Langmuir 2002, 18, 7853-7860
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Absolute Heteroaggregation Rate Constants by Multiangle Static and Dynamic Light Scattering W. L. Yu and E. Matijevic´ Center for Advanced Materials Processing and Department of Chemistry, Clarkson University, Potsdam, New York 13699
M. Borkovec* Department of Inorganic, Analytical, and Applied Chemistry, University of Geneva, Sciences II, 30 Quai Ernest-Ansermet, 1211 Geneva 4, Switzerland Received April 9, 2002 The analysis of early-stage heteroaggregation (or heterocoagulation) in binary colloidal systems composed of oppositely charged latex particles of different size in the submicrometer size range with multiangle static and dynamic light scattering is presented. The solution conditions were adjusted to exclude any significant homoaggregation. The apparent rate obtained from static light scattering mostly strongly decreases with increasing scattering angle regardless of the number fractions, while the rate from dynamic light scattering varies in a more complicated manner. The light scattering data could be interpreted quantitatively on the basis of the Rayleigh-Gans-Debye approximation. The values of absolute heteroaggregation rate constants were found to be the same within experimental error when evaluated from static or dynamic light scattering, and they were independent of the mixing ratio. The average hydrodynamic radius of the doublet obtained from the dynamic light scattering was in good agreement with theoretical estimates based on an exact hydrodynamic treatment at low Reynolds numbers. A simple formula is proposed to estimate the hydrodynamic radius of the asymmetric particle doublet, and this formula is shown to agree well with experimental data and with theory. The new conclusion from this study is that multiangle dynamic light scattering represents the method of choice for the determination of absolute heteroaggregation rate constants.
1. Introduction Stability of colloidal dispersions is an essential aspect of the science of fine particles. While much fundamental research has been focused on homoaggregation processes,1,2 that is, on interaction of like particles, fewer studies dealt with dispersions containing different kinds of particles, that is, heteroaggregation. However, the latter is generally of much greater relevance in many applications and in natural environments. Furthermore, the studies of heteroaggregation are more challenging from both experimental and theoretical points of view. In most studies of the kinetics of heteroaggregation at an early stage, an averaged stability ratio was reported,3-6 which is a poorly defined quantity, yet a useful one in practical applications. Heteroaggregation in various systems, including different metal oxides,4,5 anionic or cationic latexes and alumina,7 silica and latex,8 metal oxides with latex,9 and amphoteric lattices,3 has been so investigated. Recent studies have also dealt with the heteroaggregation at the later stages of the process both theoretically * To whom correspondence should be addressed. Phone: + 41 22 702 6405. Fax: + 41 22 702 6069. E-mail: michal.borkovec@ cabe.unige.ch. (1) Zhang, J. W.; Buffle, J. J. Colloid Interface Sci. 1995, 174, 500. (2) Behrens, S. H.; Christl, D. I.; Emmerzael, R.; Schurtenberger, P.; Borkovec, M. Langmuir 2000, 16, 2566. (3) James, R. O.; Homola, A.; Healy, T. W. J. Chem. Soc., Faraday Trans. 1977, 173, 1436. (4) Wiese, G. R.; Healy, T. W. J. Colloid Interface Sci. 1975, 52, 458. (5) Healy, T. W.; Wiese, G. R.; Yates, D. E.; Kavanagh, B. V. Environ. Sci. Technol. 1967, 1, 647. (6) Maroto, J. A.; de las Nieves, F. J. Colloid Surf., A 1995, 96, 121. (7) Sasaki, H.; Matijevic´, E.; Barouch, E. J. Colloid Interface Sci. 1980, 76, 319. (8) Bleier, A.; Matijevic´, E. J. Chem. Soc., Faraday Trans. 1978, 74, 1346. (9) Kihira, H.; Matijevic´, E. Langmuir 1992, 8, 2855.
