Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-06T08:31:25.389Z Has data issue: false hasContentIssue false

Modelling a hydrodynamic instability in freely settling colloidal gels

Published online by Cambridge University Press:  12 October 2018

Zsigmond Varga
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
Jennifer L. Hofmann
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
James W. Swan*
Affiliation:
Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
*
Email address for correspondence: [email protected]

Abstract

Attractive colloidal dispersions, suspensions of fine particles which aggregate and frequently form a space-spanning elastic gel are ubiquitous materials in society with a wide range of applications. The colloidal networks in these materials can exist in a mode of free settling when the network weight exceeds its compressive yield stress. An equivalent state occurs when the network is held fixed in place and used as a filter through which the suspending fluid is pumped. In either scenario, hydrodynamic instabilities leading to loss of network integrity occur. Experimental observations have shown that the loss of integrity is associated with the formation of eroded channels, so-called streamers, through which the fluid flows rapidly. However, the dynamics of growth and subsequent mechanism of collapse remain poorly understood. Here, a phenomenological model is presented that describes dynamically the radial growth of a streamer due to erosion of the network by rapid fluid back flow. The model exhibits a finite-time blowup – the onset of catastrophic failure in the gel – due to activated breaking of the inter-colloid bonds. Brownian dynamics simulations of hydrodynamically interacting and settling colloids in dilute gels are employed to examine the initiation and propagation of this instability, which are in good agreement with the theory. The model dynamics is also shown to accurately replicate measurements of streamer growth in two different experimental systems. The predictive capabilities and future improvements of the model are discussed and a stability-state diagram is presented providing insight into engineering strategies for avoiding settling instabilities in networks meant to have long shelf lives.

Type
JFM Papers
Copyright
© 2018 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

