Algorithms for the computer simulation of two-dimensional projections from structures determined by dividing surfaces

Abstract

Transmission electron microscopy (TEM) data come in the form of two-dimensional projections of specimens with finite width. For visual interpretation of such data it is valuable, and for quantitative comparison with models essential, to have available a method for generating simulated projections from model structures. We present three computer algorithms that simulate TEM micrographs, from structures determined by contrast between two materials separated by complicated dividing surfaces. Three very general forms used to represent dividing surfaces are treated: traditional coordinate systems, finite element representations, and tesselations adapted to complex line integration schemes. Because surface tension drives dividing surfaces to minimize their area, particular attention is paid to minimal surfaces, surfaces of nonzero constant mean curvature, and parallel surfaces that may form in specimens constrained to small dimensions. Besides the model structure itself, the other inputs are the orientation of the model structure, the location of (parallel) upper and lower truncating planes representing the finite specimen thickness or the form of the bounding surface for microdroplet samples, and the direction and magnitude of a linear deformation representing distortion due to the microtoming process. The values of these registration parameters providing the optimal fit with digitized TEM data are found by a relaxation method. Remarkable matches are obtained between micrographs of block copolymer morphologies and model structures determined by surfaces of constant mean curvature.