ELSEVIER
`
`Advances in Colloid and Interface Science
`67 (1996) 1-118
`
`ADVANCES IN
`COLLOID AND
`INTERFACE
`SCIENCE
`
`Electrokinetic properties, colloidal stability and
`aggregation kinetics of polymer colloids1
`
`R. Hidalgo-Alvareza*, A. Martina, A. Fernandeza, D. Bastosa,
`F. Martfneza, F.J. de las Nievesb
`
`aBiocolloid and Fluid Physics Group, Department of Applied Physics, Faculty of Sciences,
`Campus Fuentenueua, University of Granada, Granada 18071, Spain
`bGroup of Complex Fluid Physics, Department of Applied Physics, Faculty of Sciences,
`University of Almeria, Almeria, Spain
`
`Abstract
`
`The purpose of this article is to present some important advanc,es in the electroki(cid:173)
`netic and colloidal characterization of polymer colloids. Special attention is paid to the
`new electrokinetic techniques: diffusiophoresis, dielectric dispersion and electro-acous(cid:173)
`tic. Also the most recent theoretical approaches are reviewed with respect to the
`electrokinetic properties of polymer colloids. Recently there has been intense discussion
`concerning electrokinetic processes and the theories used for data interpretation.
`Several concerns have been raised relating to the inability of the different processes and
`theories to yield the same electrokinetic potential. The most important explanations
`(shear plane expansion, preferential ion adsorption, osmotic swelling, crossing of the
`mobili ty/(,-potential and anomalous surface conductance) to the electrokinetic behaviour
`of polymer colloids are discussed and analyzed. Also the effect of heat treatment on the
`electrokinetic properties of different types of polymer colloids is extensively considered.
`With regard to the coll<><idal stability of polymer colloids, three- and two-dimensional
`aggregations are presented. First, the stability factor Wis introduced using the classical
`theory DL VO and the values obtained of Hamaker constant compared with the theoreti(cid:173)
`cal values estimated from the Lifshitz theory. The differences usually found by several
`authors are explained as due to the hydrodynamic interaction. Special attention is paid
`
`1 This review is dedicated to Professor Johannes Lyklema on the occasion of his retirement from
`the Physical and Colloid Chemistry Department of the Agricultural University, Wageningen,
`The Netherlands
`* Corresponding author
`
`© 1996 - Elsevier Science B.V. All rights reserved.
`0001-8686/96/$32.00
`PII: 80001-8686(96)00297-7
`
`