Acrivos, A. & Goddard, J. 1965 Asymptotic expansions for laminar forced-convection heat and mass transfer. J. Fluid Mech. 23 (2), 273291.Google Scholar
Allain, C., Cloitre, M. & Wafra, M. 1995 Aggregation and sedimentation in colloidal suspensions. Phys. Rev. Lett. 74 (8), 14781481.Google Scholar
Asakura, S. & Oosawa, F. 1958 Interaction between particles suspended in solutions of macromolecules. J. Polym. Sci. 33 (126), 183192.Google Scholar
Bai, B., Liu, Y., Coste, J.-P. & Li, L. 2007 Preformed particle gel for conformance control: transport mechanism through porous media. SPE Res. Eval. Engng 10 (02), 176184.Google Scholar
Bailey, A. et al. 2007 Spinodal decomposition in a model colloid-polymer mixture in microgravity. Phys. Rev. Lett. 99 (20), 205701.Google Scholar
Bartlett, P., Teece, L. J. & Faers, M. A. 2012 Sudden collapse of a colloidal gel. Phys. Rev. E 85 (2), 021404.Google Scholar
Beavers, G. S. & Joseph, D. D. 1967 Boundary conditions at a naturally permeable wall. J. Fluid Mech. 30 (1), 197207.Google Scholar
Binks, B. P. & Horozov, T. S. 2006 Colloidal Particles at Liquid Interfaces. Cambridge University Press.Google Scholar
Blijdenstein, T. B., van der Linden, E., van Vliet, T. & van Aken, G. A. 2004 Scaling behavior of delayed demixing, rheology, and microstructure of emulsions flocculated by depletion and bridging. Langmuir 20 (26), 1132111328.Google Scholar
Boniello, G., Blanc, C., Fedorenko, D., Medfai, M., Mbarek, N. B., In, M., Gross, M., Stocco, A. & Nobili, M. 2015 Brownian diffusion of a partially wetted colloid. Nature Mater. 14 (9), 908911.Google Scholar
Brambilla, G., Buzzaccaro, S., Piazza, R., Berthier, L. & Cipelletti, L. 2011 Highly nonlinear dynamics in a slowly sedimenting colloidal gel. Phys. Rev. Lett. 106 (11), 118302.Google Scholar
Buscall, R. & White, L. R. 1987 The consolidation of concentrated suspensions. Part 1. The theory of sedimentation. J. Chem. Soc. Faraday Trans. 83 (3), 873891.Google Scholar
Clark, J. I. & Carper, D. 1987 Phase separation in lens cytoplasm is genetically linked to cataract formation in the Philly mouse. Proc. Natl Acad. Sci. USA 84 (1), 122125.Google Scholar
Falconer, K. 2004 Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons.Google Scholar
Fiore, A. M., Balboa Usabiaga, F., Donev, A. & Swan, J. W. 2017 Rapid sampling of stochastic displacements in Brownian dynamics simulations. J. Chem. Phys. 146 (12), 124116.Google Scholar
Fiore, A. M. & Swan, J. W.2018a Fast Stokesian dynamics (in preparation).Google Scholar
Fiore, A. M. & Swan, J. W. 2018b Rapid sampling of stochastic displacements in Brownian dynamics simulations with stresslet constraints. J. Chem. Phys. 148 (4), 044114.Google Scholar
Gaponik, N., Herrmann, A.-K. & Eychmüller, A. 2011 Colloidal nanocrystal-based gels and aerogels: material aspects and application perspectives. J. Phys. Chem. Lett. 3 (1), 817.Google Scholar
Goddard, J. & Acrivos, A. 1966 Asymptotic expansions for laminar forced-convection heat and mass transfer. Part 2. Boundary-layer flows. J. Fluid Mech. 24 (2), 339366.Google Scholar
Gopalakrishnan, V., Schweizer, K. & Zukoski, C. 2006 Linking single particle rearrangements to delayed collapse times in transient depletion gels. J. Phys.: Condens. Matter 18 (50), 11531.Google Scholar
Harich, R., Blythe, T., Hermes, M., Zaccarelli, E., Sederman, A., Gladden, L. F. & Poon, W. 2016 Gravitational collapse of depletion-induced colloidal gels. Soft Matt. 12 (19), 43004308.Google Scholar
Heyes, D. & Melrose, J. 1993 Brownian dynamics simulations of model hard-sphere suspensions. J. Non-Newtonian Fluid Mech. 46 (1), 128.Google Scholar
Hu, Z., Liao, M., Chen, Y., Cai, Y., Meng, L., Liu, Y., Lv, N., Liu, Z. & Yuan, W. 2012 A novel preparation method for silicone oil nanoemulsions and its application for coating hair with silicone. Intl J. Nanomed. 7, 57195724.Google Scholar
Huh, J. Y., Lynch, M. L. & Furst, E. M. 2007 Microscopic structure and collapse of depletion-induced gels in vesicle-polymer mixtures. Phys. Rev. E 76 (5), 051409.Google Scholar
Ide, K. & Sornette, D. 2002 Oscillatory finite-time singularities in finance, population and rupture. Physica A 307 (1), 63106.Google Scholar
Kamp, S. W. & Kilfoil, M. L. 2009 Universal behaviour in the mechanical properties of weakly aggregated colloidal particles. Soft Matt. 5 (12), 24382447.Google Scholar
Kilfoil, M. L., Pashkovski, E. E., Masters, J. A. & Weitz, D. 2003 Dynamics of weakly aggregated colloidal particles. Phil. Trans. R. Soc. Lond. A 361 (1805), 753766.Google Scholar
Kim, J. M., Fang, J., Eberle, A. P., Castañeda-Priego, R. & Wagner, N. J. 2013 Gel transition in adhesive hard-sphere colloidal dispersions: the role of gravitational effects. Phys. Rev. Lett. 110 (20), 208302.Google Scholar
