pubs.acs.org/Langmuir © 2010 American Chemical Society
Application of Asymmetric Flow-Field Flow Fractionation to the Characterization of Colloidal Dispersions Undergoing Aggregation Marco Lattuada,† Carlos Olivo,† Cornelius Gauer,† Giuseppe Storti,‡ and Massimo Morbidelli*,† †
Department of Chemistry and Applied Biosciences, Institute for Chemical- and Bioengineering, Wolfgang-Pauli-Strasse 10, CH-8093 Z€ urich, Switzerland, and ‡Dipartimento di Chimica, Materiali e Ingegneria Chimica “Giulio Natta”, I-20131 Milano, Italy Received November 19, 2009. Revised Manuscript Received January 22, 2010
The characterization of complex colloidal dispersions is a relevant and challenging problem in colloidal science. In this work, we show how asymmetric flow-field flow fractionation (AF4) coupled to static light scattering can be used for this purpose. As an example of complex colloidal dispersions, we have chosen two systems undergoing aggregation. The first one is a conventional polystyrene latex undergoing reaction-limited aggregation, which leads to the formation of fractal clusters with well-known structure. The second one is a dispersion of elastomeric colloidal particles made of a polymer with a low glass transition temperature, which undergoes coalescence upon aggregation. Samples are withdrawn during aggregation at fixed times, fractionated with AF4 using a two-angle static light scattering unit as a detector. We have shown that from the analysis of the ratio between the intensities of the scattered light at the two angles the cluster size distribution can be recovered, without any need for calibration based on standard elution times, provided that the geometry and scattering properties of particles and clusters are known. The nonfractionated samples have been characterized also by conventional static and dynamic light scattering to determine their average radius of gyration and hydrodynamic radius. The size distribution of coalescing particles has been investigated also through image analysis of cryo-scanning electron microscopy (SEM) pictures. The average radius of gyration and the average hydrodynamic radius of the nonfractionated samples have been calculated and successfully compared to the values obtained from the size distributions measured by AF4. In addition, the data obtained are also in good agreement with calculations made with population balance equations.
Introduction The characterization of complex dispersions of colloidal particles is one of the great challenges in colloidal science, with profound implications in several fields such as polymer manufacturing and processing, food science, production of ceramic materials, pharmaceutical and drug delivery applications, wastewater treatments, and so on.1-4 There is no doubt that one of the most important properties of a colloidal dispersion is the particle size distribution (PSD), which affects its rheological behavior, its processability, its stability, as well as the final physical and mechanical characteristics of any material somehow obtained from a dispersion.3
*To whom correspondence should be addressed. E-mail: massimo.morbidelli@ chem.ethz.ch. (1) Hunter, R. Introduction to Modern Colloid Science; Oxford University Press: New York, 2001. (2) Buffle, J.; Leppard, G. G. Environ. Sci. Technol. 1995, 29, 2176–2184. (3) Russel, W.; Saville, D.; Schowalter, W. Colloidal Dispersions; Cambridge University Press: London, 1989. (4) Brittain, H. G.; Bogdanowich, S. J.; Bugay, D. E.; Devincentis, J.; Lewen, G.; Newman, A. W. Pharm. Res. 1991, 8, 963–973. (5) Barth, H. G.; Sun, S. T. Anal. Chem. 1993, 65, R55–R66. (6) Salama, A. I. A.; Mikula, R. J. Suspensions: Fundamentals and Applications in the Petroleum Industry; Schramm, L. L., Ed.; American Chemical Society: Washington, DC, 1996; Vol. 251; pp 45-106. (7) Williams, D.; Carter, C. Transmission Electron Microscopy: A Textbook for Materials Science; Plenum Press: New York, 2009. (8) Goldstein, J.; Newbury, D.; Joy, D.; Lyman, C. Scanning Electron Microscopy and X-ray Microanalysis; Springer Science: New York, 2009. (9) Giddings, J. C. Science 1993, 260, 1456–1465. (10) Finsy, R.; Degroen, P.; Deriemaeker, L.; Gelade, E.; Joosten, J. Part. Part. Syst. Charact. 1992, 9, 237–251. (11) Dinsmore, A. D.; Weeks, E. R.; Prasad, V.; Levitt, A. C.; Weitz, D. A. Appl. Opt. 2001, 40, 4152–4159.
7062 DOI: 10.1021/la904390h
The determination of PSDs is a classical problem, which has led to the development of a multitude of techniques.2,5-16 Microscopy, both optical (often confocal) and electron, is one of the most important techniques developed for this purpose and is widely utilized to determine particle sizes and size distributions.1,7,8,11 While optical and confocal microscopy can only be applied to objects with a size larger than about half a micrometer, electron microscopy allows one to analyze particles with almost atomic resolution. Its main drawback, however, is that it requires dry samples, and drying can severely alter the size, shape, and structure of nanosized objects dispersed in solution. For this reason, several optical techniques have been developed to perform in situ measurements of size distributions.10,13,17,18 Light scattering is the most commonly used optical method to measure the size of nanoparticles. The major drawback of light scattering techniques is that they can reliably only provide average sizes, and in particular z-average sizes, that is, averages calculated by weighting the contribution of each particle by the square of its volume. This leads to average size values strongly affected by the presence of large particles in the system. On top of this, when a mixture of particles having a distribution of shapes (12) Morgan, H.; Hughes, M. P.; Green, N. G. Biophys. J. 1999, 77, 516–525. (13) Yan, Y D.; Clarke, J. H. R. Adv. Colloid Interface Sci. 1989, 29, 277–318. (14) Challis, R. E.; Povey, M. J. W.; Mather, M. L.; Holmes, A. K. Rep. Prog. Phys. 2005, 68, 1541–1637. (15) Haskell, R. J. J. Pharm. Sci. 1998, 87, 125–129. (16) Ehrl, L.; Soos, M.; Morbidelli, M. Part. Part. Syst. Charact. 2007, 23, 438– 447. (17) Wang, J. H.; Hallett, F. R. Appl. Opt. 1996, 35, 193–197. (18) Neutrons, X-rays and Light: Scattering Methods Applied to Soft Condensed Matter; Lindner, P., Zemb, T., Eds.; Elsevier Science: Amsterdam, 2002.
