Download A Simple Statistical Mechanical Approach to the free Energy of

A Simple Statistical Mechanical Approach to the free
Energy of the Electric Double Layer Including the
Excluded Volume Effect
Veronika Kralj-Iglič, Aleš Iglič
To cite this version:
Veronika Kralj-Iglič, Aleš Iglič. A Simple Statistical Mechanical Approach to the free Energy
of the Electric Double Layer Including the Excluded Volume Effect. Journal de Physique II,
EDP Sciences, 1996, 6 (4), pp.477-491. <10.1051/jp2:1996193>. <jpa-00248310>
HAL Id: jpa-00248310
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Submitted on 1 Jan 1996
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Phys.
J.
II
France
(1996)
6
477-491
Mechanical
Approach to the
Layer Including the Excluded
Statistical
A Simple
Double
the
Electric
Kralj-Iglit (~)
Veronika
APRIL
and
1996,
PAGE
477
Free Energy of
Elsect
Volume
Iglit (~,*)
Ale§
(~) Institute of Biophysics, Medical Faculty, Lipiteva 2, 61000 Ljubljana, Slovenia
Slovenia
Engineering,
Trialka
25, 61000 Ljubljana,
(~) Faculty of Electrical
(Received
29
May 1995,
A
free
the
simple analytic
energy of a single
of
solutions
of
surfaces
1995,
accepted
and
interfaces
January1996)
9
theory
ensemble
Classical
Abstract.
for
November
14
Thermodynamics
Thermodynamics
PACS.82.60.Lf
PACS.82.65.Dp
PACS.05.20.Gg
sion
revised
statistical
electric
approach
mechanical
double
layer.
is
applied
Electrostatic
to
interactions
derive
are
an
expres-
considered
field while the finite size of particles
constituting the elecby including the excluded
effect.
calculated
free
volume
The
univalent
electrolyte is compared to the corresponding free energy
obtained
by the
energy of an
Poisson-Boltzmann
theory. It is shown that the excluded
considerably
volume
effect
increases
the energy- of the
field while the corresponding
electrostatic
decrease of the entropic
contribution
contribution
pronounced. As the entropic
the
of the
electrostatic
is less
is larger than
energy
field and as the effects
both
partially cancel, the free energy
contributions
obtained
by the
on
Poisson-Boltzmann
approximation regarding to the excluded
volume
effect.
theory is an excellent
by
of the
means
trolyte
solution
electrostatic
mean
considered
is
Introduction
1.
electrolyte
solution is in
with the charged plane,
accumulated
counterions
contact
are
plane and coions are depleted from this region, thereby creating a diffuse electric
double layer. The electric
double layer is used to describe
electrostatics
of liquid crystals,
the
cellular
membranes, phospholipid bilayers and metals in contact with the electrolyte solution.
The simplest description of the electric
double layer is obtained by the
Poisson-Boltzmann
theory [1,2j and its extensions
such as the
Stern-Gouy-Chapman
model [3j which
introduces
the
of closest approach of the ions to the charged plane.
distance
these
theories
Within
ions
considered
be
dimensionless
surface
considered
uniformly
point
charges,
charge
is
be
to
to
are
smeared
described
with
the plane, the electrolyte
solution
uniform
is
continuum
over
as
a
If the
the
near
dielectric
originating
neglected.
the
system.
[4,5],
and
quantity
©
Author
phase
Les
free
for
(ditions
of ions
which
free
The
electrostatic
(*)
charges
from
A
potential of
the
constant,
energy
transitions
energy
of
correspondence
de
Physique
mean
force is
the
charged
and
is also of
is
used
of the
the
interest
to
lipid
electric
describe
to
surface
the
scope
interaction
[6].
Within
layer has
(e-mail: [email protected])
1996
be the
while
and is the
molecules
double
taken
been
electrostatic
mean
direct
of this
ion-ion
work is the
potential
interactions
free
energy
are
of
layers
the
Poisson-Boltzmann
theory
from
derived
thermodynamic
of the
two
electric
double
JOURNAL
478
PHYSIQUE
DE
applying either charging process or
the
or by summing
energy of the
expressions being equivalent [5j.
grounds,
three
order
In
to
charged plane
the
ions,
between
were
approach
mech~nical
was
electric
field
integration of the energy of
the entropic
contribution,
the
and
all
theoq-, finite size of ions, image effects, direct
interactions, and discreteness of charge forming
included in the description of the electric
double layer. A
statistical
[7-9j.
initiated
basing on the expression for a molecular
Hamiltonian
the
surlrount
interactions
N°4
temperature
field
electric
II
Poisson-Boltzmann
solvent
and
structure
and of the-b,alk
electrolytes were
double
extended
to the
development of the modified
Poisson-Boltzmann
theory [10-12j,
cluster
expansion theory of the double layer [13j and integral equation theories [14-17j.
predictions with computer
simulations
Comparison of theoretical
[18-29] indicates that the
Poisson-Boltzmann
interface
theory satisfactorily describes the
ion density and
monotonous
electrolyte over a wide range of surface charge and electrolyte
potential pmfiles of univalent
j12;16,18,19,22-24j while oscillatory behavior of the ion
profile
concentration
concentration
and charge
which
inversion
in highly charged surfaces, highly charged electrolyte
may
occur
[20-22,24j were explained by applying the theories based
and high electrolyte
concentrations
of
theories
The
nonuniform
layer problem resulting
on
the
fluids
in the
[11,12,16,22,23j.
