Charging and Aggregation Properties of Carboxyl Latex Particles

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Charging and Aggregation Properties of Carboxyl Latex Particles: Experiments versus DLVO Theory Sven Holger Behrens,†,‡ Daniel Iso Christl,‡ Rudi Emmerzael,§ Peter Schurtenberger,| and Michal Borkovec*,† Department of Chemistry, Clarkson University, Potsdam, New York 13699-5814, and Swiss Federal Institute of Technology, ETHZ-ITO, Grabenstrasse 3, 8952 Schlieren, Switzerland, and Leiden Institute of Chemistry, Leiden University, Gorlaeus Laboratories, 2300 RA Leiden, The Netherlands, and Physics Department, University of Fribourg, CH-1700 Fribourg, Switzerland Received August 26, 1999. In Final Form: November 16, 1999 Carboxyl latex particles of two different sizes were used to study the early stages of aggregation in dilute colloidal suspensions. The charging behavior as a function of solution pH was characterized in acid-base titration and electrophoresis experiments at fixed ionic strength; absolute aggregation rate constants were measured by combined static and dynamic light scattering as a function of pH and ionic strength. Up to an ionic strength of 10 mM in a KCl solution, the classical DLVO theory of colloidal stability is seen to work quantitatively. At higher ionic strength, however, well-known discrepancies between theory and experiment are observed. An analysis of the theoretical pair interaction energy suggests that quantitative agreement can be achieved when the energy barrier for reaction-limited aggregation lies at surface separations of at least 1-2 nm. This result is consistent with recent measurements of colloidal forces and interaction energies, as well as with earlier aggregation and deposition studies typically carried out in the unfavorable situation of barriers at subnanometer distances. The theoretical discussion further considers the appropriate choice of a Hamaker constant, the effect of nonlinearity in the Poisson-Boltzmann equation on stability predictions, as well as the role of charge regulation and the error introduced by the Derjaguin approximation.

Introduction Charge-stabilized colloidal suspensions play an important role in environmental and biochemical sciences, as well as in materials engineering and a variety of industrial applications. For many processes of scientific or technological interest, it is essential to have precise control over the suspension stability. Understanding the charging properties of colloid particles, their mutual interaction, and the resulting aggregation behavior has been considered one of the great challenges in colloid science throughout the history of this discipline.1-3 Many recent studies have been inspired by progresses in experimental techniques: with single particle light scattering and combined static and dynamic light scattering it is now possible to measure accurately aggregation rates of particles in a wide range of sizes,4 the underlying colloidal forces can be investigated by atomic force microscopy5 or direct force measurements using the surface force apparatus,6 and the corresponding interac* Corresponding author. E-mail: [email protected]. † Clarkson University. ‡ Swiss Federal Institute of Technology. § Leiden University. | University of Fribourg. (1) Hunter, R. J. Foundations of Colloid Science; Clarendon: Oxford, 1987. (2) Sonntag, H.; Strenge, K. Coagulation Kinetics and Structure Formation; VEB Deutscher Verlag der Wissenschaften: Berlin, 1987. (3) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1989. (4) Holthoff, H.; Schmitt, A.; Fernańdez-Babero, A.; Borkovec, M.; Cabrerizo-Vıćhez, M.; Schurtenberger, P.; Hidalgo-A Ä lvarez, R. J. Colloid Interface Sci. 1997, 192, 463. (5) Hartley, P. G.; Larson, I.; Scales, P. J. Langmuir 1997, 13, 2207. (6) Israelachvili, J. N.; Adams, G. E. J. Chem. Soc., Faraday Trans. 1978, 74, 975; Pashley, R. M. J. Colloid Interface Sci. 1980, 80, 153; Israelachvili, J. N. Adv. Colloid Interface Sci. 1982, 16, 31.

tion energies are accessible to optical tweezers techniques7 or total internal reflection microscopy.8 At the same time, a wealth of uniform synthetic particles with finely tunable surface chemical properties has become available. These particles, in addition to their many practical applications, finally allow a more stringent test of the existing aggregation theory through experiments on well-characterized colloid systems. The widely used theory of Derjaguin, Landau, Verwey, and Overbeek (DLVO)9 readily explains the qualitative features of colloid stability as resulting from the interplay of van der Waals forces and the electrostatic double layer interaction. But the sensitivity of aggregation and deposition rates to variations in the ionic strength or pH of the solution is grossly overestimated; this is the unanimous conclusion of previous investigations.10-13 Furthermore, the effect of double layer repulsion should be more pronounced for large particles than for small ones according to DLVO theory;14 yet no such influence of the particle size has been seen in earlier aggregation (7) Crocker, D.; Grier, D. Phys. Rev. Lett. 1994, 73, 352; Crocker, D.; Grier, D. Phys. Rev. Lett. 1996, 77, 1897; Sugimoto, T.; Takahashi, T.; Itoh, H.; Sato, S. I.; Muramatsu, A. Langmuir 1997, 13, 5528. (8) Prieve, D. C.; Bike, S. G.; Frey, N. A. Faraday Discuss. Chem. Soc. 1990, 90, 209; Frey, N. A.; Prieve, D. C. J. Chem. Phys. 1993, 98, 7552; Liebert, R. B.; Prieve, D. C. Biophys. J. 1995, 69, 66. (9) Derjaguin, B. V.; Landau, L. Acta Physicochim. U.S.S.R. 1941, 14, 633; Verwey, E. J. W.; Overbeek, J. Th. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (10) Lips, A.; Willis, E. Trans. Faraday Soc. 1973, 69, 1226. (11) Zeichner, G. R.; Schowalter, W. R. J. Colloid Interface Sci. 1979, 71, 237. (12) Bowen, B. D.; Epstein, N. J. Colloid Interface Sci. 1979, 72, 81. (13) Elimelech, M.; O’Melia, C. R. Environ. Sci. Technol. 1990, 24, 1528. (14) Reerink, H.; Overbeek, J. Th. G. Discuss. Faraday Soc. 1954, 18, 74.

