Electrolyte-Induced Aggregation of Acrylic Latex. 1. Dilute Particle

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Electrolyte-Induced Aggregation of Acrylic Latex. 1. Dilute Particle Concentrations Leo H. Hanus,† Robert U. Hartzler, and Norman J. Wagner* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716 Received June 30, 2000. In Final Form: March 26, 2001 We measure the aggregation kinetics of model, aqueous polymer latices of varying particle size as a function of added NaCl using dynamic light scattering to evaluate the available theoretical models for predicting the aggregation rate (stability ratio) and critical coagulation concentration (CCC). Our focus is on dilute colloidal suspensions of fixed surface chemistry but varying particle size. A master curve for the growth of the aggregate size is observed, with an intermediate regime that follows predictions for diffusion-limited cluster-cluster aggregation. Theoretical predictions based on DLVO theory are found to be in quantitative agreement for all but the largest particle size, when the particle surface potentials are determined by matching the experimentally determined CCCs. Thus, we conclude that for sufficiently smooth, nearly monodisperse particles such as those investigated here, DLVO theory can provide accurate predictions of colloidal stability for the range of parameters explored here down to truly atomic dimensions. The particle potential determined from phase analysis light scattering measurements of the zeta potential overpredicted colloidal stability but can be brought into agreement by assigning a Stern layer thickness equal to the hydrodynamic size of the counterion.

Introduction Colloidal stability has been the subject of considerable research interest because of the prevalence of colloids in many industrially important and naturally occurring products and processes. Ceramics, food, paint, paper, pharmaceutical production, mineral processing, and water treatment are a few examples.1,2 Colloidal aggregation and the significant change in colloidal properties that accompany aggregation are exploited or prevented in the above examples to achieve desired end-product characteristics. Of particular interest in this series of papers is aggregation that leads to the formation of dense bodies (such as in ceramics)2 or films (such as in paints and coatings).1 Colloidal stability and aggregation behavior can be understood in terms of the forces acting on the particles (dispersion, electrostatic, and forces due to absorbed polymers) and the mode of particle motion (Brownian motion (perikinetic) or fluid shear (orthokinetic)).3,4 In this study, the attractive dispersion forces, which drive particles suspended in a dielectrically different medium to coalesce, are counteracted by repulsive electrostatic forces due to particle surface charge. Here, we explore the established Derjaguin-Landau-Verwey-Overbeek (DLVO)3,5,6 theory by comparing with experiments on flocculation kinetics. A number of textbooks3,4,7 and the * To whom correspondence may be addressed: (302) 831-8079 phone; (302) 831-1048 fax; [email protected]. † Current address: Clariant Corp., Charlotte, NC 28216. (1) Technological Applications of Dispersions; McKay, R. B., Ed.; Surfactant Science Series 52; Marcel Dekker: New York, 1994. (2) Bergstroem, L. Ceram. Trans. 1995, 51, 341-348. (3) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989. (4) Hunter, R. J. Foundations of Colloid Science; Clarendon Press: Oxford, 1987; Vol. 1. (5) Behrens, S. H.; Borkovec, M.; Schurtenberger, P. Langmuir 1998, 14, 1951-1954. (6) Behrens, S. H.; Christl, D. I.; Emmerzael, R.; Schurtenberger, P.; Borkovec, M. Langmuir 2000, 16, 2566-2575. (7) Hiemenz, P. C.; Rajagopalan, R. Principles of Colloid and Surface Chemistry, 3rd ed.; Marcel Dekker: New York, 1997.

review of Hidalgo-Alvarez et al.8 provide a thorough discussion of the theory, the salient portions of which will be covered. Of contemporary interest is the validity of the continuum DLVO model to accurately predict colloidal stability and its failure as atomic dimensions are investigated. There have been many previous studies of electrolyteinduced aggregation of electrostatically stabilized colloids. For recent reviews, see ref 8 or the introductions of refs 5, 6, 9, 10, and 11. Numerous types of colloidal systems (e.g., colloidal gold, polystyrene latex, and silica) have been investigated9,10 using several experimental techniques (e.g., microscopy, particle counters, turbidity, light scattering, and electric birefringence).11,12 A majority of the reported studies focus on dilute aggregation and use standard light scattering methods to measure the extent of aggregation. Not all of the studies have produced consistent results. However, in this work we refer to the experimental and theoretical work of Lin et al.,13 who established two universal regimes of irreversible aggregation, diffusion14 and reaction15 limited, by studying colloidal gold, polystyrene latex, and silica using static and dynamic light scattering. Their results agree well with computer simulations of the aggregation behavior for the two limiting regimes and provide a basis for comparison. (8) Hidalgo-Alvarez, R.; Martin, A.; Fernandez, A.; Bastos, D.; Martinez, F.; de las Nieves, F. J. Adv. Colloid Interface Sci. 1996, 67, 1-118. (9) Burns, J. L.; Yan, Y.; Jameson, G. J.; Biggs, S. Langmuir 1997, 13, 6413-6420. (10) Kyriakidis, A. S.; Yiatsios, S. G.; Karabelas, A. J. J. Colloid Interface Sci. 1997, 195, 299-306. (11) Hothoff, H.; Egelhaaf, S. U.; Borkovec, M.; Schurtenberger, P.; Sticher, H. Langmuir 1996, 12, 5541-5549. (12) Sonntag, H. In Coagulation and Flocculation: Theory and Applications; Dobias, B., Ed.; Surfactant Science Series 47; Marcel Dekker: New York, 1993. (13) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin P. Nature 1989, 339, 360-362. (14) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Klein, R.; Ball, R. C.; Meakin P. J. Phys.: Condens. Matter 1990, 2, 3093-3113. (15) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin P. Phys. Rev. A 1990, 41, 2005-2020.

10.1021/la000927c CCC: $20.00 © 2001 American Chemical Society Published on Web 05/05/2001

