advances in colloid and interface science and stability of nano-colloids: history, theory and...

Advances in Colloid and Interface Science 211 (2014) 77–92

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Advances in Colloid and Interface Science

j ourna l homepage: www.e lsev ie r .com/ locate /c i s

Electrophoresis and stability of nano-colloids: History, theory andexperimental examples

C. Felix a, A. Yaroshchuk b, S. Pasupathi a, B.G. Pollet a, M.P. Bondarenko c, V.I. Kovalchuk c,⁎, E.K. Zholkovskiy c

a South African Institute for Advanced Materials Chemistry (SAIAMC), University of the Western Cape, Modderdam Road, Bellville 7535, Cape Town, South Africab ICREA and Department d'Enginyeria Química (EQ) Universitat Politècnica de Catalunya Av. Diagonal, 647, Edifici H, 4a planta, 08028, Barcelona, Spainc Institute of Biocolloid Chemistry of Ukrainian Academy of Sciences, Vernadskogo, 42, 03142 Kiev, Ukraine

⁎ Corresponding author. Tel./fax: +380 44 424 8078.E-mail address: [email protected] (V.I. Kovalchuk)

http://dx.doi.org/10.1016/j.cis.2014.06.0050001-8686/© 2014 Elsevier B.V. All rights reserved.

a b s t r a c t
a r t i c l e i n f o
Available online 20 June 2014

Keywords:ElectrophoresisCoagulation dynamicsNano-suspensionsStandard Electrokinetic ModelDLVO theory

The paper contains an extended historical overview of research activities focused on determining interfacialpotential and charge of dispersed particles fromelectrophoretic and coagulation dynamicmeasurements. Particularattention is paid to nano-suspensions for which application of Standard Electrokinetic Model (SEM) to analysis ofexperimental data encounters difficulties, especially, when the solutions contain more than two ions, the particlecharge depends on the solution composition and zeta-potentials are high. Detailed statements of Standard Electro-kinetic and DLVO Models are given in the forms that are capable of addressing electrophoresis and interaction ofparticles for arbitrary ratios of the particle to Debye radius, interfacial potentials and electrolyte compositions.The experimental part of the study consists of two groups of measurements conducted for Pt/C nano-suspensions, namely, the electrophoretic and coagulation dynamic studies, with various electrolyte compositions.The obtained experimental data are processed by using numerical algorithms based on the formulated modelsfor obtaining interfacial potential and charge. While analyzing the dependencies of interfacial potential and chargeon the electrolyte compositions, conclusions are made regarding the mechanisms of charge formation. It isestablished that the behavior of system stability is in a qualitative agreement with the results computed from theelectrophoretic data. The verification of quantitative applicability of the employedmodels is conducted by calcu-lating the Hamaker constant from experimental data. It is proposed how to explain the observed variations ofpredicted Hamaker constant and its unusually high value.

© 2014 Elsevier B.V. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782. Standard electrokinetic model and coagulation theory. Historical overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

2.1. Relationships between interfacial potential and electrophoretic velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.2. Interactions of particles and dynamics of coagulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3. Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.1. Extracting interfacial potential from electrophoretic measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.1.1. Standard Electrokinetic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.1.2. Scalarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.1.3. Numerical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.2. Coagulation dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.2.1. Electrostatic repulsion: general problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.2.2. Numerical computation of electric field distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.2.3. Obtaining Hamaker constant from electrophoretic and coagulation dynamic data . . . . . . . . . . . . . . . . . . . . . . . 86

4. Experimental example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.1. Pt/C nano-catalytic dispersions and their practical importance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.2. Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.3. Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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78 C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92

5. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 875.1. Correlation between electrophoretic and stability data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.2. Influence of electrolyte composition on surface potential and charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.3. Applicability of Standard Electrokinetic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

6. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

Nomenclature

Latin lettersa radius of particle;Ck concentration of the kth ion in the solution bulk;ck local concentration of the kth ion;Dk the kth ion diffusion coefficient;E external electric field strength;en unity vector of Cartesian coordinate system;er, eθ, eφ unit vectors of spherical coordinate system;eβ and eν unit vectors of bispherical coordinate system;F Faraday constant;Gel(h) electrostatic free energy;GW(h) van der Waals free energy;H Hamaker constant;h minimum distance between particle surfaces;I ionic strength;I unit tensor;n outward normal vector to a closed surface;p local pressure;p∞ pressure in the solution bulk;q surface charge density;R gas constant;r vector coordinate;rBA vectorwhose origin and end coincidewith the centers of

particles B and A, respectively;r polar radius in spherical coordinate system;T absolute temperature;u local velocity of liquid;Ueph electrophoretic velocity;Udph diffusiophoretic velocity;X(h) interaction force;xn Cartesian coordinate;Y(r) radial part of the streaming function;Zk electric charge of the kth ion in Faraday units;z axial coordinate;

Greek letters(β, ν) bispherical coordinates;ξk ¼ C∞

k zk=∑n

z2nC∞n dimensionless coefficient;

η viscosity;θ spherical polar coordinate;ε dielectric permittivity;κ Debye parameter;μk the kth ion electrochemical potential;Μk(r) the function describing radial dependency of perturba-

tion of kth ion electrochemical potential;Π effective pressure;σ stress tensor;τ coagulation time;τSm Smoluchowski time scale parameter;Ψ equilibrium electric potential;ψ = ΨF/RT normalized equilibrium electric potential;ζ electric potential at the interface in equilibrium state

(zeta potential);ζexp measured zeta potential;

1. Introduction

Nano-particle systems have become one of the most importantobjects in modern science and technology because of highly specific andversatile properties of nano-particles, which are determined by theirsmall, close to molecular and atomic, size. Moreover, their propertiescan be precisely tuned and functionalized by changing their size and com-position and modifying their surface. In the recent less than two decadesinnumerable amount of various types of nano-particles were obtainedhaving specific electronic, magnetic, optic, catalytic, biological and otherproperties [1–7]. This opens many opportunities for their use in numer-ous important applications in technology and biomedicine. The studiesin this area are focused not only on the synthesis and tuning of nano-particles properties but also on their interactions and behavior in variousmedia.

Liquid systems with dispersed nano-particles are particularly im-portant. Nano-particle dispersions are widely used in technologicalprocesses, e.g. for obtaining substrates covered by nano-particles ornano-porous media. They are also very important for various bio-medical applications. For most applications it is necessary to havestable nano-dispersions, which do not change their properties withtime due to particle aggregation or chemical processes. The problemof aggregative stability of solid-in-liquid dispersions iswidely studied incolloid science where it was shown that the stability is controlled bysurface forces acting between the particles [8,9]. In particular, attractive(e.g. van der Waals) interactions facilitate aggregation of particles,whereas repulsive (e.g. electrostatic or steric) forces tend to preventthem from aggregation. Quantitative description of particle interactionsand aggregation in liquids is usually based on the approach, pioneeredby Derjaguin, Landau, Verwey and Overbeek (DLFO theory) [10,11].After their foundational studies many efforts were devoted to thederivation of equations describing the dependence of attractive andrepulsive forces on the distance between the particles under variousconditions. However, the application of these equations to the case ofnano-particle systems is not that straightforward. This is a consequenceof commensurability of the particles size and the characteristic dis-tances where the surface forces are acting, which makes inapplicablesome common approximations.

The most important forces, stabilizing the dispersions, are repulsiveelectrostatic forces that arise because the particle surfaces are usuallycharged. The particle charge depends on the ionic composition of solutionsurrounding them, and near the so-called iso-electric point the dispersionloses its stability. Therefore, obtaining information on the particle chargeunder various conditions is very important for understanding andcontrolling the dispersion stability. A common approach to the relevantinformation about the particle charge is via electrokinetic measurements,in particular via the measurements of their electrophoretic mobility insolutions. However, the interpretation of such measurements for nano-sized particles is especially complicated.

Electrophoretic transport is driven by the electric forces acting onthe interfacial charges. The role of interfacial charges in the suspensionaggregative stability amounts to the electrostatic repulsion between thesimilarly charged particles, which decreases the frequency of “successful”collisions of particles participating in Brownian motion and thus the rateof coagulation. Thus, both electrophoretic transport and aggregative sta-bility of suspensions are controlled by the interfacial electric charge.

79C. Felix et al. / Advances in Colloid and Interface Science 211 (2014) 77–92

This charge arises at liquid/solid interfaces in solutions because of disso-ciation of interfacial acid or basic groups or preferential adsorption ofions.

Both electrophoresis and electrostatic interactions are essentiallycontrolled by the properties of screening layers that surround chargedparticles in electrolyte solutions. Having opposite charge signs, theinterfacial and screening (diffuse) layer charges form the interfacialElectric Double Layer (EDL). Due to the thermal motion, the screeninglayer has a non-zero thickness that can be estimated as the inversevalue, κ−1, of the Debye parameter, κ, given as [12]

κ ¼ F

ffiffiffiffiffiffiffiffi2IεRT

r: ð1Þ

In Eq. (1), F is the Faraday constant; R is the gas constant; T is the ab-solute temperature; ε is the electrolyte solution dielectric permittivity;and I is the electrolyte ionic strength

I ¼ 12

Xk

Ckz2k ð2Þ

where zk and Ck are, respectively, the kth ion valence and bulk (far awayfrom the interface) concentration.

As is clear from Eq. (1), the EDL thickness decreases with increasingionic strength, I, and thus can be regulated by adding electrolyte to thesystem. The latter is important since the thickness of diffuse part ofEDL strongly affects both electrophoresis and electrostatic interactions.

Since the thickness of EDL and the interfacial charge depend on theelectrolyte composition one can control electrophoretic transport andaggregative stability by varying it. At the same time, parallel investiga-tion of electrophoresis and coagulation as functions of solution compo-sitions and establishing correlation between them can yield importantinformation for optimizing processes where particles simultaneouslyexperience electrophoresis and coagulation (for example, electropho-retic deposition, see below). Such integrated approaches have beenemployed for obtaining information about the interfacial properties ina number of studies [13–23].

The aim of the present study is to critically evaluate the existingapproaches to the investigations of electrophoresis and coagulationwith a particular emphasis on their applicability to the case of nano-particle systems and the problems arising in this case. Based on thestatements of Standard Electrokinetic and DLVO Models we propose astrategy that allows addressing electrophoresis and interaction of parti-cles for arbitrary ratios of particles sizes to corresponding Debye radii.As a practical example, this approach is illustrated by the applicationof the strategy to the analysis of electrical interfacial properties of Pt/Cnano-particle suspension. Catalytic Pt/C nano-particles are used forobtaining Membrane Electrode Assemblies (MEA) in Proton ExchangeMembrane Fuel Cells (PEMFC) [24–27].

