Skip to main content Accesibility Help

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

  • Michael E. Cates (a1) and Elsen Tjhung (a1)

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.

Corresponding author
Email address for correspondence:
Hide All
Andelman, D., Cates, M. E., Roux, D. & Safran, S. A. 1987 Structure and phase equilibria of microemulsions. J. Chem. Phys. 87, 72297241.
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, 1120.
Aranson, I. S. 2016 Physical Models of Cell Motility. Springer.
Aveyard, R. 2012 Can Janus particles give thermodynamically stable Pickering emulsions? Soft Matt. 8, 52335240.
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.
Beris, A. N. & Edwards, B. J. 1994 Thermodynamics of Flowing Systems with Internal Microstructure. Oxford University Press.
Bibette, J., Leal-Calderon, F., Schmitt, V. & Poulin, P. 2002 Emulsion Science. Springer.
Binks, B. P. & Horozov, T. S.(Eds) 2006 Colloidal Particles at Liquid Interfaces. Cambridge University Press.
Bouteiller, L. & Lebarny, P. 1996 Polymer-dispersed liquid crystals: preparation, operation and application. Liquid Crystals 21, 157174.
Brady, J. F. & Bossis, G. 1988 Stokesian dynamics. Annu. Rev. Fluid Mech. 20, 111157.
Bray, A. J. 1994 Theory of phase-ordering kinetics. Adv. Phys. 43, 357459.
Brugues, J. & Needleman, D. 2014 Physical basis of spindle self-organization. Proc. Natl Acad. Sci. USA 111, 1849618500.
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.
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, 1477014775.
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.
Cates, M. E. & Clegg, P. S. 2008 Bijels: a new class of soft materials. Soft Matt. 4, 21322138.
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, 37913800.
Cates, M. E. & Tailleur, J. 2015 Motility-induced phase separation. Annu. Rev. Condens. Matter Phys. 6, 219244.
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, 2092320928.
Chaikin, P. M. & Lubensky, T. C. 1995 Principles of Condensed Matter Physics. Cambridge University Press.
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, 9981008.
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.
Doi, M. & Ohta, T. 1991 Dynamics and rheology of complex interfaces. J. Chem. Phys. 95, 12421248.
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.
Fielding, S. M. 2008 Role of inertia in nonequilibrium steady states of sheared binary fluids. Phys. Rev. E 77, 021504.
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.
Fryd, M. M. & Mason, T. G. 2012 Advanced nanoemulsions. Annu. Rev. Phys. Chem. 63, 493518.
Furukawa, H. 1985 Effect of inertia on droplet growth in a fluid. Phys. Rev. A 31, 11031108.
de Gennes, P. G. & Prost, J. 2002 The Physics of Liquid Crystals, 2nd edn. Oxford University Press.
de Gennes, P.-G. & Taupin, C. 1982 Microemulsions and the flexibility of oil–water interfaces. J. Phys. Chem. 86, 22942304.
Giomi, L., Bowick, M. J., Ma, X. & Marchetti, M. C. 2013 Defect annihilation and proliferation in active nematics. Phys. Rev. Lett. 110, 228101.
Gompper, G. & Schick, M. 1994 Self assembling amphiphilic systems. In Phase Transitions and Critical Phenomena (ed. Domb, C. & Lebowitz, J. L.), vol. 16. Academic.
Gonnella, G., Orlandini, E. & Yeomans, J. M. 1999 Phase separation in two-dimensional fluids: the role of noise. Phys. Rev. E 59, R4741R4744.
Hatwalne, Y., Ramaswamy, S., Rao, M. & Simha, R. A. 2004 Rheology of active-particle suspensions. Phys. Rev. Lett. 92, 118101.
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, 10411045.
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.
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, 966971.
Hohenberg, P. C. & Halperin, B. I. 1977 Theory of dynamic critical phenomena. Rev. Mod. Phys. 49, 435479.
Huse, D. A. & Leibler, S. 1988 Phase behaviour of an ensemble of nonintersecting random fluid films. J. Phys. France 49, 605621.
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, 401431.
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, 147203.
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, 516.
Kung, W., Marchetti, M. C. & Saunders, K. 2006 Hydrodynamics of polar liquid crystals. Phys. Rev. E 73, 031708.
Landau, L. V. & Lifshitz, I. M. 1959 Fluid Mechanics. Pergamon.
Landfester, K. 2003 Miniemulsions for nanoparticle synthesis. Topics Curr. Chem. 227, 75123.
Larson, R. G. 1999 The Structure and Rheology of Complex Fluids. Oxford University Press.
Lattuada, M. & Hatton, T. A. 2011 Synthesis, properties and applications of Janus nanoparticles. Nano Today 6, 286308.
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, 417423.
