Investigation of Particle Interactions in Dense Colloidal Suspensions

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Investigation of Particle Interactions in Dense Colloidal Suspensions Using Frequency Domain Photon Migration: Bidisperse Systems Zhigang Sun†,‡ and Eva M. Sevick-Muraca*,† Department of Chemical Engineering, Texas A&M University, College Station, Texas 77843-3122, and School of Chemical Engineering, Purdue University, West Lafayette, Indiana 47907-1283 Received September 21, 2001. In Final Form: November 13, 2001 The frequency domain photon migration (FDPM) technique was employed for investigation of particle interactions of dense, bidisperse, polystyrene samples of differring volume ratios of small to large particle sizes. The particle interactions of bidisperse systems were evaluated from the angular integrated structure factors, 〈S(q)〉 , which were extracted from FDPM measurements of scattering properties at high volume fractions ranging from 0.1 to 0.3. The FDPM-measured 〈S(q)〉 were then compared with theoretical predictions using the full binary hard sphere Percus-Yevick (HSPY) model, as well as the decoupling approximation (DA) and the local monodisperse approximation (LMA) models. The influence of differing volume ratios of colloid sizes on structure factors at differing wavelengths was also investigated. Results show that the interactions between small and large particles significantly impact the structure factors as well as the scattering properties in dense bidisperse colloidal suspensions. Furthermore, the binary HSPY model is most suitable for accounting for particle interactions which predominately arise from volume exclusion effects. Upon use of the binary HSPY model, the particle sizes of the two modes of the bidisperse polystyrene suspensions were recovered from dual wavelength FDPM measurements at high volume fractions (up to 0.3). The FDPM-derived particle sizes are comparable with DLS measurements of individual modes at diluted concentrations (∼0.001).

Introduction An understanding of asymmetric binary particle interactions and the resulting local microstructure is important in both academic and industrial research. Although considerable efforts have been undertaken to experimentally investigate particle interactions of monodisperse dense colloidal suspensions, few literature reports are aimed at bidisperse systems that contain colloids with two types of different particle sizes.1-3 Bidisperse systems are somewhat more complicated than monodisperse systems since both the relative size ratio and volume ratio of the two populations of particles significantly influence local microstructure. In addition, investigation of the particle interactions between colloids with two different sizes provides the basis for study of particle interactions within polydisperse suspensions. For industrial applications, direct particle sizing of bimodal colloidal suspensions at high volume fractions is difficult due to confounding effects arising from multiple scattering and particle interactions in highly concentrated suspensions. Consequently, the study of binary particle interactions is key for particle sizing of bimodal colloidal suspensions at high volume fractions. A suitable approach to study particle interaction is to investigate the static structure factor, which is the direct measure of local microstructure of particles in concen* To whom correspondence should be addressed. Phone: 979458-3206. Fax: 979-845-6446. E-mail: [email protected]. † Texas A&M University. ‡ Purdue University. (1) Barlett, P.; Ottewill, R. H. Geometric interactions in binary colloidal dispersions. Langmuir 1992, 8, 1919. (2) Rodriguez, B. E.; Kaler E. W. Binary mixtures of monodisperse latex dispersions. 1. Equilibrium structure. Langmuir 1992, 8, 2376. (3) Kaplan, P. D.; Yodh, A. G. Diffusion and structure in dense binary suspensions. Phys. Rev. Lett. 1992, 68, 393.

