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2010 | OriginalPaper | Chapter

40. Mass and Heat Transport in BS and EFG Systems

Authors : Thomas F. George, Stefan Balint, Liliana Braescu

Published in: Springer Handbook of Crystal Growth

Publisher: Springer Berlin Heidelberg

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Abstract

In this chapter several mathematical models describing processes which take place in the Bridgman–Stockbarger (BS) and edge-defined film-fed growth (EFG) systems are presented. Predictions are made concerning the impurity repartition in the crystal in the framework of each of the models. First, a short description of the real processes which are modeled is given, along with the equations, boundary conditions, and initial values defining the mathematical model. After that, numerical results obtained by computations in the framework of the model are provided, making a comparison between the computed results and those obtained in other models, and with the experimental data.

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Literature
40.1.
go back to reference J.A. Burton, R.C. Prim, W.P. Slichter: The distribution of solute on crystals grown from the melt, J. Chem. Phys. 21, 1987–1996 (1953)ADSCrossRef J.A. Burton, R.C. Prim, W.P. Slichter: The distribution of solute on crystals grown from the melt, J. Chem. Phys. 21, 1987–1996 (1953)ADSCrossRef
40.2.
go back to reference C. Wagner: Theoretical analysis of diffusion of solutes during the solidification of alloys, J. Met. 6, 154–160 (1954) C. Wagner: Theoretical analysis of diffusion of solutes during the solidification of alloys, J. Met. 6, 154–160 (1954)
40.3.
go back to reference K.M. Kim, A.F. Witt, M. Lichtensteiger, H.C. Gatos: Quantitative analysis of the thermo-hydrodynamic effects on crystal growth and segregation under destabilizing vertical thermal gradients: Ga-doped germanium, J. Electrochem. Soc. 125, 475–480 (1974)CrossRef K.M. Kim, A.F. Witt, M. Lichtensteiger, H.C. Gatos: Quantitative analysis of the thermo-hydrodynamic effects on crystal growth and segregation under destabilizing vertical thermal gradients: Ga-doped germanium, J. Electrochem. Soc. 125, 475–480 (1974)CrossRef
40.4.
go back to reference J.J. Favier: Macrosegregation-I: Unified analysis during non-steady state. Solidification, Acta Metall. 29, 197–204 (1981)CrossRef J.J. Favier: Macrosegregation-I: Unified analysis during non-steady state. Solidification, Acta Metall. 29, 197–204 (1981)CrossRef
40.5.
go back to reference J.J. Favier: Macrosegregation-II: A comparative study of theories, Acta Metall. 29, 205–214 (1981)CrossRef J.J. Favier: Macrosegregation-II: A comparative study of theories, Acta Metall. 29, 205–214 (1981)CrossRef
40.6.
go back to reference D.E. Holmes, H.C. Gatos: Convective interference and "effective" diffusion-controlled segregation during directional solidification under stabilizing vertical thermal gradients; Ge, J. Electrochem. Soc. 128, 429–437 (1981)CrossRef D.E. Holmes, H.C. Gatos: Convective interference and "effective" diffusion-controlled segregation during directional solidification under stabilizing vertical thermal gradients; Ge, J. Electrochem. Soc. 128, 429–437 (1981)CrossRef
40.7.
go back to reference C.J. Chang, R.A. Brown: Radial segregation induced by natural convection and melt/solid interface shape in vertical Bridgman growth, J. Cryst. Growth 63, 343–364 (1983)ADSCrossRef C.J. Chang, R.A. Brown: Radial segregation induced by natural convection and melt/solid interface shape in vertical Bridgman growth, J. Cryst. Growth 63, 343–364 (1983)ADSCrossRef
40.8.
go back to reference W.A. Tiller, K.A. Jackson, J.W. Rutter, B. Chalmers: The redistribution of solute atoms during the solidification of metals, Acta Metall. 1, 428–437 (1953)CrossRef W.A. Tiller, K.A. Jackson, J.W. Rutter, B. Chalmers: The redistribution of solute atoms during the solidification of metals, Acta Metall. 1, 428–437 (1953)CrossRef
40.9.
