Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-23T17:08:36.943Z Has data issue: false hasContentIssue false
  • English
  • Français

Theory of tracer diffusion in concentrated hard-sphere suspensions

Published online by Cambridge University Press:  15 May 2019

S. S. L. Peppin Open the ORCID record for S. S. L. Peppin [Opens in a new window]
S. S. L. Peppin*
Affiliation:
Mississauga, L5G 2K5 Canada
*
†Email address for correspondence: speppin@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

A phenomenological theory of diffusion and cross-diffusion of tracer particles in concentrated hard-sphere suspensions is developed. Expressions for the diffusion coefficients as functions of the host particle volume fraction are obtained up to the close-packing limit. In concentrated systems the tracer diffusivity decreases because of the reduced pore space available for diffusion. The tracer diffusivity can be modelled by a Stokes–Einstein equation with an effective viscosity that depends on the pore size. Tracer diffusion and segregation during sedimentation cease at a critical trapping volume fraction corresponding to a tracer glass transition. The tracer cross-diffusion coefficient, however, increases near the glass transition and diverges in the close-packed limit.

JFM classification

Complex Fluids: Colloids Low-Reynolds-number flows: Porous media Complex Fluids: Suspensions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 870 , 10 July 2019 , pp. 1105 - 1126
Copyright
© 2019 Cambridge University Press 

