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Journal of Colloid and Interface Science 277 (2004) 248–254www.elsevier.com/locate/jcis

Analysis of the electrophoretic mobility and viscosity of dilute Ludosolutions in terms of a spherical gel layer model

Stuart Allison

Department of Chemistry, Georgia State University, Atlanta, GA 30303, USA

Received 1 March 2004; accepted 16 April 2004

Available online 20 May 2004

Abstract

A spherical gel layer model of colloidal particles is used to analyze the electrophoretic mobility and viscosity of a dilute suspension of thsilica sol, Ludox, reported previously by Laven and Stein (J. Laven,H.N. Stein, J. Colloid Interface Sci. 238 (2001) 8–15). The colloimodeled as a sphere with a solid inner core surrounded by a diffuse gel layer of uniform thickness and comprising a specific fracf , ofthe colloidal particle’s mass. The gel layer is accessible to solvent and ions, but the gel layer retards the flow of solvent, which isto obey the Brinkman equation. The colloidal charge is assumed to be spherically symmetric, but its disposition on the surface oparticle and gel layer (α = fraction of charge in the gel layer) is left as an adjustable parameter. Experiments at pH 5.7 and 8.7 over aconcentration range of 0.3 to 80 mM are examined. At high salt and/or low pH, the thickness of the gel layer is estimated to be 1.0depending on the assumed fraction of silica present in the gel layer. At low salt (0.3 mM) and high pH, where the net absoluteLudox is large, the thickness of the gel layer is estimated to be 3.7 to 4.1 nm. Thus, the thickness of the gel layer appears to incdecreasing salt at high pH. The net charge required to simultaneously match experimental and model mobilities and viscosities isthe choice off andα. Nonetheless, for reasonable choices of these parameters (f = 0.13 andα ∼= 1.0), the estimated net absolute chargof Ludox from present modeling are in good agreement with the titration charges of Bolt (G.H. Bolt, J.Phys. Chem. 61 (1957) 1166–1169and Milonjic (S.K. Milonjic, Colloids Surf. 23 (1987) 301–311) over the entire salt concentration range at pH 8.7. At pH 5.7, howeestimated net absolute charge from current modeling exceeds the Bolt values by about 50%. 2004 Elsevier Inc. All rights reserved.

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1. Introduction

The combined and complementary techniques of etrophoresis and viscosity are potentially useful in the qutitative characterization of charge, size, and other strucfeatures of highly charged colloidal particles in solution.the present work, we shall focus on particles that are spically symmetric with respect to both their structure adistribution of charge. With regards to electrophoresis,theory is well developed for “hard” particles[1–4], “hard”particles with dynamic Stern layers (“hard” particles wspecial surface conductance properties)[5,6], and “soft” par-ticles that contain a porous gel layer that surrounds ainner core[7,8]. For dilute suspensions, viscosity theorywell developed for “hard” particles[9–11], and “hard” parti-cles that contain dynamic Stern layers[12–14]. As discussed

E-mail address: chesaa@panther.gsu.edu.

0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2004.04.050

in some detail previously[15], however, theory tends to underestimate the enhanced viscosity due to colloidal chaor the electroviscous effect, observed experimentally. Thas been observed in a number of investigations citeRef. [15]. One example is the combined viscosity and etrophoresis study of Ludox, a spherical silica sol, by Laand Stein[16]. At low ionic strength, the ratio of the observed electroviscous effect to that predicted for a “haparticle[11] is about 10. Those authors attributed this dcrepancy not necessarily to a breakdown in the underltheory of the primary electroviscous effect, but to the preence of a gel layer that is not included in modeling[16].

