Title:

Triply periodic minimal surfaces in chemical physics

Mathematically triply periodic minimal surfaces (TPMS) are easy to define but extremely hard to calculate. We have described the exact computation of three surfaces most commonly found in nature, designated P, D and G. We give analytical equations for their coordinates in terms of special functions, as well as for their metric properties such as surface to volume ratios. Nodal surfaces are simply surfaces over which any function f(x, y, z) = 0, for example Cos(x) + Cos(y) + Cos(z) = 0. We have obtained extremely accurate and far more practicable models for TPMS by fitting exact surfaces to a Fourier expansion in nonorthogonal basis sets of the nodal surfaces of crystallographic structure factor expressions. These have been compared with existing models in the literature by a novel technique of colouring their surface curvatures, and found to be far superior. The contours of zero electrostatic potential within an arrangement of positive and negative electric charges in ionic crystals, as for example in the lattice of CsCl, can be plotted as a surface which separates space into domains of positive and negative potential. These surfaces correspond topologically to TPMS. We give new analytical expressions for 11 zero equipotential surfaces (ZEPS) in terms of Jacobi theta functions. The numerical integration of these now standard expressions is extremely slow, taking several hours to obtain a surface. We give nodal expressions for the most important electrostatic surfaces reducing the computation time to mere seconds. Using this technique we have analysed the scalar curvature and vector field properties for a large range of crystal structures, producing the most comprehensive quantitative comparison of triply periodic surfaces to date. By considering the purely classical motion of a charged particle over the zero equipotential surfaces of CsCl, we have proved that quantization of field and energy is solely a function of the metric defined by charge, and not a function of scale. We have further proved that triply periodic minimal surfaces define this metric. This work shows the importance and relevance of TPMS as potential universal descriptors of all charge and matter.
