What Makes a Boundary Less Accessible

For the growth and transport processes driven by Laplacian fields, the accessibility of an interface for Brownian motion is characterized by the harmonic measure. Its multifractal properties help one to understand how the irregular geometry of biological membranes, metallic electrodes, porous catalysts, or growing aggregates is “seen” by diffusing particles. To clarify this point, we performed an extensive numerical study of the harmonic measure on two families of self-similar triangular Koch curves of variable Hausdorff dimension which may represent branched pore networks or fjordlike rough interfaces. Although these structures are apparently different, the multifractal properties of the harmonic measure in two cases are found to be very close for curves of small Hausdorff dimensions and to differ for higher irregularity. This provides new insight into optimization problems in chemical engineering.