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Theories of binary fluid mixtures: from phase-separation kinetics to active emulsions

Published online by Cambridge University Press:  18 December 2017

Michael E. Cates Open the ORCID record for Michael E. Cates [Opens in a new window]  and
Elsen Tjhung
Michael E. Cates*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
Elsen Tjhung
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WA, UK
*
†Email address for correspondence: [email protected]
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Abstract

Binary fluid mixtures are examples of complex fluids whose microstructure and flow are strongly coupled. For pairs of simple fluids, the microstructure consists of droplets or bicontinuous demixed domains and the physics is controlled by the interfaces between these domains. At continuum level, the structure is defined by a composition field whose gradients – which are steep near interfaces – drive its diffusive current. These gradients also cause thermodynamic stresses which can drive fluid flow. Fluid flow in turn advects the composition field, while thermal noise creates additional random fluxes that allow the system to explore its configuration space and move towards the Boltzmann distribution. This article introduces continuum models of binary fluids, first covering some well-studied areas such as the thermodynamics and kinetics of phase separation, and emulsion stability. We then address cases where one of the fluid components has anisotropic structure at mesoscopic scales creating nematic (or polar) liquid-crystalline order; this can be described through an additional tensor (or vector) order parameter field. We conclude by outlining a thriving area of current research, namely active emulsions, in which one of the binary components consists of living or synthetic material that is continuously converting chemical energy into mechanical work. Such activity can be modelled with judicious additional terms in the equations of motion for simple or liquid-crystalline binary fluids. Throughout, the emphasis of the article is on presenting the theoretical tools needed to address a wide range of physical phenomena. Examples include the kinetics of fluid–fluid demixing from an initially uniform state; the result of imposing a steady macroscopic shear flow on this demixing process; and the diffusive coarsening, Brownian motion and coalescence of emulsion droplets. We discuss strategies to create long-lived emulsions by adding trapped species, solid particles, or surfactants; to address the latter, we outline the theory of bending energy for interfacial films. In emulsions where one of the components is liquid-crystalline, ‘anchoring’ terms can create preferential orientation tangential or normal to the fluid–fluid interface. These allow droplets of an isotropic fluid in a liquid crystal (or vice versa) to support a variety of topological defects, which we describe, altering their interactions and stability. Addition of active terms to the equations of motion for binary simple fluids creates a model of ‘motility-induced’ phase separation, where demixing stems from self-propulsion of particles rather than their interaction forces, altering the relation between interfacial structure and fluid stress. Coupling activity to binary liquid crystal dynamics creates models of active liquid-crystalline emulsion droplets. Such droplets show various modes of locomotion, some of which strikingly resemble the swimming or crawling motions of biological cells.

JFM classification

Biological Fluid Dynamics: Biological Fluid Dynamics Complex Fluids: Complex Fluids Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows
Type
JFM Perspectives
Information
