Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-29T07:59:21.320Z Has data issue: false hasContentIssue false
  • English
  • Français

Quasichemical Models of Multicomponent NonlinearDiffusion

Published online by Cambridge University Press:  10 August 2011

A.N. Gorban ,
H.P. Sargsyan  and
H.A. Wahab
A.N. Gorban*
Affiliation:
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK
H.P. Sargsyan
Affiliation:
UNESCO Chair – Life Sciences International Postgraduate Educational Center (LSIPEC), Yerevan, Republic of Armenia
H.A. Wahab
Affiliation:
Department of Mathematics, University of Leicester, Leicester, LE1 7RH, UK
*
⋆ Corresponding author. E-mail: [email protected]
Get access

Abstract

Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusionshould be nonlinear if there exist non-diagonal terms. The vast variety of nonlinearmulticomponent diffusion equations should be ordered and special tools are needed toprovide the systematic construction of the nonlinear diffusion equations formulticomponent mixtures with significant interaction between components. We develop anapproach to nonlinear multicomponent diffusion based on the idea of the reaction mechanismborrowed from chemical kinetics.

Chemical kinetics gave rise to very seminal tools for the modeling of processes. This isthe stoichiometric algebra supplemented by the simple kinetic law. The results of thisinvention are now applied in many areas of science, from particle physics to sociology. Inour work we extend the area of applications onto nonlinear multicomponent diffusion.

We demonstrate, how the mechanism based approach to multicomponent diffusion can beincluded into the general thermodynamic framework, and prove the corresponding dissipationinequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementaryprocess cannot have an arbitrary form. For the general kinetic law (the generalized MassAction Law), additional conditions are proved. The cell–jump formalism gives anintuitively clear representation of the elementary transport processes and, at the sametime, produces kinetic finite elements, a tool for numerical simulation.

Keywords

diffusionreaction mechanismentropy productiondetailed balancecomplex balancetransport equation
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 6 , Issue 5: Complex Fluids , 2011 , pp. 184 - 262
Copyright
© EDP Sciences, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barenblatt, G.I.. On some unsteady motions of a liquid or a gas in a porous medium. Prikl. Mat. Mekh., 16 (1952), 67–78. Google Scholar
Barenblatt, G.I., ZelŠdovich, Y.B.. Self-similar solutions as intermediate asymptotics. Ann. Rev. Fluid Mech., 4 (1972), 285–312. CrossRefGoogle Scholar
Bertini, L., Landim, C., Olla, S.. Derivation of Cahn–Hilliard Equations from Ginzburg-Landau Models. J. Stat. Phys., 88 (1997), Nos. 1/2, 365–381. CrossRefGoogle Scholar
Blesgen, T., Weikard, U.. Multi-component Allen-Cahn equation for elastically stressed solids. Electron. J. Diff. Eqns., 89 (2005), 1–17. Google Scholar
Boudart, M.. From the century of the rate equation to the century of the rate constants: a revolution in catalytic kinetics and assisted catalyst design. Catal. Lett., 65 (2000), 1–3. CrossRefGoogle Scholar
