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Aggregation of Colloidal Silica by n-Alkyl Sulfates Steven 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 series of n-alkyl sulfates has been studied. Coagulation concentrations were measured for a simple salt, NaCl, and electrolyte/surfactants NaCnSO4 with chain lengths n ) 1, 2, 6, 8, 10, or 12 carbons. The C6 and shorter homologues had coagulation concentrations equivalent to that of NaCl, while C8 and higher homologues had coagulation concentrations at lower ionic strengths. Calculations of the Ludox-Ludox interaction potential show that the coagulation concentration results are consistent with the action of a screened repulsion plus a depletion attraction induced by the presence of surfactant micelles. Small angle neutron scattering measurements were made of mixtures with sodium dodecyl sulfate (SDS) under contrast-match conditions that isolated the silica-silica interactions. The silica-silica interactions indeed progressed from repulsive to attractive as more SDS micelles were added, a trend consistent with the observed aggregation. Silica in the presence of NaCl with an ionic strength equivalent to that of 0.40 M SDS showed hard sphere interactions, whereas the sample containing SDS micelles showed strong long-range attractive interactions. Thus we show how solvent microstructure influences the stability of a colloidal dispersion.
Introduction Understanding and controlling the stability of a colloidal dispersion are essential for its successful use. Specific applications may require the dispersion to be well behaved over a wide range of temperatures and chemical conditions. It may also be desirable to have other colloidal-sized components present in the mixture in addition to the original dispersion. The behavior of these mixed colloidal systems can be much more complex than that of a dispersion with only a single colloidal component. Charge-stabilized dispersions are described classically by DLVO theory,1 which shows how the total interparticle potential is the sum of the van der Waals attraction and the Coulomb repulsion. As a result of this balance of potentials, a charge-stabilized dispersion can be made unstable by screening the electrostatic repulsion, typically by the addition of electrolyte. Sufficiently large amounts of electrolyte can cause the particles to aggregate. Dispersions can also be destabilized by the addition of ‘microstructure’ to the solvent that surrounds the dispersed colloid. This microstructure is most commonly due to a soluble polymer2 but can also be self-assembling, as occurs when surfactant micelles destabilize emulsions.3-5 In these colloidal mixtures, there is now an osmotic (depletion) attraction6-8 in addition to the DLVO-type potential. The thermodynamics of colloidal mixtures is a rich field, and there have been many recent investigations into the phase behavior9-12 and stability13,14 of these mixtures. * To whom correspondence should be addressed. † Present address: National Institute of Standards and Technology, 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.
Here we investigate the changes in interparticle interactions caused by added electrolyte as the electrolyte progresses from a simple salt, to a weak amphiphile, to ultimately a true anionic surfactant. Our model chargestabilized dispersion is the commercial silica dispersion Ludox TM (DuPont). The surface chemistry of silica has been well studied,15 and the stability and phase behavior of silica in mixtures with weak nonionic amphiphiles16 and nonionic surfactants17,18 have been investigated. A wide variety of equilibrium behavior has been observed, including surfactant adsorption and colloidal phase separations. In some cases, the interparticle interactions between the silica particles undergo a transition from charge repulsion to attractive interactions that drive aggregation as solution conditions change. Interactions on the colloidal length scale can be determined with properly analyzed small angle neutron scattering (SANS) experiments.19 To determine interactions in a colloidal mixture (i.e., silica plus micelles), a contrast variation technique20,21 is necessary to isolate the contributions from individual colloidal populations. Here we will use this method to determine the effect that a homologous series of n-alkyl sulfates has on the interparticle potential of a model silica dispersion. These microscopic measurements will then be related to the macroscopic properties of the dispersion as determined by coagulation measurements. Ludox stability will be determined after addition of simple electrolyte, weak amphiphiles, and true ionic surfactants. Anionic sulfate surfactants were chosen for study because they do not adsorb to the highly negatively charged silica surface.22 The results show how interactions in a charge-stabilized (14) Giordano-Palmino, F.; Denoyel, R.; Rouquerol, J. J. Colloid Interface Sci. 1994, 165, 82. (15) Iler, R. K. The Chemistry of Silica; John Wiley and Sons: New York, 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; pp 329. (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 Polymeric Systems; Lindner, P., Zemb, T., Eds.; North-Holland: New York, 1991. (22) Huang, Z.; Yan, Z.; Gu, T. Colloids Surf. 1989, 36, 353.