go back to reference P.R. Griffin, S. Motakef: Influence of non-steady gravity on natural convection during micro-gravity solidification of semiconductors. Part I. Time Scale Analysis, J. Appl. Microgravity II 3, 121–127 (1989) P.R. Griffin, S. Motakef: Influence of non-steady gravity on natural convection during micro-gravity solidification of semiconductors. Part I. Time Scale Analysis, J. Appl. Microgravity II 3, 121–127 (1989)
40.10.
go back to reference P.R. Griffin, S. Motakef: Influence of non-steady gravity on natural convection during micro-gravity solidification of semiconductors. Part II. Implications for crystal growth experiments, J. Appl. Microgravity II 3, 128–132 (1989) P.R. Griffin, S. Motakef: Influence of non-steady gravity on natural convection during micro-gravity solidification of semiconductors. Part II. Implications for crystal growth experiments, J. Appl. Microgravity II 3, 128–132 (1989)
40.11.
go back to reference A.F. Witt, H.C. Gatos, M. Lichtensteiger, M.C. Lavine, C.J. Herman: Crystal growth and steady-state segregation under zero gravity, J. Electrochem. Soc. 122, 276–283 (1975)ADSCrossRef A.F. Witt, H.C. Gatos, M. Lichtensteiger, M.C. Lavine, C.J. Herman: Crystal growth and steady-state segregation under zero gravity, J. Electrochem. Soc. 122, 276–283 (1975)ADSCrossRef
40.12.
go back to reference P.M. Adornato, R.A. Brown: Convection and segregation in directional solidification of dilute and non-dilute binary alloys, J. Cryst. Growth 80, 155–190 (1987)ADSCrossRef P.M. Adornato, R.A. Brown: Convection and segregation in directional solidification of dilute and non-dilute binary alloys, J. Cryst. Growth 80, 155–190 (1987)ADSCrossRef
40.13.
go back to reference P.A. Clark, W.R. Wilcox: Influence of gravity on thermocapillary convection in floating zone melting of silicon, J. Cryst. Growth 50, 461–469 (1980)ADSCrossRef P.A. Clark, W.R. Wilcox: Influence of gravity on thermocapillary convection in floating zone melting of silicon, J. Cryst. Growth 50, 461–469 (1980)ADSCrossRef
40.14.
go back to reference S.A.I. Nikitin, V. Polezhayev, A.I. Fedyushkin: Mathematical simulation of impurity distribution in crystals prepared under microgravity conditions, J. Cryst. Growth 52, 471–477 (1981)ADSCrossRef S.A.I. Nikitin, V. Polezhayev, A.I. Fedyushkin: Mathematical simulation of impurity distribution in crystals prepared under microgravity conditions, J. Cryst. Growth 52, 471–477 (1981)ADSCrossRef
40.15.
go back to reference S.R. Coriell, R.F. Sekerka: Lateral solute segregation during unidirectional solidification of a binary alloy with a curved solid-liquid interface, J. Cryst. Growth 46, 479–482 (1979)ADSCrossRef S.R. Coriell, R.F. Sekerka: Lateral solute segregation during unidirectional solidification of a binary alloy with a curved solid-liquid interface, J. Cryst. Growth 46, 479–482 (1979)ADSCrossRef
40.16.
go back to reference S.R. Coriell, R.F. Boisvert, R.G. Rehm, R.F. Sekerka: Lateral solute segregation during unidirectional solidification of a binary alloy with a curved solid-liquid interface II. Large departures from planarity, J. Cryst. Growth 54, 167–175 (1981)ADSCrossRef S.R. Coriell, R.F. Boisvert, R.G. Rehm, R.F. Sekerka: Lateral solute segregation during unidirectional solidification of a binary alloy with a curved solid-liquid interface II. Large departures from planarity, J. Cryst. Growth 54, 167–175 (1981)ADSCrossRef
40.17.
go back to reference H.M. Ettouney, R.A. Brown: Effect of heat transfer on melt/solid interface shape and solute segregation in edge-defined film-fed growth: Finite element analysis, J. Cryst. Growth 58, 313–329 (1982)ADSCrossRef H.M. Ettouney, R.A. Brown: Effect of heat transfer on melt/solid interface shape and solute segregation in edge-defined film-fed growth: Finite element analysis, J. Cryst. Growth 58, 313–329 (1982)ADSCrossRef
40.18.
