Theorie der Electrophorese -- Het Relaxatie-Effect

Amitesh S. Jayaraman; Derek Y.C. Chan); J. Th. G. Overbeek (translated by Evert Klaseboer

arxiv: 1907.05542 · v1 · pith:ZUDZCAULnew · submitted 2019-07-12 · ⚛️ physics.hist-ph · cond-mat.stat-mech· physics.chem-ph

Theorie der Electrophorese -- Het Relaxatie-Effect

J. Th. G. Overbeek (translated by Evert Klaseboer , Amitesh S. Jayaraman , Derek Y.C. Chan) This is my paper

Pith reviewed 2026-05-24 22:24 UTC · model grok-4.3

classification ⚛️ physics.hist-ph cond-mat.stat-mechphysics.chem-ph

keywords electrophoresisrelaxation effectdouble layercolloidal particleselectrophoretic mobilityspherical particlezeta potentialionic atmosphere

0 comments

The pith

The electrophoretic velocity of a charged spherical particle is related to its double layer potential by a complete analysis that includes the relaxation effect.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The thesis develops a theoretical treatment connecting the velocity of a colloidal particle in an electric field to the electrostatic potential in its surrounding double layer. It solves the coupled problem of fluid flow, ion transport, and electrostatics for a sphere while accounting for how the particle's motion distorts the ionic atmosphere. A reader would care because the resulting relation supplies the basis for determining particle charge and size from observed motion in suspensions. The work shows how the relaxation of the double layer modifies the effective driving force and the drag on the particle.

Core claim

The electrophoretic velocity of a charged spherical particle under an external electric field is related to the potential of the double layer through a mathematical solution that incorporates the relaxation effect, in which the motion of the particle causes a distortion in the surrounding ionic distribution that in turn alters the local electric field and the hydrodynamic resistance.

What carries the argument

The relaxation effect, the distortion of the ionic double layer caused by the particle's motion that produces an additional force opposing the electrophoretic drive.

If this is right

The mobility can be calculated as a function of the zeta potential and the thickness of the double layer.
Surface charge density can be inferred from measured velocity once the double-layer potential is known.
The same framework supplies the hydrodynamic and electrostatic corrections needed to interpret mobility data across different electrolyte concentrations.
It supplies the theoretical foundation for using electrophoresis to characterize particle size and charge in colloidal suspensions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same relaxation mechanism would be expected to influence other electrokinetic phenomena such as sedimentation potential or streaming current in similar geometries.
Modern numerical methods could solve the governing equations for non-spherical shapes or higher field strengths where the original analytic approach reaches its limits.
The derived relation remains the reference point against which experimental deviations at high surface potentials can be tested for additional nonlinear effects.

Load-bearing premise

The fluid around the particle and the distribution of ions can be treated with continuum equations from fluid mechanics and equilibrium statistical mechanics.

What would settle it

Precise measurements of electrophoretic mobility for well-characterized spherical particles at controlled ionic strengths that deviate systematically from the velocity predicted by the double-layer potential relation.

Figures

Figures reproduced from arXiv: 1907.05542 by Amitesh S. Jayaraman, Derek Y.C. Chan), J. Th. G. Overbeek (translated by Evert Klaseboer.

**Figure 1.** Figure 1: The forces on the sphere. k1 = force excerted by the electric field on the charge of the sphere. k2 = drag (friction) according to Stokes. k3 = electrophoretic drag. water layer, that has a radius of a and a charge of +Q then, according to H¨uckel23 k1 = +QE k2 = −6πηaU k3 = (4π0rζa − Q)E . If the particle is moving with uniform velocity during electrophoresis, the total force acting on the particle must… view at source ↗

**Figure 2.** Figure 2: a) H¨uckel b) Insulator. c) Conductor [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

**Figure 3.** Figure 3: f(κa, µ) from U X = Dζ 6πη f(κa, µ) larger force than those of Fig.2a and thus result in a larger friction, can only compensate the first effect partially, such that in total a lower friction results according to the calculations of H¨uckel. In the opposite case, a conducting particle, where the electric field line density in front and behind the particle is on the contrary larger, will have to experience … view at source ↗

