Biomedical Engineering Reference
In-Depth Information
Theentropyoftheidealchainisgivenby
S
(
r
)
=
k
B
ln
p
(
r
)
+
C
where
k
B
is Boltzmann's constant and
C
is a constant. As ideal chain has no internal
energy, the force appearing as a result of stretching the ideal chain is given by
T
∂
S
F
IC
=−
∂
x
T
This force has purely entropic origin. If the stretching of polymer ideal chain is
made in direction of the end-to-end vector
r
, the force exerted on the chain is given
by
F
IC
=−
T
d
S
d
r
=
k
B
T
3
r
2
Nl
2
The above result gives the equation of state of the ideal chain: the force is linearly
proportional to the temperature. Since the result was derived using the central limit
theorem, it is exact for polymers containing a large number of monomers (known
as the
thermodynamic limit
).
1.3.9.3
Wormlike Chain
The ideal chain model provides a starting point for the investigation of more
complex systems.
An important model for polymers in solution is the Kratky- Porod model,
sometimes referred to as the wormlike chain (WLC). The WLC model is used
to describe the behavior of semiflexible polymers. The model replaces a polymer
macromolecule by an isotropic rod that is continuously flexible, in contrast to
the freely joined chain model that is flexible only between discrete segments
[245-247].
For a polymer chain of length
L
, parametrize the path along the polymer chain
with variable
s
,0
≤
≤
t
(
s
) being the unit tangent vector to the chain
at
s
. It can be shown that the orientation correlation function for a WLC follows an
exponential decay law
t
s
L
.Consider
t
(0)
≡
cos
θ
(
s
)
=
e
−
s
/ξ
(
s
)
In this expression, the quantity
is a characteristic constant for a given polymer,
known as
persistence length
, regarded as a mechanical property describing the
stiffness of a macromolecule.
The stretching force
F
WLC
acting on a WLC with contour length
L
and persistence
length
ξ
ξ
is described by the following interpolation formula:
k
B
T
ξ
1
1
4
+
r
L
F
WLC
(
r
)
=
−
4
1
L
2
r
−
For small values of fraction
r
/
L
,oneobtains
1
4
1
+
k
B
T
ξ
2
r
L
1
4
+
r
L
=
k
B
T
3
r
2
F
WLC
(
r
)
≈
−
ξ
L