Hamiltonian
molecular
layers pr~dicted by
The
of the
interaction
electric
two
double
theory agrees well with the results of the Monte
for dilute
electrolyte, large separations and low surface charge [27j,
Carlo
simulations
univalent
while oscillatory effects and
between like-charged surfaces
which may
in hi)hly
attraction
occur
surfaces,
highly
charged
electrolyte
and
high
electrolyte
[27,29,30]
charged
concentrations
were
obtained by integral equation
theories [29,32j.
based on the
molecular
Hamiltonian
often give» a numerical,
However, the theories
are
little
form.
transparent
simplicity
providing
theory can
and
Poisson-Boltzmann
the
Therefore
still
transparency
remarkably good
a
describe
fit
features
the
the
Poisson-Boltzmann
widely used
[33-37j. Also,
of the
and
interpretation
for
it
was
that
are
experimental
that
determined
layer providing
double
formalisms
related
of the
to
often
Poisson-Boltzmann
the
the
due
data,
fitted
parameters
are
physical parameters [38j.
not
as
If the surfa~~ charge density is high, the
the
and the
accumulate
counterions
surface
near
Poisson-Boltzmann
theory may there
the
overestimate
counterion
concentration
to an
extent
for
that it becomes
unphysical [37j. In this work the corresponding
accounted
corrections
were
field
approach.
electrostatic
A
simple
by including an excluded volume effect within the
mean
free
statistical
mechanical
procedure is proposed in which the expression for the
electrostatic
the
and
consistently
related
electostatic
potential
and
solvent
distribution
ion
to
to
energy,
contributing terms is transparent as the analytical
functions, is derived. The origin ofindividual
form of the free energy is retained
along the derivation.
The
obtained
free energy is compared
with the free energy of the
Poisson-Boltzmann
theory and the conditions for the validity of the
considered
latter
2.
studied.
are
Theory
MINIMIZATION
2.I.
wITii
over
real
the
THE
are~L
an
density a
species of
extends
OF
CHARGED
A
is in
ions.
in the
and
THE
bearing
positive
ENERGY
A
at
its
system
surface
OF
is
THE
described
uniformly
ELECTROLYTE
in
which
distributed
a
SOLUTION
plane
charge
at
x
with
IN
CONTACT
0 extending
surface charge
=
the
molecules
electrolyte
solution
consisting of solvent
and M
bounded by the area A in the x
occupies a volume
0 plane and
direction.
Boundary effects are neglected. It is taken that the charged
with
contact
The
FREE
PLANE.
solution
x
electrostatic
plane and the charges of ions in the solution
create
a
mean
occupied by the solution, while on the other side of the charged plane ix
=
field
<
0)
in
the-
the
volume
electrostatic
N°4
ENERGY
FREE
field is
OF
ELECTRIC
THE
LAYER
DOUBLE
479
zero.
The
excluded
effect is
volume
included
description by requiring
the
in
that
the
volume
of the
system is a sum of the volumes of all the constituting particles. Thereby in calculation of
introduced.
All sites in this lattice
is
entropy a lattice with an adjustable lattice
constant
whole
the
occupied.
deriving the expression for the free energy of the system F subject to the local thermodyequilibrium (Appendix A) the methods of statistical
used [39j. Starting
namic
mechanics
are
from energies of individual
particles and treating the particles as independent and indistinguishable the free energy of the system F is obtained (A.16),
are
In
W~'
F
where
W~~
)eeo
=
=
il E~(x)A
+
/~ l~j~
kT
dx is the
ix)
nj
In
A
ii)
dx,
nsq
of the
energy
~~ ~j~
field
electrostatic
IA. ii),
k is the
Boltz-
nj is the density of the number of particles of the j-th
species (A.13), where j
solvent
0 denotes
molecules
and j
,M ions of the j-th
1,2,.
species, ns is the density of the number of lattice sites (A.14), q) is the partition
function of
the particle of the j-th species subject to no
electrostatic
variable (A.6-A.7), e is the permitT is the
constant,
mann
temperature,
=
tivity of
strength.
solution,
Integrations
=
the
permittivity of the free space and E is the
electrostatic
field
performed over the extension of the system in the x direction (d).
At the
distance
d from the charged plane the effects of the charged plane can
longer be
no
perceived by virtue of the screening of its
electrostatic
field by the
counterions.
However, the explicit expressions for the functions nj(x), j
JI and E(x) are
0,1,2,..
known.
