Local orderings in the melting process of a square crystal and in the resulting liquid.

Using Brownian dynamics simulations we investigate the melting processes of a square crystalline lattice of colloidal particles interacting via an isotropic potential, which comprises both a hard-core repulsion and an additional softened square-well potential. For temperatures slightly lower than the transition one, we found a proliferation of small liquid clusters surrounded by the square lattice. These clusters are not static, quite the opposite, they have an intense dynamics and are continuously formed and destroyed over time. However, no unbound topological defects are observed. At the transition temperature, one of these liquid clusters starts to grow, until the entire system becomes in the liquid phase, then, characterizing a first-order phase transition. The tetratic intermediate phase, as given by the KTHNY theory, was not observed. Moreover, the liquid phase exhibits a considerable number of crystalline clusters having square and triangular orderings, which remain present even when increasing temperature by an order of magnitude. As the temperature increases, structural changes within the liquid phase are analyzed by evaluating the number and sizes of the square and triangular clusters. A transition of the dominant clusters is observed.

2D melting of confined colloids with a mixture of square and triangular order.

We implement Brownian dynamics to investigate the melting processes of colloidal particles confined isotropically and interacting via a potential which can be tailored in a repulsive-attractive-respulsive fashion as the interparticle distance increases. The stable configurations of such a system is composed of a large diversity of structures, which includes quasicrystalline, triangular, square, and mixed orderings, as well as the presence of fringes and holes, which are located, respectively, at the border and interior of the clusters. Our simulations demonstrate that during the melting process particles are able to swing between different micro phases. This intermediary stage, present in a finite range of temperature, precedes the melting in all cases investigated and is different from the hexatic phase of the KTNHY framework. We also test the fringes stability and find it to be higher than the one found in compact clusters. Finally, we show that, at the high temperature regime, the system loses its angular ordering while still preserves its radial interparticle confinement, which, ultimately, causes the proliferation of small subclusters.

Complex structures generated by competing interactions in harmonically confined colloidal suspensions.

We investigate the structural properties of colloidal particle systems interacting via an isotropic pair potential and confined by a three-dimensional harmonic potential. The interaction potential has a repulsive-attractive-repulsive profile that varies with the interparticle distance (also known as a 'mermaid' potential). We performed Langevin dynamics simulations to find the equilibrium configurations of the system. We show that particles can self-assemble in complex structural patterns, such as compact disks, fringed disks, rods, spherical clusters with superficial entrances among others. Also, for particular values of the parameters of the interaction potential, we could identify that some configurations were formed by quasi two-dimensional (2D) structures which are stable for 2D systems.

Structural orderings of anisotropically confined colloids interacting via a quasi-square-well potential.

We implement Brownian dynamics to investigate the static properties of colloidal particles confined anisotropically and interacting via a potential which can be tailored in a repulsive-attractive-respulsive fashion as the interparticle distance increases. A diverse number of structural phases are self-assembled, which were classified according to two aspects, that is, their macroscopic and microscopic patterns. Concerning the microscopic phases we found the quasicrystalline, triangular, square, and mixed orderings, where this latter is a combination of square and triangular cells in a 3×2 proportion, i.e., the so-called (3(3),4(2)) Archimedian lattice. On the macroscopic level the system could self-organize in a compact or perforated single cluster surrounded or not by fringes. All the structural phases are summarized in detailed phases diagrams, which clearly show that the different phases are extended as the confinement potential becomes more anisotropic.

Structural properties of anisotropically confined binary Coulomb balls.

We investigate the structural properties of a small binary system of dusty plasma subjected to an anisotropic external confinement. We have found that the ground state can form various symmetrical configurations, which generally correspond to variations of a multiple-ring structure. The presence of such structures is given through a detailed phase diagram constructed for the systems with N=18 and 24 particles. Furthermore, we show that the configurations of multiple rings can present a persistent behavior for the case in which the number of less-charged particles is equal to a multiple of the number of particles per ring. Finally, we discuss the main characteristics of the transitions of first and second orders since these are related to the main configurational changes of the ground states.

Angular melting scenarios in binary dusty-plasma Coulomb balls: magic versus normal clusters.

Molecular-dynamic simulations were performed in order to investigate the melting processes of isotropically confined binary systems. We considered two species of particles, which differ by their amount of electric charge. A Lindemann type of criterion was used to determine the angular melting temperature. We demonstrate that the magic-to-normal cluster transition can evolve in two distinct ways, that is, through a structural phase transition of the first order or via a smooth transition where an increase of the shells' width leads to a continuous decreasing mechanical stability of the system. Moreover, for large systems, we demonstrate that the internal cluster exerts a minor effect on the mechanical stability of the external shell. Furthermore, we show that highly symmetric configurations, such as those found for multiple ring structures, have large mechanical stability, i.e., high angular melting temperature.

Structural ordering of trapped colloids with competing interactions.

The structure of colloids with competing interactions which are confined in a harmonic external trap potential is analyzed numerically by energy minimization in two spatial dimensions. A wealth of different cluster structures is found to be stable including clusters with a fringed outer rim (reminiscent to an ornamental border), clusters perforated with voids, as well as clusters with a crystalline core and a disordered rim. All cluster structures occur in a two-dimensional parameter space. The structural ordering can therefore be efficiently tuned by changing few parameters only providing access to a controlled fabrication of colloidal clusters.

