An Analysis of the Boundary Layer in the 1D Surface Cauchy-Born Model

K. Jayawardana, C. Mordacq, C. Ortner and H.S. Park
ESAIM: Mathematical Modelling and Numerical Analysis 2013; 47:109-123.

Abstract

The Surface Cauchy-Born (SCB) method is a computational multiscale 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.

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Surface Stress Effects on the Critical Buckling Strains of Silicon Nanowires

H.S. Park
Computational Materials Science 2012; 51:396-401.

Abstract

The objective of this paper is to quantify how nanoscale surface stresses impact the critical buckling strains of silicon nanowires. These insights are gained by using nonlinear finite element calculations based upon a multiscale, finite deformation constitutive model that incorporates nanoscale surface stress and surface elastic effects to study the buckling behavior of silicon nanowires that have cross sectional dimensions between 10 and 25 nm under axial compressive loading. The key finding is that, in contrast to existing surface elasticity solutions, the critical buckling strains are found to show little deviation from the classical bulk Euler solution. The present results suggest that accounting for axial strain relaxation due to surface stresses may be necessary to improve the accuracy and predictive capability of analytic linear surface elastic theories.

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Bridging the Gap Between Experimental Measurements and Atomistic Predictions of the Elastic Properties of Silicon Nanowires using Multiscale Modeling

G. Yun and H.S. Park
Finite Elements in Analysis and Design 2012; 49:3-12.
(Invited paper: Special Issue on Analysis and Design of MEMS/NEMS).

Abstract

In the present work, we have applied recently developed nonlinear multiscale finite element techniques which account for nanoscale surface stress and surface elastic effects to investigate the elastic properties of silicon nanowires as obtained through bending deformation. The numerical results are used to clarify the factors underlying the current disconnect between atomistic simulations and experiments as to the nanowire sizes at which deviation from bulk elastic properties due to surface effects are observed. In particular, we demonstrate that when nanowires with aspect ratios (defined as the axial length divided by the square cross sectional length) larger than about 15 are considered, the elastic softening that has been observed experimentally for larger (i.e. > 20 nm diameter) nanowires is observed. In contrast, when smaller aspect ratios are considered, very little deviation from the bulk elastic properties are observed, in agreement with existing atomistic calculations. Furthermore, we demonstrate that the elastic softening is strongly boundary condition dependent, where fixed/fixed silicon nanowires exhibit a strong aspect ratio-dependent softening, while little variation in the elastic properties of fixed/free nanowires are observed. Comparisons are made with existing surface elastic theories and experiments to bring further insights into the boundary condition dependence in elastic properties.

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A Continuum Model for the Mechanical Behavior of Nanowires Including Surface and Surface-Induced Initial Stresses

F. Song, G.L. Huang, H.S. Park and X.N. Liu
International Journal of Solids and Structures 2011; 48:2154-2163.

Abstract

The continuum modeling of the mechanical behavior of nanowires has recently attracted much attention due to its simplicity and efficiency. However, there are still some critical issues to be solved. In this paper, we demonstrate the importance of accounting for the effects of initial stresses in the nanowires that are caused by deformation due to surface stresses; we note that such initial stresses have previously been neglected in most existing continuum models. By considering the local geometrical nonlinearity of strains during the incremental flexural motion, a new formulation of the Euler-Bernoulli beam model for nanowires is developed through the incremental deformation theory, in which effects of the surface stress, the surface-induced initial stress and surface elasticity are naturally incorporated. It is found through comparisons to existing experimental and computational results for both fcc metal and ceramic nanowires that the surface-induced initial stresses, which are neglected in the Young-Laplace model, can significantly influence the overall mechanical properties of nanowires. We additionally demonstrate and quantify the errors induced by using the Young-Laplace model due to its approximation of surface stresses acting on only the top and bottom surfaces of nanowires.

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An Extended Finite Element/Level Set Method to Study Surface Effects on the Mechanical Behavior and Properties of Nanomaterials

M. Farsad, F.J. Vernerey and H.S. Park
International Journal for Numerical Methods in Engineering 2010; 84:1466-1489.

Abstract

We present a new approach based upon 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 nano-plate, we further validate the proposed approach by comparing the results to 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.

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A Multiscale Finite Element Method for the Dynamic Analysis of Surface-Dominated Nanomaterials

H.S. Park
International Journal for Numerical Methods in Engineering 2010; 83:1237-1254.
(Invited paper: Special Issue on Atomistic and Multiscale Analysis).

