Atomistic simulation of stress evolution during island growth

We report the results of a series of hybrid molecular dynamics simulations of the growth of islands on a substrate for several different island/substrate interface energies. When the interface energy is small, the islands tend to be thin and broad and the magnitude of the compressive stress-thickness product is relatively large. As the interface energy increases, the islands become taller and thinner and the magnitude of the compressive stress-thickness product decreases. This trend is consistent with experimental observations. The island aspect ratio dependence on interface energy follows from consideration of the equilibrium wetting angle. The effect of interface energy on the stress-thickness product shows that the island shape, surface/interface stresses and island stresses are self-equilibrated. A simple theory is developed that shows that the stress-thickness product is simply proportional to the substrate coverage and the substrate surface stress. The present simulations yield a simple, accurate, validated theory for stress development during the pre-coalescence stage of film growth.     

Critical thickness for misfit twinning in an epilayer

A Somigliana dislocation dipole model is developed to determine the critical thickness for misfit twin formation in an epilayer with different elastic constants from its substrate. The critical dipole arm length is determined by minimizing the twin formation energy for a given epilayer thickness and lattice mismatch strain, while a zero value of the minimum formation energy determines the critical thickness for misfit twinning. The results obtained by the Somigliana dislocation dipole model are roughly consistent with those by the previous dislocation-based twinning model.     

Elastic fields of 2D and 3D misfit particles in an infinite medium

In micromechanics, accurate quantification of the elastic field (stress, strain, and displacement) caused by the presence of an inclusion in an infinite body is desired for both the particle and matrix materials. Ideally, the solution should be applicable to any particle geometry or shape and for any distribution of misfit along the interface (i.e. misfit profile). This work presents a dislocation-based numerical method, that is an extension to earlier work in this journal [Lerma, J.D., Khraishi, T., Shen, Y.L., Wirth, B.D., 2003. The elastic fields of misfit cylindrical particles: a dislocation-based numerical approach. Mech. Res. Commun. 30, 325–334], for determining the elastic fields of volume misfit particles with arbitrary misfit distribution or particle shape.     

On flat thin elastic rods with rapidly varying periodic characteristics

A thin elastic rod is considered on a plane. The free shape of the rod is described by a periodic curve. It is shown that, under constant loads, its equilibrium shape tends to the equilibrium shape of a thin rectilinear rod when the frequency of the function describing the free shape increases infinitely. The problem under study is solved on the basis of modeling the three-dimensional shapes of circular DNA molecules by a thin rectilinear elastic rod.     

On equilibrium domains in superelastic NiTi tubes — helix versus cylinder

Macroscopic cylinder- and helix-shaped deformation domains were observed in NiTi polycrystalline shape memory alloy tubes during the phase transition under uniaxial quasi-static isothermal stretching. Further experiments showed that the occurrence of cylinder or helix domain and its subsequent isothermal evolution strongly depend on the domain volume, tube wall-thickness and loading history. This paper studies the energetics of the cylindrical and helical domains using an elastic inclusion model. It is demonstrated that the total misfit strain energy of an equilibrium domain in tube essentially depends on two nondimensional length scales: the normalized effective domain length and the normalized wall-thickness. Based on such understanding, we quantify the length scale dependence in the energy of the equilibrium cylindrical and helical domains. The energetic preference of each type of domain is predicted and the critical condition for the helix → cylinder domain transition is established.     

Profile and departure size of condensation drops on vertical surfaces

The problem of the equilibrium shape and departure size of two-dimensional dropwise condensation drops on a vertical surface is considered. The equation of the surface of the drop is obtained by minimizing (for a given volume) the total energy of the drop, consisting of surface and gravitational energy, using the techniques of variational calculus. The solution is tractable once the advancing contact angle is known, and is taken as an approximation to the axial meridian profile of a three-dimensional drop. The receding contact angle is obtained as part of the solution. The drop size is specified by imposing its vertical length. Upon increasing this vertical length, a point is reached at which no real solution exists, and this is taken as the departure size of the drop. Comparison with measured departure sizes under various body forces from standard to 100 times earth gravity are good.     

Morphological evolution of heteroepitaxial islands during Stranski–Krastonov growth

Three-dimensional finite element method is used to simulate the formation, self-assembly and shape transition of heteroepitaxial islands during Stranski–Krastonov growth. In the formulation, strain energy, surface energy, surface anisotropy and elastic anisotropy of a cubic lattice structure are taken into account. In the simulations, the SiGe/Si material system is used as a model system. An empirical surface energy as a function of surface orientation is proposed. The minimum energy surfaces are identified based on existing experimental observations. The simulation results show that the coupling of elastic energy relaxation, surface energy anisotropy and elastic anisotropy strongly influences the surface roughening morphology, self-assembly and shape transition of epitaxial islands, resulting in diverse evolution pathways.     