In the next section we give a historical overview of the main state-ments of Standard Electrokinetic Model (SEM) and coagulation theory.In Section 3.1 we present the fundamentals of SEM and a numericalscheme of extracting the interfacial potential and charge from the elec-trophoreticmobilitymeasured for various electrolyte compositions. Thenumerical scheme of addressing coagulation dynamics is discussed inSection 3.2. In this section, we also propose a method of verifying theapplicability of SEM. The proposed method consists in obtaining theHamaker constant, H, from the value of the Fuchs factor, W, which isdetermined from the coagulation dynamics data. A short informationon Pt/C nano-catalytic dispersions and the details of experiments arepresented in Section 4. In Sections 5.1 and 5.2, we analyze the mecha-nisms of charge formation. Final discussion about the applicability ofSEM to our system is given in Section 5.3.

2. Standard electrokinetic model and coagulation theory.Historical overview

2.1. Relationships between interfacial potential and electrophoretic velocity

Themostwidely usedmodel for describing Electrokinetic Phenomenais referred to as Standard Electrokinetic Model (SEM) [28,29]. Accordingto thismodel, one considers the electrolyte solution surrounding colloidalparticle as a continuous medium, which is described by applying ap-proaches of electro- and hydrodynamics of continuous media combinedwith the equations of physical macro-kinetics, which are mathematicalimplementations of conservation laws for individual ionic species. Thefluxes of such species are expressed as superposition of convective, diffu-sion and electro-migration fluxes. Macroscopic equations are assumed tobe applicable up to the solid/liquid interface, which is considered amath-ematical surface.

For these two limiting cases, κa → 0 and κa → ∞ (a is the particleradius), the complex balance of forces acting on the charges in theparticle-electrolyte solution system was considered in classical papersof, respectively, Debye and Hückel [12] and Smoluchowski [30,31].These studies yielded simple results for electrophoretic velocity, Ueph,in external electric field E

Ueph ¼ 2εζ3η

E κa→ 0 að Þ

Ueph ¼ εζηE κa→∞ bð Þ

ð3Þ

where ζ is the interfacial electric potential, which is defined in thethermodynamic equilibrium state, with reference to the solution bulk; ηis the viscosity of electrolyte solution.

However, for intermediate values of κa, the relationship betweenelectrophoretic velocity and ζ-potential is much more complex. Duringthe 20th century, a number of theoretical approaches have been devel-oped for addressing electrophoresis in various situations where the con-ditions of Eq. (3) are not satisfied [32–47]. For obtaining the surfacepotential, one should use these theoretical results for extracting thevalue of ζ from the experimental data.

Analysis of electrophoresis within the frameworks of SEM wasconducted by Henry [32] who determined the linear term in the ex-pansion of electrophoretic velocity in the powers of ζ. The obtainedrelationship enables one to address electrophoresis for low zeta poten-tials, eζ ¼ Fζ=RT≪1. In terms of apparent experimental quantity ζexp,corresponding to a given ζ, the Henry result can be represented as

ζexp ¼ ζ 1−eκa 5E7 κað Þ−2E5 κað Þ½ �� ð4Þ

where E5(x) and E7(x) are the integral exponents of the fifth andseventh order, respectively. Here ζexp is an apparent quantity, obtainedfrom the measured electrophoretic velocity, Ueph, by using theSmoluchowski relationship, Eq. (3b). Note that, at given ζ, the electro-phoretic velocity and thus the apparent value of zeta potential, ζexp,depend on the parameter κa which, according to Eq. (1), depends onthe ionic strength, I. Thus, by using Eqs. (1) and (4), one can extract thesurface potential ζ from the experimentally measured quantity ζexp forgiven particle radius, a, and ionic strength of solution, I. Importantly,the electrolyte composition enters this relationship only via I (seeEq. (2)). This simple scheme is applicable for eζ ¼ Fζ=RT≪1, only.

The case of higher zeta potentialswas considered by various authors.In particular, Overbeek [33,34] and Booth [35,36] determined severalleading terms in the expansion of electrophoretic velocity in the powersof normalized zeta potential eζ ¼ Fζ=RT. In such an expansion, Overbeekobtained the first, second and third order terms for a binary electrolytesolution of general type. Booth considered symmetric electrolyte forwhich he additionally obtained the fourth order term which turnedout to be non-zero for the case of ions of different mobilities.

Fig. 1. Illustration to the explanation of non-linear dependency of electrophoretic velocityon zeta potential (see the text for more detail).


Foreζ≪1, the predictions of Henry [32] and Overbeek-Booth [33–36]theories coincide. At higher ζ, the Overbeek–Booth results reveal aslower increase of electrophoretic velocity with increasing ζ than thatdefined by Eqs. (3) and (4). Moreover, at sufficiently large κa, theelectrophoretic mobility depends on ζwith a maximum that is reachedat eζN3. Numerical results of computations based on SEM have beenreported by Wiersema et al. [37], and O'Brien and White [38]. Theseresults confirmed the behavior predicted by Booth and Overbeek, qual-itatively, but yielded some important quantitative corrections. In partic-ular, the authors of refs. [37,38] demonstrated that the expansionsdeduced in refs. [33–36] underestimate both the electrophoretic velocityand the value of ζ corresponding to themaximumwhich is observed at κaN 3 and eζ≅5–7 for 1:1 electrolyte.

Approximate analytical results describing electrophoresis forarbitrary zeta potentialswere obtained for the case of κa N 1 and symmet-ric electrolyte solution in the pioneering study of Dukhin and Semenikhin[39,40], and, later, in the papers of O'Brien and Hunter [41,42] andOhshima et al. [43]. Qualitatively, all the analytical expressions fromrefs. [39–43] demonstrate similar behavior: at sufficiently high κa, theelectrophoretic mobility as a function of eζ passes through a maximumand, for the limiting case when, simultaneously, κa → ∞ and eζ→∞,approaches an asymptotic value. In terms of eζexp ¼ Fζexp=RT such acommon limit is given by eζexp→2 ln 2ð Þ. According to the comparativeanalysis presented for κa = 50 in ref. [43], the analytical predictionsof refs. [39–43] nearly coincide for eζb4. At higher potentials, the predic-tion by Ohshima et al. [43] reveals the highest accuracy, with deviationfrom the numerical result by less than 1%.

To understand better the above discussed behavior, let us consider adisperse systemwith κa≫ 1 and a simple case of symmetric electrolytesolution with equal diffusion coefficients of ions. In such a system,applying an external electric field produces a steady state distributionof electrolyte concentration in the electro-neutral zone (i.e. outsideEDL) around the particle. Importantly, in this concentration field,always, the higher concentration zone is adjacent to the rear partwhereas the lower concentration zone is adjacent to the front part(according to the direction of particle movement). There are tworeasons for such a structure of concentration distribution:

(i) The counterions and coions are transported by the electric fieldthrough the EDL in, respectively, higher and lower proportionsthan through the electrolyte solution bulk;

(ii) The coions and counterions are transported by the electric field,respectively, in the same and opposite directions relative to theelectrophoresis of particle.

Consequently, the coions delivered by electric field to the rear zoneof moving particle are not withdrawn from there in equivalent amountsthrough the EDL region. At the same time, the counterions beingtransported against the particle motion are delivered in excess amountsthrough the EDL to the rear zone, as well. Hence, both the co- andcounter-ion concentrations (i.e., the electrolyte concentration) becomehigher than in the solution bulk, δc N 0. Similarly, coions and counterionsare withdrawn from the front zone toward the solution bulk and rearzone, respectively. Accordingly, the electrolyte concentration in thefront part becomes lower than in the solution bulk, δc b 0 (Fig. 1).

Under the influence of external electrolyte concentration gradient,a solid particle is involved into motion which is referred to asdiffusiophoresis [40,48–51]. The rate of diffusiophoresis,Udph, is propor-tional to the imposed concentration gradient anddirected toward higherelectrolyte concentration. Hence, while applying electric field, addition-ally to pure electrophoresis, the particle is driven by diffusiophoresis inthe concentration field produced by the applied electric field. Since theproduced concentration gradient is proportional to applied electricfield, the diffusiophoresis of this type is also proportional to the electricfield. Being directed toward the larger concentration, which alwaysoccurs close to rear part of the moving particle, such a field-induceddiffusiophoresis is always directed oppositely to the electrophoretic

transport thereby decreasing the total electrophoretic velocity com-pared to the purely electrophoretic motion. At sufficiently high zetapotentials diffusiophoresis results in decreasing the overall electro-phoretic velocity with increasing ζ.

The asymptotic case κa → ∞ and eζ→∞, when the electrophoreticvelocity approaches a limiting value corresponding to eζexp→2 ln 2ð Þ,corresponds to the situation when the particle is surrounded by avanishingly thin EDL that is infinitely permeable to the counterionsbut absolutely impermeable to the coions. In such a case the electro-chemical potential of counterions at the external surface of EDL is con-stant. By taking into account the approach developed in refs. [43–47],electrophoresis can be considered a superposition of particle drifts dueto the gradients of electrochemical potentials of counterions and coions.As the electrochemical potential of counterions is constant at thesurface, the particle is driven by the gradient of coion electrochemicalpotential, alone. For κa→∞, the coion electrochemical potential gradientgives rise to the particle velocity equal to that given by the Smoluchowskiformula for the normalized zeta potential whose magnitude is equal to2 ln(2).

By using the theoretical approaches of refs. [33–47], one can extractfrom electrophoretic mobility the value of ζ for an arbitrary binary elec-trolyte solution of known concentration. As the zeta-potential is the in-terfacial electric potential in the thermodynamic equilibrium state, onecan use the equilibriumdistribution of electric field and the electrostaticboundary condition interrelating the interfacial charge density and thenormal electric displacement for determining the former one. Conse-quently, a set of zeta potentials obtained from electrophoretic measure-ments in solutionswith different electrolyte compositions enable one toobtain information on the isotherm of adsorption, which defines thevalue of interfacial charge density.

In the caseswhen the electric charge is controlled by the adsorption ofH+ and OH− ions (or dissociation of interfacial basic or acidic groups),one can study adsorption isotherm by using the above theoretical predic-tions for binary electrolyte solution [33–47] only within sufficientlynarrow ranges of pH values where one can neglect the presence ofH+ and OH− ions in the transport equations. For addressing the systemwithin a broad pH range one should consider at least ternary ionicsystem. Such an analysis can be carried out numerically on the basisof formalism developed in refs. [33–47].

For some systems, using the SEMmodel yields a perfect description.A recent careful study [52] gives an example of that.While assuming thesurface charge density to be constant, the author demonstrates that themaximum of the dependency of electrophoretic velocity on the zeta po-tentialmanifests itself as aminimumon the curve displaying the depen-dency of electrophoretic velocity on salt concentration. This conclusionis supported by the perfect fitting of experimental data performed byusing the analytical result reported in [43]. However, in many cases


for properly addressing experimental data SEM requires modifications[18,28,40].

