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Micromechanical Behavior of Adhesive Granular Silica Layers: Structure Deformation V. I. Uricanu and M. H. G. Duits* Physics of Complex Fluids Group, UniVersity of Twente, Faculty of Science and Technology, associated with the J.M. Burgerscentrum for Fluid Mechanics, and Institute of Mechanics, Processes and Control-Twente (IMPACT), P.O. Box 217, 7500 AE Enschede, The Netherlands ReceiVed March 21, 2006. In Final Form: June 12, 2006 We studied the mechanical behavior of packed layers of 1-µm-sized silica particles immersed in liquids, upon indentation with a 10-µm glass sphere, attached to the cantilever of an atomic force microscope (AFM). Simultaneously, a confocal scanning laser microscope (CSLM) was used to study the deformations in the material. Our liquids consisted of (nearly) refractive-index-matching water-DMSO mixtures. Particle layers were formed by sedimentation in normal gravity. In the absence of (added) electrolyte, the collective behavior of the layer is reminiscent of that of a simple liquid. Crystal-like structures were observed, with the individual particles showing positional fluctuations. Carefully adding 2 wt % LiCl to this system leads to the formation of a weakly aggregated network, in which the crystal-like order gets lost and the particles lose their mobility. On indenting into these aggregated layers, the CSLM recordings showed imprints that closely resembled the size and shape of the indenter. A more accurate inspection of the structural changes was allowed after localizing all silica particles in three dimensions. Calculated local concentrations and coordination numbers showed that even at the level of these highly local quantities, no deformation gradients could be observed in the vicinity of the probe. Particle image velocimetry analysis suggested that deformation occurs mostly in the lateral directions. On pulling the indenter out, adhesion between the silica particles and the glass indenter became manifest via a distortion of the initially spherical dent and lower coordination numbers under the dent. Together all these behaviors indicate that the aggregated layers behave like yield-stress materials, which are solidlike up to a critical stress and liquidlike above it. The results of this study also illustrate the potential of the AFM-CSLM combination to study the detailed 3D deformation in other types of systems, like granular packings or more open particle networks.
1. Introduction Understanding the mechanical behavior of aggregated colloidal particle systems is of importance for several areas in materials research, from the rheology of aggregated suspensions to the mechanical stiffness of concentrated solids, like aggregated sediments on a filter. In many cases the practical interest is in the macroscopic mechanical behavior, but to understand and eventually control this, one has to study in more detail how the material deforms, in relation to the structure of the network and the forces acting between the particles. Up till now, still relatively few papers have appeared1-7 that address the relation between forces, structure, and deformations in microscopic detail, the more so for large deformations (see ref 8 for a review). In the present paper we explore the large-deformation behavior of an experimental model system, built up from almost monodisperse spherical particles packed into compact layers at a substrate surface. These layers were created by letting the particles sediment in normal gravity and subsequently making them adhesive by adding a flocculation agent. Such aggregated networks in the form of thin slabs provide interesting new possibilities for studying mechanical behavior. Their degree of * To whom correspondence should be addressed. E-mail: m.h.g.duits@ utwente.nl. (1) Dinsmore, A. D.; Prasad, V.; Wong, I. Y.; Weitz, D. A. Phys. ReV. Lett. 2006, 96, Art. No. 1185502. (2) Meyer, A.; Marshall, A.; Bush, B. G.; Furst, E. M. J. Rheol. 2006, 50, 77. (3) Pantina, J. P.; Furst, E. M. Phys. ReV. Lett. 2005, 94, Art. No. 138301. (4) Weeks, E. R.; Crocker, J. C.; Levitt, A. C.; Schofield, A.; Weitz, D. A. Science 2000, 287, 627. (5) Varadan, P.; Solomon, M. J. J. Rheol. 2003, 47, 943. (6) Filip, D.; Uricanu, V. I.; Duits, M. H. G.; van den Ende, D.; Mellema, J.; Agterof, W. G. M.; Mugele, F. Langmuir 2006, 22, 560. (7) Filip, D.; Duits, M. H. G.; Uricanu, V. I.; Mellema, J. Langmuir 2006, 22, 4558. (8) Van Blaaderen, A. MRS Bull. 2004, 29, 85.
