6
Free Energies o f Electrical D o u b l e Layers at the O x i d e - S o l u t i o n Interface Derek Y. C. Chan Department of Mathematics, University of Melbourne, Parkville, Victoria 3052, Australia
A. simple graphical method is used to illustrate the roles of the charge-potential relationship of surfaces bearing ionizable groups and the charge-potential relationship of the diffuse double layer in determining the free energy of formation of a charged oxide/solution interface as well as the double layer interaction free energy involving such surfaces. In all physical and chemical processes, and in particular those of relevance to geochemistry, that involve the oxide/aqueous solution interface, it is important to understand the general, non-specific characteristics of that interface before focussing on those specific processes or mechanisms of interest. Due to the structure of mineral surfaces, the mineral oxide/aqueous solution interface will invariably acquire a net charge or electrostatic potential relative to the bulk solution. The electrical state of the interface will depend in part on the chemical reactions that can take place on the mineral surface, and in part on the electrolytic composition of the aqueous environment. From the preceding contributions in this volume it is evident that the techniques of modelling the electrical double layer properties at the oxide/electrolyte interface have been well developed (2_ 11). However, the problem s t i l l contains a certain amount of 'art form* in the sense that there is more than one school of thought as to how the various modelling techniques should be applied. $
The aim of this paper is not to add to the current debate but to present a simple graphical method of analysing the free energy of formation of the electrical double layer at the oxide/solution interface (J_). This will provide a simple way of visualizing the complementary roles of chemical reactions or surface properties of 0097-6156/ 86/ 0323-0099S06.00/ 0 © 1986 American Chemical Society
G E O C H E M I C A L PROCESSES AT M I N E R A L S U R F A C E S
100
the
mineral
and
the
thermodynamics being
used
concepts other
of
specific
double
physical
algorithms
Equations
a.
The
Let
us
of
Oxide
a
For
the
the
at
is
constant
voltage
(within
reasonable does
(at
surface
also
be
can
be
calculate
the
dropping
determining
the
surface
the
key
ideas,
can
be
applied
is the to
used
to
energy
used
illustrate
and
to
to
the
bring
check
out
numerical
interaction free
energy.
can
In
a
good
say
that
an
to
external
case,
limits)
charge
electrode,
relative
this
physical allow
to
mercury
by
a
surface
as
equation
be
the or
approximation
the
the
can
electrical
bulk
region
potential
can
transfer
the
whereby
potential.
surface
controlled
not
we
surface
metal
set
source
to
any
of -
a
value
mercury/electrolyte
chemical for
of
the
reactions
case
state
of
of
the
to
NaF). mercury
is ψ
where
and
p h y s i c a l mechanisms or
source.
least
Therefore,
basic
charge
the
electrolyte
occur
general
can
which
to
in
amphoteric
illustrate
interaction free
result
familiar
ψ
interface
an
semiconductors.
method
layer
to
quite as
layer
While
Surface
surface
potential
diffuse
State
consider
develop
are such
commonly u s e d
The
the
example
here
graphical
the
of
interface.
surfaces
same
of
simple
the
a
nature
the
presented
The a
as
types
nature
of
the
constant
*
c a n be
(1)
constant
adjusted
by
changing
the
constant
voltage
source. In charge ions
a
typical
by at
the
specific
consequence parameters may
also
of
the that
allow
supporting this
of
the
to
of
the
the
surface surface
such
in
or
do
not
of
state
anions
In and
paper
clear.
alter
acquires
the
key
As
will
addition,
we
a
involve
cations
Such
a
determining
sites.
surfaces
this
discussion
surface
surface
potential
reactions. of
However, the
of
groups
of
adsorption
keep
oxide
adsorption state
characterize for
electrolyte.
model
oxide, or
amphoteric
equation
possibility the
inorganic
dissociation
of
shall
one the
ignore
embellishments ideas
presented
here. We
derive
the
considering
the
determining
ions:
equation
generic
dissociation
AH
2
AH
+
t
AH +
t A" +
H H
+
of
an
amphoteric
reactions
involving
surface
by
potential
+
(2)
6.
CHAN where
Electrical we h a v e
ions.