L. Boltzmann. Lectures on gas theory. U. of California Press, Berkeley, CA, 1964.
Briggs, G.E., Haldane, J.B.S.. A note on the kinetics of enzyme action. Biochem. J., 19 (1925), 338–339. CrossRefGoogle ScholarPubMed
Brownlee, R.A., Gorban, A.N., Levesley, J.. Nonequilibrium entropy limiters in lattice Boltzmann methods. Physica A, 387 (2008), 385–406. CrossRefGoogle Scholar
Bykov, V.I., Gilev, S.E., Gorban, A.N., Yablonskii, G.S.. Imitation modeling of the diffusion on the surface of a catalyst. Dokl. Akad. Nauk SSSR, 283 (1985), 1217–1220. Google Scholar
Bykov, V.I., Gorban, A.N., Yablonskii, G.S.. Description of non-isothermal reactions in terms of Marcelin-De-Donder Kinetics and its generalizations. React. Kinet. Catal. Lett. 20 (1982), 261–265. CrossRefGoogle Scholar
Cahn, J.W.. Free energy of a nonuniform system. II. Thermodynamic basis. J. Chem. Phys., 30 (1959), 1121–1124. CrossRefGoogle Scholar
Cahn, J.W., Hilliard, J.E.. Free energy of a nonuniform system. I. Interfacial energy. J. Chem. Phys., 28 (1958), 258–266. CrossRefGoogle Scholar
Cahn, J.W., Hilliard, J.E.. Spinodal decomposition: A reprise. Acta Metallurgica, 19 (1971), 151–161. CrossRefGoogle Scholar
H.B. Callen. Thermodynamics and an introduction to themostatistics (2nd ed.). John Wiley & Sons, NY, 1985.
Cercignani, C., Lampis, M.. On the H-theorem for polyatomic gases. J. Stat. Phys., 26 (1981), 795–801. CrossRefGoogle Scholar
B. Chopard, M. Droz. Cellular automata modeling of physical systems. Cambridge University Press, Cambridge, UK, 1998.
Clausius, R.. Über verschiedene für die Anwendungen bequeme Formen der Hauptgleichungen der Wärmetheorie. Poggendorffs Annalen der Physic und Chemie, 125 (1865), 353–400. CrossRefGoogle Scholar
Chorin, A.J., Hald, O.H., Kupferman, R.. Optimal prediction with memory. Physica D, 166 (2002), 239–257. CrossRefGoogle Scholar
F. Coester. Principle of detailed balance. Phys. Rev., 84, 1259 (1951)
S.R. De Groot, P. Mazur. Non-equilibrium Thermodynamics. North-Holland, Amsterdam, 1962.
K. Denbigh. The principles of chemical equilibrium. Cambridge University Press, Cambridge, UK, 1981.
Dushman, S., Langmuir, I.. The diffusion coefficient in solids and its temperature coefficient. Phys. Rev., 20 (1922), 113. Google Scholar
P. Ehrenfest, T. Ehrenfest-Afanasyeva. Begriffliche Grundlagen der statistischen Auffassung in der Mechanik. In: Mechanics Enziklopädie der Mathematischen Wissenschaften, Vol. 4. Leipzig, 1911. (Reprinted in: Ehrenfest, P., Collected Scientific Papers. North–Holland, Amsterdam, 1959, pp. 213–300.)
Einstein, A.. Über die von der molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden Flüssigkeiten suspendierten Teilchen. Ann. Phys., 17 (1905), 549–560. CrossRefGoogle Scholar
A. Einstein. Strahlungs-Emission und -Absorption nach der Quantentheorie. Verhandlungen der Deutschen Physikalischen Gesellschaft, 18 (1916), No. 13/14, Braunschweig, Vieweg, 318–323.
Elliott, C.M., Songmu, Z.. On the Cahn-Hilliard equation. Arch. Rat. Mechan. Anal., 96 (1986), 339–357. Google Scholar
Elliott, C.M., Stuart, A.M.. The global dynamics of discrete semilinear parabolic equations. SIAM J. Numer. Anal., 30 (1993), 1622–1663. CrossRefGoogle Scholar
Eyring, H.. The activated complex in chemical reactions. J. Chem. Phys., 3 (1935), 107–115. CrossRefGoogle Scholar
Feinberg, M.. On chemical kinetics of a certain class. Arch. Rat. Mechan. Anal., 46 (1972), 1–41. Google Scholar
Feinberg, M.. Complex balancing in general kinetic systems. Arch. Rat. Mechan. Anal., 49 (1972), 187–194. Google Scholar
Feynman, R.F.. Simulating physics with computers. Internat. J. Theor. Phys., 21 (1982), 467–488. CrossRefGoogle Scholar
Fick, A.. Über Diffusion. Poggendorff’s Annalen der Physik und Chemie, 94 (1855), 59–86. CrossRefGoogle Scholar
R.A. Fisher. The genetical theory of natural selection. Oxford University Press, Oxford, 1930.
