203 research outputs found
Nanoindentation of silicon surfaces: Molecular-dynamics simulations of atomic force microscopy
We investigate the atomic-scale details of atomic force microscopy through a quasistatic molecular dynamics simulation together with a density-functional-based tight-binding method. The changes in the AFM tip shape, the size of the tip-sample contact area, as well as the microscopic hardness and Young’s moduli of silicon {111},{110},{100} surfaces are studied. Furthermore, the effects of hydrogen termination of the surface and of subsurface vacancies on hardness and Young’s modulus are discussed.Peer reviewe
Properties of small carbon clusters inside the C60 fullerene
We present the results of an atomic-scale simulation of the confinement of small carbon clusters inside icosahedral C60 fullerene. We carefully investigate the incorporation of various clusters into C60 including chains, rings, and double ring configurations, and have analyzed both the energetics and the resulting geometries. The calculations have been performed employing the density-functional-based tight-binding methodology within the self-consistent charge representation. We find that certain carbon cluster configurations that are unstable as free molecules become stabilized inside C60. By adding single atoms into random positions inside the fullerene shell we establish an upper limit for the filling of C60 with carbon. When the number of atoms inside the fullerene exceeds ten we observe bonding to the surrounding shell and, hence, a gradual transition of the fullerene towards an sp3 rich but locally disordered carbon system.Peer reviewe
Simulations of diamond nucleation in carbon fullerene cores
Recent experiments have shown that heavy ion or electron irradiation induces the nucleation of diamond crystallites inside concentric nested carbon fullerenes, i.e., bucky onions. This suggests that the fullerene acts as a nanoscopic pressure shell. In this paper we study the formation of tetrahedrally bonded carbon inside a prototype icosahedral two-shell fullerene by means of atomic-scale computer simulations. After the simulated irradiation, we can identify regions in which almost all carbon atoms become sp3 bonded. Additionally, we observe a counteracting tendency for the carbon atoms to form shell-like substructures. To shift the balance between these two processes towards diamond nucleation strongly nonequilibrium conditions are required.Peer reviewe
Mappings of least Dirichlet energy and their Hopf differentials
The paper is concerned with mappings between planar domains having least
Dirichlet energy. The existence and uniqueness (up to a conformal change of
variables in the domain) of the energy-minimal mappings is established within
the class of strong limits of homeomorphisms in the
Sobolev space , a result of considerable interest in the
mathematical models of Nonlinear Elasticity. The inner variation leads to the
Hopf differential and its trajectories.
For a pair of doubly connected domains, in which has finite conformal
modulus, we establish the following principle:
A mapping is energy-minimal if and only if
its Hopf-differential is analytic in and real along the boundary of .
In general, the energy-minimal mappings may not be injective, in which case
one observes the occurrence of cracks in . Nevertheless, cracks are
triggered only by the points in the boundary of where fails to be
convex. The general law of formation of cracks reads as follows:
Cracks propagate along vertical trajectories of the Hopf differential from
the boundary of toward the interior of where they eventually terminate
before making a crosscut.Comment: 51 pages, 4 figure
Probe method and a Carleman function
A Carleman function is a special fundamental solution with a large parameter
for the Laplace operator and gives a formula to calculate the value of the
solution of the Cauchy problem in a domain for the Laplace equation. The probe
method applied to an inverse boundary value problem for the Laplace equation in
a bounded domain is based on the existence of a special sequence of harmonic
functions which is called a {\it needle sequence}. The needle sequence blows up
on a special curve which connects a given point inside the domain with a point
on the boundary of the domain and is convergent locally outside the curve. The
sequence yields a reconstruction formula of unknown discontinuity, such as
cavity, inclusion in a given medium from the Dirichlet-to-Neumann map. In this
paper, an explicit needle sequence in {\it three dimensions} is given in a