and experimentally.10,11 The aggregate structures and the time evolution of their size distribution were investigated in systems composed of oppositely charged particles with similar sizes,11-13 or the adsorption of small particles onto oppositely charged large particles was reported.14 In the latter case, a thin-film, freeze-dry scanning electron microscopy was used to observe the morphology of the adhered layer of the small particles and to determine the particle adsorption isotherms. The effects of the background electrolyte concentration, as well as of the size ratio of the small to large particles, were studied. Another approach to investigate heteroaggregation phenomena is by deposition of particles onto different substrates, which can be analyzed theoretically in a similar fashion. A popular approach is the chromatographic packed column technique,15-17 in which collector beads, much larger than the adhering particles, represent the substrate. Alternatively, one can follow the deposition process on a planar substrate by reflectometry18 or direct imaging.19-21 Such methods allow for not only the study of particle attachments on the surface of different parti(10) Shih, W. Y.; Shih, W. H.; Aksay, I. A. J. Am. Ceram. Soc. 1996, 79, 2587. (11) Stoll, S.; Pefferkorn, E. J. Colloid Interface Sci. 1993, 160, 149. (12) Kim, A. Y.; Berg, J. C. J. Colloid Interface Sci. 2000, 229, 607. (13) Ferna´ndez-Barbero, A.; Vincent, B. Phys. Rev. E 2001, 6301, 1509. (14) Harley, S.; Thompson, D. W.; Vincent, B. Colloid Surf., A 1992, 62, 163. (15) Elimelech, M. J. Colloid Interface Sci. 1994, 164, 190. (16) Kihira, H.; Ryde, N.; Matijevic´, E. Colloid Surf., A 1992, 64, 317. (17) Kretzschmar, R.; Barmettler, K.; Grolimund, D.; Yan, Y. D.; Borkovec, M.; Sticher, H. Water Resour. Res. 1997, 33, 1129. (18) Bo¨hmer, M. R.; van der Zeeuw, E. A.; Koper, G. J. M. J. Colloid Interface Sci. 1998, 197, 242. (19) Semmler, M.; Mann, E. K.; Ricka, J.; Borkovec, M. Langmuir 1998, 14, 5127.
10.1021/la0203382 CCC: $22.00 © 2002 American Chemical Society Published on Web 08/30/2002
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cles but also particle removal, which corresponds to the peptization process. Further studies have focused on the early stages of heteroaggregation in binary colloidal systems composed of latex particles of similar sizes.22,23 Since aggregation kinetics is conveniently followed by light scattering,3,4,9,11,12,24,25 choosing particles of similar scattering power greatly simplifies the interpretation of data. It appears that only recently was a direct attempt made to measure the heteroaggregation rate constant in a system composed of oppositely charged latex particles of different size.26 The novel multiangle static and dynamic light scattering has emerged as a powerful tool to probe particle aggregation processes in situ. The ability to obtain the aggregation rate constant in homoaggregation processes has been successfully demonstrated.27,28 In the present study, we use this technique to investigate the early stage of heteroaggregation in systems composed of oppositely charged latex particles of different size. In contrast to previous studies, from the present analysis of timedependent multiangle static and dynamic light scattering data, one can obtain absolute aggregation rate constants for heteroaggregation. The consistency between the results from static and dynamic light scattering as well as for different number fractions of particles indicates the reliability of the method. In the present study, the solution composition is chosen such that any significant homoaggregation between like particles can be excluded. The more general situation, where homoaggregation and heteroaggregation occur simultaneously, is currently under investigation and will be addressed elsewhere. 2. Theory If heteroaggregation occurs exclusively at the early stages in a binary colloidal system composed of primary particles A and B, the reaction reads
