Polymer Physics

AssumingN 2 /V¼cNa/M, wherecis the mass concentration of polymers, then

P

c

¼RTð

1

M

þ

Nau

2 M^2

cÞ (8.41)

One can see that the second Virial coefficient for the interactions between
polymer coils in dilute solutions is directly determined by the self-repulsion volume
uof each polymer coil.
The thermodynamic excluded volumeuof each coil is related with the free
energy changeDFoupon the overlapping of two coils, thus

u

ð^1

0

ð 1 eDFo=kTÞ 4 pa^2 da (8.42)

whereais the distance between mass centers of polymer coils. By assuming the
Gaussian distribution of monomers along the radius direction of the coil starting
from its mass center, the Flory-Krigbaum theory proved a linear relation between
DFoand 1y/T. At the theta point, the excluded volume of coils will be zero, and
such an unperturbed state leads to zero second Virial coefficient between two coils.
The numerical result (4.44) derived from the Flory-Krigbaum theory has been
discussed in Sect.4.2.4.
In a rough approximation, Flory-Huggins equation also gives the practical
expression for the measurement of molecular weightM,as

P

c

¼

RT

M

þ

RTv^2

v 1

ð

1

2

wÞc¼C 1 þC 2 c (8.43)

whereC 1 andC 2 are constants for a specific polymer solution. This implies that
under the limit of dilute solution, the mean-field theory seems to work well. When
w¼1/2, polymers in solutions reach their unperturbed states.
In 1959, Maron considered polymer coils as a whole with an effective hydrody-
namic volumeef 2 in dilute solutions (Maron 1959 ; Maron and Nakajima 1960 ).
Hereeis a prefactor for the effective polymer volume, which exhibits an empirical
dependence on polymer concentrations as

1

e

¼

1

e 0

þð

e 0 e 1

e 0

Þf 2 (8.44)

where for infinitely diluted solutions,e 0 ¼[]/2; whenf 2 increases till viscosity
!1,e 1 4. The total volume of the system also changes fromV 0 (before
mixing) toV(after mixing), and the mixing interactions are not independent of the
entropy. Accordingly, the Flory-Huggins equation has been developed into

DGm¼RTðX 0 þwN 1 ’ 2 Þ (8.45)

8.3 Developments of Flory-Huggins Theory 161