A statistical mechanical theory for the origin of rigidity in crystalline solids

It is known that if local atomic rearrangements leading to exchange of neighbours are allowed, no crystalline solid can be rigid in the thermodynamic limit. The usual rigid, elastic, response to shape changes, a defining character of a crystalline solid, arises because the time needed for such rearrangements diverge as the magnitude of deformation approaches zero. Starting from these general considerations, we had shown that crystal rigidity arises as a consequence of a first order phase transition in an expanded parameter space [1]. The solid is subjected to not only elastic deformation but also to a fictitious external field, which penalises rearrangements, with the understanding that properties of real crystals are recovered by first taking the thermodynamic limit and then letting this field go to zero. Within this picture, we obtain a first order phase transition between a rigid crystal, N, and a crystal, M, where atoms rearrange to eliminate internal stress for any given deformation while maintaining crystalline order. The N-M phase boundary, in the thermodynamic limit, passes through the origin, where both field and deformation is zero. This picture gives us a fundamentally new viewpoint on the phenomenon of yielding, i.e. the loss of rigidity of a crystal when deformed beyond a limit, viz. the yield point. The phenomenon of yielding is now simply the nucleation of bubbles of the thermodynamically stable M phase within the metastable, rigid N crystal. An outcome of this theory is that the yield point is always a weak function of the rate of deformation and vanishes in the true quasistatic limit. The analytic form derived by us for the yield point as a function of the rate of deformation is able to explain experimental data over 15 orders of magnitudes in time [2]. We also discuss several details of yielding in Lennard Jones solids in 2 and 3 dimensions.
[1] P. Nath et al., Proc. Natl. Acad. Sci. USA 115, E4322 (2018).
[2] V. S. Reddy et al. arXiv:1908.08829.