go back to reference J.P. Kalejs, L.Y. Chin, F.M. Carlson: Interface shape studies for silicon ribbon growth by the EFG technique I. Transport phenomena modeling, J. Cryst. Growth 61, 473–484 (1983)ADSCrossRef J.P. Kalejs, L.Y. Chin, F.M. Carlson: Interface shape studies for silicon ribbon growth by the EFG technique I. Transport phenomena modeling, J. Cryst. Growth 61, 473–484 (1983)ADSCrossRef
40.19.
go back to reference C.E. Chang, W.R. Wilcox: Control of interface shape in the vertical Bridgman-Stockbarger technique, J. Cryst. Growth 21, 135–140 (1974)ADSCrossRef C.E. Chang, W.R. Wilcox: Control of interface shape in the vertical Bridgman-Stockbarger technique, J. Cryst. Growth 21, 135–140 (1974)ADSCrossRef
40.20.
go back to reference T.W. Fu, W.R. Wilcox: Influence of insulation on stability of interface shape and position in the vertical Bridgman-Stockbarger technique, J. Cryst. Growth 48, 416–424 (1980)ADSCrossRef T.W. Fu, W.R. Wilcox: Influence of insulation on stability of interface shape and position in the vertical Bridgman-Stockbarger technique, J. Cryst. Growth 48, 416–424 (1980)ADSCrossRef
40.21.
go back to reference T.W. Clyne: Heat flow in controlled directional solidification of metals I. Experimental investigation, J. Cryst. Growth 50, 684–690 (1980)ADSCrossRef T.W. Clyne: Heat flow in controlled directional solidification of metals I. Experimental investigation, J. Cryst. Growth 50, 684–690 (1980)ADSCrossRef
40.22.
go back to reference T.W. Clyne: Heat flow in controlled directional solidification of metals II. Mathematical model, J. Cryst. Growth 50, 691–700 (1980)ADSCrossRef T.W. Clyne: Heat flow in controlled directional solidification of metals II. Mathematical model, J. Cryst. Growth 50, 691–700 (1980)ADSCrossRef
40.23.
go back to reference P.C. Sukanek: Deviation of freezing rate from translation rate in the Bridgman-Stockbarger technique I. Very low translation rates, J. Cryst. Growth 58, 208–218 (1982)ADSCrossRef P.C. Sukanek: Deviation of freezing rate from translation rate in the Bridgman-Stockbarger technique I. Very low translation rates, J. Cryst. Growth 58, 208–218 (1982)ADSCrossRef
40.24.
go back to reference P.C. Sukanek: Deviation of freezing rate from translation rate in the Bridgman-Stockbarger technique II. Moderate translation rates, J. Cryst. Growth 58, 219–228 (1982)ADSCrossRef P.C. Sukanek: Deviation of freezing rate from translation rate in the Bridgman-Stockbarger technique II. Moderate translation rates, J. Cryst. Growth 58, 219–228 (1982)ADSCrossRef
40.25.
go back to reference T. Jasinski, W.M. Rohsenow, A.F. Witt: Heat transfer analysis of the Bridgman-Stockbarger configuration for crystal growth I. Analytical treatment of the axial temperature profile, J. Cryst. Growth 61, 339–354 (1983)ADSCrossRef T. Jasinski, W.M. Rohsenow, A.F. Witt: Heat transfer analysis of the Bridgman-Stockbarger configuration for crystal growth I. Analytical treatment of the axial temperature profile, J. Cryst. Growth 61, 339–354 (1983)ADSCrossRef
40.26.
go back to reference L.R. Morris, W.C. Winegard: The development of cells during the solidification dilute Pb-Sb alloy, J. Cryst. Growth 5, 361–375 (1969)ADSCrossRef L.R. Morris, W.C. Winegard: The development of cells during the solidification dilute Pb-Sb alloy, J. Cryst. Growth 5, 361–375 (1969)ADSCrossRef
40.27.
go back to reference A.F. Witt, H.C. Gatos, M. Lichtensteiger, C.J. Herman: Crystal growth and segregation under zero gravity, J. Electrochem. Soc. 125, 1832–1840 (1978)CrossRef A.F. Witt, H.C. Gatos, M. Lichtensteiger, C.J. Herman: Crystal growth and segregation under zero gravity, J. Electrochem. Soc. 125, 1832–1840 (1978)CrossRef
40.28.
