Hostname: page-component-745bb68f8f-l4dxg Total loading time: 0 Render date: 2025-01-24T21:48:06.582Z Has data issue: false hasContentIssue false
  • English
  • Français

Structure and rheology of suspensions of spherical strain-hardening capsules

Published online by Cambridge University Press:  25 January 2021

Othmane Aouane Open the ORCID record for Othmane Aouane [Opens in a new window] ,
Andrea Scagliarini Open the ORCID record for Andrea Scagliarini [Opens in a new window]  and
Jens Harting Open the ORCID record for Jens Harting [Opens in a new window]
Othmane Aouane*
Affiliation:
Helmholtz Institute Erlangen-Nürnberg for Renewable Energy, Forschungszentrum Jülich, Fürther Straße 248, 90429Nürnberg, Germany
Andrea Scagliarini
Affiliation:
Istituto per le Applicazioni del Calcolo ‘M. Picone’, IAC-CNR, Via dei Taurini 19, 00185Roma, Italy INFN, sezione Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133Rome, Italy
Jens Harting
Affiliation:
Helmholtz Institute Erlangen-Nürnberg for Renewable Energy, Forschungszentrum Jülich, Fürther Straße 248, 90429Nürnberg, Germany Department of Chemical and Biological Engineering and Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Fürther Straße 248, 90429Nürnberg, Germany
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We investigate the rheology of strain-hardening spherical capsules, from the dilute to the concentrated regime under a confined shear flow using three-dimensional numerical simulations. We consider the effect of capillary number, volume fraction and membrane inextensibility on the particle deformation and on the effective suspension viscosity and normal stress differences of the suspension. The suspension displays a shear-thinning behaviour that is a characteristic of soft particles such as emulsion droplets, vesicles, strain-softening capsules and red blood cells. We find that the membrane inextensibility plays a significant role on the rheology and can almost suppress the shear-thinning. For concentrated suspensions a non-monotonic dependence of the normal stress differences on the membrane inextensibility is observed, reflecting a similar behaviour in the particle shape. The effective suspension viscosity, instead, grows and eventually saturates, for very large inextensibilities, approaching the solid particle limit. In essence, our results reveal that strain-hardening capsules share rheological features with both soft and solid particles depending on the ratio of the area dilatation to shear elastic modulus. Furthermore, the suspension viscosity exhibits a universal behaviour for the parameter space defined by the capillary number and the membrane inextensibility, when introducing the particle geometrical changes at the steady state in the definition of the volume fraction.

JFM classification

Biological Fluid Dynamics: Capsule/cell dynamics Complex Fluids: Suspensions Non-Newtonian Flows: Rheology
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 911 , 25 March 2021 , A11
Check for updates
Copyright
© The Author(s), 2021. Published by 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

REFERENCES

Abramowitz, M. & Stegun, I.A. 2012 Handbook of Mathematical Functions. Springer Science & Business Media.Google Scholar
Aouane, O., Xie, Q., Scagliarini, A. & Harting, J. 2018 Mesoscale simulations of Janus particles and deformable capsules in flow. In High Performance Computing in Science and Engineering ’17, pp. 369–385. Springer.CrossRefGoogle Scholar
Bagchi, P. & Kalluri, R.M. 2009 Dynamics of nonspherical capsules in shear flow. Phys. Rev. E 80 (1), 016307.CrossRefGoogle ScholarPubMed
