Mathematical Physics in Diffusion Problems

Takahisa Okino

Department of Applied Mathematics, Faculty of Engineering, Oita University, Oita City, Japan.

DOI: 10.4236/jmp.2015.614217                

Abstract

Using the divergence theorem and the coordinate transformation theory for the general Fickian second law, fundamental diffusion problems are investigated. As a result, the new findings are obtained as follows. The unified diffusion theory is reasonably established, including a self-diffusion theory and an N (N ≥ 2) elements system interdiffusion one. The Fickian first law is incomplete without a constant diffusion flux corresponding to the Brown motion in the localized space. The cause of Kirkendall effect and the nonexistence of intrinsic diffusion concept are theoretically revealed. In the parabolic space, an elegant analytical method of the diffusion equation is mathematically established, including a nonlinear diffusion equation. From the Schr?dinger equation and the diffusion equation, the universal expression of diffusivity proportional to the Planck constant is reasonably obtained. The material wave equation proposed by de Broglie is also derived in relation to the Brown motion. The fundamental diffusion theories discussed here will be highly useful as a standard theory for the basic study of actual interdiffusion problems such as an alloy, a compound semiconductor, a multilayer thin film, and a microstructure material.

Keywords

Brown Particle, Boltzmann Factor, Markov Process, Parabolic Law, Error Function

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	Markov, A.A. (1960) American Mathematical Society Translations Series, 15, 1-14.
[2]	Brown, R. (1828) Philosophical Magazine N. S., 4, 161-173.
[3]	Einstein, A. (1905) Annals of Physics, 18, 549-560. http://dx.doi.org/10.1002/andp.19053220806
[4]	Skvortsov, A.A., Orlov, A.M. and Muradov, V.E. (2009) Technical Physics Letters, 35, 606-609. http://dx.doi.org/10.1134/S1063785009070062
[5]	Ting, P.Y. (1994) IEEE Transactions on Neural Networks, 5, 622-638. http://dx.doi.org/10.1109/72.298232
[6]	You, Y.L., Xu, W.Y., Tannenbaum, A. and Kaveh, M. (1996) IEEE Transaction on Image Processing, 11, 1539-1553.
[7]	Das, I., Agrawal, N.R., Gupta Sanjeev, K., Gupta Sujeet, K. and Rastogi, R.P. (2009) Journal of Physical Chemistry A, 113, 5296-5301. http://dx.doi.org/10.1021/jp8064147
[8]	Raji, A., Kuceyeski, A. and Weiner, M. (2012) Neuron, 73, 1204-1215. http://dx.doi.org/10.1016/j.neuron.2011.12.040
[9]	Zhou, Q. (2015) International Journal of Multimedia and Ubiquitous Engineering, 10, 173-182. http://dx.doi.org/10.14257/ijmue.2015.10.1.16
[10]	Gauss, C.F. (1840) Res Beob Magn Vereins, 4, 1.
[11]	Fourier, J.B.J. (1822) Theorie analytique de la chaleur. Didot, Paris, 499-508.
[12]	Mehrer, H. (2007) Diffusion in Solid. Springer-Verlag, Berlin Heidelberg. http://dx.doi.org/10.1007/978-3-540-71488-0
[13]	Fick, A. (1855) Philosophical Magn J Sci, 10, 31-39.
[14]	Okino, T. (2012) Journal of Modern Physics, 3, 1388-1393. http://dx.doi.org/10.4236/jmp.2012.310175
[15]	Okino, T. (2013) Journal of Modern Physics, 4, 1495-1498. http://dx.doi.org/10.4236/jmp.2013.411180
[16]	Okino, T. (2014) Applied Physics Research, 6, 1-7. http://dx.doi.org/10.5539/apr.v6n2p1
[17]	Smigelskas, A.D. and Kirkendall, E.O. (1947) Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers, 171, 130-142.