structural uniformity is relatively high (e.g., compared to fractal aggregates), which improves the perspectives for obtaining representative information from a microscopic experiment without the need to average over numerous observations. In addition, there are fewer characteristic length scales, since (contrary to a single aggregate) the lateral dimensions do not play a role anymore. Provided that the layer can be made thicker than the characteristic length scale of the internal network structure, also the contribution of the substrate can be made negligible. Hence, these thin slabs could open up new possibilities for measuring bulk mechanical properties on a microscale. While in principle also other methods for imposing strain on such systems are available,9-12 we have here chosen to indent the particle network with a glass sphere, attached to an AFM cantilever. The sphere size was taken as larger than the correlation length of the network but still smaller than the overall layer thickness. To visualize the deformations with the maximum relevant spatial resolution, we have used 1.2-µm-sized spherical particles with fluorescent cores and used a confocal scanning laser microscope (CSLM) to image them in a refractive-index-matching solvent mixture. This combination of material’s choice and instrumentation allows resolving the individual positions of all particles in the 3D matrix,13-16 with a typical resolution of 0.1 times our particle diameter.17,18 (9) Tolpekin, V. A.; Duits, M. H. G.; van den Ende, D.; Mellema, J. Langmuir 2004, 20, 2614. (10) Derks, D.; Wisman, H.; van Blaaderen, A.; Imhof, A. J. Phys.: Condens. Matter 2004, 16, S3917. (11) Solomon, T.; Solomon, M. J. J. Chem. Phys. 2006, 124, Art. No. 134905. (12) Grier, D. G. Nature 2003, 424, 810. (13) Varadan, P.; Solomon, M. J. Langmuir 2003, 19, 509. (14) Solomon, M. J.; Varadan, P. Phys. ReV. E 2001, 63, art. No 0.051402. (15) Bevan, M. A.; Lewis, J. A.; Braun, P. V.; Wiltzius, P. Langmuir 2004, 20, 7045. (16) Dullens, R. P. A.; Kegel, W. K. Phys. ReV. E 2005, 71, Art. No. 011405. (17) Crocker, J. C.; Grier, D. G. J. Colloid Interface Sci. 1996, 179, 298.
10.1021/la060753k CCC: $33.50 © 2006 American Chemical Society Published on Web 08/01/2006
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The study reported in this paper was part of a larger investigation: besides the deformation patterns inside the network, also the corresponding (vertical components of the) forces exerted by the indenter were measured [via the bending of the cantilever of an atomic force microscope (AFM)]. These force measurements will be discussed in a separate (forthcoming) paper.19 2. Experimental Section 2.1. Sample Preparation. Silica particles with fluorescent cores were prepared according to a protocol described in earlier works.20-22 A small amount of Rhodamine-B isothiocyanate (RITC), coupled to (aminopropyl)triethoxysilane, was incorporated into 250-nm-size particles, which were further grown by covering with nonfluorescent silica. Hereafter, the medium was exchanged with pure water. Particle size and shape (distributions) were measured with (transmission and scanning) electron microscopy. A fairly good sphericity and an average radius of a ) 600 nm were found, with a polydispersity of 6%. The particles were suspended in mixtures of (twice distilled) water and dimethyl sulfoxide (DMSO). For each sample, 1.0 mL of the stock suspension was added to 11.0 mL of preequilibrated water/ DMSO. After vigorous shaking and cooling at 25 °C, the resulting dispersion (with particle volume fraction ≈ 0.003) was sonicated for 15 s before further use. Precisely 1.0 mL from the so-prepared dispersion was deposited into custom-made cylindrical AFM liquid cells (diameter 23 mm) and left to sediment overnight (covered with a lid). Hereafter the sediments were inspected with CSLM. To induce aggregation, 0.5 mL from the supernatant was carefully pipetted off and replaced with 1.0 mL of a 3 wt % LiCl solution at the same water/DMSO ratio. Thus, LiCl and particle concentrations were identical in all final samples. Another night was given, to let the LiCl diffuse throughout the whole sample and to let the particle network adapt to the modified colloidal interactions. Hereafter the sediments were again inspected with CSLM. 2.2. Solvent Mixture. In the present study, we chose to study silica layers at near-refractive index matching conditions. Herewith two purposes were served: (1) small van der Waals attractions, which allow making weakly aggregated systems, and (2) good optical access with the CSLM. To set these conditions, we measured the refractive index for water/DMSO mixtures at several molar ratios, using an Abbe refractometer at 25 °C. As shown by Figure 1, the particle refractive index is matched near xwater ≈ 0.55 in the absence of salt. 2.3. Microscopy Techniques. Our CSLM was an UltraView LCI10 system (Perkin-Elmer), containing a Nikon Eclipse TE200 inverted microscope and a microlens-enhanced Nipkow disk. Observations were made through a 100×, NA 1.30 oil immersion objective, which was mounted into a high-speed computer-controlled piezoscanner. This allowed for reproducible vertical displacements of the objective, with submicron precision. The fluorescent RITC dye inside the particles is excited with a Kr laser (λ ) 568 nm). The confocal images (672 × 512 pixels, each pixel corresponding to 130 × 130 nm2) were recorded with a Hamamatsu CCD camera (typical exposure time 400 ms) and stored onto the computer hard drive. This CSLM was combined with a homemade AFM. In the combined setup, a piezoscanner (Physik Instrumente) with XYZ position feedback is fixed to the CSLM table and the AFM head is mounted on top of it (see Figure 2a). This allows a good visualization of both the sediment layer and the AFM cantilever. For compression experiments, we used a cantilever with spring constant k ) 0.02 N/m and an attached borosilicate glass sphere (diameter 2R ) 10 µm). Before experiments, the cantilever was rinsed thoroughly with ethanol and left to dry at room temperature. (18) Dinsmore, A. D.; Weeks, E. R.; Prasad, V.; Levitt, A. C.; Weitz, D. A. Appl. Opt. 2001, 40, 4152. (19) Uricanu, V. I.; Duits, M. H. G. Manuscript in preparation. (20) Verhaegh, N. A. M.; Van Blaaderen, A. Langmuir 1994, 10, 1427. (21) Van Blaaderen, A.; Vrij, A. AdV. Chem. Ser. 1994, 234, 83. (22) Tolpekin, V. A.; Duits, M. H. G.; van den Ende, D.; Mellema, J. Langmuir 2003, 19, 4127.
