Nematic phase in stripe-forming systems within the self-consistent screening approximation

Orientational ordering of lamellar structures on closed surfaces

Self-assembly of particles with short-range attraction and long-range repulsion interactions on a flat and on a spherical surface is compared. Molecular dynamics simulations are performed for the two systems having the same area and the density optimal for formation of stripes of particles. Structural characteristics, e.g., a cluster size distribution, a number of defects, and an orientational order parameter (OP), as well as the specific heat, are obtained for a range of temperatures. In both cases, the cluster size distribution becomes bimodal and elongated clusters appear at the temperature corresponding to the maximum of the specific heat. When the temperature decreases, orientational ordering of the stripes takes place and the number of particles per cluster or stripe increases in both cases. However, only on the flat surface, the specific heat has another maximum at the temperature corresponding to a rapid change of the OP. On the sphere, the crossover between the isotropic and anisotropic structures occur in a much broader temperature interval; the orientational order is weaker and occurs at significantly lower temperature. At low temperature, the stripes on the sphere form spirals and the defects resemble defects in the nematic phase of rods adsorbed at a sphere.

Two-dimensional pattern formation in ionic liquids confined between graphene walls

We perform molecular dynamics simulations of ionic liquids confined between graphene walls under a large variety of conditions (pure ionic liquids, mixtures with water and alcohols, mixtures with lithium salts and defective graphene walls). Our results show that the formation of striped and hexagonal patterns in the Stern layer can be considered as a general feature of ionic liquids at electrochemical interfaces, the transition between patterns being controlled by the net balance of charge in the innermost layer of adsorbed molecules. This explains previously reported experimental and computational results and, for the first time, why these pattern changes are triggered by any perturbation of the charge density at the innermost layer of the electric double layer (voltage and composition changes, and vacancies at the electrode walls, among others), which may help tuning electrode-ionic liquid interfaces. Using Monte Carlo simulations we show that such structures can be reproduced by a simple two-dimensional lattice model with only nearest-neighbour interactions, governed by highly screened ionic interactions and short-range and excluded volume interactions. We also show that the results of our simulations are consistent with those inferred from the Landau-Brazovskii theory of pattern formation in self-assembling systems. The presence of these patterns at the ionic liquid graphene-electrode interfaces may have a strong impact on the process of ionic transfer from the bulk mixtures to the electrodes, on the differential capacitance of the electrode-electrolyte double layer or on the rates of redox reactions at the electrodes, among other physicochemical properties, and is therefore an effect of great technological interest.

Self-consistent theory for systems with mesoscopic fluctuations

We have developed a theory for inhomogeneous systems that allows for the incorporation of the effects of mesoscopic fluctuations. A hierarchy of equations relating the correlation and direct correlation functions for the local excess [Formula: see text] of the volume fraction of particles ζ has been obtained, and an approximation leading to a closed set of equations for the two-point functions has been introduced for the disordered inhomogeneous phase. We have numerically solved the self-consistent equations for one-dimensional (1D) and three-dimensional (3D) models with short-range attraction and long-range repulsion. Predictions for all of the qualitative properties of the 1D model agree with the exact results, but only semi-quantitative agreement is obtained in the simplest version of the theory. The effects of fluctuations in the two 3D models considered are significantly different, despite the very similar properties of these models in the mean-field approximation. In both cases we obtain the sequence of large-small-large compressibility for increasing ζ. The very small compressibility is accompanied by the oscillatory decay of correlations with correlation lengths that are orders of magnitude larger than the size of particles. In one of the two models considered, the small compressibility becomes very small and the large compressibility becomes very large with decreasing temperature, and eventually van der Waals loops appear. Further studies are necessary in order to determine the nature of the strongly inhomogeneous phase present for intermediate volume fractions in 3D.

