Morphological instabilities in thin films: Evolution maps Mohsen Asle Zaeem a,⇑ , Sinisa Dj. Mesarovic b a Center for Advanced Vehicular Systems, Mississippi State University, Starkville, MS 39759, USA b School of Mechanical and Materials Engineering, Washington State University, Pullman, WA 99164, USA article info Article history: Received 30 September 2010 Received in revised form 27 October 2010 Accepted 29 October 2010 Available online 20 November 2010 Keywords: Phase-field model Elasticity Multilayer thin films Instability abstract We consider morphological instabilities in binary multilayers and the post-instability evolution of the system. The alloys with and without intermediate phase are considered, as well as the cases with stable and meta-stable intermediate phase. Using the Galerkin finite element formulation for coupled Cahn–Hilliard – elasticity problem, maps of different evolution paths are developed in the parameter space of relative thicknesses of initial phases. We consider the relative importance of elastic and chemical energy of the system and develop maps for different cases. The systems exhibit rich evolution behavior. Depending on the initial configuration (which determines the mass conservation condition), the final equilibrium varies, but even greater variety is observed in evolution paths. The paths may consist of multiple evolution steps, which may proceed at different rates. Except for few special circumstances, the instabilities are to perturbations non-homogeneous in the film plane. Post-instability evolution is essentially two-dimensional, and cannot be reduced to the one- dimensional model. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction Multilayer thin films have been a very active research area in the past decade. When the thickness of an individual layer is re- duced to micro- or nano-scale, novel mechanical [1–5], optical [6], electronic [7,8], and magnetic [6–8] properties emerge, which make such materials attractive for a number of potential applica- tions. However, to maintain the desirable properties, the stability of phases is required, and this is affected by temperature and stress. In addition to externally applied load, internal stresses ap- pear as a result of enforced lattice continuity and compositional strain [9–14]. For example – a significant drop of hardness occurs after annealing of Ni/Ru multilayers at 600 °C [15], Ni layers break- down in Ni/Ag multilayer is annealed at 600 °C [16], and pinch-off of Co layers in Co/Cu multilayers is observed after creep test at 830 °C [17]. Greer [18,19] classified the changes that can occur in thin film multilayers as their microstructures evolve. He divided these pro- cesses to the following categories: inter-diffusion (same phases coexist with changed compositions) (Ag/Au [20], Ni/Al, Ag/Zn [21], Al/Ni [22]), interfacial reaction (nucleation and growth of a new phase) (Si/Ni [23], Ni/Zr [24], Al/Mn [25], Ni/Al [26]), transfor- mation in one phase of the multilayer without any changes in other phase(s) (Si/Al [27]), and, coarsening or spinodal decomposition of the layers (Ni/Ag [15], Ni/C [28]). All of these can be modeled using the diffuse interface (phase-field) model. Following the initial Cahn–Hilliard [29] formulation of the dif- fuse interface model, many numerical phase-field studies [30–43] have been reported. Very few considered thin films, and these are either based one-dimensional models [39,40] with severe restrictions on possible instabilities, or, are focused on film-fluid interactions [41–43]. The exception is the recent work by Chir- ranjeevi et al. [14]. They consider a special two-phase multilayer with one phase much stiffer than the other. Here, we present a comprehensive study of binary multilayer thin films, including two-phase systems and systems with an inter- mediate phase (meta-stable or stable). Following the dimensional analysis, we develop the maps of the evolution behavior of multi- layers, in the parameter space describing the initial geometry, and considering the cases of different relative importance of elastic and chemical energy density. In addition to having different (final) sta- ble configurations, multilayered thin films may reach those stable configurations following different paths in configurational space. We classify the paths and map the initial configuration space cor- responding to each path. We use the recently developed Galerkin finite element formula- tion [44,45] for the Cahn–Hilliard diffuse interface (phase-field) model [29], coupled with elasticity [40]. The paper is organized as follows. In Section 2, formulation, dimensional analysis, and numerical solver for the coupled phase filed-elasticity equations are presented. In Section 3, the maps of 0927-0256/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2010.10.043 ⇑ Corresponding author. Tel.: +1 662 325 0126. E-mail address: mohsen@cavs.msstate.edu (M.A. Zaeem). Computational Materials Science 50 (2011) 1030–1036 Contents lists available at ScienceDirect Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci