Electric-Field-Induced Yielding of Colloidal Gels in Microfluidic

Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136. Langmuir , 2010, 26 (2), pp 1207–1213 ... We find that t...
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Electric-Field-Induced Yielding of Colloidal Gels in Microfluidic Capillaries Michael Kogan and Michael J. Solomon* Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136 Received July 1, 2009. Revised Manuscript Received August 18, 2009 We introduce a method to generate a purely internal rupture of colloidal particle gels by application of an electric field as they are confined in a microfluidic device. Characterization of the local, microstructural effect of yielding made possible by the device avoids the complication of shear banding that often occurs in attempts to generate yielding of colloidal gels. The gels are comprised of spherical sterically stabilized poly(methyl methacrylate) particles suspended in a density and refractive index matched organic solvent mixture. Because the particles are charged, application of an electric field imposes a force on the gel body that results in homogeneous internal rupture and yielding. After cessation of the electric field, the gel network rapidly reforms. The structure of the reformed gel differs significantly from the one present prior to the application of the electric field. The microstructural changes that accompany the yielding transition are quantified by comparing confocal microscopy image volumes acquired before and after rupture. We find that the local structure of the colloidal gel after recovery, as quantified by the contact number distribution, is negligibly affected by the yielding transition; however, the long-range structure of the gel, as quantified by spatial fluctuations in number density, is significantly impacted. The result highlights the effect of the small number of short-range bond-breaking events that induce the observed changes in collective, long-range structure.

Introduction Suspensions of colloidal particles that interact through shortrange attractive forces of sufficient strength may undergo a dynamical transition called gelation.1 Colloidal gels are characterized by a sample-spanning network structure, slow colloidal dynamics, and linear viscoelasticity with a significant solidlike component. Such gels play a role in the chemical processing of ceramics, in the stabilization of complex fluid formulations such as foods and consumer products, and in understanding the phase behavior of protein solutions.2-5 In these applications, it is often the response of colloidal gels to large, nonlinear deformation that is of interest. Such large deformations can result in yielding, a process in which the sample-spanning network structure is ruptured and there is a transition from a solidlike to a liquidlike rheological response. Understanding the relationship between gel microstructure and the yielding transition is of fundamental interest. At low volume fractions (φ < ∼0.1), gel microstructure is well described by a network of fractal clusters,6 and yielding has been modeled as the rupture of the stress-bearing backbone of the fractal clusters.7 Flow can also affect aggregation at these conditions.8 For high volume fraction gels and glasses (φ > ∼0.35), the question has been investigated by means of mode coupling theory and trap *Corresponding author. E-mail: [email protected]. (1) Zaccarelli, E. J. Phys.: Condens. Matter 2007, 19, 323101. (2) Lewis, J. A. J. Am. Ceram. Soc. 2000, 83, 2341–2359. (3) Dickinson, E. An Introduction to Food Colloids; Oxford University Press: New York, 1992. (4) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: New York, 1999. (5) Cardinaux, F.; Gibaud, T.; Stradner, A.; Schurtenberger, P. Phys. Rev. Lett. 2007, 99, 118301. (6) Carpineti, M.; Giglio, M. Phys. Rev. Lett. 1992, 68, 3327–3330. (7) Mohraz, A.; Solomon, M. J. J. Rheol. 2005, 49, 657–681. (8) Tolpekin, V. A.; Duits, M. H. G.; Van den Ende, D.; Mellema, J. Langmuir 2004, 20, 2614–2627. (9) Fielding, S. M.; Sollich, P.; Cates, M. E. J. Rheol. 2000, 44, 323–369. (10) Gopalakrishnan, V.; Zukoski, C. F. Langmuir 2007, 23, 8187–8193. (11) Brader, J. M.; Voigtmann, T.; Cates, M. E.; Fuchs, M. Phys. Rev. Lett. 2007, 98, 058301.