Figure 1. Measured refractive index for water/DMSO mixtures, with (triangles) or without (squares) 2.0 wt % LiCl. Circles indicate the refractive index of the supernatant obtained after particle sedimentation. Wavelength and temperature conditions were close to those of the CSLM experiments (see the text). Stars indicate literature data.23
Figure 2. (a) Schematics of the AFM-CSLM experimental setup (not drawn to scale). (b) Signal (voltage) imposed on the piezoscanner during a complete AFM cycle. In our AFM experiments, the sample was kept immobile while the piezoscanner made controlled vertical movements. The maximum up-down amplitude applied was 20 µm. The compression and retraction speeds were always 0.1 µm/s. In each experiment cycle, the piezoscanner was driven vertically by a truncated saw tooth (see Figure 2b). Cycles began with a compression down to a preset amplitude value (i.e. depth). Once this value was attained, the piezoscanner was kept immobile during a certain time interval. Thereafter, the cantilever (base) was retracted to the initial position. In the case of aggregated sediments, adhesive forces were not only operative between the silica particles but also between the particles and the (borosilicate glass) indenter. As a consequence, during retraction, the cantilever dragged and finally ruptured pieces
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Langmuir, Vol. 22, No. 18, 2006 7785 Table 1. Sample ID and Characteristics: H2O Mole Fraction (xwater), Mass-Density (G), Viscosity (η), and Refractive Index (ns) for the Continuous Phase Liquid and Average Bond Length 〈Lb〉 (see the text) for the Particles without salt
Figure 3. Histogram for the “track length”, i.e., the number of Z-slices in which the same object is identified by the particle tracking software. The solid line marks the expected number of times that a single particle should be identified, considering the particle size and the Z-distance between subsequent images. from the sediment. Under these conditions, one cannot perform consecutive AFM cycles on the same spot (since the sample is locally “altered”) and also the pieces sticking on the indenter must be removed. We found that the particle adhesion to the glass was weak enough to allow complete particle removal by rinsing with solvent. This made it possible to do repeated experiments with the same indenter. Different types of CSLM measurements were performed. Z-series (i.e., stacks of images taken at consecutive Z-positions inside the samples) or time series (i.e., images recorded at a fixed Z-location, but at consecutive moments in time) were chosen, depending on the aspects to be evidenced. As a rule, Z-scans were taken at 0.2 µm intervals, while the time steps were dictated by the minimal exposure time of 400 ms to get sufficient image quality. All experiments were done at 24 ( 1°.
