Geoscience Reference
In-Depth Information
1.1.1.2 The general case
A complete electrical double layer model for silica is now
discussed avoiding most of the assumption used previ-
ously. The drawbacks of such approach, however, are
that there are no analytical solutions of the system of
equation and we have to use a numerical approach to
determine the zeta potential and the surface charge for
a given set of environmental conditions. We consider
again silica grains in contact with a binary symmetric
electrolyte like NaCl for the simplicity of the presentation
and comparison with the experimental data. In the pH
range 4
The surface charge density in the diffuse layer is calcu-
lated using the classical Gouy
Chapman relationship in
the case of a symmetric monovalent electrolyte:
-
φ
d
2
k
b
T
e
Q
S
=
−
8
ε
k
b
TC
f
sinh
1 40
where
C
f
is the salinity in the free electrolyte (in mol l
−
1
),
T
is the temperature (in K),
ε
f
is the permittivity of the pore
ε
0
~8.85×10
−
12
Fm
−
1
),
e
represents the
elementary charge (taken positive,
e
=1.6 × 10
−
19
C), and
k
b
is the Boltzmann constant (1.381 × 10
−
23
JK
−
1
). The
electrical potential
water (
ε
f
=81
ε
0
,
10, the surface mineral reactions at the silanol
surface sites can be written as
-
φ
d
(in V) is the electrical potential at
the OHP (see Figure 1.1). We make again the assumption
that the electrical potential
> SiO
−
+H
+
φ
d
is equal to the zeta poten-
> SiOH,
K
1
1 34
tial
placed at the shear plane. The shear plane is the
hydrodynamic surface on which the relative velocity
between the mineral grains and the pore water is null.
The continuity equation for the surface sites yields
ζ
> SiOH+H
+
> SiOH
2
+
,
K
2
1 35
> SiO
−
+Na
+
> SiONa,
K
3
1 36
The symbol
“
>
”
refers to the mineral framework, and
K
1
,
K
2
,
K
3
are the associated equilibrium constants for
the different reactions reported earlier (see Table 1.1).
Additional reactions for a multicomponent electrolyte
can be easily incorporated by adding reactions similar
to Equation (1.36) or exchange reactions. Therefore,
the present model is not limited to a binary salt. The pro-
tonation of surface siloxane groups >SiO
2
is extremely
low, and these groups can be considered as inert. We
neglect here the adsorption of anion Cl
−
at the surface
of the >SiOH
2
+
sites which occurs at pH < pH (pzc)
≈
3,
where pzc denotes the point of zero charge of silica:
0
1
=
0
SiO
+
0
SiOH
+
0
SiOH
2
+
0
SiONa
Γ
Γ
Γ
Γ
Γ
1 41
1
(in sites m
−
2
) is the total surface site density
of the mineral. We use the equilibrium constants associ-
ated with the half reactions to calculate the surface
site densities
Γ
where
Γ
i
. Solving Equation (1.41) with the expres-
sions of the equilibrium constants defined through
Equations (1.34)
-
(1.36) yields
0
SiO
=
A
0
1
Γ
Γ
1 42
φ
0
k
b
T
e
0
SiOH
=
A
0
1
K
1
C
f
H
+
exp
Γ
Γ
−
1 43
pH pzc =
1
2
log
K
1
+ log
K
2
1 37
φ
0
k
b
T
2
e
0
SiOH
2
+
=
A
0
1
K
1
K
2
C
f
H
+
2
exp
Γ
Γ
−
1 44
Consequently, the value of
K
2
is determined from the
value of
K
1
and pH (pzc)
≈
3. The surface charge density
Q
0
(in Cm
−
2
) at the surface of the minerals can be
expressed as follows:
φ
β
k
b
T
e
0
SiONa
=
A
0
1
K
3
C
f
Na
+
exp
Γ
Γ
−
1 45
φ
0
k
b
T
e
2
e
φ
0
k
b
T
A
=1+
K
1
C
f
H
+
exp
+
K
1
K
2
C
f
H
+
2
exp
−
−
SiOH
2
SiO
SiONa
Q
0
=
e
Γ
−Γ
−Γ
1 38
φ
β
k
b
T
e
+
K
3
C
f
Na
+
exp
−
1 46
i
denotes the surface site density of species
i
(in
sites m
−
2
). The surface charge density
Q
β
where
Γ
in the Stern
layer is determined according to
where
φ
0
and
φ
β
are, respectively, the electrical potential
at the o-plane corresponding to the mineral surface and
the electrical potential at the
β
-plane corresponding to
0
SiONa
Q
β
=
e
Γ
1 39
