Mathematical modelling of laser materials processing

International Journal of Materials and Product Technology

Volumn 11, Issue 3-4, 1996, Pages 193-252

  Mazumder, J ^a   Mohanty, P S ^a   Kar, A ^a,b  

^a UNIVERSITY OF ILLINOIS AT URBANA CHAMPAIGN   (United States)

^b UNIVERSITY OF CENTRAL FLORIDA   (United States)

Author keywords

Ablation; Alloying; Chemical vapour deposition; Cladding; Cutting; Heat treatment; Keyhole welding; Laser drilling; Modelling; Non equilibrium synthesis; Transport phenomena

Indexed keywords

ALLOYING; BOUNDARY CONDITIONS; CHEMICAL VAPOR DEPOSITION; CLADDING (COATING); DRILLING; HEAT TREATMENT; LASER ABLATION; LASER BEAM CUTTING; LASER BEAM WELDING; MATHEMATICAL MODELS;

KEYHOLE WELDING; NONEQUILIBRIUM SYNTHESIS;

LASER APPLICATIONS;

EID: 0030409692     PISSN: 02681900     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (135)

1. Mazumder, J. (1991) 'Overview of melt dynamics in laser processing', Optical Engineering, Vol. 30, No. 8, pp. 1208-1219.
2. Jaeger, J.C. (1942) 'Moving sources of heat and temperature at sliding contacts', Proceedings of Royal Society of N.S.W., Vol. 76, pp. 203-224.
3. Patankar, S.V. (1980) Numerical Heat Transfer and Fluid Flow, Hemisphere Publishing Corporation, New York.
4. Rosenthal, D. (1946) 'The theory of moving sources of heat and its application to metal treatments', Trans. ASME., Vol. 68, pp. 849-866.
5. Carslaw, H.S. and Jaeger, J.C. (1959) 'Conduction of Heat in Solids', (2nd Edn.)., Oxford University Press, Oxford.
6. Cline, H.E. and Anthony, T.R. (1977) 'Heat treating and melting material with a scanning laser or electron beam', J. Appl. Physics, Vol. 48, pp. 3895-3900.
7. Lax, M. (1977), J. Appl. Physics, Vol. 48, p. 3919.
8. Lax, M. (1978), J. Appl. Physics Letters, Vol. 33, p. 786.
9. Hess, L.D., Forber, R.A., Kokorowski, S.A. and Olson, G.L. (1980), in Proceeding of the Society of Photo-Optical Instrumentation Engineering, Ready, J.F. (ed.) (Washington, DC) Vol. 198, pp.31-34.
10. Chen, I. and Lee, S. (1983) 'Transient temperature profiles in solids heated with scanning laser', J. Appl. Physics, Vol. 54, pp. 1062-1066.
11. Sanders, D.J. (1984) 'Temperature distributions produced by scanning Gaussian laser beam', Appl. Optics, Vol. 23, pp. 30-35.
12. Modest, M.F. and Abakians, H. (1986) 'Heat conduction in a moving seni-infinite solid subjected to pulsed laser irradiation ', J. Heat Transfer, Vol.108, pp. 597-601.
13. Nissim, Y.I., Lietoila, A., Gold, R.B and Gibbons, J.F. (1980) 'Temperature distributions produced in semiconductors by a scanning elliptical or circular CW laser beam', J. Appl. Physics, Vol. 51, pp. 274-279.
14. Moody, J.E. and Hendel, R.H. (1982) 'Temperature profiles induced by a scanning CW laser beam', J. Appl. Physics, Vol. 53, pp. 4364-4371.
15. Pittaway, L.G. (1964) 'The temperature distribution in thin foil and semi-infinite targets bombarded by an electron beam', British J. Appl. Physics, Vol. 15, pp. 967-082.
16. Lovov, N. (1987) 'Temperature field with distributed moving heat source', International Institute of Welding, Study Group 212, Doc. 212-682-87.
17. Tsai, C.L. and Hou, C.A. (1988) 'Theoretical analysis of weld pool behavior in the pulsed current GTAW process', J. Heat Transfer, Vol. 110, pp. 160-165.
18. Kar, A. and Mazumder, J. (1989) 'Three-dimensional transient thermal analysis for laser chemical vapor deposition on uniformly moving finite slabs', J. Appl. Physics, Vol. 65, No. 8, pp. 2923-2934.
