Environmental Engineering Reference
In-Depth Information
e
2
ʓ
=
/(ʺ
B
T
0
r
R
c
)
where
Z
i
is the valence of ion
i
,
e
is the electron charge,
,
and
ʲ
∗
=
3
,
N
A
is the number of Avogadro,
N
m
is the
number of real water molecules inside of a DPD particle,
ˁ
∗
is the reduced density
of DPD particles and
V
m
is the molar volume of water. The magnitude of reduced
force between two charge distributions is
R
c
/ʻ
;
R
c
=
(ˁ
∗
N
m
V
m
/
1
/
N
A
)
4
ˀ
F
∗
e
ij
Z
i
Z
j
r
∗
2
e
−
2
ʲ
∗
r
∗
2
ʲ
∗
r
∗
(
+
ʲ
∗
r
∗
)
]}
.
=
{
1
−
[
(
1
+
1
(14)
ʓ
The Coulombic term appearing in the equations can be obtained in a simulation
using the standard Ewald scheme. The charge distribution is included by removing
the divergency of the Coulomb interactions at
r
∗
0. The energy and the force
between two charged distributions are finite quantities in the limit
r
∗
ₒ
=
0 and are
given by:
4
ˀ
u
∗
(
r
)
Z
i
Z
j
ʲ
∗
,
lim
r
=
(15)
ʓ
ₒ
0
and
4
ˀ
F
∗
e
ij
lim
r
∗
ₒ
=
0
,
(16)
ʓ
0
respectively.
The various parameters introduced, viz.
a
ij
, ˃, ʳ, ʸ
ij
, contain all the information
of the particular system being considered. It is therefore crucial to be able to establish
these parameters faithfully in order to make the DPD methodology to work.
3 Parametrisation for Realistic Systems
By far the most important parameter is the one defining the conservative force,
a
ij
,
because it contains all the physicochemical information for each component in the
system. In contrast, the noise and dissipative parameters correspond to the tempera-
ture and fluid viscosity, respectively. In a mono-component system the conservative
force parameter for equal species
a
AA
≡
a
relates to the inverse isothermal com-
pressibility (Groot and Warren
1997
)
1
nk
B
T
ʺ
T
=
1
ʺ
−
1
=
)
T
,
(17)
k
B
T
(∂
p
/∂
n
where
n
is the number density of molecules and
ʺ
T
)
T
is the usual isother-
mal compressibility. The pressure
p
in the system may be obtained using the virial
theorem such that
p
=
(∂
p
/∂
n
2
, where
ˁ
is the density and
ʱ
=
=
ˁ
k
B
T
+
ʱ
a
ˁ
0
.
101 for
2. We then have
ʺ
−
1
ˁ >
=
1
+
2
ʱ
a
ˁ/
k
B
T
1
+
0
.
2
a
ˁ/
k
B
T
.If
N
m
is the number
(ʺ
−
1
N
m
−
of molecules contained in a DPD particle, then
a
=
k
B
T
1
)/
2
ʱˁ
DPD
,
where
ˁ
DPD
is the DPD number density for the system and is usually set to
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