Simulation Model

Weak statement

    There are basically two concomitant processes on the surface: Evaporation-condensation process exchange with the surrounding vapor and interface diffusion relocate the solid matter on the surface.

    Let j be the volume of matter added to per unit area of the solid surface per unit time and P be the free energy reduction with respect to per unit volume of matter added to per unit surface area of the solid. The kinetic law for evaporation-condensation can be expressed as j=LP where L is the specific evaporation-condensation rate. Let F be the driving force and J be the volumetric diffusion flux, then the kinetic law for interface diffusion is J=MF where M represent the diffusion rate

    Then we can get the weak statement including both surface diffusion and evaporation condensation. In the equation δi is the volume of matter added to per unit area of the solid surface and δI is the mass displacement:

Finite element method

    As is shown in figure 1, to calculate the concurrent interface Migration and Diffusion in every surface of the particles, we use finite element method to break the particles into small segments.

Figure 1: Interface migration of a random segment of a solid particle

    Then we get the equations:

    Where f1, f2, f3, f4 represent the forces applied on x1, y1, x2, y2 directions. Since there is no elastic/electrostatic energy applied on the particle, we do not have the elastic/electrostatic energy density term.

    To specific the shape of a particle, we use δrn of the element as a function of δx1, δy1, δx2 and δy2, where (x1, y1) and (x2, y2) are the coordinates for the two nodes of the element.

    We also get the equation of the flux along the elements caused by the mass displacement in each segment, where Q1, Q2, Qm are also the shape functions:

    Then we assemble contributions from all elements, we get the equation as the following, where [Hij] is a 7 x 7 symmetric matrix and μ = Ll M

    Thus by taking the inverse of the H matrix, we get the velocities in x and y directions in each segment point. Then we used Euler’s approximation method to calculate the exact position of segment point with respect to a specific time.

    In our model, the chemical potential g plays an important role because it can decide the shrink rate or growth rate of the particle. So we also make a simulation of two particles in different growth rates merged together.