With regard to the theory of CVI and ESA, there was an important observation made by O'Brien, who linked these measured parameters with dynamic electrophoretic mobility μ.
where
: A is calibration constant, depending on frequency, but not particles properties;
: ρ is particle density,
: ρ density of the fluid,
: φ is volume fraction of dispersed phase,
Dynamic electrophoretic mobility is similar to electrophoretic mobility that appears in electrophoresis theory. They are identical at low frequencies and/or for sufficiently small particles.
There are several theories of the dynamic electrophoretic mobility. Their overview is given in the Ref.5. Two of them are the most important.
The first one corresponds to the Smoluchowski limit. It yields following simple expression for CVI for sufficiently small particles with negligible CVI frequency dependence:
where:
: ε is vacuum dielectric permittivity,
: ε is fluid dielectric permittivity,
: ζ is electrokinetic potential
: η is dynamic viscosity of the fluid,
: K is conductivity of the system,
: K is conductivity of the fluid,
: ρ is density of the system.
This remarkably simple equation has same wide range of applicability as Smoluchowski equation for electrophoresis. It is independent on shape of the particles, their concentration.
Validity of this equation is restricted with the following two requirements.
First, it is valid only for a thin double layer, when the Debye length is much smaller than particle's radius a:
Secondly, it neglects the contribution of the surface conductivity. This assumes a small Dukhin number:
Restriction of the thin double layer limits applicability of this Smoluchowski type theory only to aqueous systems with sufficiently large particles and not very low ionic strength. This theory does not work well for nano-colloids, including proteins and polymers at low ionic strength. It is not valid for low- or non-polar fluids.
There is another theory that is applicable for the other extreme case of a thick double layer, when
This theory takes into consideration the double layer overlap that inevitably occurs for concentrated systems with thick double layer. This allows introduction of so-called "quasi-homogeneous" approach, when overlapped diffuse layers of particles cover the complete interparticle space. The theory becomes much simplified in this extreme case, as shown by Shilov and others. Their derivation predicts that surface charge density σ is a better parameter than ζ-potential for characterizing electroacoustic phenomena in such systems. An expression for CVI simplified for small particles follows: