Theories of surface elasticity for nanoscale objects

The emergence of nanotechnology has driven recent interest in systems having surface atoms as a significant fraction of all atoms present, in particular nano-sheets (ultra-thin slabs), nano-wires, and nano-particles. In these systems, the bulk (i.e. non-surface region or interior) is typically strained in response to the stress of the surface. This elastic strain of the bulk in turn changes the surface lattice constants. Since the bulk and the surface are coupled, the problem must be solved self-consistently. Solving this problem requires a quantitative model of the surface elastic properties which are different from the bulk. In this paper we consider various models that have been proposed for surface elasticity. Our goal is to elucidate the relationship between two contrasting approaches: (1) the Shuttleworth equation which defines a surface stress based on the strain derivative of the surface energy and (2) the Gurtin-Murdoch (GM) theory which considers the surface layer as a membrane with residual strain and with elastic constants different from the bulk. The GM theory is analogous to the 2-D Frenkel-Kontorova (FK) model and can be used to obtain quantitative parameters for the FK model. We present an embedded atom method calculation of the surface elastic constants of Cu(1 1 1) using the GM theory with the surface represented by a membrane one atomic layer thick. This quantitative approach describes the elastic properties of surfaces in a physically appealing way. Just as the bulk elastic constants provide direct information regarding the stress/strain relationship in a bulk material, the surface elastic constants provide similar information for a surface monolayer. This theory will allow elasticity analysis and atomistic calculations of properties of nano-scale objects.     

Size-dependent resonance and buckling behavior of nanoplates with high-order surface stress effects

This work presents a theoretical study of the resonance frequency and buckling load of nanoplates with high-order surface stress model. A classical thin plate theory based on Kirchhoff–Love assumption is implemented with surface effects. Circular and rectangular nanoplates with simply supported end conditions are exemplified. The size-dependent solutions are compared with the simplified solutions based on simple surface stress model, and also on the classical theory of elasticity. We aim to explore the scope of applicability so that the modified continuum mechanics model could serve as a refined approach in the prediction of mechanical behavior of nanoplates.     

The effect of surface stress on the propagation of Lamb waves

This work investigates the possibility of the propagation of Lamb waves in thin solid layers with external traction free surfaces, in the presence of surface elasticity, inertia and residual stress. It is demonstrated that such waves do exist and that their characteristics can be quite different from their classical counterparts. The governing equations with non-classical boundary conditions involving the bulk and surface stress are solved exactly in the frequency-wavenumber domain. This solution is utilized to compute the Lamb wave modes for different layer thicknesses. An efficient strategy to capture all the modes of Lamb waves within a given frequency window is outlined. It is shown that the effect of surface elasticity and inertia becomes significant with increasing frequency and decreasing layer thickness, where the number of modes participating within a given frequency window is more than that permitted by the classical theory. Further, it is observed that the nature of the Lamb wave modes (in terms of negative dispersion) in the presence of surface stress is similar to what predicted by the nonlocal theory and microstructure based continuum theory.     

A non-singular continuum theory of point defects using gradient elasticity of bi-Helmholtz type

In this paper, we develop a non-singular continuum theory of point defects based on a second strain gradient elasticity theory, the so-called gradient elasticity of bi-Helmholtz type. Such a generalised continuum theory possesses a weak nonlocal character with two internal material lengths and provides a mechanics of defects without singularities. Gradient elasticity of bi-Helmholtz type gives a natural and physical regularisation of the classical singularities of defects, based on higher order partial differential equations. Point defects embedded in an isotropic solid are considered as eigenstrain problem in gradient elasticity of bi-Helmholtz type. Singularity-free fields of point defects are presented. The displacement field as well as the first, the second and the third gradients of the displacement are derived and it is shown that the classical singularities are regularised in this framework. This model delivers non-singular expressions for the displacement field, the first displacement gradient and the second displacement gradient. Moreover, the plastic distortion (eigendistortion) and the gradient of the plastic distortion of a dilatation centre are also non-singular and are given in terms of a form factor (shape function) of a point defect. Singularity-free expressions for the interaction energy and the interaction force between two dilatation centres and for the interaction energy and the interaction force of a dilatation centre in the stress field of an edge dislocation are given. The results are applied to calculate the finite self-energy of a dilatation centre.     

