Hard Structured Particles: Suspension Synthesis, Characterization

Langmuir 2004, 20, 11191-11200

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Hard Structured Particles: Suspension Synthesis, Characterization, and Compressibility William E. Smith and Charles F. Zukoski* Department of Chemical and Biomolecular Engineering, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801 Received April 21, 2004. In Final Form: September 8, 2004 Hard interactions are developed on three grades of fumed silica by eliminating interparticle forces and sterically stabilizing the particles by attaching an organic coating to the surface of the particles, suspending them in an index-matching solvent and screening the electrostatics. These hard-structured particles are studied to understand the effects of the particle’s microstructure on suspension properties without the influence of interparticle forces other than volume exclusion, Brownian, and hydrodynamic interactions. Light and X-ray scattering studies of low-volume-fraction suspensions suggest that the fumed silicas consist of primary particle of radius of gyration Rg1 ≈ 16 nm and aggregate size Rg2 ≈ 50 nm and mass fractal dimension Df ≈ 2.2. Osmotic compressibilities of these suspensions are measured as a function of particle concentration exploring the packing mechanism of fumed silica. While there is minimal detectable change in the primary particle size, Rg2 varies by ∼15%, providing insight into how suspension properties are related to particle size. As expected of hard particles with the same microstructure, the concentration dependence on the osmotic pressure superimposes with volume fraction of solids. The comparison of fumedsilica-suspension measurements to the known behavior of hard-sphere suspensions demonstrates the effects of particle geometry on suspension properties with indications of interpenetration of the fumed silica due to their open geometry.

Introduction Current understanding of suspension mechanics and microstructure has been greatly enhanced by the development of experimental hard-sphere suspensions. Hard spheres experience volume exclusion, thermal, and hydrodynamic interactions.1 The development of experimental systems composed of hard or near-hard spheres provides an opportunity to systematically explore the effects of particle size, solvent viscosity, and volume fraction without the often overwhelming and confounding influence of interparticle forces. These systems have been used to confirm the existence of the computer-predicted order-disorder phase transition,2,3 the osmotic compressibility, and structure of hard-sphere suspensions,4-6 as well as the low-volume-fraction prediction of transport properties such as diffusivity and viscosity.7-11 In the dense suspension limit, experimental hard-sphere suspensions have been used to explore glass transitions.12-15 With this * Author to whom correspondence should be addressed. E-mail: [email protected]. (1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: Cambridge, 1995. (2) Paulin, S. E.; Ackerson, B. J. Phys. Rev. Lett. 1990, 64, 26632666. (3) Hachisu, S.; Kobayashi, Y. J. Colloid Interface Sci. 1974, 46, 470476. (4) Vrij, A.; Jansen, J. W.; Dhont, J. K. G.; Pathmamanoharan, C.; Kops-Werkhoven, M. M.; Fijnaut, H. M. Faraday Discuss. 1983, 76, 19-36. (5) Ramakrishnan, S.; Fuchs, M.; Schweizer, K. S.; Zukoski, C. F. J. Chem. Phys. 2002, 116, 2201-2212. (6) Phan, S.-E.; Russel, W. B.; Cheng, Z.; Zhu, J.; Chaikin, P. M.; Dunsmuir, J. H.; Ottewill, R. H. Phys. Rev. E 1996, 54, 6633-6645. (7) van Megen, W.; Underwood, S. M. Langmuir 1990, 6, 35-42. (8) de Kruif, C. G.; Jansen, J. W.; Vrij, A. In Physics of Complex and Supramolecular Fluids; Safran, S. A., Clark, N. A., Eds.; WileyInterscience: New York, 1987; pp 315-346. (9) de Kruif, C. G.; Van Iersel, E. M. F.; Vrij, A.; Russel, W. B. J. Chem. Phys. 1985, 83, 4717-4725. (10) Krieger, I. M. Adv. Colloid Interface Sci. 1972, 3, 111-136. (11) van der Werff, J. C.; de Kruif, C. G. J. Rheol. 1989, 33, 421-454. (12) van Megen, W.; Pusey, P. N. Phys. Rev. A 1991, 43, 5429-5441.

Figure 1. Scanning electron microscopy image of a fumed silica aggregate composed of primary particles.

basic understanding, the effects of attractions and repulsions can be systematically introduced. The resulting predictions and the experimental advances laid the foundation for much of the modern understanding of suspension behavior. This paper describes the development of suspensions of particles that experience only volume-exclusion interactions but have complex shapes and a size distribution. The particles of this study are fumed silica composed of fused aggregates of ∼20 nm particles (Figure 1). Similar systems include fumed alumina and carbon blacks and comprise a class of materials with numerous technological applications. These applications often take advantage of the open-particle microstructure to alter suspension and composite flow properties. There is considerable interest in distinguishing the properties provided by particle (13) van Megen, W.; Underwood, S. M.; Pusey, P. N. Phys. Rev. Lett. 1991, 67, 1586-1589. (14) van Megen, W.; Underwood, S. M. Phys. Rev. E 1994, 49, 42064220. (15) Pusey, P. N.; van Megen, W. Phys. Rev. Lett. 1987, 59, 20832086.

10.1021/la0489864 CCC: $27.50 © 2004 American Chemical Society Published on Web 11/02/2004

