Surface Cauchy Born Research Articles

Surface Stress Effect on Silicon Nanowire Mechanical Behavior: Size and Orientation Dependence

Abstract As truly nanoscale building blocks, silicon nanowires find applications in electromechanical and electronic devices. A deeper understanding of their transport properties is of utmost importance for improving the performance of such devices or for introducing new functionalities. Due to this reason, the size dependence of properties has become a major field of study. The determination of mechanical properties of silicon nanowires has proved to be specifically challenging with contrasting conclusions drawn from experimental and computational studies regarding the surface stress effect. In this respect a comprehensive study on the crystal orientation and geometrical parameters can shed light on the role played by the surface stress in determining silicon nanowire mechanical behavior. This is accomplished by linking the local surface stress at the atomic scale to the overall behavior of the continuum system. The study starts by a primary surface analysis through atomistic simulations for silicon {100} and {110} surfaces. The implication of the surface stress on the mechanical behavior of a continuum system — nanowire in this work — is predicted through basic elasticity problems. Surface stress is then transferred to the continuum system using the Surface Cauchy-Born approach. Deformation associated with the surface stress is computed for two different cantilevers with longitudinal orientations in and having {100} and {100}/{110} transverse surfaces, respectively. Results indicate a twist deformation at the free end of silicon nanowires with rectangular cross-sections. Similarly, a bending deformation is observed for silicon nanowires. A case study is carried out to emphasize the importance of the surface stress in the operation of a nanowire electromechanical switch. Due to associated deformations, the surface stress is observed to cause variations in the electrostatic force along the nanowire cross-section by as much as 50%. The presented approach indicates a proper pathway to analyze the surface stress effect on the mechanical behavior of nanowires. Addressing the existing controversy regarding the contribution of the surface stress in the mechanical behavior of silicon nanowires, results in this work can provide a guideline for the interpretation of existing and the design of future experimental investigations.

Selecting the optimum engineering model for the frequency response of fcc nanowire resonators

The full potential of the nanoelectromechanical systems, NEMS, as one of the leading examples among the new-generation sensing technologies, is yet to be realized. One of the main challenges on the road is the mechanical modeling of their core elements, the tiny mechanical building blocks such as the nanowire resonators. The success of the engineering design of such miniaturized systems will depend heavily on the availability of accurate mechanistic models with the least possible computational cost. Although a variety of models are available for this purpose, the boundaries between their admissible domains remain rather vague. For example, analytical approaches including Euler–Bernoulli and Timoshenko beam theories provide closed-form solutions and work reasonably well for moderate nanowire geometries, and hence, they are frequently utilized in the literature. However, their validity in the case of extreme surface-to-volume ratios remains questionable. Classical finite element method can partially be used to address these deficiencies. On the other hand, molecular dynamics provide accurate results, while nanowire geometries studied with this computationally demanding technique usually remain confined to dimensions below those of practical interest. To address these issues, a benchmarking study among analytical and numerical techniques is carried out, where Surface Cauchy–Born theory serves as the reference. Using gold nanowires with different dimensions and boundary conditions, it is observed that analytical models are applicable within a length-to-thickness ratio range of 7–11 in the fixed–fixed configuration, whereas they can be used safely within a length-to-thickness ratio range of less than 25 in the fixed–free configuration. Deviations as high as 50% are encountered for length-to-thickness ratios exceeding 11 for both the analytical approach and the classical finite element method in the fixed–fixed structure. The deviations are quantitatively linked to the dominance of the surface effect through the use of the Surface Cauchy–Born model. For length-to-thickness ratios less than 7, the lack of cross-sectional deformations in analytical treatment is also observed to lead to high deviations for the fixed–fixed configuration through the comparison with higher-order beam theories. Results are verified with silver nanowires as well. The work provides a guideline for selecting the optimum mechanical model given the nanowire resonator dimensions and boundary conditions.

Thermo-coupled Surface Cauchy–Born theory: An engineering finite element approach to modeling of nanowire thermomechanical response

There are remarkable studies geared towards developing thermomechanical analyses of nanowires based on quasiharmonic and Molecular Dynamics simulations. These methods exhibit limited applicability due to the associated computational cost. In this study an engineering finite-temperature model based on Surface Cauchy–Born theory is developed, where surface energy is accounted for in the prediction of the thermomechanical response. This is achieved by using a temperature-dependent interatomic potential in the standard Cauchy–Born theory with a surface energy contribution. Simultaneous calculation of thermal and mechanical stresses is achieved by eliminating the diagonalization matrix of entropy in the quasiharmonic system. This leads to a reduction in the degrees of freedom by more than 99% in comparison with equivalent Molecular Dynamics models. For the purpose of validation, results obtained on copper and nickel nanowires through the proposed method are compared with those of the more involved Molecular Dynamics simulations. This comparison verifies the significant reduction in the computational process with an acceptable accuracy. Hence, the proposed method provides a promising engineering tool without compromising the underlying physics of the problem and has potential implications in the effective modeling of the nanoscale thermomechanical behavior.

