Streaming potential studies of colloid, polyelectrolyte and protein deposition

Advances in Colloid and Interface Science 153 (2010) 1–29

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Advances in Colloid and Interface Science

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

Streaming potential studies of colloid, polyelectrolyte and protein deposition

Z. Adamczyk a,⁎, K. Sadlej b, E. Wajnryb b, M. Nattich a, M.L. Ekiel-Jeżewska b, J. Bławzdziewicz c

a Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek 8, 30-239 Cracow, Polandb Institute of Fundamental Technological Research, Polish Academy of Sciences, ul. A. Pawińskiego 5B, 02-106 Warsaw, Polandc Department of Mechanical Engineering, Department of Physics, Yale University, New Haven, Connecticut 06520-8286, USA

⁎ Corresponding author.E-mail address: [email protected] (Z. Adamcz

0001-8686/$ – see front matter © 2009 Elsevier B.V. Aldoi:10.1016/j.cis.2009.09.004

a b s t r a c t
a r t i c l e i n f o
Available online 2 October 2009

Keywords:Colloid depositionNanoparticle depositionParticle covered surfacesPolyelectrolyte depositionProtein depositionStreaming potential of covered surfaces

Recent developments in the electrokinetic determination of particle, protein and polyelectrolyte monolayersat solid/electrolyte interfaces, are reviewed. Illustrative theoretical results characterizing particle transport tointerfaces are presented, especially analytical formulae for the limiting flux under various deposition regimesand expressions for diffusion coefficients of various particle shapes. Then, blocking effects appearing forhigher surface coverage of particles are characterized in terms of the random sequential adsorption model.These theoretical predictions are used for interpretation of experimental results obtained for colloid particlesand proteins under convection and diffusion transport conditions. The kinetics of particle deposition and thestructure of monolayers are analyzed quantitatively in terms of the generalized random sequentialadsorption (RSA) model, considering the coupling of the bulk and surface transport steps. Experimentalresults are also discussed, showing the dependence of the jamming coverage of monolayers on the ionicstrength of particle suspensions. In the next section, theoretical and experimental results pertaining toelectrokinetics of particle covered surfaces are presented. Theoretical models are discussed, enabling aquantitative evaluation of the streaming current and the streaming potential as a function of particlecoverage and their surface properties (zeta potential). Experimental data related to electrokineticcharacteristics of particle monolayers, mostly streaming potential measurements, are presented andinterpreted in terms of the above theoretical approaches. These results, obtained for model systems ofmonodisperse colloid particles are used as reference data for discussion of experiments performed forpolyelectrolyte and protein covered surfaces. The utility of the electrokinetic measurements for a precise,in situ determination of particle and protein monolayers at various interfaces is pointed out.

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Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. Particle transfer and deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1. The convective-diffusion theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. Surface blocking effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3. Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3. Electrokinetics of particle covered surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.1. The theoretical model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.2. Theoretical results for particle covered surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3. Experimental results for particle covered surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.1. Characteristics of bare surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.2. Particle covered surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.3.3. Polyelectrolyte and protein covered surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

4. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2528

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2 Z. Adamczyk et al. / Advances in Colloid and Interface Science 153 (2010) 1–29

1. Introduction

Understanding deposition (irreversible adsorption) mechanismsof colloids and bioparticles at solid/liquid interfaces is of major sig-nificance for a variety of fields ranging from geophysics, materialand food sciences, pharmaceutical and cosmetic industries, medicalsciences, electrophoresis, chromatography, catalysis, etc.

Especially significant seem protein adsorption processes, whichare involved in blood coagulation, artificial organ failure, plaque for-mation, fouling of contact lenses, heat exchangers, ultrafiltration andmembrane filtration units. On the other hand, controlled proteindeposition on various surfaces is a prerequisite of their efficient se-paration and purification by chromatography, filtration, for biosen-sing, bioreactors, immunological assays, etc.

Besides practical importance in the fields mentioned above, ir-reversible adsorption of particles at various interfaces is of majorsignificance for basic colloid science, because interesting clues on thedynamic interactions can be extracted from the kinetics of theseprocesses, from the maximum coverage and the topology of particlelayers [1–14].

Owing to its significance, particle deposition has been extensivelystudied both theoretically [15–23] and experimentally, using mostlydirect methods such as optical microscopy [24–28] AFM [4,5,7–9],reflectometry [10,29–34], and othermethods [35–39]. Because of highprecision of measurements, these results, obtained mostly for mono-disperse latex or silica particles, gold particles or dendrimers, can beused as convenient reference data for protein deposition studies,which are also considerably more complicated [40–50].

However, despitenumerous studies devoted to the subject of proteindeposition kinetics there still appear discrepancies, even conflictingreports in the literature dealing with this subject. This situations islargely caused by the limited availability of direct experimental tech-niques working in situ, under dynamic conditions, e.g., in systemsexposed to laminar flows, applied to enhance mass transfer rates.

Often the solution depletionmethods have been used to determinethe amount of proteins adsorbed at solid/liquid interfaces [40].Although convenient to apply, these methods can become inaccurate,because of possible adsorption of solute molecules on glass walls andentrapment in pores. Also, transport conditions in such systems con-sisting of protein solutions in contact with stirred suspension of largerparticles (substrate surfaces) are rather poorly defined. Moreover, noinsight into the local coverage density or the structure of the adsorbedlayer can be gained because the actual surface area of the substrate isnot well known.

More precise are the gravimetric methods, especially the quartzmicrobalance (QCM) technique [40,51,52], although in this method theforce is determined rather than themass of adsorbed protein layer. As aresult the signal depends not only on themass of the protein but also onthe structure of the adsorbed layer and its hydrodynamic resistance.Therefore, the amount of adsorbed protein alone cannot be uniquelydetermined, and tedious calibration procedures are needed. Last but notleast, the substrate surfaces are practically limited to silver coatedquartzplates.

More reliable are optical methods like ellipsometry and reflec-tometry [45,53,54], or surface plasmon resonance (SPR) [40], whichcan be applied for in situ measurements. However, they also requirecareful calibration and are most effective for a high coverage range,preferably multilayer coverages.

Very precise and effective for the low coverage range are theisotope labeling [42,55,56] and the fluorescence methods, like thetotal internal fluorescence (TIRF) method [46–48]. However, they arerather tedious for application and expensive because specific tagsmust be attached to proteins, which may change their properties incomparison with native proteins.

One of few precise and efficient methods for determining particleand protein deposition is based on electrokinetic measurements, most

frequently the streaming potential [43,44,57,58] changes induced bydeposition of particles at solid/liquid interfaces. The method allowsfor in situ measurement of deposition kinetics, and the precision ofcoverage determination, comparable with 1% of a monolayer, which,is unprecedented by any other method. Usually experiments of thistype involving colloid particles and proteins have been accomplishedexploiting the Poiseuille flow, either in the circular [57,58] or parallel-plate channel [43,44,59–64] configurations. A characteristic feature ofsuch a flow pattern is that the adsorbed particles are effectivelyimmersed in a simple shear ambient flow, prevailing at channel walls[59,60].

Recently, an efficient approach aimed at complex characterizationof protein monolayer formation at polymeric substrates has beendeveloped [65]. The method is based on combination of electrokinetic(streaming potential and streaming current) measurements carriedout in the microslit setup with in situ reflectometric interfacespectroscopy and QCM.

Although the electrokinetic techniques proved successful indetermining protein adsorption kinetics [43,44,57,58,65], their wide-spread use was hindered because of the lack of appropriate theoreticalbackground. Therefore, they remained relative methods, requiringcalibration. Basic experimental studies with the aim of performingsuch calibrations have been performed using colloid particle suspen-sions [59–64]. These experiments have been interpreted theoreticallyin terms of the hydrodynamicmodel developed in Refs. [59,60], whichconsidered in an exact manner the damping of convection currents ofions in the vicinity of adsorbed particles. Within the framework of thismodel, which is in principle the first order extension of theSmoluchowski approach for bare surfaces [66], two constants havebeen calculated, characterizing additive effects stemming fromparticles and the interface. However, this linear model is exact inthe limit of the low particle coverage only. For describing highersurface coverage effects, fitting functions of exponential type havebeen proposed, whose validity was justified empirically, by a goodagreement with experimental data [59,60,64]. Quite recently a majorprogress has been achieved in this field since exact numericalcalculations have been performed, which allowed one to determinetheoretically the dependence of the streaming potential of particlecovered surfaces for the entire range of coverage met in practice, i.e.,up to 0.5 [67]. Also, interesting theoretical results for nonsphericalparticles (linear aggregates) have been reported in the literature [68].

Therefore, the goal of this paper is to review theoretical and ex-perimental results pertaining to electrokinetic measurements of par-ticle deposition at solid/liquid interfaces, with the emphasis focusedon the possibility of using these data to evaluate quantitatively theprotein and polyelectrolyte coverage, e.g., their concentration for theppb range.

Accordingly, the organization of the paper is the following. In thefirst section illustrative theoretical results characterizing particletransport to interfaces are presented, especially analytical formulaefor the limiting flux under various deposition regimes and expressionsfor diffusion coefficients of various particles. Then, the blocking effectsappearing for higher particle coverage are shortly characterized interms of the random sequential model (RSA). The coupling of the bulkand surface transport steps is discussed. Experimental results illus-trating characteristic features of particle deposition under convectionand diffusion transport conditions are discussed.

In the next section, theoretical and experimental results pertainingto electrokinetics of particle covered surfaces are presented. Theoret-ical models, both analytical and numerical, are discussed, enabling aquantitative evaluation of the streaming current and the streamingpotential as a function of particle coverage and their surfaces prop-erties (zeta potential). Experimental data applicable to electrokineticcharacteristics of particle monolayers, mostly the streaming potentialmeasurements, are presented and interpreted in terms of the abovetheoretical approaches. These results, obtained for model systems of

3Z. Adamczyk et al. / Advances in Colloid and Interface Science 153 (2010) 1–29

monodisperse colloid particles are used as reference data for dis-cussion of the more complicated experiments performed for poly-electrolyte and protein covered surfaces.

The utility of the electrokinetic measurements for a precise, in situdetermination of particle and protein monolayers at interfaces ispointed out.

2. Particle transfer and deposition

2.1. The convective-diffusion theory

As discussed previously [14] a typical kinetic run, i.e., the depen-dence of the surface concentration of particles N on the depositiontime t, can be schematically illustrated by the curve shown in Fig. 1.After the short transition time t1, a quasi-stationary state is attained,characterized by a linear increase in Nwith the deposition time. Then,the kinetic curve levels off and N attains its maximum value N∞ (oftenreferred to as the jamming, saturation, limiting or plateau coverage).The characteristic time of attaining the maximum coverage is denotedby t2. It is to mention that in the case of diffusion controlled transportto planar interfaces the kinetic curves show no linear region in respectto the deposition time [14,69]. However, by expressing particledeposition kinetics in terms of the square root of the time variable, t½

rather than the time, one can obtain kinetic runs similar to that shownin Fig. 1.

The first relaxation time t1 can be estimated from the simpledependence

t1 =δ2dD

ð1Þ

where δd is the diffusion boundary layer thickness and D is theparticle diffusion coefficient.

Because for the colloid particle size range, D varies between 10−8

and 10−7cm2s−1 and δd is of the order of 10−4cm (1μm)under typicalconvective transport conditions [14], t1 is typically of the order of 0.1–1s. This is a negligible value in comparison with the time of particledeposition experiments varying usually between 102 and 105s [13,35–38].

Fig. 1. Schematic representation of particle deposition kinetics, i.e., the dependence ofsurface concentration N on the time t; t1 is the relaxation time of establishing thesteady-state transport conditions and t2 is the relaxation time of forming the particlemonolayer, characterized by the surface concentration of N∞ (maximum coverage).

On the other hand, the second relaxation time t2 can be estimatedfrom the formula

t2 =N∞kcnb

ð2Þ

where kc is the mass transfer rate under the steady-state conditionsand nb is the uniform concentration of particles in the bulk.

Note that according to Eq. (2) the relaxation time t2 decreasesproportionally to the bulk concentration of particles.

It can be estimated from Eq. (1) that for typical parameters de-scribing colloid suspensions N∞=109cm−2, kc=10−4cms−1, nb=1010cm−3, the second relaxation time is 103s.

In the case of diffusion-controlled transport, t2 can be calculatedfrom the formula [69]

t2 =N2∞

Dn2b

: ð3Þ

In this case the relaxation time t2 decreases proportionally to thesquare of the bulk concentration of particles.

It can be estimated from Eq. (3) that for typical parameterscharacterizing colloid suspensions, N∞=109cm−2, nb=1010cm−3,D=10−8cm2s−1, the second relaxation time is 106s, which is muchlarger than the convection relaxation time. This indicates unequivo-cally that the diffusion transport conditions are by orders of mag-nitude less effective than the convection deposition regime.

As can be deduced from these estimations there are two essentialparameters, which quantitatively characterize particle deposition kineticsat interfaces:

(i) the initial deposition rate (governed by the mass transfer rateconstant kc), when the blocking effects stemming from thepresence of adsorbed particles are negligible,

(ii) the maximum or jamming coverage N∞ governed by theadsorbed layer topology.

Hence, the major goal of a successful theoretical approach wouldbe to predict values of kc and N∞ as a function of particle size, shape,concentration in the suspension, flow rate and configuration, interfaceshape, ionic strength and pH, and other physicochemical parameters.

Usually, the initial deposition problem is analyzed in terms of theconvective-diffusion theory, formulated by Levich [70] and thenextended by others [11,12,14,71,72] to incorporate specific forcefields generated by interfaces.

On the other hand, the considerably more complicated problem ofpredicting the maximum, mono- and multilayer coverage, as well asthe structure of particle monolayers, can be effectively treated interms of various mutations of the random sequential adsorption(RSA) model [14–23].

The convective-diffusion theory is based on the continuity (massconservation) equation, which assumes the form [14]

∂n∂t = −∇⋅j = ∇⋅ D⋅∇n +

1kT

ðD⋅∇ϕÞn−Uhn� �

ð4Þ

where n is the particle number concentration (local quantity), j is theparticle flux vector given explicitly by the formula

j = −D⋅∇n +1kT

ðD⋅FÞ = −D⋅∇n− 1kT

ðD⋅∇ϕÞn + Uhn: ð5Þ

D is the diffusion tensor, k is the Boltzmann constant, T is the absolutetemperature, ϕ is the interaction potential of the net conservative forceF=−∇ϕ incorporating the specific interaction potential ϕs with the


interface, external force potential, etc., and Uh=M·Fh+Mr·Tohis the

particle velocity resulting from hydrodynamic forces Fh and torques Toh.

It is to remember, however, that by formulating Eq. (4) all hydro-dynamic and specific interactions among particles were neglected.Moreover, the diffusion tensor occurring in Eq. (4) was assumed tobe independent of the particle concentration n and its gradient. As aconsequence of these simplifying assumptions, Eq. (4) remains strict-ly valid for diluted suspensions of non-interacting particles only.However, it still may be a useful approximation for calculating thetransfer and deposition rates of particles because the volume fractionof particles in these processes Φv=nv1 (where v1 is the volume of asingle particle) is usually of the order of 0.01 and less.

It is useful to derive the limiting forms of Eqs. (4) and (5) becausetheir analytical solutions are impractical in the general case due to acomplex dependence of the specific interaction potential on the dis-tance from the interface.

If all specific interactions are neglected as well as hydrodynamicboundary effects, Eq. (4) simplifies to the form called the Smolu-chowski–Levich equation [14]

∂n∂t = D∇2n− D

kT∇⋅ðFnÞ−V⋅∇n ð6Þ

where V is the unperturbed (macroscopic) fluid velocity vector.If the external force F and the flow vanish (diffusion controlled

transport conditions), the Smoluchowski equation is reduced to thesimple form

∂n∂t = D∇2n: ð7Þ

The non-stationary Eq. (7), which is linear in respect to n, can besolved analytically (using for example the Laplace transformationmethod) for many situations of practical interest, e.g., adsorption on aspherical interface from a finite or infinite volume [73–75].

