A classical equation that accounts for observations of non-Arrhenius and cryogenic grain boundary migration

www.nature.com/npjcompumats ARTICLE OPEN A classical equation that accounts for observations of non-Arrhenius and cryogenic grain boundary migration Eric R. Homer 1✉ , Oliver K. Johnson 1 , Darcey Britton1, James E. Patterson2, Eric T. Sevy2 and Gregory B. Thompson 3 Observations of microstructural coarsening at cryogenic temperatures, as well as numerous simulations of grain boundary motion that show faster migration at low temperature than at high temperature, have been troubling because they do not follow the expected Arrhenius behavior. This work demonstrates that classical equations, that are not simplified, account for all these oddities and demonstrate that non-Arrhenius behavior can emerge from thermally activated processes. According to this classical model, this occurs when the intrinsic barrier energies of the processes become small, allowing activation at cryogenic temperatures. Additional thermal energy then allows the low energy process to proceed in reverse, so increasing temperature only serves to frustrate the forward motion. This classical form is shown to reconcile and describe a variety of diverse grain boundary migration observations. 1234567890():,; npj Computational Materials (2022)8:157 ; https://doi.org/10.1038/s41524-022-00835-2 INTRODUCTION Migration of interfaces plays an important role in the final microstructure and resulting properties of crystalline materials. Decades of experiments and theory demonstrate that these processes are thermally activated and follow an Arrhenius relationship. However, in the last two decades experiments have demonstrated that structure evolution can be observed under cryogenic conditions1–5. A traditional thermally activated model of interface migration does not support these observations because the coarsening is more dramatic at cryogenic temperatures than at room temperature, as shown by Zhang et al.1,2. Recent molecular dynamics simulations of grain boundary migration revealed that select grain boundary types exhibit non-Arrhenius behavior, where the boundaries are mobile at cryogenic temperatures and migration rates frequently decrease with increasing temperature6–15. This inverse temperature dependence is evident in other material processes and across all classes of materials16–22. Processing recipes that exploit such antithermal behavior could enable new avenues for microstructure control and lead to major advances in catalysis, nanocrystalline alloys, and high efficiency engines23. In this work we will show that while these non-Arrhenius observations are inconsistent with the traditional model used to describe grain boundary migration, they are entirely consistent with a forgotten classical model that accounts for both Arrhenius and non-Arrhenius observations of grain boundary migration. Furthermore, we will illustrate that the reason the traditional model fails in these cases is that it is a simplification of the classical model for which the assumptions are not always satisfied. We will then detail several observations from the last two decades that are now reconciled by the classical model, but which could not adequately be explained with the traditional model. Finally, the manuscript concludes with a discussion of factors to keep in mind when applying the classical model, as well as a discussion of other models that also predict non-Arrhenius behaviors but not cryogenic motion. In the traditional model of grain boundary migration the velocity of a grain boundary v is proportional to an applied driving force p used to induce grain boundary migration according to v ¼ Mp (1) where M is the grain boundary mobility, which is the kinetic property most frequently used to describe grain boundary migration. In the traditional model, the mobility M follows an Arrhenius temperature dependence   Q M ¼ Mo exp  (2) kBT where Mo is a pre-exponential constant and Q is an intrinsic barrier for the migration of the grain boundary. When mobility measurements follow an Arrhenius temperature dependence, estimates for Mo and Q can be obtained. To define the classical model, we estimate the velocity of a grain boundary by considering a combination of forward and backward atomic jumps that are thermally activated24–26. Figure 1a illustrates the potential energy landscape for an atom that can jump from grain 1 to grain 2 to allow the boundary to migrate forward (or from grain 2 to grain 1 leading to backward migration). The two states are separated by an intrinsic barrier of magnitude Q, which can be biased by a driving force p. The activation energy barrier for the two jumps is then defined as Ea = Q ± p/2. When p is small relative to Q, Ea and Q are nearly identical, but when p is large relative to Q, one must take care in distinguishing between the activation energy Ea and the intrinsic barrier height Q. In microstructure evolution, a number of different forces p can drive migration, including reduction in grain boundary area, difference in elastic energy across the boundary, stored strain energy from cold work, etc.26,27. In considering a combination of forward and backward jumps of N atoms at the grain boundary, it can be shown that the velocity is approximated by     Q p 2 sinh (3) v ¼ Nbν exp  kB T 2k B T 1 Department of Mechanical Engineering, Brigham Young University, Provo, UT 84602, USA. 2Department of Chemistry and Biochemistry, Brigham Young University, Provo, UT 84602, USA. 3Department of Metallurgical and Materials Engineering, University of Alabama, Tuscaloosa, AL 35401, USA. ✉email: Published in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Sciences E.R. Homer et al. 2 1234567890():,; Fig. 1 Examination of classical model. a Potential energy landscape with intrinsic barrier Q and driving force bias p. b Plot of Eq. (3) for values of Q equal to 0.03, 0.1, and 1 eV. c Arrhenius plot (log(v) vs. 1/T) of Eq. (3) overlaid with lines of velocity predictions based on traditional expectations following Eqs. (1) and (2). Note the large temperature range for the inset, which highlights the Arrhenius nature of curves with low values of Q at the very lowest temperature, but which obscures the non-Arrhenius nature of the same curves at high temperature, which is highlighted in the main plot. d Contour plot of Eq. (5) for different combinations of p and Q. where N is the number of atoms involved in the migration at the boundary, b is the distance of a single atom jump, ν is the attempt frequency, kB is Boltzmann’s constant, and T is absolute temperature. Note that Eq. (3) is a variation of a form in ref. 26; a more complete description of this derivation, as well as variations thereof, is provided in the Supplementary Discussion, including Supplementary Figs. 1–3. The Supplementary Discussion also examines a similar model by Ivanov and Mishin where the forward and backward activation energies Ea are not symmetric when p/2 approaches the magnitude of Q25; however (...truncated)