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models of glassy dynamics.9-12 In the volume fraction range between these two limits, gels can adopt a number of different microstructures, including those with cluster and stringlike morphology.13 Each of these gel microstructures may potentially exhibit different dynamical and rheological signatures,14,12,15,16 including yielding. Yielding is a material response; however, experimentally, investigating the yielding transition is complicated by the fact that in simple geometries such as shear and squeezing flow, instabilities such as shear banding and strain localization often accompany yielding.17 Thus, under typical flows, it is difficult to access a regime in which the microstructural effect of yielding can be studied as a material response because of the inter-relationship among yielding, shear banding, and strain localization. Although the phenomena of shear banding and strain localization are of fundamental interest in their own right, the experimental conundrum is that the rheology of the microstructures that give rise to these hydrodynamic instabilities cannot be fundamentally studied when the instabilities themselves dominate. To address the lack of fundamental understanding of the internal rupture process that gel microstructures experience during yielding, here we describe results in which yielding is accomplished homogeneously throughout the gel by application of an electric field to a gel confined in a microfluidic device. Because the poly(methyl methacrylate) colloidal particles are charged, the application of an electric field produces forces on the gel body that result in homogeneous gel rupture that can then be studied independently of wall-slip or shear-banding effects. The microstructural changes in the gel that result are quantified by analysis of confocal microscopy image volumes acquired before and after (12) Pham, K. N.; Petekidis, G.; Vlassopoulos, D.; Egelhaaf, S. U.; Poon, W. C. K.; Pusey, P. N. J. Rheol. 2008, 52, 649–676. (13) Dibble, C. J.; Kogan, M.; Solomon, M. J. Phys. Rev. E 2006, 74, 041403. (14) Dibble, C. J.; Kogan, M.; Solomon, M. J. Phys. Rev. E 2008, 77, 050401. (15) Yin, G.; Solomon, M. J. J. Rheol. 2008, 52, 785–800. (16) Purnomo, E. H.; van den Ende, D.; Mellema, J.; Mugele, F. Europhys. Lett. 2006, 76, 74–80. (17) Varadan, P.; Solomon, M. J. J. Rheol. 2003, 47, 943–968.

Published on Web 09/08/2009

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Figure 1. (a, b) Schematic design of the microfluidic channel for electric-field-induced yielding of gels. (c) Stereomicroscopy image of the bonded glass channel after microfabrication. (d) Low-magnification confocal fluorescence microscopy image of the colloidal gel formed within the glass microchannel. (e) High-resolution confocal microscopy image (NA = 1.4) of the gel. The top and bottom of the image correspond to the microchannel boundaries shown in (c, d).

the application of the electric field. This technique to generate yielding allows the effect of internal gel rupture on local microstructure to be studied for the first time. The results thus complement previous studies of yielding and flow-induced microstructural evolution by light and X-ray scattering18-21 as well as recent direct visualization studies in two-dimensional colloidal suspensions.22,23 By comparing the effect of electric-field-induced rupture on distributions of particle contact number and numberdensity fluctuations, we find that the principal effect of yielding is on the long-range structural heterogeneity of the colloidal gel.

Materials and Methods Colloidal Particle Gel Preparation. The system studied is a suspension of sterically stabilized poly(methyl methacrylate) (PMMA) colloidal particles (φ = 0.20) gelled by a short-range depletion interaction. The PMMA colloids were synthesized as per ref 24. The steric stabilizer used is the copolymer poly(22%-25% diphenylsiloxane)-(78%-75% dimethylsiloxane) (DPDM). The steric stabilizer thickness is 7.4 nm.24 Nile Red dye was incorporated into the particles during synthesis to enable the use of fluorescence confocal microscopy. A single batch of particles with a mean diameter of 1.03 μm, as measured by scanning electron microscopy, was used for all the experiments in this work. This value was confirmed by the location of the contact value of the measured radial distribution function, g(r), which was found to be 1.05 μm. These two values agree to within the error of our image processing algorithms ((35 nm13). The SEM value was used for calculations and plots. The relative standard deviation of the particle size distribution, from scanning electron microscopy, was 5.5%. Gelation was induced by addition of nonabsorbing polystyrene with Mw = 900 000 g/mol (Pressure Chemical, Pittsburgh, PA; Mw/Mn = 1.10). A mixture of cyclohexyl bromide (Sigma(18) Vermant, J.; Solomon, M. J. J. Phys.: Condens. Matter 2005, 17, R1–R30. (19) Hoekstra, H.; Mewis, J.; Narayanan, T.; Vermant, J. Langmuir 2005, 21, 11017–11025. (20) Rueb, C. J.; Zukoski, C. F. J. Rheol. 1997, 41, 197–218. (21) Varadan, P.; Solomon, M. J. Langmuir 2001, 17, 2918–2929. (22) Hoekstra, H.; Vermant, J.; Mewis, J.; Fuller, G. G. Langmuir 2003, 19, 9134–9141. (23) Masschaele, K.; Fransaer, J.; Vermant, J. J. Rheol. 2009, submitted. (24) Kogan, M.; Dibble, C. J.; Rogers, R. E.; Solomon, M. J. J. Colloid Interface Sci. 2008, 318, 252–263.

