Thermodynamic Theory of Diffusion and Thermodiffusion Coefficients in Multicomponent Mixtures

Alexander A. Shapiro 1
  • 1 CERE-Center for Energy Resources Engineering, Department of Chemical and Biochemical Engineering, Technical University of Denmark, DTU b. 229, Kgs. Lyngby, Denmark
Alexander A. Shapiro
  • Corresponding author
  • CERE-Center for Energy Resources Engineering, Department of Chemical and Biochemical Engineering, Technical University of Denmark, DTU b. 229, 2800, Kgs. Lyngby, Denmark
  • Email
  • Search for other articles:
  • degruyter.comGoogle Scholar

Abstract

Transport coefficients (like diffusion and thermodiffusion) are the key parameters to be studied in non-equilibrium thermodynamics. For practical applications, it is important to predict them based on the thermodynamic parameters of a mixture under study: pressure, temperature, composition, and thermodynamic functions, like enthalpies or chemical potentials. The current study develops a thermodynamic framework for such prediction. The theory is based on a system of physically interpretable postulates; in this respect, it is better grounded theoretically than the previously suggested models for diffusion and thermodiffusion coefficients. In fact, it translates onto the thermodynamic language of the previously developed model for the transport properties based on the statistical fluctuation theory. Many statements of the previously developed model are simplified and amplified, and the derivation is made transparent and ready for further applications. The n(n+1)/2 independent Onsager coefficients are reduced to 2n+1 determining parameters: the emission functions and the penetration lengths. The transport coefficients are expressed in terms of these parameters. These expressions are much simplified based on the Onsager symmetry property for the phenomenological coefficients. The model is verified by comparison with the known expressions for the diffusion coefficients that were previously considered in the literature.

  • [1]

    C. Peters, J. Thien, L. Wolff, H. J. Koẞ and A. Bardow, Quaternary Diffusion Coefficients in Liquids from Microfluidics and Raman Microspectroscopy: Cyclohexane + Toluene + Acetone + Methanol, J. Chem. Eng. Data 65 (2020), no. 3, 1273–1288.

    • Crossref
    • Export Citation
  • [2]

    C. Peters, L. Wolff, S. Haase, J. Thien, Th. Brands and H. J. Koẞ, Multicomponent diffusion coefficients from microfluidics using Raman microspectroscopy, Lab Chip 17 (2017), no. 16, 2768–2776.

    • Crossref
    • PubMed
    • Export Citation
  • [3]

    E. D. Chisholm and D. C. Wallace, Dynamics of monatomic liquids: Topical review, J. Phys. Condens. Matter 13 (2001), no. 37, R739–R769.

    • Crossref
    • Export Citation
  • [4]

    X. Liu, S. K. Schnell, J. -M. Simon, P. Krüger, D. Bedeaux, S. Kjelstrup, et al., Diffusion coefficients from molecular dynamics simulations in binary and ternary mixtures, Int. J. Thermophys. 34 (2013), 1169–1196.

    • Crossref
    • Export Citation
  • [5]

    G. Galliero, M. Bugel, B. Duguay and F. Montel, Mass effect on thermodiffusion using molecular dynamics, J. Non-Equilib. Thermodyn. 32 (2007), no. 3, 251–258.

  • [6]

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

  • [7]

    A. Vignes, Diffusion in binary solutions – variation of diffusion coefficient with composition, Ind. Eng. Chem. Fundam. 5 (1966), no. 2, 189–199.

    • Crossref
    • Export Citation
  • [8]

    G. D. Moggeridge, Prediction of the mutual diffusivity in binary non-ideal liquid mixtures from the tracer diffusion coefficients, Chem. Eng. Sci. 71 (2012), 226–238.

    • Crossref
    • Export Citation
  • [9]

    G. D. Moggeridge, Prediction of the mutual diffusivity in binary liquid mixtures containing one dimerizing species, from the tracer diffusion coefficients, Chem. Eng. Sci. 76 (2012), 199–205.

