Biomedical Engineering Reference
In-Depth Information
on particle
i
are given by
F
i
=
∑
j
F
ij
,
T
i
=
−
∑
j
λ
ij
r
ij
×
F
ij
.
(10.7)
Here, the factor
λ
ij
(introduced in [47]) is included as a weight to account for the
different contributions from the particles in different species (solvent or colloid)
differentiated in sizes while still conserving the angular momentum. It is defined as
R
i
R
i
+
λ
ij
=
R
j
,
(10.8)
where
R
i
and
R
j
denote the radii of the particles
i
and
j
, respectively. The force
exerted by particle
j
on particle
i
is given by
F
ij
+
F
ij
+
F
ij
+
F
ij
.
F
ij
=
(10.9)
The radial conservative force
F
ij
can be that of standard DPD and is given in
Eq. (10.2). The
translational force
is given by
F
ij
=
−
γ
ij
f
2
e
ij
e
ij
·
+(
γ
ij
−
γ
ij
)
f
2
(
r
)
1
(
r
)
v
ij
(10.10)
r
ij
)
v
ij
−
(
e
ij
.
=
−
γ
ij
f
2
e
ij
−
γ
ij
f
2
(
r
ij
)(
v
ij
·
e
ij
)
(
v
ij
·
e
ij
)
It accounts for the drag due to the relative translational velocity
v
ij
of particles
i
and
j
. This force is decomposed into two components: one along and the other perpen-
dicular to the lines connecting the centers of the particles. Correspondingly, the drag
coefficients are denoted by
γ
ij
and
γ
ij
for a
central
and a
shear
components, respec-
tively. We note that the central component of the force is identical to the dissipative
force of standard DPD (Eq. (10.1)).
The
rotational force
is defined by
r
ij
)
r
ij
×
(
λ
ij
Ω
i
+
λ
ji
Ω
j
)
,
F
ij
=
−
γ
ij
f
2
(
(10.11)
while the
random force
is given by
1
√
3
σ
σ
ij
d
W
ij
√
2
ij
tr
F
ij
dt
=
f
(
r
ij
)
[
d
W
ij
]
1
+
·
e
ij
,
(10.12)
2
k
B
T
2
k
B
T
σ
ij
=
γ
ij
and
γ
ij
are chosen to satisfy the fluctuation-
dissipation theorem,
d
W
ij
is a matrix of independent Wiener increments, and
d
W
ij
is defined as
d
W
A
μν
ij
σ
ij
=
where
d
W
μν
d
W
νμ
1
=
2
(
ij
−
ij
)
. We can also use the generalized weight
r
k
as in the previous section with
k
(
)=(
−
r
c
)
=
.
function
f
25 [48] in equations
(10.10)- (10.12). The numerical results in previous studies [45, 49] showed higher
r
1
0
