About zeta potential and electrokinetic phenomena

About zeta potential

Electrokinetic phenomena

The movement of charged colloidal particles in electric field is termed electrophoresis. When the charged solid surface is fixed, the electric field causes a movement of the liquid called electroosmosis. Forcing a liquid through a capilary or porous plug induces a difference of electric potentials called streaming potential. Forced movement of charged solid particles in a liquid, e.g., due to gravitation induces a difference of electric potentials, the sedimentation potential.

The velocity in electrophoresis and the difference of pressures between two ends of capilary or porous plug in electroosmosis is proportional to the applied field strength. The streaming potential is proportional to the applied pressure. The sedimentation potential is proportional to the velocity of the particles. The electrokinetic phenomena often accompany one another, e.g., electrophoresis is always accompanied by electroosmosis due to electric charge of the wall of electrophoretic cell.

Zeta potential

The shear plane (slipping plane) is an imaginary surface separating the thin layer of liquid bound to the solid surface and showing elastic behavior from the rest of liquid showing normal viscous behavior. The electric potential at the shear plane is called zeta potential. In the first, rough approximation, the electrophoretic mobility (the ratio of the velocity of particles to the field strength), induced pressure difference in electroosmosis, streaming potential and sedimentation potential are proportional to the zeta potential. The stability of hydrophobic colloids depends on the zeta potential: when the absolute value of zeta potential is above 50 mV the dispersions are very stable due to mutual electrostatic repulsion and when the zeta potential is close to zero the coagulation (formation of larger assemblies of particles) is very fast and this causes a fast sedimentation. Even when the surface charge density is very high but the zeta potential is low, the colloids are unstable. Also the velocity of heterocoagulation (coagulation of different particles) depends on the zeta potentals of both kinds of particles. Therefore, the zeta potential is an important parameter characterizing colloidal dispersion.

Smoluchowski equation

In the past, electroosmosis was the most popular method to determine the zeta potential. Nowadays, electrophoresis has the greatest practical meaning and many commercial fully automatic instruments are available. Marian Smoluchowski was the first to properly derive an equation to calculate the zeta potential from electrokinetic mobility:

where mu is the electrophoretic mobility, epsilon is the electric permitivity of the liquid and eta is the viscosity. Here we use the SI system, in old literature CGS units were used and all equations relating to electricity have a different form.

The applicability limit of Smoluchowski equation

Smoluchowski equation is still very popular. In this simplified approach, the electrostatic driving force is opposed by the frictional force and the other effects are neglected. Only when the zeta potential is not too high and for large colloidal particles and high ionic strengths this equation gives good results. For the values of zeta potentials with a practical meaning (zeta < 120 mV) the error is negligible when kappa*a > 100, where a is the particle radius and the Debye parameter kappa is defined as:
kappa ^2 = F ^2 * Sum over i (c sub i* (z sub i) ^2) / (epsilon*RT)
where F is the Faraday constant, c_i is the concentration of the i-th ion (in mol/m³) and z is the valency of this ion, R is the gas constant and T is the absolute temperature. The reciprocal kappa is often termed as the thickness of the electric double layer so Smoluchowski equation applies for thin double layers (as compared with the particle radius). Unfortunately, for many colloids of practical importance kappa*a < 100 and Smoluchowski equation may lead to serious errors.

Mobility and Winmobil

For electrophoresis at high zeta potentials and thick double layers additional braking forces must be considered. The force exerted on the particle by the excess of counterions moving in opposite direction is called retardation. Moreover, when the particle moves fast enough the centres of the particle charge and countercharge (diffuse layer) do not coincide and thus the effective electric field strength felt by the particle is lower than the external field (relaxation). Accurate differential equations accounting for these effects have been published by O'Brien and White. These equations do not lead to accurate analytical expressions to calculate mobility from zeta potential. This problem can be solved numerically and desired accuracy can be achieved when the number of iterations is high enough. Commercial (and not open-source) programs Mobility and Winmobil are available from University of Melbourne. The solutions obtained by means of these programs are valid for any value of the zeta potential and kappa*a. Both programs use the same algorithm, Winmobil has a more user friendly user's interface for Windows. At this moment you will probably ask:

Why do we need Zeta when Mobility and Winmobil are available?

For systems with kappa*a < 10 Zeta is useless. However in many systems with practical meaning 10 < kappa*a < 100 and here Zeta can help a lot (with kappa*a > 100 one can simply use Smoluchowski equation). With Mobility and Winmobil one can only calculate mobilities from zeta potentials as input data but it is mobility that can be experimentally determined. So the user has to select proper zeta potential values to cover the entire interesting range of mobilities, calculate mobilities for the selected zeta potentials and make a sort of calibration curve. The final step is to read zeta potentials corresponding to the experimentally determined mobilities from this curve (manually or by means of some computer program). With Zeta you simply introduce mobilities as input data and get zeta potentials as output data (no calibration curve is necessary). Unfortunately only for kappa*a > 10 the results will be reasonably accurate. This limit follows from the validity limit of the equation first published by Ohshima, H., Healy, T.W. & White, L.R., J.Colloid Interface Sci. 90, 17 (1982).

Many commercial zetameters have Smoluchowski equation built in the software, very often with the electric permitivity and viscosity data for water at 25 degrees Celsius so the results are displayed as (apparent) zeta potentials. When kappa*a < 100 the users of Winmobil need an additional step before reading from the calibration curve, namely they have to convert the apparent zeta potentials into mobilities. In Zeta, apparent zeta potentials calculated from Smoluchowski equation can be automatically converted into more accurate values if desired.

Last but not least, in contrast to the above mentioned programs, Zeta is free software. Not only can you download it from this page at no cost, but you also have access to the full source code. This way you know what the program you're running really does and can modify it to best suit your needs. Zeta can be compiled for many different platforms and operating systems, providing both a simple yet powerful command line interface as well as a graphical user interface for Windows. Interfaces for other systems can easily be written by other programmers using the Zeta Library.