Aggregation of Colloidal Silica by n-Alkyl SulfatesSteven R. Kline† and Eric W. Kaler*

Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering,University of Delaware, Newark, Delaware 19716

Received August 25, 1995. In Final Form: March 1, 1996X

The coagulation behavior of aqueous colloidal silica (Ludox TM) in the presence of a homologous seriesof n-alkyl sulfates has been studied. Coagulation concentrations were measured for a simple salt, NaCl,and electrolyte/surfactantsNaCnSO4with chain lengthsn) 1, 2, 6, 8, 10, or 12 carbons. TheC6 and shorterhomologues had coagulation concentrations equivalent to that of NaCl, while C8 and higher homologueshad coagulation concentrations at lower ionic strengths. Calculations of the Ludox-Ludox interactionpotential show that the coagulation concentration results are consistent with the action of a screenedrepulsion plus a depletion attraction induced by the presence of surfactant micelles. Small angle neutronscatteringmeasurements weremade ofmixtures with sodium dodecyl sulfate (SDS) under contrast-matchconditions that isolated the silica-silica interactions. The silica-silica interactions indeed progressedfrom repulsive to attractive as more SDS micelles were added, a trend consistent with the observedaggregation. Silica in the presence of NaClwith an ionic strength equivalent to that of 0.40MSDS showedhard sphere interactions,whereas the sample containingSDSmicelles showed strong long-range attractiveinteractions. Thus we show how solvent microstructure influences the stability of a colloidal dispersion.

Introduction

Understandingandcontrolling thestability ofa colloidaldispersion are essential for its successful use. Specificapplicationsmayrequire thedispersion tobewell behavedoverawiderangeof temperaturesandchemical conditions.It may also be desirable to have other colloidal-sizedcomponents present in the mixture in addition to theoriginal dispersion. The behavior of thesemixed colloidalsystems can be much more complex than that of adispersion with only a single colloidal component.Charge-stabilized dispersions are described classically

byDLVOtheory,1which showshow the total interparticlepotential is the sum of the van der Waals attraction andthe Coulomb repulsion. As a result of this balance ofpotentials, a charge-stabilized dispersion can be madeunstableby screening theelectrostatic repulsion, typicallyby the addition of electrolyte. Sufficiently large amountsof electrolyte can cause the particles to aggregate.Dispersions can also be destabilized by the addition of‘microstructure’ to the solvent that surrounds the dis-persed colloid. Thismicrostructure ismost commonlydueto a soluble polymer2 but can also be self-assembling, asoccurswhen surfactantmicelles destabilize emulsions.3-5

In these colloidal mixtures, there is now an osmotic(depletion) attraction6-8 in addition to the DLVO-typepotential. The thermodynamics of colloidal mixtures isa rich field, and there have been many recent investiga-tions into the phase behavior9-12 and stability13,14 of thesemixtures.

Here we investigate the changes in interparticle in-teractions caused by added electrolyte as the electrolyteprogresses from a simple salt, to a weak amphiphile, toultimately a true anionic surfactant. Our model charge-stabilized dispersion is the commercial silica dispersionLudox TM (DuPont). The surface chemistry of silica hasbeen well studied,15 and the stability and phase behaviorof silica in mixtures with weak nonionic amphiphiles16and nonionic surfactants17,18 have been investigated. Awide variety of equilibrium behavior has been observed,including surfactant adsorption and colloidal phase sepa-rations. In some cases, the interparticle interactionsbetween the silica particles undergo a transition fromcharge repulsion to attractive interactions that driveaggregation as solution conditions change.Interactions on the colloidal length scale can be deter-

mined with properly analyzed small angle neutron scat-tering (SANS) experiments.19 To determine interactionsin a colloidalmixture (i.e., silica plusmicelles), a contrastvariation technique20,21 is necessary to isolate the con-tributions from individual colloidal populations. Herewewill use this method to determine the effect that ahomologous series of n-alkyl sulfates has on the inter-particle potential of a model silica dispersion. Thesemicroscopic measurements will then be related to themacroscopic properties of the dispersion as determinedby coagulation measurements. Ludox stability will bedetermined after addition of simple electrolyte, weakamphiphiles, and true ionic surfactants. Anionic sulfatesurfactants were chosen for study because they do notadsorb to the highly negatively charged silica surface.22The results show how interactions in a charge-stabilized* To whom correspondence should be addressed.

† Present address: National Institute of Standards and Technol-ogy, Bldg. 235/E151.