The stability of fcc (1 1 1) metal surfaces

The results of a theoretical study on the stability of fcc (1 1 1) metal surfaces to certain commensurate-incommensurate reconstructive phase transformations is presented. Specifically, we have performed computer simulation studies of the 22×√3 surface reconstruction of Au(1 1 1). This reconstruction involves a uniaxial contraction of the top monolayer corresponding to a surface strain of about 4.3% and has been observed to be the stable structure for the clean surface at low temperatures. The driving force for the reconstruction has been associated with the quantity (f−γ), where f is the surface stress and γ is the surface free energy, while the opposing force is due to the disregistry with the underlying lattice. A continuum model yields a stability criterion that depends on the knowledge of a small number of physical quantities: f, γ, the equilibrium nearest-neighbor spacing r₁ and the shear modulus G. We have performed molecular dynamics simulations as a general stability analysis of these types of reconstructions. The results are in excellent agreement with the continuum model. The simulations using embedded-atom method potentials also accurately reproduce many observed features of the reconstruction on Au(1 1 1).     

Microscopic approach to many-exciton complexes in self-assembled InGaAs/GaAs quantum dots

We present a numerical calculation of many-exciton complexes in self-assembled InAs/GaAs quantum dots. We apply continuum elasticity theory and atomistic valence-force-field method to calculate strain distribution, and make use of various methods, ranging from a quasi-atomistic tight-binding approach to the single-band effective-mass approximation, to obtain single-particle energy levels. The effect of strain is incorporated by the deformation potential theory. We expand multiexciton states in the basis of Slater determinants and solve the many-body problem by the configuration-interaction method. The dynamics of multiexcitons is studied by solving the rate equations, from which the excitation–power dependence of emission spectrum is obtained. The emission spectra calculated by the microscopic tight-binding approach are found to be in good agreement with those obtained by the simple effective-mass method.     

An isotropic dynamically consistent gradient elasticity model derived from a 2D lattice

This paper presents a derivation of a second-order isotropic continuum from a 2D lattice. The derived continuum is isotropic and dynamically consistent in the sense that it is unconditionally stable and prohibits the infinite speed of energy propagation. The Lagrangian density of the continuum is obtained from the Lagrange function of the underlying lattice. This density is used to obtain the expressions for standard and higher-order stresses in direct correspondence with the equations of the continuum motion. The derived continuum is characterized by two additional parameters relative to the classical elastic continuum. These are the characteristic lengthscale and a dimensionless continualization parameter, which characterizes indirectly the timescale of the derived continuum. The margins for the latter parameter are found from the stability analysis. It is envisaged that the continualization parameter could be measured employing a high-frequency pulse propagating along the surface of the continuum. Excitation and propagation of such pulse is studied theoretically in this paper.     

Surface deformation induced by a variation in surface stresses in anisotropic half-regions

The deformations and the stresses in anisotropic half-regions taking into account surface stresses originating from surface energy, which exists originally at surfaces and interfaces dividing phases, are analysed theoretically. In the present paper, the equilibrium equation of force considering surface stresses is used to calculate the inelastic deformation induced by a variation in surface stresses. The problem of varying surface stresses in a half-surface of a half-infinite anisotropic domain is analysed using the theory of elasticity. This problem is related to the occurrence of cracks in contaminated, oxidized or chemisorbed surfaces. Stress analysis on the basis of continuum mechanics is performed precisely under the boundary condition taking into account surface stresses. The Fourier transform technique is applied to perform the analysis, and the components of stress and displacement are expressed in an explicit form. The shear component of bulk stress attains infinity at the edge of discontinuity of the surface stresses, and the free surface deforms like an edge dislocation. This result suggests that cracking in a chemically contaminant surface is easier than in a clean surface.     

Photonic bandgap structures for long-range surface plasmon polaritons

Propagation of long-range surface plasmon polaritons (LR-SPPs) along periodically thickness-modulated metal stripes embedded in dielectric is studied both theoretically and experimentally for light wavelengths in the telecom range. We demonstrate that symmetric (with respect to the film surface) nm-size thickness variations result in the pronounced band gap effect, and obtain very good agreement between measured and simulated (transmission and reflection) spectra. This effect is exploited to realize a compact wavelength add-drop filter with the bandwidth of 20 nm centered at 1550 nm. The possibilities of achieving a full bandgap (in the surface plane) for LR-SPPs are also discussed.     

