Power‐law exponent in the central region of a polymer layer adsorbed from dilute solution: Comparison between scaling and SCF results

Abstract

AbstractDe Gennes predicted the self‐similar structure φ(z) ∞ z−α for adsorbed polymer layers in the semi‐dilute (central) region of the adsorption profile of homopolymers. This power‐law behaviour is recovered in mean‐field SCF calculations. In this case the exponent is α = 2 in good solvents provided D «z «d, where D is the proximal length and d the distal length. We use a ground‐state approximation (GSA) to derive expressions for the two lengths D and d, and show that in the central region the profile is in good approximation given by φ = 1/3(z + D)−2exp(‐(z + D)2/3d2). Unless the chains are extremely long the condition D«z«d is difficult to obtain and corrections on the exponent are necessary. For most chain lengths in the experimental range, the central region is quite narrow. It is shown that for high adsorption energies (small D) α = 2 + 2d−1 in leading order, where d = R/, with R the radius of gyration and φb the bulk solution concentration. For weak adsorption the proximal length D is larger, which leads to a smaller exponent α. The d−1 correction is in excellent agreement with numerical self‐consistent‐field calculations. In poor solvents we have φ = 1/2(z + D)−1exp(‐2(z + D)2/3d2) and α = 1 + 4d−1 in the strong adsorption limit, which implies a larger correction in this case. Our analysis suggests that in a polymer adsorption profile with excluded‐volume correlations (where α = 4/3) non‐universal aspects would also be present if the chain length is finite.