Biomedical Engineering Reference
In-Depth Information
Because of the very small size of the magnetic particles, the integration in (9.4) may
be replaced by the value of the fields at the center of the particle multiplied by the
volume of the particle
v
p
1
� �
E
= -
µ
v MH
.
(9.6)
m
0
p
2
9.3.1 Paramagnetic Microparticles
Most of the time, particles used in biotechnologies are paramagnetic (one excep-
tion will be discussed in Section 9.13 on magnetic membranes), the magnetization
is then aligned with the external field [7]
�
χ
�
M
=
H
1
+
D
χ
m
where
D
m
is the demagnetization coefficient (
D
m
= 1/3 for a sphere). Because
c
<< 1
for the considered paramagnetic particles, magnetization can be approximated by
=
� �
M H
χ
and the energy of interaction is
1
2
E
= -
µ χ χ
v
(
-
)
H
(9.7)
m
0
p
p
f
2
where
c
p
is the magnetic susceptibility of the particle and
c
f
is that of the carrier
fluid. Thus, the magnetic force on a paramagnetic microparticle is given by the
gradient of (9.7)
�
1
2
F
=
µ χ χ
v
(
-
)
Ñ
H
(9.8)
m
0
p
p
f
2
In the literature another equivalent expression is often found [7, 8]. Indeed, if no
electric field is associated to the magnetic field (or if the electric field is constant),
Maxwell's equation implies that
�
Ñ ´
H
=
0
and (9.8) can be written
�
� �
F
=
µ χ χ
v
(
-
)(
H H
. )
Ñ
(9.9)
m
0
p
p
f
9.3.2 Ferromagnetic Microparticles
In the case of ferromagnetic particles, we start from (9.5) and (9.6) [9] to obtain
µ
� �
µ
�
�
0
0
F
=
v M H
Ñ
(
.
)
=
v
Ñ
((
M M H
-
).
)
(9.10)
m
p
p
p
f
2
2