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Aldrich, 66.9 vol %) and decalin (Sigma-Aldrich, 33.1 vol %) solvents was used to approximately match the refractive index and density of the solvent and PMMA particles. From the similar solvent conditions of ref 13 the radius of gyration of the polymer is Rg = 41 nm and its overlap concentration c* = 0.0053 mg/mL. The polymer concentration was 0.0057 mg/mL (c/c* = φp = 1.10). Gels were formed by vortex mixing and sonication as previously described13 and then loaded into the microfluidic device. As discussed in the Results section, confocal microscopy and associated structural characterization of the colloidal gel indicated that its microstructure is one of immobilized clusters. The ζ-potential of the particles was measured in a densitymatched mixture of cyclohexyl bromide and decalin solvents (composition: 66.9:33.1 on a volumetric basis) at 25 C using phase analysis light scattering (Malvern Instruments Zetasizer Nano, ZEN 3600). The H€ uckel approximation was used to convert the electrophoretic mobilities characterized by phase analysis light scattering to zeta potential values. Twenty-five measurements were performed on each of two samples and yielded values of þ26.3 and þ30.7 mV, with an average value of þ28.5 mV. Given the measured conductivity of the solvent mixture (estimated Debye length, κ-1 = 184 nm), the mean charge of the colloids is Qe = þ217e at infinite dilution. This charge value agrees with prior reports for similar particles and solvent systems.25

Fabrication of Microchannel Device for Electric-Field Rupture. The device design is as per Figure 1a,b. Briefly, an electric field was applied between two electrode wires in electrical contact with the organic solvent mixture in a microchannel device. The distance between centers of the ports through which the electrodes acted was 0.7 cm. The field is applied through a microchannel etched in glass that is of roughly trapezoidal cross section (Figure 1b). The channel depth is 35 μm. The device width varies from a maximum of 96 μm to a minimum of 74 μm. The glass microchannels were prepared by procedures described in refs 26 and 27. Briefly, 700 μm thick, 100 mm diameter, glass (25) Royall, C. P.; Leunissen, M. E.; van Blaaderen, A. J. Phys: Condens. Matter 2003, 15, S3581–S3596. (26) Pal, R.; Yang, M.; Lin, R.; Johnson, B. N.; Srivastava, N.; Razzacki, S. Z.; Chomistek, K. J.; Heldsinger, D. C.; Haque, R. M.; Ugaz, V. M.; Thwar, P. K.; Chen, Z.; Alfano, K.; Yim, M. B.; Krishnan, M.; Fuller, A. O.; Larson, R. G.; Burke, D. T.; Burns, M. A. Lab Chip 2005, 5, 1024–1032. (27) Srivastava, N.; Davenport, R. D.; Burns, M. A. Anal. Chem. 2005, 77, 383–392.

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Kogan and Solomon wafers (Precision Glass & Optics, CA) were cleaned in a H2O2/ H2SO4 piranha solution. A 500 A˚ thick layer of chrome and a 3500 A˚ layer of gold were deposited onto the glass wafers. Photoresist (PR 1827, Hoest Celanese) was spun-coated onto the wafer at 3000 rpm and soft baked at 100 C for 5 min. The photomask pattern was exposed onto the photoresist using a mask aligner (cannon PLA 501FA, 405 nm wavelength, 350 light integral), and the wafer was developed in MF319 (Shipley Microposit). The exposed metal patterns were etched away in commercial gold etchant (Gold Etchant TFA, Transene Co., Danvers, MA) and chrome etchant (CR-14, Cyantek Inc., Fremont, CA). The glass was etched in a 49% solution of hydrofluoric acid (49% HF, CMOS grade; J.T. Baker, Philipsburg, NJ) to obtain the desired channel depth. The trapezoidal shape of the device was caused by the isotropic nature of the HF wet etch. After the glass was etched to the desired thickness, the photoresist was removed using photoresist stripper (PRS 2000, J.T. Baker, Philipsburg, NJ). The chrome and gold coatings were removed using their respective etchants, and the wafer was diced on a dicing saw to obtain the individual glass slides, each with its own microchannel. Ports to the microchannel for the electrodes were electrochemically drilled by applying a 37 V potential to a metal point touching the glass surface in a 50 wt % sodium hydroxide solution, as described in ref 28. The microfluidic device was assembled by bonding the glass channel to a #1.5 thickness glass cover slide using optical cement (SK-9, Summers Optical, Hatfield, PA) and curing under a UV lamp for 2 h. Glass capillaries of inner diameter 4 mm and outer diameter 6 mm were UV-bonded to the glass around the connections (GB217UV, Dymax Corp., Torrington, CT). A bright-field microscopy image of the resulting device is shown in Figure 1c. Device Operation. Typically 60-120 μL of the colloid polymer mixture described above was loaded into one of the glass tubes (ports) of the device. Once the solution entirely filled the microchannel, and was visible exiting at the bottom outlet of the other glass tube, the second glass tube was also filled with the same amount of sample. Then, layers of deionized water (∼50 μL) and silicone oil (∼50 μL) were added above the colloidal sample in each port to prevent evaporation. Of particular importance was verification that the colloidal gel filled the microchannel homogeneously. Figure 1d reports a typical low-magnification fluorescence microscopy image of the capillary of the device (the unfilled capillary is shown in Figure 1c). Homogeneous filling of the capillary is apparent. At higher resolution, Figure 1e shows a typical image of one of the quiescently formed gels between the two capillary walls. This image, in which individual colloids are resolved, shows that the structure of this gel is one of immobilized clusters, as discussed in refs 13 and 14. In Figure 1e, the top and bottom of the image are the two boundaries of the capillary, separated by ∼80 μm, and also shown in Figure 1d. To induce an electric field in the device, 22 gauge copper electrodes (Anchor Wire Co., Goodlettsville, TN) were attached to an adjustable dc power supply (∼255 V maximum, Lambda LP-415A-FM, Lambda Electronics, Melville, NY). The electrodes were connected to two separate micropositioners (S-725-PRV and S-725-PLV, Signatone, Gilroy, CA), a probe tip holder (U-S, Signatone, Gilroy, CA) and a probe tip (SE-T, Signatone, Gilroy, CA) attached to each micropositioner. The electric field was typically applied for ∼130 s.

