Geometric frustration arises when lattice structure prevents simultaneous minimization of local interactions. It leads to highly degenerate ground states and, subsequently, complex phases of matter such as water ice, spin ice and frustrated magnetic materials. Here we report a simple geometrically frustrated system composed of closely packed colloidal spheres confined between parallel walls. Diameter-tunable microgel spheres are self-assembled into a buckled triangular lattice with either up or down displacements analogous to an antiferromagnetic Ising model on a triangular lattice. Experiment and theory reveal single-particle dynamics governed by in-plane lattice distortions that partially relieve frustration and produce ground-states with zigzagging stripes and subextensive entropy, rather than the more random configurations and extensive entropy of the antiferromagnetic Ising model. This tunable soft matter system provides an uncharted arena in which the dynamics of frustration, thermal excitations and defects can be directly visualized.
We study the analogy between buckled colloidal monolayers and the triangular-lattice Ising antiferromagnet. We calculate free volume-induced Ising interactions, show how lattice deformations favor zigzag stripes that partially remove the Ising model ground-state degeneracy, and identify the Martensitic mechanism prohibiting perfect stripes. Slowly inflating the spheres yields jamming as well as logarithmically slow relaxation reminiscent of the glassy dynamics observed experimentally.
Using numerical simulations we show that a variety of novel colloidal crystalline states and multi-step melting phenomena occur on square and triangular two-dimensional periodic substrates. At half-integer fillings different kinds of frustration effects can be realized. A two-step melting transition can occur in which individual colloidal molecules initially rotate, destroying the overall orientational order, followed by the onset of interwell colloidal hopping, in good agreement with recent experiments.
We show how including attraction in interparticle interactions reverses the effect of fluctuations in ordering of a prototypical artificial frustrated system. Buckled colloidal monolayers exhibit the same ground state as the Ising antiferromagnet on a deformable triangular lattice, but it is unclear if ordering in the two systems is driven by the same geometric mechanism. By a real-space expansion we find that for buckled colloids bent stripes constitute the stable phase, whereas in the Ising antiferromagnet straight stripes are favored. For generic pair potentials we show that attraction gove…
Using simulations and theory, we show that the cubatic phase is metastable for three model hard platelets. The locally favored structures of perpendicular particle stacks in the fluid prevent the formation of the columnar phase through geometric frustration resulting in vitrification. Also, we find a direct link between structure and dynamic heterogeneities in the cooperative rotation of particle stacks, which is crucial for the devitrification process. Finally, we show that the life time of the glassy cubatic phase can be tuned by surprisingly small differences in particle shape.
We study the phase behaviour and the collective dynamics of interacting paramagnetic colloids assembled above a honeycomb lattice of triangular shaped magnetic minima. A frustrated colloidal molecular crystal is realized when filling these potential minima with exactly two particles per pinning site. External in-plane rotating fields are used to anneal the system into different phases, including long range ordered stripes, random fully packed loops, labyrinth and disordered states. At higher amplitude of the annealing field, the dimer lattice displays a two step melting transition where the in…
Artificial spin ice systems, namely lattices of interacting single domain ferromagnetic islands, have been used to date as microscopic models of frustration induced by lattice topology, allowing for the direct visualization of spin arrangements and textures. However, the engineering of frustrated ice states in which individual spins can be manipulated in situ and the real-time observation of their collective dynamics remain both challenging tasks. Inspired by recent theoretical advances, here we realize a colloidal version of an artificial spin ice system using interacting polarizable particle…
We study the structure of clusters in a model colloidal system with competing interactions using Brownian dynamics simulations. A short-ranged attraction drives clustering, while a weak, long-ranged repulsion is used to model electrostatic charging in experimental systems. The former is treated with a short-ranged Morse attractive interaction, the latter with a repulsive Yukawa interaction. We consider the yield of clusters of specific structure as a function of the strength of the interactions, for clusters with m=3,4,5,6,7,10 and 13 colloids. At sufficient strengths of the attractive interac…
Using numerical simulations that mimic recent experiments on hexagonal colloidal ice, we show that colloidal hexagonal artificial spin ice exhibits an inner phase within its ice state that has not been observed previously. Under increasing colloid-colloid repulsion, the initially paramagnetic system crosses into a disordered ice-regime, then forms a topologically charge ordered state with disordered colloids, and finally reaches a three-fold degenerate, ordered ferromagnetic state. This is reminiscent of, yet distinct from, the inner phases of the magnetic kagome spin ice analog. The differenc…
We study the phase behavior of hard spheres confined between two parallel hard plates using extensive computer simulations. We determine the full equilibrium phase diagram for arbitrary densities and plate separations from one to five hard-sphere diameters using free energy calculations. We find a first-order fluid-solid transition, which corresponds to either capillary freezing or melting depending on the plate separation. The coexisting solid phase consists of crystalline layers with either triangular or square symmetry. Increasing the plate separation, we find a sequence of crystal structur…
Artificial particle ices are model systems of constrained, interacting particles. They have been introduced theoretically to study ice-manifolds emergent from frustration, along with domain wall and grain boundary dynamics, doping, pinning-depinning, controlled transport of topological defects, avalanches, and memory effects. Recently such particle-based ices have been experimentally realized with vortices in nano-patterned superconductors or gravitationally trapped colloids. Here we demonstrate that, although these ices are generally considered equivalent to magnetic spin ices, they can acces…
Recovery of ground-state degeneracy in two-dimensional square ice is a significant challenge in the field of geometric frustration with far-reaching fundamental implications, such as realization of vertex models and understanding the effect of dimensionality reduction. We combine experiments, theory, and numerical simulations to demonstrate that sheared square colloidal ice partially recovers the ground-state degeneracy for a wide range of field strengths and lattice shear angles. Our method opens an avenue to engineer a novel class of frustrated micro- and nano-structures based on sheared mag…
Emergent charge crystallization and frustration in a particle anti-spin Ice
Universitat de Barcelona（巴塞罗那大学）；Institute of Complex Systems (UBICS), Universitat de Barcelona（巴塞罗那大学复杂系统研究所）；Theoretical Division, Center for NonLinear Studies, Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室非线性研究理论部）
Folding a sheet of paper along a curve can lead to structures seen in decorative art and utilitarian packing boxes. Here we present a theory for the simplest such structure: an annular circular strip that is folded along a central circular curve to form a three-dimensional buckled structure driven by geometrical frustration. We quantify this shape in terms of the radius of the circle, the dihedral angle of the fold and the mechanical properties of the sheet of paper and the fold itself. When the sheet is isometrically deformed everywhere except along the fold itself, stiff folds result in crea…
We study the phase ordering dynamics of a two dimensional model colloidal solid using molecular dynamics simulations. The colloid particles interact with each other with a Hamaker potential modified by the presence of equatorial "patches" of attractive and negative regions. The total interaction potential between two such colloids is, therefore, strongly directional and has three-fold symmetry. Working in the canonical ensemble, we determine the tentative phase diagram in the density-temperature plane which features three distinct crystalline ground states viz, a low density honeycomb solid fo…