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dispersion change as the surrounding electrolyte begins to form microstructure, which thereby transforms the system into a colloidal mixture. Theory A. Small Angle Neutron Scattering. Small angle scattering can provide a measure of interparticle interactions. The powerful technique of contrast variation, which can be applied to isolate the Ludox-Ludox interactions, and the theory describing the scattering from multicomponent mixtures have been described elsewhere.19,23 In the present experiments, attention is focused on the interactions between the silica particles in the bimodal mixture of surfactant micelles and silica. In this case, contrast variation20,21 is used to eliminate the contribution of the micelles to the scattering. Since the micelles have a core and shell structure, they can be matched in contrast to the solvent (H2O/D2O) approximately by an appropriate level of deuteration of the micelle core. The scattering length density of the shell of hydrated headgroups is nearly the same as that of the solvent, so the shell scattering contributes less than 1% of the total scattered 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. The multicomponent scattering equations then reduce to the convenient form
I(q) ) np〈f 2(q)〉S(q)
(1)
Here, q ) (4π/λ) sin(θ/2) is the magnitude of the scattering vector. The number density of particles is np, and S(q) is the structure factor, f(q) is the scattering amplitude, and the angular brackets denote an average over the size distribution. We use a Schultz distribution24 to describe the size polydispersity of the Ludox particles, which is determined from SANS experiments on Ludox dispersions alone. With the form factor, 〈f2(q)〉, known, the fitting procedure determines the type (attractive or repulsive) of interaction and its strength. The Ludox-Ludox interactions change from repulsive to attractive as the charge-stabilized Ludox dispersion is destabilized. A Coulombic potential (RC) is used to 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 available using the RMSA closure.25,26 Attractive interactions are modeled with a structure factor calculated for a square well (SW) potential27 with depth U0 and range λσL, where σL is the diameter of the Ludox particle. At the low volume fractions and well depths encountered in this study the solution of Sharma and Sharma27 compares well to Monte Carlo simulations.28 For more attractive square wells or higher volume fractions a thermodynamically consistent scheme such as HMSA is more appropriate.28 For hard sphere (HS) interactions we describe the data with an analytic expression29 for the total scattered intensity rather than the form of eq 1. B. Depletion Potential. DLVO theory describes the total interparticle interaction potential as the sum of van der Waals attractions and electrostatic repulsions. The DLVO potential is characterized by a repulsive energetic barrier of several kT that prevents two particles from coming into close approach and aggregation. In a binary (or bimodal) mixture of particles, there is an additional attractive component to the interparticle potential called the depletion attraction.6-8 For a mixture of Ludox and micelles, the depletion attraction is
Udep 4π 3 2 1 ) - Πm σ j + r3 j 3 - rσ kT 3 4 16
[
]
σL < r < 2σ j (2)
(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.
where Πm ) n* j) mkT is the osmotic pressure of the micelles. σ (σL + σeff m )/2 is the interaction diameter, where σL is the diameter of the Ludox particles and σeff m is the effective interaction diameter of the micelles. The attractive potential between the larger particles arises when they are in close approach and exclude the micelles from the volume between them. This results in a net unbalanced osmotic pressure and an effective attractive force between the larger pair. The magnitude of the depletion attraction is proportional to the number density, n*m, of micelles, and its range depends on the micelle interaction diameter. These two quantities must be calculated correctly when evaluating the depletion potential. The number density of micelles must be calculated on the basis of the free volume available to the micelles not the total solution volume. To a first approximation, the free volume available to the micelles is 1 - φL, and more exact calculations can be done for specific size ratios of interest.11,30,31 A charged micelle excludes a larger volume than its hydrated volume. The appropriate interaction diameter of the charged surfactant micelle is the effective hard sphere diameter σeff m and is determined by calculating the second virial coefficient for the repulsive Coulombic potential and setting it equal to the hard 3 sphere virial coefficient, B2HS(T) ) 2πσHS /3. For most interaction potentials, the second virial coefficient must be determined by numerical integration. The effective hard sphere dimensions are then used to calculate the effective hard sphere volume fraction of micelles. All micelle and surfactant concentrations reported here have been calculated on the basis of the available free volume.
