Application of Irreversible Thermodynamics to Diffusion in Solids with Internal Surfaces

Anna G. Knyazeva 1
  • 1 Institute of Strength Physics and Materials Science SB RAS, pr. Akademicheskii, 2/4, Tomsk, Russia
Anna G. Knyazeva
  • Corresponding author
  • 104737Institute of Strength Physics and Materials Science SB RAS, pr. Akademicheskii, 2/4, Tomsk, Russia, 634055
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar


Two types of additional variables were included in the set of state variables and were used for a thermodynamic description of diffusion in an ordinary thermodynamic system. Vacancies are included in the mass balance. Internal surfaces are massless but are characterized by some energy, which is included in the energy balance of the thermodynamic system. Fluxes of components, vacancies, and surfaces were expressed via two groups of thermodynamic constitutive equations of with cross effects. The first group follows from the Gibbs equation. These are state equations in a differential form. The second group relates generalized thermodynamic fluxes to generalized thermodynamic forces. It was shown for a binary system that only three of six transfer coefficients are independent even if the mass transfer mechanism caused by the stress gradient is taken into account.

  • [1]

    S. Whitaker, Mechanics and thermodynamics of diffusion, Chem. Eng. Sci. 68 (2012), 362–375.

    • Crossref
    • Export Citation
  • [2]

    H. Mehrer, Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes, Springer Verlag, Berlin, Heidelberg, 2007.

  • [3]

    I. Kaur, Y. Mishin and W. Gust, Fundamentals of Grain and interphase boundary diffusion, Wiley, Chichester, West Sussex, 1995.

  • [4]

    G. D. C. Kuiken, Thermodynamics of Irreversible Processes: Applications to Diffusion and Rheology, Wiley, 1994.

  • [5]

    R. O’Hayre, Materials kinetic fundamentals. Principles, processes and applications, John Wiley & Sons, 2015.

  • [6]

    L. M. C. Sagis, Dynamic behavior of interfaces: Modeling with nonequilibrium thermodynamics, Adv. Colloid Interface Sci. 206 (2014), 328–334.

    • Crossref
    • PubMed
    • Export Citation
  • [7]

    A. I. Rusanov, Interfacial thermodynamics: Development for last decades, Solid State Ion. 75 (1995), 275–279.

    • Crossref
    • Export Citation
  • [8]

    G. G. Lang, Basic interfacial thermodynamics and related mathematical background, Chem Texts, Springer International Publishing, 2015.

  • [9]

    C. F. Gurtiss and R. B. Bird, Multicomponent Diffusion (review), Ind. Eng. Chem. Res. 38 (1999), 2515–2522.

    • Crossref
    • Export Citation
  • [10]

    A. G. Knyazeva, Cross effects in solid media with diffusion, J. Appl. Mech. Tech. Phys. 44 (2003), no. 3, 373–384.

    • Crossref
    • Export Citation
  • [11]

    V. V. Sychev, Complex Thermodynamic Systems, 1st ed., Springer, Boston, MA, 1973.

  • [12]

    M. F. Horstemeyer and D. J. Bammann, Historical review of internal state variable theory for inelasticity, Int. J. Plast. 26 (2010), 1310–1334.

    • Crossref
    • Export Citation
  • [13]

    G. A. Maugin and W. Muschik, Thermodynamics with Internal Variables. Part I. General Concepts, J. Non-Equilib. Thermodyn. 19 (1994), no. 3, 217–249.

  • [14]

    G. A. Maugin and W. Muschik, Thermodynamics with Internal Variables Part II. Applications, J. Non-Equilib. Thermodyn. 19 (1994), no. 3, 250–289.

  • [15]

    A. Berezovski and P. Ván, Internal Variables in Thermoelasticity, Solid Mechanics and Its Applications, 243, Springer, 2017.

  • [16]

    P. Van, A. Berezovski and J. Engelbrecht, Internal variables and dynamic degrees of freedom, J. Non-Equilib. Thermodyn. 33 (2008), no. 3, 235–254.

  • [17]

    J. Engelbrecht and A. Berezovski, Internal structures and internal variables in solids, J. Mech. Mater. Struct. 7 (2012), no. 10, 983–996.

    • Crossref
    • Export Citation
  • [18]

    V. V. Sychev, The Differential Equations of Thermodynamics, CRC Press, 1991.

  • [19]

    I. GyarmatiI, Non-equilibrium Thermodynamics. Field Theory and Variational Principles, Springer-Verlag, Berlin, Heidelberg, 1970.

  • [20]

    P. Glansdorff and I. Prigogine, Thermodynamic theory of structure, stability and fluctuations, 1st ed., Wiley-Interscience, 1971.