`

`2
`
`R. Hidalgo-Aluarez et al. I Adu. Colloid Interface Sci. 67 (1996) 1-118
`
`to the extended DLVO theory for studying homocoagulation of polymer colloids in three
`dimensions and to the new expressions for the van der Waals, electrostatic and struc(cid:173)
`tural forces that must be deduced to study the colloidal stability of polymer colloids in
`two dimensions. Also, the heterocoagulation of polymer colloids with different sign of
`surface charge density and particle size is reviewed, and a new definition of the
`heterocoagulation stability factor is given.
`The aggregation kinetics of polymer colloids in three dimensions is analyzed using
`the Smoluchowski theory (in terms of the reaction kernels kij) in the cases where the
`Smoluchowski's equation is analytically solvable (constant, sum, product kernel and
`linear combinations thereoD. The dynamic scaling in aggregation phenomena with
`polymer colloids is studied using the classification scheme for homogeneous kernels due
`to Van Dongen and Ernst based on the relative probabilities of large clusters sticking
`to large clusters, and small clusters sticking to large clusters. The techniques (multi(cid:173)
`particle and single particle detection) enabling us to provide cluster-size distribution of
`aggregating polymer colloids are a lso presented.
`Finally, the aggregation kinetics of two dimensional aggregation of polymer colloids
`is studied on the basis of the fractal dimension of the aggregates.The different scaling
`theories for two-dimensional aggregation a re also considered.
`
`Contents
`
`. . . . . . . . . . . . . .
`Abstract
`Introduction
`. . . . . . . . . .
`1.
`Purification of polymer colloids
`2.
`3. Surface characterization of polymer colloids
`4. Electrokinetic background . . .
`. . . . . . .
`5. Calculation of /;-potential
`5.1. Electrokinetic phenomena
`. . . . . . .
`5.2. Possible explanations for electrokinetics behaviour
`5.3. Effect of heat treatment on the electrokinetic behaviour
`6. Colloidal stability of polymer colloids in 3-D
`6.1. Classical stability t heory: DL VO
`6.2. Extended DLVO theory
`. . . . . . . . .
`6.3. Heterocoagulation with polymer colloids
`7. Aggregation kinetics of polymer colloids in 3-D
`7.1. Smoluchowski theory . . . . . . . . . . . .
`7.2. Dynamic scaling in aggregation phenomena: homogeneous kernels
`processes . . . . . . . . . . . . . . . . . . . . . . . . . . .
`7.3. Monitoring aggregation
`. . . . . . . . . . . . . . . . . .
`7.4. Experimental results on aggregation of polymer colloids
`7.5. Effect of the particle surface charge density
`8. Aggregation of polymer colloids in 2-D
`8. L Stability . . . . . . .
`8.2. Aggregation kinetics
`9. Conclusions . .
`Acknowledgements .
`References
`. . . . .
`
`1
`3
`4
`6
`7
`9
`9
`. 38
`. 47
`. 52
`. 53
`. 58
`. 62
`. 70
`. 71
`
`. 75
`. 77
`. 83
`. 93
`. 96
`. 96
`. 99
`104
`105
`106
`
`

`

`R. Hidalgo-Alvarezet al. !Adu. Colloid Interface Sci. 67 (1996)1-118
`
`3
`
`I. Introduction
`
`Polymer colloids play an important role in many industrial processes.
`These include the manufacture of synthetic rubber, surface coatings,
`adhesives, additives in paper, textiles and many others. The rapid increase
`in the utilisation oflatices over the last two decades is due to a number of
`factors. Water-based systems avoid many of the environmental problems
`associated with organic-solvent based systems; latices can be designed to
`meet a wide range of application problems; emulsion polymerization on a
`large scale proceeds smoothly for a wide range of monomers.
`The spherical shape of many polymer beads and their narrow size
`distribution makes them most suitable for fundamental studies requir(cid:173)
`ing well-defined monodisperse systems. The particles can be useful e.g.,
`as size standards for instrumental calibration and as carriers in anti(cid:173)
`body diagnostic tests. They provide valuable experimental systems for
`the study of many colloidal phenomena and recently have been used as
`model systems for the simulation of molecular phenomena, including
`nucleation, crystallization and the formation of glasses. The long time
`scale of the motion of polymer colloids enables us to make real time
`observations on various phenomena, which have been impossible for
`atomic and molecular systems. Since the polymer colloids are much
`smaller than the bubbles, and can be dispersed "monomolecularly" in
`liquids (namely, not in contact with each other as was the case with the
`bubbles), the thermal motion can be visualized. Thus, the polymer
`colloids are a much more realistic model for atoms and molecules than
`the bubble rafts.
`Polymer colloids are often treated as "model" particles: monodisperse,
`amorphous microspheres witp smooth, uniform surfaces and rigidly
`attached, well-defined surface functional groups. Their sphericity, mono(cid:173)
`dispersity, and virtually zero dielectric constant in comparison to water
`made them particularly suitable for fundamental electrokinetic and
`colloidal aggregation studies. Electrokinetic processes are widely used
`to determine the electrical charge on the slipping plane of a polymer
`colloid. Electrokinetic is closely related to the slipping process, the
`s-potential being the potential at the slipping plane. Very recently,
`Lyklema [1] considered this very aspect in a very interesting paper.
`The preparation of uniform polymer microspheres via emulsion poly(cid:173)
`merization has been extensively reviewed by Ugelstad [2]. Parameters
`such as particle size, surface charge density and type of charge group
`can be controlled by varying the conditions of the polymerization,
`
`