[18]	Schrödinger, E. (1926) Annals of Physics, 79, 361-376. http://dx.doi.org/10.1002/andp.19263840404
[19]	Okino, T. (2011) Materials Transactions, 52, 2220-2227. http://dx.doi.org/10.2320/matertrans.M2011137
[20]	Okino, T. (2012) Journal of Modern Physics, 3, 255-259. http://dx.doi.org/10.4236/jmp.2012.33034
[21]	Boltzmann, L. (1894) Annual Review of Physical Chemistry, 53, 959-964. http://dx.doi.org/10.1002/andp.18942891315
[22]	Okino, T. (2013) Journal of Modern Physics, 4, 612-615. http://dx.doi.org/10.4236/jmp.2013.45088
[23]	de Broglie, L. (1923) Nature, 112, 540. http://dx.doi.org/10.1038/112540a0
[24]	Dirac, P.A.M. (1958) Principles of Quantum Mechanics. Clarendon Press, Oxford.
[25]	Manning, J.R. (1972) Theory of Diffusion. In Diffusion American Society for Metals, 1-23.
[26]	Matano, C. (1933) Japanese Journal of Physics, 8, 109-113.
[27]	Langevin, P. (1908) Comptes Rendus de l’Académie des Sciences (Paris), 146, 530-533.
[28]	Cornet, J.F. and Calais, D (1972) Journal of Physics and Chemistry of Solids, 33, 1675-1684. http://dx.doi.org/10.1016/S0022-3697(72)80463-3
[29]	Cornet, J.F. (1974) Journal of Physics and Chemistry of Solids, 35, 1247-1252. http://dx.doi.org/10.1016/S0022-3697(74)80148-4
[30]	Iijima, Y., Hirano, K. and Kikuchi, M. (1982) Transactions of the Japan Institutetute of Metals, 23, 19-23. http://dx.doi.org/10.2320/matertrans1960.23.19
[31]	Alizadeh, R., Mahmudi, R., Ngan, A.H.W. and Langdon, T.G. (2015) Journal of Materials Science, 50, 4940-4951. http://dx.doi.org/10.1007/s10853-015-9041-x
[32]	Saad, A., Fedotov, A., Velichko, O., Pachynin, V. and Davydko, A. (2006) Physica Status Solidi (b), 12, 2665-2671. http://dx.doi.org/10.1002/pssb.200541272
[33]	Wang, C.W., Yiu, P., Chu, J.P., Shek, C.H. and Hsueh, C.H. (2015) Journal of Materials Science, 50, 2085-2092. http://dx.doi.org/10.1007/s10853-014-8770-6
[34]	Gong, M.M., Castro, R.H.R. and Liu, F. (2015) Journal of Materials Science, 50, 4610-4621. http://dx.doi.org/10.1007/s10853-015-9010-4
[35]	Darken, L.S. (1948) Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers, 175, 184-201.
[36]	Xin, L., Schnell, S.K., Simon, J.M., Krgüer, P., Bedeaux, D., Kjelstrip, S., Bardow, A. and Vlugt, T.J.H. (2013) International Journal of Thermophysics, 34, 1169-1196. http://dx.doi.org/10.1007/s10765-013-1482-3
[37]	Meyer, A. and Kargl, F. (2013) Int J Microgravity Sci Appl, 30, 30-35.
[38]	Perrin, J. (1909) Annales de Chimie et de Physique, 18, 5-114.
[39]	Shewmon, P.G. (1963) Diffusion in Solids. Series in Materials Science and Engineering. McGraw-Hill, New York.
[40]	Onishi, M., Ikeda, T., Wakamatu, Y. and Shimozaki, T. (1988) Transactions of the Japan Institutetute of Metals, 29, 383-387. http://dx.doi.org/10.2320/matertrans1960.29.383
[41]	da Silva, L.C.C. and Mehl, R.F. (1951) Transactions of the American Institute of Mining, Metallurgical and Petroleum Engineers, 191, 155-157.
[42]	Hoshino, H., Iijima, Y. and Hirano, K. (1982) Transactions of the Japan Institutetute of Metals, 21, 674-682. http://dx.doi.org/10.2320/matertrans1960.21.674
[43]	Onishi, M. and Miura, H. (1977) Transactions of the Japan Institutetute of Metals, 18, 107-112. http://dx.doi.org/10.2320/matertrans1960.18.107
[44]	Boltzmann, L. (1872) Wiener Berichte, 66, 275-370.
[45]	Ohya, M. and Watanabe, N. (2010) Entropy, 12, 1194-1245. http://dx.doi.org/10.3390/e12051194
[46]	John, L. and Haller, J. (2013) Journal of Modern Physics, 4, 1393-1399. http://dx.doi.org/10.4236/jmp.2013.410167