Surface bound states in the continuum

We introduce a novel concept of surface bound states in the continuum, i.e., surface modes embedded into the linear spectral band of a discrete lattice. We suggest an efficient method for creating such surface modes and the local bounded potential necessary to support the embedded modes. We demonstrate that the surface embedded modes are structurally stable, and the position of their eigenvalues inside the spectral band can be tuned continuously by adding weak nonlinearity.     

Theory of field effects on transport properties of polyatomic gases in the transition regime

A simple method is presented for describing the effects of external magnetic or electric fields on the transport properties of polyatomic gases over the entire range from the continuum to the Knudsen regime. Instead of treating bulk and boundary-layer effects separately, both molecular and surface scattering are included from the beginning in the collisional part of the Boltzmann equation, and the surface is treated as one component of a multicomponent mixture. A simple first-order solution of this problem is sufficient to account for the dependence of the transport coefficients on the Knudsen number in the presence of a field. Detailed results for the longitudinal and transverse viscomagnetic effects in a single gas are presented, and shown to be in good agreement with experimental data for CO and N₂.     

Modeling of a nanoscale flexoelectric energy harvester with surface effects

This work presents the modeling of a beam energy harvester scavenging energy from ambient vibration based on the phenomenon of flexoelectricity. By considering surface elasticity, residual surface stress, surface piezoelectricity and bulk flexoelectricity, a modified Euler-Bernoulli beam model for the energy harvester is developed. After deriving the requisite energy expressions, the extended Hamilton's principle and the assumed-modes method are employed to obtain the discrete electromechanical Euler-Lagrange's equations. Then, the expressions of the steady-state electromechanical responses are given for harmonic base excitation. Numerical simulations are conducted to show the output voltage and the output power of the flexoelectric energy harvesters with different materials and sizes. Particular emphasis is given to the surface effects on the performance of the energy harvesters. It is found that the surface effects are sensitive to the beam geometries and the surface material constants, and the effect of residual surface stress is more significant than that of the surface elasticity and the surface piezoelectricity. The axial deformation of the beam is also considered in the model to account for the electromechanical coupling due to piezoelectricity, and results indicate that piezoelectricity will diminish the output electrical quantities for the case investigated. This work could lead to the development of flexoelectric energy harvesters that can make the micro- and nanoscale sensor systems autonomous.     

In situ measurements of InAs and InP (0 0 1) surface stress changes induced by surface reconstruction transitions

The anisotropic surface stress changes associated with the transition between different surface reconstructions of InAs and InP (0 0 1) surfaces are measured in situ and in real time in a molecular beam epitaxy (MBE) system. Reflectivity anisotropy of the surface measured at 1.96 eV, together with reflection high energy electron diffraction (RHEED) pattern, are used in order to identify the surface reconstructions, and the monitoring of the substrate curvature evolution to determine the variations in surface stress. Our results show the important contribution to the surface stress of the dimers present in these reconstructed surfaces. Furthermore, we provide for the first time quantitative values of the surface stress changes due to the transition between surface reconstructions for these III-V semiconductors compounds. We obtain values for these changes up to 0.7 Nm⁻¹, that is, of the same magnitude as the stress induced by deposition of one monolayer during growth of lattice-mismatched III-V semiconductor heteroepitaxial systems. This points out the great importance of surface stress evolution in this kind of processes.     

基于连续弹性理论分析量子线线宽对应变分布和带隙的影响

基于连续弹性理论分别采用数值方法和格林函数法讨论了量子线的应变分布.格林函数法可以得到应变分布的解析表示式,对规则形状的量子线的应变分布计算比较方便;连续弹性理论采取的是数值解法,结果精度不如格林函数法,但是能方便计算任意形状量子线的应变分布情况, 并可以考虑不同材料的弹性常数的影响.文章还具体讨论了量子线线宽对应变分布和带隙的影响,结果表明：沿线宽方向,应变的绝对值逐渐减小,并随线宽的增加而变大;带隙则随线宽的减小而增大. 关键词： 连续弹性理论 格林函数法 应变 带隙     

Computational study of the surface properties of aluminum nanoparticles

We calculate the surface energy, surface stress, and lattice contraction of Al nanoparticles using ab initio density functional and empirical computational techniques. Ab initio calculations are carried out using the siesta pseudopotential method combined with the generalized gradient approximation. Empirical calculations are conducted using the embedded atom method. The ab initio density functional approach predicts the surface energies of Al nanoclusters to be in the range of 0.9-2.0 J/m². These values are consistent with the surface energy of bulk aluminum and are close to the surface energies of silver nanoparticles calculated in our previous study. In contrast to our previous results for Ag nanoparticles, we found a significant discrepancy between the theoretical values of surface energy and stress for Al nanoclusters. This result could be explained by a greater degree of surface reconstruction in Al clusters than in Ag clusters.     