X Abstract published in Advance ACS Abstracts, April 15, 1996.(1) Verwey, E. J. W.; Overbeek, J. T. G. Theory of the Stability of

Lyophobic Colloids; Elsevier: Amsterdam, 1948.(2) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal

Dispersions; Cambridge University Press: New York, 1989.(3) Aronson, M. P. Langmuir 1989, 5, 494.(4) Bibette, J.; Roux, D.; Nallet, F. Phys. Rev. Lett. 1990, 65, 2470.(5) Bibette, J.; Roux, D.; Pouligny, B. J. Phys. II 1992, 2, 401.(6) Asakura, S.; Oosawa, F. J. Chem. Phys. 1954, 22, 1255.(7) Asakura, S.; Oosawa, F. J. Polym. Sci. 1958, 33, 183.(8) Vrij, A. Pure Appl. Chem. 1976, 48, 471.(9) Patel, P. D.; Russel,W. B.J.Colloid Interface Sci. 1989, 131, 192.(10) Dey, D.; Hirtzel, C. S. Colloid Polymer Sci. 1991, 269, 28.(11) Ilett, S. M.; Orrock, A.; Poon, W. C. K.; Pusey, P. N. Phys. Rev.

E 1995, 51, 1344.(12) Piazza, R.; Pietro, G. D. Europhys. Lett. 1994, 28, 445.(13) Ma, C. Colloids Surf. 1987, 28, 1.

(14) Giordano-Palmino, F.; Denoyel, R.; Rouquerol, J. J. ColloidInterface Sci. 1994, 165, 82.

(15) Iler, R. K. The Chemistry of Silica; John Wiley and Sons: NewYork, 1979.

(16) Kline, S. R.; Kaler, E. W. Langmuir 1994, 10, 412.(17) Cummins, P. G.; Staples, E.; Penfold, J. J. Phys. Chem. 1990,

94, 3740.(18) Cummins, P. G.; Staples, E.; Penfold, J. J. Phys. Chem. 1991,

95, 5902.(19) Kaler, E. W. In Modern Aspects of Small-Angle Scattering;

Brumberger, H., Ed.; Kluwer Academic Publishers: Boston, 1995; pp329.

(20) Small Angle X-Ray Scattering; Glatter, O., Kratky, O., Eds.;Academic Press: New York, 1982.

(21) Williams, C. E. In Neutron, X-Ray and Light Scattering:Introduction to an Investigation Tool for Colloidal and PolymericSystems; Lindner, P., Zemb, T., Eds.; North-Holland: New York, 1991.

(22) Huang, Z.; Yan, Z.; Gu, T. Colloids Surf. 1989, 36, 353.

2402 Langmuir 1996, 12, 2402-2407

S0743-7463(95)00716-5 CCC: $12 00 © 1996 American Chemical Society

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dispersion change as the surrounding electrolyte beginsto form microstructure, which thereby transforms thesystem into a colloidal mixture.

TheoryA. Small Angle Neutron Scattering. Small angle scat-

tering can provide a measure of interparticle interactions. Thepowerful technique of contrast variation, which can be appliedto isolate the Ludox-Ludox interactions, and the theory describ-ing the scattering from multicomponent mixtures have beendescribed elsewhere.19,23 In the present experiments, attentionis focused on the interactions between the silica particles in thebimodal mixture of surfactant micelles and silica. In this case,contrast variation20,21 is used to eliminate the contribution of themicelles to the scattering. Since the micelles have a core andshell structure, they can be matched in contrast to the solvent(H2O/D2O) approximately by an appropriate level of deuterationof the micelle core. The scattering length density of the shell ofhydrated headgroups is nearly the same as that of the solvent,so the shell scattering contributes less than 1% of the totalscattered intensity even at the high micelle volume fractions(>10%) used in this study.With the micelles contrast-matched, the scattered intensity

reflects only the contributions from the Ludox particles. Themulticomponent scattering equations then reduce to the con-venient form

Here,q) (4π/λ) sin(θ/2) is themagnitude of the scattering vector.The number density of particles is np, and S(q) is the structurefactor, f(q) is the scattering amplitude, and the angular bracketsdenote an average over the size distribution. We use a Schultzdistribution24 to describe the size polydispersity of the Ludoxparticles,which is determined fromSANSexperiments onLudoxdispersionsalone. With the form factor, ⟨f2(q)⟩, known, the fittingprocedure determines the type (attractive or repulsive) ofinteraction and its strength. The Ludox-Ludox interactionschange fromrepulsive toattractiveas thecharge-stabilizedLudoxdispersion is destabilized. A Coulombic potential (RC) is usedto describe repulsive interactions between particles of charge z,with electrolyte screening characterized by the Debye length,κ-1. An analytic solution for this structure factor is availableusing theRMSAclosure.25,26 Attractive interactionsaremodeledwith a structure factor calculated for a square well (SW)potential27withdepthU0 and range λσL,whereσL is the diameterof theLudoxparticle. At the lowvolume fractionsandwell depthsencountered in this study the solution of Sharma and Sharma27compareswell toMonteCarlo simulations.28 Formoreattractivesquare wells or higher volume fractions a thermodynamicallyconsistent scheme such as HMSA is more appropriate.28 Forhard sphere (HS) interactions we describe the data with ananalytic expression29 for the total scattered intensity rather thanthe form of eq 1.B. Depletion Potential. DLVO theory describes the total