**Figure 4.** Figure 4: Drag due to the relaxation force k4. Left: at rest. Right: in motion. Direction of the electric field ”veldrichting” from left to right. If the cylinder is placed perpendicular to the field and the double layer is thin, then for an insulating particle (µ = 0) the formula of Smoluchowski is still valid, while the velocity becomes half as big if µ = 1 and 0 if µ = ∞. However, in the calculations of Von Smolu… view at source ↗

**Figure 5.** Figure 5: ”deeltjesstraal” = particle radius, ”Kolloiden” = colloids. interaction between the electrophoretic drag and the relaxation effect, which is especially important for large κa (see chapter V, (the discussion around (89aII) and (89bII)).). It is of course true that our calculations are only valid for spherical particles, but in any case from these calculations approximate conclusions can be drawn for other s… view at source ↗

**Figure 6.** Figure 6: Parallelopiped used to explain the divergence theorem. B - THE FLUID MOTION AND THE FORCES ON THE SPHERE Our treatment of the fluid motion and the by this motion exerted force on the sphere completely connects to the treatments of H¨uckel and Henry for the electrophoresis effect. without much effort these considerations can be made much more general, such that they don’t only take into account the original… view at source ↗

**Figure 7.** Figure 7: Spherical coordinates. ”veldrichting” = direction of the field. A - CALCULATION OF THE FLUID MOTION We assume (14a) and (14b) from chapter II are valid, where we have reasoned that ρmDu/Dt is equal to zero55 η∇ × ∇ × u + ∇p + ρ∇(ψ 0 + δψ) = 0 (14a) ∇ · u = 0 (14b) For the charge density ρ we can write according to Poisson ρ = −0r∇2 (ψ 0 + δψ) = ρ 0 + δρ (15) Here ψ 0 and ρ 0 are functions of r alone. For… view at source ↗

**Figure 8.** Figure 8: Only that part of the change of uθ with r must be taken into account, which leads to torsion, thus the change of the angular velocity uθ/r (and not the linear velocity uθ); see Fig.8b. p = 0r Z r ∞ ∆ψ 0 dψ0 dx dx + cos θ n3ηaU 2r 2 − 0rEχ − 0rEa r 2 Z a ∞ ξ dxo (43) ur = cos θ n − 1 + 3a 2r − a 3 2r 3 U + 20rE 3η Z r ∞ ξ dx + 1 r 3 Z a r x 3 ξ dx − 0rE η a r − a 3 3r 3 Z a ∞ ξ dxo uθ = s… view at source ↗

**Figure 9.** Figure 9: Values of f2(κa), f3(κa) and f4(κa) from equation (89). TABLE III Values of the various correction terms from the electrophoresis formula (89) κa 0.1 0.3 1 3 5 10 20 50 100 f1(κa) 1.000545 1.00398 1.0267 1.1005 1.163 1.25 1.34 1.424 1.458 f2(κa) 0.0125 0.0279 0.0411 0.053 0.057 0.056 0.04 0.0188 0.0102 f3(κa) 0.00090 0.0044 0.0116 0.020 0.022 0.021 0.0145 0.00796 0.00444 f4(κa) 0.0107 0.0218 0.0387 0.00515… view at source ↗

**Figure 10.** Figure 10: f(κa, ζ) for symmetrical electrolytes [PITH_FULL_IMAGE:figures/full_fig_p070_10.png] view at source ↗

**Figure 11.** Figure 11: f(κa, ζ) for non symmetrical electrolytes at eζ kT = 2 (ζ = 50 mV). 70 [PITH_FULL_IMAGE:figures/full_fig_p070_11.png] view at source ↗

**Figure 12.** Figure 12: The electrophoretic velocity as a function of [PITH_FULL_IMAGE:figures/full_fig_p072_12.png] view at source ↗