The objectives of the present study are twofold. First,tend modeling of the primary electroviscous effect of sphical particles to include the presence of a gel layer. Secmodeling Ludox as a sphere containing a gel layer, ipossible to account simultaneously for the electrophormobilities and viscosities observed experimentally? Theobjective is straightforward and is achieved by extend

S. Allison / Journal of Colloid and Interface Science 277 (2004) 248–254 249

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the procedure developed by O’Brien and White[4] (elec-trophoresis) and Watterson and White[11] (viscosity) ap-plicable to “hard” particles to gel layer particles. Hill, Saille, and Russel[8], recently followed this procedure, but thwork treated electrophoresis but not viscosity. A compuprogram has been developed that simultaneously compthe electrophoretic mobility and primary electroviscous effect of a gel layer model sphere. The second objectivachieved by then using this program to identify possmodels that are consistent with the transport measuremof Laven and Stein on Ludox[16].

The outline of this paper is as follows. InSection 3, thegel layer model is discussed and the significant model pmeters identified. This is followed by a brief discussionhow mobilities and viscosities are obtained from modeland how theoretical and experimental parameters associatwith viscosity are related.Section 4presents an analysof the experimental mobilities and viscosities of Ludox interms of the gel layer model. Specific models are identiwhich fit the experimental data very well. InSection 5, wediscuss what the gel layer modeling is telling us aboutcharge and structural characteristics of Ludox as a funcof salt and pH. The extension of this work to other systeincluding nonspherical particles is mentioned at the enthe paper.

2. Model

The particles are assumed to be spherical in shapebecome partially hydrated when dissolved in solvent. Ialso assumed that in solution, the particles are sufficiedilute so that particle–particle interactions are negligibThe “dry” particle radius and volume equalb∗ andV ∗

p , re-spectively. Letρ denote the weight density of dry matercomprising the core of the particle. For silica, this equ2.2 gm/cm3. Also let ρs denote the number, or “segmendensity of dry material. For silica, a “segment” is a Si2moiety, and it is straightforward to determineρs from ρ.When dissolved in solvent, a fraction,f , of the segmentsbecome solvated and a gel layer of thickness,d , is formed.Let a denote the radius of the unhydrated inner core anb = a + d denote the radius of the fully hydrated particlevolumeVp. The degree of solvation and segment densitthe gel layer are assumed to be uniform. The degree ofvation,SV, is defined

(1)SV = (Vp − V ∗

p

)/V ∗

p

and it is straightforward to show

(2)a = b∗(1− f )1/3,

(3)b = b∗(1+ SV)1/3.

In the gel layer, the presence of hydrated segmscreens the hydrodynamic flow of solvent and one canfine a hydrodynamic screening parameter,λ, which has units

s

s

of length−1. The fluid surrounding the particle is treatas a hydrodynamic continuum that is assumed to obeyBrinkman equation[17] for an incompressible fluid. Inreference frame chosen to be stationary with respect tocore of the particle,

(4)η0∇2�ν − �∇p = −�se + η0λ2�ν,

(5)�∇ · �ν = 0,

whereη0 is the solvent viscosity,v is the local fluid velocity,p is the pressure,se is the local external force/unit volumacting on the fluid due to electrical interactions, and theterm on the right hand side represents the external forcevolume due to frictional drag on the gel layer. Outside ofgel layer, this last term is absent. At the surface of the incore (r = a), “stick” hydrodynamic boundary conditions aassumed so thatv = 0. As discussed previously[18], λ2 isdirectly proportional to the segment density in the gel layFor the gel layer model of a sphere considered in this wit is straightforward to show that

(6)λ = λ∗/√

SV/f + 1,

whereλ∗ is the screening parameterof dry material. For sil-ica, this is approximately 12 nm−1 [18].

Finally, the total charge on the particle (in units of the ptonic charge) is taken to beQt. A fraction of this charge,α,is distributed uniformly throughout the gel layer, and themaining fraction, 1−α, is uniformly distributed on the innecore surface of the particle at radiusa. In summary, the seveparameters of the gel layer sphere are:a, b, λ, λ∗, f , Qt,andα. The parametersSV andb∗ can substitute fora andb according toEqs. (2) and (3). The number ofindependentparameters is six due toEq. (6).

