arXiv:0706.3321v3 [cond-mat.mtrl-sci] 1 Mar 2008 Elastoplastic theory for the dynamics of solid-solid transformations : role of non-affine deformation in microstructure selection Jayee Bhattacharya 1 , Arya Paul 1 , Surajit Sengupta 1 1 Unit for Nano-Science and Technology, S.N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Calcutta 700 098, India Madan Rao 2,3 2 Raman Research Institute, C.V. Raman Avenue, Bangalore 560 080, India 3 National Centre for Biological Sciences (TIFR), Bellary Road, Bangalore 560 065, India We study the nucleation dynamics of a model solid state transformation and the criterion for microstructure selection using a coarse-grained molecular dynamics (MD) simulation. Our simu- lations show a range of microstructures depending on the depth of quench. We closely follow the dynamics of the solid and find that transient non-affine zones (NAZ) are created at and evolve with the rapidly moving transformation front. The dynamics of these plastic regions determines the selection of microstructure. We formulate an elastoplastic theory which couples the elastic strain to the non-affine deformation, and recover all the qualitative features of the MD simulation. Using this theory, we construct a dynamical phase diagram for microstructure selection, in addition to making definite testable predictions. PACS numbers: I. INTRODUCTION The dynamics following a quench across a solid state structural transition, rarely takes the solid to its equi- librium state[1]. Severe dynamical constraints experi- enced by the product inclusion within the parent crys- tal, determine the mode of nucleation and of subsequent growth. Often, solids get stuck in long-lived microstruc- tures, which depend on the depth of quench and cool- ing rate[2]. For example, transformations occurring at high temperatures are typically accompanied by large- scale rearrangements of atoms; in this case the elastic- ity of the solid plays only a minor role in determining microstructure[1, 2, 3]. On the other hand, at low tem- peratures, only local rearrangements of atoms are possi- ble; the resulting microstructures are largely determined by elasticity[1, 2]. These are just two of the myriad pos- sibilities explored by the transforming solid. Which of these is actually selected; in other words, can we con- struct a dynamical phase diagram? In a set of papers[4, 5, 6], we had explored these is- sues using an MD simulation[7] of a model system un- dergoing a two dimensional square to rhombic structural transformation. We found that when the transforma- tion proceeds at a high temperature, the resulting prod- uct nucleus is isotropic and polycrystalline, while a low transformation temperature induces the formation of an anisotropic nucleus, roughly elliptical, consisting of a pair of twin-related crystallites[6]. The two modes of nucle- ation may be denoted Ferrite and Martensite, borrowing terminology from the microstructure of steel[1]. By fol- lowing the nucleation dynamics in ‘microscopic’ detail, we had established that the ferrite nucleus is formed fol- lowing extensive rearrangements of atomic coordinates, large extent, preserved. This is consistent with the two paradigms commonly described in real materials. How- ever, these two limits are not mutually exclusive[6]; in- deed for intermediate temperatures, the transformation proceeds such that both mechanisms may operate at dif- ferent spatial and temporal locations[6], a feature ob- served in real materials[8]. Further, the different mi- crostructures (twinned and un-twinned) were obtained simply by tuning appropriate kinetic parameters[6], as observed in the heterogeneous nucleation of colloidal crystals[9]. These observations underline the need for a unified theory of microstructure selection describing the dynamics of nucleation of both the ferrite and martensite and the conditions in which these microstructures obtain. Our preliminary attempts at this unifying picture[6] were based on the recognition (from the MD simula- tion) of the role played by non-elastic variables, which we identified with local density fluctuations. We showed that the coupled dynamics of density fluctuations and elastic strain determined the microstructure of the grow- ing nucleus[6]. Here, we provide a more refined the- ory of solid-state nucleation and microstructure selec- tion. Based on our MD simulations, we formulate an elastoplastic theory which interpolates between ferrite and martensite as some relevant parameter is tuned. We discuss the dynamical origin of microstructure selection and exhibit a dynamical phase diagram[6] of the final microstructure. A. Average compatibility : geometrical versus strain-only theories Geometrically, a martensite results from the require-