Showing papers by "Vincenzo Vitelli published in 2010"

Pleats in crystals on curved surfaces

Abstract: Hexagons can easily tile a flat surface, but not a curved one. Introducing heptagons and pentagons (defects with topological charge) makes it easier to tile curved surfaces; for example, soccer balls based on the geodesic domes of Buckminster Fuller have exactly 12 pentagons (positive charges). Interacting particles that invariably form hexagonal crystals on a plane exhibit fascinating scarred defect patterns on a sphere. Here we show that, for more general curved surfaces, curvature may be relaxed by pleats: uncharged lines of dislocations (topological dipoles) that vanish on the surface and play the same role as fabric pleats. We experimentally investigate crystal order on surfaces with spatially varying positive and negative curvature. On cylindrical capillary bridges, stretched to produce negative curvature, we observe a sequence of transitions-consistent with our energetic calculations-from no defects to isolated dislocations, which subsequently proliferate and organize into pleats; finally, scars and isolated heptagons (previously unseen) appear. This fine control of crystal order with curvature will enable explorations of general theories of defects in curved spaces. From a practical viewpoint, it may be possible to engineer structures with curvature (such as waisted nanotubes and vaulted architecture) and to develop novel methods for soft lithography and directed self-assembly.

Anharmonic and quasi-localized vibrations in jammed solids—Modes for mechanical failure

Abstract: We study harmonic and anharmonic properties of the vibrational modes in 3-dimensional jammed packings of frictionless spheres interacting via repulsive, finite-range potentials. A crossover frequency is apparent in the density of states, the diffusivity and the participation ratio of the normal modes of vibration. At this frequency, which shifts to zero at the jamming threshold, the vibrational modes have a very small participation ratio implying that the modes are quasi-localized. The lowest-frequency modes are the most anharmonic, with the strongest response to pressure and the lowest-energy barriers to mechanical failure.

Vortices on curved surfaces

Abstract: Topological defects in thin films coating a deformed substrate interact with the underlying curvature. This coupling mechanism influences the shape of biological structures and provides a new strategy for the design of interfaces with prescribed functionality. In this article, a mathematical formalism based on the method of conformal mapping that is presented permits the calculation of the energetics of disclinations, dislocations, and vortices on rigid substrates of spatially varying Gaussian curvature. Special emphasis is placed on determining the geometric force exerted on vortices in curved superfluid films. This force, which attracts (repels) vortices towards regions of negative (positive) Gaussian curvature, is an illustration of how material shape can influence quantum mechanical degrees of freedom.

Heat transport in model jammed solids

Abstract: We calculate numerically the normal modes of vibrations in three-dimensional jammed packings of soft spheres as a function of the packing fraction and obtain the energy diffusivity, a spectral measure of transport that controls sound propagation and thermal conductivity. The crossover frequency between weak and strong phonon scattering is controlled by the coordination and shifts to zero as the system is decompressed toward the critical packing fraction at which rigidity is lost. We present a scaling analysis that relates the packing fraction dependence of the crossover frequency to the anomalous scaling of the shear modulus with compression. Below the crossover, the diffusivity displays a power-law divergence with inverse frequency consistent with Rayleigh law, which suggests that the vibrational modes are primarily transverse waves, weakly scattered by disorder. Above it, a large number of modes appear whose diffusivity plateaus at a nearly constant value before dropping to zero above the localization frequency. The thermal conductivity of a marginally jammed solid just above the rigidity threshold is calculated and related to the one measured experimentally at room temperature for most glasses.

Attenuation of shear sound waves in jammed solids

Abstract: We study the attenuation of long-wavelength shear sound waves propagating through model jammed packings of frictionless soft spheres interacting with repulsive springs. The elastic attenuation coefficient, α(ω), of transverse phonons of low frequency, ω, exhibits power law scaling as the packing fraction ϕ is lowered towards ϕc, the critical packing fraction below which rigidity is lost. The elastic attenuation coefficient is inversely proportional to the scattering mean free path and follows Rayleigh’s law with α(ω) ∼ ω4(ϕ − ϕc)−5/2 for ω much less than ω* ∼ (ϕ − ϕc)1/2, the characteristic frequency scale above which the energy diffusivity and density of states plateau. This scaling of the attenuation coefficient, consistent with numerics, is obtained by assuming that a jammed packing can be viewed as a mosaic composed of domains whose characteristic size * ∼ (ϕ − ϕc)−1/2 diverges at the transition.

Attenuation of shear sound waves in jammed solids

Abstract: We study the attenuation of long-wavelength shear sound waves propagating through model jammed packings of frictionless soft spheres interacting with repulsive springs. The elastic attenuation coefficient, $\alpha(\omega)$, of transverse phonons of low frequency, $\omega$, exhibits power law scaling as the packing fraction $\phi$ is lowered towards $\phi_c$, the critical packing fraction below which rigidity is lost. The elastic attenuation coefficient is inversely proportional to the scattering mean free path and follows Rayleigh law with $\alpha(\omega)\sim \omega^4 (\phi - \phi_c)^{-5/2}$ for $\omega$ much less than $\omega^* \sim (\phi - \phi_c)^{1/2}$, the characteristic frequency scale above which the energy diffusivity and density of states plateau. This scaling of the attenuation coefficient, consistent with numerics, is obtained by assuming that a jammed packing can be viewed as a mosaic composed of domains whose characteristic size $\ell^ * \sim (\phi-\phi_c) ^{-1/2}$ diverges at the transition.