To derive explicit expressions for these functions, we use the
condition for the free
not
minimum
thermodynamic equilibrium of the whole systein, so that
in the
energy to be at its
the
eo
is
are
=
,
bF
taking
.
into
the
is
conditions
the
that
while
constant
following
the
account
the
total
(2)
0,
=
conditions:
number
particles of each
of the
species
in
the
whole
system
performed,
variation
is
~
/nj
x)A
dx
=
Aj,
J
=
j3j
M,
o,1, 2,.
,
.
the
validity of the
Gauss
law
at
any
x,
~
~~°
"
e0
~ Vjnj(X),
(~)
J"1
where
.
the
eo
is
the
elementary charge
and
vj
valence
of the
ion
of the
j-th
species,
condition, that all of the lattice sites are occupied (A.2). Using the density of the
of particles and lattice
sites (A.13-A.14), it follows
that for any x
number
M
~s
~llj(X).
"
j=0
(5)
JOURNAL
480
problem
isoparametric
Above
reduced
is
/
~
F +
~j
the
to
functional
a
ljAj
PHYSIQUE
DE
II
ordinary
N°4
problem by constructing
variational
~
L(n(x), E(x))A dx,
=
(6)
o
~=o
~~~~~
~~
LjnjX), EjX)
lj, j
=
M,
0,1, 2,.
j660E~fi)
"
of
from
variables
the
~~ l~~ ~~
~~~~
~
)~)~
~~~
nM). Eliminating two
Lagrange multipliers and n
(no ni,
(4) and (5) the variation is performed by solving a system
the
are
=
.,
of the
~~
+ ~~
constraints
equations
Euler
)
j
0,
=
(8)
M,
2, 3,.
=
llJ
,
~
~/~Ej
lx
~~'
~~~
0~
At
the
d the
distance
charge
average
density
irolume
well
as
the
as
electrostatic
mean
field
are
zero
Ed
~~
=
=
dx
(10)
0,
~
and
the
potential 4ld
is
constant.
chose
We
4ld
Taking
into
gives after
account
some
the
above
the
calculation
no(x)
=
ill)
0.
condition
the
particle
distribution
solution
of the
equations
of
system
~~
=
(8)
and
(9)
functions,
(12)
~
,
~(n~d/nod) exp(-v~eo4l(x) /kT)
1 +
z=1
i~j
ix)
~~(~~~~°~~
=
1 +
~~~~
"~~°~~~~~~~~
j
=
M,
1, 2,.
,
(13)
,
~j(n~d/nod) exp(-v~eo4l(x) /kT)
z=1
and
the
differential
equation for 4l(x)
M
-eon~
~2@
~jvj(njd/nod) exp(-eovj4l(x)/kT)
~~~
~~~~
$
M
eeo
1 +
~j(njd/nod) exp(-eovj4l(x)/kT)
J"~
'
FREE
N°4
njd is the
of the
where
OF
density of number
density
the
ENERGY
particles of
of
molecules
solvent
far
DOUBLE
ELECTRIC
THE
from
the
j-th species
the
charged plane
LAYER
at
x
481
According
d.
=
(5)
to
is
M
n0d
~njd.
ns
"
(~5)
j=1
order
In
distance x,
M and 4l on the
the explicit dependencies of nj, j
0,1, 2,.
equation (14) subject to the conditions (10) and ill is solved. The charged
boundary condition at x
by an additional
0,
is taken into
account
obtain
to
=
.,
differential
the
plane
at
x
0
=
=
~~
~~~~
~o
~
EQUILIBRIUM
2.2.
order
In
should
to
be
defined.
reference
there
It
chosen
were
seems
system
are
hitherto
appropriate
electrostatic
diminished
be in
(which
field
to
to
which
extent
an
=
0, 1, 2,.
free
Helmholtz
the
formalism
the
energy
of each
species
as
the
reference
electric
solution,
field in the
electrostatic
an
If the
coions.
Therefore
reservoir.
of the
of the
excess
the
in
reference
counterions
over
the
the
even
system
coions)
thermal
reservoir
in
system and the
are
that the
temperature of the
temperature,
so
system
well.
in
There
is
no
derived
previous
in
solution
over
n~.
obtain
we
the
The
reservoir; nj,r~~
solution
in
Applying the
sections.
(1),
field
electrostatic
mean
uniformly
number density
distributed
electrolyte
of the
counterions
such
M, occupying the sites with the
.,
by using
of
negligible.
is
the
described
is
system
particles
with a large
reservoir
contact
system to be in
dimensions
much larger than
phase whose
are
Both, the electric double layer and the reference
consequence
describing the electric double layer
surroundings having a constant
and of the
reservoir
is kept
constant
as
and
all
reservoir
species of particles are
j
the
of
the
d.
In
with
over
a
amount
bulk
a
with
contact
is the
same
reference
distributed.
corresponding to
double layer
electric
of the
the
of particles
still be
consider
to
contain
LAYER.
