Characterization of optimal carbon nanotubes under stretching and validation of the Cauchy-Born rule

Carbon nanotubes are modeled as point configurations and investigated by minimizing configurational energies including two-and three-body interactions. Optimal configurations are identified with local minima and their fine geometry is fully characterized in terms of lower-dimensional problems. Under moderate tension, we prove the existence of periodic local minimizers, which indeed validates the so-called Cauchy-Born rule in this setting.Comment: The final publication is available at springerlink.co

Evolution of crystalline thin films by evaporation and condensation in three dimensions

The morphology of crystalline thin films evolving on flat rigid substrates by condensation of extra film atoms or by evaporation of their own atoms in the surrounding vapor is studied in the framework of the theory of Stress Driven Rearrangement Instabilities (SDRI). By following the SDRI literature both the elastic contributions due to the mismatch between the film and the substrate lattices at their theoretical (free-standing) elastic equilibrium, and a curvature perturbative regularization preventing the problem to be ill-posed due to the otherwise exhibited backward parabolicity, are added in the evolution equation. The resulting Cauchy problem under investigation consists in an anisotropic mean-curvature type flow of the fourth order on the film profiles, which are assumed to be parametrizable as graphs of functions measuring the film thicknesses, coupled with a quasistatic elastic problem in the film bulks. Periodic boundary conditions are considered. The results are twofold: the existence of a regular solution for a finite period of time and the stability for all times, of both Lyapunov and asymptotic type, of any configuration given by a flat film profile and the related elastic equilibrium. Such achievements represent both the generalization to three dimensions of a previous result in two dimensions for a similar Cauchy problem, and the complement of the analysis previously carried out in the literature for the symmetric situation in which the film evolution is not influenced by the evaporation-condensation process here considered, but it is entirely due to the volume preserving surface-diffusion process, which is instead here neglected. The method is based on minimizing movements, which allow to exploit the the gradient-flow structure of the evolution equation.Comment: 23 pages, 0 figure

Solutions for a free-boundary problem modeling multilayer films with coherent and incoherent interfaces

In this paper we introduce a variational model for the study of multilayer films that allows for the treatment of both coherent and incoherent interfaces between layers. The model is designed in the framework of the theory of Stress Driven Rearrangement Instabilities, which are characterized by the competition between elastic and surface energy effects. The surface of each film layer is assumed to satisfy an ''exterior graph condition'', under which in particular bulk cracks are allowed to be of non-graph type. By applying the direct method of calculus of variations under a constraint on the number of connected components of the cracks not connected to the surface of the film layers the existence of energy minimizers is established in dimension 2. As a byproduct of the analysis the state of art on the variational modeling of single-layered films deposited on a fixed substrate is advanced by letting the substrate surface free, by addressing the presence of multiple layers of various materials, and by including the possibility of delamination between the various film layers.Comment: 27 pages, 2 figure

Existence of minimizers for a two-phase free boundary problem with coherent and incoherent interfaces

A variational model for describing the morphology of two-phase continua by allowing for the interplay between coherent and incoherent interfaces is introduced. Coherent interfaces are characterized by the microscopical arrangement of atoms of the two materials in a homogeneous lattice, with deformation being the solely stress relief mechanism, while at incoherent interfaces delamination between the two materials occurs. The model is designed in the framework of the theory of Stress Driven Rearrangement Instabilities, which are characterized by the competition between elastic and surface effects. The existence of energy minimizers is established in the plane by means of the direct method of the calculus of variations under a constraint on the number of boundary connected components of the underlying phase, whose exterior boundary is prescribed to satify a graph assumption, and of the two-phase composite region. Both the wetting and the dewetting regimes are included in the analysis.Comment: 62 pages, 6 figure

Existence of minimizers for the SDRI model

The SDRI model introduced in [Sh. Kholmatov, P. Piovano, Arch. Rational Mech. Anal., in press] in the framework of the theory on Stress-Driven Rearrangement Instabilities for the morphology of crystalline material under stress is considered. The existence of solutions is established in dimension two in the absence of graph-like assumptions and of the restriction to a finite number $m$ of connected components for the free boundary of the region occupied by the crystalline material, thus extending previous available results for epitaxially-strained thin films and material cavities. Due to the lack of compactness and lower semicontinuity even for sequences of $m$-minimizers, i.e., energy minimizers among configurations with a fixed number $m$ of connected boundary-components, the minimizing candidate of the SDRI model is directly constructed. By means of uniform density estimates for the local decay of the energy at the $m$-minimizers' boundaries, such candidate is then shown to be a minimizer also in view of the convergence of the energy at $m$-minimizers to the energy infimum as $m\to\infty$. Finally, regularity properties for the morphology of any minimizer are deduced.Comment: 31 pages, 1 figure

Extreme Love in the SPA: constraining the tidal deformability of supermassive objects with extreme mass ratio inspirals and semi-analytical, frequency-domain waveforms

We estimate the accuracy in the measurement of the tidal Love number of a supermassive compact object through the detection of an extreme mass ratio inspiral~(EMRI) by the future LISA mission. A nonzero Love number would be a smoking gun for departures from the classical black hole prediction of General Relativity. We find that an EMRI detection by LISA could set constraints on the tidal Love number of a spinning central object with dimensionless spin $\hat a=0.9$ ($\hat a=0.99$) which are approximately four (six) orders of magnitude more stringent than what achievable with current ground-based detectors for stellar-mass binaries. Our approach is based on the stationary phase approximation to obtain approximate but accurate semi-analytical EMRI waveforms in the frequency-domain, which greatly speeds up high-precision Fisher-information matrix computations. This approach can be easily extended to several other tests of gravity with EMRIs and to efficiently account for multiple deviations in the waveform at the same time.Comment: 8 pages + appendices and references; 2 tables and 1 figur