interparticle interaction potential as the sum of van der Waalsattractions and electrostatic repulsions. The DLVO potential ischaracterized by a repulsive energetic barrier of several kT thatprevents two particles from coming into close approach andaggregation. In a binary (or bimodal)mixture of particles, thereisanadditionalattractive component to the interparticlepotentialcalled the depletion attraction.6-8 For a mixture of Ludox andmicelles, the depletion attraction is

where Πm ) n*mkT is the osmotic pressure of the micelles. σj )(σL + σm

eff)/2 is the interaction diameter, where σL is the diameterof the Ludox particles and σm

eff is the effective interactiondiameter of the micelles. The attractive potential between thelarger particles arises when they are in close approach andexclude themicelles fromthevolumebetween them. This resultsin a net unbalanced osmotic pressure and an effective attractiveforce between the larger pair.The magnitude of the depletion attraction is proportional to

the number density,n*m, ofmicelles, and its range depends on themicelle interaction diameter. These two quantities must becalculated correctlywhenevaluating thedepletionpotential. Thenumber density ofmicellesmust be calculated on the basis of thefreevolumeavailable to themicellesnot the total solutionvolume.Toa first approximation, the free volumeavailable to themicellesis 1 - φL, and more exact calculations can be done for specificsize ratios of interest.11,30,31A charged micelle excludes a larger volume than its hydrated

volume. The appropriate interaction diameter of the chargedsurfactant micelle is the effective hard sphere diameter σm

eff andis determined by calculating the second virial coefficient for therepulsive Coulombic potential and setting it equal to the hardsphere virial coefficient, B2HS(T) ) 2πσHS

3 /3. For most interac-tion potentials, the second virial coefficient must be determinedby numerical integration. The effective hard sphere dimensionsare then used to calculate the effective hard sphere volumefraction of micelles. All micelle and surfactant concentrationsreported here have been calculated on the basis of the availablefree volume.

Experimental Section

Ludox TM was supplied by DuPont as a 50 wt %dispersion of discrete silica spheres of approximately 22nm diameter and was used as received. Sodium methylsulfate and ethyl sulfate were purchased from TCIAmerica. Sodium n-hexyl and n-dodecyl sulfate werepurchased from Lancaster Synthesis; sodium n-octyl andn-decyl sulfatewere purchased fromKodak. Ethyl Violetand tolueneused for the surfactant assaywere purchasedfromAldrich. D2Owaspurchased fromCambridge IsotopeLabs, 99.9 mol % enriched, and perdeuterated sodiumdodecyl sulfate,C12D25SO4

-Na+ (MSDIsotopes)hada98.6mol % enrichment of the dodecyl tails.Hexyl and longer chain alkyl sulfates were purified by

passing a concentrated (∼10-20 wt %) solution of thesurfactant through a C18 reverse phase column (WatersandAssociates) that retainshydrophobic impurities.32Thesurfactant concentration in the eluent was measured bya spectrophotometric dye assay.33,34 Methyl and ethylsulfate were used as received. Critical micelle concentra-tions of sodium octyl, decyl, and dodecyl sulfate weredetermined in 0.05 M NaCl at 25 °C by surface tensionusing aKrussDigital TensiometerK10Twith aWilhelmyplate. Therewere nominima in the surface tension plots.The Ludox dispersion, as supplied, is at pH ∼ 9 with

a background electrolyte concentration of 0.05 M sodiumsalts. All samples were prepared maintaining the pH ∼9andbackgroundelectrolyte conditions of the stockLudoxdispersion. At these pHandelectrolyte conditions, Ludoxdispersionsare stable formore thanayear. Alkyl sulfate-Ludox mixtures were prepared by weight from stocksolutions of Ludox and the desired surfactant. Concen-trationsandvolume fractionswere calculated fromweightfractions and measured densities of the solutions. All ofthesurfactant concentrationsandmicellevolume fractions

(23) Chen, S. H.; Lin, T. L. Methods Exp. Phys. 1987, 23B, 489.(24) Kotlarchyk, M.; Chen, S. H. J. Chem. Phys. 1983, 79, 2461.(25) Hayter, J. B.; Penfold, J. Mol. Phys. 1981, 42, 109.(26) Hansen, J. P.; Hayter, J. B. Mol. Phys. 1982, 46, 65.(27) Sharma, R. V.; Sharma, K. C. Physica 1977, 89A, 213.(28) Bergenholtz, J.;Wu,P.;Wagner,N. J.;D’Aguanno,B.Mol. Phys.

1996, 87, 331.(29) Griffith, W. L.; Triolo, R.; Compere, A. L. Phys.Rev. A 1987, 35,

2200.