2.2. Interactions of particles and dynamics of coagulation

According to the classical Smoluchowski theory [53], the dynamicsof coagulation process, which results in changes of size distribution ofparticles, in particular, can be characterized by a time scale-parameter,τ, given by

τ ¼ τSmW ð5Þ

where τSm=3η/4kBTn (kB≈ 1.4 ⋅ 10−23J/K is the Boltzmann constant;n is the initial concentration of particles) is the time-scale parametercorresponding to the so-called rapid or Smoluchowski coagulation,which takes placewhen 100% of particle collisions lead to the formationof doublets. In Eq. (5), the Fuchs factor,W, [54] describes the increase inτ when, due to the repulsive forces between the particles, efficiency ofparticle collisions becomes smaller than 100%

W ¼Z∞0

exp G hð Þ=kBT½ �1þ hð Þ2 dh: ð6Þ

In Eq. (6), h= (rAB − 2a)/2a, where rBA is the distance between theparticle centers (Fig. 2); G(h) is the free energy of a system of twointeracting particles defined with reference to the state where theparticles are separated by an infinitely large distance. The functionG(h) can be obtained by calculating the mechanical work performedby interaction forces while particles are displaced from the state whenthe distance between their centers is rAB = 2a(h + 1) to infinitelylarge distance [8,55,56].

We consider interactions due to the electrostatic and van der Waalsforceswhose contributions, Gel(h) andGW(h), to G(h) are assumed to beadditive

G hð Þ ¼ Gel hð Þ þ GW hð Þ: ð7Þ

Study of interaction forces via coagulation dynamics consists inmeasuring the time scale parameter τ and determining from it theFuchs factor,W, which, according to Eq. (6), is defined by the interactionfree energy as a function of distance separating the particles, G(h). Byusingmodels for obtaining eachof the contributions into the interactionfree energy, Gel(h) and GW(h), one can determine the unknown param-eters of these models. These parameters should be obtained by fittingthe experimental dependencies of the measured parameter, W, on theelectrolyte composition. The fitting is conducted with the help of theo-retical dependencies that are deduced by substituting the predictedG(h) into Eq. (6).

Fig. 2. Two interacting spherical particles (see the text for more detail).

Obtaining the electrostatic contribution, Gel(h), is based on the theoryindependently developed by Derjaguin and Landau [10] and Verwey andOverbeek [11]. According to their approach, the force acting on either ofthe particles is determined via integration of the Maxwell stress tensorover arbitrary closed surfaces enveloping the respective particle. The dis-tributions of electric field and pressure, which are required for obtainingthe Maxwell tensor, are determined by considering thermodynamic andmechanic equilibrium in the system within the frameworks of Poisson–Boltzmann (P–B) equation, which is subject to the electrostatic boundaryconditions at the particle surfaces, and the hydrostatic momentumbalance conditions (also referred to as stress balance). The obtainedforce is a function of the distance between the particles and is employedfor computing the minimum work which is expended while displacingeither of the particles far away from another. The latter work yields therequired function, Gel(h).

An equivalent method of obtaining Gel(h) consists in the calculationof interface charge of particles at imposed surface potential by using thesolution of the P–B equation. Next, the charge obtained as a function ofsurface potential should be integrated over the potential from the surfacepotential corresponding to a single particle to the potential, which occursfor a given distance between the particles [57].

Several approximate theoretical approaches have been used whileimplementing the aforementioned schemes analytically. According tothe Derjaguin approximation [55,58,59], each of the particle surfaces isrepresented as a set of quasi flat segments whose interaction is de-scribed by making use of the results obtained for two parallel infiniteplanes. Ultimately, the contributions from separate couples of thesegments are added up. The Derjaguin approximation is valid for thecase of κa ≫ 1.

Another approximation, which is valid for arbitrary κa, is based onthe linearization of electric field distribution in terms of normalized sur-face potential [60]. Consequently, these results are valid for sufficientlylow surface potentials. One should also mention an approach based onthe assumption that the electric field in the system of two particleswith overlapped EDLs is a superposition of fields within the EDL of eithersingle particle [61]. The latter approach is valid when, simultaneously,κa b 1 and κh N 1.

All the existing analytical results are inapplicable to the systemof in-terest of this study because they are unable to address the case of κa ≅ 1and high surface potentials. Thus, interpretation of our experimentaldata requires numerical approaches that have been employed for ad-dressing electrostatic interactions between two spheres [62–66].

There are somedifficulties in addressing the contribution into the in-teraction free energy due to the van der Waals forces, GW(h). Rigoroustheoretical results for the interaction of two spheres in electrolyte solu-tion still have not been obtained. Therefore, the most popular approachdescribed in the literature is based on the superposition approximation.In this approximation, each of two interacting bodies is represented as aset of infinitely small elements. Consequently, the interaction energy iscomputed by adding up the energies of interaction between each of theelement pairs. Such a calculation is based on the assumptions that thecontributions of interaction of various pairs are additive. For the systemof two spheres, the above calculation scheme yields the classical resultof Hamaker [67]

GW ¼ −H6�

4 � 2þ h=að Þ2−2h i

2þ h=að Þ2 � 2þ h=að Þ2−4� �þ ln 1− 4

2þ h=að Þ2" #8<:

9=; ð8Þ

where H is the Hamaker constant, which depends on the materials ofboth the bodies and the surrounding medium. The negative sign ofenergy in the right hand side of Eq. (8) corresponds to the attraction.The result given by Eq. (8) does not take into account the complex coop-erative interactionswithin each of the bodies and the retardation effect,which is a result of finite length of electromagnetic waves [8,55,56].

image of Fig.�2


Studies of surface forces by means of measuring the Fuchs factor,W,have a long history that has been started by the work of Reerlink andOverbeek [68], who determined both the surface potential and theHamaker constant by analyzing thedependency of experimentallymea-suredW on the concentration of symmetrical electrolyte. The analyzedsystem and the theoretical relationships employed for the interpreta-tion are valid in the case of thin EDL. In the numerous studies, whichfollowed [68], the authors used optical methods, in particular, LightScattering for the determination of time evolution of particle-size distri-bution in the suspension and, having obtained the kinetic constant forcoagulation, determined W and extracted information on the surfaceforces by using various modifications of the DLVO theory. A review ofthese studies can be found in [18,69–71].

3. Theory

3.1. Extracting interfacial potential from electrophoretic measurements

Extraction of the surface potential, ζ, from the apparent value ζexp, isconducted by means of SEM according to the most consistent andconvenient formalism developed in refs. [39–43]. Below, we followthe version of this formalism stated in review [72].

3.1.1. Standard Electrokinetic ModelTo determine the electrophoretic velocity, one should solve two

boundary value problems consequently. The first of these problemsdescribes the distribution of electric potential around a particle in thethermodynamic equilibrium state, Ψ = Ψ(r), i.e., in the absence of ex-ternally applied electric field. This problem includes the Poisson–Boltzmann (P–B) equation

∇ �∇Ψ ¼ − Fε

Xk

Ckzk exp −Ψzk F=RTð Þ ð9Þ

where∇= en∂/∂xn; en and xn are the unit vector and the coordinate of aCartesian coordinate system. The P–B equation, Eq. (9), is subject tothese boundary conditions

Ψ ¼ ζ at the particle surfaces ð10Þ

Ψ→0 at infinity : ð11Þ

Thus, by solving the boundary-value problem given by Eqs. (9)–(11),one obtains the spatial distribution of electric potential, Ψ(r), in thethermodynamic-equilibrium state. By using the obtained functionΨ(r, ζ), the interfacial charge density, q, is determined from the electro-static condition written in the form

q ¼ −εn �∇Ψ at the particle surface ð12Þ

where n is the unit outward vector normal to the particle surface. Thelatter equation yields the required relationship between the surface po-tential and charge density.

The second of the aforementioned problems is formulated for thenon-equilibrium mode when an external electric field is applied. Theproblem formulation involves the function, Ψ(r), which is supposed tobe known from the solution of the first problem. The set of governingequations of the second problem includes the continuity equations forindividual ionic fluxes and liquid flow and a version of Stokes equation,which accounts for the presence of electric force acting on the EDL spacecharge.

The flux-continuity equation can be represented in the followingform

∇ �∇μk−zk FRT

∇μk �∇Ψ ¼ − zk FDk

u �∇Ψ: ð13Þ

In Eq. (13), dμk = zkFdΦ+ dμkch is the differential of electrochemicalpotential of kth ion;Φ is the local electric potential and μkch is the chemicalpotential of kth ion. Generally, μkch depends on all the ion concentrations.However, for ideal electrolyte solutions usually postulated in electroki-netic studies, dμkch = RTdck/ck where ck = ck(r). When r→ ∞, ck(r)→ Ck.

The Stokes equation accounting for the electric body force, aftersome transformations, can be represented, as [72]

η∇�∇� u ¼ −∇Π−Xk

Ck exp −ΨFzk=RTð Þ−1½ �∇μk ð14Þ

where the effective pressure, Π ¼ p−RT∑k

Ck exp −ΨFzk=RTð Þ−1½ �, isthe deviation of local pressure, p, from its value in the thermodynamicequilibrium state relative to the solution bulk (the second term in theright hand side of latter equality).

The continuity equation for liquid velocity iswritten in its usual form

∇ � u ¼ 0: ð15Þ

The governing equations, Eqs. (13)–(15), are subject to boundaryconditions at the particle surface and infinity. The particle surface isassumed to be impermeable for ions. This condition looks obvious forthe case of dielectric particles. It is also true for conducting but ideallypolarizable particles. As for small metallic particles, according to ref.[73], one can approximately consider them as ideally polarizable sincethe effective resistance of electrochemical reactions is much largerthan the resistance of electrolyte solution even in the case of catalyticplatinum. The latter enables one to set the kth normal flux to be zeroat the particle surface.We consider the problem in the reference systemlinked to the particle. Consequently, the respective conditions takethese forms

n �∇μk ¼ 0 at the particle surfaces ð16Þ

u ¼ 0 at the particle surfaces : ð17Þ

At infinity, we impose a uniform external field strength, E, and zeroconcentration gradients. These two physical conditions are expressedthis way

∇μk ¼ −FzkE at infinity : ð18Þ

Onemore boundary condition should be set to impose zero total forceexerted on the particle. Such a force is a sum of electrical andmechanicalones and is obtained via integration of the sum of the Maxwell andviscous-stress tensors over any closed surface surrounding the particle.It is convenient to choose such a surface as a sphere with infinitelylarge radius. In this case, the electrical force acting on the totallyelectro-neutral volume inside the surface is zero. Accordingly, the integralof Maxwell stress tensor over the chosen surface, S∞ turns out to be zero,too. Consequently, the required boundary condition takes this form

∮S∞

ΠIþη ∇uþ ∇uð Þ�� n∞dS ¼ 0 ð19Þ

where I= enen, n∞ is the unit vector normal to the surface S∞. Since thefunctionΨ(r) is known as a solution of the first boundary-value problemgiven by Eqs. (9)–(11) and (13)–(15) subject to the boundary conditionsof Eqs. (16)–(19) yield a closed problem formulation enabling one tofindthe unknown functions μk(r), u(r) andΠ(r). By considering the limit ofr → ∞ for the velocity field, u(r), and transforming to the reference


system linked to the liquid at infinity, one obtains the electrophoreticvelocity, as

Ueph ¼ −U ¼ − limr→∞

u rð Þ: ð20Þ

Thus, by solving consequently the boundary value problems givenby Eqs. (9)–(11) and (13)–(19) and by using the limiting transition ofEq. (20), we can interrelate the electrophoretic velocityUeph and surfacepotential ζ for any electrolyte solution used in experiment.

