Zeta potential

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Diagram showing the ionic concentration and potential difference as a function of distance from the charged surface of a particle suspended in a dispersion medium Diagram of zeta potential and slipping planeV2.svg
Diagram showing the ionic concentration and potential difference as a function of distance from the charged surface of a particle suspended in a dispersion medium

Zeta potential is the electrical potential at the slipping plane. This plane is the interface which separates mobile fluid from fluid that remains attached to the surface.

Contents

Zeta potential is a scientific term for electrokinetic potential [1] [2] in colloidal dispersions. In the colloidal chemistry literature, it is usually denoted using the Greek letter zeta (ζ), hence ζ-potential. The usual units are volts (V) or, more commonly, millivolts (mV). From a theoretical viewpoint, the zeta potential is the electric potential in the interfacial double layer (DL) at the location of the slipping plane relative to a point in the bulk fluid away from the interface. In other words, zeta potential is the potential difference between the dispersion medium and the stationary layer of fluid attached to the dispersed particle.

The zeta potential is caused by the net electrical charge contained within the region bounded by the slipping plane, and also depends on the location of that plane. Thus, it is widely used for quantification of the magnitude of the charge. However, zeta potential is not equal to the Stern potential or electric surface potential in the double layer, [3] [4] [5] [6] because these are defined at different locations. Such assumptions of equality should be applied with caution. Nevertheless, zeta potential is often the only available path for characterization of double-layer properties.

The zeta potential is an important and readily measurable indicator of the stability of colloidal dispersions. The magnitude of the zeta potential indicates the degree of electrostatic repulsion between adjacent, similarly charged particles in a dispersion. For molecules and particles that are small enough, a high zeta potential will confer stability, i.e., the solution or dispersion will resist aggregation. When the potential is small, attractive forces may exceed this repulsion and the dispersion may break and flocculate. So, colloids with high zeta potential (negative or positive) are electrically stabilized while colloids with low zeta potentials tend to coagulate or flocculate as outlined in the table. [7]

[A] pH-dependant ionisation of a weak acid [HA] and its conjugated base [A-] drawn using Henderson-Hasselbalch equation; [B] Ionisation and solubility of a pH-responsive polymer as a function of pH [C] A schematic showing the potential difference as a function of distance from the charged surface of a particle in a medium; [D] Dissolution mechanism of pH-responsive polymers reproduced with permission from. The encircled numbers in [D] represent (1) Diffusion of water and hydroxyl ions into the polymer matrix to form a gel layer, (2) Ionization of polymer chains in the gel layer, (3) Disentanglement of polymer chains out of the gel layer to the polymer-solution interface, (4) Further ionization of polymer chains at the polymer interface, (5) Diffusion of disentangled polymer chains away from the interface toward the bulk solution. Zeta potential and pka.jpg
[A] pH-dependant ionisation of a weak acid [HA] and its conjugated base [A-] drawn using Henderson-Hasselbalch equation; [B] Ionisation and solubility of a pH-responsive polymer as a function of pH [C] A schematic showing the potential difference as a function of distance from the charged surface of a particle in a medium; [D] Dissolution mechanism of pH-responsive polymers reproduced with permission from. The encircled numbers in [D] represent (1) Diffusion of water and hydroxyl ions into the polymer matrix to form a gel layer, (2) Ionization of polymer chains in the gel layer, (3) Disentanglement of polymer chains out of the gel layer to the polymer-solution interface, (4) Further ionization of polymer chains at the polymer interface, (5) Diffusion of disentangled polymer chains away from the interface toward the bulk solution.

Zeta potential can also be used for the pKa estimation of complex polymers that is otherwise difficult to measure accurately using conventional methods. This can help studying the ionisation behaviour of various synthetic and natural polymers under various conditions and can help in establishing standardised dissolution-pH thresholds for pH responsive polymers. [8]

Stability behaviour of a colloid depending on zeta potential [9]
Magnitude of Zeta potential (mV)Stability behavior
0 to 5Rapid coagulation or flocculation
10 to 30Incipient instability
30 to 40Moderate stability
40 to 60Good stability
>61Excellent stability

Measurement

Some new instrumentations techniques exist that allow zeta potential to be measured. The Zeta Potential Analyzer can measure solid, fibers, or powdered material. The motor found in the interment creates an oscillating flow of electrolyte solution through the sample. Several sensors in the interment monitor other factors, so the software attached is able to do calculations to find the zeta potential. Temperature, pH, conductivity, pressure, and streaming potential are all measured in the interment for this reason.