10.1021/la991154z CCC: $19.00 © 2000 American Chemical Society Published on Web 01/21/2000

Charging and Aggregation Properties of Carboxyl Latex Particles

experiments.15-17 Various reasons for this supposed failure of the theoretical description have been proposed;18 the arguments involve relaxation effects,19 additional nonDLVO forces,20 the influence of surface roughness,21 or the effect of heterogeneity in the distribution of surface charge.22,23 Some error is also expected to arise from the neglect of ion-ion correlations in the common expression for the electrostatic interaction.24 While the search for a conclusive explanation of the failure of DLVO theory in stability predictions is going on,25 recent measurements of forces and interaction energies have confirmed the theory quantitatively for surface separations down to a few nanometers.7,26,27 This naturally raises the question why aggregation kinetics might not be predicted correctly by a theory which accurately describes particle interactions. The present work shows systematic light scattering studies of aggregation in monodisperse suspensions of well-characterized carboxylated latex particles. For particles of two different sizes, aggregation rates were measured in aqueous solutions of different ionic strengths, i.e., different screening lengths, and for a wide range of surface charge densities, adjusted via the solution pH. The charging behavior was characterized by potentiometric acid-base titration and electrophoretic mobility measurements. Experimental data will be compared with DLVO predictions based on the full nonlinear solution of the Poisson-Boltzmann equation. The discussion will also consider the effect of linearization, the accuracy of the Derjaguin approximation for surface curvature, and the role of charge regulation necessary to guarantee surface chemical equilibrium at all particle-particle distances. Results of the aggregation experiments have partly been published earlier.28 Samples and Experimental Methods Materials. Two aqueous stock suspensions of monodisperse, carboxylated latex particles, synthesized in surfactant-free emulsion polymerization were purchased from Interfacial Dynamics Corporation, Portland. Prior to all experimental steps the suspensions were extensively dialyzed against deionized water from a Nanopure apparatus. The total latex concentration (15) Ottewill, R. H.; Shaw, J. N. Discuss. Faraday Soc. 1966, 42, 154. (16) Matthews, B. A.; Rhodes, C. T J. Colloid Interface Sci. 1968, 28, 71; Kotera, A.; Furusawa, K.; Kudo, K. Kolloid Z. Z. Polym. 1970, 240, 837; Joseph-Petit, A. M.; Dumont, F.; Watillon, A. J. Colloid Interface Sci. 1973, 43, 649; Penners, N. H. G.; Koopal, L. K. Colloids Surf. 1987, 28, 67; Elimelech, M.; O’Melia, C. R. Langmuir 1990, 6, 1153. (17) Tsuruta, L. R.; Lessa, M. M.; Carmona-Ribeiro, A. M. J. Colloid Interface Sci. 1995, 175, 470. (18) Swanton, S. W. Adv. Colloid Interface Sci. 1995, 54, 129. (19) Frens, G.; Engel, D. J.; Overbeek, J. Th. G. Trans. Faraday Soc. 1967, 63, 418; Lyklema, J. Pure Appl. Chem. 1980, 52, 1221; Dukhin, S. S.; Lyklema, J. Faraday Discuss. Chem. Soc. 1990, 90, 261; Shulepov, S. Yu.; Dukhin, S. S.; Lyklema, J. J. Colloid Interface Sci. 1995, 171, 340; Shulepov, S. Yu. J. Colloid Interface Sci. 1997, 189, 199. (20) Ke´kicheff, P.; Spalla, O. Phys. Rev. Lett. 1995, 75, 1851; Israelachvili, J.; Wennerstro¨m, H. Nature 1996, 379, 219; Larsen, A. E.; Grier, D. G. Nature 1997, 385, 230. (21) Shulepov, S. Yu.; Frens, G. J. Colloid Interface Sci. 1995, 170, 44; Shulepov, S. Yu.; Frens, G. J. Colloid Interface Sci. 1996, 182, 388. (22) Kihira, H.; Ryde, N.; Matijevic, E. Chem. Soc. Faraday Trans. 1992, 88, 2379. (23) Litton, G. M.; Olson, T. M. J. Colloid Interface Sci. 1994, 165, 522. (24) Kjellander, R.; Marcelja, S. Chem. Phys. Lett. 1986, 127, 402; Kjellander, R.; Marcelja, S. J. Phys. Chem. 1986, 90, 1230; Kjellander, R.; Akesson, T.; Jo¨nsson, B.; Marcelja, S. J. Chem. Phys. 1992, 97, 1424. (25) Molina-Bolı´var, J. A.; Galisteo-Gonza´lez, F.; Hidalgo-A Ä lvarez, R. J. Colloid Interface Sci. 1997, 195, 289. (26) Israelachvili, J. N.; Pashley, R. M. Nature 1983, 306, 249. (27) Horn, R. G.; Clarke, D. R.; Clarkson, M. T. J. Mater. Res. 1988, 3, 413. (28) Behrens, S. H.; Borkovec, M.; Schurtenberger, P. Langmuir 1998, 14, 1951.