Bagchi, P. & Kalluri, R.M. 2010 Rheology of a dilute suspension of liquid-filled elastic capsules. Phys. Rev. E 81 (5), 056320.CrossRefGoogle ScholarPubMed
Bagchi, P. & Kalluri, R.M. 2011 Dynamic rheology of a dilute suspension of elastic capsules: effect of capsule tank-treading, swinging and tumbling. J. Fluid Mech. 669, 498–526.CrossRefGoogle Scholar
Barthès-Biesel, D. 1980 Motion of a spherical microcapsule freely suspended in a linear shear flow. J. Fluid Mech. 100 (4), 831–853.CrossRefGoogle Scholar
Barthès-Biesel, D. 2016 Motion and deformation of elastic capsules and vesicles in flow. Ann. Rev. Fluid Mech. 48, 25–52.CrossRefGoogle Scholar
Barthès-Biesel, D., Diaz, A. & Dhenin, E. 2002 Effect of constitutive laws for two-dimensional membranes on flow-induced capsule deformation. J. Fluid Mech. 460, 211–222.CrossRefGoogle Scholar
Barthès-Biesel, D. & Rallison, J.M. 1981 The time-dependent deformation of a capsule freely suspended in a linear shear flow. J. Fluid Mech. 113, 251–267.CrossRefGoogle Scholar
Batchelor, G.K. 1970 The stress system in a suspension of force-free particles. J. Fluid Mech. 41 (3), 545–570.CrossRefGoogle Scholar
Batchelor, G.K. & Green, J.T. 1972 The determination of the bulk stress in a suspension of spherical particles to order $C^{2}$. J. Fluid Mech. 56 (3), 401–427.CrossRefGoogle Scholar
Benzi, R., Succi, S. & Vergassola, M. 1992 The lattice Boltzmann equation: theory and applications. Phys. Rep. 222 (3), 145–197.CrossRefGoogle Scholar
Bhatnagar, P.L., Gross, E.P. & Krook, M. 1954 A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev. 94 (3), 511.CrossRefGoogle Scholar
Boedec, G., Leonetti, M. & Jaeger, M. 2017 Isogeometric FEM–BEM simulations of drop, capsule and vesicle dynamics in Stokes flow. J. Comput. Phys. 342, 117–138.Google Scholar
Buxton, G.A., Verberg, R., Jasnow, D. & Balazs, A.C. 2005 Newtonian fluid meets an elastic solid: coupling lattice Boltzmann and lattice-spring models. Phys. Rev. E 71 (5), 056707.Google ScholarPubMed
Carin, M., Barthès-Biesel, D., Edwards-Lévy, F., Postel, C. & Andrei, D.C. 2003 Compression of biocompatible liquid-filled HSA-alginate capsules: determination of the membrane mechanical properties. Biotechnol. Bioengng 82 (2), 207–212.CrossRefGoogle ScholarPubMed
Casquero, H., Bona-Casas, C., Toshniwal, D., Hughes, T.J.R., Gomez, H. & Zhang, Y.J. 2020 The divergence-conforming immersed boundary method: application to vesicle and capsule dynamics. arXiv:2001.08244.Google Scholar
Chang, K.-S. & Olbricht, W.L. 1993 a Experimental studies of the deformation and breakup of a synthetic capsule in steady and unsteady simple shear flow. J. Fluid Mech. 250, 609–633.CrossRefGoogle Scholar
Chang, K.-S. & Olbricht, W.L. 1993 b Experimental studies of the deformation of a synthetic capsule in extensional flow. J. Fluid Mech. 250, 587–608.CrossRefGoogle Scholar
Clausen, J.R., Reasor, D.A & Aidun, C.K. 2011 The rheology and microstructure of concentrated non-colloidal suspensions of deformable capsules. J. Fluid Mech. 685, 202–234.CrossRefGoogle Scholar
De Cock, L.J., De Koker, S., De Geest, B.G., Grooten, J., Vervaet, C., Remon, J.P., Sukhorukov, G.B. & Antipina, M.N. 2010 Polymeric multilayer capsules in drug delivery. Angew. Chem. Intl Ed. Engl. 49 (39), 6954–6973.CrossRefGoogle ScholarPubMed
Dodson, W.R. & Dimitrakopoulos, P. 2009 Dynamics of strain-hardening and strain-softening capsules in strong planar extensional flows via an interfacial spectral boundary element algorithm for elastic membranes. J. Fluid Mech. 641, 263–296.CrossRefGoogle Scholar
Dodson, W.R. III & Dimitrakopoulos, P. 2010 Tank-treading of erythrocytes in strong shear flows via a nonstiff cytoskeleton-based continuum computational modeling. Biophys. J. 99 (9), 2906–2916.CrossRefGoogle Scholar
