Minimal periodic surfaces morphology

Next we consider the effect of the block copolymer composition /= NfJN on the ordered morphology. In the limit of very strong segregation, that is, zero interface width, the natural idea is to let the stable ordered phase correspond to the phase with the minimal interface surface. To illustrate this principle and to obtain a semiquantitative estimate of the values of/for which the transitions between the three classical stmctures occur, we consider an LxLxL volume of the self-assembled diblock copolymer system. The ordered states that will be compared are the lamellar phase, a square lattice of cylinders, and spheres on a simple cubic (SC) lattice. L is the periodicity length scale of the layers, the square, and the cubic lattice (Figure 19). The LxLxL volirme element contains one cylinder resp. one sphere. Volirme conservation (Figure 20), therefore, requires fL = 7tRcL = 4n/SRs, where Rc and Rs are the radii of the cylinder and the sphere, respectively. 

Unlike the bulk morphology, block copolymer thin films are often characterized by thickness-dependent highly oriented domains, as a result of surface and interfacial energy minimization [115,116]. For example, in the simplest composition-symmetric (ID lamellae) coil-coil thin films, the overall trend when t>Lo is for the lamellae to be oriented parallel to the plane of the film [115]. Under symmetric boundary conditions, frustration cannot be avoided if t is not commensurate with L0 in a confined film and the lamellar period deviates from the bulk value by compressing the chain conformation [117]. Under asymmetric boundary conditions, an incomplete top layer composed of islands and holes of height Lo forms as in the incommensurate case [118]. However, it has also been observed that microdomains can reorient such that they are perpendicular to the surface [ 119], or they can take mixed orientations to relieve the constraint [66]. 

The structure of the PLB has been related to that of cubic phcises[7], discussed in Chapters 4 and 5. However, as we shall see, a description of these membrane morphologies as equilibrium phases seems to be applicable, if at all, in only a few cases that we have encountered. Independently of Larsson et al. [7], Linder and Staehelin [14] also suggested that a certain "membrane lattice" in a parasitic protozoa did indeed correspond to an infinite periodic minimal surface. However, no further structural details, such as the symmetry or form of IPMS, were deduced or discussed. Some ten additional examples of membrane assemblies displaying cubic symmetries have been pointed out [15,16] but no structural details were inferred. To the best of our knowledge, the above references ([7, 14-16]) are the only reports in which membrane assemblies have been related to the structure of IP. There are. 

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