DOUBLE
layer
double
solution.
amount
would
ELECTRIC
THE
electric
to
of
excess
same
there
the
over
an
particles
considered
OF
of the
energy
homogeneously
therefore
for all species of
extension
is
the
contain
to
layer described
particles were
ENERGY
system is required
the
if these
is
The
FREE
free
distributed
double
the
electrostatic
homogeneously
electric
double layer
system
the
ELECTROSTATIC
the
species
each
In
obtain
free
energy
the
contact
=
reservoir
expression
of the
the
njd,
for
reservoir
~
F~~~
=
~j
kT
v,e~
where
The
of the
V~~~ is the
system and
supersystem
electrostatic
field
volume
of the
the
reservoir
F~P
consists
and
the
distribution
electrostatic
the
functions
whole
(12)
together
of the
J"0
referred
are
contribution
contribution
of the
J
=
energy
of
the
should
be
taken
and
the
as
a
supersystem
solution
under
[39j.
the
Free
influence
energy
of the
(18)
F + F~~~
free
(13)
to
of the
reservoir,
system
and
nsq
reservoir.
F~P
While
deriving
equilibrium of
Ii 7)
~~§ dV,
njd In
system,
into
equilibrium
requirement of thermodynamic
by inserting the equilibrium
electrostatic
potential (obtained by
the
account
JOURNAL
482
solving
the
PHYSIQUE
DE
II
N°4
equation (14)) into the expression iii.
differential
After
rearranging
some
we
obtain
M
W~'
F~P
/
kT
+
=
I +
~j
~~~
M
d
~
°
4l(x) /kT
njvj eo
nod
~_~
+ n~ In
~
~"~
~j
l +
~~~
A dx
exp(-vjeo4l(~) /kT)
nod
~_~
M
kT~jNjT In
+
~~§,
where
taken
it is
supersystem
(19)
~~QJ
j=0
into
that
account
the
numbers
of each of the
species of the particles
M
the
in
constant,
are
NjT
/~ nj(x)
"
/
A dx +
njddv,
j
0,1, 2,.
=
(20)
M.
,
v~m
The
free
electrostatic
between
the free
with
the
the
electrostatic
energy
the
electric
supersystem
the free
is
same
defined
of the
energy
species NjT> j
that 4l
0,
of
field is
layer F~'
double
and
F~P
particles of the
negligibly small so
number
same
of
energy
of the
supersystem
M,
0,1, 2,.
=
difference
the
as
reference
but
which
in
,
=
jlel
j~~j
j/sp
j/sp
ref'
the
Since
reservoir
density of the
taken
is
particles
of
redistribution
the
particles
of
number
of the
electrostatic
field causing
very large, the
presence
solution in the original system only negligibly changes the
be
to
in
the
in
reservoir
njd>
j
,M,
0,I,2,.
=
that
so
follows
it
(19)
from
M
l~$
~
kT
=
NjT
nsq~
J= ~
£$~
By inserting
of the
solution
vjnj(x)
from
the
influence
under
(4)
it
F~~
where
W~~ is the
fi4l(x)
eeo
dx
order
to
calculate
4l ix
F~'
Electrostatic
free
potential depend
the
particles
in
the
solution
free
energy
(23)
+
~j
entropic
contribution
~~~
~_ ~
kTn~ In
dependence of the
be
electrostatic
energy
electrostatic
(24)
A dx.
M
F~' the
should
the
1l0d
+
~j
J=1
charged plane
that
is
and F~~t is the
field
1 +
In
field
W~~ + F~~t
=
electrostatic
~2q~
d
=
electrostatic
~M
of the
energy
F~~~
(19)-(22)
from
follows
of the
(22)
~~§
In
known.
well
as
on
n~.
This
as
the
Note
the
is
~ ~
exp(-vjeo4l(x) /kT)
~°~
potential
by solving the
electrostatic
obtained
particle
density of
that
~
the
distribution
the
number
contribution
f/
distance
(12)
which
kTn~ In
can
from
the
equation (14).
differential
functions
of sites
the
on
and
(13)
and
be occupied
(I + £$1
~)
0
the
by
Adx
FREE
N°4
originates
reflects
entropic
electrostatic
2.3.
VERY
field, the
DILUTE
If it is
assumed
for
x
It
should
solution
and
that
choice
the
of
the
The
thermic
WITH
very
PLANE.
CHARGED
THE
everywhere
dilute
particles.
of
motion
CONTACT
IN
is
483
electrostatic
energy of the
energy.
field in the system while
of the
electrostatic
influence of
of particles under the
distribution
the
SOLUTION
electrolyte
the
LAYER
emphasized
be
free
the
effect
volume
ELECTROLYTE
that
for
accounts
excluded
DOUBLE
ELECTRIC
THE
supersystem.
F~~t
contribution
the
any
reference
OF
important for the expression for
the strength and the
distribution
is
state
field W~'
the
the
from
reference
ENERGY
the
in
that
I-e-
system,
M
~jnj(x)
<
no(x),
(25)
j=i
the
distribution
ion
functions
taking
denominator
and
distribution
functions
nj(x)
the
account
njd
=
from
equation (13) by neglecting the
approximation nod Gt n~, to yield the
obtained
are
into
exp(-vjeo4l(x)/kT),
j
1, 2,.
