F. Lado

Effects of patch size and number within a simple model of patchy colloids.

We report on a computer simulation and integral equation study of a simple model of patchy spheres, each of whose surfaces is decorated with two opposite attractive caps, as a function of the fraction chi of covered attractive surface. The simple model explored--the two-patch Kern-Frenkel model--interpolates between a square-well and a hard-sphere potential on changing the coverage chi. We show that integral equation theory provides quantitative predictions in the entire explored region of temperatures and densities from the square-well limit chi=1.0 down to chi approximately 0.6. For smaller chi, good numerical convergence of the equations is achieved only at temperatures larger than the gas-liquid critical point, where integral equation theory provides a complete description of the angular dependence. These results are contrasted with those for the one-patch case. We investigate the remaining region of coverage via numerical simulation and show how the gas-liquid critical point moves to smaller densities and temperatures on decreasing chi. Below chi approximately 0.3, crystallization prevents the possibility of observing the evolution of the line of critical points, providing the angular analog of the disappearance of the liquid as an equilibrium phase on decreasing the range for spherical potentials. Finally, we show that the stable ordered phase evolves on decreasing chi from a three-dimensional crystal of interconnected planes to a two-dimensional independent-planes structure to a one-dimensional fluid of chains when the one-bond-per-patch limit is eventually reached.

Integral equations for fluids of linear molecules

A general procedure is described that puts the practice of integral equation theory for molecular fluids on a par with that of simple fluids: any integral equation approximation can be solved for any intermolecular potential with no additional approximations beyond those inherent in numerical analysis. The essential elements are expansions in spherical harmonics and numerical evaluation of the spherical harmonic coefficients of the pair distribution function. An explicit formula is derived giving the Helmholtz free energy from the computed coefficients.

An integral equation study of a simple point charge model of water

We present an extensive integral equation study of a simple point charge model of water for a variety of thermodynamic states ranging from the vapor phase to the undercooled liquid. The calculations are carried out in the molecular reference-hypernetted chain approximation and the results are compared with extensive molecular dynamics simulations. Use of a hard sphere fluid as a reference system to provide the input reference bridge function leads to relatively good thermodynamics. However, at low temperatures the computed microscopic structure shows deficiencies that probably stem from the lack of orientational dependence in this bridge function. This is in marked contrast with results previously obtained for systems that, although similarly composed of angular triatomic molecules, do not tend to the tetrahedral coordinations that are characteristic of water.

Choosing the reference system for liquid state perturbation theory

Minimization of an approximate free energy functional yields the Andersen-Weeks-Chandler approximation y(r) ≡ g(r) exp [βφ(r)] ≈ yd(r) along with a new criterion for choosing the reference hard sphere diameter d. The new prescription yields thermodynamic consistency and improved numerical results.

Charged hard spheres in a uniform neutralizing background using ' mixed' integral equations

The structure and thermodynamic properties of a collection of charged hard spheres immersed in a uniform neutralizing background are studied using ‘mixed’ integral equations, wherein the Percus-Yevick approximation is used with the hard-sphere part of the potential and the hypernetted-chain approximation for the correction due to the Coulomb tail. Numerical solutions are presented along a particular isotherm and a comparison is made with the results of the Mean Spherical Model for the same system.

Bounds on the Conductivity of a Random Array of Cylinders

We consider the problem of determining rigorous third-order and fourth-order bounds on the effective conductivity σe of a composite material composed of aligned, infinitely long, equisized, rigid, circular cylinders of conductivity σ2 randomly distributed throughout a matrix of conductivity σ1. Both bounds involve the microstructural parameter ξ2 which is an integral that depends upon S3, the three-point probability function of the composite (G. W. Milton, J. Mech. Phys. Solids 30, 177-191 (1982)). The key multidimensional integral ξ2 is greatly simplified by expanding the orientation-dependent terms of its integrand in Chebyshev polynomials and using the orthogonality properties of this basis set. The resulting simplified expression is computed for an equilibrium distribution of rigid cylinders at selected ϕ2 (cylinder volume fraction) values in the range 0 ≼ ϕ2 ≼ 0.65. The physical significance of the parameter ξ2 for general microstructures is briefly discussed. For a wide range of ϕ2 and α = σ2/σ1, the third-order bounds significantly improve upon second-order bounds which only incorporate volume fraction information; the fourth-order bounds, in turn, are always more restrictive than the third-order bounds. The fourth-order bounds on σe are found to be sharp enough to yield good estimates of σe for a wide range of ϕ2, even when the phase conductivities differ by as much as two orders of magnitude. When the cylinders are perfectly conducting (α = ∞), moreover, the fourth-order lower bound on σe provides an excellent estimate of this quantity for the entire volume-fraction range studied here, i. e. up to a volume fraction of 65%.

Improved Bounds on the Effective Elastic Moduli of Random Arrays of Cylinders

Improved rigorous bounds on the effective elastic moduli of a transversely isotropic fiber-reinforced material composed of aligned, infinitely long, equisized, circular cylinders distributed throughout a matrix are evaluated for cylinder volume fractions up to 70 percent. The bounds are generally shown to provide significant improvement over the Hill-Hashin bounds which incorporate only volume-fraction information. For cases in which the cylinders are stiffer than the matrix, the improved lower bounds provide relatively accurate estimates of the elastic moduli, even when the upper bound diverges from it (i.e., when the cylinders are substantially stiffer than the matrix). This last statement is supported by accurate, recently obtained Monte Carlo computer-simulation data of the true effective axial shear modulus.

Integral equation algorithm for fluids of fully anisotropic molecules

We outline a practical algorithm for the solution of liquid‐state integral equations for fluids of fully anisotropic rigid molecules requiring three Euler angles for their configurational description and leading to pair functions of five angular variables. The method is suitable for all potentials. We illustrate the technique with sample results for SO2.

Phase diagram and structural properties of a simple model for one-patch particles

We study the thermodynamic and structural properties of a simple, one-patch fluid model using the reference hypernetted-chain (RHNC) integral equation and specialized Monte Carlo simulations. In this model, the interacting particles are hard spheres, each of which carries a single identical, arbitrarily oriented and attractive circular patch on its surface; two spheres attract via a simple square-well potential only if the two patches on the spheres face each other within a specific angular range dictated by the size of the patch. For a ratio of attractive to repulsive surface of 0.8, we construct the RHNC fluid-fluid separation curve and compare with that obtained by Gibbs ensemble and grand canonical Monte Carlo simulations. We find that RHNC provides a quick and highly reliable estimate for the position of the fluid-fluid critical line. In addition, it gives a detailed (though approximate) description of all structural properties and their dependence on patch size.

Trapping constant, thermal conductivity, and the microstructure of suspensions of oriented spheroids

The n‐point probability function Sn(rn)  is fundamental to the study of the macroscopic properties of two‐phase random heterogeneous media. This quantity gives the probability of finding n points with positions rn ≡{r1,...,rn} all in one of the phases, say phase 1. For media composed of distributions of oriented, possibly overlapping, spheriods of one material with aspect ratio e in a ‘‘matrix’’ of another material, it is shown that there is a scaling relation that maps results for the Sn for sphere systems (e=1) into equivalent results for spheriod systems with arbitrary aspect ratio e. Using this scaling relation it is then demonstrated that certain transport and microstructural properties of spheriodal systems generally depend upon purely shape‐dependent functions and lower‐order spatial moments of S2 (minus its long‐range value) of the equivalent spherical system. Specifically, the following three distinct calculations are carried out for both hard, oriented spheroids and overlapping (i.e., spatially ...