Geochimica et Cosmochimica Acta, Vol. 65, No. 21, pp. 3643–3655, 2001
Copyright © 2001 Elsevier Science Ltd
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Interpretation and prediction of triple-layer model capacitances and the structure of the
oxide– electrolyte–water interface
DIMITRI A. SVERJENSKY
Morton K. Blaustein Department of Earth and Planetary Sciences, Johns Hopkins University, Baltimore, MD 21218, USA
(Received November 9, 2000; accepted in revised form June 25, 2001)
Abstract—The interpretation of mineral–water interactions in processes such as surface-charge development,
adsorption of aqueous ions onto surfaces, and the kinetics of dissolution at the scale of the oxide– electrolyte–
water interface has been greatly facilitated by models of the electric double layer. In models that explicitly
account for adsorption of the electrolyte ions, a critical parameter is the integral electric capacitance that
expresses the charge at the surface relative to the drop in electric potential at some distance away from the
surface where the electrolyte ions adsorb. Despite the widespread application of such surface complexation
models, much uncertainty surrounds the choice of values of the integral capacitance because it appears to
depend on the specific oxide and type of electrolyte, yet it cannot be directly measured.
In the present study, it is shown that triple-layer model capacitances (C1), obtained in a consistent manner
from regression of surface-charge data referring to a wide range of ionic strengths, electrolyte types, and
mineral surfaces, fall into two groups: on rutile, anatase, and magnetite, values of C1 increase with decreasing
crystallographic radius of the electrolyte cation from Cs! to Li!; on quartz, amorphous silica, goethite,
hematite, and alumina, values of C1 increase with decreasing hydrated electrolyte cation radius from Li! to
Cs!. The triple-layer model capacitances on both groups of solids can be described by a model of the
mineral–water interface with physically reasonable parameters consistent with X-ray standing-wave studies of
the rutile–water interface (Fenter et al., 2000). The model specifies a layer of chemisorbed water molecules
at the surface and a layer of adsorbed electrolyte cations. On rutile, anatase, and magnetite, the layer of
chemisorbed water molecules is interpenetrated by the layer of electrolyte cations that adsorb close to the
surface as dehydrated, inner-sphere complexes. On quartz, amorphous silica, goethite, hematite, and alumina,
the layer of chemisorbed water molecules varies from 0.0 to as much as 6.0 Å and the electrolyte cations form
hydrated, outer-sphere complexes. The model capacitances are consistent with interfacial dielectric constants
ranging from 20 to 62. Triple-layer model capacitances can now be predicted for oxides in either alkali or
alkaline earth electrolyte solutions that have not been studied experimentally. In addition, predictions can be
made of the structure of the oxide– electrolyte–water interface for many oxides and electrolytes. Copyright
© 2001 Elsevier Science Ltd
plane of adsorption (!0) to the drop in potential at a distance "
according to
1. INTRODUCTION
Recent interpretations of mineral–water interactions at the
scale of the oxide– electrolyte–water interface have employed
models of the electric double layer to provide a quantitative
understanding of processes such as surface-charge development, adsorption of aqueous ions onto mineral surfaces, and the
kinetics of dissolution of oxides and silicates (Hiemstra et al.,
1989a,b; Davis and Kent, 1990; Dzombak and Morel, 1990;
Stumm and Wieland, 1990; Hayes et al., 1991; Goldberg, 1992;
Dove, 1994; Katz and Hayes, 1995; Lützenkirchen et al., 1995;
Crawford et al., 1996a,b,c; Hayes and Katz, 1996; Hiemstra
and van Riemsdijk, 1996; Kosmulski, 1996, 1999; Ludwig et
al., 1996; Sverjensky and Sahai, 1996; Venema et al., 1996a,b;
Dove and Nix, 1997; Lützenkirchen, 1997, 1998; Robertson
and Leckie, 1997; Sahai and Sverjensky, 1997a,b; Stumm,
1997; Felmy and Rustad, 1998; Criscenti and Sverjensky,
1999). Most such models contain an integral electrical capacitance parameter that expresses the charge at the surface relative to the drop in electrical potential at some distance away
from the surface. For example, in the triple-layer model, the
inner-layer capacitance (C1) relates the charge at the innermost
C1 " !0/(#0 # #")
(1)
In Eqn. 1, #0 and #" refer to the potentials at the 0-plane (the
innermost plane, where protons and hydroxyls adsorb) and the
"-plane (where the electrolyte ions adsorb), respectively. Because the surface potentials in Eqn. 1 cannot be directly measured, the value of the capacitance C1 is critical to relate the
potentials to the proton surface charge (!0).
Despite the widespread application of surface complexation
models of the oxide– electrolyte–water interface, much uncertainty surrounds the choice of values of the integral capacitance
(Westall and Hohl, 1980; Hayes et al., 1991; Lützenkirchen,
1998, 1999). There are two main reasons for this. First, in
contrast to the classical studies of the Hg-water interface
(Bockris and Reddy, 1970), capacitances at the oxide– electrolyte–water interface cannot be directly measured (capacitances
have been inferred from the combination of multiple kinds of
experimental data: Sprycha, 1984; however, from surfacecharge data alone, capacitances are generally obtained as described below). Second, even though the capacitance can be
interpreted by analogy with a parallel-plate capacitor (Davis et
al., 1978), the capacitor parameters involved are also poorly
*Author to whom correspondence should be addressed ([email protected]).
3643
3644
D. A. Sverjensky
known. For example, when the 0-plane and the "-plane of the
triple-layer model are treated as the plates of a capacitor,
C1 " 8.854$int/"
(2)
where C1 has units of %F ! cm#2, &int refers to the interfacial
dielectric constant of the water between the 0-plane and the
"-plane, and " (Å) refers to the distance separating the two
planes. Although it is likely that 6 $ &int $ 78 and that 2 $ " $
4 Å, respectively (Hayes et al., 1991), the actual magnitudes of
&int and " are not known, which has prevented the use of Eqn.
2 as a means of predicting capacitances. Recent progress in the
application of X-ray standing waves to the mineral–water interface is quantifying interfacial distances relevant to electric
double-layer models for the first time (e.g., Fenter et al., 2000).