closed form. It is an application of a Carleman function introduced by
Yarmukhamedov. Furthermore, an explicit needle sequence in the probe method
applied to the reduction of inverse obstacle scattering problems with an {\it
arbitrary} fixed wave number to inverse boundary value problems for the
Helmholtz equation is also given.Comment: 2 figures, final versio
Full-wave invisibility of active devices at all frequencies
There has recently been considerable interest in the possibility, both
theoretical and practical, of invisibility (or "cloaking") from observation by
electromagnetic (EM) waves. Here, we prove invisibility, with respect to
solutions of the Helmholtz and Maxwell's equations, for several constructions
of cloaking devices. Previous results have either been on the level of ray
tracing [Le,PSS] or at zero frequency [GLU2,GLU3], but recent numerical [CPSSP]
and experimental [SMJCPSS] work has provided evidence for invisibility at
frequency . We give two basic constructions for cloaking a region
contained in a domain from measurements of Cauchy data of waves at \p
\Omega; we pay particular attention to cloaking not just a passive object, but
an active device within , interpreted as a collection of sources and sinks
or an internal current.Comment: Final revision; to appear in Commun. in Math. Physic
Incompatible sets of gradients and metastability
We give a mathematical analysis of a concept of metastability induced by
incompatibility. The physical setting is a single parent phase, just about to
undergo transformation to a product phase of lower energy density. Under
certain conditions of incompatibility of the energy wells of this energy
density, we show that the parent phase is metastable in a strong sense, namely
it is a local minimizer of the free energy in an neighbourhood of its
deformation. The reason behind this result is that, due to the incompatibility
of the energy wells, a small nucleus of the product phase is necessarily
accompanied by a stressed transition layer whose energetic cost exceeds the
energy lowering capacity of the nucleus. We define and characterize
incompatible sets of matrices, in terms of which the transition layer estimate
at the heart of the proof of metastability is expressed. Finally we discuss
connections with experiment and place this concept of metastability in the
wider context of recent theoretical and experimental research on metastability
and hysteresis.Comment: Archive for Rational Mechanics and Analysis, to appea
Doubly connected minimal surfaces and extremal harmonic mappings
The concept of a conformal deformation has two natural extensions:
quasiconformal and harmonic mappings. Both classes do not preserve the
conformal type of the domain, however they cannot change it in an arbitrary
way. Doubly connected domains are where one first observes nontrivial conformal
invariants. Herbert Groetzsch and Johannes C. C. Nitsche addressed this issue
for quasiconformal and harmonic mappings, respectively. Combining these
concepts we obtain sharp estimates for quasiconformal harmonic mappings between
doubly connected domains. We then apply our results to the Cauchy problem for
minimal surfaces, also known as the Bjorling problem. Specifically, we obtain a
sharp estimate of the modulus of a doubly connected minimal surface that
evolves from its inner boundary with a given initial slope.Comment: 35 pages, 2 figures. Minor edits, references adde
Quasisymmetric graphs and Zygmund functions
A quasisymmetric graph is a curve whose projection onto a line is a
quasisymmetric map. We show that this class of curves is related to solutions
of the reduced Beltrami equation and to a generalization of the Zygmund class
. This relation makes it possible to use the tools of harmonic
analysis to construct nontrivial examples of quasisymmetric graphs and of
quasiconformal maps.Comment: 21 pages, no figure
Limiting Carleman weights and anisotropic inverse problems
In this article we consider the anisotropic Calderon problem and related
inverse problems. The approach is based on limiting Carleman weights,
introduced in Kenig-Sjoestrand-Uhlmann (Ann. of Math. 2007) in the Euclidean
case. We characterize those Riemannian manifolds which admit limiting Carleman
weights, and give a complex geometrical optics construction for a class of such
manifolds. This is used to prove uniqueness results for anisotropic inverse
problems, via the attenuated geodesic X-ray transform. Earlier results in
dimension were restricted to real-analytic metrics.Comment: 58 page
- …