    • Crossref
    • Export Citation
  • [10]

    E. K. Cussler, Diffusion: Mass transfer in fluid systems, Cambridge University Press, 1997.

  • [11]

    J. C. Bosma and J. A. Wesselingh, Estimation of diffusion coefficients in dilute liquid mixtures, Chem. Eng. Res. Des. 77 (1999), no. A6, 325–328.

    • Crossref
    • Export Citation
  • [12]

    O. O. Medvedev and A. A. Shapiro, Modeling diffusion coefficients in binary mixtures, Fluid Phase Equilib. 225 (2004), 13–22.

    • Crossref
    • Export Citation
  • [13]

    O. O. Medvedev and A. A. Shapiro, Modeling diffusion coefficients in binary mixtures of polar and non-polar compounds, Fluid Phase Equilib. 236 (2005), 111–124.

    • Crossref
    • Export Citation
  • [14]

    O. O. Medvedev, Diffusion Coefficients in Multicomponent Mixtures, Ph.D. Thesis, DTU, Kgs. Lyngby, Denmark, 2004.

  • [15]

    Y. -D. Hsu and Y. -P. Chen, Correlation of the mutual diffusion coefficients of binary liquid mixtures, Fluid Phase Equilib. 152 (1998), 149–168.

    • Crossref
    • Export Citation
  • [16]

    G. D. Moggeridge, A local composition model for prediction of mutual diffusion coefficients in binary liquid mixtures from the tracer diffusion coefficients, Chem. Eng. Sci. 132 (2015), 250–258.

    • Crossref
    • Export Citation
  • [17]

    M. Zhou, X. Yuan, Y. Zhang and K. T. Yu, Local composition based Maxwell-Stefan diffusivity model for binary liquid systems, Ind. Eng. Chem. Res. 52 (2013), 10845–10852.

    • Crossref
    • Export Citation
  • [18]

    J. Li, H. Liu and Y. Hu, A mutual-diffusion-coefficient model based on local composition, Fluid Phase Equilib. 187–188 (2001), 193–208.

  • [19]

    D. Bosse and H. -J. Bart, Prediction of diffusion coefficients in liquid systems, Ind. Eng. Chem. Res. 45 (2006), 1822–1828.

    • Crossref
    • Export Citation
  • [20]

    D. Bosse and H. -J. Bart, Diffusion in associating liquid systems, Chem. Eng. Technol. 26 (2003), 1184–1188.

    • Crossref
    • Export Citation
  • [21]

    J. A. Wesselingh and A. M. Bollen, Multicomponent diffusivities from the free volume theory, Chem. Eng. Res. Des. 75A (1997), 590–602.

  • [22]

    H. Liu, C. M. Silva and E. A. Macedo, Generalised free-volume theory for transport properties and new trends about the relationship between free volume and equations of state, Fluid Phase Equilib. 202 (2002), no. 1, 89–107.

    • Crossref
    • Export Citation
  • [23]

    J. H. Dymond and M. A. Awan, Correlation of high-pressure diffusion and viscosity coefficients for n-alkanes, Int. J. Thermophys. 10 (1989), no. 5, 941–951.

    • Crossref
    • Export Citation
  • [24]

    R. Qian, F. Yiqun, S. Meiren and S. Jun, Predictive equation of tracer liquid diffusion coefficient from viscosity, Chin. J. Chem. Eng. 4 (1996), no. 3, 203–208.

  • [25]

    V. V. Obukhovsky, A. M. Kutsyk, V. V. Nikonova and O. O. Ilchenko, Nonlinear diffusion in multicomponent liquid solutions, Phys. Rev. E 95 (2017), 022133.

    • PubMed
    • Export Citation
  • [26]

    E. A. Mason and A. P. Malinauskas, Gas transport in porous media: The dusty gas model, Elsevier, Amsterdam/New York, 1983.

  • [27]

    J. O. Hirschfelder, C. F. Curtiss and R. B. Bird, The molecular theory of gases and liquids, John Wiley, New York, 1954.

  • [28]

    M. B. Romero and R. M. Velasco, Onsager’s symmetry in the Burnett regime, Physica A 222 (1995), no. 1-4, 161–172.

    • Crossref
    • Export Citation
  • [29]

    V. I. Roldugin, The Chapman-Enskog theory and non-equilibrium thermodynamics, J. Non-Equilib. Thermodyn. 9 (1984), no. 1, 71–80.