Capacitances have generally been obtained along with other
surface complexation parameters by fitting experimentally derived surface-charge data. Where sufficient data are not available, it has been suggested that the capacitances be set to a
value of 80 %F ! cm#2 (e.g., Hayes et al., 1991). However, it
has also been demonstrated that capacitances at the oxide–
electrolyte–water interface are a systematic function of the size
of the electrolyte ions (Davis et al., 1978; Jang and Fuerstenau,
1986; Colic et al., 1991; Kallay et al., 1994; Sahai and Sverjensky, 1997b; Kitamura et al., 1999), and even that systematic
trends with electrolyte ion size can be different on different
oxide surfaces (Kallay et al., 1994). For example, on hematite
(Colic et al., 1991), model capacitances increase with decreasing size of the hydrated ions. This trend is the same (although
the changes are much larger) as the trends for electrolyte ion
adsorption on Hg (Bockris and Reddy, 1970) and AgI (Lyklema and Overbeek, 1961), trends that have been termed lyotropic. However, on rutile, a systematic trend opposite to the
lyotropic series has been inferred (Jang and Fuerstenau, 1986;
Kallay et al., 1994). On rutile, model capacitances increase
with increasing size of the hydrated ions. Such behavior is
inconsistent with Eqn. 2. This inconsistency can be resolved by
suggesting that the adsorbed electrolyte ions on rutile have
crystallographic ionic radii. The latter would vary inversely
with capacitance, as required by Eqn. 2. In turn, this suggests
that the ions adsorbed on rutile behave as though they are
dehydrated. Taken together, all of the above systematics suggest that model capacitances depend on the specific oxide and
the type of electrolyte (Dumont et al., 1990; Kallay et al., 1994;
Johnson et al., 1999a,b) and on the state of hydration of the
electrolyte ions.
In the present study, triple-layer model capacitances are
interpreted taking into account dependencies on the type of
oxide, the type of electrolyte, and the state of hydration of the
electrolyte ions. The interpretation of model capacitances requires a set of capacitance values and associated double-layer
parameters obtained with a single model in a consistent manner. Numerous capacitances for the oxide– electrolyte–water
interface have been reported by fitting surface charge and/or
electrokinetic data with a variety of electric double-layer models and a great variety of assumptions (e.g., Hayes et al., 1991).
However, in most studies, only a small number of solids and
electrolyte types have been investigated. As a consequence, the
capacitance values from different studies are generally not
comparable. In the investigation reported here, capacitances for
Fig. 1. A diagrammatic representation of the proposed structure of
the oxide– electrolyte–water interface. The distance of separation of the
0- and the "-planes is assumed to be made up of a layer of water
molecules and a layer of electrolyte cations. Both layers contribute to
the overall integral capacitance (C1) between the 0- and the "-planes.
As a first approximation, it is assumed that the interfacial dielectric
constant of water (&int) is the same for the two layers.
a wide variety of oxides and electrolyte types are analyzed by
building on and adding to results previously obtained by use of
a consistent formulation of the triple-layer model (Sahai and
Sverjensky, 1997a). A preliminary interpretation of triple-layer
model capacitances resulted in a very approximate correlation
for estimating values of C1 based on the properties of the
electrolyte alone (Sahai and Sverjensky, 1997b) but did not
specifically account for the type of oxide or the state of hydration of the adsorbed ions. The new interpretation of model
capacitances presented below is facilitated by recent X-ray
reflectivity studies of the rutile– electrolyte–water interface
(Fenter et al., 2000). The latter provide fundamental constraints
on distances of adsorbed species from mineral surfaces, which
helps to ensure that the interpretation of model capacitances is
consistent with physically reasonable values of the interfacial
dielectric constant and the distance " (Eqn. 2). The interpretation of model capacitances presented below enables prediction
of capacitances for systems that have not been studied experimentally. It also enables prediction of the distance " for the
triple-layer model, which quantifies an important model parameter for the oxide– electrolyte–water interface— one that can be
tested with further experimental X-ray studies.
2. MODEL FOR INNER-LAYER CAPACITANCES AT THE
OXIDE–ELECTROLYTE–WATER INTERFACE
According to the triple-layer model, ions at the oxide– electrolyte–water interface are specifically bound at two planes: the
0-plane and the "-plane (Fig. 1). These two planes are planes of
uniform electric potential (#0 and #", respectively). The locations of these two planes relative to the surface of the oxide are
not required for application of the model. However, in the
Prediction of electric double-layer capacitances
present study, the separation of the planes is of importance to
interpret the capacitance. It should be noted that the surfaces of
oxides are not expected to be planar. Instead, they will contain
oxygens at different distances from a given reference plane
(Koretsky et al., 1998). Because surface charge may arise from
and be associated with these different types of oxygens, the
0-plane represents a planar approximation to the average contributions from the different surface species. The "-plane represents the locations of adsorbed electrolyte ions complexed to
surface sites at the 0-plane (Davis et al., 1978).
It is assumed here that the distance separating the 0- and
"-planes and the overall capacitance (C1) is influenced by two
factors: the size and state of hydration of the adsorbing electrolyte ions at the "-plane, and the presence of water molecules
between the 0- and "-planes (Fig. 1). First, the size and the state
of hydration of the adsorbing electrolyte ions can be expected
to vary more strongly for the electrolyte cations than for the
anions (Helgeson and Kirkham, 1976). This expectation is
supported by recent systematic electrokinetic studies of a series
of 1:1 electrolytes with a common cation or a common anion
(Johnson et al., 1999a,b). These studies show that changes in
the zeta potential are primarily associated with cation properties rather than anion properties. Consequently, as a first approximation, it is assumed that variation of C1 with electrolyte
type is attributed to variations in the radius and hydration state
of the electrolyte cation. The radius rM of the electrolyte cation
in Figure 1 will represent either a hydrated cation radius or a
crystallographic radius (i.e., a dehydrated cation), depending on
the mineral. In the calculations described below, hydrated
cation radii will be used for cations adsorbing onto those
minerals behaving according to the lyotropic series (e.g., hematite). Crystallographic radii will be used for cations adsorbing onto those minerals behaving opposite to the lyotropic
series (e.g., rutile). The choice of radii is discussed below.