Madelung energy of Yukawa lattices.

We propose a method to obtain an approximate closed form expression for the Madelung energy (ME) of Yukawa lattices. Such a method is applied for lattices of different topologies and dimensions. The obtained Madelung energies have a satisfactory accuracy for all ranges of the screening parameter κ of the Yukawa potential, and it becomes exact in the asymptotic limits κ→0 and κ→+∞. For instance, for the triangular lattice, the maximum relative error of the ME given by the method is about 0.0047. Also, satisfactory results are obtained for the one-component plasma limit. The Madelung constants of the two-dimensional hexagonal BN and square NaCl and the three-dimensional cubic NaCl crystals are estimated with a relative error of 0.004, 0.006, and 0.03, respectively. Finally, different ways to improve the method are presented and discussed.

Structural phases of colloids interacting via a flat-well potential.

Using Langevin dynamics simulations we investigate the self-assembly of colloidal particles in two dimensions interacting via an isotropic potential, which comprises both a hard-core repulsion and an additional softened square-well potential of controllable width α. In dilute concentrations, the particles assemble in small clusters with a well-defined crystalline order. For small values of α the particles form triangular lattices. As α is increased, more particles can be captured by the potential well giving rise to different crystalline symmetries and the structural phase transitions between them. The main structures observed are triangular, square, and a mixture of square and triangular cells forming an Archimedean tiling. In the concentrated regime the particles form a single percolated cluster with essentially the same orderings at the same ranges of α values as observed in the dilute regime, thus showing that cluster boundary effects have a minor influence on the cluster crystal symmetry. By using energy analysis and geometry arguments we discuss how the different observed structures minimize the system energy at different values of α.

Binary dusty plasma Coulomb balls.

We investigated the mixing and segregation of a system consisting of two different species of particles, having different charges, interacting through a pure Coulomb potential, and confined in a three-dimensional parabolic trap. The structure of the cluster and its normal mode spectrum are analyzed as a function of the relative charge and the relative number of different types of particles. We found that (a) the system can be in a mixed or segregated state depending on the relative charge ratio parameter and (b) the segregation process is mediated by a first or second order structural phase transition which strongly influences the magic cluster properties of the system.

Melting transitions in isotropically confined three-dimensional small Coulomb clusters.

Molecular dynamic simulations are performed to investigate the melting process of small three-dimensional clusters (i.e., systems with one and two shells) of classical charged particles trapped in an isotropic parabolic potential. The confined particles interact through a repulsive potential. We find that the ground-state configurations for systems with N=6 , 12, 13, and 38 particles interacting through a Coulomb potential are magic clusters. Such magic clusters have an octahedral or icosahedral symmetry and are found to have a large stability against intrashell diffusion leading to an intershell melting transition prior to the intrashell and radial melting process. For systems with two shells a local radial melting of subshells is found at low temperatures resulting in a structural transition leading to an increased symmetry of the ordered system. Using Lindemann's criterion the different melting temperatures are determined and the influence of the screening of the interparticle interaction was investigated. A normal mode analysis is performed and some of the normal modes are found to be determinantal for the melting process.

Inhomogeneous melting in anisotropically confined two-dimensional clusters.

Molecular dynamic simulations are performed to investigate the melting process of two-dimensional clusters of classical charged particles trapped in an anisotropic parabolic potential. The confined particles interact through a repulsive potential. We find that the eccentricity of the confinement potential strongly affects the melting pattern of such clusters. Increasing the eccentricity of the confinement potential drives the system through three different melting regimes. Inhomogeneous melting is the typical melting process for anisotropically confined clusters and its appearance in small systems occurs in a distinct form called here internal intershell melting. The latter involves only particles in the center of the cluster while particles on the far left and right of the cluster are still ordered having a much higher melting temperature. Using the Lindemann's criterion the melting temperatures are determined as a function of the different parameters. The internal intershell melting process is found for both long-range (i.e., logarithmic) and short-range (i.e., screened Coulomb) interparticle interaction. Decreasing the range of the interparticle interaction increases the eccentricity of the confinement potential for which internal intershell melting can occur.

Structure and spectrum of anisotropically confined two-dimensional clusters with logarithmic interaction.

We studied the structural and spectral properties of a classical system consisting of a finite number of particles, moving in two dimensions, and interacting through a repulsive logarithmic potential and held together by an anisotropic harmonic potential. Increasing the anisotropy of the confinement potential can drive the system from a two-dimensional (2D) to a one-dimensional (1D) configuration. This change occurs through a sequence of structural transitions of first and second order which are reflected in the normal mode frequencies. Our results of the ground state configurations are compared with recent experiments and we obtained a satisfactory agreement. The transition from the 1D line structure to the 2D structure occurs through a zigzag transition which is of second order. We found analytical expressions for the eigenfrequencies before the zigzag transition, which allowed us to obtain an analytical expression for the anisotropy parameter at which the zigzag transition occurs as a function of the number of particles in the system.