3.1.2. ScalarizationWe assume that the particle is a sphere having radius a.While using the spherical coordinate system shown in Fig. 3 and

taking into account that the system has spherical symmetry at equi-librium, Eqs. (9)–(12) are rewritten as

1r2

ddr

r2dψdr

¼ − κað Þ2

Xk

ξk exp −ψzkð Þ ð21Þ

ψ að Þ ¼ eζψ ∞ð Þ ¼ 0

ð22Þ

q ¼ − εRTFa

� dψ∂r 1ð Þ ð23Þ

where

ξk ¼ Ckzk=Ckz2k að Þ ψ ¼ ΨF=RT bð Þ eζ ¼ ζF=RT cð Þ : ð24Þ

In the presence of a uniform electric field E at infinity the system hasan axial symmetry. We choose the spherical coordinate system withunit vectors er, eθ, eφ and the polar axis directed along the vector E(Fig. 3). The problem symmetry dictates the following angular depen-dencies

μk r; θð Þ ¼ zkFEaΜk rð Þ cos θð Þ ð25Þ

u r; θð Þ ¼ EεRTηF

ur rð Þ cos θð Þer þ uθ rð Þ sin θð Þ eθ½ �: ð26Þ

Fig. 3. Particle in external electric field. Spherical coordinate system (see the text for moredetail).

In Eqs. (25) and (26), we also suggest a convenient normalization ofunknown functions. By combining Eqs. (13), (25) and (26), we obtain

d2Μk

dr2þ 2

rdΜk

dr− 2

r2Μk ¼ zk

dΜk

dr−3

2mkur

dψdr

ð27Þ

where the electrokinetic parameter,mk, is given by

mk ¼23

εηDk

RTF

2: ð28Þ

By using Eqs. (25) and (26), the boundary conditions of Eqs. (16)and (18) take these forms

dΜk

dr1ð Þ ¼ 0 ð29Þ

dΜk

dr∞ð Þ ¼ −1: ð30Þ

To derive a convenient form of the Stokes equation, Eq. (14), oneshould apply operator ∇ × to both sides of Eq. (14) and substitute theelectrochemical potential, μk(r, θ), and velocity, u(r, θ) in the formsgiven by Eqs. (25) and (26). While making use of such a substitution,the functions ur(r) and uθ(r) can be represented in a form that followsfrom Eq. (15)

ur rð Þ ¼ − 2r2

Y ð31Þ

uθ rð Þ ¼ 1rdYdr

: ð32Þ

The transformation scheme described above leads to the followingform of Eq. (14)

d2

dr2− 2

r2

!2

Y ¼ − κað Þ2Xk

ξkd exp −zkψð Þ

drΜk: ð33Þ

Further, Eq. (33) is subject to boundary conditions that are obtainedby combining the boundary conditions of Eqs. (17) and (19) withEqs. (26), (31) and (32). Vector boundary condition Eq. (17) transformsinto two scalar conditions

Y 1ð Þ ¼ 0dYdr

1ð Þ ¼ 0:ð34Þ

For specifying the boundary condition of Eq. (19), which imposeszero value of the total force exerted on the particle, we will use the re-sults of refs. [72,74] where this condition was obtained at the sphericalcell border. By considering the limiting case of infinitely large cell radiusthat corresponds to a single particle or very dilute suspension, theboundary condition of Eq. (19) is rewritten in the form

r2ddr

þ 2r

d2

dr2− 2

r2

!Y j

r→∞→0: ð35Þ

Thus, the governing Eqs. (21), (27) and (33) subject to the boundaryconditions (22), (23), (29)–(32), (34) and (35) make up a closed prob-lem formulation that enables one to determine N+ 2 (N is the numberof ions) unknown functions ψ(r), Μk(r), and Y(r).

By using the function Y(r) to be obtained and combining Eqs. (20),(26) and (31), one can determine the electrophoretic velocity, Ueph.

image of Fig.�3


Then the measured value of zeta potential, ζexp, normalized by RT/F isrepresented as

eζexp ¼ 2 limr→∞

Yr2

: ð36Þ

Since Y ¼ Y r; eζ ;mk; κa; zk; ξk� �

, Eq. (36) yields the required relation-ship between the measured and actual surface potentials, eζexp and eζ forany given composition of electrolyte. Next, we will discuss numericalalgorithm of determining eζ from a given value of eζexp by using theboundary value problem stated above.

3.1.3. Numerical analysisFor given functions ψ(r) and Μk(r), the distribution Y(r) is obtained

by solving the boundary value problem Eqs. (33)–(35). It can be shownthat the solution of this problem can be represented in the followingform

ur ¼ −2 κað Þ29

"Zr1

− 32r

x2 þ 32xþ 3x4

10r3− 3r2

10x

!f xð Þdx

þ 32r

−1− 12r3

Aþ 3

10r2−1

2þ 15r3

B

# ð37Þ

where

f rð Þ ¼Xk

ξkd exp −zkψ rð Þ½ �

drΜk rð Þ ð38Þ

A ¼Z∞1

x2 f xð Þdx ð39Þ

B ¼Z∞1

f xð Þx

dx: ð40Þ

While using Eqs. (31), (36) and (38)–(40), the expression for theapparent zeta potential, eζexp, takes the form

eζexp ¼ − κað Þ29

Xk

ξk

Z∞1

1rþ 2r2−3r

∂ exp −zkψ r; eζ� �h i∂r Μk r; eζ� �

dr:

ð41Þ

Thus, Eq. (41) yields the interrelation between the apparent zeta po-tential, eζexp, and the surface potential, eζ . The dependency on the lattervalue is contained in the normalized distributions of equilibriumpoten-tial,ψ r; eζ� �

, and the functionΜk r; eζ� �attributed to the kth ionwith the

help of Eq. (25). In ref. [45], an integral relationship similar to Eq. (41)was derived for z:z electrolyte solution. Hence, Eq. (41) yields a general-ization of the result of ref [45] for the case of mixed electrolyte solution.

As stated above, for interrelating eζexp and eζ with the help of Eq. (41),one should know the functionsψ r; eζ� �

andΜk r; eζ� �. The functionψ r; eζ� �

is determined separately by solving the boundary value problem given byEqs. (21) and (22). ObtainingΜk r; eζ� �

requires a more complex schemesince the functionsΜk r; eζ� �

appear in Eq. (27) together with urwhich, inturn, depends on all the functionsΜk r; eζ� �

via the integral relationship ofEq. (37). However, in two limiting cases, Eq. (27) does not contain ur thatallows obtaining the required set of functionsΜk r; eζ� �

by solving Eq. (27)subject to boundary conditions (29) and (30), separately. These twolimiting cases corresponding to κa → ∞ (Smoluchowski limit [30,31])and eζ≪1 (Henry case [32]) were mentioned in Section 2.

While addressing either Smoluchowski or Henry case, the right handside of Eq. (27) can be omitted. For the Smoluchowski case, it differsfrom zero within the vanishingly thin (κa → ∞) EDL region, only. Thelatter is a result of the presence of dψ/dr in the right hand side ofEq. (27). For the Henry case [31], which yields the linear term in theexpansion of electrophoretic mobility in the powers of eζ , one shouldsubstitute into Eq. (41) the function corresponding to eζ ≡ 0, Μk(r, 0),since Eq. (27) already contains factors proportional to eζ besides Μk(r).Hence, for both the limiting cases, Eq. (21) transforms into a homoge-neous equation whose solution,Μk ¼ Μk r; eζ ; κa� �

, satisfying boundaryconditions (29) and (30) has this form

Μk r; eζ ;∞� �¼ Μk r;0; κað Þ ¼ Μ rð Þ ¼ −r 1þ 1

2r3

: ð42Þ

Thus, for both the Smoluchowski and Henry cases, the functionΜk(r) is independent of eζ and κa and turns out to be the same for allthe ions.

Now, we substitute Eqs. (42) into (41) and combine the derivedequation with Eq. (11). The integral obtained in this manner shouldbe taken by parts three times in series while accounting for Eq. (22).Finally, we arrive at the following expression

eζexp ¼ eζ−Z∞1

5r6

− 2r4

ψdr: ð43Þ

For the Smoluchowski limiting case, κa→∞, the integral in the righthand side of Eq. (43) approaches zero. This can be understood by con-sidering that |ψ|/rn ≤ |ζ| exp[−(r − 1)κa]. Consequently, substitutingthe right hand side of the later inequality into the integral of Eq. (37)one can see that the integral approaches zero when κa → ∞. Hence,Eq. (43) and, thus, Eq. (41) lead to the expected result for theSmoluchowski limit, eζexp ¼ eζ .

While dealing with the Henry case, from the boundary value problemgiven by Eqs. (21) and (22), one obtains:

ψ ¼ eζ exp − r−1ð Þκa½ �r

þ O ζ2� �

: ð44Þ

By substituting Eqs. (44) into (43), we arrive at Eq. (4) which isequivalent to the Henry's result [32]. When the parameter κa isknown, Eq. (4) allows determining eζ for any measured value eζexp.

For a given value of κa, the relationship betweeneζexp andeζ is indepen-dent of electrolyte composition for each of the two limiting casesdiscussed above. Such an independency takes place since the distribu-tionsΜk(r) given by Eq. (42) become the same for all the ions. Generally,the functionsΜk(r) corresponding to various ions differ from each otherand can be determinedwith the help of computational scheme describedbelow.

Let us solve Eq. (27) with respect to Μk(r) by considering the righthand side of Eq. (27) as a known function. After satisfying the boundaryconditions (29) and (30), the obtained solution can be represented inthis form

Μk ¼ − 1þ Hkð Þ 1þ 12r3

r þ 1

3

Zr1

r− x4

r

!zk

dΜk

dx−3

2mkur

dψdx

dx

ð45Þ

where the integration constants Hk are given by

Hk ¼13

Z∞1

zkdΜk

dr−3

2mkur

dψdr

dr: ð46Þ


Now, we consider the set of integral relationships given by Eqs. (37)and (45). According to these relationships, at the point with coordinater = r∗, the functions ur(r) and Μk(r) take values, ur(r∗) and Μk(r∗), thatare expressed through the distributionsΜk(r) for 1 b r b r∗. Consequently,both ur(r) and Μk(r) can be determined numerically by graduallyincreasing r in the integrals on the right hand sides of Eqs. (31) and(41). Recall that, when r → ∞, the asymptotic value approached byur(r) is (−eζexp).