`

`4
`
`R. Hidalgo-Aluarez et al.I Ad.u. Colloid. Interface Sci. 67 (1996) 1-118
`
`allowing latices to be "designed" for specific end uses. For these reasons,
`polymer colloids have been widely used as model systems in investiga(cid:173)
`tions into electrokinetic and colloidal stability phenomena. However, it
`has been in the area of electrokinetics that the polymer colloids have
`failed to live up to much of the initial expectation. One particular
`disappointment has been the failure to find a convincing explanation for
`the behaviour of the ~-potential of polymer colloids as a function of
`electrolyte concentration, which has brought the ideality of the system
`into serious question.
`The aggregation of polymer colloids is a phenomenon which underlies
`many chemical, physical, and biological processes. Currently, it is be(cid:173)
`lieved that there are two limiting regimes of irreversible colloidal, or
`cluster aggregation (3). Diffusion-limited cluster aggregation (DLCA)
`occurs when every collision between diffusing clusters results in the
`formation of a bond. The rate of aggregation is then limited by the time
`it takes the clusters to diffuse towards one another. Reaction-limited
`cluster aggregation (RLCA) occurs when only a small fraction of colli(cid:173)
`sions between clusters results in the formation of a bond. Here, the
`aggregation rate is limited not by diffusion, but by the time it takes for
`the clusters to form a bond. The precise range of experimental conditions
`which result in DLCA and RLCA is a topic of current research [4-7).
`Little experimental work has been done to determine the form and
`time dependence of the cluster-size distributions tha t arise during
`colloidal aggregation [8]. Determining the detailed form of these size
`distributions is important since many of the physical properties of a
`colloidal suspension of a polymer matrix depend on this distribution.
`Moreover, it is essential to know the form of the size distribution in order
`to properly interpret static and dynamic light scattering measurements
`on colloidal aggregation. In this review, we describe measurements of
`cluster-size distribution and dynamic scaling which arise during salt(cid:173)
`induced aggregation of polymer colloids. The topic is advancing rapidly
`at the present time, synthetically with the preparation of new materials
`and physically with the development of new techniques for their char(cid:173)
`acterization. These recent advances will be reviewed and extensively
`discussed in this paper.
`
`2. Purification of polymer colloids
`
`Polymer colloids prepared by emulsion polymerization can have differ(cid:173)
`ent electrokinetic and stability properties according to the type of mono-
`
`

`

`R. Hidalgo-Alvarez et al. I Adv. Colloid Interface Sci. 67 (1996) 1-118
`
`5
`
`mers and procedures used during synthesis. However, the development
`of methods for the preparation of polymer latices containing monodis(cid:173)
`perse particles, particularly in the absence of surface active agents, has
`led to their widespread use in the testing of theories of colloidal phenom(cid:173)
`ena. After preparation, a cleaning procedure is required to remove salts,
`oxidation products, oligomeric materials and any remaining monomers
`from the latex. A worrying feature of the cleaning and characterization
`procedures is that different authors, using similar recipes for the latex
`preparation, have observed different surface groups. For example, Van(cid:173)
`derhoff [9-10) using mixed-bed ion-exchange resins to remove ionic
`impurities have concluded that only strong-acid groups, i.e., sulphate
`groups arising from the initiator fragments, are present on the latex
`surface; their conclusion was based on the single end point observed in
`conductometric titration. Other authors [11-12), particularly those
`using dialysis as a cleaning procedure, have detected the presence on
`the surface of weak-acid groupings in addition to the strong-acid group(cid:173)
`ings. In fact, weak-acid groups could be produced by the hydrolysis of
`surface sulphate groups to alcohol (hydroxyl) groups followed by oxida(cid:173)
`tion to carboxyl groups [11). Moreover, Lerche and Kretzschmar [13)
`have shown that the surface charge density of several latex samples
`depended on the cleaning method used, and that the ion-exchange and
`dialysis was not able to remove charged oligomeric material from the
`particle surfaces completely. Thus, a very key question has arisen as to
`which set of results is correct and which cleaning procedure should be
`used by preference when preparing clean latices for fundamental inves(cid:173)
`tigations. It has been found that serum replacement is an as reliable
`and easy method to clean latex suspensions [14). Nevertheless, the ion
`form of the latices cleaned by serum replacement is only achieved after
`ion-exchange with resins. A cycle of several centrifugation/redispersion
`might probably be the only method able to remove oligomer chains from
`the surface of polymer colloids [15). The different methods for preparing
`clean latices do provide polymer colloids with quite different electroki(cid:173)
`netic and stability properties. Cleaning of polymer colloids is of para(cid:173)
`mount importance for electrokinetic and stability studies. The removal
`of polymeric impurities is essential in order to have both control over
`the surface charge and of the supporting electrolyte concentration.
`Also, the deionization of latex suspensions plays a decisive role in the
`formation of fluid-like, crystalline, or amorphous interparticle structure
`[16J.
`
`

