Phase diagram of hard colloidal platelets: a theoretical account

We construct the complete liquid crystal phase diagram of hard plate-like cylinders for variable aspect ratio using Onsager's second virial theory and employing the Parsons–Lee decoupling approximation to account for higher-body interactions in the isotropic and nematic fluid phases. The stability of the solid (columnar) state at high packing fraction is included by invoking a simple equation of state based on a Lennard–Jones–Devonshire cell model which has proven to be quantitatively reliable over a large range of packing fractions. By employing an asymptotic analysis based on the Gaussian approximation we are able to show that the nematic–columnar transition is universal and independent of particle shape. The predicted phase diagram is in qualitative agreement with simulation results.     

Three-spin interaction Ising model with a nondegenerate ground state at zero applied field

The field-temperature phase diagram of a two-dimensional, three-spin interaction Ising model is studied using two different methods: mean field approximation and numerical transfer matrix techniques. The former leads to a rather rich phase diagram in which two separate phases with different symmetries can be found, and which presents first-order transition lines, a triple point, and a critical end point, like the solid-liquid-gas phase diagram of a pure compound. The numerical transfer matrix study confirms part of these results, but does not clearly evidence the existence of the less symmetric phase.     

Freezing of Classical Fluids in a Quenched Random Potential

The phase diagram of a hard-sphere fluid in the presence of a random pinning potential is studied analytically and numerically. In the analytic work, replicas are introduced for averaging over the quenched disorder, and the hypernetted chain approximation is used to calculate density correlations in the replicated liquid. The freezing transition of the liquid into a nearly crystalline state is studied using a density-functional approach, and the liquid to glass transition is studied using a phenomenological replica symmetry breaking approach. In the numerical work, local minima of a discretized version of the Ramakrishnan-Yussouff free-energy functional are located and the phase diagram in the density-disorder plane is obtained from an analysis of the relative stability of these minima. Both approaches lead to similar results for the phase diagram. The first-order liquid to crystalline solid transition is found to change to a continuous liquid to glass transition as the strength of the disorder is increased above a threshold value.     

Study of the solid-liquid-vapour phase equilibria of flexible chain molecules using Wertheim's thermodynamic perturbation theory

The phase diagram of flexible molecules formed by freely-jointed tangent spheres is studied using the first-order thermodynamic perturbation theory of Wertheim for both fluid and solid phases. A mean-field term is added to the free energy of the fluid and solid phase in order to account for attractive dispersion forces. The approach is used to determine the global (solid-liquid-vapour) phase diagrams and triple points of chain molecules of increasing chain length. It is found that the triple point temperature is not affected strongly by the length of the chain, whereas the gas-liquid critical temperature increases dramatically. The asymptotic limits of the phase diagram for infinitely long chains are discussed. The reduced critical temperature of infinitely long chains as given by the mean-field theory is 2/3, and the reduced triple point temperature is 0.048 56, so that an asymptotic value of T _t/T _c = 0.07284 for the ratio of the triple to critical point temperatures is obtained. This indicates that fully-flexible tangent chains present an enormous liquid range. The proposed theory, while being extremely simple, provides a useful insight into the phase behaviour of chain molecules, showing the existence of finite asymptotic limits for the triple and critical point temperatures. However, since n-alkanes present an asymptotic limit of about T _t/T _c, = 0.40, the agreement With experiment is not quantitative. This suggests that fully flexible models may not be appropriate to model the solid phases of real chain molecules.     

Further application of the statistical thermodynamics of curved surfaces to scaled particle theory

The thermodynamics of curved boundary layers is combined with scaled particle theory to determine the rigid-sphere equation of state. In particular, the boundary analog of the Gibbs-Tolman-Koenig-Buff equation is solved for a rigid-sphere fluid, using the approximation that the distance between the surface of a cavity and its surface of tension is a function of the density only (the first-order approximation). This, in conjunction with several exact conditions onG, the central function of scaled particle theory, leads to an approximate rigid-sphere fluid equation of state and a qualitatively correct rigid-sphere solid equation of state. The fluid isotherm compares favorably with previous results (2.9 % error in the fourth virial coefficient), but due to the inaccuracy of the solid isotherm, no phase transition is obtained. The theory described here is to be contrasted with previous approaches in that a less arbitrary functional form forG is assumed, and the surface of tension and cavity surface are not assumed to be coincident. The cycle equation of Reiss and Tully-Smith is rederived by a simpler route and shown to be correct to all orders of cavity curvature, rather than only first order as was originally thought. A new exact condition, obtained from the compressibility equation of state, is used as a boundary condition for the cycle equation to determine the location of the equimolecular surface. This molecular calculation compares favorably (discrepancy of <2 %) with a thermodynamic calculation based on the boundary analog of the Gibbs adsorption equation and indicates the accuracy and consistency of the first-order approximation.Research supported under NSF Grant #GP-12408.     

Phase diagram of trivalent and pentavalent patchy particles

We compute the equilibrium phase diagram of two simple models for patchy particles with three and five patches in a very broad range of pressure and temperature. The phase diagram presents low-density crystal structures which compete with the fluid phase. The phase diagram of the five-patch model shows re-entrant melting, in analogy with the previously studied four-patch case, a metastable gas-liquid critical point and a stable, high-density liquid. The three-patch model shows a stable gas-liquid critical point and, in the region of temperatures where equilibration is numerically feasible, a stable liquid phase, suggesting the possibility that in this small valence model the liquid retains its thermodynamic stability down to the vanishing range limit.     