Published on Web 02/09/2010
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in addition to sizes needs to be analyzed, light scattering measurements provide average values that are difficult to interpret quantitatively. In order to overcome these limitations, the single particle light scattering technique has been introduced.19,20 This technique is capable of measuring the light scattering by a single particle, with a very high resolution power. It has in fact been used to investigate the very early stages of coagulation processes.20 Its major limitation is the minimum size of particles that can be detected, which amounts to about 200 nm, but can be even higher for particles with low optical contrast. Additionally, there are to the best of our knowledge no commercial instruments implementing this technique, thus limiting its accessibility. For all these reasons, chromatographic techniques, which are capable of fractionating a sample, have been investigated with growing interest.1,9,17,21-24 Chromatography is nowadays a standard technique in chemistry where gas chromatography and high pressure liquid chromatography, coupled with different spectroscopic and analytical methods, are commonly used to fractionate and analyze mixtures of components.25 In polymer science, gel permeation chromatography is commonly utilized to determine the molecular weight distribution of linear polymers.26 Size exclusion chromatography and gel electrophoresis are methods used to fractionate complex mixtures of components in biology, which are then separately analyzed.26 Similar techniques are not that commonly used in the case of particles. One major problem is that one needs a technique capable of separating particles in a relatively broad range of sizes in order to be of any interest. A few techniques are currently capable of doing this. For example, in centrifuge-based techniques, where centrifugal forces provide the driving force for separation, size is estimated from settling times, and concentration is usually determined by measuring the turbidity of the fractionated samples.21-23 Another set of techniques that are used to fractionate particles are flow-field fractionations (FFFs).9,27 These techniques are based on a flow field which carries the sample into a channel, where some additional field provides the driving force for fractionation. Several fields have been used for this purpose, including thermal gradients, gravitational and centrifugal fields, electric fields, and flow fields.27 Given the aforementioned variety of separation driving forces, FFFs are widely used to analyze (19) Lichtenfeld, H.; Stechemesser, H.; Mohwald, H. J. Colloid Interface Sci. 2004, 276, 97–105. (20) Pelssers, E. G. M.; Stuart, M. A. C.; Fleer, G. J. J. Colloid Interface Sci. 1990, 137, 362–372. (21) Takayasu, M. M.; Galembeck, F. J. Colloid Interface Sci. 1998, 202, 84–88. (22) Rasa, M.; Schubert, U. S. Soft Matter 2006, 2, 561–572. (23) Colfen, H.; Pauck, T.; Antonietti, M. Anal. Ultracentrifugation 1997, 107, 136–147. (24) Small, H.; Langhorst, M. A. Anal. Chem. 1982, 54, A892. (25) Encyclopedia of Chromatography; Cazes, J., Ed.; CRC Press: Boca Raton, FL, 2002. (26) Handbook Of Size Exclusion Chromatography And Related Techniques: Revised And Expanded; Wu, C.-S., Ed.; CRC Press: Boca Raton, FL, 2005. (27) Colfen, H.; Antonietti, M. New Dev. Polym. Anal. I 2000, 150, 67–187. (28) Litzen, A.; Wahlund, K. G. J. Chromatogr. 1989, 476, 413–421. (29) Litzen, A.; Wahlund, K. G. Anal. Chem. 1991, 63, 1001–1007. (30) Litzen, A. Anal. Chem. 1993, 65, 461–470. (31) Fraunhofer, W.; Winter, G.; Coester, C. Anal. Chem. 2004, 76, 1909–1920. (32) von der Kammer, F.; Forstner, U. Water Sci. Technol. 1997, 37, 173–180. (33) von der Kammer, F.; Baborowski, M.; Friese, K. Anal. Chim. Acta 2005, 552, 166–174. (34) Wang, X. B.; Yang, J.; Huang, Y.; Vykoukal, J.; Becker, F. F.; Gascoyne, P. R. C. Anal. Chem. 2000, 72, 832–839. (35) Andersson, M.; Wittgren, B.; Wahlund, K. G. Anal. Chem. 2001, 73, 4852– 4861. (36) Bouby, M.; Geckeis, H.; Geyer, F. W. Anal. Bioanal. Chem. 2008, 392, 1447–1457. (37) Christian, P.; Von der Kammer, F.; Baalousha, M.; Hofmann, T. Ecotoxicology 2008, 17, 326–343. (38) Dubascoux, S.; Von Der Kammer, F.; Le Hecho, I.; Gautier, M. P.; Lespes, G. J. Chromatogr., A 2008, 1206, 160–165.
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polymers, particles, and more recently protein solutions.28-40 The flexibility of FFF techniques is enhanced by combining them with multiangle light scattering detectors, which allow one to perform static light scattering measurements on fractionated and therefore narrowly distributed samples, as clearly shown in the literature.33,41-43 In this work, we use an asymmetric flow field-flow fractionation (AF4) device to characterize aggregating colloidal dispersions of polymeric nanoparticles. Aggregating colloidal dispersions are chosen because they represent the prototype of complex colloidal dispersions, and can produce clusters with wide size distributions and complex morphology, for which however a good knowledge exists on both clusters structure and aggregation kinetics.44-49 Characterization of the fractionated samples is not carried out by lengthy calibrations of the elution times, as it is usually done in the literature, but solely based on the use of a two-angle static light scattering detector coupled to the AF4 device. Two colloidal systems made of electrostatically stabilized particles are analyzed: one standard polystyrene latex, which undergoes reaction-limited aggregation (RLCA) under high ionic strength,44,46 and one latex made of fluorinated elastomer particles, which undergo coalescence upon aggregation. Conventional light scattering measurements, both static and dynamic, are performed on the same sample before fractionation, in order to measure the average gyration and hydrodynamic radii. These average quantities are then compared to the corresponding values calculated from the size distributions measured via the AF4 device. In addition, the size distributions obtained are also compared to the predictions of population balance equation (PBE) calculations and, in the case of elastomer colloids, also to measured size distributions obtained from image analysis of cryo-scanning electron microscopy (SEM) pictures.