Donbrow, M. 1991 Microcapsules and Nanoparticles in Medicine and Pharmacy. CRC Press.Google Scholar
Dupont, C., Delahaye, F., Barthès-Biesel, D. & Salsac, A.-V. 2016 Stable equilibrium configurations of an oblate capsule in simple shear flow. J. Fluid Mech. 791, 738–757.CrossRefGoogle Scholar
Edwards-Lévy, F. & Lévy, M.-C. 1999 Serum albumin–alginate coated beads: mechanical properties and stability. Biomaterials 20 (21), 2069–2084.CrossRefGoogle ScholarPubMed
von Eilers, H. 1941 Die Viskosität von Emulsionen Hochviskoser Stoffe als Funktion der Konzentration. Kolloidn. Z. 97 (3), 313–321.CrossRefGoogle Scholar
Einstein, A. 1906 Eine neue Bestimmung der Moleküldimensionen. Ann. Phys. 324 (2), 289–306.CrossRefGoogle Scholar
Einstein, A. 1911 Berichtigung zu meiner Arbeit:Eine neue Bestimmung der Moleküldimensionen. Ann. Phys. 339 (3), 591–592.CrossRefGoogle Scholar
Evans, K.E., Nkansah, M.A., Hutchinson, I.J. & Rogers, S.C. 1991 Molecular network design. Nature 353, 124.CrossRefGoogle Scholar
Farutin, A., Biben, T. & Misbah, C. 2014 3D numerical simulations of vesicle and inextensible capsule dynamics. J. Comput. Phys. 275, 539–568.CrossRefGoogle Scholar
Fery, A. & Weinkamer, R. 2007 Mechanical properties of micro- and nanocapsules: single-capsule measurements. Polymer 48 (25), 7221–7235.CrossRefGoogle Scholar
Frankel, N.A. & Acrivos, A. 1967 On the viscosity of a concentrated suspension of solid spheres. Chem. Engng Sci. 22 (6), 847–853.CrossRefGoogle Scholar
Frijters, S., Günther, F. & Harting, J. 2012 Effects of nanoparticles and surfactant on droplets in shear flow. Soft Matt. 8 (24), 6542–6556.CrossRefGoogle Scholar
Gallier, S., Lemaire, E., Lobry, L. & Peters, F. 2016 Effect of confinement in wall-bounded non-colloidal suspensions. J. Fluid Mech. 799, 100–127.CrossRefGoogle Scholar
Gallier, S., Lemaire, E., Peters, F. & Lobry, L. 2014 Rheology of sheared suspensions of rough frictional particles. J. Fluid Mech. 757, 514–549.CrossRefGoogle Scholar
Glowinski, R., Pan, T.-W., Hesla, T., Joseph, D.D. & Periaux, J. 2001 A fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies: application to particulate flow. J. Comput. Phys. 169 (2), 363–426.CrossRefGoogle Scholar
Gross, M., Krüger, T. & Varnik, F. 2014 Rheology of dense suspensions of elastic capsules: normal stresses, yield stress, jamming and confinement effects. Soft Matt. 10 (24), 4360–4372.CrossRefGoogle ScholarPubMed
Guazzelli, E. & Pouliquen, O. 2018 Rheology of dense granular suspensions. J. Fluid Mech. 852, P1.CrossRefGoogle Scholar
Gubspun, J., Gires, P.-Y., De Loubens, C., Barthes-Biesel, D., Deschamps, J., Georgelin, M., Leonetti, M., Leclerc, E., Edwards-Lévy, F. & Salsac, A.-V. 2016 Characterization of the mechanical properties of cross-linked serum albumin microcapsules: effect of size and protein concentration. Colloid Polym. Sci. 294 (8), 1381–1389.CrossRefGoogle Scholar
Guckenberger, A. & Gekle, S. 2016 A boundary integral method with volume-changing objects for ultrasound-triggered margination of microbubbles. arXiv:1608.05196.CrossRefGoogle Scholar
Häner, E., Heil, M. & Juel, A. 2020 Deformation and sorting of capsules in a T-junction. J. Fluid Mech. 885, A4.CrossRefGoogle Scholar
Hardeman, M.R., Goedhart, P.T., Dobbe, J.G.G. & Lettinga, K.P. 1994 Laser-assisted optical rotational cell analyser (LORCA); I. A new instrument for measurement of various structural hemorheological parameters. Clin. Hemorheol. 14 (4), 605–618.Google Scholar
Hecht, M. & Harting, J. 2010 Implementation of on-site velocity boundary conditions for D3Q19 lattice Boltzmann simulations. J. Stat. Mech. 2010 (01), P01018.CrossRefGoogle Scholar
