Hostname: page-component-848d4c4894-mwx4w Total loading time: 0 Render date: 2024-07-02T18:54:20.746Z Has data issue: false hasContentIssue false
  • English
  • Français

Rheological evaluation of colloidal dispersions using the smoothed profile method: formulation and applications

Published online by Cambridge University Press:  03 March 2016

John J. Molina Open the ORCID record for John J. Molina [Opens in a new window] ,
Kotaro Otomura ,
Hayato Shiba ,
Hideki Kobayashi ,
Masaki Sano  and
Ryoichi Yamamoto
John J. Molina*
Affiliation:
Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto 606-8103, Japan Department of Chemical Engineering, Kyoto University, Kyoto 615-8510, Japan
Kotaro Otomura
Affiliation:
Department of Physics, University of Tokyo, Tokyo 133-0033, Japan
Hayato Shiba
Affiliation:
Institute for Solid State Physics, University of Tokyo, Chiba 277-8581, Japan
Hideki Kobayashi
Affiliation:
Theoretical Soft Matter and Biophysics, Institute for Advanced Simulation, Forschungszentrum Jülich, 52425 Jülich, Germany
Masaki Sano
Affiliation:
Department of Physics, University of Tokyo, Tokyo 133-0033, Japan
Ryoichi Yamamoto
Affiliation:
Department of Chemical Engineering, Kyoto University, Kyoto 615-8510, Japan
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The smoothed profile method is extended to study the rheological behaviour of colloidal dispersions under shear flow by using the Lees–Edwards boundary conditions. We start with a reformulation of the smoothed profile method, a direct numerical simulation method for colloidal dispersions, so that it can be used with the Lees–Edwards boundary condition, under steady or oscillatory-shear flow. By this reformulation, all the resultant physical quantities, including local and total shear stresses, become available through direct calculation. Three simple rheological simulations are then performed for (1) a spherical particle, (2) a rigid bead chain and (3) a collision of two spherical particles under shear flow. Quantitative validity of these simulations is examined by comparing the viscosity with that obtained from theory and Stokesian dynamics calculations. Finally, we consider the shear-thinning behaviour of concentrated colloidal dispersions.

JFM classification

Complex Fluids: Colloids Complex Fluids: Complex Fluids Mathematical Foundations: Computational methods
Type
Papers
Information
Journal of Fluid Mechanics , Volume 792 , 10 April 2016 , pp. 590 - 619
Check for updates
Copyright
© 2016 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.)

Footnotes

‡

Present address: Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan.

