Stress relaxation models with polyconvex entropy in Lagrangean and Eulerian coordinates

The embedding of the equations of polyconvex elastodynamics to an augmented symmetric hyperbolic system provides in conjunction with the relative entropy method a robust stability framework for approximate solutions \cite{LT06}. We devise here a model of stress relaxation motivated by the format of the enlargement process which formally approximates the equations of polyconvex elastodynamics. The model is endowed with an entropy function which is not convex but rather of polyconvex type. Using the relative entropy we prove a stability estimate and convergence of the stress relaxation model to polyconvex elastodynamics in the smooth regime. As an application, we show that models of pressure relaxation for real gases in Eulerian coordinates fit into the proposed framework

Advanced trajectory engineering of diffraction-resisting laser beams

We introduce an analytical technique for engineering the trajectory of diffraction-resisting laser beams. The generated beams have a Bessel-like transverse field distribution and can be navigated along rather arbitrary curved paths in free space, thus being an advanced hybrid between accelerating and non-accelerating diffraction-free optical waves. The method involves phase-modulating the wavefront of a Gaussian laser beam to create a continuum of conical ray bundles whose apexes define a prespecified focal curve, along which a nearly perfect circular intensity lobe propagates without diffracting. Through extensive numerical simulations, we demonstrate the great flexibility in the design of a gamut of different beam trajectories. Propagation around obstructions and self-healing scenarios are also investigated. The proposed wave entities can be used extensively for light trajectory control in applications such as laser microfabrication, optical tweezers and curved plasma filamentation spectroscopy

Molecular Dynamics of Polyisoprene/Polystyrene Oligomer Blends: The role of self-concentration and fluctuations on blend dynamics

The effect of self-concentration and intermolecular packing on the dynamics of polyisoprene (PI)/polystyrene (PS) blends is examined by extensive atomistic simulations. Direct information on local structure of the blend system allows a quantitative calculation of self- and effective composition terms at various length scales that are introduced to proposed models of blend dynamics. Through a detailed statistical analysis, the full distribution of relaxation times associated with reorienation of carbon-hydrogen bonds was extracted and compared to literature experimental data. A direct relation between relaxation times and local effective composition is found. Following an implementation of a model involving local composition as well as concentration fluctuations the relevant length scales characterizing the segmental dynamics of both components were critically examined. For PI the distribution of times becomes narrower for the system with the lowest PS content and then broadens as more PS is added. This is in contrast to the slow component (PS), where an extreme breadth is found for relaxation times in the 25/75 system prior to narrowing as we increase PI concentration. The chain dynamics was directly quantified by diffusion coefficients as well as the terminal (maximum) relaxation time of each component in the mixed state. Strong coupling between the friction coefficients of the two components was predicted that leads to very similar chain dynamics for PI and PS, particularly for high concentrations of PI. We anticipate this finding to the rather short oligomers (below the Rouse regime) studied here as well as to the rather similar size of PI and PS chains. The ratio of the terminal to the segmental relaxation time, τterm/τseg,c, presents a clear qualitative difference for the constituents: for PS the above ratio is almost independent of blend composition and very similar to the pure state. In contrast, for PI this ratio depends strongly on the composition of the blend; i.e. the terminal relaxation time of PI increases more than its segmental relaxation time, as the concentration of PS increases, resulting into a larger terminal/segmental ratio. We explain this disparity, based on the different length scales characterizing dynamics. The relevant length for the segmental dynamics of PI is about 0.4-0.6 nm, smaller than chain dimensions which are expected to characterize terminal dynamics, whereas for PS associated length scales are similar (about 0.7-1.0 nm) rendering a uniform change with mixing

Effect of Solvent on the Self-Assembly of Dialanine and Diphenylalanine Peptides

Diphenylalanine (FF) is a very common peptide with many potential applications, both biological and technological, due to a large number of different nanostructures which it attains. The current work concerns a detailed study of the self assembled structures of FF in two different solvents, an aqueous (H2O) and an organic (CH3OH) through simulations and experiments. Detailed atomistic Molecular Dynamics (MD) simulations of FF in both solvents have been performed, using an explicit solvent model. The self assembling propensity of FF in water is obvious while in methanol a very weak self assembling propensity is observed. We studied and compared structural properties of FF in the two different solvents and a comparison with a system of dialanine (AA) in the corresponding solvents was also performed. In addition, temperature dependence studies were carried out. Finally, the simulation predictions were compared to new experimental data, which were produced in the framework of the present work. A very good qualitative agreement between simulation and experimental observations was found

Observation of self-accelerating Bessel-like optical beams along arbitrary trajectories

