Sedimentation potential

There are detailed descriptions of this effect in many books on colloid and interface science.

Surface energy

Background related to phenomenon

Electrokinetic phenomena are a family of several different effects that occur in heterogeneous fluids or in porous bodies filled with fluid. The sum of these phenomena deals with the effect on a particle from some outside resulting in a net electrokinetic effect.

The common source of all these effects stems from the interfacial 'double layer' of charges. Particles influenced by an external force generate tangential motion of a fluid with respect to an adjacent charged surface. This force may consist of electric, pressure gradient, concentration gradient, gravity. In addition, the moving phase might be either the continuous fluid or dispersed phase.

Sedimentation potential is the field of electrokinetic phenomena dealing with the generation of an electric field by sedimenting colloid particles.

History of models

This phenomenon was first discovered by Dorn in 1879. He observed that a vertical electric field had developed in a suspension of glass beads in water, as the beads were settling. This was the origin of sedimentation potential, which is often referred to as the Dorn effect.

Smoluchowski built the first models to calculate the potential in the early 1900s. Booth created a general theory on sedimentation potential in 1954 based on Overbeek's 1943 theory on electrophoresis. In 1980, Stigter extended Booth's model to allow for higher surface potentials. Ohshima created a model based on O'Brien and White 's 1978 model used to analyze the sedimentation velocity of a single charged sphere and the sedimentation potential of a dilute suspension.

Generation of a potential

As a charged particle moves through a gravitational force or centrifugation, an electric potential is induced. While the particle moves, ions in the electric double layer lag behind creating a net dipole moment behind due to liquid flow. The sum of all dipoles on the particle is what causes sedimentation potential. Sedimentation potential has the opposite effect compared to electrophoresis where an electric field is applied to the system. Ionic conductivity is often referred to when dealing with sedimentation potential.

The following relation provides a measure of the sedimentation potential due to the settling of charged spheres. First discovered by Smoluchowski in 1903 and 1921. This relationship only holds true for non-overlapping electric double layers and for dilute suspensions. In 1954, Booth proved that this idea held true for Pyrex glass powder settling in a KCl solution. From this relation, the sedimentation potential, E_S, is independent of the particle radius and that E_S → 0, Φ _p → 0 (a single particle).

<span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt22" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"E_{s} = - \\frac{\\varepsilon\\zeta (\\rho -\\rho _{0})\\phi _{p}g }{\\sigma ^{\\infty }\\eta }"}}' id="mwOw"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{s}=-{\frac {\varepsilon \zeta (\rho -\rho _{0})\phi _{p}g}{\sigma ^{\infty }\eta }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ε<!-- ε --></mi> <mi>ζ<!-- ζ --></mi> <mo stretchy="false">(</mo> <mi>ρ<!-- ρ --></mi> <mo>−<!-- − --></mo> <msub> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>ϕ<!-- ϕ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mi>g</mi> </mrow> <mrow> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msup> <mi>η<!-- η --></mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{s}=-{\frac {\varepsilon \zeta (\rho -\rho _{0})\phi _{p}g}{\sigma ^{\infty }\eta }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d31a9a5654cba1f8ec8f20767b1f747c2656bb6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:22.308ex; height:6.176ex;" alt="{\displaystyle E_{s}=-{\frac {\varepsilon \zeta (\rho -\rho _{0})\phi _{p}g}{\sigma ^{\infty }\eta }}}"/></span>

Smoluchowski's sedimentation potential is defined where ε₀ is the permitivity of free space, D the dimensionless dielectric constant, ξ the zeta potential, g the acceleration due to gravity, Φ the particle volume fraction, ρ the particle density, ρ_o the medium density, λ the specific volume conductivity, and η the viscosity.

Smoluchowski developed the equation under five assumptions:

1. Particles are spherical, nonconducting, and monodispersed.
2. Laminar flow around the particles occurs (Reynolds number <1).
3. Interparticle interactions are negligible.
4. Surface conduction is negligible.
5. The double-layer thickness 1/κ is small compared to the particle radius a (κa>>1).

<span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt31" data-mw='{"name":"math","attrs":{},"body":{"extsrc":"\\sigma ^{\\infty} = \\frac{e^{2}}{k_{B}T}\\sum z_{i}^{2}D_{i}n_{i\\infty }"}}' id="mwSw"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma ^{\infty }={\frac {e^{2}}{k_{B}T}}\sum z_{i}^{2}D_{i}n_{i\infty }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>B</mi> </mrow> </msub> <mi>T</mi> </mrow> </mfrac> </mrow> <mo>∑<!-- ∑ --></mo> <msubsup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{\infty }={\frac {e^{2}}{k_{B}T}}\sum z_{i}^{2}D_{i}n_{i\infty }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccfb3c686408d229b08bc338e58bee1eec9b2d4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:24.303ex; height:6.176ex;" alt="{\displaystyle \sigma ^{\infty }={\frac {e^{2}}{k_{B}T}}\sum z_{i}^{2}D_{i}n_{i\infty }}"/></span>

Where D_i is the diffusion coefficient of the ith solute species, and n_i∞ is the number concentration of electrolyte solution.

Ohshima's model was developed in 1984 and was originally used to analyze the sedimentation velocity of a single charged sphere and the sedimentation potential of a dilute suspension. The model provided below holds true for dilute suspensions of low zeta potential, i.e. eζ/κ_BT ≤2

<span class="mwe-math-element" typeof="mw:Extension/math" about="#mwt33" data-mw='{"name":"math","attrs":{},"body":{"extsrc":" E_{s} = - \\frac{\\varepsilon\\zeta (\\rho -\\rho _{0})\\phi _{p} }{\\sigma ^{\\infty }\\eta } gH(\\kappa \\alpha )+\\vartheta(\\zeta ^2)"}}' id="mwWg"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{s}=-{\frac {\varepsilon \zeta (\rho -\rho _{0})\phi _{p}}{\sigma ^{\infty }\eta }}gH(\kappa \alpha )+\vartheta (\zeta ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mo>−<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>ε<!-- ε --></mi> <mi>ζ<!-- ζ --></mi> <mo stretchy="false">(</mo> <mi>ρ<!-- ρ --></mi> <mo>−<!-- − --></mo> <msub> <mi>ρ<!-- ρ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <msub> <mi>ϕ<!-- ϕ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mrow> <mrow> <msup> <mi>σ<!-- σ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞<!-- ∞ --></mi> </mrow> </msup> <mi>η<!-- η --></mi> </mrow> </mfrac> </mrow> <mi>g</mi> <mi>H</mi> <mo stretchy="false">(</mo> <mi>κ<!-- κ --></mi> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>ϑ<!-- ϑ --></mi> <mo stretchy="false">(</mo> <msup> <mi>ζ<!-- ζ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{s}=-{\frac {\varepsilon \zeta (\rho -\rho _{0})\phi _{p}}{\sigma ^{\infty }\eta }}gH(\kappa \alpha )+\vartheta (\zeta ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c11f0e682a6c6f7f870aff1bfb7091692053a419" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:37.204ex; height:6.176ex;" alt="{\displaystyle E_{s}=-{\frac {\varepsilon \zeta (\rho -\rho _{0})\phi _{p}}{\sigma ^{\infty }\eta }}gH(\kappa \alpha )+\vartheta (\zeta ^{2})}"/></span>