19. Manca, O., Morrone, B. and Naso, V. (1995) 'Quasi-steady-state three-dimensional temperature distribution induced by a moving circular Gaussian heat source in a finite depth solid', Int. J. Heat mass Transfer, Vol. 38, No. 7, pp. 1305-1315.
20. Beck, J.V. (1984) 'Green's function solution for transient heat conduction problems', Int. J. Heat Mass Transfer, Vol. 27, pp. 1235-1244.
21. Beck, J.V., Cole, K.D., Haji-Sheikh, A. and Litkouhi, B. (1992) Heat Conduction Using Green's Functions, Hemisphere, Washington, DC.
22. Bokota, A. and Iskierka, S. (1996) 'Numerical prediction of the hardened zone in laser treatment of carbon steel', Acta. Mater., Vol. 44, No. 2, pp. 445-450.
23. Maier, Ch., Schaaf, P. and Gonser, U. (1992), Mater. Sci. Eng., Vol. A150.
24. Koinstinen, D.P. and Marburger, R.E. (1959), Acta. Metall. Vol. 7.
25. El-Adawi, M.K. (1986) 'Laser melting of solids-an exact solution for time intervals less or equal to the transit time', J. Appl. Physics, Vol. 60, No. 7, pp. 2256-2259.
26. El-Adawi, M.K., Abdel-Naby, M.A. and Shalaby, S.A. (1995) 'Laser heating of a two-layer system with constant surface absorption: an exact solution', Int. J. Heat mass Transfer, Vol. 38, No. 5, pp. 947-952.
27. Elperin, T. and Rudin, G. (1995) 'Temperature field in a multilayer assembly affected by a local laser heating', Int. J. Heat mass Transfer, Vol. 38, No. 17, pp. 3143-3147.
28. Mazumder, J. and Steen, W.M. (1980) 'Heat transfer model for CW laser material processing', J. Appl. Phys., Vol. 51, No. 2, pp. 941-947.
29. Chande, T. and Mazumder, J. (1981) 'Heat flow during CW laser Materials processing', Proc., Lasers in Metallurgy, K. Mukherjee and J. Mazumder, eds., published by TMS-AIME, Warrendale, PA, pp. 165-177.
30. Mazumder, J. (1978) 'Laser Welding of titanium and tin plate', London University.
31. 2 laser', Met. technol., Vol. 6, pp. 456-462.
32. Steen, P.H., Ehrhard, P. and Schüssler, A. (1994) 'Depth of melt-pool and heat affected zone in laser surface treatments', Metall. Trans., Vol. 25A, pp. 427-435.
33. Gadag, S.P., Srinivasan, M.N. and Mordike, B.L. (1995) 'Effect of laser processing parameters on the structure of ductile iron', Mater. Sci. Eng., Vol. A196, pp. 145-151.
34. Gu, B., Ma, T., Brown, S.K. and Mannik, L. (1994) 'Three-dimensional numerical model for laser transformation hardening of metals', Mater. Sci. Tech., Vol. 10, No. 5, pp. 425-430.
35. Chan, C.L., Mazumder, J. and Chen, M.M. (1984) 'A Two-dimensional Transient Model for Convection in Laser Melted Pool', Metall. Trans. A, Vol. 15A, pp. 2175-2184.
36. Chande, T. and Mazumder, J. (1982) 'Mass transport in laser surface alloying: iron-nickel system', Applied Physics Letters, Vol. 41, No. 1, pp. 42-43.
37. Heiple, C.R. and Rooper, J.R. (1982) 'Mechanism for minor element effect in GTA fusion zone gometry', Welding J., Vol. 61, pp. 97s-102s.
38. Heiple, C.R., Rooper, J.R., Stagner, and Ande, R.J. (1986) 'Mechanism for minor element effect in GTA fusion zone gometry', Welding J., Vol. 61, pp. 97s-102s.
39. Copley, S.M., Beck, D., Esquivel, O., and Bass, M. (1979) 'Laser-Solid Interaction and Laser Processing-1978', Eds. Ferris, S.D., Leamy, H.J., and Poate, J.M., Conf. Proc. No. 50, Amer. Inst. of Phys., N.Y. pp.161-172.
