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Direct Visualization of Long-Range Heterogeneous Structure in Dense Colloidal Gels Priya Varadan and Michael J. Solomon* Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136 Received July 26, 2002. In Final Form: September 14, 2002 By means of confocal laser scanning microscopy, we directly visualize and quantify the local spatial structure of dense colloidal gels of varying volume fraction formed by short-range attractive interparticle interactions. We find that the short-range structure of the gels is similar to that of dense amorphous hard-sphere systems. However, the long-range structure is characterized by voids and density correlations that differ remarkably from amorphous hard spheres and glasses formed through repulsive interactions. Quantification of the volume fraction dependence of the gel isothermal compressibility and Voronoi volume distribution establishes that long-range structural heterogeneity is a distinguishing feature of the static structure of dense colloidal gels formed by short-range attractions.
Amorphous colloidal suspensions with attractive interparticle interactions of sufficient strength often exhibit the dynamic transition of gelation. Measurements of gel structure and dynamics are relevant to the glass transition1,2 and have implications for the sol-gel processing of ceramic, coating, and membrane materials.3 Observations of the slowing down of dynamics in colloidal gels have led to comparisons with the glass transition in dense particulate materials.4-8 To explain these observations, recent mode coupling theory calculations4,6 have postulated gel structure consistent with equilibrium statistical mechanics. This crucial assumption is unproven because scattering measurements have variously identified weak gel formation with nonequilibrium aggregation of fractal clusters,9 with spinodal decomposition,10 or with frustrated phase separation due to cluster nucleation and growth.11 Previous studies have detected long-range critical density fluctuations in colloidal gels by low-angle light scattering.12 Here we resolve additional structural features of dense colloidal gels by direct visualization with confocal laser scanning microscopy (CLSM). We identify long-range density fluctuations and spatial structural heterogeneity as distinguishing features of the static structure of dense colloidal gels. We examine the extent of agreement of these quantities with hypotheses of equilibrium structure. * To whom correspondence may be addressed. E-mail: mjsolo@ umich.edu. (1) Anderson, V. J.; Lekkerkerker, H. N. W. Nature 2002, 416, 811815. (2) Debenedetti, P. G.; Stillinger, F. H. Nature 2001, 410, 259-267. (3) Brinker, C. J.; Scherer, G. W. Sol-gel science: the physics and chemistry of sol-gel processing; Academic Press: Boston, 1990. (4) Pham, K. N.; Puertas, A. M.; Bergenholtz, J.; Egelhaaf, S. U.; Moussaid, A.; Pusey, P. N.; Schofield, A.; Cates, M. E.; Fuchs, M.; Poon, W. C. K. Science 2002, 296, 104-106. (5) Solomon, M. J.; Varadan, P. Phys. Rev. E 2001, 63, 051402-1051402-10. (6) Fabbian, L.; Gotze, W.; Sciortino, P.; Tartaglia, P.; Thiery, F. Phys. Rev. E 1999, 59, R1347. (7) Puertas, A. M.; Fuchs, M.; Cates, M. E. Phys. Rev. Lett. 2002, 88, 098301 098301-098304. (8) Cipelletti, L.; Manley, S.; Ball, R. C.; Weitz, D. A. Phys. Rev. Lett. 2001, 86, 6042-6044. (9) Carpineti, M.; Giglio, M. Phys. Rev. Lett. 1992, 68, 3327-3330. (10) Verhaegh, N. A. M.; Asnaghi, D.; Lekkerkerker, N. W.; Giglio, M.; Cipelletti, L. Physica A 1997, 242, 104-118. (11) Grant, M. C.; Russel, W. B. Phys. Rev. E 1993, 47, 2606-2614. (12) Poon, W. C. K.; Haw, M. D. Adv. Colloid Interface Sci. 1997, 73, 71-126.
CLSM offers the new possibility of directly visualizing local colloidal properties such as crystallization,1,13,14 real space structure,15 and dynamic heterogeneity16 of hard sphere glasses.17,18 Here we report our use of CLSM to characterize the extent of long-range density correlations and spatial structural heterogeneity in colloidal gels formed through short-range attractive interactions. Structural heterogeneity refers to a broad distribution in measures of local structure such as voids and clusters. By visualization of the full distribution of structures, CLSM provides new information that can be used to identify rare structures that may have important implications for gel dynamics. For example, theoretical studies suggest a possibly significant role for certain void structures in mediating the mechanical stability and glass-forming capability of dense systems.19 In addition, practical implications of structural heterogeneity are that a small number of voids and agglomerates can compromise the end-use properties of materials formed by sol-gel processing due to the participation of such defects in stressinduced cracking and mechanical failure.20 Alternatively, if structural heterogeneity can be controlled, materials with useful porous structure may be engineered.3 By means of CLSM, we investigate dense gels of colloidal particles interacting through short-range attractive forces that are induced by a change in temperature.5,11,21-24 Earlier light scattering studies with such so-called sticky or adhesive spheres demonstrated that their low volume fraction (φ < 0.10) structure is consistent with that of (13) Gasser, U.; Weeks, E. R.; Schofield, A.; Pusey, P. N.; Weitz, D. A. Science 2001, 292, 258-261. (14) de Hoog, E. H. A.; Kegel, W. A.; Blaaderen, A. v.; Lekkerkerker, N. W. Phys. Rev. E 2001, 64, 02147 02141-02149. (15) Blaaderen, A. v.; Wiltzius, P. Science 1995, 270, 1177-1179. (16) Kob, W.; Donati, C.; Plimpton, S. J.; Poole, P. H.; Glotzer, S. C. Phys. Rev. Lett. 1997, 79, 2827-2830. (17) Kegel, W. A.; Blaaderen, A. v. Science 2000, 287, 887-888. (18) Weeks, E. R.; Crocker, J. C.; Levitt, A. C.; Schofield, A.; Weitz, D. A. Science 2000, 287, 627-630. (19) Sastry, S.; Debenedetti, P. G.; Stillinger, F. H. Phys. Rev. E 1997, 56, 5533-5543. (20) Lange, F. F. J. Am. Ceram. Soc. 1989, 72, 3-15. (21) de Kruif, C. G.; Rouw, P. W.; Briels, W. J.; Duits, M. H. G.; Vrij, A.; May, R. P. Langmuir 1989, 5, 422-428. (22) Rouw, P. W.; Wouterson, A. T. J. M.; Ackerson, B. J.; de Kruif, C. G. Physica A 1989, 156, 876-898. (23) Verduin, H.; Dhont, J. K. G. J. Colloid Interface Sci. 1995, 172, 425-437. (24) Rueb, C. J.; Zukoski, C. F. J. Rheol. 1997, 41, 197-217.
10.1021/la026303j CCC: $25.00 © 2003 American Chemical Society Published on Web 10/17/2002
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percolating fractal aggregates.5,25 In this study, we extend the range of volume fraction studied from the fractal to the dense regime (φmax ) 0.50). Fluorescent core-shell colloidal silica (diameter obtained by scanning electron microscopy, 832 nm; core diameter, 420 nm; polydispersity, 4.2%; dye, fluorescein isothiocyanate) were synthesized by the procedure of Van Blaaderen and Vrij26 and grafted with octadecyl chains by the procedure of Van Helden and co-workers.27 The particles were dried under a continuous stream of nitrogen for 2 days at 60 °C prior to use. The suspensions were prepared by dispersing known amounts of dried fluorescent silica in hexadecane. In addition to minimal van der Waals interactions due to the approximate refractive index match between silica (1.4527) and hexadecane (1.434), the short-range interaction is determined by the temperaturedependent conformation of the surface grafted chains. The contour length of these chains (∼2 nm) is small relative to the colloid size (ratio ∼1:200). The fluorescent particles underwent thermoreversible gelation at ambient temperature (25 °C) when dispersed in hexadecane at high volume fractions (φ > 0.25). Quiescent gelation of the silica-hexadecane suspensions was achieved by cooling from T > 50 °C and allowing the sample to equilibrate at ambient temperature for 15 min. This procedure ensured reproducible gel microstructure as determined by CLSM measurements. CLSM was performed using a Biorad MRC 600 in fluorescence mode, mounted on a Nikon inverted microscope with an oil-coupled 100 × 1.4 N.A. objective. The excitation wavelength was 0.488 µm. The objective was focused in the sample at least 10 µm away from the glass coverslip, to avoid wall effects. Images were acquired at a rate of 1 frame/s in a viewing volume of 62 × 41 × 20 µm3. Typically, stacks of images at step intervals of 100 nm in the z-direction were acquired in 3-4 min. Particle tracking and density profiling demonstrated that these dense gels were of sufficient mechanical strength that no sedimentation occurred for intervals (t ∼ 2 h) much larger than the CLSM measurement times. Quantitative 3D image processing was performed on the image volumes to extract the centroids of the particles by adaptation of the 2D methods of Crocker and Grier.28 Briefly, this procedure involved background subtraction and Gaussian filtering of the images and location of the particle centers in 3D based on regional intensity maxima. Subsequently, the locations of the centers were refined to subpixel accuracy based on moments of the local intensity distribution. Particle centroids were resolved with an error of (20 nm in the lateral (xy) axes and (30 nm in the axial (z) direction. Here, only gel measurements are reported. Characterization of equilibrium fluid state properties at high temperature was not possible due to sedimentationinduced displacement of the dispersed colloids. Such submicrometer displacements were absent in gels but detectable in fluids given the precise spatial resolution of the quantitative image processing ((20 nm). However, the equilibrium fluid structure of dilute thermoreversible gels of smaller particles ( 5) for varying φ by reporting in Figure 3 the ratio of meansquared fluctuation in the number of particles to the mean number of particles, (〈N2〉 - 〈N〉2)/〈N〉. This quantity, when determined in the macroscopic limit, is related to the integral over all space of the pair-correlation function g(r), or equivalently, the low-scattering vector limit of the structure factor. It is also directly related to the gel isothermal compressibility.29 The low-scattering vector limit of the structure factor for model colloid-polymer systems has been recently reported by Zukoski and coworkers by scattering turbidity measurements.30 Although scattering and osmometry measurements can be used to determine this measure of compressibility, we pursue the direct measurement of (〈N2〉 - 〈N〉2)/〈N〉 made possible by (30) Ramakrishnan, S.; Fuchs, M.; Schweizer, K. S.; Zukoski, C. F. Langmuir 2002, 18, 1082-1090.
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the CLSM visualizations. The mean square of particle number fluctuations was determined by subdividing six independent 62 × 41 × 20 µm3 image volumes into bins of cubic volume 5 × 5 × 5 µm3 or 10 × 10 × 10 µm3. (Application to bins of different cubic volume ascertains the extent to which the extracted compressibility conforms to the macroscopic limit.) To obtain satisfactory statistics, 60-100 cubic volumes were used. To better understand the large magnitude and φ-dependence of the observed fluctuations, we compare the experimental results to theoretical predictions for Baxter’s adhesive sphere model.31 This model incorporates shortrange attractive interactions through the interaction parameter, τ. This simple, single parameter representation of the interaction potential of spheres possessing excluded volume, and surface adhesion has been used to model gels with short-range attractive interactions.11 The curves show the effect of τ on the volume fraction dependence of (〈N2〉 - 〈N〉2)/〈N〉 computed from the Percus-Yevick equation.32 Note that the solid curve corresponds to the hard sphere limit (τ f ∞).Comparison of theory and experiment indicates the profound difference between the long-range structure of gels formed through net attractive interactions and that of amorphous hard sphere fluids. The gels studied here differ substantially from hard sphere glasses, for which previous CLSM studies reported no long-range correlations.15 Simulations of polymer colloid mixtures with attractive interactions, using a different measure of density fluctuations, have quantified enhanced fluctuations that were associated with a possible transition to a metastable colloidal liquid.33 The magnitude of τ consistent with our fluctuation data at this temperature agrees well with a previous light scattering study for dilute suspensions in the stable, fluid phase.11 Given that the Baxter adhesive sphere model has recently been shown to yield poor predictions of the equilibrium phase behavior of polymer colloid mixtures,34 quantitative comparison of the measurements to the model is not warranted. Nevertheless, our qualitative comparison demonstrates that nonequilibrium processes need not be invoked to explain the dramatic increase in number density fluctuations observed at low φ. Figure 3 thus supports recent theoretical and computational studies of gel dynamics that assume gels formed from short-ranged interactions adopt equilibrium structure at the ergodicity transition.4,6,7 A direct measure of the heterogeneity of the gel structure is the distribution of Voronoi polyhedra (VP) volume obtained via Voronoi tessellation. Voronoi tessellation is a powerful tool of computational geometry that has been broadly applied in diverse areas such as protein bioinformatics, geology, and astronomy. 2D and 3D Voronoi analyses have been applied to characterize the structure of amorphous systems.35-37 Here, Voronoi tessellation is performed in 3D using the program for generation of convex hulls by Ken Clarkson.38 The Voronoi diagram uniquely partitions space into volumes associated with each particle center. The Voronoi polyhedron associated with a particular particle consists of all points less distant from that particular particle than any other particle. The (31) Baxter, R. J. J. Chem. Phys. 1968, 49, 2770-2774. (32) Regnaut, C.; Ravey, J. C. J. Chem. Phys. 1989, 91, 1211-1221. (33) Soga, G. K.; Melrose, J. R.; Ball, R. C. J. Chem. Phys. 1999, 110, 2280-2288. (34) Ramakrishnan, S.; Fuchs, M.; Schweizer, K. S.; Zukoski, C. F. J. Chem. Phys. 2002, 116, 2201-2212. (35) Cohen, M. H.; Grest, G. G. Ann. N.Y. Acad. Sci. 1981, 198-209. (36) Earnshaw, J. C.; Robinson, D. J. Phys. Rev. Lett. 1994, 72, 36823685. (37) Voloshin, V., P.; Naberukhin, Y., I.; Medvedev, N., N. J. Chem. Phys. 1995, 102, 4981-4986. (38) Clarkson, K. Available at http://cm.bell-labs.com/netlib/voronoi.
