Given any configuration of atomic lattices, we seek to model the time evolution of the layer step edges. We begin with a simply layered step train and derive both explicit and implicit formulae for its time evolution.
Our first task is to determine the behavior of a layer particle near a single step edge. Let us first assume no particles travel across an edge, giving perfect reflection for all incident particles. Let us further assume particles are under completely random motion within a layer. We now have the following picture:
The particle has complete freedom to move on a given side of the edge. However, upon collision with the edge it is reflected back into the layer, making all paths that cross the step edge forbidden. To model such collisions, take an image of the particle on the opposite side of the step edge. Whenever a collision occurs, our particle's path follows the image trajectory instead:
Let us now calculate the probability density of finding such a randomly moving particle a given distance "x" from the step edge (see Figure 4 for detailed variable definitions). Were the particle completely free to move throughout the plane, its probability density would be given by the one dimensional diffusion equation:
where n(x,t) is the probability density of locating a particle a distance x from the edge at time t. The solution to this one dimensional equation is:
where D is a positive constant controlling the rate of diffusion. However, our particle is not free to travel in the entire plane; we require the edge restrictions. Using the image construction method described above, the actual probability density will be the sum of both the particle and the image trajectory solutions:
This density allows us to calculate our particle's average distance from the step edge as a function of time:
So the average particle distance from the edge increases with time due to the random motion of the particles of a step layer. Since there is a change in particle position with respect to time, we create an "entropic force" that pushes the particle to follow this trajectory. Since each particle undergoes many collisions with edges and other particles, our test particle loses a large percentage of energy to the outside environment, giving an overdamped system such that:
We now have the force generated by a single edge on a layer particle a distance "x" from the edge. If we superimpose another such force from the edge on the other side of a step, we can use these "forces" to model the behavior of step trains over time. Using an explicit differencing scheme that relates applies the forces generated on layer particles by each adjacent step's edges. The particles are free to diffuse within a step so the width of the step may grow in time. An experimental run of this type yielded the following results:
Since the y-axis shows 1/width, the largest steps are those lowest on the graph. This system started as a step train of 100 evenly spaced steps with width 2 (.5 on the graph). As time progressed, the system expanded from the edges first, gradually pulling more and more inner steps to larger widths. Thus, our model predicts a "pyramid" shape for the evolving train where the outer tiers have the largest widths.
This page is created and maintained by Adam D. Smith.
Last Updated 8/14/98