Frank, F.C., Turnbull, D.. Mechanism of diffusion of copper in Germanium. Phys. Rev., 104 (1956), 617–618. CrossRefGoogle Scholar
Fratzl, A., Penrose, O., Lebowitz, J.L.. Modelling of phase separation in alloys with coherent elastic misfit. J. Stat. Phys., 95 (1999), 1429–1503. CrossRefGoogle Scholar
Frenkel, J.. Theorie der Adsorption und verwandter Erscheinungen. Zeitschrift für Physik, 26 (1924), 117–138 CrossRefGoogle Scholar
Frenkel, J.. Über die Wärmebewegung in festen und flüssigen Körpern. Zeitschrift für Physik, 35 (1925), 652–669. CrossRefGoogle Scholar
G.F. Gause. The struggle for existence. Williams & Wilkins, Baltimore, 1934.
J.W. Gibbs. On the equilibrium of heterogeneous substance. Trans. Connect. Acad., 1875–1876, 108–248; 1877–1878, 343–524.
Gillespie, D.T.. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem., 81 (1977), 2340–2361. CrossRefGoogle Scholar
Gillespie, D.T.. Stochastic simulation of chemical kinetics. Annu. Rev. Phys. Chem., 58 (2007), 35–55. CrossRefGoogle ScholarPubMed
A.N. Gorban. Equilibrium encircling. Equations of chemical kinetics and their thermodynamic analysis. Nauka, Novosibirsk, 1984.
A.N. Gorban. Singularities of transition processes in dynamical systems: qualitative theory of critical delays. Electron. J. Diff. Eqns., Monograph 05, 2004. E-print: http://arxiv.org/abs/chao-dyn/9703010, 1997.
A.N. Gorban. Basic types of coarse-graining. In: Model reduction and coarse–graining approaches for multiscale phenomena, Ed. by A.N. Gorban, N. Kazantzis, I.G. Kevrekidis, H.C. Öttinger, C. Theodoropoulos. Springer, Berlin-Heidelberg-New York, 2006, 117–176. E-print: http://arxiv.org/abs/cond-mat/0602024, 2006.
Gorban, A.N., Bykov, V.I., Yablonskii, G.S.. Macroscopic clusters induced by diffusion in catalytic Oxidation Reactions. Chem. Eng. Sci., 35 (1980), 2351–2352. CrossRefGoogle Scholar
A.N. Gorban, V.I. Bykov, G.S. Yablonskii. Essays on chemical relaxation. Novosibirsk, Nauka Publ., 1986.
Gorban, A.N., Gorban, P.A., Judge, G.. Entropy: The Markov ordering approach. Entropy, 12 (2010), 1145–1193. E-print: http://arxiv.org/abs/1003.1377, 2010. CrossRefGoogle Scholar
Gorban, A.N., Karlin, I.V., Öttinger, H.C., Tatarinova, L.L.. Ehrenfest’s argument extended to a formalism of nonequilibrium thermodynamics. Phys. Rev. E, 63 (2001), 066124. CrossRefGoogle Scholar
Gorban, A.N., Karlin, I.V.. Uniqueness of thermodynamic projector and kinetic basis of molecular individualism. Physica A, 336 (2004), 391–432. E-print: http://arxiv.org/abs/cond-mat/0309638, 2003. CrossRefGoogle Scholar
Gorban, A.N., Karlin, I.V.. Method of invariant manifold for chemical kinetics. Chem. Eng. Sci., 58 (2003), 4751–4768. CrossRefGoogle Scholar
A.N. Gorban, I.V. Karlin. Invariant manifolds for physical and chemical kinetics. Lect. Notes Phys. 660, Springer, Berlin, Heidelberg, 2005.