3. Transport

The electrophoretic mobility,µ, and suspension viscoity, η, are determined for the model particle following tgeneral procedure of O’Brien and White[4] for mobility, andWatterson and White[11], for viscosity of “hard” particleslacking a gel layer. Recently, Hill, Saville, and Russel[8],generalized this approach to the electrophoresis of sphcontaining a gel layer. The present work extends thesinclude the viscosity of gel sphere models. We have wten a Fortran program, available upon request to the auwhich carries out the calculation of mobility and viscossimultaneously. A fourth order Runge–Kutta algorithm[19]is employed to numerically solve the differential equatiofor the equilibrium electrostatic potential, fluid velocity, aperturbed ion densities. Since the theory and methodologhave been discussed in detail previously[4,8,11], the inter-ested reader is referred to these earlier studies.

For a suspension of spheres, Laven and Stein[16] definea coefficient,p′, by the relation

(7)η − 1 = 5

(1+ p′)φ∗,

η0 2

250 S. Allison / Journal of Colloid and Interface Science 277 (2004) 248–254

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whereη0 is the solvent viscosity andφ∗ is the volume frac-tion of the suspension in the absence of hydration. Ifc is thenumber concentration of dissolved particles,φ∗ = cV ∗

p . Be-cause of hydration, however, a departure ofp′ from 0 is notdue only to the primary electroviscous effect as the folloing analysis shows. In modeling, it is convenient to deshape factors,ξ andξ0, for fully hydrated charged and uncharged model particles, respectively.

(8)η

η0− 1 = ξφ = ξ0(1+ p)φ = ξ0(1+ p)cVp,

whereφ is the volume fraction ofhydrated particles andpis the primary electroviscous coefficient. In the absence oa primary electroviscous effect,p equals zero regardlesshydration effects. Given thatVp = V ∗

p (1+ SV), p andp′ arerelated by

(9)p′ = 2ξ0

5(1+ p)(1+ SV) − 1.

Thus, when electroviscous effects are absent or small,p′ canshow significant departure from 0 if the degree of hydrais large. This can explain why Laven and Stein see relatilarge values ofp′ of Ludox at high salt concentration or lopH where primary electroviscous effects are expected tosmall.

4. Analysis of experimental results on Ludox

Laven and Stein[16] report zeta potentials,ζ , and vis-cosityp′ coefficients of Ludox in KCl solutions rangingconcentration,I , from 0.3 to 80 mM. Experiments were caried out at pH 2.7, 5.7, and 8.7. The authors extract thepotential, in turn, from electrophoretic mobility,µ, by mod-eling the particles as “hard” particles and employingmethod of O’Brien and White[4]. At pH 2.7, where theabsolute charge on Ludox is very low and electroviscousfects expected to be small,p′ ∼= 0.3 independent ofI . If weassumep = 0 in Eq. (9), then it is clear that there is considerable solvation of Ludox which, in terms of our modmeans the presence of a significant gel layer. Summarizein Table 1are transport properties at pH 5.7 and 8.7 ov

Table 1Transport properties of Ludox from Laven and Stein[16]

pH KCl (mM) −ζ (mV) −µ (cm2/kV s)a p′

5.7 1 36 0.167 0.66“ 10 24 0.120 0.38“ 100 10.5 0.059 –8.7 0.3 79.5 0.340 2.27“ 1 68 0.291 1.69“ 3 58 0.255 1.06“ 10 52 0.236 0.67“ 30 46 0.221 0.38“ 80 36 0.180 0.29

a Corrected to 20◦C.

range of different salt concentrations. These values aretracted from Figs. 4 and 5 of Laven and Stein[16].

