Chemistry Reference
In-Depth Information
1
10
0
(b)
1
N=256
0.9
(a)
N=32
N=64
N=128
N=256
ε
/k
B
T= 4.0
0.8
ε
/k
B
T =2.5
0.7
Simulations
y=1-5.07 X exp(-x)
ε
/k
B
T =3.0
0.6
ε
/k
B
T =4.0
0.5
2
4
6
8
10
ε
/k
B
T =5.0
ε
/k
B
T
ε
/k
B
T=10.0
0.1
10
-1
10
5
10
0
10
-1
4
10
10
1
10
2
10
3
N
10
-2
10
9
10
10
10
11
10
12
t X 8.5 X 10
4
*(1-13.7 X exp(
ε
/k
B
T))
10
-1
0.01
10
2
10
4
10
6
10
2
10
3
10
4
10
5
10
6
10
7
t [MCS]
t [MCS]
Fig. 8.6
(
a
) Time evolution of the order parameter (fraction of adsorbed segments) for four dif-
ferent chain lengths
N
=
32, 64, 128, and 256 at surface potential
/
k
B
T
=
4
.
0. The slope of
the
N
=
256 curve is 0
.
56. The
inset
shows the scaling of the adsorption time with chain length,
N
1
.
51
.Thetime
τ
∝
τ
is determined from the intersection point of the late time plateau with the
tangent
t
0
.
56
of the
surface potential. The variation of the plateau height (i.e., the fraction of adsorbed monomers at
equilibrium) with
is depicted in the
upper inset
where the
solid line n
t
→∞
=
to the respective
n
(
t
)
curve. (
b
) Adsorption kinetics for different strengths
)
describes the equilibrium number of defects (vacancies). The
lower inset
shows a collapse of the
adsorption transients on a single “master curve,” if the time axis is rescaled appropriately
1
−
5exp
(
−
/
k
B
T
due to the presence of defects (vacancies) for any given value of
k
B
T
.Forthe
transients which collapse on a master curve, cf. the second inset in Fig.
8.6
b, one
may view the rescaling of the time axis by the expression
t
/
7exp
−
k
B
T
]
as a direct confirmation of (
8.20
) where the time variable
t
may be rescaled with the
driving force of the process (i.e., with the expression in square brackets). The factor
≈
→
t
[
1
−
13
.
μ
3
/μ
2
of the effective coordination numbers in three
and two dimensions of a polymer chain with excluded volume interactions.
13
.
7 gives then the ratio
μ
3
and
μ
2
are model dependent and characterize, therefore, our off-lattice model.
The more complex adsorption kinetics, shown in Fig.
8.7
a for regular multiblock
copolymers of block size
M
and in Fig.
8.7
b for random copolymers, suggests,
however, that the power-law character of the order parameter variation with time is
retained except for a characteristic “shoulder” in the adsorption transients. Indeed,
one should bear in mind that the zipping mechanism, assumed in our theoretical
treatment, is by no means self-evident when the file of sticking
A
-monomers is
interrupted by neutral
B
segments. The characteristic shoulder in the adsorption
transients of regular multiblock copolymers manifests itself in the early stage of
adsorption and lasts progressively longer when
M
grows. The temporal length of
this shoulder reflects the time it takes for a segment from the
second
adsorptive
A
-block in the polymer chain to be captured by the attractive surface, once the
first
A
-block has been entirely adsorbed. For sufficiently large blocks one would
therefore expect that this time interval,
τ
1
, associated with the capture event, will
scale as the Rouse time,
M
1
+
2
ν
, of a non-adsorbing tethered chain of length
M
.The
observed
τ
1
vs
M
relationship has been shown in the upper left inset in Fig.
8.7
a.
The slope of
≈
1
.
49 is less than the Rouse time scaling exponent, 2
.
18, which one