`

`6
`
`H. Hid.algo-Alvarl!2 et al. !Adu. Colloid lnte1face Sci. 67 (1996) 1- 118
`
`3. Surface c haracterization of polymer colloids
`
`Once clean polymer colloids have been prepared, it is necessary to
`determine their surface characteristics. The surface structure and charac(cid:173)
`teristics of polymer colloids are important for many reasons. They deter(cid:173)
`mine the stability of the colloidal dispersion, the adsorption characteristics
`of surfactants, latex film formation mechanisms, and the properties of
`these films obtained from latex. They also provide information on th e
`emulsion polymerization mechanisms, especially when structured beads
`are synthesized. The techniques suitable for surface analysis of polymer
`colloids are numerous. Conductometric titration of polymer colloids is
`considered a basic technique for surface-charge determination [17- 24].
`Other classical methods include soap titration (25-27], and contact
`angle measurements {28- 30]_ Moreover, surface topography of polymer
`beads is usually studied by electron microscopy (transmission electron
`microscopy) (30]_ The problems associated with the surface char ac(cid:173)
`terization of polymer colloids h ave been studied by several authors
`[17-18,20,31-36]. Labib and Robertson (321 have shown that condu c(cid:173)
`tometric titration of polymer beads is more difficult to interpr et than
`conductometric titr ation of free acids, and proposed a method to deter(cid:173)
`mine the stoichiometric end-points in an appropriate and r eliable man(cid:173)
`ner . Hlavacek et aL [37] have demonstrated that the variation in the
`composition of the liquid phase which occurs during acid-base titr ations
`of polymer colloid suspensions can be explained by a mechanism involv(cid:173)
`ing weak acid or base ion-exchange reactions coupled with surface
`ionization. Identification of the sites involved and their th ermodyna mic
`constants allows a good quantitative prediction of experimental results ,
`gives an explanation for the influence of ionic strength on the pH curves
`and an indication of the state of the solid surface. Also, Gilany [38]
`determin ed the surface charge density of polystyrene beads by using th e
`concentration and activity of a binary electrolyte added to the la tex
`dispersion. The distribution of ions was calculated by means of th e
`non-linearized Poisson-Boltzmann equation and the cell model. The
`effective charge oflatex beads was found to be smaller than the analyti(cid:173)
`cal charge. It was concluded that a small effective charge cannot be
`explained with specific binding of counter ions to the polymer colloids.
`The hydrophobic or hydrophilic character of the polymer surface may
`have a certain influence on the surface structure of the polymer colloids
`[30]_ The hydrophobic surface of polymer colloids plays a crucial role in th e
`ion distribution in the interfacial region. Contact angle is a measur ement
`
`