The discussion above defines the steps of numerical scheme to beused: (i) the function ψ r; eζ ; κa� �

is determined by solving the P–Bboundary value problem given by Eqs. (21) and (22); (ii) certain initialvalues of the integration constants A, B and Hk appearing in Eqs. (37)and (45) are assumed; (iii) while using Eqs. (31), (32) and (41), thefunctions ur(r) and Μk(r) are computed by gradually increasing r; (iv)the obtained distributions are used for recalculation of A, B and Hk bymeans of Eqs. (39), (40) and (46), respectively.

3.2. Coagulation dynamics

In this present section, we consider the interpretation scheme forthe experimental rate of disperse system coagulation. As it was statedin Section 2.2, we will extract the Hamaker constant, H, from theFuchs factor, W, which is estimated from the coagulation dynamics.

Next, we consider the scheme of obtaining, Gel(h) for given zetapotential and electrolyte composition. Since we deal with moderateκa, eζN1 and mixed electrolyte solution, the problem will be solvednumerically for a rather general case.

3.2.1. Electrostatic repulsion: general problem formulationWe consider two particles separated by a distance, l, and bearing

either constant surface potential, ζ, or constant surface charge, q, thatare determined from the electrophoretic mobility measurements fol-lowing the scheme described in the previous sections. Both the particlecharge and potential are assumed to be the same for two particles.

The system containing two particles and the infinite volume ofsurrounding electrolyte solution is considered to be in thermodynamicand mechanic equilibrium. Consequently, the distribution of electricpotential, Ψ, is obtained as a solution of P–B problem given byEqs. (9)–(11) with a reservation that, in the limiting case of constantsurface potentials, the same potential ζ is set at the surfaces of each ofthe particles. For analyzing the case of constant surface charge at thesurface of each of the particles, one should use Eq. (12) instead ofEq. (10) for setting the electrostatic boundary condition.

By using the solution of P–B problem, Ψ(r), one can determine theforce, acting on either of two particles, X. To this end, the stress tensor,σ, should be integrated over the particle surface Sp, as

X ¼ ∮Sp

σ � n dS ð47Þ

where σ ¼ ε∇Ψ∇Ψ− ε2 Ι∇Ψ �∇Ψ−Ιp is the stress tensor. On the right

hand side of the latter expression, the first two terms represent theMaxwell tensor, and the third term gives the contribution of pressure,p, into the total stresses. The local pressure can be interrelated withthe local value of the potential by means of mechanical-equilibriumcondition, which can be written in the form

∇ �σ ¼ 0: ð48Þ

By combining Eqs. (9), (47) and (48), after some transformations,one obtains

p−p∞ ¼ RTXk

Ck exp −ΨFzk=RTð Þ−1½ � ð49Þ

where p∞ is the pressure far away from the particles.

By using Eq. (48) and the tensor version of the Gauss theorem, onecan prove the following equality

∮Sp

σ � nA dS ¼ − 12a 1þ hð Þ

ZSsym

σ � rBA dS ð50Þ

where Ssym is the symmetry plane; rBA is the vector whose origin andend coincide with the centers of particles B and A, respectively.

Now, we consider the forceXA and XB exerted on the particles A andB respectively. This is obtained by combining Eqs. (46), (47), (49) and(50) and by using symmetry considerations

XA ¼ rBA2a 1þ hð ÞX ¼ − rAB

2a 1þ hð ÞX ¼ −XB ð51Þ

where the force magnitude, X, which is obviously the same for bothparticles, is expressed as an integral over the symmetry plane,which is perpendicular to the line connecting the sphere centers,Ssym

X ¼ZSsym

ε2

∇Ψ− rBA4a2 1þ hð Þ2 rBA �∇Ψð Þ

" #2þ RT

Xk

Ck exp −zkΨF=RTð Þ−1½ �( )

dS:

ð52Þ

Thus, Eqs. (51) and (52) enable one to compute the electrostatic inter-action force exerted on the interacting particles when the equilibriumelectric potential distribution, Ψ(r), is known. At a given particle radius,this force is a function of the distance between the particles, X = X(h).Consequently, the contribution of electrostatic forces into the interactionfree energy, Gel(h), is determined as

Gel hð Þ ¼Z∞h

X hð Þdh: ð53Þ

In summary, the electrostatic contribution to the system free energy isobtained by integral of Eq. (53) where the interaction force magnitude,X(h), is computed by using Eq. (52), which depends on the potentialdistribution, Ψ(r). The latter distribution is obtained as a solution of thenon-linear boundary value problem given by governing Eq. (9) subjectto the boundary condition (12) (for constant surface potential) or (10)(for constant surface charge). Both the latter conditions are set at the sur-face of each of the particles. One more boundary condition is given byEq. (11). Numerical solution of this problem for two particle system isconsidered next.

3.2.2. Numerical computation of electric field distributionFor κa≫ 1, the above outlined scheme of obtaining Gel(h) is simpli-

fied by using the Derjaguin approximation [56,62,63] discussed inSection 2.2. In this approximation, the first term in curly brackets inEq. (52) is omitted for being small at κa ≫ 1. For moderate κa, thisterm yields a noticeable contribution. In such a case, one should solvecomplete P–B problem and take into account all the terms representedin Eq. (52). The analysis of this type is given in ref. [73], which will beused in our calculations.

By following ref. [66], we will compute the distribution Ψ(r)and the electrostatic interaction force magnitude in a bi-sphericalcoordinate system. Taking into account the problem axial symmetry,one can represent the ∇ operator in terms of new coordinates (β, ν)as

∇¼ cosh νð Þ− cos βð Þaffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih hþ 2ð Þ

p eβ∂∂β þ eν

∂∂ν

ð54Þ


where eβ and eν are the unit vectors of the bi-spherical coordinatesystem. Differentiating eβ and eν satisfies the following rules:

∂eβ∂β ¼ eν

sinh νð Þcosh νð Þ− cos βð Þ að Þ ∂eβ

∂ν ¼ −eνsin βð Þ

cosh νð Þ− cos βð Þ bð Þ∂eν∂ν ¼ eβ

sin βð Þcosh νð Þ− cos βð Þ

cð Þ ∂eν∂β ¼ −eβ

sinh νð Þcosh νð Þ− cos βð Þ

dð Þ:

ð55Þ

By combining Eqs. (9), (24a,b), (54) and (55), we arrive at thefollowing dimensionless version of the P–B equation written inbi-spherical coordinates

cosh νð Þ− cos βð Þ½ �3h hþ 2ð Þ � sinβ

∂∂β

sinβcosh νð Þ− cos βð Þ �

∂ψ∂β

þ ∂∂ν

sinβcosh νð Þ− cos βð Þ �

∂ψ∂ν

�¼ − κað Þ2

Xk

ξk exp −ψzkð Þ:

ð56Þ

Boundary conditions at the particle surface are rewritten as

ψ β;ν0ð Þ ¼ eζ constant potentialð Þ ð57Þ

or

cosh ν0ð Þ− cos βð Þκa

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffih hþ 2ð Þ

p ∂ψ∂ν β;ν0ð Þ ¼ eq constant chargeð Þ ð58Þ

where eq ¼ qF=εκRT . The coordinate surface, ν = ν0, coincides with thesurface of one of the particles.

Instead of setting the same condition at the surface of another particle,we will use the system symmetry which allows us to set the followingconditions at the symmetry plane ν= 0

∂ψ∂ν β;0ð Þ ¼ 0 ð59Þ

and at the axis β= 0, π

∂ψ∂β 0;νð Þ ¼ 0

∂ψ∂β π;νð Þ ¼ 0:

ð60Þ

Thus, Eq. (56) subject to boundary conditions (57) (or (58)), (59) and(60) forms a closed problem formulation for obtaining the functionψ(β, ν). This problem is numerically solved by conducting a discretizationof the second order differential equation for obtaining equations to besolved with the help of an iteration scheme. We used the iterationmethod of Newton–Raphson that enabled us to reduce the non-linearproblem to several linear iterations. Finally the obtained functionψ(β, 0) is substituted into integral (52) which is rewritten in the form

X hð Þ ¼ πεRTF

2Zπ0

κað Þ2h hþ 2ð Þ1− cos βð Þ½ �2

Xk

ξkzk

exp −zkψ β;0ð Þð Þ−1½ � þ ∂ψ∂β β;0ð Þ �( )

sin βð Þdβ:

ð61Þ

The obtained function X(h) is substituted into the integral of Eq. (53)for obtaining Gel(h).

3.2.3. Obtaining Hamaker constant from electrophoretic and coagulationdynamic data

The above stated scheme allows one to determine the Fuchs factor,W, as a function of ζ, the electrolyte composition, particle radius andthe Hamaker constant. To this end, we will use the integral expressiongiven by Eq. (6) to substitute there the interaction free energy as a

sum of electrostatic, Gel(h), and van der Waals, GW(h), parts, Eq. (7).The function Gel(h) is obtained by using calculations presented inSections 3.2.1 and 3.2.2 for a surface potential ζ and electrolyte composi-tion. By using numerical calculations based on the SEM, ζ is determinedfrom ζexp obtained from the electrophoretic measurements. The function,GW(h), which contains, H, is substituted in the form given by Hamaker'sEq. (8) [67]. For a set of electrolyte concentrations and zeta potentials,the Hamaker constant is fitted to make the calculated Fuchs factor, W,match its value estimated from the coagulation-dynamic experiments.

4. Experimental example

4.1. Pt/C nano-catalytic dispersions and their practical importance

The experimental part of this study deals with a system of consider-able practical interest namely colloidal suspensions of composite Pt/Cnano-particles. The Pt/C composites are commonly employed as catalystmaterials in Proton ExchangeMembrane Fuel Cells (PEMFC) for both an-odic and cathodic catalytic reactions. Fuel cells are considered to be themost technically viable solution for clean and sustainable future energyscenarios.While consuming fuel (hydrogen or hydrogen rich substances)and oxidant (oxygen or air), the fuel cell generates electrical energy andproduce water as the waste [75].

The PEMFC are an especially interesting type of fuel cells due to theirinherent advantages such as high power density, reduced systemweight, simplified construction and quick startup. PEMFC are suitablefor portable, transport and stationary applications [76,77]. The maincomponent of the PEMFC is the Membrane Electrode Assembly (MEA)which consists of a proton exchange membrane located between twoporous electrodes (anode and cathode) [78]. Electrochemical reactions,both anodic and cathodic, take place at the electrodes and are promotedby the use of a catalyst. Pure Pt or Pt in combinationwith other Pt groupmetals (PGM), either supported or unsupported, are most suitable forelectrochemical reactions in PEMFC. Because of the use of Pt and PGM,theMEA represents themost expensive component of the PEMFC. There-fore active research is carried out for improving catalyst utilization.