Zeta potential can also be calculated using theoretical models, and an experimentally-determined electrophoretic mobility or dynamic electrophoretic mobility.

Electrokinetic phenomena and electroacoustic phenomena are the usual sources of data for calculation of zeta potential. (See Zeta potential titration.)

Electrokinetic phenomena

Electrophoresis is used for estimating zeta potential of particulates, whereas streaming potential/current is used for porous bodies and flat surfaces. In practice, the zeta potential of dispersion is measured by applying an electric field across the dispersion. Particles within the dispersion with a zeta potential will migrate toward the electrode of opposite charge with a velocity proportional to the magnitude of the zeta potential.

This velocity is measured using the technique of the laser Doppler anemometer. The frequency shift or phase shift of an incident laser beam caused by these moving particles is measured as the particle mobility, and this mobility is converted to the zeta potential by inputting the dispersant viscosity and dielectric permittivity, and the application of the Smoluchowski theories. [10]

Electrophoresis

Electrophoretic mobility is proportional to electrophoretic velocity, which is the measurable parameter. There are several theories that link electrophoretic mobility with zeta potential. They are briefly described in the article on electrophoresis and in details in many books on colloid and interface science. [3] [4] [5] [11] There is an IUPAC Technical Report [12] prepared by a group of world experts on the electrokinetic phenomena. From the instrumental viewpoint, there are three different experimental techniques: microelectrophoresis, electrophoretic light scattering, and tunable resistive pulse sensing. Microelectrophoresis has the advantage of yielding an image of the moving particles. On the other hand, it is complicated by electro-osmosis at the walls of the sample cell. Electrophoretic light scattering is based on dynamic light scattering. It allows measurement in an open cell which eliminates the problem of electro-osmotic flow except for the case of a capillary cell. And, it can be used to characterize very small particles, but at the price of the lost ability to display images of moving particles. Tunable resistive pulse sensing (TRPS) is an impedance-based measurement technique that measures the zeta potential of individual particles based on the duration of the resistive pulse signal. [13] The translocation duration of nanoparticles is measured as a function of voltage and applied pressure. From the inverse translocation time versus voltage-dependent electrophoretic mobility, and thus zeta potentials are calculated. The main advantage of the TRPS method is that it allows for simultaneous size and surface charge measurements on a particle-by-particle basis, enabling the analysis of a wide spectrum of synthetic and biological nano/microparticles and their mixtures. [14]

All these measuring techniques may require dilution of the sample. Sometimes this dilution might affect properties of the sample and change zeta potential. There is only one justified way to perform this dilution – by using equilibrium supernatant. In this case, the interfacial equilibrium between the surface and the bulk liquid would be maintained and zeta potential would be the same for all volume fractions of particles in the suspension. When the diluent is known (as is the case for a chemical formulation), additional diluent can be prepared. If the diluent is unknown, equilibrium supernatant is readily obtained by centrifugation.

Streaming potential, streaming current

The streaming potential is an electric potential that develops during the flow of liquid through a capillary. In nature, a streaming potential may occur at a significant magnitude in areas with volcanic activities. [15] The streaming potential is also the primary electrokinetic phenomenon for the assessment of the zeta potential at the solid material-water interface. A corresponding solid sample is arranged in such a way to form a capillary flow channel. Materials with a flat surface are mounted as duplicate samples that are aligned as parallel plates. The sample surfaces are separated by a small distance to form a capillary flow channel. Materials with an irregular shape, such as fibers or granular media, are mounted as a porous plug to provide a pore network, which serves as capillaries for the streaming potential measurement. Upon the application of pressure on a test solution, liquid starts to flow and to generate an electric potential. This streaming potential is related to the pressure gradient between the ends of either a single flow channel (for samples with a flat surface) or the porous plug (for fibers and granular media) to calculate the surface zeta potential.

Alternatively to the streaming potential, the measurement of streaming current offers another approach to the surface zeta potential. Most commonly, the classical equations derived by Maryan Smoluchowski are used to convert streaming potential or streaming current results into the surface zeta potential. [16]

Applications of the streaming potential and streaming current method for the surface zeta potential determination consist of the characterization of surface charge of polymer membranes, [17] biomaterials and medical devices, [18] [19] and minerals. [20]

Electroacoustic phenomena

There are two electroacoustic effects that are widely used for characterizing zeta potential: colloid vibration current and electric sonic amplitude. [5] There are commercially available instruments that exploit these effects for measuring dynamic electrophoretic mobility, which depends on zeta potential.