=
sum
in
the
Boltzmann
(26)
M.
,
Considering
above
the
(25), equation (14)
condition
equation
~ ~j
~
~
(/~
"
"jnjd
exPl
transforms
-vj eo
Poisson-Boltzmann
the
into
41x)/kT)
127)
o
The
in
free
electrostatic
energy
F~'
is
obtained
by considering
In(I
approximation
the
+
x)
Gt
x
expressions (23, 24),
the
F~'
=
W~~ +
/~ oco ~
~
j
kT£(nj
4l(x)
ix)
(28)
A dx.
njd
J"
The
f/ £$
term
calculate
F~' the
kTnjd
~
A d~ in
dependence
(28) originates
of the
from
the
potential
electrostatic
reference
on
the
supersystem.
In
from
the
distance
order
to
charged
plane 4l(x) should be known. This is obtained by solving the
Poisson-Boltzmann
equation (27).
The expression (28) is equivalent to the expressions proposed by Marcus [5].
(23, 24), particle
In
electrostatic
free
distribution
expressions for the
contrast
to
energy
functions (12) and (13) and
differential
electrostatic
equation for the
potential (14), the corresponding
Poisson-Boltzmann
expressions (26-28) do not depend on the density of the number
of sites n~.
This
that in the
Poisson-Boltzmann
theory the particles in the solution
means
of particles.
considered
dimensionless
there is no
limit of the
and
concentration
are
as
upper
It should be kept in mind
that this is strictly true only within
the
assumption (25) that the
of ions is very
If the
surface charge density
small everywhere in the system.
concentration
high,
attracted
of
is
the
the
charged
plane and therefore
counterions
in
vicinity
(a(
many
are
(25)
condition
plane is
very
CASE
2.4.
for
UNIVALENT
OF
illustration.
coions.
of the
solvent
Far
be
there
may
violated
even
if the
of
concentration
ions
far
charged
the
from
low.
from
number
of
molecules
the
are
charged plane
the
solution
coions
is
ELECTROLYTE.
there
In the
are
denoted
equal and
by nod
solvent
A
of
system
molecules,
density of the
denoted by
are
univalent
univalent
number
nd
while
of
electrolyte
counterions
counterions
the
density of
considered
is
and
and
the
univalent
the
density
number
of
JOURNAL
484
DE
PHYSIQUE
N°4
II
8o
70
<
3
60
no
40
'
30
20
10
net
o
0.0
0.1
X
[arm
(full lines) and the correFig. I.
Density profile of
counterions
and of solvent
molecules
net
no
Poisson-Boltzmann
sponding
density profile calculated using the
theory (broken line). The
counterion
density of number of available
lattice
sites n~ (calibrated to the
of pure water) is marked
concentration
(dotted line). The values of model parameters are (a(
oA As/m~, T
310 K,e
78.5, MH~O
=
kg/kmol,
18
pH~o
results
The
compared
are
trolyte where
=
=
"
kg/m~
1000
#
results
the
to
of the
theory for univalent elecyields analytical expressions
Poisson-Boltzmann
simplicity that d
cc.
ion density profiles [4j as
it is taken
for the sake of
-
This
free
potential and
well as for the
electrostatic
profiles
The
density
of
the
and
of
the
solvent
molecules
presented
counterions
in
[6].
are
energy
calculated by using
Figure I (full lines). The corresponding density profile of the
counterions
theory is also shown (broken line). Close to the charged plane there is a
the
Poisson-Boltzmann
considerable
excluded
volume effect on the density profile of the
and on the solvent
counterions
The
of the
comparable to the
of
molecules.
concentration
counterions
is there
concentration
for
the
the
electrostatic
molecules
solvent
that
so
the
concentration
of the
solvent
molecules
deviates
significantly
charged plane. By using the
Poisson-Boltzmann
theory the countewhere the
excluded
rion
concentration
is higher than the corresponding
volume
concentration
effect is
considered.
that in the vicinity of the charged plane the
It can be noted
counterion
calculated
by using the
Poisson-Boltzmann
theory exceeds the
of
concentration
concentration
lattice sites thereby proving to be unphysical. The corresponding coion density profile
available
the values which are due to strong repulsion by the charged plane at least two orders
attains
of magnitude
smaller in the region
considered
that the
of the
contribution
coions to the
so
of the ion and solvent density profiles as
effect is negligible.
The
deviation
excluded
volume
field from the
theory can therefore be attributed
well as of the
electrostatic
Poisson-Boltzmann
mainly to steric effect of counterions
and
solvent
molecules
in a small region in the vicinity of
the charged plane.
from
its
It
can
value
be
far
effective
the
in
seen
plane strongly
screen
thickness
number
of
from
the
Figure
its
of the
counterions
I
that
the
electrostatic
which
counterions
field.
