Flow-Induced Anisotropy and Reversible Aggregation in Two-Dimensional Suspensions
Post on 27-Jan-2017
Flow-Induced Anisotropy and Reversible Aggregation inTwo-Dimensional Suspensions
H. Hoekstra, J. Vermant,* and J. Mewis
Department of Chemical Engineering, W. de Croylaan 46,B-3001 Leuven, K.U. Leuven, Belgium
G. G. Fuller
Department of Chemical Engineering, Stanford University, Stanford, California 95305-5025
Received April 4, 2003. In Final Form: August 19, 2003
The time evolution of the flow-induced changes in structure of two-dimensional suspensions have beenstudied by means of video microscopy. The interparticle forces were tailored to produce a two-dimensional(2D) particulate network that could be reversibly broken down by means of a shear flow. Two types ofsuspensions have been investigated: systems with fairly strong attractive potentials in which essentiallyrigid bonds develop and systems with weaker attractions in which particles can still slide over each otherat contact. For both suspensions interfacial shear flow causes the flocs and their spatial organization tobecome anisotropic at various length scales. The FFT of the real space images is used to characterize theanisotropy at large length scales. Structural anisotropy at smaller length scales is deduced from theorientational dependence of the pair distribution function and from an harmonic expansion of g(r). Themechanism leading to the anisotropy during shear flow is shown to be related to a directional dependenceof breakup and re-formation of flocs. Interfacial flow also affects the density of the flocs. The evolution ofthe distribution of coordination numbers with shear rate indicates that shear flow densifies the rigid flocswhereas the opposite occurs for the mobile ones.
Colloidal suspensions with reversible attractive particleinteractions can form a colloidal gel at sufficiently highvolume fractions.1 The particles then reside in a fairlyshallow minimum of the interaction potential and moder-ate mechanical forces suffice to break such a particle geland to initiate flow in the suspension. When the materialis subsequently allowed to rest, flocculation will againoccur. The capability of such gels to sustain shear forceswhile still ensuring adequate flow behavior is exploitedin various technological and biological applications. Thesubstantial changes in structure that are induced byhydrodynamic stresses entail a highly nonlinear flowbehavior, including phenomena such as yielding and time-dependent viscosity, i.e., thixotropy. The underlyingstructural changes are poorly understood. One significantresult, deduced from small-angle light scattering experi-ments on various suspensions, is that a pronouncedanisotropy develops during flow.2-5
Direct observations of the changing particulate micro-structure could help to elucidate the mechanisms thatare responsible for initiating flow and inducing anisotropy.Such measurements become quite difficult, especially formore concentrated systems. Confocal microscopy experi-ments on model colloidal systems are promising,6 but time-resolved measurements remain difficult. Here, reversiblyflocculated two-dimensional colloidal suspensions are
considered. These can be generated by trapping particlesat a gas/liquid or liquid/liquid interface by respectivelysurface tension or interfacial tension.7 For particle mono-layers, structural information in a single plane suffices toprovide a complete description of the interparticle struc-ture.
2D suspensions display a great variety of structuresthat mimic those of their 3D counterparts; crystalline,7,8fractal,9 or even frothlike structures have been reported.10The type of structure that develops depends on variousparameters, including the physical properties (surfacecharacteristics, size, and density) and the concentrationof the particles as well as the nature of the surroundingbulk phases and their interfacial tension. Adding elec-trolyte to the aqueous phase is one of the possible waysto induce aggregation in particles adsorbed at an air/waterinterface,9,11 but the actual mechanism is more complexthan in 3D suspensions. The attractive interparticle forcesthat cause flocculation have been attributed to varioussources, ranging from van der Waals interactions tocapillary or electrocapillary forces;7,9,10,12,13 their detailednature is still under debate.
In the present case, flow-induced changes in reversiblyflocculated 2D suspensions are considered. This requiresthe application of surface flows, which can easily be
* Corresponding author: e-mail firstname.lastname@example.org.(1) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal
Dispersions; Cambridge University Press: Cambridge, 1989.(2) De Groot, J. V.; Macosko, C. W.; Kume, T.; Hashimoto, T. J. Colloid
Interface Sci. 1994, 166, 404.(3) Pignon, F.; Magnin, A.; Piau, J. M. Phys. Rev. Lett. 1997, 79,
4689.(4) Varadan, P.; Solomon, M. J. Langmuir 2001, 17, 2918.(5) Vermant, J. Curr. Opin. Colloid Interface Sci. 2001, 6, 489.(6) Varadan, P.; Solomon, M. J. Langmuir 2003, 19, 509.