A + B f AB
suspension is a time-dependent quantity and is further a function of the magnitude of the scattering vector b q. Its magnitude is given by q ) (4π/λ) sin(θ/2), where λ is the wavelength of the light in the sample and θ is the scattering angle. In a dilute system, the scattering intensity is given by the sum of three contributions
I(q,t) ) IA(q) NA(t) + IB(q) NB(t) + IAB(q) NAB(t) (3) where IA(q), IB(q), and IAB(q) are the static light scattering intensities of monomers A and B and of the doublet AB, respectively. At the early stage of aggregation, the relative change in the static light scattering intensity can be obtained by differentiating eq 3 with respect to time and combining it with eq 2. Taking the limit t f 0 yields
|
1 dI(q,t) I(q,0) dt
dNA(t) dNB(t) dNAB(t) ))) kABNA(t) NB(t) (2) dt dt dt where NA(t) and NB(t) are the particle number concentrations of A and B, respectively, NAB(t) is the number concentration of the doublet AB, and kAB is the heteroaggregation rate constant. In the present study, we focus only on the early stage of heteroaggregation and do not consider any aggregates higher than doublets. 2.1. Static Light Scattering. The intensity of the scattered light I(q,t) originating from an aggregating (20) Lu¨thi, Y.; Ricka, J.; Borkovec, M. J. Colloid Interface Sci. 1998, 206, 314. (21) Wit, P. J.; Poortinga, A.; Noordmans, J.; van der Mei, H. C.; Busscher, H. J. Langmuir 1999, 15, 2620. (22) Puertas, A. M.; Fernandez-Barbero, A.; De Las Nieves, F. J. J. Chem. Phys. 2001, 114, 591. (23) Maroto, J. A.; de las Nieves, F. J. Colloid Surf., A 1998, 132, 153. (24) McLaughlin, W.; White, J. L.; Hem, S. L. J. Colloid Interface Sci. 1993, 157, 113. (25) Sunkel, J. M.; Berg, J. C. J. Colloid Interface Sci. 1996, 179, 618. (26) Ryde, N.; Matijevic´, E. J. Chem. Soc., Faraday Trans. 1994, 90, 167. (27) Holthoff, H.; Borkovec, M.; Schurtenberger, P. Phys. Rev. E 1997, 56, 6945. (28) Holthoff, H.; Egelhaaf, S. U.; Borkovec, M.; Schurtenberger, P.; Sticher, H. Langmuir 1996, 12, 5541.
kABN0xAxB[IAB(q) - IA(q) - IB(q)] IA(q)xA + IB(q)xB
tf0
(4)
in which I(q,0) ) IA(q) NA(0) + IB(q) NB(0), N0 is the initial total particle number concentration, and xj ) Nj/N0 are the number fractions of particle j where j ) A, B and xA + xB ) 1. Within the Rayleigh-Gans-Debye (RGD) approximation, the scattered intensity I(q) by an object is given by the integral of the phase shift over the particle volume V. This term is proportional to the particle volume squared29
I(q) ∝ |
∫Ve-iqb‚rb d3r|2 ∝ V2P(q)
(5)
where i is the imaginary unit. The angular dependence is described by the form factor P(q), which is defined such that P(0) ) 1. For a homogeneous sphere of species j with radius rj, the form factor is29
Pj(q) )
(1)
The concentrations of all the components in such an aggregating suspension are time dependent and can be described as
)
{
}
3 [sin(qrj) - qrj cos(qrj)] (qrj)3
2
(6)
For a doublet composed of two spherical particles with radii rA and rB, evaluation of eq 5 yields the scattered light intensity26
IAB(q) ) IA(q) + IB(q) + 2xIA(q) IB(q)
sin[q(rA + rB)] q(rA + rB)
(7)
Substituting eqs 5 and 7 into eq 4 yields26
|
1 dI(q,t) I(q,0) dt
tf0
) kABN0F
(8)
where
F)
2xAxBrA3rB3xPAPB sin[q(rA + rB)] rA6PAxA + rB6PBxB
q(rA + rB)
(9)
This equation reveals a proportionality relation between the apparent rate obtained from static light scattering and the factor F. The prefactor yields the aggregation rate constant kAB. Note that this relation only applies within the RGD approximation. (29) Kerker, M. The Scattering of Light and Other Electromagnetic Radiation; Academic Press Inc.: New York, 1969.
