**Electric Double Layer**

The double layer model is used to visualize the ionic environment in the vicinity of a charged surface. It can be either a metal under potential or due to ionic groups on the surface of a dielectric. It is easier to understand this model as a sequence of steps that would take place near the surface if its neutralizing ions were suddenly stripped away.

One of the first principles which must be recognized is that matter at the boundary of two phases possesses properties which differentiates it from matter freely extended in either of the continuous phases separated by the interface. When talking about a solid-solution interface, it is perhaps easier to visualize a difference between the interface and the solid than it is to visualize a difference between the interface and the extended liquid phase. Where we have a charged surface, however, there must be a balancing counter charge, and this counter charge will occur in the liquid. The charges will not be uniformly distributed throughout the liquid phase, but will be concentrated near the charged surface. Thus, we have a small but finite volume of the liquid phase which is different from the extended liquid. This concept is central to electrochemistry, and reactions within this interfacial boundary that govern external observations of electrochemical reactions. It is also of great importance to soil chemistry, where colloidal particles with different surface charges play a crucial role.

There are several theoretical treatments of the solid-liquid interface. We will look at a few common ones, not so much from the position of needing to use them, but more from the point of what they can tell us.

**A. Helmholtz Double Layer**

This theory is a simplest approximation that the surface
charge is neutralized by opposite sign counterions placed at an
increment of *d* away from the surface.

The surface charge potential is linearly dissipated from the
surface to the contertions satisfying the charge. The distance, *d*,
will be that to the center of the countertions, i.e. their
radius. The Helmholtz theoretical treatment does not adequately
explain all the features, since it hypothesizes rigid layers of
opposite charges. This does not occur in nature.

**B. Gouy-Chapman Double Layer**

Gouy suggested that interfacial potential at the charged
surface could be attributed to the presence of a number of ions
of given sign attached to its surface, and to an equal number of
ions of opposite charge in the solution. In other words,
counter ions are not rigidly held, but tend to diffuse into the
liquid phase until the counter potential set up by their
departure restricts this tendency. The kinetic energy of the
counter ions will, in part, affect the thickness of the resulting
diffuse double layer. Gouy and, independently, Chapman
developed theories of this so called *diffuse double layer*
in which the change in concentration of the counter ions near a
charged surface follows the Boltzman distribution

n = n_{o}exp(-zeY/kT)

where n_{o } = bulk concentration

z = charge on the ion

e = charge on a proton

k = Boltzmann constant

Already, however, we are in error, since derivation of this
form of the Boltzman distribution assumes that activity is equal
to molar concentration. This may be an OK approximation for the
bulk solution, but will not be true near a charged surface.

Now, since we have a diffuse double layer, rather than a rigid
double layer, we must concern ourselves with the volume charge
density rather than surface charge density when studying the
coulombic interactions between charges. The volume charge
density, r , of any volume, i, can be
expressed as

r_{i} = Sz_{i}en_{i}

The coulombic interaction between charges can, then, be expressed by the Poisson equation. For plane surfaces, this can be expressed as

d^{2}Y/dx^{2}
= -4pr/d

where Y varies from Y_{o} at the surface to 0 in bulk
solution. Thus, we can relate the charge density at any given
point to the potential gradient away from the surface.

Combining the Boltzmann distribution with the Poisson equation
and integrating under appropriate limits, yields the electric
potential as a function of distance from the surface. The
thickness of the diffuse double layer:

l_{double} = [e_{r}kT/(4pe^{2}Sn_{io}z_{i}^{2})]^{1/2}

at room temperature can be simplified as

l_{double} = 3.3*10^{6}e_{r}/(z*c*^{1/2})

in other words, the double layer thickness decreases with increasing valence and concentration.

The Gouy-Chapman theory describes a rigid charged surface, with a cloud of oppositely charged ions in the solution, the concentration of the oppositely charged ions decreasing with distance from the surface. This is the so-called diffuse double layer.

This theory is still not entirely accurate. Experimentally, the double layer thickness is generally found to be somewhat greater than calculated. This may relate to the error incorporated in assuming activity equals molar concentration when using the desired form of the Boltzman distribution. Conceptually, it tends to be a function of the fact that both anions and cations exist in the solution, and with increasing distance away from the surface the probability that ions of the same sign as the surface charge will be found within the double layer increase as well.

**C. Stern Modification of the Diffuse double Layer**

The Gouy-Chapman theory provides a better approximation of reality than does the Helmholtz theory, but it still has limited quantitative application. It assumes that ions behave as point charges, which they cannot, and it assumes that there is no physical limits for the ions in their approach to the surface, which is not true. Stern, therefore, modified the Gouy-Chapman diffuse double layer. His theory states that ions do have finite size, so cannot approach the surface closer than a few nm. The first ions of the Gouy-Chapman Diffuse Double Layer are not at the surface, but at some distance d away from the surface. This distance will usually be taken as the radius of the ion. As a result, the potential and concentration of the diffuse part of the layer is low enough to justify treating the ions as point charges.

Stern also assumed that it is possible that some of the ions
are specifically adsorbed by the surface in the plane d, and this layer has become known as the
Stern Layer. Therefore, the potential will drop by Y_{o} - Y_{d}
over the "molecular condenser" (i.e., the Helmholtz
Plane) and by Y_{d} over the
diffuse layer. Y_{d} has
become known as the zeta (z)
potential.

This diagram serves as a visual comparison of the amount of counterions in each the Stern Layer and the Diffuse Layer.

Thus, the double layer is formed in order to neutralize the charged surface and, in turn, causes an electrokinetic potential between the surface and any point in the mass of the suspending liquid. This voltage difference is on the order of millivolts and is referred to as the surface potential. The magnitude of the surface potential is related to the surface charge and the thickness of the double layer. As we leave the surface, the potential drops off roughly linearly in the Stern layer and then exponentially through the diffuse layer, approaching zero at the imaginary boundary of the double layer. The potential curve is useful because it indicates the strength of the electrical force between particles and the distance at which this force comes into play. A charged particle will move with a fixed velocity in a voltage field. This phenomenon is called electrophoresis. The particle’s mobility is related to the dielectric constant and viscosity of the suspending liquid and to the electrical potential at the boundary between the moving particle and the liquid. This boundary is called the slip plane and is usually defined as the point where the Stern layer and the diffuse layer meet. The relationship between zeta potential and surface potential depends on the level of ions in the solution.The figure above represents the change in charge density through the diffuse layer. One shows considered to be rigidly attached to the colloid, while the diffuse layer is not. As a result, the electrical potential at this junction is related to the mobility of the particle and is called the zeta potential. Although zeta potential is an intermediate value, it is sometimes considered to be more significant than surface potential as far as electrostatic repusion is concerned.