3. CSLM Image Analysis Postrecording processing of the CSLM (Z-scans and time series) was done using IDL software. To localize particle coordinates in three dimensions, routines from particle tracking software, as available on the Internet24 and as described in the literature,25 were used. First, the individual Z-slices were analyzed. A spatial filtering and a pixel-intensity threshold were applied, resulting in the rejection of noise and the survival of only those features that could correspond to particles. For some Z-stacks the intensity threshold was adapted to the Z-position, to account for the lower intensities at larger optical depths. An objectfinding algorithm was used to identify “potential particles”, using a criterion for the minimum allowed distance between two objects. Next, the objects were subjected to selection criteria for the size, elongation, and the area-integrated intensity, after which the “surviving objects” were fitted with 2D Gaussian profiles, to localize their XY-centers with subpixel accuracy.17 All these operations are routinely used in multiple particle tracking microrheology (MPTM),25-28 in which the diffusive motions of (many) individual particles are studied by localizing the particles for each image in a time-series and subsequently (23) Cowie, M. G.; Topowski, P. M. Can. J. Chem. 1961, 39, 2240. (24) http://www.physics.emory.edu/∼weeks/idl/piv.html. (25) Mason, T. G.; Ganesan K.; vanZanten, J. H.; Wirtz, D.; Kuo, S. C. Phys. ReV. Lett. 1997, 3282. (26) MacKintosh, F. C.; Schmidt, C. F. Curr. Opin. Colloid Interface Science 1999, 4, 300. (27) Breedveld, V.; Pine, D. J. J. Mater. Sci. 2003, 38, 4461. (28) Tseng, Y.; Kole, T. P.; Wirtz, D. Biophys. J. 2002, 83, 3162.
xwater
ID
nS
F (g/mL)
0.6845 0.7127 0.7276 0.7431 0.7764 0.8127
1 2 3 4 5 6
1.4334 1.4291 1.4260 1.4235 1.4158 1.4060
1.0898 1.0869 1.0850 1.0825 1.0766 1.0693
with 2% LiCl
η (mPa)
〈Lb〉 (µm)
nS
〈Lb〉 (µm)
3.715 3.635 3.562 3.467 3.178 2.854
1.64 1.67 1.62 1.59 1.62 1.63
1.4400 1.4359 1.4337 1.4314 1.4256 1.4190
1.36 1.35 1.36 1.37 1.39 1.39
finding their displacements via a comparison between subsequent images. The result hereof is a set of proposed tracks, where a track represents a (2D) trajectory assigned to an object (i.e. a particle). A few differences between a typical MPTM application and our 3D localization are noteworthy. Instead of correlating between images separated by time, we have compared images taken at different heights. This only works if the Z-steps are taken small enough to ensure that each particle is detected in several images and if the Z-scanning is performed fast enough to avoid significant displacement of a particle while its intensity profile is being scanned. Similar to the MPTM analysis, one has to set a maximum value (or XY-tolerance δxy) for the detected particle displacement between subsequent Z-images. The number of times the same particle is identified within the different images (in MPTM terms, the track length) should peak around the number of Z-steps needed to scan the vertical intensity profile of a single particle. To achieve this, one has to vary the XY-tolerance: for too small δxy, tracks belonging to the same particle will be split up, while for too large δxy different particles will be lumped into one track. Figure 3 gives a typical example of an optimized track length distribution. Tracks of “double length” are always found. They correspond to particle strings having a (nearly) vertical orientation. Very short tracks correspond mostly to tracks that were split up while still belonging to a single particle. We note here that not only positional data are stored in the tracks but also the intensities along the track. The next step in the localization was to calculate the (most probable) XYZ locations from the intensity profiles. For tracks belonging to the main peak of Figure 3, this was done by fitting I(Z) to a parabolic intensity profile, taking the location of the top as the Z-coordinate of the particle. The corresponding X and Y were refined through linear interpolation between the points measured in the Z-slices above and below the fitted Z-coordinate. Too long tracks (n > 10 in Figure 3) were analyzed for multiple maxima, split up, and subsequently fitted with parabola. Too short tracks (n < 4 in Figure 3) were analyzed for the intensity maximum, after which the XYZ of this maximum was taken as the location of a particle. Obviously this procedure leads to a double counting for particles whose tracks were split in two. To rectify this problem, all “potential particles” were in a last step inspected for other candidate particles that were closer to them than the smallest particle diameter. In case such an unphysical situation was encountered, the candidate particle with the lowest intensity was discarded. To our knowledge, this method has not been used before to localize particles in 3D. As an indicator for the quality of the 3D localization, we have taken the number of too short tracks that could not be recombined and divided by the number of (undisputed) particles in the main peak. This number could be as small as a few percent.
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Figure 4. Representative CSLM images (scale 87.4 × 66.6 µm2) for nonaggregated sediments from sample 1 (images A1, A2) and sample 5 (B1, B2). These images were selected from complete Z-stacks. The A1 and B1 images correspond to the first layer of particles settled onto the bottom. The corresponding height for images A2 and B2 is 15 µm.
Figure 5. Particle concentration profiles for sample 1 (see Table 1) before (left) and after (right) adding 2 wt % LiCl. The ordinate represents the number of particles counted within a volume defined by the chosen XY rectangle and Z-binning (different for the two cases). In the system without LiCl, a crystal-like order (periodicity 1.5 µm) extends almost till the top of the layer. In the system with 2 wt % LiCl, the electrostatic repulsions are suppressed, resulting in a shrinkage of the layer and a loss of order (except near Z ) 0, due to the ordering effect of the substrate30). On the basis of a particle diameter of 1.2 µm, calculated (average) volume fractions are 0.29 (left) and 0.39 (right).
