Bending of a Bimetallic Beam Due to the Kirkendall Effect

Boettinger, William J.; McFadden, Geoffrey B.

doi:10.1007/s11669-009-9609-8

Cited by 10 publications

(7 citation statements)

References 7 publications

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“…Because of the accommodation/supply of excess/depleted vacancies, the solid locally expands/shrinks [54,55,56,57,58] when maintaining the vacancy mole fraction at its thermal-equilibrium value. We treat the solid as a very viscous fluid [59,60,61,62,63] with a much larger viscosity than that of the surrounding environment. In this case, we solve the Navier-Stokes-Cahn-Hilliard equations to update the shape of the material as follows [64]:…”

Section: Kirkendall-effect-induced Deformation Modeled By Navier-stok...mentioning

confidence: 99%

Extended smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries

Chen

Thornton

2012

Modelling Simul. Mater. Sci. Eng.

109

113

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In this article, we describe an approach for solving partial differential equations with general boundary conditions imposed on arbitrarily shaped boundaries. A continuous function, the domain parameter, is used to modify the original differential equations such that the equations are solved in the region where a domain parameter takes a specified value while boundary conditions are imposed on the region where the value of the domain parameter varies smoothly across a short distance. The mathematical derivations are straightforward and generically applicable to a wide variety of partial differential equations. To demonstrate the general applicability of the approach, we provide four examples herein: (1) the diffusion equation with both Neumann and Dirichlet boundary conditions; (2) the diffusion equation with both surface diffusion and reaction; (3) the mechanical equilibrium equation; and (4) the equation for phase transformation with the presence of additional boundaries. The solutions for several of these cases are validated against corresponding analytical and semi-analytical solutions. The potential of the approach is demonstrated with five applications: surface-reaction-diffusion kinetics with a complex geometry, Kirkendall-effect-induced deformation, thermal stress in a complex geometry, phase transformations affected by substrate surfaces, and a self-propelled droplet.

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Section: Kirkendall-effect-induced Deformation Modeled By Navier-stok...mentioning

confidence: 99%

Extended smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries

Chen

Thornton

2012

Modelling Simul. Mater. Sci. Eng.

109

113

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“…is determined using an error function based similarity solution (see [27]) where K is the solution to…”

Section: Dimensionless Equations and Timescalesmentioning

confidence: 99%

Modeling the early stages of reactive wetting

2010

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Recent experimental studies of molten metal droplets wetting high temperature reactive substrates have established that the majority of triple-line motion occurs when inertial effects are dominant. In light of these studies, this paper investigates wetting and spreading on reactive substrates when inertial effects are dominant using a thermodynamically derived, diffuse interface model of a binary, three-phase material. The liquid-vapor transition is modeled using a van der Waals diffuse interface approach, while the solid-fluid transition is modeled using a phase field approach. The results from the simulations demonstrate an O t − 1 /2 spreading rate during the inertial regime and oscillations in the triple-line position when the metal droplet transitions from inertial to diffusive spreading. It is found that the spreading extent is reduced by enhancing dissolution by manipulating the initial liquid composition. The results from the model exhibit good qualitative and quantitative agreement with a number of recent experimental studies of hightemperature droplet spreading, particularly experiments of copper droplets spreading on silicon substrates. Analysis of the numerical data from the model suggests that the extent and rate of spreading is regulated by the spreading coefficient calculated from a force balance based on a plausible definition of the instantaneous interface energies. A number of contemporary publications have discussed the likely dissipation mechanism in spreading droplets. Thus, we examine the dissipation mechanism using the entropy-production field and determine that dissipation primarily occurs in the locality of the triple-line region during the inertial stage, but extends along the solid-liquid interface region during the diffusive stage. * Electronic address: daniel.wheeler@nist.gov 1 arXiv:1006.4881v1 [cond-mat.mtrl-sci]

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“…For example, the imbalance of atomic diffusion fluxes during the Kirkendall effect results in net vacancy flux and concomitant lattice drift and shape changes due to the climb of edge dislocations [25][26][27][28]. The scarcity of internal vacancy sinks may result in the generation of high internal elastic stresses and viscous material flow [25,29]. In fact, in the past decade Kirkendall effect has been widely utilized in the synthesis of hollow metallic nanostructures [30][31][32].…”

Section: Mainmentioning

confidence: 99%

“… 25 − 28 The scarcity of internal vacancy sinks may result in the generation of high internal elastic stresses and viscous material flow. 25 , 29 In fact, in the past decade, Kirkendall effect has been widely utilized in the synthesis of hollow metallic nanostructures. 30 − 32 It is also worth noting that the chemical diffusion-induced stresses in nanostructured electrodes of rechargeable Li-ion batteries represent a formidable technological problem.…”

mentioning

confidence: 99%

Plastic Forming of Metals at the Nanoscale: Interdiffusion-Induced Bending of Bimetallic Nanowhiskers

Richter

Suadiye

et al. 2020

ACS Nano

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Controlled plastic forming of nanoscale metallic objects by applying mechanical load is a challenge, since defect-free nanocrystals usually yield at near theoretical shear strength, followed by stochastic dislocation avalanches that lead to catastrophic failure or irregular, uncontrolled shapes. Herein, instead of mechanical load, we utilize chemical stress from imbalanced interdiffusion to manipulate the shape of nanowhiskers. Bimetallic Au–Fe nanowhiskers with an ultrahigh bending strength were synthesized employing the molecular beam epitaxy technique. The one-sided Fe coating on the defect-free, single-crystalline Au nanowhisker exhibited both single- and polycrystalline regions. Annealing the bimetallic nanowhiskers at elevated temperatures led to gradual change of curvature and irreversible bending. At low homological temperatures at which grain boundary diffusion is a dominant mode of mass transport this irreversible bending was attributed to the grain boundary Kirkendall effect during the diffusion of Au along the grain boundaries in the Fe layer. At higher temperatures and longer annealing times, the bending was dominated by intensive bulk diffusion of Fe into the Au nanowhisker, accompanied by a significant migration of the Au–Fe interphase boundary toward the Fe layers. The irreversible bending was caused by the concentration dependence of the lattice parameter of the Au(Fe) alloy and by the volume effect associated with the interphase boundary migration. The results of this study demonstrate a high potential of chemical interdiffusion in the controlled plastic forming of ultrastrong metal nanostructures. By design of the thickness, microstructure, and composition of the coating as well as the parameters of heat treatment, bimetallic nanowhiskers can be bent in a controlled manner.

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Bending of a Bimetallic Beam Due to the Kirkendall Effect

Cited by 10 publications

References 7 publications

Extended smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries

Extended smoothed boundary method for solving partial differential equations with general boundary conditions on complex boundaries

Modeling the early stages of reactive wetting

Plastic Forming of Metals at the Nanoscale: Interdiffusion-Induced Bending of Bimetallic Nanowhiskers

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