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Non-equilibrium pair interactions in colloidal dispersions

Published online by Cambridge University Press:  12 December 2017

Benjamin E. Dolata  and
Roseanna N. Zia Open the ORCID record for Roseanna N. Zia [Opens in a new window]
Benjamin E. Dolata
Affiliation:
Robert Frederick Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14850, USA
Roseanna N. Zia*
Affiliation:
Department of Chemical Engineering, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: [email protected]
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Abstract

We study non-equilibrium pair interactions between microscopic particles moving through a model shear-thinning fluid. Prior efforts to model pair interactions in non-Newtonian fluids have largely focused on constitutive models derived from polymer-chain kinetic theories focusing on conformational degrees of freedom, but neglecting the details of microstructural evolution beyond a single polymer length scale. To elucidate the role of strong structural distortion in mediating pair interactions in Brownian suspensions, we formulate and solve a Smoluchowski equation describing the detailed evolution of the particle configuration between and around a pair of microscopic probes driven at fixed velocity by an external force through a colloidal dispersion. To facilitate analysis, we choose a model system of Brownian hard spheres that do not interact hydrodynamically; while simple, this ‘freely draining’ model permits insight into connections between microstructure and rheology. The flow induces a non-equilibrium particle density gradient that gives rise to both viscous drag and an interactive force between the probes. The drag force acts to slow the centre-of-mass velocity of the pair, while the interactive force arising from osmotic pressure gradients can lead to attraction or repulsion, as well as deterministic reorientation of the probes relative to the external force. The degree to which the microstructure is distorted, and the shape of that distortion, depend on the arrangement of the probes relative to one another and their orientation to the driving force. It also depends on the magnitude of probe velocity relative to the Brownian velocity of the suspension. When only thermal fluctuations set probe velocity, the equilibrium depletion attraction is recovered. For weak forcing, long-ranged interactions mediated via the bath-particle flux give rise to entropic forces on the probes. The linear response is a viscous drag that slows forward motion; only the weakly nonlinear response can produce relative motion–attraction, repulsion or reorientation of the probes. We derive entropic coupling tensors, similar in ethos to pair hydrodynamic tensors, to describe this behaviour. The structural symmetry that permits this analogy is lost when forcing becomes strong, revealing instabilities in system behaviour. Far from equilibrium, the interactive force depends explicitly on the initial probe separation, orientation and strength of forcing; widely spaced probes interact through the distorted microstructure, whereas the behaviour of closely spaced probes is largely set by excluded-volume effects. In this regime, a pair of closely spaced probes sedimenting side-by-side tend to attract and reorient to permit alignment of their line-of-centres with the flow, while widely spaced probes fall without reorienting. Our results show qualitative agreement with experimental observations and provide a potential connection to the observed column instability in shear-thinning fluids.

JFM classification

Complex Fluids: Colloids Micro-/Nano-fluid dynamics: Non-continuum effects Complex Fluids: Suspensions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , pp. 694 - 739
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Copyright
© 2017 Cambridge University Press 

