The electrical double layer
Air, D=1
+
Water, D=80
+
r
D is the static dielectric constant
From Coulombs equation,
q1q2
Fc 
2
4 D 0 r
By integration we can estimate the energy to separate a
charge from the surface.
In water W~1.9x10-20 J,
In Air W~1.5x10-18 J
Compare this with thermal energy 1 kT ~4.0x10-21 J
Clearly it is the high dielectric constant or polar nature of
water which causes dissociation. In air or hexane (D~2),
no dissociation is expected.
This is why NaCl dissolves in water but not in oil.
The real situation is more complicated as a large
number of ions will be dissociated from each surface
This generates a high electric field and a stronger
attraction between the surface and the dissociated
ions. Additionally, the solvated ions repel each other.
In fact the dissociated ions do not leave the surface
region completely.
They form a “diffuse double layer”.
Diffuse Layer
Bulk Electrolyte
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Solid
Aqueous Solution
The charged surface and the diffuse ion layer of
counterions form a double-layer (diffuse) capacitor.
Quantitative treatment of the
electrical double layer is an
extremely difficult problem. In
order to treat the problem we
make use of several
assumptions and simplifications.
Let us examine the case of an
infinite, flat, charged planar
surface. x is the distance normal
to the surface.
see Hunter, R Foundations of Colloid Science I & II, Oxford, 1989
Quantitative treatment of the electrical double layer is
an extremely difficult problem. In order to treat the problem
we make use of several assumptions and simplifications.
x
Let us examine the case of an infinite, flat, charged planar
surface. x is the distance normal to the surface.
Y0
Y(x)
Y0 is the electric potential at the surface and Y(x) is the electric potential
at a distance x from the surface
For a planar interface the potential is related to the
charge density by Poisson’s equation
d Y(x)
(x)

2
dx
 oD
2
(1)
Where  x is the net charge density in Cm-3 at
distance
 x (ions per unit volume is equivalent to
charge per unit volume).
In order to describe the decay of the potential from the
surface we would like to determine  x . The density (or
population per unit volume) of any ion of charge Ziq must
depend on it’s potential energy at that position. (Note Z is the
valency). The potential energy is by definition given by
ZiqY(x).
Note that q is the positive value of the electron charge, or the
charge on a proton. Since any ion next to a charged surface
must be in equilibrium with the corresponding ions in the bulk
solution, it follows that the electrochemical potential of an ion at
distance x from the surface must be equal to its bulk value.
Thus:
 ib   ix   i0  kT lnCi (B)   i0  Zi q(x) kT lnCi (x)
where Ci(B) and Ci(x) are the ion concentrations in bulk
and at distance x from the charged surface, and it is
assumed that these are dilute solutions (ie ψ(B) = 0).
This equation leads directly to the Boltzmann
distribution, which can be used to obtain the
concentration at any other electrostatic potential energy
by the familiar relationship:
In bulk solution we define the concentration of ion i as i(bulk).
We would expect a Boltzmann distribution of ions determined
by the ratio of the potential to the kinetic energies, therefore.
 Zi qY(x) 
i (x)  i (bulk) exp 


kT 
Note the sign of the ion charge is important
(2)
Why a Boltzmann Distribution?
• Boltzmann’s law
– the probability of finding molecules in a particular
spatial arrangement decreases exponentially
(negative sign) with the ratio of the potential
energy of that arrangement compared with kT.
If the potential is positive then positively charged ions
(co-ions) will be at a lower concentration near the surface
than in the bulk. Ions of the opposite sign (counterions)
will be at a higher concentration near the surface than
in the bulk
The net total charge density is given by
(x)  i() (x)  i() (x)
(3)
Assuming symmetric electrolytes (1:1, 2:2, 3:3 etc) and
combining with equation (2) we obtain.
  Z iqY(x) 
Z iqY(x) 
(x)  Zq(bulk)exp
 exp







kT
kT


(4)
This can be simplified to
 (x)  2Zq (bulk )sinh (ZqY(x) / kT)
(5)
Combining with equation (1) we obtain the
Poisson-Boltzmann equation
d Y 2Zq(bulk)
sinh(ZqY(x) /kT) (6)
2 
dx
 0D
2
After double integration and some algebra an exact
analytical result is obtained, but if we assume low
surface potential then equations become linear.
These equations can be greatly simplified by assuming
low potentials. (ie values of Y(x)<25mV). The equation
(6) reduces to ;
so
d 2Y(x)
2
  Y(x)
2
dx
(7)
Y(x)  Y 0 exp(x)
(8)
From inspection of equation (8) the physical meaning of the
-1 (The Debye Length) is made clear. It is a measure of how
the potential decays with distance from the surface.
2



