Position Dependent Chemical Potential

    In the previous calculations,  and  (the chemical potential of pure component B and A on the substrate) are assumed material constants.  However, when the substrate is pre-patterned with some atoms,  and  become position dependent.  The following figure demonstrates the idea.  When an A-atom is deposited on different regions of the substrate, it is attached to a local surface composed of different kinds of materials.  The A atom interacts with different surface atoms and hence causes  to be spatial dependent.

It is found only  enters the diffusion equation.

Define .

It is also found the difference of  at various positions, rather than the value of , influences the diffusion.

In following calculation,  is given a spatial function that looks like the following figure.  In the blue region,   is assumed 0.1, in other regions  is assumed 0.

In following sequence, all the other parameters are the same as previous calculations with isotropic surface stress for average concentration 0.4.

(The average concentration is taken to be 0.4.  The initial condition is set to fluctuate randomly within 0.001 from the average.  The time unit in the figures is t.  The size of each figure is 256b´256b.  In the simulation, W and Q are taken to be 2.2 and 1, respectively)

During the evolution, the six pre-patterned lines attract A atoms and repel B atoms.  Ordered lattice of dots formed within each small cell defined by the pre-patterned lines.