Theoretical Evidence for Revision of Fickian First Law and New Understanding of Diffusion Problems

Takahisa Okino

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

DOI: 10.4236/jmp.2012.310175                

Abstract

Based on the divergence theorem, we reveal that the Fickian first law relevant to the diffusion flux |J(t,x,y,z) > in the time and space is incomplete without an integral constant |J₀(t) > for the integral of Fickian second law. The new diffusion flux (NDF) taking it into account shows that we can systematically understand the problems of one-way diffusion, impurity diffusion and self-diffusion as a special case of the interdiffusion. Applying the NDF to the interdiffusion problem between metal plates, it is clarified that the Kirkenkall effect is caused by |J₀(t) > and also that the interdiffusion coefficients in alloy can be easily obtained. The interdiffusion problems are reasonably solved regardless of the intrinsic diffusion conception. Thus the NDF to replace the Fickian first law is an essential equation in physics.

Keywords

Diffusion Equation; Fickian First Law; Kirkendall Effect

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. B. J. Fourier, “Theorie Analytique de la Chaleur,” Chez Firmin Didot, Paris, 1822.
[2]	A. Fick, “On Liquid Diffusion,” Philosophical Magazine Journal of Science, Vol. 10, 1855, pp. 31-39.
[3]	L. Boltzmann, “Weitere Studien uber das Warmegleichgewicht unter Gasmolek?ulen,” Wiener Berichte, Vol. 66, 1872, pp. 275-370.
[4]	A. Einstein, “Die von der Molekularkinetischen Theorie der Warme Geforderte Bewegung von in Ruhenden Flussiigkeiten Suspendierten Teilchen,” Annalen der Physik, Vol. 18, No. 8, 1905, pp. 549-560. doi:10.1002/andp.19053220806
[5]	R. Brown, “A Brief Account of Microscopical Observations Made in the Months of June, July and August, 1827, on the Particles Contained in the Pollen of Plants; and on the General Existence of Active Molecules in Organic and Inorganic Bodies,” Philosophical Magazine, Vol. 4, 1828, pp. 161-173.
[6]	J. Perrin, “Mouvement Brownien et Realite Moleculare,” Annales de chimie et de Physique, Vol. 18, No. 8, 1909, pp. 5-114.
[7]	L. Boltzmann, “Zur Integration der Diffusionsgleichung bei Variabeln Diffusionscoefficienten,” Annual Review Physical Chemistry, Vol. 53, No. 2, 1894, pp. 959-964.
[8]	C. Matano, “On the Relation between Diffusion-Coefficients and Concentrations of Solid Metals,” Japanese Journal of Physics, Vol. 8, 1933, pp. 109-113.
[9]	A. D. Smigelskas and E. O. Kirkendall, “Zinc Diffusion in Alpha Brass,” Transactions of AIME, Vol. 171, 1947, pp. 130-142.
[10]	L. S. Darken, “Diffusion, Mobility and Their Interrelation through Free Energy in Binary Metallic System,” Transactions of AIME, Vol. 175, 1948, pp. 184-201.
[11]	Y. Iijima, K. Funayama, T. Kosugi and K. Fukumichi, “Shift of Multiple Markers and Intrinsic Diffusion in Gold Iron Alloys,” Philosophical Magazine Letters, Vol. 74, No. 6, 1996, pp. 423-428. doi:10.1080/095008396179959
[12]	T. Okino, “New Mathematical Solution for Analyzing Interdiffusion Problems,” Materials Transactions, Vol. 52, No. 12, 2011, pp. 2220-2227. doi:10.2320/matertrans.M2011137
[13]	T. Okino, “Brownian Motion in Parabolic Space,” Journal of Modern Physics, Vol. 3, No. 3, 2012, pp. 255-259. doi:10.4236/jmp.2012.33034