Investigation of surface stress effect in 3D complex nano parts using FEM and modified boundary Cauchy–Born method

In this paper the surface effects in 3D complex parts are investigated via classical FEM and the modified boundary Cauchy–Born method, which are continuum based theories, and the results are compared with molecular dynamics simulations. The boundary Cauchy–Born covers the weakness of the surface Cauchy–Born in modeling edge and corner elements, so the results are in better agreement with molecular dynamics simulations. Using super convergent patch recovery method, MBCB shows more precise results in comparison with BCB as it can model acute and obtuse angles which cannot be modeled via BCB. The flexibility of MBCB in modeling 3D complex and non-right angle geometries are utilized to model industrial examples under free relaxation condition. Unlike samples in previous works, the complexity of geometry proposed here results in highly nonlinear problems where the MBCB method can no longer guarantee convergency of the solutions. Therefore, a new and modified arc-length method is developed and utilized to overcome this nonlinearity.

An analysis of the boundary layer in the 1D surface Cauchy–Born model

The surface Cauchy–Born (SCB) method is a computational multi-scale method for the simulation of surface-dominated crystalline materials. We present an error analysis of the SCB method, focused on the role of surface relaxation. In a linearized 1D model we show that the error committed by the SCB method is O(1) in the mesh size; however, we are able to identify an alternative approximation parameter – the stiffness of the interaction potential – with respect to which the relative error in the mean strain is exponentially small. Our analysis naturally suggests an improvement of the SCB model by enforcing atomistic mesh spacing in the normal direction at the free boundary. In this case we even obtain pointwise error estimates for the strain.

Surface stress effects may induce softening: Euler–Bernoulli and Timoshenko buckling solutions

Surface elasticity effects may be significant for small scale structures. In this paper, the effect of surface elasticity effects is investigated for the buckling of Euler–Bernoulli and Timoshenko beams. Engesser and Haringx theory of Timoshenko shear beams are studied. The surface elasticity effects are considered for small scale beam structures based on the Laplace–Young equation, which results in an equivalent distributed loading term in the beam equation. We show that these effects are explained by their non-conservative nature that can be essentially modelled as a follower tensile loading for inextensible beams. As a consequence the usual paradigm that smaller is stiffer is not necessarily found for these structural problems. The buckling small scale shear beams in presence of surface elasticity effects is studied for various boundary conditions. For clamped-free boundary conditions, we show that the buckling load is reduced compared to the one without this surface effect. This result is consistent with some recent numerical results based on surface Cauchy–Born model and with experimental results available in the literature. For other boundary conditions such as hinge–hinge and clamped–clamped boundary conditions, the results are identical to the ones already published. We explain in this paper the surprising results observed in the literature that surface elasticity effects may soften a nanostructure for some specific boundary conditions (due to the non-conservative nature of its loading application). Furthermore, self-instability is theoretically noticed for small shear beams.

Surface Elasticity Effects Can Apparently Be Explained Via Their Nonconservativeness

Abstract Recently, considerable attention has been given to investigating the surface effects on nanoscale materials. These effects can be predominant for small-scale structures, such as nanobeams, nanoplates, and nanoshells. In this paper, surface elasticity effects are considered for small scale beam structures based on the Laplace–Young equation, which results in an equivalent distributed loading term in the beam equation. We show that these effects are explained by their nonconservative nature that can be essentially modeled as a follower tensile loading for inextensible beams. The buckling and vibrations of small scale beams in the presence of surface elasticity effects is studied for various boundary conditions. It is shown that the surface elasticity effects may significantly affect the buckling and vibrations behavior of small scale beams. For clamped-free boundary conditions, we show that the buckling load is reduced compared to the one without this surface effect. This result is consistent with some recent numerical results based on surface Cauchy–Born model and with experimental results available in the literature. It appears that this result cannot be obtained if surface elasticity effects are modeled as a conservative-type loading. For other boundary conditions such as hinge–hinge and clamped–clamped boundary conditions, the results are identical to the ones already published. We explain in this paper the surprising results observed in the literature that surface elasticity effects may soften a nanostructure for some specific boundary conditions (due to the nonconservative nature of its loading application). The same conclusions are obtained for the vibrations of small scale beams with surface elasticity effects, where the natural frequency tends to decrease with surface elasticity effects for clamped-free conditions.