Experimental Section Materials. Monodisperse polystyrene particles with an average diameter of 60 nm (surfactant stabilized, sulfate latex, lot no. 30479, coefficient of variation of diameter = 13.3%) were purchased from Duke Scientific. Monodisperse polystyrene particles with a diameter of 143 nm (surfactant stabilized, sulfate latex, product no. Z-PST-POS-001-0,143-8) and 190 nm (surfactant stabilized, sulfate latex, product no. Z-PST-POS021-0,18) were purchased from Postnova Analytics (Germany). Monodisperse polystyrene particles with the following sizes were all purchased from Invitrogen: 120 nm (surfactant free, sulfate latex, product no. 1-100, batch no. 1614,1, coefficient of variation of diameter = 8.4%), 200 nm (surfactant free, carboxyl latex, product no. 7-200, batch no. 2029, coefficient of variation of diameter = 6.0%), 310 nm (surfactant stabilized, sulfate latex, product no. 16-300, batch no. S20-42-23,2, coefficient of variation (39) Pease, L. F.; Lipin, D. I.; Tsai, D. H.; Zachariah, M. R.; Lua, L. H. L.; Tarlov, M. J.; Middel-berg, A. P. J. Biotechnol. Bioeng. 2009, 102, 845–855. (40) Thunemann, A. F.; Rolf, S.; Knappe, P.; Weidner, S. Anal. Chem. 2009, 81, 296–301. (41) Thielking, H.; Roessner, D.; Kulicke, W M. Anal. Chem. 1995, 67, 3229– 3233. (42) Korgel, B. A.; van Zanten, J. H.; Monbouquette, H. G. Biophys. J. 1998, 74, 3264–3272. (43) Wyatt, P. J. J. Colloid Interface Sci. 1998, 197, 9–20. (44) Ball, R. C.; Weitz, D. A.; Witten, T. A.; Leyvraz, F. Phys. Rev. Lett. 1987, 58, 274–277. (45) Meakin, P.; Family, F. Phys. Rev. A 1987, 36, 5498–5501. (46) Lin, M. Y.; Lindsay, H. M.; Weitz, D. A.; Ball, R. C.; Klein, R.; Meakin, P. Phys. Rev. A 1990, 41, 2005–2020. (47) Hidalgo Alvarez, R.; Martin, A.; Fernandez, A.; Bastos, D.; Martinez, F.; delasNieves, F. J. Adv. Colloid Interface Sci. 1996, 67, 1–118. (48) Behrens, S. H.; Christl, D. I.; Emmerzael, R.; Schurtenberger, P.; Borkovec, M. Langmuir 2000, 16, 2566–2575. (49) Lattuada, M.; Wu, H.; Sandkuhler, P.; Sefcik, J.; Morbidelli, M. Chem. Eng. Sci. 2004, 59, 1783–1798.
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Article of diameter = 1.2%), 420 nm (surfactant free, sulfate latex, product no. 1-400, batch no. 1034.2, coefficient of variation of diameter = 4.0%), 600 nm (surfactant free, sulfate latex, product no. 1-600, batch no. 2042,1, coefficient of variation of diameter = 5.2%), and 810 nm (surfactant free, sulfate latex, Product no. 1-800, batch no. 642.4, coefficient of variation of diameter = 2.0%). All the aforementioned standard reference particles have been used only for calibration and test experiments. All polystyrene latexes have a refractive index equal to 1.59. The following two types of particles have been instead used for aggregation experiments. Surfactant-free polystyrene particles with covalently bound carboxyl charges, having a diameter of 92 nm as measured by dynamic light scattering, were purchased from Invitrogen (declared diameter = 90 nm, coefficient of variation of diameter = 11.8%, product no. 7-80, batch no. BM 603, solid % = 4.3, refractive index = 1.59). Fluorinated elastomer polymer nanoparticles, stabilized by a fluorinated ionic surfactant bearing carboxyl groups, with an average diameter of about 120 nm (refractive index equal to 1.37 as reported by the manufacturer) were kindly donated by Solvay Solexis (Italy). The glass transition temperature (Tg) of the polymer material is approximately -20 C, which makes it soft at room temperature. Sodium chloride (NaCl) with 99.5% purity was purchased by J.T. Baker. Sodium dodecyl sulfate (SDS) with 98% purity was purchased from Fluka. H2SO4, with 65% concentration in water, was purchased from Merck. Ion exchange resins, MTO-Dowex Marathon MR-3, were purchased from Supelco. Instruments. Light Scattering Instruments. Static light scattering measurements have been performed using a Brookhaven BI-200SM goniometer equipped with a solid state laser Ventum LP532 from Laser Quantum (U.K.). The wavelength of the laser is 532 nm. Some other dynamic light scattering measurements have been performed with a ZetaNano ZS instrument (Malvern U.K.), which operates in backscattering mode at an angle of 173, with a solid state laser having a wavelength of 633 nm. For cryo-SEM analysis, samples with a volume fraction of 0.009%, containing 8 wt % ethanol as ice protective, were frozen at about -160 C in the high pressure freezing machine HPM 010 from BAL-TEC (Liechtenstein). Sample storage and fracture under liquid nitrogen were followed by approximately 40 min freeze-drying and coating with tungsten (e3 nm) under high vacuum at -80 C in BALTEC’s freeze-etch system BAF 060. After transfer via the airlock shuttle system VCT 100 (BALTEC),50 images were taken on a Zeiss (Germany) Gemini 1530 FEG scanning electron microscope equipped with a cold stage. Asymmetric Flow Field-Flow Fractionation Equipment. The asymmetric flow field-flow fractionation equipment used for this work is a Postnova AF 10’000, manufactured by Postnova Analytics (Germany), equipped with a two-angle static light scattering detector (Precision Dynamics), capable of measuring the intensity of the scattered light at 90 and 15 and operating at a wavelength of 808 nm. In addition, the AF4 instrument is also equipped with a UV detector, capable of operating at different wavelengths. As eluent, an aqueous solution containing 0.1wt % SDS has been used. Briefly, our setup consists of three pumps, connected to a control module unit including two switch valves that control the flow configuration in all of the equipment. The sample is first injected manually into the AF4 device using a syringe through an injection valve, where the first injection pump carries the sample into the fractionation channel. Before starting with the fractionation, the sample is focused into a small region of the channel via a predefined flow-field configuration. When the fractionation starts, the second channel pump pushes the fluid along the channel, while the third cross-flow pump applies a cross-flow, that is, withdraws a predefined flow rate from the main channel through a semipermeable membrane located on (50) Ritter, M.; Henry, D.; Wiesner, S.; Pfeiffer, S.; Wepf, R. Microscopy and Microanalysis; Bailey, G. W., Ed.; Springer: New York, 1999.