Assuming the steady-state and neglecting external forces, Eq. (6)reduces to the simple form

∇― 2n−Pe

―V⋅∇

−n = 0 ð8Þ

where Pe = VchLchD∞

is the dimensionless Peclet number, ∇− =

Lch∇;−V = 1

VchV , Lch is the characteristic length scale and Vch is the

characteristic convection velocity.Eq. (8), often called the convective-diffusion equation, was ex-

ploited widely for the description of the transfer of particles to col-lectors of various geometry [3,24–26,70–72].

The boundary conditions for Eqs. (4)–(8) are specified in the formof the perfect sink model, expressed as

n = 0 at z = δmn→nb for z→∞ ð9Þ

where δm is the primaryminimumdistance, z is the separation in distancefrom the interface and nb is the uniform bulk particle concentration.

Interesting analytical solutions describing particle deposition ratecan be derived upon solving analytically the above limiting equations.For example, in the case of diffusion-controlled transfer of particles toa spherically shaped interface (collector) of radius R (representingeither a liquid drop, a gas bubble or a solid particle (see Table 1)) theflux of particles is described by the following equation, first derived bySmoluchowski [76]

−ja =D12

a + R+

D12

πt

� �1=2� �nb ð10Þ

where ja is the particle adsorption flux, D12=D+Dc is the diffusioncoefficient of the particle relative to the collector (whichmay undergo

diffusion in the general case) Dc is the collector diffusion coefficientand a is the particle radius. Obviously, for stationary collectors, Dc=0.

As can be deduced from Eq. (10), for longer times, when (a+R)2/D12t1<1, the flux attains a steady-state value equal to

−ja =D12

a + Rnb = kcnb ð11Þ

where kc =D12

a + R is the mass-transfer rate constant.

It can be estimated by assuming a=R=100 nm that the relaxa-tion time t1 equals 4×10−2s. For a=1000 nm the relaxation timebecomes 1s.

From Eq. (11) one can deduce that for R→∞ (adsorption on aplanar interface), the flux remains unsteady for all times and is givenby the well-known expression

−ja = kcnb =Dπt

� �1=2nb: ð12Þ

As can be noticed, the flux vanishes with time proportionally tot−1/2, which means that particle adsorption on planar interfaces,driven by diffusion alone, becomes a very inefficient mode of trans-port for long times.

By integrating Eq. (12), one obtains the expression for surfaceconcentration of adsorbed particles

N = 2Dtπ

� �1=2nb: ð13Þ

This equation can be exploited to determine the particle diffusioncoefficient by experimentally measuring the surface concentration Nof irreversibly adsorbed particles as a function of time.

Analogous solutions for a two flat interface system (plates) shownschematically in Table 1 are the following

kc =2Dh

∑∞

l=1e−ð2l− 1Þ2π2

4Dth2

N = h 1− 8π2 ∑

∞

l=1

e−ð2l− 1Þ2π2Dt

4h2

ð2l− 1Þ2

26664

37775nb

: ð14Þ

Because the diffusion transport in the case of planar interfacesbecomes very ineffective for long times, one often applies fluid con-vection in experiments, either in the form of stirring or in a morecontrolled manner, e.g., by creating flows of a desired configurationand intensity. In this way, particles are brought to the region close tointerfaces, whichmakes their concentration uniform, except for a thinlayer called the diffusion boundary layer δd [14].

In the case of convection-driven transport, t1 is often negligible incomparison with typical experimental times, thus one can use fordescribing particle transfer rates the stationary convective-diffusionequation, Eq. (8). This equation can be further simplified for manysituations of practical interest to one-dimensional forms, with theperpendicular fluid component independent of the position over thecollector, as is the case for the rotating disk [69] or impinging-jetcollectors [14]. Interfaces exposed to such kind of flows are calleduniformly accessible surfaces. In this case, one can express the masstransfer coefficient in the general form [14]

kc = −ja = nb =D

LchΓ43

� � C⊥Pe3

� �1=3ð15Þ

where Γ 43

� �= 0:893 is the Euler gamma function value for 4/3 and C⊥

is the dimensionless constant depending on the flow configuration.

Table 1Bulk transfer rate constants (reduced flux) kc and surface concentration expressions for uniformly accessible surfaces.

Surface and flow configuration Transfer rate constantkc=|j| /nb[cms−1]

Surface concentration ofparticles N [cm−2]

Solid sphere in a quiescent suspension

D + Dca + R

� + D + Dc

πt

� 1=2 D + Dca + R

� nbt + 2 D + Dc

π t� 1=2 nb

Planar interface in a quiescent suspension

Dπt

� 1=2 2 Dtπ

� 1=2nb

Two plate system in a quiescent suspension

2Dh ∑

∞

l=1e−

ð2l�1Þ2π24

Dth2 h 1� 8

π2 ∑∞

l=1

e−ð2l�1Þ2π2

4Dth2

ð2l� 1Þ2

24

35nb

Sphere in uniform suspension flow⁎⁎ (near stagnation point)

0:889A1 = 3f

V1 = 3∞ D2 = 3

R1 = 3 valid for ϑ < π4 kcnbt

The rotating disk

0:620Ω1 = 2D2 = 3

ν1 = 6 kcnbt

Radial impinging-jet ⁎⁎⁎

0:776α1 = 3

r V1 = 3∞ D2=3

R2=3

0:530α1 = 3

r Q1 = 3D2=3

R4=3

kcnbt

Remarks; definitions.The above expressions were derived by neglecting all specific and hydrodynamic (short-range) interactions, as wall as surface blocking effect.D — diffusion coefficient of particles.Dc — diffusion coefficient of the collector.nb — number concentration of the suspension in the bulk [cm−3].⁎⁎The flow parameter Af is given by the expression.

Af ðReÞ =32

1 +316Re

1 + 0:249Re0:56

!for Re =

2RV∞ν

< 300

⁎⁎⁎αr flow parameter (dimensionless).αr=4.03+0.628Re−1.89×10−3Re2 for h /R=1, Re<20αr=1.78+0.186Re+3.4×10−2Re2 for h /R=1.6, Re<20αr=1.438+0.337Re+1.25×10−2Re2 for h /R=2, Re<20αr for other Re number and (h /R) are given in Ref. [14].V∞=Q /πR2

Re=2V∞R /vQ — volumetric flow rate in the capillary tube.



Analytical expressions for kc are collected in Table 1 for somesurfaces of practical interest. They have been calculated usingEq. (15), the definition of the Peclet number, and by substitutingLch=a [14].

As can be noticed, in all cases kc increases proportionally to D∞2/3

rather than to D∞ as intuitively expected. Because D∞ is inverselyproportional to particle size, this means that the convective fluxdecreases as a−2/3 with particle radius. It is also interesting to observethat kc is rather insensitive to the fluid velocity V∞, increasing in allcases proportionally to V∞

1/3, except for the case of the rotating diskwhere kc remains proportional to V∞=0.886 (Ων)1/2, where Ω is thedisk angular velocity.

In the case of non-uniformly accessible surfaces, expressions for kccan be derived for symmetric flows, using the two-dimensional formof Eq. (8) [14]. For small colloidal particles, when the interceptioneffect can be neglected, one can express kc, which is now a localquantity, in the general form

kcð−xÞ =DLch

Pe1=3

Γ 43

� �gcð−xÞ

ð16Þ

where gc(x−) is the dimensionless function of the tangential

coordinate x−.The integration of the localmass transfer rate over the entire collector

surface gives the averaged expression for rate constant in the form:

⟨kc⟩ =1Sc

∫Sc

kcð−xÞdSð−xÞ =D

LchΓ43

� �⟨gc⟩

Pe1=3 =DLch

⟨Sh⟩

⟨gc⟩ =1Sc

∫Sc

dSð−xÞgcðxÞ

: ð17Þ

where ⟨Sh⟩ = ⟨kc⟩LchD is the averaged mass transfer Sherwood number.

As can be deduced from Eq. (17), the averaged flux, which isindependent of the tangential coordinate, increases proportionally toPe1/3 for all collectors, except for the fluid sphere.

For the sake of convenience the local and average flux expressionsfor various collectors are given in Table 2.

It is interesting to observe that in the case of parallel-plate andcylindrical channels, widely used for protein deposition studies by thestreaming potential method, the expressions for the local mass trans-fer rates are [60]:

kc = 0:776⟨V⟩1=3D2=3

b2=3−x 1 =3 = 0:678V1 = 3m D2=3

b2=3−x1 =3 ðparallel�platechannelÞ ð18Þ

kc = 0:854⟨V⟩1=3D2=3

R2=3−x1 =3 = 0:678V1 = 3m D2=3

R2=3−x1=3 ðcylindricalchannelÞ ð19Þ

where b is the half width of the parallel-plate channel, x−=x /L is thedimensionless distance from the inlet to the channel, L is its length, Ris the cylindrical channel radius,

⟨V⟩ =23Vm =

Pb2

3ηLð20Þ

is the averaged fluid velocity in the channel, P is hydrostatic pressuredrop along the channel and η is the dynamic viscosity of the fluid.

In the case of the cylindrical channel one has

⟨V⟩ =12Vm =

PR2

8ηL: ð21Þ

Note that the local flux for both the parallel-plate and the cylindricalchannel diverges for x−→0 (entrance to channels) proportionally tox−−1/3. This non-physical behavior is caused by the postulate that theinitial particle concentration distribution in the channel was uniform upto the interface. In reality, at the entrance area, the suspension is devoid

of particles in the thin layer adjacent to the interface. By considering this,the singularity occurring in Eq. (18) is eliminated [77].

The flux averaged over the entire channel area is

kc = ⟨kc⟩ = 1:1646⟨V⟩1 =3D2=3

b1=3L1=3: ð22Þ

In the case of the cylindrical channel one has

kc = ⟨kc⟩ = 1:281⟨V⟩1=3D2=3

R1=3L1=3: ð23Þ

It is interesting to note that the local flux in this channel attains avalue equal to the average flux at the distance −x = 8

27 = 0:296 fromthe entrance.

It is to mention, however, that the above analytical expressions forthemass transfer rates are strictly valid for the nanoparticle size range.For larger particles, of the size of micrometers, the kinetics of theirtransfer to interfaces can be predicted more accurately by solvingthe governing mass-balance equation, Eq. (4), which incorporatesthe effects of interception, specific, external and hydrodynamicforces in an exact manner. Numerical solutions of this equation forcollectors of practical interest are discussed in detail elsewhere[14,72].

Implementation of the above equations also requires the knowl-edge of the diffusion coefficient of particles, which can be calculatedfrom the Stokes–Einstein relationship

D =kT6πη

⟨RH⟩−1 ð24Þ

where the quantity ⟨RH⟩ is defined as the orientation averaged hydro-dynamic radius of a particle of arbitrary shape.

Obviously for solid spheres ⟨RH⟩ is equal to the sphere radiusa. For a solid sphere doublet ⟨RH⟩=1.39a [78], for a triplet (linearaggregate) ⟨RH⟩=1.73a [79,80] and for a linear aggregate composedof ns equal sized spheres one has [81]

⟨RH⟩ =ns

ln2ns−0:25a =

1ln2λ−0:25

L2

� �ð25Þ

where λ = L2a

� = ns and L=2ns a is the length of the aggregate (see

Table 3).For particles having the shape of prolate spheroids, one can derive

the analytical expression [14,82]

⟨RH⟩ =1− b2

a2

� �12

cosh ab

� a ð26Þ

where a is the spheroid longer axis and b is the shorter axis (seeTable 3.).

In the limit of very large axis ratio parameterλ = ab, Eq. (26) becomes

⟨RH⟩ =ns

ln2λa =

1ln2λ

L2

� �: ð27Þ

On the other hand, for oblate spheroids, their hydrodynamic radiusis given by the expression [14,82]

⟨RH⟩ =1− b2

a2

� �12

cos�1 ba

� a ð28Þ

where a is the spheroid longer axis and b is the shorter axis.In the limit of large axis ratio parameter λ = a

b, Eq. (28) becomes

⟨RH⟩ =2πa: ð29Þ

Table 2Pe definitions and bulk transfer rates (local and averaged) for non-uniformly accessible surfaces.

Collector and flow configuration Local rate constant kc(x−) Averaged rate constant ⟨kc⟩

Sphere in uniform flow

0:776f1ð�xÞA1 = 3f V1 = 3

∞ D2=3

R2=3

f1 =sinϑ

ϑ−12sin2ϑ

� �1=3

ϑ = �x = x= R

0:624A1 = 3f

V1 = 3∞ D2 = 3

R2 = 3

Radial impinging-jet RIJ

0:776f1ð�xÞα1 = 3r V1 = 3

∞ D2=3

R2=3

�x =πr2R

f1ð�xÞ = sin�x�x � 1

2sin2�x

� �1=3

0:389 α1 = 3r V1 = 3

∞ D2 = 3

R2 = 3

Parallel-plate channel

0:776⟨V⟩1=3D2=3

b2=3�x1=3�x = x = b

⟨V⟩ =23Vm =

P b2

3ηL

1:165 ⟨V⟩1 = 3D2 = 3

b1 = 3 L1 = 3

Plate in uniform flow

0:3395V

1 = 2∞ D2=3

ν1=6L1=2�x1=2�x = x = L

0:678 V1 = 2∞ D2 = 3

ν1 = 6L1 = 2

Cylinder in uniform flow

0:678f2ðϑÞA

1=3f V

1 = 3∞ D2=3

R2=3

f2ðϑÞ = sin1=2ϑ

∫ϑ

o

sin1=2ϑ′dϑ′

" #1=3

ϑ = �x = x= R

0:582A

1 = 3f

V1 = 3∞ D2 = 3

R2 = 3

Cylindrical channel

0:854⟨V⟩1=3D2=3

R2=3�x1=3�x = x = R

⟨V⟩ =12Vm =

PR2

8ηL

1:281 ⟨V⟩1 = 3D2 = 3

R1 = 3L1 = 3


For the sake of convenience, expressions for RH for these and otherparticle shapes, including bent spheroids and sphere aggregatesforming semi-circles and circles, are collected in Table 3.

It is to mention that the limitation of the expressions for kc is thatthey are valid for initial stages of particle deposition only, whendisturbances to their transport stemming from previously adsorbedparticles remain negligible. With the progress of particle adsorption,there appear deviations from linearity, in respect to the time, causedby the following main reasons:

(i) particle desorption phenomena(ii) volume exclusion effects induced by pre-adsorbed particles.

The former effect is important for reversible systems when theenergy minimum depth is not too long, which is usually the case forparticle size below 10 nm. The effect of desorption can be considered ina quite simple way by analyzing the specific particle-interface energyprofile, which enables one to evaluate explicitly the desorption rateconstant kd, either analytically or numerically. Then, the kineticboundary conditions for bulk transport equations in a linear form withrespect to bulk suspension concentration and surface coverage can beformulated. Such boundary value problems can be solved analyticallyfor many cases of practical interest, e.g., for diffusion transport tospherical or planar interfaces in contact with suspensions of particles[6,69,73,74].


In contrast, a proper description of the volume exclusion effect(often called less accurately the surface blocking effect) is morecomplicated. The direct cause of the volume exclusion effects is thedynamic interactions, among adsorbed and moving particles.These interactions depend not only on the particle coverage but

Table 3Averaged hydrodynamic radius <RHN values for various particle shapes.

