Biomedical Engineering Reference
In-Depth Information
ensembles
of particles is quite predictable, as we experience it in our everyday life when we mix
luids with luids and luids with solutes. Difusion is quantitatively embodied in Fick's three
laws, or equations, of which we will touch only on the irst two here.
∂
∂
C
x
i
J
= −
D
Fick's first law of diffusion
(
3.25
)
i
In plain words, the irst law states that difusing species low down their concentration gradients
at a rate that is proportional to the gradient present. he quantity
J
i
is the low or lux of the solute
i
(in units of mass or moles) that crosses 1 cm
2
of surface in 1 second.
C
i
is the concentration of solute
i
in mass units per unit volume. he quantity
D
, which is one of the most important in microluid-
ics, is called the
difusion coeicient
, as deined in the Stokes-Einstein relationship below:
kT
R
H
D
=
6
πη
Stokes Einstein relation
-
(
3.26
)
where:
k
= Boltzmann constant
T
= temperature
η = viscosity of the solution
R
H
= hydrodynamic radius of the particle.
Note that because the viscosity of a solvent is almost always dependent on temperature, this
expression depends doubly on temperature. he hydrodynamic radius is roughly related to the
geometric radius of the particle, but depends also on how much solvent moves with the particle,
and is diicult to calculate for a nonspherical particle. Overall, the value of
D
is characteristic of
a given particle in a given solvent at a given temperature. Some examples of difusion coeicients
are given in
Table 3.1
. Note that the difusion of hydronium ion is anomalously fast, in that the
proton can hop from water molecule to water molecule.
Fick's second law of difusion
predicts how difusion causes the concentration ield to change
with time:
2
∂
∂
C
t
∂
∂
C
x
i
i
=
D
Fick's second law of diffusion
(
3.27
)
2
A simple case of difusion in one dimension (taken as the
x
axis) is the introduction of a step
concentration
C
(0) (constant from −∞ to 0) at position
x
= 0 at time 0, which acts as the bound-
Table 3.1
Important Diffusion Coeficients
Diffusion Coeficient,
D,
in Water (
μ
m
2
/s)
Molecule
Molecular Weight
H
3
O
+
19
9000
Na
+
23
2000
O
2
32
1000
Glycine
75
1000
Hemoglobin
60,000
70
Tobacco.mosaic.virus
40,000,000
5
