Chemistry Reference
In-Depth Information
On average, for each atom in a polymer chain, there are
166:5 N
a
0:01402 N
a
water atoms
surrounding the polymer chain. Since one chain contains 1402 atoms, the total number of
water molecules bombarding the polymer chain is given by:
166:5 N
a
0:01402 N
a
3
1
3
5
10
5
1402
3
55
:
5
3
The above result shows that considering water molecules explicitly would
introduce a large number of degrees of freedom, which makes it mathematically
impossible to handle the fluctuating forces.
The next step is to correlate
Eq. (6-33)
to the mean square displacement of the
Brownian particle, thereby the self-diffusion coefficient. Multiplying both sides of
Eq. (6-33)
by x(t) and utilizing the relation that x v
5
xe
0
0308
d
dt
ð
x x
Þ 2
x
2
,
;
x
5
Eq. (6-33)
becomes
m
d
x
2
dt
ð
x
x
_
Þ 5
m
_
2γ
x
x
_
1
x
ξ
(6-34)
Dividing the above equation by m and taking the average of all terms, the fol-
lowing equation is obtained:
d
dt
h
i 2
m
h
1
m
x
x
2
_
i 5 h _
_
i 1
ξ
x
x
x
x
(6-35)
The last term on the right-hand side would vanish due to the fact that the
mean values of x and
ξ
are zero (
h
x
ξi 5 h
x
ihξi 5
0). The reason that
h
x
ξi 5 h
x
ihξi
is that there is no correlation between x and
ξ
. This independence is only valid at
ξ
(t)
− γν
ν
FIGURE 6.5
Schematic illustrating the forces acting on a Brownian particle.
