Some coupled diffusion problems for ternary systems
DOI:
https://doi.org/10.7242/1999-6691/2025.18.4.33Keywords:
multicomponent diffusion, implicit finite-difference scheme, concentration profiles, non-ideal contactAbstract
Diffusion plays an important role in creating of new composite materials because it and accompanying phenomena participate in the formation of so-called transition layers. This paper presents possible boundary conditions for diffusion problems involving non--ideal contact between materials. A discontinuity in concentrations is also possible in the presence of ideal contact, due to differences in the mobilities of elements in different materials. Non-ideal contact leads to singularities in the boundary conditions, which may be due to various physical causes and leads to the concept of diffusion resistance at the boundary. It is shown that, within the framework of thermodynamics, the condition with an exponential dependence of the discontinuity in concentrations on temperature is substantiated. An algorithm for numerically solving the conjugate diffusive problem was developed based on component-wise separation of equations and an implicit second--order difference scheme, which is analogous to separation by physical processes. The boundary conditions were also approximated to second order using a Taylor series representation of the unknown quantities over small spatial steps (different for different regions) in the vicinity of the boundary. Using the problem of concentration redistribution between two materials as an example, the convergence of the algorithm and the consistency of the resulting concentration distributions at different points in time are illustrated. The parameters chosen for illustrations comply with thermodynamic constraints. Diffusion resistance at the interface between materials affects the concentration distribution and the rate at which equilibrium is established. Cross-diffusion coefficients do not alter the qualitative influence of imperfect contact on concentration redistribution. Such problems and the proposed algorithm can be useful in modeling the synthesis of composite materials, as well as in problems involving welding, soldering, and coating.
Downloads
References
Geguzin Y.E. Diffuzionnaya zona. Moscow: Nauka, 1977. 343 p.
Barah O.O., Natukunda F., Bori I., Ukagwu K.J. Mechanisms and modelling of diffusion in solids: a multiscale framework with industrial case studies and AI enhancements. Discover Sustainability. 2025. Vol. 6. P. 804. DOI: 10.1007/s43621-025-01746-0
Mehrer H. Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes. Springer Berlin Heidelberg, 2007. 654 p. . DOI: 10.1007/978-3-540-71488-0
Wei Y., Zhu L., Li Y., Chen Y., Guo B. Formation mechanism and microstructure evolution of Cu/Ti diffusion bonding interface and its influence on joint properties. Vacuum. 2023. Vol. 213. 112167. DOI: 10.1016/j.vacuum.2023.112167
Wei S., Li Y., Zhang R., Chen H., Liang T., Yin W. Microstructure and mechanical properties of Zr-W and Zr-Ta-W interface fabricated by hot isostatic pressing diffusion welding. Journal of Nuclear Materials. 2025. Vol. 606. 155627. DOI: 10.1016/j.jnucmat.2025.155627
Li G., Akbar A., Zhang L.-W., Rosei F., Liew K.M. Surface modification strategy for controlling wettability and ionic diffusion behaviors of calcium silicate hydrate. Applied Surface Science. 2023. Vol. 622. 156993. DOI: 10.1016/j.apsusc.2023.156993
Zhao S., Zhu J., Yang Z., Zhu Y., Sun H., Zhao L. Interface and crystallization evolution induced by reactive nitrogen and oxygen sputtering in Ni/Ti multilayer. Surface and Coatings Technology. 2023. Vol. 472. 129941. DOI: 10.1016/j.surfcoat.2023.129941
Bose S. Oxidation- and Corrosion-Resistant Coatings. High Temperature Coatings / ed. by S. Bose. 2nd ed. Elsevier, 2018. P. 97–198. DOI: 10.1016/B978-0-12-804622-7.00006-1
Cui Y., Zhao J., Zhao Y., Shao J. Diffusion of metal ions from a substrate into oxide coatings. Optical Materials Express. 2016. Vol. 6, no. 10. P. 3119–3126. DOI: 10.1364/OME.6.003119
Tret’yakov Y.D. Tverdofaznyye reaktsii. Moscow: Khimiya, 1978. 360 p.
