Hard spheres at a planar hard wall: Simulations and density functional theory

Hard spheres are a central and important model reference system for both homogeneous and inhomogeneous fluid systems. In this paper we present new high-precision molecular-dynamics computer simulations for a hard sphere fluid at a planar hard wall. For this system we present benchmark data for the density profile ρ(z) at various bulk densities, the wall surface free energy γ, the excess adsorption Γ, and the excess volume vex, which is closely related to Γ. We compare all benchmark quantities with predictions from state-of-the-art classical density functional theory calculations within the framework of fundamental measure theory. While we find overall good agreement between computer simulations and theory, significant deviations appear at sufficiently high bulk densities

New Langevin and Gradient Thermostats for Rigid Body Dynamics

We introduce two new thermostats, one of Langevin type and one of gradient (Brownian) type, for rigid body dynamics. We formulate rotation using the quaternion representation of angular coordinates; both thermostats preserve the unit length of quaternions. The Langevin thermostat also ensures that the conjugate angular momenta stay within the tangent space of the quaternion coordinates, as required by the Hamiltonian dynamics of rigid bodies. We have constructed three geometric numerical integrators for the Langevin thermostat and one for the gradient thermostat. The numerical integrators reflect key properties of the thermostats themselves. Namely, they all preserve the unit length of quaternions, automatically, without the need of a projection onto the unit sphere. The Langevin integrators also ensure that the angular momenta remain within the tangent space of the quaternion coordinates. The Langevin integrators are quasi-symplectic and of weak order two. The numerical method for the gradient thermostat is of weak order one. Its construction exploits ideas of Lie-group type integrators for differential equations on manifolds. We numerically compare the discretization errors of the Langevin integrators, as well as the efficiency of the gradient integrator compared to the Langevin ones when used in the simulation of rigid TIP4P water model with smoothly truncated electrostatic interactions. We observe that the gradient integrator is computationally less efficient than the Langevin integrators. We also compare the relative accuracy of the Langevin integrators in evaluating various static quantities and give recommendations as to the choice of an appropriate integrator.Comment: 16 pages, 4 figure

Direct calculation of the hard-sphere crystal/melt interfacial free energy

We present a direct calculation by molecular-dynamics computer simulation of the crystal/melt interfacial free energy, $\gamma$, for a system of hard spheres of diameter $\sigma$. The calculation is performed by thermodynamic integration along a reversible path defined by cleaving, using specially constructed movable hard-sphere walls, separate bulk crystal and fluid systems, which are then merged to form an interface. We find the interfacial free energy to be slightly anisotropic with $\gamma$ = 0.62$\pm 0.01$, 0.64$\pm 0.01$ and 0.58$\pm 0.01 k_BT/\sigma^2$ for the (100), (110) and (111) fcc crystal/fluid interfaces, respectively. These values are consistent with earlier density functional calculations and recent experiments measuring the crystal nucleation rates from colloidal fluids of polystyrene spheres that have been interpreted [Marr and Gast, Langmuir {\bf 10}, 1348 (1994)] to give an estimate of $\gamma$ for the hard-sphere system of $0.55 \pm 0.02 k_BT/\sigma^2$, slightly lower than the directly determined value reported here.Comment: 4 pages, 4 figures, submitted to Physical Review Letter

The anisotropic hard-sphere crystal-melt interfacial free energy from fluctuations

This is the publisher's version, also available electronically from http://scitation.aip.org/content/aip/journal/jcp/125/9/10.1063/1.2338303.We have calculated the interfacial free energy for the hard-sphere system, as a function of crystal interface orientation, using a method that examines the fluctuations in the height of the interface during molecular dynamics simulations. The approach is particularly sensitive for the anisotropy of the interfacial free energy. We find an average interfacial free energy of γ=0.56±0.02kBTσ−2. This value is lower than earlier results based upon direct calculations of the free energy [R. L. Davidchack and B. B. Laird, Phys. Rev. Lett.85, 4751 (2000)]. However, both the average value and the anisotropy agree with the recent values obtained by extrapolation from direct calculations for a series of the inverse-power potentials [R. L. Davidchack and B. B. Laird, Phys. Rev. Lett.94, 086102 (2005)]