40. Esquivel, O., Mazumder, J., Bass, M. and Copley, S.M. (1980) 'Rapid Solidification Processing, Principles and Technologies, II, Eds. Mehrabian, R., Kear, B.H. and Cohen, M., Claitor's Publishing Division , Baton Rouge, LA., pp. 180-188.
41. Copley, S.M., Beck, D.G., Esquivel, O. and Bass M. (1981) Proc. Conf. of Laser in Metallurgy, Chicago, Il., Eds. Mukherji and Mazumder, TMS-AIME, Warrendale, PA., pp. 1-20.
42. Anthony, T.R. and Cline, H.E. (1977) 'Surface rippling induced by surface tension gradients during laser surface melting and alloying', Appl. Phys., Vol. 48, pp. 3888-3894.
43. Oreper, G.M., Eager, T.W. and Szekely, J. (1983) 'Convection in arc weld pools', Welding J., Vol. 63, pp.307s-312s.
44. Oreper, G.M. and Szekely, J. (1984) 'Heat and fluid flow phenomena in weld pools', J . Fluid. Mech., Vol. 147, pp. 53-80.
45. Kuo, S., and Wang, Y.H. (1986) 'Three-dimensional convection in laser melted pools', Metall. Trans., Vol. 17A, pp. 2265-2270.
46. Chan, C.L., Zehr, R., Mazumder, J. and Chen, M.M. (1986) 'Three-dimensional Model for Convection in Laser Weld Pool," Proc. of 3rd Eng. Foundation Conference on Modeling and Control of Casting and Welding Processes, TMS-AIME, S. Kou and R. Mehrabian, eds., pp. 229-246.
47. Chan, C.L. (1986) 'Thermocapillary convection during laser surface heating', Ph. D. Thesis, University of Illinois at Urbana-Champaign.
48. Chann, C.L., Mazumder, J. and Chen, M.M. (1987) 'Three-Dimensional Axisymmetric Model for Convection in Laser Melted Pool', Materials Science and Technology, Vol. 3, No. 4, pp. 306-311.
49. Chan, C.L., Mazumder, J. and Chen, M.M. (1988) 'Asymptotic solution for thermocapillary flow at high and low prandtl numbers due to concentrated surface heating', ASME J. of Heat Transfer, Vol. 110, No. 1, pp. 140-146.
50. Chan, C.L., Mazumder, J. and Chen, M.M. (1988) 'Effect of surface tension gradient driven convection in laser melt pool: three-dimensional perturbation model', J. of Appl. Physics, Vol. 64, No. 11, pp. 6166-6174.
51. Thomson, M.E., and Szekely, J. (1989) 'The transient behavior of weld pools with deformed free surface', Int. J. Heta Mass Transfer, Vol. 32, pp. 1007-1019.
52. Ravindran, K., Srinivasan, J. and Marathe, A.G. (1994) 'Finite element study on the role of convection in laser surface melting', Numer. Heat Transfer, Vol. 26, No. 5, pp. 601-618.
53. Morvan, D., Cipriani, F.D. and Bournot, P. (1994) 'Thermocapillary convection during laser surface melting', Int. J. Heat mass Transfer, Vol. 37, No. 14, pp. 1973-1983.
54. Zebib, A., Homsy, G.M. and Meiburg. E. (1985) 'High Marangoni number convection in a square cavity.' Phys. Fluids, Vol. 28, pp. 3467-3476.
55. Sen, A.K. and Davis, S.H. (1982) ' Steady thermocapillary flows in two-dimensional slots', J. Fluid Mech. Vol. 121, pp. 163-186.
56. Chen, J.C., Sheu, J.C., and Jwu, S.S. (1990) 'Numerical computation of thermocapillary convection in a recatngular cavity', Numer. Heat Transfer, Vol. 17, pp. 287-308.
57. Hsieh, S.S. and Wen, M.Y. (1993) 'Numerical computation of thermocapillary convection in an open cylindrical cavity with different aspect ratio', Int. J. Heat mass Transfer, Vol. 36, No. 18, pp. 4351-4363.
58. Cohen, M.I. (1972) 'Material processing' in Laser Handbook, Vol. 2, Eds. Arechi, F.T. and Schulz-Dubois, pp. 1577-1647.
59. Ready, J.F. (1971) Effects of High-Power Laser Radiation, Academic Press, New York, London.