Electroacoustic techniques have the advantage of being able to perform measurements in intact samples, without dilution. Published and well-verified theories allow such measurements at volume fractions up to 50%. Calculation of zeta potential from the dynamic electrophoretic mobility requires information on the densities for particles and liquid. In addition, for larger particles exceeding roughly 300 nm in size information on the particle size required as well.[ citation needed ]

Calculation

The most known and widely used theory for calculating zeta potential from experimental data is that developed by Marian Smoluchowski in 1903. [21] This theory was originally developed for electrophoresis; however, an extension to electroacoustics is now also available. [5] Smoluchowski's theory is powerful because it is valid for dispersed particles of any shape and any concentration. However, it has its limitations:

The model of the "thin double layer" offers tremendous simplifications not only for electrophoresis theory but for many other electrokinetic and electroacoustic theories. This model is valid for most aqueous systems because the Debye length is typically only a few nanometers in water. The model breaks only for nano-colloids in a solution with ionic strength approaching that of pure water.

The development of electrophoretic and electroacoustic theories with a wider range of validity was a purpose of many studies during the 20th century. There are several analytical theories that incorporate surface conductivity and eliminate the restriction of the small Dukhin number for both the electrokinetic and electroacoustic applications.

Early pioneering work in that direction dates back to Overbeek [22] and Booth. [23]

Modern, rigorous electrokinetic theories that are valid for any zeta potential, and often any , stem mostly from Soviet Ukrainian (Dukhin, Shilov, and others) and Australian (O'Brien, White, Hunter, and others) schools. Historically, the first one was Dukhin–Semenikhin theory. [24] A similar theory was created ten years later by O'Brien and Hunter. [25] Assuming a thin double layer, these theories would yield results that are very close to the numerical solution provided by O'Brien and White. [26] There are also general electroacoustic theories that are valid for any values of Debye length and Dukhin number. [5] [11]

Henry's equation

When κa is between large values where simple analytical models are available, and low values where numerical calculations are valid, Henry's equation can be used when the zeta potential is low. For a nonconducting sphere, Henry's equation is , where f1 is the Henry function, one of a collection of functions which vary smoothly from 1.0 to 1.5 as κa approaches infinity. [12]

Related Research Articles

<span class="mw-page-title-main">Colloid</span> Mixture of an insoluble substance microscopically dispersed throughout another substance

A colloid is a mixture in which one substance consisting of microscopically dispersed insoluble particles is suspended throughout another substance. Some definitions specify that the particles must be dispersed in a liquid, while others extend the definition to include substances like aerosols and gels. The term colloidal suspension refers unambiguously to the overall mixture. A colloid has a dispersed phase and a continuous phase. The dispersed phase particles have a diameter of approximately 1 nanometre to 1 micrometre.

<span class="mw-page-title-main">Electro-osmosis</span>

Electroosmotic flow is the motion of liquid induced by an applied potential across a porous material, capillary tube, membrane, microchannel, or any other fluid conduit. Because electroosmotic velocities are independent of conduit size, as long as the electrical double layer is much smaller than the characteristic length scale of the channel, electroosmotic flow will have little effect. Electroosmotic flow is most significant when in small channels. Electroosmotic flow is an essential component in chemical separation techniques, notably capillary electrophoresis. Electroosmotic flow can occur in natural unfiltered water, as well as buffered solutions.

<span class="mw-page-title-main">Electrophoresis</span> Motion of charged particles in electric field

Electrophoresis is the motion of dispersed particles relative to a fluid under the influence of a spatially uniform electric field. Electrophoresis of positively charged particles (cations) is sometimes called cataphoresis, while electrophoresis of negatively charged particles (anions) is sometimes called anaphoresis.

Capillary electrophoresis (CE) is a family of electrokinetic separation methods performed in submillimeter diameter capillaries and in micro- and nanofluidic channels. Very often, CE refers to capillary zone electrophoresis (CZE), but other electrophoretic techniques including capillary gel electrophoresis (CGE), capillary isoelectric focusing (CIEF), capillary isotachophoresis and micellar electrokinetic chromatography (MEKC) belong also to this class of methods. In CE methods, analytes migrate through electrolyte solutions under the influence of an electric field. Analytes can be separated according to ionic mobility and/or partitioning into an alternate phase via non-covalent interactions. Additionally, analytes may be concentrated or "focused" by means of gradients in conductivity and pH.