Gorban, A.N., Karlin, I.V., Ilg, P., Öttinger, H.C.. Corrections and enhancements of quasi-equilibrium states. J. Non-Newtonian Fluid Mech., 96 (2001), 203–219. CrossRefGoogle Scholar
Gorban, A.N., Sargsyan, H.P.. Mass action law for nonlinear multicomponent diffusion and relations between its coefficients. Kinetics and Catalysis, 27 (1986), 527. Google Scholar
A.N. Gorban, M. Shahzad. QE+QSS for derivation of kinetic equations and stiffness removing. E-print: http://arxiv.org/abs/1008.3296, 2010.
Graham, T.. The Bakerian lecture: on the diffusion of liquids. Phil. Trans. R. Soc. Lond., 140 (1) (1850), 1–46; doi: 10.1098/rstl.1850.0001. CrossRefGoogle Scholar
Grmela, M., Öttinger, H.C.. Dynamics and thermodynamics of complex fluids. I. Development of a general formalism. Phys. Rev. E, 56 (1997), 6620–6632. CrossRefGoogle Scholar
Gurney, W.S.C., Nisbet, R.M.. A note on nonlinear population transport. J. Theor. Biol., 56 (1976), 249–251. CrossRefGoogle Scholar
Gurtin, M.E.. Generalized Ginzburg-Landau and Cahn-Hilliard equations based on a microforce balance. Physica D, 92 (1996), 178–192. CrossRefGoogle Scholar
I. Gyarmati. Non-equilibrium thermodynamics. Field theory and variational principles. Springer, Berlin, 1970.
W. Heitler. Quantum Theory of Radiation. Oxford University Press, London, 1944.
R. Hengeveld. Dynamics of biological invasions. Chapman and Hall, London, 1989.
Horn, F., Jackson, R.. General mass action kinetics. Arch. Rat. Mechan. Anal., 47 (1972), 81–116. Google Scholar
W.G. Hoover. Computational statistical mechanics. Elsevier, Amsterdam, 1991.
Karlin, I.V., Gorban, A.N., Succi, S., Boffi, V.. Maximum entropy principle for lattice kinetic equations. Phys. Rev. Lett., 81 (1998), 6–9. CrossRefGoogle Scholar
L.B. Kier, P.G. Seybold, Ch-K. Cheng. Modeling chemical systems using cellular automata. Dordrecht, The Netherlands, 2005.
Kincaid, J.F., Eyring, H., Stearn, A.E.. The theory of absolute reaction rates and its application to viscosity and diffusion in the liquid State. Chem. Rev., 28 (1941), 301–365. CrossRefGoogle Scholar
Kirkendall, E.O.. Diffusion of zinc in alpha brass. Trans. Am. Inst. Min. Metall. Eng., 147 (1942), 104–110. Google Scholar
A.B. Kudryavtsev, R.F. Jameson, W. Linert. The law of mass action. Springer, Berlin – Heidelberg – New York, 2001.
K.J. Laidler, A. Tweedale. The current status of Eyring’s rate theory. In: Advances in Chemical Physics: Chemical dynamics: Papers in honor of Henry Eyring, Volume 21 (eds J. O. Hirschfelder and D. Henderson). John Wiley & Sons, Inc., Hoboken, NJ, USA, 2007.
L.D. Landau, E.M. Lifshitz. Fluid mechanics: Volume 6 (Course of theoretical physics). Butterworth-Heinemann, Oxford–Woburn, 1987.
Langer, J.S., Bar-on, M., Miller, H.D.. New computational method in the theory of spinodal decomposition. Phys. Rev. A, 11 (1975), 1417–1429. CrossRefGoogle Scholar
G. Lebon, D. Jou, J. Casas-Vázquez. Understanding non-equilibrium thermodynamics: Foundations, applications, Frontiers. Springer, Berlin, 2008.
A.J. Lotka. Elements of physical biology. Williams and Wilkins, Baltimore, 1925.
