Abstract
The path probability method (PPM) of irreversible statistical mechanics has been successfully applied to various diffusion problems. The major advantage of this atomistic treatment over the phenomenological approach of irreversible thermodynamics is that all Onsager matrix coefficients can be derived analytically so that relations among measurable quantities can be clearly understood in terms of microscopic parameters. This review article attempts to present the PPM in the simplest possible form. The importance of the PPM as an atomistic technique is illustrated using a simple example. The applicability and limitations of the technique are also emphasized.
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References
N. L. Peterson, in “Solid State Physics”, Vol. 22, edited by F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1968) p. 409.
A. D. Le Claire, in “Physical Chemistry (Solid State)”, Vol. 10, edited by H. Eyring, D. Henderson and W. Jost (Academic Press, New York, 1970) p. 261.
J. R. Manning, “Diffusion Kinetics for Atoms in Crystals” (Van Nostrand, New York, 1968).
A. B. Lidiard,Phil. Mag. 46 (1955) 1218.
N. G. vanKampen, in “Fluctuation Phenomena in Solids”, edited by R. E. Burgess (Academic Press, New York, 1965) p. 139.
G. E. Murch andS. J. Rothman, in “Nontraditional Methods in Diffusion”, edited by G. E. Murch, H. K. Birnbaum and J. R. Cost (TMS, Warrendale, PA, 1984) p. 281.
R. Kikuchi,Prog. Theor. Phys. (Kyoto) Suppl. 35 (1966) 1.
H. Sato, in “Nontraditional Methods in Diffusion”, edited by G. E. Murch, H. K. Birnbaum and J. R. Cost (TMS, Warrendale, PA, 1984) p. 203.
R. Kikuchi,Phys. Rev. 81 (1951) 988.
T. Ishikawa, K. Wada, H. Sato andR. Kikuchi,ibid. A33 (1986) 4164.
R. Kikuchi, private communication (1989).
H. Sato andR. Kikuchi,Phys. Rev. B28 (1983) 648.
H. Sato, S. A. Akbar andG. E. Murch, in “Diffusion in Solids: Recent Developments”, edited by M. A. Dayananda and G. E. Murch (TMS, 1985) p. 67.
R. Kikuchi, T. Ishikawa andH. Sato,Physica 123A (1984) 227.
T. Ishikawa, S. A. Akbar, W. Zhu andH. Sato,J. Amer. Ceram. Soc. 71 (1988) 513.
S. A. Akbar, M. Kaburagi, H. Sato andR. Kikuchi,ibid. 70 (1987) 246.
S. A. Akbar andH. Sato, in “Oxidation of Metals and Related Mass Transport”, edited by M. A. Dayananda, S. J. Rothman and W. E. King (TMS, Warrendale, PA, 1987) p. 49.
S. A. Akbar, M. Kaburagi andH. Sato,J. Phys. Chem. Solids 48 (1987) 579.
K. Wada, A. Suzuki, H. Sato andR. Kikuchi,ibid. 46 (1985) 1195.
T. Ishikawa, H. Sato, R. Kikuchi, S. A. Akbar, W. Zhu andA. S. Datta, in preparation.
H. Sato andR. Kikuchi,J. Chem. Phys. 55 (1971) 677.
R. Kikuchi andH. Sato,ibid. 55 (1971) 702.
H. Sato andR. Kikuchi, in “Superionic Conductors”, edited by G. D. Mahan and W. L. Roth (Plenum Press, New York, 1976) p. 135.
R. Kikuchi andH. Sato, in “Superionic Conductors”, edited by G. D. Mahan and W. L. Roth (Plenum Press, New York, 1976) p. 167.
H. Sato andR. Kikuchi,J. Physique 38 C7 (1977) 159.
H. Sato, in “Solid Electrolytes”, edited by S. Geller (Springer-Verlag, 1977) p. 3.
H. Sato andR. Kikuchi, in “Fast Ion Transport in Solids”, edited by P. Vashista, J. N. Mundy and G. K. Shenoy (North Holland, Amsterdam, 1979) p. 337.
H. Sato andK. Gschwend,Phys. Rev. B22 (1980) 4626.
H. Sato,Solid State Ionics 5 (1981) 183.
A. Suzuki, H. Sato andR. Kikuchi,Phys. Rev. B29 (1984) 3550.
T. Ishikawa andH. Sato, in “Superionic Solids and Solid Electrolytes: Recent Trends”, edited by A. L. Laskar and S. Chandra (Academic Press, New York, 1989) p. 439.
R. Kikuchi andH. Sato,J. Chem. Phys. 51 (1969) 161.
Idem., ibid. 53 (1970) 2702.
Idem., ibid. 57 (1972) 4962.
H. Sato andR. Kikuchi, in “Mass Transport Phenomena in Ceramics, Materials Science Research”, edited by A. R. Cooper and A. H. Heuer (Plenum Press, New York, 1975) p. 149.
H. Sato, T. Ishikawa andR. Kikuchi,J. Phys. Chem. Solids 46 (1985) 1361.
T. Ishikawa, H. Sato, R. Kikuchi andS. A. Akbar,J. Amer. Ceram. Soc. 68 (1985) 1.
S. A. Akbar andH. Sato,J. Phys. Chem. Solids 50 (1989) 729.
H. Sato, A. Suzuki andR. Kikuchi,Solid State Ionics 9,10 (1983) 725.
H. Sato, K. Wada, A. Suzuki andS. A. Akbar,ibid. 18,19 (1986) 178.
H. Sato, S. A. Akbar andT. Ishii,ibid. 28–30 (1988) 138.
J-S. Choi, M. Sarikaya, I. A. Aksay andR. Kikuchi,Phys. Rev. B42 (1989) 4244.