Particle shape <RHN expression

Solid sphere

⟨RH⟩=a

Fluid sphere

⟨RH⟩ =1 + 2η = 3ηf

1 + η = ηfa

Solid sphere doublet

⟨RH⟩=1.39a

Solid sphere triplet

⟨RH⟩=1.73a

Solid sphere aggregate ns=10

⟨RH⟩=3.61a

Solid sphere linear aggregate

⟨RH⟩ = 1ln2λ−0:25λa

Prolate spheroid

⟨RH⟩ =1−b2

a2

� �12

cosh�1 ab

a

Oblate spheroids

⟨RH⟩ =1−b2

a2

� �12

cos�1ba

a

Prolate spheroid (slender)

⟨RH⟩ = 1ln2a

ba = 1

ln2ab

L2

�

Slender cylinder

⟨RH⟩ = 1ðln2λ−0:11Þ

L2

�

also on their distribution over interfaces, which is in turn de-pendent on mechanisms of particle transfer like diffusion, flow,migration, etc. As a result of this non-linear coupling, a rigorousanalysis of surface exclusion phenomena becomes impractical inthe general case. However, approximate results of quite general

Remarks, Ref.

Analytical (Stokes law)

Analytical expressionη

f

— fluid viscosityη — medium viscosity

Analytical, Ref.[78]

Numerical solution of the Stokes equation [81]

Numerical solution of the Stokes equation [81]

Asymptotic expression for 100≤ns≤300 λ=ns [81]

aNb, λ=a/bN1 analytical solution

aNb, λ=a /bN1 analytical solution

λ=L/2b approximate solution [14], valid for λ≫1

λ=L/2b approximate solution, valid for λ≫1 [14]

Table 3 (continued)

Particle shape <RHN expression Remarks, Ref.

Bent spheroid

⟨RH⟩ = 11112 ln 2λ−0:31ð Þ

L2

� Approximate solution, λ=L/2b valid for λ≫1 [128]

Half circle approximated by ns spheres

⟨RH⟩ = 11112ln2λ + 0:084

λa Numerical solution valid for 100≤λ≤400 [81]

Bent spheroid

⟨RH⟩ = 10:95ln2λm0:02

L2

� L=2πd,λ=L/2b≫1 approximate solution [128]

Circle (ring) made of ns spheres

⟨RH⟩ = 11112ln2λ + 0:67

λa λ=ns numerical solution valid for 100<λ<450 [81]

Disk

⟨RH⟩ = 2π a Analytical, valid for d/a≪1, d — disk thickness [14]


validity can be derived using various RSA approaches discussed inthe next section.

2.2. Surface blocking effects

A limitation of the convective-diffusion theory discussed aboveis lack of the possibility of deriving information on surface blockingeffects, maximum (jamming) coverages, and the structure of particlemonolayers. This can be done within the framework of other ap-proaches, exploiting statistical, rather than deterministic concepts.The most efficient in this respect seems to be the above mentionedRSA approach, whose main advantages are simplicity, flexibility andefficiency in generating large particle populations. This enablesone to derive blocking functions, jamming limits for particles ofvarious shape, pair correlation functions, and so forth. This kindof information can be then exploited as boundary conditions forthe bulk transport equations, which makes it possible to solve

the coupled bulk and surface transport problem in a proper way[14,69].

The general rules of theMonte Carlo type simulation scheme basedon the RSA approach are [15–19,21].

(i) an adsorbing (virtual) particle is created, whose position andorientation is selected at random within prescribed limitsdefining the adsorption domain,

(ii) if the virtual particle fulfills prescribed adsorption criteria, it isadsorbed with unit probability and its position remainsunchanged during the entire simulation process (localizedand irreversible adsorption postulate),

(iii) if the adsorption criteria are violated, a new attempt is madethat is fully uncorrelated with previous attempts.

It is interesting to note that the RSA model is applicable for par-ticles of various shape and various dimensionality like a quasi one


dimensional (1D) adsorption on a line segment, a two-dimensional(2D) adsorption on planar surfaces of finite (surface features) orinfinite extension and three-dimensional (3D), adsorption (referredto as random addition) in the space.

It is advantageous for analyzing these results to introduce thedimensionless coverage,which for the 2D case (adsorption at a surface),can be defined as

Θ = ðSg =ΔSÞN ð30Þ

where Sg is the characteristic cross-section of the particle andΔS is thesurface area of the adsorbing domain, for example a surface featureand N is the number of particles adsorbed at ΔS.

For spherical particles one can define Sg unequivocally as πa2 so

Θ = πa2N =ΔS: ð31Þ

On the other hand, in simulations based on the RSA scheme,particle adsorption probability for a given coverage, defined as theavailable surface function ASF [14,16–18,21] or less properly as theblocking function [2,70,72], can be calculated as

ASFðΘÞ = BðΘÞ = Nsucc =Natt ð32Þ

for Natt→ ∞, where Nsucc is the number of successful adsorptionevents and Natt is the number of virtual adsorption attempts.

Usually, in these simulations, the hard particle interaction potential[14] is used for describing particle–particle interactions. On the otherhand, the particle collector interactions are described by the perfectsink interaction potential, originally introduced by Smoluchowski[76]. These idealized potentials are especially suitable for colloidparticles, when the double layer thickness remains smaller thanparticle dimensions. For nanoparticles, the exponentially decayingYukawa repulsive potential is used [11,14], and the particle collectorinteractions are described by the attractive Yukawa potential.

In the case of surfaces bearing isolated adsorption centers of the sizecomparable with the adsorbing particles, a modified RSA simulationscheme is used whose first step consists in the deposition of sitesaccording to the classical RSA of desired surface density Ns /ΔS and thecoverageΘs=πas2Ns /ΔS (whereNs is the number of centers and as is thesite radius). The simulations of random site adsorption can be carriedout either for point-like sites distributed randomly over a homogeneoussurface with the zero minimum distance [83], or for sites having finitedimensions, e.g., hard disks of the diameter 2as incorporated into thesubstrate [84], or hard spheres attached to the surface [85].

Because of its major significance, adsorption on continuous surfaces(bearing uniformly distributed centers of the size much smaller thanparticle dimensions) haswidely been studied in terms of the RSAmodeldescribedabove.Most results concernhard spherical particle adsorptionon planar interfaces of infinite [15–23] or finite extension [86,87].

There also exist results for polydisperse spherical particles [88] andfor anisotropic hard particles of a convex shape like squares [89],rectangles (cylinders), spherocylinders (disk rectangles) and ellipses(spheroids) [17]. Results are available for particles interacting via a shortrange repulsive potential resulting from the electric double layers[1,11,12,14,22].

In these simulations one can derive not only the number of adsorbedparticles in the jamming state but also under transient states, which canbe physically linked to quasi-time variable. In this way the kinetics ofparticle deposition can be determined. For very long simulation times,the final state is achieved when no more particles can be adsorbedand the jamming coverage Θ∞ is attained. This is the most importantparameter, derived from RSA simulations by extrapolation, because itcharacterizes the interface capacity to accommodate particles. Forhard spherical particles it has been determined thatΘ∞=0.547 [15,18],which is markedly smaller than the maximum hexagonal packingof spheres in 2D, equal π = 2

ffiffiffi3

p= 0:9069 or the regular packing, equal

π/4=0.7854 [14]. For randomly oriented squares Θ∞=0.53 and forellipses characterized by axis ratio 2:1, Θ∞=0.583 [17]. Values of Θ∞ forparticles of other shapes can be found in Ref. [14]. It is to mention thatthese values correspond to the side-on adsorption of particles. Forelongated particles, their adsorption on quasi continuous surfaces canalso occur at arbitrary orientations, which can be described in terms ofthe unoriented adsorption regime, analyzed in detail in Refs. [19,90,91].

It was shown in Ref. [14] that the above results can also be used forinterpretation of experimental results for particles interacting via theshort range Yukawa potential upon defining the effective interactionrange h⁎. Using this concept one can calculate the jamming coverage forinteracting particles (referred to as the maximum coverage) from thesimple relationship valid for both spherical and anisotropic particles

Θmx = Θ∞1

ð1 + h*Þ2 : ð33Þ

As discussed [11,12,22], the effective interaction range h⁎ remainsproportional to the thickness of the electric double layer Le = εkT

2e2 I

� �1=2(where ε is the dielectric permittivity of the solvent, e is the ele-mentary charge and I is the ionic strength of the electrolyte), with theproportionality constant of about two, for particle size range of 100–500 nm [14].

Further details on the dependence of the effective interactionrange on the ionic strength and particle size for various particleshapes can be found in Ref. [14].

In addition to the jamming coverage, from RSA simulations one canalso determine the available surface function (blocking function)whichhas major significance for predicting particle adsorption kinetics. It wasdemonstrated [16] that for not too high coverage, one can approximatethe blocking function by the second order series expansion

BðΘÞ = 1−C1Θ + C2Θ2 + 0ðΘ3Þ: ð34Þ

For spheres C1=4 and C2 = 6ffiffiffi3

p= π = 3:31. For other particle

shapes, the C1 and C2 constants are given in Ref. [14].Results calculated from Eq. (34) reflect well the exact data derived

from simulations for particle coverage Θ<0.4.On the other hand, for Θ approaching the jamming coverage, the

blocking function for spheres can be well approximated by the ex-pression [16]

BðΘÞ = 2:31 1− ΘΘ∞

� �3: ð35Þ

For non-spherical particles, the blocking function, for a coverageclose to the jamming limit, assumes an analogous form [17]

BðΘÞ = C∞ 1− ΘΘ∞

� �4ð36Þ

where the dimensionless constant C∞ varies between 2.8 and 3.2 forspheroids [14].

In the case of spheres, one can also formulate an approximateanalytical expression which fits well the blocking function for theentire range of coverage [16]

BðΘÞ = 1+0:812Θ

Θmx+ 0:4258

ΘΘmx

� �2+ 0:0716

ΘΘmx

� �3� �1− Θ

Θmx

� �3

ð37Þ

where Θmx=0.547.It is to mention that the above results have been obtained within

the framework of the classical RSA model, postulating a two-dimensional adsorption of particles. In more refined approaches,referred to as the generalized RSA model, a more realistic, three-dimensional motion of particles in the adsorption layer is considered


[23]. It was demonstrated that under the conditions of a quasi-stationary transport (convective diffusion) the flux to particle coveredsurfaces can be expressed as [14,23]

jðΘÞ = j0Κ−ΒðΘÞ

1 + ðΚ−1Þ−ΒðΘÞð38Þ

where j0=−kcnb is the initial flux for bare surfaces analyzed above,K=ka /kc is the coupling constant, ka is the adsorption constantdescribing the rate of particle transfer through the adsorption layer[14] and B

−(Θ) is the generalized blocking function describing the

effect of three dimensional particle deposition process.Because of mathematical difficulties, the generalized blocking

function B−(Θ) has been evaluated in an exact way for the low

coverage range only [23]. Since the deviation from the 2D functiondescribed by Eq. (34) was rather minor, it was suggested that forhigher coverage range, close to jamming one can as well approximateB−(Θ) by Eq. (37).On the other hand, the deposition rate constant ka can be evaluated

as [14]

ka =D2a

1

1 + 12 ln

2aδm

� � ð39Þ

where δm is the minimum distance between the particle and theinterface.

Using Eq. (39), the expression for the coupling constant K becomes

K =D

2akc

1

1 + 12 ln

2aδm

� � =1

2Sh 1 + 12 ln

2aδm

� � ð40Þ

where values of the bulk transfer rate constant kc are given inTables 1–2.

It was shown in Ref. [14] that for a particle size below 100 nm,K≫1 under typical forced convection transport conditions.

Using these data one can calculate particle deposition kinetics forthe entire range of coverage by numerically integrating Eq. (38).

In the case of diffusion controlled transport the situation is morecomplicated since particle deposition kinetics can only be evaluatedvia numerical solutions of governing transport equations with theboundary condition at the interface having the form [69]

jðΘ;tÞ = kanðδaÞ−ΒðΘÞ ð41Þ

where n(δa) is the local particle concentration at the edge of theadsorption layer δa (subsurface concentration).

2.3. Experimental results

The validity of the above theoretical predictions has beenconfirmed by a variety of experimental results concerning particledeposition in model systems. The most reliable results were obtainedformonodisperse colloid particles using direct experimental methods,such as optical microscope observations [1,2,6,23–25,28] or AFMtechniques for latex [4,5,10] colloid gold [8,34] and dendrimersuspensions [9,33]. One can also apply the electron microscopy todetermine the particle coverage [39].

Most of these experimental studies are devoted to kinetic aspectsof particle deposition, which allow one to determine the two mostrelevant parameters, namely the initial flux (deposition rate constantkc0), and the maximum coverage Θmx as a function of particle size and

ionic strength of suspensions.Typical examples of depositionkinetics determinedbyBrouwer et al.

[34] for colloid gold particles (averaged diameter 13.4 nm) at a siliconsubstrate covered bya25 nmthick silica layer, are shown in Fig. 2. In this

study, the impinging jet cell was used, combined with a reflectometricmethod of particle coverage determination using the thin island filmtheory. As canbe seen, for short deposition time, all kinetic curves followthe same straight line dependence, irrespectively of the ionic strengthofparticle suspension, which indicates that the blocking effects werenegligible. On the other hand, for longer deposition time, particlecoverage attained its maximum value, which decreased with thedecrease in the ionic strength. Hence, the maximum coverage forI=13.6 mMwas equal to 0.22 and for I=3.6 mM it was equal to 0.15.This effect can be interpreted in terms of increased blocking effectsstemming from the lateral repulsion among deposited particles.

It is interesting to mention that the experimental kinetic runsshown in Fig. 2 were quantitatively interpreted in terms of the RSAmodel using the integrated form of Eq. (38) for K=45 (these theo-retical results are depicted by solid lines in Fig. 2).

Analogous kinetic runs were reported for other convection con-trolled systems, for example by Kleimann et al. [10], who studieddeposition of positively charged amidine latex particles (averaged sizeof 22.5 nm) at oxidized silicon, covered by a 100 nm of thickness silicalayer in the impinging jet cell, using the reflectometric method. Particlecoverage was determined using the AFM measurements. However, noattempt was undertaken to analyze the kinetic runs theoretically.

Similar kinetics of particle deposition were reported by Kozlovaand Santore [38] who studied deposition of silica particles (averageddiameter 460 nm, fluorescently labeled by fluorescein) on a glasssubstrates covered by a thin (10 nm) silica layer and modified bypositively charged polyelectrolyte, poly(dimethylaminoethylmetha-crylate) (pDMAEMA).

Analogous deposition kinetics under convective transport havebeen observed for proteins as well. For example Lin and Hlady [46]have determined, using the TIRF method, adsorption kinetics of HSA(human serum albumin) at silica gradient surfaces, using a parallelplate channel flow cell.

On the other hand, Wertz and Santore [48] measured deposition ofBSA and fibrinogen using the TIRF method in the parallel-plate channelcell made of microscope glass slides modified by silane adsorption, atwall shear rate of 5 s−1 and various bulk concentrations of the protein(25–100 ppm). Their experimental results (points) for fibrinogen areshown in Fig. 3 as the dependence of the surface concentration of theprotein expressed in mg/m2 on the dimensionless time of depositionτ=kccbt/Nch (where Nch is the characteristic coverage equal to 10 mg/m2). The solid and the dashed lines denote the theoretical resultscalculated by numerical integration of Eq. (38) for K=2×103 andK=103, respectively. As can be seen in Fig. 3, the theoretical resultsdescribe properly the experimental data for the entire range ofdeposition time and the bulk protein concentration. It is interestingto observe, however, that the maximum coverage of fibrinogenincreased slightly with its bulk concentration. This suggests that eithersome of protein molecules were forming a second layer due to surfaceaggregation or they were adsorbing end on at the surface.

It is to mention, however, that protein deposition experiments areoften carried out under diffusion controlled transport. Therefore,numerous studies of colloid particle deposition under diffusion-con-trolled transport have been carried out using the AFM [4,5,10] oroptical microscope observation methods [6] with the aim of collectingreliable reference data for interpreting protein deposition kinetics.Typical examples of kinetic runs obtained in this case of positivelycharged amidine latex particles (average diameter 800 nm) adsorbingon bare mica are shown in Fig. 4 [92].

