Chemistry Reference
In-Depth Information
hx
2
ðtÞi 52Dt
Thus, the root mean square distance that the Brownian particles travel is:
h
p
2
p
p
2Dt
10
211
10
23
m
x
2
ð
t
Þi
5
5
3
7
:
28
3
3
24
3
3600
5
0
:
72
3
5
0
:
72 mm
The Langevin dynamics (i.e., the fluctuation dissipation theorem) can be
applied to describe the diffusion of polymer coils in dilute polymer solutions as
well. This is simply because polymer coils are generally much larger than the sol-
vent molecules so that the solvent molecules can be treated as a continuum
medium. The only difference here is that there is internal structure in these
Brownian particles that depends on the chemical composition of the polymer. In
general, the bead-spring model is used to describe the interaction between poly-
mer segments within each particle.
6.5.3
Rouse Model
The model that is used to describe the self-diffusion of polymer coils was first
proposed by Prince E. Rouse.
Figure 6.7
shows the bead-spring model represent-
ing a polymer coil. The chemical bonds are described by the well-known har-
monic oscillator approximation
[16,17]
. Each bead in the coil experiences drag
forces, proportional to its velocity, exerted by the neighboring solvent molecules.
The Langevin equation holds for each bead.
dR
n
dt
52
1
γ
@
U
R
n
1 ξ
n
(6-43)
@
R
0
R
1
R
N
FIGURE 6.7
Schematic representation of the bead-spring model for a polymer coil.