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References

Allen, E. & Uhlherr, P. H. T. 1989 Nonhomogeneous sedimentation in viscoelastic fluids. J. Rheol. 33 (4), 627–638.CrossRefGoogle Scholar
Asakura, S. & Oosawa, F. 1954 On interaction between two bodies immersed in a solution of macromolecules. J. Chem. Phys. 22 (7), 1255–1256.Google Scholar
Batchelor, G. K. & Van Rensburg, R. W. J. 1986 Structure formation in bidisperse sedimentation. J. Fluid Mech. 166, 379–407.Google Scholar
Bergenholtz, J., Brady, J. F. & Vicic, M. 2002 The non-Newtonian rheology of dilute colloidal suspensions. J. Fluid Mech. 456, 239–275.Google Scholar
Bergougnoux, L., Ghicini, S., Guazzelli, E. & Hinch, J. 2003 Spreading fronts and fluctuations in sedimentation. Phys. Fluids 15 (7), 1875–1887.CrossRefGoogle Scholar
Bird, R. B., Armstrong, R. C., Hassager, O. & Curtiss, C. F. 1977 Dynamics of Polymeric Liquids: Volume 2 Kinetic Theory. Wiley.Google Scholar
Bobroff, S. & Phillips, R. J. 1998 Nuclear magnetic resonance imaging investigation of sedimentation of concentrated suspensions in non-Newtonian fluids. J. Rheol. 42 (6), 1419–1436.Google Scholar
Brady, J. F., Khair, A. S. & Swaroop, M. 2006 On the bulk viscosity of suspensions. J. Fluid Mech. 554, 109–123.Google Scholar
Brady, J. F. & Vicic, M. 1995 Normal stresses in colloidal suspensions. J. Rheol. 39 (3), 545–566.Google Scholar
Brenner, H. 1964 The Stokes resistance of an arbitrary particle II: an extension. Chem. Engng Sci. 19 (9), 599–629.CrossRefGoogle Scholar
Brunn, P. 1977 Interaction of spheres in a viscoelastic fluid. Rheol. Acta 19 (16), 461–475.Google Scholar
Carpen, I.2005 Studies of suspension behavior. I. Instabilities of non-Brownian suspensions. II. Microrheology of colloidal suspensions. PhD thesis, California Institute of Technology.Google Scholar
Chu, H. C. W. & Zia, R. N. 2016 Active microrheology of hydrodynamically interacting colloids: Normal stresses and entropic energy density. J. Rheol. 60 (4), 755–781.Google Scholar
Chu, X. L., Nikolov, A. D. & Wasan, D. T. 1996 Effects of interparticle interactions on stability, aggregation and sedimentation in colloidal suspensions. Chem. Engng Commun. 148 (1), 123–142.Google Scholar
Cox, B. J., Thamwattana, N. & Hill, J. M. 2006 Maximising the electrorheological effect for bidisperse nanofluids from the electrostatic force between two particles. Rheol. Acta 45 (6), 909–917.CrossRefGoogle Scholar
Daugan, S., Talini, L., Herzhaft, B., Peysson, Y. & Allain, C. 2004 Sedimentation of suspensions in shear-thinning fluids. Oil Gas Sci. Technol. 59 (1), 71–80.Google Scholar
Diamant, H. 2007 Long-range hydrodynamic response of particulate liquids and liquid-laden solids. Israel J. Chem. 47 (2), 225–231.Google Scholar
Durlofsky, L. & Brady, J. F. 1987 Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30 (11), 3329–3341.CrossRefGoogle Scholar
Dzubiella, J., Löwen, H. & Likos, C. 2003 Depletion forces in nonequilibrium. Phys. Rev. Lett. 91 (24), 248301.Google Scholar
Feng, J., Huang, P. Y. & Joseph, D. D. 1996 Dynamic simulation of sedimentation of solid particles in an Oldroyd-B fluid. J. Non-Newtonian Fluid Mech. 63 (1), 63–88.Google Scholar
Gheissary, G. & Van den Brule, B. H. A. A. 1996 Unexpected phenomena observed in particle settling in non-Newtonian media. J. Non-Newtonian Fluid Mech. 67, 1–18.CrossRefGoogle Scholar
Goyal, N. & Derksen, J. J. 2012 Direct simulations of spherical particles sedimenting in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 183, 1–13.Google Scholar
Gumulya, M. M., Horsley, R. R., Pareek, V. & Lichti, D. D. 2011a The effects of fluid viscoelasticity on the settling behaviour of horizontally aligned spheres. Chem. Engng Sci. 66 (23), 5822–5831.CrossRefGoogle Scholar
