Polymer Physics

whereWis the weight of the network chains,ris the dry polymer density, andMcis
the average molecular weight of the network chains. Thel 1 l 2 l 3 -fold increase of
each chain volume brings an additional translation entropy. The total free energy of
the swelling system is thusF¼DFmþDFe. Taking the minimum of the total free
energy with respect toN 1 , one can obtain the equilibrium total chemical potential of
polymer chains, as given by

Dm¼DmmþDme¼ 0 (8.28)

where the molecular weight of the cross-linked polymer can be regarded as
infinity, then

Dmm¼

@Fm

@N 1

¼kTðlnf 1 þf 2 þwf^22 Þ (8.29)

On the other hand, since

f 2 ¼

V 0

V 0 þN 1 v 1

¼ð 1 þ

N 1 v 1

V 0

Þ^1 (8.30)

and v 1 is the molar volume of the solvent, one further reaches

Dme¼

@Fe

@N 1

¼

@

@N 1



3 rV 0 kT½ð 1 þ

N 1 v 1

V 0

Þ

2 = 3

 1 

1

3

lnð 1 þ

N 1 v 1

V 0

ފ

2 Mc

¼

rv 1 kTðf^12 =^3 

f 2

2

Þ

Mc

(8.31)

Then one obtains

lnð 1 f 2 Þþf 2 þwf^22 þ

rv 1

Mc

ðf

1 = 3

2 

f 2

2

Þ¼ 0 (8.32)

This equation is known as Flory-Rehner equation (Flory and Rehner 1943 ; Flory
1950 ). Given the mixing interaction parameterwand the bulk polymer densityr,by
measuring the swelling ratioQ¼f 2 ^1 upon equilibrium swelling, one can calculate
the average molecular weight of the network chains according to the Flory-Rehner
equation.
For a polyelectrolyte gel, the charge interactions can be further added into the
total chemical potentials according to Donnan equilibrium (Donnan and
Guggenheim 1932 ), and one can obtain

lnð 1 f 2 Þþf 2 þwf^22 þ

rv 1

Mc

ðf^12 =^3 

f 2

2

Þff 2 ¼ 0 (8.33)

158 8 Statistical Thermodynamics of Polymer Solutions