Dybkov V.I. Reaction Diffusion and Solid State Chemical Kinetics. Trans Tech Publications, 2010. 334 p. . DOI: 10.4028/www.scientific.net/MSFo.67-68
Reiss H. Diffusion-Controlled Reactions in Solids. Journal of Applied Physics. 1959. Vol. 30. P. 1141–1152. DOI: 10.1063/1.1735284
Katona G.L., Safonova N.Y., Ganss F., Mitin D., Vladymyrskyi I.A., Sidorenko S.I., Makogon I.N., Beddies G., Albrecht M., Beke D.L. Diffusion and solid state reactions in Fe/Ag/Pt and FePt/Ag thin-film systems. Journal of Physics D: Applied Physics. 2015. Vol. 48. 175001. DOI: 10.1088/0022-3727/48/17/175001
Ding F., Wang Q., Liang C., Zhang Y. Diffusion behavior and microstructural evolution of bonding interface between AuSn20 and tungsten-copper alloy. Soldering & Surface Mount Technology. 2025. Vol. 37, no. 5. P. 333–341. DOI: 10.1108/SSMT-11-2024-0066
Wang Y., Huang Y., Liu W., Chen B., Liu J., Zhang L., Liu W., Ma Y. Investigation of diffusion reaction mechanism between W 20Ta solid solution and Ni. Materials Characterization. 2023. Vol. 200. 112894. DOI: 10.1016/j.matchar.2023.112894
Xu D., Chen P., Fu K., Sang C., Chen R., Hong T., Cheng J., Xu K. Connection reinforcement design of ODS-W/Cu joint: Transforming immiscible interface into dual reaction diffusion interface. Materials Characterization. 2025. Vol. 228. 115402. DOI: 10.1016/j.matchar.2025.115402
Ding Y., Wen D., Wang X., et al. Interface Evolution and Mechanical Properties of Al/Ta Laminated Composites Fabricated by Vacuum-Embedded Diffusion Welding. International Journal of Refractory Metals and Hard Materials. 2025. Vol. 133. 107302. DOI: 10.2139/ssrn.5239598
Ma J., Xu W., Zheng C., Li Y., Feng X., Yang Y. Effect of trace Al and Ti elements on borosilicate glass corrosion resistance of Inconel 690 alloy. Journal of Nuclear Materials. 2025. Vol. 606. 155626. DOI: 10.1016/j.jnucmat.2025.155626
Straumal A.B., Mazilkin I.A., Tsoi K.V., Baretzky B., Straumal B.B. "Wetting" Phase Transitions by the Second Solid Phase for Linear Defects (Grain Boundary Triple Junctions). JETP Letters. 2020. Vol. 112. P. 257–261. DOI: 10.1134/S0021364020160031
Bokshteyn B.S., Yaroslavtsev A.B. Diffuziya atomov i ionov v tverdykh telakh. Moscow: MISiS, 2005. 362 p.
Voroshnin L.G., Vityaz’ P.A., Nasybulin A.K.H., Khusid B.M. Mnogokomponentnaya diffuziya v geterogennykh splavakh. Minsk: Vysheyshaya shkola, 1984. 142 p.
De Groot S.R., Mazur P. Non-Equilibrium Thermodynamics. North-Holland Publishing Company, 1962. 510 p.
Gurov K.P., Kartashkin B.A., Ugaste Y.E. Vzaimnaya diffuziya v mnogofaznykh metallicheskikh sistemakh. Moscow: Nauka, 1981. 350 p.
Gurov K.P. Fenomenologicheskaya termodinamika neobratimykh protsessov. Moscow: Nauka, 1978. 128 p.
Gyarmati I. Non-equilibrium thermodynamics: Field theory and variational principles. Berlin Heidelberg: Springer-Verlag, 1970. 184 p.