trated colloidal suspensions. Several techniques based upon radiative scattering from single colloidal particle, such as small-angle scattering of light, X-ray, and neutron (SALS, SAXS, and SANS, respectively), have been extensively used to assess the static structure factor in colloidal suspensions. SALS has been proven to be a useful tool for probing particle interactions in colloidal suspensions. However, it is restricted to dilute suspensions due to the requirement that the light passing through the sample scatters no more than once. Recently, crosscorrelation techniques4 were proposed to isolate singly scattered light and suppress multiply scattered light. However, this approach is not stable and requires sophisticated alignment procedures to maximize the amplitude of the cross correlation. Compared to the SALS method, SAXS requires a synchrotron source and SANS requires a nuclear source. The requirement of a neutron source or X-ray synchrotron source prevents their ubiquitous implementation. Furthermore, colloidal particles must be very small (less than 100 nm) to obtain full information of the structure factor, S(q), using SAXS and SANS. In recent years, new experimental methods based on multiple light scattering, such as diffuse transmission spectroscopy5 and diffusing wave spectroscopy (DWS),6 have been developed to probe the structure of concentrated colloidal suspensions. Frequency domain photon migration (FDPM) is yet another new method for probing the structure factor of dense colloidal suspensions using (4) Schatzel, K. Suppression of multiple-scattering by photon crosscorrelation techniques. J. Mod. Opt. 1991, 38, 1849. (5) Kaplan, P. D.; Dinsmore, A. D.; Yodh, A. G.; Pine, D. J. Diffuse transmission spectroscopy-a structural probe of opaque colloidal mixtures. Phys. Rev. E 1994, 50, 4827. (6) Weitz, D. A.; Zhu, J. X.; Durian, D. J.; Gang, H.; Pine, D. J. Diffusing-wave spectroscopy: the technique and some applications. Phys. Scr. 1994, T49B, 610.

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multiply scattered light.7 In this method, the angleintegrated structure factor, instead of structure factor itself, was derived from FDPM measurement for study of particle interactions. The advantages of FDPM can be enumerated: (i) Since FDPM depends on multiply scattered light, it is more suitable than light, X-ray, and neutron scattering approaches for assessment of particle interactions within concentrated suspensions. (ii) Since FDPM measures time-dependent propagation characteristics rather than the amount of light detected, it is selfcalibrating and no external calibration is required. (iii) Since FDPM allows determination of absorption and scattering independently, characterization of colloidal scatterers is not biased by changes in the light absorption or the color of suspending fluids.8,9 The primary limitation of FDPM is the requirement for multiple light scattering or scatterer concentrations nominally about 1%. Using the FDPM technique, we have successfully studied the interactions of volume exclusion in monodisperse suspensions,10 depletion in colloid and soluble polymer mixtures,11 and electrostatic interactions within dense colloidal suspensions of varying ionic strength.12 In this work, we employed the FDPM technique to investigate the particle interactions of dense bidisperse polystyrene samples, which were produced by mixing two monodisperse samples with different particle sizes at different volume ratios. The angle-integrated structure factors, 〈S(q)〉, were derived from FDPM-measured scattering properties at high volume fractions ranging from 0.1 to 0.3 and were then compared with theoretical predictions by several structure factor models for forward model validation. The influence of the volume ratio of small to large colloid sizes on angle-integrated structure factor was also investigated. Next, the validated structure factor model was used to recover particle sizes of bidisperse samples from dual wavelength FDPM measurements at high volume fractions (up to 0.3), and the inversion results were compared to DLS measurements in diluted samples of each individual colloidal component (∼0.001 volume fraction). Finally, the influences of size polydispersity on FDPM-measured 〈S(q)〉 were studied. Theory Light Scattering in Bidisperse Dense Suspensions. In a dense colloidal suspension, the interference of light due to closely spaced particles becomes significant, and the scattering efficiency of a single particle in the ensemble is substantially reduced when compared to a particle within a dilute suspension. For a monodisperse dense system, the isotropic scattering coefficient, µ′s, can be determined by the following scattering equation: (7) Shinde, R.; Balgi, G.; Richter, S.; Banerjee, S.; Reynolds, J.; Pierce, J.; Sevick-Muraca, E. Investigation of static structure factor in dense suspensions by use of multiply scattered light. Appl. Opt. 1999, 38, 197. (8) Sevick, E. M.; Chance, B.; Leigh, L.; Nioka, S.; Maris, M. Quantitation of time- and frequency- resolved optical spectra for the determination of tissue oxygenation. Anal. Biochem. 1991, 195, 330. (9) Pham, T. H.; Conquoz, O.; Fishkin, F. B.; Anderson, E.; Tromberg, B. J. Broad bandwidth frequency domain instrument for quantitative tissue optical spectroscopy. Rev. Sci. Instrum. 2000, 71, 2500. (10) Sun, Z. G.; Clint, C. D.; Sevick-Muraca, E. M. Investigation of particle interactions in dense colloidal suspensions using frequency domain photon migration: monodisperse system. J. Colloid Interface Sci. 2002, in press. (11) Banerjee, S.; Shinde, R.; Sevick-Muraca, E. M. Probing static structure of colloid-polymer suspensions with multiply scattered light. J. Colloid Interface Sci. 1999, 209, 142. (12) Huang, Y. Q.; Sun, Z. G.; Sevick-Muraca, E. M. Assessment of electrostatic interactions in dense colloidal suspensions with multiple scattered light. Langmuir 2002, in press.

