polymer melt: $$\label{mu_FH}
    \mu_{BW} = \frac{ 1}{x} \ln \frac{\theta_b}{x} +\chi_f \ln r_f - \ln (1 - \theta_b) -  2\chi  \theta_b.$$ where $\chi = z (\eta - 1)/2$ is the Flory-Huggins chi-parameter, $\chi_f \equiv -x^{-1}\ln(0.5zr_f^{-2})$ is a chain stiffness parameter.

The Scheutjens-Fleer (SF) theory [@Scheutjens-Fleer] is recovered by employing the Brag-Williams approximation and using the condition of layer independent chemical potential. When these conditions are applied to (\[p\]) supplemented by (\[x-mer eqs bB\]), one finds that $$\begin{aligned}
\label{Delta_U}
 \ln p &=& \frac{1}{x} \ln \frac{\theta_b}{x} +\chi_f -\Delta U, \\
 \Delta U &=& -\ln \left( \frac{1-\theta}{1-\theta_b} \right) - 2\chi(\theta-\theta_b) - (\chi_s+\lambda_1 \chi) \delta_{i,1},\end{aligned}$$ where $\chi_s = -U_s/k_B T$ is the adsorption energy with $\delta_{i,1}$ being the Kronecker delta selecting the first layer near the surface. Performing further simplifications of (\[theta\]) with the help of (\[x-mer eqs bB\]), we obtain an expression identical to the corresponding expression in the SF theory: $$\label{theta_SF}
 \theta = \theta_b e^{\Delta U} x^{-1} \underset{k=0}{ \overset{x-1}{\sum }}P _{k}P _{x-k-1}.$$

Glass transition {#GlasTransSec}
================

Theoretically, the problem can be approached from the prospective of thermodynamic theory of glass transition by Gibbs and DiMarzio [@Gibbs-DiMarzio] and the dynamic extension of Adam and Gibbs [@Adam_Gibbs]. The thermodynamic origin of the glass transition was 