This page is mostly for personal reference so that I can find these expressions quickly. Maybe I’ll write this up properly, maybe not.

Expressions

Total entropy of a two-polymer system: \[\begin{equation*} S = - k_\text{B} n_0 \left[-\log\left(\frac{z-1}{\mathrm e}\right) + \sum_{i\in\{\text A, \text B\}} \frac{\phi_i}{N_i} \log\left( \frac{2^{s_i} (z-1)\phi_i}{\mathrm e N_i} \right)\right] \end{equation*}\]

Total internal energy of a two-polymer system: \[\begin{equation*} U = n_0 \left[ \frac{z}{2} \sum_{i, j\in\{\text A, \text B\}} u_{ij} \phi_i \phi_j - \sum_{i\in\{\text A, \text B\}} u_{ii} \phi_i\right] \end{equation*}\] (Here, I have subtracted the interaction energy of molecules within chains with other chain-links, as these should not be counted. Note that what I have subtracted is not the energy of a pure polymer system, which would have an additional leading factor of \(z/2\).)

Total Helmholtz free energy of a two-polymer system: \[\begin{align*} F &= U - TS \notag \\ &= n_0 k_\text{B} T \left[ -\log\left(\frac{z-1}{\mathrm e}\right) + \sum_{i} \frac{\phi_i}{N_i} \log\left( \frac{2^{s_i} (z-1)\phi_i}{\mathrm e N_i} \right) + \frac{z}{2} \sum_{i, j} \frac{u_{ij}}{k_\text{B}T} \phi_i \phi_j - \sum_{i} \frac{u_{ii}}{k_\text{B}T} \phi_i\right] \end{align*}\]