60. Chan, C.L. and Mazumder, J. (1987) 'One-dimensional steady-state modle for damage by vaporization and liquid expulsinon due to laser material interaction', J. appl. Phys., Vol. 62, pp. 4579-4586.
61. Aden, M., Beyer, E., Herziger, G. (1990) 'Laser induced vaporization of metals as a Riemann problem' J. Phys. D: Appl. Phys., Vol. 23, pp. 655-661.
62. Antaki, P.J. (1990) 'Asymptotoc analysis of laser induced sublimation for small Stefan number', ASME Heat Transfer DIV Publ. HTD V135, Heat Transfer in Space Systems, pp. 109-115.
63. Hassan, A.F., El-Nicklaway, M.m., El-Adawi, M.K., and Hemida, A.A. (1993) 'A general problem of pulse heating of slab, Opt. Laser. Technol. Vol.25.
64. Dabby, F.W. and Paek, U.C. (1972) 'High-intensity laser-induced vaporization and explosion of solid material', IEEE J. Quantum Electron. Vol. QE-8, pp. 106-111.
65. Wagner, R.E. (1974) 'Laser drilling mechanics', J. Appl. Phys. Vol. 45, pp. 4631-4637.
66. Schuöcker, D. and Abel, W. (1983) 'Material removal mechanism of laser cutting', Proceedings of the SPIE, pp. 88-95.
67. Zweig, A.D. (1991), J. Appl. phys. Vol. 70, p. 1684.
68. Chryssolouris, G. (1991) Laser Machining: Theory and Practice, Springer New York.
69. Armon, E., Zvirin, Y., Laufer, G., and Solan, A. (1989) 'Metal drilling with a CO2 laser beam II. Analysis of aluminum drilling experiments, J. Appl. Phys., Vol. 65, pp. 4995-5001.
70. Armon, E., Hill, M., Splading, I.J., and Zvirin, Y. (1989) 'Metal drilling with a CO2 laser beam II. Analysis of aluminum drilling experiments, J. Appl. Phys., Vol. 65, pp. 5003-5006.
71. Kar, A., Rockstroh, T., and Mazumder. J. (1990) 'Two-dimensional model for laser induced material damage: effects of assist gas and mulitple reflections inside the cavity', J. Appl. Phys., Vol. 71, pp. 2560-2569.
72. Kar, A., and Mazumder. J. (1992) 'Two-dimensional model for material damage due to melting and vaporization during laser irradiation', J. Appl. Phys., Vol. 68, pp. 3884-3891.
73. Qui, W., Watson, J, Thompson, D.S., and Deans, W.F. (1994) 'Modelling the interaction bewteen laser and target material in laser microspectral analysis', Opt. Laser Technol., Vol. 26, No. 3, pp. 157-166.
74. Modest, M.F. and Abakians, H. (1986) 'Heat conduction in moving seni-infinite solid subject to pulsed laser irradiation', J. Heat Transfer, Vol. 108, pp. 597-601.
75. Modest, M.F. and Abakians, H. (1986) 'Evaporative cutting of a semi-infinite body with a moving CW laser', J. Heat Transfer, Vol. 108, pp. 602-607.
76. Modest, M.F. and Abakians, H. (1988) 'Evaporative cutting of a semi-transparent body with a moving CW laser', J. Heat Transfer, Vol. 110, pp. 924-930.
77. Ramanathan, S. and Modest, M.F. (1990) 'Effect of variable properties on evaporative cutting with a moving CW laser', Heat Transfer in Space Systems, Vol. HTD-135, ASME, pp. 924-930.
78. Roy, S. and Modest, M.F. (1990) 'Three-dimensional conduction effects during evaporative scribing with a moving CW laser', J. Thermoph. Heat Transfer, Vol. 4, No. 2, pp. 199-203.
79. Bang, S.Y. and Modest, M.F. (1991) 'Multiple reflection effects on evaporative cutting with a moving CW laser', J..Heat Transfer, Vol. 113, pp. 663-669.
80. Roy, S. and Modest, M.F. (1993) 'Evaporative cutting with a moving CW laser-I. Effects of three-dimensional conduction and variable properties', Int. J. Heat Mass Transfer, Vol. 36, No. 14, pp. 3515-3528.