References

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics, 1st edn. Cambridge University Press.Google Scholar
Batchelor, G. K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41 (3), 545–570.Google Scholar
Batchelor, G. K. 1977 The effect of Brownian motion on the bulk stress in a suspension of spherical particles. J. Fluid Mech. 83 (1), 97–117.Google Scholar
Batchelor, G. K. & Green, J. T. 1972 The hydrodynamic interaction of two small freely-moving spheres in a linear flow field. J. Fluid Mech. 56 (2), 375–400.Google Scholar
Bender, J. W. & Wagner, N. J. 1995 Optical measurement of the contributions of colloidal forces to the rheology of concentrated suspensions. J. Colloid Interface Sci. 172 (1), 171–184.CrossRefGoogle Scholar
Boek, E. S., Coveney, P. V., Lekkerkerker, H. N. W. & van der Schoot, P. 1997 Simulating the rheology of dense colloidal suspensions using dissipative particle dynamics. Phys. Rev. E 55 (3), 3124–3133.Google Scholar
Brady, J. F. 1993 The rheological behavior of concentrated colloidal dispersions. J. Chem. Phys. 99 (1), 567–581.CrossRefGoogle Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20 (1), 111–157.Google Scholar
Bretherton, F. P. 1962 The motion of rigid particles in a shear flow at low Reynolds number. J. Fluid Mech. 14 (2), 284–304.Google Scholar
Brucker, K. A., Isaza, J. C., Vaithianathan, T. & Collins, L. R. 2007 Efficient algorithm for simulating homogeneous turbulent shear flow without remeshing. J. Comput. Phys. 225 (1), 20–32.CrossRefGoogle Scholar
Bybee, M. D.2009 Hydrodynamic simulations of colloidal gels: microstructure, dynamics, and rheology. PhD thesis, University of Illinois at Urbana-Champaign.Google Scholar
Carroll, S. M. 2004 Spacetime and Geometry: An Introduction to General Relativity, 1st edn. Addison-Wesley.Google Scholar
Cates, M. E., Stratford, K., Adhikari, R., Stansell, P., Desplat, J.-C., Pagonabarraga, I. & Wagner, A. J. 2004 Simulating colloid hydrodynamics with lattice Boltzmann methods. J. Phys.: Condens. Matter 16 (38), S3903–S3915.Google Scholar
Chwang, A. T. & Wu, T. Y. 1974 Hydromechanics of low-Reynolds-number flow. Part 1. Rotation of axisymmetric prolate bodies. J. Fluid Mech. 63 (3), 607–622.CrossRefGoogle Scholar
Corte, L., Gerbode, S. J., Man, W. & Pine, D. J. 2009 Self-organized criticality in sheared suspensions. Phys. Rev. Lett. 103 (24), 248301.CrossRefGoogle ScholarPubMed
Cox, R. G. 2006 The motion of long slender bodies in a viscous fluid. Part 2. Shear flow. J. Fluid Mech. 45 (4), 625–657.Google Scholar
Durlofsky, L., Brady, J. F. & Bossis, G. 1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180 (1), 21–49.CrossRefGoogle Scholar
Einstein, A. 1911 Berichtigung zu meiner Arbeit: ‘Eine neue Bestimmung der Moleküldimensionen’. Ann. Phys. 339 (3), 591–592.Google Scholar
Evans, D. J. & Morriss, G. P. 2008 Statistical Mechanics of Nonequilibrium Liquids, 2nd edn. Cambridge University Press.Google Scholar
Foss, D. R. & Brady, J. F. 2000 Structure, diffusion and rheology of Brownian suspensions by Stokesian dynamics simulation. J. Fluid Mech. 407 (1), 167–200.Google Scholar
Furukawa, A. & Tanaka, H. 2009 Inhomogeneous flow and fracture of glassy materials. Nature Mater. 8 (7), 601–609.Google Scholar
Glowinski, R., Pan, T. W., Hesla, T. I. & Joseph, D. D. 1998 A fictitious domain method with distributed Lagrange multipliers for the numerical simulation of particulate flow. Contemp. Maths 218 (1), 121–137.Google Scholar
Glowinski, R., Pan, T. W., Hesla, T. I., Joseph, D. D. & Périaux, J. 2000 A distributed Lagrange multiplier/fictitious domain method for the simulation of flow around moving rigid bodies: application to particulate flow. Comput. Meth. Appl. Mech. Engng 184 (2–4), 241–267.Google Scholar
Guo, M., Ehrlicher, A. J., Jensen, M. H., Renz, M., Moore, J. R., Goldman, R. D., Lippincott-Schwartz, J., Mackintosh, F. C. & Weitz, D. A. 2014 Probing the stochastic, motor-driven properties of the cytoplasm using force spectrum microscopy. Cell 158 (4), 822–832.Google Scholar
Head, D. A., Ikebe, E., Nakamasu, A., Zhang, P., Villaruz, L. G., Kinoshita, S., Ando, S. & Mizuno, D. 2014 High-frequency affine mechanics and nonaffine relaxation in a model cytoskeleton. Phys. Rev. E 89 (4), 042711.Google Scholar
Hildebrand, F. B. 1987 Introduction to Numerical Analysis, 2nd edn. Dover.Google Scholar
Hinch, E. J. & Leal, L. G. 1972 The effect of Brownian motion on the rheological properties of a suspension of non-spherical particles. J. Fluid Mech. 52 (4), 683–712.Google Scholar
