Triply periodic level surfaces as models for cubic tricontinuous block copolymer morphologies

The domains of microphase separated block copolymers develop interfacial surfaces of approximately constant mean curvature in response to thermodynamic driving forces. Of particular recent interest are the tricontinuous triply periodic morphologies and their mathematical representations. Level surfaces are represented by certain real functions which satisfy the expression F(x,y,z) = t, where t is a constant. In general, they are non-self-intersecting and smooth, except at special values of the parameter t. We construct periodic level surfaces according to the allowed reflections of a particular cubic space group; such triply periodic surfaces maintain the symmetries of the chosen space group and make attractive approximations to certain recently computed triply periodic surfaces of constant mean curvature. This paper is a study of the accuracy of the approximations constructed using the lowest Fourier term of the Pm3m, Fd3m and I4132 space groups, and the usefulness of these approximations in analysing experimentally observed tricontinuous block copolymer morphologies at a variety of volume fractions. We numerically compare surface area per unit volume of particular level surfaces with constant mean curvature surfaces having the same volume fraction. We also demonstrate the utility of level surfaces in simulating projections of tricontinuous microdomain morphologies for comparison with actual transmission electron micrographs and determination of block copolymer microstructure.

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