Knyazeva A.G. Cross Effects in Solid Media with Diffusion. Journal of Applied Mechanics and Technical Physics. 2003. Vol. 44. P. 373–384. DOI: 10.1023/A:1023485224031
Knyazeva A.G. Diffuziya i reologiya v lokal’no-ravnovesnoy termodinamike. Bulletin of Perm State Technical University. Mathematical Modeling of Systems and Processes. 2005. No. 13. P. 45–60.
Knyazeva A.G. Nonlinear models of deformable media with diffusion. Physical mesomechanics. 2011. Vol. 14, no. 6. P. 35–51.
Knyazeva A.G., Demidov V.N. Factors of carrying over for the three-componental deformable alloy. PSTU Mechanics Bulletin. 2011. No. 3. P. 84–99.
Knyazeva A.G. Thermodynamic generalization of the theory of thermoelastic diffusion for a medium with changing density. PNRPU Mechanics Bulletin. 2025. No. 3. P. 101–113. DOI: 10.15593/perm.mech/2025.3.09
Mikolaychuk M.A., Knyazeva A.G. Model’ diffuzii primesi v strukturno-neodnorodnoy deformiruyemoy srede. Russian Physics Journal. 2012. Vol. 55, no. 5–2. P. 74–80.
Mikolaychuck M.A., Knyazeva A.G., Grabovetskaya G.P., Mishin I.P. Research of the stress influence on the diffusion in the coating plate. PNRPU Mechanics Bulletin. 2012. No. 3. P. 120–134.
Mikolaichuk M.A., Knyazeva A.G. Effect of stresses and strains on impurity redistribution in a plate under uniaxial loading. Journal of Applied Mechanics and Technical Physics. 2010. Vol. 51. P. 422–430. DOI: 10.1007/s10808-010-0057-3
Knyazeva A.G., Mikolaychuk M.A. Saturation of a plate with an environmental impurity under mechanical loading conditions. Mechanics of Solids. 2011. Vol. 46. P. 692–704. DOI: 10.3103/S0025654411050050
Zhang J., Zhao Q., Zhang L., Wang J., Sun C. Molecular dynamics simulations based on the diffusion interface of solid-phase Ti–Al system. Chemical Physics. 2025. Vol. 597. 112797. DOI: 10.1016/j.chemphys.2025.112797
Fan S., Peng M., Duan Y., Yu Q., Zhou X., Bu H., Li J., Yang Z., Li M. Molecular dynamics simulation of diffusion mechanisms of Al–Mg interface. Physica B: Condensed Matter. 2025. Vol. 715. 417620. DOI: 10.1016/j.physb.2025.417620
Zvyagintseva A.V. Mathematical modeling of hydrogen diffusion in a medium with impurity and structural traps in the conditions of the formation and decomposition of hydrogen-containing complexes. International Journal of Hydrogen Energy. 2025. Vol. 101. P. 112–119. DOI: 10.1016/j.ijhydene.2024.12.047
Akimova E.N., Gorbachev I.I., Popov V.V. Solving the multicomponent diffusion problems by parallel matrix sweep algorithm. Mathematical Models and Computer Simulations. 2005. Vol. 17, no. 9. P. 85–92.
