Skip to main content Accessibility help
×
Home

Equilibrium structure and diffusion in concentrated hydrodynamically interacting suspensions confined by a spherical cavity

  • Christian Aponte-Rivera (a1), Yu Su (a1) and Roseanna N. Zia (a2)

Abstract

The short- and long-time equilibrium transport properties of a hydrodynamically interacting suspension confined by a spherical cavity are studied via Stokesian dynamics simulations for a wide range of particle-to-cavity size ratios and particle concentrations. Many-body hydrodynamic and lubrication interactions between particles and with the cavity are accounted for utilizing recently developed mobility and resistance tensors for spherically confined suspensions (Aponte-Rivera & Zia, Phys. Rev. Fluids, vol. 1(2), 2016, 023301). Study of particle volume fractions in the range $0.05\leqslant \unicode[STIX]{x1D719}\leqslant 0.40$ reveals that confinement exerts a qualitative influence on particle diffusion. First, the mean-square displacement over all time scales depends on the position in the cavity. Additionally, at short times, the diffusivity is anisotropic, with diffusion along the cavity radius slower than diffusion tangential to the cavity wall, due to the anisotropy of hydrodynamic coupling and to confinement-induced spatial heterogeneity in particle concentration. The mean-square displacement is anisotropic at intermediate times as well and, surprisingly, exhibits superdiffusive and subdiffusive behaviours for motion along and perpendicular to the cavity radius respectively, depending on the suspension volume fraction and the particle-to-cavity size ratio. No long-time self-diffusive regime exists; instead, the mean-square displacement reaches a long-time plateau, a result of entropic restriction to a finite volume. In this long-time limit, the higher the volume fraction is, the longer the particles take to reach the long-time plateau, as cooperative rearrangements are required as the cavity becomes crowded. The ordered dynamical heterogeneity seen here promotes self-organization of particles based on their size and self-mobility, which may be of particular relevance in biophysical systems.