  • [21]

    K. Annamalai and I. K. Puri, Advanced thermodynamics engineering, CRC Press, 2002.

  • [22]

    A. A. Vakulenko and A. V. Proskura, On the diffusion creep of metals (in Russian), Izvestiya RAN Proceedings of the Russian Academy of Sciences, Mech. Solids 2 (1997), 133–144.

  • [23]

    B. S. Bokstein and B. B. Straumal (Eds.), Diffusion in solids – past, present and future, Scitec Publications Ltd., Uetikon-Zuerich, 2006.

  • [24]

    Yu. R. Kolobov, R. Z. Valiev and M. B. Ivanov, Grain Boundary Diffusion and Properties of Nanostructured Materials, Cambridge International Science Publishing Ltd., 2007.

  • [25]

    A. G. Knyazeva, Thermodynamics with additional Parameters for polycrystals, Int. J. Nanomech. Sci. Technol. 6 (2015), no. 4, 1–25.

  • [26]

    J. F. Nay, Physical properties of Crystals: their representation by Tensors and Matrices, Clarendon Press, Oxford, 1957.

  • [27]

    B. S. Bokshtein, S. Z. Bokshtein and A. A. Zhukhovitskii, Thermodyamics and kinetics of diffusion in solid bodies (in Russian), Metallurgia, Moscow, 1974.

  • [28]

    A. G. Knyazeva, On modeling irreversible processes in materials with a large area of internal surfaces (in Russian), Phys. Mesomech. 6 (2003), no. 5, 11–27.

  • [29]

    A. G. Knyazeva, Effective diffusion coefficients for materials containing internal surfaces, Zh. Funkts. Mater. (in Russian) 2 (2008), no. 2, 45–55.

  • [30]

    V. S. Eremeev, Diffusion and stresses (in Russian), Energoatomizdat, Moscow, 1984.

  • [31]

    S. Kurasch, J. Kotakoski, O. Lehtinen, V. Skákalová, Ju. Smet, C. E. Krill, et al., Atom-by-Atom Observation of Grain Boundary Migration in Graphene, Nano Lett. 12 (2012), 3168–3173.

    • Crossref
    • PubMed
    • Export Citation
  • [32]

    H. Hallberg and V. V. Bulatov, Modeling of grain growth under fully anisotropic grain boundary energy, Model. Simul. Mater. Sci. Eng. 27 (2019), 045002.

  • [33]

    A. Rajabzadeh, Experimental and theoretical study of the shear-coupled grain boundary migration mechanism. Materials. Université Paul Sabatier (Toulouse 3), 2013.

  • [34]

    N. Combe, F. Mompiou and M. Legros, Shear-coupled grain-boundary migration dependence on normal strain/stress, Phys. Rev. Mater. 1 (2017), 033605.

  • [35]

    M. Aouadi, Classic and Generalized Thermoelastic Diffusion Theories, in: Encyclopedia of Thermal Stresses, R. B. Hetnarski (Ed.), Springer, 2014.

  • [36]

    T. Kansal, Generalized theory of thermoelastic diffusion with double porosity, Arch. Mech. 70 (2018), no. 3, 241–268.

  • [37]

    M. Aouadi, Generalized Theory of Thermoelastic Diffusion for Anisotropic Media, Journal of Thermal Stresses 31 (2008), no. 3, 270–285.

    • Crossref
    • Export Citation
  • [38]

    H. H. Sherief, F. A. Hamza and H. A. Saleh, The theory of generalized thermoelastic diffusion, Int. J. Eng. Sci. 42 (2004), 591–608.

    • Crossref
    • Export Citation
  • [39]

    E. S. Parfenova and A. G. Knyazeva, The influence of vacancy generation at the initial stage of ion implantation, AIP Conf. Proc. 1623 (2014), 479–482.

    • Crossref
    • Export Citation
  • [40]

    E. S. Parfenova and A. G. Knyazeva, The Features of Diffusion and Mechanical Waves Interaction at the Initial Stage of Metal Surface Treatment by Particle Beam under Nonisothermal Conditions, Key Eng. Mater. 712 (2016), 99–104.

    • Crossref
    • Export Citation
  • [41]

    A. G. Knyazeva and E. S. Parfenova, Interrelation diffusion and mechanical waves at the initial stage of ion beam action on the metallic surface, AIP Conf. Proc. 1893 (2017), 030109.

Purchase article
Get instant unlimited access to the article.
Log in
Already have access? Please log in.

Log in with your institution

Journal + Issues

The Journal of Non-Equilibrium Thermodynamics serves as an international publication organ for new ideas, insights and results on non-equilibrium phenomena in science, engineering and related natural systems. The central aim of the journal is to provide a bridge between science and engineering and to promote scientific exchange on non-equilibrium phenomena and on analytic or numeric modeling for their interpretation.