Gumulya, M. M., Horsley, R. R., Wilson, K. C. & Pareek, V. 2011b A new fluid model for particles settling in a viscoplastic fluid. Chem. Engng Sci. 66 (4), 729–739.Google Scholar
Hobson, E. W. & Tuinier, R. 1931 The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press.Google Scholar
Hoh, N. J.2013 Effects of particle size ratio on single particle motion in colloidal dispersions. PhD thesis, California Institute of Technology.Google Scholar
Hoh, N. J. & Zia, R. N. 2016 Force-induced diffusion in suspensions of hydrodynamically interacting colloids. J. Fluid Mech. 795, 739–783.CrossRefGoogle Scholar
Jeffrey, D. J. 1973 Conduction through a random suspension of spheres. Proc. R. Soc. Lond. A 335, 355–367.Google Scholar
Jeffrey, D. J. & Onishi, Y. 1984 Calculation of the resistance and mobility functions for two unequal rigid spheres in low-Reynolds-number flow. J. Fluid Mech. 139, 261–290.Google Scholar
Joseph, D. D., Liu, Y. J., Poletto, M. & Feng, J. 1994 Aggregation and dispersion of spheres falling in viscoelastic liquids. J. Non-Newtonian Fluid Mech. 54, 45–86.Google Scholar
Keh, H. J. & Chen, S. H. 1995 Particle interactions in thermophoresis. Chem. Engng Sci. 50 (21), 3395–3407.Google Scholar
Khair, A. S. & Brady, J. F. 2005 “Microviscoelasticity” of colloidal dispersions. J. Rheol. 49 (6), 1449–1481.Google Scholar
Khair, A. S. & Brady, J. F. 2006 Single particle motion in colloidal dispersions: a simple model for active and nonlinear microrheology. J. Fluid Mech. 557, 73–117.Google Scholar
Khair, A. S. & Brady, J. F. 2007 On the motion of two particles translating with equal velocities through a colloidal dispersion. Proc. R. Soc. Lond. A 463 (2077), 223–240.Google Scholar
Khair, A. S. & Squires, T. M. 2010 Active microrheology: A proposed technique to measure normal stress coefficients of complex fluids. Phys. Rev. Lett. 105, 156001.Google Scholar
Kim, S. & Russel, W. B. 1985 The hydrodynamic interactions between two spheres in a Brinkman medium. Phys. Fluids 154, 253–268.Google Scholar
Krüger, M. & Rauscher, M. 2007 Colloid–colloid and colloid–wall interactions in driven suspensions. J. Chem. Phys. 127 (3), 034905.Google Scholar
Krüger, M. & Rauscher, M. 2009 Diffusion of a sphere in a dilute solution of polymer coils. J. Chem. Phys. 131 (9), 094902.Google Scholar
Long, D. & Ajdari, A. 2001 A note on the screening of hydrodynamic interactions, in electrophoresis, and in porous media. Eur. Phys. J. E 4 (1), 29–32.Google Scholar
Mejía-Monasterio, C. & Oshanin, G. 2011 Bias- and bath-mediated pairing of particles driven through a quiescent medium. Soft Matt. 7 (3), 993–1000.Google Scholar
Meyer, A., Marshall, A., Bush, B. G. & Furst, E. M. 2006 Laser tweezer microrheology of a colloidal suspension. J. Rheol. 50 (1), 77–92.Google Scholar
Mohanty, R. P. & Zia, R. N. 2017 The impact of hydrodynamics on viscosity evolution in colloidal dispersions: transient, nonlinear microrheology. AIChE J. (under review).Google Scholar
Mora, S., Talini, L. & Allain, C. 2005 Structuring sedimentation in a shear-thinning fluid. Phys. Rev. Lett. 95 (8), 088301.Google Scholar
Padmanabhan, P. & Zia, R. N.2017 Gravitational collapse of reversible colloidal gels: non-equilibrium phase transition driven by osmotic pressure. (Unpublished).Google Scholar
Phillips, R. J. 2010 Structural instability in the sedimentation of particulate suspensions through viscoelastic fluids. J. Non-Newtonian Fluid Mech. 165 (9), 479–488.Google Scholar
Phillips, R. J. & Talini, L. 2007 Chaining of weakly interacting particles suspended in viscoelastic fluids. J. Non-Newtonian Fluid Mech. 147 (3), 175–188.Google Scholar
