Environmental Engineering Reference
In-Depth Information
=
j
=
1
F
ji
C
where
F
i
C
is the total conservative force on DPD particle
i
:
F
i
C
with
F
ji
C
the force applied by particle
j
on particle
i
; and from
E
k
we may calculate
the fluid pressure tensor
N
N
1
V
P
ʱʲ
=
m
i
v
i
ʲ
v
i
ʱ
+
F
i
ʲ
ʱ
i
.
(36)
i
=
1
i
=
1
Here,
m
i
is the mass of particle
i
(which we set equal to 1 in DPD-units) and
v
i
ʱ
is
the
ʱ
-component of the velocity of particle
i
in the volume
V
. Similarly,
F
i
ʲ
is the
ʲ
-component of the force
F
i
on particle
i
,
ʱ
i
is the
ʱ
-coordinate of particle
i
,etc.
Equation (
4
) may then be used to calculate directly the interfacial tension
ʳ
at the
volume boundary, with
ʳ
=
(
r
c
) ʳ
∗
.
ʳ
depends on temperature. From the mechanical work needed to increase a surface
area,
dW
k
B
T
/
=
ʳ
dA
,wehave
∂
G
∂
A
ʳ
=
n
,
(37)
T
,
P
,
with
G
the Gibbs free-energy and
A
the surface area. Given that all spontaneous
thermodynamic processes follow
0, it is easy to understand why the liquid
tries to minimise its surface area. From its definition,
G
ʔ
G
<
=
H
−
TS
with
H
the
enthalpy and
S
the entropy of the system. Thus
∂ʳ
∂
T
S
A
P
=−
(38)
A
,
so that the normal behaviour of
ʳ
is to decrease with temperature.
Results concerning the study of the interfacial tension between immiscible mix-
tures such as benzene/water and ciclohexane/water at different temperatures, using
the parametrisation mentioned above and performing DPD simulations, can be found
in Mayoral and Gama-Goicochea (
2013
). Results taken from this reference are shown
in Fig.
3
, where the interfacial tension obtained by DPD simulations are compared
with experimental data. The excellent agreement confirms that the parametrisation
via the use of solubility parameters at different temperatures to obtain the repulsive
DPD parameters
a
ij
as functions of
T
is appropriate for introducing the effect of
temperature in DPD simulations.
Additionally, the interfacial tension between two species will change when an
electrolyte is added at different concentrations, since the cohesive forces between
neighbouring molecules will be altered. Its behaviour with concentration will depend
strongly on the type of electrolyte. Figure
4
(top) shows the behaviour of the interfa-
cial tension
ʳ
∗
between
n
-dodecane and water with sodium chloride
NaCl
added,
obtained by DPD electrostatic simulations. In this figure
M
denotes the num-
ber of DPD ions added as molar concentration. The increase with salt concentration
is expected, and the same behaviour is observed when several other inorganic salts
are added (Mayoral and Nahmad-Achar
2012
). The opposite behaviour is observed,
[
NaCl
]
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