First-principles calculation of mechanical properties of Si <001> nanowires and comparison to nanomechanical theory

We report the results of first-principles density functional theory calculations of the Young's modulus and other mechanical properties of hydrogen-passivated Si nanowires. The nanowires are taken to have predominantly {100} surfaces, with small {110} facets according to the Wulff shape. The Young's modulus, the equilibrium length and the constrained residual stress of a series of prismatic beams of differing sizes are found to have size dependences that scale like the surface area to volume ratio for all but the smallest beam. The results are compared with a continuum model and the results of classical atomistic calculations based on an empirical potential. We attribute the size dependence to specific physical structures and interactions. In particular, the hydrogen interactions on the surface and the charge density variations within the beam are quantified and used both to parameterize the continuum model and to account for the discrepancies between the two models and the first-principles results.Comment: 14 pages, 10 figure

A statistical mechanics framework for static granular matter

The physical properties of granular materials have been extensively studied in recent years. So far, however, there exists no theoretical framework which can explain the observations in a unified manner beyond the phenomenological jamming diagram [1]. This work focuses on the case of static granular matter, where we have constructed a statistical ensemble [2] which mirrors equilibrium statistical mechanics. This ensemble, which is based on the conservation properties of the stress tensor, is distinct from the original Edwards ensemble and applies to packings of deformable grains. We combine it with a field theoretical analysis of the packings, where the field is the Airy stress function derived from the force and torque balance conditions. In this framework, Point J characterized by a diverging stiffness of the pressure fluctuations. Separately, we present a phenomenological mean-field theory of the jamming transition, which incorporates the mean contact number as a variable. We link both approaches in the context of the marginal rigidity picture proposed by [3, 4].Comment: 21 pages, 15 figure

Thermodynamics of non-local materials: extra fluxes and internal powers

The most usual formulation of the Laws of Thermodynamics turns out to be suitable for local or simple materials, while for non-local systems there are two different ways: either modify this usual formulation by introducing suitable extra fluxes or express the Laws of Thermodynamics in terms of internal powers directly, as we propose in this paper. The first choice is subject to the criticism that the vector fluxes must be introduced a posteriori in order to obtain the compatibility with the Laws of Thermodynamics. On the contrary, the formulation in terms of internal powers is more general, because it is a priori defined on the basis of the constitutive equations. Besides it allows to highlight, without ambiguity, the contribution of the internal powers in the variation of the thermodynamic potentials. Finally, in this paper, we consider some examples of non-local materials and derive the proper expressions of their internal powers from the power balance laws.Comment: 16 pages, in press on Continuum Mechanics and Thermodynamic

Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua

We compare and contrast two types of deformations inspired by mixing applications -- one from the mixing of fluids (stretching and folding), the other from the mixing of granular matter (cutting and shuffling). The connection between mechanics and dynamical systems is discussed in the context of the kinematics of deformation, emphasizing the equivalence between stretches and Lyapunov exponents. The stretching and folding motion exemplified by the baker's map is shown to give rise to a dynamical system with a positive Lyapunov exponent, the hallmark of chaotic mixing. On the other hand, cutting and shuffling does not stretch. When an interval exchange transformation is used as the basis for cutting and shuffling, we establish that all of the map's Lyapunov exponents are zero. Mixing, as quantified by the interfacial area per unit volume, is shown to be exponentially fast when there is stretching and folding, but linear when there is only cutting and shuffling. We also discuss how a simple computational approach can discern stretching in discrete data.Comment: REVTeX 4.1, 9 pages, 3 figures; v2 corrects some misprints. The following article appeared in the American Journal of Physics and may be found at http://ajp.aapt.org/resource/1/ajpias/v79/i4/p359_s1 . Copyright 2011 American Association of Physics Teachers. This article may be downloaded for personal use only. Any other use requires prior permission of the author and the AAP

Atomistic calculations of interface elastic properties in noncoherent metallic bilayers

