Thermodynamic Limit of the Transition Rate of a Crystalline Defect
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Thermodynamic Limit of the Transition Rate of a Crystalline Defect Julian Braun , Manh Hong Duong & Christoph Ortner Communicated by F. Otto
Abstract We consider an isolated point defect embedded in a homogeneous crystalline solid. We show that, in the harmonic approximation, a periodic supercell approximation of the formation free energy as well as of the transition rate between two stable configurations converge as the cell size tends to infinity. We characterise the limits and establish sharp convergence rates. Both cases can be reduced to a careful renormalisation analysis of the vibrational entropy difference, which is achieved by identifying an underlying spatial decomposition of the entropy.
1. Introduction The presence of defects in crystalline materials significantly affects their mechanical and chemical properties, hence determining defect geometry, energies, and mobility is a fundamental problem of materials modelling. The inherent discrete nature of defects requires that any “ab initio” theory should start from an atomistic description. The purpose of the present work is to extend the model of crystalline defects of [15] (cf. Section 2) to incorporate vibrational entropy, in order to describe the thermodynamic limit of transition rates (mobility) of point defects. As an intermediate step we will also discuss the thermodynamic limit of defect formation free energy. Apart from being interesting in their own right, our results provide the analytical foundations for a rigorous derivation of coarse-grained models [5,19,33,36], and of numerical and multi-scale models at finite temperature [3,4,22,32,33] which JB and CO are supported by ERC Starting Grant 335120 and by EPSRC Grant EP/R043612/1. MHD was supported by ERC Starting Grant 335120
J. Braun, M. H. Duong & C. Ortner
entirely lack the solid foundations that static zero-temperature multi-scale schemes enjoy [23,24,26]. Precise definitions will be given in Section 2 but, for the purpose of a purely formal motivation, we consider a crystalline solid with an embedded defect described by an energy landscape E N : (Rm )Λ N → R, based on a set of reference atoms Λ N ⊂ Rd . We then consider a local minimizer u¯ min N of E N representing a defect state. In transition state theory (TST) [16,39], the transition rate K N from u¯ min N to 2 is given by comparing the equilibrium density, given by the a nearby state u¯ min N Boltzmann probability distribution, in a basin A ⊂ (Rm )Λ N around u¯ min N to the density on a hyper-surface S ⊂ (Rm )Λ N separating A from a similar basin around 2 . That is, u¯ min N −β E (u) N e du TST K N = S −β E (u) , N du Ae with inverse temperature β. The transition state is an index-1 saddle point u¯ saddle ∈ N most likely transition path between the two minima. S of E N representing the . Similarly, For sufficiently large β, S e−β E N (u) du is concentrated close to u¯ saddle N −β E (u) min N du is concentrated around the local minimum u¯ N . Therefore, it is Ae reasonable to consider the harmonic approximations ) + 21
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