(30) Lekkerkerker, H. N. W. Colloids Surf. 1990, 51.(31) Lekkerkerker,H.N.W.;Poon,W.C.K.; Pusey,P.N.; Stroobants,

A.; Warren, P. B. Europhys. Lett. 1992, 20, 559.(32) Rosen, M. J. J. Colloid Interface Sci. 1981, 79, 587.(33) Motomizu, S.; Fujiwara, S.; Fujiwara, A.; Toei, K. Anal. Chem.

1982, 54, 392.(34) Nakamae, M.; Ogino, K.; Abe, M.Colloid Polym. Sci. 1988, 266,

475.

I(q) ) np⟨f2(q)⟩S(q) (1)

Udep

kT) - 4π

3Πm[σj3 - 3

4rσj2 + 1

16r3] σL < r < 2σj (2)

Aggregation of Colloidal Silica by n-Alkyl Sulfates Langmuir, Vol. 12, No. 10, 1996 2403

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have been corrected for the free volume available to themicelles as described above. Alkyl sulfate-Ludox mix-tures were stored in a 25 °C water bath, and thecoagulation stability of the Ludox TM was determinedafter a period of 24 h. The stability of the Ludox wasquantified by measuring a consistent coagulation con-centration for each sodiumalkyl sulfate. The coagulationconcentration was defined as the concentration of elec-trolyte required to cause a10% increase in the absorbanceat 500 nm after a period of 24 h. Absorbance measure-mentswere performed onaPerkin-ElmerLambda2UV-VIS spectrophotometer with the samples held in 2 mLscrew-capped vials.Neutron scattering experimentswere performed on the

30m spectrometer at theNational Institute of Standardsand Technology in Gaithersburg, MD. Neutrons ofwavelength λ ) 5 Å with a spread of ∆λ/λ ) 15% werefocused on samples held in 2 mm quartz cells. Twodifferent sample to detector distances of 1.6 and 12 mwere used to give q overlap between data sets and anoverall q range of 0.005 Å-1 < q < 0.22 Å-1. Scatteringfromthe sampleswas corrected for backgroundandemptycell scattering. Thesensitivityof individualdetectorpixelswas normalized by comparing it to the incoherent scat-tering of water. The corrected data sets were circularilyaveraged and placed on an absolute scale by use of apolystyrene/deuterated polystyrene standard and NISTsoftware.35 Instrumental smearing was simulated36 forthe instrument configurationandwavelength spreadusedandwas found to be negligible, except for the low-q regionof the 1.6mdata (0.03Å-1 <q<0.05Å-1). These smeareddatapointswerenot included in themodel fitting. Asinglespectra, Figure 3, was collected at detector distances of3.4m (25 cm detector offset) and 12m. The same q rangewas obtained, but now smearing effects were negligiblefor the entire q range. Intensity models were fit to theexperimental data by using a nonlinear least-squaresoptimization procedure. The goodness of fit criteria wasthe ø2 error between model and data.

ResultsFigure 1 shows an example of the behavior of Ludox

withananionic surfactantand isapseudoternary stabilitydiagram for the surfactant sodium dodecyl sulfate (SDS),

LudoxTM, and 0.05MNaCl. At lowSDS concentrations,the Ludox is stable and does not aggregate, and thedispersions appear transparent and blue. At highersurfactant concentrations, after a period of 24 h, thesolutions appear noticeably turbid. At still higher SDSconcentrations, the Ludox forms a gel or a thick whiteprecipitate. The turbid and gel regions on the diagramgrowlargerandmove to lowerSDSconcentrationsat timeslonger than the initial 24 h observation. These observa-tions suggest that the stability of Ludox with respect toSDS is a kinetic rather than an equilibrium process overthe experimental time scale of 24 h. For this reason, thestability boundary is defined by a coagulation concentra-tion. The coagulation concentration corresponds to aspecific rate of aggregation, which in turn corresponds toa specific potential energy barrier that the particlesmustovercome to coagulate.Coagulation concentrations were measured for NaCl

and for the homologous series of n-alkyl sulfates. Thecoagulation concentration of NaCl occurs at a high ionicstrength (Figure 2), and the coagulation concentrationsof the shorter chain alkyl sulfates (C1, C2, and C6homologues) coincide with the value for the simpleelectrolyte. In contrast, the longer chain homologuessodium octyl, decyl, and dodecyl sulfate, which all formmicelles in solution, display coagulation concentrationsat significantly lower ionic strengths. Onan ionic strengthbasis, the micelle-forming alkyl sulfates are far moreeffective in coagulatingLudox than is a simple electrolyte.The SANS spectrum for Ludox with added electrolyte

is shown as the open circles in Figure 3 and serves tocharacterize the Ludox population. The line through thedata is a fitted model of polydisperse hard spheres. TheLudox volume fraction is φL ) 0.06, and NaCl was addedto give an ionic strength of I ) 0.098 M. The volumefraction of Ludox was calculated from the known weightfraction of silica in the sample. The datawas fit well withonly two adjustable parameters, the Ludox radius, RL, of118 Å and a Schultz polydispersity σR/RL ) 0.18. Ascattering length density of 3.65× 1010 cm-2 was used forLudox, calculatedusing abulkdensity of silica of 2.3 g/mLdetermined from the measured density of dilute Ludoxdispersions. These fittedparameters forLudoxwereheldconstant for all subsequent modeling.When the added electrolyte is amphiphilic enough to

formmicelles, the solution contains a bimodal populationof scatterers. When the scattering length density of the

(35) NIST “SANS Data Reduction and Imaging Software”, 1993.(36) Barker, J. G.; Pedersen, J. S.J.Appl.Crystallogr. 1995, 28, 105.

