Continuum Surface Energy from a Lattice Model

We investigate some connections between the continuum and atomistic descriptions of de- formable crystals, using some interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. When the crystal shape is a lattice polygon, we show that the energy equals the bulk elastic energy, plus the boundary integral of a surface energy density, plus the sum over the vertices of a corner energy function. This is an exact result when the interatomic potential has finite range; for infinite-range potentials it is asymptotically valid as the lattice parameter zero. The surface energy density is obtained explicitly as a function of the deformation gradient and boundary normal. The corner energy is found as an explicit function of the deformation gradient and the normals of the two facets meeting at the corner. For more general convex domains with possibly curved boundary, the surface energy density depends on the unit normal in a striking way. It is continuous at irrational directions, discontinuous at rational ones and nowhere differ- entiable. This pathology is alarming since it renders the surface energy minimization problem (under domain variations) ill-posed. An alternative approach of defining the continuum region is introduced, that restores continuity of the surface energy density function

Continuum Surface Energy from a Lattice Model

We investigate connections between the continuum and atomistic descriptions of deformable crystals, using certain interesting results from number theory. The energy of a deformed crystal is calculated in the context of a lattice model with general binary interactions in two dimensions. A new bond counting approach is used, which reduces the problem to the lattice point problem of number theory. The main contribution is an explicit formula for the surface energy density as a function of the deformation gradient and boundary normal. The result is valid for a large class of domains, including faceted (polygonal) shapes and regions with piecewise smooth boundaries.Comment: V. 1: 10 pages, no fig's. V 2: 23 pages, no figures. Misprints corrected. Section 3 added, (new results). Intro expanded, refs added.V 3: 26 pages. Abstract changed. Section 2 split into 2. Section (4) added material. V 4, 28 pages, Intro rewritten. Changes in Sec.5 (presentation only). Refs added.V 5,intro changed V.6 address reviewer's comment

Electronic structure near an impurity and terrace on the surface of a 3-dimensional topological insulator

Motivated by recent scanning tunneling microscopy experiments on surfaces of Bi$_{1-x}$Sb$_{x'}$\cite{yazdanistm,gomesstm} and Bi$_2$Te$_3$,\cite{kaptunikstm,xuestm} we theoretically study the electronic structure of a 3-dimensional (3D) topological insulator in the presence of a local impurity or a domain wall on its surface using a 3D lattice model. While the local density of states (LDOS) oscillates significantly in space at energies above the bulk gap, the oscillation due to the in-gap surface Dirac fermions are very weak. The extracted modulation wave number as a function of energy satisfies the Dirac dispersion for in-gap energies and follows the border of the bulk continuum above the bulk gap. We have also examined analytically the effects of the defects by using a pure Dirac fermion model for the surface states and found that the LDOS decays asymptotically faster at least by a factor of 1/r than that in normal metals, consistent with the results obtained from our lattice model.Comment: 7 pages, 5 figure

Applicability Of Continuum Fracture Mechanics In Atomistic Systems

By quantitating the amplitude of the unbounded stress, the continuum fracture mechanics defines the stress intensity factor K to characterize the stress and displacement fields in the vicinity of the crack tip, thereby developing the relation between the stress singularity and surface energy (energy release rate G). This G-K relation, assigning physical meaning to the stress intensity factor, makes these two fracture parameters widely used in predicting the onset of crack propagation. However, due to the discrete nature of the atomistic structures without stress singularity, there might be discrepancy between the failure prediction and the reality of nanostructured materials. Defining the local atomistic stress with convergence within one lattice ensures the near-tip stress in the discrete systems displays the detailed stress concentration. Through comparison to the equivalent continuum finite element models, although these atomistic near-tip stress distributions preserve the trend of inverse square root singularity, the corresponding fracture toughness in terms of critical stress intensity factor (or energy release rate) is size dependent (i.e., varying with the size of the singular stress zone, K-dominance zone). Consequently, the failure load predicted by constant fracture toughness deviates from what a nanostructure can sustain if the singular stress is not dominant. The two-parameter model, including the contributions from both singular and non-singular terms, is utilized to improve the inadequacy of continuum fracture mechanics. On the other hand, since the magnitude of the atomistic near-tip stress is finite, the maximum stress criterion is valid in atomistic systems, proven by the close match between the peak stress and the theoretic strength under the failure condition. Furthermore, the surface energy determined by the overall energy balance over the crack growth within several sizes of lattice constant is shown to be size-independent, in contrast to the size-dependent results obtained by the G-K relatio

Lattice Model of an Ionic Liquid at an Electrified Interface

We study ionic liquids interacting with electrified interfaces. The ionic fluid is modeled as a Coulomb lattice gas. We compare the ionic density profiles calculated using a popular modified Poisson-Boltzmann equation with the explicit Monte Carlo simulations. The modified Poisson-Boltzmann theory fails to capture the structural features of the double layer and is also unable to correctly predict the ionic density at the electrified interface. The lattice Monte Carlo simulations qualitatively capture the coarse-grained structure of the double layer in the continuum. We propose a convolution relation that semiquantitatively relates the ionic density profiles of a continuum ionic liquid and its lattice counterpart near an electrified interface