Polymer Physics

thus they are not geometrically identical in the lattice space. In 1991, Dudowicz and
Freed proposed a cluster of interconnected lattice sites to reflect the various
geometries of chain units (Dudowicz and Freed 1991 ), and the results gave

DFm

NkT

¼

f 1

r 1

lnf 1 þ

f 2

r 2

lnf 2 þf 1 f 2 fð

g 1 g 2

q

Þ^2 þ

B

kT

½q 2 

2

q

ðh 1 f 2 þh 2 f 1 ފg

(8.51)

whereg 1 andg 2 were determined by the structures of chain units,h 1 andh 2 were the
combinatorial methods for three consecutive bonds passing through chain units of
the corresponding species. Such an equation can be reduced to the Flory-Huggins
equation. The asymmetry in the molecular geometries of two components in
polymer blends may lead to the LCST-type phase diagram, see more introductions
about LCST in Sect.9.1.

8.3.6 Semi-Flexible Polymers

In 1956, Flory introduced the semi-flexibility into the classical lattice statistical
thermodynamic theory of polymer solutions (Flory 1956 ). From the classical lattice
statistics of flexible polymers, we have derived the total number of ways to arrange
polymer chains in a lattice space, as given by (8.15). The first two terms on the
right-hand side of that equation are the combinational entropy between polymers
and solvent molecules, and the last three terms belong to polymer conformational
entropy. Thus the contribution of polymer conformation in the total partition
function is

Zconf¼½

qðq 1 Þðr^2 Þ

2 eðr^1 Þ

ŠN^2 (8.52)

Here, 1/2 is the symmetric factor for the two chain ends, i.e. the first putting
monomer can be either one of the two chain ends; putting the second monomer like
random walks, which hasqchoices; starting from putting the third monomer, there
are q1 choices for non-reversing random walks; the natural numberecan be
regarded as a correction to each step of random walks due to the coexistence of
other chains. The Huggins’ surface fraction (8.11) derives the correction term as
(12/q)^1 q/2, which approaches the natural number whenq!1(Flory 1982 ). For
semi-flexible chains, Flory assumed that starting from putting the third monomer,
each step of random walks is no longer random, but rather, follows a partition
function according to Boltzmann’s distributions to assign the probabilities of
conformations to their conformational energy. Assuming a collinear connection
of the bonds corresponds to thetrans-conformation at the ground state with
zero energy, and the non-collinear connection of the bonds corresponds to

8.3 Developments of Flory-Huggins Theory 163