Chen L.- Q., Zhao Y. From classical thermodynamics to phase-field method. Progress in Materials Science. 2022. Vol. 124. 100868. DOI: 10.1016/j.pmatsci.2021.100868
Mathew C.C., Song J., Adu-Gyamfi E., Fu Y. Phase field numerical model for simulating the diffusion controlled stress corrosion cracking phenomena in anisotropic material. Computational Materials Science. 2025. Vol. 247. 113528. DOI: 10.1016/j.commatsci.2024.113528
Ngiam Y., Cao Z.H., Huang M.X. Understanding hydrogen embrittlement in press-hardened steel by coupling phase field and hydrogen diffusion modeling. Materials Science and Engineering: A. 2022. Vol. 834. P. 142523. DOI: 10.1016/j.msea.2021.142523
Chen Q., Ma N., Wu K., Wang Y. Quantitative phase field modeling of diffusion-controlled precipitate growth and dissolution in Ti–Al–V. Scripta Materialia. 2004. Vol. 50, no. 4. P. 471–476. DOI: 10.1016/j.scriptamat.2003.10.032
Wu X.- W., Chen M., Ke L.-L. An electro-thermo-mechanical coupling phase-field model of defect evolution induced by electromigration in interconnects. International Journal of Mechanical Sciences. 2025. Vol. 285. 109792. DOI: 10.1016/j.ijmecsci.2024.109792
Jacobsson E., Hallberg H., Hektor J., Ristinmaa M. Modelling diffusive phase transformations in multiphase systems using the Voronoi implicit interface method. Modelling and Simulation in Materials Science and Engineering. 2025. Vol. 33. 025006. DOI: 10.1088/1361-651X/ada818
Kainuma R., Ichinose M., Ohnuma I., Ishida K. Formation of γ’/β interface morphologies in Ni–Al–X ternary diffusion couples. Materials Science and Engineering: A. 2001. Vol. 312. P. 168–175. DOI: 10.1016/S0921-5093(00)01873-6
Brechet Y., Kirkaldy J.S. Parabolic periodic solutions of precipitation-modified ternary diffusion equations. Canadian Journal of Physics. 1992. Vol. 70. P. 193–198. DOI: 10.1139/p92-033
Kulkarni K.N. Analytical solution for interdiffusion in multicomponent systems and its application in high entropy alloys. AIP Advances. 2021. Vol. 11. 015116. DOI: 10.1063/5.0032837
Knyazeva A.G., Savitskii A.P. Estimate of volume changes in the diffusion zone. I. Isothermal interaction of two semi-infinite media. Russian Physics Journal. 1997. Vol. 40. P. 420–427. DOI: 10.1007/BF02508770
Knyazeva A.G., Savitskii A.P. Estimating volume changes in the diffusion zone. 2. Interaction of two finite media. Russian Physics Journal. 1997. Vol. 40. P. 546–553. DOI: 10.1007/BF02766386
Mohanty R.R., Sohn Y. Phase-field investigation of multicomponent diffusion in single-phase and two-phase diffusion couples. Journal of Phase Equilibria and Diffusion. 2006. Vol. 27, no. 6. P. 676–683. DOI: 10.1007/BF02736572
Morino T., Ode M., Hirosawa S. Direct CALPHAD coupling phase-field model: Closed-form expression for interface composition satisfying equal diffusion potential condition. Physical Review E. 2024. Vol. 109. 065303. DOI: 10.1103/PhysRevE.109.065303
Knyazeva A.G. Some diffusion problems involved in analyzing the properties of coatings. Physical Mesomechanics. 2001. Vol. 4, no. 1. P. 49–65.
Butov V.G., Gubarkov D.V., Knyazeva A.G. Distribution of the diffusing element concentration in a three-layered system and estimation of the diffusion coefficient based on the solution of an inverse problem. Physical Mesomechanics. 2000. Vol. 3, no. 6. P. 105–112.
Knyazeva A.G., Pobol I.L., Romanova V.A. Stress field in the diffusion zone of an electron-beam brazed joint. Physical Mesomechanics. 2001. Vol. 4, no. 5. P. 41–53.
Knyazeva A.G., Anisimova M.A., Korosteleva E.N. Features of diffusion-controlled processes of regulated volumetric synthesis from powder mixtures Ti-Al-Fe-Fe2O3. PNRPU Mechanics Bulletin. 2022. No. 3. P. 125–134. DOI: 10.15593/perm.mech/2022.3.13
Knyazeva A.G., Pobol’ I.L., Oleshuk I.G. Pereraspredeleniye legiruyushchikh elementov mezhdu soyedinyayemymi materialami v usloviyakh izotermicheskoy payki i soputstvuyushchiye mekhanicheskiye napryazheniya. Russian Physics Journal. 2013. Vol. 56, no. 7–2. P. 14–24.
