Lattice Model Results for Pattern Formation in a Mixture with Competing Interactions.

A monolayer consisting of two types of particles, with energetically favored alternating stripes of the two components, is studied by Monte Carlo simulations and within a mesoscopic theory. We consider a triangular lattice model and assume short-range attraction and long-range repulsion between particles of the same kind, as well as short-range repulsion and long-range attraction for the cross-interaction. The structural evolution of the model upon increasing temperature is studied for equal chemical potentials of the two species. We determine the structure factor, the chemical potential-density isotherms, the specific heat, and the compressibility, and show how these thermodynamic functions are associated with the spontaneous formation of stripes with varying degrees of order.

Entropy driven incommensurate structures in the frustrated kagome staircase Co3V2O8.

Co3V2O8features spin-3/2 moments arrayed on a kagome staircase lattice. A spin density wave with a continuously evolving propagation vector ofk⃗=(0,δ,0), showing both incommensurate states and multiple commensurate lock-ins, is observed at temperatures above the ferromagnetic ground state. Previous work has suggested that this changing propagation vector could be driven by changes in exchange interactions due to Co atom displacements. We present a straightforward model showing that a Hamiltonian with competing (but temperature independent) interactions can semi-quantitatively reproduce this behavior using a mean field approximation. The simulated spin density wave magnetic structures feature buckled kagome planes that are either ferromagnetically or antiferromagnetically ordered. Propagation vectors that differ fromδ=1/2will have multiple different ways of arranging these ferromagnetic layers that have very similar energies. This classical stacking entropy appears to be crucial in stabilizing the temperature-dependent propagation vector.

Structural phases of classical 2D clusters with competing two-body and three-body interactions.

In modeling systems of interacting particles, many-body terms beyond pairwise interactions are often overlooked. Nevertheless, in certain scenarios, even small contributions from three-body or higher-order terms can disrupt significant changes in their collective behavior. Here we investigate the effects of three-body interactions on the structure and stability of 2D, harmonically confined clusters. We consider clusters with three distinct pairwise interactions:logr,1/r, ande-κr/r, thus covering a wide range of condensed and soft matter systems, such as vortices in mesoscopic superconductors, charged colloids, and dusty plasma. In each case, we evaluate the energetics and normal mode spectra of equilibrium and metastable configurations as the intensity of an attractive, Gaussian three-body potential is varied. We demonstrate that, above a threshold value of the three-body energy strength, the cluster shrinks and eventually becomes self-sustained, that is, it remains cohesive after the confinement potential is shut down. Depending on the strengths of the two-body and three-body interaction terms, this compaction can be continuous or abrupt. The latter case is characterized by a discontinuous jump in the particle density and coexsitence of the compact and non-compact phases as metastable states, as in a first-order phase transition. For some values of the particle number, the compaction is preceded by one or more structural changes, resulting in configurations not usually seen in purely pairwise-additive clusters.

Pattern Formation in Two-Component Monolayers of Particles with Competing Interactions.

Competing interactions between charged inclusions in membranes of living organisms or charged nanoparticles in near-critical mixtures can lead to self-assembly into various patterns. Motivated by these systems, we developed a simple triangular lattice model for binary mixtures of oppositely charged particles with additional short-range attraction or repulsion between like or different particles, respectively. We determined the ground state for the system in contact with a reservoir of the particles for the whole chemical potentials plane, and the structure of self-assembled conglomerates for fixed numbers of particles. Stability of the low-temperature ordered patterns was verified by Monte Carlo simulations. In addition, we performed molecular dynamics simulations for a continuous model with interactions having similar features, but a larger range and lower strength than in the lattice model. Interactions with and without symmetry between different components were assumed. We investigated both the conglomerate formed in the center of a thin slit with repulsive walls, and the structure of a monolayer adsorbed at an attractive substrate. Both models give the same patterns for large chemical potentials or densities. For low densities, more patterns occur in the lattice model. Different phases coexist with dilute gas on the lattice and in the continuum, leading to different patterns in self-assembled conglomerates ('rafts').

Competing effects of rotational diffusivity and activity on finite-sized clusters.

Colloidal particles interacting via short-range attraction and long-range repulsion are known to stabilize finite-sized clusters under equilibrium conditions. In this work, the effect of self-propulsion speed (activity) and rotational diffusivity (Dr) on the phase behavior of such particles is investigated using Brownian dynamics simulations. The system exhibits rich phase behavior consisting of clusters of different kinds. The cluster size varies non-monotonically with activity: increasing first and decreasing at higher activity, thus driving cluster-to-fluid phase transition. Rotational diffusivity also facilitates the formation of clusters. Larger clusters could be stabilized at lowDrvalues while at highDrvalues, clusters are stable even at higher activities. The analysis of the static structure factor of the system confirms that rotational diffusivity delays the cluster-to-fluid transition driven by activity.

Thermodynamics, static properties and transport behaviour of fluids with competing interactions.