go back to reference A.M. Balint, D.G. Baltean, T. Levy, M. Mihailovici, A. Neculae, S. Balint: The dopant fields in uniform-diffusion-layer, global-thermal-convection and precrystallization-zone models, Mater. Sci. Semicond. Process. 3, 115–121 (2000)CrossRef A.M. Balint, D.G. Baltean, T. Levy, M. Mihailovici, A. Neculae, S. Balint: The dopant fields in uniform-diffusion-layer, global-thermal-convection and precrystallization-zone models, Mater. Sci. Semicond. Process. 3, 115–121 (2000)CrossRef
40.29.
go back to reference D.T.J. Hurle, E. Jakeman, C.P. Johnson: Convective temperature oscillations in molten gallium, J. Fluid Mech. 64, 565–576 (1974)ADSCrossRef D.T.J. Hurle, E. Jakeman, C.P. Johnson: Convective temperature oscillations in molten gallium, J. Fluid Mech. 64, 565–576 (1974)ADSCrossRef
40.30.
go back to reference C.A. Wang, A.F. Witt: Annual Report Material Processing Center (Massachusetts Institute of Technology, Massachusetts 1984) C.A. Wang, A.F. Witt: Annual Report Material Processing Center (Massachusetts Institute of Technology, Massachusetts 1984)
40.31.
go back to reference M.M. Mihailovici, A.M. Balint, S. Balint: The axial and radial segregation due to the thermo-convection, the decrease of the melt in the ampoule and the effect of the precrystallization-zone in the semiconductor crystals grown in a Bridgman-Stockbarger system in a low gravity environment, J. Cryst. Growth 237–239, 1752–1756 (2002)CrossRef M.M. Mihailovici, A.M. Balint, S. Balint: The axial and radial segregation due to the thermo-convection, the decrease of the melt in the ampoule and the effect of the precrystallization-zone in the semiconductor crystals grown in a Bridgman-Stockbarger system in a low gravity environment, J. Cryst. Growth 237–239, 1752–1756 (2002)CrossRef
40.32.
go back to reference M.M. Mihailovici, A.M. Balint, S. Balint: On the controllability of the level of the dopant concentration and of the compositional uniformity of a doped crystal, grown in a low gravity environment by Bridgman-Stockbarger method, Int. J. Theor. Physics, Group Theory Nonlin. Opt. 10, 425–436 (2003) M.M. Mihailovici, A.M. Balint, S. Balint: On the controllability of the level of the dopant concentration and of the compositional uniformity of a doped crystal, grown in a low gravity environment by Bridgman-Stockbarger method, Int. J. Theor. Physics, Group Theory Nonlin. Opt. 10, 425–436 (2003)
40.33.
go back to reference A.M. Balint, M.M. Mihailovici, D.G. Baltean, S. Balint: A modified Chang-Brown model for the determination of the dopant distribution in a Bridgman-Stockbarger semiconductor crystal growth system, J. Cryst. Growth 230, 195–201 (2001)ADSCrossRef A.M. Balint, M.M. Mihailovici, D.G. Baltean, S. Balint: A modified Chang-Brown model for the determination of the dopant distribution in a Bridgman-Stockbarger semiconductor crystal growth system, J. Cryst. Growth 230, 195–201 (2001)ADSCrossRef
40.34.
go back to reference A.M. Balint, M.M. Mihailovici, D.G. Baltean, S. Balint: Interface structure in the growth of semiconductor crystals using the Bridgman-Stockbarger method, Thin Solid Films 380, 108–110 (2000)ADSCrossRef A.M. Balint, M.M. Mihailovici, D.G. Baltean, S. Balint: Interface structure in the growth of semiconductor crystals using the Bridgman-Stockbarger method, Thin Solid Films 380, 108–110 (2000)ADSCrossRef
40.35.
go back to reference M.M. Mihailovici, A.M. Balint, S. Balint: The dopant distribution computed in the modified Chang-Brown model using quasi-steady state approximation, Comput. Mater. Sci. 24, 262–267 (2002)CrossRef M.M. Mihailovici, A.M. Balint, S. Balint: The dopant distribution computed in the modified Chang-Brown model using quasi-steady state approximation, Comput. Mater. Sci. 24, 262–267 (2002)CrossRef
40.36.
go back to reference D.G. Baltean, T. Levy, S. Balint: Transport de masse par convection et diffusion dans un milieu multiporeux, C. r. Acad. Sci. Paris, Ser. IIB 326, 821–826 (1998)ADSMATH D.G. Baltean, T. Levy, S. Balint: Transport de masse par convection et diffusion dans un milieu multiporeux, C. r. Acad. Sci. Paris, Ser. IIB 326, 821–826 (1998)ADSMATH
40.37.