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, 12571266.
Lober, J., Ziebert, F. & Aranson, I. S. 2015 Collisions of deformable cells lead to collective migration. Sci. Rep. 5, 9172.
Lopez-Leon, T. & Fernandez-Nieves, A. 2011 Drops and shells of liquid crystal. Colloid Polym. Sci. 289, 345359.
Loudet, J. C., Barois, P. & Poulin, P. 2000 Colloidal ordering from phase separation in a liquid-crystalline continuous phase. Nature 407, 611613.
Lubensky, T. C., Pettey, D., Currier, N. & Stark, H. 1998 Topological defects and interactions in nematic emulsions. Phys. Rev. E 57, 610625.
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.
Mogilner, A. 2009 Mathematics of cell motility: have we got its number? J. Math. Biol. 58, 105134.
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.
Nazarenko, V. G., Nych, A. B. & Lev, B. I. 2001 Crystal structure in nematic emulsion. Phys. Rev. Lett. 87, 075504.
Onuki, A. 2002 Phase Transition Dynamics. Cambridge University Press.
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, 19431948.
Poulin, P. 1999 Novel phases and colloidal assemblies in liquid crystals. Curr. Opin. Colloid Interface Sci. 4, 6671.
Poulin, P., Stark, H., Lubensky, T. C. & Weitz, D. A. 1997 Novel colloidal interactions in anisotropic fluids. Science 275, 17701773.
Prinsen, P. & van der Schoot, P. 2003 Shape and director-field transformation of tactoids. Phys. Rev. E 68, 021701.
Roux, D, Coulon, C. & Cates, M. E. 1992 Sponge phases in surfactant solutions. J. Phys. Chem. 96, 41744187.
Safran, S. A. 2003 Statistical Thermodynamics of Surfaces, Interfaces and Membranes. Westview Press.
Safran, S. A. & Turkevich, L. A. 1983 Phase diagrams for microemulsions. Phys. Rev. Lett. 50, 19301933.
Saha, S., Golestanian, R. & Ramasawmy, S. 2014 Clusters, asters and collective oscillations in chemotactic colloids. Phys. Rev. E 89, 062316.
Saintillan, D. & Shelley, M. J. 2007 Orientational order and instabilities in suspensions of self-locomoting rods. Phys. Rev. Lett. 99, 058102.
Sanchez, T., Chen, D. T. N., Decamp, S. J., Heymann, M. & Dogic, Z. 2009 Spontaneous motion in hierarchically assembled active matter. Nature 491, 431435.
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.
Schnitzer, M. J. 1993 Theory of continuum random walks and application to chemotaxis. Phys. Rev. E 48, 25532568.
Shimuzu, R. & Tanaka, H. 2015 A novel coarsening mechanism of droplets in immiscible fluid mixtures. Nat. Commun. 6, 7407.
Siggia, E. 1979 Late stages of spinodal decomposition in binary mixtures. Phys. Rev. A 20, 595605.
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, 673678.
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.
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.
Stenhammar, J., Wittkowski, R., Marenduzzo, D. & Cates, M. E. 2016 Light-induced self-assembly of active rectification devices. Sci. Adv. 2, e1501850.
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, 21982201.
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).
Subramaniam, A. B., Abkarian, M. & Stone, H. A. 2005 Controlled assembly of jammed colloidal shells on fluid droplets. Nat. Mater. 4, 553556.
Sulaiman, N., Marenduzzo, D. & Yeomans, J. 2006 Lattice Boltzmann algorithm to simulate isotropic-nematic emulsions. Phys. Rev. Lett. 74, 041708.
Tiribocchi, A., Da Re, M., Marenduzzo, D. & Orlandini, E. 2016 Shear dynamics of an inverted nematic emulsion. Soft Matt. 12, 81958213.
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.
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, 1238112386.
Tjhung, E., Tiribocchi, A., Marenduzzo, D. & Cates, M. E. 2015 A minimal physical model captures the shapes of crawling cells. Nat. Commun. 6, 5420.
Verkhovsky, A. B., Svitkina, T. M. & Borisy, G. G. 1999 Self-polarization and directional motility of cytoplasm. Curr. Biol. 9, 1120.
Wagner, A. & Cates, M. E. 2001 Phase ordering of two-dimensional symmetric binary fluids: a droplet scaling state. Europhys. Lett. 56, 556562.
Weaire, D. & Hutzler, S. 1999 The Physics of Foams. Oxford University Press.
Webster, A. J. & Cates, M. E. 1998 Stabilization of emulsions by trapped species. Langmuir 14, 20682079.
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.
Wolff, K., Marenduzzo, D. & Cates, M. E. 2012 Cytoplasmic streaming in plant cells: the role of wall slip. J. R. Soc. Interface 71, 1398.
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, 12071221.
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.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

JFM classification


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