We experimentally demonstrate self-accelerating Bessel-like optical beams propagating along arbitrary trajectories in free space. With computer generated holography, such beams are designed to follow different controllable trajectories while their main lobe transverse profiles remain nearly invariant and symmetric. Examples include parabolic, snake-like, hyperbolic, hyperbolic secant, and even three-dimensional spiraling trajectories. The self-healing property of such beams is also demonstrated. This new class of optical beams can be considered as a hybrid between accelerating and non-accelerating nondiffracting beams that may find a variety of applications

Hierarchical multiscale modeling of polymer-solid interfaces: atomistic to coarse-grained description, and structural and conformational properties of polystyrene-gold systems

A hierarchical simulation approach was developed in order to study polystyrene films sandwiched between two parallel Au(111) surfaces. The coarse-grained potentials describing the interaction of polystyrene with the gold surface were developed systematically using constrained all-atom molecular simulations of a styrene trimer on the Au(111) surface. The model was validated by studying a 5 nm film of short (10mer) polystyrene chains using all-atom and coarse-grained molecular dynamics simulations. The density, structure and conformational properties of coarse-grained films were found to be in excellent agreement with all-atom ones. The coarse-grained model was then used to study the structural and conformational properties of roughly 10 nm and 20 nm thick films with 10, 50, 100 and 200mer chains. The width of the interphase region of the polymer films is property specific. The density profiles reached the bulk value around 1.5 nm from the interface, for all chain lengths. An estimate of the width of the interphase region based on the conformation tensor profile indicates that the interphase width is proportional to the square root of the chain length (number of monomers) and for 200mer chains the interphase width is approximately 6-9 nm

Spectral theory of some non-selfadjoint linear differential operators

We give a characterisation of the spectral properties of linear differential operators with constant coefficients, acting on functions defined on a bounded interval, and determined by general linear boundary conditions. The boundary conditions may be such that the resulting operator is not selfadjoint. We associate the spectral properties of such an operator $S$ with the properties of the solution of a corresponding boundary value problem for the partial differential equation $\partial_t q \pm iSq=0$. Namely, we are able to establish an explicit correspondence between the properties of the family of eigenfunctions of the operator, and in particular whether this family is a basis, and the existence and properties of the unique solution of the associated boundary value problem. When such a unique solution exists, we consider its representation as a complex contour integral that is obtained using a transform method recently proposed by Fokas and one of the authors. The analyticity properties of the integrand in this representation are crucial for studying the spectral theory of the associated operator

Spectral gap in stationary non-equilibrium processes

In this paper we study the spectral gap for a family of interacting particles systems on $[-N,N]$, proving that it is of the order $N^{-2}$. The system arises as a natural model for current reservoirs and Fick's law

A mathematical model for mechanotransduction at the early steps of suture formation

Growth and patterning of craniofacial sutures are subjected to the effects of mechanical stress. Mechanotransduction processes occurring at the margins of the sutures are not precisely understood. Here, we propose a simple theoretical model based on the orientation of collagen fibres within the suture in response to local stress. We demonstrate that fibre alignment generates an instability leading to the emergence of interdigitations. We confirm the appearance of this instability both analytically and numerically. To support our model, we use histology and synchrotron x-ray microtomography and reveal the fine structure of fibres within the sutural mesenchyme and their insertion into the bone. Furthermore, using a mouse model with impaired mechanotransduction, we show that the architecture of sutures is disturbed when forces are not interpreted properly. Finally, by studying the structure of sutures in the mouse, the rat, an actinopterygian (\emph{Polypterus bichir}) and a placoderm (\emph{Compagopiscis croucheri}), we show that bone deposition patterns during dermal bone growth are conserved within jawed vertebrates. In total, these results support the role of mechanical constraints in the growth and patterning of craniofacial sutures, a process that was probably effective at the emergence of gnathostomes, and provide new directions for the understanding of normal and pathological suture fusion

Nonparaxial accelerating Bessel-like beams

A new class of nonparaxial accelerating optical waves is introduced. These are beams with a Bessel-like profile that are capable of shifting laterally along fairly arbitrary trajectories as the wave propagates in free space. The concept expands on our previous proposal of paraxial accelerating Bessel-like beams to include beams with subwavelength lobes and/or large trajectory angles. Such waves are produced when the phase at the input plane is engineered so that the interfering ray cones are made to focus along the prespecified path. When the angle of these cones is fixed, the beams possess a diffraction-free Bessel profile on planes that stay normal to their trajectory, which can be considered as a generalized definition of diffractionless propagation in the nonparaxial regime. The analytical procedure leading to these results is based on a ray optics interpretation of Rayleigh-Sommerfeld diffraction and is presented in detail. The evolution of the proposed waves is demonstrated through a series of numerical examples and a variety of trajectories