(7) Pieranski, P. Phys. Rev. Lett. 1980, 45, 569.(8) Aveyard, R.; Binks, B. P.; Clint, J. H.; Fletcher, P. D. I.; Horozov,
T. S.; Neumann, B.; Paunov, V. N.; Annesly, J.; Botchway, S. S.; Nees,D.; Parker A. W.; Ward, A. D.; Burgess, A. N. Phys. Rev. Lett. 2002, 88,246102-1.
(9) Robinson, D. J.; Earnshaw, J. C. Phys. Rev. A 1992, 46, 2045.(10) Ruiz-Garcia, J.; Gamez-Corrales, R.; Ivlev, B. I. Phys. Rev. E
1998, 58, 660.(11) Robinson, D. J.; Earnshaw, J. C. Phys. Rev. A 1992, 46, 2055.(12) Williams, D. F.; Berg, J. C. J. Colloid Interface Sci. 1991, 52,
218.(13) Nikolaides, M. G.; Bausch, A. R.; Hsu, M. F.; Dinsmore, A. D.;
Brenner, M. P.; Weitz, D. A. Nature (London) 2002, 420, 299.
9134 Langmuir 2003, 19, 9134-9141
10.1021/la034582k CCC: $25.00 2003 American Chemical SocietyPublished on Web 09/26/2003
generated by inserting a suitable geometry with movingwalls through the interface. Such surface flow experimentson 2D monolayers of stable suspensions provide insightinto the changes in lattice structure caused by either shearor extensional flow.14,15 Some data are also available onthe behavior of large aggregates or large particles innonhomogeneous flows.12,16 Here, special attention willbe given to the flow-induced anisotropy in flocculatedsystems. It will be characterized by calculating the fastFourier transform (FFT) from the images and the orien-tational dependence of a pair distribution function.Additionally, the coordination number will be used todiscuss density changes in the flocs, i.e., structural changesat the smallest length scales.
Materials and Methods
Repulsive electrostatic interactions between particles arestrongly enhanced at interfaces.7,8 Hence, the surface charge ofthe particles should not be too high if one wants to induceflocculation. Polystyrene particles with suitable surface chargeshave been synthesized using an emulsion polymerization reactionwitharadical-forming initiatorbutwithouta terminatingagent.17In this manner particles with a diameter of 2.5 m and with a potential of only -3 mV in 10 mM NaCl solution were obtained.They were rigorously washed by repeated centrifugation anddecantation to remove all residual surfactant remaining fromthe synthesis. Subsequently, the particles were dispersed inmethanol at a volume fraction of 0.0093. The actual 2D suspensionis prepared in a clean transparent Petrie dish. First, the aqueousphase is introduced. Two different compositions were used. Theyboth consist of a mixture of glycerol (70 wt %) and deionizedwater, with either CaCl2 added at an overall concentration of 0.4M or a small amount of SDS (sodium dodecyl sulfate), i.e., aconcentration of 1 mM. When spreading the suspension ofparticles inmethanoloneitherof the aqueous phases,amonolayerof particles develops while the methanol evaporates.
A homogeneous simple shear flow was generated in the 2Dsuspension with a parallel band apparatus, shown schematicallyin Figure 1a. This flow cell consists of two counter-rotating bands,made of polypropylene, that are inserted through the surface ofthe liquid layer. PTFE spacer blocks were placed around theshear band apparatus, at a suitable distance from the bands, toensure that the material outside the region between the twobands experienced roughly the same shear rate as the materialbetween the bands. Extensional flows were generated using afour-roll mill apparatus, which essentially consists of four rotatingcylinders that can be inserted axially into an interface.14 Theevolution of the floc structure after applying flow was observedwith a Nikon Microphot SA microscope. An objective with amagnification of 40 and a numerical aperture of 0.6 was used,centered on the stagnation line of the shear flow field or on thestagnation point of the extensional flow field. The images werecollected with a Sony XC-77 CCD camera and a frame grabber.