The seven parameters of the gel layer model considin the present work area, b, f , Qt, α, λ, andλ∗; or, alterna-tively, b∗, SV , f , Qt, α,λ, andλ∗. As mentioned previouslyλ∗ is taken to be 12 nm−1 for silica. FromEq. (6), λ is a sim-ple function ofλ∗, SV, andf which reduces the numberindependent unknowns to five. On the basis of the discuson pp. 14–15 of Ref.[16], we assign a value ofb∗ = 7.5 nm(radius of a “dry” Ludox particle). This value is significantsmaller than “hard” particle estimates from sedimentatstatic, and dynamic light scattering[20]. However, these experiments are expected to yield a particle radius closea solvated particle radius at low salt and high pH whesubstantial swelling of the colloid is likely to occur as dcussed below. With this assignment ofb∗, the number ofindependent unknowns is reduced to four (SV, f , Qt, andα). Beyond this point, it becomes impossible to assignther parameters with certainty. Although the value off forLudox is not known, it has been estimated to be 0.20 forica sols of a similar size[21]. Consequently, values off inthis range shall be used. In the present work, we shalllow the strategy of considering models with different valuof f (fraction of silica in the gel layer) andα (fraction ofnet charge in the gel layer) which, combined with the preous parameter assignments, leaveSV andQt as unknownsThese remaining two parameters are then unambiguodetermined by finding models which simultaneously fitmobility and viscosity data of Laven and Stein[16].

For the present, let us focus on the high salt behawhere electroviscous effects are expected to be small ap ∼= 0. With b∗ andf assigned, the parametera is fixed byEq. (2). Also, ξ0 ∼= 5/2 for a sphere with a thin gel layer. Aan initial estimate, setp = 0 andξ0 = 5/2. UsingEq. (9)and the known value ofp′, an initial estimate ofSV is deter-mined.Equations (3) and (6)then yield initial estimates ofbandλ. At this point, a better estimate ofξ0 can be obtainedby modeling the uncharged model gel layer sphere withrent values ofb andλ. If necessary, this procedure canrepeated to yield self-consistent values ofb, λ, andξ0 for anuncharged model particle. In practice,ξ0 is fairly insensitiveto the model parameters for the cases of interest in this wand lies close to its hard-sphere value of 5/2. The only re-maining parameter isQt, and this is obtained by modelina range of model particles with variable charge (but idecala, b, f , α, andλ) until the modelµ matches experimenOnceQt is determined, we also have a better estimate op.If, for example,p deviates significantly from the estima(p = 0 in the above example), the whole procedure canrepeated until initial and final values ofp agree to a goodapproximation. This procedure can also be applied to losalt, but the initial estimates ofp andξ0 can be varied anmay be fairly far removed from their correct value. Becaof this, more iterations are required to achieve convergeEven at the lowest salt concentrations where electrovis

S. Allison / Journal of Colloid and Interface Science 277 (2004) 248–254 251

3 to

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effects are largest, however, the procedure converges in4 iterative cycles.

As an example, consider Ludox in 80 mM KCl at pH 8where experiment yieldsp′ = 0.29 andµ = −0.180 cm2/

kV s (from Table 1). Assumef = 0.13 andα = 1.00. Fol-lowing the procedure of the previous paragraph, we estima = 7.16 nm,b = 8.31 nm, andλ = 6.18 nm−1. This gellayer model givesp′ = 0.29 ignoring primary electroviscoueffects (p = 0). Next, this model is used with a range ofQt’sin order to find that charge which matches the experimtal mobility. The solid line inFig. 1 gives mobility versusQt for this model, andQt ∼= −290 yields a match betweemodel and experiment. This model also yieldsp = 0.009,p′ = 0.292 in good agreement with experiment. Also shoin Fig. 1 are corresponding mobilities for a solid sphere(a = b = 7.5 nm), which is the top dashed line, and the sagel layer model described previously, but withα = 0.20(lower dashed line). In order to match the experimental mbility, Qt ∼= −116 for the solid sphere and∼= −700 for thegel sphere model withα = 0.20. The overall shape of ththree curves inFig. 1 is similar, but at the same absolucharge, the absolute mobilities of the gel sphere modelslower than that of solid sphere. Also, shifting more of tcharge from the gel layer to the inner core surface ofotherwise same gel layer model reduces the absolutebility even more. In the particular example considered hthe absolute mobility of the solid sphere reaches a maximaroundQt ∼= −850, and such behavior is expected[4]. How-ever, no maximum is seen for either gel layer model overcharge range−1000< Qt < 0. Nonetheless, theµ versusQtcurves of both gel layer models inFig. 1appear to approac“plateau” behavior at high absolute charge. For the gel lamodel withα = 0.20, the absolute value of the “plateau” mbility is low. This accounts for the unusually large absolcharge required to match the experimental mobility forgel layer model withα = 0.20. In this case, a small changin mobility requires a large change in charge. For the ssphere and gel layer model withα = 1.0, a small change inmobility does not require as large a change in charge.