Effects of confinement on pattern formation in two dimensional systems with competing interactions

Template-assisted pattern formation in monolayers of particles with competing short-range attraction and long-range repulsion interactions (SALR) is studied by Monte Carlo simulations in a simple generic model [N. G. Almarza et al., J. Chem. Phys., 2014, 140, 164708]. We focus on densities corresponding to formation of parallel stripes of particles and on monolayers laterally confined between straight parallel walls. We analyze both the morphology of the developed structures and the thermodynamic functions for broad ranges of temperature T and the separation L2 between the walls. At low temperature stripes parallel to the boundaries appear, with some corrugation when the distance between the walls does not match the bulk periodicity of the striped structure. The stripes integrity, however, is rarely broken for any L2. This structural order is lost at T = TK(L2) depending on L2 according to a Kelvin-like equation. Above the Kelvin temperature TK(L2) many topological defects such as breaking or branching of the stripes appear, but a certain anisotropy in the orientation of the stripes persists. Finally, at high temperature and away from the walls, the system behaves as an isotropic fluid of elongated clusters of various lengths and with various numbers of branches. For L2 optimal for the stripe pattern the heat capacity as a function of temperature takes the maximum at T = TK(L2).

Density functional theory for systems with mesoscopic inhomogeneities

We study the effects of fluctuations on the mesoscopic length scale on systems with mesoscopic inhomogeneities. Equations for the correlation function and for the average volume fraction are derived in the self-consistent Gaussian approximation. The equations are further simplified by postulating the expression for the structure factor consistent with scattering experiments for self-assembling systems. Predictions of the approximate theory are verified by a comparison with the exact results obtained earlier for the one-dimensional lattice model with first-neighbor attraction and third-neighbor repulsion. We find qualitative agreement for the correlation function, the equation of state and the dependence of the chemical potential μ on the volume fraction ζ. Our results confirm also that strong inhomogeneities in the disordered phase are found only in the case of strong repulsion. The inhomogeneities are reflected in an oscillatory decay of the correlation function with a very large correlation length, three inflection points in the [Formula: see text] curve and a compressibility that for increasing ζ takes very large, very small and again very large values.

Nematic phase in the J(1)-J(2) square-lattice Ising model in an external field

The J(1)-J(2) Ising model in the square lattice in the presence of an external field is studied by two approaches: the cluster variation method (CVM) and Monte Carlo simulations. The use of the CVM in the square approximation leads to the presence of a new equilibrium phase, not previously reported for this model: an Ising-nematic phase, which shows orientational order but not positional order, between the known stripes and disordered phases. Suitable order parameters are defined, and the phase diagram of the model is obtained. Monte Carlo simulations are in qualitative agreement with the CVM results, giving support to the presence of the new Ising-nematic phase. Phase diagrams in the temperature-external field plane are obtained for selected values of the parameter κ=J(2)/|J(1)| which measures the relative strength of the competing interactions. From the CVM in the square approximation we obtain a line of second order transitions between the disordered and nematic phases, while the nematic-stripes phase transitions are found to be of first order. The Monte Carlo results suggest a line of second order nematic-disordered phase transitions in agreement with the CVM results. Regarding the stripes-nematic transitions, the present Monte Carlo results are not precise enough to reach definite conclusions about the nature of the transitions.

Nature of long-range order in stripe-forming systems with long-range repulsive interactions

We study two dimensional stripe forming systems with competing repulsive interactions decaying as r(-α). We derive an effective Hamiltonian with a short-range part and a generalized dipolar interaction which depends on the exponent α. An approximate map of this model to a known XY model with dipolar interactions allows us to conclude that, for α<2 long-range orientational order of stripes can exist in two dimensions, and establish the universality class of the models. When α≥2 no long-range order is possible, but a phase transition in the Kosterlitz-Thouless universality class is still present. These two different critical scenarios should be observed in experimentally relevant two dimensional systems like electronic liquids (α=1) and dipolar magnetic films (α=3). Results from Langevin simulations of Coulomb and dipolar systems give support to the theoretical results.