Journal of Fluid Mechanics , Volume 836 , 10 February 2018 , P1
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Copyright
© 2017 Cambridge University Press 

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References

Andelman, D., Cates, M. E., Roux, D. & Safran, S. A. 1987 Structure and phase equilibria of microemulsions. J. Chem. Phys. 87, 7229–7241.CrossRefGoogle Scholar
Anderson, V. J., Terentjev, E. M., Meeker, S. P., Crain, J. & Poon, W. C. K. 2001 Cellular solid behaviour of liquid crystal colloids – 1. Phase separation and morphology. Eur. Phys. J. E 4, 11–20.Google Scholar
Aranson, I. S. 2016 Physical Models of Cell Motility. Springer.CrossRefGoogle Scholar
Aveyard, R. 2012 Can Janus particles give thermodynamically stable Pickering emulsions? Soft Matt. 8, 5233–5240.CrossRefGoogle Scholar
Barnhart, E. L., Lee, K.-C., Keren, K., Mogilner, A. & Theriot, J. A. 2011 An adhesion-dependent switch between mechanisms that determine motile cell shape. PLoS Biol. 9, e1001059.CrossRefGoogle ScholarPubMed
Beris, A. N. & Edwards, B. J. 1994 Thermodynamics of Flowing Systems with Internal Microstructure. Oxford University Press.CrossRefGoogle Scholar
Bibette, J., Leal-Calderon, F., Schmitt, V. & Poulin, P. 2002 Emulsion Science. Springer.CrossRefGoogle Scholar
Binks, B. P. & Horozov, T. S.(Eds) 2006 Colloidal Particles at Liquid Interfaces. Cambridge University Press.Google Scholar
Bouteiller, L. & Lebarny, P. 1996 Polymer-dispersed liquid crystals: preparation, operation and application. Liquid Crystals 21, 157–174.CrossRefGoogle Scholar
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111–157.Google Scholar
Bray, A. J. 1994 Theory of phase-ordering kinetics. Adv. Phys. 43, 357–459.CrossRefGoogle Scholar
Brugues, J. & Needleman, D. 2014 Physical basis of spindle self-organization. Proc. Natl Acad. Sci. USA 111, 18496–18500.CrossRefGoogle ScholarPubMed
Buttinoni, I., Bialke, J., Kummel, F., Lowen, H., Bechinger, C. & Speck, T. 2013 Dynamical clustering and phase separation in suspensions of self-propelled colloidal particles. Phys. Rev. Lett. 110, 238301.Google Scholar
Camley, B. A., Zhang, Y., Zhao, Y., Li, B., Ben-Jacob, E., Levine, H. & Rappel, W.-J. 2014 Polarity mechanism such as contact inhibition of locomotion regulate persistent rotational motion of mammalian cells on micropatterns. Proc. Natl Acad. Sci. USA 111, 14770–14775.CrossRefGoogle ScholarPubMed
Cates, M. E.2012 Complex fluids: the physics of emulsions, arXiv:1209.2290; chap. 10 in Soft Interfaces (Proceedings of les Houches 2012 Summer School, Session XCVIII) (ed. L. Bocquet, D. Quéré, T. A. Witten, L. F. Cugliandolo et al.), Oxford University Press, 2017.Google Scholar
Cates, M. E. & Clegg, P. S. 2008 Bijels: a new class of soft materials. Soft Matt. 4, 2132–2138.Google Scholar
Cates, M. E., Henrich, O., Marenduzzo, D. & Stratford, K. 2009 Lattice Boltzmann simulations of liquid crystalline fluids: active gels and blue phases. Soft Matt. 5, 3791–3800.CrossRefGoogle Scholar
Cates, M. E. & Tailleur, J. 2015 Motility-induced phase separation. Annu. Rev. Condens. Matter Phys. 6, 219–244.CrossRefGoogle Scholar
Cavallaro, M., Botto, L., Lewandowski, E. P., Wang, M. & Stebe, K. J. 2011 Curvature-driven capillary migration and assembly of rod-like particles. Proc. Natl Acad. Sci. USA 108, 20923–20928.Google Scholar
Chaikin, P. M. & Lubensky, T. C. 1995 Principles of Condensed Matter Physics. Cambridge University Press.Google Scholar
Clegg, P. S., Tavacoli, J. W. & Wilde, P. J. 2016 One-step production of multiple emulsions: microfluidic, polymer-stabilized and particle-stabilized approaches. Soft Matt. 12, 998–1008.CrossRefGoogle ScholarPubMed