Huang, H., Yang, X., Krafczyk, M. & Lu, X.-Y. 2012 Rotation of spheroidal particles in Couette flows. J. Fluid Mech. 692 (1), 369–394.Google Scholar
Hwang, W. R., Hulsen, M. A. & Meijer, H. E. H. 2004a Direct simulation of particle suspensions in sliding bi-periodic frames. J. Comput. Phys. 194 (2), 742–772.Google Scholar
Hwang, W. R., Hulsen, M. A. & Meijer, H. E. H. 2004b Direct simulations of particle suspensions in a viscoelastic fluid in sliding bi-periodic frames. J. Non-Newtonian Fluid Mech. 121 (1), 15–33.Google Scholar
Iwashita, T. & Yamamoto, R. 2009 Direct numerical simulations for non-Newtonian rheology of concentrated particle dispersions. Phys. Rev. E 80 (6), 061402.Google Scholar
Jeffery, G. B. 1922 The motion of ellipsoidal particles immersed in a viscous fluid. Proc. R. Soc. Lond. A 102 (715), 161–179.Google Scholar
Ji, S., Jiang, R., Winkler, R. G. & Gompper, G. 2011 Mesoscale hydrodynamic modeling of a colloid in shear-thinning viscoelastic fluids under shear flow. J. Chem. Phys. 135 (13), 134116.Google Scholar
Kim, S. & Karrila, S. J. 2005 Microhydrodynamics: Principles and Selected Applications, 1st edn. Dover.Google Scholar
Kobayashi, H. & Yamamoto, R. 2011 Implementation of Lees–Edwards periodic boundary conditions for direct numerical simulations of particle dispersions under shear flow. J. Chem. Phys. 134 (6), 064110.Google Scholar
Kumar, A.2010 Microscale dynamics in suspensions of non-spherical particles. PhD thesis, University of Illinois at Urbana-Champaign.Google Scholar
Kumar, A. & Higdon, J. J. L.2012 Orientation and microstructure in sheared Brownian suspensions of anisotropic dicolloidal particles. arXiv:1207.5158v1.Google Scholar
Ladd, A. J. C. 1988 Hydrodynamic interactions in a suspension of spherical-particles. J. Chem. Phys. 88 (8), 5051–5063.CrossRefGoogle Scholar
Ladd, A. J. C. 1990 Hydrodynamic transport-coefficients of random dispersions of hard-spheres. J. Chem. Phys. 93 (5), 3484–3494.CrossRefGoogle Scholar
Ladd, A. J. C. 1993 Short-time motion of colloidal particles – numerical-simulation via a fluctuating lattice-Boltzmann equation. Phys. Rev. Lett. 70 (9), 1339–1342.Google Scholar
Lees, A. W. & Edwards, S. F. 1972 The computer study of transport processes under extreme conditions. J. Phys. C: Solid State Phys. 5 (15), 1921–1928.Google Scholar
Lin, C. J., Peery, J. H. & Schowalter, W. R. 1970 Simple shear flow round a rigid sphere – inertial effects and suspension rheology. J. Fluid Mech. 44 (1), 1–17.Google Scholar
Luo, H. & Bewley, T. R. 2004 On the contravariant form of the Navier–Stokes equations in time-dependent curvilinear coordinate systems. J. Comput. Phys. 199 (1), 355–375.Google Scholar
Luo, X., Maxey, M. R. & Karniadakis, G. E. 2009 Smoothed profile method for particulate flows: error analysis and simulations. J. Comput. Phys. 228 (5), 1750–1769.Google Scholar
Marchetti, M. C., Joanny, J. F., Ramaswamy, S., Liverpool, T. B., Prost, J., Rao, M. & Simha, A. 2013 Hydrodynamics of soft active matter. Rev. Mod. Phys. 85 (3), 1143–1189.Google Scholar
Mari, R., Seto, R., Morris, J. F. & Denn, M. M. 2014 Shear thickening, frictionless and frictional rheologies in non-Brownian suspensions. J. Rheol. 58 (6), 1693–1724.Google Scholar
Mason, T. G., Bibette, J. & Weitz, D. A. 1995 Optical measurements of frequency-dependent linear viscoelastic moduli of complex fluids. Phys. Rev. Lett. 74 (7), 1250–1253.Google Scholar
Maxey, M. R. & Patel, B. K. 2001 Localized force representations for particles sedimenting in Stokes flow. Intl J. Multiphase Flow 27 (9), 1603–1626.Google Scholar
Mewis, J. & Wagner, N. J. 2012 Colloidal Suspension Rheology, 1st edn. Cambridge University Press.Google Scholar
Mikulencak, D. R. & Morris, J. F. 2004 Stationary shear flow around fixed and free bodies at finite Reynolds number. J. Fluid Mech. 520 (1), 215–242.Google Scholar
Milner, S. 1993 Dynamical theory of concentration fluctuations in polymer solutions under shear. Phys. Rev. E 48 (5), 3674–3691.Google Scholar
Mizuno, D., Head, D. A., MacKintosh, F. C. & Schmidt, C. F. 2008 Active and passive microrheology in equilibrium and nonequilibrium systems. Macromolecules 41 (19), 7194–7202.CrossRefGoogle Scholar
Molina, J. J. & Yamamoto, R. 2013 Direct numerical simulations of rigid body dispersions. I. Mobility/friction tensors of assemblies of spheres. J. Chem. Phys. 139 (23), 234105.Google Scholar
