Biomedical Engineering Reference
In-Depth Information
defines the length scale in the DPD system. The conservative force is given by
a
ij
(
1
−
r
ij
/
r
c
)
for r
ij
≤
r
c
,
F
ij
(
r
ij
)=
(10.2)
0
for r
ij
>
r
c
,
where
a
ij
is the conservative force coefficient between particles
i
and
j
.
The random and dissipative forces form a thermostat and must satisfy the fluctua-
tion-dissipation theorem in order for the DPD system to maintain equilibrium tem-
perature
T
[41]. This leads to:
r
ij
)=
ω
r
ij
)
2
D
R
2
ω
(
(
,
σ
=
2
γ
k
B
T
,
(10.3)
where
k
B
is the Boltzmann constant. The choice for the weight functions is as follows
k
(
1
−
r
ij
/
r
c
)
for r
ij
≤
r
c
,
R
ω
(
r
ij
)=
(10.4)
0
for r
ij
>
r
c
,
where
k
25)
for these envelopes have been used [42, 43] in order to increase the viscosity of the
DPD fluid.
The time evolution of velocities and positions of particles is determined by the
Newton's second law of motion
=
1 for the original DPD method. However, other choices (e.g.,
k
=
0
.
d
r
i
=
v
i
dt
,
(10.5)
m
i
j
=
i
F
ij
+
F
ij
+
F
ij
dt
.
1
d
v
i
=
(10.6)
The above stochastic equations of motion can be integrated using a modified velocity-
Verlet algorithm [39]; for systems governed by mixed hard-soft potentials sub-
cycling techniques similar to the ones presented in [44] can be employed.
10.2.2 DPD method for colloidal particles
To simulate colloidal particles by single DPD particles, we use a new formulation
of DPD, in which the dissipative forces acting on a particle are explicitly divided
into two separate components:
central
and
shear
(non-central) components. This
allows us to redistribute and hence, balance the dissipative forces acting on a single
particle to obtain the correct hydrodynamics. The resulting method was shown to
yield the quantitatively correct hydrodynamic forces and torques on a single DPD
particle [45], and thereby produce the correct hydrodynamics for colloidal particles
[46]. This formulation is reviewed below.
We consider a collection of particles with positions
r
i
and angular velocities
Ω
i
.
We define
r
ij
=
r
i
−
r
j
,
r
ij
=
|
r
ij
|
,
e
ij
=
r
ij
/
r
ij
,
v
ij
=
v
i
−
v
j
. The force and torque