Figure 1. Pseudoternary stability diagram for Ludox TM/SDS/0.05 M NaCl. Regions were determined visually after aperiod of 24 h at 25 °C. Only the brine-rich corner is shownbecause the Ludox is supplied as 50 wt % in 0.05 M NaCl, and30 wt % SDS is near its solubility limit.

Figure 2. Coagulation concentrations determined for LudoxTM in a homologous series of alkyl sulfates and NaCl.Coagulation concentration curves for NaCl (a), sodiummethyl(b), ethyl (c), andhexyl sulfate (d) all overlap. The longer chainalkyl sulfates that formmicelles, octyl (e), decyl (f), and dodecyl(g), all have coagulation concentrations at significantly lowerionic strengths. The ×’s at φL ) 0.06 mark the compositionsof the SANS samples shown in Figure 4.

2404 Langmuir, Vol. 12, No. 10, 1996 Kline and Kaler

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solvent isdifferent than that of bothof the twopopulations,the scattered intensity has contributions from bothpopulations. The SANS spectrum for a mixture of SDSmicelles and Ludox is also shown in Figure 3. The muchsmaller micelles (Rm ) 25 Å, φm

eff ) 0.44) display acorrelation peak near q) 0.09 Å-1 and dominate the highq scattering,while the scatteringat lowq is predominantlyfrom the larger Ludox particles. In contrast to the SANSspectrum fromLudox particles alone, the low q scatteringis reduced in the bimodal mixture. This is a result of thecontributionof thecross-structure factor (which isnegativeat low q) to the total scattered intensity.37The scattering from the SDS micelles has been mini-

mized by using SDS with deuterated tails. Isotopic H/Dsubstitution is usually assumed to have no effect onsurfactant microstructures, although the substitution ofH2O/D2O does affect the micellization properties of sur-factants.38 Isotopic substitution D/H in the surfactanttails has a lesser effect on the micellization properties.MixturesofDTABandDDABandthedeuteratedanalogueof DTABmix ideally and show nomeasurable differencesdue to the isotopic substitution in the alkyl tails.39Furthermore, the SANS spectra fromSDS-hmicelles andSDS-d micelles were measured and analyzed to give thesame micelle properties.40 Experimentally, the contrastmatch point occurs when 72 mol % of the hydrogenatedtails are replaced with deuterated tails, giving both theSDS micelle core and this H2O/D2O solvent (14% H2O byweight) equal scattering length densities of 5.35 × 1010cm-2.SANS data for contrast-match conditions are shown in

Figure 4, for a Ludox concentration of φL ) 0.06 and fivedifferent micelle volume fractions approaching the co-agulation concentration frombelow. The compositions ofthese samples are indicated by the x’s in Figure 2. Thelines through theSANSdata aremodel fits using eq 1 andthe appropriate form of the potential (RC, HS, or SW),and the results are shown in Table 1. The interactionstrength was the only significant adjustable parameter

in the fitting. The incoherent background and a multi-plicative scale factor to correct forminor errors inabsolutescaling (<10%) were also fitted parameters but did notinfluence the shape of the spectra. The Ludox-Ludoxinteraction progresses from repulsive, to hard sphere, tostrongly attractive as the micelle volume fraction in-creases. The spectrum from the sample with the lowestmicelle volume fraction shows repulsive interactions andwas fit with the structure factor for the RC potential. Atthese conditions the dimensionless screening parameter,κR ) 8.9, and the RC potential are a poor approximation.Less emphasis thanusual shouldbeplacedon theabsolutevalue of the charge, z, but the interactions still arise fromcharge repulsion. The highest micelle volume fractionφmeff ) 0.44 or 0.40 M SDS (which has the same ionicstrength as the case with added NaCl alone, Figure 3) isfittedwith a squarewellmodel. The strong upturn in thelow-q scattering indicates, and the model fitting resultsconfirm, that there is an additional attractive componentto the interparticle potential when micelles are present.