H. Sato, K. Gschwend andR. Kikuchi,J. Physique 38 C7 (1977) 357.
K. Gschwend, H. Sato andR. Kikuchi,J. Chem. Phys. 69 (1978) 5006.
K. Gschwend, H. Sato, R. Kikuchi, H. Iwasaki andH. Maniwa,ibid. 71 (1979) 2844.
J. K. McCoy, R. Kikuchi, K. Gschwend andH. Sato,Phys. Rev. B25 (1982) 1734.
S. Fukuda, PhD thesis, University of Washington, Seattle, WA (1988).
K. Wada, M. Kaburagi, T. Uchida andR. Kikuchi,J. Stat. Phys. 53 (1988) 1081.
K. Wada, H. Tsuchinaga andT. Uchida, in “Dynamics of Ordering Processes in Condensed Matter”, edited by S. Komura and H. Furukawa (Plenum, New York, 1988) p. 29.
T. Uchida, F. Sato andK. Wada, “Kinetics of Crystal Growth on the SOS Model”, preprint (1990).
H. Schmalzried, W. Laqua andP. L. Lin,Z. Naturforsch 34a (1978) 192.
H. Schmalzried andW. Laqua,Oxid. Metals 15 (1981) 339.
H. K. Bowen, in “Materials Science in Energy Technology”, edited by G. G. Libowitz and S. M. Whitingham (Academic Press, New York, 1979).
J. Mizusaki, W. R. Cannon andH. K. Bowen,J. Amer. Ceram. Soc. 63 (1980) 391.
M. N. Menon, S. M. Copley andB. A. Troesch,Acta Metall. 23 (1975) 199.
W. T. Petuskey andH. K. Bowen,J. Amer. Ceram. Soc. 64 (1981) 611.
G. E. Murch, in “Diffusion in Crystalline Solids”, edited by G. E. Murch and A. S. Nowick (Academic Press, 1984).
G. E. Murch andJ. C. Dyre,CRC Crit. Rev. Solid State Mater. Sci. 15 (1989) 345.
S. A. Akbar,J. Mater. Sci. (1992) in press.
T. Ishii, H. Sato andR. Kikuchi,Phys. Rev. B34 (1986) 8335.
J. O. Isard,J. Non-Cryst. Solids 1 (1969) 235.
D. E. Day,ibid. 21 (1976) 343.
J. A. Bruce, R. A. Howie andM. D. Ingram,Solid State Ionics 18,19 (1986) 1129.
L. M. Foster, M. P. Anderson, G. V. Chandrasekhar, G. Burns andR. B. Bradford,ibid. 5 (1981) 215.
M. P. Anderson andL. M. Foster,ibid. 5 (1981) 219.
J. L. Briant andG. C. Farrington,ibid. 5 (1981) 207.
J. A. Bruce andM. D. Ingram,ibid. 9,10 (1983) 717.
M. A. Dayananda andC. W. Kim,Metall. Trans. 10A (1979) 1333.
C. W. Kim andM. A. Dayananda,ibid. 14A (1983) 857.
M. A. Dayananda, in “Diffusion in Solids: Recent Developments”, edited by M. A. Dayananda and G. E. Murch (TMS, Warrendale, PA, 1985) p. 195.
H. Sato,JIM 32 (1991) 509.
Y. Okamura andA. R. Allnatt,Phil. Mag. A 54 (1986) 773.
L. K. Moleko, Y. Okamura andA. R. Allnatt,J. Chem. Phys. 88 (1988) 2706.
L. K. Moleko, A. R. Allnatt andE. L. Allnatt,Phil. Mag. A 59 (1989) 141.
A. D. Le Claire,J. Nucl. Mater. 69/70 (1978) 70.
G. E. Murch, “Atomic Diffusion Theory in Highly Defective Solids, Diffusion and Defects”, Monograph Series, No. 6, edited by Y. Adda, A. D. Le Claire, L. M. Slifkin and F. H. Wohlbier (Trans Tech., Aedermansdorf, 1980).
K. Wada, T. Ishikawa, H. Sato and R. Kikuchi,Phys. Rev. A33 (1986) 4171.
L. Zhang, M. Gurr andG. E. Murch, in “Ceramic Developments, Materials Science Forum”, Vols34–36 edited by C. C. Sorrell and B. Ben-Nissan (Trans Tech., Aedermansdorf, 1988) p. 807.
G. E. Murch andL. Zhang,J. Aust. Ceram. Soc. 25 (1989) 9.
L. Zhang andG. E. Murch,Phil. Mag. A62 (1990) 267.
G. E. Murch andL. Zhang, in “Diffusion in Materials”, Vol. 179, edited by A. L. Laskar, L. J. Bocquet, G. Brebeck and C. Monty, NATO ASI Series (Kluwer Academic, Boston, 1990) p. 251.
L. Zhang andG. E. Murch,Solid State Ionics (1991) in press.
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Akbar, S.A. The path probability method: An atomistic technique of diffusion. J Mater Sci 27, 3125–3132 (1992). https://doi.org/10.1007/BF01116002
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DOI: https://doi.org/10.1007/BF01116002