Electrolyte-Induced Aggregation of Acrylic Latex

Recently, Behrens et al.5,6 explored the limiting conditions of applying classical DLVO theory to model perikinetic colloidal aggregation. They showed that the proximity of the stability barrier to the particle surface is a controlling factor in determining whether DLVO theory can adequately describe the rates of perikinetic aggregation. Their experiments on model polystyrene latices demonstrate that conditions leading to stability barriers greater than 2 nm from the surface, such as low salt and low surface charge densities, are adequately described by DLVO theory. However, high salt and high surface charges, which result in stability barriers within 2 nm of the surface, show deviations that Behrens et al. attribute to a breakdown in the continuum description. The latter conditions are typical of more concentrated latices and, as such, are of consequence to the ceramics, paints, and coatings industries. A thorough investigation of perikinetic aggregation as a function of particle size, electrolyte, and surface charge in this regime is warranted and has not been fully executed in the literature. The objective of this series of papers is to ascertain the similarities and differences in colloidal aggregation in semidilute and concentrated suspensions relative to the dilute suspensions that are typically studied and for which a significant amount of theory has been developed. Many of the practical applications that involve aggregation (such as in ceramics, paints, and coatings) employ concentrated (very high solids loadings) dispersions. The applicability of aggregation kinetics and fractal structure information obtained from studies of dilute dispersions is of questionable relevance for these highly concentrated dispersions. In this the first of the series, we begin with an investigation of electrolyte-induced perikinetic aggregation. Four, wellcharacterized suspensions of acrylic latex particles of varying size, but similar surface chemistries, are studied. The particles were specifically synthesized to provide very smooth surfaces with low, uniform charge densities to critically test aggregation theories. We employ dynamic light scattering (DLS)16,17 to measure the limiting aggregation kinetics of the dilute dispersions for a range of electrolyte concentrations and evaluate the effectiveness of the current theoretical models for predicting the aggregation behavior. We also consider the effects of different measures of the surface charge, as well as surface roughness on the results. A master curve of perikinetic aggregation for varying particle size, ionic strength, and particle concentration is presented for rapid aggregation. Theoretical Background We briefly review the theoretical developments of direct interest to this study and refer the reader to a recent review8 and modern textbooks3,4 for more information. Three fundamental issues govern the type of clusters formed during aggregation: the mode of motion that brings the particles into contact, the probability of the particles sticking upon contact, and the mobility of the subsequent aggregate. Here, in the absence of shear, we focus on Brownian, or perikinetic, aggregation, with clustercluster aggregation18 being the dominant mode. The particle sticking probability leads to two universal limiting aggregation regimes: diffusion and reaction limited, where diffusion-limited implies a high sticking probability (i.e., (16) Hanus, L. H.; Ploehn, H. J. In Surface Characterization Methods: Principles, Techniques, and Applications; Milling, A. J., Ed.; Surfactant Science Series 87; Marcel Dekker: New York, 1999. (17) Finsy, R. Adv. Colloid Interface Sci. 1994, 52, 79-143. (18) Hemker, D. J.; Frank, C. W. Macromolecules 1990, 23, 44044410.

Langmuir, Vol. 17, No. 11, 2001 3137

the rate of diffusion not the sticking probability limits the aggregation process) and vice versa. Experimental studies and computer simulations indicate that large Brownian aggregates have a fractal structure3,13 over a range of measurable length scales. The number of particles in an aggregate N is related to the aggregate radius of gyration Rg and single particle radius a through

N ) (Rg/a)df

(1)

where df is the fractal dimension of the aggregate with 1 e df e 3 (where df ) 3 is the limit of a completely compact spherical aggregate). The mode of aggregation (particlecluster or cluster-cluster) can be distinguished from the fractal dimension of the resultant aggregate. Rapid, irreversible particle-cluster aggregation leads to a denser aggregate than cluster-cluster aggregation with fractal dimensions of around 2.5 and 1.8, respectively.3 Lin et al.13-15 established that perikinetic colloidal aggregation occurs predominately via cluster-cluster aggregation with two universal limiting regimes based on the particle sticking probability. In diffusion-limited colloid aggregation (DLCA),14 the absence of a strong repulsive barrier allows particles to bind upon contact. Thus, under DLCA conditions, clusters are more likely to attach to the exterior of a growing aggregate than to the interior, and as a result, the fractal dimension df has a relatively low value of about 1.8. In reaction-limited colloid aggregation (RLCA),15 a repulsive barrier limits the efficacy of collisions, allowing some clusters to enter the interior of the growing aggregate. This produces a denser fractal and a larger fractal dimension df of about 2.1. Note that these DLCA and RLCA fractal dimensions are for irreversible aggregation; i.e., the aggregates must not restructure internally once formed. Light Scattering Analysis of Aggregation. Light scattering is well suited to investigate the kinetics and structure of dilute Brownian aggregates. Static light scattering yields a direct measure of the fractal dimension for both limiting aggregation regimes via the angular dependence of the scattered intensity (valid for qRg . 1)14,15

IS(q) ∝ (qRg)-df ∝ q-df

(2)

where q ) (4πn/λ) sin(θ/2) is the magnitude of the scattering vector (with solvent refractive index n, scattering angle θ, and laser wavelength in a vacuum λ). Dynamic light scattering (DLS), which we use in this study, can also yield information about the fractal dimension and is more suited for measuring the aggregation kinetics. For self-beating (homodyne) DLS experiments, the scattered intensity autocorrelation function (ACF) is measured.16,17

G(2)(τ) ) 〈IS(0)IS(τ)〉 ) A + B|g(1)(τ)|2 ) A + B[

(3)

∫0∞f(Γ) exp(-Γτ) dΓ]2

where the angular brackets indicate an ensemble average, τ is the delay time of the ACF, A ) limτf∞G(2)(τ) is the baseline constant of the ACF, B ) limτf0G(2)(τ) - A is a correction factor related to the coherence area of the photodetector, and f(Γ) is the intensity-weighted decay constant distribution (the intensity-weighted fraction of particles with characteristic decay constant Γ). The last equality holds in the absence of particle interactions and

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multiple scattering, which is achieved in conventional DLS instruments by diluting the analysis sample to volume fractions φ on the order of 10-4. Particle size, shape, and composition polydispersities all influence the decay constant Γ and thus f(Γ). For optically homogeneous particles, the decay constant Γ in eq 3 can be related to an effective particle diffusivity DDLS ) Γq-2, which includes size and shape polydispersity contributions. An effective, spherical particle radius follows from the Stokes-Einstein equation as

kBT q2kBT ) RDLS ) 6πηΓ 6πηDDLS

(4)

where kB is Boltzmann’s constant and η and T are the solvent viscosity and absolute temperature, respectively. For DLCA kinetics, the number density of aggregates C at time t can be modeled using an approach proposed by Smoluchowski

k11 C0 t )1+ C t)1+ C 2W 0 tp

(5)

which assumes a constant (size-independent) stability ratio W for all aggregate clusters and that aggregate collisions occur predominately between clusters of approximately equal size.3,11 In eq 5, C0 ) 3φ0/4πa3 is the initial single particle concentration, k11 ) 8kBT/3η is the aggregation rate constant for doublet formation, and tp ) 2W/k11C0 ) ηπa3W/φkBT is the characteristic time for Brownian aggregation. As the ratio C0/C equals the average number of single particles N h per aggregate, eqs 1 and 5 can be used to derive a relationship between the average radius of gyration Rg and the aggregation kinetics14

N h )

()

Rg C0 t )1+ ) C tp a

df

(6)

For long aggregation times (t/tp , 1), eq 6 simplifies to

(

) ()

Rg t ) 1+ a tp

1/df

)

t tp

1/df

(7)

No equivalent expression relating the radius of gyration to the fractal dimension has been derived for RLCA aggregates. However, kinetic information can be obtained via the exponential growth functionality

Rg ∝ exp(t/t0)