4.1. Layer Structure before Indentation. A first observation was that the quality of the CSLM images appeared somewhat asymmetric with respect to the refractive index of 1.45 (the typical average value for such particles, np9,20,29). For water/DMSO mixtures with refractive index ns < 1.45, particles could still be observed through the ocular up to ≈40 µm (even though the light then has to pass through many silica layers). But for ns g 1.46 a substantial loss of image quality was found already for heights
exceeding 10 µm. In the latter case, particles near the top of the layer could not be identified in a reliable manner anymore by the software. Similar observations were also made for layers in which the particles had been made to aggregate. For this reason we have limited our refractive index range to 1.406-1.440. An overview of all prepared samples is given in Table 1. The structures formed after sedimentation (still before adding the LiCl) turned out to be ordered, as could be expected (see ref 30 for a detailed study of crystallization under gravity at a planar
(29) Philipse, A. P.; Smits, C.; Vrij, A. J. Colloid Interface Sci. 1989, 129, 335.
(30) Hoogenboom, J. P.; Derks, D.; Vergeer, P.; van Blaaderen, A. J. Chem. Phys. 2002, 117, 11320.
4. Results and Discussion
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Figure 6. Histograms for the particle intercenter distances in the stable and the aggregated state for sample 4. Both curves have a zero baseline, but one of them has been shifted vertically for clarity.
wall). Figure 4 shows direct camera images, taken at the bottom of the cell (upper images) and at a height of 15 µm (lower images). As illustrated in Figure 4B2, for samples with a “large” refractive index mismatch (∆n ≈ 0.03), the crystal-like order gets lost after a number of silica layers. This could also be seen from the concentration vs Z profile (not shown) calculated after 3D localization. Probably this is due to the larger contribution of the van der Waals attraction, which makes the total pair potential less repulsive and hence reduces the driving force for crystallization. In all layers created through sedimentation from stable suspensions, the particles showed positional fluctuations that were easily visible in real time. This is the expected behavior for particles that are stabilized by electrostatic repulsions and trapped in a soft repulsive cage formed by their neighbors. Adding LiCl via partial replacement of the continuous phase (see section 2.1) resulted in clear changes in both the structure and the dynamics. While we did not monitor the process over time, the result after 1 day of equilibration was that the (partial) crystal-like order was lost for all samples. For the two samples that were closest to the refractive-index-matching condition, this transition also involved shrinkage of the layer. An example hereof is given in Figure 5, which shows concentration profiles (see section 3 for the calculation) for the same sample before and after the addition of the LiCl. For the other samples the height of the layer remained more or less constant. Another effect of adding the LiCl was that practically all particles lost their mobility. Clearly these findings are in accordance with a reduction of the repulsive force between the particles due to the LiCl. This allows the attractive van der Waals forces to become dominant, which in turn allows the particles to come close, stick together, and lose their mobility. The fact that in most samples a small fraction of the particles still showed positional fluctuations suggests that the aggregation was still weak. This is in agreement with a small Hamaker constant, as expected under near refractive-index matching conditions. The transition from a crystal-like structure to an aggregated network also became apparent from an analysis of the shortest interparticle distance. Using the XYZ-coordinate lists, all interparticle distances within the set were calculated and mapped into a histogram. The distributions of the nearest neighbor distances obtained from such histograms are illustrated in Figure 6 for a repulsive and an attractive network. Clearly, the aggregation of the particles is accompanied by a reduction of the nearest neighbor distance. The location of the peak (found by fitting Gaussians after a baseline subtraction) was seen to shift from ≈1.63 to ≈1.37 µm, depending on the sample. The precise results
Figure 7. (a) Average coordination number as a function of height, for a typical aggregated layer (sample 3 in Table 1). (b) Distribution of the coordination number for the (different points of the) layer. See the text for further details.
