Chemistry Reference
In-Depth Information
30
(a)
(b)
t=0
t=4.1*10
4
t=8.2*10
4
t=1.2*10
4
t=2.0*10
5
t=2.9*10
5
t=3.7*10
5
25
20
15
10
R(t)
m(t)
v(t)
n(t)
5
0
0
50
100
150
200
250
300
Monomer index
Fig. 8.5
(
a
) Chain conformation at successive time moments during the adsorption process for a
polymer with
N
256. The
z
-coordinate of the
i
th monomer is plotted against monomer index
i
.
(
b
) Stem-flower picture of the adsorption dynamics. The total number of adsorbed monomers
at time
t
is denoted by
n
=
. The tail which, contains all non-adsorbed monomers, consists of
a stretched part, a “stem,” of length
m
(
t
)
(
t
)
, and of a nonperturbed part - a “flower.” The rate
of adsorption is
v(
t
)
. The distance between the surface and the front of the tension propagation
is
R
(
t
)
)
ν
d
n
a
2
then follows
n
t
1
/(
1
+
ν)
≈
t
0
.
62
which is in good
ζ
0
n
(
t
(
t
)/
d
t
=
f
dr
/
(
t
)
∝
agreement with simulation results [
28
,
31
,
32
].
8.4.2 Time Evolution of the Distribution Functions - Theory
Consider the instantaneous number of adsorbed monomers
n
at time
t
(i.e., the total
train length) distribution function
P
. Using the “master equation” method [
38
],
one may derive a system of coupled kinetic equations for
P
(
n
,
t
)
by treating the
zipping dynamics as a
one-step
adsorption/desorption process within an elementary
time interval. Assuming that the corresponding rate constants
(
n
,
t
)
w
+
(
), w
−
(
of
monomer attachment/detachment are related by the detailed balance condition [
38
]
(which is an approximation for a non-equilibrium process), one can fix their ratio
w
+
(
n
n
)
)/w
−
(
, and even fully specify them by introducing a
friction-dependent
transmission
coefficient
q
n
−
1
n
)
=
exp
(
F
dr
/
k
B
T
)
a
2
a
2
)
(whereby the stem length
m
depends on the total train length
n
, according to
n
(
m
)
=
k
B
T
/(
ζ)
=
k
B
T
/(
ζ
0
m
m
1
/ν
−
≈
m
). Then the one-step master equation reads [
38
]
d
d
t
P
)
=
w
−
(
)
+
w
+
(
(
n
,
t
n
+
1
)
P
(
n
+
1
,
t
n
−
1
)
P
(
n
−
1
,
t
)
−
w
+
(
)
−
w
−
(
n
)
P
(
n
,
t
n
)
P
(
n
,
t
)
(8.17)
which along with the boundary conditions