Sun and Sevick-Muraca

µ′s(λ) ) 6φ k2x3

∫0πF(n, x, λ, θ) S(q, φ) sin θ (1 - cos θ) dθ

(1)

Here the form factor, F(n, x, λ, θ), describes the scattering intensity of a single particle at infinite dilution, the static structure factor, S(q, φ), describes the correction factor of coherent scattering due to particle interactions occurring within dense suspensions, x is the diameter of particles, n is the relative refractive index of particle to medium, λ is wavelength, θ is the scattering angle, φ is the volume fraction of particles in the suspension, q is the magnitude of the wave vector, q ) 2k sin(θ/2), k is given by 2πm/λ, and m is the refractive index of medium. The structure factor is a direct measure of the local ordering of colloidal particles, and the value of S(q, φ) is equal to unity in the absence of particle interaction (e.g. in a dilute suspension). For an interacting bidisperse system, particle interactions occur between colloids not only with same particle size but also with different particle sizes. Correspondingly, the isotropic scattering coefficient for bidisperse systems can be obtained by extending the scattering equation for a monodisperse system (i.e., eq 1) to a binary system, which is given by

µ′s(λ) ) 6φ k (yLxL3 + ySxS3) 2

∫0π(FS) sin θ (1 - cos θ) dθ

(2)

where

FS ) ySFSSSS + yLFLSLL + 2xySyLFLSSLS

(3)

FLS ) Re(f1,Lf /1,S + f2,Lf /2,S) FL ) Re(f1,Lf /1,L + f2,Lf /2,L)

(4)

FS + Re(f1,Sf /1,S + f2,Sf /2,S) Here, yL and yS are fractional number densities of the large and small particles, with yL + yS ) 1. The parameters SLL, SSS, and SLS are partial structure factors, FL, FS, and FLS are corresponding binary form factors, f1 and f2 are the scattering amplitudes into two orthogonal polarization states arising from a particle, which can be calculated by Mie scattering theory,13 and f /1 and f /2 are conjugates of f1 and f2, respectively. It is noted that calculation of the binary form factor FLS by Mie theory of light scattering (i.e., eq 4) is not the same as that for X-ray or neutron scattering. Since the Rayleigh-Gans-Debye (RGD) approximation of Mie scattering theory is used for X-ray or neutron scattering, the scattered wave has no phase delay so that the scattering amplitudes are real. Consequently, the binary form factor can be calculated as FLS ) xFLFS. This is because the wavelengths of neutrons and X-rays are much longer than the effective sizes of the nuclei and electrons they scatter from, which makes the GRD approximation appropriate. However, Mie scattering theory must be employed in this study due to the fact that the wavelengths used for our experiments are comparable to the particle sizes of colloids. Since the Mie scattering amplitudes are complex, eq 4 must be used to calculate the binary form factor FLS. (13) Bohren, C.; Huffman, D. Absorption and Scattering of light by small particles; John Wiley & Sons: New York, 1983.

Particle Interactions in Colloidal Suspensions

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Since it is difficult to extract partial structure factors directly from FDPM measurements, we extracted the angle-integrated static structure factor, 〈S(q)〉, from isotropic scattering coefficients using eq 2. The angleintegrated structure factor 〈S(q)〉 is defined as the ratio of scattering coefficients when particle interactions are accounted for (i.e., µ′s,interacting) vs when they are not (i.e., µ′s,noninteracting). The angle-integrated structure factor is given by µ′s,interacting 〈S(q)〉 ) ) µ′s,non-interacting