shown in Table 1 indicate a clear distinction between nonaggregated and aggregated layers but only minor differences within these categories. The 1.37-µm particle intercenter distance in the bonded state is significantly larger than the 1.2-µm diameter found with TEM; this could suggest that particles are bonded in the network without permanently touching each other. For calculating volume fractions, we have used the TEM diameter throughout this paper. On the basis of the nearest neighbor peak as displayed in Figure 6, we have defined a “bonding distance” for the aggregated network: all particle pairs left of the dashed line were assumed to be “bonded”. Obviously, this chosen location of the line is somewhat arbitrarily, and the criterion does not take into account the slight polydispersity of the particles. However, when studying the properties of many bonded particle pairs, these simplifications are not expected to play an important role. We have used the bonding distance criterion to define coordination numbers for particles in the aggregated networks, by counting the number of neighbors within the bonding distance of a given particle. Since a coordination number can be assigned to each individual particle, it is a highly local variable (e.g., compared to a number density). Figure 7a shows the average coordination number as a function of the vertical position in a typical aggregated network. A gradual decrease from 8 to 6 is found in the main part of the layer (between 4 and 22 µm). For other samples the profiles were somewhat flatter, showing a decrease from 7 to 6. Inspection of the distribution of the
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Figure 8. Reconstruction of the layer structure for sample 3 by projection along Y onto the XZ-plane. This is an “isotropic plot” in which the Z-coordinate has been expressed in units of pixels (length of a pixel 130 nm). Dimensions: X-range, 48 µm; Z-range, 0-26 µm. Projected Y-ranges, 7.8 µm (right), 31.2 µm (left). In the picture on the right, a different color was assigned to each coordination number (see the color bar). In the picture on the left, coordination numbers deviating by more than the square root of the variance are highlighted with blue (low) and red (high).
Figure 9. Structure of the same material as illustrated in Figure 8, but now with the spherical indenter inside (stationary at its lowest point). Both images correspond to a Y-range of 7.8 µm, centered on the middle of the sphere. Color schemes as in Figure 8.
coordination number for the whole layer (see Figure 7b) reveals that there is an appreciable spread, extending to values well below the so-called (average) caging number (4.7-4.8 for monodisperse hard spheres31) and to values above the so-called (average) parking number (8.7 for monodisperse HS31). To analyze this in more detail, we have also calculated the distributions for the lowest 4 µm and the highest 7 µm of the 25-µm-high layer (see Figure 7b). The occurrence of low coordination numbers in the distribution for the whole layer is largely due to the particles near the interface between the layer and the supernatant. The high coordination numbers might be due to the preparation history: starting with a crystal-like structure might have caused some local configurations to be preserved during the compaction of the layers that took place after adding the electrolyte. We remark here that a similar analysis as for Figure 7 was performed for the corresponding crystal-like sample, defining a maximum “bond length” of 1.8 µm based on Figure 6. This resulted in a graph (not shown) in which the probability for the coordination number showed a continuous rise from 0 to 12. An even more detailed picture of the internal structure of the layer is obtained by plotting the particle coordinates on a map, using a color scale to indicate the coordination number. The right-hand side of Figure 8 shows the result for an XZ-map, covering the entire Z-range of the layer. on the left-hand side of Figure 8 data are plotted for the same layer, but highlighting the (31) Wouterse, A.; Plapp, M.; Philipse, A. P. J. Chem. Phys. 2005, 123, Art. No. 054507.
largest deviations from the average coordination number. Not only the (expected) lower coordination numbers near the top and at the bottom of the layer are clearly visible now, but also it can be seen how the highest coordination numbers are distributed with a bias toward the lower Z-regions. It indicates a slight height dependence in the way the layer restructured after adding the LiCl that was too subtle to show up in the concentration vs Z profile. 4.2. Restructuring during Indentation/Retraction. When a dense layer of 1.2-µm-sized silica particles is indented with a 10-µm-large glass sphere, the particles in the network are forced to rearrange their positions. For nonaggregated networks, in which the individual particles were highly mobile, the response was very reminiscent of a pure liquid: the silica particles were simply pushed away, after which a similar structure (to the one before indentation) was restored, even in the direct vicinity of the “intruder”. After the glass sphere was pulled out, the dent was very quickly filled with particles again (in this context we also like to mention a very recent paper33 in which indentation experiments into a similar particle/liquid system are reported). Networks that were aggregated through the addition of the LiCl showed a remarkably similar deformation behavior during the formation of the dent. Despite the formation of a “sticky network” and the corresponding drastic losses in particle mobility that had occurred, the particulate material was still able to flow (32) Manley, S.; Skotheim, J. M.; Mahadevan, L.; Weitz, D. A. Phys. ReV. Lett. 2005, 94, Art. 218302. (33) Schall, P.; Cohen, I.; Weitz, D. A.; Spaepen, F. Nature 2006, 440, 319.
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Figure 10. Local volume fractions in the layer, while the indenter is inside the material. Each data point corresponds to a cylindrical element, at a certain radial distance from the vertical symmetry axis going through the center of the probe sphere. On the right this is illustrated as a cross section.
Figure 11. Structure of the same material as illustrated in Figure 8, but now with the spherical indenter retracted out of the layer. Both images correspond to a Y-range of 7.8 µm, centered on the middle of the sphere.