On the Bending and Twisting of Rods with Misfit

We derive a one-dimensional variational problem representing the elastic energy of a rod with misfit, starting from a nonlinear, three-dimensional elastic energy with nontrivial preferred strain. Our approach to dimension reduction is to find a Gamma-limit as the thickness of the rod tends to 0. The limiting energy is a quadratic function of the rates at which the rod bends and twists, and we give explicit expressions for the preferred curvature and twist in the special case of isotropic elastic moduli.     

Stress-driven interfacial instability of a bilayer structure: The interaction between free surface and interface

The morphological evolution of strained films is of technological importance to microelectronics and nanotechnology. The morphological instability of a bilayer system is analyzed, consisting of an elastic film and an elastic substrate with a misfit strain on the coherent interface. A kinetic model is derived by considering the morphological fluctuations of different perturbation amplitudes along both the free surface and the interface and the coupling effect between the film and the substrate. The couplings include the misfit strain, surface/interface energy, and surface/interface diffusion, which determine the morphological instability of the system. A quadratic dispersion relationship is established for the growth rate of the longitudinal surface and interfacial perturbations along the free surface and the interface, respectively. The propagation of the surface perturbations is revealed from the free surface to the interface, and the characteristic frequencies are identified for the initiation of the morphological instability.     

NUMERICAL STUDY OF THE BEHAVIOR OF A BUBBLE ATTACHED TO A TIP IN A NONUNIFORM ELECTRIC FIELD

In order to investigate the effects of a nonuniform electric field on the behavior of a bubble, a numerical study on the shape of a bubble attached to a conducting tip on a supporting wall is performed. The equilibrium bubble shape is determined by solving the free boundary problem that consists of the governing equation for electric field and the normal stress condition at the bubble surface. A numerically generated composite orthogonal coordinate system is employed to solve the free boundary problem. A bubble on a tip is found to be extended in the direction parallel to the applied electric field. The elongation increases steeply with an increase of the electric field strength and the height of the tip. It is also observed that a highly elongated bubble has a shape with slender waist. The bubble shape obtained from numerical studies are qualitatively similar to the shapes observed in experiments. If the contact radius is maintained during bubble deformation, the contact angle and the aspect ratio increase with the increase of the electric field strength and the tip height. On the other hand, if the contact angle is fixed during bubble deformation, the contact radius decreases as the electric field strength increases. In order to estimate the effect of electric field on the bubble departure volume, the surface tension force and the downward electric force exerted on a bubble are also computed for a bubble of fixed volume under the fixed contact angle condition. The sum of the two forces is found to decrease with increasing strength of nonuniform electric field. This fact suggests that the bubble departure volume decreases in a nonuniform electric field.     

Shapes of Epitaxially Grown Quantum Dots

A variational model introduced by Spencer and Tersoff (Appl. Phys. Lett. 96:073114, 2010) to describe optimal faceted shapes of epitaxially deposited films is studied analytically in the case in which there are a non-vanishing crystallographic miscut and a lattice incompatibility between the film and the substrate. The existence of faceted minimizers for every volume of the deposited film is established. In particular, it is shown that there is no wetting effect for small volumes. Geometric properties including a faceted version of the zero contact angle are derived, and the explicit shapes of minimizers for small volumes are identified.     

Energetics of misfit dislocation dipoles in anisotropic epitaxial films with nanoscale compositional modulation

The elastic strain and stress fields associated with nanoscale compositional modulation in an anisotropic epitaxial film on an anisotropic substrate are obtained by using Stroh formalism and the Eshelby-type inclusion method. The composition of the epitaxial film is considered to periodically fluctuate in a surface soft mode, with the amplitude of the composition modulation maximal near the growing surface and decreasing exponentially into the film. It has been experimentally observed that the composition modulation affects the formation of a new type of crystal defects, i.e., misfit dislocation dipoles, in III–V compound semiconductor materials. The formation energy of a misfit dislocation dipole under the elastic fields due to the composition modulation is calculated in this study. It is composed of the core and self energies of two dislocations, the interaction energy between two dislocations, and the interaction energies between the composition modulation and two dislocations. Numerical calculations are performed for a dislocation dipole in a lattice-matched Ga_0.5In_0.5P film on a GaAs substrate.     

The cutting of soil by narrow blades

The available models for predicting the forces acting on a narrow soil cutting blade have required separate measurements of the shape of the three-dimensional soil failure pattern ahead of the blade. It is proposed that a three-dimensional model consisting of straight line failure patterns in the soil can be used to predict both the draft forces and the volume of soil disturbed in front of a narrow blade. Limit equilibrium mechanics equations are written for the soil wedges in terms of an unknown angle of the failure zone and the theoretical draft force is minimized with respect to this angle. Force factors are thus found which are of the type to fit Reece's general earthmoving equation, but which vary with the width to depth ratio of the blade as well as with the rake angle of the blade and the friction angle of the soil. In addition the approximate geometry of the three-dimensional failure pattern in the soil is predicted for varying blade shapes and soil strengths. This allows the design of simple tools on the basis of their draft force requirements and their soil cutting efficiency. The draft force predictions and failure geometry calculations are shown to have considerable verification by experimental results.     