Fig. 1. Mobility versus charge at 80 mM KCl for three models. The(dashed) curve is for a solid sphere witha = b = 7.5 nm. The middle (solid)curve corresponds to a gel sphere model witha = 7.16 nm,b = 8.31 nm,λ = 6.18 nm−1, andα = 1.00. The lower (dashed) curve is for a gel sphmodel with the samea, b, andλ, butα = 0.20.

-

the basis of the experimental mobility measurement aloeach of the three models matches experiment providedare free to varyQt. If we also considerp′ from the viscos-ity measurement, we can rule out the solid sphere mowhich givesp = p′ = 0.010 which is much smaller than eperiment. From the mobility and viscosity data combinhowever, the two gel layer models with differentQt ’s andα’s cannot be distinguished.

For pH 5.7 and 8.7 calculations are carried out forf setequal to 0.05, 0.13, and 0.30. For each of these, two chmodels are considered in which 20% of the charge is ingel layer (α = 0.20) and all of the charge is placed in the glayer (α = 1.0). Results are summarized inTable 2(pH 5.7)andTable 3(pH 8.7). As discussed previously, oncef andα

are defined, it is possible to determineSV andQt (or a, b, λ,andQt) that match experimentalµ andp′. If the assumptionis made thatf andα remain constant asI varies, the chargon the Ludox increases and the gel layer thickness decreas the salt concentration,I , increases.

This trend of increasing colloidal charge with increaing ionic strength is consistent with titration charge msurements on silica[22–28]. At a particular ionic strengthconsiderable variation in the surface charge density ofica has been observed. Surface charge densities depethe type of monovalent salt as well as the ionic stren[23,28]. Surface charge density measurements are alsositive to the particular silica under study[27]. In comparingthe charge on Ludox estimated in the present work withdictions based on titration charge data, we shall considework of Bolt [22] and Milonjic[28] since the conditions theused are similar to those employed by Laven and Stein[16].The measurements of Bolt were carried out on Ludox,in NaCl rather than KCl salt solutions (used by Laven aStein). Milonjic investigated different monovalent salt sotions including both NaCl and KCl, but the silica sol was nLudox. However, the radius of the silica sol particles eployed by Milonjic was about 4.5 nm which is comparain size to Ludox. The absolute surface charge density ofica sol in 1 mM salt is about 25% higher in KCl than in Naat pH 8.5 from the study of Milonjic. However, at 100 msalt, the absolute surface charge densities are about thein both KCl and NaCl.

Shown in Fig. 2 are colloidal charge estimates fromBolt [22] (solid line) and Milonjic[28] (dashed line, KClsolution) and gel layer model values at pH 8.7. For the sof brevity, only thef = 0.13 gel layer model case shall bconsidered. Triangles correspond toα = 0.20 (20% of thecharge in the gel layer), and squares toα = 1.0 (all chargein the gel layer). From this figure, the gel-layer model vues withα = 1.0 are in qualitative agreement with estimafrom both Bolt and Milonjic.Fig. 3 compares the same glayer model charges considered inFig. 2with the Bolt chargeestimates (diamonds) at pH 5.7. The Milonjic study didgo below pH 6.5 and consequently no charge estimamade based on that work. At pH 5.7, the (absolute) Bcharges are about 2/3 those obtained by transport mod

