Transition pathways connecting crystals and quasicrystals

Due to structural incommensurability, the emergence of a quasicrystal from a crystalline phase represents a challenge to computational physics. Here, the nucleation of quasicrystals is investigated by using an efficient computational method applied to a Landau free-energy functional. Specifically, transition pathways connecting different local minima of the Lifshitz–Petrich model are obtained by using the high-index saddle dynamics. Saddle points on these paths are identified as the critical nuclei of the 6-fold crystals and 12-fold quasicrystals. The results reveal that phase transitions between the crystalline and quasicrystalline phases could follow two possible pathways, corresponding to a one-stage phase transition and a two-stage phase transition involving a metastable lamellar quasicrystalline state, respectively.

DNA self-organization controls valence in programmable colloid design

Just like atoms combine into molecules, colloids can self-organize into predetermined structures according to a set of design principles. Controlling valence—the number of interparticle bonds—is a prerequisite for the assembly of complex architectures. The assembly can be directed via solid “patchy” particles with prescribed geometries to make, for example, a colloidal diamond. We demonstrate here that the nanoscale ordering of individual molecular linkers can combine to program the structure of microscale assemblies. Specifically, we experimentally show that covering initially isotropic microdroplets with N mobile DNA linkers results in spontaneous and reversible self-organization of the DNA into Z(N) binding patches, selecting a predictable valence. We understand this valence thermodynamically, deriving a free energy functional for droplet–droplet adhesion that accurately predicts the equilibrium size of and molecular organization within patches, as well as the observed valence transitions with N. Thus, microscopic self-organization can be programmed by choosing the molecular properties and concentration of binders. These results are widely applicable to the assembly of any particle with mobile linkers, such as functionalized liposomes or protein interactions in cell–cell adhesion.

Cubic microlattices embedded in nematic liquid crystals: a Landau-de Gennes study

We consider a Landau-de Gennes model for a connected cubic lattice scaffold in a nematic host, in a dilute regime. We analyse the homogenised limit for both cases in which the lattice of embedded particles presents or not cubic symmetry and then we compute the free effective energy of the composite material. In the cubic symmetry case, we impose different types of surface anchoring energy densities, such as quartic, Rapini-Papoular or more general versions, and, in this case, we show that we can tune any coefficient from the corresponding bulk potential, especially the phase transition temperature. In the case with loss of cubic symmetry, we prove similar results in which the effective free energy functional has now an additional term, which describes a change in the preferred alignment of the liquid crystal particles inside the domain. Moreover, we compute the rate of convergence for how fast the surface energies converge to the homogenised one and also for how fast the minimisers of the free energies tend to the minimiser of the homogenised free energy.

On the Kirchhoff-Love Hypothesis (Revised and Vindicated)

The Kirchhoff-Love hypothesis expresses a kinematic constraint that is assumed to be valid for the deformations of a three-dimensional body when one of its dimensions is much smaller than the other two, as is the case for plates. This hypothesis has a long history checkered with the vicissitudes of life: even its paternity has been questioned, and recent rigorous dimension-reduction tools (based on standard $\varGamma $ Γ -convergence) have proven to be incompatible with it. We find that an appropriately revised version of the Kirchhoff-Love hypothesis is a valuable means to derive a two-dimensional variational model for elastic plates from a three-dimensional nonlinear free-energy functional. The bending energies thus obtained for a number of materials also show to contain measures of stretching of the plate’s mid surface (alongside the expected measures of bending). The incompatibility with standard $\varGamma $ Γ -convergence also appears to be removed in the cases where contact with that method and ours can be made.

Coarse graining of a Fokker–Planck equation with excluded volume effects preserving the gradient flow structure

The propagation of gradient flow structures from microscopic to macroscopic models is a topic of high current interest. In this paper, we discuss this propagation in a model for the diffusion of particles interacting via hard-core exclusion or short-range repulsive potentials. We formulate the microscopic model as a high-dimensional gradient flow in the Wasserstein metric for an appropriate free-energy functional. Then we use the JKO approach to identify the asymptotics of the metric and the free-energy functional beyond the lowest order for single particle densities in the limit of small particle volumes by matched asymptotic expansions. While we use a propagation of chaos assumption at far distances, we consider correlations at small distance in the expansion. In this way, we obtain a clear picture of the emergence of a macroscopic gradient structure incorporating corrections in the free-energy functional due to the volume exclusion.

Shape equations for two-dimensional manifolds with nonempty boundary based on a variational method

A smooth surface is considered which has a curved boundary. A system of exterior differential forms is introduced which describes the surface and boundary curves completely in the moving frame approach. A total free energy functional is defined based on these forms for which an equilibrium equation and boundary conditions of the surface are derived by calculating the variation of the total free energy. These results can be applied to a surface with several freely exposed edges.