Surface charge is a two-dimensional surface with non-zero electric charge. These electric charges are constrained on this 2-D surface, and surface charge density, measured in coulombs per square meter (C•m−2), is used to describe the charge distribution on the surface. The electric potential is continuous across a surface charge and the electric field is discontinuous, but not infinite; this is unless the surface charge consists of a dipole layer. In comparison, the potential and electric field both diverge at any point charge or linear charge.

A streaming current and streaming potential are two interrelated electrokinetic phenomena studied in the areas of surface chemistry and electrochemistry. They are an electric current or potential which originates when an electrolyte is driven by a pressure gradient through a channel or porous plug with charged walls.

Electroacoustic phenomena arise when ultrasound propagates through a fluid containing ions. The associated particle motion generates electric signals because ions have electric charge. This coupling between ultrasound and electric field is called electroacoustic phenomena. The fluid might be a simple Newtonian liquid, or complex heterogeneous dispersion, emulsion or even a porous body. There are several different electroacoustic effects depending on the nature of the fluid.

The Dukhin number is a dimensionless quantity that characterizes the contribution of the surface conductivity to various electrokinetic and electroacoustic effects, as well as to electrical conductivity and permittivity of fluid heterogeneous systems. The number was named after Stanislav and Andrei Dukhin.

<span class="mw-page-title-main">Double layer (surface science)</span> Molecular interface between a surface and a fluid

In surface science, a double layer is a structure that appears on the surface of an object when it is exposed to a fluid. The object might be a solid particle, a gas bubble, a liquid droplet, or a porous body. The DL refers to two parallel layers of charge surrounding the object. The first layer, the surface charge, consists of ions which are adsorbed onto the object due to chemical interactions. The second layer is composed of ions attracted to the surface charge via the Coulomb force, electrically screening the first layer. This second layer is loosely associated with the object. It is made of free ions that move in the fluid under the influence of electric attraction and thermal motion rather than being firmly anchored. It is thus called the "diffuse layer".

<span class="mw-page-title-main">Surface conductivity</span>

Surface conductivity is an additional conductivity of an electrolyte in the vicinity of the charged interfaces. Surface and volume conductivity of liquids correspond to the electrically driven motion of ions in an electric field. A layer of counter ions of the opposite polarity to the surface charge exists close to the interface. It is formed due to attraction of counter-ions by the surface charges. This layer of higher ionic concentration is a part of the interfacial double layer. The concentration of the ions in this layer is higher as compared to the ionic strength of the liquid bulk. This leads to the higher electric conductivity of this layer.

Electrokinetic phenomena are a family of several different effects that occur in heterogeneous fluids, or in porous bodies filled with fluid, or in a fast flow over a flat surface. The term heterogeneous here means a fluid containing particles. Particles can be solid, liquid or gas bubbles with sizes on the scale of a micrometer or nanometer. There is a common source of all these effects—the so-called interfacial 'double layer' of charges. Influence of an external force on the diffuse layer generates tangential motion of a fluid with respect to an adjacent charged surface. This force might be electric, pressure gradient, concentration gradient, or gravity. In addition, the moving phase might be either continuous fluid or dispersed phase.

<span class="mw-page-title-main">Colloid vibration current</span>

Colloid vibration current is an electroacoustic phenomenon that arises when ultrasound propagates through a fluid that contains ions and either solid particles or emulsion droplets.

Electric sonic amplitude or electroacoustic sonic amplitude is an electroacoustic phenomenon that is the reverse to colloid vibration current. It occurs in colloids, emulsions and other heterogeneous fluids under the influence of an oscillating electric field. This field moves particles relative to the liquid, which generates ultrasound.

Electrophoretic light scattering is based on dynamic light scattering. The frequency shift or phase shift of an incident laser beam depends on the dispersed particles mobility. With dynamic light scattering, Brownian motion causes particle motion. With electrophoretic light scattering, oscillating electric field performs this function.

