Higher order Morita approximation and its validity for random copolymer adsorption onto homogeneous and periodic heterogeneous surfaces

Adsorption of a single AB random copolymer (RC) chain onto homogeneous and inhomogeneous ab surfaces with a regular periodic pattern is studied theoretically. For the averaging over disorder in the RC sequence, the constrained annealed approximation, known as the Morita approximation is employed. A general scheme for constructing the Morita approximation of an arbitrary order m is proposed; it is based on representation of the RC monomer sequence as a Markov chain of overlapping (m-1)-ads. The problem is solved within the framework of the generating functions approach for the two-dimensional partially directed walk model of the polymer on a square lattice. Temperature dependences of various adsorption characteristics are obtained. The accuracy of the Morita approximation is assessed by comparison with the numerical results for RC with quenched sequences obtained by the transfer matrix approach. It is shown that for RC adsorption on a homogeneous surface for AB-copolymers with adsorbing A and neutral B blocks, it is sufficient to use the Morita approximation of second or third order. If the non-adsorbing B block is repelled from the surface, then a higher order of the approximation (5th-6th) is required. An important indicator of validity of the Morita approximation is the entropy: if the order of the approximation is not high enough, the entropy becomes negative in the low-temperature regime which means that the Morita approximation is wrong in this order. In the case when the surface is periodically inhomogeneous, the Morita approximation works worse and very high orders are required, which can be beyond the computational capabilities. Moreover, the Morita approximations become degenerate: approximations of two or more consecutive orders give the same result.