Hochmuth, R.M. 2000 Micropipette aspiration of living cells. J. Biomech. 33 (1), 15–22.CrossRefGoogle ScholarPubMed
Koleva, I. & Rehage, H. 2012 Deformation and orientation dynamics of polysiloxane microcapsules in linear shear flow. Soft Matt. 8 (13), 3681–3693.CrossRefGoogle Scholar
Krieger, I.M. & Dougherty, T.J. 1959 A mechanism for non-Newtonian flow in suspensions of rigid spheres. Trans. Soc. Rheol. 3 (1), 137–152.CrossRefGoogle Scholar
Krüger, T. 2012 Computer Simulation Study of Collective Phenomena in Dense Suspensions of Red Blood Cells under Shear. Springer Science & Business Media.CrossRefGoogle Scholar
Krüger, T., Kusumaatmaja, H., Kuzmin, A., Shardt, O., Silva, G. & Viggen, E.M. 2017 The Lattice Boltzmann Method. Springer.CrossRefGoogle Scholar
Krüger, T., Varnik, F. & Raabe, D. 2011 Efficient and accurate simulations of deformable particles immersed in a fluid using a combined immersed boundary lattice Boltzmann finite element method. Comput. Maths Applics 61 (12), 3485–3505.CrossRefGoogle Scholar
Lac, E., Barthès-Biesel, D., Pelekasis, N.A. & Tsamopoulos, J. 2004 Spherical capsules in three-dimensional unbounded Stokes flows: effect of the membrane constitutive law and onset of buckling. J. Fluid Mech. 516, 303–334.CrossRefGoogle Scholar
Lévy, M.-C. & Edwards-Lévy, F. 1996 Coating alginate beads with cross-linked biopolymers: a novel method based on a transacylation reaction. J. Microencapsul. 13 (2), 169–183.CrossRefGoogle ScholarPubMed
Li, X. & Sarkar, K. 2008 Front tracking simulation of deformation and buckling instability of a liquid capsule enclosed by an elastic membrane. J. Comput. Phys. 227 (10), 4998–5018.CrossRefGoogle Scholar
Loewenberg, M. & Hinch, E.J. 1996 Numerical simulation of a concentrated emulsion in shear flow. J. Fluid Mech. 321, 395–419.CrossRefGoogle Scholar
de Loubens, C., Deschamps, J., Boedec, G. & Leonetti, M. 2015 Stretching of capsules in an elongation flow, a route to constitutive law. J. Fluid Mech. 767, R3.Google Scholar
de Loubens, C., Deschamps, J., Edwards-Levy, F. & Leonetti, M. 2016 Tank-treading of microcapsules in shear flow. J. Fluid Mech. 789, 750–767.CrossRefGoogle Scholar
de Loubens, C., Deschamps, J., Georgelin, M., Charrier, A., Edwards-Levy, F. & Leonetti, M. 2014 Mechanical characterization of cross-linked serum albumin microcapsules. Soft Matt. 10 (25), 4561–4568.CrossRefGoogle ScholarPubMed
MacMeccan, R.M. III 2007 Mechanistic effects of erythrocytes on platelet deposition in coronary thrombosis. PhD thesis, Georgia Institute of Technology.Google Scholar
Maron, S.H. & Pierce, P.E. 1956 Application of Ree–Eyring generalized flow theory to suspensions of spherical particles. J. Colloid Sci. 11 (1), 80–95.CrossRefGoogle Scholar
Matsunaga, D., Imai, Y., Yamaguchi, T. & Ishikawa, T. 2016 Rheology of a dense suspension of spherical capsules under simple shear flow. J. Fluid Mech. 786, 110–127.CrossRefGoogle Scholar
Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37, 239–261.CrossRefGoogle Scholar
Miyazawa, K., Yajima, I., Kaneda, I. & Yanaki, T. 2000 Preparation of a new soft capsule for cosmetics. J. Cosmet. Sci. 51 (4), 239–252.Google Scholar
Mooney, M. 1951 The viscosity of a concentrated suspension of spherical particles. J. Colloid Sci. 6 (2), 162–170.CrossRefGoogle Scholar
Mueller, S., Llewellin, E.W. & Mader, H.M. 2010 The rheology of suspensions of solid particles. In Public Relations Society of America, vol. 466, pp. 1201–1228. The Royal Society.Google Scholar
Munarin, F., Petrini, P., Fare, S. & Tanzi, M.C. 2010 Structural properties of polysaccharide-based microcapsules for soft tissue regeneration. J. Mater. Sci. Mater. Med. 21 (1), 365–375.Google ScholarPubMed
