On molecular statics and surface-enhanced continuum modeling of nano-structures
Highlights
► Surface-enhanced continuum. ► Molecular statics. ► Atomistic fields obtained via Eulerian or Lagrangian averaging. ► Comparison of continuum and atomistic simulation.
Introduction
Continuum mechanics has been proven to be a reliable and efficient tool to solve many problems at different length and time scales. Modern technology, however, operates at length-scales much lower than usually considered in the continuum formulation. At that scale it is not enough to consider only bulk energy, however, surface energy becomes more and more dominant with a decrease of size. Thus, the solution to a boundary value problem is not any longer independent of the dimensions, on the contrary – pronounced size effects are observed. This can be modeled by enhancing the continuum theory with a surface energy, entropy, etc.
Phenomenological models that endow the surface with its own energy date back to the pioneering work of Gibbs [1]. More recently, Gurtin and Murdoch [2] described surface effects using tensorial surface stresses (see also [3], [4] for the case of thermomechanical interfaces). Daher and Maugin [5] invoked the method of virtual power to endow the surface or interface with its own thermodynamic constituents. The continuum approach of this work takes as its point of departure the surface elasticity formulation detailed by Javili and Steinmann [6]. The validation of such SEC models as compared to atomistic simulations shall be carried out.
With the continuing increase in computational power, nanoscale problems can be solved with (discrete) atomistic models such as MS. In MS a finite number of (infinitesimal) particles usually endowed only with translational degrees of freedom are considered. The primary input to such models is the interparticle potential, which is often obtained from first principles calculations. Unlike in their continuum counterparts, nothing needs to be added to the atomistic model to capture surface effects. They arise naturally due to the termination of long-range interparticle forces. MS simulations, however, are computationally very demanding. Thus, a distinct advantage of continuum models over their atomistic counterparts is the increased computational efficiency.
Since both the discrete and the continuum approach can be used to model structures at the nano-scale, the question on the correspondence between the two approaches arises. In particular, kinematic and kinetic continuum quantities (such as displacements, the deformation gradient, and Cauchy stress) should be expressible in terms of the MS quantities. That is important not only for the validation of the computationally efficient continuum simulations against discrete models, but also for enhancing and developing methods that couple discrete and continuum formulations. That is, in different parts of a domain different formulations can be used. In order to bridge these domains, several methods have recently been proposed. Among such methods are the Quasi-continuum method [7], [8], [9], [10], the Bridging-domain method [11], [12], the Bridging-scale-method [13], methods based on the averaging procedure used here [14], [15] and others [16], [17], to name a few. For an overview on coupling methods we refer to [18], [19]. Park et al. [20], Park and Klein [21] developed an alternative continuum framework based on the surface Cauchy–Born model, an extension of the classical Cauchy–Born model, to include surface stresses (see also [22]).
The framework we use here to link the discrete and continuum models dates back to works of Irving and Kirkwood [23] and Noll [24]. This linkage is achieved by spatial averaging in the current (Eulerian) configuration, followed by statistical (probability density) averaging. In the last few decades many publications were devoted to this field, see among others [25], [26], [27], [28], [29], [30] and references therein. It was shown in [31], that the integral of the Cauchy stress obtained by this type of averaging over space is consistent with the Virial pressure, which is commonly used as a macroscopic measure of stress in discrete systems, thus indirectly proving the validity of the Eulerian averaging approach. As an alternative to the averaging in the current (Eulerian) configuration, it was recently proposed to average instead in the reference (Lagrangian) configuration [32]. The link between the two averaging methods is not trivial, the validity of the Lagrangian approach was numerically illustrated [32] only for the case of homogeneous solutions.
In this contribution we focus on two different averaging approaches – Eulerian and Lagrangian. The kinematic and kinetic local fields evaluated from the MS simulations are compared to the results of calculations based on a SEC formulation. Note that Yvonnet et al. [33] have recently studied the applicability of the SEC formulation in modeling Wurtzite structured nano-wires by fitting data from ab initio calculations (see also [34]).
The paper is organized as follows: In Section 2 we briefly introduce the notation adopted here and review the theoretical background for a SEC formulation as well as the link between the continuum and discrete formulation of MS. In Section 3 we study different numerical examples on FCC-type copper crystal. First, we discuss a procedure to obtain the bulk elastic properties from the molecular statics simulations. Then, both Lagrangian and Eulerian averaging procedures are benchmarked in two simple homogeneous solution cases. Finally, we study an example with a non-homogeneous solution and compare the results obtained by the discrete and continuum formulation. By changing the dimensions of the problem, the size effect is highlighted and studied. The discussion and conclusions are presented in Section 4.
Section snippets
Theory
The purpose of this preliminary section is to summarize certain key concepts in continuum mechanics and atomistic modeling, as well as to introduce the notation adopted here. Detailed expositions on non-linear continuum mechanics can be found in Truesdell and Noll [35], Marsden and Hughes [36], among others.
Numerical examples
In what follows we consider a FCC crystal of copper. On the atomistic side it is modeled using the EAM potential by Foiles et al. [42], which has the lattice parameter at zero temperature and the cut-off radius 4.94 A. Atomistic simulations are performed using the LAMMPS open-source software [43]. The quartic averaging kernel of radius R was used to evaluate the continuum fields from the atomistic simulations:with r = ∣xα − x∣/R and .
Since the total potential energy U
Discussion and conclusions
In this contribution we compared the continuum mechanics approach enhanced with a surface energy to atomistic (MS) simulations. Lagrangian and Eulerian approaches for particle averaging were applied to the numerical examples of a FCC crystal. The size effect observed in the macroscopic stresses as evaluated from the MS simulations was well captured by the SEC formulation.
By investigating the MS simulation of the homogeneous RVE, the non-linearity of the stress–strain relationships in the FCC
Acknowledgments
The first author is grateful to the German Science Foundation (Deutsche Forschungs–Gemeinschaft, DFG), Grant STE 544/46-1, for the financial support. The support of this work by the ERC Advanced Grant MOCOPOLY is gratefully acknowledged by the second and third authors.
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