Weighted-density approximation for general nonuniform fluid mixtures

In order to construct a general density-functional theory for nonuniform fluid mixtures, we propose an extension to multicomponent systems of the weighted-density approximation (WDA) of Curtin and Ashcroft [Phys. Rev. A 32, 2909 (1985)]. This extension corrects a deficiency in a similar extension proposed earlier by Denton and Ashcroft [Phys. Rev. A 42, 7312 (1990)], in that that functional cannot be applied to the multi-component nonuniform fluid systems with spatially varying composition, such as solid-fluid interfaces. As a test of the accuracy of our new functional, we apply it to the calculation of the freezing phase diagram of a binary hard-sphere fluid, and compare the results to simulation and the Denton-Ashcroft extension.Comment: 4 pages, 4 figures, to appear in Phys. Rev. E as Brief Repor

Calculation of the interfacial free energy of a fluid at a static wall by Gibbs–Cahn integration

This is the publisher's version, also available electronically from http://scitation.aip.org/content/aip/journal/jcp/132/20/10.1063/1.3428383.The interface between a fluid and a static wall is a useful model for a chemically heterogeneous solid-liquid interface. In this work, we outline the calculation of the wall-fluid interfacial free energy(γwf) for such systems using molecular simulation combined with adsorptionequations based on Cahn’s extension of the surface thermodynamics of Gibbs. As an example, we integrate such an adsorptionequation to obtain γwf as a function of pressure for a hard-sphere fluid at a hard wall. The results so obtained are shown to be in excellent agreement in both magnitude and precision with previous calculations of this quantity, but are obtained with significantly lower computational effort

Geometric integrator for Langevin systems with quaternion-based rotational degrees of freedom and hydrodynamic interactions

We introduce new Langevin-type equations describing the rotational and translational motion of rigid bodies interacting through conservative and non-conservative forces, and hydrodynamic coupling. In the absence of non-conservative forces the Langevin-type equations sample from the canonical ensemble. The rotational degrees of freedom are described using quaternions, the lengths of which are exactly preserved by the stochastic dynamics. For the proposed Langevin-type equations, we construct a weak 2nd order geometric integrator which preserves the main geometric features of the continuous dynamics. The integrator uses Verlet-type splitting for the deterministic part of Langevin equations appropriately combined with an exactly integrated Ornstein-Uhlenbeck process. Numerical experiments are presented to illustrate both the new Langevin model and the numerical method for it, as well as to demonstrate how inertia and the coupling of rotational and translational motion can introduce qualitatively distinct behaviours

Determination of the solid-liquid interfacial free energy along a coexistence line by Gibbs–Cahn integration

This is the publisher's version, also available electronically from http://scitation.aip.org/content/aip/journal/jcp/131/11/10.1063/1.3231693.We calculate the solid-liquid interfacial free energyγsl for the Lennard-Jones (LJ) system at several points along the pressure-temperature coexistence curve using molecular-dynamics simulation and Gibbs–Cahn integration. This method uses the excess interfacial energy(e) and stress (τ) along the coexistence curve to determine a differential equation for γsl as a function of temperature. Given the values of γsl for the (100), (110), and (111) LJ interfaces at the triple-point temperature (T∗=kT/ϵ=0.618), previously obtained using the cleaving method by Davidchack and Laird [J. Chem. Phys. 118, 7657 (2003)], this differential equation can be integrated to obtain γsl for these interfaces at higher coexistence temperatures. Our values for γsl calculated in this way at T∗=1.0 and 1.5 are in good agreement with those determined previously by cleaving, but were obtained with significantly less computational effort than required by either the cleaving method or the capillary fluctuation method of Hoyt, Asta, and Karma [Phys. Rev. Lett. 86, 5530 (2001)]. In addition, the orientational anisotropy in the excess interfaceenergy, stress and entropy, calculated using the conventional Gibbs dividing surface, are seen to be significantly larger than the relatively small anisotropies in γsl itself

Nucleation and Bulk Crystallization in Binary Phase Field Theory

We present a phase field theory for binary crystal nucleation. In the one-component limit, quantitative agreement is achieved with computer simulations (Lennard-Jones system) and experiments (ice-water system) using model parameters evaluated from the free energy and thickness of the interface. The critical undercoolings predicted for Cu-Ni alloys accord with the measurements, and indicate homogeneous nucleation. The Kolmogorov exponents deduced for dendritic solidification and for "soft-impingement" of particles via diffusion fields are consistent with experiment.Comment: 4 pages, 4 figures, accepted to PR