There are several MEA preparation techniques which alter the waythat the catalyst layer is formed. Each technique is aimed at improvingMEA performance and reducing the catalyst loading and thereby overallcost. The catalyst layer can be deposited either onto the gas diffusionlayer known as a catalyst coated substrate (CCS) or directly onto themembrane known as a catalyst-coated membrane (CCM) [79]. Electro-phoretic deposition (EPD) is a highly efficient process for the productionof films and coatings. EPD is easy to implement, low cost, fast and appli-cable to awide variety ofmaterials [80]. EPDhas already been successful-ly demonstrated for the deposition of catalytic layers in MEAs [24–27].Deposition occurs when the particles collect via coagulation at the elec-trode (or membrane) surface and form a relatively compact and homo-geneous film [81,82]. The knowledge of both electrophoretic mobilityand stability of catalyst suspensions is of paramount importance for theoptimization of EPD catalyst-layer formation. Usually, EPD is accompa-nied by considerable pH changes close to the electrode (membrane)surface. Therefore, the electrophoretic mobility and stability should beunderstood within a broad pH range for various salt concentrations, i.e.we face the necessity to extract the values of zeta potential for ternaryelectrolyte system. This can be done by using the algorithms describedabove in the theoretical sections.

4.2. Materials

HiSpec 4000, 40 wt.% Pt/C (Johnson Matthey, United Kingdom) wasused as received as catalyst material for all experiments. Ultrapure H2O(18.3 MΩ cm) was obtained via a Zeneer Power III water purificationsystem (Human Corporation, South Korea). Ionic strength of the sus-pensions was controlled by addition of NaCl (KIMIX, South Africa)while the pH was adjusted by the addition of NaOH (KIMIX, South

Fig. 5.Malvern Zetasizer Nano ZS instrument and semi-disposable capillary cell (inset).


Africa) and HClO4 (KIMIX, South Africa). Catalyst suspensions wereprepared by mixing the Pt/C powder (0.1 or 0.03 mg) with 10 ml NaClelectrolyte solution (0.1 to 40 mM). The pH of the suspensions wasmonitored using the Metrohm 827 pH lab (Metrohm, Switzerland)equipped with a Primatrode pH electrode. Homogeneous suspensionswere obtained by means of ultrasonic dispersion for 5 min via theBiologics 3000 ultrasonic homogenizer (Biologics, Inc., USA) fittedwith micro tip ultrasonic finger. The power of the homogenizer wasset at 40% with pulser set to the off position (0%). The initial diameterof particles in suspension was about 280 nm.

4.3. Methods

Transmission Electron Micrograph of the dry Pt/C powder wereobtained using a Tecnai G2 F20 X-Twin Mat200 kV Field EmissionTEM, operating at 200 kV (Fig. 4).

Measurement of electrophoretic mobility and particle size of Pt/Cparticles in aqueous solutions were obtained using the Zetasizer NanoZS (Malvern Instrument Ltd., United Kingdom) as shown in Fig. 5.

The instrument was fitted with a production standard 532 nm,50 mW diode laser source. The Zetasizer instrument measures electro-phoretic mobility via a 3 M-PALS technique which is a combination oflaser doppler velocimetry (LDV) and phase analysis light scattering(PALS). Particle size was measured via Dynamic Light Scattering (DLS)also known as Photon Correlation Spectroscopy (PCS). The instrumentis capable ofmeasuring particle size between 0.6 nm–6 μmand the elec-trophoretic mobility of particles with a size range of 3 nm–10 μm [83,84]. A syringe was used to fill a semi-disposable capillary cell with thesample which was then immersed into a temperature controlled blockholder to avoid thermal gradients in the absence of the applied electricfield [85]. Electrophoretic mobility was measured by applying a fixedvoltage of 100 V and programming the instrument to record 3 electro-phoretic mobility values for each sample. Particle size was obtained byaveraging 10 size values obtained over a 600 s time interval with amea-surement recorded every 60 s. All measurements were performed at25 °C.

While studying aggregation, we compare the coagulation rates forsystemswith different salt concentrations and different pHs. To check thepossibility of addressing all the studied cases in terms of a single timescale parameter, τ, given by Eq. (5), to which we will refer as the coagu-lation time, we analyze each of the time dependencies of the “particlesize” (as the device displays it) that have been obtained for various solu-tion compositions. For each of the dependencies, dimension vs. time, wedetermine its own τ by considering the initial stage of coagulation.

Fig. 4. TEM image of 40% Pt/C (JM HiSpec 4000).

Finally, we redraw all the experimental curves by representing the“particle size” as a function of time normalized by the coagulationtime, τ, determined for each of the curves separately.

The results of implementing the above described scheme arerepresented in Fig. 6, where all the experimental points could be col-lapsed to a single smooth curve. Such behavior reveals that obtaining τfor each of the solution composition yields the required information onthe system aggregative behavior.

5. Results and discussion

As discussed above, for finite values of κa, the predictions of electro-phoretic velocity can noticeably deviate from the results given byEq. (3). However, the devices measuring electrophoretic velocity oftendisplay the experimental data in terms of zeta potential by assumingthat the Smoluchowski relationship given by Eq. (3b) is valid. The lattermeans that the displayed quantity, ζexp, is obtained from the measuredelectrophoretic velocity, Ueph, by using Eq. (3b), as

ζexp ¼ UephηEε

: ð62Þ

1E-3 0,01 0,1 1

300

400

500

600

700

800

900

1000

c=3 pH=10 c=3 pH=11 c=3 pH=11.5 c=10 pH=10 c=10 pH=11 c=10 pH=11,5 c=15 pH=10 c=15 pH=11 c=15 pH=11,5 c=20 pH=10 c=20 pH=11 c=20 pH=11,5 c=25 pH=10 c=25 pH=11 c=25 pH=11,5 c=40 pH=10 c=40 pH=11 c=40 pH=11,5

2a, n

m

t/τ

Fig. 6. Size of particles vs. normalized time for various electrolyte solution compositions(NaCl concentration 3, 10, 15, 20, 25 and 40 mM; pH 10, 11 and 11.5).

image of Fig.�4

image of Fig.�5

image of Fig.�6

2 4 6 8 10 12 14

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

10

20Volume fraction 10-6

exp,

Volume fraction 4*10-6

exp, , run 1

exp, , run 2

, mV

pH

(a)

2 4 6 8 10 12 14-70

-60

-50

-40

-30

-20

-10

0

10Volume fraction 10-6

exp, , run 1

exp, , run 2


exp, , run 1

exp, , run 2

exp, , run 3

, mV

pH

(b)

ζζ

ζ ζ

ζ ζζ ζ

ζ ζζ ζ

ζ ζζ ζζ ζ

Fig. 7. Dependency of apparent (dashed lines) and actual (solid lines) zeta potentials onpH for salt concentration 10−4 M (a) and 10−3 M (b); the volume fractions are 10−6

and 4 ∗ 10−6.


For sufficiently high κa, the displayed and actual values of zetapotentials should coincide, ζexp = ζ. However, for finite κa, the latterequality does not hold, and ζexp turns out to be a function of ζ, theparticle radius, a, and the electrolyte composition. Consequently, in thegeneral case, to determine the value of ζ from the measured ζexp oneshould know the particle radius, a, the electrolyte solution compositionand the relationships which express ζexp through ζ, a and the parametersdescribing the solution composition. Such a relationship should theoreti-cally be established on the basis of amodel, as discussed above. Belowweconsider a particular example where the proposed methodology isapplied to the analysis of electrical interfacial properties of Pt/C nano-particles suspension.

5.1. Correlation between electrophoretic and stability data

The calculation scheme described in Section 3.1 enables us to obtainthe apparent value of zeta potential ζexp from its actual value ζ for arbi-trary values of ζ and κa. By solving the inverse problem, one can estimateboth the surface potential, ζ, and charge density, q, from ζexp and κa thatare known from experiment. The parameter κa is obtained by determin-ing particle radii and specifying Eq. (3) for ternary electrolytes employedin experiments, namely, the mixtures of NaCl with either NaOH or HCl.The calculation scheme of Section 3.1 is also specified for these electrolytesolutions. Below,we present the results of such calculations for the exper-imental system described above.

The results of the first group were obtained while measuring theelectrophoretic mobility for sufficiently low concentrations of salt,10−4 M and 10−3 M, within a wide pH range which includes boththe low and high pHs where the suspension becomes unstable.

The curves in Fig. 7a and b display the behavior of both the apparentand actual surface potentials, ζexp and ζ, as functions of solution pH for the10−4 M and 10−3 M concentrations of salt. The presented data were ob-tained for two solid-phase volume fractions, (10−6 and 4 ∗ 10−6), forwhich the suspension can be considered infinitely diluted in terms ofelectrophoresis (but not coagulation). Accordingly, the data, except for afew points, are close to each other.

In both the graphs, while increasing pH within the acidic range, thepositive surface potential decreases and reaches zero at rather low pHs.With the further increase of pH, the potential becomes negative and in-creases in absolute value until reaching a maximum magnitude withinthe alkaline range but close to the neutral pH. The final decrease of po-tential is observed within the alkaline range. Importantly, ζexp and ζnearly coincide for acidic and alkaline pHs, but, within the neutralrange of pH, the actual potential magnitude, |ζ|, exceeds that of theapparent value, |ζexp|, by a factor of about 2, for the salt concentration10−4 M, and 1.2, for 10−3 M.

These results correlate with our observation of system stability, ac-cording to which the system remains relatively stable at neutral pHs,but addition of acid significantly accelerate coagulation which reachesthe maximum rate at about pH ~ 3 ÷ 4. At such pHs, as it follows fromthe electrophoreticmobilitymeasurements, the particle charge dramat-ically decreases, and thus the electrostatic repulsion weakens, whichleads to the acceleration of coagulation.

5.2. Influence of electrolyte composition on surface potential and charge

One can suggest two mechanisms whose simultaneous action canlead to the behavior of zeta potential displayed by curves in Fig. 7 a andb, namely, (i) changes of surface charge due to the binding-release ofH+ and OH− ions, and (ii) the decrease of potential due to the EDLcompressing,which occurswhile increasing the ionic strength and there-by decreasing the Debye length. The latter mechanism manifests itselfwhen the base, NaOH, concentration becomes sufficiently high.