252 S. Allison / Journal of Colloid and Interface Science 277 (2004) 248–254

Table 2Model parameters for Ludox at pH 5.7

f α I (mM) a b SV λ −Qt p p′ −µ (cm2/kV s)

0.05 0.20 1 7.37 9.06 0.76 2.98 37 0.033 0.62 0.165“ “ 10 “ 8 .60 0.51 3.59 55 0.010 0.37 0.120“ “ 100 “ 8 .45 0.43 3.87 87 0.002 0.30 0.059“ 1.0 1 “ 9.06 0.76 2.98 37 0.034 0.62 0.167“ “ 10 “ 8 .60 0.51 3.59 53 0.010 0.37 0.121“ “ 100 “ 8 .45 0.43 3.87 74 0.001 0.30 0.0590.13 0.20 1 7.16 8.93 0.69 4.78 38 0.034 0.62 0.168“ “ 10 “ 8 .51 0.46 5.63 57 0.010 0.38 0.120“ “ 100 “ 8 .34 0.38 6.09 96 0.001 0.30 0.059“ 1.0 1 “ 8.93 0.69 4.78 37 0.034 0.62 0.166“ “ 10 “ 8 .51 0.46 5.63 55 0.010 0.38 0.121“ “ 100 “ 8 .34 0.38 6.09 81 0.001 0.30 0.0590.30 0.20 1 6.66 8.86 0.65 6.74 38 0.034 0.62 0.167“ “ 10 “ 8 .45 0.43 7.69 60 0.010 0.38 0.120“ “ 100 “ 8 .31 0.36 8.10 124 0.001 0.30 0.059“ 1.0 1 “ 8.86 0.65 6.74 38 0.034 0.62 0.167“ “ 10 “ 8 .45 0.43 7.69 56 0.010 0.38 0.120“ “ 100 “ 8 .31 0.36 8.10 92 0.001 0.30 0.059

Table 3Model parameters for Ludox at pH 8.7

f α I (mM) a b SV λ −Qt p p′ −µ (cm2/kV s)

0.05 0.20 0.3 7.37 11.08 2.23 1.78 134 0.199 2.25 0.340“ “ 1 “ 10 .63 1.85 1.95 188 0.114 1.69 0.291“ “ 3 “ 9 .78 1.22 2.38 206 0.069 1.05 0.255“ “ 10 “ 9 .14 0.81 2.89 340 0.044 0.67 0.236“ “ 30 “ 8 .58 0.50 3.63 390 0.025 0.38 0.221“ “ 80 “ 8 .41 0.41 3.96 380 0.010 0.29 0.180“ 1.0 0.3 “ 11.08 2.25 1.78 108 0.199 2.25 0.340“ “ 1 “ 10 .63 1.85 1.95 129 0.113 1.69 0.291“ “ 3 “ 9 .78 1.22 2.38 142 0.068 1.05 0.255“ “ 10 “ 9 .14 0.81 2.89 190 0.041 0.67 0.235“ “ 30 “ 8 .58 0.50 3.63 226 0.023 0.38 0.219“ “ 80 “ 8 .41 0.41 3.96 242 0.009 0.29 0.1800.13 0.20 0.3 7.16 10.86 2.04 2.94 146 0.199 2.29 0.341“ “ 1 “ 10 .42 1.68 3.22 210 0.112 1.71 0.290“ “ 3 “ 9 .60 1.10 3.90 235 0.068 1.06 0.254“ “ 10 “ 8 .99 0.72 4.69 470 0.044 0.67 0.236“ “ 30 “ 8 .45 0.43 5.78 680 0.026 0.38 0.221“ “ 80 “ 8 .31 0.36 6.18 700 0.010 0.29 0.180“ 1.0 0.3 “ 10.86 2.04 2.94 112 0.197 2.28 0.340“ “ 1 “ 10 .42 1.68 3.22 135 0.111 1.70 0.290“ “ 3 “ 9 .60 1.10 3.90 150 0.067 1.05 0.255“ “ 10 “ 8 .99 0.72 4.69 215 0.041 0.66 0.236“ “ 30 “ 8 .45 0.43 5.78 265 0.024 0.37 0.221“ “ 80 “ 8 .31 0.36 6.18 285 0.009 0.29 0.1790.30 0.20 0.3 6.66 10.73 1.93 4.40 170 0.198 2.28 0.340“ “ 1 “ 10 .27 1.57 4.81 270 0.113 1.68 0.291“ “ 3 “ 9 .53 1.05 5.66 360 0.069 1.07 0.255“ “ 10 “ 8 .93 0.69 6.61 920 0.045 0.67 0.236“ “ 30 “ 8 .41 0.41 7.80 1750 0.028 0.38 0.221“ “ 80 “ 8 .28 0.35 8.16 2250 0.012 0.30 0.180“ 1.0 0.3 “ 10.73 1.93 4.40 115 0.197 2.28 0.340“ “ 1 “ 10 .27 1.57 4.81 143 0.113 1.68 0.292“ “ 3 “ 9 .53 1.05 5.66 165 0.067 1.06 0.255“ “ 10 “ 8 .93 0.69 6.61 250 0.042 0.67 0.235“ “ 30 “ 8 .41 0.41 7.80 348 0.025 0.38 0.221“ “ 80 “ 8 .28 0.35 8.16 390 0.010 0.30 0.181

