grad partition function of a single layer $Z^{\ast}_{i}$ that corresponds to the equilibrium $\theta_i^{\ast}$ in the layer $i$ is given by $$Z_i^{\ast}=\sum_{\theta_{11},\,\theta_{g},\, \theta_{s}} \Omega (\theta_{i}^{\ast},\theta_{11},\theta_{g},\theta_{s}) \eta^{\theta_{c,i}L} w_{f}^{\theta_{g} L} w_{s} ^{\theta_{s} L},
\label{GPF}$$ where $\Omega $ is the number of arrangements on a layer of $L$ sites that have the same densities $\theta_{11}(\theta_i^{\ast})$, $\theta_{g}(\theta_i^{\ast})$, $\theta_{s}(\theta_i^{\ast})$.

The standard method to calculate the equilibrium grand partition function is to use the Bethe-Peierls approximation. [@Chang1939] The starting point is to take the Cayley tree embedded between 3 consecutive layers with the central site site placed in the layer $i$ surrounded by $z$ first generation sites situated in the same layer. This structure is thought to represent an aggregate of segments in the sea of molecules constituting the polymer melt. For the sake of tractability, we assume that the Cayley tree structure is maintained outside the aggregate meaning that this structure possesses no closed loops at least for the $x$ number of generations sufficient to embed a single chain. In order to denote the state of occupation of the above $z+1$ sites, we define a set of numbers $\{\delta _{j}\}=\delta _{0},\delta _{1},...,\delta _{z}$ that assume the values of 0 or 1 for empty and occupied 