David, F. 2004 Geometry and field theory of random surfaces and membranes. In Statistical Mechanics of Membranes and Surfaces (ed. Nelson, D. R., Piran, T. & Weinberg, S.), World Scientific.Google Scholar
Doi, M. & Ohta, T. 1991 Dynamics and rheology of complex interfaces. J. Chem. Phys. 95, 1242–1248.Google Scholar
Fernandez-Nieves, A., Link, D. R., Marquez, M. & Weitz, D. A. 2007 Topological changes in bipolar nematic droplets under flow. Phys. Rev. Lett. 98, 087801.CrossRefGoogle ScholarPubMed
Fielding, S. M. 2008 Role of inertia in nonequilibrium steady states of sheared binary fluids. Phys. Rev. E 77, 021504.Google Scholar
Fodor, E., Nardini, C., Cates, M. E., Tailleur, J., Visco, P. & van Wijland, F. 2016 How far from equilibrium is active matter? Phys. Rev. Lett. 117, 038103.Google Scholar
Fryd, M. M. & Mason, T. G. 2012 Advanced nanoemulsions. Annu. Rev. Phys. Chem. 63, 493–518.Google Scholar
Furukawa, H. 1985 Effect of inertia on droplet growth in a fluid. Phys. Rev. A 31, 1103–1108.CrossRefGoogle Scholar
de Gennes, P. G. & Prost, J. 2002 The Physics of Liquid Crystals, 2nd edn. Oxford University Press.Google Scholar
de Gennes, P.-G. & Taupin, C. 1982 Microemulsions and the flexibility of oil–water interfaces. J. Phys. Chem. 86, 2294–2304.CrossRefGoogle Scholar
Giomi, L., Bowick, M. J., Ma, X. & Marchetti, M. C. 2013 Defect annihilation and proliferation in active nematics. Phys. Rev. Lett. 110, 228101.Google Scholar
Gompper, G. & Schick, M. 1994 Self assembling amphiphilic systems. In Phase Transitions and Critical Phenomena (ed. Domb, C. & Lebowitz, J. L.), vol. 16. Academic.Google Scholar
Gonnella, G., Orlandini, E. & Yeomans, J. M. 1999 Phase separation in two-dimensional fluids: the role of noise. Phys. Rev. E 59, R4741–R4744.Google ScholarPubMed
Hatwalne, Y., Ramaswamy, S., Rao, M. & Simha, R. A. 2004 Rheology of active-particle suspensions. Phys. Rev. Lett. 92, 118101.CrossRefGoogle ScholarPubMed
Hawkins, R. J., Poincloux, R., Benichou, O., Piel, M., Chavrier, P. & Voituriez, R. 2011 Spontaneous contractility-mediated cortical flow generates cell migration in three-dimensional environments. Biophys. J. 101, 1041–1045.CrossRefGoogle ScholarPubMed
Hemingway, E. J., Maitra, A., Banerjee, S., Marchetti, M. C., Ramaswamy, S., Fielding, S. M. & Cates, M. E. 2015 Active viscoelastic matter: from bacterial drag reduction to turbulent solids. Phys. Rev. Lett. 114, 098302.Google Scholar
Herzig, E. M., White, K. A., Schofield, A. B., Poon, W. C. K. & Clegg, P. S. 2007 Bicontinuous emulsions stabilized solely by colloidal particles. Nat. Mater. 6, 966–971.Google Scholar
Hohenberg, P. C. & Halperin, B. I. 1977 Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435–479.CrossRefGoogle Scholar
Huse, D. A. & Leibler, S. 1988 Phase behaviour of an ensemble of nonintersecting random fluid films. J. Phys. France 49, 605–621.CrossRefGoogle Scholar
Ishikawa, T., Locsei, J. T. & Pedley, T. J. 2008 Development of coherent structures in concentrated suspensions of swimming model micro-organisms. J. Fluid Mech. 615, 401–431.Google Scholar
Kendon, V. M., Cates, M. E., Pagonabarraga, I., Desplat, J.-C. & Bladon, P. 2001 Inertial effects in three-dimensional spinodal decomposition of a symmetric binary fluid mixture: a lattice Boltzmann study. J. Fluid Mech. 440, 147–203.CrossRefGoogle Scholar
Kruse, K., Joanny, J. F-, Julicher, F., Prost, J. & Sekimoto, K. 2005 Generic theory of active polar gels: a paradigm for cytoskeletal dynamics. Eur. Phys. J. E 16, 5–16.Google ScholarPubMed
Kung, W., Marchetti, M. C. & Saunders, K. 2006 Hydrodynamics of polar liquid crystals. Phys. Rev. E 73, 031708.Google Scholar
Landau, L. V. & Lifshitz, I. M. 1959 Fluid Mechanics. Pergamon.Google Scholar
Landfester, K. 2003 Miniemulsions for nanoparticle synthesis. Topics Curr. Chem. 227, 75–123.Google Scholar