An extended finite element/level set method to study surface effects on the mechanical behavior and properties of nanomaterials

AbstractWe present a new approach based on coupling the extended finite element method (XFEM) and level sets to study surface and interface effects on the mechanical behavior of nanostructures. The coupled XFEM‐level set approach enables a continuum solution to nanomechanical boundary value problems in which discontinuities in both strain and displacement due to surfaces and interfaces are easily handled, while simultaneously accounting for critical nanoscale surface effects, including surface energy, stress, elasticity and interface decohesion. We validate the proposed approach by studying the surface‐stress‐driven relaxation of homogeneous and bi‐layer nanoplates as well as the contribution from the surface elasticity to the effective stiffness of nanobeams. For each case, we compare the numerical results with new analytical solutions that we have derived for these simple problems; for the problem involving the surface‐stress‐driven relaxation of a homogeneous nanoplate, we further validate the proposed approach by comparing the results with those obtained from both fully atomistic simulations and previous multiscale calculations based upon the surface Cauchy–Born model. These numerical results show that the proposed method can be used to gain critical insights into how surface effects impact the mechanical behavior and properties of homogeneous and composite nanobeams under generalized mechanical deformation. Copyright © 2010 John Wiley & Sons, Ltd.

Surface stress effects on the bending properties of fcc metal nanowires

The major purpose of this work is to investigate surface stress effects on the bending behavior and properties of $⟨100⟩/{100}$ gold nanowires with both fixed/fixed and fixed/free boundary conditions. The results are obtained through utilization of the recently developed surface Cauchy-Born model, which captures surface stress effects on the elastic properties of nanostructures through a three-dimensional, nonlinear finite element formulation. There are several interesting findings in the present work. First, we quantify the stress and displacement fields that result in the nanowires due to bending deformation. In doing so, we find that regardless of boundary condition, the stresses that are present in the nanowires due to deformation induced by surface stresses prior to any applied bending deformation dominate any stresses that are generated by the bending deformation unless very large $(\ensuremath{\approx}5%)$ bending strains are applied. In contrast, when the stresses and displacements induced by surface stresses prior to bending are subtracted from the stress and displacement fields of the bent nanowires, we find that the bending stresses and displacements do match the solutions expected from bulk continuum beam theory, but only within the nanowire bulk, and not at the nanowire surfaces. Second, we find that the deformation induced by surface stresses also has a significant impact on the nanowire Young's modulus that is extracted from the bending simulations, where a strong boundary-condition dependence is also found. By comparing all results to those that would be obtained using various linear surface-elastic theories, we demonstrate that a nonlinear, finite deformation formulation that captures changes in both bulk- and surface-elastic properties resulting from surface stress-induced deformation is critical to reproducing the experimentally observed boundary-condition dependence in Young's modulus of metal nanowires. Furthermore, we demonstrate that linear surface-elastic theories based solely on the surface energy erroneously predict an increase in Young's modulus with decreasing nanowire size regardless of boundary condition. In contrast, while the linear surface-elastic theories based upon the Gurtin and Murdoch formalism can theoretically predict elastic softening with decreasing size, we demonstrate that, regardless of boundary condition, the stiffening due to the surface stress dominates the softening due to the surface stiffness for the range of nanowire geometries considered in the present work. Finally, we determine that the nanowire Young's modulus is essentially identical when calculated via either bending or resonance for both boundary conditions, indicating that surface effects have a similar impact on the elastic properties of nanowires for both loading conditions.

Quantifying the size-dependent effect of the residual surface stress on the resonantfrequencies of silicon nanowires if finite deformation kinematics are considered

There are two major objectives to the present work. The first objective is to demonstratethat, in contrast to predictions from linear surface elastic theory, when nonlinear, finitedeformation kinematics are considered, the residual surface stress does impact the resonantfrequencies of silicon nanowires. The second objective of this work is to delineate,as a function of nanowire size, the relative contributions of both the residual(strain-independent) and the surface elastic (strain-dependent) parts of the surface stress tothe nanowire resonant frequencies. Both goals are accomplished by using the recentlydeveloped surface Cauchy–Born model, which accounts for nanoscale surface stressesthrough a nonlinear, finite deformation continuum mechanics model that leads to thesolution of a standard finite element eigenvalue problem for the nanowire resonantfrequencies. In addition to demonstrating that the residual surface stress does impact theresonant frequencies of silicon nanowires, we further show that there is a strong sizedependence to its effect; in particular, we find that consideration of the residual surfacestress alone leads to significant errors in predictions of the nanowire resonant frequency,with an increase in error with decreasing nanowire size. Correspondingly, thestrain-dependent part of the surface stress is found to have an increasingly importanteffect on the resonant frequencies of the nanowires with decreasing nanowire size.

