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: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.

We report the synthesis and morphological characterization of a miktoarm block copolymer architecture:  (PSα M−PIM) n −(PSM−PIα M) n , where M ∼ 20 000, n = 1, 2, and the arm asymmetry parameter α = 1, 2, or 4 (α is the ratio of the outer block molecular weight to that of the inner block). These block copolymers are symmetric in overall composition and exhibit n- and α-dependent microdomain morphologies. Alternating lamellae are observed for linear tetrablocks (n = 1), α = 1, 2, 4, and for inverse starblock (n = 2), α = 1, 2. An architecturally-induced morphological transition from lamellae to a tricontinuous cubic structure is observed with n = 2 and α = 4. The formation of the tricontinuous cubic microdomain structure in this compositionally symmetric system is thought to relieve the overcrowding of the four peripheral PS−PI junctions by providing a curved intermaterial dividing surface with a triply periodic microdomain structure, allowing some bridging by the interior blocks of the miktoarm star.

Interference lithography holds the promise of fabricating large-area, defect-free photonic structures on the sub-micrometer scale both rapidly and cheaply. There is a need for a procedure to establish a connection between the structures that are formed and the parameters of the interfering beams. There is also a need to produce self-supporting three-dimensional bicontinuous structures. A generic technique correlating parameters of the interfering beams with the symmetry elements present in the resultant structures by a level-set approach is developed. A particular space group is ensured by equating terms of the intensity equation to a representative level surface of the desired space group. Single- and multiple-exposure techniques are discussed. The beam parameters for certain cubic bicontinuous structures relevant to photonic crystals, viz.,the diamond(D), the simple cubic (P), and the chiral gyroid (G) are derived by utilizing either linear or elliptically polarized light.

The gyroid is a triply periodic minimal surface in the associated family of the Schwarz P-and D-surface. We prove it is embedded and find constant mean curvature companions of the gyroid with small constant mean curvature. We also discuss a surface similar to the gyroid in the associated family of the Schwarz H-surface. AMS-classification: 53A10, 49Q05Embedded surfaces of constant mean curvature are used to model the behaviour of molecular interface surfaces and further microstructure phenomena (see [N, p.240] and [DP] for an overview and references). As a striking example we mention the interface surfaces of diblock copolymers [TAHH]. Three embedded triply periodic surfaces, all in the same associated family, are most frequently mentioned in the applications: The well-known P-and D-surface discovered by Schwarz, and the more recent gyroid. In the special case of block copolymers evidence pointing to the gyroid is announced in [HGT] [FKZ].The gyroid was discovered by the cristallographer Alan Schoen IS]. A plastic chip model of the gyroid is depicted on the frontispiece of Osserman's book [O] (and [DHKW, p.217]). In the mathematical literature the only account on the gy= roid known to us is a paragraph in the article by Karcher on Schoen's surfaces [K, 5.1.2] that does not address embeddedness. It is known that countably many surfaces associated to P and D are immersed (and not dense) in R3; however, only P, D, and the gyroid are embedded.The lack of reflectional symmetries makes the gyroid difficult to visualize. We suggest to view the gyroid as a tesselation of gyrating ribbons associated to the square catenoids of the P-surface (Sect. 1). In the embeddedness proof, however, we adopt another point of view that better suits the symmetry group of the gyroid.