Swell-induced surface instability of confined hydrogel layers on substrates

In response to external stimuli, polymeric hydrogels can change volume and shape dramatically. Experimental studies have observed a variety of instability patterns of hydrogels, due to swelling or shrinking, many of which have not been well understood. The present paper considers swell-induced surface instability of a hydrogel layer on a rigid substrate. Based on a recently developed theoretical framework for neutral polymeric gels, a linear perturbation analysis is performed to predict the critical condition for the onset of the surface instability. Using a nonlinear finite element method, numerical simulations are presented to show the swelling process, with the evolution of initial surface perturbations followed by the formation of crease-like surface patterns. In contrast to previously suggested critical conditions for surface creasing, the present study suggests a material-specific condition that predicts a range of critical swelling ratios from about 2.5 to 3.4 and quantitatively relates the critical condition to material properties of the hydrogel system. A stability diagram is constructed with two distinct regions for stable and unstable hydrogels depending on two dimensionless material parameters.     

Micro-to-macro transitions for continua with surface structure at the microscale

A geometrically non-linear framework for micro-to-macro transitions is developed that accounts for the effect of size at the microscopic scale. This is done by endowing the surfaces of the microscopic features with their own (energetic) structure using the theory of surface elasticity. Following a standard first-order ansatz on the microscopic motion in terms of the macroscopic deformation gradient, a Hill-type averaging condition is used to link the two scales. The surface elasticity theory introduces two additional microscopic length scales: the ratio of the bulk volume to the energetic surface area, and the ratio of the surface and bulk Helmholtz energies. The influence of these microscopic length scales is elucidated via a series of numerical examples performed using the finite element method.     

Development of new triangular elements for free surface flows

New finite elements have been developed to simulate steady and unsteady two-dimensional free surface flows. The depth-averaged velocity components with the free surface elevation have been used as independent variables in the model. The differences between the various elements presented lie in the choice of velocity approximation. The Newton–Raphson method has been used to solve the non-linear system of equations. Emphasis is put on bench-mark examples to assess the accuracy and efficiency of the elements. A simple stable new element tested herein shows promising advantages for industrial finite element codes.     

Application of the reciprocity theorem to scattering of surface waves by a cavity

Scattering of surface waves by a cylindrical cavity at the surface of a homogenous, isotropic, linearly elastic half-space is analyzed in this paper. In the usual manner, the scattered field is shown to be equivalent to the radiation from a distribution of tractions, obtained from the incident wave on the surface of the cavity. For the approximation used in this paper, these tractions are shifted to tractions applied to the projection of the cavity on the surface of the half-space. The radiation of surface waves from a normal and a tangential line load, recently determined by the use of the reciprocity theorem, is employed to obtain the field scattered by the cavity from the superposition of displacements due to the distributed surface tractions. The vertical displacement at some distance from the cavity is compared with the solution of the scattering problem obtained by the boundary element method (BEM) for various depths and widths of the cavity. Comparisons between the analytical and BEM results are graphically displayed. The limitations of the approximate approach are discussed based on the comparisons with the BEM results.     

Curvature-dependent surface energy and implications for nanostructures

At small length scales, several size-effects in both physical phenomena and properties can be rationalized by invoking the concept of surface energy. Conventional theoretical frameworks of surface energy, in both the mechanics and physics communities, assume curvature independence. In this work we adopt a simplified and linearized version of a theory proposed by Steigmann–Ogden to capture curvature-dependence of surface energy. Connecting the theory to atomistic calculations and the solution to an illustrative paradigmatical problem of a bent cantilever beam, we catalog the influence of curvature-dependence of surface energy on the effective elastic modulus of nanostructures. The observation in atomistic calculations that the elastic modulus of bent nanostructures is dramatically different than under tension – sometimes softer, sometimes stiffer – has been a source of puzzlement to the scientific community. We show that the corrected surface mechanics framework provides a resolution to this issue. Finally, we propose an unambiguous definition of the thickness of a crystalline surface.     

Resonant reflection of a surface acoustic wave from strip waveguides

A numerical study is performed of the oblique reflection of a surface acoustic wave from a strip of finite width deposited on the surface of a half-infinite substrate. The finite element method is used. If the strip–substrate contact supports waveguide modes with the velocity exceeding the surface wave velocity on the free surface of the substrate, then an interval of angles of incidence exists where the surface wave efficiently excites a waveguide mode. The excitation of the waveguide mode is accompanied by a singular behavior of the reflection and the transmission coefficients. The dependence of the magnitude and the phase of the coefficients on the angle of incidence, the frequency, the width and the thickness of the strip is examined. In particular, it is found that the magnitude of the reflection coefficient abruptly almost vanishes and abruptly increases almost to unity within the resonance interval of angles of incidence.     