Figure 12. XZ-Projection showing how particles stick to the indenter and the cantilever after it has been pulled out of the layer.
like a liquid and “homogenize” itself, once it had been set into motion by the intruding sphere. This is illustrated by the XZplots shown in Figure 9, which were obtained by analyzing an image Z-stack with the indenter stationary inside the material. The particulate fluid smoothly follows the geometry of the big glass sphere. No cracks (a hallmark of elasticity32) appear to have been formed and neither is there clear evidence for local densification of the material in the direct vicinity of the probe (as was found in other weakly adhesive systems6). A more quantitative inspection for changes in local particle concentration was carried out by particle counting in vertically oriented cylindrical shells with their Z-symmetry axis passing through the center of the indenter. These cylinders were taken as 10.0-µm-high (equal to the diameter of the indenter), while the shell thickness was taken as 1.30 µm (roughly one particle diameter). By varying the (average) radius of the shell (Rshell), measuring the number of particles (centers) found between its inner and outer radii (typically 100-600), and taking into account
Figure 13. Schematic view to illustrate (for a time-series experiment) how the glass ball is indented into an aggregated layer and at which Z-position particle displacements are observed.
the cylinder and single particle volumes, local particle volume fractions were obtained. Figure 10 shows the results obtained for two different Z-ranges: from 12.4 to 22.4 µm above the bottom of the cell (12.4 µm corresponding to the lowest point of the probe) and from 2.0 to 12.0 µm on the same scale. From the figure it becomes apparent that only a small amount of excess material (i.e., particles) was accumulated under the probe. And lateral to the probe the number density very quickly reaches its “bulk” value again, without an “excess corona”, as was found for a similar system of emulsion droplets.6 We have also compared the local particle concentrations (for Rshell > 10 µm) for the two Z-ranges with their values before the indentation: both concentrations turned out to be the identical
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Figure 14. PIV vector map of the deformation within a layer at constant Z, while a 10-µm-large ball is being indented into the layer along the Z-direction (see Figure 13). Only the most prominent displacements between subsequent time frames are plotted. They are displayed as green vectors, overlaid on the intensity difference map of the corresponding pair of images. The length of the vectors has been magnified 7 times to make them visible. In time frame 109, the average vector length is 40 nm (0.3 pixels), while the maximum vector length is 220 nm (1.7 pixels).
(within accuracy) for the indented and nonindented sample. These observations confirm that the deformations caused by the indenter are highly local. Moreover, they indicate a material behavior which cannot (yet) be distinguished from that of an ordinary liquid; i.e., it flows easily, thus homogenizing the structure and erasing its memory for the previous deformation history. When the glass sphere was pulled out of the layer, adhesions between the silica particles and the indenter became apparent. At the end of the pullout stage, many particles were found attached to the glass sphere, while the dent had undergone a substantial deformation. Figure 11 illustrates how the dent has become shallower and has lost its spherosymmetrical shape. The decreased coordination numbers just under the dent indicate that some
stretching of the particle network took place. Also particles sticking out of the layer are observed; these are remnants of stretched particle chains that were formed between the layer and the indenter (see Figure 12). Observing the structure after the pullout revealed that the (deformed) dent remained for a long time (at least several hours). This “frozen state” demonstrates that besides the liquidlike flow behavior during indentation, the material can also show a solidlike behavior. The latter behavior was already suggested by the time independence of the set of particle coordinates before indentation, but it is now also demonstrated to apply quickly after releasing mechanical stress. Taken together, these findings point to a mechanical behavior of a so-called “yield-stress fluid”. Such materials behave like a solid up to a certain (yield) stress,
AdhesiVe Granular Silica Layers
Figure 15. PIV-analysis of a time step from the same series as in Figure 14. To obtain more clearly visualized displacements, a time step of two frames has been taken (images 110-112). Also the minor displacements are shown now. Numbers along both axes indicate pixel units; 1 pixel corresponds to 130 nm.