7064 DOI: 10.1021/la904390h
Lattuada et al. the bottom of the channel. The cross-flow provides the driving force for fractionation. In this work, cellulose-based hydrophilic membranes with a molecular weight cutoff of 10 kDa have been used. All fractionations have been performed using a constant flow rate of 1 mL/min along the channel. Our instrument allows one to program the cross-flow rate, and we have used a cross-flow rate linearly decreasing over a time span of 30-80 min from 40% to 10% of the main channel flow rate. Sample Preparation. All standard polystyrene particles used for calibration and subsequent test experiments have been diluted with deionized water to particle volume fraction values between 5 10-7 and 2 10-4 before being injected in the AF4 device. The diameter of all latexes was confirmed to agree with that declared by the manufacturer according to both dynamic and static light scattering (fitting the scattered intensity profile using Mie theory) measurements, except for the 420 nm latex, the diameter of which was found to be 410 nm. The aggregation experiments have been carried out according to the following procedures. For the aggregation of polystyrene particles, the latex was first diluted 20 times, then all traces of surfactant were removed by washing it for 24 h with ion exchange resins. Then, the critical coagulation concentration (CCC) of NaCl was determined and found equal to about 380 mM. Then, a salt concentration smaller than the CCC was chosen for the experiments, in order to have a sufficiently slow aggregation process, that would allow us to comfortably monitor the initial phases of the aggregation kinetics, which are the most suitable to be analyzed with the AF4 device. The conditions chosen for the experiments were as follows: particle volume fraction 0.02% and salt concentration 300 mM NaCl, neutral pH. Significant care has been used in the preparation of the final aggregating solution, in order to avoid spurious and uncontrolled aggregation while mixing the polystyrene latex solution with the salt solution. After the washing with ion exchange resins, a solution containing all the salt required, having a volume equal to 95% of the final solution volume, was rapidly mixed with the latex solution, which consisted only of the remaining 5% of the total volume. The large volume difference between the two solutions is needed in order to expose the latex to a salt concentration similar to the final one chosen for the experiment, so as to avoid high local concentration peaks due to mixing limitations. At fixed times, samples were withdrawn from the solution undergoing aggregation and then diluted four times in an aqueous solution containing 0.1 wt % SDS (the same solution used as eluent in the AF4 device), to quench the aggregation process, before injecting them in the AF4 device for fractionation. Different dilutions (10 and 50 times) have been used to perform dynamic light scattering and static light scattering measurements, respectively, on the same nonfractionated samples. In the case of the fluorinated rubbery colloidal particles, a similar procedure was used for the preparation of the samples. Each experiment was started by pouring 9 parts prediluted acid (H2SO4) onto 1 part prediluted latex to end up with the desired particle volume fraction of 0.5%. The H2SO4 concentration was chosen equal to 0.055 mol/L, which is substantially smaller than the CCC of (H2SO4), which was found to be equal to 0.4 mol/L. The samples were again diluted four times in a water solution containing 0.1 wt % SDS before fractionation, in order to quench the aggregation. The overall particle concentrations used for dynamic light scattering and static light scattering measurements with these elastomer colloids were higher than those used for polystyrene, due to the lower refractive index of the fluorinated polymer and consequently lower intensity of the scattered light.
Theory Light Scattering. In this section, the most important equations utilized in analyzing the experimental data obtained from the AF4 device will be briefely discussed. For a more thorough discussion of this topic, the reader can consult the Supporting Information and the many textbooks on the topic.18 Langmuir 2010, 26(10), 7062–7071
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When performing static light scattering measurements on a polydisperse sample of spherical particles, the average radius of gyration ÆRgæ can be determined. This quantity is defined as18 jP max
ÆRg æ ¼ 2
j ¼1
Nj Vp, j 2 Rg, j 2
jP max j ¼1
ð1Þ Nj Vp, j 2
where Vp,j is the volume of a spherical particle belonging to the jth size class, having a radius of gyration Rg,j, while Nj is the number of particles belonging to the jth size class. A similar expression can be derived in the case of a population of fractal clusters made of identical spherical primary particles:49 iP max
ÆRg æ ¼ 2
mi i i ¼1 iP max i ¼1
2
Rg, i 2 ð2Þ
mi i2
where mi is the number of clusters each made of i primary particles and having a radius of gyration Rg,i. When dynamic light scattering measurements are performed on a polydisperse samples, a different average size can be determined, that is, the average hydrodynamic radius ÆRhæ.18 In the case of a dilute population spherical particles, ÆRhæ is given by18 jP max
ÆRh æ ¼
j ¼1
Nj Vp, j 2 Pj ðqÞ
jP max
Nj Vp, j 2 Pi ðqÞ Rh , j j ¼1
ð3Þ
iP max
ÆRh æ ¼
ð4Þ
ð5Þ
where λ0 is the wavelength of the radiation in vacuum and n is the refractive index of the solvent. In the case of clusters, and in particular of fractal clusters, the contribution of each cluster is also affected by the cluster rotational motion, which contributes in a rather complicated manner to the overall depolarization of light.51 Therefore, in eq 4, each cluster contributes not with its hydrodynamic radius Rh,i but with its effective hydrodynamic radius Reff h,i, which is proportional to the hydrodynamic radius (51) Lattuada, M.; Wu, H.; Morbidelli, M. Langmuir 2004, 20, 5630–5636.
