Share:


Difference methods to one and multidimensional interdiffusion models with Vegard rule

    Bogusław Bożek   Affiliation
    ; Lucjan Sapa Affiliation
    ; Marek Danielewski Affiliation

Abstract

In this work we consider the one and multidimensional diffusional transport in an s-component solid solution. The new model is expressed by the nonlinear parabolic-elliptic system of strongly coupled differential equations with the initial and the nonlinear coupled boundary conditions. It is obtained from the local mass conservation law for fluxes which are a sum of the diffusional and Darken drift terms, together with the Vegard rule. The considered boundary conditions allow the physical system to be not only closed but also open. We construct the implicit finite difference methods (FDM) generated by some linearization idea, in the one and two-dimensional cases. The theorems on existence and uniqueness of solutions of the implicit difference schemes, and the theorems concerned convergence and stability are proved. We present the approximate concentrations, drift and its potential for a ternary mixture of nickel, copper and iron. Such difference methods can be also generalized on the three-dimensional case. The agreement between the theoretical results, numerical simulations and experimental data is shown.

Keyword : interdiffusion, Darken method, Vegard rule, parabolic-elliptic nonlinear differential system, implicit finite difference method, existence and uniqueness of solutions to difference scheme, convergence, stability

How to Cite
Bożek, B., Sapa, L., & Danielewski, M. (2019). Difference methods to one and multidimensional interdiffusion models with Vegard rule. Mathematical Modelling and Analysis, 24(2), 276-296. https://doi.org/10.3846/mma.2019.018
Published in Issue
Mar 18, 2019
Abstract Views
73
PDF Downloads
78
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

P. Biler. Existence and asymptotics of solutions for a parabolic-elliptic system with nonlinear no-flux boundary conditions. Nonlinear Anal. Theor., 19(12):1121–1136, 1992. https://doi.org/10.1016/0362-546X(92)90186-I

P. Biler, W. Hebisch and T. Nadzieja. The debye system: existence and large time behavior of solutions. Nonlinear Anal. Theor., 23(9):1189–1209, 1994. https://doi.org/10.1016/0362-546X(94)90101-5

L. Boltzmann. About the integration of the diffusion equation with variable coefficients. Annalen der Physik und Chemie, 53:959–964, 1894. https://doi.org/10.1002/andp.18942891315

B. Bożek, M. Danielewski, K. Tkacz-Śmiech and M. Zajusz. Interdiffusion: compatibility of darken and onsager formalisms. Materials Science and Technology, 31(13):1633–1641, 2015. https://doi.org/10.1179/1743284715Y.0000000077

H. Brenner. Fluid mechanics revisited. Physica A, 370(2):190–224, 2006. https://doi.org/10.1016/j.physa.2006.03.066

H. Brenner. Diffuse volume transport in fluids. Phys. A, 389(19):4026–4045, 2010. https://doi.org/10.1016/j.physa.2010.06.010

M. Danielewski, R. Filipek, K. Holly and B. Bożek. Interdiffusion in multicomponent solid solutions. the mathematical model for thin films. Phys. Stat. Sol.(a), 145(2):339–350, 1994. https://doi.org/10.1002/pssa.2211450214

M. Danielewski, K. Holly and W. Krzyżański. Interdiffusion in r-component (r≥2) one dimensional mixture showing constant concentration. Computer Methods in Materials Science, 8:31–46, 2008. https://doi.org/10.1103/PhysRevB.50.13336

M. Danielewski and H. Leszczyński. Computation of trajectories and displacement fields ina three-dimensional ternary diffusion couple: parabolic transform method. Math. Probl. Eng., 2015:1–11, 2015. https://doi.org/10.1155/2015/650452

L.S. Darken. Diffusion, mobility and their interrelation through free energy in binary metallic systems. Trans. AIME, 175:184–201, 1948.

M.A. Dayananda. Determination of eigenvalues, eigenvectors, and interdiffusion coefficients in ternary diffusion from diffusional constraints at the matano plane. Acta Mater., 129:474–481, 2017. https://doi.org/10.1016/j.actamat.2017.03.012

A.R. Denton and N.W. Ashcroft. Vegard’s law. Phys. Rev. A, 43:3161–3164, 1991. https://doi.org/10.1103/PhysRevA.43.3161

C.R. Doering and J.D. Gibbon. Applied Analysis of the Navier–Stokes Equations. Cambridge University Press, New York, 2004.

R. Filipek, P. Kalita, L. Sapa and K. Szyszkiewicz. On local weak solutions to nernst-planck-poisson system. Appl. Anal., 96(13):2316–2332, 2017. https://doi.org/10.1080/00036811.2016.1221941

H. Gajewski. On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors. Z. Angew. Math. Mech., 65(2):101–108, 1985. https://doi.org/10.1002/zamm.19850650210

K. Holly and M. Danielewski. Interdiffusion and free-boundary problem for r-component (r ≥ 2) one-dimensional mixtures showing constant concentration. Phys. Rev. B, 50:13336–13346, 1994. https://doi.org/10.1103/PhysRevB.50.13336

M. Malec and L. Sapa. A finite difference method for nonlinear parabolicelliptic systems of second order partial differential equations. Opuscula Math., 27(2):259–289, 2007.

C. Matano. On the relation between the diffusion-coefficients and concentrations of solid metals. Jpn. J. Phys., 8:109–113, 1933.

L. Sapa. Implicit difference methods for differential functional parabolic equations with dirichlet’s condition. Z. Anal. Anwend., 32(3):313–337, 2013. https://doi.org/10.4171/ZAA/1487

L. Sapa. Difference methods for parabolic equations with robin condition. Appl. Math. Comput., 321:794–811, 2018. https://doi.org/10.1016/j.amc.2017.10.061

L. Sapa, B. Bożek and M. Danielewski. Existence, uniqueness and properties of global weak solutions to interdiffusion with vegard rule. Topol. Methods Nonlinear Anal., 52(2):423–448, 2018. https://doi.org/10.12775/TMNA.2018.008

L. Sapa, B. Bożek and M. Danielewski. Weak solutions to interdiffusion models with vegard rule. In American Institute of Physics(Ed.), 6th International Eurasian Conference on Mathematical Sciences and Applications (IECMSA -2017), volume 1926 of Book Series: AIP Conference Proceedings, pp. 020039–1 – 020039–9, 2018. https://doi.org/10.1063/1.5020488

A.D. Smigelskas and E.O. Kirkendall. Zinc diffusion in alpha brass. Trans. AIME., 171:130–142, 1947.

B. Wierzba and M. Danielewski. The lattice shift generated by two dimensional diffusion process. Comp. Mater. Sci., 95:192–197, 2014. https://doi.org/10.1016/j.commatsci.2014.07.015

B. Wierzba and W. Skibiński. The intrinsic diffusivities in multi component systems. Phys. A, 440:100–109, 2015. https://doi.org/10.1016/j.physa.2015.08.009

A.D. Wilkinson. Mass Transport in Solids and Fluids. Cambridge University Press, Cambridge, 2000. https://doi.org/10.1017/CBO9781139171267

M. Zajusz, J. Dąbrowa and M. Danielewski. Determination of the intrinsic diffusivities from the diffusion couple experiment in multicomponent systems. Scripta Mater., 138:48–51, 2017. https://doi.org/10.1016/j.scriptamat.2017.05.031