Size-dependent properties of two-dimensional solids

Ordering of two-dimensional crystals confined in strips of finite width

Monte Carlo simulations are used to study the effect of confinement on a crystal of point particles interacting with an inverse power law potential proportional, variantr;{-12} in d=2 dimensions. This system can describe colloidal particles at the air-water interface, a model system for experimental study of two-dimensional melting. It is shown that the state of the system (a strip of width D ) depends very sensitively on the precise boundary conditions at the two "walls" providing the confinement. If one uses a corrugated boundary commensurate with the order of the bulk triangular crystalline structure, both orientational order and positional order is enhanced, and such surface-induced order persists near the boundaries also at temperatures where the system in the bulk is in its fluid state. However, using smooth repulsive boundaries as walls providing the confinement, only the orientational order is enhanced, but positional (quasi-)long range order is destroyed: The mean-square displacement of two particles n lattice parameters apart in the y direction along the walls then crosses over from the logarithmic increase (characteristic for d=2 ) to a linear increase with n (characteristic for d=1 ). The strip then exhibits a vanishing shear modulus. These results are interpreted in terms of a phenomenological harmonic theory. Also the effect of incommensurability of the strip width D with the triangular lattice structure is discussed, and a comparison with surface effects on phase transitions in simple Ising and XY models is made.

Computer simulations of two-dimensional melting with dipole-dipole interactions

We perform molecular dynamics and Monte Carlo simulations of two-dimensional melting with dipole-dipole interactions. Both static and dynamic behaviors are examined. In the isotropic liquid phase, the bond orientational correlation length xi 6 and susceptibility chi 6 are measured, and the data are fitted to the theoretical ansatz. An algebraic decay is detected for both spatial and temporal bond orientational correlation functions in an intermediate temperature regime, and it provides an explicit evidence for the existence of the hexatic phase. From the finite-size scaling analysis of the global bond orientational order parameter, the disclination unbinding temperature Ti is estimated. In addition, from dynamic Monte Carlo simulations of the positional order parameter, we extract the critical exponents at the dislocation unbinding temperature Tm. All the results are in agreement with those from experiments and support the Kosterlitz-Thouless-Halperin-Nelson-Young (KTHNY) theory.

Large-scale simulations of the two-dimensional melting of hard disks

Large-scale computer simulations with more than four million particles have been performed to study the melting transition in a two-dimensional hard disk fluid. The van der Waals loop previously observed in the pressure-density relationship of smaller simulations is shown to disappear systematically with increase in sample size, but even with these large system sizes, the freezing transition still exhibits what appears to be weakly first-order behavior, though the scaling of the bond orientation order is consistent with the Halperin-Nelson-Young picture. Above this freezing transition region, scaling analysis of the translational order yields a lower bound for the melting density that is much higher than previously thought and provides compelling evidence that the solid phase first melts into a hexatic phase via a continuous transition, before it goes into the isotropic phase.

Direct test of defect-mediated laser-induced melting theory for two-dimensional solids

We investigate by direct numerical solution of appropriate renormalization flow equations the validity of a recent dislocation unbinding theory for laser-induced freezing and melting in two dimensions. The bare elastic moduli and dislocation fugacities are obtained for three different two-dimensional systems namely, the hard disk, inverse 12th power, and Derjaguin-Landau-Verwey-Overbeek potentials. A restricted Monte Carlo simulation sampling only configurations without dislocations is used to obtain these quantities. These are then used as inputs to the flow equations. Numerical solution of the flow equations then yields the phase diagrams. We conclude that (a) the flow equations need to be correct at least up to third order in defect fugacity to reproduce meaningful results, (b) there is excellent quantitative agreement between our results and earlier conventional Monte Carlo simulations for the hard disk system, and (c) while the qualitative form of the phase diagram is reproduced for systems with soft potentials there is some quantitative discrepancy which we explain.

Mean-field cage theory for the freezing of hard-sphere fluids

Using some observations and some mean-field approximations, we develop a mean-field cage theory for the freezing of hard-sphere fluids with v(f) > or =a(d) and obtain the freezing densities as functions of the closest-packing densities and the spatial densities, which are in good agreement with the experimental and simulation results.

Should "lane formation" occur systematically in driven liquids and colloids?

We report on nonequilibrium molecular dynamics simulations of binary mixtures of particles in a color field. Both nonequilibrium molecular dynamics and Brownian dynamics generally assume that the mechanical noise is of thermal origin only and that, at a given temperature, its amplitude remains constant however strong the applied field is. We show that this postulate systematically results in the strong ordering of particles into lanes. By applying a nonequilibrium molecular dynamics method which does not exert any constraint on the noise amplitude, we show that releasing this constraint prevents the systematic "lane formation" from occurring. We observe the onset of density inhomogeneities and jamming instead. This behavior is reminiscent of the shear-thickening regime observed experimentally on colloidal suspensions and in simulations taking into account hydrodynamic interactions.

Constrained deformation of a confined solid: anomalous failure by nucleation of smectic bands

We report results of computer simulations of the deformation and failure behavior of a thin crystalline strip of "hard disks" in two dimensions confined within a quasi-one-dimensional "hard-wall" channel of fixed width corresponding to a few disk diameters. Starting from a commensurate triangular solid, stretching the strip along its length introduces a rectangular distortion. This, beyond a critical strain, leads to failure of the solid by "phase separation" into alternating bands of solid and smectic-like phases. The critical strain is inversely proportional to the channel width, i.e., thinner strips are stronger. The large plastic deformation which precedes failure is observed to be reversible.

Critical exponents of isotropic-hexatic phase transition in the hard-disk system

The hard-disk system is studied by observing the nonequilibrium relaxation behavior of a bond-orientational order parameter. The density dependence of characteristic relaxation time tau is estimated from the finite-time scaling analysis. The critical point between the fluid and the hexatic phase is refined to be 0.899 (1) by assuming the divergence behavior of the Kosterlitz-Thouless transition. The value of the critical exponent eta is also studied by analyzing the fluctuation of the order parameter at the criticality and estimated as eta=0.25 (2). These results are consistent with the prediction by the Kosterlitz-Thouless-Halperin-Nelson-Young theory.

Nonequilibrium relaxation analysis of two-dimensional melting

The phase diagram of a hard-disk system is studied by observing nonequilibrium relaxation functions of a bond-orientational order parameter using particle dynamics simulations. From a finite-time scaling analysis, two Kosterlitz-Thouless transitions can be observed when the density is increased from the isotropic fluid phase to closest packing. The transition densities are estimated to be 0.901(2) and 0.910(2), where the density denotes the fraction of area occupied by the particles, the density is normalized to one for the quadratic packing configuration. These observations are consistent with the predictions of the Kosterlitz-Thouless-Halperin-Nelson-Young theory.

Influence of vacancies on the melting transition of hard disks in two dimensions

We present the results of molecular dynamics simulations of two-dimensional (2D) hard disk systems in the vicinity of melting. The simulations are used to calculate the elastic constants, which can be used to estimate the location of the Kosterlitz-Thouless dislocation unbinding transition. Simulations on defect-free lattices indicate that this transition is expected to occur at essentially the same density as a first-order solid-isotropic transition and so it is not possible to rule out either a one step weak first-order transition between the solid and the isotropic fluid or a two step transition via a hexatic phase. Simulations performed on systems with vacancies indicate that the elastic constants are essentially unchanged at constant density. This result implies that vacancies have little influence on the melting of 2D hard disk solids.