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geometry and those conferred by particle-interaction potentials. In the same manner that the development of hard spheres provided opportunities for understanding the role of hydrodynamic and thermal forces on suspension properties, the hard-structured particles discussed here can be studied to understand the effects of the particle’s microstructure on suspension properties without the influence of other interparticle forces. Hard interactions are developed by minimizing van der Waals attractions and electrostatic repulsions and by keeping the extent of steric forces to a small fraction of the particle size. van der Waals interactions are minimized by suspending particles in a continuous phase with the same index of refraction as the particles. As it is the highfrequency contributions that dominate the magnitude of the Hamaker constant, matching indices greatly reduces the van der Waals interactions.1 While there remain zero frequency contributions for sub-micrometer particles that are well-index-matched, these contributions are found to result in attractions that are small compared to the average thermal energy in the system. These criteria have been met in two standard systems, one composed of poly(methyl methacrylate) with short-range steric forces provided by grafted polymer chains16 and the second consisting of silica particles with grafted short alkane chains on the surface.17,18 Two varieties of the silica system have been discussed in-depth. The first is octadecyl silica, where C18 chains are grafted on the particle surface and the suspending fluid is Decalin or a short chain alkane.18 The second system is composed of silica reacted with trimethoxysilanes.17 These suspensions are suspended in an ethanol-toluene mixture or tetrahydrofurfuryl alcohol (THFA) to provide index-matching. In this case, the grafting does not eliminate all particle charge, and slight adjustments of the pH and ionic strength to eliminate electrostatic repulsions must be made.17,19 This study uses 3-(trimethoxysilyl) propyl methacrylate-grafted particles suspended in THFA adjusted by nitric acid with the focus on the characterization and microstructure of three different grades of fumed silica particles treated in this manner. The thermodynamic and dynamic properties of hardsphere suspensions are a function of particle radius, a, particle number density, F, average thermal energy in the system, kT, and continuous-phase shear viscosity, ηc. This results in mechanical and thermodynamic properties depending on a single dimensionless group characterizing particle concentration: particle volume fraction, φ ) (4/3)πa3F. For the hard-structured particles discussed in this paper, there are two length scales, one associated with the primary particles which have a radius of gyration of Rg1 and a second characterizing the clusters which have a radius of gyration of Rg2, resulting in an added complexity of the scaling of suspension properties. Below, the low-volume-fraction properties of structured hard-particle suspensions are studied. These studies provide good estimates of Rg1, Rg2, and mass fractal dimension Df, which characterizes the structure of the aggregates. After the characterization of single-particle properties, the particle concentration dependence of suspension compressibilities is studied. The relative (16) Antl, L.; Goodwin, J. W.; Hill, R. D.; Ottewill, R. H.; Owens, S. M.; Papworth, S. Colloids Surf. 1986, 17, 67-78. (17) Philipse, A. P.; Vrij, A. J. Colloid Interface Sci. 1989, 128, 121136. (18) van Helden, A. K.; Jansen, J. W.; Vrij, A. J. Colloid Interface Sci. 1981, 81, 354-368. (19) Maranzano, B. J.; Wagner, N. J.; Fritz, G.; Glatter, O. Langmuir 2000, 16, 10556-10558.

Smith and Zukoski

importance of the thermodynamic properties based on different measures of particle volume fraction of these suspensions is explored because three particle types that have different Rg2 but the same Rg1 and Df. Materials and Methods Suspension Synthesis. Fumed silica particles (Cab-O-Sil M-5, HS-5 and EH-5) were dispersed in ethanol through the use of ultrasonication, Artek Model 300 Ultrasonic Dismembrator, to render the particles into their most-reduced form.20 The amount of ultrasonication was determined by monitoring the average size using Brookhaven Instruments Fiber Optics Quasi-Elastic Light Scattering (FOQELS). In dilute solutions, FOQELS measured the autocorrelation function, thus providing a method for extracting the effective spherical hydrodynamic diameter following the Stokes-Einstein relationship. After the agglomerates were dispersed, the fumed silica particles were treated with 3-(trimethoxysilyl) propyl methacrylate (TPM) as described by Philipse and Vrij17 to render the particles nearly hard sphere. The reaction of TPM with the silica has been shown to eliminate nearly all (>98%) surface silanol groups.19 After treatment with TPM, the particles were rinsed repeatedly, followed by centrifugation and decantation to remove any excess TPM or ammonium hydroxide from the coating procedure. After addition of fresh solvent following the decantation, the particles were dispersed and ultrasonicated to ensure no agglomerates were present. The TPM-coated particles were then transferred into an indexmatching solvent, tetrahydrofurfuryl alcohol (THFA), through repeated centrifugation, decantation, and particle dispersion with ultrasonication. The centrifugation was performed in a Beckman J-21 Centrifuge with JA-10 fixed-angle rotor. After the final dispersion through ultrasonication, the suspension was centrifuged at a low speed for 3 h with the centrifugal force varying from 27 to 113G from the top to the bottom of the bottle. The sediment was discarded in an attempt to remove any large entangled agglomerates or contaminants. The effect of ultrasonication was also investigated by measuring the suspension viscosity as a function of ultrasonication time. The time for dispersion of the particles was set according to when the viscosity ceased to change with further ultrasonication. While the refractive index-matching by the THFA eliminates van der Waals attractions and allows the use of light scattering on concentrated systems, neutralization of the remaining surface charge was performed by adding aqueous nitric acid, as suggested by Philipse and Vrij,17 and employed by Maranzano and Wagner.19,21 Addition of aqueous nitric acid to the weakly charged dispersion after the TPM coating reduced the dissociation of the surface silanol groups which had not been terminated with TPM and thus eliminated repulsive interactions. Solution concentrations were determined by dilution from a stock solution with the solvent, 0.02 M HNO3 in THFA. The concentration of the suspensions, c, was converted to a volume fraction based on the silica concentration using the known density of fumed silica, 2.2 g/mL. Density measurements were made using a Mettler/KEM DA-100 density-specific gravity meter. Characterization Methods. Electrophoretic mobilites were measured using a Malvern Instruments Zetasizer 3000 HSA, 633-nm helium-neon laser and capillary cell. Five measurements were taken, and the reported data represent the mean values. Viscosity measurements were performed on a Bohlin C-VOR digital rheometer, utilizing a cup and bob geometry with a 14mm bob diameter and outer cup diameter of 15.4 mm, yielding a tool gap of 0.7 mm. A solvent trap was used to prevent solvent evaporation, and the temperature was monitored using a CP180 rapid Peltier temperature control module with coaxial cylinders. A constant sample volume of 3 mL was used in the experiments. Light and X-ray scattering measurements provide an experimental technique allowing for study of length scales ranging from the primary particle size to the aggregate size by measuring (20) Eisenlauer, J.; Killmann, E. J. Colloid Interface Sci. 1980, 74, 108-119. (21) Maranzano, B. J.; Wagner, N. J. J. Rheol. 2001, 45, 1205-1222.