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

Anderson, J. L. 1989 Colloid transport by interfacial forces. Annu. Rev. Fluid Mech. 21, 61–99.Google Scholar
Anderson, J. L. & Quinn, J. A. 1974 Restricted transport in small pores: a model for steric exclusion and hindered particle motion. Biophys. J. 14, 130–150.Google Scholar
Annunziata, O. 2008 On the role of solute solvation and excluded-volume interactions in coupled diffusion. J. Phys. Chem. B 112 (38), 11968–11975.Google Scholar
Annunziata, O., Buzatu, D. & Albright, J. G. 2012 Protein diffusiophoresis and salt osmotic diffusion in aqueous solutions. J. Phys. Chem. B 116 (42), 12694–12705.Google Scholar
Annunziata, O., Rard, J. A., Albright, J. G., Paduano, L. & Miller, D. G. 2000 Mutual diffusion coefficients and densities at 298.15 K of aqueous mixtures of NaCl and Na2SO4 for six different solute fractions at a total molarity of 1. 500 mol dm-3 and of aqueous Na2SO4 . J. Chem. Engng Data 45 (5), 936–945.Google Scholar
Annunziata, O., Vergara, A., Paduano, L., Sartorio, R., Miller, D. G. & Albright, J. G. 2009 Quaternary diffusion coefficients in a protein-polymer-salt-water system determined by Rayleigh interferometry. J. Phys. Chem. B 113, 13446–13453.Google Scholar
Ariza, M. J. & Puertas, A. M. 2009 Colloidal permeability of liquid membranes consisting of hard particles by nonequilibrium simulations. J. Chem. Phys. 131 (16), 164903.Google Scholar
Auzerais, F. M., Jackson, R. & Russel, W. B. 1988 The resolution of shocks and the effects of compressible sediments in transient settling. J. Fluid Mech. 195, 437–462.Google Scholar
Batchelor, G. K. 1976 Brownian diffusion of particles with hydrodynamic interaction. J. Fluid Mech. 74, 1–29.Google Scholar
Batchelor, G. K. 1983 Diffusion in a dilute polydisperse system of interacting spheres. J. Fluid Mech. 131, 155–175; and Corrigendum 137, 1983, 467–469.Google Scholar
Batchelor, G. K. 1986 Note on the Onsager symmetry of the kinetic coefficients for sedimentation and diffusion in a dilute bidispersion. J. Fluid Mech. 171, 509–517.Google Scholar
Bird, R. B., Stuart, W. E. & Lightfoot, E. N. 2002 Transport Phenomena, 2nd edn.. Wiley.Google Scholar
Brouwers, H. J. H. 2010 Viscosity of a concentrated suspension of rigid monosized particles. Phys. Rev. E 81 (5), 051402.Google Scholar
Bruna, M. & Chapman, S. J. 2012 Diffusion of multiple species with excluded-volume effects. J. Chem. Phys. 137, 204116.Google Scholar
Bruna, M. & Chapman, S. J. 2015 Diffusion in spatially varying porous media. SIAM J. Appl. Maths 75, 1648–1674.Google Scholar
Buzzaccaro, S., Rusconi, R. & Piazza, R. 2007 ‘Sticky’ hard spheres: equation of state, phase diagram, and metastable gels. Phys. Rev. Lett. 99, 098301.Google Scholar
Buzzaccaro, S., Tripodi, A., Rusconi, R., Vigolo, D. & Piazza, R. 2008 Kinetics of sedimentation in colloidal suspensions. J. Phys.: Condens. Matter 20 (49), 494219.Google Scholar
Carman, P. C. 1939 Permeability of saturated sands, soils and clays. J. Agricultural Sci. 29, 262–273.Google Scholar
Carnahan, N. F. & Starling, K. E. 1969 Equation of state for nonattracting rigid spheres. J. Chem. Phys. 51 (2), 635–636.Google Scholar
De Kruif, C. G., Jansen, J. W. & Vrij, A. 1987 A sterically stabilized silica colloid as a model supramolecular fluid. In Physics of Complex and Supramolecular Fluids (ed. Safran, S. A. & Clark, N. A.), pp. 315–346. Wiley Interscience.Google Scholar
de Groot, S. R. & Mazur, P. 1962 Non-Equilibrium Thermodynamics. North-Holland Publishing Co.Google Scholar
Delgado, J. M. P. Q. 2006 A simple experimental technique to measure tortuosity in packed beds. Canadian J. Chem. Engng 84 (6), 651–655.Google Scholar
Edward, J. T. 1970 Molecular volumes and the Stokes–Einstein equation. J. Chem. Educ. 47 (4), 261–270.Google Scholar
Einstein, A. 1956 Investigations on the Theory of Brownian Movement. Dover.Google Scholar
Fan, T.-H., Dhont, J. K. G. & Tuinier, R. 2007 Motion of a sphere through a polymer solution. Phys. Rev. E 75, 011803.Google Scholar
Faroughi, S. A. & Huber, C. 2015 A generalized equation for rheology of emulsions and suspensions of deformable particles subjected to simple shear at low Reynolds number. Rheol. Acta 54, 85–108.Google Scholar
Fiore, A. M., Wang, G. & Swan, J. W. 2018 From hindered to promoted settling in dispersions of attractive colloids: simulation, modeling, and application to macromolecular characterization. Phys. Rev. Fluids 3, 063302.Google Scholar
Ghanbarian, B., Hunt, A. G., Ewing, R. P. & Sahimi, M. 2013 Tortuosity in porous media: a critical review. Soil Sci. Soc. Am. J. 77, 1461–1477.Google Scholar
