n - PATh :.: Process and Product Applied Thermodynamics research ...

( ρ)
ρ
, 0 ≤ ρ
( ρ)
2
Modeling
Ω n
max
da n ( ρ ) = −K
n ln
≤
(III.21)
l
Ω
n
s
ρ max
da n ( ρ ) = 0, ≤ ρ ≤ ρ max
(III.22)
2
where ρmax is the maximum possible molecular density and it depends on the selected
model. In this work it was defined as
ρ
max
1
= (III.23)
3
mN σ
A
When the density is greater than half the maximum density, the long-wavelength
fluctuations are not relevant compared to the mean-field value, since the critical region is
still far away. Ω s and Ω l represent the density fluctuations for the short-range and long-
range attraction, respectively, and Kn is a coefficient,
k T
= (III.24)
B
K n 3n
3
2 L
where T is the temperature and L the cut-off length.
( ρ,
x)
⎤
dx
β
β
ρ ⎡− Gn
Ω n ( ρ ) = ∫ exp ⎢ ⎥ (III.25)
0
⎣ K n ⎦
G
β
n
( ρ,
x)
β
β
β
( ρ + x)
+ a n ( ρ + x)
− 2a
n ( ρ)
a n
= (III.26)
2
The superindex β refers to both long (l) and short (s) range attraction, respectively, and
β
G is a function that depends on the evaluation of the function a , calculated as
( ) ( ) ( ) 2
ρ a ρ + α mρ
l
a n
n 1
= − (III.27)
2
φw
ρ (III.28)
+
2 L
2
( ) = an−
1(
ρ)
+ α(
mρ)
2 1 2
s
a n
n
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