Copyright

Corresponding author

Email address for correspondence: rzia@stanford.edu

References

Hide All
Ahlrichs, P., Everaers, R. & Dünweg, B. 2001 Screening of hydrodynamic interactions in semidilute polymer solutions: a computer simulation study. Phys. Rev. E 64 (4 Pt 1), 040501.
Ando, T. & Skolnick, J. 2010 Crowding and hydrodynamic interactions likely dominate in vivo macromolecular motion. Proc. Natl Acad. Sci. USA 107 (43), 1845718462.
Aponte-Rivera, C. & Zia, R. N. 2016 Simulation of hydrodynamically interacting particles confined by a spherical cavity. Phys. Rev. Fluids 1 (2), 023301.
Banchio, A. J. & Brady, J. F. 2003 Accelerated Stokesian dynamics: Brownian motion. J. Chem. Phys. 118 (22), 1032310332.
Batchelor, G. K. 1976 Brownian diffusion of particles with hydrodynamic interaction. J. Fluid Mech. 74, 129.
Batchelor, G. K. 1982 Sedimentation in a dilute polydisperse system of interacting spheres. Part 1. General theory. J. Fluid Mech. 119, 379407.
Bhattacharya, S. 2008 Cooperative motion of spheres arranged in periodic grids between two parallel walls. J. Chem. Phys. 128 (7), 074709.
Bhattacharya, S., Mishra, C. & Bhattacharya, S. 2010 Analysis of general creeping motion of a sphere inside a cylinder. J. Fluid Mech. 642, 295328.
Bickel, T. 2007 A note on confined diffusion. Physica A 377 (1), 2432.
Brady, J. F. 1994 The long-time self-diffusivity in concentrated colloidal dispersions. J. Fluid Mech. 272, 109133.
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111157.
Brangwynne, C. P., Koenderink, G. H., MacKintosh, F. C. & Weitz, D. A. 2008 Cytoplasmic diffusion: molecular motors mix it up. J. Cell Biol. 183 (4), 583587.
Brangwynne, C. P., Koenderink, G. H., MacKintosh, F. C. & Weitz, D. A. 2009 Intracellular transport by active diffusion. Trends in Cell Biology 19 (9), 423427.
Brenner, H. 1961 The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Engng Sci. 16 (3–4), 242251.
Chow, E. & Skolnick, J. 2015 Effects of confinement on models of intracellular macromolecular dynamics. Proc. Natl Acad. Sci. USA 112 (48), 1484614851.
Colby, R. H. 2010 Structure and linear viscoelasticity of flexible polymer solutions: comparison of polyelectrolyte and neutral polymer solutions. Rheol. Acta 49 (5), 425442.
Crocker, J. C. & Hoffman, B. D. 2007 Multiple-particle tracking and two-point microrheology in cells. Meth. Cell Biol. 83 (7), 141178.
Cunningham, E. 1910 On the velocity of steady fall of spherical particles through fluid medium. Proc. R. Soc. Lond. A 83 (563), 357365.
Daniels, B. R., Masi, B. C. & Wirtz, D. 2006 Probing single-cell micromechanics in vivo: the microrheology of C. elegans developing embryos. Biophys. J. 90 (12), 47124719.
Durlofsky, L. & Brady, J. F. 1987 Analysis of the Brinkman equation as a model for flow in porous media. Phys. Fluids 30 (11), 33293341.
Durlofsky, L., Brady, J. F. & Bossis, G. 1987 Dynamic simulation of hydrodynamically interacting particles. J. Fluid Mech. 180, 2149.
Foss, D. R. & Brady, J. F. 2000 Structure, diffusion and rheology of Brownian suspensions by Stokesian dynamics simulation. J. Fluid Mech. 407, 167200.
de Gennes, P. G. 1976 Dynamics of entangled polymer solutions. Part II. Inclusion of hydrodynamic interactions. Macromolecules 9 (4), 594598.
Golden, A. 2000 Cytoplasmic flow and the establishment of polarity in C. elegans 1-cell embryos. Curr. Opin. Genetics Develop. 10 (4), 414420.
Golding, I. & Cox, E. 2006 Physical nature of bacterial cytoplasm. Phys. Rev. Lett. 96 (9), 098102.
Gönczy, P. & Rose, L. S.2005 Asymmetric cell division and axis formation in the embryo. In WormBook, The C. elegans Research Community.
González, A., White, J. A., Román, F. L. & Evans, R. 1998 How the structure of a confined fluid depends on the ensemble: hard spheres in a spherical cavity. J. Chem. Phys. 109 (9), 36373650.
Henderson, G. P., Gan, L. & Jensen, G. J. 2007 3-D ultrastructure of O. tauri: electron cryotomography of an entire eukaryotic cell. PLoS ONE 2 (8).
Hoh, N. J. & Zia, R. N. 2016a Force-induced diffusion in suspensions of hydrodynamically interacting colloids. J. Fluid Mech. 795, 739783.
Hoh, N. J. & Zia, R. N. 2016b The impact of probe size on measurements of diffusion in active microrheology. Lab on a Chip 16, 31143129.
Hoogerbrugge, P. J. & Koelman, J. M. V. A. 1992 Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Eur. Phys. Lett. 19 (3), 155160.
Hunter, G. L., Edmond, K. V. & Weeks, E. R. 2014 Boundary mobility controls glassiness in confined colloidal liquids. Phys. Rev. Lett. 112 (21), 218302.
Jaensch, S., Decker, M., Hyman, A. A. & Myers, E. W. 2010 Automated tracking and analysis of centrosomes in early Caenorhabditis elegans embryos. Bioinformatics 26 (12), 1320.
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, 261290.
Jones, R. B. 2009 Dynamics of a colloid in a spherical cavity. In Theoretical Methods for Micro Scale Viscous Flows (ed. Feuillebois, F. & Sellier, A.), chap. 4, pp. 61104. Transworld Research Network.
Keys, A. S., Iacovella, C. R. & Glotzer, S. C. 2011 Characterizing complex particle morphologies through shape matching: descriptors, applications, and algorithms. J. Comput. Phys. 230 (17), 64386463.
Kim, S. H., Park, J. G., Choi, T. M., Manoharan, V. N. & Weitz, D. A. 2014 Osmotic-pressure-controlled concentration of colloidal particles in thin-shelled capsules. Nat. Commun. 5, 3068.
Ladd, A. J. C. 1994 Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. J. Fluid Mech. 271, 285309.
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow, 2nd edn. Gordon and Breach Science Publishers.
