Abstract
This study deals with the problem of diffusion for polydisperse colloids. The resolution of this complex problem usually requires computationally expensive numerical models. By considering the number of colloidal particles and their mass as independent variables, the equations of state for a dilute polydisperse colloid are derived on a statistical mechanics basis. Irreversible thermodynamics is then applied to obtain a simple two-moment diffusion model. The validity of the model is illustrated by comparing its results with those obtained by a classical size spectrum approach, in a sedimentation equilibrium problem and in an unsteady one-dimensional diffusion problem in Stokes–Einstein regime, and under the hypothesis that the size spectrum distribution is stochastic. In the first problem, the two-moment diffusion problem allows to represent rigorously the vertical size segregation induced by gravity, while in the second one, it allows a convenient description of the diffusion of polydisperse colloids by using two coupled diffusion equations, with an accuracy comparable with that of the classical size spectrum approach. The contribution of our work lies primarily in the application of a non-equilibrium thermodynamics methodology to a challenging issue of colloid modeling, namely, polydispersity, by going from statistical mechanics to the derivation of phenomenological coefficients, with the two-moment approach as a guideline.
References
[1] S. C. James and C. V. Chrysikopoulos, Transport of polydisperse colloid suspensions in a single fracture, Water Resour. Res.35 (1999), no. 3, 707–718.10.1029/1998WR900059Search in Google Scholar
[2] S. C. James and C. V. Chrysikopoulos, Monodisperse and polydisperse colloid transport in water-saturated fractures with various orientations: Gravity effects, Adv. Water Resour.34 (2011), no. 10, 1249–1255.10.1016/j.advwatres.2011.06.001Search in Google Scholar
[3] G. M. Hidy and J. R. Brock, The Dynamics of Aerocolloidal Systems: International Reviews in Aerosol Physics and Chemistry, Elsevier, 1970.Search in Google Scholar
[4] G. K. Batchelor, Diffusion in a dilute polydisperse system of interacting spheres, J. Fluid Mech.131 (1983), 155–175.10.1017/S0022112083001275Search in Google Scholar
[5] M. Fasolo, P. Sollich and A. Speranza, Phase equilibria in polydisperse colloidal systems, React. Funct. Polym.58 (2004), no. 3, 187–196. Frontiers of Polymer Colloids.10.1016/j.reactfunctpolym.2003.12.005Search in Google Scholar
[6] L. Onsager, The effects of shape on the interaction of colloidal particles, Ann. N.Y. Acad. Sci.51 (1949), no. 4, 627–659.10.1201/9780203022658.ch4eSearch in Google Scholar
[7] Y. Monovoukas and A. P. Gast, The experimental phase diagram of charged colloidal suspensions, J. Colloid Interface Sci.128 (1989), no. 2, 533–548.10.1016/0021-9797(89)90368-8Search in Google Scholar
[8] S. S. L. Peppin, Theory of tracer diffusion in concentrated hard-sphere suspensions, J. Fluid Mech.870 (2019), 1105–1126.10.1017/jfm.2019.270Search in Google Scholar
[9] S. Kjelstrup and D. Bedeaux, Non-equilibrium thermodynamics of heterogeneous systems, 16, World Scientific, 2008.10.1142/6672Search in Google Scholar
[10] D. Bedeaux and S. Kjelstrup, Irreversible thermodynamics—a tool to describe phase transitions far from global equilibrium, Chem. Eng. Sci.59 (2004), no. 1, 109–118.10.1016/j.ces.2003.09.028Search in Google Scholar
[11] W. G. Gray and C. T. Miller, Introduction to the thermodynamically constrained averaging theory for porous medium systems, Springer, 2014.10.1007/978-3-319-04010-3Search in Google Scholar
