Chemistry Reference
In-Depth Information
(2) The rods are long enough. The molecules can be cylindrical, semi-
sphero-cylindrical or even lathe. The basic conclusions for these
molecular shapes are quite similar. In the text below, only the solu-
tions of cylindrical molecules are described. The calculation can
be easily extended to deal with disc-like molecules by varying the
length-to-breadth ratio.
(3) The solution is dilute. In a dilute solution, the density of the system,
ρ
=
N/V
, with
N
being the total number of rods in a solution of volume
V
, is small. Thus the volume fraction of rods in the solution is
Φ=
πD
2
4
Lρ.
(2.2)
2.1.1.
The partition function
The orientational distribution function of rods with long axis
a
is designated
by
f
(
a
).
ρf
(
a
)dΩ
a
is the number of rods which point to the solid angle dΩ
centered at
.
First of all, we examine the partition function
Z
— an important func-
tion in thermodynamics and statistics, and calculate the free energy of the
system according to the formula
a
−
k
B
T
ln
Z,
(2.3)
where
T
is the Kelvin temperature and
k
B
is the Boltzmann constant. The
thermodynamical quantities associated with the equilibrium state can then
be obtained. The calculation is similar to that for a simple fluid with
N
identical spherical particles, however it is more complicated because of the
anisotropic shape. The partition function is the volume in the phase space or
the summation over all the possible configurations that the system adopts.
For the system of cylindrical molecules, the partition function is given by
F
=
N
!
h
6
N
exp(
1
···
−
βU
)
dτ
1
dτ
2
···
dτ
N
,
(2.4)
Z
=
where
β
=1
/
(
k
B
T
). The Planck constant
h
is introduced in the denom-
inator to make
Z
dimensionless. The factor of 1/
N
! arises from the
indistinguishability of
N
molecules. A rigid rod has 6 degrees of free-
dom, three from translational motion and the rest from rotational motion.
The infinitesimal volume element
τ
i
of the phase space is
dτ
i
=
d
r
i
·
d
p
i
·
dθ
i
·
dφ
i
·
dψ
i
·
d
p
θi
·
dp
φi
·
dp
ψi
,
