Chemistry Reference
In-Depth Information
If we consider a harmonic oscillator (see
Figure 6.8
) in a solvent, the position
of the oscillator as a function of time is (see Problem 6-5)
ð
t
k
t
ð
t
2
t
_Þ
e
2
x
ð
t
Þ 5
ξð
t
Þ
dt
(6-50)
2
N
The mean square displacement of such an oscillator is (see Problem 6-5)
2
k
B
T
γ
2
hð
x
ð
t
Þ2
x
ð
0
ÞÞ
i 5
t
5
2Dt
(6-51)
Solution of the partial differential equations in
Eq. (6-48)
is not easy as the
positions of beads are dependent on each other. To overcome this problem, the
motion of beads is required to decompose into independent modes using normal-
ized coordinates. The procedure is complicated and not presented here. The result
of such mathematical exercise leads to
Eq. (6-52)
:
k
B
T
N
D
G
5
(6-52)
γ
As you can see from the above equation, the self-diffusion coefficient of the
center of mass of a polymer coil, D
G
, in a dilute polymer solution has an inverse
relationship of the number of monomers. However, such a relationship is not
observed experimentally. In fact, experiment shows that D
G
of polymers is pro-
portional to molecular weight (number of monomers) with an exponent
ν
differ-
ent from unity:
M
2ν
D
G
~
(6-53)
3/5. The discrepancy
between the scaling factor obtained from the Rouse model and that of the experi-
ment is attributed to the fact that in the Rouse model the so-called hydrodynamic
interactions (i.e., the motion of the solvent molecules being dragged along by the
motion of the connected beads) are ignored. Bruno H. Zimm extended the Rouse
model by including the hydrodynamic interactions.
In a
θ
solvent, v
5
1/2 and in a good solvent, v
5
6.5.4
Zimm Model
Hydrodynamic interactions do not change the potential energy, U, but they do
alter the frictions experienced by the beads. The detailed analysis of the hydrody-
namic interactions is beyond the scope of this volume and is not presented here
[16]
. According to the Zimm model, the following equation for the self-diffusion
coefficient of the center of mass of a chain in dilute polymer solution is given by
k
B
T
η
s
D
G
5
0
:
196
p
b
(6-54)
