Polymer Physics

Let us firstly pickjchains from the fully ordered initial state, and randomly put them
into the lattice space with a volumeN, without any overlapping. Then we look at how
to put the representative (jþ1)th chain. According to the incompressible mixing
assumption, the first monomer can be randomly put into any one of the restN-rjvacant
sites, so there areN-rjpossible ways; the second monomer can only be put into any one
ofqcoordination sites surrounding the first monomer, and according to the assumption
of concentrated solutions, the vacant probability of each coordination site is the
concentration of vacant sites 1(rjþ1)/N, thus the number of possible ways to put
the second monomer isq[1(rjþ1)/N]; according to the flexible chain assumption,
the third monomer can be put into any one ofq1 coordination sites surrounding the
second monomer, each with the vacant probability 1(rjþ2)/N, so the number of
possible ways to put the third monomer is (q1)[1(rjþ2)/N]; so on so forth, ther-th
monomer can be put into any one ofq1 coordination sites surrounding the previous
monomer, each with the vacant probability 1(rjþr1)/N, so the number of possible
ways to put this monomer is (q1)[1(rjþr1)/N]. Therefore, the total number of
ways to put the (jþ1)th chain is

Wjþ 1 ¼qðq 1 Þr^2 ðNrjÞP

r 1

i¼ 1

ð 1 

rjþi

N

Þ¼

qðq 1 Þr^2

Nr^1



ðNrjÞ!

ðNrjrÞ!

(8.5)

It should be noted that the probability of a vacant coordination site surrounding
ith monomer ispij¼ 1 (rjþi)/N, which is on the basis of assumption two, i.e. the
so-calledrandom-mixing approximation. On this point, Flory and Huggins made
different treatments. Let’s consider theith monomer that has been put into a
previously vacant site, the (iþ1)th monomer has to be put into a vacant coordina-
tion site surrounding the previously vacant site. Therefore,Pijshould be the fraction
of two consecutively connected vacant sites in the total pairs of two neighboring
sites containing one vacant site. The total vacant sites areNrji, and their total
coordination number isq(Nrji), each with the vacant probability 1(rjþi)/N,
so the total number of two consecutively connected vacant sites is

Nvoidvoid¼

1

2

qðNrjiÞð 1 

rjþi

N

Þ (8.6)

Here 1/2 is the symmetric factor for estimating the vacant site pairs twice.
Similarly, the total amount of coordination site pairs in the lattice space isqN/2.
In 1942, Flory did not consider the consecutive occupation of vacant site pairs
(Flory 1942 ), and calculated only those neighboring site pairs containing one vacant
site, as

Nvoid¼

1

2

qNð 1 

rjþi

N

Þ (8.7)

8.2 Flory-Huggins Lattice Theory of Polymer Solutions 153