Influence of surface effects on fatigue of microcracks nucleation

Cyclic plastic slips in persistent slip bands (PSBs) at the free surface and in the bulk are evaluated for high-cycle fatigue of polycrystals. Analytical results are obtained for PSBs in the bulk and for surface type A PSBs, but the finite element method is used for surface type B PSBs. For practical applications, the results are presented in the form of amplification factors, which are almost insensitive to the PSB aspect ratio and other material parameters. These values are then used to include surface effects in an evaluation of the post-saturation number of cycles to crack nucleation. Some environmental effects are also taken into account. An energy balance criterion is used, and the effect of microslip irreversibilities is evaluated on a statistical basis.     

Propagation of acoustic surface waves on a phononic surface investigated by transient reflecting grating spectroscopy

We present a study of surface acoustic waves (SAW) propagation on a 1D phononic surface (PS) by mean of an heterodyne-detected transient reflecting grating experiment. We excited and detected coherent stationary SAWs characterized by variable wave-vectors. The measured SAW frequencies enables the characterization of the band diagram of this PS sample beyond the first Brillouin zone (BZ). Four different SAW frequencies have been revealed, whose band diagram show articulated dispersion phenomena. In order to address the nature of the investigated SAWs, the experimental results are compared with a numerical simulation of elastic modes based on a finite element model. The observed SAWs are addressed to four Bloch waves characterized by different frequencies and surface energy localization. Moreover, we measured the SAW propagation on a flat non-phononic part of the sample surface and compared it with results from the PS.     

Enrichment of surface charge contributes to the stability of surface nanobubbles

Surface nanobubbles are spontaneously formed at the interface between hydrophobic surfaces and aqueous solutions, which show extraordinarily longer lifetime than that was predicted by the classical thermodynamics model. In the present work, by using a surface plasmon resonance microscopy (SPRM) to quantitatively measure the dissolution kinetics of individual surface nanobubbles in real time, we explored the effects of ionic strength and pH value on the dissolution rates (lifetime) of nanobubbles. The results revealed that nanobubbles could exist stably for a long time in low-concentration electrolyte solutions or high-concentration non-electrolyte solutions, while they dissolved quickly in high-concentration electrolyte solutions. With the increase of ionic strength, the dissolution rates were accelerated by 2–3 orders of magnitude, and thus the lifespan of these surface nanobubbles was significantly shortened. In addition to ionic strength, it was further found that, with the increase of acidity or alkalinity of the solution, the dissolution rates of the surface nanobubbles were faster than that in neutral solution. These results demonstrated that the interfacial charge enrichment significantly contributed to the extraordinary stability of the surface nanobubbles.     

Spinodal surface instability of soft elastic thin films

When the thicknesses of thin films reduce to microns or even nanometers, surface energy and surface interaction often play a significant role in their deformation behavior and surface morphology. The spinodal surface instability induced by the van der Waals force in a soft elastic thin film perfectly bonded to a rigid substrate is investigated theoretically using the bifurcation theory of elastic structures. The analytical solution is derived for the critical condition of spinodal surface morphology instability by accounting for the competition of the van der Waals interaction energy, elastic strain energy and surface energy. Detailed examinations on the effect of surface energy, thickness and elastic properties of the film show that the characteristic wavelength of the deformation bifurcation mode depends on the film thickness via an exponential relation, with the power index in the range from 0.749 to 1.0. The theoretical solution has a good agreement with relevant experiment results.     

Analytical formulations of image forces on dislocations with surface stress in nanowires and nanorods

The interaction between dislocations and surfaces is usually characterized by image forces. Most analytical solutions to image forces could be found in literatures for two-dimensional (2D) solids with or without the consideration of surface stress. This work provides alternative analytical formulations of image forces for nanowires which are in more flexible forms compared with the infinite power series solutions from complex variable method. Moreover, this work proposes analytical formulations of image forces for nanorods (3D) by approximating the 3D shape effect as a height-dependent shape function, which is obtained through curve fitting of the finite element results of image forces without surface stress. The results of nanowires are demonstrated to be acceptable compared with the classical solution and complex variable method. More importantly, the analytical formulation of nanorods has not been found in other literatures so far. This work could contribute to nanostructure design and provide guidance for the fabrication of high quality nanostructures.     