Larson, R. G. 1999 The Structure and Rheology of Complex Fluids. Oxford University Press.Google Scholar
Lattuada, M. & Hatton, T. A. 2011 Synthesis, properties and applications of Janus nanoparticles. Nano Today 6, 286–308.Google Scholar
Lee, M. N., Thijssen, J. H. J., Witt, J. A. & Clegg, P. S. 2013 Making a robust interfacial scaffold: bijel rheology and its link to processability. Adv. Funct. Mater. 23, 417–423.CrossRefGoogle Scholar
Leoni, M., Manyuhina, O. V., Bowick, M. J. & Marchetti, M. C. 2017 Defect driven shapes in nematic droplets: analogies with cell division. Soft Matt. 13, 1257–1266.Google Scholar
Lober, J., Ziebert, F. & Aranson, I. S. 2015 Collisions of deformable cells lead to collective migration. Sci. Rep. 5, 9172.CrossRefGoogle ScholarPubMed
Lopez-Leon, T. & Fernandez-Nieves, A. 2011 Drops and shells of liquid crystal. Colloid Polym. Sci. 289, 345–359.Google Scholar
Loudet, J. C., Barois, P. & Poulin, P. 2000 Colloidal ordering from phase separation in a liquid-crystalline continuous phase. Nature 407, 611–613.Google Scholar
Lubensky, T. C., Pettey, D., Currier, N. & Stark, H. 1998 Topological defects and interactions in nematic emulsions. Phys. Rev. E 57, 610–625.Google Scholar
Marchetti, M. C., Joanny, J.-F., Ramaswamy, S., Liverpool, T. B., Prost, J. R. M. & Simha, R. A. 2013 Hydrodynamics of soft active matter. Rev. Mod. Phys. 85, 1143.Google Scholar
Mogilner, A. 2009 Mathematics of cell motility: have we got its number? J. Math. Biol. 58, 105–134.Google Scholar
Nardini, C., Fodor, E., Tjhung, E., van Wijland, F., Tailleur, J. & Cates, M. E. 2017 Entropy production in field theories without time reversal symmetry: quantifying the non-equilibrium character of active matter. Phys. Rev. X 7, 021007.Google Scholar
Nazarenko, V. G., Nych, A. B. & Lev, B. I. 2001 Crystal structure in nematic emulsion. Phys. Rev. Lett. 87, 075504.CrossRefGoogle ScholarPubMed
Onuki, A. 2002 Phase Transition Dynamics. Cambridge University Press.Google Scholar
Poincloux, R., Collin, O., Lizarraga, F., Romao, M., Debray, M., Piel, M. & Chavrier, P. 2011 Contractility of the cell rear drives invasion of breast tumor cells in 3D Matrigel. Proc. Natl Acad. Sci. USA 108, 1943–1948.Google Scholar
Poulin, P. 1999 Novel phases and colloidal assemblies in liquid crystals. Curr. Opin. Colloid Interface Sci. 4, 66–71.CrossRefGoogle Scholar
Poulin, P., Stark, H., Lubensky, T. C. & Weitz, D. A. 1997 Novel colloidal interactions in anisotropic fluids. Science 275, 1770–1773.Google Scholar
Prinsen, P. & van der Schoot, P. 2003 Shape and director-field transformation of tactoids. Phys. Rev. E 68, 021701.Google Scholar
Roux, D, Coulon, C. & Cates, M. E. 1992 Sponge phases in surfactant solutions. J. Phys. Chem. 96, 4174–4187.Google Scholar
Safran, S. A. 2003 Statistical Thermodynamics of Surfaces, Interfaces and Membranes. Westview Press.Google Scholar
Safran, S. A. & Turkevich, L. A. 1983 Phase diagrams for microemulsions. Phys. Rev. Lett. 50, 1930–1933.CrossRefGoogle Scholar
Saha, S., Golestanian, R. & Ramasawmy, S. 2014 Clusters, asters and collective oscillations in chemotactic colloids. Phys. Rev. E 89, 062316.Google Scholar
Saintillan, D. & Shelley, M. J. 2007 Orientational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett. 99, 058102.Google Scholar
Sanchez, T., Chen, D. T. N., Decamp, S. J., Heymann, M. & Dogic, Z. 2009 Spontaneous motion in hierarchically assembled active matter. Nature 491, 431–435.CrossRefGoogle Scholar
Sanz, E., White, K. A., Clegg, P. S. & Cates, M. E. 2009 Colloidal gels assembled via a temporary interfacial scaffold. Phys. Rev. Lett. 103, 255502.CrossRefGoogle Scholar
Schnitzer, M. J. 1993 Theory of continuum random walks and application to chemotaxis. Phys. Rev. E 48, 2553–2568.Google Scholar