`

`R. Hidalgo-Aluarezet al. /Adu. Colloid Interface Sci. 67 (1996) 1-118
`
`7
`
`of the hydrophobicity of the polymer-solution interface, and has been
`used to obtain information on the surface structure of core-shell and
`block polymer colloids [28-30,39]. To determinate the water of hydra(cid:173)
`tion around charged polymer beads, Grygiel and Starzak [40] have
`studied the interfacial properties of carboxylated polymer beads using
`environment-sensitive laser excitation spectroscopy of the Eu3+ ion. This
`ion spectroscopy technique uses the changes in the electronic properties
`of the ion in different molecular environment to elucidate the structure
`and properties of those environments. Lifetime measurements show
`when binding to a highly charged surface (32.3 µC cm-2) that the ion
`loses about half its waters of hydration while energy transfer from Eu3+
`for these highly charged surfaces gives an ion separation (7.1 A) that is
`consistent with the known average separation of the surface sites (7.1
`A). For lesser charged beads (15.1 and 2.6 µC cm-2 , respectively), the
`energy transfer separation distance is smaller than the surface site-site
`separation indicating energy transfer between surface-bound and inter(cid:173)
`facial ions. For lesser charged beads an ion separation of about 9.6 and
`9. 7 A is found, indicating that bound ions retain most of their water of
`hydration. Using osmotic pressure measurements Rymden [35] has
`observed that the ion binding in aqueous polymer colloids depends on
`the surface charge density of carboxylated latex beads.
`
`4. Electrokinetic background
`
`Electrokinetic phenomena is a generic term applied to effects associ(cid:173)
`ated with the movement of ionic solutions near charged interfaces.
`Determination of the detailed structure of the electric double layer
`(e.d.l.) of polymer colloids is of primary importance in problems of
`stability and rheology of disperse systems, electrokinetic processes,
`filtration and electrofiltration, desalting of liquids on organic mem(cid:173)
`branes, etc. Calculating t;-potential of the polymer-solution interface is
`important when looking for an accurate microscopic explanation of
`electrokinetic phenomena.
`To describe the structure of the e.d.l., information is needed on three
`potentials: the surface potential ('1'0 ), the potential of the Stern layer
`('l's) and the diffuse potential ('I'd). In the absence of organic impurities
`and polyelectrolytes adsorbed on the latex surface the \I'd-potential can
`be equated to the potential in the electrokinetic slipping plane (~-poten­
`tial). In some cases, one can take '¥0 to be approximately equal to 'l's
`with indifferent electrolyte, and thus a detailed study of the structure
`
`

`

`8
`
`R. Hidalgo-Aluarez et al.I Adu. Colloid interface Sci. 67 (1996) 1- 118
`
`of the e.d.L only requires a knowledge of the '¥0 - and s-potentials.
`Extensive reviews (41-49) testify to the strong interest which has been
`shown in the electrokinetic phenomena during the past few decades.
`Typical electrokinetic phenomena used to characterize polymer col(cid:173)
`loids are:
`(1) Electrophoresis: where a uniform electric field is applied and the
`particle velocity is measured [17-20,49-107].
`(2) Streaming potential: where a liquid flux is allowed to pass through
`a porous medium and the resulting electric potential difference is
`measured [103,108-114).
`(3) Electro-osmosis: where an electric field is applied to a porous medium
`and the resulting volumetric flow of fluid is measured [54,115, 116].
`( 4) Diffusiophoresis: where a gradient of a solute in solution is applied
`and the migration of suspended colloid particles is measured [43,117-
`127]. Much of the early theoretical and experimental work on diffusio(cid:173)
`phoresis was on gaseous systems. Recent work, however, has focused on
`diffusiophoresis in liquid systems involving charged particles and elec(cid:173)
`trolytes in solution.
`(5) Dielectric dispersion: this technique involves the measurement of
`the dielectric response of a sol as a function of the frequency of an applied
`electric field. The complex dielectric constant [62,83,128-140] and/or
`electrical conductivity (141- 147] of a suspension are measured as a
`function of frequency. The presence of dispersed particles generally
`causes the conductivity of this dispersion to deviate from the conductiv(cid:173)
`ity of the equilibrium bulk electrolyte solution
`(6) Electro-acoustic phenomena: where alternating pressure fields are
`applied and the resulting electrical fields are measured [85,148-154].
`When an alternating voltage is applied t o a colloidal dispersion, the
`particles move back a nd forth at a velocity that depends on their size,
`s-potential and the frequency of the applied field. As they move, the
`particles generate sound waves. This effect was predicted by Debye [155]
`in 1933.
`(7) Electroviscous effects in colloidal suspensions and electrolyte flows
`through electrically capillaries under a pressure gradient. The presence
`of an e.d.l. exerts a pronounced effect on the flow behaviour of a fluid.
`These effects are grouped together under the name of electroviscous
`effects [156-162] .
`In all cases there is a relative motion between the charged surface and
`the fluid containing the diffuse double layer. There is a strong coupling
`between velocity, pressure, electric, and ion concentration fields.
`
`