∫ (y F S π

0

S S

SS + yLFLSLL + 2xySyLFLSSLS) sin θ (1 - cos θ) dθ

∫ (y F π

0

S S

+ yLFL) sin θ (1 - cos θ) dθ

Table 1. Particle Size Information of Samples A and B Measured by the DLS Method in Diluted Suspensions sample

mean (nm)

deviation (nm)

polydispersity (%)

A B

146.5 ( 0.9 223.5 ( 0.6

23.4 ( 2.9 21.0 ( 1.5

16.0 9.4

Table 2. Total Volume Fractions Measured by Evaporation Experiments for All Experimental Bidisperse Samples sample label

C:D ) 1:3

vol ratio C:D ) 1:1

C:D ) 3:1

10% 15% 20% 25% 30%

0.1000 0.1476 0.2082 0.2574 0.3024

0.0997 0.1455 0.2004 0.2540 0.3019

0.0966 0.1503 0.1977 0.2531 0.3020

(5)

The values of µ′s,non-interacting can be calculated directly from Mie theory, while the values of µ′s,interacting can be obtained in two ways: (i) experimental measurement of µ′s in dense suspensions using the FDPM technique; (ii) theoretical prediction of µ′s from Mie theory and a model of particle interactions or S(q, φ). Experimental values of angle-integrated structure factor 〈S(q)〉 are determined from measured µ′s,interacting normalized by the Mie theory prediction of µ′s,non-interacting, whereas theoretical values of 〈S(q)〉 are determined from predicted values of both µ′s,interacting and µ′s,non-interacting. By comparing angle-integrated structure factors obtained experimentally with those predicted theoretically, one can examine the structure factor models to investigate fundamental particle interactions in dense bidisperse colloidal suspensions. Structure Factor Models of Bidisperse Suspensions. For a surfactant-stabilized, bidisperse, dense colloidal suspension, there are different particle interactions that contribute to local microstructure and the associated structure factor. Since hard sphere volumeexcluded interactions predominately impact structure factor at high volume fractions when compared to electrostatic forces, van der Waals forces, and depletion forces, etc., particle interactions can be modeled to a first-order approximation as an effective volume-excluded effect. In previous studies, we found that the monodisperse hard sphere Percus-Yevick (HSPY) model is suitable to account for particle interactions in monodisperse suspensions at high volume fractions.10 For bidisperse systems, Ashcroft and Langreth14 derived the analytical solutions of the partial structure factors for an asymmetric binary hardsphere mixture by using the HSPY approximation. Due to the complexity of the solution to the binary HSPY model, two approximations for the effective structure factor can be used for analysis of scattering data. The first approximation neglects particle interactions between small and large particles, which means SLS ) 0. Since a similar approach termed as local monodisperse approximation (LMA) has been proposed for polydisperse systems,15 we call this approximation as the LMA model for binary systems. The second approximation assumes that the asymmetric binary system can be effectively considered as a monodisperse system, where the particle size is the average particle size in the mixture. Thus, the structure factor is calculated using the monodisperse HSPY model with the average particle size as the parameter. We call (14) Ashcroft, N. W.; Langreth, D. C. Structure of binary liquid mixture. Phys. Rev. 1967, 156, 685. (15) Pedersen, J. S. Determination of size distributions from smallangle scattering data for systems with effective hard-sphere interactions. J. Appl. Crystallogr. 1994, 27, 595.