Second, the presence of water molecules between the 0- and
"-planes and an explicit role for adsorbed water molecules is
included in the present model. The presence of water molecules
adsorbed to the surface is to be expected from considerations of
the termination of the bulk crystal structure (Koretsky et al.,
1998). Abundant infrared evidence also suggests that water
molecules chemisorb to the surface of oxides and metals (Little,
1966; Koretsky et al., 1997). At the oxide– electrolyte–water
interface, the details of the configuration of the water molecules
and the cations are not known. However, on metal surfaces in
water, the nature of the adsorbed water is established from
experiments (Toney et al., 1994) and interpretations of measured capacitances (Bockris and Reddy, 1970). For example, in
interpretations of the capacitances at the Hg-water interface
(Bockris and Reddy, 1970), it is assumed that the distance " is
made up of two layers: a layer of water molecules chemisorbed
to the surface and a layer of electrolyte cations and anions. One
possible configuration of the two layers is represented in Figure
1. Here, only one species in each layer has been depicted, an
electrolyte cation with radius rM and a single water molecule
with radius rH2O. It can be seen in Figure 1 that the cation is
quite close to the surface of the solid—that is, that " $ 2rH2O !
rM. Other possible configurations might correspond to a much
thicker layer of water and the situation where " % 2rH2O ! rM.
Given the uncertainties surrounding the coordination state of
the cations and the amount of water at the interface on oxides,
3645
it is convenient to define the distance parameter r1 (Fig. 1)
where
" " r1 ! rM
(3)
The parameter r1 expresses the distance of closest approach of
the cation to the 0-plane, which is presumably related to the
thickness of the layer of water, the orientations of the water
molecules in the layer, and the coordination of the cation
relative to the water molecules with which it is in contact.
It is next assumed that the two layers depicted in Figure 1
can be treated as capacitors in a series. As a consequence, the
capacitance C1 can be expressed (Bockris and Reddy, 1970) by
!
" !
2rH20
r1 ! rM # 2rH20
1
"
!
C1
(8.854)$int,1
(8.854)$int,2
"
(4)
In Eqn. 4, the numerator of each term on the right-hand side of
the equation represents the thickness of each layer. The denominator of each term contains the effective dielectric constant of
water in each layer, &int,1 and &int,2. Although it might be
expected that &int,1 and &int,2 would be different (e.g., Bockris
and Reddy, 1970), independent values of such dielectric constants are not available. Consequently, as a first approximation,
it is assumed in the present study that &int,1 " &int,2 " &int. This
assumption eliminates the dependence in Eqn. 4 on the radius
of the water molecules, which reduces the number of unknown
parameters resulting in
1
rM
r1
"
!
C1 (8.854)$int (8.854)$int
(5)
Eqn. 5 provides a basis for regression of model capacitances
(1/C1) in terms of the electrolyte cation size (rM). In the present
study, the parameters &int and r1 are parameters to be determined by regression of the capacitances for a single mineral (or
a small group of related minerals) and a wide range of electrolyte types.
3. APPLICATION TO TRIPLE-LAYER MODEL
CAPACITANCES
3.1. Selection of Crystallographic and Hydrated Radii
The application of Eqn. 5 requires radii for the adsorbing
electrolyte cations. Crystallographic radii (rM,cr) for monovalent and divalent electrolyte cations (Shannon and Prewitt,
1969; Shock and Helgeson, 1988) are listed in Table 1. It
should be noted that these radii refer specifically to sixfold
coordination. The effects of coordination state on the radius of
adsorbed cations at the oxide–water interface are as yet unknown. Hydrated radii (rM,hyd) listed in Table 1 for Li!, Na!,
Be!!, Mg!!, Ca!!, Sr!!, and Ba!! were assumed equal to
the Stokes radii computed from mobilities (Robinson and
Stokes, 1959). For the cations K!, Rb!, Cs!, and NH!
4 , the
hydrated radii were assumed to be equal to the crystallographic
radii, reflecting the lack of a permanent hydration shell (Robinson and Stokes, 1959; Kitamura et al., 1999). Although other
choices of radii could be made, the values in Table 1 permit a
simple interpretation of triple-layer model capacitances, resulting in the integration of a wide variety of experimentally
derived data. Refinement of the radii chosen here will no doubt
3646
D. A. Sverjensky
Table 1. Crystallographic and hydrated radii (Å) for electrolyte
cations.
Ion
rM.cra
rM.hydb
Li!
Na!
K!
Rb!
Cs!
Tl!
NH4!
N(CH4)3!
Be2!
Mg2!
Ca2!
Sr2!
Ba2!
Ra2!
0.74
1.02
1.38
1.49
1.70
1.50
1.47c
3.47d
0.27e
0.72
1.00
1.16
1.36
1.43c
2.37
1.83
1.38
1.49
1.70
1.50
1.47
3.47
4.08
3.46
3.09
3.09
2.88
1.43
a
Crystallographic radii for VI-fold coordination from Shannon and
Prewitt (1969) unless otherwise stated.
b
Hydrated radii calculated from limiting equivalent conductivities
(Robinson and Stokes, 1959, p. 126) or set equal to the crystallographic
radius.
c
Shock and Helgeson (1988).
d
Estimated by Robinson and Stokes (1959, p. 122).
e
Radius for IV-fold coordination from Shannon and Prewitt (1969).
result from further experiments involving X-ray reflectivity
measurements.