**Figure 13.** Figure 13: Electrophoresis velocities of oils (”olie”, Powis) and AgJ (Troelstra) at various electrolyte concentrations. valued counter ions, usually the first decrease is missing, such that we will have to assume that the electrophoretic velocity of the clean sol is roughly found in the minimum of the curves of Figs.10 and 11. We then observe curves with an increasing section (increasing f1(κa) and decreasing of th… view at source ↗

read the original abstract

In this thesis, a theoretical treatment of the relation between electrophoretic velocity and the potential of the double layer of colloidal particles is presented. Translators' note: The theory of electrophoresis is one of the foundational topics that underpinned the development of colloid and surface science and ranks with the famous Derjaguin-Landau-Verwey-Overbeek (DLVO) theory of colloidal stability. J. Th. G. Overbeek ("Theo" to all who knew him) was the first to develop a complete theoretical analysis of the electrophoretic motion of a charged spherical particle under the influence of an external electric field. This provided the theoretical framework for a widely used experimental method to characterize the state of charge and particle size of small colloidal particles. The solution of this problem required mastery of fluid mechanics, colloidal electrostatics, statistical thermodynamics and transport theory in addition to solid applied mathematics. Theo carried out this study as his doctoral thesis under H.R. Kruyt at Utrecht University. The thesis, in Dutch, was later published as a monograph. Given the important pedagogic value and historical status of this work, we felt that it deserved to enjoy a wide readership.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This is a clean English translation of Overbeek's 1940s thesis that makes the original derivation of electrophoretic mobility accessible, but it contains no new results.

read the letter

This preprint is an English translation of Overbeek's doctoral thesis from the 1940s on the theory of electrophoresis. The core contribution is the translators' decision to render the full derivation into English so that the work can reach readers who do not handle Dutch. The original analysis treats the motion of a charged sphere in an electric field by combining Stokes flow, Poisson-Boltzmann electrostatics, and convective diffusion of ions. It arrives at an expression for velocity in terms of the double-layer potential without relying on thin-layer approximations alone. That framework later fed into routine zeta-potential measurements and sits alongside DLVO theory as a standard reference point in colloid science. The translators' note correctly flags the range of tools required: fluid mechanics, statistical thermodynamics, and applied mathematics. No circular reasoning appears in the setup; the equations are solved from the governing transport and force-balance statements. The main limitation is obvious and not hidden: nothing here is new. The mathematics and assumptions are those of the 1940s, and modern readers will still need later numerical or experimental checks for quantitative accuracy under specific conditions. The paper is therefore useful for historians of surface chemistry and for students who want to see the original line of reasoning laid out step by step. It is not material that a current research group would cite for fresh predictions. A serious editor should not route it to peer review as original science; the appropriate home is an archive or a history-of-science outlet where the translation itself can be noted.

Referee Report

0 major / 1 minor

Summary. The manuscript is an English translation of J. Th. G. Overbeek's 1940s doctoral thesis on the theory of electrophoresis, focusing on the relaxation effect. It derives the electrophoretic velocity of a charged spherical colloidal particle in an external electric field in terms of the double-layer potential, using continuum fluid mechanics, colloidal electrostatics, statistical thermodynamics, and transport theory.

Significance. If the translation is faithful, the work has substantial historical and pedagogic value as the first complete theoretical treatment of electrophoretic motion for charged spheres. It provided the framework for a key experimental technique in colloid science and is noted as ranking with the DLVO theory in foundational importance.

minor comments (1)

[Translators' note] The translators' note would be strengthened by adding the precise original publication details (journal or monograph series, year, and publisher) of the Dutch thesis.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation of the manuscript and their recommendation to accept. The referee accurately highlights the historical importance of Overbeek's thesis as the first complete theoretical treatment of electrophoretic mobility for charged spheres and its foundational role alongside the DLVO theory.

Circularity Check

0 steps flagged

No significant circularity; derivation from first principles

full rationale

The document is a 1940s doctoral thesis (translated) deriving electrophoretic velocity from continuum fluid mechanics, electrostatics, statistical thermodynamics and transport theory applied to a charged sphere. No self-citations appear as load-bearing premises, no fitted parameters are relabeled as predictions, and no uniqueness theorems or ansatzes are imported from the authors' own prior work. The central result is obtained by solving the coupled differential equations under stated boundary conditions; the derivation chain is self-contained against external physical laws and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on abstract only; no specific free parameters or invented entities mentioned. The work relies on established domain assumptions in colloidal science from the 1940s.

axioms (1)

standard math Standard mathematical techniques for solving differential equations in fluid mechanics and electrostatics
Invoked in the theoretical treatment as per the translators' note.

pith-pipeline@v0.9.0 · 5760 in / 1144 out tokens · 36789 ms · 2026-05-24T22:24:52.664096+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The derivation of an electrophoresis equation, expressing the total sum of hydrodynamic and electric forces... relaxation effect, due to the distortion of the double layer
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

assumptions of continuum fluid mechanics, colloidal electrostatics, statistical thermodynamics and transport theory