(8)

where t0 is the RLCA aggregation time constant related to the initial particle concentration and particle sticking probability.15,19 The difficulty in applying eqs 7 or 8 for DLCA (power law growth) and RLCA (exponential growth), respectively, is that quadratic cumulants20 (or other model)16 fits of the intensity autocorrelation function (eq 3) measured by DLS produce an effective decay constant Γ that includes contributions due to particle size and shape polydispersities. Substitution of Γ into eq 4 leads to an effective (19) Carpineti, M.; Giglio, M. Adv. Colloid Interface Sci. 1993, 46, 73-90. (20) The quadratic cumulant model G(2) ) A + B exp(-2Γ h τ + (∆Γ)2τ2) is simply the solution of eq 3 for a Gaussian distribution of Γ values [f ) 1/[(2π)1/2(∆Γ)] exp(-(Γ - Γ)2/2(∆Γ)2)].

hydrodynamic radius RDLS, which must be related to the average of the particles’ radius of gyration Rg for eq 7 or 8 to be applied correctly. Hereafter, since we observe power law aggregate growth, we focus on DLCA only. For DLCA, Torres et al.21 derived the following relationship between the average DLS-measured radius and the average radius of gyration: RDLS/Rg ) 1/Γ/(2 - df-1) (where Γ/ is the gamma function) by assuming a monodisperse particle size distribution and qRg . 1. Substituting this relationship into eq 7 leads to

(

)

RDLS t 1 ) 1+ a tp Γ/(2 - df-1)

1/df

()

≈ 1.129

t tp

1/df

(9)

for df ≈ 1.86. Using different assumptions (valid for all Rgq as long as aq , 121), Lin et al.14 derived values for the coefficient in eq 9 that vary between 0.64 and 1.44. For our conditions, aq is greater than 1 for all particle sizes, so we use eq 9 in the analyses that follow. DLVO Theory. We employ DLVO theory to model the interaction energy between the electrostatically stabilized particles. There is no fully satisfactory analytical expression for the electrostatic repulsion over the entire range of parameter space that our experiments investigate. As performed by a number of investigators (see refs 3, 5, 6, 22, and 23), a full numerical solution of the nonlinear Poisson-Boltzmann equation is feasible and gives guidance for choosing an analytic form that is accurate to within a given confidence interval. The applicability of various approximate, analytical forms as a function of suspension conditions have been mapped out by Glendinning and Russel3,22 and Carnie et al.23 for constant surface potential and constant surface charge boundary conditions. As will be shown, our zeta potential measurements can be rationalized by assuming a constant surface charge for all of the suspensions with κa > 100 and ψe/kBT between 1 and 2. Under the conditions of our experiments the values of κh are on the order of 1, with h ) r - 2a the sphere surface-to-surface separation. For a constant charge boundary condition, none of the analytical approximations3 is valid for these conditions. A full numerical solution or the numerical Derjaguin approximation of Carnie et al.23 should be used to accurately model the electrostatic repulsion. Instead of employing a numerical solution, we employ the nonlinear superposition approximation for constant potential boundary conditions, which is only strictly valid for κa . 1 and κh . 1. Here, we will employ this analytical form throughout the analysis. The total interaction energy (the sum of the dispersion and electrostatic terms) is24

Φ ) kBT Aeff h(h + 4a) 2a2 2a2 + ln + + 2 6kBT h(h + 4a) (h + 2a) (h + 2a)2 8(a + ∆) ezψ tanh2 exp(-k(h - 2∆)) (10) LB 4kBT

(

( )

(

))

(21) Torres, F. E.; Russel, W. B.; Schowalter, W. R. J. Colloid Interface Sci. 1991, 142, 554-574. (22) Glendinning, A. B.; Russel, W. B. J. Colloid Interface Sci. 1983, 93, 95-104. (23) Carnie, S. L.; Chan, D. Y. C.; Stankovich, J. J. Colloid Interface Sci. 1994, 165, 116-128. (24) Peula, J. M.; Santos, R.; Forcada, J.; Hidalgo-Alvarez, R.; de las Nieves, F. J. Langmuir 1998, 14, 6377-6384.


Langmuir, Vol. 17, No. 11, 2001 3139

In the above equation, Aeff is the Hamaker constant, z is the electrolyte valence, ∆ is the Stern layer thickness, LB ) e2/4kBTπ0 is the Bjerrum length (in m), and ψ is the particle potential at the Stern layer ∆ (at ∆ ) 0 nm, ψ ) ψS the particle surface potential). Also, κ ) 10-9{(2/0)(NAv/kBT)[(2Csalt + Ccounterion)/(1 - φ)]}1/2 is the inverse Debye screening length in (nm-1) with Avogadro’s number NAv, bulk electrolyte concentration Csalt (in mM), colloid counterion contribution to the electrolyte concentration Ccounterion ) 1027(3Qφ/4πa3eNAv) (in mM), and particle charge Q (in C). To predict colloidal stability using eq 10, we require independent measures of the Hamaker constant Aeff, single particle radius a, Stern layer thickness ∆, and particle potential ψ. The Hamaker constant was estimated from literature values for the nonretarded Hamaker constant Aeff(0) for poly(methyl methacrylate) (PMMA) in water and the predicted electrolytic screening of the zerofrequency term in the Hamaker constant summation (see ref 4): At 20 °C, Aeff(0) ) 1.05 × 10-20 J for PMMA in water,3 Aeff0 ) 0.294 × 10-20 J, and Aeff ) Aeff(0) - Aeff0 ) 0.756 × 10-20 J if the first eight terms (s ) 1 through 8) are included in the sum.4 The Stern layer thickness was estimated from the counterion (Na+) radius. The Stokes radius of Na+ is 0.18 nm (ref 3, Appendix A), and the radii for hydrated and dehydrated Na+ ions are approximately 0.372 and 0.096 nm, respectively.24,25 In this work, we explore values within this range. The particle radius a was determined using the intensity-average radius obtained from DLS measurements of the diluted suspensions (φ ) 10-4) with no added NaCl. The particle potential ψ was determined in two ways: by setting ψ so that the DLVO theory predicted the experimentally observed critical coagulation concentrations (CCCs), and by relating ψ to the zeta potential obtained from electrophoretic mobility measurements. The electrophoretic mobilities µ of the diluted sample suspensions (φ ) 10-4) were measured as a function of added NaCl using phase analysis light scattering26,27 (PALS), and the particle zeta potentials ζ were calculated using,3,16

ζ ) f(ηµ/0)

(11)

with f ) 1 for κa . 1 and exp(ezζ/2kBT)/κa , 1 (the Smoluchowski equation) and f ) 3/2 for κa , 1 (the Hu¨ckel equation).4 For the DLVO calculations, the particle zeta potential must be related to the particle potential. This is a nontrivial task, and we used the common assumption of ψ ≈ ζ. The particle surface potentials ψ were converted into particle charges Q using the following

Q)

( )

2a(1 + aκ) ezψ sinh LB 2kBT

(12)

This equation gives nearly identical results (for the conditions investigated) to the Loeb-Overbeek-Wiersema relationship (eq 4.8.7 of ref 3), which can predict Q to within 5% for κa > 1/2 and any ψ. (25) Israelachvili lists similar but slightly different values (0.36 and 0.095 nm) for hydrated and dehydrated Na+ ions: Israelachvili, J. Intermolecular and Surface Forces, 2nd ed.; Academic Press: New York, 1992. (26) Tscharnuter, W. W.; McNeil-Watson, F.; Fairhurst, D. In Particle Size Distribution III; Provder, T., Ed.; ACS Symposium Series 693; American Chemical Society: Washington, DC, 1998. (27) McNeil-Watson, F.; Tscharnuter, W.; Miller, J. Colloid Surf., A 1998, 140, 53-57.