81. Bang, S.Y., Roy, S. and Modest, M.F. (1993) 'CW laser machining of hard ceramics-II. Effects of multiple reflection', Int. J. Heat Mass Transfer, Vol. 36, No. 14, pp. 3529-3540.
82. Modest, M.F. (1996) 'Three-dimensional, transient model for laser machining of ablating/decomposing materials', Int. J. Heat Mass Transfer, Vol. 39, No. 2, pp. 221-234.
83. Arata, Y., Miyamoto, T. (1978), Technocrat. Vol.11.5, pp.33-42
84. Klemens, P.G., (1976) 'Heat balance and flow conditions for electron beam and laser welding', J. Appl. Phys., Vol. 47, N0. 5, pp. 2165-2174.
85. Metzbower, E.A. (1993) 'Keyhole formation', Metall. Trans., Vol. 24B, pp. 875-880.
86. Andrew, j.G. and Atthey, D.R. (1976) 'Hydrodynamics limits to penetration of a material by a high power laser beam', J. Phys. D: Appl. Phys., vol. 9, pp. 2181-2194.
87. Dowden, J.M., Postacioglu, N., Davis, M.P., and Kapadia, P. (1987) 'A keyhole model in penetration welding with a laser', J. Phys. D: Appl. Phys., Vol. 20, pp.36-44.
88. Postacioglu, N, Kapadia, P., Davis, M., and Dowden, J.M. (1987) 'Upwelling in the liquid region surrounding the keyhole in penetration welding with a laser', J. Phys. D: Appl. Phys., Vol. 20, pp. 340.345.
89. Steen, W.M., Dowden, J.M., Davis, M., and Kapadia, P. (1988) 'A point and line source model of laser keyhole welding' J. Phys. D: Appl. Phys., Vol. 21, pp. 1255-1260.
90. Akhter, R., Davis, M., Dowden, J., and Kapadia, P., Ley, M. and Steen, W.M. (1989), J. Phys. D: Appl Phys., Vol. 21, p. 23.
91. Dowden, J., Kapadia, P., and Postacioglu, N. (1989), J. Phys. D: Appl. Phys., Vol. 22, p. 741.
92. Dowden, J., Chang, W.S., Kapadia, P., and Strange, C. (1991), J. Phys. D: Appl. Phys., Vol. 24, p. 519.
93. Tix, C. and Simon, G. (1993), J. Phys. D: Appl. Phys., Vol. 25, p.2066.
94. Kroos, J. Gratzke, U., and Simon, G. (1993), J. Phys. D: Appl. Phys., Vol. 25, p. 2066.
95. Kroos, J. Gratzke, U., Vicanek, M., and Simon, G. (1993) 'Dynamic behaviour of the keyhole in laser welding', J. Phys. D: Appl. Phys., Vol. 26, pp. 481-485.
96. Klein, T. Vicanek, M. , Krros, J. Decker, I., and Simon, G. (1994) 'Oscillations of the keyhole in penetration laser beam welding', J. Phys. D: Appl. Phys. Vol. 27, pp. 2023-2030.
97. Lambrakos, S.G., Metzbower, E.A., Moore, P.G., Hunn, J.H. and Monis, A. (1993), 'A numerical model for deep penetration welding process', J. Materials Eng. and Performance, Vol. 2, No. 6, pp. 819-838.
98. Duchrme, R., Wlliams, K., Kapadia, P., Dowden, J., Steen, B., and Glowack, M. (1994), J. Phys. D: Appl. Phys. Vol. 27, pp. 1619-1627.
99. Kaplan, A. (1994) 'A model of deep penetration laser welding based on calculation of the kethole profile', J. Phys. D: Appl. Phys. Vol. 27, pp. 1805-1814.
100. Kim, C., Kauh, S., Ro, S.T. and Lee, J. S. (1994) 'Parametric study of the two dimensional keyhole model for high power density welding process', J. Heat Transfer, Vol. 116, pp. 209-214.
101. Lim, J. and Chan, C.L. (1995) 'Modeling deep penetration laser welding using BEM sensitivity scheme', Eng. Analysis Boundary Elements, vol. 16, pp.93-98.
102. Knight, C.J. (1979) 'Theoretical modeling of rapid surface vaporization with back pressure', AIAA, J, Vol. 17, No. 5, pp. 519-523.
103. Kar, A., and Mazumder, J. (1995) 'Mathematical modeling of key-hole laser welding', J. Appl. Phys. , Vol. 78 (11)