Finite interaction range spin glass in the Ising model

Using the Ising model and the random-field approximation, it is demonstrated that, in amorphous diluted magnets and crystalline metal alloys, the nearest neighbor interaction promotes a superparamagnetic state, which is followed by “freezing” of the magnetic moments of clusters and by transition into a macrospin glass state (cluster spin glass) as the temperature decreases. A theoretical magnetic phase diagram is constructed.     

On the theory of vapor — Liquid — Crystal phase transitions

An approach is discussed which allows one to describe within the framework of the single model the aggregative states of a system of atoms obeying the Fermi statistics. A phase diagram is obtained containing both the critical point where the distinctions between the liquid and vapor vanish and the triple point where the liquid, vapor, and crystal are in equilibrium.     

Non-uniformities in polymer/liquid crystal mixtures

We theoretically model the nucleation of nematic droplets during phase ordering in mixtures of a flexible polymer and a low-molecular-weight liquid crystal. By appealing to classical nucleation theory (CNT), we calculate the energy barrier to nucleation and the size of a critical nucleus. We study the influence of a metastable intermediate phase on the nucleation of the nematic. Below a triple point in the phase diagram, there are two distinct mechanisms for the formation of a nematic nucleus: 1) direct nucleation from the isotropic phase and 2) nucleation via a precursor metastable isotropic phase. We calculate the crossover concentration as a function of temperature, delineating the regions of the phase diagram in which each mechanism prevails. In the latter case, the presence of a hidden metastable isotropic-isotropic binodal may either promote or delay the nucleation of a nematic phase. Received 9 August 2002 RID="a" ID="a"e-mail: matuyama@chem.mie-u.ac.jp     

Electrical transport and specific heat of a Cr+2.2 at% Al single crystal

Electrical resistivity, thermoelectric power and specific heat measurements are reported for a Cr+2.2 at% Al alloy single crystal, having an Al content close to the triple point on the magnetic phase diagram of the Cr–Al alloy system. A weak resistivity component of spin-density-wave origin, not previously observed in alloys near the triple point concentration, is isolated at low temperatures. Its presence is confirmed by the thermoelectric measurements. The specific heat measurements suggest a maximum in the density of states at the Fermi energy for alloy concentrations close to the triple point, a point that appears to be a particular type of critical point for the Cr–Al alloy system.     

Mixed spin Ising model with four-spin interaction and random crystal field

The effects of fluctuations of the crystal field on the phase diagram of the mixed spin-1/2 and spin-1 Ising model with four-spin interactions are investigated within the finite cluster approximation based on a single-site cluster theory. The state equations are derived for the two-dimensional square lattice. It has been found that the system exhibits a variety of interesting features resulting from the fluctuation of the crystal field interactions. In particular, for low mean value D of the crystal field, the critical temperature is not very sensitive to fluctuations and all transitions are of second order for any value of the four-spin interactions. But for relatively high D, the transition temperature depends on the fluctuation of the crystal field, and the system undergoes tricritical behaviour for any strength of the four-spin interactions. We have also found that the model may exhibit reentrance for appropriate values of the system parameters.     

Small bipolarons in the 2-dimensional Holstein-Hubbard model. II. Quantum bipolarons

We study the effective mass of the bipolarons and essentially the possibility to get both light and strongly bound bipolarons in the Holstein-Hubbard model and some variations in the vicinity of the adiabatic limit. Several approaches to investigate the quantum mobility of polarons and bipolarons are proposed for this model. First, the quantum fluctuations are treated as perturbations of the mean-field (or adiabatic) approximation of the electron-phonon coupling in order to calculate the bipolaron bands. It is found that the bipolaron mass generally remains very large except in the vicinity of the triple point of the phase diagram (see [1]), where the bipolarons have several degenerate configurations at the adiabatic limit (single site (S0), two sites (S1) and quadrisinglet (QS)), while the polarons are much lighter. This degeneracy reduces the bipolaron mass significantly. Next we improve this result by variational methods (modified Toyozawa Exponential Ansatz or TEA) valid for larger quantum perturbations away from the adiabatic limit. We first test this new method for the single polaron. We find that the triple point of the phase diagram is washed out by the lattice quantum fluctuations which thus suppress the light bipolarons. Further improvements of the method by hybridization of several TEA states do not change this conclusion. Next we show that some model variations, for example a phonon dispersion may increase the stability of the (QS) bipolaron against the quantum lattice fluctuations. We show that the triple point of the phase diagram may be stable to quantum lattice fluctuations and a very sharp mass reduction may occur, leading to bipolaron masses of the order of 100 bare electronic mass for realistic parameters. Thus we argue that such very light bipolarons could condense as a superconducting state at relatively high temperature when their interactions are not too large, that is, their density is small enough. This effect might be relevant for understanding the origin of the high superconductivity of doped cuprates far enough from half filling. Received 15 September 1999     

The phase behaviour of a fluid in a slitlike pore with permeable walls

The confinement of a lattice fluid in a set of slitlike pores separated by semipermeable walls with a finite width has been studied. The walls are modelled by a square-well repulsive potential with a finite height. The thermodynamic properties and the phase behaviour of the system are evaluated by means of Monte Carlo simulations. For some states theoretical calculations have been made using a mean-field-type theory. These investigations confirm previous findings for confined Lennard-Jones fluids, obtained from a density functional approach. For intermediate and low potential barriers that separate the pores, the isotherms exhibit two hysteresis loops and the liquid-vapour coexistence curve divides into two branches describing condensation inside the pore and inside the permeable wall. These two branches are separated by a triple point. At temperatures lower than the triple point temperature, the condensation takes place instantaneously in both the pore and inside the permeable wall. It was found that when the temperature is scaled by the bulk critical temperature, the phase diagram emerging from this simple mean-field treatment is close to the phase diagram obtained from simulation.