Critical Coagulation Concentration (CCC). The critical coagulation concentration (CCC) of electrolyte induces rapid aggregation by screening the electrostatic repulsion so as to eliminate the kinetic barrier to aggregation. With eq 10, the CCC is defined as Φmax ) 0, such that the stability barrier is eliminated. Stability Ratio. The stability ratio W is the ratio of the measured aggregation rate to the rate of perikinetic aggregation in the absence of interparticle and hydrodynamic interactions, W ) kfast/k. The stability ratio can be calculated from the particle interaction energy Φ, separation distance h ) r - 2a, and hydrodynamic interaction G(h) as3

W ) 2a

∫2a∞

exp[Φ(r)/kBT] r2G(r)

dr ) exp[Φ(h)/k T]

B ∫0∞ (h + 2a)2G(h)

2a

dh (13)

A common assumption4,24 is to neglect the hydrodynamic interactions by setting G(h) ) 1. This assumption becomes valid as h becomes large but is not appropriate for the range of h values of interest here, making a numerical calculation28 for G(h) necessary. We employ the QROMO subroutine from Numerical Recipes29 to numerically evaluate the integral in eq 13 using G(h) from ref 28 and eq 10 for Φ. An estimate for the value of the stability ratio can be made directly from the maximum height of the interaction energy Φmax using the approximation of Prieve and Ruckenstein3

W ≈ W∞ + 0.25[exp(Φmax/kBT) - 1]

(14)

where W∞ is the rapid coagulation value of the stability ratio (with a value generally slightly greater than 1). In this paper the stability ratio is extracted from the fractal growth rate and compared to the more traditional method that uses the doublet aggregation rate. As will be demonstrated, for our system the stability ratio can be determined accurately by creating a master curve of dimensionless aggregate size as a function of dimensionless aggregation time for systems with varying particle size and electrolyte concentrations. log-log plots of the measured stability ratio versus electrolyte concentration typically yield two linear regions, and the intersection of the linear regions is typically taken to be the CCC.3,4,7 This method is also used with W estimated from the initial aggregation rates, W′ ) kfast/k,30,31 where the rates k are estimated from the slope of the growth of average particle size for the first 10-40 min of aggregation (depending on the aggregation rate). Finally, plots of the initial aggregation rates k as a function of electrolyte concentration also provide an indication of the CCC as the electrolyte concentration at which the rate plateaus. Experimental Section Acrylic Latex Preparation. Four nearly monodisperse acrylic latex suspensions (r100, r150, r200, and r300) were supplied by Rohm & Haas Corp. The suspensions were synthesized using a monomer mixture of butyl acrylate and methyl methacrylate, with 1 wt % methacrylic acid added, using sodium dodecylbenzene sulfonate surfactant and ammonium persulfate (28) Batchelor, G. K. J. Fluid Mech. 1976, 74, 1-26. (29) Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; Vetterling, W. T. Numerical Recipes; Cambridge University Press: New York, 1988. (30) Kim, A. Y.; Berg, J. C. Langmuir 2000, 16, 2101-2104. (31) Klyubin, V. V.; Kruglova, L. A.; Sokolov, V. N. Colloid J. USSR 1988, 50, 738-45.

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Table 1. Acrylic Latex SuspensionssPhysical Characteristics Cinit.a label

(g/100 g)

Csurf.b (g/100 g)

pHc

r300 r200 r150 r100

0.8 0.4 0.4 0.4

0.61 0.29 0.25 0.23

7.9 7.7 8.4 7.6

radius, a (nm) DLSd CHFe 167 ( 35 111 ( 28 94 ( 23 57 ( 15

Tge (°C)

viscosity,e η (cP)

wt fraction,e s (%)

59 ( 2 58 ( 2 60 ( 2 63 ( 2

15 26 41 144

47 47 48 47

149 90 75 53

a Initiator concentration in g per 100 g monomer. b SDBS concentration in g per 100 g monomer. c pH increased after synthesis using NH4OH. d Measured with our homemade instrument at the conditions listed in Table 2 with φ ) 10-4 and CNaCl ) 0 M. e As reported by Rohm & Haas, CHF, capillary hydrodynamic fractionation, Tg determined using differential scanning calorimetry.

initiator. The residual monomers in the suspensions were determined to be less than 100 ppm by gas chromatography. The relevant physical characteristics of the suspensions are summarized in Table 1. Note that the synthesis conditions were at Tg + 20 °C and so the particles are expected to be very smooth, spherical, and nearly monodisperse. The suspensions were dialyzed in deionized water using Spectra/Por 2 dialysis tubes until the conductivity of the dialyzate was close to that of the pure water, which typically took several days.32 Deionized water from a Barnstead NANOpure type D4700 system was used for the dialysis and for all other preparations that follow. The resulting dispersions were stored in sealed plastic bottles and were observed to be stable over many months. The irreversibility of the aggregation was tested by inducing aggregation using a NaCl concentration near the CCC (e.g., for the r300 particles, using a suspension and NaCl concentration of 0.3% and 1.2 M, respectively). The aggregate size was monitored using dynamic light scattering. When the aggregate size reached 10 times the single particle radius a, the aggregation was quenched by diluting the sample by 100 times. Dilution alone did not break up the aggregates, but dilution followed by 15 min of sonication broke the aggregates down to single particles. All the results reported here are primary minimum aggregation. Acrylic Latex Characterization. The particle densities were measured in solution at 20 °C using an AP Paar DMA 48 density meter. Suspension solution density was measured as a function of particle mass fraction s for a concentration series (five concentrations with s ranging from 5 to 40%) for each suspension. The particle mass fractions were determined gravimetrically by heating in a vacuum oven at 130 °C for at least 2 h. The particle density was determined using the additive volume relationship33 Fpart ) s[Fsolution-1 + Fsolv-1(s - 1)]-1 with Fsolv ) 0.9982 g/mL for water at 20 °C. The measured particle density for each suspension was Fpart ) 1.158 ( 0.001 g/mL on average, and this value was used to convert the measured particle mass fractions s to volume fractions φ using φ ) [(s-1 - 1)Fpart/Fsolv + 1]-1. The particle electrophoretic mobilities µ were measured at 25 °C as a function of NaCl concentration at a constant neutral pH and the same particle concentration (φ ) 10-4) used in the aggregation studies using a Brookhaven Instruments Zeta-PALS instrument. Five measurements (15-30 cycles per measurement) were made for each suspension at NaCl concentrations of 0, 0.1, 0.2, 0.4, and 0.6 M depending on the stability of the suspension. Equation 11 was used to convert the measured µ values to zeta potentials ζ. Aggregation Studies. Two different dynamic light scattering (DLS) instruments, a homemade instrument and commercial instrument from Brookhaven Instruments (BI) Corp.34 were used to measure particle size before and during aggregation. The homemade instrument consisted of a JDS-Uniphase35 1144P (17 mW) HeNe laser, Electron Tubes 9863-100 photomultiplier tube (PMT) detector, and ALV-500036 digital correlator. The Brookhaven instrument consisted of a Lexel-95 (2 W) argon-ion laser, BI-200SM goniometer, Electron Tubes 9863-250 PMT, and BI(32) Wilkinson, M. C.; Hearn, J.; Steward, P. A. Adv. Colloid Interface Sci. 1999, 81, 77-165. (33) Bodnar, I.; Silva, A.; Deitcher, R. W.; Weisman, N. E.; Wagner, N. J.; Kim, Y. H. J. Polym. Sci., B: Polym. Phys. 2000, 38, 857-873. (34) Brookhaven Instruments Corporation (www.bic.com): Holtsville, New York, 1993. (35) JDS-Uniphase, San Jose, CA, www.uniphase.com, (800) 6448674. (36) ALV-Laser Vertriebsgesellschaft G.m.b.H., Langen, Germany, www.alvgmbh.de, +49-6103-78094/5.