Kramers, H. A. 1940 Brownian motion in a field of force and the diffusion model of chemical reactions. Physica 7 (4), 284304.Google Scholar
Lu, P. J., Conrad, J. C., Wyss, H. M., Schofield, A. B. & Weitz, D. A. 2006 Fluids of clusters in attractive colloids. Phys. Rev. Lett. 96 (2), 028306.Google Scholar
Lu, P. J., Zaccarelli, E., Ciulla, F., Schofield, A. B., Sciortino, F. & Weitz, D. A. 2008 Gelation of particles with short-range attraction. Nature 453, 499503.Google Scholar
MacMinn, C. W., Dufresne, E. R. & Wettlaufer, J. S. 2016 Large deformations of a soft porous material. Phys. Rev. Appl. 5 (4), 044020.Google Scholar
Manley, S., Skotheim, J., Mahadevan, L. & Weitz, D. A. 2005 Gravitational collapse of colloidal gels. Phys. Rev. Lett. 94 (21), 218302.Google Scholar
Mezzenga, R., Schurtenberger, P., Burbidge, A. & Martin, M. 2005 Understanding foods as soft materials. Nature Mater. 4 (10), 729740.Google Scholar
Noro, M. G. & Frenkel, D. 2000 Extended corresponding-states behavior for particles with variable range attractions. J. Chem. Phys. 113 (8), 29412944.Google Scholar
Northcott, K. A., Snape, I., Scales, P. J. & Stevens, G. W. 2005a Contaminated water treatment in cold regions: an example of coagulation and dewatering modelling in Antarctica. Cold Reg. Sci. Technol. 41 (1), 6172.Google Scholar
Northcott, K. A., Snape, I., Scales, P. J. & Stevens, G. W. 2005b Dewatering behaviour of water treatment sludges associated with contaminated site remediation in Antarctica. Chem. Engng Sci. 60 (24), 68356843.Google Scholar
Padmanabhan, P. & Zia, R. 2018 Gravitational collapse of colloidal gels: non-equiliibrium phase separation driven by osmotic pressure. Soft Matt. 14 (17), 32653287.Google Scholar
Poon, W. 2002 The physics of a model colloid–polymer mixture. J. Phys.: Condens. Matter 14 (33), R859.Google Scholar
Poon, W., Pirie, A., Haw, M. & Pusey, P. 1997 Non-equilibrium behaviour of colloid-polymer mixtures. Physica A 235 (1), 110119.Google Scholar
Poon, W., Starrs, L., Meeker, S., Moussaid, A., Evans, R., Pusey, P. & Robins, M. 1999 Delayed sedimentation of transient gels in colloid–polymer mixtures: dark-field observation, rheology and dynamic light scattering studies. Faraday Discuss. 112, 143154.Google Scholar
Razali, A., Fullerton, C. J., Turci, F., Hallett, J. E., Jack, R. L. & Royall, C. P. 2017 Effects of vertical confinement on gelation and sedimentation of colloids. Soft Matt. 13 (17), 32303239.Google Scholar
Rotne, J. & Prager, S. 1969 Variational treatment of hydrodynamic interaction in polymers. J. Chem. Phys. 50 (11), 48314837.Google Scholar
Russel, W. B., Saville, D. A. & Schowalter, W. R. 1989 Colloidal Dispersions. Cambridge University Press.Google Scholar
Secchi, E., Buzzaccaro, S. & Piazza, R. 2014 Time-evolution scenarios for short-range depletion gels subjected to the gravitational stress. Soft Matt. 10 (29), 52965310.Google Scholar
Senis, D., Talini, L. & Allain, C. 2001 Settling in aggregating colloidal suspensions. Oil Gas Sci. Technol. 56 (2), 153159.Google Scholar
Starrs, L., Poon, W., Hibberd, D. & Robins, M. 2002 Collapse of transient gels in colloid-polymer mixtures. J. Phys.: Condens. Matter 14 (10), 2485.Google Scholar
Swan, J. W. & Wang, G. 2016 Rapid calculation of hydrodynamic and transport properties in concentrated solutions of colloidal particles and macromolecules. Phys. Fluids 28 (1), 011902.Google Scholar
Teece, L. J., Faers, M. A. & Bartlett, P. 2011 Ageing and collapse in gels with long-range attractions. Soft Matt. 7 (4), 13411351.Google Scholar
Teece, L. J., Hart, J. M., Hsu, K. Y. N., Gilligan, S., Faers, M. A. & Bartlett, P. 2014 Gels under stress: The origins of delayed collapse. Colloids Surf. A 458, 126133.Google Scholar
Varga, Z. & Swan, J. 2016 Hydrodynamic interactions enhance gelation in dispersions of colloids with short-ranged attraction and long-ranged repulsion. Soft Matt. 12 (36), 76707681.Google Scholar
Varga, Z. & Swan, J. W. 2018a Large scale anisotropies in sheared colloidal gels. J. Rheol. 62 (2), 405418.Google Scholar
Varga, Z. & Swan, J. W. 2018b Normal modes of weak colloidal gels. Phys. Rev. E 97 (1), 012608.Google Scholar
Varga, Z., Wang, G. & Swan, J. 2015 The hydrodynamics of colloidal gelation. Soft Matt. 11 (46), 90099019.Google Scholar
Weeks, J. R., van Duijneveldt, J. S. & Vincent, B. 2000 Formation and collapse of gels of sterically stabilized colloidal particles. J. Phys.: Condens. Matter 12 (46), 9599.Google Scholar
Yang, H.-H., Zhang, S.-Q., Yang, W., Chen, X.-L., Zhuang, Z.-X., Xu, J.-G. & Wang, X.-R. 2004 Molecularly imprinted sol-gel nanotubes membrane for biochemical separations. J. Am. Chem. Soc. 126 (13), 40544055.Google Scholar
Zaccarelli, E. 2007 Colloidal gels: equilibrium and non-equilibrium routes. J. Phys.: Condens. Matter 19 (32), 323101.Google Scholar