Characterization of Hard Structured Particles

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the elastic scattering from the colloids

I ) φVS(∆Ω)2P(q)S(q) + Bi q)

( ) () 4πnc Θ sin λ 2

(1) (2)

where VS is the scattering volume, ∆Ω is the scattering length distance between the particle and solvent, P(q) is the single particle form factor, S(q) is the static structure factor, Bi is the incoherent scattering term, q is the scattering vector, nc is the refractive index of the medium, λ is the wavelength of the radiation, and Θ is the scattering angle. By construction, S(q) goes to one in the limit when φ goes to zero. Thus, P(q) can be determined in the dilute limit from measurement of the scattering intensity. S(q) is then calculated by dividing the concentrationnormalized intensity of the sample by the normalized intensity of the single-particle scattering function. Ultra-small-angle X-ray scattering measurements were performed at the UNICAT facility on the 33-ID beamline, Advanced Photon Source facility at Argonne National Laboratory. The samples were loaded into custom-made aluminum liquid cell holders between two Kapton polyimide film windows with a beam path length of 1 mm. Two different instrument setups were used for the measurements: slit-smeared (USAXS) and pinhole collimated (SBUSAXS). USAXS measurements22 were made using a Bonse-Hart camera using Si(111) optics with an experimental q range of 10-4-0.1 Å-1. The beam size was 0.4 mm vertical and 1.5 mm horizontal, through which approximately 2 × 1013 photons/s were incident at 10 keV. The scattering beam was analyzed with a rotating Si(111) channel-cut crystal and measured with a photodiode detector. The detector operates with a slit length of Qh ) 0.05 Å-1 (in q space). The measured intensity at a given q was thus an average intensity over a slit area, resulting in a slit-smeared intensity, Is(q)

IS(q) )

1 Qh

∫

Qh

0

IDSxQ2 + q2dQ

2π h Qh ) λ L

( )( )

(3)

Figure 2. Electrophoretic mobilities of TPM-coated HS-5 fumed silica in ethanol as a function of nitric acid. At zero acid concentration, ψ ) -0.82 (µm cm)/(s V). bath. Light intensity was measured by a photomultiplier tube and processed by a BI-9000AT digital correlator. A correction was made for the absorbance, A, of the solution by multiplying the scattering by 10A in accordance with Beer’s Law. The transmission loss to scattering is assumed to be negligible because the particles are index-matched with the solvent. The absorbance at 514 nm was measured using an HP 8453 general purpose UV-visible spectrophotometer with a 1-cm path length. Absolute intensities were obtained by calibration with the scattering of toluene. Dynamic light scattering (DLS) measurements were performed using the same setup used for SLS. The fluctuations in scattered intensity provide insight into the translational diffusion of the particles. The measured time autocorrelation function W2(τ) of the scattered intensity I(t) is an average of the product of the intensity at time t and t + τ. This can be related to the intermediate scattering function f(q,τ), which decays from a value of 1 at τ ) 0 to a value of 0 at τ f ∞. In the dilute limit, the decay is a single exponentional

〈I(q,t)I(q,t + τ)〉 ) W2(q,τ) ) C + K[f(q,τ)]2 (4)

where IDS is the desmeared intensity, h is defined as a variable along the slit length, and L is the distance from the sample to detector (300 mm). The USAXS measurements were corrected for slit-smearing following the method described by Lake.23 SBUSAXS measurements also use a Bonse-Hart camera, but side-reflection Si(111) stages are employed, allowing for effective pinhole collimation.24 There is a slight loss in q range and intensity resulting from the use of the SBUSAXS setup. Detailed comparisons on uniform hard spheres demonstrate the desmearing process yields data equivalent to the pinhole measurements.25 Small-angle X-ray scattering (SAXS) measurements were also performed at Argonne National Laboratory, where the samples were placed in glass capillary tubes. Once mounted on the cold stage, the samples were placed in the monochromatic X-ray beam of sector 34ID-C at UNICAT. The SAXS patterns were measured using a CCD detector, located 1090 mm from the sample. The collected images were then integrated using the Fit2D program.26 Static light scattering (SLS) measurements were performed using Brookhaven Instruments BI-200 SM goniometer with a Lexel Argon-Ion Model 95 laser, λ ) 514 nm from Θ ) 40°-140°. The samples were placed in glass tubes with a path length of 1 cm and kept at 25 °C through the use of a constant-temperature (22) Long, G. G.; Allen, A. J.; Ilavsky, J.; Jemian, P. R.; Zschack, P. In Proc. 11th US National SRI Conf.; Pianetta, P., Arthur, J., Brennan, S., Eds.; AIP: Stanford, 1999; Vol. CP521, pp 183-187. (23) Lake, J. A. Acta Crystallogr. 1967, 23, 191-194. (24) Ilavsky, J.; Allen, A. J.; Long, G. G.; Jemian, P. R. Rev. Sci. Instrum. 2002, 73, 1660-1662. (25) Shah, S. A.; Ramakrishnan, S.; Chen, Y.-L.; Schweizer, K. S.; Zukoski, C. F. Langmuir 2003, 19, 5128-5136. (26) Hammersley, A. FIT2D V. 10.27, Reference Manual; European Synchrotron Radiation Facility: Grenoble, France, 1998.

f(q,τ) )

∫

∞

0

exp(-g1τ)dτ

(5) (6)

where C is the baseline, K is an instrumental constant, and g1 is the exponential decay constant. The above equation is valid when the scattering volume is much larger than the volume over which the particles are correlated, which is the case for the measurements reported here.

Results and Discussion Neutralization of Charge. As noted above, after the coating process, there are residual surface charges that need to be eliminated to produce hard particles. Electrophoresis measurements demonstrate the neutralization of the dispersion in ethanol by demonstrating that the electrophoretic mobility, ψ, goes through zero during titration with HNO3 (Figure 2). Note that at a concentration of 0.02 M, the particle electrophoretic mobility passes through zero and reaches an asymptotic value at higher acid concentrations. Evidence that the neutralization of particle charge alters particle interactions is found by measuring the viscosity of suspensions with different nitric acid concentrations at a fixed volume fraction. After a concentrated stock solution of TPM-coated particles in THFA was made, nitric acid was added and the viscosity at various shear stresses was measured. This process was repeated until further addition of nitric acid no longer changed the viscosity. With the addition of nitric acid, the flow curves shift to lower viscosity values and cease to change at nitric acid concentrations above 0.02 M. Figure 3 demonstrates that