3.2. Regression of Capacitances
Model capacitances derived from regression of surface protonation data as a function of pH and ionic strength are given
in Table 2. It should be emphasized that the capacitances refer
to experimental studies of surface protonation from many investigators over a wide range of ionic strengths (Bolt, 1957;
Bérubé and Bruyn, 1968a,b; Abendroth, 1970; Breeuwsma and
Lyklema, 1973; Huang and Stumm, 1973; Yates, 1975; Riese,
1982; Blesa et al., 1984; Sprycha, 1984, 1989; Fokkink et al.,
1987; Hayes, 1987; Liang, 1988; Hayes et al., 1991; Casey,
1994; Kallay et al., 1994; Lumsden and Evans, 1995; Huang,
1996; Venema et al., 1996a; Machesky et al., 1998; Criscenti
and Sverjensky, 1999; Criscenti, 2000). The derivation of many
of the capacitances given in Table 2 by an internally consistent
method of regression of surface-charge data has already been
described (Sahai and Sverjensky, 1997a,b; Criscenti and Sverjensky, 1999; Criscenti, 2000). Additional capacitances referring to rutile and silica in N(CH3)4Cl, rutile in LiCl, NaCl, and
CsCl, and quartz in KNO3 and CaCl2 were included in the
present study (Appendix) to extend the ranges of minerals and
electrolyte types reported previously. It should be noted that the
Table 2. Inner-layer capacitances (C1, %F.cm#2; taken from Sahai and Sverjensky, 1997a, unless otherwise stated) consistent with the triple-layer
model derived from regression of surface-charge experiments referring to wide ranges of ionic strength.
Electrolyte
Solid
C1
Source of surface charge data
N(CH3)4Cl
CsCl
KNO3
KNO3
KNO3
NaCl
NaClO4
NaNO3
NaCl
LiCl
LiNO3
LiCl
LiCl
NaCl
CsCl
KNO3
KNO3
CaCl2
NaCl
KNO3
N(CH3)4Cl
NaCl
NaCl
KCl
NaCl
NaNO3
NaNO3
NaCl
NaNO3
NaCl
'-TiO2
'-TiO2
'-TiO2
Fe3O4
'-TiO2
'-TiO2
'-TiO2
'-TiO2
"-TiO2
"-TiO2
'-TiO2
'-TiO2
Fe2O3
Fe2O3
Fe2O3
Fe2O3
Fe2O3
quartz
quartz
quartz
am. SiO2
am. SiO2
am. SiO2
am. SiO2
FeOOHd
FeOOHd
FeOOHd
(-Al2O3
'-Al2O3
(-Al2O3
55
95
110
120
125
120
125
130
130
140
145
155
80
90
90
95
95
81
100
105
50
95
100
120
60
60
60
90
100
110
Yates (1975)a
Kallay et al. (1994)a
Yates (1975)a
Blesa et al. (1984)
Fokkink et al. (1987)
Machesky et al. (1998)a
Bérubé and de Bruyn (1968b)a
Bérubé and de Bruyn (1968a)
Sprycha (1984)
Sprycha (1984)
Yates (1975)
Kallay et al. (1994)a
Breeuwsma and Lyklerna (1973)
Liang (1988)
Breeuwsma and Lyklema (1973)
Yates (1975)
Fokkink et al. (1987)
Riese (1982)a
Riese (1982)
Huang (1996)a
Casey (1994)a
Casey (1994)
Bolt (1957)
Abendroth (1970)
Lumsden and Evans (1995)
Hayes (1987)b
Venema et al. (1996a)c
Huang and Stumm (1973)
Hayes et al. (1991)b
Sprycha (1989)
a
Fitted in the present study (Appendix).
Criscenti and Sverjensky (1999).
c
Criscenti (2000).
d
All goethites included here have BET surface areas greater than 50 m2.g#1.
b
Prediction of electric double-layer capacitances
3647
1
rM,cr
0.75
"
!
C1 (8.854)26.1 (8.854)26.1
Fig. 2. The inverse of the triple-layer model capacitances (1/C1) of
rutile, anatase, and magnetite as a function of the crystallographic
radius of the electrolyte cation (rM).
capacitances obtained in this manner are linked to the equilibrium constants for cation and anion adsorption that were obtained during the regression analyses. Only surface-charge data
referring to a wide range of ionic strengths were used. Such
data enable separate evaluation of the capacitance and the
electrolyte ion equilibrium constants. Consequently, when the
site density for an oxide is known from tritium exchange
experiments, and when the values of the surface protonation
constants are obtained independently (e.g., Sverjensky and
Sahai, 1996), a unique capacitance and pair of electrolyte
equilibrium constants can be obtained (see also Hayes et al.,
1991).
3.2.1. Rutile, Anatase, and Magnetite
It can be seen in Table 2 that the first group of solids—rutile,
anatase, and magnetite— have model capacitances that increase
!
from 55 to 155 %F ! cm#2 in the series N(CH3)!
$
4 $ Cs
!
!
!
K $ Na $ Li , which is in order of decreasing crystallographic radius of the electrolyte cation. As already noted above,
this order is in agreement with the results of Kallay et al.
(1994). It is the opposite of the lyotropic series. Because the
variation in model capacitances reflects crystallographic radii,
it seems reasonable to postulate as a first approximation that the
monovalent cations are dehydrated when adsorbed to rutile,
anatase and magnetite. In fact, dehydration of cations adsorbed
on these minerals might be expected to be favored. The work
required to remove waters of solvation from cations near mineral surfaces is smaller when the minerals have higher dielectric constants (James and Healy, 1972). Although the dielectric
constant of anatase is not known, rutile and magnetite have
such high dielectric constants that the calculated work of dehydration for cations near these minerals is negligible (James
and Healy, 1972).
Taking account of the trend with crystallographic radius in
Table 2, the model capacitances for rutile, anatase, and magnetite have been regressed with Eqn. 5 in terms of the crystallographic radius (rM,cr) from Table 1 yielding the solid line in
Figure 2 and the equation
(6)
It can be seen in Figure 2 that the model capacitances are
clearly consistent with Eqn. 6 for rutile, anatase, and magnetite
in a range of 1:1 electrolyte types. The discrepancies between
calculated and experimentally derived model capacitances are
mostly less than &10 %F ! cm#2 (Fig. 4). Furthermore, the
model capacitances are consistent with physically reasonable
parameters (&int and r1) for the oxide– electrolyte–water interface. The effective interfacial dielectric constant near the
"-plane (&int " 26.1) is physically reasonable because it lies
between the value of the dielectric constant at the interface (6) and
the bulk water value of 78 (Bockris and Khan, 1993; Boily, 1999).