Gilleland, W. T., Torquato, S. & Russel, W. B. 2011 New bounds on the sedimentation velocity for hard, charged and adhesive hard-sphere colloids. J. Fluid Mech. 667, 403–425.Google Scholar
Gimel, J.-C. & Taco, N. 2011 Self-diffusion of non-interacting hard spheres in particle gels. J. Phys.: Condens. Matter 23, 234115.Google Scholar
Guan, J., Wang, B. & Granick, S. 2014 Even hard-sphere colloidal suspensions display Fickian yet non-Gaussian diffusion. ACS Nano 8 (4), 3331–3336.Google Scholar
Hannam, S. D. W., Daivis, P. J. & Bryant, G. 2017 Dramatic slowing of compositional relaxations in the approach to the glass transition for a bimodal colloidal suspension. Phys. Rev. E 96, 022609.Google Scholar
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83 (4), 695–720.Google Scholar
Hodgdon, J. A. & Stillinger, F. H. 1993 Stokes–Einstein violation in glass-forming liquids. Phys. Rev. E 48, 207–213.Google Scholar
Holyst, R., Bielejewska, A., Szymaski, J., Wilk, A., Patkowski, A., Gapiski, J., ywociski, A., Kalwarczyk, T., Kalwarczyk, E., Tabaka, M. et al. 2009 Scaling form of viscosity at all length-scales in poly(ethylene glycol) solutions studied by fluorescence correlation spectroscopy and capillary electrophoresis. Phys. Chem. Chem. Phys. 11, 9025–9032.Google Scholar
Hunter, G. L. & Weeks, E. R. 2012 The physics of the colloidal glass transition. Rev. Mod. Phys. 75, 066501.Google Scholar
Kalwarczyk, T., Sozanski, K., Jakiela, S., Wisniewska, A., Kalwarczyk, E., Kryszczuk, K., Hou, S. & Holyst, R. 2014 Length-scale dependent transport properties of colloidal and protein solutions for prediction of crystal nucleation rates. Nanoscale 6, 10340–10346.Google Scholar
Kaye, B. H. & Davies, R. 1972 Experimental investigation into the settling behaviour of suspensions. Powder Technol. 5 (2), 61–68.Google Scholar
Kim, I. C. & Torquato, S. 1992 Diffusion of finite sized Brownian particles in porous media. J. Chem. Phys. 96 (2), 1498–1503.Google Scholar
Koch, D. L., Hill, R. J. & Sangani, A. S. 1998 Brinkman screening and the covariance of the fluid velocity in fixed beds. Phys. Fluids 10 (12), 3035–3037.Google Scholar
Kops-Werkhoven, M. M. & Fijnaut, H. M. 1981 Dynamic light scattering and sedimentation experiments on silica dispersions at finite concentrations. J. Chem. Phys. 74, 1618–1625.Google Scholar
Kumar, S. K., Szamel, G. & Douglas, J. F. 2006 Nature of the breakdown in the Stokes–Einstein relationship in a hard sphere fluid. J. Chem Phys. 124 (21), 214501.Google Scholar
Ladd, A. J. C. 1990 Hydrodynamic transport coefficients of random dispersions of hard spheres. J. Chem. Phys. 93, 3484–3494.Google Scholar
Lekkerkerker, H. N. W. & Stroobants, A. 1993 On the spinodal instability of highly asymmetric hard sphere suspensions. Physica A 195, 387–397.Google Scholar
Lockett, M. J. & Al-Habbooby, H. M. 1974 Relative particle velocities in two-species settling. Powder Technol. 10, 67–71.Google Scholar
Mackowiak, S. A., Noble, J. M. & Kaufman, L. J. 2011 Manifestations of probe presence on probe dynamics in supercooled liquids. J. Chem. Phys. 135 (21), 214503.Google Scholar
Malusis, M. A., Shackelford, C. D. & Maneval, J. E. 2012 Critical review of coupled flux formulations for clay membranes based on nonequilibrium thermodynamics. J. Contam. Hydrol. 138, 40–59.Google Scholar
van Megen, W., Ottewill, R. H., Owens, S. M. & Pusey, P. N. 1985 Measurement of the wave-vector dependent diffusion coefficient in concentrated particle dispersions. J. Chem. Phys. 82, 508–515.Google Scholar
Metzler, R., Jeon, J.-H., Cherstvy, A. G. & Barkai, E. 2014 Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16, 24128–24164.Google Scholar
Minton, A. P. 2007 The effective hard particle model provides a simple, robust, and broadly applicable description of nonideal behavior in concentrated solutions of bovine serum albumin and other nonassociating proteins. J. Pharma. Sci. 96, 3466–3469.Google Scholar
Neale, G. H. & Nader, W. K. 1973 Prediction of transport processes within porous media: diffusive flow processes within an homogeneous swarm of spherical particles. AIChE J. 19 (1), 112–119.Google Scholar
Oatley-Radcliffe, D. L., Williams, S. R., Ainscough, T. J., Lee, C., Johnson, D. J. & Williams, P. M. 2015 Experimental determination of the hydrodynamic forces within nanofiltration membranes and evaluation of the current theoretical descriptions. Separation Purification Technol. 149, 339–348.Google Scholar
O’Hern, C. S., Silbert, L. E., Liu, A. J. & Nagel, S. R. 2003 Jamming at zero temperature and zero applied stress: the epitome of disorder. Phys. Rev. E 68, 011306.Google Scholar
Peppin, S. S. L., Elliott, J. A. W. & Worster, M. G. 2005 Pressure and relative motion in colloidal suspensions. Phys. Fluids 17 (5), 053301.Google Scholar