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was determined with a carbon analyzer. The size distribution of the particles was studied by transmission electron microscopy. Particles look perfectly spherical; their average radius and its coefficient of variation obtained from the micrographs are given in the table. All experiments were carried out at 25 ( 1 °C. Potentiometric Titrations. The charge on the latex particles was measured through potentiometric titration using the Wallingford titrator.29 A Microlink MF18 interface (Biodata, Manchester) was used to connect a personal computer to a glass electrode, a separate silver chloride reference electrode, and four burets containing deionized water (Nanopure quality), and solutions of 0.050 M HCl, 0.042 M KOH, and 2.0 M KCl, respectively. Great care was taken to prevent CO2 from entering the system at any stage of the experiment. A titration cycle always started with a forward titration of 50 mL of an acidified solution of low ionic strength with base, followed by a backward titration with acid, and ended with the addition of salt solution to adjust the ionic strength to the next higher level. After the addition of titrant the potential was recorded as soon as it drifted by less than 0.02 mV/min or after a maximum waiting time of 60 min. In each cycle the ionic strength was held constant within 1% by automatically balancing the addition of acid or base with an addition of water or salt solution. The precise base concentrations and electrode parameters were determined from blank titrations of the pure electrolyte solution using a least-squares fitting routine; the analysis included activity corrections and diffusion potentials.30 Results of the latex titration were converted to surface charge densities after subtraction of blank titration curves from the actual particle titration. The specific surface area was determined from the electron micrographs, and the total latex concentration measured with a carbon analyzer. Forward and backward titration resulted in almost identical curves. Around pH 7, however, they showed a small but significant hysteresis (e4 mC/ m2), which has previously been reported for titrations of oxide particles as well.31-33 Electrode response appeared extremely slow in this pH region, and the potential values sometimes still drifted considerably while they were recorded. Because of these difficulties, we will report averages of forward and backward titration curves. Electrophoresis. Electrophoretic mobility34 was measured on a laser Doppler velocimetry setup (ZetaSizer 3, Malvern) operating with electric fields of 30 V/cm at a modulation frequency of 1 kHz. Data were collected as a function of pH at different ionic strengths (adjusted with KCl, KOH, and HCl), the pH was measured with a combination electrode (6.0204.100, Metrohm) in test tubes sealed to prevent the penetration of CO2. The suspensions used contained volume fractions of latex particles around 0.04; the results were insensitive to changes in the particle concentration. Light Scattering. Colloid stability was measured by conventional dynamic light scattering on a standard goniometer setup (ALV-SP-125 S/N 30) with classical pinhole detection optics using a 1 W Krypton Laser (Innova 300, Coherent) operating at a wavelength of 647.1 nm. The temperature in the scattering cell was stabilized within 0.3 °C. All glassware used in the solution preparation had been cleaned in a heated 1:1 mixture of concentrated sulfuric acid (98%, Fluka, p.a.) and hydrogen peroxide (30%, p.a., Merck) and extensively been rinsed with Nanopure water. Prior to experiments, the required amounts of acid or base, salt solution, and water (same as for the titration) were injected into the cuvette. Just before a measurement was started, the stable latex suspension was added, the cuvette was (29) Kinniburgh, D. G.; Milne, C. J.; Venema, P. Soil Sci. Soc. Am. J. 1995, 59, 417. (30) Baes, C. F.; Mesmer, R. E. The Hydrolysis of Cations; Wiley: New York, 1981. (31) Schudel, M.; Behrens, S. H.; Holthoff, H.; Kretzschmar, R.; Borkovec, M. J. Colloid Interface Sci. 1997, 196, 241. (32) Yates, D. E. The Structure of the Oxide/Aqueous Electrolyte Interface. Ph.D. Thesis, University of Melbourne, Melbourne, Australia, 1975. (33) Penners, N. H. G.; Koopal, L. K.; Lyklema, J. Colloids Surf. 1986, 21, 457. (34) Hunter, R. J. Zeta Potential in Colloid Science; Academic Press: London, 1981.

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sealed, quickly shaken to guarantee complete mixing, and immediately inserted into the scattering cell. The scattered light was detected under a scattering angle of 90°, the signal of a photomultiplier (Thorn Emi 9130/100SB03) was processed by an ALV-5000 correlator (ALV-Laser, Germany). From the homodyne autocorrelation function the intensity-averaged diffusion constant and the corresponding hydrodynamic radius were determined by standard cumulant analysis.35 Light scattering data presented below for the 52 nm particles were produced several months later than the ones for 155 nm particles and were obtained on a similar experimental setup in a different laboratory. At this time, no clean sample of the batch of larger particles was available anymore, therefore a repetition of the experiments with these particles was not possible. A time-resolved series of measurements was performed to monitor the increase of the hydrodynamic radius rh during the aggregation process. Fast aggregation was measured at particle concentrations around 108 mL-1, corresponding to an aggregation half-time of about 2 h. For studies in the slow aggregation regime the particle concentration was progressively increased up to a factor of about 10. Initial radii were identical to the radii in a stable suspension. Reported experimental stability ratios W were calculated as W ) (drh/dt)fasttf0/(drh/dt)tf0, where tf0 means an extrapolation to the moment when aggregation was initiated.36 If the initial particle concentration n0 is different from the reference case of fast aggregation (denoted by the superscript), then the expression for W has to be modified by an additional factor of n0/n0fast. Here, n0 is the particle concentration and the superscript “fast” indicates values pertaining to a comparative measurement in the fast aggregation regime.36 The stability ratio is defined as W)kfast/k, where k is the actual dimer formation rate and kfast corresponds to fast aggregation. Absolute fast rate constants kfast were measured by timeresolved simultaneous static and dynamic measurements under different scattering angles in 1 M KCl. In contrast to the classical static measurements or to dynamic measurements only, this combined technique permits us to determine dimer formation rates without any simplifying assumptions concerning unknown optical or hydrodynamic properties of the dimers. Details of the method are given elsewhere.4,37 These measurements were carried out on a fiber optic multiangle light scattering instrument with an argon ion laser operating at a wavelength of 488 nm, the instrumental details have also been described before.38

