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torting the helical structure of the ) 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 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 . 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) · " the cholesteric phase into the nematic alloys or . 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 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 —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 or liquid , 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 in form electric which is comparable to the 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). and , but small enough the Debye screening length λD. that usually dominates In their studies of Brownian motion 90 Binary 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 in a colloid mixtures and colloid- Solid colloidal particles (to be consid­ liquid and atoms in a . 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 , 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 of charged colloids, tions are almost invariably additive. In persed in 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 of lead to flocculation or coagulation of the depletion effect (explained later). two species of colloid, modelled as hard colloids into -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 , 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 . 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 ( of the initially well- within each cube. phase diagrams (figures 3a and 3b) of a mixed can take several weeks). Figure 2b shows the of colloid-colloid mixture and a colloid- Four kinds of colloidal 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 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, 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 , 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 occupied by each species, defined AB2 (see figure 2) is a layered structure tence of two 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 , 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 “” 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 , 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 of ulated by Monte Carlo or Molecular Dy­ the particles, providing a nearly namics methods and, for ξ ≈ 0.58, such transparent suspension. The sample is simulations have led to phase diagrams illuminated from behind by white light. involving AB2 and AB13 alloys which agree Bragg diffraction of the light by small quite well with experiment (see figure 3a). crystallites causes the rainbow-like Intriguingly, the crystallization of hard scattering spheres, both one- and two-component, is driven by entropic contributions to their Photograph by A.H. Krall free energies. At a high enough concen­ tration the particles have greater freedom for local motions in the ordered struc­ Fig 2 Scanning electron micrograph of tures than in the metastable from a dried AB2 crystal comprising sub­ which they grow: the gain in as­ micron particles of two sizes. The sociated with local “free volume” more crystals were formed in suspension and than offsets the loss of entropy resulting the liquid was then allowed to from the adoption of long-ranged order. evaporate slowly over several weeks. Colloid-polymer mixtures have also The resulting dry "compact" of particles been investigated by theory—in the sim­ was fractured and sputter-coated with plest models the polymer molecules are gold for study in the electron assumed to be totally interpenetrable— and by simulation, modelling the poly­ microscope. While some particles at the mer by a lattice random walk. Again, rea­ surface have been displaced, long- sonable agreement with experiment is ranged binary order is clearly evident found, and again the phase separations throughout the picture. Layers of large are entropically driven. The polymer mol­ particles, viewed almost edge-on, are ecules cannot penetrate the particles, so separated by layers of small particles there is a shell, or “depletion zone”, of outer radius Ra + Rg around each Micrograph by A.B. Schofield from which the centres of the polymer molecules are excluded (inset, figure 3b).

82 europhysics news may/june 1999 liquids

Fig 3a Fig 3b Fig 3c

Fig 3a Phase diagram of a binary mixture Fig 3b Experimental phase diagram of a depletion zones, indicated by the black of hard spheres at radius ratio mixture of colloid and non-adsorbing circles, around the particles. When the RB/RA = 0.58. The axes are the volume polymer (PMMA particles and polystyrene depletion zones of two particles overlap, fractions, ØΑ and ØΒ, of the individual polymer) at polymer-to- ratio the uneven distribution of polymer around species. The lines demarcate different ~ 0.57. The x-axis is the volume fraction of the particles results in an osmotic force regions of the phase diagram, predicted by colloid; the left-hand y-axis gives the pushing them together theory and computer simulation. weight fraction of polymer (CP, in mg ml-1) At low total concentrations colloidal fluid and the right-hand y-axis the polymer Fig 3c Schematic of the colloid-polymer is found, whereas at higher concentrations volume fraction ØP. The symbols indicate phase diagram of Fig 3b replotted with the two- and three-phase coexistence of fluid the observed phase behaviour: circles, one- polymer μP as the y-axis. and/or the four possible crystal structures phase fluid; , two-phase gas- Apart from being inverted, the diagram now —pure A, pure B, AB2, and AB13—are liquid coexistence; plus signs, two-phase resembles the -concentration predicted. The data points indicate the fluid-crystal coexistence; crosses, three- projection of the phase diagram of a simple PMMA samples studied experimentally. phase gas-liquid-crystal coexistence; atomic material (Gas, Liquid, Crystal). Overall, reasonable agreement between squares, gas-crystal coexistence; triangles, Increasing the polymer chemical potential experiment and theory was found. colloidal gel. Note the striking difference (by increasing its concentration) increases Inset Schematic of sterically-stabilised between Figs 3a and 3b, reflecting the the strength of the interparticle depletion PMMA particles. The thickness of the effect of non-additive colloid-polymer attraction at constant thermal kinetic polymer brushes is exaggerated. Typically interactions. Inset Illustration of the energy. In an atomic material, reducing the the thickness is ~ 10 nm, to be compared depletion effect. The centres of the temperature reduces the at to particle radii of 200 to 300 nm polymer molecules are excluded from constant interparticle attraction

When two particles are close enough that crystal, similar (except for being upside ing possibility of an attraction between their depletion zones overlap, more free down) to the familiar temperature-densi­ like charges. Finally, there is , the volume in the sample as a whole is avail­ ty representation of an atomic liquid. flow behaviour of colloidal systems, a able to the polymer, giving the suspen­ complex and technologically important sion higher entropy. By integrating out Colloid is a vigorous and grow­ subject. the degrees of freedom associated with ing activity, exploiting close interaction the polymer molecules, one can also use­ between experiment, theory and simula­ The authors are professors at British universi­ fully describe this tendency for particles tion. There have been many exciting de­ ties; Jean-Pierre Hansen works at Cambridge to in terms of an attractive “deple­ velopments, some of them unique to col­ University and Peter N. Pusey works at the Uni­ tion force” between the particles, caused loidal systems. For example, their dynam­ versity of Edinburgh by the unbalanced osmotic ex­ ics are slow, allowing detailed studies of erted by the polymer on their surfaces. and the kinetics of phase Further reading The phase diagram of figure 3b can be re­ transitions. New statistical mechanical Observation. Prediction and Simulation of Phase plotted (figure 3c) so that the vertical axis approaches have been developed to de­ Transitions in Complex Fluids edited by M. Baus, is the chemical potential of the polymer, scribe the effects of polydispersity, the in­ L.F. Rull and J.-P. Ryckaert (Kluwer, Dordrecht, which must be the same in all coexisting evitable distribution of size of colloidal 1995) · Binary Hard-Sphere Mixtures by M.D. phases. Now we see a phase diagram con­ particles. And there is the behaviour of Eldridge et al, Molecular Physics 84 395 (1995) taining a gas-liquid critical point and a charged colloids, for which recent experi­ Colloid-Polymer Mixtures by S.M. Ilett et al, triple line of coexisting gas, liquid and ments and theory have raised the intrigu­ Physical Reiew E51 1344 (1995) europhysics news may/june 1999 83