Stable Equilibria of Anisotropic Particles on Substrates: A Generalized Winterbottom Construction

Abstract

We present a new approach for predicting stable equilibrium shapes of two-dimensional crystalline islands on flat substrates, as commonly occur through solid-state dewetting of thin films. The new theory is a generalization of the widely used Winterbottom construction (i.e., an extension of the Wulff construction for particles on substrates). This approach is equally applicable to cases where the crystal surface energy is isotropic, weakly anisotropic, strongly anisotropic and "cusped". We demonstrate that, unlike in the classical Winterbottom approach, multiple equilibrium island shapes may be possible when the surface energy is strongly anisotropic. We analyze these shapes through perturbation analysis, by calculating the first and second variations of the total free energy functional with respect to contact locations and island shape. Based on this analysis, we find the necessary conditions for the equilibria to be stable to two-dimensional perturbations and exploit this through a generalization of the Winterbottom construction to identify all possible stable equilibrium shapes. Finally, we propose a dynamical evolution method based on surface diffusion mass transport to determine whether all of the stable equilibrium shapes are dynamically accessible. Applying this approach, we demonstrate that islands with different initial shapes may evolve into different stationary shapes and show that these dynamically-determined stationary states correspond to the predicted stable equilibrium shapes, as obtained from the generalized Winterbottom construction.

A Parametric Finite Element Method for Solid-State Dewetting Problems in Three Dimensions

“…Condition (ii) prescribes a contact angle condition along the moving contact line. In order to understand this condition, we may consider two limiting cases as η = 0 and η = ∞: (i) when η = 0, the contact line moving velocity is zero, and we prescribe a fixed boundary condition such that the contact line does not move; and (ii) when η → ∞, as we always assume that the moving velocity should be finite, condition (ii) will reduce to the so-called anisotropic Young equation [30,4] (2.10) c γ Γ · n Γ − σ = 0. which prescribes an equilibrium contact angle condition. Therefore, condition (ii) actually allows a relaxation process for the dynamic contact angle evolving to its equilibrium contact angle [49,28].…”

“…The above sharp-interface model (2.3)-(2.4) with boundary conditions (2.6)-(2.7) are derived based on the consideration of thermodynamic variation [4,30], and therefore, it naturally satisfies the thermodynamic-consistent physical law. More precisely, the total (dimensionless) free energy of the system, including the interface energy W int and substrate energy W sub , can be written as [4,30] (2.11)…”

“…Other related works for anisotropic flows in the literature can be found in [22,40,23] and references therein. Furthermore, the PFEMs have also been designed for simulating the evolution of solid-thin films on a substrate in 2D [4,29] and 3D case with axisymmetric geometry [58]. But how to design a PFEM for simulating solid-state dewetting problems in the full 3D is still urgent and challenging.…”

“…2 depicts equilibrium shapes under different mesh sizes, where σ = cos(15π/36), Comparisons of the cross-section profiles along the x-direction of the numerical equilibrium shapes under different meshes with its theoretical equilibrium shape, where the initial shape is chosen as a (1, 2, 1) cuboid, the surface energy γ(n) = 1 + 0.25(n 4 1 + n 4 2 + n 4 3 ), and σ = cos(15π/36). The theoretical equilibrium shape (black line) is constructed by the Winterbottom construction[50,4].…”

Programmable Atom Equivalents: Atomic Crystallization as a Framework for Synthesizing Nanoparticle Superlattices

“…In atomic systems, multiple mechanisms have been explored as a means of controlling crystal habit, and many of these have yet to be examined in PAE superlattices, indicating a wide area of potential research opportunities. For example, limited tuning of superlattice shape could theoretically be achieved by nucleating crystals at a DNA‐functionalized surface, possibly allowing for tunable Winterbottom constructions that effectively truncate the thermodynamically preferred Wulff polyhedra at specific lattice planes, taking advantage of DNA's programmability to tune substrate–PAE interactions . Extending this strategy of imposing boundary conditions on PAE crystallite growth, the Summertop formalism (where nucleation and growth are restricted by multiple interfaces) could also be investigated by assembling PAEs at surfaces with concave or convex shapes.…”