`

`R. Hidalgo-Aluarez et al. I Adu. Colloid Interface Sci. 67 (1996) 1- 118
`
`9
`
`5. Calculation of zeta-potential
`
`5.1. Electrokinetic phenomena
`
`The literature pertaining to the study of electrokinetic properties of
`polymer colloids has a long and confusing history. We note specifically:
`(a) experimental electrokinetic data performed in different laboratories
`on ostensibly identical systems often conflict; (b)minor changes (clean(cid:173)
`ing procedure, surface charge, and particle size) may result in major
`differences in the measured electrokinetic data and (c) the ~ potentials
`obtained using the various electrokinetic processes on the same disper(cid:173)
`sions are quite different in values. These studies are difficult due to the
`complex interactions involved.
`
`5.1.1. Electrophoretic mobility
`Recent development of laser-based instrumentation for electropho(cid:173)
`retic mobility experiments has made it possible to determine the zeta
`potential (<'.;;) of particles suspended in liquid media for systems that were
`difficult or impossible to study using classical techniques. The new
`instruments use electrophoretic light scattering (ELS) to measure electro(cid:173)
`phoretic mobilities. ELS allows direct velocity measurements for particles
`moving in an applied electric field by analyzing the Doppler shift oflaser
`light scattered by the moving particles (65]. Recently Kontush et al. [163]
`designed a setup for studying nonlinear electrophoresis.
`It is found that electrophoretic mobility curves pass through a mini(cid:173)
`mum (anionic latex beads) or a maximum (cationic latex beads) as a
`function of increasing ionic strength. From a theoretical point of view,
`calculation of the ?;;-potential from electrophoretic mobility data encoun(cid:173)
`ters a number of difficulties as a result of the polarization of e.d.l. The
`term "polarization" implies that the double layer around the particles is
`r egarded as being distorted from its equilibrium shape by the motion of
`the particle. In general, for Ka ~ 30 it is necessary to account for e.d.l.
`pola rization when calculating~ (50,81,83,107]. There a re several theo(cid:173)
`retical treatments to convert electrophoretic mobility (µe) data into
`/;-potential under polarization conditions. Monodisperse spherical poly(cid:173)
`mer latices have proved to be a very useful model system for testing the
`most recent theoretical approaches [50-51,68,70,107}. As most of the
`theories deal with spherically sha ped particles of identical size, the
`introduction ofmonodisperse latices appeared to offer excellent chances
`for experimental verification of these theories. However, growing evidence
`
`