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ð6Þ
where Rh,r,i is the rotational hydrodynamic radius of a cluster with i primary particles. In general, the extraction of the size distribution from scattering data is a difficult task. From the measured static light scattering spectra, for example, if the functional forms of a particle form factor and scattering structure factor are known, one could extract the size distribution.18 However, the inevitably present measurement error makes the exact inversion of this ill posed problem impossible, since no unique solution can be found. This means that a size distribution estimated through this procedure can easily contain artifacts. However, in the case of monodisperse samples, such as those obtained after AF4 fractionation, the knowledge of the form factors and scattering structure factors is sufficient to accurately determine the size of particles or clusters in the sample. For example, from the measurement of the scattered intensity at two angles, one can compute the ratio between the two intensities, which is a function of the size of the particles or clusters. When the two angles are 15 and 90, which are those measured by the detector of our AF4, the ratios for a monodisperse sample can be determined from the following expression for spherical particles: ratio ¼
K1 Pj ðq1 Þ K2 Pj ðq2 Þ
ð7Þ
and in the case of fractal clusters from K1 Sj ðq1 Þ K2 Sj ðq2 Þ
ð8Þ
where q1 and q2 are the scattering wave vectors and K1 and K2 are setup constants calculated at 15 and 90, respectively. Note that there might be a difference in setup constants if two different detectors (with different properties, geometries, and sensitivities) are used for the two different angles. From eq 7 (or 8), the size of the particles (or clusters, respectively) can be uniquely determined. Once the size of a sample is known, then eq 9 IðqÞ ¼ I0 K1 Nj Vp, j 2 Pj ðqÞ
where Si(q) is the scattering structure factor of a cluster made of i primary particles. The quantity q, called the scattering wave vector, is defined as a function of the scattering angle θ: 4πn θ sin q ¼ λ0 2
! Rh , i Rg, i 2 Rh, i 3D ln Si ðqÞ ¼ 1þ 1þ Reff 2Rh, r, i 3 DðqRg Þ2 h, i
ratio ¼
where Pj(q) is the form factor of a particle belonging to the jth size class, while in the case of a dilute population of clusters the above expression becomes49 mi i2 Si ðqÞ i ¼1 iP max m i 2 S ðqÞ i i Reff i ¼1 h, i
according to51
ð9Þ
for spherical particles and eq 10 IðqÞ ¼ I0 K1 Vp 2 mi i2 PðqÞ Si ðqÞ
ð10Þ
for clusters, evaluated at one of the two angles, can be used to determine the concentration Nj of particles belonging to the jth size class or the concentration mi of clusters made of i primary particles, respectively. For this work, we have used Mie theory52 (using the routine MIEV0, developed by Wiscombe53) to calculate the form factor of spherical particles. Instead, Raileigh-Debye-Gans theory18 was used to calculate the scattering structure factor of RLCA clusters. The calculation of structure factors was done using the (52) Bohren, C.; Huffman, D. Absorption and Scattering of Light by Small Particles; Wiley-Interscience: New York, 1983. (53) Wiscombe, W J. Appl. Opt. 1980, 19, 1505–1509.
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knowledge of particle-particle correlation function described in a previous work.54 Using a similar approach, all cluster radii of gyration, hydrodynamic radii,55 and effective hydrodynamic radii51 have also been calculated, always assuming a cluster fractal dimension of 2.15, which is well in the range of values typically observed in the literature for RLCA clusters.46 Treatment of AF4 Data. It is well-known that the fractionation of a monodisperse sample carried out with any chromatographic technique generates a peak, the broadness of which depends on the fractionation mechanism, the channel geometry, and the Peclet number of the eluate, defined as the ratio between the convective and the diffusive contributions to its transport.25 In the case of AF4, there are different examples of mathematical models published in the literature developed to predict the elution times of samples and the corresponding peak shape, given the physical properties of the sample and the hydrodynamic conditions inside the channel.29,30 However, the capability of these models to interpret experimental results is in some cases limited by intrinsic variations in the elution times observed in real devices (which, in the particular case of our instrument, can be as high as 10-20%). Given the difficulty to account for such variations, we have chosen a different approach, which does not rely on elution times and does not require any detailed elution model. Namely, we made a universal calibration based entirely on the scattering signals, which does not depend on the elution time. In order to proceed, the following assumption is made: at any time, any scattering signal is dominated by only one fraction of the sample having one size. This corresponds to assuming that diffusion and other hydrodynamic effects in the channel, which are responsible for peak broadening, are negligible, or in other words we consider for each sample only the average size. This same assumption is typically made in analyzing chromatographic data,56 and it has already been adopted in analyzing light scattering data from AF4 fractionations.41-43 In the case of the scattering signals, this implies that the intensity of the light scattered at a given instant is assumed to be only due to particles or clusters with a given average size. The static light scattering detector coupled to our AF4 instrument provides data of the intensity of light scattered at two angles every second. Due to intrinsic fluctuations in local concentrations in the measurement cell, the intensity of the scattered light produces a quite irregular pattern over short times, and therefore, it is necessary to average it over a time interval in order to observe a reproducible trend. For our experiments, we found out that the best compromise was to split the signal into time intervals having a duration of 1 min, during which the average is computed. We found that this was the best compromise in terms of reproducibility and resolution. We have also verified that the results obtained by applying this procedure do not depend strongly on the duration of the time interval used to average the data, as long as it is between 0.5 and 1.5 min. The universal calibration has been carried out as follows. First of all, for each measured signal, the baseline needs to be estimated, which is easily accomplished by flowing pure eluent through the scattering cell. Then, the intensity of the baseline is subtracted from the measured scattered intensity of a sample to obtain the net intensity due only to the sample. The ratio between the
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scattered intensities at the two angles can be subsequently determined, and from eqs 7 or 8, depending on whether the system is made of particles or clusters, the size of a sample in each bin can be determined. There is one additional warning: the ratio between K1 and K2 needs to be determined by calibration. Since our instrument is using two different detectors to measure the scattered intensity at two different angles, the two constants appearing in eqs 7 and 8 are in fact different. Therefore, we have used a standard (monodisperse) polystyrene sample, made of spherical particles with a diameter of 90 nm, to compute the ratio between the two signals. Then, given the particle size, the ratio between the form factor at the two angles has been computed using Mie theory, and the ratio between the two constants has been finally determined. Once this ratio is known, all sample sizes can be estimated using eqs 7 and 8, independent of the elution time. In particular, this has been performed by first calculating the ratio values for a broad range of particle sizes with a resolution of 1 nm in particle diameter and for cluster masses with resolution of one primary particle. Then, each measured ratio value has been compared with the theoretical ones, and the size or mass is assigned by finding the size or mass for which the theoretical ratio is closest to the measured one. Once the sizes were determined, the particle or cluster concentrations were determined from eq 9 (or 10) evaluated at 90, since the signal at this angle is more stable and less susceptible to background noise with respect to that at 15. Population Balance Equation Modeling. Some of the most powerful tools to model the kinetics of aggregation processes are the population balance equations. These are material balances for all the particles (or clusters) composing the particle size (or cluster mass) distribution, where all the spatial and temporal correlations among the different clusters are neglected and aggregation is modeled as a second order kinetic event. In the study of aggregation processes under stagnant conditions, the aggregation events are always considered as irreversible, and no breakage mechanism of clusters is taken into account. Furthermore, the systems where aggregation takes place are uniform, that is, without concentration gradients. In those conditions, the PBE takes the following form:49 ¥ i -1 X dNi ðtÞ 1X ¼ Ki, j Ni ðtÞ Nj ðtÞ þ Ki -j, j Ni -j ðtÞ Nj ðtÞ dt 2 j ¼1 j ¼1
ð11Þ where Ki,j is the aggregation rate constant (called kernel) between two clusters with mass i and j and Ni(t) is the number concentration of clusters with mass i. The first term on the right-hand side of eq 11 is the consumption rate of clusters with mass i, which is given by the rate of collision with clusters of any mass. The second term is instead the rate of production of clusters, given by the rate of all the possible collisions among smaller clusters that result in a cluster of mass i. In the PBE, all physical information about the aggregation mechanism is lumped into the kernels. In the case of noncoalescing particles, which undergo reaction-limited cluster aggregation, the following kernel is used:49 Ki , j ¼
(54) Lattuada, M.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2003, 268, 106–120. (55) Lattuada, M.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2003, 268, 96–105. (56) Hagel, L.; Jagschies, G.; Sofer, G. Handbook of Process Chromatography, Second Edition: Development, Manufacturing, Validation and Economics; Academic Press: London, 2007.