Confocal Microscopy Image Volume Acquisition and Analysis. The 3D microstructure of the gel before and after yielding was interrogated by a spinning disk confocal microscope (Nipkow Disk Confocal Scanner Unit supplied by Solamere Technology Group, Salt Lake City, UT). The device consisted of a confocal spinning disk unit (Yokagawa CSU10, Tokyo, Japan) attached to a Nikon TE1000 (Tokyo, Japan) inverted (28) Walker, P. A.; Morris, M. D.; Burns, M. A.; Johnson, B. N. Anal. Chem. 1998, 70, 3766–3769.

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Article microscope. Imaging was with an oil immersion objective (NA = 1.4). The Nile red dye in the colloids was excited by the 488 nm line of an argon ion laser beam. A dichroic mirror filtered the excitation/reflection wavelength and fluorescent images were collected by an XR Mega-10 EX ICCD camera (Stanford Photonics Inc., Palo Alto, CA). QED InVivo software (Media Cybernetics Inc., Bethesda, MD) was used to acquire and store the images on a RAID hard disk drive. Image volumes were acquired before and after the breakage of the gel. A piezo-Z stage (Applied Scientific Instrumentation, Eugene, OR) was used to rapidly acquire 130 2D images within the channel at a step size of 90 nm. The microchannel was typically imaged at array size approximately 600  1000, and such images were cropped to 512  512 for image processing. The pixel size of all images was 69 nm. The time to collect one 48  68  12 μm3 image volume was ∼5 s. Image volumes were collected prior to the application of the electric field, shortly after the cessation of the field (t ∼ between 2 and 40 min postfield) and then again between 235 and 320 min after the cessation of the field (referred to as ∼4 h after the cessation of the field in the figure captions). Centroids of particles in the confocal microscopy image volumes were determined to an estimated precision of (35 nm in the objective plane of the microscope and (45 nm along the axis perpendicular to the object plane by image processing methods described in the literature.13,29 To characterize the microstructure, two statistical quantities were computed from the centroid locations. The first is the distribution of particle contact (or coordination) numbers in the system. A particle’s contact number, z, is its number of nearest neighbors. Particles are nearest neighbors if they are separated by a distance less than or equal to the first minimum of the system’s radial distribution function (r = 1.6 μm for this system). The second is a measure of number density fluctuations, (ÆN2æ - ÆNæ2)/ÆNæ, that is proportional to the isothermal compressibility in the long wavelength limit.30 This quantity is computed by dividing the confocal image volume into cubes of size L. The mean, ÆNæ, and variance, ÆN2æ, of particle number in these subregions are then computed. The quantity is plotted against the inverse length scale a/L, where a is the colloid radius. Additional details of these two structural measures are available in ref 13.

Results Figure 2a-f is a time series of images that shows the internal rupture and recovery of the colloidal gel. Initially, the colloidal gel microstructure is one of interconnected clusters (Figure 2a). This structure is typical of quiescently formed gels at moderate pair interaction strength in this colloid volume fraction range. Its properties include a relatively high contact number distribution (due to the clusters) and a structure characteristic of spinodal decomposition (due to the broad distribution of voids). At t = 3 s, a horizontally oriented electric field gradient was applied across the microchannel. Upon application of the field, the gel body began to internally rupture at multiple points. Voids were initiated in the gel, and distinct mobile clusters appeared. Although the breakage of the gel is spatially uniform, the yielding process itself was observed to be gradual. As shown in Figure 2, rather than instantaneously fluidizing, the gel structure progressively evolved over time until, for t ∼ 50 s, an apparent steady state was achieved. At steady state the clusters resulting from the rupture were continuously advected under the effect of the applied electric field. This cluster advection was rapid, with a velocity that was presumably a function of the electro-osmotic flow and the electrophoretic mobility of each cluster. Note that even though (29) Crocker, J. C.; Grier, D. G. J. Colloid Interface Sci. 1996, 179, 298–310. (30) Hansen, J. P.; McDonald, I. R. Theory of Simple Liquids; Academic Press: San Diego, 1990.

