In order to calculate the system free energy, the statistical thermodynamics needs to
account for all the possible spatial arrangements of certain amount of molecules
distributed on various energy levels. This task appears to be an impossible mission
because of the huge amount of molecules involved, as high as on the order
of Avogadro constant. The mean-field assumption is often used to simplify the
calculation by assuming that each molecule experiences an averaged force field.
Therefore, the possible spatial arrangements for a molecule at each energy level are
presumably independent of the number of molecules experiencing that energy.
According to the Boltzmann’s relationship, the summation of the former corresponds
to the entropy, while the summation of the latter corresponds to the internal heat. One
can thus make separate statistics on the contributions of mixing entropyDSmand
mixing heatDUmto the total free energy change upon mixing.
In the following, we will start from the fully ordered and phase separated state as the
ground state of free energy, and aim at the randomly mixing state, to calculate first
the mixing entropy and then the mixing heat, as illustrated in Fig.8.5. We will derive the
expression of mixing free energy, as the so-called Flory-Huggins equation.
8.2.3 Calculation of Mixing Entropy
The mixing entropy normally contains the contributions of translation, rotation,
vibration and combination, as
Sm¼StranslationþSrotateþSvibrateþScombinate (8.4)Since the fourth term is obtained from the combination of chain units and solvent
molecules, the first three terms for the whole chain are far less than the fourth one if
considering their global contributions in the lattice space. Therefore, only the
combination entropyScombinateis calculated. According to the Boltzmann’s relation
Scombinate¼klnO, we only need to calculate the total amount of arrangement of
molecules in the lattice space.
Fig. 8.5 Illustration of the route to calculate the mixing free energy of polymer solutions
152 8 Statistical Thermodynamics of Polymer Solutions