Abstract

The purpose of this article is to present a multiscale finite element method that captures nanoscale surface stress effects on the dynamic mechanical behavior of nanomaterials. The method is based upon arguments from crystal elasticity, i.e. the Cauchy-Born rule, but significantly extends the capability of the standard Cauchy-Born rule by accounting for critical nanoscale surface stress effects, which are well-known to have a significant effect on the mechanics of crystalline nanostructures. We present the governing equations of motion including surface stress effects, and demonstrate that the methodology is general and thus enables simulations of both metallic and semiconducting nanostructures. The numerical examples on elastic wave propagation and dynamic tensile and compressive loading show the ability of the proposed approach to capture surface stress effects on the dynamic behavior of both metallic and semiconducting nanowires, and demonstrate the advantages of the proposed approach in studying the deformation of nanostructures at strain rates and time scales that are inaccessible to classical molecular dynamics simulations.

This paper is available in PDF form .


Surface Stress Effects on the Bending Properties of FCC Metal Nanowires

G. Yun and H.S. Park
Physical Review B 2009; 79:195421.

Abstract

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 (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 the bending-derived 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.

This paper is available in PDF form .


Quantifying the Size-Dependent Effect of the Residual Surface Stress on the Resonant Frequencies of Silicon Nanowires if Finite Deformation Kinematics are Considered

H.S. Park
Nanotechnology 2009; 20:115701.

Abstract

There are two major objectives to the present work. The first objective is to demonstrate that, in contrast to predictions from linear surface elastic theory, when nonlinear, finite deformation kinematics are considered, the residual surface stress does impact the resonant frequencies 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 surface elastic (strain-dependent) parts of the surface stress to the nanowire resonant frequencies. Both goals are accomplished by using the recently developed surface Cauchy-Born model, which accounts for nanoscale surface stresses through a nonlinear, finite deformation continuum mechanics model that leads to the solution of a standard finite element eigenvalue problem for the nanowire resonant frequencies. In addition to demonstrating that the residual surface stress does impact the resonant frequencies of silicon nanowires, we further show that there is a strong size-dependence to its effect; in particular, we find that consideration of the residual surface stress alone leads to significant errors in predictions of the nanowire resonant frequency, with an increase in error with decreasing nanowire size. Correspondingly, the strain-dependent part of the surface stress is found to have an increasingly important effect on the resonant frequencies of the nanowires with decreasing nanowire size.

This paper is available in PDF form .


Surface Stress Effects on the Resonant Properties of Metal Nanowires: The Importance of Finite Deformation Kinematics and the Impact of the Residual Surface Stress

H.S. Park and P.A. Klein
Journal of the Mechanics and Physics of Solids 2008;56:3144-3166.

Abstract

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 <100> gold nanowires with {100} 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.

This paper is available in PDF form .


Strain Sensing Through the Resonant Properties of Deformed Metal Nanowires

H.S. Park
Journal of Applied Physics 2008;104:013516.
(Also selected for publication in the Virtual Journal of Nanoscale Science and Technology, July 21, 2008).

Abstract

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 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.

This paper is available in PDF form .


Surface Stress Effects on the Resonant Properties of Silicon Nanowires

H.S. Park
Journal of Applied Physics 2008; 103:123504.
(Also selected for publication in the Virtual Journal of Nanoscale Science and Technology, June 30, 2008.)

Abstract

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.

This paper is available in PDF form .


A Multiscale, Finite Deformation Formulation for Surface Stress Effects on the Coupled Thermomechanical Behavior of Nanomaterials

G. Yun and H.S. Park
Computer Methods in Applied Mechanics and Engineering 2008; 197:3337-3350
(Invited paper: Special Issue on Computational Methods of Nanostructures).

Abstract

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.

This paper is available in PDF form .


A Finite Element Formulation for Nanoscale Resonant Mass Sensing Using the Surface Cauchy-Born Model

G. Yun and H.S. Park
Computer Methods in Applied Mechanics and Engineering 2008; 197:3324-3336
(Invited paper: Special Issue on Computational Methods of Nanostructures).

Abstract

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.

This paper is available in PDF form .


A Surface Cauchy-Born Model for Silicon Nanostructures

H.S. Park and P.A. Klein
Computer Methods in Applied Mechanics and Engineering 2008; 197:3249-3260
(Invited paper: Special Issue on Computational Methods of Nanostructures).

Abstract

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.

This paper is available in PDF form .


Surface Cauchy-Born Analysis of Surface Stress Effects on Metallic Nanowires

H.S. Park and P.A. Klein
Physical Review B 2007; 75:085408
(Also selected for publication in the Virtual Journal of Nanoscale Science and Technology, Feb. 19, 2007.)

Abstract

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 to 280 nm with square cross sectional lengths ranging from 6 to 35 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.

This paper is available in PDF form .


A Surface Cauchy-Born Model for Nanoscale Materials

H.S. Park, P.A. Klein and G.J. Wagner
International Journal for Numerical Methods in Engineering 2006; 68:1072-1095.

Abstract

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 modeling 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.

This paper is available in PDF form .