To understand the role of the first mechanism, we consider thebehavior of surface charge density as a function of pH at constant saltconcentrations, 10−4 M and 10−3 M (Fig. 8 a and b). At some quite

low pH value, the particle charge turns from positive to negative and,then, increases in magnitude with increasing pH and reaches a valueof about 0.02 C/m2 in a concentrated base solution. In the alkalinerange, the curve behavior resembles a Langmuir isotherm.

At neutral and acidic pH, we do not observe such adsorption satura-tion. Instead, there is a slow linear dependency on pH, while the concen-trations of hydrogen (and hydroxyl) ions change substantially. Note thatthe pH axis in Fig. 8 is decimal logarithmic with respect to the OH− ionconcentration. Perhaps, such behavior is a result of existence of twotypes of surface ionic groups with different properties. Saturation ofgroups that belong to one of the types can coincide at, approximately,pH=7,with start of ion binding (or releasing) by groups of another type.

For better understandingwhich ions take part in forming the surfacecharge,we determine all the ion concentrations in the immediate vicinityof surface, CkS, by using the Boltzmann distribution

CSk ¼ Ck exp −eζzk� �

: ð63Þ

We plot these concentrations against the surface charge density, q,which is calculated for each of the points by using the electrolyte com-position and the calculated value of surface potential corresponding tothis point. Generally, at given ζ, any relation between Ck

S and q, shouldalso depend on the electrolyte composition. Therefore, the set of pointsCkS vs. q, plotted for different electrolyte compositions in different exper-

iments should be spread over a certain area in the graph. However,while assuming that the surface charge is formed due to the interaction(dissociation or adsorption) of surface groups with the kth sort of ionsonly, the surface charge is completely defined by the concentration Ck

S

2 4 6 8 10 12 14-0,030

-0,025

-0,020

-0,015

-0,010

-0,005

0,000

0,005

Volume fraction 10-6

run 1Volume fraction 4*10-6

run 1 run 2

q, C

/m2

pH

(a)

2 4 6 8 10 12 14-0,030

-0,025

-0,020

-0,015

-0,010

-0,005

0,000

0,005

Volume fraction 10-6

, run 1, run 2


, run 1, run 2, run 3

q, C

/m2

pH

(b)

Fig. 8. Dependency of surface charge density on pH at salt concentration (a) 10−4 M and(b) 10−3 M; the volume fractions are 10−6 and 4 ∗ 10−6.


and the parameters of adsorption (dissociation-binding) isotherm andis independent of electrolyte composition. In such a case, the corre-sponding points in the graph Ck

S(q) are expected to lie on a smoothline, the isotherm, or to be close to it (taking into account the experi-mental error).

Let us now consider the positions of points plotted in the graphof Fig. 9 for H+, Na+ and Cl− ions. As it is clear from Fig. 9, Na+ andCl− ions do not form the surface charge because the points corresponding

-0,030 -0,025 -0,020 -0,015 -0,010 -0,005 0,000 0,0051E-9

1E-8

1E-7

1E-6

1E-5

1E-4

1E-3

0,01

0,1

1

10

100

1000

H+

Na+

Cl-

c, m

M

q, C/m2

Fig. 9. Concentrations of H+, Na+ and Cl− ions that correspond to given values of surfacecharge in different experiments.

0 10 20 30 40

-65

-60

-55

-50

-45

-40

-35

, pH=10, pH=11, pH=11.5exp

, pH=10

exp, pH=11

exp, pH=11.5

, mV

CNaCl

, mM

ζζζζζζ

ζ

Fig. 10. Dependency of surface potential on salt concentration for different pHs: ζ— solidlines, ζexp — dashed lines.

to these ions approach to a smooth line at a relatively large charge,|q| N 0, 005C/m2, only. Note that the points corresponding to theH+ ions make up a set which can be approximated by a smooth line.Consequently, we conclude that the charge is formed either by hydrogenor by hydroxyl ions.

Importantly, by using the present approach, it is impossible to distin-guish whether the charge is formed by binding of OH− ions or releasingof H+ ions. The charge in this region strongly depends onpH (see Fig. 8 aand b).

Near the maximum charge, |q|≈ 0, 02C/m2, we have a set of pointsplotted for the Na+ ions that nearly form a smooth line. However, itdoes not mean that the Na+ ions form the charge. This can be under-stood by considering that the Na+ ion is the only counterion (cation)in the solution. Consequently, under conditions of locally flat doublelayer, which is satisfiedwhen ζexp and ζ are close to each other, and suf-ficiently high ζ, one can establish this relationship between the charge qand the concentration Cs

Naþ

q ¼ − εRTF

� ∂eΨ∂r

��r¼a

≈−ffiffiffiffiffiffiffiffiffiffiffi2εRT

p ffiffiffiffiffiffiffiffiffiffiCsNaþ

q: ð64Þ

Thus, Eq. (64) strictly interrelates the charge density and the con-centration Cs

Naþ , and, hence, is independent of electrolyte composition.However, this interrelation is not an adsorption isotherm.

The second group of experiments has been conducted to elucidate themechanisms of decrease of the surface potential magnitude at high pHs,as shown in Fig. 7 a and b.

The principal purpose of these experimentswas tofigure outwhetherthis decrease is due to the compression of EDL or there are changes in theion adsorption or binding. To answer this questionwe increased the ionicstrength by adding a salt, NaCl, instead of the base, NaOH, i.e., we main-tained pH in each of experiments.

The curves of Fig. 10 show the dependency of surface potential onthe salt concentration. The difference between the behaviors of curvesplotted for ζ and ζexp is noticeable but smaller than in the case of lowsalt concentrations. One can see that the absolute value of surface po-tential decreases except for two points in high concentration rangewhich deviate from the decreasing trend within the limits of experi-mental error.

Obtaining the surface charge density corresponding to the surfacepotential, ζ, and the ion concentrations presented in Fig. 10 yields a re-markable result. Although ζ decreases inmagnitudewith increasing saltconcentration, the surface charge magnitude increases (Fig. 11).

The opposite behavior of potential and charge demonstrated bycurves in Figs. 10 and 11 is explained by the increase of concentrationof hydroxyl ions and decrease of that of hydrogen ions in the immediatevicinity of particle surface, see Eq. (63). Accordingly, the adsorption of

0 10 20 30 40-0,025

-0,020

-0,015

-0,010

pH=10 pH=11 pH=11.5

q, C

/m2

CNaCl

, mM

Fig. 11. Dependencies of surface charge density on salt concentration for different fixedpHs.

Table 1Measured quantities: zeta potential, ζexp, coagulation time, τ, ionic strength, I. Calculatedquantities: surface potential, ζ, surface charge density, q, and Hamaker constant, H.

pH ζ Coagulation Ionic ζ q H

(exp) time, s strength

mV mM mV C/m^2 J

0.03 mg/ml Pt/C in 0.1 mM NaCl, τSm = 300 s4.7 −23.7 1064.3 0.120 −32.00 −0.00101 1.03E-184.8 −24.7 2003 0.116 −33.50 −0.00104 1.13E-184.9 −24.4 12,801 0.113 −33.10 −0.00102 1.03E-185 −26.4 47,170 0.110 −36.20 −0.00111 1.24E-1812 −54.3 39,018 10.100 −60.85 −0.01768 1.66E-1912.1 −54.7 11,620 12.689 −60.70 −0.01972 1.46E-1912.2 −49.9 2792.3 15.949 −54.30 −0.01894 1.07E-19

0.03 mg/ml Pt/C in 1 mM NaCl, τSm = 300 s4.8 −23.7 1171.85846 1.016 −27.85 −0.00216 1.46E-194.9 −24.65 1792.53255 1.013 −29.10 −0.00237 1.59E-195 −25.6 2777.72166 1.010 −30.30 −0.00247 1.71E-195.1 −27.55 3742.66225 1.008 −32.80 −0.0027 2.04E-1912.1 −48.5 2458.15181 13.589 −52.90 −0.01691 1.12E-1912.2 −48.1 1831.41644 16.849 −51.96 −0.01836 9.63E-2012.3 −48.9 1387.91573 20.953 −52.50 −0.0207 8.72E-20

0.03 mg/ml Pt/C in 1 mM NaCl repeats, τSm = 300 s


potential-defining OH−-ions (and/or desorption of H+ ions) increases.Thus, the behavior of charge as a function of salt concentration addition-ally confirms that the surface charge is formed due to the adsorption ordesorption of OH− or H+ ions, respectively. Clearly, the charge magni-tude is higher for higher pHs that are illustrated in Fig. 11 by the positionsof the respective curves.

It can be expected that, at a given ionic strength, the charge will behigher for systems having higher pH. To verify that, in Fig. 12, we con-structed a graph similar to plot of Fig. 11, but, as the horizontal axis, weused the ionic strength of mixed electrolyte solution, NaCl and NaOH, in-stead of the salt concentration.

In Fig. 12, at the same ion strength, the points corresponding tohigher pHs, at the same time, correspond to larger charge magnitudes.Accordingly, the lowest curve, which corresponds to the largest chargemagnitude, was plotted for the lowest salt concentration when almostthe whole solution ionic strength is due to the NaOH ions.

In summary, binding of OH− ions (and/or release of H+ ions) is theprincipal mechanism of charging the particle surface. While increasingthe electrolyte solution pH, one observes an increase of the negativesurface charge by absolute value. Nevertheless, the negative surfacepotential magnitude decreases due to the compression of EDL whichoccurs when the ionic strength increases.

5.3. Applicability of Standard Electrokinetic Model

While considering coagulation, the free energy of particle interactionshould be compared with the energy of thermal motion kT, which, forthe room temperature, is about 4 ⋅ 10−21J. By using the Hamakerconstant of about 10−20J, the Van der Waals energy is estimated to be

0 10 20 30 40-0,025

-0,020

-0,015

-0,010

-0,005

pH=10 pH=11 pH=11.5 C

NaCl=0.1 mM

q, C

/m2

I, mM

Fig. 12. Dependency of surface charge density on ionic strength, I, at different pHs.

significant at distances of the order of magnitude of particle radius,Eq. (8). Clearly, at such distances Derjaguin approximation is not appli-cable. As for the electrostatic repulsion, its contribution is important upto several Debye lengths. When EDL is sufficiently thin, the rapid coag-ulation can occur in the secondary minimum.

In our experiments, for all samples, we always observed an initiallylinear dependency of aggregate size on time. The latter allows us toestimate the time-scale parameter of coagulation dynamics, τ. Byusing the estimated value of τ and Eq. (5), we determined theFuchs factor, W, for each experiment. The Smoluchowski time, τSm,has been estimated at τSm = 300s and τSm = 100s for the 10−6 and4 ∗ 10−6 particle volume fractions, respectively. By using the numericalscheme outlined in Section 3.2.3, the Hamaker constant, H, was deter-mined for each of the employed electrolyte compositions and the surfacepotential, ζ, extracted from the electrophoretic data. In the interpretationthis potential was considered independent of inter-particle distance. Toevaluate the deceleration of coagulation, from which the Hamaker con-stant is determined, we divide the coagulation time by τSm.