this model the decouple approximation (DA) model for binary systems similar to the same approximation proposed for polydisperse systems.16 In this work, the HSPY model, as well as the DA and LMA models, for bidisperse systems were examined by FDPM measurements in wellcharacterized, bidisperse colloidal suspensions. Experimental Section Materials. Samples of bidisperse colloidal suspensions were prepared by mixing two polystyrene samples with different mean particle sizes, A and B, which were provided by Dow Chemical, Midland, MI, and used as received. The dilutions were made with distilled deionized water. The mean particle size and deviation and polydispersity of samples A and B were measured by using DLS (Zetasizer 3000, Malvern Instruments, U.K.) of diluted suspensions (∼0.01%), and the results are shown in Table 1. The polydispersity is defined as the ratio of size deviation vs mean particle size. Normally, colloidal suspensions with narrow PSD (e.g., polydispersity less than ∼10%) are often assumed as the monodisperse systems since the ideal monodisperse systems are not available. In previous work, we have shown that suspensions with polydispersity below 16% can be assumed as monodisperse systems for FDPM measurements.10 Correspondingly, the samples A and B are assumed as monodisperse systems in this study, and the polydispersity effects are addressed in the section below. Three different volume ratios of small to large particles were mixed at fixed total volume fractions ranging from 10% to 30%, and then the exact total volume fraction in each bidisperse sample was computed from weight measurements before and after evaporation of the fluid. Volume fraction measurements are listed in Table 2. FDPM Measurement. FDPM measurements were performed on all the bidisperse samples at two different wavelengths, 687 and 828 nm. The details of FDPM instrumentation and theory are described elsewhere.10,17 Briefly, the measurements consisted of measuring the phase shift (PS) and attenuation of average of intensity (DC) and of amplitude (AC) at twelve different sourcedetector distances ranging from 7 to 11 mm in response to the source modulation at six different frequencies ranging from 50 to 100 MHz. Measurements were conducted on samples of approximately 100 mL. The optical parameters and their uncertainties were then calculated using AC and PS data as a function of source-detector position and averaged across all modulation frequencies, as outlined in ref 17. This procedure yielded isotropic scattering coefficients with high accuracy and precision. The FDPM-measured isotropic scattering coefficients for each sample are shown in Figure 1, which was used to extract the angle-integrated structure factors for investigation of particle interactions in bidisperse systems. We found that the precision (16) Kotlarchyk, M.; Chen, S. H. Analysis of small-angle neutronscattering spectra from polydisperse interacting colloids. J. Chem. Phys. 1983, 79, 2461. (17) Sun, Z. G.; Huang, Y. Q.; Sevick-Muraca, E. M. Precise analysis of frequency domain photon migration for characterization of concentrated colloidal suspensions. Rev. Sci. Instrum. 2002, in press.

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Figure 1. Isotropic scattering coefficient as a function of total volume fractions for three different volume ratios of small to large particles (diamond, 1:3; triangle, 1:1; square, 3:1) and two different wavelengths (solid symbols, 687 nm; open symbols, 828 nm). of FDPM-measured isotropic scattering coefficients is less than 1% for all measurements in this study.

Results and Discussion Comparison of Bidisperse Structure Factor Models. To find the appropriate structure factor model for the bidisperse systems, the isotropic scattering coefficients µ′s predicted from Mie theory with different structure factor models (i.e., noninteraction, HSPY, DA, and LMA models) were compared with the experimental data as a function of total volume fraction. If it is assumed that no interaction between particles exists, the structure factor S(q) should be equal to unity. Figure 2 illustrates an example of the results measured at 687 nm for bidisperse systems where the volume ratio of small to large particles is 1:1. Correspondingly, the experimentally measured angleintegrated structure factors 〈S(q)〉 were compared with theoretical values predicted by the HSPY, DA, and LMA models for the same samples, which are shown in Figure 3. For these model predictions, the hard sphere diameters necessary to calculate the partial structure factors were taken to be equal to the mean particle sizes measured by DLS. The differences between the HSPY predictions and the experimentally measured scattering coefficients and values for 〈S(q)〉 could result from the inaccuracies of the DLS particle size. One can see that the bidisperse HSPY model gives the closest estimates of the FDPM-measured isotropic scattering coefficients and angle-integrated structure factors at high volume factions. At low concentrations, the angleintegrated structure factor matches the HSPY prediction with less accuracy than at volume fractions greater than 0.2. The reason may be explained in terms of eq 5. In contrast to the measured scattering coefficient, the scattering coefficient predicted in the absence of interactions, µ′s,non-interacting, increases linearly with volume fraction. Consequently, any error in the measurement of the actual, interacting scattering coefficient at high volume fraction is minimized by the large value of the denominator in eq 5. Due to the lack of sensitivity in scattering measurements to volume exclusion interactions at low


Figure 2. Isotropic scattering coefficient obtained at a wavelength of 687 nm as a function of total volume fraction for bidisperse samples where the volume ratio of small to large particles is 1:1. The symbols denote FDPM measurement data. The lines denote values predicted using Mie theory with noninteraction (semidashed), DA (dotted), LMA (dashed), and HSPY (solid) models.

