Environmental Engineering Reference
In-Depth Information
=
a volume
V
L
x
L
y
L
z
. Charges interact according to Coulomb's law and the total
electrostatic energy for the periodic system is given by
⊡
⊣
i
⊤
1
4
ˀʵ
0
ʵ
r
q
i
q
j
r
N
⊦
,
U
(
)
=
(10)
|
r
ij
+
(
n
x
L
x
,
n
y
L
y
,
n
z
L
z
)
|
j
>
i
n
x
n
y
n
z
where
n
and
n
z
are non-negative integer numbers, and
ʵ
0
and
ʵ
r
are the dielectric constants of vacuum and water at room temperature,
respectively. According to Ewald's approach it is convenient to separate this long-
range electrostatic interaction into real and reciprocal space, getting a short-ranged
sum which may be written as
=
(
n
x
,
n
y
,
n
z
)
,
n
x
,
n
y
,
1
4
ˀʵ
0
ʵ
r
erfc
(ʱ
ʵ
r
)
r
N
U
(
)
=
q
i
q
j
r
j
>
i
i
∞
N
q
i
2
2
V
ʱ
ʵ
√
ˀ
+
Q
(
k
)
S
(
k
)
S
(
−
k
)
−
,
(11)
k
=
0
i
=
1
with
e
−
k
2
2
ʵ
N
/
4
ʱ
2
L
(
q
i
e
i
k
·
r
ij
Q
(
k
)
=
,
S
(
k
)
=
,
k
=
m
x
,
m
y
,
m
z
).
k
2
i
=
1
Here,
ʱ
ʵ
is the parameter that controls the contribution of the real space,
k
is the
magnitude of the reciprocal vector
k
,
m
x
,
)
is the complementary error function (cf. Mayoral and Nahmad-Achar
2012
). Equa-
tion (
11
) is a good approach to 1
m
y
,
m
z
are integer numbers, and erfc
(ʱ
ʵ
r
/
r
including the full long-range characteristic of
electrostatic interactions.
In the DPD approach the conservative force
F
c
is mathematically well defined
at
r
0, making possible the overlap between particles, but the electrostatic con-
tribution diverges at
r
=
0, giving rise to non-natural ionic pairs. A solution of
this problem was proposed by Groot (
2003
) by using charge distributions on DPD
particles such as a Slater-type charge density distribution expressed as:
=
q
ˀʻ
3
e
−
2
r
/ʻ
,
ˁ(
r
)
=
(12)
where
ʻ
is the decay length of the charge. For this distribution, well-known approx-
imated expressions for the force are available. The reduced interaction potential
between two charged distributions separated by a distance
r
∗
from center to center
is given by:
4
ˀ
u
∗
(
r
∗
)
Z
i
Z
j
r
∗
e
−
2
ʲ
∗
r
∗
+
ʲ
∗
r
∗
)
=
[
1
−
(
1
]
,
(13)
ʓ
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