go back to reference K. Moutsopoulos, S. Bories: Dispersion en milieux poreux hétérogènes, C. R. Acad. Sci. Paris, Ser. IIB 316, 1667–1672 (1993), in FrenchMATH K. Moutsopoulos, S. Bories: Dispersion en milieux poreux hétérogènes, C. R. Acad. Sci. Paris, Ser. IIB 316, 1667–1672 (1993), in FrenchMATH
40.38.
go back to reference J.C. Maxwell: Electricity and Magnetism (Clarendon, Oxford 1873)MATH J.C. Maxwell: Electricity and Magnetism (Clarendon, Oxford 1873)MATH
40.39.
go back to reference V.I. Avetisov, Mendeleev Institute Moscow (personal communication) V.I. Avetisov, Mendeleev Institute Moscow (personal communication)
40.40.
go back to reference M.M. Mihailovici, A.M. Balint, S. Balint: Way to improve the compositional uniformity of doped crystals grown by Bridgman-Stockbarger method in a low gravity environment, Int. J. Theor. Phys. Group Theory Nonlinear Opt. 11, 109–119 (2004)MATH M.M. Mihailovici, A.M. Balint, S. Balint: Way to improve the compositional uniformity of doped crystals grown by Bridgman-Stockbarger method in a low gravity environment, Int. J. Theor. Phys. Group Theory Nonlinear Opt. 11, 109–119 (2004)MATH
40.41.
go back to reference D.J. Larson, J. Bethin, B.S. Dressler: Shuttle Mission 51-G, Experiment MRS77F055, Flight Sample Characterization (Grumman Corporate Research Center, Bethpage 1988), NASA Report RE-753 D.J. Larson, J. Bethin, B.S. Dressler: Shuttle Mission 51-G, Experiment MRS77F055, Flight Sample Characterization (Grumman Corporate Research Center, Bethpage 1988), NASA Report RE-753
40.42.
go back to reference P.S. Dutta, A.G. Ostrogorsky: Segregation of Ga in Ge and InSb in GaSb, J. Cryst. Growth 217, 360–365 (2000)ADSCrossRef P.S. Dutta, A.G. Ostrogorsky: Segregation of Ga in Ge and InSb in GaSb, J. Cryst. Growth 217, 360–365 (2000)ADSCrossRef
40.43.
go back to reference L.L. Zheng, D.J. Larson Jr., H. Zhang: Role of thermotransport (Sorret effect) in macrosegregation during eutectic/off-eutectic directional solidification, J. Cryst. Growth 191, 243–251 (1998)ADSCrossRef L.L. Zheng, D.J. Larson Jr., H. Zhang: Role of thermotransport (Sorret effect) in macrosegregation during eutectic/off-eutectic directional solidification, J. Cryst. Growth 191, 243–251 (1998)ADSCrossRef
40.44.
go back to reference L. Braescu: The dependence of the dopant distribution on the pulling rate and on the capillary channel radius in the case of a Nd:YVO4 cylindrical bar grown from the melt by the EFG method, Mater. Sci. Eng. B 146, 41–44 (2008)CrossRef L. Braescu: The dependence of the dopant distribution on the pulling rate and on the capillary channel radius in the case of a Nd:YVO4 cylindrical bar grown from the melt by the EFG method, Mater. Sci. Eng. B 146, 41–44 (2008)CrossRef
40.45.
go back to reference E. Tulcan-Paulescu, A.M. Balint, S. Balint: The effect of the initial dopant distribution in the melt on the axial compositional uniformity of a thin doped crystal grown in strictly zero-gravity environment by Bridgman-Stockbarger method, J. Cryst. Growth 247, 313–319 (2003)ADSCrossRefMATH E. Tulcan-Paulescu, A.M. Balint, S. Balint: The effect of the initial dopant distribution in the melt on the axial compositional uniformity of a thin doped crystal grown in strictly zero-gravity environment by Bridgman-Stockbarger method, J. Cryst. Growth 247, 313–319 (2003)ADSCrossRefMATH
40.46.
