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

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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.