Muldowney, G. P. & Higdon, J. J. L. 2006 A spectral boundary element approach to three-dimensional Stokes flow. J. Fluid Mech. 298 (1), 167–192.Google Scholar
Mussler, M., Rafaï, S., Peyla, P. & Wagner, C. 2013 Effective viscosity of non-gravitactic Chlamydomonas Reinhardtii microswimmer suspensions. Eur. Phys. Lett. 101 (5), 54004.Google Scholar
Nakayama, Y., Kim, K. & Yamamoto, R. 2008 Simulating (electro) hydrodynamic effects in colloidal dispersions: smoothed profile method. Eur. Phys. J. E 26 (4), 361–368.CrossRefGoogle ScholarPubMed
Nakayama, Y. & Yamamoto, R. 2005 Simulation method to resolve hydrodynamic interactions in colloidal dispersions. Phys. Rev. E 71 (3), 036707.CrossRefGoogle ScholarPubMed
Nguyen, N. Q. & Ladd, A. J. C. 2002 Lubrication corrections for lattice-Boltzmann simulations of particle suspensions. Phys. Rev. E 66 (4), 046708.Google Scholar
Onuki, A. 1997 A new computer method of solving dynamic equations under externally applied deformations. J. Phys. Soc. Japan 66 (6), 1836–1837.Google Scholar
Pine, D. J., Gollub, J. P., Brady, J. F. & Leshansky, A. M. 2005 Chaos and threshold for irreversibility in sheared suspensions. Nature 438 (7070), 997–1000.Google Scholar
Poblete, S., Wysocki, A., Gompper, G. & Winkler, R. G. 2014 Hydrodynamics of discrete-particle models of spherical colloids: a multiparticle collision dynamics simulation study. Phys. Rev. E 90 (3), 033314.Google Scholar
Rogallo, R. S.1981 Numerical experiments in homogeneous turbulence. NASA Tech. Memorandum 81315.Google Scholar
Russel, W. B., Saville, D. A. & Schowalter, W. R. 1992 Colloidal Dispersions, 1st edn. Cambridge University Press.Google Scholar
Seto, R., Botet, R. & Briesen, H. 2011 Hydrodynamic stress on small colloidal aggregates in shear flow using Stokesian dynamics. Phys. Rev. E 84 (4), 041405.Google Scholar
Seto, R., Mari, R., Morris, J. F. & Denn, M. M. 2013 Discontinuous shear thickening of frictional hard-sphere suspensions. Phys. Rev. Lett. 111 (21), 218301.Google Scholar
Shikata, T. & Pearson, D. S. 1994 Viscoelastic behavior of concentrated spherical suspensions. J. Rheol. 38 (3), 601–616.Google Scholar
Sierou, A. & Brady, J. F. 2001 Accelerated stokesian dynamics simulations. J. Fluid Mech. 448 (1), 115–146.Google Scholar
Sierou, A. & Brady, J. F. 2004 Shear-induced self-diffusion in non-colloidal suspensions. J. Fluid Mech. 506 (1), 285–314.CrossRefGoogle Scholar
Singh, A. & Nott, P. R. 2003 Experimental measurements of the normal stresses in sheared Stokesian suspensions. J. Fluid Mech. 490 (1), 293–320.Google Scholar
von Smoluchowski, M. 1906 Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Ann. Phys. 326 (14), 756–780.Google Scholar
Sokolov, A. & Aranson, I. S. 2009 Reduction of viscosity in suspension of swimming bacteria. Phys. Rev. Lett. 103 (14), 148101.Google Scholar
Succi, S. 2001 The Lattice Boltzmann Equation: For Fluid Dynamics and Beyond, 1st edn. Clarendon.Google Scholar
Tanaka, H. & Araki, T. 2000 Simulation method of colloidal suspensions with hydrodynamic interactions: fluid particle dynamics. Phys. Rev. Lett. 85 (6), 1338–1341.CrossRefGoogle ScholarPubMed
Todd, B. D. & Daivis, P. J. 2007 Homogeneous non-equilibrium molecular dynamics simulations of viscous flow: techniques and applications. Mol. Simul. 33 (3), 189–229.Google Scholar
Venturi, D. 2009 Convective derivatives and Reynolds transport in curvilinear time-dependent coordinates. J. Phys. A: Math. Theor. 42 (12), 125203.Google Scholar
Villone, M. M., D’Avino, G., Hulsen, M. A., Greco, F. & Maffettone, P. L. 2011 Numerical simulations of particle migration in a viscoelastic fluid subjected to Poiseuille flow. Comput. Fluids 42 (1), 82–91.Google Scholar
Wagner, N. J. & Brady, J. F. 2009 Shear thickening in colloidal dispersions. Phys. Today 62 (10), 27–32.Google Scholar
van der Werff, J. C. & de Kruif, C. G. 1989 Hard-sphere colloidal dispersions – the scaling of rheological properties with particle-size, volume fraction, and shear rate. J. Rheol. 33 (3), 421–454.Google Scholar
Yeo, K. & Maxey, M. R. 2010 Simulation of concentrated suspensions using the force-coupling method. J. Comput. Phys. 229 (6), 2401–2421.CrossRefGoogle Scholar
Youngren, G. K. & Acrivos, A. 1974 Stokes flow past a particle of arbitrary shape: a numerical method of solution. J. Fluid Mech. 69 (02), 377.Google Scholar
Zhang, D., Jack, D. A., Smith, D. E. & Montgomery-Smith, S. 2011 Numerical evaluation of single fiber motion for short-fiber-reinforced composite materials processing. J. Manuf. Sci. Engng 133 (5), 051002.Google Scholar