Periodic chain constraint

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]. 

In a number of papers [16-23,25], the discrete variant of the PCAO-model is considered the chain is modeled by a random walk on the lattice with spacing a and the topological constraints are placed on the dual lattice with period c. 

Effects of the first factor, release of constraints by retraction during the equilibration period, can be estimated with the bond flip model as long as the strain is not too large. The average frequency of retraction-induced jumps per chain during equilibration is... 

The first term in Eq. (4.6) is the entropic stretching penalty derived under the assumption that the chains are uniformly stretched to a length of one-half of the lamellar domain period A. The second term represents the repulsive energetic interactions confined to the (sharp) A-B interface. It is given by the product of the contact area per chain derived under volume filling constraint, A = Na /(X/2), and the interfacial... 

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