`

`10
`
`R. Hidalgo-Aluarez et al. I Adu. Colloid interface Sci. 67 (1996) 1-118
`
`of anomalous behaviour of the ~-potential as a function of 1:1 electrolyte
`concentration has appeared in the literature (17-19,21,50,68-71, 79,83-
`91] . The standard electrophoretic theories used for the conversion of
`mobility into ~-potential give rise to a maximum in ~potential as well.
`This behaviour contradicts the Gouy-Chapman model which predicts a
`continuous decrease in potential. Various explanations for this maxi(cid:173)
`mum have been proposed [17,18,22,28,50,68,70,71,74,83,97,107), and
`some authors [112] have even pointed out that a maximum rp.obility
`value does not necessarily imply a maximum in ~-potential, indicating
`that the conversion of mobility into (,-potential of polystyrene micro(cid:173)
`spheres/electrolyte solution interface should be done by means of a
`theoretical approach which takes into account all possible mechanisms
`of double layer polarization. Other authors, on the contrary, have
`pointed out that the appearance of a minimum (or maximum) in the
`s -potential is unimportant since the e.d.l. around polymer colloids, even
`with 1: 1 electrolytes, cannot be explained on the basis of the Gouy(cid:173)
`Chapman model. They proposed the use of a dynamic Stern layer [55]
`or an electric triple layer model [152) instead.
`Overbeek (104) and Booth (105] were the first to incorporate polari(cid:173)
`zation of the e.d.l. into the theoretical treatment. They assumed that
`the transfer and charge redistribution processes only involved the
`mobile part of the e.d.l. Also, O'Brien and White (57], starting with the
`same set of equations as Wiersema [106), have more recently published
`a theoretical approach to electrophoresis, which takes into account any
`combination of ions in solution with the possibility of very high (,-poten(cid:173)
`tials (up to 250 mV), far enough from the values to be expected in most
`experimental conditions. In simple terms, the theory of O'Brien and
`White predicts the measured electrophoretic mobility of a colloidal
`particle in an applied electric field to be the sum of three forces, viz: (1)
`an electric force propelling the particle, due to the charged nature of the
`particle, (2) a drag force due to hydrodynamic drag, and (3) a relaxation
`force due to an electric field induced in the opposite direction to the
`applied field as a res ult of the induced polarization within the diffuse
`layer of ions surrounding the particle.
`It predicts the electrical force propelling the particle to be propor(cid:173)
`
`tional to s and the retarding forces to be proportional to ~2. A maximum
`
`in the conversion of mobility to zeta potential is thus predicted for
`particles size and ionic strength conditions such that 5 < Ka ~ 100.
`The most striking features of O'Brien and White's theoretical treat(cid:173)
`ment results are:
`
`

`

`R. Hidalgo-Aluarez et al. / Adu. Colloid Interface Sci. 67 (1996) 1-118
`
`11
`
`(a) For all values of Ka 2': 3, the mobility function has a maximum
`which becomes more pronounced at high Ka values.
`(b) The maximum occurs at s = 5-7 (i.e. s ,,,, 125-175 mV).
`(c) Their computer solution is much more rapid than the earlier
`theoretical treatments used in the conversion of mobility into s-potential
`(up to 250 mV).
`A simplified version and analytical form for the mobility equation,
`accurate to order llKa and valid for our purposes for Ka > 10, can be
`expressed as [45):
`
`( zs)
`3m exp -2
`Ka
`2+
`1 + -
`
`z2
`
`(1)
`
`where mis the dimensionless ion drag coefficient given by
`
`m =
`
`2E0t. Ni kT
`3T] zi\0
`All the above cited theoretical approaches to convert mobility into
`~-potential assume the absence of ionic conduction inside the shear
`plane. In an attempt to account for this phenomenon theoreticaUy,
`Semenikhin and Dukhin [61) developed an equation incorporating both
`the dimensionless s-potential, and the dimensionless diffused 'I'd-poten(cid:173)
`tial. The mobility µe for a spherical particle with a thin e.d.l. (Ka > 25)
`in a 1: 1 electrolyte is then a function of:
`µ e = f(s,Ka,m,g1,g2)
`where
`
`(2)
`
`(3)
`
`p is the ratio of the counterion diffusion coefficient near the particle to
`its value in the bulk solution and m is t he dimensionless ionic drag
`coefficient.
`
`(4)
`
`