The Model for Charging and Aggregation A theoretical approach to the charging properties and the aggregation kinetics of the studied latex particles shall be discussed separately in the following sections. The Surface Charge Density. The charging state of the particle surfaces is determined by the deprotonation of homogeneously distributed carboxylic surface headgroups

-COOH h -COO- + H+

(1)

The surface charge density σ is then given by

σ)-

eΓtot 1 + aH(S)/K

The charge of a latex surface can be related to the solution properties with the so-called diffuse layer model (DLM).39,40 This model assumes that all the surface charge is located at the solution interface, which is characterized by the electrostatic surface potential ψ0. The proton activity at the surface is evaluated as

aH(S) ) aH × exp(-βeψ0)

where pH ) -log aH, and β-1 ) kBT the thermal energy. Equations 2 and 3 define a relation between the surface charge and the surface potential. Equilibrium requires the simultaneous fulfillment of a second charge-potential relation that follows from the distribution of mobile ions in the diffuse layer.41 In a description based on the Poisson-Boltzmann equation for 1-1 electrolytes, the surface charge density σ of an isolated particle with radius R, can be expressed in terms of the surface potential ψ0 as3

σ)

20κ 2 tanh (βeψ0/4) sinh (βeψ0/2) + βe κR

where e is the protonic charge, Γtot the total density of chargeable sites, aH(S) denotes the surface activity of the protons,39 and K is the dissociation constant. (35) Berne, B. J.; Pecora, R. Dynamic Light Scattering; John Wiley: New York, 1981. (36) Holthoff, H.; Egelhaaf, S. U.; Borkovec, M.; Schurtenberger, P.; Sticher, H. Langmuir 1996, 12, 5541. (37) Holthoff, H.; Borkovec, M.; Schurtenberger, P. Phys. Rev. E. 1997, 56, 6945. (38) Egelhaaf, S. U.; Schurtenberger, P. Rev. Sci. Instrum. 1996, 67, 2. (39) Healy, T. W.; White, L. R. Adv. Colloid Interface Sci. 1978, 9, 303.

[

]

(4)

where 0 is the total permittivity of the solution and κ-1

) x0/(2NAβe2I) the Debye length, further involving the ionic strength I (NA is Avogadro’s number). Without the second term on the right-hand side, this is just the classical Gouy-Chapman result. The additional term was proposed by Loeb, Overbeek, and Wiersema,42 and gives a first order correction for the surface curvature, accurate to within 5% of the true charge density for any surface potential whenever the Debye length is smaller than the particle diameter. For the larger particles of this study the correction is insignificant, but for the smaller ones it can be as large as 13% for the electrostatic potential at the lowest ionic strength. Together, eq 4 and the σ-ψ0 relation from the adsorption isotherm (eqs 2 and 3) determine the equilibrium value of the surface charge σ and potential ψ0 at a given pH. For comparison of the resulting theoretical charging curves with titration measurements, the total site density Γtot and the dissociation parameter pK ) -logK were fitted to the experimental data. The value of Γtot fixes the saturation value of the charge density, and a variation of pK shifts the modeling curves horizontally without changing their actual shape. Electrophoretic Mobility and Zeta Potential. For a flat surface in a monovalent electrolyte, the electrostatic potential at a distance x from the surface is related to the surface potential ψ0 via

ψ(x) ) (2)

(3)

4 arctanh[exp(-κx) tanh (βeψ0/4)] βe

(5)

as follows from integration of the Poisson-Boltzmann equation (eq 3). The zeta potential was computed as the electrostatic potential ζ ) ψ(xs) at some distance xs from the surface, corresponding to the thickness of an immobile fluid layer adjacent to the particle surface. The outer end of this immobile layer, where the motion of fluid relative to the particle sets in, is commonly referred to as the surface of shear. Equation 5 with the curvature correction (eq 4) already incorporated in ψ0 is expected to be rather (40) Westall, J.; Hohl, H. Adv. Colloid Interface Sci. 1980, 12, 265. (41) Chan, D. Y. C.; Mitchell, D. J. J. Colloid Interface Sci. 1983, 95, 193. (42) Loeb, A. L.; Overbeek, J. Th. G.; Wiersema, P. H. The Electrical Double-Layer Around a Spherical Colloid Particle; MIT Press: Boston, 1961.


accurate even for the smaller particles at low ionic strength when x , κ-1 < R. The distance xs was taken as a fit parameter. Zeta potentials were translated into electrophoretic mobilities u according to the widely used method of O’Brien and White.43,44 For the values of κR in question, this method predicts a maximum in the mobility as a function of the zeta potential. The Kinetics of Aggregation. Aggregation rates k for the process of dimer formation were calculated according to the modified Fuchs expression3

{

}

B(h)

(6)

where ks ) 8/(3β) is the Smoluchowski result for pure diffusion (η being the dynamic viscosity of the solution). The hydrodynamic drag at a surface to surface distance h is taken into account through the approximating formula45

B(h) )

6(h/R)2 + 13(h/R) + 2

(7)

6(h/R)2 + 4(h/R)

DLVO theory further considers a total pair interaction energy

V(h) ) VvdW(h) + Vel(h)

(8)

composed of an attractive van der Waals potential and an electrostatic repulsion. When retardation in the dispersion force is neglected, the van der Waals term can be expressed as

[

]

s2 - 4 2 A 2 + + ln VvdW(h) ) 6 s2 - 4 s2 s2

(9)

where s ) 2 + h/R, and A denotes the Hamaker constant. In the Derjaguin approximation

∫h∞ Vplane-plane(h′) dh′

Vsphere-sphere(h) ) πR

(10)

it simply reads

VvdW(h) ) -

AR 12h

∫h∞ ∫y∞ {cosh[βeψm(x)] - 1} dx dy

2πRn β

σ)

