this condition reads $$\begin{aligned}
 \label{cond2}
Z^{-1}c\Psi ^{z} &=& Z^{-1}\{c\Psi^{z-1} + c (z-1) \Psi _{x} ^{z-2} \lambda
^{2} \eta \chi _{x-1}  \nonumber \\
&&+ c\frac{z-2}{2} r_{f} \Psi _{x} ^{z-3}\lambda ^{3}\eta
^{2}\underset{k=1}{ \overset{x-2}{\sum }}\chi _{k}\chi _{x-k-1}\},\end{aligned}$$ where the quantity in the curly brackets is the partial partition function for the first generation sites calculated under the condition that one such site is empty.

Using (\[x-mer eqs bA\]), we rewrite (\[cond2\]) in a more tractable form $$\label{x-mer eqs bB}
    \varepsilon = \left( X\delta -1 + \sqrt{(X\delta-1)^2+4\eta X\delta} \right)/2\eta.$$ where $\varepsilon \equiv \lambda \xi$ and $X \equiv \theta/(1-\theta)$ and $\delta$ is given by $$\label{delta_eq}
    \delta =  \frac{z-1}{z} -
\frac{1}{z} \underset{k=1}{ \overset{x-2}{\sum }} P_k P_{x-k-1} \left/ \underset{k=0}{ \overset{x-1}{\sum }} P_k P_{x-k-1} \right.$$

In the limit of large distances from the surface the bulk properties should be attained. The latter provides the value of the bulk segment activity $\lambda$ obtained in Ref. [@2011] and reproduced here for convenience: $$\begin{aligned}
    \lambda &=& p_b(r_f \eta)^{-1} (\Psi^{z-1}/\Psi_x^{z-2})_b, \label{lambda_eq_st} \\
    p_b &=& [2^{-1} z r_f^{-2} \eta^{-1} x X_b^{-1} (\Psi_x^2/\Psi)_b]^{-1/x}\end{aligned}$$ where the subscript “b” stands for the bulk and the $\Psi$-factors are determined from (\[x-mer eqs bB\]) with $\delta_b = z^{\prime}/zx$. Substituting the above bulk chemical potential into (\[x-mer eqs bA\]) 