Ionization of atoms in strong low-frequency electromagnetic field

The ionization of atoms in a low-frequency linearly polarized electromagnetic field (the photon energy is much lower than the ionization potential of an atom) is considered under new conditions, in which the Coulomb interaction of an electron with the atomic core in the final state of the continuum cannot be considered in perturbation theory in the interaction of the electron with the electromagnetic field. The field is assumed to be much weaker that the atomic field. In these conditions, the classical motion of the electron in the final state of the continuum becomes chaotic (so-called dynamic chaos). Using the well-known Chirikov method of averaging over chaotic variations of the phase of motion, the problem can be reduced to non-linear diffusion on the energy scale. We calculate the classical electron energy in the final state, which is averaged over fast chaotic oscillations and takes into account both the Coulomb field and the electromagnetic field. This energy is used to calculate the probability of ionization from the ground state of the atom to a lower-lying state in the continuum using the Landau-Dykhne approximation (to exponential accuracy). This ionization probability noticeably depends on the field frequency. Upon a decrease in frequency, a transition to the well-known tunnel ionization limit with a probability independent of the field frequency is considered.     

Quantum bound states embedded in a continuum and their classical analogs

We show that both a rigid and a nonrigid dipole can be trapped by an external uniform magnetic field in classical mechanics. The trapped states of a dipole present a nontrivial example of classical bound states embedded in a continuum (BSEC) that can be treated as analogs of quantum BSECs. For example, the classical motion of a dipole is confined to a finite region in space, though there are no classical turning points. We also examine the quantum motion of a dipole in a magnetic field and show that, for the most natural choices of the parameters (the rigid rotating dipole or the one bound by oscillator potential, uniform time-independent magnetic field, etc.), there are no quantum BSEC solutions.     

Size-dependent dispersion characteristics in piezoelectric nanoplates with surface effects

In this paper, surface effects on the dispersion characteristics of elastic waves propagating in an infinite piezoelectric nanoplate are investigated by using the surface piezoelectricity model. Based on the surface piezoelectric constitutive theory, the presence of surface stresses and surface electric displacements exerting on the boundary conditions of the piezoelectric nanoplate is taken into account in the modified mechanical and electric equilibrium relations. The partial wave technique is employed to obtain the general solutions of governing equations, and the dispersion relations with surface effects are expressed in an explicit closed form. The impacts of surface piezoelectricity, residual surface stress and plate thickness on the propagation properties of elastic waves are analyzed in detail. Numerical results show that the dispersion behaviors in piezoelectric nanoplates are size-dependent, and there exists a critical plate thickness above which the surface effects may vanish.     

Modeling of surface stress effects on bending behavior of nanowires: Incremental deformation theory

The surface stress effects on bending behavior of nanowires have recently attracted a lot of attention. In this letter, the incremental deformation theory is first applied to study the surface stress effects upon the bending behavior of the nanowires. Different from other linear continuum approaches, the local geometrical nonlinearity of the Lagrangian strain is considered, therefore, the contribution of the surface stresses is naturally derived by applying the Hamilton's principle, and influence of the surface stresses along all surfaces of the nanowires is captured. It is first shown that the surface stresses along all surfaces have contribution not only on the effective Young's modulus of the nanowires but also on the loading term in the governing equation. The predictions of the effective Young's modulus and the resonance shift of the nanowires from the current method are compared with those from the experimental measurement and other existing approaches. The difference with other models is discussed. Finally, based on the current theory, the resonant shift predictions by using both the modified Euler-Bernoulli beam and the modified Timoshenko beam theories of the nanowires are investigated and compared. It is noticed that the higher vibration modes are less sensitive to the surface stresses than the lower vibration modes.