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8kT ði1=Df þ j 1=Df Þði -1=Df þ j -1=Df Þ ðijÞλ 3ηW 4
ð12Þ
where W is the Fuchs stability ratio, a quantity that accounts for particle-particle repulsive interactions, Df is the cluster fractal dimension, which for RLCA processes is approximately equal to 2.15, and λ is a reactivity exponent, accounting for the higher Langmuir 2010, 26(10), 7062–7071
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reactivity of large clusters compared to small ones. For many nonpolymeric colloidal particles, such as silica and gold colloids, λ = 0.5.57 However, for polystyrene, this value is changing over a broad range of values within 0 and 0.5.49 Therefore, in the following, we treat λ as a fitting parameter. In the case of coalescing particles, the RLCA kernel is given by the following expression: Ki , j ¼
2kT ðRi þ Rj ÞðRi -1 þ Rj -1 Þ 3ηWi, j
ð13Þ
where Ri is the radius of a particle with mass i, calculated assuming full coalescence and therefore spherical shape, and Wi,j is the Fuchs stability ratio between a particle with mass i and a particle with mass j. Since coalescence is changing the overall particle surface, the surfactant that stabilizes the elastomer particles is redistributed among the particles, leading to an increase in surface charge density. This means that the stability ratio values increase during the aggregation process and differ for different pairs of particles. The details of the calculations of Wi,j are reported elsewhere.58 Once the cluster mass (or particle size) distribution is known, one can calculate the average quantities measured by light scattering, such as the average radius of gyration and the average hydrodynamic radius, given by eqs 2 and 4 for clusters and by eqs 1 and 3 for particles, respectively.
Results and Discussion The motivation of this work was the development of a methodology to extract size distributions without relying on retention times calibration of the equipment. Such a calibration procedure has the disadvantage that it needs to be repeated every time that the operating conditions are changed. One of the strengths of the AF4 is its flexibility in terms of operating conditions, which translates into the possibility to change the crossflow flow rates and profile in order to find the optimal conditions necessary to achieve a satisfactory fractionation. Any change in the crossflow profile would immediately change the elution times and would require a new calibration. Therefore, the availability of a procedure to extract sizes without relying on elution times is desirable. The assumption which our procedure is based on is that the chromatograms can be discretized and divided into equal sized bins, in each of which the signal is dominated by a sample of a given average size. The first test aimed at showing the reliability of this approach was based on verifying eq 7 on a collection of standard polystyrene particles, the properties of which are listed in the Materials subsection, each one characterized by a very monodisperse distribution, spanning more than 1 order of magnitude in size. The procedure followed was the following. All samples were fractionated separately. The corresponding chromatograms are shown in the Supporting Information section. The analysis of the scattering signals was done by first searching the maximum value in the elution peak and then integrating over a 1 min interval centered at the maximum value. Increasing the width of the interval resulted in no change at all, since the uniformity of size of these standard particles is reflected in uniform values of the ratio between the two light scattering signals over the entire peak area. The time dependence of the ratios is also shown in the Supporting Information, from which it can be seen that constant (57) Sandkuhler, P.; Lattuada, M.; Wu, H.; Sefcik, J.; Morbidelli, M. Adv. Colloid Interface Sci. 2005, 113, 65–83. (58) Gauer, C.; Wu, H.; Morbidelli, M. Langmuir 2009, 25, 9703–9713.
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Figure 1. Ratio between the intensities of light scattered at 15 and 90 for standard stable polystyrene nanoparticles as a function of the particle diameter Dp. Data points were obtained from AF4, while the continuous line represents a calculation performed with Mie theory.
(albeit sometimes noisy) trends are observed for elution times close to the peaks of the chromatograms. Figure 1 shows the experimental values of the ratio between the 15 and 90 scattered intensities, compared to the predictions of Mie theory for polystyrene monodisperse spherical particles. It is observed that the comparison between Mie theory calculations and experimental data is satisfactory over the entire particle size range. Note in particular that the ratio between the two scattered intensities increases monotonically as the particle size increases until a value that depends on the optical properties of the particles, and for polystyrene corresponds to a diameter of about 550 nm for a wavelength of 808 nm. For larger sizes, the behavior of the ratio is oscillatory with the size. The experimentally measured values of the ratio compare well with the calculation also in this regime, even though the ratio is almost 100 times larger than that of small particles, implying that the 15 scattered intensity is close to its upper limit of detection and the 90 intensity is close to its lower limit of detection. This non-monotonic trend implies that, from the sole knowledge of the ratio between the scattered intensities of spherical particles at two angles, one cannot determine uniquely the size of spherical particles larger than 550 nm in the case of this instrument. For this reason, the measurements of coalescing particles performed in this work involve only particles much smaller than this threshold. Let us now turn our attention to the systems undergoing aggregation. For both systems, we have focused our attention on the initial stages of the aggregation process, since the detection and fractionation of very large clusters or particles fall beyond the optimal regime in which the instrument works as discussed above. Figures 2 and 3 show the 15 and 90 chromatograms, respectively, for the 90 nm polystyrene nanoparticles used for the RLCA experiments. The different chromatograms correspond to samples taken at different times during the aggregation process, as indicated in the legend, and diluted to quench the aggregation before injecting them in the AF4 device. From the two figures, one can observe that a rather narrow and well-defined peak can be observed at short times, corresponding mostly to primary particles, while as the aggregation proceeds the chromatograms show the progressive development of a tail on the right-hand side of the DOI: 10.1021/la904390h
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Figure 2. Chromatograms of samples taken from a polystyrene 90 nm nanoparticles dispersion undergoing reaction-limited aggregation at different times, as indicated in the legend. The signal shown is the scattered intensity at 15.