The
screening
layer xi/2,
electric
double
net
(calculated
relative
which
to
its
accumulated
are
can
be
is the
value
described
distance
far
from
near
the
charged
by introducing
the
density of
the charged plane nd)
where
the
ENERGY
FREE
N°4
drops
to
half of its
one
value
at
x
OF
act
calculated
~~~
~~~
while
(0) is density
according to
the
~
of the
the
nd
of
number
incti°j
#
counterions
potential
is
~~~~
~
at
x
to
2n~~n~~~~~~l~~o4l/kT)
~~
by solving
calculated
is
(Eq. (30))
is
free
energy
is
/kT))
~ ~~'
~~~~
equation
differential
the
~~~~)~~~~~~~~~)~~~)~)
~~~~
'
(15)
nod
Equation (31)
electrostatic
The
0
=
eeonod
according
129j
Rdj/2>
l +
~
where
485
expressions (23, 24),
~ ~~°
~~ ~~~~
electrostatic
LAYER
0,
=
nctiXi/2)
where
DOUBLE
ELECTRIC
THE
#
n~
(32)
2nd.
numerically using the Itunge Kutta method, while the integration of
of the
performed numerically by using the Simpson method.
The
extension
determined
by the condition
that its twofold
increase
system in the ~ direction d had been
The density of the
did not
the value of F~~ beyond the prescribed
number of
increase
error.
of pure
determined
by the
pwNA /MH~O, where pw is the
sites n~ was
concentration
water n~
molecules
and NA is the Avogadro
number.
density of water, MH~O is molar mass of water
Within the
Poisson-Boltzmann
theory the effective thickness of the electric double layer can
analytically
obtained
be
F~~
solved
=
~~~~
~
~
where
I
/~z is the
(~/(1
+
(~/(1
+
exp(-eo4l(0)/kT)) /2
exp(-eo4l(0) /kT)) /2
+
exp(eo4l(0)/2kT))
1)(1
1)(1
+
exp(eo4l(0) /2kT)
~'
Debye length
~~~~
~
4l(0)
~ ~~
~~~
=
In
I +
((a( /c)~
+
(a( /c)
(35)
,
eo
c
=
8kTeeoNAnd
(36)
thickness of the electric
double layer ~i/2 in dependence on the absolute
value of
(full
line).
density
of
shown
The
corresponding
the
plane
Figure
0
is
in
2
(a(
area
x
Poisson-Boltzmann
theory is also shown
dependence of ~i/2 on (a(, calculated by using the
(broken line). Both
higher charge
for
small
the
plane
bearing
decreasing
since
(a(
curves
are
effective.
ofcounterions
and the screening is therefore
in its vicinity larger number
attracts
more
excluded
volume
effect is taken
the
Within
the presented theory where the
into
account,
Poisson-Boltzmann
Howthickness
of the diffuse layer is
increased
with
respect to the
case.
excluded
volume imposes an upper limit on the density of the number of counterions,
ever, the
especially significant near the charged plane where
accumulated
which
becomes
counterions
are
(Fig. 1). This imposes a limit also on the screening of the electrostatic field. As more
couninfluence of the
the charged plane with
further
terions
attracted
increase of (a(, the
are
near
excluded
effect
minimal
volume
with increasing (a(, so that ~i/2> after reaching its
increases
model, there
value, begins to increase with increasing (a( (Fig. 2). In the
Poisson-Boltzmann
The
the
effective
charge
=
JOURNAL
486
PHYSIQUE
DE
00
0.5
The
2.
effective
thickness
of the
electric
10
is
=
upper
no
layer x1j2
double
charge area density of the x
0 plane (a( for
effect (full line) and the corresponding dependence
(broken line). The values of other parameters
the
are
the
of
N~4
lAs/m2j
Id
Fig.
II
nd
0.I
=
calculated
same
dependence
in
the
on
absolute
mol/I considering the excluded
using the
Poisson-Boltzmann
Figure
in
as
value
volume
theory
I.
density of the number of ions and there is a
decrease
monotonous
of (a(
For high values of (a( the use of
Poisson-Boltzmann
theory
profile is not justified. Namely, the condition (25) is not
concentration
the vicinity of the charged plane.
limit
of
the
of xi /2 with the
increase
for calculating the ion
fulfilled
in
Figure 2 that xi/2 using both models are very small meaning that the
charged plane is almost
insensitive
details
the charged plane.
to the
near
(with or without
electrostatically
condense
the
the
charged
plane
counterions
near
way
finite
constraint) does not affect the effective potential far from the surface. This
volume
difference in
be also seen in Figure I where the
density profiles calculated with
counterion
models rapidly
decreases.
It
can
behavior
The
the
can
both
also
be
far
from
in
seen
the
free energy F~' (a) and both
contributions
to F~'
energy
W~'
F~~~ (b) in dependence on the
of the
electrostatic
field
and the entropic
contribution
absolute
value of the charge area density of the x
0 plane (~( (full lines). The corresponding
dependencies calculated by using the
Poisson-Boltzmann
theory are also shown (broken lines).
