Chemistry Reference
In-Depth Information
2
ln
1
l
2
l
0
l
max
−
−
U
FENE
=−
K
(
1
−
l
0
)
−
(8.1)
l
0
with
K
4.
The nonbonded interactions are described by the Morse potential:
=
20
,
l
max
=
1
,
l
0
=
0
.
7
,
l
min
=
0
.
U
M
(
)
M
=
r
exp
(
−
2
α(
r
−
r
min
))
−
2exp
(
−
α(
r
−
r
min
))
(8.2)
with
1.
We use periodic boundary conditions in the
x
α
=
24
,
r
min
=
0
.
8
,
M
/
k
B
T
=
y
-directions and impenetra-
ble walls in the
z
-direction. We study homopolymer chains, regular multiblock
copolymers, and random copolymers (with a fraction of attractive monomers,
p
−
=
0
.
25
,
0
.
5
,
0
.
75) of length 32, 64, 128, 256, and 512. The size of the
box is 64
64 in all cases except for the 512 chains where we use
a larger box size of 128
×
64
×
128. The standard Metropolis algorithm is
employed to govern the moves with self-avoidance automatically incorporated in
the potentials. In each Monte Carlo update, a monomer is chosen at random and
a random displacement attempted with
×
128
×
x
,
y
,
z
chosen uniformly from the
−
.
≤
,
,
≤
.
interval
0
5
x
y
z
0
5. The transition probability for the attempted
move is calculated from the change
U
of the potential energies before and after
the move as
W
. As for a standard Metropolis algorithm, the
attempted move is accepted if
W
exceeds a random number uniformly distributed
in the interval
=
exp
(
−
U
/
k
B
T
)
.
Typically, the polymer chains are originally equilibrated in the MC method for
a period of about 10
6
MCS (depending on degree of adsorption
(
0
,
1
)
and chain length
N
this period is varied) whereupon one performs 200 measurement runs, each of
length 8
10
6
MCS. In the case of random copolymers, for a given composition, i.e.,
percentage
p
of the
A
-monomers, we create a new polymer chain in the beginning
of the simulation run by means of a randomly chosen sequence of segments. This
chain is then sampled during the course of the run and replaced by a new sequence
in the beginning of the next run.
×
8.2.2 Coarse-Grained Lattice Model with PERM
The adsorption of a diblock
AB
copolymer with one end (monomer
A
) grafted to
a flat impenetrable surface and with only the
A
-monomers attractive to the surface
is described by self-avoiding walks (SAW) of
N
−
1 steps on a simple cubic lattice
with restriction
z
≥
0. The partition sum may be written as
Z
(
1
N
(
q
N
s
q
)
=
A
N
(
N
s
)
(8.3)
N
s
where
A
N
(
N
s
)
is the number of configurations of SAWs with
N
steps having
N
s
sites on the wall, and
q
e
/
k
B
T
=
>
is the Boltzmann factor (
0 is the attractive