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Figure 2. Confocal microscopy images show the time dependence of gel rupture and recovery for an applied electric field of 364 V/cm. Imaging began at t = 0 s. The electric field was activated at t = 3 s and deactivated at t = 133 s. The images shown are at (a) t = 0 s, (b) t = 24 s, (c) t = 82 s, (d) t = 109 s, (e) t = 360 s, and (f) t = 5 h.

the field strength applied was not sufficient to break bonds down to the level of individual colloids, the suspension was fluidized in the sense that the microstructure was ruptured to the point where clusters were no longer immobilized and jammed. After cessation of the electric field gradient, cluster mobility rapidly decayed and the gel structure recovered. An image series that documents the rupture process is available in the Supporting Information (movie S1). Images acquired shortly after cessation of the field (Figure 2e) and ∼4 h after the cessation of the field (Figure 2f) show that the recovered gel microstructure differs significantly from the initial, quiescently formed structure of Figure 2a. Figure 3 reports that the colloidal gel rupture is dependent on the strength of the applied electric field gradient. After long times (t > 100 s), applied fields of 71 V/cm (Figure 3b) and 114 V/cm (Figure 3c) do not rupture the gel. Their steady-state microstructure is not significantly different than the initial, quiescently formed microstructure (Figure 3a). Greater electric field gradients (179 V/cm, Figure 3d; 264 V/cm, Figure 3e; 364 V/cm, Figure 3f) result in internal rupture that is qualitatively similar to the discussion of Figure 2. The clusters and voids shown in Figure 3 are typically of dimensions that are a significant fraction of the image volume. Although the 2D images shown in Figure 3d-f are representative, it is important to recognize that the rupture process yields a highly heterogeneous structure. Thus, to discuss mean changes in microstructure, as is important for rheology for example, in subsequent experiments we average results from 9 or 10 image volumes collected at randomly selected locations in the microchannel. Note that although in Figure 3 particle clusters appear isolated and disconnected in the 2D images, nearly all clusters are part of a sample-spanning network. This statement is apparent from analysis of the full 3D volume acquired by the confocal microscope. An example image stack of such a gel volume acquired after gel recovery is available in the Supporting Information (movie S2). 1210 DOI: 10.1021/la9023635

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Figure 3. Confocal microscopy images show the effect of electric field strength on gel structure shortly after cessation of the applied electric field. The results are for (a) a quiescent sample not exposed to an electric field, and samples subjected to electric fields of (b) 71 V/cm, (c) 114 V/cm, (d) 179 V/cm, (e) 264 V/cm, and (f) 364 V/cm. Samples at 71 and 114 V/cm did not undergo electric-field-induced rupture. Samples at higher electric fields did yield.

An important parameter in mode coupling theories of gel rheology is the localization length, a measure of the degree to which colloids fluctuate about their mean positions.31,32 In this view, yielding is a consequence of delocalization;33 upon nonlinear deformation, particles escape their bonds and change their local coordination structure. Inspection of the image series shows that few of the colloids change their local coordination structure as a result of the gel rupture. Instead, the fluidization that accompanies yielding produces large, mobile clusters. These clusters are formed by destruction of just a small number of pair bonds in the colloidal gel. Thus, the local coordination structure of most particles is hardly affected by the nonlinear deformation. Figure 4 shows that the net result of the electric-field-induced rupture on the coordination number distribution of the recovered colloidal gel is negligible. The initial, quiescently formed gel has a coordination number distribution characteristic of a cluster gel: the mean contact number at this field strength is about six, and there is an abundance of colloids with a large coordination number. Interestingly, the rupture process does not significantly change the coordination number distribution, as shown in Figure 4 for an applied electric field gradient of 364 V/cm. Immediately after cessation of the field, as well as long after cessation of the field, the coordination number distribution is not significantly different than prior to rupture. Figure 5 shows that the small effect of electric field rupture on coordination number extends to each electric field gradient magnitude studied. Differences among the three conditions are within z ∼ 1, which is the normal sample variation, as shown by comparing measurements at 71 and 114 V/ cm, the two cases in which the gel was not ruptured. (31) Chen, Y.-L.; Schweizer, K. S. J. Chem. Phys. 2004, 120, 7212–7222. (32) Bergenholtz, J.; Poon, W. C. K.; Fuchs, M. Langmuir 2003, 19, 4493–4503. (33) Kobelev, V.; Schweizer, K. S. Phys. Rev. E 2005, 71, 021401.