Peppin, S. S. L., Elliott, J. A. W. & Worster, M. G. 2006 Solidification of colloidal suspensions. J. Fluid Mech. 554, 147–166.Google Scholar
Philipse, A. P. & Pathmamanoharan, C. 1993 Liquid permeation (and sedimentation) of dense colloidal hard-sphere packings. J. Colloid Interface Sci. 159, 96–107.Google Scholar
Pryamitsyn, V. & Ganesan, V. 2005 A coarse-grained explicit solvent simulation of rheology of colloidal suspensions. J. Chem. Phys. 122 (10), 104906.Google Scholar
Rallison, J. M. 1988 Brownian diffusion in concentrated suspensions of interacting particles. J. Fluid Mech. 186, 471–500.Google Scholar
Rintoul, M. D. & Torquato, S. 1996 Computer simulations of dense hard-sphere systems. J. Chem. Phys. 105, 9258–9265.Google Scholar
Russel, W. B., Seville, D. A. & Schowalter, W. R. 1989 Colloidal Dispersions. Cambridge University Press.Google Scholar
Russel, W. B., Wagner, N. J. & Mewis, J. 2013 Divergence in the low shear viscosity for Brownian hard-sphere dispersions: At random close packing or the glass transition? J. Rheol. 57 (6), 1555–1567.Google Scholar
Santos, A. & Rorhmann, R. D. 2013 Chemical-potential route for multicomponent fluids. Phys. Rev. E 87, 052138.Google Scholar
Schneider, C. P. & Trout, B. L. 2009 Investigation of cosolute-protein preferential interaction coefficients. J. Phys. Chem. B 113 (7), 2050–2058.Google Scholar
Segrè, P. N., Meeker, S. P., Pusey, P. N. & Poon, W. C. K. 1995 Viscosity and structural relaxation in suspensions of hard-sphere colloids. Phys. Rev. Lett. 75, 958–961.Google Scholar
Sentjabrskaja, T., Zaccarelli, E., Michele, C., De Sciortino, F., Tartaglia, P., Voigtmann, T., Egelhaaf, S. U. & Laurati, M. 2016 Anomalous dynamics of intruders in a crowded environment of mobile obstacles. Nature Commun. 7 (1–7), 11133.Google Scholar
Smit, J. A. M., Eijsermans, J. C. & Staverman, A. J. 1975 Friction and partition in membranes. J. Phys. Chem. 79 (20), 2168–2175.Google Scholar
Snabre, P., Pouligny, B., Metayer, C. & Nadal, F. 2009 Size segregation and particle velocity fluctuations in settling concentrated suspensions. Rheol. Acta 48 (8), 855–870.Google Scholar
Spiegler, K. S. & Kedem, O. 1966 Thermodynamics of hyperfiltration (reverse osmosis): criteria for efficient membranes. Desalination 1, 311–326.Google Scholar
Sung, B. J. & Yethiraj, A. 2008 The effect of matrix structure on the diffusion of fluids in porous media. J. Chem. Phys. 128 (5), 054702.Google Scholar
Sutherland, W. 1905 Lxxv. A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin. The London, Edinburgh, and Dublin Philosophical Magazine and J. Sci. 9 (54), 781–785.Google Scholar
Thies-Weesie, D. M. E. & Philipse, A. P. 1994 Liquid permeation of bidisperse colloidal hard-sphere packings and the Kozeny–Carman scaling relation. J. Colloid Interface Sci. 162, 470–480.Google Scholar
Vergara, A., Annunziata, O., Paduano, L., Miller, D. G., Albright, J. G. & Sartorio, R. 2004 Multicomponent diffusion in crowded solutions. 2. Mutual diffusion in the ternary system tetra(ethylene glycol)-NaCl-water. J. Phys. Chem. B 108 (8), 2764–2772.Google Scholar
Vergara, A., Paduano, L., Vitagliano, V. & Sartorio, R. 2001 Multicomponent diffusion in crowded solutions. 1. Mutual diffusion in the ternary system poly(ethylene glycol) 400-NaCl-water. Macromolecules 34, 991–1000.Google Scholar
Voigtmann, Th. 2011 Multiple glasses in asymmetric binary hard spheres. Europhys. Lett. 96 (3), 36006.Google Scholar
Wang, B., Kuo, J., Bae, S. C. & Granick, S. 2012 When Brownian diffusion is not Gaussian. Nat. Mater. 11, 481–485.Google Scholar
Wang, M. & Brady, J. F. 2015 Short-time transport properties of bidisperse suspensions and porous media: A Stokesian dynamics study. J. Chem. Phys. 142 (9), 094901.Google Scholar
Woodcock, L. V. 1981 Glass transition in the hard-sphere model and Kauzmann’s paradox. Ann. N.Y. Acad. Sci. 371 (1), 274–298.Google Scholar
Wu, G.-W. & Sadus, R. J. 2005 Hard sphere compressibility factors for equation of state development. AIChE J. 51, 309–313.Google Scholar
Xia, X. & Wolynes, P. G. 2001 Diffusion and the mesoscopic hydrodynamics of supercooled liquids. J. Phys. Chem. B 105 (28), 6570–6573.Google Scholar
Xue, J.-Z., Herbolzheimer, E., Rutgers, M. A., Russel, W. B. & Chaikin, P. M. 1992 Diffusion, dispersion, and settling of hard spheres. Phys. Rev. Lett. 69, 1715–1718.Google Scholar
Zaccarelli, E., Liddle, S. M. & Poon, W. C. K. 2015 On polydispersity and the hard sphere glass transition. Soft Matt. 11, 324–330.Google Scholar
Zhang, H. & Nagele, G. 2002 Tracer-diffusion in binary colloidal hard-sphere suspensions. J. Chem. Phys. 117 (12), 5908–5920.Google Scholar