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2.2. Dynamic Light Scattering. In dynamic light scattering from a dilute suspension, analysis of the temporal intensity fluctuations yields the translational diffusion coefficient of the colloidal particles. In an aggregating suspension, this parameter is an average of the diffusion coefficients of different aggregates weighted by their scattered light intensity. For a dilute binary colloidal system in which only doublets AB are formed, the average diffusion coefficient reads
(or to determine) the average diffusion coefficient of the particle doublet DAB, or equivalently its apparent hydrodynamic radius rAB, as defined by the Stokes-Einstein relation (cf. eq 12). This diffusion coefficient can be expressed in terms of the trace of the diffusion tensor D of the particle doublet by
D(q,t) ) IA(q) NA(t)DA + IB(q) NB(t)DB + IAB(q) NAB(t)DAB
The diffusion tensor can be in turn obtained from the generalized Stokes-Einstein relation
DAB ) 1/3 Tr D
IA(q) NA(t) + IB(q) NB(t) + IAB(q) NAB(t)
D ) kTZ-1
(10) where DA, DB, and DAB are the diffusion coefficients of primary particles A and B and of the doublet AB, respectively. Differentiating eq 10 with respect to time, combining with eq 2, and taking the short time limit, one obtains the initial relative change of the average diffusion coefficient
|
1 dD(q,t) D(q,0) dt
[
)
tf0
]
IABDAB - IADA - IBDB IAB - IA - IB kABN0xAxB IADAxA + IBDBxB IAxA + IBxB (11) Instead of reporting a diffusion coefficient D, it is customary to refer to an apparent hydrodynamic radius rh. These quantities are related according to the Stokes-Einstein relation
D)
kT kT ) σ 6πηrh
|
1 drh(q,t) rh(q,0) dt
|
tf0
) kABN0(F - G) (13)
where F is defined by eq 9, and
sin q(rA + rB) 2rA6PA + 2rB6PB + 4rA3rB3xPAPB q(rA + rB)
xAxB
(17)
This problem has been studied at low Reynolds numbers in much detail.30-32 The total force can be obtained from the sum of the individual hydrodynamic forces on each particle. Forces on spherical particles in the presence of a second one at a given distance in a uniform flow field are known to good accuracy. In the present case, we are interested in the case of touching spheres. Specializing the expressions given by Jeffrey and Onishi31 for two spheres in contact, and using eqs 15-17, we express the apparent hydrodynamic radius in terms of the dimensionless quantity
R)
2rAB ) rA + r A φ
-1
6 + 2φY-1
(18)
where
φX )
λ 1 A X + A12X + A X 1 + λ 11 1 + λ 22
(19)
φY )
1 λ A Y + A12Y + A Y 1 + λ 11 1 + λ 22
(20)
and
1 dD(q,t) D(q,0) dt
G)
F ) ZU
X
) tf0
(16)
where Z is the mobility matrix of the doublet. The latter can be obtained by calculating the total drag force F on the doublet in a uniform flow velocity field U from the relation
(12)
where kT is the thermal energy, σ is the particle mobility (friction coefficient), and η is the viscosity of the medium. Substituting this relation in eq 11 and invoking the RGD approximation (cf. eqs 5 and 7), one obtains the initial relative change of the average hydrodynamic radius
(15)
R(rA + rB)(rA5PAxA + rB5PBxB)
-
PArA5 + PBrB5 (14) xAxB 5 (rA PAxA + rB5PBxB) where R ) 2rAB/(rA + rB) is the dimensionless relative hydrodynamic doublet radius, which is determined by the apparent hydrodynamic radius rAB. Again, this relation only applies within the RGD approximation. 2.3. Evaluation of the Hydrodynamic Radius of the Doublet. To extract the aggregation rate constant from the dynamic light scattering data, one has to know
We have introduced λ ) rB/rA and the resistance functions A11j, A12j, and A22j (j ) X, Y), which are tabulated for 0 < λ e 1 (see Tables 2 and 4 in ref 31 or Table 11.1 in ref 32). For λ ) 1, one obtains the result for the symmetric doublet, namely R ) 1.392. Note that this value differs slightly from the previously given value28 of R = 1.38. This minor discrepancy originates from the use of an approximate expression for the mobility functions given in ref 30. As λ f 0, the diffusion of the doublet is dominated by the larger sphere, and R f 2. We can devise a simple approximate formula by realizing that the average diffusion coefficient must be the same when one exchanges the labels of the spheres. To invoke this symmetry fully, we define a dimensionless asymmetry parameter (30) Happel, J.; Brenner, H. Low Reynolds Number Hydrodynamics; Noordhoff International Publishing: Englewood Cliffs, NJ, 1965. (31) Jeffrey, D. J.; Onishi, Y. J. Fluid Mech. 1984, 139, 261. (32) Kim, S.; Karilla, S. J. Microhydrodynamics: Principles and Selected Applications, Butterworth-Heinemann; Stoineham, MA, 1991.
