vganjali/Atomic-Deposition

A 3D atomic deposition simulation model written entirely in Python. PytOpenGL library is used for 3D visualization of individual atoms being deposited on a flat surface. This was done as a course project (EE216: Nanomaterials and Nanometer-scale Device) at UCSC and was done with a group of three PhD students:

Yucheng Li
Md Nafiz Amin
Vahid Ganjalizadeh (https://github.com/vganjali)

Basic Physics: L-J potential

The Lennard-Jones potential (L-J potential) is a mathematical model to approximate the interaction between a pair of neutral atoms or molecules. $V_{LJ}=4\epsilon[(\frac{\sigma}{r})^{12}-(\frac{\sigma}{r})^{6}]=\epsilon[(\frac{r_{m}}{r})^{12}-2(\frac{r_{m}}{r})^{6}]$ where

$\epsilon$: Depth of the potential well
$\sigma$: Distance at wich $V_{LJ}=0$
$r_m$: Distance at wich $V_{LJ}$ is minimum ($=-\epsilon$)
($r_m=2^{1/6}\sigma\approx1.122\sigma$)
$r^{-12}$: The repulsive term, describes Pauli repulsion at short ranges due to overlapping electrin orbitals
$r^{-6}$ The attractive term, describes attraction at long ranges (for example, Van de Waals force)
The same equation is used in many-particle environment

Algorithm Flowchart

Key Considerations

Bottom Layer

A well-arranged bottom layer is assumed to be pre-formed when the simulation starts
Exerts attractive/repulsive force on incoming particles according to L-J potential
Provides a general model for the simulation

Adaptive Time-step

Particle slows down when L-J potential changes sharply
Increases the stablity around the equilibrium point
$r_{k+1}=r_{k}+\Delta t \nu_{k}+{\Delta t}^2\frac{F_{r_k}}{m}$
$r_{k+1}=\nu_{k}+\frac{\Delta t}{2m}(F_{r_k}+F_{r_{k+1}})$
$\Delta t_{k+1}={1}/{[\nu_k \times exp(r_c-r_n)]}$
$r_c$ is critical distance to look for neighbors
$r_n$ is the distance from the closest neighbor

Timeout

A counter is the triggered whenever the first change in the sign of the force $F=-\frac{\partial V_{LJ}}{\partial r}$ is detected
After a reasonable number of time-steps, timeout happens and the particle reaches the equilibrium
Simplifies complexity due to large force from the bottom surface

Velocity Damping

Velocity of the particle is dampled if there is a change in sign of velocity or force
Velocity sign change indicates collision, and damping emulates inelastic collision
Force sign change indicates crissing the equilibrium and damping velocity around it simplifies the motion

Grids

1/0 indicate presence/absence of old atoms
Submatrix dimension $(2 critical distance+1) \times (2 critical distance+1)$

Force Submatrix

At the beginning, components of F are calculated for each point within the submatrix
A new particle uses a look-up table (LUT) of pre-calculated force values
This increases the computation speed even when number of particles is large

Real Physical Parameters

For Cu atoms:

vganjali / Atomic-Deposition