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
Accepted for publication in Journal of the Mechanics and Physics of Solids 2008.
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.
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
.