Surface stress effects on the resonant properties of metal nanowires: The importance of finite deformation kinematics and the impact of the residual surface stress

We utilize the recently developed surface Cauchy–Born model, which extends the standard Cauchy–Born theory to account for surface stresses due to undercoordinated surface atoms, to study the coupled influence of boundary conditions and surface stresses on the resonant properties of 〈 1 0 0 〉 gold nanowires with { 1 0 0 } surfaces. There are two major purposes to the present work. First, we quantify, for the first time, variations in the nanowire resonant frequencies due to surface stresses as compared to the corresponding bulk material which does not observe surface effects within a finite deformation framework depending on whether fixed/free or fixed/fixed boundary conditions are utilized. We find that while the resonant frequencies of fixed/fixed nanowires are elevated as compared to the corresponding bulk material, the resonant frequencies of fixed/free nanowires are reduced as a result of compressive strain caused by the surface stresses. Furthermore, we find that for a diverse range of nanowire geometries, the variation in resonant frequencies for both boundary conditions due to surface stresses is a geometric effect that is characterized by the nanowire aspect ratio. The present results are found to agree well with existing experimental data for both types of boundary conditions. The second major goal of this work is to quantify, for the first time, how both the residual (strain-independent) and surface elastic (strain-dependent) parts of the surface stress impact the resonant frequencies of metal nanowires within the framework of nonlinear, finite deformation kinematics. We find that if finite deformation kinematics are considered, the strain-independent surface stress substantially alters the resonant frequencies of the nanowires; however, we also find that the strain-dependent surface stress has a significant effect, one that can be comparable to or even larger than the effect of the strain-independent surface stress depending on the boundary condition, in shifting the resonant frequencies of the nanowires as compared to the bulk material.

Strain sensing through the resonant properties of deformed metal nanowires

In this article, we study the potential of gold nanowires as resonant nanoscale strain sensors. The sensing ability of the nanowires is determined by calculating the variations in resonant frequency that occur due to applied uniaxial tensile and compressive strain. The resonant frequencies are obtained using the surface Cauchy–Born model, which captures surface stress effects on the nanowires through a nonlinear continuum mechanics framework; due to the continuum formulation, the strain-dependent nanowire resonant frequencies are calculated through the solution of a standard finite element eigenvalue problem, where the coupled effects of the applied uniaxial strain and surface stress are naturally included through the finite element stiffness matrix. The nanowires are found to be more sensitive to compressive than tensile strain, with resonant frequency shifts around 200–400 MHz with the application of 1% tensile and compressive strain. In general, the strain sensitivity of the nanowires is found to increase with decreasing cross-sectional size, with additional dependencies on their aspect ratio.

Surface stress effects on the resonant properties of silicon nanowires

The purpose of the present work is to quantify the coupled effects of surface stresses and boundary conditions on the resonant properties of silicon nanowires. We accomplish this by using the surface Cauchy–Born model, which is a nonlinear, finite deformation continuum mechanics model that enables the determination of the nanowire resonant frequencies including surface stress effects through solution of a standard finite element eigenvalue problem. By calculating the resonant frequencies of both fixed/fixed and fixed/free ⟨100⟩ silicon nanowires with unreconstructed {100} surfaces using two formulations, one that accounts for surface stresses and one that does not, it is quantified how surface stresses cause variations in nanowire resonant frequencies from those expected from continuum beam theory. We find that surface stresses significantly reduce the resonant frequencies of fixed/fixed nanowires as compared to continuum beam theory predictions, while small increases in resonant frequency with respect to continuum beam theory are found for fixed/free nanowires. It is also found that the nanowire aspect ratio, and not the surface area to volume ratio, is the key parameter that correlates deviations in nanowire resonant frequencies due to surface stresses from continuum beam theory.