Discussion

The coagulation behavior of Ludox shows a distinctpattern as the added electrolyte progresses from a simpleelectrolyte to aweak amphiphile to a true surfactant. Theshortest chain alkyl sulfates, sodium methyl and ethylsulfate, are not hydrophobic enough to formmicelles andasa result have coagulationproperties equivalent to thoseof simple electrolytes. Hexyl sulfate is aweakamphiphilewith a high cmc (>0.4M). The coagulation concentrationfor simple electrolytes (Figure 2, curve a) is less than 0.36M for all Ludox volume fractions. Thus hexyl sulfate actsas a simple electrolyte (Figure 2, curve d), since the

(37) Kline, S. R.; Kaler, E. W. To appear in J. Appl. Crystallogr.(38) Chang, N. J.; Kaler, E. W. J. Phys. Chem. 1985, 89, 2996.(39) Lusvardi, K. M.; Full, A. P.; Kaler, E. W. Langmuir 1995, 11,

487.(40) Kline, S. R. Ph.D. Thesis, University of Delaware, 1996.

Figure 3. SANS spectra from Ludox at φL ) 0.06 and addedNaCl (open circles) at an ionic strength equivalent to that of0.40 M SDS (I ) 0.098 M). The line through the data pointsis a model fit to the data for hard spheres of radius 118 Å with18% polydispersity. The filled triangles are the SANS spectrafrom a binary mixture of Ludox (φL ) 0.06) and hydrogenatedSDS micelles (φm

eff ) 0.44). The larger Ludox particles scatterstrongly at lowq, while themicelles display a strong correlationpeaknearq)0.09Å-1. Both spectra are at a solvent scatteringlength density of 5.35 × 1010 cm-2.

Figure 4. SANS spectra for Ludox at φL ) 0.06 and SDSconcentrations as indicated on Figure 2. Surfactant tails aredeuterated to contrast match the micellar cores. The effectivevolume fractions of micelles appear alongside the model fits tothe data and are described in the text. The lowest curve (φSDS) 0.014) is on an absolute scale, with higher micelle concentra-tion curves offset bymultiplicative factors of 4, 16, 64, and 256,respectively. The Ludox-Ludox interaction potential progressesfrom repulsive, to hard sphere, to square well as the volumefraction of SDS micelles increases.

Table 1. Fitted Parameters for SANS Data fromMixtures of Ludox (OL ) 0.06) and an Increasing Volume

Fraction of SDS Micelles

micelleφeff

interactionmodel

interactionstrength

incoherentbkgd (cm-1)

scalefactor xø2/N

0.014 RC Z ) 81e- 0.096 0.88 6.80.076 HS 0 0.098 0.92 4.90.26 SW U0/kT ) 0.27 0.16 1.02 4.20.36 SW U0/kT ) 0.60 0.17 1.07 5.30.44 SW U0/kT ) 1.68 0.18 1.14 5.9

Aggregation of Colloidal Silica by n-Alkyl Sulfates Langmuir, Vol. 12, No. 10, 1996 2405

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coagulation concentration is less than the cmc. Sodiumoctyl sulfate also has a high cmc (measured to be 0.113M in 0.05 M NaCl), so while it does form some micellesat its coagulation concentration of 0.175 M, there is alsoa high concentration of unmicellized surfactant in thesolution. This free surfactant contributes significantly tothe electrostatic screening, but the effect of the micellesmoves the coagulation concentration to a lower ionicstrength.Sodiumdecylanddodecyl sulfatearestrongamphiphiles

and have cmc’s of 0.0141 and 0.001 45M, respectively, in0.05 M NaCl. As a result, the majority of the surfactantforms micelles and the ionic strength at the coagulationconcentration is approximately 0.1 M for each. Ap-proximately half of this ionic contribution is from thebackground electrolyte that is present in the Ludox stockdispersion. Inaddition, the large concentrationofmicellesprovides a significant depletion attraction that balancesthe reduced amount of electrostatic screening.The total interaction potential is the sum of the DLVO

anddepletion contributions, and their interplay gives riseto the energetic barrier to aggregation. Because all of thedetails of the sample composition are known the totalinteraction potential can be calculated at the coagulationconcentration. In particular, the height of the barrier tocoagulation,Vmax, can be determined. The van derWaalscontribution is readily calculated using a value for theHamaker constant of 1.5kT for silica in water.41 Theelectrostatic repulsion is calculated using a constantsurface potential of -68 mV for Ludox16 and the knownionic screening. Micellar aggregates do not contribute tothe ionic strength of the solution, but their dissociatedcounterions do;42 so the ionic strength of the solutions isgiven by

where Cs is the surfactant concentration and a constantbackground of [NaCl]) 0.05M is present from the Ludoxstock solution. The fractional dissociation of themicelles,δ, is estimated to be 0.2 frommodeling of SANSdata frommicellar solutions over the same concentration range asused in the coagulation measurements. The micelleaggregation numbermust be known for the calculation ofthe effective hard sphere diameter, which sets the rangeof the depletion attraction. Aggregation numbers of 40and 70 for sodium octyl and decyl sulfate, respectively,are estimated from literature values,43-45 and an ag-gregation number of 90 for SDSmicelles was determinedfromour ownSANSexperiments. Themicellar propertiesshown in Table 2 are used to calculate the depletionpotential. The magnitude of the depletion potential is