Experimental Section Ludox TM was supplied by DuPont as a 50 wt % dispersion of discrete silica spheres of approximately 22 nm diameter and was used as received. Sodium methyl sulfate and ethyl sulfate were purchased from TCI America. Sodium n-hexyl and n-dodecyl sulfate were purchased from Lancaster Synthesis; sodium n-octyl and n-decyl sulfate were purchased from Kodak. Ethyl Violet and toluene used for the surfactant assay were purchased from Aldrich. D2O was purchased from Cambridge Isotope Labs, 99.9 mol % enriched, and perdeuterated sodium dodecyl sulfate, C12D25SO4-Na+ (MSD Isotopes) had a 98.6 mol % enrichment of the dodecyl tails. Hexyl and longer chain alkyl sulfates were purified by passing a concentrated (∼10-20 wt %) solution of the surfactant through a C18 reverse phase column (Waters and Associates) that retains hydrophobic impurities.32 The surfactant concentration in the eluent was measured by a spectrophotometric dye assay.33,34 Methyl and ethyl sulfate were used as received. Critical micelle concentrations of sodium octyl, decyl, and dodecyl sulfate were determined in 0.05 M NaCl at 25 °C by surface tension using a Kruss Digital Tensiometer K10T with a Wilhelmy plate. There were no minima in the surface tension plots. The Ludox dispersion, as supplied, is at pH ∼ 9 with a background electrolyte concentration of 0.05 M sodium salts. All samples were prepared maintaining the pH ∼ 9 and background electrolyte conditions of the stock Ludox dispersion. At these pH and electrolyte conditions, Ludox dispersions are stable for more than a year. Alkyl sulfateLudox mixtures were prepared by weight from stock solutions of Ludox and the desired surfactant. Concentrations and volume fractions were calculated from weight fractions and measured densities of the solutions. All of the surfactant concentrations and micelle volume fractions (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.
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Figure 1. Pseudoternary stability diagram for Ludox TM/ SDS/0.05 M NaCl. Regions were determined visually after a period of 24 h at 25 °C. Only the brine-rich corner is shown because the Ludox is supplied as 50 wt % in 0.05 M NaCl, and 30 wt % SDS is near its solubility limit.
have been corrected for the free volume available to the micelles as described above. Alkyl sulfate-Ludox mixtures were stored in a 25 °C water bath, and the coagulation stability of the Ludox TM was determined after a period of 24 h. The stability of the Ludox was quantified by measuring a consistent coagulation concentration for each sodium alkyl sulfate. The coagulation concentration was defined as the concentration of electrolyte required to cause a 10% increase in the absorbance at 500 nm after a period of 24 h. Absorbance measurements were performed on a Perkin-Elmer Lambda 2 UVVIS spectrophotometer with the samples held in 2 mL screw-capped vials. Neutron scattering experiments were performed on the 30 m spectrometer at the National Institute of Standards and Technology in Gaithersburg, MD. Neutrons of wavelength λ ) 5 Å with a spread of ∆λ/λ ) 15% were focused on samples held in 2 mm quartz cells. Two different sample to detector distances of 1.6 and 12 m were used to give q overlap between data sets and an overall q range of 0.005 Å-1 < q < 0.22 Å-1. Scattering from the samples was corrected for background and empty cell scattering. The sensitivity of individual detector pixels was normalized by comparing it to the incoherent scattering of water. The corrected data sets were circularily averaged and placed on an absolute scale by use of a polystyrene/deuterated polystyrene standard and NIST software.35 Instrumental smearing was simulated36 for the instrument configuration and wavelength spread used and was found to be negligible, except for the low-q region of the 1.6 m data (0.03 Å-1 < q < 0.05 Å-1). These smeared data points were not included in the model fitting. A single spectra, Figure 3, was collected at detector distances of 3.4 m (25 cm detector offset) and 12 m. The same q range was obtained, but now smearing effects were negligible for the entire q range. Intensity models were fit to the experimental data by using a nonlinear least-squares optimization procedure. The goodness of fit criteria was the χ2 error between model and data. Results Figure 1 shows an example of the behavior of Ludox with an anionic surfactant and is a pseudoternary stability diagram for the surfactant sodium dodecyl sulfate (SDS), (35) NIST “SANS Data Reduction and Imaging Software”, 1993. (36) Barker, J. G.; Pedersen, J. S. J. Appl. Crystallogr. 1995, 28, 105.