Lyon, R.J.P.. Time aspects of geothermometry. Mining Eng., 11 (1959), 1145–1151. Google Scholar
Mahan, B.H.. Microscopic reversibility and detailed balance. An analysis. J. Chem. Educ., 52 (1975), 299–302. CrossRefGoogle Scholar
Maier-Paape, S., Stoth, B., Wanner, T.. Spinodal decomposition for multicomponent cahnŰhilliard systems. J. Stat. Phys., 98 (2000), 871–896. CrossRefGoogle Scholar
McLaughlin, E.. The Thermal conductivity of liquids and dense gases. Chem. Rev., 64 (1964), 389–428. CrossRefGoogle Scholar
H. Mehrer. Diffusion in solids – fundamentals, methods, materials, diffusion-controlled processes. Textbook, Springer Series in Solid-State Sciences, Vol. 155, Springer, Berlin – Heidelberg – New York, 2007.
Mehrer, H., Stolwijk, N.A.. Heroes and highlights in the history of diffusion. Diffusion Fundamentals, 11 (2009), 1–32. Google Scholar
Michaelis, L., Menten, M.. Die Kinetik der Intervintwirkung. Biochemistry Zeitung, 49 (1913), 333–369. Google Scholar
Nakajima, H.. The discovery and acceptance of Kirkendall effect: The result of a short research career. JOM, 49 (1997), 15–19. CrossRefGoogle Scholar
Narasimhan, T.N.. Energetics of the Kirkendall effect. Current Science, 93 (2007), 1257–1264. Google Scholar
Onsager, L.. Reciprocal relations in irreversible processes. I. Phys. Rev., 37 (1931), 405–426. CrossRefGoogle Scholar
Onsager, L.. Reciprocal relations in irreversible processes. II. Phys. Rev., 38 (1931), 2265–2279. CrossRefGoogle Scholar
H.C. Öttinger. Beyond equilibrium thermodynamics. Wiley-Blackwell, Hoboken, NJ, 2005.
Öttinger, H.C.. Constraints in nonequilibrium thermodynamics: General framework and application to multicomponent diffusion. J. Chem. Phys., 130 (2009), 114904. CrossRefGoogle ScholarPubMed
Öttinger, H.C., Grmela, M.. Dynamics and thermodynamics of complex fluids. II. Illustrations of a general formalism. Phys. Rev. E, 56 (1997), 6633–6655. CrossRefGoogle Scholar
K. Oura, V.G. Lifshits, A.A. Saranin, A.V. Zotov, M. Katayama. Surface science: An introduction. Springer, Berlin – Heidelberg, 2003.
S.V. Petrovskii, B.-L. Li. Exactly solvable models of biological invasion. Chapman & Hall / CRC Press, Boca–Raton–London–New York–Washington D.C., 2006.
Philibert, J.. One and a half century of diffusion: Fick, Einstein, before and beyond. Diffusion Fundamentals, 2 (2005), 1.1–1.10. Google Scholar
W.C. Roberts-Austen. Bakerian lecture on the diffusion in metals. Phil. Trans. Roy. Soc. A 187, (1896). Part I: Diffusion of Molten Metals. 383–403; Part II: Diffusion of Solid Metals. 404–415.
Rothman, D., Zaleski, S.. Lattice-gas models of phase separation: interfaces, phase transitions and multiphase flow. Rev. Mod. Phys., 66 (1994), 1417–1480. CrossRefGoogle Scholar
Schelling, P.K., Phillpot, S.R., Keblinski, P.. Comparison of atomic-level simulation methods for computing thermal conductivity. Phys. Rev. B, 65 (2002), 144306. CrossRefGoogle Scholar
N.N. Semenov. Some problems relating to chain reactions and to the theory of combustion. Nobel Lecture, December 11, 1956. In: Nobel lectures in chemistry 1942–1962. World Scientific, Hackensack, NJ, 1999.
