Biomedical Engineering Reference
In-Depth Information
always exists an orbital effect regardless of particle distribution in
ensembles.
Considering the helical motion given by Eq. (4.15a-b) the
collision period for the ion to transverse the mean free path is
modified, increasing
a
with an increasing B field, compared with
the unperturbed path. This is obtained by considering the radial
displacement in the
XY
plane and using Eq. (4.15a-b) to solve for
the modified time
t
m
.
c
t
col
−
1
1
ω
c
π
−
ar
cos(
1
t
m
=
2
ω
(4.17)
Expanding the perturbed collision period as a Taylor series
where
ω
c
t
col
<<
1, we get
t
m
=
t
col
+
ω
c
t
col
c
t
col
c
t
col
3
ω
5
ω
24
+
640
+
7168
+
...
(4.18)
Analogous to free-space cyclotron motion (Chen, 1974), and
similar to the findings of Kinouchi et al. (1988), the
ρ
offsets of the
B field are thus found to retard diffusion in the transverse plane.
This effect modifies the Nernst-Planck equation relating ionic flux
to Fick'sdiffusion andE field mobility(Plonsey andBarr, 1988).
j
T
=−
μ
Ca
E
(4.19)
since in the hard-sphere model Fick's constant is dependent upon
the mean ionic total velocity,
v
th
, and the mean free path,
d
:D
=
v
th
d
/3 (Reif, 1965; Plonsey and Barr, 1988). The total particle flux,
j
D
,duetobothFick'sdiffusion,
j
F
,andtheretardationeffectoftheB
field,
j
B
, is written as
j
D
=
j
F
+
j
B
where
j
B
=
D
μ
D
∇
C
+
C
Ca
B
2
24
∇
C
•
B
×
n
(4.20)
where D is Fick's constant,
∇
C
is the concentration gradient in
Mm
−
1
and the orbital effect upon ions in the
ˆ
θ
direction is assumed
negligibleatmacroscopic levels.
The retardation effect of the B field upon the diffusion rate may
be estimated. For example, if
B
is 60
μ
T, a typical LGF strength, an
estimate of relative difference in D (
μ
2
Ca
B
2
24
)iscomputedas6.9
×
a
Infact,ifthecollisionperiodswereallconstant,andsynchronouswiththecyclotron
frequency, there would be no transverse diffusion, because the ion would orbit one
complete transverse circularpath.
