The electrical double layer

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The electrical double layer
D is the static dielectric constant
2
From Coulombs equation,
By integration we can estimate the energy to
separate a charge from the surface. In water
W1.9x10-20 J, In Air W1.5x10-18 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 (D2), no dissociation is
expected. This is why NaCl dissolves in water
but not in oil.
3
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.
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The charged surface and the diffuse ion layer of
counterions form a double-layer (diffuse)
capacitor.
5
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
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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.
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Y
0
Y
x)
(
?0 is the electric potential at the surface and
?(x) is the electric potential at a distance x
from the surface
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For a planar interface the potential is related
to the charge density by Poissons equation
(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).
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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 its
potential energy at that position. (Note Z is the
valency). The potential energy is by definition
given by Ziq?(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
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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 ?(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.
(2)
Note the sign of the ion charge is important
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Why a Boltzmann Distribution?

• Boltzmanns 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.

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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
(3)
Assuming symmetric electrolytes (11, 22, 33
etc) and combining with equation (2) we obtain.
(4)
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This can be simplified to
(5)
Combining with equation (1) we obtain the
Poisson-Boltzmann equation
(6)
After double integration and some algebra an
exact analytical result is obtained, but if we
assume low surface potential then equations
become linear.
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These equations can be greatly simplified by
assuming low potentials. (ie values of
?(x)lt25mV). The equation (6) reduces to
(7)
so
(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.
(9)
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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.
(10)
we obtain for a 11 electrolyte (using eqn 5),
(11)
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At low potentials equation (11) reduces to
(12)
The surface potential is therefore related to the
surface charge density and the ionic composition
of the solution.
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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

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When two colloids interact
We have already obtained a simple expression for
the VdWs 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. 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
(1)
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Recall that the Debye length is

• A natural length that the potential diminishes
  1/e
• Dependant only on the solution, not the surface
  charge

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Non-interacting surfaces xgtgt?-1
Interacting Surfaces xlt?-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
??0
??m
?0
?0
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We therefore obtain for the potential at the
midplane
(2)
At the midpoint the gradient of the potential is
zero, d?(x)/dx0. 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 double-layer 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.
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The osmotic pressure ? of an ideal solution is
given by
(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
(4)
where i refers to each ionic solute and
and are the concentrations of each ionic solute
in the mid-plane and in bulk solution,
respectively.
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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
(5)
where the double-layer repulsive energy decays
exponentially with distance and depends on the
Debye length (ie electrolyte concentration) and
the surface potential.
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The interaction energy between two planar
surfaces is obtained by integrating the osmotic
pressure from infinity to x, which gives
(6)
By using a geometric factor the interaction
energy between two spheres is obtained.
(7)
r
r
x
(1) (2) (1)
Again, the interaction decays with separation and
depends on the surface potential and electrolyte
concentration.
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