E. Seneta. Nonnegative matrices and Markov chains. Springer, New York, 1981.
N. Shigesada, K. Kawasaki. Biological invasions: theory and practice. Oxford University Press, Oxford, 1997.
Stueckelberg, E.C.G.. Théorème H et unitarité de S. Helv. Phys. Acta, 25 (1952), 577–580. Google Scholar
S. Succi. The lattice Boltzmann equation for fluid dynamics and beyond. Clarendon Press, Oxford, UK, 2001.
Succi, S., Karlin, I., Chen, H.. Role of the H theorem in lattice Boltzmann hydrodynamic simulations (Colloquium). Rev. Mod. Phys., 74 (2002), 1203–1220. CrossRefGoogle Scholar
S. Succi, “Lattice Boltzmann at all-scales: from turbulence to DNA translocation”, Mathematical Modelling Centre Distinguished Lecture, University of Leicester, Leicester, UK, 15 November 2006.
Teorell, T.. Studies on the “diffusion effect” upon ionic distribution–I Some theoretical considerations. Proc. N. A. S. USA, 21 (1935), 152–161. CrossRefGoogle Scholar
Teorell, T.. Studies on the diffusion effect upon ionic distribution–II Experiments on ionic accumulation. The Journal of General Physiology, 21 (1937), 107–122. CrossRefGoogle ScholarPubMed
T. Toffoli, N. Margolus. Cellular automata machines: A new environment for modeling. MIT Press, Cambridge, MA, 1987.
Tuijn, C.. On the history of models for solid–state diffusion. Defect and Diffusion Forum, 143-147 (1997), 11–20. CrossRefGoogle Scholar
Van Kampen, N.G.. Nonlinear irreversible processes. Physica, 67 (1973), 1–22. CrossRefGoogle Scholar
P. Van Mieghem. Performance analysis of communications networks and systems. Cambridge University Press, Cambridge, 2006.
J.H. Van’t Hoff. Etudes de dynamique chimique. Frederic Muller, Amsterdam, 1884.
J.L. Vázquez. The porous medium equation. Mathematical Theory. Oxford University Press, Oxford, 2007.
A.I. Volpert, S.I. Khudyaev. Analysis in classes of discontinuous functions and equations of mathematical physics. Nijoff, Dordrecht, 1985.
Volterra, V.. Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei, 2 (1926), 31–113. Google Scholar
J. Von Neumann, A.W. Burks. Theory of self-reproducing automata. University of Illinois Press, Urbana, 1966.
Watanabe, S.. Symmetry of physical laws. Part I. Symmetry in space-time and balance theorems. Rev. Mod. Phys., 27 (1955), 26–39. CrossRefGoogle Scholar
Wegscheider, R.. Über simultane Gleichgewichte und die Beziehungen zwischen Thermodynamik und Reactionskinetik homogener Systeme. Monatshefte für Chemie / Chemical Monthly, 32 (1911), 849–906. CrossRefGoogle Scholar
D.A. Wolf-Gladrow. Lattice-gas cellular automata and lattice Boltzmann models. Springer, 2000.
S. Wolfram. A new kind of science. Wolfram Media, Champaign, IL, 2002.
Wynne-Jones, W.F.K., Eyring, H.. The absolute rate of reactions in condensed phases. J. Chem. Phys., 3 (1935), 492–502. CrossRefGoogle Scholar
G.S. Yablonskii, V.I. Bykov, A.N. Gorban, V.I. Elokhin. Kinetic models of catalytic reactions. Series “Comprehensive Chemical Kinetics", Vol. 32, Compton R.G. (ed.), Elsevier, Amsterdam, 1991.
Y.B. Zeldovich. Proof of the uniqueness of the solution of the equations of the law of mass action. In: Selected Works of Yakov Borisovich Zeldovich; Volume 1, Ostriker, J.P., Ed. Princeton University Press, Princeton, NJ, USA, 1996; pp. 144–148.