`

`12
`
`R. Hidalgo-Alua rez et al. I Ad u. Colloid Interface Sci. 67 ( 1996) 1-118
`-
`-
`If s and 'I'd are larger than 2, Eq. ( 3) can be simplified as:
`
`- -~ -[l +Rel ( 4 In ct i;i-4 lj
`
`µe - 2 S ·
`
`1 + 2 Rel
`
`where
`
`exp(\iii2) + 3m exp (S/2)
`A5
`Rel = - = ---"----- - -
`A.a
`Ka
`
`(S)
`
`(6)
`
`Dukhin [49] introduced the dimensionless relaxation parameter Rel
`as a measure of the effect of surface conductance on electrokinetic
`phenomena. It is noted that Rel can be used with two meanings, viz. in
`indicating the degree of e.d.l. polarization (non-equilibrium degree) for
`curved surfaces and in indicating the relative contribution of surface
`conductance to the total conductance in non-polarized systems (equilib(cid:173)
`rium states). An increase in the surface conductance and/or decrease in
`the radius results in an increase in Rel and thus in the polarization field
`in the direction of the induced electromigration current. We can distin(cid:173)
`guish two different mechanisms of electrical conduction: surface conduc(cid:173)
`tion associated with tangential charge transfer through the mobile
`portion of the e.d.l. (normal conduction taken into account in the
`Overbeek-Booth-Wiersema theory); and anomalous surface conduc(cid:173)
`tion, which is related to the tangential charge transfer between the
`slipping plane and the particle surface. The Semenikhin and Dukhin
`theory [61] considers only a particular case of anomalous conduction
`associated with the presence of a boundary layer. Figure 1 shows the
`variation of adimensional electrophoretic mobility as a function of
`adimensional s-potential for different electrokinetic radius values ob(cid:173)
`tained by Eq. (5) for cationic latex particles. The differences with the
`O'Brien and White theory are quite clear.
`The induced tangential ionic flow near the surface has to be provided
`for by radial ionic migration, diffusion and convection from beyond the
`e.d.l. where co- and counterion concentration can differ considerably
`from those near the surface. The concentration polarization results in
`an angll!lar dependence of the ion concentration and the potential.
`Semenikhin and Dukhin [73] derived analytical formulae that express
`these dependencies for the cross section of the thin diffuse e.d.l. of a
`spherical particle. Contrary to the mathematical procedures employed
`
`

`

`R . Hidalgo-Aluarez et al. I Adu. Colloid Interface Sci. 67 (1996) 1-118
`
`13
`
`~obility (dimensionless)
`
`7~~~~~~~~~~~~~~~~~~~~~~~~-,
`
`6
`
`k" a = 100
`
`k " a= SO
`
`Zeta Potential (dimensionless)
`
`Fig. 1. Variation of dimensionless electrophoretic mobility as a function of dimensionless
`/;-potential for different electrokinetic radius obtained by Eq. (5).
`
`by Overbeek [104], Booth [105) and Wiersema [106], the main advantage
`of Dukhin and Semenikhin approach is the possibility of taking into
`account the effect of anomalous conduction on polarization.
`The Semenikhin-Dukhin theory assumes that there is no contribu(cid:173)
`tion to the electrical conduction by any Stern layer ions, and that the
`PBE applies up to the outer Helmholtz plane. The diffuse layer ion s
`between the shear plane and the outer Helmholtz plane do conduct a
`current and this anomalous surface conductance (inside the shear
`plane) dramatically reduces the mobility for a given value of ~-potential.
`Consequently, for a description of electrophoresis under condition Rel >
`1, not only the effect of diffusion flows must be taken into consideration,
`but also the change in the polarization potential across the thin diffuse
`layer [49) .
`Comparative studies using different th