Comparisons of the distances derived above from analyzing
model capacitances can be made with the results of X-ray
scattering studies for Rb and Sr on rutile (Fenter et al., 2000).
Although triple-layer model capacitances for the adsorption of
Rb! and Sr!! on rutile are not available experimentally, the
results of the regression described above permit prediction of
these capacitances and of the distance ". For example, from
Eqns. 3 and 6 and the crystallographic radii in Table 1, it
follows that values of " for Rb! on rutile can be calculated
with
" (Rb! on TiO2) " 0.75 ! rRb!,cr " 2.24 Å
(7)
This value of " " 2.24 Å is a predicted separation of the 0- and the
"-planes in Figure 1. It also represents a minimum distance of the Rb!
from the rutile surface. The total distance can be estimated by adding
on the distance of the 0-plane from a surface reference plane. Following
Fenter et al. (2000), the reference plane is selected to be the last Ti-O
lattice plane parallel to (110). Although the precise location of the
0-plane relative to the reference plane is subject to uncertainty, crystal
chemical considerations give rise to two different oxygen distances
relative to the oxygens in the last Ti-O lattice plane (Koretsky et al.,
1998): bridging oxygens (%Ti2O, at '1.3 Å), and oxygens on chemisorbed water attached to %O5Ti groups (i.e., %O5Ti(H2O) with H2O at
'1.9 Å). Assuming that all of these oxygens could contribute to
surface charge on the 0-plane (i.e., the oxygens at 0, 1.3, and 1.9 Å), the
average distance of the 0-plane from the surface is '1.1 Å. Combining
this estimate with the prediction of " given by Eqn. 7 results in an
estimate for the distance of Rb! from the surface of rutile in a
Rb-electrolyte solution:
Rb! distance " 2.24 ! 1.1 " 3.3 Å.
(8)
!
X-ray studies have established that the Rb ion is 3.4 Å from
the (110) surface of rutile in a Rb-electrolyte solution (Fenter et
al., 2000), which agrees well with the estimate in Eqn. 8.
Another check on the use of Eqns. 3 and 6 for prediction of
values of " is provided by X-ray standing wave results for
Sr!! adsorption on rutile in Na-salt solutions (Fenter et al.,
2000). Assuming that Sr!! adsorbs on the "-plane and that the
distance " is determined primarily by the dominant electrolyte
cation in solution (i.e., Na!), it can be predicted with Eqn. 6
and the radius of the Na! ion (Table 1) that
" (Na! on TiO2) " 0.75 ! 1.02 " 1.77 Å
(9)
for rutile in solutions of Na-salts. Combining this estimate with
the estimate of the distance of the 0-plane used above (1.1 Å),
results in an estimate for the distance of Na! or Sr!! from the
surface of rutile in a Na-electrolyte solution given by
Na! or Sr!! distance " 1.77 ! 1.1 " 2.9 Å
(10)
3648
D. A. Sverjensky
Fig. 3. The inverse of the triple-layer model capacitances (1/C1) of hematite, quartz, silica, goethite, and alumina as a
function of the hydrated radius of the electrolyte cation (rM).
The result in Eqn. 10 compares very favorably with the experimentally derived distance of 2.8 Å for the Sr!! ion adsorbed
to the (110) surface of rutile in a Na!-electrolyte solution
(Fenter et al., 2000).
The above results strongly support the use of crystallographic radii (Table 1) as a first approximation to the sizes of
monovalent and divalent cations adsorbed on rutile. The results
suggest that the structures of the interfaces of rutile, anatase and
Prediction of electric double-layer capacitances
3649
rutile (e.g., for hematite, quartz, and rutile, values of the dielectric constant are 25, 4.6, and 121, respectively; Sverjensky
and Sahai, 1996). Consequently, hydration of cations adsorbed
on these minerals might be favored because the work required
to remove the waters of solvation near low dielectric constant
mineral surfaces is substantial (James and Healy, 1972).
Taking account of the hydrated radii (rM,hyd) in Table 1, the
model capacitances for hematite, quartz, and amorphous silica
have been regressed with Eqn. 5, yielding the solid lines in
Figure 3 and the equations
Hematite:
1
rM,hyd
3.5
"
!
C1 (8.854)53 (8.854)53
(11)
Quartz:
rM,hyd
1
3.7
"
!
C1 (8.854)62 (8.854)62
(12)
rM,hyd
0.0
1
"
!
C1 (8.854)19.7 (8.854)19.7
(13)
Amorphous silica:
Fig. 4. Comparison of triple-layer model capacitances derived by
fitting experimental surface-charge data with theoretically calculated
capacitances.
It can be seen in Figure 3 that the model capacitances are clearly
consistent with Eqns. 11 through 13 for hematite, quartz, and amorphous silica in a range of 1:1 and 2:1 electrolytes. The discrepancies
between calculated and experimentally derived model capacitances are
again mostly less than &10 %F ! cm#2 (Fig. 4), and the model
capacitances are again consistent with physically reasonable parameters (&int and r1) for the oxide– electrolyte–water interface. However, no
independent X-ray results are available yet to test these results.
It can be noted from Eqns. 11 through 13 that the regression results
for r1 for hematite and quartz (3.5 and 3.7 Å, respectively) are much
larger than obtained above for rutile (0.75 Å). This result suggests that
the hydrated cations are much further from the surfaces of hematite and
quartz than the cations adsorbed to rutile. However, it should also be
noted that in the case of quartz, this result is heavily dependent on the
data point for CaCl2, which largely determines the slope of the calculated line in Figure 3. The much larger interfacial dielectric constants
for hematite and quartz compared with rutile suggest that the water near
these interfaces, on average, is less oriented and has a greater similarity
to bulk water than the water bound to rutile.
Capacitances for goethites and aluminas are also plotted in Figure 3.