Micro-plasticity of surface steps under adhesive contact: Part I—Surface yielding controlled by single-dislocation nucleation

Mechanics of nano- and meso-scale contacts of rough surfaces is of fundamental importance in understanding deformation and failure mechanisms of a solid surface, and in engineering fabrication and reliability of small surface structures. We present a micro-mechanical dislocation model of contact-induced deformation of a surface step or ledge, as a unit process model to construct a meso-scale model of plastic deformations near and at a rough surface. This paper (Part I) considers onset of contact-induced surface yielding controlled by single-dislocation nucleation from a surface step. The Stroh formalism of anisotropic elasticity and conservation integrals are used to evaluate the driving force on the dislocation. The driving force together with a dislocation nucleation criterion is used to construct a contact-strength map of a surface step in terms of contact pressure, step height, surface adhesion and lattice resistance. Atomistic simulations of atomic surface-step indentation on a gold (1 0 0) surface have been also carried out with the embedded atom method. As predicted by the continuum dislocation model, the atomistic simulations also indicate that surface adhesion plays a significant role in dislocation nucleation processes. Instabilities due to adhesion and dislocation nucleation are evident. The atomistic simulation is used to calibrate the continuum dislocation nucleation criterion, while the continuum dislocation modeling captures the dislocation energetics in the inhomogeneous stress field of the surface-step under contact loading. Results show that dislocations in certain slip planes can be easily nucleated but will stay in equilibrium positions very close to the surface step, while dislocations in some other slip planes easily move away from the surface into the bulk. This phenomenon is called contact-induced near-surface dislocation segregation. As a consequence, we predict the existence of a thin tensile-stress sub-layer adjacent to the surface within the boundary layer of near-surface plastic deformation. In the companion paper (Part II), we analyze the surface hardening behavior caused by multiple dislocations.     

Evolution of material voids for highly anisotropic surface energy

In this paper we consider the evolution by surface diffusion of material voids in a linearly elastic solid, focusing on the evolution of voids with large surface energy anisotropy. It is well known that models for the time evolution of similar material surfaces can become mathematically ill-posed when the surface energy is highly anisotropic. In some cases, this ill-posedness has been associated with the formation of corners along the interface. Here the ill-posedness is removed through a regularization which incorporates higher order terms in the surface energy. Spectrally accurate numerical simulations are performed to calculate the steady-state solution branches and time-dependent evolution of voids, with a particular emphasis on inferring trends in the zero regularization (c→0) limit. For steady voids with large anisotropy we find that apparent corners form as c→0. In the presence of elastic stresses σ the limiting corner angles are most often found to differ from angles found on the (σ=0) Wulff shape. For large elastic stresses we find that steady solutions no longer exist; instead the void steadily lengthens via a filamenting instability referred to as tip streaming.     

Variational formulation of three-dimensional viscous free-surface flows: Natural imposition of surface tension boundary conditions

We present a new surface-intrinsic linear form for the treatment of normal and tangential surface tension boundary conditions in C⁰-geometry variational discretizations of viscous incompressible free-surface flows in three space dimensions. The new approach is illustrated by a finite (spectral) element unsteady Navier-Stokes analysis of the stability of a falling liquid film.     

Topology optimization of piezoelectric nanostructures

We present an extended finite element formulation for piezoelectric nanobeams and nanoplates that is coupled with topology optimization to study the energy harvesting potential of piezoelectric nanostructures. The finite element model for the nanoplates is based on the Kirchoff plate model, with a linear through the thickness distribution of electric potential. Based on the topology optimization, the largest enhancements in energy harvesting are found for closed circuit boundary conditions, though significant gains are also found for open circuit boundary conditions. Most interestingly, our results demonstrate the competition between surface elasticity, which reduces the energy conversion efficiency, and surface piezoelectricity, which enhances the energy conversion efficiency, in governing the energy harvesting potential of piezoelectric nanostructures.     

Determinations of both length scale and surface elastic parameters for fcc metals

In the present research, a systematical study of trans-scale mechanics theory is performed. The surface/interface energy density varying with material deformation is considered, and the general surface/interface elastic constitutive equations are derived. New methods to determine the material length scale parameter and the surface elastic parameters based on a simple quasi-continuum method, i.e. the Cauchy–Born rule, are developed and applied to typical fcc metals. In the present research, the material length parameters will be determined through an equivalent condition of the strain energy density calculated by adopting the strain gradient theory and by adopting the Cauchy–Born rule, respectively. Based on the surface constitutive equations obtained in the present research, the surface elastic parameters are calculated by using the Gibbs definition of surface energy density and the Cauchy–Born rule method.     

The attenuation of monochromatic surface waves due to the presence of an inextensible cover

The attenuation of surface gravity waves is an important process associated with air–sea and wave–current interactions. Here we investigate experimentally the attenuation of monochromatic surface gravity waves due to the presence of various surface covers. The surface covers are fixed in space such that they do not advect with the wave motion and are selected such that the bending modulus is negligible for the wave frequencies used in the experiment in order to minimize any flexural effects. Wave attenuation rates are found to be independent of wave steepness and the type of cover used over the tested parameter range. Results are consistent with the theoretical attenuation rate for an inextensible surface cover.