0κ exp(2βeψm) - 1 sn(v|m) × × βe exp(βeψm/2) cn(v|m) dn(v|m)

(13)

and

ψ0 ) ψm +

2 ln cd(v|m) βe

(14)

The functions sn(v|m), cn(v|m), dn(v|m), and cd(v|m) are Jacobian elliptic functions of argument v and parameter m in standard notation,50 at v ) κh/[4 exp(βeψm/2)] and m ) exp(2βeψm). The above procedure yields the electrostatic interaction with the surface chemical equilibrium maintained at all particle separations, i.e., full charge regulation. The popular approximations of interaction at constant charge or constant potential are recovered when eq 2 is replaced by the simpler boundary conditions σ ) σ∞ ) const or ψ0 ) ψ0∞ ) const, which will give an upper and lower boundary to the true interaction energy.41,49 The surface charge σ∞ ) limhf∞σ and potential ψ0∞ ) limhf∞ψ0 of the isolated particle are calculated via eqs 2-4. When the surface potential is low (ψ0 < 25 mV), the Poisson-Boltzmann equation may be replaced by its linearized version, the Debye-Hu¨ckel equation. The pair interaction energy then has the analytical form51

(ψ0∞)2 ln[1 + ∆ exp(-κh)] Vel(h) ) 2πR0 ∆

(15)

with ∆ ) (Creg - Cdl)/(Creg + Cdl) taking values between -1 and 1 depending on the ability of the surfaces to adjust their charge density upon approach. The regulation capacity Creg ) [(dσ)/(dψ0)]ψ0)ψ0∞ is the derivative of the σ(ψ0) relation given by eqs 2 and 3; Cdl ) 0κ is the diffuse layer capacity in the Debye-Hu¨ckel approximation. While for Creg . Cdl (∆ ) 1), the surfaces interact at constant potential, the expression for constant charge interaction is recovered in the opposite limit (Creg , Cdl, ∆ ) -1).

(11) Results and Discussion

which will be accurate when radii are large compared to the particle separation.46 No analytical expression exists for the electrostatic interaction on the Poisson-Boltzmann level. While computation of the exact double layer energy still requires a considerable numerical effort,47 the Derjaguin approximation is much more straightforwardly obtained as48

Vel(h) )

where n is the number density of particles, and ψm(h) is the electrostatic potential in the midplane between flat surfaces with the same density of chargeable sites Γtot. This midplane potential can be calculated by solving the set of equations for σ, ψ0, and ψm that is given by eq 2 and the two transcendental equations48,49

-1

∫0∞ (2R + h)2 exp[βV(h)] dh

k ) ks 2R

Langmuir, Vol. 16, No. 6, 2000 2569

(12)

(43) O’Brien, R. W.; White, L. R. J. Chem. Soc., Faraday Trans. 2 1978, 77, 1607. (44) Program WINMOBIL, Mangelsdorf, C. S.; White, L. R. J. Chem. Soc. Faraday Trans. 2 1990, 86, 2856. (45) Honig, E. P.; Roebersen, G. J.; Wiersema, P. H. J. Colloid Interface Sci. 1971, 36, 97. (46) J. N. Israelachvili Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1991. (47) Warszyn´ski, P.; Adamczyk, Z. J. Colloid Interface Sci. 1997, 187, 283. (48) Ninham, B. W.; Parsegian, V. A. J. Theor. Biol. 1971, 31, 405.

The Particle Charge. In Figure 1, measured charge densities (markers) are compared to calculations within the diffuse layer model. A pK value of 4.9 and a maximum charge density of 98 mC/m2 for the large and 78 mC/m2 for the small particles were obtained from a fit to the experimental data and served as input parameters for the calculation. (Values of 92 and 74 mC/m2 for the maximum charge had been reported by the manufacturer after conductometric titration.) The model slightly overestimates the dependence of the surface charge on the ionic strength, although less so for the larger particles. Excellent agreement is found for ionic strengths up to 100 mM and pH values below 6, which will be a regime of particular interest for the understanding of colloid stability. The pK of 4.9 found for the carboxyl groups on the (49) Behrens, S. H.; Borkovec, M. J. Phys. Chem. B 1999, 103, 2918. (50) Abramowitz, M.; Stegun, A. Handbook of Mathematical Functions, 9th ed.; Dover Publications: New York, 1972. (51) Carnie, S. L.; Chan, D. Y. C. J. Colloid Interface Sci. 1993, 161, 260.

2570


Figure 1. The particle charge as a function of pH for two species of different size. Markers represent the result of potentiometric titrations at fixed ionic strength I with KCl as background electrolyte. The solid lines were calculated within the diffuse layer model for a pK of 4.9 and the total site densities given in Table 1.

surface is expectedly higher than the literature value for soluble carboxylic acids, as can be rationalized by a low dielectric constant within the particle. The Electrophoretic Mobility. Measured and calculated electrophoretic mobilities are shown in Figure 2. The experimental data scatter more strongly than the titration results, but qualitatively show the expected behavior. In particular, there exists a pH at which a crossing of the mobilities for 1 and 10 mM solution occurs. For pH values below 6, the particle mobility is described fairly well by the theoretical curves. They were generated assuming a distance of xs ) 0.25 nm from the particle surface to the plane of shear. Variation of this parameter strongly affects the saturation value of the mobility at high ionic strength, but has very little influence on the theoretical curve for the 1 mM solution. The minimum in that curve reflects the minimum in the mobility as a function of zeta potential as predicted by the standard electrokinetic model (inset). At high pH and low ionic strength, measured mobilities lie below this predicted minimal value, manifesting a limitation of the electrokinetic model. Similar observations reported by other

Behrens et al.