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Figure 1. Comparison of the experimental results for the apparent hydrodynamic radius of the dimer obtained from the dynamic light scattering with theoretical estimates. The dimensionless hydrodynamic radius R is plotted as a function of the asymmetry parameter β (cf. eq 21). The solid line is the exact hydrodynamic calculation, while the dashed line is the simple parabolic relation given in eq 22. The inset shows the entire range of the calculated results.
β)
rB - rA λ - 1 ) rB + r A λ + 1
(21)
with -1 < β < 1. The relative hydrodynamic radius of the doublet R must be therefore a symmetric function of β. The result is plotted in Figure 1. Particularly from the inset, it is evident that a simple parabola can be used as a working approximation
R ≈ 1.392 + 0.608β2
(22)
Figure 1 also shows the experimental data, but they will be discussed in detail later. 3. Experiments 3.1. Instrumentation. The optical properties of dilute dispersions were monitored with a multiangle simultaneous static and dynamic light scattering setup with eight fiber-optic detectors mounted on a goniometer (ALV/CGS8, Langen, Germany). The instrument is fully computer controlled and uses a Krypton ion laser (Innova 301, Coherent, CA) operating at the wavelength 647.1 nm. The fibers are connected to photomultipliers, and their output is fed into digital correlators. The static and dynamic light scattering measurements can thus be performed simultaneously at eight different angles, and the angular resolution can be adjusted by rotating the goniometer during the run. The sample temperature was kept at 25.0 ( 0.1 °C. A phase sensitive light scattering setup (ZetaPALS, Brookhaven, NY) was used to carry out the electrophoretic
mobility measurement. All measurements were carried out at pH 4 and ionic strength 10-4 M. The sample temperature was again kept at 25.0 ( 0.1 °C. 3.2. Materials. Surfactant free polystyrene latex particles were manufactured by Interfacial Dynamics Corporation, Portland, OR. The original particle suspensions were dialyzed with deionized water (Barnstead Easypure UV-UF) until the conductivity decreased to that of the freshly deionized water. The particle concentration of the suspension after dialysis was measured with static light scattering against that of the original suspension of known solid content. The samples used for light scattering were prepared by adding a certain volume of the stock dispersion into an HCl solution of pH 4, resulting in an ionic strength of 10-4 M. The particles were characterized in stable suspensions by static and dynamic light scattering at particle concentrations near 1014 m-3. The RGD approximation did fit the angular dependence of the experimental static light scattering data with properly adjusted particle radii very well (cf. eqs 5 and 6). The resulting values are given in Table 1. In the dynamic light scattering measurements, the average hydrodynamic radii of the monomers were measured at different scattering angles and were found to be essentially constant. The resulting hydrodynamic radii are summarized in Table 1. The results are mutually consistent and in favorable agreement with the information supplied by the manufacturer, as determined by transmission electron microscopy. The best radii chosen for the interpretation of the aggregation data are summarized in Table 1, together with the electrophoretic mobilities of all latex particles at pH 4 and an ionic strength of 10-4 M, and the values of the critical coagulation concentration (CCC). The CCC was determined at pH 4 and in a KCl electrolyte by time-resolved dynamic light scattering, as discussed elsewhere.2,28 The individual particles are perfectly stable with typical stability ratios of at least several hundreds at ionic strengths of 10-4 M at particle concentrations below 1015 m-3. Homoaggregation can be therefore safely excluded in the following experiments. 3.3. Heteroaggregation Experiments. The glass cuvettes used for the light scattering experiments were cleaned in a boiling mixture of H2O2/H2SO4. The appropriate amount of the negatively charged particle suspension was diluted in the HCl solution, and the experiment was then initiated by adding an appropriate amount of the amidine latex suspension. The cell was then sealed, gently shaken, and monitored by light scattering. The total particle concentration was in the range 1013 to 1014 m-3, and the data were accumulated with a time resolution of 25 s for up to 30-60 min. This time is shorter than the typical half-time of the aggregation 1/(kABN0), which lies in the range 1-5 h. Average hydrodynamic radii were obtained by second-order cumulant analysis. The aggregation experiments were
Table 1. Properties of Primary Latex Particles particle batch amidine latex sulfate latex I sulfate latex II carboxyl latex
SLSa
(nm)
67 144 84 105
DLSb
(nm)
67 140 84 105
TEMc (nm)
CVc
best radiusd (nm)
CCCe (M)
mobilityf (m2 V-1 s-1)
60 135 85 110
14 3.3 2.3 4.6
67 142 84 105
0.10 0.25 0.15 0.09
+3.3 × 10-8 -3.9 × 10-8 -3.5 × 10-8 -2.3 × 10-8
a Intensity weighted radius obtained from static light scattering (SLS). The error bar is about (0.5 nm. b Intensity weighted hydrodynamic radius from dynamic light scattering (DLS). Error bar of about (1.0 nm follows from measurements at different scattering angles. c Number weighted radius and coefficient of variation from transmission electron microscopy (TEM) obtained by the manufacturer. d Best radius used in further data analysis. e Critical coagulation concentration (relative error is about 10%). c Electrophoretic mobility (relative error is about 10%).
