liquids torting the helical structure of the phase) symmmetry in a cholesteric. This chirali­ have a cholesteric texture in the cell nu­ and the nematic phase develops when the ty-induced morphological transition is cleus? equilibrium pitch of the cholesteric phase not yet completely understood. becomes larger than the sample thick­ In conclusion, these examples show the The author is a research director at the Ecole ness. This transition is first order and can variety of instabilities and pattern for­ Normale Supérieure in Lyon in France easily be observed when approaching a mation that arise during the growth of smectic phase, because the cholesteric liquid crystals. Most phenomena ob­ Further Reading pitch diverges at this transition. In this served in liquid crystals are generic and Dynamics of Curved Fronts edited by P. Pelcé way, it is possible to observe the growth of present in other systems such as metals, (Academic Press, New York, 1988) · "Mesophase the cholesteric phase into the nematic alloys or polymers. In particular, the Growth" by J. Bechhöfer in Pattern Formation in one. The main observation is that the tex­ questions concerning morphological Liquid Crystals edited by A. Buka and L. Kramer ture of the cholesteric phase varies with transitions, confinement effects, wave­ (Springer, 1996) . J.C. Géminard, P. Oswald, D. the front velocity. An example of direc­ length selection, secondary instabilies Temkin, J. Malthête Europhys. Lett. 22 69 (1993) tional growth is given in figure 3. This and transition to chaos or turbulent states • P. Oswald, J. Bechhoefer, A. Libchaber Phys. Rev. transition is due to a π-rotation of the are quite general. By contrast, problems Lett. 58 2318 (1987) · F. Melo, P. Oswald Phys. cholesteric fingers (stripes) whose ends relating to chirality are more specific but Rev. Lett. 64 1381 (1990) · J. Baudry's Thèse de are different due to the absence of mirror could play a role in biology—does DNA l'Ecole Normale Supérieure de Lyon, 1999 Colloidal dispersions— ink, paints, lubricants, cosmetics and pharmaceuticals, and foods such as milk and mayonnaiseare— are ubiquitous in everyday life and play a key role in many industrial processes. The dispersions are essentially two-phase systems, involving mesoscopic solid or liquid particles, suspended in a liquid Jean-Pierre Hansen, England and Peter N. Pusey, Scotland Phase Behaviour of Colloidal Systems he sizes of colloidal particles are typ­ and counter ions in solution form electric which is comparable to the colloid lattice Tically in the range 10 to 103 nanome­ double-layers that repel strongly whenev­ spacing, often giving them a beautiful tres—they are thus much larger than er neighbouring surfaces get closer than opalescent appearance (figures 1 and 2). atoms and molecules, but small enough the Debye screening length λD. that Brownian motion usually dominates In their studies of Brownian motion 90 Binary mixtures gravitational settling, allowing thermody­ years ago, Einstein and Perrin exploited Recent research has focused on colloid- namic equilibrium to be reached. already the analogy between colloids in a colloid mixtures and colloid-polymer Solid colloidal particles (to be consid­ liquid and atoms in a gas. There are, how­ mixtures. The two key parameters are the ered here) may be mineral crystallites, ever, significant differences between size ratio ξ = RB/RA, where RA and RB de­ like the gold solution studied by Faraday atoms and colloids, apart from the obvi­ note the radii of the two species, and the 150 years ago, or synthetic polymeric par­ ous change in spatial scale. In particular, degree of non-additivity of their interac­ ticles, like polystyrene spheres suspended the interactions between colloidal parti­ tions. In simple molecular systems, ξ is in water, or amorphous polymethyl­ cles may be tuned, eg by the addition of rarely smaller than ~ 0.5, while interac­ methacrylate (PMMA) particles dis­ salt to a dispersion of charged colloids, tions are almost invariably additive. In persed in hydrocarbon liquids. The im­ which leads to a reduction of λD, or by the colloidal systems, by contrast, ξ can take penetrable mesoscopic particles usually addition of free (non-adsorbing) poly­ rather extreme values (as small as 0.1 or interact via strong, attractive, short- mer, which leads to an effective attraction less) and colloid-polymer interactions are ranged van der Waals forces, which may between the colloids due to the osmotic highly non-additive. Thus in a mixture of lead to flocculation or coagulation of the depletion effect (explained later). two species of colloid, modelled as hard colloids into gel-like structures—this led These tuneable repulsive and attractive spheres, the centres of two particles can­ Graham to coin their name from the interactions between colloidal particles not come closer than the sum of their Greek κολλα for “glue”. Flocculation may lead to a rich variety of phase behaviour