Abstract. 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.

Spatially confined rigid membranes reorganize their morphology in response to the imposed constraints. A crumpled elastic sheet presents a complex pattern of random folds focusing the deformation energy 1 , whereas compressing a membrane resting on a soft foundation creates a regular pattern of sinusoidal wrinkles with a broad distribution of energy [2][3][4][5][6][7][8] . Here, we study the energy distribution for highly confined membranes and show the emergence of a new morphological instability triggered by a period-doubling bifurcation. A periodic selforganized focalization of the deformation energy is observed provided that an up-down symmetry breaking, induced by the intrinsic nonlinearity of the elasticity equations, occurs. The physical model, exhibiting an analogy with parametric resonance in a nonlinear oscillator, is a new theoretical toolkit to understand the morphology of various confined systems, such as coated materials or living tissues, for example wrinkled skin Several theoretical approaches have been proposed to describe the wrinkling instability for very small compression ratio, that is, near the instability threshold 2,3,7 . However, the large-compression domain remains largely unexplored, with the notable exception of the wrinkle-to-fold transition observed in ref. 8 for an elastic membrane on liquid and the self-similar wrinkling patterns in skins 14 . In the former case, the deformation of the membrane is progressively focalized into a single fold, concentrating all the bending energy. In contrast, for thin rigid membranes on elastomers, large compression induces perturbations of the initial wrinkles but the elasticity of the soft foundation maintains a regular periodic pattern whose complexity increases with the compression ratio.A polydimethylsiloxane (PDMS) film, stretched and then cured with ultraviolet radiation-ozone, or a thin polymer film bound to an elastomer foundation, remains initially flat. Under a slight compression, δ = (L 0 − L)/L 0 , these systems instantaneously form regular (sinusoidal) wrinkles with a well-defined wavelength, λ 0 . Increasing δ generates a continuous increase of the amplitude of the wrinkles and a continuous shift to lower wavelength (λ = λ 0 (1 − δ); see Fig. 1g). By further compression of the sheet, more complex patterns emerge. Above some threshold, δ > δ 2 0.2, we observe a dramatic change in the morphology leading to a pitchfork bifurcation: one wrinkle grows in amplitude at the expense of its neighbours (Fig. 1). The profile of the membrane is no longer described by a single cosinusoid but requires a combination of two periodic functions, cos(2π x/λ) and cos(2π x/2λ). The amplitude of the 2λ mode increases with the compression ratio, whereas the λ mode vanishes. This effect is similar to period-doubling bifurcations in dynamical systems 15,16 observed in Rayleigh-Bernard convections 17 , dynamics of the heart tissue 18-20 , oscillated granular matter 21,22 or bouncing droplets on soap film 23 . In contrast to previous works, we describe here a sp...

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. The drive towards miniaturization in technology is demanding for increasingly thinner components, raising new mechanical challenges [1]. Thin films are, however, unstable to boundary or substrate-induced compressive loads: moderate compression results in regular wrinkling [2][3][4][5][6] while further confinement can lead to crumpling [7,8]. Regions of stress focusing can be a hindrance, acting as nucleation points for mechanical failure. Conversely, these deformations can be exploited constructively for tunable thin structures. For example, singular points of deformation dramatically affect the electronic properties of graphene [9].Here, we show that thin sheets under boundary confinement spontaneously generate a universal self-similar hierarchy of wrinkles, from strained suspended graphene to ordinary hanging curtains. We develop a formalism based on wrinklons, a localized transition zone in the merging of two wrinkles, as building blocks to describe these wrinkled patterns.To illustrate this hierarchical pattern, in Fig. 1(a), we show a wrinkled graphene sheet along with an ordinary hanged curtain. These patterns are also similar to the selfsimilar circular patterns first reported by Argon et al. for the blistering of thin films adhering on a thick substrate [10]. The diversity and complexity of those systems, characterized by various chemical and physical conditions, could suggest, a priori, that the underlying mechanisms governing the formation of these patterns are unrelated. However, these systems can be depicted, independently from the details of the experiments, as a thin sheet constrained at one edge while the others are free to adapt their morphology. These constraints can take the form of an imposed wavelength at one edge or just the requirement that it should remain flat.

Chemical gardens are mineral aggregates that grow in three dimensions with plant-like forms and share properties with selfassembled 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.C hemical gardens, discovered more than three centuries ago (1), are attracting nowadays increasing interest in disciplines as varied as chemistry, physics, nonlinear dynamics, and materials science. Indeed, they exhibit rich chemical, magnetic, and electrical properties due to the steep pH and electrochemical gradients established across their walls during their growth process (2). Moreover, they share common properties with structures ranging from nanoscale tubes in cement (3), corrosion filaments (4) to larger-scale brinicles (5), or chimneys at hydrothermal vents (6). This explains their success as prototypes to grow complex compartmentalized or layered self-organized materials, as chemical motors, as fuel cells, in microfluidics, as catalysts, and to study the origin of life (7-18). However, despite numerous experimental studies, understanding the properties of the wide variety of possible spatial structures and developing theoretical models of their growth remains a challenge.In 3D systems, only a qualitative basic picture for the formation of these structures is known. Precipitates are typically produced when a solid metal salt seed dissolves in a solution containing anions such as silicate. Initially, a semipermeable membrane forms, across which water is pumped by osmosis from the outer solution into the metal salt solution, further dissolving the salt. Above some internal pressure, the membrane breaks, and a buoyant jet of the generally less dense inner solution then rises and further precipitates in the outer solution, producing a collection of mineral shapes that resembles a garden. The growth of chemical gardens is thus driven in 3D by a complex coupling between osmotic, buoyancy, and reaction-diffusion processes (19,20).Studies have attempted to generate reproducible micro-and nanotubes by reducing the erratic nature of the 3D growth of chemical gardens (10,11,13,15,21). They have for instance been studied in microgravity to suppress buoyancy (22, 23), or by injecting aqueous ...

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.