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Licensed Unlicensed Requires Authentication Published by De Gruyter December 14, 2017

Fractional Effects on the Light Scattering Properties of a Simple Binary Mixture

  • Rosalío F. Rodríguez EMAIL logo , Jorge Fujioka and Elizabeth Salinas-Rodríguez

Abstract

A fractional generalized hydrodynamic (GH) model of the concentration fluctuations correlation function is analyzed. Our analysis is based on a previously proposed irreversible thermodynamics non-fractional model that describes the first GH deviations in a finite frequency and wavelength regime where the local equilibrium assumption is not valid. Our basic purpose is to generalize this theory to investigate the time fractional effects on the intermediate scattering function (ISF) of a model of a binary mixture. We discuss two different forms of introducing the fractional derivatives into the hydrodynamic time evolution equation of the ISF and examine their consistency in the non-fractional limit. We calculate analytically the fractional intermediate scattering function (FISF) and compare it with the non-fractional one. We find that although the FISFs are in general less than ISF, the FISF may be a significant fraction of the ISF (∼36%) and might be measurable by light or neutron scattering techniques. We show that fractional time derivatives provide a consistent description of this correlation function in the GH regime and reduce to its well-known behavior in the Navier–Stokes limit (NS) where local equilibrium is restored. We also suggest an experimental verification of our results for near equilibrium states. Finally, we summarize the main results of our work and make some further physical remarks.

References

[1] L. Onsager, Reciprocal relations in irreversible processes. I., Phys. Rev. 37 (1931), 405, 2265.10.1103/PhysRev.37.405Search in Google Scholar

[2] S. R. De Groot and P. Mazur, Nonequilibrium Thermodynamic, North Holland, Amsterdam, 1962.Search in Google Scholar

[3] L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed., Pergamon, Oxford, 1987.Search in Google Scholar

[4] H. J. M. Hanley editor, Transport Phenomena in Fluids, M. Dekker, New York, 1971.Search in Google Scholar

[5] E. L. Cussler, Multicomponent Diffusion, Elsevier, New York, 1976.10.1016/B978-0-444-41326-0.50008-3Search in Google Scholar

[6] R. D. Mountain and J. M. Deutch, Light scattering from binary solutions, J. Chem. Phys. 50 (1969), 1103–1108.10.1063/1.1671163Search in Google Scholar

[7] J. P. Boon and S. Yip, Molecular Hydrodynamics, McGraw-Hill, New York, 1980.Search in Google Scholar

[8] R. P. Meilanov and R. A. Magomedov, Thermodynamics in fractional calculus, J. Eng. Phys. Thermophysics87 (2014), 1521–1531.10.1007/s10891-014-1158-2Search in Google Scholar

[9] S. L. Sobolev, Locally nonequilibrium transfer processes, Usp. Fiz. Nauk. 167 (1997), 1095–1106.10.3367/UFNr.0167.199710f.1095Search in Google Scholar

[10] D. Jou, C. Pérez-García, L. S. García-Colín, M. López De Haro and R. F. Rodríguez, Generalized hydrodynamics and extended irreversible thermodynamics, Phys. Rev. A31 (1985), 2502–2508.10.1103/PhysRevA.31.2502Search in Google Scholar

[11] R. F. Rodríguez, L. S. García-Colín, M. López De Haro, D. Jou and C. Pérez-García, The underlying thermodynamic aspects of generalized hydrodynamics, Phys. Lett. A107 (1985), 17–20.10.1016/0375-9601(85)90237-3Search in Google Scholar

[12] R. F. Rodríguez, L. S. García-Colín and M. López De Haro, Extended thermodynamic description of diffusion in an inert binary mixture, J. Chem. Phys. 83 (1985), 4099–4102.10.1063/1.449075Search in Google Scholar

[13] R. F. Rodríguez, M. López De Haro and L. S. García-Colín, Mutual diffusion in a binary mixture, Lect. Notes Phys. 253 (1986), 343–348.10.1007/3-540-16489-8_45Search in Google Scholar

[14] B. J. West and S. Picozzi, Fractional Langevin model of memory in financial time series, Phys. Rev. E65 (2002), Article 037106.10.1103/PhysRevE.65.037106Search in Google Scholar PubMed

[15] P. Pramukkul, A. Svenkeson, P. Grigolini, M. Bologna and B. J. West, Complexity and the fractional calculus, Adv. Math. Phys. 2013 (2013), Article 498789.10.1155/2013/498789Search in Google Scholar

[16] B. J. West, Physiology, Promiscuity and Prophecy at the Millennium: A Tale of Tails, Studies of Nonlinear Phenomena in the Life Sciences, Vol. 7, World Scientific, Singapore, 1999.10.1142/4069Search in Google Scholar