Figure 2. The particle mobility u as observed in electrophoresis experiments and theoretical curves from the diffuse layer model for a distance xs of 0.25 nm between the surface and the plane of shear. Theoretical zeta potentials ζ were translated into mobilities according to the standard electrokinetic theory embodied in the computational method of O’Brien and White.43 The corresponding u-ζ relation for the given values of κR are shown in the inset.

groups52 have been interpreted as an indication of a “hairy” particle surface. Refinements of the standard theory that include the effect of surface conductance would even aggravate this type of discrepancy.53 Due to the predicted mobility minimum as a function of zeta potential, a discussion of the experimental data in terms of the zeta potentials is difficult: not only can there be some ambiguity as to whether mobilities slightly above the expected minimum correspond to lower or higher potentials, but measured mobilities below the minimum obviously cannot be converted into zeta potentials this way at all. For those mobilities of Figure 2 that can be converted, the corresponding zeta potential are displayed in Figure 3. Aggregation Behavior. Figure 4 shows the stability ratios W ) kfast/k for different ionic strengths, measured in dynamic light scattering, and the prediction of DLVO theory (eqs 6, 8). The theoretical curves were obtained (52) Rosen, L. A.; Saville, D. A. J. Colloid Interface Sci. 1990, 140, 82; Rosen, L. A.; Saville, D. A. J. Colloid Interface Sci. 1992, 149, 542. (53) Kijlstra, J.; van Leeuwen, H. P.; Lyklema, J. J. Chem. Soc., Faraday Trans. 1992, 88, 3441.


Figure 3. The zeta potentials corresponding to the mobilities of Figure 2. Some of the experimental mobilities lie outside the range of values expected by electrokinetic theory and cannot be converted.

using the nonlinear Poisson-Boltzmann equation with full charge regulation, and applying the Derjaguin approximation consistently in both the electrostatic contribution and the van der Waals term (eqs 11, 12). The electrostatic interaction between particles were computed from the model without any adjustable parameters, since the pK and the total site density were taken from a fit to the titration curves. The only remaining free parameter in the calculation of stability ratios was the Hamaker constant. A value of 1.9 × 10-20 J for the larger and 0.7 × 10-20 J for the smaller particles provided the best fit to the experimental data for 1, 2, and 10 mM. (A previously reported experimental value of 1.8 × 10-20 J for the larger particles was corrected after a slightly higher site density was found by titration than had initially been assumed.28) Theoretical estimates for the Hamaker constant are available from Lifshitz theory; the literature value for the unretarded interaction of polystyrene across water is (1.31.4) × 10-20 J;54 very recently a value of 0.9 × 10-20 J has been calculated.55 (54) Parsegian, V. A.; Weiss, G. H. J. Colloid Interface Sci. 1981, 81, 285; Prieve, D. C.; Russel, W. R. J. Colloid Interface Sci. 1988, 125, 1. For a review, see also Bowen, W. R.; Jenner, F. Adv. Colloid Interface Sci. 1995, 56, 201. (55) Bevan, M. A.; Prieve, D. C. Langmuir 1999, 15, 7925.

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Stability Ratios. The DLVO theory is seen to work quantitatively for several situations of aggregation at (moderately) low ionic strength. Clearly, the predicted steep increase in stability with pH above some critical value is confirmed by all experiments at ionic strengths up to 10 mM. To our knowledge, such a sensitive dependence of stability on the magnitude of the electrostatic interaction has been reported, apart from the communication of preliminary results on the same particles,28 in only one earlier study, which focused on the heterocoagulation of amphoteric latex particles.56 In a close-up (Figure 5), the experiments clearly show a greater slope of the stability curves for the large particles than for the small ones, just as claimed by the theory14 and quite in contrast to previous reports.15,16 At higher ionic strengths though, the aggregation data show a much more gradual increase of stability than predicted by DLVO theory, which is the far more common observation.10-13 For sufficiently high ionic strength (300 mM in the case of the larger particles), even the charge density of the fully deprotonated particle surfaces is sufficiently screened for aggregation to take place, and the charge saturation at high pH is reflected by a saturation in the stability. Theory predicts a saturation in the experimentally accessible regime only for even higher ionic strengths. A schematic summary of the aggregation behavior is given for the larger particles in Figure 6. It represents stability as a function of pH and ionic strength in a contour plot of equistability lines, extracted in a semiquantitative way from Figure 4 (top). At ionic strengths above 10 mM, the stability shows a much weaker sensitivity to variations in both the pH and the ionic strength than is predicted by DLVO theory. Absolute Rates of Fast Aggregation. Above some critical ionic strength (∼480 mM for the larger particles according to the theory), aggregation takes place at the fast rate kfast (W ) 1) for every pH. The absolute values of this fast rate for the two different species are similar to previously reported ones57 but smaller than the calculated values by a factor of 2-3 (see Table 1). While stability in the regime of slow aggregation is essentially determined by particleparticle forces in a small range of separations (around the energy barrier) only, the absolute rate constant in the fast regime is substantially influenced by details of the interaction at all distances and is thus harder to predict. The neglected retardation of the dispersion force certainly accounts to some degree for the overestimation of rate constants. In the present case the calculated values are substantially larger than the measured ones (see table), but previously reported studies indicate a better agreement for larger particles.4,37 It has also been pointed out that impurities can strongly influence aggregation in the fast regime.36 So far, reliable data are available only for very few systems; further investigations are certainly necessary for a better understanding of the observed absolute rates of fast aggregation. The Limitations of DLVO Theory. Figure 7 compares the theoretical pair interaction energy profiles at an ionic strength of 10 mM with the energy profiles at 100 mM; it may help to understand why the aggregation model gives a satisfactory description of colloid stability for low ionic strengths (e10 mM) only. The dotted line is the pure van der Waals energy (W ) 1.0), the two full lines include (56) James, R. O.; Holmola, A.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1977, 173, 1436. (57) van Zanten, J. H.; Elimelech, M. J. Colloid Interface Sci. 1992, 154, 1.

