We study the macroscopic behavior of ideal polymer chains of various architectures in the vicinity of a surface, by considering that at least one of the units of the chains is in contact with the surface. The probability that any unit will be in contact with the surface is employed and the macroscopic properties are determined as averages over this probability. The present model is an amendment to the model used previously with the one end of a linear chain fixed at the surface, and though it is a simplification to the full problem where the chains can move freely in the whole space, it is free from the necessity to include the volume and the polymer concentration in order to describe chains in the vicinity of the surface. We estimate the mean number of contacts between a chain and the surface for the cases of linear, ring, regular star, and regular comb polymers as the basic quantity for the description of both the thermodynamics of chains at a surface and the degree of adsorption of polymers of various architectures.

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The effects of chain size and architectural asymmetry on the miscibility of chemically identical polymer blends composed of star/star, ring/ring, and ring/linear chains are studied by means of an analytical theory. The free energy of these blends is obtained through the summation of the series of the one-loop diagrams at any dimensionality d. From the spinodal equation, we find that for the same total molecular weights the effective repulsions between ring/ring species are higher than those of linear/linear blends and higher than those of the symmetrical star/star blends of four branches but smaller than the respective interactions of star/star blends where both species have five arms. At d = 3 no phase separation is found for blends composed of ring/ring and ring/linear chains at any size disparity or volume fraction. The same is true in the case of blends composed of star/star or star/linear chains with a few or a moderate number of arms in agreement with recent experimental and theoretical results.

We present a rigorous derivation of a scaling theory of the thermodynamics of polymer solutions at finite concentrations. The derivation proceeds directly in 3(2, 4, etc.) dimensions from the expression for the partition function for a solution of monodisperse continuous Gaussian chains with excluded volume. There is no need for the use of renormalization group methods or for the extrapolation of results from calculations near four-dimensions. Nethertheless, the importance of four dimensions emerges directly from the scaling theory as does a proof that only the binary excluded volume parameter appears in the equation of state, average chain dimensions, etc., in the limit of long enough chains. The irrelevance of the range of the binary segment interaction and the sufficiency of the two parameter theory is thus established by the scaling theory in a straightforward and elementary fashion. We demonstrate that the power law dependence, ν in 〈R2〉∝L2ν, of the mean square end-to-end distance 〈R2〉 on the chain length L in good solutions can be calculated directly in three (two, etc.) dimensions from the scaling relations. Applications are presented to good solutions at finite concentrations, to confined chains in good solvents, and to the properties of solutions of block copolymers.