to look more deeply into the coupled conduction–radiation heat transfer, the model is now restricted to the central part of the Bridgman furnace. The cylindrical crucible containing the BaF2 sample exchanges heat by radiation with an infinite surrounding medium whose axial temperature distribution, TA ¼ f ðZÞ; is chosen in agreement with the numerical results obtained in the global modelling. A steady-state solution of the heat transport equation is computed for several positions of the crucible in the furnace, considering crystals with different geometrical characteristics, i.e. a ¼ D=L ¼ 0:16 and 0.34. The semi-transparent properties of BaF2 are considered by using the diffusive approximation of Rosseland [14], i.e. the optically thick material has a thermal conductivity dependent on temperature (for both phases) given by ktot ¼ k þ
Table 2 Influence of some thermo-physical properties on the S–L interface deflection at a solidified fraction f ¼ 0:4 k ¼ kL =kC
Opaque d (mm)
Transparent d (mm)
0.1 1 10
4.7 1.4 4
3.2 1.2 3.5
if seen from the melt, thus k ¼ kL =kC o1: In order to study the effect of the heat transfer by internal radiation on the interface deflection, d; the BaF2 sample is considered either as opaque or fully transparent material. The numerical results for a 24 mm diameter charge are summarised in Table 2, showing that an opaque sample displays the most deflected S–L interfaces, for the considered values of k: In the case of a fully transparent charge, the axial heat transport is increased by radiation, leading to a higher axial heat flux than that in an opaque sample, thus preventing the increase of the S–L interface curvature.
16n2 sT 3 ; 3a
ð6Þ
where n is the refractive index, s is the Stefan– Boltzmann constant and a is the Rosseland mean absorption coefficient. In the case of BaF2, the melt absorption coefficient is about 10 times higher than that of the crystal [8,9]. Since the Rosseland model does not solve an extra transport equation for the incident radiation (as the P-1 model does, [15]), it is faster and requires less memory. 4.2. Influence of the semi-transparency of the sample on the interface shape It is common knowledge that the interface convexity changes when growth proceeds, due to the modification of the boundary conditions seen by the crucible. S–L interface deflections associated with various geometrical characteristics of the charge are given in Table 3, both for opaque and semi-transparent models. It can be observed that the most deflected interfaces are obtained when the sample is supposed to be opaque, for
F. Barvinschi et al. / Journal of Crystal Growth 237–239 (2002) 1762–1768 Table 3 Influence of the semi-transparency on the interface deflection. The solidified fraction is f ¼ 0:3 a ¼ D=L
Opaque d (mm)
Semi-transparent a ¼ aL =aC ¼ 10 d (mm)
1.37 3.48
1.02 2.94
Fig. 4. Isotherms in opaque (a) and semi-transparent (b) BaF2 sample (a ¼ 0:16), for the same position of the crucible in the furnace. Isotherm increment is 5 K.
both aspect ratio values. This behaviour, as it is shown in the Fig. 4 as well, it is in concordance with the reported results of [6,7,11]. We conclude this study by choosing the ratio between the absorption coefficient of the melt and of the crystal a ¼ aL =aC ¼ 10: With this ratio, the computed interface deflection values obtained by the local model are in good agreement with the experimental results (compare the S–L interface deflections in Fig. 1a. and Fig. 4b), suggesting that
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the diffusive Rosseland model of radiation is accurate enough for this material.
5. Conclusions Control strategies for generating a desired temperature distribution during a crystal growth process are difficult to obtain if only experimental attempts are used. The modelling of the heat transfer in the system may be useful in the design of the real growth configuration. However, it is important to take into account all the relevant physical phenomena. The importance of semitransparency of both crystal and melt is studied by comparing the experimental and the computed results corresponding to the growth of BaF2 crystals. By using an appropriate experimental procedure, the solid–liquid interface shape during the Bridgman growth of BaF2 crystals is revealed by quenching the solid–liquid interface during a few minutes. A steady-state simulation of the global Bridgman furnace was performed in order to calculate the temperature distribution in the equipment. The effect of a molybdenum screen on the S–L interface shape and position is determined, resulting in the optimisation of the furnace design. A melt thermal conductivity 10 times smaller than that of the solid is validated for BaF2. The coupled radiative–conductive heat transfer in crystal and melt has been studied in a local model on the basis of the diffusive radiation model. The good agreement between calculated and experimental solid–liquid interface deflections shows that our models work can be used for the design of a furnace and for clarifying some aspects of the growth process.
Acknowledgements This work was supported in part by the Ministry of Education, Romania, through Grant No. 5032/ 17.1/1996 and in part by the CE, through a TEMPUS grant IMG-RO-2095.
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