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Figure 4. Measured and calculated stability ratios at different fixed ionic strengths. Experimental data points were obtained from the increase in the hydrodynamic radius measured by dynamic light scattering; theoretical values were calculated in the Derjaguin approximation from the solution of the Poisson-Boltzmann equation including full charge regulation. The chosen Hamaker constant of 1.9 × 10-20 J for the larger and 0.7 × 10-20 J for the smaller particles allows the best fit to the measurements at ionic strengths up to 10 mM.

the electrostatic interaction at the pH values corresponding to a stability ratio of W ) 10 and W ) 1000. By increasing the pH (and correspondingly the surface charge and potential) an energy barrier is built up, which particles have to overcome in order to aggregate. For stability predictions it is vital to know the shape of the energy curve accurately in the vicinity of the maximum, since these particle separations give the dominant contribution to the integral in eq 6. Now at an ionic strength of up to 10 mM the energy barrier lies at surface separations of at least 1-2 nm, where DLVO theory was confirmed by measurements of colloidal forces and energies, too.7,26,27 On the other hand, at higher ionic strengths (and correspondingly small Debye lengths), the energy goes through the maximum at surface separations of only a few Å (Figure 7), where non-DLVO forces have been observed.26 In any event, a model ignoring the structure of both the solution and the particle surface cannot be expected to perform well at such minute distances. While it has been recognized early on that the height of the interaction energy barrier in the regime of slow aggregation governs the predicted stability,14 not much attention has been given to the role of the barrier position

as a criterion for the validity of aggregation theory. The requirement of an energy barrier at several nm implies a requirement of very low surface charge: for colloid particles of the given Hamaker constant and surface charge densities above 3 mC/m2, the large ionic strength needed in order to induce aggregation will imply an energy barrier in the sub nanometer range and likely a breakdown of the DLVO model. A recent study on weakly charged sulfate latex corroborates this idea.58 (In earlier investigations on systems with constant surface charge, the charge density was typically much higher17,59 as can sometimes only be inferred from the observed critical coagulation concentrations.10,11,60) Analysis of Selected Further Aspects. The Hamaker Constant. The influence of the Hamaker constant on theoretical stability curves is illustrated in Figure 8. All curves were obtained from the nonlinear treatment in the (58) Behrens, S. H.; Semmler, M.; Borkovec, M. Prog. Colloid Polym. Sci. 1932, 110, 66. (59) Peula, J. M.; Fernańdez-Babero, A.; Hidalgo-A Ä lvarez, R.; de las Nieves, F. J. Langmuir 1997, 17, 33 938. (60) Watillon, A.; Joseph-Petit, A. M. Discuss. Faraday Soc. 1966, 42, 143.


Langmuir, Vol. 16, No. 6, 2000 2573 Table 1. Characterization of Samples

Ac Ra site densityd batch (nm) CVb (10-20J) (nm-2) I II

155 52

0.03 0.11

1.9 0.7

0.60 0.48

exp. ratee theor. ratef (10-18m3/s) (10-18m3/s) 2.4 3.0

7.7 6.8

a Number averaged mean radius obtained from transmission electron microscopy. b The corresponding coefficient of variation. Preliminary values given in ref 28 were slightly modified after more extensive studies. c The effective Hamaker constant chosen to fit the experimental data at 2 mM and 10 mM solution for the large particles and the data at 1 and 10 mM for the small particles. d Density of chargeable headgroups according to the saturation value of charge obtained in potentiometric titration. Good agreement is found with values from conductometric titrations given by the producer.28 e Absolute fast aggregation rate constant measured by simultaneous static and dynamic light scattering under different angles; values are similar to previously reported ones.57 f Absolute fast rate constant as predicted by eq 6 with eq 9 for the interaction energy.

Figure 5. The effect of charge regulation and linearization on stability predictions. Solid lines represent predictions based on the Poisson-Boltzmann equation (PB) for boundary conditions of constant charge (CC), constant potential (CP), and full charge regulation (REG); dotted curves were generated using the Debye-Hu¨ckel equation (DH). The Derjaguin approximation was applied throughout. Note the general agreement between theory and experiment with respect to the different slopes for smaller and larger particles.

Figure 7. The DLVO pair interaction energy profile for both types of particles at an ionic strength of (a) 10 mM and (b) 100 mM. Each of the curves corresponds to one point on the predicted stability curves of Figure 4. The pH values chosen correspond to a stability ratio of W ) 10 and W ) 1000; the pure van der Waals interaction is shown as a dotted line. Note the shift of the energy barrier from about 2 nm to subnanometer distances as the ionic strength is increased from 10 mM to 100 mM.

the stability curves, but did not indicate a substantial difference in the Hamaker constant for the two particle species seen in Figure 8. The data presented in Figure 4 (bottom) were obtained several months later in a different laboratory. Although the same batch of particles was studied and great care was taken in order to ensure equivalent experimental conditions and the best achievable cleanliness of the system, a slight alteration of the particles or a difference in the purity of the used chemicals cannot be excluded. Figure 6. Theoretical equistability lines for the larger particles and their experimental analogue extracted in a semiquantitative way from the data shown in Figure 4. According to DLVO theory, the regime of fast aggregation (hatched area) is divided from a domain of considerable stabilization (W > 1000) by a very narrow region only. In practice, this region appears much broader for ionic strengths above 10 mM.