while they flow like a viscous liquid above this stress. The mechanistic explanation for this behavior is that above the yield stress, the bonds that hold the network together get broken. Remarkably enough, the transition between the two regimes cannot be very clearly recognized from discontinuities in the static structure (like yielded zones near the indenter and stillsolid zones further away). 4.3. Particle Image Velocimetry. To inspect for changes in the structure during indentation, we also recorded several time series at a fixed Z-position as indicated in Figure 13. Figure 14 shows a time series recorded in a plane 3 µm under the lowest point of the probe at maximum indentation. These recordings were analyzed with a particle image velocimetry routine (written in IDL) available on the Internet.24 This routine divides a pixel intensity map (e.g. the direct camera image) into small pixel areas and finds for each area in the image at time t1 the shift in X,Y-pixels that gives the best fit in the subsequent image at t2. Besides the length and direction of the shift vector, also the quality of the fit is obtained as an “error” parameter. Thus, a vector map is created for the whole image. To clarify the obtained maps, we rejected all vectors whose associated “fitting errors” were above a certain value. In the whole Z-plane indicated in Figure 13, virtually no particle displacements were detected, until the lower surface of the indenter came within a range of 3.5 µm. In other words, at a depth of 10 µm under the layer-supernatant interface, only the last 0.5 µm of the 7.0-µm-long indentation trajectory gave rise to particle displacements. Typical displacement patterns within this 0.5-µm Z-range are illustrated in Figure 14. While initially the lateral displacements are small and the patterns are somewhat scattered, in later stages the vector maps show apparently organized patterns, around a circular area (under the glass ball) in which there are hardly any XY-displacements. Figure 15 shows a larger image taken at the very last stage of the compression, which shows an “immobile area” with diameter 4.6 µm, surrounded by a “mobile area” with diameter 12 µm. The distinction between these two areas is remarkable, in that it does not correspond to a gradually changing deformation field. The “immobile area” could correspond to particles moving (downward) along the Z-direction; that this happens to some extent is suggested by the underlying difference map in Figure
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15, which shows both bright and dark spots: particles that are being pushed into/out of the focal plane should show up as bright/ dark spots in the difference map. When a volume of liquid containing randomly dispersed particles is pushed down, the two types of events will happen simultaneously. The range of these vertical displacements is estimated to be a fraction of the particle diameter. The (components of the) XY-displacements in the “mobile area” are measured by the PIV software. In the last stage of the compression shown by Figure 15, the largest (typical) XYdisplacement amounts to ≈1 pixel (130 nm) per two frames (0.8 s). For comparison, the Z-displacement of the ball amounts to approximately 80 nm over this time span. These numbers illustrate that compression along the vertical direction is more strongly opposed than a lateral expansion of the network. The outer envelope of the mobile area can be used to estimate what is the lateral extent of the deformation pattern in the sense of the displacement (or velocity) field. Clearly, the extent of this field (Figure 15) is larger than that of the particle concentration (Figure 10). Finally, it is also interesting to study the dynamics of the particle network at the end of the compression step. The last two time frames in Figure 14 show that, on stopping the motion of the cantilever base, the deformation pattern disappears almost completely within the time of 0.4 s between subsequent frames. This suggests that any mechanical relaxation time longer than this time should be not be related to the particle network itself.
5. Conclusions and Outlook A simple procedure was explored to make thin layers of silica particles, supported on a glass cover slip, starting from a dilute suspension. Using 1.2-µm-sized silica particles with fluorescent cores and near refractive-index-matching conditions allowed localizing (essentially) all individual particles in three dimensions, up to 36 µm deep into the material with a CSLM. This allowed measuring local concentrations and the pair correlation function. The coordination number was found to be a sensitive measure for spatial variations in the structure. PIV was showed to be a very sensitive way of detecting particle displacements in two dimensions. Our principal interest was to study how the structure of adhesive particle networks changes upon indentation with a large rigid sphere. Our 3D structure data analysis clearly showed that the particle network accommodated the indenter by spatially redistributing the particles in a rather homogeneous manner. A spherical hole was made, while the top of the layer was kept flat. This behavior is reminiscent of a simple liquid, although prior to the indentation the material behaved like a solid. Adhesion between the particles and the AFM probe only became manifest when the probe was pulled out of the material. The failure of the particle layer to repair the dent after complete pullout indicated solidification. This behavior is indicative of a plastic material, which liquefies above a critical (i.e., yield) stress and solidifies below this stress. Evidently, this yield stress must be above the gravitational stress but below the stress applied with the AFM. Another reason for performing this study was to further explore AFM-CSLM as a method for studying the relation between structure, deformation, and mechanical properties. We see many possibilities for that. Experiments could be extended to other deformations, and possibly even to 3D particle tracking. Also other structures could be studied: e.g., more open particle networks, in which force chains can be recognized, or wet granular
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packings. Also, one could think of composite (e.g. polymer) networks in which particles are embedded and act as displacement sensors. Acknowledgment. This research was performed within the Softlink program and sponsored by The Netherlands Organization
Uricanu and Duits
for Scientific Research NWO-CW, with financial support from Akzo-Nobel. We thank Jorrit Mellema for his stimulating involvement in the early stages of the project. M.D. thanks Victor Breedveld for discussions on particle tracking and providing IDL codes. Erik Weeks is acknowledged for the PIV code. LA060753K
