Predicting the dislocation nucleation rate as a function of temperature and stress
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Keonwook Kang Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Maxico 87545
Wei Caia) Department of Mechanical Engineering, Stanford University, Stanford, California 94305 (Received 16 May 2011; accepted 5 August 2011)
Predicting the dislocation nucleation rate as a function of temperature and stress is crucial for understanding the plastic deformation of nanoscale crystalline materials. However, the limited time scale of molecular dynamics simulations makes it very difficult to predict the dislocation nucleation rate at experimentally relevant conditions. We recently develop an approach to predict the dislocation nucleation rate based on the Becker–Döring theory of nucleation and umbrella sampling simulations. The results reveal very large activation entropies, which originated from the anharmonic effects, that can alter the nucleation rate by many orders of magnitude. Here we discuss the thermodynamics and algorithms underlying these calculations in greater detail. In particular, we prove that the activation Helmholtz free energy equals the activation Gibbs free energy in the thermodynamic limit and explain the large difference in the activation entropies in the constant stress and constant strain ensembles. We also discuss the origin of the large activation entropies for dislocation nucleation, along with previous theoretical estimates of the activation entropy.
I. INTRODUCTION
Dislocation nucleation is essential to the understanding of ductility and plastic deformation of crystalline materials with submicrometer dimensions1–3 or under nanoindentation.4–6 It is and also essential to the synthesis of high quality thin films for microelectronic, optical, and magnetic applications.7,8 The fundamental quantity of interest is the dislocation nucleation rate I as a function of stress r and temperature T. Continuum9–11 and atomistic models12–14 have been used to predict the dislocation nucleation rate, and they both have limitations. The applicability of continuum models may be questionable because the size of the critical dislocation nucleus can be as small as a few lattice spacings. In addition, the continuum models are often based on linear elasticity theory, whereas dislocation nucleation typically occurs at high strain conditions in which the stress–strain relation becomes nonlinear. These difficulties do not arise in molecular dynamics (MD) simulations, which can reveal important mechanistic details of dislocation nucleation. Unfortunately, the time step of MD simulations is on the a)
Address all correspondence to this author. e-mail: [email protected] This paper has been selected as an Invited Feature Paper. DOI: 10.1557/jmr.2011.275 J. Mater. Res., Vol. 26, No. 18, Sep 28, 2011
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order of a femtosecond, so that the time scale of MD simulations is typically on the order of a nanosecond, given existing computational resources. Therefore, the study of dislocation nucleation via direct MD simulation has been limited to extremely high st
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