Quantitative phase-field simulation

The quantitative phase-field model proposed by Bragard et al.²⁷ was used for the simulation of WS₂ growth under the modification of the description for the N-time symmetry. An order parameter ϕ, which takes a value of + 1 for the sold crystalline WS₂ and − 1 for the liquid phase, was employed. ϕ changed from − 1 to + 1 continuously inside the interface. The time-evolution equation²⁹ is given by

Here n is the unit vector normal to the interface; W₀ is the interfacial thickness; λ is the coupling constant given by λ = a₁W₀/d₀, with a1=52/8, d₀ is the capillary length defined as d₀ = σ₀ (T_mc_p/L²); β₀ is the kinetic coefficient; u_int is the dimensionless undercooling at the interface; and ε_c and ε_k are anisotropy parameters of the interfacial energy and mobility, respectively. The thermodynamic and interfacial parameters are listed in Table ​Table1.1. Equation (⁴) was discretized using a second-order finite-difference scheme with grid spacing Δx = 3.698 μm and solved using a first-order Euler scheme with a time step Δ of t = 1 × 10⁻³ s. We set u_int to be 3.1 × 10⁻⁴, assuming that the temperature of the interface is close to that of the substrate because it is not straightforward to specify the temperature of the interface during crystal growth.

Input parameters for Q-PFS.