relatively insensitive to the aggregation number, as avariation of (5 in aggregation number gives less than a1% change in the magnitude of the depletion potential.This is not surprising, as the micelles interact witheffective hard sphere dimensions.The total interactionpotentialwas calculated, including

the depletion attraction, and for all cases was typical ofDLVO-type potentials, displaying amaximum at r/2RL )1.01. For clarity, only the contributions at Umax arereported and shown in Table 3. Note that while theindividually calculatedcontributionsvary, the totalbarrieris equivalent for simple electrolytes and micelle formers.That this independent calculation (free of fitting param-eters) returns the same barrier height for each surfactantis comforting, because the definition of the coagulationconcentrationdeterminedexperimentallyalso consistentlydefines a barrier height.This tradeoff between micellar depletion and electro-

static screening under various conditions is illustrated inthe contributions to the total interactionpotential inTable3. The results show quantitatively that the depletionattraction is the correct formofattractivepotential. Thereis an increasing contribution of the depletion attractionas theamphiphile becomes stronger. Thedecreasing levelof electrostatic screening (stronger repulsion) as the alkylchain is increased from 8 to 12 carbons is largely due tothe difference in their cmc’s,whichdecrease by two ordersof magnitude as the amphiphile becomes a stronger ionicsurfactant.From the detailed calculation of the total interaction

potential and the aggregation measurements, the rate ofdoublet formation can be quantified and compared totheoretical predictions. Theslowrateofdoublet formationis calculated46 fromthe initial rate of changeofabsorbance,dA/dt, and at the coagulation concentration the slow rateconstant ks is found to be 1.4 × 10-28 m3 s-1. Since thepotential is known, the stability ratio is calculated as2

including the hydrodynamic correction,47,48

with u) (r- 2RL)/RL. Here kf is the rate constant for fastaggregation. The productWks ) kf can then be comparedto the prediction of Smoluchowski,49 kf ) 4kT/3η ) 6.2 ×10-18 m3 s-1 for water at 25 °C. Integration of eq 4 for theelectrolyte systems in Table 2 gives stability ratios in therangeW ) (3.5-4.8) × 1010. The experimental fast rateof coagulation,Wks, thus ranges from0.8 to 1.1 times that

(41) Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.;Academic Press: New York, 1992.

(42) Shanks, P. C.; Franses, E. I. J. Phys. Chem. 1992, 96, 1794.(43) Hayashi, S.; Ikeda, S. J. Phys. Chem. 1980, 84, 744.(44) Berr, S. S.; Jones, R. R. Langmuir 1988, 4, 1247.(45) Ogino, K.; Kakihara, T.; Abe, M. Colloid Polym. Sci. 1987, 265,

604.

(46) Lichtenbelt, J.W.T.; Ras,H. J.M.C.;Weirsema,P.H.J.ColloidInterface Sci. 1974, 46, 522.

(47) Honig,E.P.;Roebersen,G.J.;Wiersema,P.H.J.Colloid InterfaceSci. 1971, 36, 97.

(48) Spielman, L. A. J. Colloid Interface Sci. 1970, 33, 562.(49) Smoluchowski, M. V. Z. Phys. 1916, 17, 557.

Table 2. Micellar Parameters Used to Calculate theTotal Interaction Potential at the Coagulation

Concentration

surfactantcoagulationconc (M) cmc (M)

aggregationnumber σm

eff (Å) φeff

NaC8SO4 0.23 0.113 40a 42 0.069NaC10SO4 0.41 0.0141 70a 59 0.37NaC12SO4 0.47 0.00145 90b 67 0.49

a Estimated from the literature. b Determined from SANS.

I ) [NaCl] + [cmc] + 12

δ[Cs - cmc] (3)

Table 3. Calculated Contributions to the TotalInterparticle Potential at a Ludox Separation of

r/2RL ) 1.010a

system UVDW (kT) Uel (kT) Udep (kT) Umax (kT)

Ludox + NaCl -5.4 30.6 25.2Ludox + NaC8SO4 -5.4 31.9 -0.6 26.0Ludox + NaC10SO4 -5.4 33.8 -2.4 26.0Ludox + NaC12SO4 -5.4 34.0 -2.9 25.8

a The energy barrier at the coagulation concentration, Umax, isthe same height for each of the surfactant or electrolyte systems.

