Analysis of etching at a solid-solid interface

We present a method to derive an analytical expression for the roughness of an eroded surface whose dynamics are ruled by cellular automaton. Starting from the automaton, we obtain the time evolution of the height average and height variance (roughness). We apply this method to the etching model in $1+1$ dimensions, and then we obtain the roughness exponent. Using this in conjunction with the Galilean invariance we obtain the other exponents, which perfectly match the numerical results obtained from simulations. These exponents are exact, and they are the same as those exhibited by the Kardar-Parisi-Zhang (KPZ) model for this dimension. Therefore, our results provide proof for the conjecture that the etching and KPZ models belong to the same universality class. Moreover, the method is general, and it can be applied to other cellular automata models.

Figure 1

The figures are the four situations in which the dynamics of the etching model can be divided. The initial configuration is at the top, where the select size, column 2, is green, and its neighbors are shown in dark blue. After the corrosion process the final situation is exhibited at the bottom.

Figure 2

Evolution of $\sigma \left(t\right)$. The data points are the results of numerical simulation of the etching model for several substrate lengths. The straight lines are the results for the uncorrelated process. (a) Expansion of $\sigma \left(t\right)$ for small values of $t$, implying small values of $w\left(t\right)/w_s$. (b) Intermediate and large times.