When deposited over a solid surface, some chemicals form patterns at the nanoscale. Two major factors cause this pattern formation. The minimums in Gibb’s free energy drive the phase separation of the chemical components. This separation increases the amount surface free energy. To minimize its total energy, the system reacts by reducing the number of phase boundaries. On the other hand, the surface stress produced by concentration variations tends to create finer patterns by increasing the number of phase boundaries. These two opposing factors cause the system to reach equilibrium and form a stable pattern.

This pattern formation is described by a set of nonlinear integral-differential diffusion equations that couple the concentrations and the surface stress. These equations are simplified using the Fourier Transformation, which converts the integral-differential equations into simpler partial differential equations in Fourier space. The Fast Fourier Transform is used to transform values between real and Fourier space.

We successfully wrote a program in C# to simulate this self-assembly process. We modified the equations to include temperature fluctuations. Our simulations agree qualitatively with the experimental results reported in literature and what we expect. We have successfully simulated the transitions between quantum dots, serpentine stripes, and quantum pits. We have shown that heterogeneous pattern formations can be guided by preexisting patterns. We have also shown that temperature can be used to control the size of patterns. This software can be used to understand and design nano pattern formation on solid surfaces. Future work will be focused on improving the numerical method and including other mechanisms for controlling these pattern formations.