Table 2. Dynamic Light Scattering Experimental Parameters instrument

T, °C

θ, deg

λ, nm

q, nm-1

I0, mW

dpin, µm

homemade BI

21 24

90 90

632.8 488.0

0.0264 0.0342

17 10-20

200 100

9000 digital correlator. Fisherbrand (14-961-27) borosilicate glass culture tubes (13 × 100 mm, cylindrical) contained samples for both instruments. Temperature was controlled in both instruments to (0.01 °C by refrigerated circulators. The experimental settings employed for the two instruments are summarized in Table 2. Reasonable reproducibility was observed in the single particle radius measurements performed on the two instruments. The homemade instrument was used for all measurements in this study except where noted. The aggregation kinetics of each acrylic latex suspension (r300, r200, r150, and r100) was measured at several NaCl concentrations (ranging from 0.4 to 1.8 M). Stock solutions of the suspensions (with volume fractions of 0.2%) were prepared by diluting the dialyzed suspensions in deionized water and then filtering through a 0.8 µm Cole Parmer syringe filter (#2915-62). Scattering samples were prepared by rapidly adding predetermined volumes of deionized water, NaCl stock solution (2 M), and particle stock to the sample cell. The NaCl solution was added to the water and mixed and then mixed with the diluted suspension stock solution. The latter mixing defined the zero time of the experiment. For most of the experiments, a particle volume fraction of 10-4 and a controlled sample temperature of 21.0 °C (near ambient) were employed unless otherwise indicated. NaCl concentrations ranging from 0.4 to 1.8 M were investigated. Typically, 50 ACFs were collected, each with a duration of 30 s, and a 30 s delay was used between each collection, giving a total experimental measurement time of 50 min for each sample. To test the reproducibility of the aggregation behavior and to slow the rate, we also studied the r200 particles at a lower particle volume fraction of 3 × 10-5. Finally, another sample of r300 particles with φ ) 10-4 and CNaCl ) 1.142 M (near to the CCC) was studied at 24 °C for a total experimental duration of greater than 24 h (much longer than our other experiments) using the Brookhaven instrument. A 2-min collection duration followed by a 2-min delay between collections was employed for this experiment. Suspending Solvent Viscosity. The viscosity of aqueous NaCl solutions was obtained from a polynomial fit to the viscosity data as a function of NaCl concentration from ref 37. To correct for temperature effects, these values (for NaCl solutions at 25 °C) were multiplied by the ratio of the viscosity of water at the temperature of interest T and at 25 °C (ηH2O,T/ηH2O,25°C). For all other solvent parameters, such as the particle refractive index, the values for water at the specified temperature were used without correction for the added NaCl.

Results and Discussion Figure 1 shows the measured aggregate diameter as a function of time and added NaCl concentration for the four acrylic latex suspensions. In absolute units, the transition from stable to rapid flocculation is evident with increasing salt. Comparison across particle sizes shows a clear size effect both on the initial rate of the rapid (37) Lide, D. R., Ed. CRC Handbook of Chemistry and Physics, 80th ed.; CRC Press: New York, 1999.


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Figure 1. DLS-measured diameter versus aggregation time as a function of NaCl concentration (symbols, in M) for the acrylic latex suspensions r300, r200, r150, and r100.

Figure 2. Master curve of the aggregation data presented in Figure 1. The line is eq 9. The arrows indicate the calculated onset of sedimentation effects for each particle type.

flocculation and on the salt concentration required to achieve rapid flocculation. Aggregation rates increase with added salt for each particle size and with decreasing particle size for a given salt concentration, as expected. Figure 2 shows the data in Figure 1 in nondimensional form. The DLS-measured aggregate radius is made nondimensional with the single particle radius a (Table 1, φ ) 10-4 with no added NaCl). The aggregation time t is reduced with the Brownian aggregation time tp (eq 5). To construct Figure 2, the stability ratio W (tp ) 2W/k11C0) was adjusted to overlap the experiments at different NaCl

concentrations onto a master curve. The coincidence of the data illustrates the commonality of the aggregation physics for this series of particles; i.e., the qualitative differences observed between systems in Figure 1 are observed to be an artifact of the limited observation window sampling only part of the master curve for each of the different particle sizes. At short nondimensional times (t/tp < 2), all of the data sets indicate an induction time with little aggregation. Single particles are beginning to combine and form aggregate clusters, but not enough clusters have formed for significant cluster-cluster aggregation to occur. Consequently, the assumptions underlying the fractal growth model (eq 7) are not valid for t/tp < 2. For intermediate times (2 < t/tp < 100), DLCA size growth is observed, and the theoretical line (RDLS/a ) 1.129(t/tp)1/df with df ) 1.86) accurately predicts the experimental data (note, there are no adjustable parameters involved). For long times (t/tp > 100), the aggregate growth rate slows. This is a consequence either of settling of the larger aggregates or of the largest aggregates becoming comparable in size to the scattering volume (a cylinder with a radius and height of 200 µm). The formation and settling of large particles within the scattering volume may lead to number fluctuations (due to clusters entering and exiting the scattering volume), which can corrupt the analysis of the intensity autocorrelation function. Gravity settling has been invoked to explain similar observations of the long time aggregation behavior of colloidal suspensions in several studies.38-40

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Figure 3. log-log plots of the calculated stability ratio as a function of NaCl concentration: (O) from the master curve; (2) W′ ) kfast/k. The insets show the measured slope (k) of the initial growth rate, normalized by the values at the highest salt concentration ([) as a function of NaCl concentration. The lines are the theory with constant charge (s) and constant potential (- - -).

