Biomedical Engineering Reference
In-Depth Information
d
V
τ
=
1
S
and
2
d
D
τ
»
2
4
then, the ratio
b
=
t
1
/
t
2
is
τ
β
τ
d D
4
k T
k T
1
B
B
1
(5.47)
=
»
=
4
=
4
2
V
D
ρ
g d Vol
D
mg d
d
2
S
p
If we introduce the characteristic Boltzmann length scale
L
=
k
B
T
/
g
D
m
[5], (5.47)
becomes
β
= 4
L
d
The ratio (5.47) represents an energy ratio between the energy of the Brownian
motion and the potential energy of the particle. If
b
<< 1, sedimentation dominates
and the particles will descend with the settling velocity. As a general rule, particles
larger than 1
m
m tend to be affected by sedimentation. For example, cells usually
sediment because their size is larger than 10
m
m. A more detailed analysis may be
found in [3].
The preceding analysis shows that it is very important not to let the particles
aggregate. If so, the Brownian motion ceases and the aggregate will sediment. This
is why surfactants or PEG (polyethylene-glycol) [10] are usually added to buffer
liquids.
5.4 Microscopic (Discrete) Approach
In the preceding sections, we have followed an approach based on the continuum:
this approach assumes that in every elementary volume of liquid there is a number
of microparticles sufficiently important to define a concentration. This approach is
very convenient because it introduces a partial differential equation (PDE) that can
be solved by usual discretization techniques, like the finite element method or the
finite volume method. However, very complex geometries of the diffusion domain
may not be easily treated with such methods.
We present here another approach very useful when dealing with microscopic
scales. Contrary to the preceding continuum approach, this approach is discrete,
meaning that the displacement of every particle is calculated. This approach is well
adapted to complex geometrical domains and small number of particles.
5.4.1 Monte Carlo Method
The Monte Carlo method is based on the mimicking of the random walk of par-
ticles. Because the mean free path is very short, it is possible without changing the
