| The class of minimal surfaces that fill space homogeneously are particularly relevant for condensed matter structures. Surfaces that are periodic in space, having a certain genus per unit cell, have been much used to model the structures of microemulsions, lipid bilayers, and even inorganic solid state systems. This thesis deals with the development of a numerical algorithm for constructing minimal surfaces of prescribed space group symmetry. Because the approach used was general enough to include the quasiperiodic space groups, a highly symmetrical icosahedral minimal surface was constructed for the first time.;At the heart of the algorithm is a simple free energy functional that models binary phase separation. A limit of the model corresponds to vanishingly small interfacial thickness. At equilibrium such interfaces are surfaces of constant mean curvature. By tuning the volume fractions of the two phases, the zero value of mean curvature (i.e. the minimal surface) is obtained. The algorithm typically finds the minimal surface of minimum area (and genus number) in the unit cell. A variety of distinct surfaces can be generated, however, by imposing different space group symmetries. In the periodic case, a new surface having Ia3d symmetry was obtained in this fashion.;The extension of the algorithm to the icosahedral space groups involved formulating the free energy functional in six-dimensional space. An additional limit is necessary that eventually removes an unphysical coupling in phason space. A surface with icosahedral space group P53m was studied in considerable detail. Cubic periodic approximants of the surface, having up to 144 genus per unit cell, were constructed and provided reasonable estimates of the surface area per unit volume of the quasiperiodic surface. Topology changes of the surface were found to be the direct analogues of atomic phason jumps in quasicrystals. A detailed analysis of these topology changes suggests that a tiling construction of the icosahedral surface may be possible. |
