Biomedical Engineering Reference
In-Depth Information
Figure 4.20
Anti-clockwise orbital offset to motion along the
x
direction
due to piece-wise helical motion.
collisions, there is (a) a radially directed but random perturbation
to the straight line of the unperturbed path that retards the walk in
the
ρ
directionand(b)aunidirectionalbutdiscontinuousorbital(
θ
)
offset. Without a B field present there are no random orbital offsets
to the direction ofmotion.
Figure 4.20 illustrates a single piece-wise helical path between
collisions.Thereisanangularchange
δθ
duetoeachcollisionandan
associated orbital offset
δλ
=
ρ
t
δθ
where
ρ
t
is the mean free path
length in the transverse plane. The initial
directions are random
in accordance with the random collisions, so the unperturbed effect
is discontinuous. Since the offset directions (clockwise or anti-
clockwise) are dependent upon the direction of the applied static
B field, they are unidirectional. When these orbital offsets are
averaged over a su
ciently long period, they result in an ionic
orbit, analogous to accumulation of offsets due to a static E field,
resulting in ionic mobility. The
θ
offsets have a radius of orbit equal
to the mean free path length. For ions (leaving aside larger sized
particles, which may be experimentally observable), these
θ
offsets
donotproduceadirectlyobservableeffect.Ontheotherhand,the
ρ
offsetscanproduceamacroscopiceffectuponFick'sdiffusioninthe
transverse plane depending upon the level of the Bfield.
It is seen from Fig. 4.16h that the B field effect is isotropic in the
transverse plane since the directions are uniformly distributed in
this plane. Thus no retarding effect should be observed where an
ensemble of ions exists in equilibrium, since the individual radial
offsetscountereachotherifthereisauniformspatialdistributionof
ionic particles. However, where a concentration gradient exists, for
example,nearasourceofions,therewillbeanensembleeffect,since
the
θ
ρ
offsets do not cancel each other but add. Conversely, there