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Langmuir, Vol. 18, No. 21, 2002 7857 Table 2. Values of Absolute Heteroaggregation Rate Constants, kAB, and of the Relative Hydrodynamic Radii of the Doublets, r kAB (10-18 m3 s-1)
R ) 2rAB/ (rA + rB)
system
SLSa
DLSb
DLSb
amidine latex/sulfate latex I amidine latex/sulfate latex II amidine latex/carboxyl latex
5.96 5.35 6.71
6.03 5.28 6.40
1.48 1.42 1.44
a Values obtained from fitting static light scattering data at all number fractions together. The relative error is about 4%. These values are used for the calculations shown in Figure 2. b Values obtained from fitting dynamic light scattering data at all number fractions together. The relative error is about 1% for the rate constants and 3% for the relative hydrodynamic radius of the doublet. These values are used for the calculation of the RGD approximation in Figures 4.
4. Results and Discussion
Figure 2. Apparent heteroaggregation rate obtained from static light scattering as a function of the scattering angle at different number fractions of amidine latex particles xA at ionic strength 10-4 M and pH 4. Points are experimental data while the solid lines are calculations based on the Rayleigh-GansDebye approximation with parameters summarized in Tables 1 and 2: (a) amidine latex and sulfate latex I; (b) amidine latex and sulfate latex II; (c) amidine latex and carboxyl latex.
followed only during the early stages of the aggregation process. It was verified that the apparent hydrodynamic radius extrapolated to time zero was in close agreement with the values expected for a nonaggregating suspension. The aggregation data were only used when the apparent hydrodynamic radius did not increase more than 25% of its initial value. No observable effects were caused by the possible adhesion of the amidine latex particles onto the glass wall, as checked by rinsing of the cuvettes with concentrated amidine suspensions prior to the experiment.
The kinetics of heteroaggregation between oppositely charged latex particles was studied in the early stages by multiangle time-resolved simultaneous static and dynamic light scattering. We shall first discuss the apparent rate and the absolute heteroaggregation rate constant determined by time-resolved multiangle static light scattering, and then we shall focus on the corresponding time-resolved multiangle dynamic light scattering data. Three types of binary mixtures were investigated, and the properties of the individual particles used are summarized in Table 1. In all experiments, the ionic strength was 10-4 M and the pH was adjusted to 4 by adding HCl. Under these conditions, only heteroaggregates are forming, and no homoaggregation takes place. 4.1. Static Light Scattering. Figure 2 shows the apparent rate obtained from static light scattering at short times for the three aggregating heterosystems at three different number fractions of the amidine latex particles, namely at xA ) 0.5, 0.7, and 0.9. The apparent rate corresponds to the initial relative rate of change of the scattering intensity (cf. eq 8) and is plotted as a function of the scattering angle. The experimental data are compared with calculations based on the Rayleigh-GansDebye (RGD) approximation. The calculations use eqs 6-9 and need only two types of parameters as input: the radii of the individual particles rA and rB, which were determined independently (see Table 1), and the aggregation rate constants. The latter were obtained by fitting the static light scattering data and are summarized in Table 2. For example, the value used in Figure 2a is kAB ) 5.96 × 10-18 m3 s-1. The fitting procedure is discussed further below. The results shown in Figure 2 reveal that, regardless of the number fraction and the natures of the components, for small scattering angles the apparent rate is positive and decreases with the increasing scattering angle. In other words, while the scattering intensity increases as aggregation proceeds, this increase is most pronounced for small scattering angles. At one particular scattering angle, the apparent rate goes through zero. Within the RGD approximation, the apparent rate is zero at the scattering angle, when the condition (rA + rB)q ) kπ is satisfied, where k is an integer (cf. eqs 8 and 9). As long as RGD remains valid, only k ) 1 is usually of relevance. Higher order zeros (k > 1) become visible only for larger particles, but the RGD approximation fails in this regime.27 In the system shown in Figure 2a, the critical scattering angle is θ ) 71°. At this particular angle, the scattered light intensity remains constant during the aggregation process. At θ < 71°, the scattering