As can be seen, these kinetic results, expressed in terms of thesquare root of adsorption time t½ resemble very closely, the depen-dencies predicted for convective transport, shown in Fig. 2. For thedeposition time shorter than ca. 10 h (when Θ<0.2), particle de-position kinetics remained a linear function of t½, independently ofthe ionic strength of the suspension. This result agrees with thelimiting analytical solution derived from Eq. (13), which was derived

Fig. 2. Deposition kinetics of gold colloid particles on oxidized silicon at different ionicstrengths determined by reflectometry using the impinging-jet flow cell. The ionicstrength of the solutions, varied by addition of NaCl was the following: 1. 13.6 mM(triangles — △), 2. 8.6 mM (diamonds — ◊), 3. 6.1 mM (squares — □), and 4. 3.6 mM(circles — ○). The coverage was calculated from the reflectometer signal using the thinisland film theory. The solid lines represent the theoretical results calculated byintegrating Eq. (38) for K=45. From Ref. [34].


by neglecting all blocking effects and specific interactions amongparticles. However, for longer times, especially at lower ionic strengthof 2×10−5M, experimental results deviate from these analyticalpredictions, attaining maximum values, increasing with the ionicstrength. This is a direct manifestation of blocking effects due torepulsion among particles. As can be seen in Fig. 4, the experimentaldata were quantitatively accounted for by theoretical results obtainedby numerical solution of the governing diffusion equation, Eq. (7),with the boundary condition described by Eq. (41). It is interesting toobserve that the theoretical results derived from the widelyused Langmuir model, where the blocking function is described by1−Θ /Θmx (dashed lines in Fig. 4) do not reflect properly the experi-mental kinetic runs.

For further analysis of streaming potential data for particle coveredsurfaces it is important to know not only the coverage of particles butalso their distribution over the substrate surface. This can bequalitatively seen in Fig. 5 (part a) showing the amidine latex mono-

Fig. 3. Surface concentration of adsorbed bovine fibrinogenN as a function of the reduceddeposition time τ=kccbt/Nch. The points denote the experimental results obtained by theTIRF method in the parallel-plate channel made of microscope glass slides modified bysilane adsorption, atwall shear rate of 5 s−1 and various bulk concentrations of theprotein[48]: 1. cb=100 ppm (full circles — ●), 2. cb=50 ppm (circles — ○), and 3. cb=25 ppm(triangles — △). The solid line denotes exact theoretical results calculated by solving thegoverning diffusion equation with the blocking effect given by the RSA model forK=2×103 and the dashed line for K=1×103.

layer formed at mica for I=2×10−5M, Θ=0.27 and I=10−3M,Θ=0.44. A short-range, liquid-like ordering of particles is clearlyvisible in this picture. It is interesting to mention that this micrographwas taken using optical microscopy under wet conditions, whichpreserved particle monolayers intact.

The structure of particle monolayers can be quantitatively char-acterized in terms of the pair correlation function g(r /a) (oftenreferred to as the radial distribution function) [15,16,18,21]. Thisfunction expresses the ensemble averaged probability of finding aparticle pair at the center to center distance r (normalized usually toparticle radius a).

As can be seen, a characteristic feature of the g(r /a) functionshown in Fig. 5a, is that there is a well pronounced maximum at thedistance r /a=3, i.e., when the gap between particle surfaces equalsone particle radius. The position of the peak is a direct indication of therange of the lateral interactions among adsorbing particles. Here, dueto a large screening distance (equal to ca. 200 nm in this case), therange is comparable with the particle dimensions.

With the increase in the ionic strength, the repulsion betweenparticles becomes smaller at short distances, because of more intense

Fig. 4. Deposition kinetics of positively charged polystyrene latex particles (averageddiameter of 800 nm) onmica, determined by optical microscopy (circles—○) and AFM(squares — ■), part “a” ionic strength 2×10−5M, pH=5.5, T=298 K, and nb=2.6×109cm−3. Part “b” ionic strength 10−3M, pH=5.5, T=298 K, and nb=2.6×109cm−3. The solid line denotes exact theoretical results calculated by solvingthe governing diffusion equation with the blocking effect given by the RSA model, andthe dashed dotted line represents the theoretical results calculated using the Langmuirblocking model.


screening. They can, therefore, form a more compact monolayer.Accordingly, the peak in the pair correlation function appears at thedistance r /a close to 2 (see Fig. 5b), which is expected for hard(noninteracting) particles.

The results shown in Fig. 5 indicate unequivocally that themaximum coverage of particles and structure of their monolayerscan be regulated to a significant extent by the change in the ionicstrength of particle suspensions.

The maximum particle coverage Θmx, which can be determinedfrom extrapolation of the kinetic runs, is of a primary experimentalsignificance. Therefore, many studies have been carried out in theliterature with the aim of determining this parameter as a functionof the double-layer parameter a / Le=(a2ekT /2e2I)1/2, which repre-sents the ratio of the particle radius to the double layer thickness Le.Most of these studies were carried out using monodisperse latexparticles [4–6,10], colloid gold particles [8,34] or dendrimers [9],under the diffusion transport condition or forced convectiontransport in the impinging-jet cells. The coverage of particles wasdetermined by optical microscopy, AFM, electron microscopy, orreflectrometry. These results obtained for various particle sizes,ionic strength and particle transport conditions are collected inFig. 6 as the dependence of the reduced maximum coverage Θmx/Θ∞(where Θ∞=0.547 is the jamming coverage for hard spheres) onthe a / Le parameter. As can be observed, the experimental datacollected under various conditions can well be reflected for a / LeN1by theoretical results derived from the RSA model, which predict asignificant reduction in Θ /Θmx for lower values of a / Le, i.e., eitherfor low ionic strength or smaller particles, because of increasingrange of the repulsive double-layer interactions. The results derivedfrom the RSA model can be approximated by the following

Fig. 5. Part “a”, micrographs showing positively charged latex particles (averaged diameΘs=0.27, and ionic strength 10−3M, pH=5.5, T=298 K, Θs=0.44 [91]. Part “b” the pair coderived from optical microscopy and the solid line denotes the theoretical results derived f

analytical formula derived in Ref. [14]

Θmx

Θ∞=

1ð1 + H*Þ2 ð42Þ

where

H* =1

2 aLe

� lnϕ0

2ϕch

� �− ln 1 +

12 a

Le

� ln ϕ0

2ϕch

� � !" #ð43Þ

and ϕ0 is the repulsive interaction energy between particle pair atclose separations (height of the energy barrier), and ϕch is thecharacteristic energy close to 1 kT [14].

On the other hand, negative deviations of experimental data fromthis theoretical model, which appeared for a/LeN10, can be explainedby the time of deposition experiments being too short. This conclu-sion is supported by the fact that theoretical results obtained forfinite simulation times, corresponding to typical experimental times(empty squares in Fig. 6), correlate better with some experimentalpoints than the theoretical results obtained for infinite times (emptytriangles).

The results presented in this section enable a quantitative de-scription of particle deposition kinetics under various transport con-ditions, especially evaluation of the maximum coverage, which is aprerequisite for a proper interpretation of electrokinetic measure-ments for particle covered surfaces discussed next.

ter 800 nm) deposited on bare mica, ionic strength 2×10−5M, pH=5.5, T=298 K,rrelation function g(r) for the same parameters. The points denote experimental resultsrom the Monte-Carlo RSA simulations.

Fig. 6. Collection of experimental data showing the dependence of the reducedmaximumcoverage of colloid particles Θmx/Θ∞ on the a/Le parameter: [●] Johnson and Lenhoffdata, AFM, natural convection [4]; [▲] Adamczyk et al. optical microscopy, gravity [27];[▼] Adamczyk and Szyk, optical microscopy, diffusion [6]; [■] Böhmer et al., radialimpinging jet RIJ cell, reflectometry, [29]; [◆] Harley et al., electronmicroscopy, diffusion[39]; [○] Kleimann et al. [10] reflectometry and AFM, impinging-jet cell; [◊]— Semmler etal. [5], [△] theoretical RSA simulations for the dimensionless time τ=105; [□] theoreticalRSA simulations for τ=10. The solid and dashed lines represent the analyticalapproximation calculated from Eqs. (42) and (43) for ϕ0=100kT.


3. Electrokinetics of particle covered surfaces

As can be deduced from the previous discussion, kinetic measure-ments of particle deposition at solid/liquid interfaces performed viadirect observation methods, although precise, are tedious and timeconsuming, especially for nanoparticle size range (proteins). One ofthe alternatives of overcoming this difficulty is the use of electroki-netic techniques, especially the streaming potential measurements,making it possible to determine under in situ conditions the kineticsof particle deposition of arbitrary size range. The utility of this methodhas recently been enhanced due to the availability of reference ex-perimental data, based on direct determination of the particle cover-age. These results can be quantitatively interpreted in terms of thenew theoretical results derived by exact numerical solution of the

Fig. 7. Schematic view of the simple shear flow past co

electrokinetic problem of particle covered surfaces. Reviewing theseresults is the goal of this section.

3.1. The theoretical model

The streaming current is the result of the convective flux of ionsfrom the thin double-layer region adjacent to solid/electrolyte inter-faces. It is induced by amacroscopic flow of the fluid, usually driven bythe hydrostatic pressure gradient, e.g., for channel and capillary flows,or motion of interfaces (circular disks arrangement, rotating disk).

Accordingly, the streaming current Is, defined as the amount ofcharge flowing per unit of time through the plane S, perpendicularto the interface (see Fig. 7), can be expressed via the constitutiveequation

Is = ∬sρeV⋅dS ð44Þ

where ρe is the electric charge density, V is the macroscopic fluidvelocity field and dS is the surface element.

Eq. (44) is valid for arbitrary interface shape provided that its localradius of curvature remains much larger than the double layer thick-ness Le and the particle dimension.

In formulating various theoretical approaches discussed in thissection it is further assumed that the flow field is laminar in the regionadjacent to the interface and that the electric charge density isgoverned by the Poisson–Boltzmann equation, unperturbed by theflow, i.e.,

ρe = −ε∇2ψ = eΣzinibe−zieψ

kT ð45Þ

where ψ is the electric potential, zi is the i-th ion valency, nib is theconcentration of this ion in the bulk (it is to be noted that the SI unitsystem has been adopted in this work).

By considering Eq. (45) the expression for the streaming currentbecomes

Is = −ε∬s∇2ψV⋅dS: ð46Þ

The explicit evaluation of Eq. (46) requires the knowledge of thescalar electrostatic potential field ψ and the hydrodynamic vectorfield V, which cannot be found analytically in the general case ofparticle covered surfaces.

However, analytical solutions can be found for shearing flows in thecase of bare interfaces and in the lowparticle coverage regime, using thecluster expansion technique (see Appendix A). These solutions are of aquite broad utility because the simple shear appears close tomost solid/

lloid particles adsorbed at a solid/liquid interface.


liquid interfaces exposed to laminar ambient flows, e.g., for channelflows, for the rotating disk, impinging-jets, for the uniform flow past asphere or cylinder, etc., (see Table 2). In all these cases the macroscopicflow in the xxdirection parallel to interfaces can be expressed as

V = GoziX ð47Þ

where Go = ∂V∂z

� �ois the shear rate at the interface, z is the coordinate

locally perpendicular to the interface and iX is the unit vector in thedirection parallel to the interface and the flow.

Substituting Eq. (47) into Eq. (46) and integrating two times byparts (assuming also that the electric potential vanishes far from theinterface) one obtains the general formula for the streaming currentof homogeneous surfaces (in the absence of particles)

−Is = εGolζ = Clζ ð48Þ

where l is the width of the interface (see Fig. 7), Cl=εGol and ζ is theelectric potential in the slip plane (where the liquid starts to move),usually referred to as the zeta potential [93–95].

Note that Eq. (48) does not involve the length of the interface.It is interesting to observe that according to Eq. (48), the streaming

current depends linearly on the zeta potential of the interface, al-though the governing Poisson–Boltzmann equation, Eq. (45) isnonlinear. Thus, Eq. (48) is valid for an arbitrary charge of the inter-face, electrolyte concentration and composition, including mixturesof electrolytes of arbitrary valency.

Eq. (48) can be evaluated explicitly for some geometries of majorpractical significance, where the shear rate at the interface is knownfrom analytical solutions of the Navier–Stokes equation.

For example, in the case of a rectangular channel of the cross-section 2b×2c (where 2b is the channel height and 2c= l is thechannel width) the expressions for the average shear rates at the baseand side walls are [14]

Gi = Go 1−bcfc

bc

� ��

G2 = Gofcbc

� � ð49Þ

where Go = PbηL and

PLis the hydrostatic pressure gradient along the

channel of the length,

fcbc

� �=

16π3 ∑

∞

n=0

1ð2n + 1Þ3 tan

ð2n + 1Þπc2b

ð50Þ

It is interesting to observe that the function fc(b/c) varies littlewith the channel shape assuming the value of 0.5 for a channel withsquare cross-section (b=c) and 0.54 in the limiting case of parallel-plate channel, where b/c≪1. Hence,

Gi = G2 =12Go ð51Þ

for channel with square cross-section and

Gi≅Go 1−0:54bc

� �G2≅0:54Go

ð52Þ

for a parallel-plate channel.Using Eqs. (48) and (49) one can formulate the following ex-

pression for the streaming current for a rectangular channel

−Is = 4εGoc ζi 1−bcf

bc

� �� +

bcζ2f

bc

� �� ð53Þ

where ζi is the zeta potential of the two base walls and ζ2 is the zetapotential of the two side walls.

It is interesting to observe that in the case when all channel wallsare of the same material (characterized by equal zeta potentials,ζi=ζ2), Eq. (53) simplifies to the limiting form independent of thechannel shape (cross-section)

−Is = 4εGocζi = 4εPbcηL

ζi = −Is0 : ð54Þ

On the other hand, for the parallel-plate channel, where b/c≪1(which is usually the case in experimental studies) the expression forIs becomes

−Is = 2εGoc 1−0:54bc

1− ζ2ζi

� �� ζi: ð55Þ

From Eq. (55) one can estimate the correction associated with thefact that the sidewalls of a parallel-plate channel are usuallymade of adifferent material than the base walls (substrate surface studies). Forb/c=1/20 (which is a typical value for many experimental works[59,60,64]) assuming ζ2

ζi= 0:5, the correction is 0.014. In the extreme

case of ζ2ζi

= 0, the correction equals 0.027. These values are sig-nificantly smaller than other experimental errors, connected mostlywith the electrode asymmetry potential and surface conductivity.

For a cylindrical plate channel (capillary) with the radius Rc, theperimeter of the capillary is l=2πR and we have Go = PRc

2ηL [14].Accordingly, Eq. (48) becomes

−Is = επPR2

c

ηLζi = −Is0 ð56Þ

wherePLis the hydrostatic pressure gradient along the capillary of

the length L.The above equations for Is are valid for Le

b << 1, in the case of theparallel plate channel and Le

Rc<< 1, in the case of the capillary.

Because of the appearance of the streaming current, electric chargeis transported away from the double layer region adjacent to theinterface, which results in the appearance of an electric potentialdifference, referred to as the streaming potential Es. This generates abackward electric current Ib=−Is due to the electric conductance ofthe cell. As discussed in Refs. [93,94] the streaming potential isconnected with the streaming current by the Ohmic dependence

Es = − IsRe = IbRe ð57Þ

where Re is the overall electric resistance of the cell governed mainlyby the specific conductivity of the electrolyte in the cell.

However, for dilute electrolytes and thin channels, the contribu-tion stemming from surface conductance starts to play a significantrole [94,95], thus special procedures are to be undertaken to correctfor this effect. Thus, in the general case Re can be expressed as

Re =L

ΔScKe=

LΔScðK

0e + KsÞ

ð58Þ

where ΔSc is the channel cross-section area, Ke′ is the specificconductivity due to electrolyte and Ks is the surface conductivity,depending in the general case on the channel shape.