Navot, Y. 1998 Elastic membranes in viscous shear flow. Phys. Fluids 10 (8), 1819–1833.Google Scholar
Neubauer, M.P., Poehlmann, M. & Fery, A. 2014 Microcapsule mechanics: from stability to function. Adv. Colloid Interface Sci. 207, 65–80.CrossRefGoogle Scholar
Peskin, C.S. 2002 The immersed boundary method. Acta Numerica 11, 479–517.CrossRefGoogle Scholar
Rachik, M., Barthes-Biesel, D., Carin, M. & Edwards-Levy, F. 2006 Identification of the elastic properties of an artificial capsule membrane with the compression test: effect of thickness. J. Colloid Interface Sci. 301 (1), 217–226.CrossRefGoogle ScholarPubMed
Ramanujan, S. & Pozrikidis, C. 1998 Deformation of liquid capsules enclosed by elastic membranes in simple shear flow: large deformations and the effect of fluid viscosities. J. Fluid Mech. 361, 117–143.CrossRefGoogle Scholar
Rosti, M.E., Brandt, L. & Mitra, D. 2018 Rheology of suspensions of viscoelastic spheres: deformability as an effective volume fraction. Phys. Rev. Fluids 3 (1), 012301.CrossRefGoogle Scholar
Sagis, L.M.C., de Ruiter, R., Miranda, F.J.R., de Ruiter, J., Schroën, K., van Aelst, A.C., Kieft, H., Boom, R. & van der Linden, E. 2008 Polymer microcapsules with a fiber-reinforced nanocomposite shell. Langmuir 24 (5), 1608–1612.CrossRefGoogle ScholarPubMed
Sarı, A., Alkan, C. & Karaipekli, A. 2010 Preparation, characterization and thermal properties of PMMA/n-heptadecane microcapsules as novel solid–liquid microPCM for thermal energy storage. Appl. Energ. 87 (5), 1529–1534.CrossRefGoogle Scholar
Sierou, A. & Brady, J.F. 2002 Rheology and microstructure in concentrated noncolloidal suspensions. J. Rheol. 46 (5), 1031–1056.CrossRefGoogle Scholar
Skalak, R. 1973 Modelling the mechanical behavior of red blood cells. Biorheology 10 (2), 229–238.CrossRefGoogle ScholarPubMed
Stickel, J.J. & Powell, R.L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37, 129–149.CrossRefGoogle Scholar
Succi, S. 2001 The Lattice Boltzmann Equation: for Fluid Dynamics and Beyond. Oxford University Press.Google Scholar
Suryanarayana, C., Rao, K.C. & Kumar, D. 2008 Preparation and characterization of microcapsules containing linseed oil and its use in self-healing coatings. Prog. Org. Coat. 63 (1), 72–78.CrossRefGoogle Scholar
Takeishi, N., Rosti, M.E., Imai, Y., Wada, S. & Brandt, L. 2019 Haemorheology in dilute, semi-dilute and dense suspensions of red blood cells. J. Fluid Mech. 872, 818–848.CrossRefGoogle Scholar
Taylor, G.I. 1934 The formation of emulsions in definable fields of flow. Proc. R. Soc. Lond. A 146 (858), 501–523.Google Scholar
Vlahovska, P.M. & Gracia, R.S. 2007 Dynamics of a viscous vesicle in linear flows. Phys. Rev. E 75 (1), 016313.CrossRefGoogle ScholarPubMed
Vlahovska, P.M., Young, Y.N., Danker, G. & Misbah, C. 2011 Dynamics of a non-spherical microcapsule with incompressible interface in shear flow. J. Fluid Mech. 678, 221–247.CrossRefGoogle Scholar
Walter, A., Rehage, H. & Leonhard, H. 2001 Shear induced deformation of microcapsules: shape oscillations and membrane folding. Colloids Surf. A 183, 123–132.CrossRefGoogle Scholar
Walter, J., Salsac, A.-V. & Barthès-Biesel, D. 2011 Ellipsoidal capsules in simple shear flow: prolate versus oblate initial shapes. J. Fluid Mech. 676, 318–347.CrossRefGoogle Scholar
Wu, J. & Shu, C. 2012 Simulation of three-dimensional flows over moving objects by an improved immersed boundary–lattice Boltzmann method. Intl J. Numer. Meth. Fluids 68 (8), 977–1004.CrossRefGoogle Scholar
Zhao, H. & Shaqfeh, E.S.G. 2013 The dynamics of a non-dilute vesicle suspension in a simple shear flow. J. Fluid Mech. 725, 709–731.Google Scholar