It can be seen in Figure 3 that the data for goethite and alumina are
restricted to the Na-salts. Also, in the case of the goethites, only those
samples that had reported BET surface areas greater than 50 m2 ! g#1
were used. Goethites analyzed in the present study with BET surface
areas less than 50 m2 ! g#1 were not used seemed to have anomalously
high surface charge and capacitance characteristics, a feature also noted
by Hiemstra and van Riemsdijk (1991). The data in Figure 3 for
goethite and aluminas serve only to define very preliminary values of
the distance (r1). Assuming that the interfacial dielectric constant for
these solids is the same as that for hematite, as a first approximation,
results in the following equations:
magnetite in electrolyte solutions are very similar. Because the
distance r1 " 0.75 Å on all these surfaces is much less than the
radius of a water molecule (1.4Å), the alkali cations are adsorbing very close to the 0-plane, and therefore very close to
the surface. Even though they may be coordinated to one or
more of the chemisorbed water molecules at the surface, the
alkali ions are probably dehydrated compared with their state in
aqueous solution.
3.2.2. Hematite, Quartz, Amorphous Silica, Goethite, and
Alumina
It can be seen in Table 2 that hematite, quartz, and amorphous silica have model capacitances that tend to increase for
a series of cations such as Li $ Na $ K and N(CH3)4 $ Na $
K, which are in order of decreasing hydrated radius (Table 1).
As already noted above, this order is in agreement with the
lyotropic series for the Hg-water and AgI-water interfaces.
Because the variation in model capacitances reflects hydrated
cation radii, it seems reasonable to postulate as a first approximation that the monovalent cations are hydrated when adsorbed to hematite, quartz, and amorphous silica. All these
minerals have dielectric constants much lower than those of
Table 3. Predicted inner-layer capacitances (C1, %F.cm#2) consistent with the triple-layer model.a
Solid
'-TiO2
"-TiO2
Fe3O4
'-MnO2
Fe2O3
FeOOH
'-Al2O3
(-Al2O3
Quartz
am. SiO2
a
Li!
Na!
K!
Rb!
Cs!
NH4!
N(CH4)3!
Mg!!
Ca!!
Sr!!
Ba!!
155
155
155
155
80
56
89
89
90
73
131
131
131
131
88
60
99
99
99
95
108
108
108
108
96
63
107
107
108
126
103
103
103
103
94
63
107
107
106
119
94
94
94
94
90
61
103
103
102
103
104
104
104
104
94
63
107
107
113
119
55
55
55
55
67
50
74
74
77
50
157
157
157
157
67
50
74
74
77
50
132
132
132
132
71
52
78
78
81
56
121
121
121
121
71
52
78
78
81
56
110
110
110
110
74
53
81
81
83
61
Calculated with Eqns. 6 and 11 through 15 and radii from Table 1.
3650
D. A. Sverjensky
Table 4. Predicted distances " (Å) consistent with the triple-layer model.a
Solid
Li!
Na!
K!
Rb!
Cs!
NH4!!
N(CH4)3!
Mg!!
Ca!!
Sr!!
Ba!!
'-TiO2
"-TiO2
Fe3O4
'-MnO2
Fe2O3
FeOOH
'-Al2O3
(-Al2O3
'-SiO2
am. SiO2
1.5
1.5
1.5
1.5
5.9
8.4
5.3
5.3
6.1
2.4
1.8
1.8
1.8
1.8
5.3
7.8
4.7
4.7
5.5
1.8
2.1
2.1
2.1
2.1
4.9
7.4
4.3
4.3
5.1
1.4
2.2
2.2
2.2
2.2
5.0
7.5
4.4
4.4
5.2
1.5
2.5
2.5
2.5
2.5
5.2
7.7
4.6
4.6
5.4
1.7
2.2
2.2
2.2
2.2
5.0
7.5
4.4
4.4
5.2
1.5
4.2
4.2
4.2
4.2
7.0
9.5
6.4
6.4
7.2
3.5
1.5
1.5
1.5
1.5
7.0
9.5
6.4
6.4
7.2
3.5
1.8
1.8
1.8
1.8
6.6
9.1
6.0
6.0
6.8
3.1
1.9
1.9
1.9
1.9
6.6
9.1
6.0
6.0
6.8
3.1
2.1
2.1
2.1
2.1
6.4
8.9
5.8
5.8
6.6
2.9
a
Calculated with Eqns. 16 through 21 and radii from Table 1.
1
rM,hyd
6.0
"
!
C1 (8.854)53 (8.854)53
(14)
rM,hyd
1
2.9
Aluminas: "
!
C1 (8.854)53 (8.854)53
(15)
Goethite:
In Eqn. 15, it can be seen that the apparent value of r1 for goethite is
larger than for any of the other minerals depicted in Figures 2 and 3, in
part because the actual capacitance value for these goethites is so small
(only 60 %F ! cm#2; Table 2). Additional experimental data for surfacecharge development on goethite, aluminas, and quartz in a range of
different electrolytes are needed.
" " 3.5 ! rM,hyd.
(17)
" " 3.7 ! rM,hyd.
(18)
" " 0.0 ! rM,hyd.
(19)
For quartz,
For amorphous silica,
For corundum and (-alumina,
" " 2.9 ! rM,hyd.
4. PREDICTION OF TRIPLE-LAYER MODEL
CAPACITANCES AND MODEL STRUCTURES OF
OXIDE–ELECTROLYTE–WATER INTERFACES
(20)
For goethite (with BET surface areas greater than 50 m2!g#1),
" " 6.0 ! rM,hyd.
4.1. Prediction of Capacitances
The close agreement between the capacitances calculated with Eqn.
6 and Eqns. 11 through 15 and the experimentally derived capacitances
can be seen more clearly in Figure 4. It should be emphasized that the
latter are a set of model capacitances that were obtained by fitting
surface-charge data over a wide range of ionic strengths. They refer to
a specific set of assumptions detailed previously (Sahai and Sverjensky,
1997a,b) and are subject to uncertainties of at least &10 %F ! cm#2,
which is of the same order as most of the discrepancies in Figure 4.
Uncertainties of this order contribute uncertainties in calculations of
surface charge of less than &2 %C ! cm#2 at an ionic strength of 0.1
mol/L and a pH '4 units from the zero point of charge.
