Phase Behaviour Of Hard Spheres Research Articles

Ordering of colloidal hard spheres under gravity: from monolayer to multilayer.

The phase behaviour of hard spheres confined by a gravitational potential to a thin layer (up to several monolayers) near a hard, flat surface is investigated using grand canonical Monte Carlo simulation. Depending on the strength of the gravitational field, the bottom monolayer of spheres may adopt uniform hexagonal order before, during, or after the growth of the second layer of particles. The crossover from ordering with a sparsely populated overlayer to ordering with almost one-third of the system's particles forming a second layer is observed upon decreasing the dimensionless Péclet number Pe = mgσ/kBT from 18 to 16. The particular sensitivity of the nature of the transition to particle size in this range is interpreted in terms of competing influences on the base layer structure by particles in the overlayer: promotion of order through increased pressure, versus stabilization of defects through occupation of low-lying sites on top of them. Simulations of grain boundaries between 2-D ordered domains of different orientation are used to correlate the degree of overlayer coverage to its effects on grain boundary stiffness as an indicator of defect free energy. Finally, we examine the structure of the ordered phases at coexistence over a range of gravitational strengths and find that orientational ordering of the second monolayer occurs along with first-order transition of the base layer at Pe = 8 but not at Pe = 10.

Phase behavior of hard spheres mixed with supramolecular polymers

We present a theory that predicts the phase behavior of hard spheres (HS) mixed with non-adsorbing supramolecular polymers in a common solvent. It combines Generalized Free Volume Theory with a mean-field model for equilibrium polymers in solution extended with scaling theory to incorporate excluded volume effects in a good solvent. The system displays similar phase stability as conventional HS-polymer mixtures, although there is a small preference for gas–liquid coexistence with respect to fluid–crystal equilibrium in the case of a HS-supramolecular polymer mixture. Phase transitions can be induced by external stimuli such as temperature or light due to the stimuli-responsive nature of the supramolecular polymer size.

Bond orientational ordering in a metastable supercooled liquid: a shadow of crystallizationand liquid–liquid transition

It is widely believed that a liquid state can be characterized by a single order parameter,density, and that a transition from a liquid to solid can be described by density ordering(translational ordering). For example, this type of theory has had great success indescribing the phase behaviour of hard spheres. However, there are some features thatcannot be captured by such theories. For example, hard spheres crystallize into either hcpor fcc structures, without a tendency of bcc ordering which is expected by theAlexander–McTague theory based on the Landau-type free energy of the density orderparameter. We also found hcp-like bond orientational ordering in a metastablesupercooled liquid, which promotes nucleation of hcp crystals. Furthermore, theoriesbased on the single order parameter cannot explain water-like thermodynamic andkinetic anomalies of a liquid and liquid–liquid transition in a single-componentliquid. Based on these facts, we argue that we need an additional order parameterto describe a liquid state. It is bond orientational order, which is induced bydense packing in hard spheres or by directional bonding in molecular and atomicliquids. Bond orientational order is intrinsically of local nature, unlike translationalorder which is of global nature. This feature plays a unique role in crystallizationand quasicrystal formation. We also reveal that bond orientational ordering isa cause of dynamic heterogeneity near a glass transition and is linked to slowdynamics. In relation to this, we note that, for describing the structuring of ahighly disordered liquid, we need a structural signature of low configurationalentropy, which is more general than bond orientational order. Finally, the water-likeanomaly and liquid–liquid transition can be explained by bond orientational orderingdue to hydrogen or covalent bonding and its cooperativity, respectively. So weargue that bond orientational ordering is a key to the physical understanding ofcrystallization, quasicrystallization, glass transition, water-like anomaly and liquid–liquidtransition. A unified description of these phenomena may be possible along thisline.

Direct imaging of temperature-sensitive core-shell latexes by cryogenic transmission electron microscopy

We present a comprehensive investigation of the volume transition in thermosensitive core-shell particles. The particles consist of a solid core of poly (styrene) (radius: 52 nm) onto which a network of crosslinked poly(N-isopropylacrylamide) (PNIPAM) is affixed. The degree of crosslinking of the PNIPAM shell effected by the crosslinker N,N ′-methylenebisacrylamide was varied between 1.25 and 5 mol%. Immersed in water, the shell of these particles is swollen at low temperatures. Raising the temperature above 32°C leads to a volume transition within the shell. Cryogenic transmission electron microscopy (Cryo-TEM) and dynamic light scattering (DLS) have been used to investigate the structure and swelling of the particles. The Cryo-TEM micrographs directly show inhomogeneities of the network. Moreover, a buckling of the shell from the core particle is evident. This buckling increases with decreasing degree of crosslinking. A comparison of the overall size of the particles determined by DLS and Cryo-TEM demonstrates that the hydrodynamic radius provides a valid measure for the size of the particles. The phase transition within the network measured by DLS can be described by the Flory–Rehner theory. It is shown that this model captures the main features of the volume transition within the core-shell particles including the dependence of the phase transition on the degree of crosslinking. All dispersions crystallize at volume fractions above 0.5. The resulting phase diagram is identical to the phase behavior of hard spheres within the limits of error. This demonstrates that the core-shell microgels can be treated as hard spheres up to volume fractions of at least 0.55.

Freezing of hard spheres confined in narrow cylindrical pores

Monte Carlo simulations for the equation of state and phase behavior of hard spheres confined inside very narrow hard tubes are presented. For pores whose radii are greater than 1.1 hard sphere diameters, a sudden change in the density and the microscopic structure of the fluid is neatly observed, indicating the onset of freezing. In the high-density structure the particles rearrange in such a way that groups of three particles fit in sections across the pore.

