Soft particles are known to overlap and form stable clusters that self-assemble into periodic crystalline phases with density-independent lattice constants. We use molecular dynamics simulations in two dimensions to demonstrate that, through a judicious design of an isotropic pair potential, one can control the ordering of the clusters and generate a variety of phases, including decagonal and dodecagonal quasicrystals. Our results confirm analytical predictions based on a mean-field approximation, providing insight into the stabilization of quasicrystals in soft macromolecular systems, and suggesting a practical approach for their controlled self-assembly in laboratory realizations using synthesized soft-matter particles. DOI: 10.1103/PhysRevLett.113.098304 PACS numbers: 82.70.Dd, 61.44.Br, 64.70.D-, 64.75.Yz Particles interacting via pair potentials with repulsive cores, which are either bounded or only slowly diverginglike those found naturally in soft matter systems [1]-can be made to overlap under pressure to form clusters [2], which then self-assemble to form crystalline phases [3]. The existence of such cluster crystals was recently confirmed in amphiphilic dendritic macromolecules using monomer-resolved simulations [4], and in certain bosonic systems [5]. They occur even when the particles are purely repulsive, and typically exhibit periodic fcc or bcc structures. Here we employ molecular dynamics (MD) simulations in two dimensions, guided by analytical insight, to show how isotropic pair potentials can be designed to control the self-assembly of the clusters, suggesting a practical approach that could be applied in the laboratory. We obtain novel phases, including a striped (lamellar) phase and a hexagonal superstructure, as well as decagonal (tenfold) and dodecagonal (twelvefold) quasicrystals.Given a system of N particles in a box of volume V, interacting via an isotropic pair potential UðrÞ with a repulsive core, a sufficient condition for the formation of a cluster crystal is a negative global minimumŨ min ¼Ũðk min Þ < 0 in the Fourier transform of the potential [6]. This condition implicitly requires the potential not to diverge too strongly, so that the Fourier transform exists. The wave number k min determines the length scale for the order in the system by setting the typical distance between neighboring clusters. Above a sufficiently high mean particle densityc ¼ N=V, a further increase ofc increases the number of overlapping particles within each cluster, but does not change the distance between their centers. It also determines the spinodal temperature k B T sp ¼ −Ũ minc [3,6], below which the liquid becomes unstable against crystallization, where k B is the Boltzmann constant.As the particles form increasingly larger clusters, the system becomes well characterized by a continuous coarse-grained density function cðrÞ. The thermodynamic behavior can then be described in a mean-field approximation, which becomes exact in the high-density hightemperature limit [7]. Using MD simulations, we ex...

Quasicrystals whose building blocks are of mesoscopic rather than atomic scale have recently been discovered in several soft-matter systems. Contrary to metallurgic quasicrystals whose source of stability remains a question of great debate to this day, we argue that the stability of certain soft-matter quasicrystals can be directly explained by examining a coarse-grained free energy for a system of soft isotropic particles. We show, both theoretically and numerically, that the stability can be attributed to the existence of two natural length scales in the pair potential, combined with effective three-body interactions arising from entropy. Our newly gained understanding of the stability of soft quasicrystals allows us to point at their region of stability in the phase diagram, and thereby may help control the self-assembly of quasicrystals and a variety of other desired structures in future experimental realizations.PACS numbers: 61.44. Br, 64.75.Yz, Quasicrystals are more common than one had originally expected when their discovery was first announced. 1More than a hundred different metallic alloys are known to form stable quasicrystalline phases of icosahedral symmetry alone, 2 with a few dozen additional stable phases exhibiting decagonal (10-fold) and possibly other symmetries.3 Yet, to this date, there is no general agreement regarding the origin of their stability and the respective roles of energy and entropy in determining the observed phases.4 These growing numbers of stable solid-state quasicrystals, whose building blocks are on the atomic scale, have been joined in recent years by a host of soft-matter systems exhibiting quasiperiodic long-range order with building blocks on a much larger scale of tens to hundreds of nanometers-micelle-forming dendrimers, 5,6 star block copolymers, 7 mesoporous silica, 8 and binary systems of nanoparticles.9 These newly discovered soft quasicrystals hold the promise for applications based on self-assembled nanomaterials, 10 with unique electronic or photonic properties that take advantage of their quasiperiodicity. 11At the same time, they provide alternative experimental platforms for the basic study of quasiperiodic longrange order, and offer the opportunity to study the thermodynamic stability of quasicrystals from a fresh viewpoint. To this date, soft quasicrystals have been observed only with dodecagonal point-group symmetry, having quasiperiodic order in the 12-fold plane and periodic order normal to the plane, whereas dodecagonal solid-state quasicrystals are rare and mostly only metastable.3 Soft quasicrystals may belong, therefore, to a distinct class of quasicrystals, whose source of stability is likely to be different from their solid-state counterparts. We propose here a simple theoretical framework to address these new systems. We use it to explain the stability of the observed structures and indicate the (surprisingly simple) minimum conditions under which quasicrystals could be stabilized. Knowledge of these conditions gives us the ability to estimate ...

The metastable-to-stable phase-transition is commonly observed in many fields of science, as an uncontrolled independent process, highly sensitive to microscopic fluctuations. In particular, self-assembled lipid suspensions exhibit phase-transitions, where the underlying driving mechanisms and dynamics are not well understood. Here we describe a study of the phase-transition dynamics of lipid-based particles, consisting of mixtures of dilauroylphosphatidylethanolamine (DLPE) and dilauroylphosphatidylglycerol (DLPG), exhibiting a metastable liquid crystalline-to-stable crystalline phase transition upon cooling from 60°C to 37°C. Surprisingly, unlike classically described metastable-to-stable phase transitions, the manner in which recrystallization is delayed by tens of hours is robust, predetermined and controllable. Our results show that the delay time can be manipulated by changing lipid stoichiometry, changing solvent salinity, adding an ionophore, or performing consecutive phase-transitions. Moreover, the delay time distribution indicates a deterministic nature. We suggest that the non-stochastic physical mechanism responsible for the delayed recrystallization involves several rate-affecting processes, resulting in a controllable, non-independent metastability. A qualitative model is proposed to describe the structural reorganization during the phase transition.

We analyze transmission electron microscopy (TEM) images of selfassembled quasicrystals, composed of binary systems of nanoparticles. We use an automated procedure that identifies the positions of dislocations and determines their topological character. To achieve this we decompose the quasicrystal into its individual density modes, or Fourier components, and identify their topological winding numbers for every dislocation. This procedure associates a Burgers function with each dislocation, from which we extract the components of the Burgers vector after choosing a basis. The Burgers vectors that we see in the experimental images are all of lowest order, containing only 0's and 1's as their components. We argue that the density of the different types of Burgers vectors depends on their energetic cost.

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