Fabian Brau

MINIMAL LENGTH UNCERTAINTY RELATION AND THE HYDROGEN ATOM

We propose a new approach to calculate perturbatively the effects of a particular deformed Heisenberg algebra on an energy spectrum. We use this method to calculate the harmonic oscillator spectrum and find that the corrections are in agreement with a previous calculation. Then, we apply this approach to obtain the hydrogen atom spectrum and we find that splittings of degenerate energy levels appear. Comparison with experimental data yields an interesting upper bound for the deformation parameter of the Heisenberg algebra.

Wrinkling Hierarchy in Constrained Thin Sheets from Suspended Graphene to Curtains

We show that thin sheets under boundary confinement spontaneously generate a universal self-similar hierarchy of wrinkles. From simple geometry arguments and energy scalings, we develop a formalism based on wrinklons, the localized transition zone in the merging of two wrinkles, as building blocks of the global pattern. Contrary to the case of crumpled paper where elastic energy is focused, this transition is described as smooth in agreement with a recent numerical work [R. D. Schroll, E. Katifori, and B. Davidovitch, Phys. Rev. Lett. 106, 074301 (2011)]. This formalism is validated from hundreds of nanometers for graphene sheets to meters for ordinary curtains, which shows the universality of our description. We finally describe the effect of an external tension to the distribution of the wrinkles.

Hierarchical wrinkling patterns

This paper reports a simple and flexible method for generating hierarchical patterns from wrinkling instability. Complex features with gradually changing topographies are generated by using the spontaneous wrinkling of a rigid membrane (titanium) on a soft foundation (polystyrene) compressed via the diffusion of a solvent. We show that the morphology of these unreported wrinkled patterns is directly related to the rheological properties of the polymer layer and the geometry of the diffusion front. Based on these ingredients, we rationalize the mechanism for the formation of hierarchical wrinkling patterns and quantify our experimental findings with a simple scaling theory. Finally, we illustrate the relevance of our structuration method by studying the mechanosensitivity of fibroblasts.

Wrinkle to fold transition: influence of the substrate response

Spatially confined rigid membranes reorganize their morphology in response to imposed constraints. Slight compression of a rigid membrane resting on a soft foundation creates a regular pattern of sinusoidal wrinkles with a broad spatial distribution of energy. For larger compression, the deformation energy is progressively localized in small regions which ultimately develop sharp folds. We review the influence of the substrate on this wrinkle to fold transition by considering two models based on purely viscous and purely elastic foundations. We analyze and contrast the physics and mathematics of both systems.

Spiral precipitation patterns in confined chemical gardens

Significance Chemical gardens are plant-like mineral forms existing in nature at various scales. Even though they have been described for more than three centuries thanks to their beauty and analogies with biological forms, their growth mechanisms remain largely not understood. To gain insight into their formation processes, we develop an innovative strategy to study these systems in which chemical gardens grow in a quasi-2D confined geometry upon injecting one reagent solution into the other. The advantage of this procedure over previous works is to allow studying, in controlled and reproducible conditions, the effects of variable concentrations and injection rates on the selected morphology. Chemical gardens are mineral aggregates that grow in three dimensions with plant-like forms and share properties with self-assembled structures like nanoscale tubes, brinicles, or chimneys at hydrothermal vents. The analysis of their shapes remains a challenge, as their growth is influenced by osmosis, buoyancy, and reaction–diffusion processes. Here we show that chemical gardens grown by injection of one reactant into the other in confined conditions feature a wealth of new patterns including spirals, flowers, and filaments. The confinement decreases the influence of buoyancy, reduces the spatial degrees of freedom, and allows analysis of the patterns by tools classically used to analyze 2D patterns. Injection moreover allows the study in controlled conditions of the effects of variable concentrations on the selected morphology. We illustrate these innovative aspects by characterizing quantitatively, with a simple geometrical model, a new class of self-similar logarithmic spirals observed in a large zone of the parameter space.

Multiple scales in streamer discharges, with an emphasis on moving boundary approximations

Streamer discharges determine the very first stage of sparks or lightning, and they govern the evolution of huge sprite discharges above thunderclouds as well as the operation of corona reactors in plasma technology. Streamers are nonlinear structures with multiple inner scales. After briefly reviewing basic observations, experiments and the microphysics, we start from density models for streamers, i.e. from reaction–drift–diffusion equations for charged-particle densities coupled to the Poisson equation of electrostatics, and focus on derivation and solution of moving boundary approximations for the density models. We recall that so-called negative streamers are linearly stable against branching (and we conjecture this for positive streamers as well), and that streamer groups in two dimensions are well approximated by the classical Saffman–Taylor finger of two fluid flow. We draw conclusions on streamer physics, and we identify open problems in the moving boundary approximations.

Capillary interactions among spherical particles at curved liquid interfaces

We study the effect of interfacial curvature on the binding energy and forces exerted on small spherical particles that adsorb on an interface between two immiscible liquids. When the interface has anisotropic curvature, the constant-contact-angle condition at the particle-fluid boundary requires a deformation of the interface. Focusing on the case of an initially cylindrical interface, we predict the shape after a spherical particle binds. We then calculate the energy of adsorption and find that it depends on the shape of the interface very far from the binding site. Turning to the problem of two adsorbed spherical particles, we predict a capillary interaction that arises purely from the deformations caused by the contact-angle condition. An analogy is made between these curvature-induced capillary forces and electrostatic forces between quadrupoles in two dimensions. We conclude with a conjectured general form for the interaction of a single spherical particle with the Gaussian curvature of the underlying fluid interface, which we compare to previous work.

Bohr-Sommerfeld quantization and meson spectroscopy

Universite´de Mons-Hainaut, Place du Parc 20, B-7000 Mons, Belgium~Received 27 May 1999; published 18 May 2000!We use the Bohr-Sommerfeld quantization approach in the context of constituent quark models. Thismethod provides, for the Cornell potential, analytical formulas for the energy spectra which closely approxi-mate numerical exact calculations performed with the Schro¨dinger or the spinless Salpeter equations. TheBohr-Sommerfeld quantization procedure can also be used to calculate other observables such as the rms radiusor wave function at the origin. The asymptotic dependence of these observables on quantum numbers is alsoobtained in the case of potentials which behave asymptotically as a power law. We discuss the constraintsimposed by these formulas on the dynamics of the quark-antiquark interaction.PACS number~s!: 12.39.Pn, 03.65.SqI. INTRODUCTION

Upper and lower limits for the number of S-wave bound states in an attractive potential

New upper and lower limits are given for the number of S-wave bound states yielded by an attractive (monotonic) potential in the context of the Schrodinger or Klein–Gordon equation.

Light meson spectra and instanton-induced forces

Universite´de Mons-Hainaut, Place du Parc 20, B-7000 Mons, Belgium~Received 27 February 1998; published 9 July 1998!The spinless Salpeter equation supplemented by an instanton-induced force is used to describe the spectra oflight mesons, including the pseudoscalar ones. The coupling constants of the instanton-induced potential, aswell as the quark constituent masses, are not treated as simple free parameters but are calculated from theunderlying instanton theory. Quite good results are obtained provided the quarks are considered as effectivedegrees of freedom with a finite size. A further test of the model is performed by calculating the electromag-netic mass differences between S-wave mesons.@S0556-2821~98!05815-9#PACS number~s!: 12.39.Ki, 12.39.Pn, 13.40.Dk, 14.40.AqI. INTRODUCTION