Experiments during shear flow on small flocs or doublets, forthe system dispersed onto the aqueous phase containing CaCl2salt, displayed only rigid body rotation, indicative of the rigidnature of the flocs. In the range of shear rates explored, internalreoganizations were rare and breakup of the small aggregateswas not observed. It should be noted that the hydrodynamicforces for dilute systems are rather weak. In the accessible rangeof shear rates, this system presented the limiting case of internallynonrearranging flocs.18 The situation is different for the systemthat contains a small amount of surfactant. Here, spheres thatseem to be touching each other can move relative to each other
in the range of shear rates explored. Addition of 1 mM SDS topure water reduces the surface tension only from 72 to 68.5 mN/m,22 which cannot explain the observed differences. The effectof the surfactant on the contact angle is much more pronounced.Contact angle measurements were made by depositing a dropletof the aqueous phase on a glass plate that was spin-coated witha dispersion containing the polystyrene beads. With SDS addedthe contact angle is observed to change from 87 to 67. Additionof 1 mM SDS causes a larger fraction of the particle surface tobe wetted with the aqueous phase which seems to weaken theattractive interaction. This is in agreement with previous work.21
Image analysis was based on the image processing routinesdeveloped by Crocker and Grier23 and implemented in thesoftware package IDL.24 First the image is inverted, and a band-pass filter is applied. As a result, the particles, but also someerroneous features such as dust on the camera or thermal noise,appear as bright spots on a dark background. Possible locationsof particles are identified on the pixel map by using a localbrightness maximum criterion. Subsequent application of a totalbrightness cutoff criterion allows one to discriminate betweenparticles and the typically less bright dust and noise. Thesestandard procedures, as developed by Crocker and Grier,23function well as long as particles are not touching each other.
(14) Stancik, E. J.; Widebrant, M. J. O.; Laschitsch, A. T.; Vermant,J.; Fuller, G. G. Langmuir 2002, 18, 4372.
(15) Stancik, E. J.; Gavranovic, G. T.; Widenbrant, M. J. O.;Laschitsch, A. T.; Vermant, J.; Fuller, G. G. Faraday Discuss. 2003,123, 145.
(16) Camoin, C.; Bossis, G.; Guyon, E.; Blanc, R.; Brady, J. F. J.Mech. Theor. Appl. 1985, spec. iss., 141.
(17) Almog, Y.; Reich, S.; Moshy, L. Br. Polym. J. 1982, 14, 131.(18) Torres, F. E.; Russel, W. B.; Schowalter, W. R. J. Colloid Interface
Sci. 1991, 142, 554.
(19) Camoin, C.; Blanc, R. J. Phys., Lett. 1985, 46, L67.(20) Herrmann, H. J.; Kolb, M. J. Phys. A.: Math. Gen. 1986, 19,
L1027.(21) Robinson, D. J.; Earnshaw, J. C. Langmuir 1993, 8, 1436.(22) Mysels, K. J. Langmuir 1986, 2, 423.(23) Crocker, S. J.; Grier, D. C. J. Colloid Interface Sci. 1996, 179,
298.(24) Interactive Data Language, Research Systems Inc.
Figure 1. (a) Shear band apparatus and the flow field showingthe compressional (C) and extensional (E) axis of the flow field;(b) optical micrograph of the suspension at rest; (c) snapshotduring flow (shear rate 0.033 s-1) at a strain ) 5.1, (d) )5.37, (e) ) 6.25, (f) ) 6.99, (g) ) 7.29, and (h) ) 7.43.(All images refer to a surface coverage of 0.33 and 0.4 M CaCl2added to the aqueous phase.)
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Some difficulties arise, however, in the case of aggregated systemsat an interface. Particles inside flocs appear less bright, whichmakes the selection of the brightness cutoff parameter moredifficult. In addition, the irregular shape of the aggregates andthe subsequent, possibly complex, local deformation of theinterface at the edges of the aggregates can give rise to steepchanges in brightness at the edges of flocs. When using only alocal brightness criterion these steep intensity changes at theedges are mistakenly identified as particles. Therefore, the imageanalysis algorithm had to be adapted for the present purpose.
A more complex criterion based on a combination of localbrightness, the size of the objects, and the size of theircorresponding Voronoi spaces was used. Spurious features thatare identified at the edge of a floc will typically have a muchlarger Voronoi area than particles inside the flocs that haveneighboring particles on most sides. Hence, when the Voronoiarea is above a certain cutoff value, the features will not beidentified as particles. It remains possible, however, that also asmall percentage of real particles, located at the edges of theflocs, would be rejected in this manner. Therefore, the surfacearea of all identified objects is also calculated. The objects thatcan be identified as real particles are found to have an area thatis smaller than those corresponding to spurious features nearthe edges, which have less localized intensity changes. Hence,a critical value for the surface area can be used to restore featuresat the edges of a floc that have been rejected on the basis of theVoronoi criterion but that are real particles. When the threecriteria are met, most noise can be rejected without deletingparticles. To test the algorithm, the number of particles identifiedfor a series of test images was checked by manually counting theparticles. For a total of 6500 particles the original routinesidentified an excess of particles of 44% whereas with themodifications proposed here this error was reduced to 13%. Allparticles within the flocs were resolved correctly, but a smallpart of the spurious features at the edges still remained. As thespurious features are randomly distributed at the edges of theflocs, they do not affect the conclusions in the remainder of thetext. It can be concluded that the routines of Crocker and Grier,extended with the Voronoi criterion, and a check of the area ofthe objects provide a suitable procedure for detecting particlepositions, even within the aggregates.