S. Allison / Journal of Colloid and Interface Science 277 (2004) 248–254 253

on-ts of

omel-

ntra-

-

, thetwont

doxtivesub-yer.-ent

thateat

estck-medaltis

eryu-delonal

ringure-n-alsons,t ago-

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theH is

Fig. 2. Ludox charge at pH 8.7 as a function of monovalent salt ccentration. Solid and dashed lines are from the titration measuremenBolt [22] and Milonjic [28], respectively. Triangles and squares are frtransport modeling withα = 1.0 and 0.20, respectively. In transport moding, f = 0.13.

Fig. 3. Ludox charge at pH 5.7 as a function of monovalent salt concetion. Diamonds are from the titration measurements of Bolt[22]. Trianglesand squares are from transport modeling withα = 1.0 and 0.20, respectively. In transport modeling,f = 0.13.

ing. Also at pH 5.7, where the absolute charges are lowgel-layer model results are relatively insensitive to thedifferent charge distribution models. At pH 8.7, a significadifference between theQt ’s is seen for theα = 0.20 and 1.00model cases at high salt. In summary, the charges on Luestimated in terms of the gel-layer model are in qualitaagreement with titration charge estimates provided astantial fraction of the silica charge resides in the gel laAs the solid and dashed lines inFig. 2demonstrate, quantitative differences are seen in the titration charge measuremof silica sols reported in independent studies.

5. Discussion

It is clear from the previous section andTables 2 and 3that it is possible to simultaneously find gel layer modelsfit the mobility and viscosity data. This analysis supports thpicture of a gel layer approximately 1.0 to 1.5 nm thickhigh salt that expands on going to low salt. At the lowsalt studied, 0.3 mM KCl and at pH 8.7, the gel layer thiness varies from 3.7 to 4.1 nm depending on the assuvalue of f . It should also be mentioned that the low shydrodynamic radius,Rh, corresponding to these models

s

10.8 to 10.9 nm. (In addition to electrophoretic mobilitiesand viscosities, our computer program yieldsRh for chargedgel layer spheres.) Dynamic light scattering studies at vlow salt, high pH (pH 9.7), and low concentration of Ldox yieldRh values that are actually higher than the movalues mentioned above, but this could be due to additiexpansion on going to a still lower salt concentration[20].The important point is that the independent light scattemeasurements as well as the mobility and viscosity measments of Laven and Stein[16] point to substantial expansioof Ludox at low salt and high pH. Also, ap′ value substantially greater than 0 at high salt shows that a gel layer ispresent at high salt and/or low pH. Under these conditiohowever, the gel layer thickness of Ludox is smaller. ApH of 5.7, the thickness of the gel layer also expands oning to low salt, but upon comparing results inTables 2 and 3,this expansion is less pronounced than at high pH.