Sedimentation potential occurs when dispersed particles move under the influence of either gravity or centrifugation in a medium. This motion disrupts the equilibrium symmetry of the particle's double layer. While the particle moves, the ions in the electric double layer lag behind due to the liquid flow. This causes a slight displacement between the surface charge and the electric charge of the diffuse layer. As a result, the moving particle creates a dipole moment. The sum of all of the dipoles generates an electric field which is called sedimentation potential. It can be measured with an open electrical circuit, which is also called sedimentation current.

<span class="mw-page-title-main">Zeta potential titration</span>

Zeta potential titration is a titration of heterogeneous systems, for example colloids and emulsions. Solids in such systems have very high surface area. This type of titration is used to study the zeta potential of these surfaces under different conditions. Details of zeta potential definition and measuring techniques can be found in the International Standard.

<span class="mw-page-title-main">Particle size</span> Notion for comparing dimensions of particles in different states of matter

Particle size is a notion introduced for comparing dimensions of solid particles, liquid particles (droplets), or gaseous particles (bubbles). The notion of particle size applies to particles in colloids, in ecology, in granular material, and to particles that form a granular material.

<span class="mw-page-title-main">Capillary electrochromatography</span>

Capillary electrochromatography (CEC) is a chromatographic technique in which the mobile phase is driven through the chromatographic bed by electroosmosis. Capillary electrochromatography is a combination of two analytical techniques, high-performance liquid chromatography and capillary electrophoresis. Capillary electrophoresis aims to separate analytes on the basis of their mass-to-charge ratio by passing a high voltage across ends of a capillary tube, which is filled with the analyte. High-performance liquid chromatography separates analytes by passing them, under high pressure, through a column filled with stationary phase. The interactions between the analytes and the stationary phase and mobile phase lead to the separation of the analytes. In capillary electrochromatography capillaries, packed with HPLC stationary phase, are subjected to a high voltage. Separation is achieved by electrophoretic migration of solutes and differential partitioning.

Dispersion Technology Inc is a scientific instrument manufacturer located in Bedford Hills, New York. It was founded in 1996 by Philip Goetz and Dr. Andrei Dukhin. The company develops and sells analytical instruments intended for characterizing concentrated dispersions and emulsions, complying with the International Standards for acoustic particle sizing ISO 20998 and Electroacoustic zeta potential measurement ISO 13099.

<span class="mw-page-title-main">Induced-charge electrokinetics</span>

Induced-charge electrokinetics in physics is the electrically driven fluid flow and particle motion in a liquid electrolyte. Consider a metal particle in contact with an aqueous solution in a chamber/channel. If different voltages apply to the end of this chamber/channel, electric field will generate in this chamber/channel. This applied electric field passes through this metal particle and causes the free charges inside the particle migrate under the skin of particle. As a result of this migration, the negative charges moves to the side which is close to the positive voltage while the positive charges moves to the opposite side of the particle. These charges under the skin of conducting particle attract the counter-ions of the aqueous solution; thus, the electric double layer (EDL) forms around the particle. The EDL sign on the surface of the conducting particle changes from positive to negative and the distribution of the charges varies along the particle geometry. Due to these variations, the EDL is non-uniform and has different signs. Thus, the induced zeta potential around the particle, and consequently slip velocity on the surface of the particle, vary as a function of local electric field. Differences in magnitude and direction of slip velocity on the surface of the conducting particle effects the flow pattern around this particle and causes micro vortices. Yasaman Daghighi and Dongqing Li, for the first time, experimentally illustrated these induced vortices around a 1.2mm diameter carbon-steel sphere under the 40V/cm direct current (DC) external electric filed. Chenhui Peng et al. also experimentally showed the patterns of electro-osmotic flow around an Au sphere when alternating current (AC) is involved . Electrokinetics here refers to a branch of science related to the motion and reaction of charged particles to the applied electric filed and its effects on its environment. It is sometimes referred as non-linear electrokinetic phenomena as well.

