Reorientational angle distribution and diffusion coefficient for nodal and cylindrical surfaces

Abstract

We present a catalogue of diffusion coefficients and reorientational angle distribution (RAD) for various periodic surfaces, such as I-WP, F-RD, S, and S1 nodal surfaces; cylindrical structures like simple, undulated, and spiral cylinders, and a three-dimensional interconnected-rod structures. The results are obtained on the basis of a simulation algorithm for a diffusion on a surface given by the general equation φ(r)=0 [Hołyst et al., Phys Rev. E 60, 302 (1999)]. I-WP, S, and S1 surfaces have a spherelike RAD, while F-RD has a cubelike RAD. The average of the second Legendre polynomial with RAD function for all nodal surfaces, except the F-RD nodal surface, decays exponentially with time for short times. The decay time is related to the Euler characteristic and the area per unit cell of a surface. This analytical formula, first proposed by B. Halle, S. Ljunggren, and S. Lidin in J. Chem. Phys. 97, 1401 (1992), is checked here on nodal surfaces, and its range of validity is determined. RAD function approaches its stationary limit exponentially with time. We determine the time to reach stationary state for all surfaces. In the case of the value of the effective diffusion coefficient the mean curvature and a connectivity between parts of surfaces have the main influence on it. The surfaces with low mean curvature at every point of the surface are characterized by high-diffusion coefficient. However if a surface has globally low mean curvature with large regions of nonzero mean curvature (negative and positive) the effective diffusion coefficient is low, as for example, in the case of undulated cylinders. Increasing the connectivity, at fixed curvatures, increases the diffusion coefficient.