Segmental concentration profiles of end-tethered polymers with excluded-volume and surface interactions

Abstract

The segmental concentration profile ρ of end-tethered flexible polymer chains is calculated for comparison with recent measurements on polymer layers having a low surface grafting density σ. Moments of ρ are also calculated as these quantities, rather than the full concentration profiles, are usually measured experimentally. Exact calculations of ρ are summarized for ideal flexible chains with an arbitrary short-range polymer–surface interaction. Though technical difficulties restrict our renormalization group (RG) calculations for swollen chains to limiting values of the polymer–surface interaction, our analysis indicates that the relatively simple, closed-form, Gaussian-chain expression for ρ provides a good approximation to the RG results if the average size of the Gaussian chain is replaced by its swollen-chain analog. This approximate concentration profile for chains with excluded volume and variable polymer–surface interaction should prove useful when interpreting experimental results for low-grafting-density polymer layers formed in good solvents. Recent measurements [Kent et al., J. Chem. Phys. 103, 2320 (1995)] of the second moment of ρ in the limit σ→0 agree well with our RG calculations, and the measured concentration profile ρ, whose determination by neutron reflection is somewhat uncertain, agrees qualitatively with our theoretical predictions.