Competing interaction fluids have become ideal model systems to study a large number of phenomena, for example, the formation of intermediate range order structures, condensed phases not seen in fluids driven by purely attractive or repulsive forces, the onset of particle aggregation under in- and out-of-equilibrium conditions, which results in the birth of reversible and irreversible aggregates or clusters whose topology and morphology depend additionally on the thermodynamic constrictions, and a particle dynamics that has a strong influence on the transport behaviour and rheological properties of the fluid. In this contribution, we study a system of particles interacting through a potential composed by a continuous succession of a short-ranged square-well (SW), an intermediate-ranged square-shoulder and a long-ranged SW. This potential model is chosen to systematically analyse the contribution of every component of the interaction potential on the phase behaviour, the microstructure, the morphology of the resulting aggregates and the transport phenomena of fluids described by competing interactions. Our results indicate that the inclusion of a barrier and a second well leads to new and interesting effects, which in addition result in variations of the physical properties associated to the competition among interactions.

Structural and Dynamical Behaviour of Colloids with Competing Interactions Confined in Slit Pores.

Systems with short-range attractive and long-range repulsive interactions can form periodic modulated phases at low temperatures, such as cluster-crystal, hexagonal, lamellar and bicontinuous gyroid phases. These periodic microphases should be stable regardless of the physical origin of the interactions. However, they have not yet been experimentally observed in colloidal systems, where, in principle, the interactions can be tuned by modifying the colloidal solution. Our goal is to investigate whether the formation of some of these periodic microphases can be promoted by confinement in narrow slit pores. By performing simulations of a simple model with competing interactions, we find that both the cluster-crystal and lamellar phases can be stable up to higher temperatures than in the bulk system, whereas the hexagonal phase is destabilised at temperatures somewhat lower than in bulk. Besides, we observed that the internal ordering of the lamellar phase can be modified by changing the pore width. Interestingly, for sufficiently wide pores to host three lamellae, there is a range of temperatures for which the two lamellae close to the walls are internally ordered, whereas the one at the centre of the pore remains internally disordered. We also find that particle diffusion under confinement exhibits a complex dependence with the pore width and with the density, obtaining larger and smaller values of the diffusion coefficient than in the corresponding bulk system.

Adsorption in Mixtures with Competing Interactions.

A binary mixture of oppositely charged particles with additional short-range attraction between like particles and short-range repulsion between different ones in the neighborhood of a substrate preferentially adsorbing the first component is studied by molecular dynamics simulations. The studied thermodynamic states correspond to an approach to the gas-crystal coexistence. Dependence of the near-surface structure, adsorption and selective adsorption on the strength of the wall-particle interactions and the gas density is determined. We find that alternating layers or bilayers of particles of the two components are formed, but the number of the adsorbed layers, their orientation and the ordered patterns formed inside these layers could be quite different for different substrates and gas density. Different structures are associated with different numbers of adsorbed layers, and for strong attraction the thickness of the adsorbed film can be as large as seven particle diameters. In all cases, similar amount of particles of the two components is adsorbed, because of the long-range attraction between different particles.

Assembly of Building Blocks by Double-End-Anchored Polymers in the Dilute Regime Mediated by Hydrophobic Interactions at Controlled Distances.

Hierarchical assembly of building blocks via competing, orthogonal interactions is a hallmark of many of nature's composite materials that do not require highly specific ligand-receptor interactions. To mimic this assembly mechanism requires the development of building blocks capable of tunable interactions. In the present work, we explored the interplay between repulsive (steric and electrostatic) and attractive hydrophobic forces. The designed building blocks allow hydrophobic forces to effectively act at controlled, large distances, to create and tune the assembly of membrane-based building blocks under dilute conditions, and to affect their interactions with cellular membranes via physical cross-bridges. Specifically, we employed double-end-anchored poly(ethylene glycol)s (DEA-PEGs)-hydrophilic PEG tethers with hydrophobic tails on both ends. Using differential-interference-contrast optical microscopy, synchrotron small-angle X-ray scattering (SAXS), and cryogenic electron microscopy, we investigated the ability of DEA-PEGs to mediate assembly in the dilute regime on multiple length scales and on practical time scales. The PEG length, anchor hydrophobicity, and molar fraction of DEA-PEG molecules within a membrane strongly affect the assembly properties. Additional tuning of the intermembrane interactions can be achieved by adding repulsive interactions via PEG-lipids (steric) or cationic lipids to the DEA-PEG-mediated attractions. While the optical and electron microscopy imaging methods provided qualitative evidence of the ability of DEA-PEGs to assemble liposomes, the SAXS measurements and quantitative line-shape analysis in dilute preparations demonstrated that the ensemble average of loosely organized liposomal assemblies maintains DEA-PEG concentration-dependent tethering on defined nanometer length scales. For cationic liposome-DNA nanoparticles (CL-DNA NPs), aggregation induced by DEA-PEGs decreased internalization of NPs by cells, but tuning the DEA-PEG-induced attractions by adding repulsive steric interactions via PEG-lipids limited aggregation and increased NP uptake. Furthermore, confocal microscopy imaging together with colocalization studies with Rab11 and LysoTracker as markers of intracellular pathways showed that modifying CL-DNA NPs with DEA-PEGs alters their interactions with the plasma and endosomal membranes.