All the
electrostatic
field in the system is stronger if
increasing with increasing (~( as
curves
are
volume
(~( is higher. Since the screening of the electric field is less effective when the excluded
Figure
3
the
shows
electrostatic
=
effect
is
taken
electrostatic
theory.
Boltzmann
When
into
solution
rearrangement
the
F~~
to
(which
this
is the
near
where
the
the
effect
sum
latter
on
field
the
of
concentration
effect
F~~~,
so
prevailing.
that
of W~~ and F~~~)
the
effect
partly
is
further
much
on
cancels
into
free
itself
is
the
much
volume
energy
the
ions
of ions is
in
of the
(Fig. 3a).
Poisson-
the
Solution.
further
However,
in
the
W~' is
electric
the
Poisson-
vicinity
the
in
on
the
extended
than
higher
effect
of the
energy
solution.
pronounced
less
excluded
the
and
by using
rearranging of
counterions
The
solution
calculated
energy
rearrangement
extends
charged plane
the
into
reflects
considered, the
is
electrostatic
ions
F~~~
contribution
effect
the
of the
charged plane,
contrary
volume
as
model,
Boltzmann
entropic
The
excluded
the
the
further
into
account, the field protrudes
field W~' is higher than the corresponding
of
therefore
double
layer
ENERGY
FREE
N°4
OF
THE
b
a
~
~
~
<
~
0.
o-o
DOUBLE
ELECTRIC
,
~
LAYER
487
JOURNAL
488
PHYSIQUE
DE
II
N°4
Acknowledgments
authors
The
indebted
are
Appendix
dr.
Vlachy
V.
fruitful
for
discussion.
A
Energy
Free
prof.
to
Electrolyte
ofthe
Solution
in
Contact
with
Charged
the
equilibrium and taking into account
expression for the free energy within the
thermodynamic
Assuming local
particles in the
solution
the
Plane
energies of the individual
field
approximation
mean
derived.
is
The
divided
system is
into
cells of
equal volume,
V~
IA-I
A A~
=
direction.
that
in the ~
It is assumed,
of
the
change
macroscopic
properties
system
over
In the particular cell chosen, there
and Nj ions of
No solvent
molecules
are
volume effect is included in the description so that
j
I, 2,..., M. The excluded
of
distributed
Nj
sites
equal volume in the cell, all sites being occupied,
are
over
A~ is
where
comparing
the
dimension
to the
of
cell
the
distance
A~
is
small
appreciably.
j-th Species,
which
particles
the
=
M
~j Nj
(A.2)
N].
=
j=o
accordance
In
(A.2)
to
assumed,
is also
The
solution
particle
in
partition
qm,j
the
volume
of the
proportional
cell is
particles
described
the cell is
using the
function q of the mj-th particle
that
=
volume
the
of the
~jexp (-e~,~~ /kT)
m~
,
methods
of the
of
particles
the
j
N].
It
[39j.
The
is
jA.3j
M,
o,1, 2,.
=
sites
mix.
mechanics
statistical
N~,
of the
number
while
j-th species
1, 2,.
=
the
to
conserved
is
,
,
~
where
k is the
energy
states
Since
the
to
Boltzmann
and T is the
constant
mj-th particle em,j~.
individualion
is charged, there
an
of the given
energy
Em,
Km,
"
jz
jz
(#(rm,j)
+ ej
to
other
contributions
the
and
potential
assumed,
It is
to
the
states
relative
calculated
chosen
its
through
runs
possible
all
ions
energy
are
infinitely
is
constant.
that
the
a
ofits
contribution
mj
"
Nj,
1, 2,.
j
,
energy
"
(A.4)
M,
1, 2,.
,
electrostatic
the
to
the
distant
We
chose
electrostatic
reference
potential
apart
that
so
o.
4l~~f
field in the
the
system
energy ej4l~~f. In the reference
field in this
vanishes
electrostatic
case
=
system
does
qm,j
q$
=
~
exp
influence
not
Therefore, by inserting (AA) into (A.3)
electrostatic
potential energy can therefore be
Km,j~.
of ions, the
potential
electrostatic
j-th species, 4l(rm,j) the potential of the
electrostatic
potential energy of the ion situated at rm,j, and
potential energy
The
electrostatic
energy of a given state.
Km,j~
ion is
is
4~~ef)
ion of the
all
I
state,
where ej is the charge of the
field, ej (4l(rm,j)
4l~ef) the
ofthe
Index
temperature.
of the
and
written
(-ej4l(rm,j)/kT)
the
summing
before
contributions
over
the
all
energy
sum,
(A.5)
,
ENERGY
FREE
N°4
OF
DOUBLE
ELECTRIC
THE
489
LAYER
where
qS,j
~jexp (-Km,j~/kT)
=
m~
,
1, 2,.
=
j
N~,
1, 2,.
=
jA.6j
M.
,
,
~
function
Partition
of the
qmoo
£exp
qs~o
=
molecule
solvent
is
j-K~z~o~/kTj,
=
mo
IA-?)
No.
1, 2,.