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Figure 3. Low-shear viscosity as a function of acid concentration showing the elimination of the excess surface charge. All samples are HS-5 at a c ) 0.13 g/mL. Table 1. Particle Sizes of HS-5 Fumed Silica Suspensions before and after TPM Coating Rg2 [nm]

RH2 [nm]

TPM Coating

Solvent

97 ( 4 93 ( 4

74 ( 5 72 ( 5

no yes

aqueous ethanol

the zero-shear-rate viscosity reaches a plateau value for nitric acid concentrations greater than 0.02 M. The plateau viscosity for the TPM-coated silica indicates that once the repulsions are fully screened, the index-matching of the THFA and the TPM coating essentially eliminate van der Waals attractions.10 Dilute Suspension Characterization. The HS-5 fumed-silica suspensions were studied before and after coating to ensure the coating process did not change the particles. A comparison before and after coating in the dilute scattering regime, where there are no particleparticle interactions, showed negligible changes in the characteristic size of the particles. USAXS measurements provided a measure of Rg2 in the Guinier regime of scattering, while FOQELS gave the hydrodynamic radius RH2 using the Stokes-Einstein equation. The samples prior to the TPM coating were dispersed through ultrasonication in 0.1% aqueous sodium lauryl sulfate solution, while the TPM-coated particles were dispersed in ethanol for comparison. No appreciable changes were seen, as indicated in Table 1. A Debye plot, utilizing the absolute scattering of the suspensions by SLS, was constructed to determine the weight-average molecular weight, MW, Rg2, and second virial coefficient, A2, by extrapolating to zero angle and zero concentration on the basis of the following relationships

( ) ( )( )( )

Rθ A2NA 2 2 ) MWP(Θ) - 2 cP (Θ) K*c MW K* ) 4π2nc2

dn dc

2

1 λ

4

1 NA

(7) (8)

where RΘ is Raleigh ratio, K* is an optical constant, P(Θ) is the form factor, NA is Avogadro’s number, and dn/dc is the refractive index increment of the suspension with respect to particle concentration. A dn/dc value of 0.0103 mL/g was obtained by measuring the refractive index with a Milton Roy refractive index meter on the bulk suspension at various particle concentrations. The Debye plot for the HS-5 grade is shown in Figure 4 and the results are printed in Table 2, where it is seen that EH-5 particles are slightly smaller and there is only a small change in molecular weight between the three grades.

Figure 4. Debye plot of HS-5, c ) 0.002 (1), 0.003 (#), 0.004 (2), 0.005(+), 0.006 (b), and 0.009 (9) g/mL, where w ) 200 is the stretch constant. The open symbols show the extrapolated values for zero concentration (O) and zero angle (0). The lines show the fits to the extrapolated data for zero concentration (solid) and zero angle (dotted). Table 2. Results from the Debye Analysis of the Three Grades of TPM-Coated Fumed Silica in THFA M-5 HS-5 EH-5

Rg2 [nm]

MW ( × 108) [g/mol]

A2( × 10-15) [cm3]

53 ( 4 51 ( 4 46 ( 4

1.6 ( 0.2 2.0 ( 0.2 2.3 ( 0.2

2.9 ( 1 1.4 ( 1 3.0 ( 2

Comparing the values of Rg2 in Tables 1 and 2 shows a significant decrease in the size. The decrease in Rg2, following the solvent change from ethanol to THFA, was expected due to the final centrifugation employed to remove any large agglomerates because scattering is heavily dominated by large particles. A study of the particle form factor, P(q), from a dilute sample provides an opportunity to further characterize the system. An analytical form of the scattering of complex systems that contain multiple structural levels was developed by Beaucage.27,28 In this unified fitting approach, each structural level contains a Guinier regime leading to the determination of Rg and a power-law regime where the fractal dimension can be determined. Each level can be described by four parameters, G, B, H and Rg, dealing with scattering cross section, cross over to fractal scattering, fractal dimension, and particle size, respectively. Because fumed silica has two structural levels, one pertaining to the primary particles and the other to the aggregates, the unified fit can be written as given below

(

I(q) ) G2 exp

)

-q2Rg22 + 3

( ){[ ( )] } ( ) {[ ( )] }

B2 exp

-q

Rg12

2

3

-q

G1 exp

erf

qRg2

x6 q

3

+

erf

2

Rg12

3 H2

+ B1

qRg1

3 H1

x6 q

(9)

where the subscripts 1 and 2 represent the influence of the primary particles and fractal aggregates, respectively. The number of parameters describing the form factor can be reduced to five by assuming the scattering objects are composed of fractal aggregates of spherical primary particles of uniform size.27-29 This allows G1 and G2 to be related through the size of the primary particles, the fractal (27) Beaucage, G. J. Appl. Crystallogr. 1995, 28, 717-728. (28) Beaucage, G. J. Appl. Crystallogr. 1996, 29, 134-146.


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dimension of the aggregate, and the scattering cross section of the primary particle. This is accomplished by setting H2 ) Df, and the power law prefactor, B2, is related to G2, Df, and Rg2

( )( ) G2Df

B2 )

Rg2Df

Γ

Df 2

(10)

where Γ is the gamma function. This assumption also allows G1 and G2 to be related to each other by the number of primary particles in the aggregate, z

G2 ) zG1

(11)

For a mass fractal, z is a function of Rg2, Df, and the primary particle size

z)

{( ) [( Rg2 b

2

1+

)(

2 2 2+ Df Df

)]}

Df/2

(12)

where b is the primary particle characteristic size. By assuming the primary particles are uniform spheres

b)2

x

5 R 3 g1

(13)

In making these assumptions, eq 9 can be reduced to a function of G1, Rg2, Df, B1, Rg1, and H1. In assuming spherical primary particles, the unified model can be further constrained by the relationship

B1 )