Considering Eq. (57) one can derive from Eq. (53) the followingformula for the streaming potential in the case of the rectangular channel

Es = Es0 1− bcf 1− ζ2

ζi

� �� ð59Þ

Fig. 8. Part “a”, the dependence of the Ci function characterizing the contribution ofstreaming potential change stemming from interfaces in the limit of low coverage onthe a /Le parameter. 1. |ζi|=100 mV, 2. |ζi|=50 mV. Part “b”, the dependence of the Cpfunction characterizing the contribution of streaming potential change stemming fromadsorbed particles in the limit of low coverage on the a /Le parameter for ζp=25 mV.


where

Es0 = 2εPblRe

ηLζi = ε

PηKe

ζi ð60Þ

is the value of the streaming potential for the parallel-plate channel(for b/a≪1).

For the cylindrical channel one obtains an identical expression

Es0 = εPπR2

c Re

ηLζi = ε

PηKe

ζi: ð61Þ

As can be noticed, when introducing the specific conductivity, theexpressions for the streaming potential are the same for arbitrarychannel cross-sections if the walls are made of the samematerial. Thisis so, however, only if the surface conductivity contribution can beneglected.

It is interesting to observe that Eq. (61) was first derived bySmoluchowski [66].

As can be noticed, the zeta potential of surfaces, which is a quantityof primary experimental interest for characterizing their electricalstate, can be calculated by measuring experimentally the slope of thedependence of Es on

PLonce the electric conductivity of the channel

is known.

3.2. Theoretical results for particle covered surfaces

It is tomention, however, that all the above formulae are valid for ahomogeneous charge distribution over interfaces and for uniform(position independent) flows.When particles are present at interfaces(see Fig. 7), both these conditions are violated. Particles disturb theelectric charge distribution near interfaces and the macroscopic flowV in their vicinity.

In principle, Eq. (44) could be used to evaluate the streamingcurrent in this case if the charge distribution ρe and the liquid velocityV were known as a function of particle position, coverage and theirdistribution over interfaces. This would require a solution of thenonlinear Poisson–Boltzmann and Navier–Stokes equations for suchmany body problems, which poses insurmountable difficulties atpresent time. Therefore, theoretical results have been derived forsome limiting cases only, for example, for the low coverage [59,60]and the thin double layer regimes [67]. It was postulated moreoverthat the ion distribution in the vicinity of the particles is not perturbedby the local flow, governed by the linear Stokes equations [59,60,67].

With this assumption, exact analytical expressions for the stream-ing current of particle covered surfaces were derived in Refs. [59,60] inthe limit of low coverage

IsðΘÞ = Is0 1−CiaLe

� �Θ + Cp

aLe

� � ζp

ζ iΘ

� �ð62Þ

where Ci, Cp are dimensionless functions of the double-layer param-eter, a /Le, characterizing separately the effects stemming from themacroscopic flow disturbances and the electric charge densitydisturbances, ζi the zeta potential of the bare interface and ζp is thezeta potential of particles.

It is interesting to mention that Eq. (62) is valid for arbitrary shearflow and the interface shape provided that the range of variation ofthe shear rate is much larger than the particle dimension.

The dependence of Ci, Cp on a/Le was calculated numerically bysolving the Poisson–Boltzmann equation and the Navier–Stokesequation for the particle-interface configuration, using the bisphericalcoordinate system [96,97]. The dependence of these functions on thea/Le parameter calculated in this way is shown graphically in Fig. 8.

For the range of 2<a/Le<∞, the Ci, Cp functions remain practicallyconstant. For thicker double layers, Ci decreases and attains the value

of 8 for a /Le=1, whereas the Cp constant slightly increases, attaining6.7 for a /Le=1 [96,97].

On the other hand for the limiting case of thin double-layers,where a /Le≫1, these functions assume the limiting values of Ci0=10.2 Cp

0=6.51 [59,60,67].Until now, there are no exact results available in the literature for

non-spherical elongated) particles, which are of considerable interestfor interpretation of streaming potential measurements of polyelec-trolyte covered surfaces. The only exception are the results obtainedin Ref. [68] for strings of rigid spherical particles forming linearaggregates obtained using the multipole expansion method. Fromthese numerical results describing the hydrodynamic force per asingle sphere, it can be predicted that for a two particle aggregate,Ci0=7.99 for the side on (parallel) orientation and 13.1 for the per-

pendicular orientation. For an aggregate composed of 10 spheres onehas, Ci

0=6.04 for the side on orientation and 30.8 for the


perpendicular orientation. As can be noticed, the difference betweenparallel and perpendicular orientations is very significant, whichsuggests that the streaming current or streaming potential measure-ments can be used as a sensitive tool for detecting polyelectrolyte orprotein orientations at interfaces. In Table 4 values of the Ci

0 constantfor various aggregates are collected for the side-on orientationaveraged in the adsorption plane, for perpendicular orientation andfor averaged end on with Sg=nsπa2 in Eq. (30).

For interpretation of experimental results it is advantageous torewrite Eq. (62) in a reduced form. For a charged interface

―Is =

―Es = 1−CiΘ + Cp

ζpζi

Θ ð63Þ

where―Is =

IIs0

;―Es =

EsEs0

. In the case of neutral interfaces,

―Is =

―Es = CpΘ ð64Þ

where―Is = I

―/ Ich ,

―Es =E / Ech and the reduced quantities are

−Ich=2εGocζp and Ech= IchRe.Weemphasize that the reduced streamingcurrent

―Is (or, equivalently,

the streaming potential―Es) does not depend on parameters describing

either the electric resistance or geometry of the cell. Hence―Is or

―Es can be

used as a universal quantity characterizing the effect of adsorbed particlesalone. In particular, it can be applied to determine the particle coverage. Itshould be noted that the slope of the dependencies of

―Is or

―Es on the

particle coverage, accessible experimentally, ismuchhigher that unity. Forexample, in the case of neutral particles it is equal to 10.2 (in the limitof thin double-layers) and equal to 16.7 in the case of particles chargedoppositely to the interface. Therefore, electrokinetic methods forevaluation of particle coverage that are based on measurements of

Table 4Values of the Ci

0 constant (upper row) and the Cp0 constant (lower row) for linear aggregat

Configuration Number of b

1

Side on (averaged over α)

10.26.51

Perpendicular to interface

10.26.51

End on (averaged over α and β)

10.26.51

the streaming current or streaming potential can achieve high sensitivity,even for low coverages of the order of 1%.

In principle, such electrokinetic methods can be used to determineparticle coverage also in the high-coverage regime. However, Eq. (63)is valid only in the limit of low coverage (approximately for Θ≪0.05),which restricts its wider use in practice. Therefore, correctionfunctions have been proposed in Refs. [59,60], accounting for thefact that for higher coverage the flow and electric charge densityperturbations resulting from separate particles are no more additive.Accordingly, Eq. (63) was generalized to the form

―Is =

―Es = 1−AiðΘÞΘ + ΑpðΘÞ

ζpζi

Θ ð65Þ

where

AiðΘÞ =1−e−CiΘ

Θ

ApðΘÞ =1−e−CpΘ

Θ

: ð66Þ

These results have been successfully used for interpretation ofexperimental data obtained for monodisperse polymeric particlesuspensions [59,60,64,96,97], silica particles [61–63] and polyelec-trolytes [98,99].

Recently, exact theoretical results havebeenderived in the limit of thindouble layers [67],which can be used to determine the range of validity ofthese semi-empirical correction functions. These resultswere obtained byevaluating numerically the flow in the vicinity of adsorbed particles usingthemultipole expansionmethod [100–102], for the coverage range up to0.5. Moreover, in the case of lower coverage range, analytical results havebeen derived by applying the cluster expansion method. As shown in the

es adsorbed at a solid interface (calculated using the results given in Ref. [68]).

eads ns

2 4 10 20 40 100

7.99 6.80 6.04 5.78 5.65 5.575.03 4.22 3.70 3.52 3.42 3.37

13.1 18.1 30.8 48.9 80.8 1648.02 10.8 17.9 28.0 45.9 92.6

10.3 12.0 17.5 25.8 40.6 79.26.48 7.43 10.7 15.6 24.5 47.5


Appendix A, the functions Ai(Θ), Ap(Θ) appearing in Eq. (65) can beexpressed, in the form of the power series expansion

AiðΘÞ = 10:2−59:43Θ + 292Θ2 + 0ðΘ3ÞApðΘÞ = 6:51−36:82Θ + 181Θ2 + 0ðΘ3Þ : ð67Þ

Eq. (67) was derived by assuming an equilibrium distribution ofparticles in the adsorption layer. In the case of the RSA configuration,the expansion has the form (cf. the Appendix A)

AiðΘÞ = 10:2−59:43Θ + 296Θ2 + 0ðΘ3ÞApðΘÞ = 6:51−36:82Θ + 184Θ2 + 0ðΘ3Þ : ð68Þ

Accordingly the power series expansion for the reduced streamingcurrent/potential assumes in the case of an equilibrium configuration

Fig. 9. Part “a”, the dependence of the Ai(Θ) function, characterizing the streamingpotential changes in the limit of negligible double-layer thickness (a /Le→∞), onparticle coverage Θ, calculated theoretically. Part “b”, the dependence of the Ap(Θ)function on particle coverage Θ calculated theoretically. The solid line denotes therational approximation of exact numerical results derived from Eq. (71), the dashedline shows the analytical results derived from the virial expansion Eq. (67) and thedashed dotted line shows the results calculated from the exponential fitting functiongiven by Eq. (66).

of particles the following form

−Is =

−Es = 1−ð10:2−6:5

−ζ ÞΘ + ð59:43−36:82

−ζ ÞΘ2

−ð292−181−ζ ÞΘ3 + 0ðΘ4Þ:

ð69Þ

In the case of the RSA configuration, the power expansion becomes

−Is =

−Es = 1−ð10:2−6:5

−ζ ÞΘ + ð59:43−36:82

−ζ ÞΘ2

−ð296−184−ζ ÞΘ3 + 0ðΘ4Þ

ð70Þ

where−ζ =

ζpζi:

It is to mention however, that the expansions, Eqs. (67)–(70) arevalid for low coverage only, approximately for Θ<0.07.

Therefore, in Ref. [67] exact numerical calculations were per-formed for particle configurations at planar interfaces characterizedby desired coverage (reaching 0.5 as mentioned above) with periodicboundary conditions. The multipole algorithm [100–102] has beenused. Results averaged over 300–400 configurations enabled one toformulate the following analytical functions, which approximated theexact numerical results with precision better than 1%.

AiðΘÞ =10:2−5:75Θ1 + 5:46Θ

ApðΘÞ =6:51−2:38Θ1 + 5:46Θ

: ð71Þ

In Fig. 9 theoretical predictions derived from Eq. (71) arecompared with the semi-empirical fitting functions given byEq. (66) and the power expansion, given by Eq. (69). As can beseen, the semi-empirical function Ai(Θ) agrees very well with theexact data derived from Eq. (71) for the entire range of coveragestudied, whereas the function Ap(Θ) systematically over predicts theexact results.

More extensive comparison of the results, stemming from the exactand the semi-empirical approaches for various particle to interface zetapotential ratio ζp/ζi is shown in Fig. 10. As can be seen, the theoreticalresults derived from both approaches agree well for ζp/ζi close to zero(when the absolute value of particle zeta potential ismuch smaller thanthe interface zetapotential).However, thedeviation increases for highervalues of particle zeta potential, where ζp/ζi<0.

It is interesting to observe that for higher coverage range, i.e., forΘN0.3, the exact values of the reduced streaming potential Esbecome practically constant. This is in accordance with the so calledSmoluchowski principle, discussed extensively in Refs. [103,104].Using this principle it can be predicted that the streaming potential ofhomogeneous disperse systems becomes independent of the concen-tration of particles and their shape in the limit of thin double-layers.As can be seen in Fig. 10, however, the Smoluchowski principle,is apparently not obeyed for the low coverage range in the case ofζp/ζi=1, corresponding to rough interface produced by deposition ofparticles of the same zeta potential. Thus, for Θ<0.2, it is predictedthat the reduced streaming potential of such composite surfacesdecreases monotonically with the coverage (roughness degree)attaining the limiting value of 0.73, irrespectively of the size of theroughness. This means that zeta potential of rough surfaces measuredby the streaming potential method is expected to be 27% smaller thanthe zeta potential for smooth surfaces. In this context, an experimen-tal confirmation of the decrease of the streaming potential with theincrease in the surface roughness would be of a major significance,although difficult to realize.

It seems, therefore, that despite a significant progress in theoreticalcalculations of the streaming potential for particle covered surfaces,there is need for theoretical calculations in the domain of the roughsurfaces, thick double-layers, non-spherical particles and multilayercoverage of spherical particles. Also, further theoretical studies areneeded in order to unequivocally determine the physical reason for thedeviation from the Smoluchowski principle for the low coverage range.


3.3. Experimental results for particle covered surfaces

Most of the results discussed in this section were obtained usingthe parallel plate channel arrangement devised originally by vanWagenen and Andrade [105] and then used extensively to determineelectrokinetic characteristics of bare surfaces (mica and silica slides)[106–108] and protein covered surfaces [43,44]. Similar cells havebeen exploited widely for measuring streaming potential of particlecovered surfaces, most often mica modified by adsorption of simpleions or cationic surfactants [59,60,64,96] or modified glass [61,62].The essential part of such a cell, described in detail in Refs.[59,60,106,107], is the parallel plate channel produced by clampingtogether two Teflon blocks and two mica sheets separated by a gasketwhose thickness determines the channel height 2b (being usually ofthe order of 100–400 μm). Since the width of such cells 2c is of theorder of 0.3–1 cm, the b /c parameter is typically much smaller than0.1, therefore the correction due to side walls made of Teflon is of theorder of a few percent, as estimated above.

The cell is completed with two compartments containingstreaming potential electrodes (Ag/AgCl) connected with an elec-trometer of high resistance. The other electrode pair (platinium) isused formeasuring the electric conductance of the cell. The electrolyteor suspension flow through the channel is driven either by gas(nitrogen) pressure [105–108], pumping [61,62] or by regulating thelevels of the two electrolyte reservoirs [59,60,64]. The pressure dropalong the channel is measured by pressure transducers or, in the caseof hydrostatic pressure driven flow, by a cathetometer.

The advantages of such a cell are large working area increasingmeasurement precision, possibility of adsorption of desired solutes orparticles, a capability for direct in situ microscope observation ofsurfaces without drying and simple theoretical interpretation of theresults. However, a disadvantage of the parallel-plate channel cells isits tedious and time consuming assembling prior to experiment,sealing problems and the necessity of determining in every experi-ment the correction due to surface conductance of the cell.

These disadvantages were partially eliminated in the micro-slitstreaming potential cell described in Refs. [94,109–111]. In this cellthe distance between substrate plates forming the parallel platechannel can be regulated between wide limits (1–50 μm), whichallows one to determine precisely the surface conductance of the cell.Another advantage of such an arrangement is that substrate plates (10

Fig. 10. The dependence of the reduced streaming potential―Es of particle covered

interfaces on the coverage Θ calculated theoretically for various particle to interfacezeta potential ratio: 1. ζp /ζi=1 (rough interfaces), 2. ζp /ζi=0 (neutral particles), 3.ζp /ζi=−0.5, 4. ζp /ζi=−1, 5. ζp /ζi=−2, the solid lines denote the exact theoreticalresults derived from Eqs. (65) and (71) and the dashed-dotted lines denote the resultsderived from the exponential fitting functions given by Eq. (66).

per 20 mm) are suitable for experimental examination by othertechniques such as XPS, AFM, ellipsometry and wetting angle.

In measurements of protein deposition kinetics, circular channelcells were also used [57,58], whose construction is much simpler thanthe above parallel plate channel cells. However, the range of capillarymaterials available for such studies is rather limited, there also appearsignificant problems with cleaning and examination of surfaces bymicroscopy or other surface oriented techniques.

