Biomedical Engineering Reference
In-Depth Information
A
ξ
, terms are described by Nguyen and Schulze [53]:
77
79
m
∞
m
3
,
3
k
B
T
4
78
−
ε
A
0
=
(9)
78
+
ε
m
=
1
hω
n
p
−
1
.
887
n
p
−
A
ξ
(H)
=−
0
.
235
¯
1
(10)
,
(n
p
+
1
.
887
)
−
1
/
2
0
.
588
×
1
/q
−
(H/
5
.
59
)
q
(H/λ
p
)
q
1
/q
[
1
+
]
[
1
+
]
π
where
ε
is the solid dielectric constant,
h
is the Planck consta
nt divided by
2
,
n
p
is
¯
9
.
499
/
n
p
+
10
16
the solid refractive index,
ω
1
.
887 is a modi-
fied London wavelength accounting for the effect of electromagnetic retardation on
the van der Waals interaction, and
H
is in nm.
The bubble surface is negatively charged (
=
2
×
rad/s,
λ
p
=
55 mV) in deionised water. The solid
surface is also negative in water at neutral pH. The electrostatic double-layer in-
teraction between the air-water and water-solid surfaces is usually repulsive. The
double-layer disjoining pressure between two planar parallel surface elements of a
water film confined by the gas phase and the solid phase can be obtained from the
solution of the Poisson-Boltzmann equation. The calculation of the double-layer
force as a function of the separation distance is recently reviewed [50, 53]. For low
surface (zeta) potentials (
<
50 mV), the Hogg-Healy-Fuerstenau approximation for
the double-layer interaction at constant surface potentials gives
−
ε
w
ε
0
κ
2
2
ψ
b
−
ψ
s
2
ψ
b
ψ
s
cosh
(κH)
−
edl
=
,
(11)
sinh
2
(κH)
where
ψ
s
and
ψ
b
are the solid and bubble surface potentials,
ε
0
is the dielectric con-
stant of vacuum and
ε
w
=
78 for water. For the double-layer interaction at constant
surface charge one obtains
ε
w
ε
0
κ
2
2
ψ
b
+
ψ
s
2
ψ
b
ψ
s
cosh
(κH)
+
edl
=
.
(12)
sinh
2
(κH)
Another useful approximation can be obtained by superposition which is valid for
high surface potentials and long separation distance and gives
32
ε
w
ε
0
κ
2
k
B
T
ez
2
tanh
ezψ
b
4
k
B
T
tanh
ezψ
s
4
k
B
T
exp
(
edl
=
−
κH),
(13)
where
k
B
is the Boltzmann constant,
T
is the absolute temperature,
z
is the valence
of the symmetric
z
z
electrolyte.
The surface (zeta or streaming) potential of the air-water and solid-water in-
terfaces can be measured by microelectrophoresis and streaming potential method.
Useful results for the bubble and solid surfaces are summarised in the topics [53].
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