Theodore Kaplan

Continuous growth model for interface motion during alloy solidification

Abstract A model is developed that predicts the steady state velocity of a planar interface and the chemical composition of the growing phase in terms of the interface temperature and the composition of the parent phase at the interface. The model is applied to solidification of a two-component melt. Solute partitioning is treated by a previously developed continuous growth model for solute trapping. The interface velocity is found by generalizing the driving force in a velocity-vs-driving force function used for solidification of one-component melts. Two different ways of generalizing the driving force are used, with and without the inclusion of a “solute drag” term. Predictions are made both with and without solute drag for an ideal solution and for AgCu, a simple eutectic system in which the terminal phases have the same crystal structure. In both cases, a transition from diffusion-controlled to diffusionless solidification and a falling interface temperature occur as the interface velocity increases. In the model without solute drag, significantly less interfacial undercooling is predicted than in the model with solute drag. The relationship to previous theoretical work, especially to the continuum treatments of Baker and Cahn, and to pertinent experiments is discussed.

Green’s Functions and Boundary Integral Analysis for Exponentially Graded Materials: Heat Conduction

Free space Green s functions are derived for graded materials in which the thermal conductivity varies exponentially in one coordinate. Closed-form expressions are obtained for the steady-state diffusion equation, in two and three dimensions. The corresponding boundary integral equation formulations for these problems are derived, and the three-dimensional case is solved numerically using a Galerkin approximation. The results of test calculations are in excellent agreement with exact solutions and finite element simulations.

Green's function for a two–dimensional exponentially graded elastic medium

The free–space Green function for a two–dimensional exponentially graded elastic medium is derived. The shear modulus Âμ is assumed to be an exponential function of the Cartesian coordinates (x,y), i.e. μ ≡ μ(x,y) = μ0e2(β1x+β2y), where μ0, β1, and β2 are material constants, and the Poisson ratio is assumed constant. The Green function is shown to consist of a singular part, involving modified Bessel functions, and a non–singular term. The non–singular component is expressed in terms of one–dimensional Fourier–type integrals that can be computed by the fast Fourier transform.

Direct Evaluation of Hypersingular Galerkin Surface Integrals

A direct algorithm for evaluating hypersingular integrals arising in a three-dimensional Galerkin boundary integral analysis is presented. The singular integrals are defined as limits to the boundary, and by integrating two of the four dimensions analytically, the coincident integral is shown to be divergent. However, the divergent terms can be explicitly calculated and shown to cancel with corresponding singularities in the adjacent edge integrals. A single analytic integration is employed for the edge and vertex singular integrals. This is sufficient to display the divergent term in the edge-adjacent integral and to show that the vertex integral is finite. By explicitly identifying the divergent quantities, we can compute the hypersingular integral without recourse to Stokess theorem or the Hadamard finite part. The algorithms are developed in the context of a linear element approximation for the Laplace equation but are expected to be generally applicable. As an example, the algorithms are applied to solve a thermal problem in an exponentially graded material.

Improved quarter-point crack tip element

We present a modification to the quarter-point crack tip element and employ this element in two-dimensional boundary integral fracture analysis. The standard singular element is adjusted so that the near-tip crack opening displacement satisfies a known constraint: the coefficient of the term which is linear in the distance to the tip must vanish. Stress intensity factors calculated with the displacement correlation technique are shown to be highly accurate, and significantly more accurate than with the standard element. The improvements are especially dramatic for mixedmode problems involving curved and interacting cracks. 2002 Published by Elsevier Science Ltd.

Evidence for a new class of solids. First-principles study of K(Al13)

Abstract The stability of the crystalline phase of a cluster-assembled solid K(Al 13 ) has been investigated using first-principles total energy calculations. We find that K(Al 13 ) may form in the CsCl structure with a lattice constant of 6.52 A. Unlike the gas phase, in which the ground state of the Al 13 cluster is icosahedral, the Al 13 becomes cuboctahedral in the solid phase due to crystal field effects. The system is metallic and is stable against lattice distortions. The calculations suggest that a new metastable solid could be made from two immiscible elements through specially designed synthesis processes.

Molecular dynamics simulations of crack propagation in Ni with defects

A series of molecular dynamics simulations using the embedded atom method is conducted to investigate crack propagation under mode I loading in a Ni single crystal with and without defects. The crack system (0 0 1)[1 0 0] in a slab of 160 000 atoms was studied. Defects consisting of lines of vacancies were introduced near the crack tip. Critical loads and strain energy distributions around the crack tip are obtained. Our results indicate that the critical strain necessary for crack propagation is dependent on the defect configuration and can either increase or decrease relative to the defect-free system.

Surface strains in epitaxial systems

The stress state of heteroepitaxial film systems is examined using a boundary integral method together with boundary conditions that allow deflections at the substrate/film interface. It is found that for geometries that deviate from planar structures significant variations in surface strain and film energy arise. These calculations explain recent important experimental results for Ge growth on Si, including observations for Ge islands on Si that show the surface lattice constant can exceed the bulk Ge value, and observations that the preferred region for growth on terraced films is not necessarily at the steps.

Temperature dependence of the elastic constants of Ni: reliability of EAM in predicting thermal properties

The temperature dependence of the elastic constants of Ni is calculated using molecular dynamics (MD) simulations in conjunction with the embedded atom method (EAM). The Parrinello - Rahman version of molecular dynamics is employed along with the fluctuation formulae in the and EhN ensembles at various temperatures from 0 K to somewhat below the melting point (experimental value 1725 K). The calculated results for the elastic constants, compressibility, linear coefficient of thermal expansion, specific heat and the melting temperature compare reasonably well to experiment.

AC response of fractal interfaces

Abstract The small signal ac impedance of the interface between a blocking electrode and a aqueous or solid electrolyte often contains a constant-phase-angle (CPA) element whose impedance has the frequency dependence according to Z ∝ (jω)−ν. Recently it has been demostrated experimentally that the exponent ν is intimately related to the roughness of the interface, with ν approaching 1 when the surface is made increasingly smooth. We propose to model the interface by a fractal called Cantor bar. The equivalent circuit of the model has the property of the CPA element with the exponent ν = 3 − ds, where 2