Finite-Stretching Corrections to the Strong-Stretching Theory of Polymer Brushes in Solvent

Grafted polymers in solvent are naturally stretched and form a brush. Earlier theoretical approach known as strong-stretching theory (SST) has been very successful in predicting fundamental properties such as parabolic density profile and broad chain end distribution. A more rigorous self-consistent-field theory (SCFT) has shown good agreement with SST but it also revealed new features. For instance, there exists a proximal layer next to the substrate ($z=0$) where the polymer concentration $\phi(z)$ vanishes. Furthermore, a brush has an exponentially decaying tail region beyond the brush height $h$ predicted by SST. Due to the complexity of numerical approach few previous studies focused on these features. We have made a systematic analysis of the proximal layer shape and its effect on the free energy. The size of the proximal region $\mu$ scales as $1/h$ and the profile has a scaling symmetry. Polymer concentration $\phi(z)$ grows linearly near the grafting surface with a slope $6/Na^2$ when the integral of $\phi(z)$ is normalized to unity. Here $a$ is the statistical segment length and $N$ is its total number of segments per chain. A universal function $\overline{\phi}(x)$ is numerically found so that $\phi(z) \approx \mu \overline{\phi}(z/\mu)$ independent of $h$. We also investigated the shape of the tail region to which entropically excited chains contribute.