Kline and Kaler
Figure 2. Coagulation concentrations determined for Ludox TM in a homologous series of alkyl sulfates and NaCl. Coagulation concentration curves for NaCl (a), sodium methyl (b), ethyl (c), and hexyl sulfate (d) all overlap. The longer chain alkyl sulfates that form micelles, octyl (e), decyl (f), and dodecyl (g), all have coagulation concentrations at significantly lower ionic strengths. The ×’s at φL ) 0.06 mark the compositions of the SANS samples shown in Figure 4.
Ludox TM, and 0.05 M NaCl. At low SDS concentrations, the Ludox is stable and does not aggregate, and the dispersions appear transparent and blue. At higher surfactant concentrations, after a period of 24 h, the solutions appear noticeably turbid. At still higher SDS concentrations, the Ludox forms a gel or a thick white precipitate. The turbid and gel regions on the diagram grow larger and move to lower SDS concentrations at times longer than the initial 24 h observation. These observations suggest that the stability of Ludox with respect to SDS is a kinetic rather than an equilibrium process over the experimental time scale of 24 h. For this reason, the stability boundary is defined by a coagulation concentration. The coagulation concentration corresponds to a specific rate of aggregation, which in turn corresponds to a specific potential energy barrier that the particles must overcome to coagulate. Coagulation concentrations were measured for NaCl and for the homologous series of n-alkyl sulfates. The coagulation concentration of NaCl occurs at a high ionic strength (Figure 2), and the coagulation concentrations of the shorter chain alkyl sulfates (C1, C2, and C6 homologues) coincide with the value for the simple electrolyte. In contrast, the longer chain homologues sodium octyl, decyl, and dodecyl sulfate, which all form micelles in solution, display coagulation concentrations at significantly lower ionic strengths. On an ionic strength basis, the micelle-forming alkyl sulfates are far more effective in coagulating Ludox than is a simple electrolyte. The SANS spectrum for Ludox with added electrolyte is shown as the open circles in Figure 3 and serves to characterize the Ludox population. The line through the data is a fitted model of polydisperse hard spheres. The Ludox volume fraction is φL ) 0.06, and NaCl was added to give an ionic strength of I ) 0.098 M. The volume fraction of Ludox was calculated from the known weight fraction of silica in the sample. The data was fit well with only two adjustable parameters, the Ludox radius, RL, of 118 Å and a Schultz polydispersity σR/RL ) 0.18. A scattering length density of 3.65 × 1010 cm-2 was used for Ludox, calculated using a bulk density of silica of 2.3 g/mL determined from the measured density of dilute Ludox dispersions. These fitted parameters for Ludox were held constant for all subsequent modeling. When the added electrolyte is amphiphilic enough to form micelles, the solution contains a bimodal population of scatterers. When the scattering length density of the
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Figure 3. SANS spectra from Ludox at φL ) 0.06 and added NaCl (open circles) at an ionic strength equivalent to that of 0.40 M SDS (I ) 0.098 M). The line through the data points is a model fit to the data for hard spheres of radius 118 Å with 18% polydispersity. The filled triangles are the SANS spectra from a binary mixture of Ludox (φL ) 0.06) and hydrogenated SDS micelles (φeff m ) 0.44). The larger Ludox particles scatter strongly at low q, while the micelles display a strong correlation peak near q ) 0.09 Å-1. Both spectra are at a solvent scattering length density of 5.35 × 1010 cm-2.