[17] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep. 339 (2000), 1–77.10.1016/S0370-1573(00)00070-3Search in Google Scholar

[18] R. F. Rodríguez, J. Fujioka and E. Salinas-Rodríguez, Fractional fluctuation effects on the light scattered by a viscoelastic suspension, Phys. Rev. E88 (2013), Article 022154.10.1103/PhysRevE.88.022154Search in Google Scholar

[19] R. F. Rodríguez, J. Fujioka and E. Salinas-Rodríguez, Fractional correlation functions in simple viscoelastic liquids, Physica A427 (2015), 326–340.10.1016/j.physa.2015.01.060Search in Google Scholar

[20] G. Stolovitzky and K. R. Sreenivasan, Kolmogorov’s refined similarity hypotheses for turbulence and general stochastic processes, Rev. Mod. Phys. 66 (1994), 229–240.10.1103/RevModPhys.66.229Search in Google Scholar

[21] R. Gorenflo, F. Mainardi and A. Vivoli, Continuous-time random walk and parametric subordination in fractional diffusion, Chaos Solitons & Fractals34 (2007), 87–103.10.1016/j.chaos.2007.01.052Search in Google Scholar

[22] M. G. Hernández, E. Salinas-Rodríguez, S. A. Gómez, J. A. E. Roa-Neri, S. Alfaro and F. J. Valdés-Parada, Helium permeation through a silicalite-1 tubular membrane, Heat Mass Transfer51 (2015), 847–857.10.1007/s00231-014-1460-8Search in Google Scholar

[23] F. J. Valdés-Parada, J. A. Ochoa-Tapia, E. Salinas-Rodríguez, S. Gómez-Torres and M. G. Hernández, Upscaled model for dispersive mass transfer in a tubular porous membrane separator, Rev. Mex. Ingen. Quím. 13 (2014), 237–257.Search in Google Scholar

[24] J. M. Van De Graaf, F. Kapteijn and J. A. Moulijn, Permeation of weakly adsorbing components through a silicalite-1 membrane, Chem. Eng. Sci. 54 (1999), 1081–1092.10.1016/S0009-2509(98)00326-1Search in Google Scholar

[25] B. J. Berne and R. Pecora, Dynamic Light Scattering, J. Wiley, New York, 1976.Search in Google Scholar

[26] B. J. West, Colloquium: Fractional calculus view of complexity: A tutorial, Rev. Mod. Phys. 86 (2014), 1169–1189.10.1103/RevModPhys.86.1169Search in Google Scholar

[27] L. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.Search in Google Scholar

[28] A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.Search in Google Scholar

[29] M. Caputo and C. Cametti, Diffusion with memory in two cases of biological interest, J. Theoret. Biol. 254 (2008), 697–703.10.1016/j.jtbi.2008.06.021Search in Google Scholar PubMed

[30] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Beach, Amsterdam, 1993.Search in Google Scholar

[31] K. S. Miller and B. Ross, An Introduction to Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.Search in Google Scholar

[32] K. B. Oldham and J. Spanier, The Fractional Calculus, Vol. 11, Academic Press, New York, 1974.Search in Google Scholar

[33] M. D. Ortigueira, J. A. T. Machado and J. S. Da Costa, Which differ integration? IEE proceedings-vision, Image Signal Process152 (2005), 846–850.10.1049/ip-vis:20045049Search in Google Scholar

[34] J. Fujioka, A. Espinosa and R. F. Rodríguez, Fractional optical solitons, Phys. Lett. A374 (2010), 1126–1134.10.1016/j.physleta.2009.12.051Search in Google Scholar

[35] P. Pramukkul, A. Svenkeson, P. Grigolini, M. Bologna and B. West, Complexity and the fractional calculus, Adv. Math. Phys. 2013 (2013), Article 498789.10.1155/2013/498789Search in Google Scholar

[36] J. P. Boon, C. Allain and P. Lallemand, Propagating thermal modes in a fluid under thermal constraint, Phys. Rev. Lett. 43 (1979), 199–203.10.1103/PhysRevLett.43.199Search in Google Scholar

[37] P. N. Segrè, R. W. Gammon, J. V. Sengers and B. M. Law, Rayleigh scattering in a liquid far from thermal equilibrium, Phys. Rev. A45 (1992), 714–724.10.1103/PhysRevA.45.714Search in Google Scholar

[38] B. M. Law, P. N. Segrè, R. W. Gammon and J. V. Sengers, Light-scattering measurements of entropy and viscous fluctuations in a liquid far from thermal equilibrium, Phys. Rev. A41 (1990), 816–824.10.1103/PhysRevA.41.816Search in Google Scholar

Received: 2017-7-11
Revised: 2017-10-20
Accepted: 2017-10-27
Published Online: 2017-12-14
Published in Print: 2018-1-26

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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