2πG1S1 V12

(14)

where V1 is the volume of the primary particle and S1 is the surface area of the primary particle. This final constraint would reduce the number of fitting parameters to five (G1, Rg2, Df, Rg1, and H1). However, extensive testing indicated a five-parameter fit does not well-capture the scattering properties of the dilute suspensions of this study. As a result, the six-parameter unified model was used in fitting the data. This provides evidence that the scattering properties are not well described as originating from aggregates of uniform spheres. Reducing the unified fit model to five parameters did, however, offer additional insights, which will be discussed below. The scattering from the three different grades of TPMcoated fumed silica in THFA was explored using SLS, SBUSAXS, and SAXS. The scattering from the three techniques was combined by matching the intensities from the overlapping q regimes of the different techniques, and the resulting intensity curve shown in Figure 5 was fit with the unified model with Rg2 held to the value determined using the Debye plot, resulting in the parameters in Table 3. In fitting the scattering data, the intensities are in arbitrary units and were normalized by the zero q scattering intensity. SLS allowed measurements on more-dilute samples than SBUSAXS. As a result, correlations between particle centers of mass become important at low q in the SBUSAXS measurements. However, in the dilute SBUSAXS samples used, S(q) increases to unity for q > 0.05 Å-1. Therefore, the scattering for q > 0.05 Å-1 for those samples would be P(q). Describing the scattering as originating from mass fractals composed of nonuniform particles results in Rg1 (29) Hyeon-Lee, J.; Beaucage, G.; Pratsinis, S. E.; Vemury, S. Langmuir 1998, 14, 5751-5756.

Figure 5. Combined scattering from M-5 (4) 0.05 g/mL, HS-5 (]) 0.03 g/mL, and EH-5 (O) 0.06 g/mL. SLS data covers the small q regime, SBUSAXS covers the intermediate q regime and the large q regime is covered by SAXS. Some data points are removed for clarity. The lines represent the Beaucage model with the parameters in Table 2, M-5 (dotted), HS-5 (solid) and EH-5 (dashed). The M-5 curves are offset by a factor of 0.5 for clarity.

≈ 16 nm. This value is larger than estimates from transmission electron microscopy measurements by Philips CM-12 TEM of R1,TEM ≈ 10 nm (Figure 6). Following the assumption of uniform spheres and using TEM estimates of primary particle size, Rg1,TEM ) x3/5 R1,TEM ≈ 8 nm, produced a value half the size of the unified fit. The lack of correspondence between Rg1 and Rg1,TEM is a result of forcing the Beaucage model to fit the experimental data. TEM images show true primary particles are neither spherical nor uniformly sized. Characterization of the particles was not possible solely through TEM due to the interpenetration of the particles during the drying process leading to uncertainty as to which particles were physically entangled, and thus the size of the aggregate could not be determined by TEM. The power law coefficient describing the scattering of the primary particles, H1, can be related to surface fractal dimension DP

DP ) 6 - H1

(15)

where smooth surfaces have DP ) 2 and rough surfaces have 2 < DP < 3. From the results of the unified fit, there is some indication that the surfaces of the primary particles are rough, as seen in the TEM images and in agreement with previous studies.30 The characteristic power-law scattering regime of fractal particles described by Df is often used to describe the physical structure of how the primary particles form the larger aggregate, with an open fractal having a small Df and a more densely packed fractal having a larger Df. The magnitude of Df has been used to describe the process by which the fractal aggregates were made. Diffusion-limited cluster aggregation has a Df ≈ 1.8, while reaction-limited cluster aggregation has a Df ≈ 2.2. (30) Hurd, A. J.; Schaefer, D. W.; Martin, J. E. Phys. Rev. A 1987, 35, 2361-2364.

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Table 3. Parameters from the Beaucage Model to Fit the Scattering Data M-5 HS-5 EH-5

G1

B1 ( × 10-9) [Å-H1]

Rg1 [nm]

H1

Rg2 [nm]

Df

z

0.081 ( 0.002 0.072 ( 0.002 0.121 ( 0.003

1.8 ( 0.2 2.7 ( 0.4 2.0 ( 0.4

16 ( 1 16 ( 1 16 ( 1

4.0 ( 0.1 3.8 ( 0.1 3.9 ( 0.1

53 51 46

2.2 ( 0.1 2.4 ( 0.1 2.2 ( 0.1

43 44 33

that the power-law scattering in these experimental systems was severely limited by the relatively small difference in sizes of the aggregates compared with the primary particles. Thus, we are reluctant to use Df to explicitly define particle-formation mechanisms. We note that our description is based completely on using the Beaucage model to fit our scattering data. As shown in Figure 1, we are able to image the open nature of the fumed silica particles. The analysis of Jullien and coworkers32,33 suggests this would not be possible for Df > 2. Our conclusion is that a combination of the distribution of our aggregates and the limited difference between Rg1 and Rg2 allow us to image the open nature of the aggregates with TEM and extract Df > 2 from scattering. The parameters determined by the unified-fit model are average values due to the size distribution inherent to the process by which fumed silica is manufactured. A qualitative look at how the size distribution affects the measured values will now be considered. The size of the aggregate measured from X-ray and light scattering is the z-average radius of gyration Rgz2

Rgz22

∫0∞ m2NR(Rg2)Rg22dRg2 ) ∫0∞ m2NR(Rg2)dRg2 m≈

(16)

( ) Rg2 Rg1

Df

(17)

where NR(Rg2) is the number of particles of size Rg2 and m is the mass of aggregate. In doing the analysis, Df for each aggregate is assumed to be constant, along with the constraints discussed above in reducing the unified model to six parameters. To reduce the unified model to five parameters, the strictly spherical relationship of eq 14 is not followed. Instead, a geometric constant, C1, is used to relate B1 and G1

B1 ) C1G1

Figure 6. TEM images of M-5 (A), HS-5 (B), and EH-5 (C) with spheres of radii 10 and 15 nm represented by solid and dashed circles, respectively.

In a previous study, Hurd et al.30 determined a Df ≈ 1.8 for the same grades of fumed silica but the conditions of the measurements were different. Portions of the scattering curves were made with dispersions by ultrasonication, and others were loosely packed dry samples. Another set of measurements of water-dispersed fumed silica by Freltoft et al.31 found a Df ) 2.34, which is closer to the values of this study. Within experimental capabilities, the measurements performed in this investigation suggest the particles have nearly the same fractal dimension, close to that predicted for reaction-limited aggregation. On the other hand, one might expect larger values of Df if sintering occurs between the primary particles. These findings should be weighed with the understanding (31) Freltoft, T.; Kjems, J. K.; Sinha, S. K. Phys. Rev. B 1986, 33, 269-275.

