  Vertically-stacked monolayers of graphene and other atomically-thin 2D
materials have attracted considerable research interest because of their
potential in fabricating materials with specifically-designed properties.
Chemical vapor deposition has proved to be an efficient and scalable
fabrication method. However, a lack of mechanistic understanding has hampered
efforts to control the fabrication process beyond empirical trial-and-error
approaches. In this paper, we develop a general multiscale Burton-Cabrera-Frank
(BCF) type model of the vertical growth of 2D materials to predict the
necessary growth conditions for vertical versus in-plane (monolayer) growth of
arbitrarily-shaped layers. This extends previous work where we developed such a
model assuming the layers were fully-faceted (Ye et al., ACS Nano, 11,
12780-12788, 2017). To solve the model numerically, we reformulate the system
using the phase-field/diffuse domain method that enables the equations to be
solved in a fixed regular domain. We use a second-order accurate, adaptive
finite-difference/nonlinear multigrid algorithm to discretize and solve the
discrete system. We investigate the effect of parameters, including the van der
Waals interaction energies between the layers, the kinetic attachment rates,
the edge-energies and the deposition flux, on layer growth and morphologies.
While the conditions that favor vertical growth generally follow an analytic
thermodynamic criterion we derived for circular layers, the layer boundaries
may develop significant curvature during growth, consistent with experimental
observations. Our approach provides a mechanistic framework for controlling and
optimizing the growth multilayered 2D materials.
