Melting in Monolayers : Hexatic and Fluid Phases

There are strong evidences that the melting in two dimensions depends crucially on the form and range of the interaction potentials between particles. We study with Monte Carlo simulations the phase diagram and the melting of a monolayer of point-particles interacting with repulsive Inverse Power Law Interactions, $V(r)=Q^2(\sigma/r)^n$ where $n$ can take any real positive value ($n$-OCP monolayer). As $n$ is varied from 0 to $\infty$ (Hard Disks), including Coulomb ($n=1$) and Dipolar ($n=3$), melting occurs with different mechanisms and the overall picture permits to understand the diversity of mechanisms found experimentally or in computer simulations for 2D melting. The empirical transition curves for $n\leq 3$ and the excellent qualitative and semi-quantitative agreements with the KTHNY theory found for the melting of $n$-OCP monolayers with $n\leq 3$ are the main results of the present work.Comment: 14 pages, 9 figures 1 Tabl

Ewald sums for Yukawa potentials in quasi-two-dimensional systems

In this note, we derive Ewald sums for Yukawa potential for three dimensional systems with two dimensional periodicity. This sums are derived from the Ewald sums for Yukawa potentials with three dimensional periodicity [G. Salin and J.-M. Caillol, J. Chem. Phys. {\bf 113}, 10459 (2000)] by using the method proposed by Parry for the Coulomb interactions [D.E. Parry, Surf. Sci. {\bf 49}, 433 (1975); {\bf 54}, 195 (1976)].Comment: 7 pages, no figure. To appear in J. Chem. Phy

The melting of the classical two dimensional Wigner crystal

We report an extensive Monte-Carlo study of the melting of the classical two dimensional Wigner crystal for a system of point particles interacting via the $1/r$-Coulomb potential. A hexatic phase is found in systems large enough. With the multiple histograms method and the finite size scaling theory, we show that the fluid/hexatic phase transition is weakly first order. No set of critical exponents, consistent with a Kosterlitz-Thouless transition and the finite size scaling analysis for this transition, have been found.Comment: 6 pages, 5 figures, 1 tabl

Mixtures of Hard Ellipsoids and Spheres: Stability of the Nematic Phase

The stability of liquid crystal phases in presence of small amount of non-mesogenic impurities is of general interest for a large spectrum of technological applications and in the theories of binary mixtures. Starting from the known phase diagram of the hard ellipsoids systems, we propose a simple model and method to explore the stability of the nematic phase in presence of small impurities represented by hard spheres. The study is performed in the isobaric ensemble with Monte Carlo simulations

Holonomic constraints : an analytical result

Systems subjected to holonomic constraints follow quite complicated dynamics that could not be described easily with Hamiltonian or Lagrangian dynamics. The influence of holonomic constraints in equations of motions is taken into account by using Lagrange multipliers. Finding the value of the Lagrange multipliers allows to compute the forces induced by the constraints and therefore, to integrate the equations of motions of the system. Computing analytically the Lagrange multipliers for a constrained system may be a difficult task that is depending on the complexity of systems. For complex systems, it is most of the time impossible to achieve. In computer simulations, some algorithms using iterative procedures estimate numerically Lagrange multipliers or constraint forces by correcting the unconstrained trajectory. In this work, we provide an analytical computation of the Lagrange multipliers for a set of linear holonomic constraints with an arbitrary number of bonds of constant length. In the appendix of the paper, one would find explicit formulas for Lagrange multipliers for systems having 1, 2, 3, 4 and 5 bonds of constant length, linearly connected.Comment: 13 pages, no figures. To appear in J. Phys. A : Math. The

Yukawa potentials in systems with partial periodic boundary conditions II : Lekner sums for quasi-two dimensional systems

Yukawa potentials may be long ranged when the Debye screening length is large. In computer simulations, such long ranged potentials have to be taken into account with convenient algorithms to avoid systematic bias in the sampling of the phase space. Recently, we have provided Ewald sums for quasi-two dimensional systems with Yukawa interaction potentials [M. Mazars, {\it J. Chem. Phys.}, {\bf 126}, 056101 (2007) and M. Mazars, {\it Mol. Phys.}, Paper I]. Sometimes, Lekner sums are used as an alternative to Ewald sums for Coulomb systems. In the present work, we derive the Lekner sums for quasi-two dimensional systems with Yukawa interaction potentials and we give some numerical tests for pratical implementations. The main result of this paper is to outline that Lekner sums cannot be considered as an alternative to Ewald sums for Yukawa potentials. As a conclusion to this work : Lekner sums should not be used for quasi-two dimensional systems with Yukawa interaction potentials.Comment: 25 pages, 5 figures and 1 tabl

Taking one charge off a two-dimensional Wigner crystal

A planar array of identical charges at vanishing temperature forms a Wigner crystal with hexagonal symmetry. We take off one (reference) charge in a perpendicular direction, hold it fixed, and search for the ground state of the whole system. The planar projection of the reference charge should then evolve from a six-fold coordination (center of a hexagon) for small distances to a three-fold arrangement (center of a triangle), at large distances $d$ from the plane. The aim of this paper is to describe the corresponding non-trivial lattice transformation. For that purpose, two numerical methods (direct energy minimization and Monte Carlo simulations), together with an analytical treatment, are presented. Our results indicate that the $d=0$ and $d\to\infty$ limiting cases extend for finite values of $d$ from the respective starting points into two sequences of stable states, with intersecting energies at some value $d_t$; beyond this value the branches continue as metastable states.Comment: 17 pages, 11 figure