Overall, the close agreement between the calculated and experimentally derived capacitances suggests that Eqn. 6 and Eqns. 11 through 15
and the distances (r1) given in Figures 2 and 3 can be used to make
predictions of triple-layer model capacitances for a wide range of
electrolytes and solids. The results of such predictions are given in
Table 3 for a selection of alkalis and other monovalent and divalent
cations. Because most of the data used to calibrate Eqn. 6 and Eqns. 11
through 15 refer to monovalent cations, the predictions in Table 3 for
these ions should be the most reliable. It must be emphasized that the
predictions for divalent cations are preliminary values only.
4.2. Prediction of Model Structure of the Oxide–Water Interface
The model developed in the present study for calculation of capacitances also has implications for the structure of the oxide– electrolyte–
water interface in the context of the triple-layer model. In particular,
calculation of the distance " for the triple-layer model can be made
with Eqn. 3 because the distance r1 is known for a number of minerals
of interest. The regression calculations depicted in Figures 2 and 3
suggest that the following relations for calculating " can be used:
For rutile, anatase, and magnetite,
" ) 0.75 * r M,cr.
For hematite,
(16)
(21)
Predicted values of ", established by use of Eqns. 16 through 21 are
given in Table 4 for a wide variety of electrolytes.
To illustrate the predicted differences between adsorption of dehydrated and hydrated ions, values of the alkali ion distances from the
surfaces of rutile and quartz are depicted in Figure 5. It should be
emphasized that the capacitance model presented above predicts only
the distance " in the triple-layer model (i.e., r1 ! rM in Fig. 5). The
alkali ions on both rutile and quartz in Figure 5 are at a total distance
from the surface equal to the predicted values of " (" r1 ! rM, from
Table 4) plus the distance of the 0-plane from a reference plane. In
Figure 5, the surface reference plane is at 0 Å. For rutile, the 0-plane
may be '1.1 Å from the reference plane (see above). For quartz,
following the same crystal– chemical reasoning previously used for
rutile, the 0-plane is depicted '0.4 Å from the reference plane (Fig. 5).
The 0-plane location for rutile depicted in Figure 5 results in a
calculated Rb! ion distance from the rutile surface very close to the
experimental value (3.4 Å; Fenter et al., 2000). The rest of the results
in Table 4 and Figure 5 represent model estimates that offer a more
detailed picture of the oxide– electrolyte–water interface than has previously been offered within the context of the triple-layer model. For
example, from the distances in Figure 5, it can be inferred that the Li!
ion should be closer to the surface of rutile than the Cs! ion. However,
for quartz, it is predicted that the Cs! ion will be much closer to the
surface than the Li! ion. Predictions such as these can be tested with
further X-ray standing-wave studies.
4.3. Implications for Surface Complexation of Metals at the
Oxide–Electrolyte–Water Interface
Taken together, the triple-layer capacitance model developed here
and the results of X-ray standing-wave experiments imply that the
alkali cations are bound very close to the surfaces of rutile, anatase, and
magnetite. The capacitance model specifies a layer of chemisorbed
water molecules at the surface and a layer of adsorbed alkali cations.
Because the distance r1 " 0.75 Å (Fig. 1) on these surfaces is much less
Prediction of electric double-layer capacitances
3651
adsorbed on the "-plane of these minerals will form inner-sphere
complexes. The results of X-ray standing-wave and EXAFS studies of
Sr2! adsorption on rutile (Fenter et al., 2000) are consistent with this
suggestion. However, on minerals such as quartz, amorphous silica,
and similar solids, trace amounts of Ca2! or Sr2! adsorbed on the
"-plane should form outer-sphere complexes. This expectation is consistent with the results of X-ray standing-wave and EXAFS studies of
Sr2! adsorption on silica and goethite (O’Day et al., 2000; Sahai et al.,
2000) and hydrous ferric oxide (Axe et al., 1998).
5. CONCLUSIONS
Fig. 5. Predicted distances of adsorption of alkali cations from the
surfaces of rutile and quartz. Note all the distances are given relative to
a mineral-surface reference plane, which is the last metal– oxygen plane
of the solid. The distance of the 0-plane was estimated from crystalchemical considerations (see text). The position of the alkalis relative
to the 0-plane (i.e., " " r1 ! rM) was calculated with the capacitance
model summarized in the text. On rutile, the electrolyte cations are
dehydrated, whereas on quartz they are hydrated.
than the radius of a water molecule (1.4 Å), it appears that the layer of
chemisorbed water molecules is interpenetrated by the layer of alkali
cations. Even though the cations may be coordinated to one or more of
the chemisorbed water molecules at the surface, they are dehydrated
compared with their state in aqueous solution. This conclusion strongly
suggests that the alkali cations form inner-sphere complexes with the
surface functional groups and/or the chemisorbed water molecules on
rutile, anatase and magnetite. In contrast, on minerals such as quartz,
amorphous silica, goethite, hematite, and alumina, the alkali cations are
probably hydrated, forming outer-sphere complexes.
If the alkali cations of a background electrolyte determine the structure of the oxide–water interface by forming inner-sphere complexes
on the "-plane of the triple-layer model for rutile and similar solids, it
follows that trace amounts of other metals (e.g., Ca2!, Sr2!), if also
(1) The triple-layer model capacitances (C1) for different mineral
surfaces fall into two groups: on rutile, anatase, and magnetite, values
of C1 increase with decreasing crystallographic radius from Cs! to Li!;
on quartz, amorphous silica, goethite, hematite, and alumina, values of
C1 increase with decreasing hydrated cation radius from Li! to Cs!.
(2) The triple-layer model capacitances on both groups of solids can
be described by a model of the interface with physically reasonable
parameters. The model specifies a layer of chemisorbed water molecules at the surface and a layer of adsorbed electrolyte cations. On
rutile, anatase, and magnetite, the layer of chemisorbed water molecules is interpenetrated by the layer of electrolyte cations. On quartz,
amorphous silica, goethite, hematite, and alumina, the layer of chemisorbed water molecules varies in thickness. On both groups of minerals, the model capacitances are consistent with interfacial dielectric
constants ranging from 20 to 62.