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intensity increases with time during the aggregation process, as revealed by the positive value of the apparent rate constant, while it decreases at θ > 71°. In two other systems, the values of the critical scattering angles are found to be θ ) 107° (Figure 2b) and θ ) 90° (Figure 2c). In summary, Figure 2 shows that, with the same amidine latex, the value of the critical scattering angle increases with the decreasing size of the negatively charged particles used in the system. The number fraction also affects rather significantly the magnitude of the apparent rate. For example, for the data shown in Figure 2a, the largest rate is obtained for the number fraction of the amidine latex of 0.9 at scattering angles < 71°. Within this angular range, the magnitude of the apparent rate decreases with the decreasing number fraction. For the data shown in Figure 2b and c, the apparent rate goes through a maximum with number fraction throughout the scattering angle window. Similar results were reported in a recent heteroaggregation study,22 where the apparent rate observed by static light scattering went also through a maximum with increasing number fraction of one component. The heteroaggregation rate constant in a binary system can be obtained from the static light scattering data by a convenient linearization procedure. From eq 8 it follows that the apparent rate obtained from static light scattering is proportional to the factor F, defined by eq 9. This factor is known, as it only depends on the particle radius and the particle number fraction. When plotting the apparent rate as a function of the factor F, the data collapse on a single straight line with zero intercept. From the slope, the absolute rate constant can be extracted. Figure 3 shows the corresponding linear plots for the three systems with the different number fractions of the components. The results are listed in Table 2. This linearization procedure yields equivalent results to those of a direct nonlinear least-squares fit of the data. This method for determining heteroaggregation rate constants by multiangle static light scattering was proposed for the first time by Ryde et al.26 and used by these authors to measure absolute aggregation rate constants in a similar system composed of oppositely charged particles. 4.2. Dynamic Light Scattering. Figure 4 shows the apparent rate observed by dynamic light scattering at short times for the same three aggregating heterosystems shown in Figure 2 at three different number fractions of amidine latex particles, namely at xA ) 0.5, 0.7, and 0.9 (see also Table 1). The apparent rate is the initial relative rate of change of the average hydrodynamic radius obtained from second cumulant analysis. The experimental data are compared with calculations based on the RGD approximation (cf. eqs 6, 9, 13, and 14). Besides the radii of the primary particles (see Table 1), the values entering the calculations are the heteroaggregation rate constant kAB and the hydrodynamic radius of the doublet rAB, characterized by the dimensionless parameter R ) 2rAB/ (rA + rB). The values used for the calculations were obtained by dynamic light scattering data, as will be discussed shortly, and are summarized in Table 2. For example, in Figure 4a one uses kAB ) 6.03 × 10-18 m3 s-1 and R ) 1.48. Figure 4 reveals that in contrast to the apparent rates determined by static light scattering, which have both positive and negative values, those obtained from the dynamic light scattering are always positive. As one expects intuitively, the average hydrodynamic radius always increases with time. The number fractions affect the dynamic light scattering data in a more complicated manner than that for the static
Yu et al.