(18)

which permits the scattering from the primary particles to be matched at the high q regime, giving insight into the effects of aggregate size polydispersity. Additionally, an assumption is made that the particle-size distribution is well-described by a log-normal distribution (LND). The LND is used to estimate the effects of size distribution on the characterization techniques, as opposed to using this distribution to definitively extract size distribution parameters. The LND is written

NR(Rg2) )

1 SL*Rg2x2π

(

[

exp

]

(ln Rg2 - ML)2 2SL2

)

(19)

SL2 µ ) exp ML + 2

(20)

σ ) xexp(SL2 + 2ML)[exp(SL2) - 1]

(21)

where µ is the arithmetic mean, σ is the standard deviation of the distribution, and ML and SL are variables relating


Langmuir, Vol. 20, No. 25, 2004 11197 Table 4. DLS Analysis for the Three Grades of TPM-Coated Fumed Silica in THFA D0 ( × 10-8) [cm2/s]

RH2 [nm]

βave

8.9 ( 0.5 9.1 ( 0.8 10 ( 0.5

42 ( 4 41 ( 5 37 ( 3

0.79 ( 0.08 0.80 ( 0.08 0.81 ( 0.08

M-5 HS-5 EH-5

from rotational diffusivity is negligible.34 As with the Rg2 measurements, RH2 is smaller after the transfer to THFA than when the TPM-coated particles were in ethanol. As mentioned above, the Rg2 and RH2 values are averages due to the inherent polydispersity of fumed silica particles. Using the same assumptions as described above, the time autocorrelation function, eq 5, can be calculated for a polydisperse system of hard spheres in the dilute limit through the following relationship

W2(τ) ) C +

Λ Figure 7. Comparison of the scattering from EH-5 (+) with the unified model using Rgz ) 46 nm, Df ) 2.2, and Rg1 ) 16 nm for different degrees of polydispersity, σ ) 0.1 (solid) and 0.4 (dashed). Some data points were removed for clarity.

µ and σ. In setting limits to the particle-size distribution, a minimum value of Rg2 ) 16 nm and a maximum value of Rg2 ) 75 nm were used. A further constraint on the distribution is the Rgz2 of the LND is held constant at the value listed in Table 2 because the value is accurately known from scattering. Thus, the average form factor can be calculated by the following equation and compared to the measured values of the EH-5 sample

〈P(q)〉 )

∫NR(Rg2)P(q,Rg2)dRg2 ∫NR(Rg2)dRg2

(22)

The results shown in Figure 7 demonstrate the lack of sensitivity of the form factor measurements to the polydispersity, as all the curves superimpose with the experimental data. DLS measurements provide another method for characterizing the suspension. The short-time diffusion coefficient Ds can be determined by

Ds )

ξ q2

(23)

where ξ is the initial decay rate. Assuming monodisperse, hard spheres in the dilute limit, Ds reduces to the StokesEinstein diffusivity, D0, which can be a measure of RH2 by the Stokes-Einstein relationship

Do )

kT 6πηcRH2

(24)

A least-squares fit on the measured autocorrelation function was used to determine ξ, and by using eq 23, Ds was calculated. D0, was extracted by extrapolating to zero concentration and was then used in the RH2 calculations using the Stokes-Einstein relationship, providing a characteristic hydrodynamic size of the aggregate listed in Table 4. The DLS measurements in this study were made at small q vectors (qRg2 < 1.5); thus, the influence

[

[

]

q2τ dRg2 ∫0∞ NR(Rg2)P(Rg2,Θ) exp - 6πηkT cβRH2

∫0

∞

NR(Rg2)P(Rg2,Θ)dRg2

]

2

(25)

where P(Rg2,Θ) is the intensity of the form factor as calculated using the LND and the unified fit model at an angle, Θ, where the DLS measurements were taken, and β is a constant that relates Rg2 to RH2

β)

RH2 Rg2

(26)

The experimental measurements of Rg2 and RH2 are averages, and their ratio is not the same as the ratio of Rg2 and RH2 of individual particles due to the weighting of the moments of the size distributions as described by Pusey et al..35 Thus, while the ratio of βave remains the same, β increases as the width of the distribution is increased

βave )

〈RH2〉 Rgz2

(27)

In this analysis, the Rgz value is held constant along with the measured RH2 value reported in Table 4. In Figure 8, it can be seen that the autocorrelation function is also not extremely sensitive to the polydispersity, allowing only a qualitative understanding that there is not a large degree of polydispersity in the system. In the absence of a large degree of polydispersity, β ≈ βave and the value measured for the fumed silica particles is consistent with the expected value for cluster aggregation, where β ) 0.765 and 0.831 for diffusion-limited and reaction-limited cluster aggregation, respectively.36 In summary, TEM micrographs demonstrate that the particles investigated consist of open networks of fused subunits. By applying the Debye analysis and Beaucage model to dilute suspension scattering, the particles have (32) Tence, M.; Chevalier, J.; Jullien, R. J. Phys. (Paris) 1986, 47, 1989-1998. (33) Sigrist, S.; Jullien, R.; Lahaye, J. Cem. Concr. Compos. 2001, 23, 153-156. (34) Lindsay, H. M.; Klein, R.; Weitz, D. A.; Lin, M. Y.; Meakin, P. Phys. Rev. A 1988, 38, 2614-2626. (35) Pusey, P. N.; Rarity, J. G.; Klein, R.; Weitz, D. A. Phys. Rev. Lett. 1987, 59, 2122. (36) Lattuda, M.; Wu, H.; Morbidelli, M. J. Colloid Interface Sci. 2003, 268, 96-105.

