Surface-Induced Phase Transformations: Multiple Scale and Mechanics Effects and Morphological Transitions Valery I. Levitas 1 and Mahdi Javanbakht 2 1 Iowa State University, Departments of Aerospace Engineering, Mechanical Engineering, and Material Science and Engineering, Ames, Iowa 50011, USA 2 Iowa State University, Department of Mechanical Engineering, Ames, Iowa 50011, USA (Received 14 June 2011; published 19 October 2011) Strong, surprising, and multifaceted effects of the width of the external surface layer and internal stresses on surface-induced pretransformation and phase transformations (PTs) are revealed. Using our further developed phase-field approach, we found that above some critical , a morphological transition from fully transformed layer to lack of surface pretransformation occurs for any transformation strain " t . It corresponds to a sharp transition to the universal (independent of " t ), strongly increasing the master relationship of the critical thermodynamic driving force for PT X c on . For large " t , with increasing , X c unexpectedly decreases, oscillates, and then becomes independent of " t . Oscillations are caused by morphological transitions of fully transformed surface nanostructure. A similar approach can be developed for internal surfaces (grain boundaries) and for various types of PTs and chemical reactions. DOI: 10.1103/PhysRevLett.107.175701 PACS numbers: 64.60.Bd, 63.70.+h, 64.60.an Reduction in the total surface energy during phase trans- formation (PT) may lead to various surface-induced phe- nomena—e.g., surface premelting, ordering or disordering, martensitic PT, PT from martensitic variant M i to variant M j , and barrierless nucleation [1–3]. Thus, transformation may start from the surface from stable in bulk to metastable phases at temperature , which may be far from the ther- modynamic equilibrium temperature e between phases, namely, below e for melting and above e for martensitic PTs. While some of our results are applicable to most of the above PTs, we will focus on PTs during cooling, which include martensitic PTs. When the thermal driving force X ¼ð1 = e Þ=ð1 c = e Þ ( c is the temperature of the loss of stability of the parent phase) for martensitic PT increases and approaches zero, a few nanometers thick transformed layer appears, grows, and looses its thermody- namic stability, and transformation propagates through the entire sample. Phase-field or Ginzburg-Landau (GL) ap- proach is widely used for simulation of the surface-induced PTs [3–6]. PT in this approach is described in terms of evolution of a single or multiple order parameter(s). The martensitic PT below is described by n order parameters i that vary from 0 for austenite A to 1 for martensitic variant M i . Melting is described by the same potential for a single order parameter [6]. Significant advances were recently achieved in generalization for multivariant martensitic PTs, formulation of a noncontradictory expression for sur- face energy versus i , coupling to advanced mechanics, and consistent expression for interface tension [5,6]. Despite this progress, two major contradictions are present in the current GL approaches to surface-induced phenomena. (a) While the GL approach resolves finite width of interfaces that are responsible for PTs, the external surface is sharp, although its width is comparable to . (b) A sharp external surface also does not permit a correct introduction of surface tension using the method that we developed for the phase interfaces [5,6]. The goal of this paper is to introduce and study the effect of the finite width of an external surface coupled to mechanics with the help of our further developed GL approach. Thus, a surface (e.g., solid-gas) layer of the width is described by a solution of the GL equation for an additional order parameter . Obtained results (Figs. 1–6) revealed multiple unexpected effects of the surface layer and mechanics, including morphological transitions in the nanostructure, which drastically change our understanding and interpre- tation of the transformation behavior and results of mea- surements. Deformation of the crystal lattice of A into the lattice of M i is described by the transformation strain tensor " ti , which in our case is taken for cubic-tetragonal PT in NiAl. To elucidate the effect of internal stress gen- erated by " ti in different materials, we considered trans- formation strain k" ti with 0 k 1. With increasing X,a stationary nanostructure i ð Þ ( is the position vector) varies (Fig. 4). The critical surface nanostructure c ðÞ corresponds to the critical driving force X c above which the entire sample transforms. For neglected mechanics, two branches on the curve X c versus the dimensionless width of the surface layer ¼ = are obtained [Fig. 1(b)]. For 1, the effect of the surface layer is negligible and X c and c are the same as for the sharp surface. However, for some critical and quite small ¼ 0:166, the slope of the curve X c ð Þ has an unexpected jump and a drastic increase in the critical driving force occurs with increasing . Critical nano- structure undergoes a morphological transition at this point, from a homogeneous layer along the surface with PRL 107, 175701 (2011) PHYSICAL REVIEW LETTERS week ending 21 OCTOBER 2011 0031-9007= 11=107(17)=175701(5) 175701-1 Ó 2011 American Physical Society