Figure 3. Apparent heteroaggregation rate obtained from static light scattering as a function of the parameter F given in eq 9. In this representation, the data for different number fractions collapse on a straight line with zero intercept. Its slope yields the absolute heteroaggregation rate constant. The results are summarized in Table 2: (a) amidine latex and sulfate latex I; (b) amidine latex and sulfate latex II; (c) amidine latex and carboxyl latex.
light scattering data. A typical example is seen in the system composed of amidine latex and sulfate latex I (Figure 4a). At the number fraction of the amidine latex xA ) 0.9, the apparent rate mostly decreases with the increasing scattering angle. At xA ) 0.7, however, a distinct minimum of the apparent rate as a function of the scattering angle appears. At xA ) 0.5, the minimum is shifted to small angles and the signal mostly increases with increasing scattering angle. In the systems shown in Figure 4b and c, the influence of the number fraction
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fitting procedure. In this fashion, we obtain the heteroaggregation rate constant kAB and the relative hydrodynamic doublet radius R. The data for different number fractions were first fitted separately and then simultaneously. The results evaluated from the latter are summarized in Table 2. Although one additional parameter must be extracted from the dynamic light scattering data, the results are highly consistent with the data obtained from static light scattering. By analyzing the errors in detail and by studying the variation of the parameters obtained from the separate fits of the individual data sets for each number fraction, we conclude that the rate constants obtained by the dynamic light scattering technique are in fact more accurate than those obtained from static light scattering. The main reason for this difference is that the time-dependent dynamic light scattering data show a wider range of linearity and are less noisy than the static light scattering data. The values of the relative hydrodynamic radius of the doublet R obtained experimentally by the dynamic light scattering for the asymmetric doublets can now be compared with the exact hydrodynamic calculations at low Reynolds numbers (cf. eqs 15-20). The results are shown in Figure 1, and one can see that the agreement is satisfactory indeed. The experimental values for the symmetric doublet (β ) 0) for particles < 200 nm in radius are taken from previous work.27,33 The approximate parabolic relation given in eq 22 performs equally well. In conclusion, the simple parabolic relation can be used to reliably estimate the hydrodynamic radii of asymmetric doublets. For this reason, one can also invoke this formula to estimate the hydrodynamic radius of the particle doublet and to eliminate this parameter from the data analysis. This approach yields comparable results to those presented above. 5. Conclusion
Figure 4. Apparent heteroaggregation rate obtained from dynamic light scattering as a function of the scattering angle at different number fractions of amidine latex particles xA at ionic strength 10-4 M and pH 4. Points are experimental data while the solid lines are calculations based on the RayleighGans-Debye approximation with parameters summarized in Tables 1 and 2: (a) amidine latex and sulfate latex I; (b) amidine latex and sulfate latex II; (c) amidine latex and carboxyl latex.
on the apparent rate is much simpler. The optical factor decreases with the increasing scattering angle regardless of the number fraction. Thus, the relative increase of the average hydrodynamic radius during the aggregation process, determined at small scattering angles, is more sensitive to the appearance of aggregates than that obtained at large scattering angles. The multiangle dynamic light scattering, which is described by eq 13, cannot be conveniently linearized, and the parameters were directly extracted using a nonlinear
Multiangle time-dependent static and dynamic light scattering was used to study early stages of heteroaggregation between oppositely charged colloidal latex particles. For the sub-micrometer sized particles considered, it was shown that the Rayleigh-Gans-Debye approximation provides an excellent description of the form factors of the particle doublet and thus can be used to extract absolute heteroaggregation rate constants. The rate constants can be extracted from static light scattering, according to a method previously suggested by Ryde et al.26 While the analysis of analogous dynamic light scattering data requires the determination of an additional parameter R (i.e., hydrodynamic radius of the doublet), it turns out that the rate constants can be determined in fact more accurately than those from static light scattering. The aggregation rate constants can also be determined by invoking the simple parabolic relation in eq 22 to estimate the hydrodynamic radius of the doublet. The main conclusion from this work is that timedependent multiangle dynamic light scattering represents the method of choice for the determination of absolute heteroaggregation rate constants in colloidal systems, which are composed of particles of different size. In the present study, homoaggregation was carefully excluded by appropriate choice of solution conditions. In future work, we shall study the applicability of multiangle timedependent static and dynamic light scattering in systems (33) Bouyer, F.; Robben, A.; Yu, W. L.; Borkovec, M. Langmuir 2001, 17, 5225.
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where homoaggregation and heteroaggregation happen at the same time. Acknowledgment. This work was supported by Grants CHE-9870965 and CTS-9820795 of the U.S.
Yu et al.
National Science Foundation and Grants 2100-066514.01 and 2160-067459.02 of the Swiss National Science Foundation. LA0203382