Lau, A. W. C., Hoffman, B. D., Davies, A., Crocker, J. C. & Lubensky, T. C. 2003 Microrheology, stress fluctuations, and active behavior of living cells. Phys. Rev. Lett. 91 (19), 198101.
Leighton, D. & Acrivos, A. 1987 The shear-induced migration of particles in concentrated suspensions. J. Fluid Mech. 181, 415439.
Marshall, W. F., Straight, A., Marko, J. F., Swedlow, J., Dernburg, A., Belmont, A., Murray, A. W., Agard, D. A. & Sedat, J. W. 1997 Interphase chromosomes undergo constrained diffusional motion in living cells. Current Biology: CB 7 (12), 930939.
McGuffee, S. R. & Elcock, A. H. 2010 Diffusion, crowding & protein stability in a dynamic molecular model of the bacterial cytoplasm. PLoS Comput. Biol. 6 (3), e1999694.
Navardi, S. & Bhattacharya, S. 2010 A new lubrication theory to derive far-field axial pressure difference due to force singularities in cylindrical or annular vessels. J. Math. Phys. 51 (4), 043102.
Navardi, S., Bhattacharya, S. & Wu, H. 2015 Stokesian simulation of two unequal spheres in a pressure-driven creeping flow through a cylinder. Comput. Fluids 121, 145163.
Németh, Z. T. & Löwen, H. 1999 Freezing and glass transition of hard spheres in cavities. Phys. Rev. E 59 (6), 68246829.
O’Neill, M. E. & Majumdar, S. R. 1970 Asymmetrical slow viscous fluid motions caused by the translation or rotation of two spheres. Part II. Asymptotic forms of the solutions when the minimum clearance. Z. Angew. Math. Phys. 21, 180187.
Oseen, C. W. 1927 Neuere Methoden und Ergebnisse in der Hydrodynamik. Akademische Verlagsgesellschaft M.B.H.
Peng, Y., Chen, W., Fischer, T. M., Weitz, D. A. & Tong, P. 2009 Short-time self-diffusion of nearly hard spheres at an oil–water interface. J. Fluid Mech. 618, 243261.
Percus, J. K. & Yevick, G. J. 1958 Analysis of classical statistical mechanics by means of collective coordinates. Phys. Rev. 110 (1), 113.
Rallison, J. M. & Hinch, E. J. 1986 The effect of particle interactions on dynamic light scattering from a dilute suspension. J. Fluid Mech. 167, 131168.
Shinar, T., Mana, M., Piano, F. & Shelley, M. J. 2011 A model of cytoplasmically driven microtubule-based motion in the single-celled Caenorhabditis elegans embryo. Proc. Natl Acad. Sci. USA 108 (26), 1050810513.
Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamics simulations. J. Fluid Mech. 448, 115146.
Snook, I. K. & Henderson, D. 1978 Monte Carlo study of a hard-sphere fluid near a hard wall. J. Chem. Phys. 68 (5), 21342139.
Steinhardt, P. J., Nelson, D. R. & Ronchetti, M. 1983 Bond-orientational order in liquids and glasses. Phys. Rev. B 28 (2), 784805.
Su, Y., Swan, J. W. & Zia, R. N. 2017 Pair mobility functions for rigid spheres in concentrated colloidal dispersions: stresslet and straining motion couplings. J. Chem. Phys. 146, 124903.
Suh, J., Wirtz, D. & Hanes, J. 2003 Efficient active transport of gene nanocarriers to the cell nucleus. Proc. Natl Acad. Sci. USA 100 (7), 37383882.
Sun, J. & Weinstein, H. 2007 Toward realistic modeling of dynamic processes in cell signaling: quantification of macromolecular crowding effects. J. Chem. Phys. 127 (15), 155105.
Swan, J. W. & Brady, J. F. 2010 Particle motion between parallel walls: hydrodynamics and simulation. Phys. Fluids 22 (10), 103301.
Swan, J. W. & Brady, J. F. 2011a Anisotropic diffusion in confined colloidal dispersions: the evanescent diffusivity. J. Chem. Phys. 135, 014701.
Swan, J. W. & Brady, J. F. 2011b The hydrodynamics of confined dispersions. J. Fluid Mech. 687, 254299.
Tabei, S. M. A., Burov, S., Kim, H. Y., Kuznetsov, A., Huynh, T., Jureller, J., Philipson, L. H., Dinner, A. R. & Scherer, N. F. 2013 Intracellular transport of insulin granules is a subordinated random walk. Proc. Natl Acad. Sci. USA 110 (13), 49114916.
Teich, E. G., van Anders, G., Klotsa, D., Dshemuchadse, J. & Glotzer, S. C. 2016 Clusters of polyhedra in spherical confinement. Proc. Natl Acad. Sci. USA 113 (6), E669E678.
Tough, R. J. A. & van den Broeck, C. 1989 Diffusion within a sphere: a non-Gaussian statistical model for particle displacements in a dense colloidal suspension. Physica A 157, 769796.
Verkman, A. S. 2002 Solute and macromolecule diffusion in cellular aqueous compartments. Trends Biochem. Sci. 27 (1), 2733.
Vogel, N., Utech, S., England, G. T., Shirman, T., Phillips, K. R., Koay, N., Burgess, I. B., Kolle, M., Weitz, D. A. & Aizenberg, J. 2015 Color from hierarchy: diverse optical properties of micron-sized spherical colloidal assemblies. Proc. Natl Acad. Sci. USA 112 (35), 1084510850.
Wachsmuth, M., Waldeck, W. & Langowski, J. 2000 Anomalous diffusion of fluorescent probes inside living cell nuclei investigated by spatially-resolved fluorescence correlation spectroscopy. J. Molecular Biol. 298 (4), 677689.
Weber, S. C. & Brangwynne, C. P. 2015 Inverse size scaling of the nucleolus by a concentration-dependent phase transition. Current Biology 25, 16.
Weber, S. C., Theriot, J. A. & Spakowitz, A. J. 2010 Subdiffusive motion of a polymer composed of subdiffusive monomers. Phys. Rev. E 82 (1), 111.
Weeks, E. R., Crocker, J. C., Levitt, A. C., Schofield, A. & Weitz, D. A. 2000 Three-dimensional direct imaging of structural relaxation near the colloidal glass transition. Science 287 (5453), 627631.
Weiss, M., Elsner, M., Kartberg, F. & Nilsson, T. 2004 Anomalous subdiffusion is a measure for cytoplasmic crowding in living cells. Biophys. J. 87 (5), 35183524.
Wodarz, A. 2002 Establishing cell polarity in development. Nature Cell Biol. 4, 3944.
Zia, R. N., Swan, J. W. & Su, Y. 2015 Pair mobility functions for rigid spheres in concentrated colloidal dispersions: force, torque, translation, and rotation. J. Chem. Phys. 143, 224901.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed