Morphological instabilities in thin films: Evolution maps
Research highlights
► Maps of transformation and evolution of binary multilayer thin films are developed using Cahn-Hilliard-elasticity model. ► Effects of the growth of a meta-stable/stable intermediate phase and compositional strain on the instability of layers are investigated. ► Mixed order FEM is used to solve the governing equations of Cahn-Hilliard-elasticity problem. ► Instabilities are to perturbations non-homogeneous in the film plane. ► Post-instability evolution is essentially 2D and cannot be captured by 1D models.
Introduction
Multilayer thin films have been a very active research area in the past decade. When the thickness of an individual layer is reduced to micro- or nano-scale, novel mechanical [1], [2], [3], [4], [5], optical [6], electronic [7], [8], and magnetic [6], [7], [8] properties emerge, which make such materials attractive for a number of potential applications. 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 appear as a result of enforced lattice continuity and compositional strain [9], [10], [11], [12], [13], [14]. For example – a significant drop of hardness occurs after annealing of Ni/Ru multilayers at 600 °C [15], Ni layers breakdown 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 processes 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]), transformation 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 diffuse interface model, many numerical phase-field studies [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [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], [42], [43]. The exception is the recent work by Chirranjeevi 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 intermediate phase (meta-stable or stable). Following the dimensional analysis, we develop the maps of the evolution behavior of multilayers, 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) stable 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 corresponding to each path.
We use the recently developed Galerkin finite element formulation [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 transformations for two-phase systems are developed. In Sections 4 Systems with stable intermediate phase, 5 Systems with meta-stable intermediate phase, similar maps are developed for systems with stable and meta-stable intermediate phase. Discussion and conclusions are given in Section 6.
Section snippets
Formulation and numerical method
Concentration of a component in a binary system is subjected to a conservation law. When used a phase-field variable, it results in a nonlinear 4th order diffusion partial differential equation (PDE), which is then coupled to the 2nd order elasticity PDEs. Consider a binary system, with components A and B, which forms up to three phases (α, β, and γ), and characterized by the molar fraction of the component B, c. The total free energy, F, of a non-homogeneous system with volume V, bounded by
Two-phase systems
We first consider binary multilayers without an intermediate phase. Denote the initial thicknesses of α and γ layers by dα and dγ (Fig. 2). In the 3-layers configuration shown in Fig. 2, the central layer breaks up into particles for . For the purpose of mapping, we will denote this type of behavior as type I. For the breakup is followed by full homogenization at concentration equal to the average composition of the system. This is of course possible only if the γ-layer is thin
Systems with stable intermediate phase
We now consider binary multilayer systems with the stable intermediate phase, β. An example is shown in Fig. 7 (type IV). First, a fast growth of the intermediate phase takes place: the whole γ-phase and a part of α-phase are consumed. Then, slow thinning and fragmentation the central α-layer follows until it disappears. A thick β-layer remains in the final equilibrium configuration.
The maps of transformations for 5-layers system (such as the one shown in Fig. 7) with stable intermediate phase,
Systems with meta-stable intermediate phase
A meta-stable intermediate phase produces a more complex behavior than the stable one. The evolution is often characterized by partial growth of the intermediate phase, which is arrested before the other phase is consumed. Moreover, the intermediate phase itself can subsequently be consumed by another phase. Consequently, the final configurations and paths are more complex than in the case of a stable intermediate phase.
An example of evolution of systems with meta-stable intermediate phase is
Conclusions and discussion
The morphological instabilities and post-instability evolution of binary multilayers, with and without intermediate phase (stable and meta-stable), is considered. Using the Galerkin finite element formulation [44], [45] for coupled Cahn–Hilliard – elasticity problem, maps of different evolution paths are developed in the parameter space of relative thicknesses of initial phases.
Dimensional analysis reveals that, for isotropic materials, governing equations depend on four non-dimensional
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