Derjaguin approximation. Previously reported preliminary results for the smaller particles at 10 mM showed the same good agreement with theory regarding the slope of

The fair agreement of the fitted Hamaker constants of (0.7-1.9) × 10-20 J with the theoretical values of (0.91.4) × 10-20 J may be delusive: both sorts of particles are hydrophobic as inferred from the contact angle of water on layers of dried particles. Recent force measurements indicate that a hypothetical hydrophobic attraction would decay with distance in a similar fashion as van der Waals forces and may thus be lumped into an effective Hamaker constant. A larger apparent Hamaker constant could

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Figure 9. The effect of surface curvature on the calculated stability for the small particles at 1 mM. Exact results from the Poisson-Boltzmann equation for two spheres under constant charge and constant potential conditions were obtained from a finite difference calculation. They are compared to the corresponding results in the Derjaguin approximation (lines), which overestimates both the van der Waals attraction and the double layer repulsion.

Figure 8. The sensitivity of the calculated stability on the Hamaker constant. Lines represent DLVO predictions for the larger particles at 10 mM solution from the nonlinear treatment under charge regulation (in the Derjaguin approximation). A Hamaker constant of 1.9 × 10-20 J was used in Figure 4 for the larger particles, the value of 0.7 × 10-20 J was used to model the smaller particles.

therefore be expected.61 The effect of hydrophobic interaction is likely counteracted by the fact that particle charges may not be located exactly (or entirely) on the surface, but reach into the solution to some extent. A situation where all the charged headgroups are separated from the particle surface by the same distance can be modeled by assuming a corresponding displacement of the plane of origin for the electrostatic interaction. Small displacements approximately result in a parallel shift of the stability curves similar to the one observed upon a decrease of the Hamaker constant. We have found that a displacement of the order of 0.4 nm can produce up to 40% smaller apparent Hamaker constants. The effect is more pronounced at higher ionic strength than at lower ionic strength and more pronounced for the smaller particles than for the larger ones. At the expense of introducing an additional fitting parameter, the model could produce a better agreement between the two Hamaker constants and a dependence on the ionic strength closer to the one observed in the experiment. (61) Yoon, R.-H.; Flinn, D. H.; Rabinovich, Y. I. J. Colloid Interface Sci. 1997, 185, 363.

Charge Regulation and Linearization. Figure 5 shows a close-up of the stability data for an ionic strength of 10 mM and compares the predictions based on the PoissonBoltzmann equation (using the Derjaguin approximation) for constant charge, charge regulation, and constant potential conditions (solid lines, eqs 12-14), with the corresponding results of the linearized theory (dotted lines, eq 15). Clearly, both the approximation of constant charge and constant potential can deviate considerably from the more realistic intermediate case of charge regulation.49 Linearization, on the other hand, causes only minor changes in the pH domain where DLVO theory works (low charge regime), because here the involved electrostatic potentials are low as well, typical values in the slow aggregation regime range from 9 to 19 mV at 1 mM and from 15 to 27 mV at 10 mM. Neither linearization nor an approximation of the charge regulation behavior strongly effects the slope of the predicted stability curve. Surface Curvature. The quality of the used Derjaguin approximation for surface curvature is illustrated in Figure 9 for the smaller particles and the lowest ionic strength considered (1 mM, κR ) 5.4), where deviations from the exact solution are most pronounced. The Derjaguin approximation for boundary conditions of constant charge and constant potential is compared to results based on eq 9 and the exact numerical result from a finite difference calculation47 of the electrostatic force. No exact solution was available for the case of charge regulation, but from the presented limiting cases, it is quite clear that corrections for surface curvature are relatively small. A more detailed analysis in fact shows that the errors caused by the Derjaguin approximation in the attractive and in the repulsive term largely cancel each other. Conclusions Classical DLVO theory of aggregation was successfully applied for a quantitative description of colloidal stability in a pH range where particles are weakly charged (e3 mC/m2) and aggregation can be observed at ionic strengths below 10 mM. At higher ionic strength, well-known discrepancies between theory and experiments are re-


covered. An analysis of interaction energy profiles indicates that such deviations are closely related to positions of the energy barrier at particle separations below one nanometer, where deviations from DLVO theory have also been established by force measurements. They might be due to surface heterogeneities, to the discrete nature of charges or the finite size of simple ions and water molecules, all of these aspects becoming most important at very short separations. Whenever the energy barrier determining the predicted stability lay at separations of 1-2 nm or more, the correct dependence of stability both on pH and on the particle size was captured by the theory. Simultaneous static and dynamic light scattering measurements however showed that the theory overestimates absolute aggregation rates for these particles by a factor of 2-3, which is possibly related to the neglect of retardation in the dispersion force. The Hamaker constant served as the only free parameter in a calculation of stability based on the Poisson-Boltzmann equation for boundary conditions of charge regulation in the Derjaguin approximation. The average value of 1.3 × 10-20 J obtained from a fit to the onset of stabilization lies within the range

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of literature results calculated from continuum theory. Linearization of the Poisson-Boltzmann equation and the effect of charge regulation prove uncritical for the general performance of the model. The Derjaguin approximation is excellent for the given ionic strengths and particle sizes, when applied consistently in the attractive and repulsive component of the interaction. Acknowledgment. The exact numerical solution of the Poisson-Boltzmann equation for two spheres was made possible by a code and generous help from P. Warszyn´ski and Z. Adamczyk. The interpretation of experimental aggregation data was greatly promoted by several discussions with J. Israelachvili. Further support by R. Kretzschmar, H. Holthoff, and H. Sticher is also gratefully acknowledged. We also thank D. Prieve for providing a preprint of ref 55. This work was financed by the Swiss National Science Foundation and the U.S. National Science Foundation (Grants CHE-9870965 and CTS9820795). LA991154Z

Charging and Aggregation Properties of Carboxyl Latex Particles

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