40.47.
go back to reference H.E. LaBelle Jr., A.I. Mlavsky, B. Chalmers: Growth of controlled profile crystals from the melt: Part I – Sapphire filaments, Mater. Res. Bull 6, 571–579 (1971)CrossRef H.E. LaBelle Jr., A.I. Mlavsky, B. Chalmers: Growth of controlled profile crystals from the melt: Part I – Sapphire filaments, Mater. Res. Bull 6, 571–579 (1971)CrossRef
40.48.
go back to reference H.E. LaBelle Jr., A.I. Mlavsky, B. Chalmers: Growth of controlled profile crystals from the melt: Part II – Edge-defined, film-fed growth (EFG), Mater. Res. Bull. 6, 581 (1971)CrossRef H.E. LaBelle Jr., A.I. Mlavsky, B. Chalmers: Growth of controlled profile crystals from the melt: Part II – Edge-defined, film-fed growth (EFG), Mater. Res. Bull. 6, 581 (1971)CrossRef
40.49.
go back to reference B. Chalmers, H.E. LaBelle Jr., A.I. Mlavsky: Edge-defined, film-fed crystal growth, J. Cryst. Growth 13/14, 84–87 (1972)ADSCrossRef B. Chalmers, H.E. LaBelle Jr., A.I. Mlavsky: Edge-defined, film-fed crystal growth, J. Cryst. Growth 13/14, 84–87 (1972)ADSCrossRef
40.50.
go back to reference J.P. Kalejs: Impurity redistribution in EFG, J. Cryst. Growth 44, 329–344 (1978)ADSCrossRef J.P. Kalejs: Impurity redistribution in EFG, J. Cryst. Growth 44, 329–344 (1978)ADSCrossRef
40.51.
go back to reference J.P. Kalejs, G.M. Freedman, F.V. Wald: Aluminium redistribution in EFG silicon ribbon, J. Cryst. Growth 48, 74–84 (1980)ADSCrossRef J.P. Kalejs, G.M. Freedman, F.V. Wald: Aluminium redistribution in EFG silicon ribbon, J. Cryst. Growth 48, 74–84 (1980)ADSCrossRef
40.52.
go back to reference B. Chalmers: Transient solute effects in shaped crystal growth of silicon, J. Cryst. Growth 82, 70–73 (1987)ADSCrossRef B. Chalmers: Transient solute effects in shaped crystal growth of silicon, J. Cryst. Growth 82, 70–73 (1987)ADSCrossRef
40.53.
go back to reference J. Cao, M. Prince, J.P. Kalejs: Impurity transients in multiple crystal growth from a single crucible for EFG silicon octagons, J. Cryst. Growth 174, 170–175 (1997)ADSCrossRef J. Cao, M. Prince, J.P. Kalejs: Impurity transients in multiple crystal growth from a single crucible for EFG silicon octagons, J. Cryst. Growth 174, 170–175 (1997)ADSCrossRef
40.54.
go back to reference J.P. Kalejs: Interface shape studies for silicon ribbon growth by the EFG technique II. Effect of die asymmetry, J. Cryst. Growth 61, 485–493 (1983)ADSCrossRef J.P. Kalejs: Interface shape studies for silicon ribbon growth by the EFG technique II. Effect of die asymmetry, J. Cryst. Growth 61, 485–493 (1983)ADSCrossRef
40.55.
go back to reference J.P. Kalejs: Modeling contribution in commercialization of silicon ribbon growth from the melt, J. Cryst. Growth 230, 10–21 (2001)ADSCrossRef J.P. Kalejs: Modeling contribution in commercialization of silicon ribbon growth from the melt, J. Cryst. Growth 230, 10–21 (2001)ADSCrossRef
40.56.
go back to reference O. Bunoiu, I. Nicoara, J.L. Santailler, T. Duffar: Fluid flow and solute segregation in EFG crystal growth process, J. Cryst. Growth 275, 799–805 (2005)ADSCrossRef O. Bunoiu, I. Nicoara, J.L. Santailler, T. Duffar: Fluid flow and solute segregation in EFG crystal growth process, J. Cryst. Growth 275, 799–805 (2005)ADSCrossRef
40.57.