11198


Figure 8. Autocorrelation function for varying widths of the LND, σ ) 0.1 (solid) and 0.4 (dashed), compared with the measured values of EH-5 (+) at c ) 0.06 g/mL and q ) 0.0025 Å-1.

grade-specific radii of gyration with subunits approximately one-third the size of the aggregate. The particles have a fractal dimension of 2.2-2.4 with RH2/Rg2 ≈ 0.8, consistent with the aggregate structure. Dynamic light scattering and the angular dependence of static-lightscattering intensity provide limited information about the particle-size distribution. Dense Suspensions: Compressibility. The scattering from the three different grades of hard, structured particles was also measured at different particle concentrations to gain insight concerning the change in the interparticle correlations as the suspension becomes denser. S(q) values were calculated as described above. Figure 9 shows that the shape of S(q) is independent of the different grades, with a low q plateau and a steplike function to an S(q) value of 1. Due to the experimental limitations discussed above, S(q) values from SBUSAXS are not reported below q ≈ 0.05 Å-1. At low particle concentrations, S(q) starts with a value of unity. As the concentration increases, there is a suppression of S(q) and an increase in the value of q characterizing where S(q) goes to one. At sufficiently high suspension density, there is an upturn in the low q region of S(q), a continued suppression of S(q) at intermediate q, and a saturation value of q where S(q) goes to 1. For uniform spheres, S(q) will show damped oscillations about S(q) ) 1. The first peak in S(q) occurs at q*. The value of q* is associated with the average particle spacing

Smith and Zukoski

and, at high density, the first peak occurs at a value near qa ≈ 3.5 where a is the particle radius. The magnitude of S(q*) increases with density and is associated with the coherence of the first shell of nearest neighbors. Figure 9 shows that, for the structured particles studied here, there are no peaks in S(q). At intermediate q, the continuous decrease of S(q) with increasing concentration indicates that in the range of qRg2 ≈ 2-6, particle density fluctuations are suppressed. However, if q* is approximated as the value of q when S(q) approaches unity, Figures 9 and 10 show q* increases with particle concentration and begins to saturate near a value of qRg2 ≈ 8-9. This value is greater than expected for dense packing of hard particles with equivalent radii. The saturation at larger qRg2 values suggests the aggregates are able to experience larger density fluctuations than hard spheres of equivalent Rg2, which could be explained by the open geometry of the particles allowing interpenetration in the more-concentrated suspensions. As q goes to zero, for suspensions of uniform particles, S(q) is a measure of the suspension compressibility

1 dΠ 1 ) S(0) kT dF

(28)

where Π is the suspension osmotic pressure. For narrow size-distribution suspensions, decreases in S(0) reflect increased crowding and a diminution of density fluctuations in the suspension. In the low φ limit, 1/S(0) ) 1 + 2A2F, where A2 is a measure of the pair-excluded volume and is equal to four times the volume of a single sphere for hard spheres, A2HS ) 4Vp. Figure 11 shows 1/S(0) as a function of silica volume fraction. The values of the three different grades of fumed silica superimpose, an indication that the particles pack in a similar manner. Figure 11 also shows the linear behavior up to a φS ≈ 0.08 by which the slope R can be determined using a least-squares fit. A thermodynamic size, RTH, of the fumed silica particles can be extracted by relating the slope to A2HS where

( ) (

2A2HSF ) R

)( )

NAc 4πRTH3 NAc )8 MW 3 MW

(29)

with the results shown in Table 5. These values compare with those reported above, indicating that the characteristic size governing the thermodynamic behavior is that

Figure 9. Static structure-factor curves plotted as a function of qRg2 at different particle concentrations for M-5 [0.18 (2); 0.34 (4) g/mL], HS-5 [0.18 (×); 0.25 (+); 0.32 (0) g/mL] and EH-5 [0.18 (-); 0.25 (#); 0.37 (O) g/mL]. The gap in the data around qRg2 ) 2 is due to interparticle correlations in the SBUSAXS measurements of P(q).


Langmuir, Vol. 20, No. 25, 2004 11199

If an assumption is made that the neutralized suspensions behave like hard spheres, a direct comparison using no adjustable parameters of the thermodynamics of the fumed silica suspensions with hard spheres is made through the Carnahan-Starling37 equation of state for monodisperse hard spheres

Π 1 + φ + φ 2 - φ3 ) FkT (1 - φ)3

(31)

The dimensionless inverse osmotic compressibility, 1/S(0) ) 1/kT(dΠ/dF), is determined by Figure 10. Experimental data showing the concentration dependence of qRg2 when S(q) ) 0.9 for M-5 (4), HS-5 (0), and EH-5 (O).

Figure 11. Osmotic compressibility of M-5 (2), HS-5 (9), and EH-5 (b) as a function of silica concentration compared with the Carnahan-Starling equation of state based on volume fraction of silica (solid line) and RTH (dashed line). The open symbols represent the minimum value in S(q) for those samples which showed an upturn. Table 5. Thermodynamic Size and Excluded Volume Comparison between Hard Spheres and Hard Structured Particles M-5 HS-5 EH-5

Rg2 [nm]

RTH [nm]

A2*,R1

A2*,R2

53 ( 4 51 ( 4 46 ( 4

40 43 46

20 9 20

0.5 0.3 0.9

of the aggregate. Table 5 also lists values for A2*,R1, and A2*,R2

A2* )

A2 16 3 πa 3 i

(30)

where A2*,R1 and A2*,R2 are the values for comparison of hard sphere behavior on the basis of primary particle and aggregate size, respectively, with the radius of hard spheres related by ai ) x5/3 Rgi. A2*,R1 shows that, in comparison with hard spheres with size Rg1, fumed silica particles experience a much larger excluded volume, which may not be surprising because the primary particles are fused together and unable to move independently of the rest of the primary particles of the aggregate. On the other hand, the excluded volume is slightly smaller than expected for hard spheres of size Rg2, as reflected in a comparison of Rg2 and RTH and in A2*,R2 being less than one. This analysis, within a fairly large experimental uncertainty, suggests that the particles may be able to interpenetrate due to the open structure.