Problems with sealing and tedious cell assembling can be de-finitively eliminated using the cell developed in Ref. [112], based onthe rotating disk principle. In this case, the liquid flow along thesurface is driven by the centrifugal force appearing due to the rotationof a circular disk with an attached substrate surface. This arrange-ments is especially suitable for measuring kinetics of streamingpotential changes although the measurement is less precise becauseof small values of the streaming potential (usually below 1 mV) and astrong dependence of the signal on the positioning of electrodesrelative to the disk center.

3.3.1. Characteristics of bare surfacesBecause of its high precision, the streaming potential method has

been widely used to characterize various substrate surfaces, mostlymica.

For example, Scales at al. [106] determined the dependence of zetapotential of muscovite mica as a function of pH, showing that it variedbetween−30 mV and−80 mV for pH=4 and pHN6, respectively (atconstant ionic strength of 0.001 M KCl). The dependence of mica zetapotential on concentration of various monovalent salts (LiCl, NaCl,KCl, CsCl) was also determined for fixed pH=5.8. For low saltconcentration the limiting value of the zeta potential of mica was−120 mV, which increased with salt addition up to −80 mV for0.01 M LiCl and NaCl solutions and −40 mV for 0.01 M KCl. Thedifference in zeta potential observed for various salts was attributedto the specific adsorption of cations (counterions). The highest zetapotential of mica equal to−20 mVwas determined for 0.001 MHCl. Itis interesting to observe that no zeta potential reversal of mica as afunction of pHwas observed, which suggests that this substrate showsno isoelectric point.

A similar behavior was observed in the case of fused silica whosezeta potential was also determined using the parallel plate streamingpotential cell [104]. For pH=3 the zeta potential of micawas−20 mVfor 0.001 M KCl, and −5 mV for 0.1 M KCl. The latter value is subjectto a considerable error because of very low signal due to the highconductivity of the electrolyte solution. For pHN7 the zeta potentialdecreased to −100 mV for 0.001 M KCl and −40 mV for 0.1 M KCl.

In Ref. [106] the case of divalent cations such as calcium Ca2+ wasalso studied. It was shown that the zeta potential of mica increased toabout −10 mV for 0.01 M CaCl2 solution (pH=5.8).

A similar behavior was observed in Ref. [64] in the case of MgCl2solutions, where the zeta potential ofmicawas increased from−70 mVfor 10−5M salt solution to −15 mV for 3×10−3M salt solution.

On the other hand, in the case of LaCl3 an inversion of the zetapotential of mica to positive values was observed for the salt concen-tration above 3×10−5M, with the maximum value attaining 35 mVfor 0.01 M solution of salt [106].

These results suggest that the zeta potential of mica can beadjustedwithin broad limits by variation of pH and composition of thesupporting electrolyte. This can be exploited in model studies aimedat verification of theoretical approaches describing zeta potential ofparticle covered surfaces.

3.3.2. Particle covered surfacesFew basic studies for particle covered surfaces have been carried

out because of considerable experimental problems associated withparticle monolayer deposition of controlled coverage and structure,which can resist vigorous shearing flows in the channel. Most of the

Fig. 11. The dependence of the reduced streaming potential of mica―Es on particle

coverage Θ. The points denote experimental results obtained for polymeric latexparticles whose zeta potential was reduced to zero by pH change, pH=7, I=10−3M,NaCl, T=293 K [64]. The solid line represents the exact theoretical results [67], thedashed-dotted line shows the results calculated from the exponential fitting function,Eq. (66).


results reported in the literature have been obtained for micasubstrate and monodisperse polymeric particles (polystyrene latexor melamine latex) using the parallel plate channel cell. The main goalof these studies was to determine the validity of the correctionfunctions Ai(Θ), Ap(Θ), to reflect the streaming potential variation ofsurfaces covered with particles. In order to do so, two series ofexperiments have been carried out [64]:

(i) for amphoteric latex particles, whose zeta potential wasreduced to zero by pH variation,

(ii) for neutral surfaces whose zeta potential became zero uponadsorption of a cationic surfactant.

In Fig. 11 results derived from the first series of experiments arepresented in the form of the dependence of the reduced streamingpotential of mica ―Es = EsðΘÞ=Es0 (where the scaling variable was thestreaming potential measured for bare substrate Es0 under the samephysicochemical conditions) on particle coverage Θ. It is interesting tomention that the particle coverage in these experiments was de-termined directly by optical microscope observations. Experimentalvalues of the particle coverage determined in this way were in a goodagreement with theoretical predictions of the convective-diffusiontheory discussed above.

The points in Fig. 11 denote experimental results obtained forpolymeric latex particles of the size 1130 nm (triangles) and 1530 nm(inverse triangles) whose zeta potential was reduced to zero by pHchange to 7 [64]. The solid line represents the exact theoretical resultsobtained from Eq. (65) with the correction functions given by Eq. (71)and the dashed-dotted line shows the results calculated using theexponential fitting function, Eq. (66). As can be seen, the theoreticalresults derived from both models, which agree with each other verywell, reflect properly the experimental data for the entire range ofparticle coverage (up to 0.45). Hence, the results shown in Fig. 11,obtained for neutral particles have a major significance, showingclearly that the hydrodynamic effects alone, associated with flowdamping in the vicinity of deposited particles, induce a considerablechange in the effective zeta potential. In other words, zeta potential ofparticle covered (heterogeneous) surfaces is determined not only bythe value of the particle zeta potential as commonly assumed, but alsoby the flow damping effects controlled by the coverage of particlesand their distribution on surfaces. This is a direct manifestation of thefact that electrokinetic phenomena are of a non-equilibrium nature,depending on hydrodynamic rather than thermodynamic conditions.

It is also interesting to observe that the slope of the dependence ofthe reduced streaming potential on the particle coverage is very large(initially equal to 10.2 as predicted by the theory) that suggests thatthe presence of adsorbed particles (even uncharged) can be preciselydetected by the streaming potential measurements, especially for thelow coverage range Θ<0.1, where other experimental methods fail.

However, additional systematic studies performed for smallerparticle sizes, and other values of ionic strength, where the a/Leparameter assumes larger values comparable with unity, are neededin order to determine precisely the range of applicability of the abovetheoretical approach.

In Fig. 12 the results derived from the second series of experimentsare presented. In this case, the charge of mica substrate was reducedto zero by reversible adsorption of the cationic surfactant DTACl (usedat the concentration of 1.2×10−4M, pH=5.5 [64]). Under theseconditions, the zeta potential of a polystyrene latex (average diameter870 nm) was 45 mV. As previously, particle coverage was determinedunder wet, in situ conditions using optical microscope observations.The solid line in Fig. 12 shows the exact theoretical results obtainedfrom Eq. (65) with the correction functions given by Eq. (71) and thedashed-dotted line shows the results calculated using the exponentialfitting function. As can be seen, the theoretical results derived fromthe exact model deviate from the approximate model, being of about

15% smaller for particle coverage 0.2. Since the experimental data liebetween both theoretical curves, it is difficult to unequivocally statewhich model is more appropriate. It seems, therefore, desirable toperform systematic experiments for a much broader range of surfacecoverage, particle size and zeta potential.

Experimental data are also available for intermediate cases ofcharged particles adsorbed at charged interfaces, characterized byvarious ζp/ζi parameter values. Such results obtained for polystyrenelatex particles 470 nm in diameter at mica modified by adsorption ofMg+2 ions are shown in Fig. 13 [64,97]. As mentioned, the addition ofMgCl2 increased monotonically the zeta potential of mica, whichattained the value of −25 mV (for 10−3M MgCl2, pH=5.5). The zetapotential of latex, remained for the same conditions at the level of67 mV, which produced a high value (in absolute terms) of theparameter ζp/ζi=−2.7. It is worthwhile mentioning that in this caseof highly charged particles adsorbing at weakly charged interfaces(full squares and curve 3 in Fig. 13) the sensitivity of the streamingpotential measurements remained especially high, for a broad rangeof coverage 0<Θ<0.3 because of the large slope of the Es vs. Θdependence. This suggests that in order to create convenientconditions for detecting particles via the streaming potentialmeasurement, the zeta potential of substrates is to be kept low (inabsolute terms) compared to the zeta potential of particles. Ascan be seen, for ζp/ζi=−2.7, the experimental results are in a betteragreement with the theoretical results derived using the exactfunctions Ai(Θ) and Ap(Θ) (described by Eq. (71)) than with theresults derived using the approximate exponential function, given byEq. (66).

On the other hand, in the case of no MgCl2 addition, the zetapotential of latex was equal to 54 mV, whereas the zeta potential ofmica −120 mV which gives ζp/ζi=−0.44. As can be seen in Fig. 13,the experimental results obtained in this case are well reflected byboth theoretical models (curve 1), which agree well with each other.

It is also interesting to observe that this holds true also for theexperimental data obtained by Michelmore and Hayes [63] who de-termined streaming current for negatively charged silica particles(trade nameMonosphere) having the diameter of 1000 nm for 10−3–

10−4M KCl solutions. As the substrate, mica sheets were used,modified by adsorption of di(aminopropyl) trimethhoxysilane, whichresulted in the inversion of the zeta potential of mica. This attained63–87 mV (for 10−3–10−4M KCl). As can be seen in Fig. 13 the results


coverageΘ. The points denote experimental results obtained for polymeric latex particles(of the size 870 nm) deposited onmica surface whose zeta potential was reduced to zeroby adsorption of cationic surfactant DTACl at concentration of 1.2×10−4M, pH=7,T=293 K [64]. The solid line represents the exact theoretical results [67], the dashed-dotted line shows the results calculated from the exponential fitting function, Eq. (66).


of Michelmore and Hayes [63] are in good agreement with the dataobtained for latex particles, and the theoretical predictions basedon Eq. (71).

However, in the case of smaller Monospher silica particles 250 and500 nm in diameter the results obtained in Ref. [63], deviated fromthese theoretical predictions, because the dependence of the reducedstreaming current on the particle coverage was less steep thanthe theory predicts. This was probably caused by the presence ofaggregates in the suspension, especially for the particle of 250 nmdiameter which were very hard to fully disperse. This is a likelyexplanation considering the theoretical predictions for the Ci constantfor aggregates, given in Table 4. One can see, that for an aggregationdegree larger than 10, the Ci constant decreased considerably,


coverage Θ. The points denote experimental results obtained for polymeric latexparticles (positively charged) whose zeta potential was regulated by Mg2+ ionadsorption [64]. The points denoted by [◊] represent the experimental resultsof Michelmore and Hayes [63] obtained in the case of negatively charged silicaparticles (diameter 1000 nm) deposited on modified glass surface, 1. ζp /ζi=−0.44, 2.ζp /ζi=−0.8, and 3. ζp /ζi=−2.72, pH=5.5, I=10−4–10−5M, NaCl, T=293 K. Thesolid line represents the exact theoretical results, the dashed-dotted line shows theresults calculated from the exponential fitting function, Eq. (66).

attaining values below 6, in comparison with the value of 10.2corresponding to single particle deposition.

Additionally, the true size of particles was not measured in theseexperiments, which could produce a higher uncertainty of the results.

Another type of electrokinetic measurements for particle coveredsurfaces has been performed by Vincent et al. [113] who determinedthe electrophoretic mobility of negatively charged latex particles(average diameter 3200 nm) covered to a controlled extent underdiffusion transport conditions by positively charged latex particles ofthe average diameter 195 nm. The positive latex coverage wasdetermined indirectly by the solution depletion method. In Fig. 14results obtained in these experiments (for ionic strength range 10−4–

10−2M NaCl, ζp/ζi=−0.86) are plotted as the dependence of thereduced electrophoretic mobility (normalized by its value for largelatex particles) on the particle coverage reaching 0.4. The surfacecoverage was recalculated for the absolute coverage Θ=πa2N,whereas in the original work it was normalized using the maximumhexagonal packing of spheres equal to 0.91. Because the theoreticalresults described by Eq. (65) apply to arbitrary simple shear flow,which prevails near larger particles undergoing an electrophoreticmotion, they can be used for interpretation of the experimentalresults of Vincent et al. [113]. As can be seen in Fig. 14, thesetheoretical results are in a reasonable accordance with the experi-mental data for the entire range of coverage, with the exponentialmodel performing slightly better than the exact theoretical model.

The analysis of these experimental data obtained for modelsystems of colloid particle covered surfaces, indicates quite unequiv-ocally that the streaming potential method can be used as a precisetool for determining particle coverage even for the range of a fewpercents, which is not feasible by other methods.

However, a more definite assessment of the applicability of thetheoretical models requires additional experiments, involving mono-disperse particles of the size below 100 nm andwell defined substratesurfaces, other than mica. Similarly, measurements for nonsphericalparticles would be of a great value. In part, this kind of information canbe derived from the streaming potential measurements performed forpolyelectrolytes and proteins, discussed next.

3.3.3. Polyelectrolyte and protein covered surfacesThe applicability of the above theoretical approach for the inter-

pretation of streaming potential measurements performed for poly-electrolyte covered surfaces has been confirmed in Refs. [98,99]. Twotypes of commonly used cationic polyelectrolytes have been studied

Fig. 14. The dependence of the reduced electrophoretic mobility of negatively chargedlatex particles (diameter 3200 nm) on the coverage Θ of positive latex particles(diameter 195 nm). The points denote experimental results obtained by Vincent et al.[113] for I=10−4–10−2M, NaCl. The solid line represents the exact theoretical resultsderived from Eqs. (65) and (71) for ζp/ζi=−0.86, the dashed-dotted line shows theresults calculated from the exponential fitting function, Eq. (66) for the same ζ/ζi.

Fig. 15. The dependence of the reduced streaming potential of mica―Es on the coverage

of PEI, ΘPEI. The points denote experimental results obtained in the parallel-platechannel for pH=6.1−6.4, T=293 K, ζi=−60 mV, ζp=40 mV [99]. The dashed andsolid lines denotes the theoretical results calculated for the ionic strength of 10−2 and10−3M, respectively using Eqs. (65) and (66). The curves calculated from the Gouy–Chapman model, Eqs. (72) and (73) are denoted GC.


(i) the poly(ethylene imine) PEI of the average molecular weightof 75kD [99] and (ii) poly(allylamine) PAH having the averagemolecular weight of 70kD [98]. Dynamic light scattering and AFMmeasurements enabled one to state that PEI molecules resemble fuzzyspheres characterized by the hydrodynamic diameter of 10.6 nm. Theaverage number of uncompensated charges per molecule and its zetapotential dependence on pH and ionic strength was derived frommicroelectrophoretic measurements. These physicochemical datawere exploited for a quantitative analysis of streaming potentialmeasurements for PEI coveredmica. Results of suchmeasurements areshown in Fig. 15 as the dependence of the reduced streaming potentialof mica, Es, on the coverage of PEI, ΘPEI. It is to mention that thecoverage was expressed in absolute terms as the geometrical fractionof the surface occupied by PEI molecules whose cross-section area(foot print) was 88.2 nm [99]. The coverage was calculated using theconvective theory discussed in the first section. The pH range of theseexperiments was 6.1–6.4, ionic strength I=10−2M, NaCl, T=293 K,ςi=−60 mV, ςp=40 mV [99]. As can be seen in Fig. 15, theexperimental data are well reflected for the entire range of ΘPEI (upto 0.4) by the theoretical model described by Eqs. (65) and (66)postulating a 3Dparticle-like adsorption of PEI. A characteristic featuredemonstrated in these experiments is that the streaming potential ofPEI coveredmica is less dependent on the ionic strength, in accordancewith theoretical predictions. Note also a high sensitivity of themeasured streaming potential to PEI coverage, since a 100% changein the signal is observed for PEI coverage as low as 0.1. This indicatesthat bulk concentrations of PEI at the level of a fraction of a ppm caneasily be detected by the streaming potential method.

