Chemistry Reference
In-Depth Information
Expanding the free energy in Equation 2.89 to the fourth power gives the
Landau-de Gennes form. One can evaluate the thermodynamic properties
accordingly.
In the elastically-jointed-rod chain, the mean square end-to-end distance
along the director tends to be a rod as
q
approaches infinity or
T
is very
small,
i.e.
,
R
z
→
N
2
l
0
as
b
→∞
.
(2.94)
2.4.4.
Discrete and continuum chain models
If we take the limit of the rod length
l
0
→
→∞
such that the polymer length is kept constant
L
=
Nl
0
, we have an integral
instead of a summation:
0 and the number of rods
N
L
L
=
i
h
i
→
ds,
(2.95)
0
where
h
i
is the length of each segment along the jointed-rods and
s
is the
contour length of a point along a continuous chain. The discrete elastically-
jointed-rod chain thus evolves into a continuous elastic chain or the worm
chain. In this case, the bending elastic energy becomes
2
qh
θ
i
+1
−
2
L
dsε
dθ
ds
2
,
U
el
→
i
h
1
θ
i
1
2
→
(2.96)
h
0
→∞
and
l
0
→
where the elastic constant
ε
=
ql
0
as
q
0. Meanwhile,
the nematic part,
U
nem
, becomes
L
U
nem
→−
i
h
p
h
SP
2
(cos
θ
i
)
→−
dsvSP
2
(cos
θ
(
s
))
,
(2.97)
0
where
v
=
p/l
0
as
l
0
→
0 and
p
→
0.
v
is the nematic coupling constant
per unit length.
These two parts constitutes the energy for a worm-like nematic poly-
mer. The first part (Equation 2.96) demands a slow reversal as a chain
evolves and favors parallel alignment, but the second term (Equation 2.97)
prefers both anti-parallel and parallel alignments, and a rapid reversal of
