Biomedical Engineering Reference
In-Depth Information
magnetic material and the total volume of the membrane, then
f V
=
N v
p
and the
magnetic force on the membrane is:
�
� �
F
f V M H
(
. )
»
µ
Ñ
(9.57)
mag
0
where
V
is the total volume of the membrane. The magnetic pressure is then given
by the expression
�
� �
F
µ
mag
0
2
(9.58)
P
=
»
µ
f h M H
(
. )
Ñ
=
f h
χ
Ñ
H
m
0
p
S
2
Using (9.53), we find that the membrane deflection is then linked to the gradient of
the square of the magnetic field by
µ
0
2
4
f
χ ν
(1
-
)
a
p
2
2
(9.59)
w
»
Ñ
H
max
2
64
Eh
The membrane maximum deflection is proportional to the content in magnetic
particles
f
, to the magnetic susceptibility of the particles
c
p
, to the foruth power
of the radius
a
4
and to the gradient of the square of the magnetic field Ñ
H
2
. It is
inversely proportional to the Young modulus
E
and to the square of the membrane
thickness
h
2
.
Because micromagnets or microelectromagnets do not deliver an important
magnetic field, and consequently an important gradient, the efficiency of a mag-
netic micromembrane depends on the use of very magnetizable particles that can
be packed in the PDMS matrix without increasing too much the Young modulus
and Poisson coefficient. To this extent it is likely that micromembranes containing
nanoparticles will be more efficient than that containing metal microplates.
9.13.3 Oscillation of Magnetic Membranes
An application of magnetic micromembranes in biotechnology is the mixing of
fluids. We analyze here briefly the principle of actuation of micromembranes by an
oscillating (or pulsating) magnetic field.
9.13.3.1 Paramagnetic Membranes
In the previous section, we saw that the deflection of a paramagnetic membrane is
proportional to the gradient of the square of the external magnetic field Ñ
H
2
. Sup-
pose now that the source of the external magnetic field is an electromagnet and that
this magnetic field is periodically reversed. Changing
H
�
into
-
�
leaves the term
Ñ
H
2
unchanged. The magnetic force on a paramagnetic membrane is always at-
tractive, and the deflection of the membrane is always directed towards the source
of the external magnetic field.
Another way of looking at this phenomenon is to consider Langevin's law for
paramagnetic media (Figure 9.40). When the field is reversed to its opposite (
H
�
is