In Fig. 15 results derived from the widely used Gouy–Chapman(GC)model based on the concept of a flat adsorption of PEI in the formof charge patches, are also plotted. Using this model, the reducedstreaming potential of PEI covered mica was calculated from thedependence [99]

−Is =

−Es = F

ln 12 j σ

σchj +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσσ ch

� �2+ 4

r� �

ln 12 j σ0

σchj +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiσ0σ ch

� �2+ 4

r� � : ð72Þ

The plus and minus sign denote the positive and the negative netcharge, where σ0 is the electrokinetic charge of bare mica determinedfrom the streaming potential measurements, σch = εkT

4πeLe

� is the

normalizing charge, and σ is the electrokinetic charge of mica coveredby PEI, calculated from the dependence

σ = σ0 + NcNPEI ð73Þ

where Nc is the number of charges per one PEI molecule determinedby microelectrophoresis and NPEI is the surface concentration ofadsorbed PEI molecules determined from the convective-diffusiontheory.

As can be seen in Fig. 15, the theoretical results derived from theGC model, which predicts a significant dependence of the reducedstreaming potential of PEI covered mica on the ionic strength, provedinadequate for interpretation of experimental data.

In view of these evidences, one can conclude that after adsorption,the PEI molecules retained their structure, i.e., three-dimensionalspheroidal shape, rather than a flat disk shape as was often postulatedin the literature.

Similar results have been obtained in the case of PAH, whosemolecules assume in electrolytes of moderate ionic strength a veryelongated, rod-like shape with the axis ratio much larger than ten[98,114]. The cross-section area of the molecule was estimated to be155 nm2. Results showing the dependence of the reduced streamingpotential ofmica on the coverage of PAH obtained for pH=6, I=10−2M,NaCl, T=293 K, ζi=−60 mV, and ζp=50mV are presented in Fig. 16

[98]. As can be seen, the theoretical results calculated from Eqs. (65) and(66) by taking the values of Ci=10 and Cp=6.5, pertaining to sphericallyshaped particles, deviate significantly from the experimental data. Assuggested in Ref. [98] this discrepancy can be attributed to an elongatedshape of the molecule. Quantitatively, this can be accounted for byexploiting the theoretical results derived in Ref. [68] (see Table 4) forrodlike particles composed of beads, which approximated quite well thereal shape of PAH molecules. It can be estimated using these theoreticalresults that the value ofCi should approach 5.6 in the limit of larger aspectratio when a random, side-on (flat) adsorption was assumed [68].Moreover, as shown in Table 4, this value is practically independent of theaspect ratiowithin the rangeof 20 to 100. Considering that theCp constantfor elongated molecules, equals 3.4 one can use Eqs. (65) and (66) tocalculate theoretical streaming potential data. Accurate values of Cp aregiven in Table 4. As can be seen in Fig. 16 the agreement of thesetheoretical predictionswithexperimental data is satisfactory for theentirerange of PAH coverage (up to 0.45).

Interestingly enough, by assuming an unoriented adsorption of PAHwith a random angle of its axis relative to the plane of adsorption, thevalue of Ci becomes 40.6 and Cp=24.5 (for aspect ratio 40 pertaining tovery elongated PAH molecule). As can be seen in Fig. 16, theoreticalresults calculated by assuming this hypothesis (depicted bydashed line)deviate completely from the experimental data, which proves quiteunequivocally that PAH adsorbed mostly side on at mica surface.

Hence, the results shown in Figs. 15 and 16 indicate that from thestreaming potentialmeasurements, important clues both on the shapeand conformations of polyelectrolyte molecules adsorbed at solid/liquid interfaces under in situ conditions can be derived. In particularone can easily distinguish between side on and unoriented (random)adsorption.

It has been further shown in Refs. [114–116] that the streamingpotential method can be exploited as a convenient tool forcharacterizing the formation of polyelectrolyte multilayers on solidsubstrates, produced in the layer by layer deposition processes.Examples of such results are shown in Figs. 17 and 18.

In Fig. 17 experimental results obtained by the streaming potentialmethodare shown in the caseof thePEI/poly(acrylic acid) (PAA)bilayer.Thefirst layer of this systemconsisted of a saturated (precursor) layer ofPEI produced under diffusion transport conditions for deposition timeof20 min, from a 2 ppm solution (pH=5.5, I=10−2M NaCl). The zetapotential of such a layer was ςi=40 mV [116]. Then, a second layer ofPAA of molecular weights 14 kD and 70 kD was deposited under the

Fig. 17. The reduced streaming potential for mica―Es covered by a saturated monolayer

of PEI vs. the coverage of PAA forming the second layer, ΘPAA. The points denoteexperimental results obtained in the parallel-plate channel for pH=5.5, I=10−2M,NaCl, T=293 K, ζi=40 mV, ζp=−19 mV [116]. 1. (▲) — PAA of molecular weight14 kD, 2. (●) — PAA of molecular weight 70 kD, the dashed line represents theoreticalresults derived from the Gouy–Chapman model, Eqs. (72) and (73) and the dashed-dotted line represent theoretical results derived from Eqs. (65) and (66), for Ci=8.5,Cp=6.

Fig. 16. The dependence of the reduced streaming potential of mica―Es on the coverage of

PAH, ΘPAH. The points denote experimental results obtained in the parallel-plate channelfor pH=6, I=10−2M, NaCl, T=293 K, ζi=−60 mV, ζp=50mV [98]. The solid linerepresents theoretical results derived for Eqs. (65) and (66) for Ci=5.6, Cp=3.4corresponding to side-on adsorption of PAH molecules (approximated by an linearaggregate composedof 40 spherical beads). Thedashed line denotes the theoretical resultsfor spherically shapedmolecules and the dashed-dotted line represents theoretical resultsderived from Eqs. (65) and (66), for Ci=40.6, Cp=24.5, corresponding to randomlyoriented adsorption of PAH molecules.


same transport conditions, from 2 ppm solutions (the bulk zetapotential value of PAA, determined by microelectrophoresis wasζp=−19 mV). The coverage of PAA was estimated using the solutionof the diffusion equation.

As can be seen in Fig. 17, the dependence of�Es of the bilayer on

the PAA coverage ΘPAA can be well described in the case of lowermolecular weight (14kD) by the Gouy–Chapmanmodel (described byEq. (72), with the bilayer charge calculated in an analogous way as forPEI covered mica). This suggests that in this case, because the PAAmolecule size (hydrodynamic radius of ca. 3 nm)was smaller than thePEI size (having the hydrodynamic radius of 5.3 nm), the adsorbingPAA chains were mostly adsorbing side-on, in a two-dimensionalform, with little tendency to loops and tails formation.

On the other hand, in the case of higher molecular weight PAAsample (70 kD), the experimental results were better reflected by thethree-dimensional model, described by Eqs. (65) and (66) with Ci=8.5and Cp=6, corresponding to PAA adsorption parameters. The positivedeviation of experimental data from this model, observed for lowercoverage range ΘPAA<0.2, can probably be attributed to penetration ofPAA molecules into the PEI monolayer. For higher coverage, when allvoids in the PEI monolayer were saturated the experimental resultsfollowed the 3D adsorption trend. This was so, because the hydrody-namic radius of PAA 70 kD molecules was 19.7 nm, which significantlyexceeded the hydrodynamic radius of PEI molecules. Therefore,formation of tails and loops was quite likely for higher coverage of PAA.

The results shown in Fig. 17 suggest that the streaming potentialmeasurements can be exploited as a sensitive tool for detectingconformations of adsorbed polyelecrolytes, in particular to discriminatebetween side-on (flat) or three-dimensional adsorption mechanism.

This technique can also be efficiently used for characteriza-tion of polyelectrolyte multilayers, which are often used for producingnanocapsule shells according to the layer-by layer mechanism,described in refs. [117–121]. Examples of such studies are shownin Fig. 18. The reduced streaming potential (reduced zeta poten-tial of the multilayer) of mica covered by a PAH/PSS (poly(sodium 4-styrenesulfonate) of the molecular weight of 70 kD [114]) is plottedthere as a function of the number of layers. The pointsdenote experimental results obtained in the parallel-plate channelfor pH=7.4, TRIS buffer, T=293 K, ζi=−115 mV (mica) and twovalues of ionic strength I=10−3M (part a) and I=0.15 (part b). As

can be seen, in both cases periodic oscillations in the apparent zetapotential of the multilayer were observed indicating the formation ofconsecutive polyelectrolyte layer. The amplitude of these oscillationswas considerably higher for the higher ionic strength (0.15 M), whichindicates quite unequivocally that the layers formed under theseconditions were much thicker, characterized by a three-dimensionalstructure composed of loops and tails of polymers. Hence, theseresults, although difficult for a quantitative interpretation within theframework of the theoretical approach discussed above, demonstrat-ed the utility of the streaming potential method to characterizeconformation of polyelectrolyte multilayers.

The electrokinetic methods have also been widely exploited for aqualitative determination of kinetics and mechanism of proteinadsorption on flat substrates using the parallel plate configuration[43,44] and the capillary arrangement [57,58]. The pioneers in this fieldwere Norde and Rouwendal [43] who determined changes in thestreaming potential of a glass substrate (microscope slides forming theparallel plate channel) induced by adsorption of lysosyme. Kinetic runshave been obtained under the forced convection transport conditionsfor a broad rangeof lysosymebulk concentrationvaried from0.1 ppm to10 ppm. It has been shown that the dependence of the streamingpotential on adsorption time (lysosyme coverage) was linear for initialstages of adsorption, what was a major finding of this work. Althoughthis is in accordance with the above theoretical predictions, aquantitative analysis of the data obtained in that work is not possiblebecause the coverage of lysosyme at the substrate surface was notdetermined.

Analogous linear dependencies of the streaming potential ofprotein covered surfaces on adsorption time have been reported inRef. [44] for bovine serum albumin (BSA), and immunogammaglobulins IgG and in Ref. [57] for fibrinogen.

Adsorption kinetics of lysosyme using the streaming potentialmethod was studied in detail by Etheve and Dejardin [58]. As thesubstrate surfaces, a bundle consisting of 1–3 silica capillaries ex-hibiting a rather scattered zeta potential ζi, varying between−23 and−45 mV was used. The electrolyte flow was forced by gas pressure.The coverage of adsorbed lysosyme, labeled using iodine isotope,was determined in situ using a radioactive detector. This allowed oneto express the relative streaming potential changes of capillaries interms of the amount of adsorbed protein in μgcm−2. A linear

Fig. 18. The reduced streaming potential for mica―Es covered by a PAH/PSS multilayers

vs. the number of layers nPE. The points denote experimental results obtained in theparallel-plate channel for pH=7.4, I=10−3M, TRIS buffer, T=293 K, ζi=−115 mV[114]. Part “a”, polyelectrolyte multilayer deposition was carried out for pH=7.4 from10−3M, TRIS buffer, part “b”, polyelectrolyte multilayer deposition was carried out forpH=5.5 and 0.15 M, NaCl.


dependence of the reduced streaming potential on the amount ofprotein adsorbed was determined in this way, although no attempt toanalyze these data quantitatively was undertaken. It is to mention,however, that the maximum coverage of adsorbed protein wasgenerally much smaller than determined in other studies reported inthe literature, which suggests incomplete monolayer formation.

The experimental data of Etheve and Dejardin [58] have beeninterpreted quantitatively in Ref. [97] by assuming that lysosyme geo-metrical cross-section area was equal to 12 nm2 and the zeta potentialof lysosyme ζp varied between 20 and 29 mV. It was found that thetheoretical results derived from Eqs. (15) and (16) reflect quite well theexperimental data for the range of the protein coverage of up to 0.2.

Osaki et al. [65] performed interesting studies of fibronectinadsorption (a massive glycoprotein with molecular weight of ca.440kD) on films of poly(octadecene-alt-maleic acid) (POMA) andpoly(propene-alt-maleic acid) (PPMA) formed on a glass substrate.The combination of streaming potential measurements carried out inthe microslit electrokinetic setup with in situ reflectometric interfacespectroscopy and QCM was used in this work. These measurementsallowed one to determine the dependence of adsorption mechanism

and the extent of conformational changes of the protein on these twosubstrates, which differed significantly in hydrophobicity. It has beenestablished that for themore hydrophobic POMA surface, fibronectionwas adsorbing irreversibly under random orientation with littletendency to changes in conformation. On the other hand, adsorptionon more hydrophilic PPMA films was more reversible and the proteinunderwent long-lasting conformational changes induced by electro-static interactions. However, a quantitative analysis of the streamingpotential changes of the substrate surfaces vs. the protein coveragehas not been undertaken. These results indicate that the electrokineticmeasurements can serve as an efficient tool not only for determiningthe amount of protein adsorption but also the protein/interfaceinteractions and the structure of the adsorption layers.

Similar regularities as discussed above for the channel flows havealso been reported in cases of protein adsorption on colloid particles,e.g., human serum albumin (HAS) on synthetic hydroxyapatitepowder [122], monoclonal IgGs and their fragments on polymericparticles (lattices) [123], anti-HSA on copolymericmicrospheres [124]and others [125]. Protein adsorption was studied by microelectro-phoresis, which allowed one to determine zeta potential of proteincovered particles. A steep, linear decrease in the relative zeta potentialof particles was observed as a function of protein concentration orcoverage as predicted by theory. A further quantitative analysis ofthese data is prevented, however, by the lack of sufficient physico-chemical data, characterizing the protein and substrate properties.

As the above examples suggest, the theoretical approach exposed inthis work can be effectively used for analyzing electrokinetic mea-surements of protein covered surfaces.However, a precise verification ofits range of applicability requires further experiments in which proteincoverage is determined via direct experimental methods, e.g., the AFMmethod, for substrates of controlled andwell defined surface properties.

4. Concluding remarks

The convective-diffusion theory makes it possible to predict initialdeposition rates of particles of various shapes, as a function of theirdiffusion coefficient, interface geometry and flow configuration.

On the other hand, particle deposition kinetics for a higher cover-age range, when the surface blocking effects play a dominant role, canbe derived from the RSA model enabling one to find the jammingcoverage and the structure of particle monolayers.

These theoretical results, confirmed by extensive experimentalmeasurements for colloid particles, can be used for a proper inter-pretation of the streaming potential measurements performed forparticle covered surfaces.

The essential finding derived from the streaming potential theoryis the expression for the normalized streaming current/potential ofparticle covered surfaces having the form

−Is =

−Es = 1−AiðΘÞΘ + ApðΘÞ

ζpζi

Θ:

This equation is valid for arbitrary shear flows, particle size andshape in the limit of thin double-layers.

For the low coverage regime Θ<0.05 and spherical particles, theabove expression of the streaming current/potential becomes

−Is =

−Es = 1−10:2Θ + 6:51

ζpζi

Θ:

It has been shown that this expression can be also used as a goodapproximation for relatively thick double-layers, provided that a/LeN1.

It has also been demonstrated theoretically that for elongatedparticles the expression for I s,

−Es in the low coverage regime,


assuming side-on orientation becomes

−Is =

−Es = 1−5:6Θ:

On the other hand, for the unoriented regime (particle orientationsaveraged between side on and perpendicular) the constant appearingin this equation was considerably higher, increasing with the axisratio of particles (number of beads), attaining the value of 40.6 for axisratio of 40 (see Table 4).

These theoretical results indicate unequivocally that the measuredstreaming potential values depend not only on the zeta potential ofinterfaces and particles but also on their orientation relative tosubstrate surfaces. This is a direct manifestation of the fact that theelectrokinetic properties of interfaces are of dynamic (irreversible)character rather than thermodynamic character, which is oftenoverlooked in theoretical interpretation of experimental data.