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Figure 4. Contact number distributions for the colloidal gel subjected to an electric field of 364 V/cm. The plot compares the distributions prior to application of the electric field, shortly after the cessation of the field, and ∼4 h after the cessation of the field.

Figure 5. Mean contact number, Æzæ, of the colloidal gel as a function of electric field strength for conditions prior to application of the field, shortly after the cessation of the field, and ∼4 h after the cessation of the field.

The measure (ÆN2æ - ÆNæ2)/ÆNæ quantifies structural heterogeneity in colloidal gels by quantifying number density fluctuations. This measure is equivalent to the low scattering vector limit of the structure factor and is an efficient metric for characterizing the structure in confocal microscopy image volumes.34 Figure 6 plots this measure of heterogeneity as a function of the dimensionless inverse length scale a/L. (Here a is the particle radius and L is the bin size over which the mean and variance in particle number N are computed.) Prior to breakage, Figure 6 (pre-E-field data set) shows that heterogeneity monotonically increases as length scales are increased. The increase plotted at low a/L is consistent with a distribution of voids on all length scales, is akin to structures characteristic of spinodal decomposition, and has previously been associated with gels of cluster morphology.13 Figure 6 shows that the electric-field-induced rupture produces a resolvable effect on number density fluctuations in the colloidal gel, particularly at intermediate length scales L/a ∼ 10. The data plotted are for an applied electric field gradient of 364 V/cm. Although the qualitative shape of the heterogeneity curve does not change upon rupture, the magnitude of number density (34) Varadan, P.; Solomon, M. J. Langmuir 2003, 19, 509–512.

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Figure 6. The a/L dependence of the number density fluctuation quantity (ÆN2æ - ÆNæ2)/ÆNæ is plotted for the colloidal gel subjected to an electric field of 364 V/cm. The plot compares the distributions prior to application of the electric field, shortly after the cessation of the field, and ∼4 h after the cessation of the field.

Figure 7. Effect of applied electric field strength on (ÆN2æ - ÆNæ2)/

ÆNæ evaluated at a/L = 0.102 is plotted for conditions prior to application of the field, shortly after the cessation of the field, and ∼4 h after the cessation of the field. Rupture was not observed in the gels for field strengths less than 150 V/cm.

fluctuations is much greater in the ruptured gels. For example, on the length scale of 10 particle radii (a/L ∼ 0.1), the recovered gel is 80% more heterogeneous 4 h after rupture than before application of the electric field. On the length scale of 20 particle radii (a/L ∼ 0.05), the change in this measure of heterogeneity is about 34% greater. This result, shown in Figure 6 for the particular electric field gradient of 364 V/cm, holds for all electric field gradient conditions that result in rupture, as shown in Figure 7. Figure 7 plots (ÆN2æ - ÆNæ2)/ÆNæ for a/L = 0.1 and thereby quantifies structural heterogeneity on the scale of 10 particle radii. For small electric fields (E < ∼150 V/cm), for which no electric-field-induced rupture is observed, the number density fluctuations are unchanged over the course of the experiment. Increasing the electric field gradient leads to a monotonic increase in structural heterogeneity. The strong upturn in Figure 7 for E > ∼150 V/cm coincides with the observation of electric-field-induced rupture. At the largest field strengths, (ÆN2æ - ÆNæ2)/ÆNæ increases by nearly a factor of 4. Although the standard error of the mean of the number density fluctuations is large in Figures 6 and 7, because the materials themselves are so heterogeneous, the differences DOI: 10.1021/la9023635

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between the curves are significant relative to the measurement errors. Thus, it appears that the long-range collective structure of the colloidal gels (as quantified by changes in number density fluctuations) is more significantly affected by rupture than shortrange pair bond structure (as quantified by changes in coordination number).

