Biomedical Engineering Reference
In-Depth Information
(
r
ij
=(
x
i
−
x
j
)
/
|
x
i
−
x
j
|
). The total force acting on particle
i
due to conser-
vative forces is
=
j
=
i
f
ij
=
j
=
i,r
ij
<r
0
f
i
S
ij
(1
−
r
ij
/r
0
)
r
ij
(7.15)
where
S
ij
is the strength of the interaction between particle
i
and particle
j
.
The dissipative particle-particle interactions are given by
f
ij
=
γW
D
(
r
ij
)(
r
ij
·
v
ij
)
r
ij
,
where
γ
is a viscosity coe
cient and
v
ij
=
v
j
−
v
i
,
for
r
ij
<r
0
and
f
ij
= 0 for
r
ij
>r
0
so that
f
i
=
j
=
i
f
ij
=
γW
D
(
r
ij
)(
r
ij
·
−
v
ij
)
r
ij
(7.16)
j
=
i,r
ij
<r
0
and the random forces are given by
f
ij
=
σW
R
(
r
ij
)
ζ
r
ij
for
r
ij
<r
0
and
f
ij
=0
for
r
ij
>r
0
, where
σ
is the fluctuation strength coecient and
ζ
is a random
variable selected from a Gaussian distribution with a zero mean and a unit
variance so that
f
i
=
j
=
i
f
ij
=
j
=
i,r
ij
<r
0
σW
R
ζ
r
ij
(7.17)
In practice,
ζ
can be selected randomly from a uniform distribution with
a zero mean and a unit variance (the sum of a quite small number of uni-
formly distributed random numbers over a number of time steps quite accu-
rately approximates a Gaussian random variable). The random and dissipative
particle-particle interactions are related through the fluctuation-dissipation
theorem (Espanol and Warren 1995), which requires that
γ
=
σ
2
/
2
k
B
T
,
where
k
B
is the Boltzmann constant,
T
is the prescribed temperature, and
W
D
(
r
)=(
W
R
(
r
))
2
, where
W
D
(
r
) and
W
R
(
r
) are
r
-dependent weight func-
tions, both vanishing for
r
r
0
. These are the detailed balance condition
required for DPD. In standard DPD models, the simple weighting function,
W
D
(
r
)=
W
R
(
r
)
2
=(1
≥
r
/
r
0
)
2
(for
r<r
0
), is used. The combination of
dissipative and fluctuating forces, related by the fluctuation-dissipation the-
orem (Kubo 1966), acts as a thermostat, which maintains the temperature of
the system, measured through the average kinetic energy of the particles at a
temperature of
T
, provided that the time step used in the simulation is small
enough. This idea can be taken one step further (Lowe 1999) by integrating
the equations of motion with only the conservative forces
−
f
i
over the time
step, ∆
t
, and then “thermalizing” the relative velocities of a fraction,
f
T
,of
the particle pairs separated by a distance of
r
0
or less by randomly selecting
the relative velocities from a Maxwell-Boltzmann distribution and multiplying
{
}
by
√
2 to convert the Maxwell-Boltzmann single particle velocity distribution
at temperature
T
to a relative velocity distribution function and preserving
the average velocity of the particle pairs to conserve momentum. The frac-
tion,
f
T
, of particles that are “thermalized” at each step in the simulation can
then be varied to change the effective viscosity of the fluid. An advantage of
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