A parametric finite element method for solid-state dewetting problems with anisotropic surface energies

Abstract

We propose an efficient and accurate parametric finite element method (PFEM) for solving sharpinterface continuum models for solid-state dewetting of thin films with anisotropic surface energies. The governing equations of the sharp-interface models belong to a new type of high-order (4th-or 6th-order) geometric evolution partial differential equations about open curve/surface interface tracking problems which include anisotropic surface diffusion flow and contact line migration. Compared to the traditional methods (e.g., marker-particle methods), the proposed PFEM not only has very good accuracy, but also poses very mild restrictions on the numerical stability, and thus it has significant advantages for solving this type of open curve evolution problems with applications in the simulation of solid-state dewetting. Extensive numerical results are reported to demonstrate the accuracy and high efficiency of the proposed PFEM.

Sharp-interface approach for simulating solid-state dewetting in two dimensions: A Cahn–Hoffman ξ-vector formulation

“…These earlier studies were focused on the isotropic surface energy, although recent experiments have demonstrated that the crystalline anisotropy could play important roles in solid-state dewetting. To include the surface energy anisotropy, many approaches have been proposed in recent years, such as a discrete model [13], a kinetic Monte Carlo model [42,15], a crystalline model [9,65] and continuum models based on partial differential equations [5,25,26,55]. From a mathematical perspective, theoretical solid-state dewetting studies can be categorized into two major problems: one focuses on the equilibrium of solid particles on substrates [4,33]; the other focuses on investigating the kinetic evolution of solid-state dewetting [25,26,55].…”

“…The resulted Wulff shape, is the inner convex region bounded by all planes that are perpendicular to orientation n and at a distance of γ(n) from the origin. The Winterbottom construction [57,5] was subsequently proposed to handle with the case about particles on substrates by truncating the Wulff shape with a flat plane, and where the Wulff shape is truncated depends on the wettability of the substrate. Meanwhile, many theories [7,8] demonstrated that the derivative of γ(n) plays an important role in investigating equilibrium and kinetic problems for solid particles with anisotropic surface energies.…”

“…1.1 depicts the γ-plot, 1/γ-plot and ξ-plot for four different types of surface energy anisotropies: (a) isotropic surface energy, i.e., γ(n) ≡ 1; (b) cubic surface energy γ(n) = 1 + a(n 4 1 + n 4 2 + n 4 3 ) with a representing the degree of anisotropy; (c) ellipsoidal surface energy γ(n) = a 2 1 n 2 1 + a 2 2 n 2 2 + a 2 3 n 2 3 ; (d) "cusped" surface energy defined as γ(n) = |n 1 | + |n 2 | + |n 3 |. In the application of materials science, the surface energy could be piecewise smooth and have some "cusped" points, where it is not differentiable [5,19,41]. A typical example is the "cusped" surface energy defined above.…”

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

“…Solid-state dewetting of thin films belongs to the evolution of an open curve/ surface governed by surface diffusion and contact line migration [27,49,28,5,30]. In earlier years, the marker-particle method was firstly presented for solving sharpinterface models of solid-state dewetting in two dimensions (2D) [51,49] and three dimensions (3D) [18].…”

“…The goal of this paper is to extend our previous works [5,29] from 2D to the 3D by using a variational formulation in terms of the Cahn-Hoffman ξ-vector for simulating solid-state dewetting of thin films. More precisely, the main objectives are as follows: (i) to derive a variational formulation of the sharp-interface model for simulating solid-state dewetting problems in 3D [30]; (ii) to develop a PFEM for simulating the solid-state dewetting of thin films in 3D; (iii) to demonstrate the capability, efficiency and accuracy of the proposed PFEM; and (iv) to investigate many of the complexities which have been observed in experimental dewetting of patterned islands on substrates, such as Rayleigh instability, pinch-off, edge retraction and corner mass accumulation.…”