Chemistry Reference
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Fig. 20 Nematic order
parameter (a) and volume
fraction
f
of lattice sites
taken by monomers (b)
plotted versus the normalized
chemical potential
mðk
B
T
1
is chosen here), for the bond
fluctuation model on the
simple cubic lattice, for
chains with
N
20 and the
bond angle potential (
9
) with
f
8. The
vertical lines
show
the value of the chemical
potential at the transition
point,
m
5,
estimated from the analysis of
the osmotic pressure of this
polymer solution (see
Fig.
21
).
Triangles
refer to
results obtained from the
densely packed starting
configuration, while
squares
correspond to the dilute
isotropic starting
conformation. These data
were omitted in (a) because
the system relaxes into a
metastable nematic
multidomain rather than into
the stable ordered
monodomain configuration.
From Ivanov et al. [
123
]
166
0
:
general [
18 20
], the nematic isotropic transition shows up via pronounced hyster-
esis in the
S
versus
m
curve (Fig.
20a
) and a (weaker) hysteresis in the corresponding
density variation (Fig.
20b
). In order to be able to locate the transition point from
the isotropic to the nematic solution precisely, a thermodynamic integration method
in the grand canonical ensemble (TI
m
VT method) was used. Denoting the chemical
potential of a ideal gas of chains as
m
id
, and defining
m
ex
¼ m m
id
, the osmotic
pressure
p
of the solution becomes
ð
k
B
T
1 here) [
290 293
]:
Z
r
0
m
ex
ðr
0
Þ
d
r
0
p ¼ rð
1
þ m
ex
Þ
;
(52)
where
r ¼N=
V
is the density of polymer chains in the system (note that
r ¼ f=ð
in our model). Of course, in practice the integral in (
52
) is discretized,
but for a very good accuracy clearly a large number of state points
8
N
Þ
need to
be simulated to render the discretization error negligible. Despite this disadvantage,
this old [
290
] method is still superior in accuracy to any other approach [
123
]. The
ðm
i
;
T
;
V
Þ
