Biomedical Engineering Reference
In-Depth Information
for one dimension, and
⎛
⎞
∂
∂
σ
∂
∂
∂
∂
σ
∂
∂
∂
∂
σ
∂
∂
∂
∂
σ
⎛
⎜
⎞
⎟
+
⎛
⎜
⎞
⎟
xyz
xyz
xyz
=
x
D
(
)
D
(
)
⎟
+
z
D
(
)
(5.26)
⎜
t
x
y
y
z
⎝
⎠
for three dimensions, where d
n
= amount of a substance diffusing,
q
= cross
section (along
x
-axis), −d
c
/d
x
= concentration gradient, d
t
= time, and
D
= dif-
fusion constant. This states that an amount of some substance dif-
n
) will dif-
fuse through a cross section (
q
) along the
x
-axis in a time (d
t
) so long as a
concentration gradient exists, and that the amount is governed by a diffusion
constant. As the concentration gradient goes to zero (i.e., complete unifor-
mity) the diffusion falls off to zero. Diffusivities in solids cover a wide range
of values and are typically quite low for dense crystalline materials such as
LiNbO
3
. A material diffusion constant (
D
) is not a universal constant but is
rather a constant for specific situations and conditions. Diffusion coefficients
are unique to individual materials and are a strong function (usually linear
for solids) of temperature. For crystalline materials such as LiNbO
3
multi-
ple diffusion coefficients exist depending on the plane of interest [57]. As
molecular diffusion is limited by collisions between molecules these planar
variations are expected.
Fick's Law is merely a general statement of diffusion theory and, although
it cannot be violated, many specific forms of the law exist and apply to dif-
ferent situations. For diffusion of Ti into LiNbO
3
the diffusion depth, as
expected from Fick's Law, is a function of the material diffusion constant, the
diffusion time, and the diffusion temperature. The diffusion profile has been
found to be roughly exponential, and more specifically the Ti concentration
has been found to follow a Gaussian distribution [58]:
τ
c x t
(
, )
=
(5.27)
3
.
5
3
(
4
π
y
D t
)
0
exp
⎣
−
x
(
4
d t
)
⎦
3
y
where
c
(
x
3
,
t
) is the Ti concentration in the LiNbO
3
crystal
t
is the diffusion time
τ is the thickness of Ti coating prior to diffusion
D
y
is the anisotropic diffusion coefficient (temperature dependent)
The general form of the temperature dependent diffusion coefficient is
=
ʹ
[
−
E kT
/ ]
D D e
(5.28)
0
where
T
is the crystal temperature
E
is the activation energy
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