liquid crystals, they occur before crystallization just because the latter requires a
large degree of supercooling, as illustrated in Fig.10.6. When the cooling curve
shows two consecutive phase transitions and the heating curve shows only phase
transition for crystal melting, it is known as the monotropic phenomenon.
10.2 Statistical Thermodynamics of Polymer Crystallization
Statistical thermodynamics are the theoretical bridge connecting microscopic
molecular parameters to macroscopic thermodynamic properties (such as the
melting-point). This approach is mainly based on the calculation of the partition
function that is the sum of all the possible microscopic states mainly following the
Boltzmann distribution. However, the number of microscopic states is so huge that
we could not count them one-by-one, but rather treat the representative states with
appropriate statistical approximations. The most often used statistical approxima-
tion is the mean-field assumption, which uses an averaged field to treat the
interactions that each molecule experiences from all other molecules. Accordingly,
both the numbers of microscopic energy levels and the molecules on each level can
be separately calculated with certain simplifications.
Flory and Huggins have derived the statistical thermodynamic theory of polymer
solutions by using a lattice model and the mean-field assumption. They obtained the
equations for mixing entropy, mixing enthalpy and then the mixing free energy. The
Flory-Huggins theory assumes the polymer chains are flexible, i.e. the same
energies can be shared by different microscopic conformational states selected
via internal rotation. However in practice, most of polymers are not so flexible,
and various microscopic conformation states contain different energies. Therefore,
Flory modified the lattice statistical thermodynamic theory by introducing the
conformational energy parameterEc(Flory 1956 ).Ecis the energy difference
between the collinear and non-collinear connections of two consecutive bonds
along the chain, and the magnitude ofEcreflects the semi-flexibility of polymer
chains. But this approach only considers the short-range interactions along the
Fig. 10.6 Illustration of monotropic phenomenon on the cooling and heating curves of the
metastable mesophases. (a) Free energy curves; (b) DSC heating and cooling curves
192 10 Polymer Crystallization