Figure 3. Chromatograms of samples taken from a polystyrene 90 nm nanoparticles dispersion undergoing reaction-limited aggregation at different times, as indicated in the legend. The signal shown is the scattered intensity at 90.
main peak, which corresponds to the formation of doublets, triplets, quadruplets, and larger clusters. The peak corresponding to primary particles decreases in height, indicating a progressive consumption of primary particles, which form increasing amounts of small and larger clusters. In order to get a more precise picture of this process, it is instructive to take a look at the ratio between the two signals, which is shown in Figure 4. From here on, unless otherwise specified, the ratio is intended to have been calculated by first subtracting the background noise from both scattering signals. In the time interval where the signals are sufficiently different from their baselines (i.e., for elution times larger than 10 min), the ratio shows a monotonically increasing trend as a function of the elution time, indicating that larger objects are eluting at later times. In addition, as the aggregation proceeds, the ratio values corresponding to longer elution times reach progressively higher values, suggesting that larger and larger clusters are eluted. When the elution peak approaches the 7068 DOI: 10.1021/la904390h
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Figure 4. Ratio between the 15 and 90 scattered intensity signals reported in Figures 2 and 3, respectively, corresponding to samples taken from a polystyrene 90 nm nanoparticles dispersion undergoing reaction-limited aggregation at different times, as indicated in the legend.
baseline, the ratio values start becoming highly noisy due to the low signal-to-noise ratio, and these were excluded from the analysis. In the case of fractal clusters, the quantity of interest is more the cluster mass distribution (CMD) rather than the size distribution. In order to quantify the CMD, we have applied the procedure of extraction of size distribution described in the previous section. The extraction of the cluster mass distribution, in the case of polystyrene nanoparticles undergoing reaction-limited aggregation, was performed by using the known ratio values of the scattering structure factor of clusters with fractal dimension equal to 2.15.44-46 For the experimental conditions probed, which are focusing on initial steps of the aggregation process, it is very important to have scattering models available for small clusters, such as those reported in our previous work.54 The CMDs for some of the samples taken at different times are shown in Figure 5. The CMDs show the expected behavior: at short times, the CMD is composed only of primary particles, doublets, triplets, and tetramers, while as the aggregation process proceeds, the amount of primary particles decreases and the number of clusters increases substantially. In order to verify the reliability of the CMD obtained from AF4, we have also measured the average radius of gyration and the average hydrodynamic radius of the same samples, and have then used the obtained CMD to back calculate the same two average sizes. The comparison is shown in Figure 6: both pairs of sizes agree quite well, at least up to measured times of 1 h. For longer measured times, instead, while the average reconstructed hydrodynamic radius still matches quite well the measured one, the average reconstructed radius of gyration underestimates the measured one. This indicates some inaccuracy in determining the amount of large clusters in the measured CMD, which can be expected since large clusters are subject to stronger interactions with the membrane in the fractionation channel. As a confirmation of this effect, we have observed that the AF4 device required more frequent membrane replacements when dealing with systems with aggregation, in particular when leading to large aggregates, which create a strong membrane fouling. Therefore, it is conceivable that large clusters tend to be retained by the Langmuir 2010, 26(10), 7062–7071
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Figure 5. Mass weighted CMDs of samples taken from a polystyrene dispersion of nanoparticles having a diameter of 90 nm, undergoing reaction-limited aggregation at different times, as indicated in the legend. The lines are the CMD calculated from the PBE.
Figure 6. Time evolution of the average radius of gyration ÆRgæ and the average hydrodynamic radius ÆRhæ for polystyrene nanoparticles undergoing reaction-limited aggregation. AF4 data have been obtained by using the CMD reported in Figure 5 to back calculate ÆRgæ and ÆRhæ. The lines are the predictions from calculations using the PBE.
membrane and their number is underestimated in the fractionated samples. Figures 5 and 6 show also the predictions of PBE simulations. The calculations have been carried out by adjusting two parameters, the Fuchs stability ratio W and the parameter λ in the RLCA kernel, so as to match the time evolution of the two average sizes ÆRgæ and ÆRhæ in Figure 6. The values of W and λ that provide the best fit are given by 5 103 and 0.25, respectively. We have shown in our previous works49,57 that W defines the initial rate of aggregation of primary particles and has the effect of Langmuir 2010, 26(10), 7062–7071
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accelerating or slowing down the overall aggregation rate, without having any effect on the shape of the CMD. On the other hand, λ defines the reactivity of large clusters as compared to small ones and has a strong effect on the shape of the CMD and on the time evolution of the observable quantities such as ÆRgæ and ÆRhæ. Large values of λ imply a broader CMD as well as a larger difference between ÆRgæ and ÆRhæ values. From Figure 6, one can see that the agreement of the PBE calculations with the time evolution of the measured quantities is quite satisfactory. In all our previous works, we have always tested our model predictions versus average sizes measured by light scattering, claiming that since ÆRgæ and ÆRhæ are two different moments of the CMD, then our model should probably be quite accurate in predicting the main features of the CMD itself. With this work, we can prove (for the first time in the case of fractal clusters, see also ref 58 for elastomer particles) that this is indeed the case, as Figure 5 shows. We can see that the calculated CMDs are in fact quite close qualitatively and also quantitatively to the ones measured through the AF4 device. This confirms that the monitoring of at least two different moments of the CMD provides a good description of the physics of the aggregation process. One important observation has to be made about the values of the parameter λ used to simulate the aggregation data. It should be pointed out that an RLCA process should be characterized by a λ value of 0.5, which has been observed for many colloidal systems, such as silica and gold colloids, and can be also motivated on physical grounds.57,59 However, we have observed experimentally for different polymer colloids λ values changing from 0 to 0.5.49,60 These changes are certainly related to surface phenomena, such as surfactant redistribution, as experiments with surfactant addition have indicated.60 A low value of λ, such as that used in our simulations, indicates that the reactivity of clusters increases much less than expected, even though the fractal dimension of the clusters at longer times reaches the predicted value of 2.15. Since our system is made of surfactant free particles that have been additionally cleaned using ion exchange resins, we speculate that this reduced reactivity of particles might be related to nonuniform distribution of surface charges on the particles. Polystyrene colloids are well-known to have patchy surfaces,61 and it is not surprising that this characteristic affects the reactivity of particles. If the surface charge density of a particle is not uniformly distributed, the potential barrier will also be nonhomogeneous. Therefore, it can be expected that particles under reaction-limited conditions will initially react through the portions of their surface with the lower surface charge densities, leaving the more charged and more stable regions free. As the aggregation proceeds, and the less reactive patches are consumed, the clusters will have particles with more and more stable portions of their surfaces exposed, leading to a higher stability compared to the initial aggregation steps. This is reflected in values of λ smaller than expected. Let us now consider the case of elastomer particles, made of a polymer with a glass transition temperature significantly lower than room temperature, which undergo coalescence. In Figure 7, a cryo-SEM picture of these particles is shown, taken for a sample undergoing aggregation. The picture shows that there are only spherical clusters, that is, particles with different size, due to (59) Lattuada, M.; Wu, H.; Sefcik, J.; Morbidelli, M. J. Phys. Chem. B 2006, 110, 6574–6586. (60) Sandkuhler, P.; Sefcik, J.; Morbidelli, M. J. Phys. Chem. B 2004, 108, 20105–20121. (61) Chen, W.; Tan, S. S.; Ng, T. K.; Ford, W T.; Tong, P. Phys. Rev. Lett. 2005, 95, 218301.