Stiffness matrices of thin-walled composite beam with mono-symmetric I- and channel-sections on two-parameter elastic foundation

For the coupled analysis of thin-walled composite beam under the initial axial force and on two-parameter elastic foundation with mono-symmetric I- and channel-sections, the stiffness matrices are derived. The stiffness matrices developed by this study are based on the homogeneous forms of simultaneous ordinary differential equations using the eigen-problem. For this, from the elastic strain energy, the potential energy due to the initial axial force and the strain energy considering the foundation effects, the equilibrium equations and force–displacement relationships are derived. The exact displacement functions for displacement parameters are evaluated by determining the eigenmodes corresponding to multiple non-zero and zero eigenvalues. Then the element stiffness matrix is determined using the force–displacement relationships. For the purpose of comparison, the finite element model based on the classical Hermitian interpolation polynomial is presented. In order to verify the accuracy and the superiority of the beam elements developed herein, the numerical solutions are presented and compared with results from the Hermitian beam elements and the ABAQUS’s shell elements. Particularly, the influence of the initial compressive and tensile forces, the fiber orientation, and the boundary conditions on the coupled behavior of composite beam with mono-symmetric I- and channel-sections is parametrically investigated.     

Thin-plate theory for large elastic deformations

Non-linear plate theory for thin prismatic elastic bodies is obtained by estimating the total three-dimensional strain energy generated in response to a given deformation in terms of the small plate thickness. The Euler equations for the estimate of the energy are regarded as the equilibrium equations for the thin plate. Included among them are algebraic formulae connecting the gradients of the midsurface deformation to the through-thickness derivatives of the three-dimensional deformation. These are solvable provided that the three-dimensional strain energy is strongly elliptic at equilibrium. This framework yields restrictions of the Kirchhoff-Love type that are usually imposed as constraints in alternative formulations. In the present approach they emerge as consequences of the stationarity of the energy without the need for any a priori restrictions on the three-dimensional deformation apart from a certain degree of differentiability in the direction normal to the plate.     

A method for overcoming the surface tension time step constraint in multiphase flows II

A method for overcoming the surface tension time step constraint is presented. The algorithm presented in this work is an improvement on the work presented by Sussman and Ohta (SIAM J Sci Comput 2009). In this work, the method of Sussman and Ohta is extended in order to treat problems with contact angle dynamics. Furthermore, this work presents a more efficient method for computing volume‐preserving motion by mean curvature than the method presented previously. The new method is tested on the following four 2D problems: (1) 3D axisymmetric (r?z) surface tension driven zero gravity droplet oscillation, (2) measurement of the magnitude of parasitic currents for a droplet on a substrate initialized in static equilibrium, (3) relaxation of a 2D droplet on a substrate to static shape, and (3) relaxation of a 2D bubble on a substrate to static shape. Copyright © 2011 John Wiley & Sons, Ltd.     

Further considerations of two-dimensional condensation drop profiles and departure sizes

The problem of the equilibrium shape and departure size of two-dimensional dropwise condensation drops on a vertical surface, presented in an earlier work, is extended to include advancing contact angles to 180°. The equation of the surface of the drop is obtained by minimizing (for a given volume) the total energy of the drop, consisting of surface and gravitational energy, using the techniques of variational calculus. The solution is tractable once the advancing contact angle is known, and is taken as an approximation to the axial meridian profile of a threedimensional drop. The receding contact angle is obtained as part of the solution. The drop size is specified by imposing its vertical length in contact with the wall. A maximum value of this length exists which provides a real solution, and this is taken as the departure size of the drop. It is shown that the general departure shape for an advancing contact angle of 180° includes the cases for all advancing contact angles.     

Analytical and numerical solutions for shapes of quiescent two-dimensional vesicles

We describe an analytic method for the computation of equilibrium shapes for two-dimensional vesicles characterized by a Helfrich elastic energy. We derive boundary value problems and solve them analytically in terms of elliptic functions and elliptic integrals. We derive solutions by prescribing length and area, or displacements and angle boundary conditions. The solutions are compared to solutions obtained by a boundary integral equation-based numerical scheme. Our method enables the identification of different configurations of deformable vesicles and accurate calculation of their shape, bending moments, tension, and the pressure jump across the vesicle membrane. Furthermore, we perform numerical experiments that indicate that all these configurations are stable minima.     

基于三维弹性杆模型的DNA拉伸响应的研究

通过Young-Laplace方程将界面张力引入Kirchhoff方程,并结合Gibbs与Langmuir吸附方程建立了受溶液浓度影响的三维DNA弹性杆模型。基于此模型,引入DNA端部的边界条件,运用打靶法来模拟计算溶液中的DNA链段受端部拉力作用下的几何构型。进一步分析了在界面能与弹性应变能的耦合作用下,DNA链段平衡构型的形状与尺寸的变化规律。