A multiscale, finite deformation formulation for surface stress effects on the coupled thermomechanical behavior of nanomaterials

We present a multiscale, finite deformation formulation that accounts for surface stress effects on the coupled thermomechanical behavior and properties of nanomaterials. The foundation of the work lies in the development of a multiscale surface Helmholtz free energy, which is constructed through utilization of the surface Cauchy–Born hypothesis. By doing so, temperature-dependent surface stress measures as well as a novel form of the heat equation are obtained directly from the surface free energy. The development of temperature-dependent surface stresses distinguishes the present approach, as the method can be utilized to study the behavior of nanomaterials by capturing the size-dependent variations in the thermoelastic properties with decreasing nanostructure size. The coupled heat and momentum equations are solved in 1D using a fully implicit, monolithic scheme, and show the importance of capturing surface stress effects in accurately modeling the thermomechanical behavior of nanoscale materials.

A finite element formulation for nanoscale resonant mass sensing using the surface Cauchy–Born model

The purpose of this work is to develop the theoretical basis needed to study nanoscale resonant mass sensing with finite elements using the surface Cauchy–Born (SCB) model. The theory is developed in 1D, where it is identified that the primary modeling issue lies in capturing inhomogeneous surface stresses arising from adsorbate/substrate interactions. By utilizing internal degrees of freedom within the SCB framework, we show that the SCB model can represent the bonding energies, and thus the inhomogeneous surface stress that arises due to interactions by atoms of dissimilar materials. A key outcome of this is that it is shown that a finite element solution using the SCB model is able to simultaneously capture both mass and stiffness variations due to adsorbate/substrate interactions, and their effects on the nanostructure resonant properties. We first verify that the SCB model accurately captures the resonant properties of monatomic 1D atomic chains, then demonstrate the approach by studying the resonant properties of 1D atomic chains that interact with adsorbates. Importantly, we demonstrate that a finite element solution using the SCB model can predict the distinct shifts in resonant frequency that occur due to the adsorption of different materials on the 1D monatomic chain.

A Surface Cauchy-Born model for silicon nanostructures

We present a Surface Cauchy-Born approach to modeling non-centrosymmetric, semiconducting nanostructures such as silicon that exist in a diamond cubic lattice structure. The model is based on an extension to the standard Cauchy-Born theory in which a surface energy term that is obtained from the underlying crystal structure and governing interatomic potential is used to augment the bulk energy. The incorporation of the surface energy leads naturally to the existence of surface stresses, which are key to capturing the size-dependent mechanical behavior and properties of nanomaterials. We present the approach in detail, then demonstrate its capabilities by calculating the minimum energy configurations of silicon nanowires due to surface stresses as compared to full scale atomistic calculations.

Surface Cauchy-Born analysis of surface stress effects on metallic nanowires

We present a surface Cauchy-Born approach to modeling FCC metals with nanometer scale dimensions for which surface stresses contribute significantly to the overall mechanical response. The model is based on an extension of the traditional Cauchy-Born theory in which a surface energy term that is obtained from the underlying crystal structure and governing interatomic potential is used to augment the bulk energy. By doing so, solutions to three-dimensional nanomechanical boundary value problems can be found within the framework of traditional nonlinear finite element methods. The major purpose of this work is to utilize the surface Cauchy-Born model to determine surface stress effects on the minimum energy configurations of single crystal gold nanowires using embedded atom potentials on wire sizes ranging in length from $6\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}280\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ with square cross sectional lengths ranging from $6\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}35\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$. The numerical examples clearly demonstrate that other factors beside surface area to volume ratio and total surface energy minimization, such as geometry and the percentage of transverse surface area, are critical in determining the minimum energy configurations of nanowires under the influence of surface stresses.

A surface Cauchy–Born model for nanoscale materials

We present an energy-based continuum model for the analysis of nanoscale materials where surface effects are expected to contribute significantly to the mechanical response. The approach adopts principles utilized in Cauchy–Born constitutive modelling in that the strain energy density of the continuum is derived from an underlying crystal structure and interatomic potential. The key to the success of the proposed method lies in decomposing the potential energy of the material into bulk (volumetric) and surface area components. In doing so, the method naturally satisfies a variational formulation in which the bulk volume and surface area contribute independently to the overall system energy. Because the surface area to volume ratio increases as the length scale of a body decreases, the variational form naturally allows the surface energy to become important at small length scales; this feature allows the accurate representation of size and surface effects on the mechanical response. Finite element simulations utilizing the proposed approach are compared against fully atomistic simulations for verification and validation. Copyright © 2006 John Wiley & Sons, Ltd.