Triply Periodic Bicontinuous Cubic Microdomain Morphologies by Symmetries

In response to thermodynamic driving forces, the domains in microphase-separated block copolymers have distinct intermaterial dividing surfaces (IMDS). Of particular interest are bicontinuous and tricontinuous, triply periodic morphologies and their mathematical representations. Level surfaces are represented by functions F: R 3 → R of points ( x, y, z ) ∈ R 3, which satisfy the equation F ( x, y, z ) = t, where t is a constant. Level surfaces make attractive approximations of certain recently computed triply periodic constant mean curvature (cmc) surfaces and they are good starting surfaces to obtain cmc surfaces by mean curvature flow. The functions F( x, y, z ) arise from the nonzero structure factors F ( hkl ) of a particular space group, such that the resulting surfaces are triply periodic and maintain the given symmetries. This approach applies to any space group and can, therefore, yield desired candidate morphologies for novel material structures defined by the IMDS. We present a technique for generating such level surfaces, give new examples, and discuss certain bicontinuous cubic IMDS in detail.