Fractionation effects in phase equilibria of polydisperse hard-sphere colloids

The equilibrium phase behavior of hard spheres with size polydispersity is studied theoretically. We solve numerically the exact phase equilibrium equations that result from accurate free energy expressions for the fluid and solid phases, while accounting fully for size fractionation between coexisting phases. Fluids up to the largest polydispersities that we can study (around 14%) can phase separate by splitting off a solid with a much narrower size distribution. This shows that experimentally observed terminal polydispersities above which phase separation no longer occurs must be due to nonequilibrium effects. We find no evidence of reentrant melting; instead, sufficiently compressed solids phase separate into two or more solid phases. Under appropriate conditions, coexistence of multiple solids with a fluid phase is also predicted. The solids have smaller polydispersities than the parent phase as expected, while the reverse is true for the fluid phase, which contains predominantly smaller particles but also residual amounts of the larger ones. The properties of the coexisting phases are studied in detail; mean diameter, polydispersity, and volume fraction of the phases all reveal marked fractionation. We also propose a method for constructing quantities that optimally distinguish between the coexisting phases, using principal component analysis in the space of density distributions. We conclude by comparing our predictions to Monte Carlo simulations at imposed chemical potential distribution, and find excellent agreement.

Phase diagram of hard-core repulsive Yukawa particles with a density-dependenttruncation: a simple model for charged colloids

Using computer simulations we study the phase behaviour of hard spheres with repulsiveYukawa interactions and with the repulsion set to zero at distances larger than adensity-dependent cut-off distance. Earlier studies based on experiments and computersimulations in colloidal suspensions have shown that the effective colloid–colloid pairinteraction that takes into account many-body effects resembles closely this truncatedYukawa potential. We present a phase diagram for the truncated Yukawa system bycombining Helmholtz free energy calculations and the Kofke integration method. Comparedto the non-truncated Yukawa system we observe (i) a radical reduction of the stability ofthe body centred cubic (BCC) phase, (ii) a wider fluid region due to instability of the facecentred cubic (FCC) phase and due to a re-entrant fluid phase and (iii) hardly anyshift of the (FCC) melting line when compared with the (BCC) melting line forthe full Yukawa potential for sufficiently high salt concentrations, i.e. truncationof the potential does not affect the location of the solid–fluid line but replacesonly the BCC phase with the FCC at the melting line. We compare our resultswith earlier results on the truncated Yukawa potential and with results fromsimulations where the full many-body Poisson–Boltzmann problem is solved.

Phase diagrams of hard-core repulsive Yukawa particles.

We determine the phase behavior of hard spheres interacting with repulsive Yukawa (screened Coulomb) interaction using computer simulations. We study the effect of the hard-core diameter on the phase behavior of repulsive Yukawa particles by comparing our phase diagrams with that of repulsive point Yukawa particles. We show that for sufficiently high contact values of the pair potential (betaepsilon=20, 39, 81, and higher), the fluid-face-centered-cubic (fcc) solid, at high screening, the fluid-body-centered-cubic (bcc) solid and the bcc-fcc coexistence for packing fractions eta less, similar 0.5 are well described by the phase boundaries of point Yukawa particles, by employing a mapping of the point Yukawa system onto a hard-core repulsive Yukawa system. While the bcc-fcc coexistence is well described by the point Yukawa limit for eta<0.5, we find a deviation at higher eta as the hard-core repulsion favors the fcc solid for eta>/=0.5, independent of the screening. Consequently, a second triple point appears in the phase diagram in the weak screening regime. In addition, we find that all the phase coexistence regions in our phase diagrams for hard-core repulsive Yukawa system are very narrow, i.e., a small density jump in the coexisting phases.

Equilibrium phase behavior of polydisperse hard spheres.

We calculate the phase behavior of hard spheres with size polydispersity, using accurate free energies for the fluid and solid phases. Cloud and shadow curves are found exactly by the moment free energy method, but we also compute the complete phase diagram, taking full account of fractionation. In contrast to earlier, simplified treatments we find no point of equal concentration between fluid and solid or reentrant melting at higher densities. Rather, the fluid cloud curve continues to the largest polydispersity that we study (14%); from the equilibrium phase behavior a terminal polydispersity can thus be defined only for the solid, where we find it to be around 7%. At sufficiently large polydispersity, fractionation into several solid phases can occur, consistent with previous approximate calculations; we find, in addition, that coexistence of several solids with a fluid phase is also possible.

Phase behavior of hard spheres with a short-range Yukawa attraction.

The phase diagram of a system consisting of hard spheres with an attractive Yukawa interaction is computed by Monte Carlo simulations. Upon decreasing the range of attraction, we find the following scenarios: (a) a stable fluid-fluid and fluid-solid transition, a triple point with fluid-fluid-solid coexistence, and a metastable isostructural solid-solid transition (b) a fluid-fluid and isostructural solid-solid transition, which are both metastable with respect to a broad fluid-solid transition, (c) a metastable fluid-fluid transition, a stable isostructural solid-solid and fluid-solid transition and a triple point with fluid-solid-solid coexistence. Our results show that a large part of the fluid-solid coexistence region is occupied by metastable fluid-fluid and/or solid-solid coexistence regions, which has repercussions on the kinetics and on the nonequilibrium behavior of the system, such as nucleation, vitrification, and gelation.