(3) It is inferred from the present study, and from the X-ray standing
wave results of Fenter et al. (2000), that the alkali and alkaline earth
cations adsorb close to the surfaces of rutile, anatase, and magnetite as
dehydrated inner-sphere complexes. In contrast, these cations adsorb
further away from the surfaces of quartz, amorphous silica, goethite,
hematite, and alumina as hydrated, outer-sphere complexes.
(4) Triple-layer model capacitances can be predicted for systems that
have not been studied experimentally—for example, for oxides immersed in all alkali and alkaline earth electrolyte solutions. Similarly,
the distance " for the triple-layer model and estimates of the total
distance from a surface reference plane can be predicted for many
systems that have not been studied experimentally.
It should be emphasized that the capacitances predicted in this article
refer specifically to the triple-layer model of the oxide– electrolyte–
water interface. It seems likely that a similar approach could be taken
with other surface complexation models that explicitly treat electrolyte
ion adsorption provided that an extensive reanalysis of all the surfacecharge data is undertaken. In addition, the predicted capacitances
summarized above refer only to single oxide and electrolyte systems.
When transition and heavy metals are also present in the system, it can
be assumed, as a first approximation, that the capacitance of the
oxide– electrolyte–water interface is unaffected (e.g., Criscenti and
Sverjensky, 1999).
Acknowledgments—Discussions with W. Casey, L. Criscenti, J. A.
Davis, P. Fenter, M. Machesky, C. Koretsky, J. Kubicki, and N. Sahai
helped the development of this work significantly, as did reviews by J.
Dyer and D. Sparks. Financial support was provided by DuPont
(through Noel Scrivner), NSF grant EAR 9526623, and DOE grant
DE-FG02-96ER-14616. I also thank Harold C. Helgeson for his everinspiring example of how to do theoretical geochemistry, as well as his
insistence on the importance of having lunch.
Associate editor: D. L. Sparks
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APPENDIX
Figures A1 and A2 contain plots of surface-charge data as functions
of pH and ionic strength. The solid curves in the figures represent
calculations carried out in the present study by use of the parameters in
Table A1 and the extended triple-layer approach described previously
(Sahai and Sverjensky, 1997a,b; Criscenti and Sverjensky, 1999). In
this approach, the fitted parameters are the capacitance (C1) and the
equilibrium constants for the binding of the electrolyte cation (log KM)
and anion (log KL) in the case of 1:1 electrolytes. In the case of quartz
in CaCl2, the only fit parameter was the capacitance (see Table A1,
footnote g).
All other parameters in the Table A1 were estimated in a manner
consistent with the extended triple-layer approach. The site densities
(Ns) were taken from Sahai and Sverjensky (1997a,b) and Criscenti and
Sverjensky (1999) and are consistent with the results of tritium exchange experiments. In the case of the surface-charge data for quartz in
CaCl2 (Riese, 1982), the site density in Table 1 refers to tritium
exchange experiments carried out by Riese (1982). The values of the
surface protonation equilibrium constants in the Table A1 were calculated from the ZPC (" 0.5[log K1 ! log K2]) reported by the authors
who had carried out the surface-charge experiments, and from the
theoretically estimated values of (pK (" log K2 # log K1) published
previously (Sverjensky and Sahai, 1996).
3654
D. A. Sverjensky
Fig. A1. Plot of surface-charge data as functions of pH and ionic strength for rutile in different electrolyte solutions. The
symbols represent experimental data. The curves were generated in the present study (see text).
Prediction of electric double-layer capacitances
3655
Fig. A2. Plot of surface-charge data as functions of pH and ionic strength for quartz and amorphous silica in different
electrolyte solutions. The symbols represent experimental data. The curves were generated in the present study (see text).
Table A1. Triple-layer model parameters derived from regression of surface-charge experiments referring to wide ranges of ionic strength using
the methods described by Sahai and Sverjensky (1997a).
Salt (mL)
Solid
Nsa
log
K1b
log
K2c
log KMd
log KLe
C1f
Source of surface charge data
N(CH3)4Cl
CsCl
KNO3
NaCl
NaClO4
LiCl
CaCl2
KNO3
N(CH3)4Cl
'-TiO2
'-TiO2
'-TiO2
'-TiO2
'-TiO2
'-TiO2
quartz
quartz
am. SiO2
12.5
12.5
12.5
12.5
12.5
12.5
4.5
11.4
4.6
2.6
2.8
2.6
2.2
2.6
2.8
#2.2
#2.2
#1.2
9.0
9.2
9.0
8.6
9.0
9.2
6.2
6.2
7.0
3.4
2.5
2.1
2.5
2.0
2.8
—g
0.2
2.0
3.4
2.4
2.2
2.3
1.9
2.2
0.2
2.0
0.7
55
95
110
120
125
155
81
105
50
Yates (1975)
Kallay et al. (1994)
Yates (1975)
Machesky et al. (1998)
Bérubé and de Bruyn (1968b)
Kallay et al. (1994)
Riese (1982)
Huang (1996)
Casey (1994)
Site density (nm#2).
Refers to the equilibrium %SOH ! H! " %SOH!
2.
c
Refers to the equilibrium %SO# ! H! " %SOH.
d
#
!
#
Refers to the equilibrium %SO ! M " %SO _M!.
e
#
#
" %SOH!
Refers to the equilibrium %SOH!
2 ! L
2 _L .
f
Capacitance (C1) has units of %F. cm#2.
g
The Ca equilibria used to fit the data for quartz in CaCl2 are based on independent fits to data for the adsorption of trace amounts of calcium in
KNO3 and NaCl solutions (D. A. Sverjensky, unpublished data) which resulted in the following equilibria and equilibrium constants (adjusted for
differences in site densities and molarity/mole fraction conversions): %SOH ! Ca!! ! H2O " %SO#_Ca(OH)! ! 2H! (log K " #14.0); and
2%SOH ! Ca!! " (%SO#)2_Ca!! ! 2H! (log K " #7.1); the value of C1 given for quartz in CaCl2 solutions was the only adjustable parameter
in the regression of the data depicted in Figure A2.
a
b