  • [30]

    K. G. Denbigh, Thermodynamics of the steady state, Methuen & Co. LDT, London, 1951.

  • [31]

    K. G. Denbigh, The heat of transport in binary regular solutions, Trans. Faraday Soc. 48 (1952), no. 1, 1–8.

    • Crossref
    • Export Citation
  • [32]

    R. Haase, Thermodynamics of Irreversible Processes, Addison Welsey, London, 1969.

  • [33]

    W. M. Rutherford and H. G. Drickamer, Theory of thermal diffusion in liquids and the use of pressure to investigate the theory, J. Chem. Phys. 22 (1954), no. 7, 1157–1165.

    • Crossref
    • Export Citation
  • [34]

    E. L. Dougherty and H. G. Drickamer, A theory of thermal diffusion in liquids, J. Chem. Phys. 23 (1955), no. 2, 295–309.

    • Crossref
    • Export Citation
  • [35]

    E. L. Dougherty and H. G. Drickamer, Thermal diffusion and molecular motion in liquids, J. Chem. Phys. 59 (1955), 443–444.

    • Crossref
    • Export Citation
  • [36]

    L. J. Tichacek, W. S. Kmak and H. G. Drickamer, Thermal diffusion in liquids: the effect of non-ideality ans association, J. Phys. Chem. 60 (1956), 660–665.

    • Crossref
    • Export Citation
  • [37]

    J. Shieh, Thermal diffusion and segmental motion in binary n-alkane systems, J. Phys. Chem. 73 (1969), no. 5, 1508–1513.

    • Crossref
    • Export Citation
  • [38]

    K. Shukla and A. Firozaabadi, A new model of thermal diffusion coefficients in binary hydrocarbon mixtures, Ind. Eng. Chem. Res. 37 (1998), 3331–3342.

    • Crossref
    • Export Citation
  • [39]

    L. J. T. M. Kempers, A thermodynamic theory of the Soret effect in a multicomponent liquid, J. Chem. Phys. 90 (1989), no. 11, 6541–6548.

    • Crossref
    • Export Citation
  • [40]

    L. J. T. M. Kempers, A comprehensive thermodynamic theory of the Soret effect in a multicomponent gas, liquid or solid, J. Chem. Phys. 15 (2001), no. 11, 6330–6341.

  • [41]

    K. Ghorayeb and A. Firoozabadi, Molecular, pressure, and thermal diffusion in non-ideal multicomponent mixtures, AIChE J. 46 (2000), no. 5, 883–891.

    • Crossref
    • Export Citation
  • [42]

    M. Bagnoli, Modeling the thermal diffusion coefficients, Ph.D. Thesis, DTU, Kgs. Lyngby, 2004.

  • [43]

    M. A. Rahman and Z. M. Saghir, Thermodiffusion or Soret effect: Historical review, Int. J. Heat Mass Transf. 73 (2014), 693–705.

    • Crossref
    • Export Citation
  • [44]

    S. Seshasai and Z. M. Saghir, Thermodiffusion Models, in: Springerbriefs in Applied Science and Technology (2013), 11–55. ISBN 978-1-4614-5598-1.

  • [45]

    K. Shukla, Statistical thermodynamics of thermal diffusion factors in binary hydrocarbon mixtures – an application, Mol. Phys. 115 (2017), no. 9-12, 1253–1263.

    • Crossref
    • Export Citation
  • [46]

    F. Montel, H. Hoang and G. Galliero, Linking up pressure, chemical potential and thermal gradients, Eur. Phys. J. E 42 (2019), 65.

    • Crossref
    • Export Citation
  • [47]

    E. Lapeira, M. Gebhardt, T. Triller, A. Mialdun, W. Köhler, V. Shevtsova, et al., Transport properties of the binary mixtures of the three organic liquids toluene, methanol, and cyclohexane, J. Chem. Phys. 146 (2017), 094507.

  • [48]

    M. Braibanti, P. A. Artola, P. Baaske, H. Bataller, J. P. Bazile, M. M. Bou-Ali, et al., European Space Agency experiments on thermodiffusion of fluid mixtures in space, Eur. Phys. J. E 42 (2019), 86.