1 1 dΠ 1 + 4φ + 4φ2 - 4φ3 + φ4 ) ) S(0) kT dF (φ - 1)4

(32)

Figure 11 shows the fumed silica suspensions in comparison with the Carnahan-Starling equation of hard spheres, where S(0) values are shown for S(q) at the minimum in S(q) for those measurements with upturns in the low q region along with an extrapolated value of S(0). Lines are drawn to indicate the behavior expected if the particles behaved like hard spheres with a volume fraction of the silica and if the particles behaved like particles that have a volume fraction based on RTH. The data lies between these two estimates, indicating again that the particles have thermodynamic properties that are intermediate between small, independent, hard particles and larger, impenetrable, hard spheres. Due to the fixed geometry of the fumed silica, the primary particles are not able to freely move, but are fused together. This does not allow the primary particles to behave like a hard-sphere suspension because they cannot pack in a manner that minimizes the free energy of the suspension. Thus, it is not surprising that these suspensions have a higher osmotic incompressibility than hardsphere suspensions of size Rg1 at the same volume fraction of silica. When the experimental compressibilities are compared with those predicted for hard spheres of size Rg2, the diminished osmotic compressibility on the comparison of the aggregate size is not expected to be due to polydispersity37-39 because the qualitative understanding of the polydispersity discussed above would not account for such a large deviation from the Carnahan-Starling equation of state. Instead, considering the open geometry of fumed silica, it would be anticipated the particles could interpenetrate, increasing the compressibility of suspensions of aggregates above the expected hard-sphere values of equivalent aggregate volume fraction. As noted above, there is an upturn in S(q) in the low q regime as the concentration of particles is increased. Because dS(0)/dq is zero or positive at low concentrations but becomes negative at larger concentrations, these findings may result from a volume-fraction-dependent interaction. A possible explanation could be that an effective attraction is developed on the basis of hard interactions, structured geometries, and interpenetration that results in clusters enhancing long-wavelength density fluctuations. Particles sensing weak attractions experience larger density fluctuations than do hard particles. As a result, S(0) takes on a value that is larger than experienced by particles of the same size experiencing only volumeexclusion interactions. For attractive systems in the zero q limit, dS(q)/dq is always negative. In repulsive or volumeexclusion situations in the zero q limit, dS(q)/dq is zero (37) Vrij, A. J. Chem. Phys. 1979, 71, 3267-3270. (38) Vrij, A. J. Chem. Phys. 1978, 69, 1742-1747. (39) van Beurten, P.; Vrij, A. J. Chem. Phys. 1981, 74, 2744-2748.

11200


or positive. In the low volume fraction limit of these fumed silica suspensions, dS(q)/dq is zero or positive, suggesting that attractions, if they exist, are very weak. Attractive suspensions modeled with square-well potentials40,41 have been shown to deviate from hard-sphere suspensions, but would not capture the experimental measurements of S(0) and A2 of this study. A second possible explanation for the upturn in S(q) could be the signature of correlations between voids in the suspension. A Debye-Beuche-type analysis has been used in other depletion-gelation colloidal systems to gain understanding of the origin of the upturn in S(q).42 The experimental limitations of this study did not allow lower q values to be explored, so it is difficult to come to a complete understanding of the upturn in the low q regime. While there is not a complete understanding of the scattering in the low q regime, the data shows a convincing correlation between the particles packing in a similar manner with respect to the concentration of fumed silica in the suspension. This can be explained by the reasoning that these three grades of fumed silica have nearly the same microstructure as measured by Df. Thus, it would be expected that the particles are able to arrange in a similar manner, despite the differences in Rg2. As with hard spheres, it is not the size of the sphere but its shape that governs the packing ability of a concentrated suspension. Conclusion Suspensions of nearly hard particles have been synthesized by TPM-coating fumed silica, dispersing the particles in THFA, and eliminating excess surface charge. The properties of dilute suspensions have been characterized in detail by TEM, X-ray scattering, and static and dynamic light scattering. Clusters of three different sizes have been characterized, while primary particle sizes in the three samples are found to be nearly identical. In addition, W2(τ) was found, within experimental uncertainty, to be a single exponential in the dilute limit. Attempts to predict W2(τ) with a particle-size distribution indicated the particle-size distributions were narrow, which was in agreement with the investigation of the effects of size distribution on the angular dependence of the form factor. (40) Heyes, D. M.; Aston, P. J. J. Chem. Phys. 1992, 97, 5738-5748. (41) Ramakrishnan, S.; Zukoski, C. F. J. Chem. Phys. 2000, 113, 1237-1248. (42) Shah, S. A.; Chen, Y.-L.; Ramakrishnan, S.; Schweizer, K. S.; Zukoski, C. F. J. Phys.: Condens. Matter 2003, 15, 4751-4778.

Smith and Zukoski

The static structure factors of suspensions of these particles were measured over a wide scattering vector range. At low volume fraction, the second virial coefficient indicated the particles had excluded volumes that were slightly smaller when compared to impenetrable spheres with sizes corresponding to the average particle radius of gyration. dS(q)/dq in the low q limit is zero or positive in this low-volume-fraction range, and as the concentration of particles is raised, the scattering at low q is enhanced such that dS(0)/dq becomes negative. The S(0) passes through a minimum that continuously decreases with increasing particle concentration and reaches a value of unity at a value of qRg2 greater than expected for impenetrable hard spheres. The value of S(0) is found to depend on the volume fraction of silica for the different grades but in a manner that differs from the expected hard sphere behavior. S(0) lies between values predicted for hard spheres with sizes of the aggregate and hard spheres of size Rg1. These results combine to suggest the particles can interpenetrate due to their open geometry but have packing characteristics that are independent of particle size. Acknowledgment. We thank Joachim Floess for his insights and discussions, along with Cabot Corporation for the use of their Cab-O-Sil fumed silica products. We thank P. R. Jemian, J. Ilavsky, S. Boutet, and I. K. Robinson for their assistance in gathering the scattering data. The UNICAT facility at the Advanced Photon Source (APS) is supported by the U. S. DOE under Award No. DEFG02-91ER45439, through the Frederick Seitz Materials Research Laboratory at the University of Illinois at Urbana-Champaign, the Oak Ridge National Laboratory (U. S. DOE Contract No. DE-AC05-00OR22725 with UT-Battelle LLC), the National Institute of Standards and Technology (U. S. Department of Commerce), and UOP LLC. The APS is supported by the U. S. DOE, Basic Energy Sciences, Office of Science under Contract No. W-31-109-ENG-38. We acknowledge the Center for Microanalysis of Materials, University of Illinois, which is partially supported by the U. S. Department of Energy under Grant No. DEFG02-91-ER45439 where imaging of particles was made possible. Finally, we acknowledge financial support from the Nanoscale Science and Engineering Initiative of the National Science Foundation under NSF Award No. DMR-0117792. LA0489864

Hard Structured Particles: Suspension Synthesis, Characterization

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