go back to reference L. Braescu, S. Balint, L. Tanasie: Numerical studies concerning the dependence of the impurity distribution on the pulling rate and on the radius of the capillary channel in the case of a thin rod grown from the melt by edge-defined film-fed growth (EFG) method, J. Cryst. Growth 291, 52–59 (2006)ADSCrossRef L. Braescu, S. Balint, L. Tanasie: Numerical studies concerning the dependence of the impurity distribution on the pulling rate and on the radius of the capillary channel in the case of a thin rod grown from the melt by edge-defined film-fed growth (EFG) method, J. Cryst. Growth 291, 52–59 (2006)ADSCrossRef
40.58.
go back to reference L. Braescu, T.F. George, S. Balint: Evaluation and control of the dopant distribution in a Nd:LiNbO3 fiber grown from the melt by the edge-defined film-fed growth (EFG) method, Photonic Fiber and Crystal Devices: Advances in Materials and Innovations in Device Applications I (Optics and Photonics 2007), SPIE Proc. 6698, 669803:1–8 (2007)CrossRef L. Braescu, T.F. George, S. Balint: Evaluation and control of the dopant distribution in a Nd:LiNbO3 fiber grown from the melt by the edge-defined film-fed growth (EFG) method, Photonic Fiber and Crystal Devices: Advances in Materials and Innovations in Device Applications I (Optics and Photonics 2007), SPIE Proc. 6698, 669803:1–8 (2007)CrossRef
40.59.
go back to reference L. Braescu, T. Duffar: Effect of buoyancy and Marangoni forces on the dopant distribution in the case of a single crystal fiber grown from the melt by the edge-defined film-fed growth (EFG) method, J. Cryst. Growth 310, 484–489 (2008)ADSCrossRef L. Braescu, T. Duffar: Effect of buoyancy and Marangoni forces on the dopant distribution in the case of a single crystal fiber grown from the melt by the edge-defined film-fed growth (EFG) method, J. Cryst. Growth 310, 484–489 (2008)ADSCrossRef
40.60.
go back to reference T.F. George, L. Braescu: Sapphire fibers grown from the melt by the EFG technique: Dependence of the impurity distribution on temperature and surface tension gradients, Photonic Fiber and Crystal Devices: Advances in Materials and Innovations in Device Applications II (Optics and Photonics 2008), SPIE Proc. 7056, 705603–1–705603–10 (2008)CrossRef T.F. George, L. Braescu: Sapphire fibers grown from the melt by the EFG technique: Dependence of the impurity distribution on temperature and surface tension gradients, Photonic Fiber and Crystal Devices: Advances in Materials and Innovations in Device Applications II (Optics and Photonics 2008), SPIE Proc. 7056, 705603–1–705603–10 (2008)CrossRef
40.61.
go back to reference L. Braescu, T.F. George: Arbitrary Lagrangian-Eulerian method for coupled Navier-Stokes and convection-diffusion equations with moving boundaries, Proc. 12nd WSEAS Int. Conf. Appl. Math. – Mathʼ07 (2007) pp.33–36 L. Braescu, T.F. George: Arbitrary Lagrangian-Eulerian method for coupled Navier-Stokes and convection-diffusion equations with moving boundaries, Proc. 12nd WSEAS Int. Conf. Appl. Math. – Mathʼ07 (2007) pp.33–36
40.62.
go back to reference F. Duarte, R. Gormaz, S. Natesan: Arbitrary Lagrangian-Eulerian method for Naver-Stokes equations with moving boundaries, Comput. Methods Appl. Mech. Eng. 193, 4819–4836 (2004)ADSMathSciNetCrossRefMATH F. Duarte, R. Gormaz, S. Natesan: Arbitrary Lagrangian-Eulerian method for Naver-Stokes equations with moving boundaries, Comput. Methods Appl. Mech. Eng. 193, 4819–4836 (2004)ADSMathSciNetCrossRefMATH
40.63.
go back to reference M. Fernandez, M. Moubachir: Sensitivity analysis for an incompressible aeroelastic system, Math. Models Methods Appl. Sci. 12, 1109–1130 (2002)MathSciNetCrossRefMATH M. Fernandez, M. Moubachir: Sensitivity analysis for an incompressible aeroelastic system, Math. Models Methods Appl. Sci. 12, 1109–1130 (2002)MathSciNetCrossRefMATH
Metadata
Title
Mass and Heat Transport in BS and EFG Systems
Authors
Thomas F. George
Stefan Balint
Liliana Braescu
Copyright Year
2010
DOI
https://doi.org/10.1007/978-3-540-74761-1_40

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