This suggests a possibility of determining via the streamingpotential measurements not only the coverage of particles, but alsotheir shape and conformation. For higher coverage range and forspherical particles, the Ai(Θ) and Ap(Θ) functions can well beapproximated by the expressions

AiðΘÞ =1−e−CiΘ

Θ≅10:2−5:75Θ

1 + 5:46Θ

ApðΘÞ =6:51−2:38Θ1 + 5:46Θ

:

These expressions are valid for Θ<0.50, i.e., for the entire range ofcoverage met in practice.

These predictions have been confirmed experimentally by mea-suring the streaming potential for monodisperse polymeric particlesusing the parallel plate channel cells. It has been proven in this waythat the particle coverage can be determined with high precision,even for weakly charged or neutral particles where othermethods fail.The utility of the streaming potential measurements for a quantitativedescription of polyelectrolyte and protein adsorption at planar andcurved interfaces has also been demonstrated.

Acknowledgments

This work was supported by the Polish Ministry of Science andHigher Education (MNiSzW) grants N205 022311112, N50704831208, the COST Action D43 Grant and by the NSF CAREERGrant No. CBET-0348175.

Appendix A

A.1. Derivation of equations governing streaming current for particlecovered surfaces

Derivation of equations presented in this Appendix is based on theconcepts developed in Refs. [59,67].

Let us consider a homogeneous interface of the area L×l with Np

particles irreversibly attached to it, immersed in an ambient laminarflow V0=G0zix. The component of the flow in the z direction isV0· ix=U0 (where ix is the unit vector in the x direction).

According to the constitutive equation, Eq. (44) the streamingcurrent in the presence of particles can be expressed as

Is = ∬S

ρe0V0⋅dS + ∬

S

ðρeV−ρe0V0Þ⋅dS = � ε∬

S

∇2ψ0V0⋅dS

�ε∬S

ð∇2ψV−∇2ψ0V0Þ⋅dS

ðA1Þ

where S is a plane perpendicular to the interface and the x axisdirection, placed somewhere over the interface (see Fig. 7), ρe is the

electric charge density in the presence of particles, ρe0 is the elec-tric charge density at bare interface (in the case of no particlespresent) and V is the actual flow near the interface in the presenceof particles.

The second term on the rhs of Eq. (A1) can be treated as the excessstreaming current due to the presence of particles.

Eq. (A1) can also be formulated in an alternative form, suitable forfurther manipulations

Is = Is0 + NpI1 = Is0 + Θ―I1 ðA2Þ

where

Is0 = � ε∬S

∇2ψ0V0⋅dS: ðA3Þ

Θ = SglL Np is the average coverage of particles over the interface

(Sg is the characteristic cross-section area of particles) and Ī1, is thereduced volume integral given by the expression:

―I1 =

lLSgNp

I1 =l

SgNp∭vðρeU−ρe0

U0Þdv ðA4Þ

where U=V· ix, dv is the volume element and I1 can be treated as anaverage contribution to the streaming current due to one of the Np

particles adsorbed on the interface.Assuming that the charge density vanishes at larger distances

from the interface, which is usually the case due to electrostaticscreening, the domain in the volume integral, Eq. (A4) can be expressedas

0 < z < ∞0 < x < L0 < y < l

: ðA5Þ

Eq. (A2), although generally valid for particles of any shape and forarbitrary flows, is rather impractical for a direct evaluation, since it isdependent on many parameters such as the particle coverage, anddistribution over interfaces, particle shape, double layer thickness,zeta potential of the particles and the interface, etc. Moreover, itsexplicit evaluation requires the knowledge of the fluid velocity andthe electric charge density fields. The latter cannot be calculated in anexact way for the system of many particles and the interface, becauseof the nonlinearity of the Poisson–Boltzmann equation.

Therefore, in order to derive from Eq. (A2) meaningful results ofpractical interest, a series of simplifying assumptions have to bemade.The first, most important is that the charge density (electrostatic field)is not too much perturbed by the velocity field, and vice versa, so theycan be evaluated independently from each other. Under thisassumption, the fluid velocity field in the case of low Reynolds numberflows (linear Stokes flows) can be evaluated using numericaltechniques for arbitrary particle distribution and coverage. However,this is not feasible at presentwith respect to the electric potential field,whose distribution is governed by the nonlinear Poisson–Boltzmann(PB) equation. The latter can be calculated for spherical particles in thetwo limiting cases (i) low coverage regime, where Θ1/2≪1, and (ii)thin double layer regime, where a/Le≫1. In the former case, one cansolve the PB equation numerically in a bipolar coordinate system,whereas in the latter case only the potential in the shear plane (particlesurface) is required.

Hence, in the case (i) assuming further the additivity of the electricfields stemming from the particle and the interface, on can express thestreaming current in the simple form:

Is = Is0 + Θ ðCiζi + CpζpÞ ðA6Þ


where the two constants Ci, Cp characterizing the contributionstemming from the interface and particles, respectively, are givenby the expressions

Ci =l

ζiSgNp∭vðρI

eU−ρe0U0Þdv

Cp = − lζpSgNp

∭v

ρIIeUdv

ðA7Þ

where ρeI , ρeII are the electric charge density fields in the case ofuncharged particles (ζp=0) and uncharged interface (ζi=0),respectively.

However, the electrostatic field needed for an explicit evaluation ofEq. (A7) can only be derived numerically, as done in Ref. [96].

In the case (ii), i.e., for thin double-layers, the change of electricfield occurs within thin layers adjacent to the interface and particlesurface, where the fluid flow becomes linear with respect to thedistance from these surfaces, because of the no-slip boundaryconditions.

Accordingly, fluid flow distributions in this region can be wellapproximated by the equation [67]

Ui = ziz⋅∇ðV−V0Þ⋅ix ðat the interfaceÞUp = ðr−aÞ⋅ikðrÞ⋅∇V⋅ix ðat each particle surfaceÞ ðA8Þ

where Ui is the fluid velocity component along the x direction in thevicinity of the interface, Up is the x-component of fluid velocity nearparticle surfaces iz is the unit vector in the z axis direction, ip is thevector perpendicular to the particle surfaces, pointing into the fluidand r is the position vector measured relative to the particle center.

Moreover, because of the thin layer assumption, the volumeintegral in Eq. (A4) can be reduced to the sum of surface integralscalculated over the interface and all particle surfaces. By consideringthis, and using Eq. (A8), one can formulate Eq. (A4) in the form

Is = Is0−ΘεlζiSgNp

∬Si

iz⋅∇ðV−V0Þ⋅ixdSi−ΘεlζpSgNp

∑Np

k=1∬Sk

ikðrÞ⋅∇V⋅ixdSk:

ðA9Þ

Thus, the expression for the streaming current for particle coveredsurfaces becomes

Is = Is0 + εlGoζiΑiðΘÞΘ−εlGoζpΑpðΘÞΘ ðA10Þ

where the constants Ai(Θ), Ap(Θ) are given by the expressions

AiðΘÞ = − 1SgGoNp

∬Si

iz⋅∇ðV−V0Þ⋅ixdSi

ApðΘÞ = − 1SgGoNp

∑Np

k=1∬Sk

ikðrÞ⋅∇V⋅ixdSk:

ðA11Þ

It is interesting to mention that Eqs. (A10) and (A11) are valid forarbitrary coverage of particles, their shape (at least if it can beapproximated by a configuration of touching spheres) and particledistribution over interfaces provided that the flow field near particlesis known.

In the low coverage limit of spherical particles, the flow field can beevaluated analytically in the form of series solution of sphericalharmonics [96] or in the form of Bessel functions [126]. However, for

higher coverage range, the flow can only be evaluated numerically[69].

In the case of ζi≠0, Eq. (A10) can be simplified to the reducedform by considering that for a simple shear ambient flow V0=G0zix,Is0=−εlG0ζi, thus

―Is =

IsIs0

= 1−ΑiðΘÞΘ + ΑpðΘÞΘζp

ζ i: ðA12Þ

On the other hand, in the case of ζi=0, Eq. (A10) becomes

―Is = ΑpðΘÞΘ ðA13Þ

where―Is = Is = ð−εlG0ζpÞ:

Limiting solution of practical interest can be derived fromEq. (A10) in the case of disk-shaped (two dimensional particles)adsorbing side-on at the interface. Since, obviously in this case,the macroscopic flow is not perturbed by particles, i.e., V=V0, thusAi(Θ)=0 and Ap(Θ)=1, because the surface integral over alladsorbed particles equals simply their surface area Sg Np. Therefore,in this case

―Is = 1 +

ζpζi

Θ: ðA14Þ

In Ref. [67] it was demonstrated that for spherical particles thefunctions Ai(Θ) and Ap(Θ) can be related to the hydrodynamic forceexerted by the flow on the particles and a force multipole. This can beexpressed as

AiðΘÞ = − 1ηSgGoNp

Fx

ApðΘÞ = − 1ηSgGoNp

Hx

ðΑ15Þ

where

Fx = F⋅ix = ½∑Np

k=1Fk�⋅ix

Hx = H⋅ix = ½∑Np

k=1

13a2

Q k � Fk

� ��⋅ixFk = ∬

Sk

ik⋅ΠdSk

Q k = 3∫½2ðr� RkÞ2Π ⋅ik−ðr� RkÞðr� RkÞ⋅Π ik�dSk

ðΑ16Þ

where Rk denotes the position of the k-th sphere over the interface, Skis the surface of k-th sphere and Π is the hydrodynamic stress tensorat the particle surface Sk, respectively.

A.2. Derivation of the streaming current expressions for the lowcoverage regime and thin double-layers using the cluster expansionmethod

Hereafter we derive the limiting expressions for Ai (Θ) and Ap (Θ)for thin double layers assuming that the electrostatic field around


each particle is the same as for an isolated particle and theelectrostatic field near the interface remains unperturbed.

The solution is sought in the form of a polynomial series expansionwritten as

AiðΘÞΘ = ∑3

n=1Cin

Θn

ApðΘÞΘ = ∑3

n=1Cpn

Θn

ðA17Þ

where Cin, Cpnare coefficients of the expansion to be determined.

The starting point of this derivation is the general expression forthe 2D cluster expansion for the system of Np particles distributedover an interface, S, which can be expressed using concepts ofstatistical mechanics in the form [16,127]

Θ―I; ðΘÞ = −∑

∞

n=1ð−1Þn Θ

n

n!∬…∫gnðr1;r2;…;rnÞ

―Ic

ðr1;r2;…;rnÞdr12dr13…dr1n

ðA18Þ

where g are the n-particle correlation functions, r12,…,r1n are therelative position vectors of particles forming the cluster and Ic arerelated to components of the cluster expansion evaluated for clusters,composed of n-particles.

For a 2D equilibrium distribution of hard (noninteracting) spheresthe two particle (pair) correlation function g2 can be approximated by[16]

g2ðr12Þ = 1 +―S12ðr12ÞΘ + 0ðΘ2Þ for 2a < r12 < 4a ðA19Þ

where―S12ðr12Þ = S12ðr12Þ= Sg and

S12ðr12Þ = 8a2 cos−1 r124a

� �−r12 4a2−

r212

4

!1=2

ðA20Þ

is the overlapping area of the two exclusion circles around eachparticle forming the pair (having the radius of 2a) and r12 is thedistance between the particle centers.

In the case of the RSA distribution of particles the pair correlationfunction becomes [16]

g2 = 1 +23―S12ðr12ÞΘ: ðA21Þ

Obviously, for r12<2a, the g2 function vanishes since particles donot penetrate into each other.

Considering the expression for g2, Eq. (A18) can be formulated upto the third order in particle coverage, in the following form

ΘI1 = εlζiGo Ii1Θ−12Ii12Θ

2 +16Ii123−

12I′i12

� �Θ3

� �

+ εlζpGo½Ip1Θ−12Ip12Θ

2 +16Ip123−

12I′p12

� �Θ3� + 0ðΘ4Þ:

ðA22Þ

The cluster integrals appearing in Eq. (A22) are explicitly given by

Ii1 = − 1πa2Go

∬Si

iz ⋅∇ðV1−V0Þ⋅ixdSi = 6Fx1

Ip1 = − 1πa2Go

∬Sp

ip⋅∇V1⋅ixdSp

Ii12 = −⟨ 1πa2Go

∬Si

iz⋅∇ðV12−V0Þ⋅ixdSi−2Ii1⟩r12 ;φ1

Ip12 = −⟨ 1πa2Go

∬Sp

ip⋅∇V12⋅ixdSp−2Ip1 ⟩r12 ;φ1

Ii123 = −⟨ 1πa2Go

∬Si

iz⋅∇ðV123−V0Þ⋅ixdSi−3Ii1 + Ii12 + Ii13 + Ii23 ⟩r12 ;r13 ;φ1 ;φ2

Ip123 = −⟨ 1πa2Go

∬Sp

ip⋅∇V123⋅ixdSp−3Ip1 + Ip12 + Ip13 + Ip23 ⟩r12 ;r13 ;φ1 ;φ2

I′i12 = −⟨―S12

πa2Go∬Si

iz⋅∇ðV12−V0Þ⋅ixdSi−2Ii1⟩r12 ;φ1

I′p12 = −⟨―S12

πa2Go∬Sp

ip⋅∇V12⋅ixdSp−2Ip1 ⟩r12 ;φ1

ðA23Þ

where V1(r) is the flow field near a singlet (r is the distance fromthe particle center), Fx1, is the dimensionless force acting on oneparticle (singlet) normalized by 6πηa2G0, V12(r, r12, φ1) is theflow near the doublet (r is the distance from one particle center,φ1 is the angle of the vector connecting particle centers measuredrelative to the ambient flow direction, the x-axis), V123 (r, r12, r13,φ1, φ2) is the flow near the triplet (r is the distance from oneparticle center, r12, r13 are the distance between particle centersand φ1,φ2 are the angles of the vectors connecting particle centersmeasured relative to the ambient flow direction, the x-axis), < >means averaging over the corresponding variables (distances andorientation angles).

As can be noticed, these integrals can be evaluated explicitly ifflow fields are known for clusters composed of one, two andthree particles, as a function of their orientations (against theambient flow direction) and mutual distances among particleforming a cluster. It is also worth while noticing, that in fact, allthese integrals, as shown in Ref. [67] are connected to the forceor its higher multipoles associated with the specific particleclusters. These quantities, and so the integrals in Eq. (A23) wereevaluated numerically in Ref. [67] using the efficient multipolemethod.

It was demonstrated in this way that for the hard sphereequilibrium distribution the expansion coefficient in the series,Eq. (A17) has the following numerical values

C1i = C01i = 10:2

C1p = C01p = 6:51

C2i = −59:4

C2p = −36:8

C3i = 292

C3p = 181

: ðA24Þ

It is interesting to mention that the coefficients C1i0 and C1p

0 havebeen calculated previously in Ref. [59].


In the case of the RSA distribution the only modification ofEq. (A24) is

C3i = 296C3p = 184 : ðA25Þ

Hence, the series expansion for the reduced streaming potential ofparticle covered surfaces with ζi≠0 is given by the expression

―Is =

―Es = 1−AiðΘÞΘ + ApðΘÞΘ

―ζ = 1−ð10:2−6:51

―ζ ÞΘ

+ ð59:43−36:82―ζ ÞΘ2−ð292−181

―ζ ÞΘ3 + 0ðΘ4Þ

ðA26Þ

where―ζ =

ζpζi:

In the case of a RSA configuration, the power expansion becomes

―Is =

―Es = 1−ð10:2−6:51

―ζ ÞΘ + ð59:43−36:82

―ζ ÞΘ2

−ð296−184―ζ ÞΘ3 + 0ðΘ4Þ:

ðA27Þ

In the case of a neutral interface, where, ζi=0 these expressionsbecome

IsðΘÞ = 6:51Θ−36:82Θ2 + 181Θ3 + 0ðΘ4Þequilibriumconfiguration

ðΑ28Þ

ΙsðΘÞ = 6:51Θ−36:82Θ2 + 184Θ3 + 0ðΘ4ÞRSAconfiguration: ðΑ29Þ

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Streaming potential studies of colloid, polyelectrolyte and protein deposition

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