Discussion The results have shown that electric fields can be applied to gels of charged colloids confined in a microfluidic capillary and that the forces generated by the electric field can induce rupture and fluidization of the gel. As opposed to shear flow, in which shear banding and strain localization are often observed to accompany yielding, the applied electric field generated yielding of the gel uniformly throughout the specimen. After the cessation of the electric field, the gel network structure re-formed. Although the local, contact number structure of the recovered gel was not significantly affected by the electric-field-induced yielding, the long-range void structure of the gel was significantly affected. To conclude the paper, we discuss the conditions under which the applied electric field generated internal rupture. We draw connections between the flow-induced structural evolution observed here and other observations reported by means of light and X-ray scattering. Finally, we consider implications of the results for theoretical descriptions of yielding, such as mode coupling theory, and for the industrial processing of gel bodies for material and complex fluid applications. In this work, electric field strengths varying from about 50 to 350 V/cm were applied to network gels of intermediate volume fraction (φc = 0.2) and moderately strong attractive depletion (Ucontact/kbT ∼ 4.3) interactions. Homogeneous rupture and internal yielding were observed for electric field strengths greater than ∼150 V/cm. The transition field strength required to break the gels is comparable to fields typically applied in other microscale applications. For example, electric field strengths for separation of DNA by gel electrophoresis in one microfluidic device was ∼30 V/cm.26 Alternatively, at the rupture field strength (V ∼ 150 V/cm), the free particle mobility of the colloids studied here would be ∼7.1 μm/s. Although the failure of the gel studied here was by homogeneous yielding, in preliminary work, other types of failure and deformation were observed. In particular, if the bonds in the gel network were strong relative to the bonds between the gel and capillary boundary, then adhesive failure at the wall was observed. However, although these other failure modes were observed for other pair potential strengths, internal yielding was the principal failure mode of the moderately strong gel network discussed in this paper. Moreover, yielding itself was observed to be a gradual process (Figure 2). This progressive cascade of sequential breakage events agrees with previous observations in systems with physical bonds, including colloidal12 and star polymer35 glasses as well as Laponite suspensions.36 Because of the complexity of the forces induced by the applied electric field, the local stresses to which the gel was subjected cannot be quantitatively assessed at this time. At least three effects of an applied electric field on the gel network must be considered. First, the charge density on the colloid surface induces an effective body force on each colloid, just as in the case of a single particle’s electrophoretic mobility. If the colloids display charge hetero(35) Helgeson, M. E.; Wagner, N. J.; Vlassopoulos, D. J. Rheol. 2007, 51, 297– 316. (36) Gibaud, T.; Barentin, C.; Taberlet, N.; Manneville, S. Soft Matter 2009, 5, 3026–3037.

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geneity, this induced force will vary from colloid to colloid. Second, the electro-osmotic flow induced by the applied field will sweep through the interstitial regions of the gel, thereby inducing a drag force on the network. Also, if the resultant of these forces is sufficient to induce transient (relative) deformation of the gel network, then an additional viscous drag will result due to relative displacement of the colloids. Third, to maintain force equilibrium, all these forces will be balanced by the induced elastic response of the gel (due to displacement of colloids from their equilibrium positions). The situation is similar to the case of electric field effects on concentrated, dispersed suspensions37-40 (in which unusual ordering can occur41,42) and on the electrorheological transition (in which orientation of dipole moments leads to sample spanning chain formation).43-45 The complexity of the electric field effects on the gel precludes a direct correspondence with any standard rheological deformation. The electric field appears to function as an applied stress, perhaps analogous to the stress induced by gravity in studies of delayed sedimentation. The effect of electro-osmosis would be to overlay an additional uniform flow. The possible relationship among electric-field-induced yielding, creep deformation, and the problem of delayed sedimentation is of potential interest for future study. Even though the local stress applied to the gel network is poorly described, the confocal microscopy visualization provides a simple method to characterize the microstructural changes that result as a consequence of yielding. The clusters characterized here in 3D seem comparable to those recently quantified in 2D for pressure driven microchannel flow.46 Changes observed in this study are important, for example, to understand the microstructual origin of common rheological phenomena such as stressinduced aging, rejuvenation, delayed sedimentation, and thixotropy. Moreover, the significant differences observed in the recovered gel structure have implications for the use of gels for complex fluid stabilization. We have shown that the recovered gel is more heterogeneous (more clustered) than the quiescently formed gel. The effect of this heterogeneity on sedimentation stability47,48 and nonlinear elasticity49 are of interest. The latter measurements may suggest a relationship between elasticity and the structural changes observed here.49 The behavior of the recovered gel upon application of a second episode of electric field induced stress would also be of interest to the problem of gel rejuvenation and stability. The simplicity of the electric-field-induced rupture method recommends it as a tool for rapid characterization of the yielding behavior of colloidal gels. Introducing the electric field into the system is straightforward, and the microscale format of the device is ideal for confocal microscopy imaging. The major challenge in executing the method is to ensure that the gel is loaded homogeneously into the microfluidic capillary. Good adhesion of the gel body to the capillary walls is also necessary. Light, neutron, and X-ray scattering have previously been applied to study the effect of steady-shear flow on colloidal gel (37) Zukoski, C. F.; Saville, D. A. J. Colloid Interface Sci. 1987, 115, 422–436. (38) Rider, P. F.; OBrien, R. W. J. Fluid Mech. 1993, 257, 607–636. (39) Ohshima, H. J. Colloid Interface Sci. 1997, 188, 481–485. (40) Kim, K.; Nakayama, Y.; Yamamoto, R. Phys. Rev. Lett. 2006, 96, 208302. (41) Trau, M.; Saville, D. A.; Aksay, I. A. Langmuir 1997, 13, 6375–6381. (42) Gong, T. Y.; Wu, D. T.; Marr, D. W. M. Langmuir 2003, 19, 5967–5970. (43) Bonnecaze, R. T.; Brady, J. F. J. Chem. Phys. 1992, 96, 2183–2202. (44) Gast, A. P.; Zukoski, C. F. Adv. Colloid Interface Sci. 1989, 30, 153–202. (45) Parthasarathy, M.; Klingenberg, D. J. Mater. Sci. Eng., R 1996, 17, 57–103. (46) Conrad, J. C.; Lewis, J. A. Langmuir 2008, 24, 7628–7634. (47) Gopalakrishnan, V.; Schweizer, K. S.; Zukoski, C. F. J. Phys.: Condens. Matter 2006, 18, 11531–11550. (48) Huh, J. Y.; Lynch, M. L.; Furst, E. M. Phys. Rev. E 2007, 76, 051409. (49) Lee, M. H.; Furst, E. M. Phys. Rev. E 2008, 77, 041408.