[12] P. Hohenberg and A. Krekhov, An introduction to the Ginzburg–Landau theory of phase transitions and nonequilibrium patterns, Phys. Rep.572 (2015), 1–42.10.1016/j.physrep.2015.01.001Search in Google Scholar
[13] M. Fabrizio, D. Grandi and L. Molari, Water evaporation and condensation by a phase-field model, J. Non-Equilib. Thermodyn.41 (2016), 295–312.10.1515/jnet-2016-0004Search in Google Scholar
[14] D. Bedeaux and J. Rubi, Nonequilibrium thermodynamics of colloids, Phys. A, Stat. Mech. Appl.305 (2002), no. 3, 360–370.10.1016/S0378-4371(01)00554-4Search in Google Scholar
[15] V. I. Roldughin, Nonequilibrium thermodynamics of colloidal systems, Russ. Chem. Rev.81 (2012), no. 10, 875–917.10.1070/RC2012v081n10ABEH004313Search in Google Scholar
[16] B. Eric and B. Alain, Colloid thermophoresis as a non-proportional response. J. Non-Equilib. Thermodyn.32 (2007), no. 3, 221–229.Search in Google Scholar
[17] R. Piazza and A. Parola, Thermophoresis in colloidal suspensions, J. Phys. Condens. Matter20 (2008), no. 15, 153102.10.1088/0953-8984/20/15/153102Search in Google Scholar
[18] D. Frenkel, Why colloidal systems can be described by statistical mechanics: some not very original comments on the Gibbs paradox, Mol. Phys.112 (2014), no. 17, 2325–2329.10.1080/00268976.2014.904051Search in Google Scholar
[19] Y. Liu, Y. Laiguang, Y. Weinong and L. Feng, On the size distribution of cloud droplets, Atmos. Res.35 (1995), no. 2, 201–216.10.1016/0169-8095(94)00019-ASearch in Google Scholar
[20] S. Edwards and R. Oakeshott, Theory of powders, Phys. A, Stat. Mech. Appl.157 (1989), no. 3, 1080–1090.10.1016/0378-4371(89)90034-4Search in Google Scholar
[21] A. D. Watson, G. C. Barker and M. M. Robins, Sedimentation in bidisperse and polydisperse colloids, J. Colloid Interface Sci.286 (2005), no. 1, 176–186.10.1016/j.jcis.2004.12.053Search in Google Scholar
[22] G. Hooyman, Thermodynamics of sedimentation in paucidisperse systems, Physica22 (1956), no. 6, 761–769.10.1016/S0031-8914(56)90028-3Search in Google Scholar
[23] X. Li and M. Li, Droplet size distribution in sprays based on maximization of entropy generation, Entropy5 (2003), no. 5, 417–431.10.3390/e5050417Search in Google Scholar
[24] P. Bartlett, Thermodynamic properties of polydisperse hard spheres, Mol. Phys.97 (1999), no. 5, 685–693.10.1080/002689799163523Search in Google Scholar
[25] P. Sollich, Predicting phase equilibria in polydisperse systems, J. Phys. Condens. Matter14 (2001), no. 3, R79–R117.10.1088/0953-8984/14/3/201Search in Google Scholar
[26] E. Ruckenstein, The origin of thermodynamic stability of microemulsions, Chem. Phys. Lett.57 (1978), no. 4, 517–521.10.1016/0009-2614(78)85311-1Search in Google Scholar
[27] C. Ngô and H. Ngô, Physique statistique-3ème édition, Dunod, 2008.Search in Google Scholar
[28] M. Hassanizadeh and W. G. Gray, General conservation equations for multi-phase systems: 2. mass, momenta, energy, and entropy equations, Adv. Water Resour.2 (1979), 191–203.10.1016/0309-1708(79)90035-6Search in Google Scholar
[29] D. Jou, J. Casas-Vázquez and G. Lebon, Extended irreversible thermodynamics, Springer, 2010.10.1007/978-90-481-3074-0Search in Google Scholar
[30] S. R. De Groot and P. Mazur, Non-equilibrium thermodynamics, Dover, 1984.Search in Google Scholar
[31] G. K. Batchelor, Brownian diffusion of particles with hydrodynamic interaction, J. Fluid Mech.74 (1976), no. 1, 1–29.10.1017/S0022112076001663Search in Google Scholar
[32] Y. Demirel and S. Sandler, Effects of concentration and temperature on the coupled heat and mass transport in liquid mixtures, Int. J. Heat Mass Transf.45 (2002), no. 1, 75–86.10.1016/S0017-9310(01)00121-1Search in Google Scholar
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