The following equation (from ref 41), based on a comparison of the diffusion and sedimentation time scales, can be used to estimate the average cluster radius when sedimentation will begin to strongly influence the aggregation process

RDLS,S )

(

RDLS 3kBTφ1/3 4 Rg 4πa ∆Fg

)

3/4df

≈ 1.129

(

3kBTφ1/3

)

3/4df

4πa4∆Fg

(12)

where ∆F is the difference between the particle and solvent densities. For the r300, r200, r150, and r100 particles, eq 12 yields RDLS,S/a ) 4.8 ( 1.6, 9.4 ( 3.8, 12.3 ( 4.8, and 27.3 ( 11.6, respectively. Other, similar derivations in (38) Carpineti, M.; Giglio, M. Phys. Rev. Lett. 1993, 68, 3327. Carpineti, M.; Giglio, M. Adv. Colloid Interface Sci. 1993, 46, 73-90. (39) Gonzalez, A. E.; Ramirez-Santiago, G. Phys. Rev. Lett. 1995, 74, 1238-1241. (40) Bremer, L. G. B.; Bijsterbosch, B. H.; Walstra, P.; Vliet, T. van Adv. Colloid Interface Sci. 1993, 46, 117-128. (41) Allain, C.; Cloitre, M.; Wafra, M. Phys. Rev. Lett. 1995, 74 (8), 1478-1481. Allain, C.; Cloitre, M. Adv. Colloid Interface Sci. 1993, 46, 129-138.

the literature (refs 38 and 40) lead to slightly different values but yield the same trend: the influence of sedimentation increases with the single particle radius a. The arrows in Figure 2 show the calculated points where sedimentation becomes important for the various particles. This comparison suggests the deviations from the fractal growth model at long times can be ascribed to sedimentation. These results are not sensitive to the method of analysis of the ACF. This was demonstrated by analyzing the slow aggregation r300 particles at a NaCl concentration of 1.142 M over a total aggregation time of 24.2 h. Good agreement for the cluster sizes was obtained via different modeling approaches: single exponential16 and quadratic cumulants.20 Further, these results are not sensitive to the particle concentration, as long as it is dilute. Identical aggregation studies were performed on the r200 particles but at a lower particle concentration of φ ) 3 × 10-5. The results are qualitatively similar, only the absolute aggregation rate is slower for the lower particle concentration. The theoretical dependence in eq 5 for tp is tp ∝ 1/φ, such that lowering the volume fraction by a factor of 10/3 should result in a slowing down of the aggregation kinetics


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Figure 4. Plots of the dimensionless aggregation diameter as a function of NaCl concentration for several aggregation times, as indicated. The concentration where the aggregate size growth plateaus indicates the CCC, as identified by the arrows.

(i.e., increasing tp) by 10/3. These data are contained in the master curve, Figure 2, demonstrating the expected behavior. Further analysis along the lines to be discussed next also demonstrated that the CCC is independent of the particle concentration in this concentration range. Having established experimentally the commonality of DLCA for our particles and the range of dimensionless aggregation time over which DLCA is the dominant mechanism, we turn our attention to comparing the results of this investigation to quantitative theoretical predictions for the stability ratio. The integration of eq 13 requires identification of all the parameters appearing in the DLVO interaction potential, eq 10. The only significant unknown in eq 10 is the value of the electrostatic pair potential at the Stern layer, ψ. This is determined both directly and self-consistently from the measured stability (CCC), as well as from independent measurements of the particles’ electrophoretic mobility, as follows. The CCC was extracted from the experiments by different methods for comparison.9,30 As noted, the stability ratios derived in forming the master curve Figure 2 were analyzed to determine the CCC. Equivalently, plots of the log of an estimate of the stability ratio given by log(RDLS/a - 1) as a function of the log of the NaCl concentration also show an inflection point, which can be denoted as the CCC. Note that this method does not require the formation of a master curve as it uses directly the instantaneous measurements of the aggregate size. However, because our aggregation data all fall onto a master

curve, this method is equivalent to using the stability ratios determined by constructing our master curve. Figure 3 shows the results of estimating the stability ratio (W′ ) kfast/k) and CCC30,31 from the initial aggregation rates k relative to the results obtained from Figure 2. The rates k (shown in the insets in Figure 3) were estimated by fitting the initial slope (for the first 10-40 min depending on the aggregation rate)31 of the measured average particle size versus time (Figure 1). We used the rate k calculated at high salt (here, 1.8 M) to estimate kfast. As in the previous method, the point of inflection in the plots provides an estimate of the CCCs. The final method of determining the CCC is to monitor the particle size as a function of added salt at a specified aggregation time.9 Figure 4 shows the measured, dimensionless aggregation diameter as a function of NaCl concentration for a selection of aggregation times. The salt concentration where the aggregate size growth plateaus is generally taken to be the CCC. The arrows in Figure 4 indicate our estimates of the CCC. Table 3 compares the values of the CCC determined by each method. As shown, for the data measured and presented here, all three methods yield CCC values in good agreement. Given the ability to formulate a master curve for all four particle sizes at all salt concentrations, we judge the first method to be superior to the others and use those values in what follows. It is encouraging, however, that measurements of the initial rates are in good agreement. The latter method, i.e., identifying the

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Figure 5. Zeta potentials (ζ) derived from PALS electrophoresis measurements as a function of NaCl concentration for the acrylic latex suspensions ([). The thick lines show the model fit (eq 12) assuming a constant particle charge. The dashed lines show the constant surface potentials ψS obtained by fitting the DLVO theory (eq 10 with ∆ ) 0 nm) to the experimentally observed CCCs. The thin lines are the measured conductivity, for reference. Table 3. Critical Coagulation Concentrations (CCCs) (in M) Determined from DLS by Various Analyses, As Noteda method

r300

r200

r150

r100

master curve W′ (short time kinetics) Figure 4 average

1.143 1.140 1.143 1.142

0.890 0.890 0.895 0.892

0.850 0.850 0.850 0.850

0.750 0.750 0.750 0.750

a The uncertainty in the CCC values is estimated to be less than 0.05 M.