    • Crossref
    • Export Citation
  • [49]

    S. R. De Groot and P. Mazur, Non-equilibrium thermodynamics, New Holland Publications, 1962.

  • [50]

    G. D. C. Kuiken, Thermodynamics of irreversible processes, John Wiley & Sons, 1994.

  • [51]

    A. A. Shapiro, Evaluation of diffusion coefficients in multicomponent mixtures by means of the fluctuation theory, Physica A 320 (2003), 211–234.

    • Crossref
    • Export Citation
  • [52]

    A. A. Shapiro, Fluctuation theory for transport for transport properties in multicomponent mixtures: Thermodiffusion and heat conductivity, Physica A 322 (2004), 151–175.

  • [53]

    G. Galliero, O. O. Medvedev and A. A. Shapiro, Molecular dynamics simulations of the penetration lengths: application within the fluctuation theory for diffusion coefficients, Physica A 350 (2005), 315–317.

    • Crossref
    • Export Citation
  • [54]

    L. Waldmann, Non-equilibrium thermodynamics of boundary conditions, Z. Naturforsch. (1967), 1269–1280.

  • [55]

    F. Sharipov and D. Kalempa, Gaseous mixture flow through a long tube at arbitrary Knudsen numbers, J. Vac. Sci. Technol. A 20 (2002), 814–822.

    • Crossref
    • Export Citation
  • [56]

    L. Onsager, Reciprocal relations in irreversible processes, Phys. Rev. 37 (1931), 405–426.

    • Crossref
    • Export Citation
  • [57]

    L. Onsager, Reciprocal relations in irreversible processes, Phys. Rev. 38 (1931), 2265–2279.

    • Crossref
    • Export Citation
  • [58]

    R. Wang, X. Xu, K. Xu and T. Qian, Onsager’s cross coupling effects in gas flows confined to microchannels, Phys. Rev. Fluids 1 (2016), 044102.

  • [59]

    G. A. Bird, Molecular gas dynamics, Oxford University Press, 1976.

  • [60]

    O. O. Medvedev and A. A. Shapiro, Verifying reciprocal relations for experimental diffusion coefficients in multicomponent mixtures, Fluid Phase Equilib. 208 (2003), no. 1-2, 291–301.

    • Crossref
    • Export Citation
  • [61]

    A. A. Shapiro, P. K. Davis and J. L. Duda, Diffusion in multicomponent mixtures, in: R. Gani and G. Kontogeorgis (eds.), Computer Aided Property Estimation, Elsevier, Amsterdam (2004), 205–228.

  • [62]

    M. Pavelka, F. Maršik and V. Klika, Consistent theory of mixtures on different levels of description, Int. J. Eng. Sci. 78 (2014), 192–217.

    • Crossref
    • Export Citation
  • [63]

    A. A. Shapiro and E. H. Stenby, Factorization of transport coefficients in macroporous media, Transp. Porous Media 41 (2000), no. 3, 305–323.

    • Crossref
    • Export Citation
  • [64]

    J. A. Wesselingh and R. Krishna, Mass transfer in multicomponent mixtures, VSSD, 2000.

  • [65]

    V. Klika, M. Pavelka and J. B. Benziger, Functional constraints on phenomenological coefficients, Phys. Rev. E 95 (2017), 022125.

    • PubMed
    • Export Citation
  • [66]

    E. Bringuier, Gauge-invariant approach to thermodiffusion in a liquid binary mixture, Physica A 390 (2011), 1861–1875.

    • Crossref
    • Export Citation
  • [67]

    M. L. Michelsen, J. Mollerup, Thermodynamic models: Fundamentals and computational aspects, Tie-Line Publications, 2004.

  • [68]

    W. Feller, An Introduction to Probability Theory and Its Applications, 1, John Wiley & Sons, 1968.

  • [69]

    W. Feller, An Introduction to Probability Theory and Its Applications, 2, John Wiley & Sons, 1968.

Purchase article
Get instant unlimited access to the article.
$42.00
Log in
Already have access? Please log in.


or
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.

Search