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# Double layer (interfacial)

Double Layer (interfacial) (DL) is a structure that appears on the surface of an object when it is placed into a liquid. This object might be a solid particle, or gas bubble, or liquid droplet, or porous body. This structure consists of two parallel layers of ions. One layer (either positive or negative) coincides with the surface of the object. It is surface charge. The other layer is in the fluid. It electrically screens the first one. It is diffuse, because it forms under the influence of electric attraction and thermal motion of free ions in fluid. It is called the diffuse layer.

DL is usually important for systems with high surface area. This might be colloids with very small sizes, on the scale of a micrometer or even nanometers. Porous bodies with small size of pores on the same scale is another example.

Interfacial Double Layer (DL) plays a very important role in real world systems. For instance, milk exists only because fat droplets are covered with Double Layers that prevent their coagulation into cheese. Double Layers exist in practically all heterogeneous fluid based systems, such as blood, paint, inks, ceramic slurries, cement slurries, etc.

There are many phenomena that exist due to Double Layer. One of the most known is electrophoresis. It belongs to a wider group of electrokinetic phenomena. Another group comprises electroacoustic phenomena.

## History

The earliest model of the electrical double layer is usually attributed to Helmholtz [1] . Helmholtz treated the double layer mathematically as a simple capacitor, based on a physical model in which a single layer of ions is adsorbed at the surface.

Later Louis Georges Gouy [2], and David Leonard Chapman [3] made significant improvements by introducing a diffuse model of the electrical double layer, in which the electric potential decreases exponentially away from the surface to the fluid bulk.

Gouy-Chapman model fails for highly charged DLs. In order to resolve this problem Stern suggested introduction of additional internal layer, that is now called the Stern layer[4].

Combined Gouy-Chapman-Stern model is most commonly used. It still has some limitations, such as

There are more recent theoretical developments studying these limitations of the Gouy-Chapman-Stern model. They are reviwed by J.Lyklema in "Fundamentals of Interface and Colloid Science" [5],

## Detailed Description of DL

There are detailed descriptions of the interfacial Double Layer in many books on Colloid and Interface Science[6], [7], [8],[9],[10], [11].

There is an IUPAC Technical Report [12] prepared by a group of most known world experts on the subject of interfacial Double Layer and related electrokinetic phenomena.

This Figure illustrates interfacial Double Layer in more detail following Lyklema book, Ref 1. : “...the reason for the formation of a “relaxed” (“equilibrium”) double layer is the non-electric affinity of charge-determining ions for a surface...”. This process leads to the build up of an electric surface charge, expressed usually in μC per cm-2. This surface charge creates an electrostatic field that then affects the ions in the bulk of the liquid. This electrostatic field, in combination with the thermal motion of the ions, creates a counter charge, and thus screens the electric surface charge. The net electric charge in this screening diffuse layer is equal in magnitude to the net surface charge, but has the opposite polarity. As a result the complete structure is electrically neutral. Some of the counter-ions might specifically adsorb near the surface and build an inner sub-layer, or so-called Stern layer. The outer part of the screening layer is usually called the diffuse layer.

The diffuse layer, or at least part of it, can move under the influence of tangential stress. There is a conventionally introduced slipping plane that separates mobile fluid from fluid that remains attached to the surface. Electric potential at this plane is called electrokinetic potential or zeta potential. It is also denoted as ζ-potential.

Electric potential on the external boundary of the Stern layer is referred to as Stern potential. Electric potential at the surface is electric surface potential.

Usually zeta potential is used for estimating the degree of DL charge. A characteristic value of this electric potential in the DL is 25 mV with a maximum value around 100 mV. Chemical composition of the sample that brings ζ-potential down to 0 is called iso-electric point. It is usually certain pH value.

Zeta potential can be measured using electrophoresis or electroacoustic phenomena.

The characteristic thickness of the Double Layer is Debye length κ-1. It is reciprocally proportional to the square root of the ion concentration C. In aqueous solution it is on scale of a few nanometers.

The theory for a flat surface and a symmetrical electrolyte has been given by Gouy and Chapman, ref.1. It is usually referred to as Gouy-Chapman theory. It yields a simple relationship between electric charge in the diffuse layer σd and the Stern potential Ψd:

$\sigma^d = -\sqrt{{8\varepsilon_0}{\varepsilon_m}CRT}sinh \frac{F\Psi^d}{2RT}$

There is no general analytical solution for mixed electrolytes, curved surfaces or even spherical particles. There is an asymptotic solution for spherical particles with low charged DLs. In the case when electric potential over DL is less than 25 mV, the so-called Debye-Huckel approximation holds. It yields the following expression for electric potential Ψ in the spherical DL as a function of the distance r from the particle center:

${\Psi}(r) = {\Psi^d}\frac{a}{r}EXP({-\kappa}(r-a))$

There are several important asymptotic models which play important roles in theoretical developments associated with interfacial DL.

The first one is "thin Double Layer", Ref.1-7. This model assumes that Double Layer is much thinner than the particle or capillary radius. This restricts the value of the Debye legth and particle radius as following:

κa > > 1

This model offers tremendous simplifications for many theories. Theory of electrophoresis is just one example, Ref.2. Theory of Electroacoustic phenomena Ref.6 is another example.

The Model of thin Double Layer is valid for most aqueous systems because the Debye length is only a few nanometers there. It breaks only for nano-colloids in solution with ionic strength close to water.

The opposite model of "thick Double Layer" assumes that Debye length is larger than particle radius:

κa < 1

This model can be useful for some nano-colloids and non-polar fluids, where Debye length is much larger.

The last model introduces "overlapped Double Layers", Ref.6. This is important in concentrated dispersions and emulsions when distances between particles become comparable with Debye length.

## References

1. ^ Helmholtz, H. Pogg.Ann. LXXXIX, 211 (1853)
2. ^ Gouy, G. Comt.Rend. 149, 654 (1909), J.Phys. 4, 9, 457 (1910)
3. ^ Chapman, D.L. Phil.Mag., 6, 25, 475 (1913)
4. ^ Stern, O. Z.Electrochem, 30, 508 (1924)
5. ^ Lyklema, J. “Fundamentals of Interface and Colloid Science”, vol.2, page.3.208, 1995
6. ^ Lyklema, J. “Fundamentals of Interface and Colloid Science”, vol.2, page.3.208, 1995
7. ^ Hunter, R.J. "Foundations of Colloid Science", Oxford University Press, 1989
8. ^ Dukhin, S.S. & Derjaguin, B.V. "Electrokinetic Phenomena", J.Willey and Sons, 1974
9. ^ Russel, W.B., Saville, D.A. and Schowalter, W.R. “Colloidal Dispersions”, Cambridge University Press,1989
10. ^ Kruyt, H.R. “Colloid Science”, Elsevier: Volume 1, Irreversible systems, (1952)
11. ^ Dukhin, A.S. and Goetz, P.J. "Ultrasound for characterizing colloids", Elsevier, 2002
12. ^ ”Measurement and Interpretation of Electrokinetic Phenomena”, International Union of Pure and Applied Chemistry, Technical Report, published in Pure Appl.Chem., vol 77, 10, pp.1753-1805, 2005