Biomedical Engineering Reference
In-Depth Information
where
m
is the mass of the particle,
b
is the friction coefficient, and
F
(
t
) is the random force of
the liquid molecules. On small time scales, inertial effects are dominant in the Langevin
equation. The friction coefficient can be calculated using Stoke's drag on a spherical particle with
aradiusof
s
p
:
b
¼
3
pms
p
;
(2.26)
where
m
is the viscosity of the surrounding liquid. The force
F
(
t
) is random in time; thus, its auto-
correlation function should represent the delta function:
t
0
Þi ¼
t
0
Þ;
(2.27)
where
d
is the Dirac function. Solving
(2.25)
for the one-dimensional case leads to the particle
velocity:
h
F
ð
t
Þ
F
ð
Ad
ð
t
Z
t
ðbt
0
=
d
t
0
:
u
ð
t
Þ¼
u
0
exp
ðbt
=
m
Þþ
exp
ðbt
=
m
Þ
exp
m
Þ
(2.28)
0
The variance of the displacement
x
(
t
) can subsequently be determined as
[8]
:
2
Z
t
0
Z
t
d
x
2
d
t
t
0
Þ
d
t
0
¼
u
2
bt
0
=
d
t
0
2
x
ð
t
Þ
u
ð
t
Þ¼
u
ð
t
Þ
u
ð
2
h
i
exp
ð
m
Þ
(2.29)
0
u
2
where
h
i
is the variance of the particle velocity. For time scale much larger than
m
/
b
,
x
2
h
ð
t
Þi ¼
2
Dt
:
(2.30)
Thus, the diffusion coefficient of the particle can be determined as:
u
2
D
¼h
i
m
=
b
:
(2.31)
The kinetic energy of the particle is related to the temperature as:
1
2
m
1
2
k
B
T
u
2
h
i¼
:
(2.32)
u
2
Substituting
<
>¼
k
B
T
/
m
in
(2.31)
results in the Stokes-Einstein equation of the diffusion
coefficient
[9]
:
k
B
T
3
pms
p
:
D
¼
(2.33)
2.2.3
Diffusion coefficient
2.2.3.1 Diffusion coefficient in gases
Using the kinetic theory discussed in
Section 2.1.1
, diffusion coefficient in meters squared per second
between two gases
i
and
j
can be formulated as
[7]
:
p
1
10
27
T
3
=
2
1
:
86
=
M
i
þ
1
=
M
j
D
¼
D
ij
¼
D
ji
¼
(2.34)
ps
ij
U
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