=
,
~
The
and
heat
solution
in
the
number
constant
cell is
chosen
considered
be
to
with
system
a
particles No> Aii,N2>
of all species of
the
NM>
immersed
is
,
bath,
that
so
correlations
its
temperature
described
are
T is
through
effect of the
effect, the particles in the cell are
considered
chosen species are also
qS,j
e
constant.
Since it is assumed,
that
field
and
electrostatic
mean
explicitly considered to
to be equal with
respect
qj,
m~
N~,
1, 2,.
=
indistinguishable. Assuming that the system
taking into account all possible nonequivalent
canonical
partition function of the cell is
'
o,1, 2,.
l~fly
jnj
'
a
excluded
the
volume
particles of
The
a
I(m,j~,
states
jA.8j
ii
,
thermodynamic
particles in
"local"
the
is in
and
p~
=
energy
,
and
Qcj~c
j
all
to
l~~
in
particle-particle
the
independent.
be
volume
constant
which
distributions
equilibrium
cell, the
of the
the
fifC(
~J jNj
~
s
gj
'
J4
J~~
fl j~~
J"°
J.
where
N
=
NM). Using (A.5)-(A.8), equation (A.9)
(No> Ni>N2>
can
be
written
as
,
M
Q~lvc, N, Tj
exp
=
~c,
j-Awe'/kT) fljqjjN>
IAioj
S'
J=°
fl
,
~
J"°
N
J.
where
~j e14l(ri),
AW~'
=
IA-it)
i
summing
over
Knowing
solution
in
the
the
all
ions
(of
canonical
chosen
cell
all
the
species)
partition
in
function
cell.
the
Q~>
we
obtain
the
Helmholtz
AF
number
of sites
of the
energy
(A.12
kT In Q~.
=
Inserting (A.10) into (A.12), using IA-I) and (h.2), applying the Stirling
introducing the density of the number of particles of the j-th species nj as
of the
free
AF,
approximation
well
as
the
and
density
n~,
'
nj
j
=
M,
2,
0. 1,
=
,
(A.13)
,
fifC
n~
IA.14)
~
=
V~
,
JOURNAL
490
PHYSIQUE
DE
II
N°4
obtain
we
M
Awe(
~ ~
~
~T
~
~
In
y~~
~~
~=o
Equation (A.15) gives the expression for the
chosen cell, where the densities of the number of
.,M.
j=0,1,2,.
The
contributions
the
over
of all the cells
of the
extension
which
system
d is
W~'
+ kT
free
Helmholtz
ions
solution
energy of the
of position, I.e. nj
functions
are
the
constitute
performed
(h.~~)
A A~.
~~~J
system
to
obtain
(xl
In
summed,
are
the
free
I.e.
the
in
the
energy
the
in
i~j(~),
=
integration
field
mean
approximation
F
=
/~
~(
o
While
created
in
a
calculating W~~,
by all other ions
medium
that
it is
and
consider
we
expression (A.11)
ions
To
in
the
be
can
into
e.
where
eo
is
free
energy
electrostatic
particles
in
permittivity
the
of
the
field
the
to
free
all
the
charges
ions
at
are
regarding
distributed
to
volume
with
the site of
assumed
the
to
infinite
a
given
ion is
be
immersed
self
energy
charge density pe(r)
of
so
into
/~ E~(~)A
~eeo
2
(A.17)
d~,
~
and
space
expressed by (A.16)
system
due
of the
=
potential
The
problem
the
transformed
the
well.
as
solution
(h.16)
A d~.
nsqj
that
account
avoid
W~~
~~~~
~=o
charged plane
the
permittivity
with
point charges,
taken
nj
in
the
E is
the
includes
and
system
electric
interactions
the
field
of
entropy
strength.
ions
of
with
mixing
Thus
the
the
mean
of all
the
system.
References
ill Gouy M.G.,
J. Phys. France
Chapman
D.L.,
Phitos. Mag.
[2j
(1910) 457-468.
(1913) 475-481.
(1924) 508.
9
25
Stern O., Z.
Etectrochem.
30
and
J,Th.G.,
Verwey E.J.W.
Overbeek
(Elsevier, Amsterdam, 1948) pp. 77-97.
[5j Marcus It-A-, J. Chem. Prigs. 23 (1955)
[3j
[4j
[6j
[7j
[8j
[9]
[10j
iii]
[12j
[13j
[14j
[isj
[16j
Theory
of
the
Stability of Lyophobic
Colloids
1057-1068.
H., Teubner M., Woolley P. and Eibl H., Biophys. Chem. 4 (1976) 319-342.
Kirkwood
J.G., J. Chem. Phys. 2 (1934) 767-781.
Loeb A.L., J. Cottoid Sci. 6 (1951) 75-91.
Buff F.P. and Stillinger F.H., J. Chem. Phys. 25 (1956) 312-318.
Levine S., Bell G.M. and Smith A.L., J. Phys. Chem. 64 (1960)
l188-l195.
Outhwaite
C.W., Bhuiyan L. and Levine S., J. C. S. Faraday II 76 (1980) 1388-1408.
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