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

This might lead to errors for sols with small particles

The Brownian motion of the particle is not taken into account. This might lead to errors for sols with small particles

work page
[2]

A sol can be considered concentrated, if the total charge of the sol particles is at least of the same order as the total charge of all other ions in the sol

The mutual interaction between sol particles is neglected, which is not allowed in highly concen- trated sols. A sol can be considered concentrated, if the total charge of the sol particles is at least of the same order as the total charge of all other ions in the sol

work page
[3]

It should be necessary, in about the same manner as Stern 101 has indicated, to take into account the ion radius and adsorption forces

The ions in the double layer are considered as point charges, while only the Coulomb energy is accounted for. It should be necessary, in about the same manner as Stern 101 has indicated, to take into account the ion radius and adsorption forces

work page
[4]

Verlende 102 ﬁnds that the conductivity of the double layer is 10-50 times larger than according to its charge

Lately indications have shown that the structure of the double layer is more complex than we imagined up to now. Verlende 102 ﬁnds that the conductivity of the double layer is 10-50 times larger than according to its charge. M. Klomp´ e 103 renders credible that a large electrophoretic velocity can exist, for cases where the potential diﬀerence in the diﬀ...

work page
[5]

The biggest objection against our calculations is the fact, that the series expansion was truncated early, such that the results are only valid for small ζ-potentials. In view of the enormously tedious calculations that are associated with higher terms, it seems to be recommendable to solve the in chapter IV given principal equations through graphical mea...

work page
[6]

We have indicated clearly, which factors are important in the calculation of the electrophoretic velocity and how this calculation can be performed. 101O. STERN, Z. Elektrochem. 30, 508 (1924). 102Ed. VERLENDE, Proc. Kon. Ned. Akad. v. Westensch., Amsterdam 42 764 (1939. PhD Thesis Ghent 1940, page 50. 103M. KLOMP ´E, PhD. Thesis Utrecht (1941). Marga Klo...

work page 1924
[7]

An electrolysis formula (48) has been deduced, which fully accounts for the electrophoretic drag and the relaxation eﬀect

work page
[8]

This conﬁrms the formula of Henry for ζ→ 0 and also conﬁrms that for ζ-potentials below 25 mV only small corrections (a few percent) are needed

From (48), after introducing several approximations, the formula (89) was deduced, which de- scribes the relationship between the electrophoretic velocity and ζ for small ζ-potentials accurately. This conﬁrms the formula of Henry for ζ→ 0 and also conﬁrms that for ζ-potentials below 25 mV only small corrections (a few percent) are needed

work page
[9]

The order of magnitude of these corrections, the dependence on κa and on the associated phenomena correspond to a large extend to experimental facts such as: a

If (89) is applied to larger ζ-potentials, it turns out that the relaxation eﬀect brings along impor- tant corrections on the electrophoretic velocity. The order of magnitude of these corrections, the dependence on κa and on the associated phenomena correspond to a large extend to experimental facts such as: a. the inﬂuence of the valence of co- and count...

work page 1931
[10]

Since inside the sphere the conductivity is very high, the ﬁeld is very low

The current is perpendicular to the sphere surface and is equal inside and outside the sphere. Since inside the sphere the conductivity is very high, the ﬁeld is very low. Thus Φ i = 0, thus (Φ)r=a = 0, thus (I)x=κa = 0. See eqs. (74), (17) and (58)

work page
[11]

The current through the surface is transferred in a (given) ratio of cations and anions. It is for example imaginable, that only the H+ ions (in the Pt-sol) or the Ag+ ions (in a Ag-sol) are being discharged at the boundary of the sphere-ﬂuid and that the anions are not participating at all

work page
[12]

Thus according to (77’) D ( dI dx ) x=κa = Di ( I x ) x=κa

The strengths of the ﬁeld inside and outside the sphere are proportional to the dielectric con- stants. Thus according to (77’) D ( dI dx ) x=κa = Di ( I x ) x=κa . Since for a good conductor the dielectric constant is very high, this condition demands that ( I)x=κa = 0, which is thus identical to the ﬁrst condition. If one would like to be exact, then th...

