Biomedical Engineering Reference
In-Depth Information
This means that the force of tension is equal to the change in Helmholtz
free energy per unit extension. In the normal unstressed state, the free en-
ergy has a minimum (
∂
A
/∂
x)
T
=
0at certain
x
=
x
0
,andforsmallstrainswe
get
∂
∂
x
2
2
A
F
=
(
x
−
x
0
)
T
where
x
denotes the length of the system under the force
F
. In this approximation,
the force is a linear function of the deformation (
x
−
x
0
). The tension force, like
free energy, may be expressed as the sum of two terms
∂
A
∂
x
∂
U
∂
x
T
∂
S
∂
x
F
=
=
τ
−
τ
In general, the force results in both the internal energy and entropy changes.
τ
1.3.9.2
Ideal Chain
Ideal chain model (Gaussian chain) or the freely joined chain assumes that there
are no interactions between chain monomers. The ideal chain is the simplest model
of a polymer. In this model, fixed length polymer segments are linearly connected,
and all bond and torsion angles are equiprobable: the polymer can be described by
the random walk statistics [242-244].
Monomers are regarded as rigid rods (segments) of a fixed length
l
,and
their orientation is independent of the orientations and positions of neighboring
monomers. This means that no interactions between monomers are considered,
the energy of the polymer is taken to be independent of its shape, and all of its
configurations are equally probable. If
N
monomers form the polymer chain, its
total length is
L
=
Nl
.
...
,
b
N
−
1
be the vectors corresponding to individual monomers, and let
r
denote the end-to-end vector of a chain, that is, the distance between the ends of the
chain, Figure 1.38. The monomer vectors have randomly distributed components
in the three directions of space. With the assumption that the number of monomers
N
is large, and the central limit theorem applies, the length
r
is described by the
following Gaussian probability density function:
Let
b
0
,
3
2
Nl
2
3
/
2
3
r
2
e
−
Nl
2
p
(
r
)
=
2
π
π
5
4
b
3
2
r
3
Figure 1.38
Chain of rigid, randomly oriented rods.
In this example, we have five rods only, with directions
described by vectors
1
b
0
,
...
,
b
4
, and with end-to-end
0
vector
r
.