Figure 4. SANS spectra for Ludox at φL ) 0.06 and SDS concentrations as indicated on Figure 2. Surfactant tails are deuterated to contrast match the micellar cores. The effective volume fractions of micelles appear alongside the model fits to the data and are described in the text. The lowest curve (φSDS ) 0.014) is on an absolute scale, with higher micelle concentration curves offset by multiplicative factors of 4, 16, 64, and 256, respectively. The Ludox-Ludox interaction potential progresses from repulsive, to hard sphere, to square well as the volume fraction of SDS micelles increases.
solvent is different than that of both of the two populations, the scattered intensity has contributions from both populations. The SANS spectrum for a mixture of SDS micelles and Ludox is also shown in Figure 3. The much smaller micelles (Rm ) 25 Å, φeff m ) 0.44) display a correlation peak near q ) 0.09 Å-1 and dominate the high q scattering, while the scattering at low q is predominantly from the larger Ludox particles. In contrast to the SANS spectrum from Ludox particles alone, the low q scattering is reduced in the bimodal mixture. This is a result of the contribution of the cross-structure factor (which is negative at low q) to the total scattered intensity.37 The scattering from the SDS micelles has been minimized by using SDS with deuterated tails. Isotopic H/D substitution is usually assumed to have no effect on surfactant microstructures, although the substitution of H2O/D2O does affect the micellization properties of surfactants.38 Isotopic substitution D/H in the surfactant tails has a lesser effect on the micellization properties. Mixtures of DTAB and DDAB and the deuterated analogue of DTAB mix ideally and show no measurable differences due to the isotopic substitution in the alkyl tails.39 Furthermore, the SANS spectra from SDS-h micelles and SDS-d micelles were measured and analyzed to give the same micelle properties.40 Experimentally, the contrast match point occurs when 72 mol % of the hydrogenated tails are replaced with deuterated tails, giving both the SDS micelle core and this H2O/D2O solvent (14% H2O by weight) equal scattering length densities of 5.35 × 1010 cm-2. SANS data for contrast-match conditions are shown in Figure 4, for a Ludox concentration of φL ) 0.06 and five different micelle volume fractions approaching the coagulation concentration from below. The compositions of these samples are indicated by the x’s in Figure 2. The lines through the SANS data are model fits using eq 1 and the appropriate form of the potential (RC, HS, or SW), and the results are shown in Table 1. The interaction strength was the only significant adjustable parameter
Table 1. Fitted Parameters for SANS Data from Mixtures of Ludox (OL ) 0.06) and an Increasing Volume Fraction of SDS Micelles
(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.
micelle interaction φeff model 0.014 0.076 0.26 0.36 0.44
RC HS SW SW SW
interaction strength
incoherent scale bkgd (cm-1) factor
Z ) 81e0 U0/kT ) 0.27 U0/kT ) 0.60 U0/kT ) 1.68
0.096 0.098 0.16 0.17 0.18
0.88 0.92 1.02 1.07 1.14
xχ2/N 6.8 4.9 4.2 5.3 5.9
in the fitting. The incoherent background and a multiplicative scale factor to correct for minor errors in absolute scaling (0.4 M). The coagulation concentration for simple electrolytes (Figure 2, curve a) is less than 0.36 M for all Ludox volume fractions. Thus hexyl sulfate acts as a simple electrolyte (Figure 2, curve d), since the
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Table 2. Micellar Parameters Used to Calculate the Total Interaction Potential at the Coagulation Concentration surfactant NaC8SO4 NaC10SO4 NaC12SO4 a
coagulation aggregation eff conc (M) cmc (M) number σm (Å) 0.23 0.41 0.47
0.113 0.0141 0.00145
40a 70a 90b
42 59 67
φeff 0.069 0.37 0.49
system
UVDW (kT)
Uel (kT)
Udep (kT)
Umax (kT)
Ludox + NaCl Ludox + NaC8SO4 Ludox + NaC10SO4 Ludox + NaC12SO4
-5.4 -5.4 -5.4 -5.4
30.6 31.9 33.8 34.0
-0.6 -2.4 -2.9
25.2 26.0 26.0 25.8
a The energy barrier at the coagulation concentration, U max, is the same height for each of the surfactant or electrolyte systems.
