Orientational order in systems with competing interactions

Abstract

Abstract. Quasi two dimensional systems with competing interactions usually display complex patterns in the relevant order parameter. In many cases these patterns are analog to liquid-crystal phases, showing smectic, nematic or hexatic order. We show that order parameters suitable for the characterization of these phases in systems with nearly isotropic competing interactions emerge naturally from an analysis of a Landau model. We describe with some detail the nematic case, which characterizes orientational order of striped domain walls. The Landau model presents an isotropic-nematic transition of the Kosterlitz-Thouless type. Although for the perfectly isotropic model long range nematic order is absent in infinite systems, we show that in real systems of finite size nematic order of domain walls can be observed. IntroductionIn many natural systems complex patterns originate due to the presence of competing interactions at different scales. Examples range from classical systems like magnetic ultrathin films [1], diblock copolymers [2] and colloidal suspensions [3], to quantum systems like quantum Hall samples [4] and high T c superconductors [5]. These pattern forming systems may present positional as well as orientational order, the origin of which depends on microscopic interactions. In the case of magnetic ultrathin films, of a few atomic monolayers thick, strong anisotropy perpendicular to the film plane forces the magnetization to be out of plane. In this case short range ferromagnetic interaction competes with antiferromagnetic dipolar interaction. As a consequence, the system shows ordered striped phases, of alternating magnetization along one direction, as is seen in Figure 1.In Figure 2 similar patterns can be observed in a completely different system, a block copolymer. In this case, elongated polymer chains with different types of building "blocks" compete between them and produce phase segregation, giving rise to complex patterns in two and three dimensions. Figure 3 shows a schematic representation of microemulsion phases. Microemulsions are mixtures of anfiphilic molecules which have an hydrophilic head and an hydrophobic tail. When in contact with water and oil, as depicted in the figure, they tend to phase separate giving rise to different kinds of ordered patterns with an order similar to liquid crystals [7,8].

Modeling spatial patterns in the visual cortex

“…In this way we can define the structure factor as an order parameter, in a form that is similar to the tensor order parameter in nematic liquid crystals [31]. In Fig.…”

Nematic phase in theJ1−J2square-lattice Ising model in an external field

“…To give even a partial answer to such a question is always a considerable challenge. This is because the nematic order parameter in stripe-forming systems amounts to compute correlation functions in different space directions searching for a breaking of isotropy characteristic of these phases [38,39]. Then the most common analytical approaches for a one-particle order parameter, namely, mean field theory, fails at detecting nematic phases and one must go beyond naive MFT to an approximation which allows one to compute anisotropic correlations.…”

Pair correlations and structure factor of theJ1-J2square lattice Ising mode

“…In these systems a nematic order parameter can be defined, in analogy with the nematic order parameter of liquid crystals, as a second-rank symmetric traceless tensor which encodes the 180 o rotational symmetry of nematic phases. In two dimensions the nematic tensor has only one independent entry which can be written as [4]:…”

“…In order to characterize these nematic-like phases from microscopic models it is necessary to go beyond naive mean field approximations. In particular, the orientational or nematic order parameter in two dimensional modulated systems is proportional to the difference between correlation functions in two orthogonal space directions [4,14]. As will be better described below, a nematic order parameter can be defined as a weighted integral of the structure factor.…”