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Conical surfaces pose an interesting challenge to crystal growth: A crystal growing on a cone can wrap around and meet itself at different radii. We use a disk-packing algorithm to investigate how this closure constraint can geometrically frustrate the growth of single crystals on cones with small opening angles. By varying the crystal seed orientation and cone angle, we find that—except at special commensurate cone angles—crystals typically form a seam that runs along the axial direction of the cone, while near the tip, a disordered particle packing forms. We show that the onset of disorder results from a finite-size effect that depends strongly on the circumference and not on the seed orientation or cone angle. This finite-size effect occurs also on cylinders, and we present evidence that on both cylinders and cones, the defect density increases exponentially as circumference decreases. We introduce a simple model for particle attachment at the seam that explains the dependence on the circumference. Our findings suggest that the growth of single crystals can become frustrated even very far from the tip when the cone has a small opening angle. These results may provide insights into the observed geometry of conical crystals in biological and materials applications. 

We investigate the ground-state configurations of two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on fixed curved surfaces. We focus on the intrinsic geometry and show that isothermal coordinates are particularly convenient as they explicitly encode a geometric contribution to the elastic potential. In the special case of a cone with half-angle β, the apex develops an effective topological charge of −χ, where 2πχ = 2π(1 − sin β) is the deficit angle of the cone, and a topological defect of charge σ behaves as if it had an effective topological charge Qeff = (σ − σ2/2) when interacting with the apex. The effective charge of the apex leads to defect absorption and emission at the cone apex as the deficit angle of the cone is varied. For total topological defect charge 1, e.g., imposed by tangential boundary conditions at the edge, we find that for a disk the ground-state configuration consists of p defects each of charge +1/p lying equally spaced on a concentric ring of radius d = ( p−1 3p−1 ) 1 2p R, where R is the radius of the disk. In the case of a cone with tangential boundary conditions at the base, we find three types of ground-state configurations as a function of cone angle: (i) for sharp cones, all of the +1/p defects are absorbed by the apex; (ii) at intermediate cone angles, some of the +1/p defects are absorbed by the apex and the rest lie equally spaced along a concentric ring on the flank; and (iii) for nearly flat cones, all of the +1/p defects lie equally spaced along a concentric ring on the flank. Here the defect positions and the absorption transitions depend intricately on p and the deficit angle, which we analytically compute. We check these results with numerical simulations for a set of commensurate cone angles and find excellent agreement. 

Conical surfaces, with a δ function of Gaussian curvature at the apex, are perhaps the simplest example of geometric frustration. We study two-dimensional liquid crystals with p-fold rotational symmetry (p-atics) on the surfaces of cones. For free boundary conditions at the base, we find both the ground state(s) and a discrete ladder of metastable states as a function of both the cone angle and the liquid crystal symmetry p. We find that these states are characterized by a set of fractional defect charges at the apex and that the ground states are in general frustrated due to effects of parallel transport along the azimuthal direction of the cone. We check our predictions for the ground-state energies numerically for a set of commensurate cone angles (corresponding to a set of commensurate Gaussian curvatures concentrated at the cone apex), whose surfaces can be polygonized as a perfect triangular or squaremesh, and find excellent agreement with our theoretical predictions. 

We study the phonon modes of interacting particles on the surface of a truncated cone resting on a plane subject to gravity, inspired by recent colloidal experiments. We derive the ground-state configuration of the particles under gravitational pressure in the small-cone-angle limit and find an inhomogeneous triangular lattice with spatially varying density but robust local order. The inhomogeneity has striking effects on the normal modes such that an important feature of the cone geometry, namely its apex angle, can be extracted from the lattice excitations. The shape of the cone leads to energy crossings at long wavelengths and frequency-dependent quasilocalization at short wavelengths.We analytically derive the localization domain boundaries of the phonons in the limit of small cone angle and check our results with numerical results for eigenfunctions. 

Polyurethane (PU) elastomers are among the most used rubberlike materials due to their combined merits, including high abrasion resistance, excellent mechanical properties, biocompatibility, and good processing performance. A PU elastomer exhibits pronounced hysteresis, leading to a high toughness on the order of 104 J/m2. However, toughness gained from hysteresis is ineffective to resist crack growth under cyclic load, causing a fatigue threshold below 100 J/m2. Here we report a fatigue-resistant PU fiber–matrix composite, using commercially available Spandex as the fibers and PU elastomer as the matrix. The Spandex fibers are stiff, strong, and stretchable. The matrix is soft, tough, and stretchable. We describe a pullout test to measure the adhesion toughness between the fiber and matrix. The test is highly reproducible, showing an adhesion toughness of 3170 J/m2. The composite shows a maximum stretchability of 6.0, a toughness of 16.7 kJ/m2, and a fatigue threshold of 3900 J/m2. When a composite with a precut crack is stretched, the soft matrix causes the crack tip to blunt greatly, which distributes high stress over a long segment of the Spandex fiber ahead the crack tip. This deconcentration of stress makes the composite resist the growth of cracks under monotonic and cyclic loads. The PU elastomer composites open doors for realistic applications of fatigue-resistant elastomers