Estimated from the literature. b Determined from SANS.
coagulation concentration is less than the cmc. Sodium octyl sulfate also has a high cmc (measured to be 0.113 M in 0.05 M NaCl), so while it does form some micelles at its coagulation concentration of 0.175 M, there is also a high concentration of unmicellized surfactant in the solution. This free surfactant contributes significantly to the electrostatic screening, but the effect of the micelles moves the coagulation concentration to a lower ionic strength. Sodium decyl and dodecyl sulfate are strong amphiphiles and have cmc’s of 0.0141 and 0.001 45 M, respectively, in 0.05 M NaCl. As a result, the majority of the surfactant forms micelles and the ionic strength at the coagulation concentration is approximately 0.1 M for each. Approximately half of this ionic contribution is from the background electrolyte that is present in the Ludox stock dispersion. In addition, the large concentration of micelles provides a significant depletion attraction that balances the reduced amount of electrostatic screening. The total interaction potential is the sum of the DLVO and depletion contributions, and their interplay gives rise to the energetic barrier to aggregation. Because all of the details of the sample composition are known the total interaction potential can be calculated at the coagulation concentration. In particular, the height of the barrier to coagulation, Vmax, can be determined. The van der Waals contribution is readily calculated using a value for the Hamaker constant of 1.5kT for silica in water.41 The electrostatic repulsion is calculated using a constant surface potential of -68 mV for Ludox16 and the known ionic screening. Micellar aggregates do not contribute to the ionic strength of the solution, but their dissociated counterions do;42 so the ionic strength of the solutions is given by
1 I ) [NaCl] + [cmc] + δ[Cs - cmc] 2
Table 3. Calculated Contributions to the Total Interparticle Potential at a Ludox Separation of r/2RL ) 1.010a
(3)
where Cs is the surfactant concentration and a constant background of [NaCl] ) 0.05 M is present from the Ludox stock solution. The fractional dissociation of the micelles, δ, is estimated to be 0.2 from modeling of SANS data from micellar solutions over the same concentration range as used in the coagulation measurements. The micelle aggregation number must be known for the calculation of the effective hard sphere diameter, which sets the range of the depletion attraction. Aggregation numbers of 40 and 70 for sodium octyl and decyl sulfate, respectively, are estimated from literature values,43-45 and an aggregation number of 90 for SDS micelles was determined from our own SANS experiments. The micellar properties shown in Table 2 are used to calculate the depletion potential. The magnitude of the depletion potential is (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.
relatively insensitive to the aggregation number, as a variation of (5 in aggregation number gives less than a 1% change in the magnitude of the depletion potential. This is not surprising, as the micelles interact with effective hard sphere dimensions. The total interaction potential was calculated, including the depletion attraction, and for all cases was typical of DLVO-type potentials, displaying a maximum at r/2RL ) 1.01. For clarity, only the contributions at Umax are reported and shown in Table 3. Note that while the individually calculated contributions vary, the total barrier is equivalent for simple electrolytes and micelle formers. That this independent calculation (free of fitting parameters) returns the same barrier height for each surfactant is comforting, because the definition of the coagulation concentration determined experimentally also consistently defines a barrier height. This tradeoff between micellar depletion and electrostatic screening under various conditions is illustrated in the contributions to the total interaction potential in Table 3. The results show quantitatively that the depletion attraction is the correct form of attractive potential. There is an increasing contribution of the depletion attraction as the amphiphile becomes stronger. The decreasing level of electrostatic screening (stronger repulsion) as the alkyl chain is increased from 8 to 12 carbons is largely due to the difference in their cmc’s, which decrease by two orders of magnitude as the amphiphile becomes a stronger ionic surfactant. From the detailed calculation of the total interaction potential and the aggregation measurements, the rate of doublet formation can be quantified and compared to theoretical predictions. The slow rate of doublet formation is calculated46 from the initial rate of change of absorbance, dA/dt, and at the coagulation concentration the slow rate constant ks is found to be 1.4 × 10-28 m3 s-1. Since the potential is known, the stability ratio is calculated as2
W)
∫
kf ) 2RL ks
∞ 2RL
D∞ D(u)
(
exp
)
Utot(r) dr kT r2
(4)
including the hydrodynamic correction,47,48
D∞ D(u)
)
6u2 + 13u + 2 6u2 + 4u
(5)
with u ) (r - 2RL)/RL. Here kf is the rate constant for fast aggregation. The product Wks ) kf can then be compared to 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 the electrolyte systems in Table 2 gives stability ratios in the range W ) (3.5-4.8) × 1010. The experimental fast rate of coagulation, Wks, thus ranges from 0.8 to 1.1 times that (46) Lichtenbelt, J. W. T.; Ras, H. J. M. C.; Weirsema, P. H. J. Colloid Interface Sci. 1974, 46, 522. (47) Honig, E. P.; Roebersen, G. J.; Wiersema, P. H. J. Colloid Interface Sci. 1971, 36, 97. (48) Spielman, L. A. J. Colloid Interface Sci. 1970, 33, 562. (49) Smoluchowski, M. V. Z. Phys. 1916, 17, 557.