salt concentration beyond which the aggregation rate is relatively constant, is clearly the least satisfactory of the different methods. Figure 5 shows the zeta potentials measured by PALS as a function of NaCl concentration, along with the measured conductivities. All ζ values were calculated using eq 11 with f ) 1 (valid for κa . 1) except the ζ values measured at near zero NaCl concentration, which were calculated using f ) 3/2. With increasing salt concentration, the conductivity increases (as expected) and the zeta potentials decrease. The lines show the model fit (eq 12) of the zeta potentials assuming a constant surface charge density. The quality of the fit suggests that these acrylic latex particles exhibit a constant surface charge density independent of the added NaCl concentration at constant pH. This is in contrast to expectations for particles stabilized with weak acids, where charge regulation would be expected.42 This observation enables extrapolating the

zeta-potential values to NaCl concentrations higher than measurement permits and, thus, comparing directly with the values of particle surface charge density determined from the observed CCCs. However, ambiguities in comparing the surface potential derived from CCC measurements to the zeta potential derived from electrophoretic mobilities include determining the location of the Stern layer and assigning the zeta potential to the potential at the Stern layer. In the following, we examine this by extracting the potential from DLVO theory and the experimentally observed CCCs. Table 4 compares the values of the Stern layer potentials determined from the measured CCC values using the DLVO eq 10 for different estimates of the Stern layer thickness D to that extrapolated from the fits of the constant charge model to the measured electrophoretic mobilities (Figure 5). Here we assume, like many others, that the zeta potential is equivalent to the value of the potential at the Stern layer. Reasonable agreement is achieved between the PALS-based and DLVO-predicted particle charges (except for the 300 nm particles) using the hydrodynamic Na+ ion size of 0.18 nm,3 which is intermediate between the estimates of the size of a bare Na+ ion (0.096 nm) and the fully hydrated species (0.372 nm).24,25 Consequently, we use the particle surface charges (42) Gisler, T.; Schulz, S. F.; Borkovec, M.; Sticher, H.; Schurtenberger, P.; D’Aguanno, B.; Klein, R. J. Chem. Phys. 1994, 101, 99249936


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Table 4. CCCs, Particle Charge Densities, and Particle Potentials for the Acrylic Latex Particles from DLVO theory from PALS-measured ζ

∆ ) 0 nm

∆ ) 0.18 nm

∆ ) 0.37 nm

label

from DLS CCC (M)

ζa at CCC (mV) eq 12

qPALSb (µC/cm2) eq 12

qDLVOc (µC/cm2) eq 12

ψSd (mV) eq 10


ψd (mV) eq 10


ψd (mV) eq 10

r300 r200 r150 r100

1.142 ( 0.1 0.892 ( 0.1 0.850 ( 0.1 0.750 ( 0.1

10.3 ( 4.3 18.7 ( 9.9 19.9 ( 9.2 15.6 ( 8.1

2.6 ( 1.1 4.2 ( 2.3 4.4 ( 2.1 3.2 ( 1.7

10.4 ( 4.3 8.4 ( 4.3 8.1 ( 3.9 7.2 ( 3.8

38.1 ( 13.4 35.4 ( 15.5 34.9 ( 14.6 33.4 ( 15.5

5.0 ( 2.1 4.4 ( 2.2 4.3 ( 2.1 4.0 ( 2.1

19.6 ( 7.8 19.7 ( 9.5 19.7 ( 9.1 19.5 ( 9.8

2.5 ( 1.0 2.4 ( 1.2 2.4 ( 1.1 2.3 ( 1.2

9.9 ( 4.1 10.8 ( 5.4 10.9 ( 5.2 11.2 ( 5.8

a Extrapolated (using eq 12 with q PALS and assuming ζ ) ψ) value of the PALS-measured ζ at the experimentally observed CCC (column 2 of the table). b Surface charge per unit area (q ) Q/4πa2) based on PALS measurements of ζ as function of NaCl concentration (Figure 5) calculated using eq 12 with the assumption ζ ) ψ. c Particle charge per unit area calculated using eq 12 and ψ for ∆ ) 0, 0.18, and 0.37 nm. d ψ value from DLVO theory (eq 10 with ∆ ) 0, 0.18, and 0.37 nm) that gives the experimentally observed CCC (column 2). Note, the average ψ values (not shown) for ∆ ) 0.096 nm are 26 mV.

Figure 6. Plots of the DLVO interaction energies for the r300, r200, r150, and r100 particles as a function of increasing NaCl concentration (ranging from 0.4 to 1.8 M) using a constant surface potential ψS, derived by fitting the experimentally observed CCCs with DLVO theory (eq 10 with ∆ ) 0 nm).

obtained using DLVO theory with ∆ ) 0.18 nm to calculate the stability ratios for comparison to the values obtained from the master curve. It is evident that assigning the zeta-potential measured at low salt concentrations to the surface potential relevant for calculating the CCC will result in a great overestimation of the true CCC, as has been reported previously for other systems.43 The error estimates reported in Table 4 (and everywhere else in the paper unless otherwise noted) were calculated using (43) Horn, F. M.; Richtering, W.; Bergenholtz, J.; Willenbacher, N.; Wagner, N. J. J. Colloid. Interface Sci. 2000, 225, 166-178.

propagation of errors, with the error in the radius estimated from the polydispersity obtained from DLS quadratic cumulants. Figure 6 shows the DLVO interaction energies (eq 10 with ∆ ) 0 nm) calculated for a range of NaCl concentrations for all four particle sizes. In constructing these plots, a constant surface potential was assumed using the ψS values from Table 4, column 6. Similar peak positions (h values) are obtained if a constant particle charge is assumed (Table 4, column 5 or 7 for different ∆ values), but the peak heights differ. Of importance here is the

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such that the DLVO theory would be expected to fail within ≈2 nm of the surface.5,6 Remarkably, our measurements demonstrate that good agreement can be achieved down to ≈0.3 nm, whereupon we observe discrepancies consistent with others in the literature. A number of hypotheses are postulated to explain this failure, including particle roughness and finite molecular sizes.5,6 The ability of the DLVO theory to describe the potential for our acrylic latex particles to distances approaching 0.3 nm, on the scale of the adsorbed counterions, is remarkable and strongly suggests that surface roughness or nonuniform surface charge45 may play a role in reports that show deviations at much larger separations (≈2 nm). The particles employed here are of lower Tg (60 °C) than the polystyrene (PS) used by Behrens et al.5,6 and are synthesized at T ≈ Tg + 20 °C. Thus, the particles used here should be substantially smoother than the PS or many of the oxide particles used in previous studies. During synthesis, plasticization by the monomer and elevated reaction temperatures will keep our particles fluid until polymerization is complete. Further, all surfactant is removed by extensive dialysis and there are no detectable oligiomeric species in the dispersion. Also, the particles studied here have a low surface charge density and should not exhibit any surface segregation or extended hairs46,47 that are typical of many other particle systems. Finally, PS is more hydrophobic than the particles examined here, and this may also play a subtle role in explaining the greater discrepancies seen in the literature for PS latex dispersions. Thus, we conclude that for sufficiently smooth, uniform, hydrophilic particles, the DLVO potential with the nonlinear analytical solution for a constant potential boundary condition can provide accurate predictions for colloidal stability when the stability barrier is greater than ∼0.3 nm from the particle surface. At this time, we can only speculate that the failure of the continuum model when the stability barrier is within twice the Stern layer thickness (≈∆ ) 0.18 nm) is a consequence of finite ion size effects.48 In fact, if one quantifies poor agreement as a deviation from the measured W by more than a factor of 2, then inspection of Figure 3 and Figure 6 indicates that, for all particle sizes, failure is accompanied by stability barrier peak positions within ≈0.3 nm of the surface. This is, of course, an estimate but, nonetheless, is on the order of the hydrated sizes of the ions in solution and adsorbed onto the surface. Note that the deviations at lower salt concentrations are such that the particles are always less stable than predicted. Recent simulations48 accounting for the finite size of the counterions shows the possibility of an attraction induced by depletion of the ions, which also has a length scale on the order of the ion size. Thus, attraction due to ion depletion is a possible explanation consistent with the experimental evidence.