Figure 3. Angle-integrated structure factor obtained at a wavelength of 687 nm as a function of total volume fraction for bidisperse samples where the volume ratio of small to large particles is 1:1. The symbols denote FDPM measurement data. The lines denote values predicted using Mie theory with DA (dotted), LMA (dashed), and HSPY (solid) models.

concentrations, and due to the possibility of unaccounted long range particle interactions that are may be nonnegligible in comparison to the volume exclusion forces at these low concentrations, it is not surprising that the angleintegrated structure factor from derived from theory and part experiment match better at volume fractions greater than 0.2. The results also show that the particle inter-


Figure 4. Isotropic scattering coefficient obtained at a wavelength of 687 nm as a function of total volume fraction for bidisperse samples with differing volume ratio of small to large particles. The lines denote values predicted by the binary HSPY model. The symbols denote FDPM measurement data.

actions between small and large particles have significant impact upon local microstructure and, consequently, affect the isotropic scattering properties in dense suspensions. This effect increases with increasing total volume fractions, which means that SLS cannot be neglected, and the LMA model fails to describe the microstructure for dense bidisperse systems. Our results are consistent with the results reported by Kaplan et al.,3 where DWS was used to study particle interactions in binary hard sphere systems. Compared to the LMA model, the values predicted by the DA model are closer to those predicted by the HSPY model as well as those values experimentally obtained. Yet the DA model overestimates structure factor and isotropic scattering properties, especially at high total volume factions. Similar results were found for differing wavelengths and differing volume ratio of small to large particles. We interpret these results as a validation of the binary HSPY model for predicting the scattering properties of dense colloidal suspensions. Effects of Volume Ratio of Small to Large Particles. The influence of the volume ratio of small to large particles on µ′s and 〈S(q)〉 is shown in Figures 4 and 5, respectively. One can see that, if the total volume fraction is kept constant, µ′s and 〈S(q)〉 decrease with increasing volume ratio of small to large particles. This result suggests that suspensions become more ordered as the fraction of small particles increases. This is may be due to the fact that (i) the scattering intensity of small particles is less than that of large particles and (ii) the structure factor of dense suspensions primarily comprised of small particles is less than that primarily comprised of large particles, consistent with previous work.10 Values predicted from the bidisperse HSPY model are also shown in Figures 4 and 5 and, again, are consistent with experiment results. The discrepancies between the experimental values and the theoretical predictions may be due to other types of particle interactions, the effects of polydispersity, measurement error owing to an incom-

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Figure 5. Angle-integrated structure factor obtained at a wavelength of 687 nm as a function of total volume fraction for bidisperse samples with differing volume ratio of small to large particles. The lines denote values predicted by the binary HSPY model. The symbols denote FDPM measurement data.

plete mixing process, and the possible discrepancies in the mean diameters reported from DLS measurement. Recovery of Bimodal Particle Size at High Volume Fractions. Since the bidisperse HSPY model is a good approximation for modeling particle interactions of bidisperse colloidal suspensions at high volume fractions, we can incorporate this model into eq 2 to recover two unknown particle sizes from FDPM measurements. By using experimental data measured at wavelengths of 687 and 828 nm, we recovered both the small and large particle sizes at each sample with differing total volume fractions and volume ratios of small to large particle (shown in Table 2). This was done by simultaneously solving eq 2 at the two wavelengths. The inversion results are shown in Figure 6. One can see that the recovered diameter of the larger particle is more accurate than the diameter of smaller particle in most samples, especially for the samples with a small fraction of small particles. This may be due to the fact that the larger particles contribute more to scattering intensity than the smaller particles at lower concentrations. Consequently, it is difficult to accurately recover the particle diameter of small particles from the experimental data. However, the accuracy of recovered diameter of small particle increases as its volume fraction increases. With the exception of the small particle size obtained at low volume fractions, the fitted particle sizes are relatively independent of the volume fractions, indicating again that the bidisperse HSPY model may be suitable to account for particle interactions in bidisperse suspensions. Since both recovered particle sizes at different total volume fractions are comparable with DLS measurements of each individual colloidal component (∼0.001 volume fraction), the FDPM method may be considered a good approach to size bidisperse systems at high volume fractions. It is noteworthy that DLS provides measurements of unimodal suspensions and is inaccurate for sizing bimodal or multimodal suspensions at diluted concentrations. Effect of Polydispersity. As described earlier, the samples A and B that were used to prepare the bidisperse systems are not true monodisperse systems. Hence, we