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structure.18,19,21 At high shear rates, scattering associated with structural heterogeneity (scale > ∼10-30 particle radii) has been observed. The effect of flow on local gel structure-nearestneighbor structure for example-has been less frequently studied; however, two studies have indicated that an applied flow produces structural anisotropy at this scale.19,50 Direct visualization studies in two-dimensional colloidal suspensions also support this view.22,23 Our 3D direct visualization studies complement such work by probing contact structure, equivalent to the high scattering vector limit of the structure factor. This study has implications for theories of yielding and nonlinear rheology that are based on collective structure, such as mode coupling theory. For example, in naive mode coupling theory, linear elasticity is a function of the collective structure factor and the localization length.51 The localization length is an average measure of the extent to which a particle can fluctuate about its mean position. In a quiescent gel or glass, the localization length reflects constraints on single-particle mobility due to bonding or caging. In a recent extension of the theory to account for activated barrier hopping, stress affects particle localization through a nonequilibrium free energy function that involves the equilibrium collective structure.33 Although this theory cannot capture the effects of large microstructural changes, the theory has been successfully applied to predict the perturbative stresses and strains necessary for yielding. The success highlights the uncertain implications of the long-range heterogeneity changes observed here for gel rheology.10 Thus, current theories of the yield stress are most sensitive to mean short-range structural measures like the localization length. However, this work suggests that this measure is an incomplete indicator of yielding. First, the short-range structure of the recovered gel is insignificantly affected by yielding (Figures 4 and 5). Second, as per the images of Figure 2, yielding itself is triggered by a relatively small number of bond-breaking events. Most pair bonds remain intact throughout the period for which the gel is fluidized (movie S1). The relatively few bond-breaking events explains why recovered structure is most significantly (50) Woutersen, A. T. J. M.; May, R. P.; de Kruif, C. G. J. Rheol. 1993, 37, 71– 88. (51) Ramakrishnan, S.; Chen, Y.-L.; Schweizer, K. S.; Zukoski, C. F. Phys. Rev. E 2004, 70, 040401.

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Article

changed on the cluster scale (Figures 6 and 7). Within each cluster bond configurations and local coordination are hardly affected by the applied stress. Instead, at the cluster boundary bonds break and re-form as yielding and fluidization occur. Thus, instead of depending on a stress-induced change in average short-range structural measures such as the localization length, yielding appears to be a function of the balance between two populations of short-range structural states. Identifying and tracking the relative abundance of these populations would be of interest for a predictive theory of yielding. These two populations are: (i) Particles that after yielding are found in the interior of clusters. Bonds of these particles are largely unaffected by the applied stress. They are as localized relative to their neighbors as in the quiescent gel. (ii) Particles that after yielding are found at the boundary of clusters. These colloids become fully delocalized relative to a fraction of their neighbors due to the applied stress. The second population includes colloids at the initial yield points as well as particles that ultimately reside at cluster boundaries due to the propagation of initial yielding to complete gel rupture. Although the second population is much smaller than the first, it is apparently this group that dominates the mean quantities that determine the elasticity, since they become fully delocalized relative to their neighbors. That is, the change in average localization that determines the onset of yielding is due to the increasing abundance of particles in the second, delocalized population, rather than a uniform delocalization of all particles in the suspension. Acknowledgment. This study was supported by a grant from the National Science Foundation (NSF CBET 0522340) and a gift from the International Fine Particles Research Institute. We acknowledge helpful discussions with Eric Furst and Jan Vermant. Supporting Information Available: Movie S1 showing breakage of the gel at 364 V/cm electric field gradient (the time step between successive images is 0.133 s, playback is at 4 the actual speed of the experiment); movie S2 showing a CLSM z-series of a gel at 264 V/cm electric field gradient imaged shortly after breakage (the 3D connectivity of the gel structure is apparent). This material is available free of charge via the Internet at http://pubs.acs.org.

DOI: 10.1021/la9023635

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