proximity of the stability barrier to the colloidal surface; for all four particle sizes the stability barrier is within 0.4 nm of the particle surface, with the barrier position moving closer with increasing particle radius. This draws into question the accuracy of the continuum DLVO model at such short distances, as previous studies (e.g., Behrens et al.5,6), surface force apparatus (SFA) measurements,44 and theoretical considerations suggest that the continuum model should break down for distances much larger (≈2 nm) than those considered here (≈0.3 nm). This failure is often judged by the inability of the DLVO model to accurately predict the stability ratio. The lines in Figure 3 are stability ratios W predicted from the Smoluchowski theory (eq 13) including the pair hydrodynamics and using the DLVO theory (eq 10). For reference, we also calculated the Prieve and Ruckenstein approximation3 (eq 14) with W∞ set by the value calculated from the full integral expression (eq 13) at the highest salt concentration. On the scale of Figure 3, the Prieve and Ruckenstein estimates of W are nearly identical to the full, exact numerical calculation and so they are not shown for clarity. In calculation of the stability ratios, two cases are considered as inputs for the salt-dependent particle potential: constant surface potential with ∆ ) 0 nm (Table 4, column 6) and constant particle charge with ∆ ) 0.18 nm (Table 4, column 7). As seen in Figure 3, for the 200, 150, and 100 nm particles, the use of a fixed surface potential yields an excellent prediction for the stability ratios, except at the very lowest salt concentrations, where the stability is overestimated. Assuming a constant particle charge density results in a systematic overestimation of the stability ratio at lower salt concentrations. The successful prediction of stability ratios suggests that the constant potential model accurately describes the electrostatic repulsion, at least down to separations approaching 04 nm. This is inconsistent with the fit of the zeta potentials obtained from PALS measurements (Figure 5) using the constant particle charge density model. The difficulty relating mobility measurements to surface or Stern layer properties (potential or charge) is a possible explanation for this discrepancy. Poorer agreement is obtained for the largest particles (300 nm), where the theory greatly underestimates the stability at higher salt but seriously overpredicts stability at lower salt concentrations, for either constant particle charge or constant surface potential. Referring to Figure 3, it is evident that for the 300 nm particles, there is a systematic deviation (by a factor of ≈2) between the values of W (at high salt) obtained from the master curve (Figure 2) and those obtained directly from the initial aggregation rates. The latter are in good agreement with the theoretically predicted values, although both measurements show W diverges more slowly than predicted at lower electrolyte concentrations. The good agreement observed between our predictions of the stability ratio and the CCC for the smaller three particle sizes and the visible inaccuracy for the larger, 300 nm particles is qualitatively consistent with the recently published conclusions of Behrens et al.5,6 The source of this discrepancy was determined to be the proximity of the stability barrier to the particle surface,

We have demonstrated that the fast aggregation kinetics of dilute dispersions of a series of model, nearly monodisperse acrylic latex particles of varying size at fixed pH can be reduced to a master curve that follows DCLA kinetics with df ) 1.86 over a range of 2 < t/tp < ≈100 where the upper limit depends on the single particle radius

(44) Israelachvilli, J. N.; Adams, G. E. Nature 1976, 262, 774. Pashley, R. M. J. Colloid Interface Sci. 1980, 80, 153. Pashley, R. M. J. Colloid Interface Sci. 1981, 83, 531. Pashley, R. M.; Israelachvilli, J. N. Colloids Surf. 1981, 2, 169. Pashley, R. M.; Israelachvilli, J. N. Nature 1983, 306, 249. Christenson, H. K.; Horn, R. G. Chem Phys. Lett. 1983, 98, 45. Horn, R. G.; Clarke, D. R.; Clarkson, M. T. J. Mater. Res. 1988, 3, 413.

(45) Walz, J. Y. Adv. Colloid Interface Sci. 1998, 74, 119-168. (46) Nabzar, L.; Duracher, D.; Elaissari, A.; Chauveteau, G.; Pichot, C. Langmuir 1998, 14, 5062-5069. (47) Peula, J. M.; Fernandez-Barbero, A.; Hidalgo-Alvarez, R.; de las Nieves, F. J. Langmuir 1997, 13, 3938-3943. (48) Allahyarov, E.; D’Amico, I.; Lo¨wen, H. Phys. Rev. Lett. 1998, 81, 1334-1337.

Conclusions


a. This limiting value is observed to decrease with increasing particle radius and is most likely due to sedimentation. Further, the measured master curve for colloidal aggregation follows the predicted dependence on particle size and concentration. Good predictions are observed for the stability ratios for the three smaller particle sizes when a constant surface potential is fit to yield the correct CCC. Plots of the DLVO potentials demonstrate that discrepancies (specifically overprediction of the stability) occur when the stability barrier peak is located within ≈0.3 nm from the particle surface. This result is much closer to the surface than that obtained in the recent work 5,6 (≈2 nm) on polystyrene carboxyl and hematite latices. We postulate that this breakdown at ≈0.3 nm may be evidence for finite ion size effects.48 The CCC cannot be predicted accurately from zeta potential measurements performed at low salt concentrations. However, fitting a series of measurements at varying electrolyte concentration to a constant charge model yielded values in reasonable agreement with those derived from the CCC measurements when the Stern layer was adjusted to ∆ ) 0.18 nm, which is the Stokes radius for the counterion Na+ (corresponding to a partially hydrated Na+ ion state). This observation is consistent with other reports in the literature suggesting that surface potentials derived from electrophoretic mobility measurements at low electrolyte concentrations systematically overpredict the CCC.

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The quality of the particles investigated here mitigated artifacts associated with surface asperities, chemical inhomogeneities, and possible hydrophobic effects leading to the validation of a quantitative master curve for DLCA and agreement with theoretical predictions. The results suggest that finite ion sizes resulting in depletion interactions may be amenable to study with such dispersions. Of interest to industrial applications and our future work on this problem is the applicability of these results to predict aggregation in concentrated dispersions (volume fractions on the order of 10-50%), such as used in ceramics, coatings, and paints, where the aggregation kinetics and microstructure will affect the final mechanical and optical properties. Acknowledgment. We gratefully acknowledge the financial support of DuPont Marshall Laboratory, the Delaware Research Partnership, and the NSF REU Program (EEC-9820322). We also acknowledge Dr. Patricia Lesko (formerly of the Rohm & Haas Corporation) for providing the model latices used in this research. We thank Dr. Dirk Eck (University of Mainz, Germany), Dr. Robert Butera (DuPont Marshall Laboratory), Dr. Tom Wilson (also formerly of Rohm & Haas Corporation), and Dr. James Brady (Hercules, Inc.) for their helpful input and advice. Finally, this paper benefited from careful reviewing, and we thank the reviewers for their effort. LA000927C

Electrolyte-Induced Aggregation of Acrylic Latex. 1. Dilute Particle

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