Vendin S.V. To the solution of issues of nonstationary diffusion in layered environments. Bulletin of BSTU named after V.G. Shukhov. 2019. No. 3. P. 100–105. DOI: 10.34031/article_5ca1f6340f3497.49776836
Popov V.M. Teploobmen v zone kontakta neraz”yemnykh soyedineniy. Moscow: Energiya, 1971. 216 p.
Zhang H., Huang D., Zhang Y. A peridynamic thermal contact model for heat conduction analysis of thermally imperfect interface and conductive crack. International Journal of Heat and Mass Transfer. 2025. Vol. 241. 126763. DOI: 10.1016/j.ijheatmasstransfer.2025.126763
Ding W., Xue T., Zhang X. A thermal-mechanical coupling modeling with imperfect interfaces: Transition from ballistic to diffusive heat transfer. Applied Mathematical Modelling. 2026. Vol. 150. 116361. DOI: 10.1016/j.apm.2025.116361
Wang J., Xue T., Zhang X. Time fractional-integer hybrid modeling for anomalous thermal contact problems. International Journal of Heat and Mass Transfer. 2025. Vol. 251. 127338. DOI: 10.1016/j.ijheatmasstransfer.2025.127338
Zheng H., Zhang L., Dong Q., Sun G. Prediction of the effective diffusion coefficient on sulfate ions in heterogeneous concrete based on Mori-Tanaka scheme. Construction and Building Materials. 2024. Vol. 449. 138326. DOI: 10.1016/j.conbuildmat.2024.138326
Wang R., Wang P., Bhatt S., et al. Phase-field modeling of diffusion bonding in 316H stainless steel: Impact of processing conditions on grain morphology and bonding quality. Materials Science and Engineering: A. 2025. Vol. 945. 149051. DOI: 10.1016/j.msea.2025.149051
Anisimova M.A., Knyazeva A.G., Korosteleva E.N., Povernov S.E. Peculiarities of sintering CuO-Al and TiO2-Al powder compositions under conditions of controlled heating. Chemical Physics and Mesoscopics. 2024. Vol. 26, no. 4. P. 457–470. DOI: 10.62669/17270227.2024.4.38
Paskonov V.M., Polezhayev V.I., Chudov L.A. Chislennoye modelirovaniye protsessov teplo- i massoobmena. Moscow: Nauka, 1984. 288 p.
Samarskiy A.A., Vabishchevich P.N. Vychislitel’naya teploperedacha. Moscow: Editorial URSS, 2003. 784 p.
Larikov L.N., Isaychev V.I. Struktura i svoystva metallov i splavov. Diffuziya v metallakh i splavakh. Kiev: Naukova Dumka, 1987. 512 p.
Toor H.L. Solution of the linearized equations of multicomponent mass transfer: I. AIChE Journal. 1964. Vol. 10, no. 4. P. 448–455. DOI: 10.1002/aic.690100408
Volpert A.I., Posvyanskii V.S. On the positivity of solutions of multicomponent diffusion and chemical kinetics equations. Soviet Journal of Chemical Physics. 1984. Vol. 3, no. 8. P. 1200–1205.
Wojewoda I., Lopez G.A., Zieba P., Mittemeijer E.I. Diffusion processes in diffusion-soldered interconnections. Archives of Metallurgy and Materials. 2004. Vol. 49, no. 2. P. 277–291.
Mincheva M., Siegel D. Nonnegativity and positiveness of solutions to mass action reaction–diffusion systems. Journal of Mathematical Chemistry. 2007. Vol. 42. P. 1135–1145. DOI: 10.1007/s10910-007-9292-0
Rios W.Q., Antunes B., Rodrigues A.E., Portugal I., Silva C.M. Accurate Effective Diffusivities in Multicomponent Systems. Processes. 2022. Vol. 10. 2042. DOI: 10.3390/pr10102042
Chen Q., Engström A., Ågren J. On Negative Diagonal Elements in the Diffusion Coefficient Matrix of Multicomponent Systems. Journal of Phase Equilibria and Diffusion. 2018. Vol. 39. P. 592–596. DOI: 10.1007/s11669-018-0648-x
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Computational Continuum Mechanics
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