The paper describes theoretical and computational studies associated with the interface elastic properties of noncoherent metallic bicrystals. Analytical forms of interface energy, interface stresses, and interface elastic constants are derived in terms of interatomic potential functions. Embedded-atom method potentials are then incorporated into the model to compute these excess thermodynamics variables, using energy minimization in a parallel computing environment. The proposed model is validated by calculating surface thermodynamic variables and comparing them with preexisting data. Next, the interface elastic properties of several fcc-fcc bicrystals are computed. The excess energies and stresses of interfaces are smaller than those on free surfaces of the same crystal orientations. In addition, no negative values of interface stresses are observed. Current results can be applied to various heterogeneous materials where interfaces assume a prominent role in the systems' mechanical behavior.open322

Diffraction microstrain in nanocrystalline solids under load - heterogeneous medium approach

This is an account of the computation of X-ray microstrain in a polycrystal with anisotropic elasticity under uniaxial external load. The results have been published in the article "Microstrain in nanocrystalline solids under load by virtual diffraction", at Europhysics Letters 89, 66002 (2010). The present information was submitted to Europhysics Letters as part of the manuscript package, and was available to the reviewers who recommended the paper for publication.Comment: Supporting online material for J. Markmann, D. Bachurin, L.-H. Shao, P. Gumbsch, J. Weissm\"uller, Microstrain in nanocrystalline solids under load by virtual diffraction, Europhys. Lett. 89, 66002 (2010

Kinetic theory of age-structured stochastic birth-death processes

Classical age-structured mass-action models such as the McKendrick-von Foerster equation have been extensively studied but are unable to describe stochastic fluctuations or population-size-dependent birth and death rates. Stochastic theories that treat semi-Markov age-dependent processes using, e.g., the Bellman-Harris equation do not resolve a population's age structure and are unable to quantify population-size dependencies. Conversely, current theories that include size-dependent population dynamics (e.g., mathematical models that include carrying capacity such as the logistic equation) cannot be easily extended to take into account age-dependent birth and death rates. In this paper, we present a systematic derivation of a new, fully stochastic kinetic theory for interacting age-structured populations. By defining multiparticle probability density functions, we derive a hierarchy of kinetic equations for the stochastic evolution of an aging population undergoing birth and death. We show that the fully stochastic age-dependent birth-death process precludes factorization of the corresponding probability densities, which then must be solved by using a Bogoliubov-–Born–-Green–-Kirkwood-–Yvon-like hierarchy. Explicit solutions are derived in three limits: no birth, no death, and steady state. These are then compared with their corresponding mean-field results. Our results generalize both deterministic models and existing master equation approaches by providing an intuitive and efficient way to simultaneously model age- and population-dependent stochastic dynamics applicable to the study of demography, stem cell dynamics, and disease evolution

Numerical computations of facetted pattern formation in snow crystal growth

Facetted growth of snow crystals leads to a rich diversity of forms, and exhibits a remarkable sixfold symmetry. Snow crystal structures result from diffusion limited crystal growth in the presence of anisotropic surface energy and anisotropic attachment kinetics. It is by now well understood that the morphological stability of ice crystals strongly depends on supersaturation, crystal size and temperature. Until very recently it was very difficult to perform numerical simulations of this highly anisotropic crystal growth. In particular, obtaining facet growth in combination with dendritic branching is a challenging task. We present numerical simulations of snow crystal growth in two and three space dimensions using a new computational method recently introduced by the authors. We present both qualitative and quantitative computations. In particular, a linear relationship between tip velocity and supersaturation is observed. The computations also suggest that surface energy effects, although small, have a larger effect on crystal growth than previously expected. We compute solid plates, solid prisms, hollow columns, needles, dendrites, capped columns and scrolls on plates. Although all these forms appear in nature, most of these forms are computed here for the first time in numerical simulations for a continuum model.Comment: 12 pages, 28 figure

Distributed optimal control of a nonstandard system of phase field equations

We investigate a distributed optimal control problem for a phase field model of Cahn-Hilliard type. The model describes two-species phase segregation on an atomic lattice under the presence of diffusion; it has been recently introduced by the same authors in arXiv:1103.4585v1 [math.AP] and consists of a system of two highly nonlinearly coupled PDEs. For this reason, standard arguments of optimal control theory do not apply directly, although the control constraints and the cost functional are of standard type. We show that the problem admits a solution, and we derive the first-order necessary conditions of optimality.Comment: Key words: distributed optimal control, nonlinear phase field systems, first-order necessary optimality condition