The charges,Qt, estimated in current modeling fall in thrange predicted by Bolt[22] and Milonjic[28] at pH 8.7. Theactual estimated charge from modeling, however, is sensto how the charge is distributed, particularly at high salt. Tcharges estimated by Bolt and Milonjic at pH 8.7 are mconsistent with a model in which a substantial fractionthe particle charge is distributed in the gel layer (α ∼= 1.0).It has been known for a long time that silica sols may hunusually large surface charge densities and this obsetion has supported the view that certain silica surfaces“porous” [23]. Some variation in the surface charge densof silica sols and silica surfaces in general are reported inliterature as discussed briefly in the previous section. Rmusson and Wall[24] report absolute charge densities oparticular silica sol that are about 50% higher than repoby Bolt. Kosmulski et al. report surface charge densitiesfumed silica[25] and quartz[26] over a concentration rangof NaCl from 0.001 to 1.0 M. The charge densities theyport tend to be higher than Bolt by 10 to 30%. It is probasafe to conclude that absolute charges somewhat largerthose reported by Bolt may be expected, but not excesslarger. On this basis, we can probably rule out the mowith f as large as 0.3 with most of the charge placed oninner core surface since such a model would have to havery large absolute charge. At pH 5.7, the estimatedQt ’sare in less satisfactory agreement with the predictionBolt. However, pH 5.7 lies close to the pKa of the silanolresidues that are responsible for the charge on Ludox infirst place[27]. It is well known that the actual chargea colloid or macromolecule varies more strongly withwhen the pH lies close to the pKa of its dissociating sitesIt should also be emphasized that most of the charge destudies were carried out in NaCl solutions whereas the wof Laven and Stein was carried out in KCl salt solutioThere could be differences in the surface charges betwthe two salt solutions, particularly when pH∼= pKa [28].There are also subtle counterion effects with regards tocoagulation behavior of silica sols that develop as the plowered[29], which is consistent with this interpretation.

254 S. Allison / Journal of Colloid and Interface Science 277 (2004) 248–254

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A few comments are in order concerning alternativethe gel layer model considered here. Dynamic Stern lamodels[12–14], in which there exists a layer of mobile ionwithin the Stern layer, have been successful in reconclow absolute mobilities and high absolute surface chadensities on colloidal particles. The same models, howehave been less successful in accounting for the viscosidilute, but highly charged spherical colloids. As in the cof a “hard” spherical particle model, dynamic Stern lamodels underestimate the primary electroviscous effecsystems such as polystyrene latex spheres[13]. The analy-sis of other systems such as alumina[30] have the additionacomplication of being nonspherical, and the primary elecviscous effect of nonspherical particles is strongly shapependent[31]. The anomalous dependence of the viscositLudox on salt concentration and its interpretation in termparticle expansion has been recognized for some time[32].In addition, the presence of a gel layer on silica sols belopH of 10.5 has been postulated in order to explain theirusual coagulation behavior[29]. In order to account for thlargep′ coefficients observed by Laven and Stein for Ludat high salt and/or low pH, we believe that some expanof the colloidal particle must occur. In many respects, thelayer model is a generalization of the dynamic Stern lamodel in which a thin conduction layer of ions on the sface of the particle is replaced with a thicker gel layer. Atmolecular level, the gel layer model allows for a zone nthe colloidal surface that consists of solvent and a loosework of colloidal material, or “polymer” strands protrudininto the solvent. Such a picture coincides with our currunderstanding of the colloidal silica surface[33].

It is hoped that the gel layer spherical model will befuture used in understanding the transport of colloidal pacles. As stated previously, the stand alone Fortran prograavailable upon request from theauthor. In addition, modeling of nonspherical gel layer models is now possible usinboundary element procedure[18]. These calculations, however, are much more computationally demanding thanprocedure used in the present study.

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