References

  1. ↑ IUPAC , Compendium of Chemical Terminology , 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) " electrokinetic potential, ζ ". doi : 10.1351/goldbook.E01968
  2. ↑ "Colloidal systems – Methods for Zeta potential determination". ISO International Standard 13099, Parts 1,2 and 3. International Organization for Standardization (ISO). 2012.
  3. 1 2 Lyklema J (1995). Fundamentals of Interface and Colloid Science. Vol. 2. pp. 3–208. ISBN   978-0-12-460529-9.
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  7. ↑ Hanaor D, Michelazzi M, Leonelli C, Sorrell CC (2012). "The effects of carboxylic acids on the aqueous dispersion and electrophoretic deposition of ZrO2". Journal of the European Ceramic Society. 32 (1): 235–244. arXiv: 1303.2754 . doi:10.1016/j.jeurceramsoc.2011.08.015. S2CID   98812224.
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  10. ↑ "Zeta Potential Using Laser Doppler Electrophoresis". Malvern.com. Archived from the original on 7 April 2012.
  11. 1 2 Hunter RJ (1989). Foundations of Colloid Science. Oxford University Press. ISBN   978-0-19-855189-8.[ page needed ]
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  13. ↑ "Zeta Potential Measurement With TRPS". Izon Science.
  14. ↑ Vogel R, Pal AK, Jambhrunkar S, Patel P, Thakur SS, Reátegui E, et al. (December 2017). "High-Resolution Single Particle Zeta Potential Characterisation of Biological Nanoparticles using Tunable Resistive Pulse Sensing". Scientific Reports. 7 (1): 17479. Bibcode:2017NatSR...717479V. doi:10.1038/s41598-017-14981-x. PMC   5727177 . PMID   29234015.
  15. ↑ Jouniaux, L.; Ishido, T. (2012). "Electrokinetics in Earth Sciences: A Tutorial". International Journal of Geophysics. 2012: e286107. doi: 10.1155/2012/286107 . ISSN   1687-885X.
  16. ↑ Luxbacher, Thomas (2014). The ZETA Guide: Principles of the streaming potential technique. Anton Paar GmbH. ISBN   978-3-200-03553-9.
  17. ↑ Elimelech, Menachem; Chen, William H.; Waypa, John J. (1994). "Measuring the zeta (electrokinetic) potential of reverse osmosis membranes by a streaming potential analyzer". Desalination. 95 (3): 269–286. doi:10.1016/0011-9164(94)00064-6. ISSN   0011-9164.
  18. ↑ Werner, Carsten; König, Ulla; Augsburg, Antje; Arnhold, Christine; Körber, Heinz; Zimmermann, Ralf; Jacobasch, Hans-Jörg (1999). "Electrokinetic surface characterization of biomedical polymers — a survey". Colloids and Surfaces A: Physicochemical and Engineering Aspects. 159 (2): 519–529. doi:10.1016/S0927-7757(99)00290-3. ISSN   0927-7757.
  19. ↑ Ferraris, Sara; Cazzola, Martina; Peretti, Veronica; Stella, Barbara; Spriano, Silvia (2018). "Zeta Potential Measurements on Solid Surfaces for in Vitro Biomaterials Testing: Surface Charge, Reactivity Upon Contact With Fluids and Protein Absorption". Frontiers in Bioengineering and Biotechnology. 6: 60. doi: 10.3389/fbioe.2018.00060 . ISSN   2296-4185. PMC   5954101 . PMID   29868575.
  20. ↑ Fuerstenau, D. W. (1956). "Streaming Potential Studies on Quartz in Solutions of Aminium Acetates in Relation to the Formation of Hemi- micelles at the Quartz-Solution Interface". The Journal of Physical Chemistry. 60 (7): 981–985. doi:10.1021/j150541a039. ISSN   0022-3654.
  21. ↑ Smoluchowski M (1903). "Przyczynek do teoryi endosm ozy elektrycznej i kilku zjawisk pokrewnych" [Contribution to the theory of electro-osmosis and related phenomena](PDF) (in Polish). Archived from the original (PDF) on August 10, 2017.
  22. ↑ Overbeek JT (1943). "Theory of electrophoresis — The relaxation effect". Koll. Bith.: 287.
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  24. ↑ Semenikhin NM, Dukhin SS (January 1975). "Polarization of a Moderately Thin Double-Layer Around Spherical-Particles and Its Influence on Electrophoresis". Colloid Journal of the USSR. 37 (6): 1013–1016.
  25. ↑ O'Brien RW, Hunter RJ (July 1981). "The electrophoretic mobility of large colloidal particles". Canadian Journal of Chemistry. 59 (13): 1878–1887. doi:10.1139/v81-280.
  26. ↑ O'Brien RW, White LR (1978). "Electrophoretic mobility of a spherical colloidal particle". Journal of the Chemical Society, Faraday Transactions 2. 74: 1607. doi:10.1039/F29787401607.
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