Varga et al. supplementary movie 1

Movie of settling gel with δ=0.1, ε=0.05, G=0.5 and φ=20% (same as in figure 2). The differently coloured layers are purely for illustrative purposes, indicate initial particle positions in the gel, and are meant to guide the eye through the breakdown of the network during free settling. The dispersion gelled over 500τ_D in the absence of gravity and has a structure characterized by d_f=2.05. At t=0τ_G gravity is turned on in the simulation and the network settles for 2500 τ_G.

Download Varga et al. supplementary movie 1(Video)
Video 13 MB

Varga et al. supplementary movie 2

Movie of settling gel with δ=0.15, ε=0.05, G=0.3 and φ=20%. The differently coloured layers are purely for illustrative purposes, indicate initial particle positions in the gel, and are meant to guide the eye through the breakdown of the network during free settling. The dispersion gelled over 500τ_D in the absence of gravity and has a structure characterized by d_f=2.11. At t=0τ_G gravity is turned on in the simulation and the network settles for 2500 τ_G.

Download Varga et al. supplementary movie 2(Video)
Video 16.3 MB

Varga et al. supplementary movie 3

Movie of settling gel with δ=0.1, ε=0.02, G=0.5 and φ=20%. The differently coloured layers are purely for illustrative purposes, indicate initial particle positions in the gel, and are meant to guide the eye through the breakdown of the network during free settling. The dispersion gelled over 500τ_D in the absence of gravity and has a structure characterized by d_f=1.96. At t=0τ_G gravity is turned on in the simulation and the network settles for 2500 τ_G.

Download Varga et al. supplementary movie 3(Video)
Video 16.4 MB

Varga et al. supplementary movie 4

Movie of settling gel with δ=0.1, ε=0.05, G=0.5 and φ=20%. The differently coloured layers are purely for illustrative purposes, indicate initial particle positions in the gel, and are meant to guide the eye through the breakdown of the network during free settling. The dispersion gelled over 500τ_D in the absence of gravity and has a structure characterized by d_f=2.08. A streamer was seeded in the centre with a radius of R_0=0.8R^* and the gel cross section is shown. At t=0τ_G gravity is turned on in the simulation and the network settles for 2500 τ_G.

Download Varga et al. supplementary movie 4(Video)
Video 9.5 MB