Physical foundations of biological complexity.

Biological systems reach hierarchical complexity that has no counterpart outside the realm of biology. Undoubtedly, biological entities obey the fundamental physical laws. Can today's physics provide an explanatory framework for understanding the evolution of biological complexity? We argue that the physical foundation for understanding the origin and evolution of complexity can be gleaned at the interface between the theory of frustrated states resulting in pattern formation in glass-like media and the theory of self-organized criticality (SOC). On the one hand, SOC has been shown to emerge in spin-glass systems of high dimensionality. On the other hand, SOC is often viewed as the most appropriate physical description of evolutionary transitions in biology. We unify these two faces of SOC by showing that emergence of complex features in biological evolution typically, if not always, is triggered by frustration that is caused by competing interactions at different organizational levels. Such competing interactions lead to SOC, which represents the optimal conditions for the emergence of complexity. Competing interactions and frustrated states permeate biology at all organizational levels and are tightly linked to the ubiquitous competition for limiting resources. This perspective extends from the comparatively simple phenomena occurring in glasses to large-scale events of biological evolution, such as major evolutionary transitions. Frustration caused by competing interactions in multidimensional systems could be the general driving force behind the emergence of complexity, within and beyond the domain of biology.

Segmentation in cohesive systems constrained by elastic environments.

The complexity of fracture-induced segmentation in elastically constrained cohesive (fragile) systems originates from the presence of competing interactions. The role of discreteness in such phenomena is of interest in a variety of fields, from hierarchical self-assembly to developmental morphogenesis. In this paper, we study the analytically solvable example of segmentation in a breakable mass-spring chain elastically linked to a deformable lattice structure. We explicitly construct the complete set of local minima of the energy in this prototypical problem and identify among them the states corresponding to the global energy minima. We show that, even in the continuum limit, the dependence of the segmentation topology on the stretching/pre-stress parameter in this problem takes the form of a devil's type staircase. The peculiar nature of this staircase, characterized by locking in rational microstructures, is of particular importance for biological applications, where its structure may serve as an explanation of the robustness of stress-driven segmentation.This article is part of the themed issue 'Patterning through instabilities in complex media: theory and applications.'

Nanophase separation in monomolecularly thin water-ethanol films controlled by graphene.

Control over nanoscale patterning of ultrathin molecular films plays an important role both in natural as well as artificial nanosystems. Here we report on nanophase separated patterns of water and ethanol within monomolecularly thin films confined between the cleavage plane of mica and single or a few layers of graphene. Employing scanning force microscopy of the graphene layers conforming to the molecular films we quantify the patterns using the ethanol-water cross correlation and the autocorrelation of domain wall directions. They reveal that lateral pattern dimensions grow and the domain walls stiffen upon increasing the thickness of the graphene multilayers. We attribute the control of the patterns through the graphene layers to the competition between the mechanical deformation energy of the graphene sheets and the electrostatic repulsion of dipoles normal to the interface. The latter results from charge transfer between graphene and the molecules confined between mica and graphene.

How to Optimize Binding of Coated Nanoparticles: Coupling of Physical Interactions, Molecular Organization and Chemical State.

One of the key challenges in the development of nano carriers for drug delivery and imaging is the design of a system that selectively binds to target cells. A common strategy is to coat the delivery device with specific ligands that bind strongly to overexpressed receptors. However such devices are usually unable to discriminate between receptors found on benign and malignant cells. We demonstrate, theoretically, how one can achieve enhanced binding to target cells by using multiple physical and chemical interactions. We study the effective interactions between a polymer decorated nano micelle or nanoparticle with three types of model lipid membranes that differ in the composition of their outer leaflet. They are: i) lipid membranes with overexpressed receptors, ii) membranes with a given fraction of negatively charged lipids and iii) membranes with both overexpressed receptors and negatively charged lipids. The coating contains a mixtures of two short polymers, one neutral for protection and the other a polybase with a functional end-group to optimize specific binding with the overexpressed receptors and electrostatic interactions with charged lipid head-groups. The strength of the binding for the combined system is much larger than the sum of the independent electrostatic or specific interactions binding. We find a range of distances where the addition of two effective repulsive interactions become an attraction in the combined case. The changes in the strength and shape of the effective interaction are due to the coupling that exists between molecular organization, physical interactions and chemical state, e.g., protonation. The predictions provide guidelines for the design of carrier devices for targeted drug and nanoparticle delivery and give insight in the competing and highly non-additive nature of the different effective interactions in nanoscale systems in constrained environments that are ubiquitous in synthetic and biological systems.