In Table 1, we present the results of such calculations conducted forexperimentswhose results are shown in Figs. 7 and 8 in terms of surfacepotential and charge as functions of pH.Wedo so for two salt concentra-tions, 10−3 M and 10−4 M, and two particle volume fractions, 10−6 and4 ∗ 10−6. While preparing Table 1, we excluded the data with coagula-tion times shorter than the Smoluchowski value, τSm, given by Eq. (5).

5 −28.9 1134.12405 1.010 −34.45 −0.00285 2.36E-195.1 −29.9 1665.31577 1.008 −35.85 −0.00298 2.54E-19

0.1 mg/ml in 0.1 mM NaCl, τSm = 100 s4.2 −14.6 388.909118 0.163 −18.75 −0.00065 2.06E-194.5 −18.0 943.820157 0.132 −23.70 −0.00076 4.23E-1912.1 −51.7 2813.28198 12.689 −56.95 −0.01805 1.31E-1912.2 −46.6 1019.51999 15.949 −50.35 −0.01715 9.30E-2013.5 −15.2 545.963125 316.328 −15.40 −0.02014 7.94E-22

0.1 mg/ml in 0.1 mM NaCl repeats, τSm = 100 s4.7 −29.1 6018.24953 0.120 −40.30 −0.0013 1.53E-184.8 −28.9 8911.66667 0.116 −40.10 −0.00128 1.54E-18

0.1 mg/ml Pt/C in 1 mM NaCl, τSm = 100 s6 −24.6 403.432586 1.001 −29.00 −0.00235 7.47E-2112.3 −43.2 1283.4414 20.953 −46.00 −0.0175 6.68E-20

0.1 mg/ml in 1 mM NaCl repeats, τSm = 100 s6 −32.8 894.712644 1.001 −39.70 −0.00335 3.12E-196.1 −32.5 601.771404 1.001 −39.20 −0.0033 3.07E-196.3 −32.4 1902.24475 1.001 −39.10 −0.00329 2.97E-196.4 −33.1 5206.99029 1.000 −40.10 −0.00338 3.06E-19


Besides, we excluded the cases where coagulation is so slow that thechanges in the aggregate size are comparable to the experimental error.

For the alkaline range, we obtain the Hamaker constant1.28 ⋅ 10−19J(±0.28 ⋅ 10−19J) (in parentheses we present the standarddeviation)when the salt concentration is 10−4 M. A close value (withinthe standard deviation) is obtained in the same alkaline range but athigher salt concentration, 10−3 M: −0.91 ⋅ 10−19J(±0.19 ⋅ 10−19J).In acidic media the average Hamaker constant turns out to be larger,especially, for the case of lower concentration of salt, 10−4 M,10.19 ⋅ 10−19J(±4.79 ⋅ 10−19J), while at 10−3 M, the deviation issmaller, and, with account for standard deviation, lies near theupper limit of the above result predicted for the base region,2.18 ⋅ 10−19J(±0.93 ⋅ 10−19J).

As possible reasons for the computed Hamaker constant variationsone can consider an error in the measured ζ-potential, that strongly in-fluences the electrostatic barrier, and accumulation of particles simulta-neously in the primary and secondary potential minimum, that isaccompanied by a slow particle transition between them [70]. Howeversometimes the discrepancy between the theory and experiment is solarge that it cannot be explained by such reasons, especially for smallparticles (of the order of 100 nm) [71]. Alternative explanation can re-late to heterogeneity of the particle surface. A higher value of Hamakerconstant obtained for the acidic range is indicative of an additionalattraction that is not taken into account within the framework of SEMand manifests itself as an apparent increase in the Hamaker constant.Such an additional attraction can be associated with heterogeneity ofsurfaces which often leads to the appearance of an attractive meanforce between the surfaces bearing mosaic charge. A stronger impactof heterogeneity can be expected for higher electrolyte concentrations.However, even at the lower ionic strengths, the EDL is relatively thincompared with the particle size (κa ≈ 5). Hence, sufficiently large re-gions with different charge densities can manifest themselves. Recallthat we deal with carbon particles modified by metallic platinum.Accordingly the surface heterogeneity is quite possible. The dynamicsof particles interaction can be also significant for the aggregation kinetics[71,86].

In Table 2, we present the coagulation times estimated from thesecond group of experiments whose results are shown in Figs. 10–12and corresponding estimates of Hamaker constant. In these calculations,we assumed the surface potential to be independent of the distance be-tween the particles.

The electrolyte concentration is the major parameter defining thecoagulation rate since it defines the ionic strength and, thus, the EDL

Table 2Initial data and results of calculation of Hamaker constants for assumptions: Pt/C of 10−6 volume

C NaClmM/l

pH ζ Coagulation time, s

(exp)

mV

3 10 −45.8 22,066.43 11 −47.9 38,687.73 11.5 −49.0 27,237.410 10 −43.2 11,371.710 11 −46.7 30,236.410 11.5 −49.3 25,450.215 10 −39.1 1407.515 11 −43.7 1770.615 11.5 −46.1 857.220 10 −37.8 352.220 11 −41.6 1251.720 11.5 −43.8 565.525 10 −37.5 355.825 11 −43.7 681.225 11.5 −45.2 368.840 10 −41.6 422.540 11 −40.7 267.940 11.5 −39.1 289.1

thickness. When the salt concentration is lower than 10−2 M, coagu-lation is nearly absent. At 1.5 · 10−2 M, the coagulation occurs at anoticeable rate that further increases at 2 · 10−2 M. At concentra-tions of 2.5 · 10−2 M and 4 · 10−2 M, the electrostatic repulsiondoes not affect the coagulation, which is characterized by theSmoluchowski rate. Computing the mean Hamaker constant for theconcentration range 1.5 · 10−2 M–2 · 10−2 M (here the result is mostreliable), we obtain H ≈ 0.71(±0.12) · 10−19 J. This value is lowerthan that estimated for the low salt concentration case which was pre-sented before. The smaller Hamaker parameter means slower coagula-tion than that expected from the theory.

6. Conclusions

Parallel investigation of electrophoresis and coagulation, bothdepending on ionic solution composition, can yield important informa-tion about the interfacial properties of particles and aggregative stabilityof their dispersions. However, in the case of nano-particle systems inter-pretation of experimental results is not easy and requires significantmodifications of traditional approaches. An overview of academic liter-ature has revealed that reasonable semi-quantitative correlations be-tween electrophoretic mobility and stability of model nano-colloidalsystems could be establishedwithin the scope of Standard ElectrokineticModel. At the same time, inmany cases application of SEM to the analysisof experimental data encounters difficulties, in particular, when the solu-tions contain more than two ions, the particle charge depends on thesolution composition and zeta-potentials are high. Such situations de-mand novel developments of SEM.

As an experimental example, we studied the influence of electrolytecomposition on the aggregative stability of a diluted suspension of280 nm carbon particlesmodified bymetallic platinum and experimen-tally established the following behavior of this disperse system. The sys-tem remains relatively stable at neutral pH but starts to coagulatewhenan acid is added.When adding a base (NaOH), the system is stable untilpH reaches the value of 12.2.With further increase of pH, the system co-agulates, and the coagulation ismore rapid than at low pH. Importantly,the coagulation threshold, observed at high pHs, is independent of thesalt concentration in contrast to that observed at low pHs.

We attempted to explain the above behavior through the variation ofelectrostatic repulsion forces that are described by the DLVO theory [10,11]. For example, the observed dependency of stability on pH can occurdue to the variation of surface charge density. Such a variation takesplace because the pH controls both the concentrations of hydrogen/

fraction at various salt concentrations and pHs, Smoluchowski coagulation time τSm¼300 s.

Ionic strengthmM

ζmV

qC/m^2

Hζ = constJ

3.1 −53.8 −0.0084 2.67E-194 −55.65 −0.0099 2.43E-196.162278 −55.65 −0.0123 1.89E-19

10.1 −47.3 −0.0126 1.05E-1911 −51.2 −0.0146 1.15E-1913.16228 −54 −0.0171 1.15E-1915.1 −41.9 −0.0133 6.78E-2016 −47 −0.0158 8.20E-2018.16228 −49.6 −0.0179 8.56E-2020.1 −40.05 −0.0145 5.50E-2021 −44.25 −0.0167 6.30E-2023.16228 −46.53 −0.0187 6.68E-2025.1 −39.55 −0.0159 4.74E-2026 −46.3 −0.0196 6.18E-2028.16228 −47.85 −0.0213 6.40E-2040.1 −43.6 −0.0226 4.40E-2041 −42.5 −0.0221 4.28E-2043.16228 −40.8 −0.0216 3.81E-20


hydroxyl ions and their adsorption or dissociation of interfacial ionogenicgroups. Another mechanism manifests itself at sufficiently large devia-tions of pH from the neutral value. In this case, the pH changes affectthe ionic strength and thus the Debye screening length, thereby changingthe electrostatic interaction forces even at constant interfacial charge. Themechanism associatedwith changing the Debye length is also expectedto be responsible for the decrease of electrostatic repulsion when in-creasing the salt concentration.

To verify if the changes in electrostatic interactions really control thedependency of stability on pH, we used a theoretical model thataccounted for electrostatic and Van der Waals interactions. The modelenables one to predict the interaction energy as a function of distancebetween the particles for arbitrary values of surface potentials andratio of Debye to particle radii. The computed energy was employedfor the prediction of coagulation time, which is usually being deter-mined in coagulation experiments.

To obtain information on the dependencies of surface potential andcharge on electrolyte composition,wemeasured electrophoreticmobilityof the Pt/C particles and extracted the surface zeta-potential from it byusing the Standard Electrokinetic Model. This calculation has demon-strated that, within certain ranges of ion concentrations, the surface po-tential, which is employed in the stability model, has noticeably largerabsolute value than the apparent zeta potential estimated from the mo-bility by using Smoluchowski formula.

The self-consistency of our approachwas checked via computing theHamaker constant for a set of experiments conducted at various electro-lyte compositions. If themodel is adequate such calculations should yieldthe same value of this constant (within the experimental error). Howev-er, we obtained the Hamaker constants varying from one experiment toanother. In the case of high solution pHs, the variation is not very signif-icant, and one can conclude that the model describes the system behav-ior, at least, semi-quantitatively. For low pHs, the computed Hamakerconstant varied substantially and sometimes took anomalously highvalues. Supposedly, this behavior is amanifestation of the surface hetero-geneity, which can be expected for the studied particles.

Acknowledgments

Financial support from European Commissionwithin the scope of FP7(project acronym “CoTraPhen”, Grant Agreement Number: PIRSES-GA-2010-269135) is gratefully acknowledged. The work was supported byCOST action CM1101. The authors would like also to thank Prof. S.S.Dukhin for valuable discussions.

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