104. Mohanty, P.S., Kar, A., and Mazumder, J. (1995) 'A modeling study on the influence of pulse shaping on keyhole laser welding', Proceedings of ICALEO '95, San Diego, California, November 13-16, 1995, Laser Institute of America, Orlando, Florida .
105. Ogmen, M., and Duley, W.W. (1989), J. Phys. Chem. solids, Vol. 50, p. 1221.
106. Kelly, R. and Braren, B. (1991), Appl. Phys. B, Vol. 53, p. 160.
107. Afanas'ev, Yu. V., Krol, V. M., Krokhin, O.N., and Nemchinov, I.V. (1966), J. Appl. Math. Mech., Vol. 30, p. 1218.
108. Kozlov, I.M., Ramanov, G.S., Stankevich, Yu. A., and Teterev, A.V. (1989), Heat Transfer- Sov. Res. Vol. 21, p.222.
109. Batanov, V.A., Bunkin, F.V., Prokhorov, A.M., and Fedorov, V.B. (1973), Sov. Phys. JETP, Vol. 36, p. 311.
110. Gai, H. and Voth, G.A., (1992), J. Appl. Phys., Vol.71, p. 1415.
111. Kools, J.C.S., Baller, T.S., De Zewart, S.T., and Dieleman, J. (1992), J. Appl. Phys., Vol.71, p. 4547.
112. Dahmani, F. (1992), Phys. Fluids B, Vol. 4, p. 1585.
113. Singh, R.K., and Narayan, J. (1990), Physical Review B, Vol. 41, p. 8843.
114. Kar, A. and Mazumder, J. (1994) 'Mathematical model for laser ablation to generate nanoscale and submicrometer-size particles', Physical Review E, Vol. 49, No. 1, pp. 410-419
115. Zel'dovich, Ya. B., and Raizer, Yu. P. (1966) 'Physics of Shock Waves and High Temperature Hydrodynamics Phenomena' Ed. Hayes, W.D., and Probstein, R.F. (Academic Press New York), Vol. I.
116. Lifshitz, I.M., and Slyozov, V.V. (1961) J. Phys. Chem. Solids., Vol. 19, p.35.
117. Roache, P.J. (1985) 'Computational Fluid Dynamics', Hermosa, Albuquerque, p. 242.
118. Chande, T., and Mazumder, J. (1985) 'Two-dimensional, transient model for mass transport in laser surface alloying', J. Appl. Phys., Vol. 57, No. 6, pp. 2226-2232
119. Chan, C.L., Mazumder, J., and Chen, M.M. (1984) 'A two-dimensional transient model for convection in laser melted pool', Metallurgical Transactions A, Vol. 15A, p. 2175.
120. Aziz, M.J. (1983), Appl. Phys. Lett., Vol. 43, p. 552.
121. Aziz, M.J., and Kaplan, T. (1988), Acta. Metall., Vol. 36, p. 2335.
122. Kar, A., and Mazumder, J. (1992) 'Model for nonequilibrium partitioning during rapid solidifcation of binary concentrated solutions.', Acta. Metall., Vol. 40, No. 8, pp. 1873-1881.
123. Trivedi, R. and Kurz, W. (1990), Metal. Trans., Vol. 21A, p. 1311.
124. Aziz, M.J. (1982), J. Appl. Phys., Vol. 53, p. 1158.
125. Baker, J.C. and Cahn, J.W. (1969), Acta. Metall., Vol. 17, p.575.
126. Wood, R.F., Wang, J.C., Giles, G.E. and Kirkpatrik, J.R. (1980) in Laser and Electron Beam Processing of Materials, p. 37, Academic, New York.
127. Piglmayer, K., Doppelbauer, J., and Bauerle, D. (1984) '' Mater. Res. Soc. Symp. Vol. 29, p. 47.
128. Ehrlich, D.J. and Tsao, J.Y. (1983) J. Vac. Sci. Technol. Vol. B1, p. 969.
129. Herman, I.P., Hyde, R.A., McWillams, B.M., Wiseberg, A.H. and Wood, L.L. (1983), Mater. Res. Soc. Symp. Vol. 17, p. 9.
130. Allen, S.D. (1981), J. Appl. Physics, Vol. 52, p. 650.
131. Skouby, D.C. and Jensen, K.F. (1988) , J. Appl. Physics, Vol. 63, p. 198.
132. Allen, S.D., Goldstone, J.A, and Jan, R.Y. (1986) J. Appl. Physics, Vol. 59, p. 1653.
133. Moylan, C.R., Baum, T.H., and Jones, C.R. (1986) , Appl. Physics, Vol. A 40, p.1.
134. 4', J. Appl. Physics, Vol. 66, No. 1, pp. 191-195.
135. Kar, A., Azer, M. and Mazumder, J. (1991) 'Three-dimensional transient mass-transfer model for LCVD of titanium on stationary finite slabs', J. Applied Physics, Vol. 69, No. 2, pp. 757-766.