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Aggregation of Colloidal Silica by n-Alkyl Sulfates
of the Smoluchowski prediction. This match with theory is consistent with other recent results.50,51 This level of agreement is expected,2 since small errors (1%) in the value of the surface potential (ψ0 ) -68 mV) lead to much larger errors in the barrier height Umax. The SANS data provide additional evidence that the presence of solvent microstructure results in a depletion potential. Ludox in the presence of NaCl at I ) 0.098 M (Figure 3) acts as a dispersion of hard spheres. At the identical ionic strength of SDS, which corresponds to 0.40 M SDS and the top curve of Figure 4, there is an attraction between Ludox particles. The van der Waals and electrostatic contributions to the total potential are identical for each case, yet the total interaction potential is different. This is direct evidence that the presence of self-assembling microstructure provides an additional attractive component to the interparticle potential. At the three highest micelle concentrations, the square well model provides a good description of the contrastmatched micelle scattering data using a long-range potential with λ ) 1.3 and well depths up to a few kT. Short-range potentials such as the sticky hard sphere potential52,53 (λ < 1.1) describe well the short-range overlap interactions of organic-coated silica particles54,55 or chargestabilized silica in the presence of a weak nonionic amphiphile.16 However, for our data, a short-range attractive potential provided an inferior fit to the data when compared to a wider square well. These results are consistent with the longer-range nature of the depletion attraction. The maximum range of the depletion attraction is σL + σeff m , so that λ in the square well model will eff equal 1 + σeff m /σL. For the SDS case, σm is 67 Å, so λ is equal to 1.3. This well width of 1.3 was held fixed for the square well model fitting. However, if the width was optimized along with the well depth, the optimal well width remained equal to 1.3. The shallow well depths (0.31.7kT) are expected, since these mixtures show no signs of phase separation or aggregation over a period of weeks. In other related situations, a square well model applied to light scattering data from an emulsion/micelle mixture4 yielded well depths of 2-5kT for samples approaching (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.
Langmuir, Vol. 12, No. 10, 1996 2407
the phase boundary. This well depth is of the same magnitude as those reported here even though the emulsion/micelle size ratio was 100:1 and the micelle volume fractions were much smaller (φm < 0.02). The calculated minimum in the interparticle potential in a charged polystyrene latex/dextran mixture9 ranged from 3kT to 6kT at the phase boundary for size ratios of 6.9 and 1.9, comparable to the size ratio of Ludox/SDS of 4.7 in our mixture. At micelle volume fractions closer to the coagulation concentration for our mixture, we expect the attractive well depths to be in this same 2-6kT range observed in phase-separating mixtures with depletion interactions. Conclusions We report a systematic investigation of the aggregation behavior of Ludox in the presence of a series of n-alkyl sulfates. A clear distinction can be drawn between true ionic surfactants and simple electrolytes or weak amphiphiles. Without the formation of microstructure, weak amphiphiles have coagulation concentrations equivalent to those of simple electrolytes. When the amphiphile has become sufficiently strong (gNaC8SO4), the presence of self-assembling microstructure (micelles) results in a depletion attraction that enhances Ludox aggregation. The use of a contrast-matching SANS method shows that attractive interactions are present, and modeling results indicate the action of a long-range attractive potential which is consistent in range and magnitude with a micelleinduced depletion attraction. Additionally, SANS from Ludox in equivalent ionic strengths of NaCl and SDS showed hard sphere interactions with the simple electrolyte but strong attractive interactions with SDS present. This is clear evidence of the destabilizing effect of microstructure on a charge-stabilized dispersion. Acknowledgment. This work was supported by E. I. duPont de Nemours & Co. The authors thank Dr. C. Glinka and Dr. J. Barker (NIST) for their assistance with the SANS measurements. We acknowledge the support of the National Institute of Standards and Technology, U.S. Department of Commerce, in providing the facilities used in this experiment. This material is based upon activities supported by the National Science Foundation under Agreement No. DMR-9122444. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the view of the National Science Foundation. LA950716L
