and making use of (\[gamma\_norm\]), the normalization factor $\gamma = p_b^{x+1}$ is found and finally this leads to $$\begin{aligned}
 X &=& 2^{-1}zr_{f}^{-2} p_b^{x+1} \eta^{-1} \Psi_x^2 \Psi^{-1} p^{-1}
 \underset{k=0}{ \overset{x-1}{\sum }}P _{k}P _{x-k-1}, \label{theta} \\
 p &=& p_b w_s (\Psi_x^{z-2}/\Psi^{z-1})(\Psi^{z-1}/\Psi_x^{z-2})_b, \label{p}\end{aligned}$$ where $w_s$ differs from unity only for the first layer near the surface. Additionally, a special care must be exercised regarding in the first layer where the reduced coordination number, $(\lambda_0+\lambda_1)z$, should be enforced due to absence of the lower layer next to the surface. This concerns quantities $p$ and $\delta$ in the equations above. The recursive relations (\[req\_rl\_P\]) together with eqs. (\[lambda\_eq\_st\]) and (\[theta\]) constitute the system that describes the density profile near the surface.

The so-called surface excess $\Gamma$, the average number of polymer segments per unit area of the surface in excess of those present in the bulk, is defined by $$\label{alpha}
     \Gamma = \sum_{i=0}^{M} (\theta-\theta_b).$$ This quantity is useful to estimate the surface tension of a polymer solution by taking the random mixing approach. [@Scheutjens-Fleer]

The Bragg-Williams approximation is attained by means of the proper limit, $z \rightarrow \infty$, $Vz=const$, that produces the following expression for the chemical potential $\mu = \frac{1}{x}\ln \lambda$ of the semiflexible 