DkT
1

   2 02

2q Z (bulk) 

1
(9)
We now have a means of determining the ion profile from
a surface and an approximate means for determining the
potential profile for low potentials. We are also often
interested in the charge density of the surface (Cm-2),
which gives rise to the potentials.
The total double layer must be electroneutral, therefore
the charge at the surface 0 must be equal and
opposite in sign to the net charge of the diffuse layer D.

 D   (x)dx    0
(10)
0
we obtain for a 1:1 electrolyte (using eqn 5),
1
q 0 
2 sinh 

 0  8 0 DkT (bulk)
2kT 
(11)
At low potentials equation (11) reduces to
 0  Y 0
(12)
The surface potential is therefore related to the surface
charge density and the ionic composition of the solution.
Several assumptions have been made
• The surface is flat, infinite and uniformly
charged
• The ions are assumed to be point
charges, distributed according to the
Boltzmann distribution
• The solvent is represented solely by a
dielectric constant
• The electrolyte is assumed to be
symmetrical
When two colloids interact
We have already obtained a simple expression for the
VdW’s interaction between two spherical particles. A
simple result can be obtained for the interaction between
diffuse electrical double layers, if we assume we are
dealing with only low electrostatic potentials.
Y(x)  25mV
This is known as the Debye-Huckel approximation.
For simplicity we will first deal with two planar surfaces.
From previous lectures recall that for low potentials the decay
of the potential is described by:
Y(x)  Y0 exp( x 
(1)
Recall that the Debye length is:

1
1
2
  0 DkT
  2 2

2q Z  (bulk)
• A natural length that the potential diminishes 1/e
• Dependant only on the solution, not the surface charge
Non-interacting surfaces
x>>-1
Interacting Surfaces
x<-1
For interacting surfaces we apply the superposition
approximation, which means the total potential at point x is
given by the sum of the unperturbed potential from each
surface
x
x
Y=Y0
Y=Ym
Y=0
Y=0
We therefore obtain for the potential at the midplane
(
Y(m)  2Y0 exp  x
2

(2)
At the midpoint the gradient of the potential is zero,
dY(x)/dx=0. Therefore no net electric field is acting on the
ions atthis point.
However, at the midpoint there is a higher total concentration
of ions than in the bulk. Osmotic pressure will then act to
dilute the ions by drawing water into the region between the
interacting surfaces. This is equal to the electrical doublelayer repulsion between the charged plates. Hence, if we can
determine the osmotic pressure at the mid-plane we can
determine the repulsive pressure between the surfaces.
The osmotic pressure  of an ideal solution is given by
   kT
(3)
where,  is the solute concentration. The repulsive
pressure is therefore given by the difference between
the osmotic pressure at the mid-plane and the bulk
solution and is given by:
 R  kT  (
m
i
bulk
 i

(4)
i
im
ibulk
are the
where i refers to each ionic solute and
and
concentrations of each ionic solute in the mid-plane and
in bulk solution, respectively.
As we know how the mid-plane potential varies with
surface separation (equation 2), we can use the Boltzmann
equation to calculate the concentrations of each ion at the
mid-plane. By this procedure, assuming low potentials we
obtain the result:
R  20D Y exp(x
2
2
0
(5)
where the double-layer repulsive energy decays exponentially
with distance and depends on the Debye length (ie electrolyte
concentration) and the surface potential.
The interaction energy between two planar surfaces is
obtained by integrating the osmotic pressure from infinity to
x, which gives
VF  20DY exp(x
2
0
(6)
By using a geometric factor the interaction energy between
two spheres is obtained.


V sphere  2R 0DY02 exp(x
r
x
(1)
(2)
(7)
r
(1)
Again, the interaction decays with separation and depends
on the surface potential and electrolyte concentration.