work page 1940

[1] [1]

This might lead to errors for sols with small particles

The Brownian motion of the particle is not taken into account. This might lead to errors for sols with small particles

work page

[2] [2]

A sol can be considered concentrated, if the total charge of the sol particles is at least of the same order as the total charge of all other ions in the sol

The mutual interaction between sol particles is neglected, which is not allowed in highly concen- trated sols. A sol can be considered concentrated, if the total charge of the sol particles is at least of the same order as the total charge of all other ions in the sol

work page

[3] [3]

It should be necessary, in about the same manner as Stern 101 has indicated, to take into account the ion radius and adsorption forces

The ions in the double layer are considered as point charges, while only the Coulomb energy is accounted for. It should be necessary, in about the same manner as Stern 101 has indicated, to take into account the ion radius and adsorption forces

work page

[4] [4]

Verlende 102 ﬁnds that the conductivity of the double layer is 10-50 times larger than according to its charge

Lately indications have shown that the structure of the double layer is more complex than we imagined up to now. Verlende 102 ﬁnds that the conductivity of the double layer is 10-50 times larger than according to its charge. M. Klomp´ e 103 renders credible that a large electrophoretic velocity can exist, for cases where the potential diﬀerence in the diﬀ...

work page

[5] [5]

The biggest objection against our calculations is the fact, that the series expansion was truncated early, such that the results are only valid for small ζ-potentials. In view of the enormously tedious calculations that are associated with higher terms, it seems to be recommendable to solve the in chapter IV given principal equations through graphical mea...

work page

[6] [6]

We have indicated clearly, which factors are important in the calculation of the electrophoretic velocity and how this calculation can be performed. 101O. STERN, Z. Elektrochem. 30, 508 (1924). 102Ed. VERLENDE, Proc. Kon. Ned. Akad. v. Westensch., Amsterdam 42 764 (1939. PhD Thesis Ghent 1940, page 50. 103M. KLOMP ´E, PhD. Thesis Utrecht (1941). Marga Klo...

work page 1924

[7] [7]

An electrolysis formula (48) has been deduced, which fully accounts for the electrophoretic drag and the relaxation eﬀect

work page

[8] [8]

This conﬁrms the formula of Henry for ζ→ 0 and also conﬁrms that for ζ-potentials below 25 mV only small corrections (a few percent) are needed

From (48), after introducing several approximations, the formula (89) was deduced, which de- scribes the relationship between the electrophoretic velocity and ζ for small ζ-potentials accurately. This conﬁrms the formula of Henry for ζ→ 0 and also conﬁrms that for ζ-potentials below 25 mV only small corrections (a few percent) are needed

work page

[9] [9]

The order of magnitude of these corrections, the dependence on κa and on the associated phenomena correspond to a large extend to experimental facts such as: a

If (89) is applied to larger ζ-potentials, it turns out that the relaxation eﬀect brings along impor- tant corrections on the electrophoretic velocity. The order of magnitude of these corrections, the dependence on κa and on the associated phenomena correspond to a large extend to experimental facts such as: a. the inﬂuence of the valence of co- and count...

work page 1931

[10] [10]

Since inside the sphere the conductivity is very high, the ﬁeld is very low

The current is perpendicular to the sphere surface and is equal inside and outside the sphere. Since inside the sphere the conductivity is very high, the ﬁeld is very low. Thus Φ i = 0, thus (Φ)r=a = 0, thus (I)x=κa = 0. See eqs. (74), (17) and (58)

work page

[11] [11]

The current through the surface is transferred in a (given) ratio of cations and anions. It is for example imaginable, that only the H+ ions (in the Pt-sol) or the Ag+ ions (in a Ag-sol) are being discharged at the boundary of the sphere-ﬂuid and that the anions are not participating at all

work page

[12] [12]

Thus according to (77’) D ( dI dx ) x=κa = Di ( I x ) x=κa

The strengths of the ﬁeld inside and outside the sphere are proportional to the dielectric con- stants. Thus according to (77’) D ( dI dx ) x=κa = Di ( I x ) x=κa . Since for a good conductor the dielectric constant is very high, this condition demands that ( I)x=κa = 0, which is thus identical to the ﬁrst condition. If one would like to be exact, then th...

work page 1940