Chemistry Reference
In-Depth Information
In the above equation, R
n
is the position of bead n (n
5
1, 2,
...
); U is the
potential energy of the springs;
ξ
n
is
the random force acting on bead n. Assuming that Hooke's law for the springs
connecting the beads (harmonic oscillator approximation) applies, the potential
energy, U, stored in the coil is:
γ
is the friction coefficient of the beads; and
2
k
X
N
n
5
1
ð
1
2
U
5
R
n
2
R
n
2
1
Þ
(6-44)
where N is the total number of beads and
3k
B
T
b
2
k
5
(6-45)
Here, b is the length of each segment. Note that in potential energy no non-
bonded interaction, bond angle, and torsion angle potential has been considered
(this can be true when segments are long enough), which reduces the complexity
of the system dramatically without considerable effect on accuracy. Combining
Eqs. (6-43) and (6-44)
yields
dR
n
dt
52
k
γ
ð
R
n
1
1
1
R
n
2
1
2
2R
n
Þ 1 ξ
n
(6-46)
To make this equation applicable to n
5
0 and n
5
N, one assumes
R
2
1
5
R
0
and R
N
1
1
5
R
N
(6-47)
If N is large enough, one can treat the variable n as continuous. As a result,
the Rouse model can be rewritten as
2
R
@
R
ð
n
;
t
Þ
k
γ
@
ð
n
;
t
Þ
t
5
1 ξð
n
;
t
Þ
(6-48)
@
@
n
2
And the conditions at n
5
0 and n
5
N simply become
@
R
ð
n
;
t
Þ
n
5
0
(6-49)
@
R
1
R
0
Origin
FIGURE 6.8
Schematic of a harmonic oscillator.