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Figure 4. Probability distribution of VP volumes for gels of varying φ. The abscissa is a dimensionless measure of the Voronoi free volume. The error bars shown are the standard deviation for analysis of five independent image volumes. The inset plot shows the measured standard deviations of the experimental data and the standard deviations of randomly packed hard spheres generated by the computations of Jullien et al.39
set of all Voronoi polyhedra fills space. We focus on the VP volume distribution since discrete voids are clearly visible in Figure 1. An alternative approach of characterizing the cluster size distribution would provide complementary information at lower volume fractions; however, it would be less revealing at higher volume fractions. Figure 4 is the first application of VP volume distributions to characterize the spatial structure of a colloidal gel. The peak value of the distribution shifts toward lower Voronoi volumes with increasing particle volume fraction since the average volume available to each particle decreases. At low volume fraction, we find surprisingly long tails in the VP volume distribution. To our knowledge, these long tails have no counterpart in prior simulations or experiments of molecular or colloidal systems. The standard deviation and skewness of the distributions quantify structural heterogeneity. The standard deviation in VP volume for the gels is compared to simulation results for randomly packed hard spheres39 in the inset of Figure 4. The gel standard deviation is significantly greater than that reported for disordered hard sphere systems, particularly at low volume fraction. While colloid polydispersity may broaden the distribution to a certain degree, the larger standard deviation is most likely reflective of real differences in spatial structure between gels with short-range attractive interactions and amorphous hard spheres. (39) Jullien, R.; Jund, P.; Caprion, D.; Quitmann, D. Phys. Rev. E 1996, 54, 6035-6041.
Letters
The low volume fraction gels also exhibit highly skewed (asymmetric) volume distributions with long tails. Recently, VP volume distributions from molecular dynamics simulations40 of glass-forming polymer melts have been shown to collapse onto a universal curve when scaled by the standard deviation. We find that the experimental VP volume distributions for φ g 0.40 obey this scaling, while gels at lower φ that display highly skewed distributions do not. This comparison demonstrates the extent to which the VP distributions can be explained by existing theory and simulation. We find that the coefficient of skewness for these distributions varies from 2.2 for φ ) 0.26 to 0.45 for φ ) 0.50. The skewness quantifies the long tails in the VP distributions, which extend to 15 particle volumes for φ ) 0.26 and are linked to structural heterogeneity. Figures 3 and 4 together provide experimental evidence for significant cooperative effects in dense colloidal gels formed through attractive interactions since voids of dimension as large as 5-10 particle diameters pervade the system. However, in contrast to hard sphere colloidal suspensions, in which repulsive excluded volume leads to caging and vitrification, here we have shown that the adhesive contact of sticky spheres leads to clusters and voids. The relationship between structural heterogeneity quantified in this study and the arrested dynamics reported in recent studies of a similar system4,7 warrants further investigation. The origin of long-range number density fluctuations could be the result of spinodally decomposed structures and mechanical instability; however, this hypothesis is inconsistent with the images and the VP volume distributions since they reveal no characteristic scale, as might be anticipated even for the earliest stages of spinodal decomposition. Moreover, we detect no evidence of particle ordering, even on local scales, the observation of which would support incipient phase instability due to crystal nucleation as the trigger for longrange inhomogeneity. Finally, the observed consistency of the long-range structure with equilibrium statistical thermodynamics is intriguing and points to a smooth evolution in structure at the transition from fluidlike properties at high temperature to the arrested dynamics and nonergodic properties observed at ambient temperature. Thus, at the gelation transition long-range density fluctuations and voids of broad volume distribution are the dominant structural features of dense colloidal gels, and it appears that mechanistic descriptions of gelation should account for them. Furthermore, the observed structural heterogeneity likely contributes to the unusual rheology and mechanical properties of colloidal gels. Acknowledgment. This work was supported by NSF CTS (9813824, 0093076, 0116331), NASA Microgravity Program (NAG3-2356), and TAPPI Foundation. We thank A. Herzog and M. Elsesser for help with synthesis, K. Ganesan for assistance with the 3D image analysis, and S.C. Glotzer, J.F. Douglas, R. Lionberger, R. Ziff, and R. Larson for valuable discussions. LA026303J (40) Starr, F. W.; Sastry, S.; Douglas, J. F.; Glotzer, S. C. Phys. Rev. Lett. 2002, 89, 125501-125504.