Shimuzu, R. & Tanaka, H. 2015 A novel coarsening mechanism of droplets in immiscible fluid mixtures. Nat. Commun. 6, 7407.CrossRefGoogle Scholar
Siggia, E. 1979 Late stages of spinodal decomposition in binary mixtures. Phys. Rev. A 20, 595–605.Google Scholar
Solon, A. P., Fily, Y., Baskaran, A., Cates, M. E., Kafri, Y., Kardar, M. & Tailleur, J. 2015 Pressure is not a state function for generic active fluids. Nat. Phys. 11, 673–678.Google Scholar
Stansell, P., Stratford, K., Desplat, J.-C., Adhikari, R. & Cates, M. E. 2006 Nonequilibrium steady states in sheared binary fluids. Phys. Rev. Lett. 96, 085701.Google Scholar
Stenhammar, J., Tiribocchi, A., Allen, R. J., Marenduzzo, D. & Cates, M. E. 2013 Continuum theory of phase separation kinetics for active Brownian particles. Phys. Rev. Lett. 111, 145702.Google Scholar
Stenhammar, J., Wittkowski, R., Marenduzzo, D. & Cates, M. E. 2016 Light-induced self-assembly of active rectification devices. Sci. Adv. 2, e1501850.CrossRefGoogle ScholarPubMed
Stratford, K., Adhikari, R., Pagonabarraga, I., Desplat, J.-C. & Cates, M. E. 2005 Colloidal jamming at interfaces: a route to fluid-bicontinuous gels. Science 309, 2198–2201.Google Scholar
Stratford, K., Desplat, J.-C., Stansell, P. & Cates, M. E. 2007 Binary fluids under steady shear in three dimensions. Phys. Rev. E 76, 030501(R).Google ScholarPubMed
Subramaniam, A. B., Abkarian, M. & Stone, H. A. 2005 Controlled assembly of jammed colloidal shells on fluid droplets. Nat. Mater. 4, 553–556.Google Scholar
Sulaiman, N., Marenduzzo, D. & Yeomans, J. 2006 Lattice Boltzmann algorithm to simulate isotropic-nematic emulsions. Phys. Rev. Lett. 74, 041708.Google ScholarPubMed
Tiribocchi, A., Da Re, M., Marenduzzo, D. & Orlandini, E. 2016 Shear dynamics of an inverted nematic emulsion. Soft Matt. 12, 8195–8213.CrossRefGoogle ScholarPubMed
Tiribocchi, A., Wittkowski, R., Marenduzzo, D. & Cates, M. E. 2015 Active model H: scalar active matter in a momentum-conserving fluid. Phys. Rev. Lett. 115, 188302.Google Scholar
Tjhung, E., Marenduzzo, D. & Cates, M. E. 2012 Spontaneous symmetry breaking in active droplets provides a generic route to motility. Proc. Natl Acad. Sci. USA. 109, 12381–12386.Google Scholar
Tjhung, E., Tiribocchi, A., Marenduzzo, D. & Cates, M. E. 2015 A minimal physical model captures the shapes of crawling cells. Nat. Commun. 6, 5420.CrossRefGoogle ScholarPubMed
Verkhovsky, A. B., Svitkina, T. M. & Borisy, G. G. 1999 Self-polarization and directional motility of cytoplasm. Curr. Biol. 9, 11–20.Google Scholar
Wagner, A. & Cates, M. E. 2001 Phase ordering of two-dimensional symmetric binary fluids: a droplet scaling state. Europhys. Lett. 56, 556–562.CrossRefGoogle Scholar
Weaire, D. & Hutzler, S. 1999 The Physics of Foams. Oxford University Press.Google Scholar
Webster, A. J. & Cates, M. E. 1998 Stabilization of emulsions by trapped species. Langmuir 14, 2068–2079.Google Scholar
Wittkowski, R., Tiribocchi, A., Stenhammar, J., Allen, R. J., Marenduzzo, D. & Cates, M. E. 2014 Scalar 𝜙4 field theory for active-particle phase separation. Nat. Commun. 5, 4351.Google Scholar
Wolff, K., Marenduzzo, D. & Cates, M. E. 2012 Cytoplasmic streaming in plant cells: the role of wall slip. J. R. Soc. Interface 71, 1398.Google Scholar
Yam, P. T., Wilson, C. A., Ji, L., Hebert, B., Barnhart, E. L., Dye, N. A., Wiseman, P. W., Danuser, G. & Theriot, J. A. 2007 Actin–myosin network reorganization breaks symmetry at the cell rear to spontaneously initiate polarized cell motility. J. Cell Biol. 178, 1207–1221.CrossRefGoogle ScholarPubMed
Ziebert, F. & Aranson, I. S. 2013 Effects of adhesion dynamics and substrate compliance on the shape and motility of crawling cells. PLoS One 8, e64511.Google Scholar