Double
assumed
The change
Layers at the Oxide-Solution the potential
i n free
are characterized Κ and Κ : +
determining
energy
associated
[ΑΗ]·Η/[ΑΗ ] s ζ
- Κ
[Α~]·Η
-
+
s
Κ
the i n t r i n s i c dissociation
terms
the activities we s h a l l
and
surface
use
consistent
If
neglect
of the diffuse
the surface
sites/unit
area,
(e
= protonic
charge)
a
We
Q
S
have
eN s
2
+
]
that
+
[AH,/"]
charge
constants
From E q u a t i o n
determining relation
groups
at i s
+
[A~]}
density K
+
(4)
(5)
of
terms
ions
σ
the
at the
a
and K
l
c a n be
potential
related
of
H
the positive and as K
5 we s e e t h a t
i s i n turn
a
2
i n
varied
determining
to the concentration
H, i n the bulk
i s given
i n
, K_ of
written
concentration ions
i s the
of the surface
determining
+
this
to describe
amphoteric
density
potential
which
coefficients
approximation
theory
these
[ K , K_ a r e sometimes
O) ]·
This
i n
surface.
+ Η /Κ + Κ /Η } s + s
surface of
the solution
potential
with
Κ /Η }/{l s
the
i n the electrolyte.
charge
[A~]}/{[AH]
the
at the surface,
assume
-
{ Η /Κ s +
groups.
changing
ions
{[AH
concentration
literature
by of
-
g
and the d i s s o c i a t i o n
negative the
eN
expressed
solution surface
-
be d e f i n e d
at
due t o a c t i v i t y
layer
the surface
should
species
instead.
i s populated
Ν
constants
constants
t h e Gouy-Chapman
double
hydrogen reactions
(3)
corrections
the use of
t o be these
-
the various
concentrations
with
behaviour
of
101
+
/ [AH]
Strictly, However,
ions
with
by t h e i n t r i n s i c d i s s o c i a t i o n
9
of
Interface
solution
by t h e Boltzmann
-
we
shall
distribution
-βψ /kT H where
is
relative
t o the bulk
the
Gouy-Chapman double
mean
model
s
(6)
= H e
electrostatic solution.
which
potential
Equation
we
shall
6 i s
use
to
at
the
interface
consistent
describe
with the
the
diffuse
layer.
Combining charge d e n s i t y o
S
Q
Equations 5 a n d 6, c a n be w r i t t e n as = eN {1
g
{δ sinh
+ 6 cosh
[β(Ψ
[β(ψ
Ν
-
the expression
Ν
Ψ
for the
surface
i ^ / k T } /
0
β
)/Κ]}
(7)
where δ
= 2(Κ /Κ_) +
ΔρΚ
=
1 / 2
Ξ ρΚ
+
2 χ 1 0
- ΔρΚ
Δ
ρ
Κ
/
2
(8)
(9)
G E O C H E M I C A L P R O C E S S E S AT M I N E R A L S U R F A C E S
102 pH ψ Ν and
pH
is
charge?
dictated is,
by
the
interface
surface the
bulk
pH -
potential surface will
is that
of the
on
(Ψ
ψ
is
determined
by
),
the
the
concentration
of
potential
and
while
by
the
with
a
7
interface groups layer
a
will
analogous
to Equation
b. We
the
The D i f f u s e shall
use of
of
that
of
the
(see
as
curve
δ
0
the
bulk
Equation
is
δ (see
7
controlled Equation
we a r e
dictated
a
depends
ψ
or
Ν
that
is
-
s
Q
Ψ
curve
reminder
-
OQ
S
of
values
by
state
the
are
of
8)
dealing
by
chemical
as
that
7 which
or
for semiconductor,
of
the
surface double
surface there
charges
will
relationship
characterizes
under
oxide/solution
the diffuse
develop
charge-potential
allowed
the
dissociation
the e l e c t r o l y t e
such
or
interface
the
be an
that
the e l e c t r i c a l
is
response
of
of
disposition surface
an in
diffuse
d
Q
layer.
potential
cause
charge
Gouy-Chapman
double
the
at
Debye
planar the
implies
that
layer at
will sinh
length
a
1:1
be g i v e n
of
model
counter-ions i f
electrolyte by
the the
and
interface. the
a
The surface
is
then
(_3)
(ei|; /2kT) 0
describe
solid/electrolyte
near
= (2Ke e kT/e)
the inverse
a
to
to this
electrolyte
density
r
(_3)
of
interface
Q
model
According
accumulation
double
the planar
o κ is
a
co-ions of
of
Layer
familiar
will
depletion potential
Double
the diffuse
application
where
a
the
surface.
behaviour
its
value
a
(pH)
this
of
equilibrium
only
this
the quantity
as
For surfaces
mechanisms state
of
equilibrium
determined
of
of
but
surface
calculated
potential ions
shape
via
at
c a n be
the
ions; the
But s i n c e
Nernst
on
Κ , K_ and
dissociation
particular S
a
which
between t h e
i f
charge
the location the
imposes
only
Thus,
is
That
density
constants
or
zero
groups.
the surface.
different
groups
determining
o^
of
which
groups.
