Nanotechnology today is a hot topic. Building devices at a nanoscale has many applications. Nanoscale patterns can be formed by self-assembly. When some chemicals are deposited over an elastic substrate, they rearrange themselves into patterns to achieve the lowest possible energy. 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 eventually form a stable pattern.

We have successfully written a program in C# the can simulate pattern formations. Our simulations agree qualitatively with experimental observations and our expectations, which validate our program. In this program, we have successfully implemented the FFT, which was originally written for C++.

Our program has simulated (1) the transitions from quantum dots to serpentine stripes and to quantum pits as the concentration increases; (2) the dependency of heterogeneous pattern formation on preexisting patterns; (3) and the effect of temperature changes on the size of the patterns formed, which is important because the physical and chemical properties of these patterns (e.g. quantum dots) depends on size.

In the future we plan to include other mechanisms to control the pattern formations. Two such mechanisms are electric and magnetic fields. In addition, we will try to solve the equations using a fully implicit method, which will hopefully improve the numerical stability. We will also look at ways to make our code even more efficient and faster, which will help our studies.