chain, and neglects the long-range interactions along the chain as well as the inter-
chain interactions. Therefore, the theoretical prediction of melting points using this
approach could not agree with experimental observations.
Recently, on the basis of the classical lattice statistical thermodynamics and
Flory’s modification for semi-flexible polymers, a statistical thermodynamic theory
for the solutions of crystallizable polymers has been developed (Hu and Frenkel
2005 ). During the crystallization process, each polymer looks for its relatively
stable conformation, and meanwhile balances the compact packing tendency
between polymer chains. Therefore, polymers most often form helical chains
with parallel stacking within the compact packing structures. We can introduce
the local anisotropic parallel packing attraction parameterEpbetween two bonds,
which characterize the potential energy rise for two neighboring bonds from
crystalline parallel packing to amorphous non-parallel packing. The mean-field
theory based onEpcould predict the properties of polymer melting points, which
have been verified by the molecular simulations of parallel systems (Hu and
Frenkel 2005 ).
Crystallization of amorphous polymers not only involves the compact packing of
chain units such as small molecules, but also improves the ordering of chain
conformation. The conformational entropy change of polymers also dictates the
critical thermodynamic ordering of crystalline polymers. The calculation of
the conformational entropy employs the lattice model of polymer solutions. In the
lattice model of polymer solutions, besides the potential energy riseBfor mixing
between chain units and solvent molecules, and the potential energy riseEcfor the
semi-flexibility of the chain, the potential energy riseEpfor non-parallel packing of
polymer bonds at the neighboring positions should also be taken into account.
Starting from the fully ordered ground state, the total potential energy rise due to
non-parallel packing is
DUp¼EpQ 22 (10.6)whereQ 22 is the total pairs of non-parallel packed neighboring bonds. According to
the mean-field assumption, each chain is estimated to have the approximated amount
of bond sites potential for parallel neighbors as (q2)*(r1). Each bond site
contains the average occupation probability equivalent to the fraction of bond
occupation in the total volume, which is the total number of bond sitesNq/2 divided
by the total number of bondsN 2 (r1), as 2N 2 (r1)/(Nq). This fraction is the
probability for a parallel occupation on a neighboring bond site. Accordingly, the
probability for a non-parallel occupation of that bond site is 1 2 N 2 (r1)/(Nq).
There are in totalN 2 chains in the solution system. Considering the symmetric factor
“2” for pair interactions between the bonds, we obtain the total amount of non-
parallel packing pairs as
Q 22 ¼
1
2
N 2 ðq 2 Þðr 1 Þð 1 2 N 2
r 1
NqÞ (10.7)
10.2 Statistical Thermodynamics of Polymer Crystallization 193