Poon, W. C. K., Starrs, L., Meeker, S. P., Moussaïd, A., Evans, R. M. L., Pusey, P. N. & Robins, M. M. 1999 Delayed sedimentation of transient gels in colloid–polymer mixtures: dark-field observation, rheology and dynamic light scattering studies. Faraday Discuss. 112, 143–154.Google Scholar
Riddle, M. J., Narvaez, C. & Bird, R. B. 1977 Interactions between two spheres falling along their line of centers in a viscoelastic fluid. J. Non-Newtonian Fluid Mech. 2 (1), 23–35.Google Scholar
Russel, W. B. 1984 The Huggins coefficient as a means for characterizing suspended particles. J. Chem. Soc. Faraday. Trans. 2 80 (1), 31–41.Google Scholar
Saintillan, D. 2008 Nonlinear interactions in electrophoresis of ideally polarizable particles. Phys. Fluids 20 (6), 067104.Google Scholar
Sonn-Segev, A., Bernheim-Groswasser, A., Diamant, H. & Roichman, Y. 2014 Viscoelastic response of a complex fluid at intermediate distances. Phys. Rev. Lett. 112 (8), 088301.Google Scholar
Squires, T. M. & Brady, J. F. 2005 A simple paradigm for active and nonlinear microrheology. Phys. Fluids 17 (7), 073101.CrossRefGoogle Scholar
Sriram, I. & Furst, E. M. 2012 Out-of-equilibrium forces between colloids. Soft Matt. 8 (12), 3335–3341.Google Scholar
Sriram, I. & Furst, E. M. 2015 Two spheres translating in tandem through a colloidal suspension. Phys. Rev. E 91, 042303.Google ScholarPubMed
Sriram, I., Meyer, A. & Furst, E. M. 2010 Active microrheology of a colloidal suspension in the direct collision limit. Phys. Fluids 22 (6), 062003.Google Scholar
Starrs, L., Poon, W. C. K., Hibberd, D. J. & Robins, M. M. 2002 Collapse of transient gels in colloid–polymer mixtures. J. Phys.: Condens. Matter 14, 2485–2505.Google Scholar
Swan, J. W. & Zia, R. N. 2013 Active microrheology: Fixed-velocity versus fixed-force. Phys. Fluids 25 (8), 083303.Google Scholar
Swan, J. W., Zia, R. N. & Brady, J. F. 2014 Large amplitude oscillatory microrheology. J. Rheol. 58 (1), 1–41.Google Scholar
Swaroop, M. & Brady, J. F. 2007 The bulk viscosity of suspensions. J. Rheol. 51 (3), 409–428.Google Scholar
Tee, S., Mucha, P. J., Cipelletti, L., Manley, S., Brenner, M. P., Segre, P. N. & Weitz, D. A. 2002 Nonuniversal velocity fluctuations of sedimenting particles. Phys. Rev. Lett. 89 (5), 054501.Google Scholar
Vishnampet, R. & Saintillan, D. 2012 Concentration instability of sedimenting spheres in a second-order fluid. Phys. Fluids 24 (7), 073302.Google Scholar
Weiland, R. H. & McPherson, R. R. 1979 Accelerated settling by addition of buoyant particles. Ind. Engng Chem. Fundam. 18 (1), 45–49.Google Scholar
Xu, W., Nikolov, A. D. & Wasan, D. T. 1997 Role of depletion and surface-induced structural forces in bidisperse suspensions. AIChE J. 43 (12), 3215–3222.Google Scholar
Yu, Z., Wachs, A. & Peysson, Y. 2006 Numerical simulation of particle sedimentation in shear-thinning fluids with a fictitious domain method. J. Non-Newtonian Fluid Mech. 136 (2), 126–139.Google Scholar
Zia, R. N.2011 Individual particle motion in colloids: microviscosity, microdiffusivity, and normal stresses. PhD thesis, California Institute of Technology.Google Scholar
Zia, R. N. & Brady, J. F. 2010 Single-particle motion in colloids: force-induced diffusion. J. Fluid Mech. 658, 188–210.Google Scholar
Zia, R. N. & Brady, J. F. 2012 Microviscosity, microdiffusivity, and normal stresses in colloidal dispersions. J. Rheol. 56 (5), 1175–1208.Google Scholar
Zia, R. N. & Brady, J. F. 2013 Stress development, relaxation, and memory in colloidal dispersions: Transient nonlinear microrheology. J. Rheol. 57 (2), 457–492.Google Scholar
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