Ground‐state description of the adsorption of homodisperse and polydisperse polymers

A ground‐state approximation (GSA) is employed to model the structure of an adsorbed layer of homodisperse and polydisperse polymer. The model uses the basic assumption that the volume fraction at a distance z from the surface of a component with chain length N can be written as the product of the square of an eigenfunction g(z) and the N‐th power of an eigenvalue eϵ. This approximation implies the neglect of end effects (tails): only loops are considered. For a homodisperse polymer, the eigenvalue is defined through ϵN = In(1/ϕ b ), where ϕ b is the bulk solution concentration. The eigenfuction can be written in terms of two parameters: a “proximal” length D which through the boundary condition may be related to the adsorption energy, and a “distal” length which is inversely proportional to √ε. For a polydisperse polymer, D is the same as for a homodisperse polymer, but ε has to be computed from an implicit equation which involves a summation over all chain lengths present. The contribution of each chain length N in a mixed adsorbed layer is obtained by weighting with e εN . This approximate analytical model gives results which are in good agreement with numerical self‐consistent‐field calculations. Examples are given to illustrate the applicability of the model to polydisperse systems. These include adsorption preference of long chains in polymer mixtures and the difference between adsorption and desorption isotherms in polydisperse systems. Simple expressions are obtained for the chain length characterising the transition between (long) adsorbed and (short) non‐adsorbed chains and for the width of the transition zone.