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Figure 6. Recovered small particle diameters (open symbols) and large particle diameters (solid symbols) as a function of volume fraction. The lines denote the DLS results of samples A (dashed) and B (solid). The symbols denote the inverse results from FDPM measurements for bidisperse samples with differing volume ratio of small to large particles (squares, 1:3; triangles, 1:1; diamonds, 3:1).


Figure 7. Isotropic scattering coefficient as a function of total volume fraction for bidisperse samples where the volume ratio of small to large particles is 1:1. The symbols are FDPM measurement data at 687 nm (diamonds) and 828 nm (triangles). The lines are values predicted using binary (solid) and polydisperse (dashed) HSPY models.

investigated the influence of polydispersity. The theory to predict the isotropic scattering coefficient and the angleintegrated structure factor in polydisperse colloidal systems has been described elsewhere,10,18 where the polydisperse HSPY model was employed to calculate the partial structure factors in polydisperse systems.19 Figures 7 and 8 illustrate the influence of polydispersity on predicted scattering properties and structure factors for bidisperse samples of varying total volume fractions when the volume ratio of small to large particles is kept constant at 1:1. The polydispersities of both samples A and B are shown in Table 1. Since the differences between bidisperse and polydisperse HSPY models are small for most of our samples with total volume fractions less than 0.25, the assumption of a bimodal system of two monodisperse populations is reasonable. However, at high total volume fractions (∼0.3), the polydisperse HSPY model appear to be more accurate for modeling the particle interactions. Conclusions In this work, the FDPM technique was employed to study particle interactions of dense, bidisperse, polystyrene samples with differing volume ratios of small to large particle sizes and differing total volume fractions. The angular integrated structure factors 〈S(q)〉 and isotropic scattering coefficients were extracted from FDPM measurements at high volume fractions ranging from 0.1 to 0.3 and were then compared with the theoretical predictions by the binary HSPY model, as well as DA and LMA models. Our results show that for bidisperse systems the (18) Sun, Z. G.; Clint, C. D.; Sevick-Muraca, E. M. An approach for particle sizing in dense polydisperse colloidal suspension using multiple scattering light. Langmuir 2001, 17, 6142. (19) Blum, L.; Stell, G. Polydisperse systems (I): scattering function for polydisperse fluids of hard or permeable spheres. J. Chem. Phys. 1979, 71, 42.

Figure 8. Angle-integrated structure factor as a function of total volume fraction for bidisperse samples where the volume ratio of small to large particles is 1:1. The symbols are FDPM measurement data at 687 nm (diamonds) and 828 nm (triangles). The lines are values predicted using binary (solid) and polydisperse (dashed) HSPY models.

interactions between small and large particles significantly impact the structure factor and scattering properties. Comparably, the binary HSPY model is most suitable for accounting for particle interactions and local microstructure in dense bidisperse colloidal suspensions. The influence of volume ratio of small to large particles on structure factors and scattering properties can be observed in FDPM measurement, which is consistent with predictions using the binary HSPY model. By combining Mie theory with the binary HSPY model to account for particle interactions, we recovered both


small and large particle sizes of bidisperse system from FDPM measurements at high volume fractions (up to 0.3). The recovered sizes are comparable to DLS measurements of individual modes at diluted concentrations (∼0.001 volume faction). With measurements conducted at multiple wavelengths, the FDPM technique could provide a potential tool for particle sizing of bidisperse colloidal suspensions at high volume fractions.

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Acknowledgment. This work is supported in parts by the National Science Foundation (Grant CTS-9876583), The Dow Chemical Co., and the DuPont Young Faculty Fellow Program. We acknowledge the undergraduate researcher, Clint Tomlin, who successfully assisted in the acquisition of FDPM data. LA0114629

Investigation of Particle Interactions in Dense Colloidal Suspensions

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