dependent
For a
ψ~ .
point
relationship
is
charge
relationship
and t h e p r o p e r t i e s near
this
potential
7.
serves
actual be
surface
determining
family
the
equation of
s
the surface
Although Equation
7, of
constants
charge-potential of
the
the surface
surface
the surface
the overall
dissociation
reactions
by
sign
The s u p e r s c r i p t
(_2).
of
association
that
see Equation
1)
of
properties.
potential
by
Figure
at
the
of
solution
Equation
surface
pH
and the charge
δ
that
ψ^, t h e s u r f a c e
g
0
of ψ^
(11)
state
potential
ensure
given
of
pH]
Ν , the d i s s o c i a t i o n
then
will
function -
groups
known
bulk
Note
surface
any other
potential
Ν
reaction
the concentration
of
groups
be
Nernst
and
(10)
dissociation
can acquire.
surface
independent
the
the p o t e n t i a l
charge of
of
)/2
-
[pH pzc
an equation
equilibrium
between
density
value 7 is
dissociation
relation the
the
+ pK
+
2.303(kT/e)
the
Equation
Z C
= (pK
pzc
d
the e l e c t r o l y t e
(12) and ^ i s t h e
6.
Electrical
CHAN
Double
relative
permittivity
function
of
interface
ψ~
which
motions
of
electrolyte accumulated as
is
a
superscript
d
denotes
that
is
and a
double
potential. which
charge-potential the
-
thermal
A
more
effects
effects $Q
double
will
(4).
relationship
diffuse
the charge
includes
and
a the
in
of
potential OQ
of
ions
applied
connecting
by
layer
amount
layer
as
state
between
fluctuation
controlled
for
of
equilibrium
diffuse
equation
103
12
equation
diffuse
the
the
size
different
the
Interface
Equation
an
interactions of
of
as
the
result
treatment ion
solvent.
by
control
a
interface
dictated
will as
in
the
regarded
coulombic
finite
result
of
be
and
sophisticated such
may
Layers at the Oxide-Solution
The of
layer
the
in
the
electrolyte. c.
The
The
Equilibrium
properties
interface surface state
the
is
given
constant,
the
or
zero
calculations been
the
the
and
by
a
oxide/aqueous these
potential
as
determining
of
a
Equations
the
ψφ v s
is
7
pH
of
range
of
the
charged
12
to
obtain
the
pH
representative
10)·
A
distinctive
interface,
illustrate
of
this for
model
the
for
surface
potential result,
the
slope
we of
curve
2
M
(13)
l
+
of
charge
solution
amphoteric
expression
point
numerical
surface
concentration
To
the
of
are
(2_, an
held value
c o s h ( e i j ; / 2 k T ) [ l + 6 cosh[e(i|> - i j O / k T ] . ^ £! î£ \~ 2 γ 6[6 + c o s h ( e ^ - i^/kT)] '
+
U
of that
with
an
to
this
the
Detailed
function
oxide/solution
n
(ι
11).
the
of
being
changing
pH r e l a t i v e
solution
between
illustration parameters
parameters that
equilibrium
by
a
amphoteric
the
other
interface
of
the
moved
as
non-Nernstian.
and
An
variations
is
oxide/solution
of
intersection
Equation
the
solution a
of
be
potential
function
ions
use
(see
calculations
development
may
the
state
12.
Given a l l
adjusting
examples for
and
point
of
of
Graphically,
point
7
2.
interface
surface
given
of
unique
Figure
state
equation layer.
Equations
the
and
feature the
in
the
double
equilibrium
of
density have
to
equivalently
charge
equilibrium both
diffuse
representing
result of
the
satisfy
corresponds
curves
of
of
must
and
Point
K
N
L
3
J
3 where the
γ
=
bulk
As ionic far
3
of
with
we
have
have
of
state
earlier the of
valid.
N^
being
άψ / d p H = the
the
the
in
of
function
of
electrolyte but
=
dm
the
general
at
and
C
For
a
298K.
di^/dpH given
the will
·
59.2mV
non-Nernstian
possibility
surface,
constant
mol
2.303(kT/e)
slope
degree
supporting the
Avogadro's
concentration
displayed
illustrate
mentioned
remain
C),
electrolyte
we
13 t o
species
equation so
g
system
Figure
Equation
N K/4N
1:1
Nernstian In
(10
by
behaviour. adsorption complicate
ideas
of the
discussed
104
GEOCHEMICAL
P R O C E S S E S AT M I N E R A L S U R F A C E S
0
Figure layer pH
pzc
pH r
curves
surface
where
for ψ
a
Gouy-Chapman
> 0
(