radii : additive interactions. Random-coil however be prevented by either steric or which has been thoroughly investigated, polymer molecules, however, are soft and electrostatic stabilization. Steric stabiliza­ both experimentally and theoretically. De­ can interpenetrate rather easily. Never­ tion is achieved by grafting polymer pending on colloid concentration, and the theless, they cannot penetrate the solid “brushes” on the surface of the colloidal concentration of added ions, polymers or colloidal particles. Thus, in a colloid- particles, providing an elastic repulsion other species, suspensions exhibit colloidal polymer mixture, the range of the colloid- when two particles come so close that analogues of the known phases of simple polymer interaction is greater than the their “brushes” are compressed. Colloidal molecular systems: gas, liquid, crystalline sum of the ranges of the self-interactions: particles in water generally acquire a solid and glass. Colloidal crystals in sus­ non-additive interactions. charge by dissociation of surface groups; pension are easily detected by the Bragg re­ The importance of non-additivity of the charged surface and microscopic co­ flection of visible light, the wavelength of inter-species interactions is illustrated europhysics news may/june 1999 81 liquids strikingly by comparing the experimental (phase separation of the initially well- within each cube. phase diagrams (figures 3a and 3b) of a mixed suspension can take several weeks). Figure 2b shows the phase diagram of colloid-colloid mixture and a colloid- Four kinds of colloidal crystal are in­ suspensions containing PMMA particles polymer mixture at almost the same size volved: pure A and pure B, and the binary of one size, radius RA = 228 nm, and ran­ ratio, ξ ~ 0.58. “colloidal alloys” AB2 and AB13. dom-coil polymer molecules with radius For a one-component hard-sphere sus­ of gyration Rg = 130 nm (we take the poly­ Experiment pension, ØΑ or ØΒ = 0 (ie on one of the ax­ mer to be the B species so that Rg = RB). Figure 3a shows the phase diagram of sus­ es of figure 3a), colloidal crystals are first Despite the similar size ratios, the phase pensions containing mixtures of un­ formed at Øα,β ~ 0.50 and crystallization behaviour of the colloid-polymer mixture charged, sterically-stabilised PMMA par­ is complete at Øα,β ~ 0.55. In the binary (figure 3b) is markedly different from that ticles of radii RA = 321 nm, RB = 186 nm. mixture, freezing into pure A or pure B is of the colloid-colloid mixture (figure 3a). Thin grafted polymer brushes ensure that also observed near to the axes of the Again at low concentrations, ØΑ or the interparticle interaction is essentially phase diagram. Away from the axes, how­ ØΒ < 0.50, a single fluid phase is observed. that of hard spheres. The axes, ØA and ØΒ, ever, the preferred crystalline structures However, now at somewhat higher con­ in figure 3a are the fractions of the sample are the remarkable AB2 and AB13 alloys. centrations there is a region of coexis­ volume occupied by each species, defined AB2 (see figure 2) is a layered structure tence of two fluid phases, dilute “colloidal by Øα,β= nA,B (4π/3)R 3a,b, where nA,B is the consisting of planes of the large A parti­ gas” and concentrated “colloidal liquid”, number of particles per unit volume. For a cles in a triangular (or hexagonal) lattice, and a gas-liquid critical point. At still total volume fraction, Øα + ØΒ, up to about separated by planes containing twice as higher concentrations we find a three- 0.50 the suspension remains in a homoge­ many of the smaller B particles. AB13 (fig­ phase “triple triangle” where colloidal neous fluid-like phase throughout which ure 1) is a yet more complex structure gas, liquid and crystal coexist (figure 3b). the particles diffuse in Brownian motion. whose unit cell contains 112 particles. The At the highest concentrations, gas-crystal For ØΑ + ØΒ > 0.50, however, a rich variety A particles are arranged on a simple cu­ coexistence or long-lived metastable of two- and three-phase regions of fluid- bic lattice; oriented clusters of 13 B parti­ “gels” are observed. solid and solid-solid coexistence is found cles with an icosahedral arrangement lie Theory and simulation The theoretical description of colloidal Fig 1 An AB13 colloidal crystal, dispersions, which are multi-component comprising microscopic PMMA (poly­ systems involving widely different length methylmethacrylate) particles of two scales, requires some coarse-graining. sizes at radius ratio RB / RA = 0.58. The liquid suspension medium is gener­ The particles are suspended in a ally treated as a continuum, characterized exclusively by macroscopic properties. mixture of hydrocarbon liquids chosen Mixtures of hard spheres are readily sim­ to nearly match the refractive index of ulated by Monte Carlo or Molecular Dy­ the particles, providing a nearly namics methods and, for ξ ≈ 0.58, such transparent suspension.