W )kfks

) 2RL∫∞

2RL

D∞

D(u)exp(Utot(r)

kT ) drr2

(4)

D∞

D(u)) 6u2 + 13u + 2

6u2 + 4u(5)

2406 Langmuir, Vol. 12, No. 10, 1996 Kline and Kaler

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of the Smoluchowski prediction. This match with theoryis consistent with other recent results.50,51 This level ofagreement is expected,2 since small errors (1%) in thevalue of the surface potential (ψ0 ) -68mV) lead tomuchlarger errors in the barrier height Umax.The SANS data provide additional evidence that the

presence of solvent microstructure results in a depletionpotential. Ludox in the presence of NaCl at I ) 0.098 M(Figure 3) acts as a dispersion of hard spheres. At theidentical ionic strength of SDS,which corresponds to 0.40MSDSand the top curve ofFigure 4, there is anattractionbetween Ludox particles. The van der Waals and elec-trostatic contributions to the total potential are identicalfor eachcase, yet the total interactionpotential isdifferent.This is direct evidence that thepresence of self-assemblingmicrostructure provides an additional attractive compo-nent to the interparticle potential.At the three highestmicelle concentrations, the square

well model provides a good description of the contrast-matched micelle scattering data using a long-rangepotential with λ ) 1.3 and well depths up to a few kT.Short-range potentials such as the sticky hard spherepotential52,53 (λ <1.1)describewell the short-rangeoverlapinteractionsof organic-coatedsilicaparticles54,55 or charge-stabilized silica in the presence of a weak nonionicamphiphile.16 However, for our data, a short-rangeattractive potential provided an inferior fit to the datawhen compared to awider squarewell. These results areconsistent with the longer-range nature of the depletionattraction. The maximum range of the depletion attrac-tion is σL + σm

eff, so that λ in the square well model willequal 1 + σm

eff/σL. For the SDS case, σmeff is 67 Å, so λ is

equal to 1.3. This well width of 1.3 was held fixed for thesquare well model fitting. However, if the width wasoptimizedalongwith thewelldepth, theoptimalwellwidthremained equal to 1.3. The shallow well depths (0.3-1.7kT) are expected, since these mixtures show no signsof phase separation or aggregation over a period ofweeks.In other related situations, a squarewellmodel applied

to light scatteringdata fromanemulsion/micellemixture4yielded well depths of 2-5kT for samples approaching

the phase boundary. This well depth is of the samemagnitude as those reported here even though theemulsion/micelle size ratio was 100:1 and the micellevolume fractions were much smaller (φm < 0.02). Thecalculated minimum in the interparticle potential in acharged polystyrene latex/dextran mixture9 ranged from3kT to 6kT at the phase boundary for size ratios of 6.9 and1.9, comparable to the size ratio of Ludox/SDS of 4.7 inour mixture. At micelle volume fractions closer to thecoagulation concentration for our mixture, we expect theattractive well depths to be in this same 2-6kT rangeobserved in phase-separating mixtures with depletioninteractions.

ConclusionsWereport a systematic investigation of the aggregation

behavior of Ludox in the presence of a series of n-alkylsulfates. A clear distinction can be drawn between trueionic surfactants and simple electrolytes or weak am-phiphiles. Without the formationofmicrostructure,weakamphiphiles have coagulation concentrations equivalentto those of simple electrolytes. When the amphiphile hasbecome sufficiently strong (gNaC8SO4), the presence ofself-assembling microstructure (micelles) results in adepletion attraction that enhances Ludox aggregation.The use of a contrast-matching SANSmethod shows thatattractive interactions are present, and modeling resultsindicate the action of a long-range attractive potentialwhich is consistent in rangeandmagnitudewithamicelle-induced depletion attraction. Additionally, SANS fromLudox in equivalent ionic strengths of NaCl and SDSshowed hard sphere interactions with the simple elec-trolyte but strong attractive interactions with SDSpresent. This is clear evidence of the destabilizing effectof microstructure on a charge-stabilized dispersion.

Acknowledgment. This work was supported by E. I.duPont de Nemours & Co. The authors thank Dr. C.Glinka andDr. J. Barker (NIST) for their assistance withthe SANS measurements. We acknowledge the supportof the National Institute of Standards and Technology,U.S. Department of Commerce, in providing the facilitiesused in this experiment. This material is based uponactivities supported by the National Science Foundationunder Agreement No. DMR-9122444. Any opinions,findings, and conclusions or recommendations expressedin this publication are those of the authors and do notnecessarily reflect the view of the National ScienceFoundation.

LA950716L

(50) Penners, N. G. H.; Koopal, L. K. Colloids Surf. 1987, 28, 67.(51) Young, W. D.; Prieve, D. C. Langmuir 1991, 7, 2887.(52) Baxter, R. J. J. Chem. Phys. 1968, 49, 2770.(53) Menon, S. V. G.; Manohar, C.; Rao, K. S. J. Chem. Phys. 1991,

95, 9186.(54) Duits, M. H. G.; May, R. P.; Vrij, A.; de Kruif, C. G. d. Langmuir

1991, 7, 62.(55) Woutersen, A. T. J. M.; May, R. P.; de Kruif, C. G. d. J. Colloid

Interface Sci. 1992, 151, 410.

Aggregation of Colloidal Silica by n-Alkyl Sulfates Langmuir, Vol. 12, No. 10, 1996 2407

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