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more complicated cases. For random copolymers the role of the block length
M
would then be played by the typical correlation length.
8.4.4 Time Evolution of the Distribution Functions - MC Data
One gains most comprehensive information regarding the adsorption process from
the time evolution of the different building blocks (trains, loops, and tails) proba-
bility distribution functions (PDF) [
39
]. From the MC simulation data, displayed
in Fig.
8.8
a, for example, one may verify that the resulting distribution
D
of
different train lengths is found to be exponential, in close agreement with the theo-
retically expected shape [
39
], predicted under the assumption that local equilibrium
of loops of unit length is established much faster than the characteristic time of
adsorption itself. When scaled with the mean train length
h
av
(
(
h
,
t
)
t
)
=
h
(
t
)
, at time
t
,
in both cases for
0 and 5.0 one finds an almost perfect straight line in
semi-log coordinates. One may thus conclude that
D
/
k
B
T
=
3
.
preserves its exponential
form during the course of the adsorption process, validating thus the conjecture
of rapid local equilibrium. The latter, however, is somewhat violated for the case
of very strong adsorption,
(
h
,
t
)
0, where the rather scattered data suggest
that the process of loop equilibration is slowed down and the aforementioned time
separation is deteriorated.
The PDF of loops
W
/
k
B
T
=
5
.
at different times after the onset of adsorption is shown
in Fig.
8.8
b. Evidently, the distribution is sharply peaked at size 1 whereas less than
the remaining 20% of the loops are of size 2. Thus the loops can be viewed as
single thermally activated defects (vacancies) comprising a desorbed single bead
with both of its nearest neighbors still attached to the adsorption plane. As the inset
(
k
,
t
)
0.06
(a)
(b)
10
-1
ε/k
B
T =3.0
10
-1
ε/k
B
T=5.0
0.05
10
-2
10
-2
t=1.64
t=3.28
t=6.55
t=9.83
t=13.1
t=16.4
t=19.7
t=1.64
t=3.28
t=6.55
t=9.83
t=13.1
t=16.4
t=19.7
10
-3
0.04
10
-4
10
-2
10
-5
0.03
10
20
N =
256
k
10
-3
ε/k
b
T = 4.00
t=0.16
t=0.82
t=1.64
t=3.28
t=4.92
y=0.05
X
exp(-0.83x)
0.02
10
-3
0.01
10
-4
10
-4
0
0
10
20
0
10
20
0
2
4
6
8
10
12
14
16
18
2 0
h(t)/〈h(t)〉
h(t)/〈h(t)〉
k
Fig. 8.8
(
a
) Distribution of train lengths during the adsorption process of a homopolymer chain
with
N
, shown in semi-log coordinates. PDFs
for different times (in units of 10
5
MCS) collapse on master curves when rescaled by the mean
train length
h
av
=
256 at two strengths of the adsorption potential
(
t
)
.(
b
) Distribution of loop lengths
W
(
k
,
t
)
for
N
=
256 and
/
k
B
T
=
4
.
0 during
ongoing polymer adsorption. In the
inset
the PDF is normalized by
n
(
t
)
and shown to be a
straight
line
in log-log coordinates
