Chemistry Reference
In-Depth Information
d
d
t
P
)
=
w
−
(
)
−
w
+
(
(
1
,
t
2
)
P
(
2
,
t
1
)
P
(
1
,
t
)
for
n
=
1
(8.18)
d
d
t
P
)
=
w
+
(
)
−
w
−
(
(
N
,
t
N
−
1
)
P
(
N
−
1
,
t
N
)
P
(
N
,
t
)
for
n
=
N
and
P
fully describe the single chain adsorption kinetics.
The equation of motion for the mean number of adsorbed segments
(
n
,
t
=
0
)
=
δ(
n
−
1
)
n
=
n
=
1
nP
(
n
,
t
)
can be obtained from (
8.17
), assuming for simplicity
P
(
N
,
t
)
=
P
(
0
,
t
)
=
0:
=−
w
−
(
)
+
w
+
(
)
d
d
t
n
n
n
(8.19)
w
+
(
), w
−
(
With the relations for the rate constants,
n
n
)
, this equation of motion
becomes
1
e
−
F
dr
/
k
B
T
d
d
t
n
k
B
T
a
2
ζ
0
m
(
t
)
(
t
)
=
−
(8.20)
where for brevity we use the notations
n
. Note that (
8.20
)
reduces to the kinetic equation [
31
], derived at the end of Sect.
8.4.1
for weak driving
force,
F
dr
(
t
)
=
n
and
m
(
t
)
=
m
k
B
T
, by neglecting fluctuations in the zipping mechanism. Evidently,
by taking fluctuations into account,
F
dr
/
a
is replaced by a kind of effective
second
virial
coefficient
. Thus, the zipping as a strongly
non-equilibrium process cannot be treated quasistatically by making use of a simple
“force balance.”
(
k
B
T
/
a
)
[
1
−
exp
(
−
F
dr
/
k
B
T
)
]
8.4.3 Order Parameter Adsorption Kinetics - MC Results
The time variation of the order parameter
n
N
(the fraction of adsorbed seg-
ments) for homopolymer chains of different length
N
and strong adhesion
(
t
)/
/
k
B
T
=
4
0 is shown in Fig.
8.6
a, b whereby the observed straight lines in double-log coor-
dinates suggest that the time evolution of the adsorption process is governed by a
power law. As the chain length
N
is increased, the slope of the curves grows steadily,
and for length
N
.
=
256 it is equal to
≈
0
.
56. This value is close to the theoretically
+
ν)
−
1
expected slope of
(
1
≈
0
.
62. The total time
τ
it takes a polymer chain to be
N
α
whereby the observed
fully adsorbed is found to scale with chain length as
τ
∝
power
59,
most probably due to finite-size effects. One may also verify from Fig.
8.6
b that for
a given length
N
the final (equilibrium) values of the transients at late times
t
α
≈
1
.
51 is again somewhat smaller than the expected one 1
+
ν
≈
1
.
→∞
grow while the curves are horizontally shifted to shorter times as the surface poten-
tial gets stronger. Nonetheless, the slope of the
n
(
)
t
curves remains unchanged when
/
k
B
T
is varied, suggesting that the kinetics of the process is well described by the
assumed zipping mechanism. The changing plateau height may readily be under-
stood as reflecting the correction in the equilibrium fraction of adsorbed monomers