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Figure 7. Cryo-SEM picture of fluorinated elastomer polymer colloids undergoing slow aggregation, showing small primary particles together with larger particles obtained as a result of cluster coalescence.
Figure 8. Chromatograms of samples taken from a dispersion of elastomer polymer colloids with a diameter of 120 nm, undergoing coalescence, at different times, as indicated in the legend. The signal shown is the scattered intensity at 90.
cluster coalescence. This is strong proof of the fact that in the conditions under examination such particles do not form fractal clusters.58 The chromatograms of the different samples taken at fixed intervals during the aggregation process have a behavior similar to that of the polystyrene samples undergoing RLCA, with initially a rather narrow size distribution, which progressively develops a tail in the large size range corresponding to the formation of larger and larger particles, as shown in Figure 8. The PSD obtained from the analysis of the AF4 chromatograms is shown in Figure 9. It confirms what we qualitatively expected, that is, that the PSD develops an increasing tail of larger particles. In order to verify the accuracy of these results, we have compared the obtained PSD with the one obtained from image analysis of the cryoSEM pictures. In Figure 10, the comparison is shown, and the PSDs are cumulative size distributions. It can be observed that the agreement between cryo-SEM data and distributions obtained from AF4 is fairly good, once more supporting our 7070 DOI: 10.1021/la904390h
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Figure 9. PSDs of samples taken from fluorinated elastomer polymer colloids undergoing slow aggregation at different times, as indicated in the legend.
Figure 10. Comparison between the cumulative particle size distributions calculated from those shown in Figure 9 (lines) and cumulative PSDs obtained form the analysis of cryo-SEM pictures (data points), taken at different times (listed in the legend) during the aggregation process.
method. In addition, similarly to what was done for the case of polystyrene particles, in Figure 11, the average hydrodynamic and gyration radii are shown, and for each average size both the measured values using light scattering on the nonfractionated samples, and the back calculated ones using the PSDs in Figure 9. The agreement is in this case very good for the time during which the aggregation was monitored and confirms once more the reliability of our approach. Finally, in Figure 11, the predictions from PBE are also shown, obtained using kernel (13), where the stability ratio values are calculated during each integration step. The progressive increase in colloidal stability of the system due to surfactant redistribution is indirectly confirmed by the very slow growth of size during the aggregation process. The model predictions are once more capable of quantitatively simulating the experimentally observed time evolution of Langmuir 2010, 26(10), 7062–7071
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Figure 11. Time evolution of the average radius of gyration ÆRgæ and the average hydrodynamic radius ÆRhæ for fluorinated elastomer polymer colloids undergoing slow aggregation and coalescence. The AF4 data have been obtained by using the PSD reported in Figure 9 to back calculate ÆRgæ and ÆRhæ. The lines are the predictions from calculations using the PBE.
both average particle sizes. Further details about this process are reported elsewhere.58
Conclusions In this work, we made use of an asymmetric flow-field flow fractionation unit, equipped with a two-angle static light scattering detector, to characterize dispersions of particles undergoing aggregation. Two different systems have been analyzed. The first one is a conventional dispersion of polystyrene nanoparticles undergoing reaction-limited cluster aggregation, while the second one is a dispersion of fluorinated elastomer particles with a glass transition temperature below room temperature which undergo full coalescence during aggregation. Our characterization relies solely on the analysis of static light scattering signals, since their ratio is only a function of the size and shape of the particles or clusters. During the experiments, samples have been taken at fixed
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time intervals during the aggregation process and fractionated with AF4. The same samples have been analyzed using both static and dynamic light scattering before fractionation. Using our knowledge of the scattering structure factor of fractal clusters formed during a conventional reaction-limited aggregation process, and of the particle form factor of spherical particles, we have been able to extract and follow the time evolution of the CMD of RLCA clusters and of the PSD of coalescing particles for the two systems. The reliability of the obtained distributions has been tested by using them to back calculate the average scattering properties measured via light scattering on the nonfractionated samples, that is, the average radius of gyration extracted from static light scattering data and the average hydrodynamic radius obtained from dynamic light scattering data. The agreement between these sets of data was very satisfactory for both systems, validating the reliability of the proposed approach. In the case of the coalescing particles, the extracted PSDs have also been successfully compared to those obtained from image analysis of cryo-SEM pictures. In addition, modeling of both aggregation processes through PBE indicated that PBE predictions compare satisfactorily with experimental data. Suspensions undergoing aggregation have been chosen as the prototypes of complex colloidal systems, in order to show the possibilities offered by the coupling of a fractionation unit to a static light scattering detector. It is worth noting that the proposed approach does not rely on calibration of the elution time, making it suitable to be used for a wide range of fractionating conditions, and therefore exhibiting high robustness. Furthermore, with the development of more powerful equipment coupled with more sophisticated multiangle scattering detectors, the proposed approach can be further extended and become a powerful tool to extract quantitative information about complex colloidal dispersions. Acknowledgment. This work was financially supported by the Swiss National Science Foundation (Grant No. 200020-126487/1). Many useful discussions with Dr. Hua Wua and Dr. Miroslav Soos are gratefully acknowledged. Supporting Information Available: Schematic of the AF4 working principle, a more detailed treatment of the light scattering theory used in this work, and chromatograms of standard particles used for calibration. This material is available free of charge via the Internet at http://pubs. acs.org.
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