Results and DiscussionFlow-Induced Anisotropy. At low surface coverage
the particulate structure for both systems consists ofdistinct, clearly separated flocs. In the accessible rangeof shear rates the flow has little or no effect on the flocstructure. At a higher surface concentration of particlesa particle network develops at rest, which will be brokendown when flow is started. The internal structure of theflocs and their spatial arrangement then become depend-ent on time and shear rate. A typical sequence of images,taken after suddenly applying a constant shear rate, isshown in Figure 1. The particulate network that developsafter the particles have been spread onto the surface ofthe aqueous phase is fairly dense, as can be seen in Figure1b. With a surface coverage of 0.33, fractal dimensions of1.71 for the system with salt and 1.86 for the system withsurfactant were obtained by counting the number ofparticles as a function of radial position. The fractal scalingholds for length scales up to at least 60 particle diameters;larger length scales could not be probed due to the limitedsize of the CCD array. These values are relatively highas a fractal dimension of 1.44 is typical for diffusion-limitedcluster aggregation (DLCA) in 2D, whereas 1.55 isexpected for reaction limited cluster aggregation (RCLA).9Hence, a more open structure would be expected afterspreading the particles. Possibly, the complex surface flowsand aggregation phenomena that occur during the evapo-ration of the methanol result in an abnormal, more densestructure. Values comparable to the ones observed in thepresent work have also been reported previously foraggregated suspensions in two dimensions.19,20 It has been
verified that this dense initial structure did not affect thereversibility and reproducibility of the steady-state results,as no hystersis was observed when increasing anddecreasing the shear rate. This could not be proved bymonitoring a fractal dimension, as the fractal scalingbreaks down during flow. The coordination numbers werecalculated and found to remain identical when alteringthe shear history. In all cases the time evolution of thecoordination number was recorded in order to ensure thatthe steady state was reached.
Shearing the structure of Figure 1b results in a typicalcascade of events illustrated in Figures 1c-h. Once theflow starts, the particle network breaks down intoindividual flocs which continuously break up and rebuild.The formation of a larger floc, through the collision of twosmaller ones, can be observed in Figures 1c,d (see theinsets). This floc then rotates (e, f) to break up again asshown in sequence (g, h). The flow field in these suspen-sions is essentially nonhomogeneous on the length scaleof the flocs; this causes a part of the floc that is beingtracked to leave the field of view during the later stages.
Under stationary flow conditions a dynamic equilibriumdevelops between the breaking up and rebuilding of flocs.Combination of two small flocs into a larger one occurspreferentially when the backbones of the constituent flocsare oriented along the compressional axis of the flow field(Figure 1a and sequence (c-e)). Breakup, on the otherhand, is favored by an orientation of the floc along theextensional axis (Figure 1, sequence (f-h)). For thedispersion containing surfactant in the aqueous phase, asimilar behavior can be observed. These flocs, however,can also rearrange their internal structure. Hence, wehave two types of suspensions that represent two limitingcases: in the salt-based system the effect of breakup andaggregation is separated from the effect of floc rearrange-ments, which does play a role in the SDS-based system.
Figure 2 shows a microscopic image, as well as anisocontour plot, of an FFT for typical structures observedduring steady-state flow in the samples containing salt.The FFT has been averaged over 25 images. Although theflocs themselves do not deform for this sample, the FFTpattern is clearly anisotropic. It is similar in nature tosome of the scattering patterns that have been observedon three-dimensional suspensions.2-5 The FFT patternindicates that the structural length scales are smaller inthe flow direction than in the direction perpendicular toit, i.e., the direction of the velocity gradient. This can berationalized by considering the directional dependency ofthe processes for floc breakup and rebuilding.
Referring to the sequence of Figure 1, it can be seenthat larger objects are mainly formed along the compres-sional axis. When two smaller flocs interlock, they obtain
Figure 2. (a) Microscopy image (snapshot) and (b) isocontourplot of a 2D-FFT (averaged over 25 microscopic images) asobserved during steady-state shear flow at a surface coverageof 0.33, a shear rate of 0.098 s-1, and a salt concentration of 0.4M.
9136 Langmuir, Vol. 19, No. 22, 2003 Hoekstra et al.
a common rotation speed (see also Figure 1). These largerflocs remain stable as long as their major axis is orientedin the quadrants following the compressional axis (denotedII and IV in Figure 1a). Upon reaching the extensionalaxis, the flocs breakup and these smaller flocs remainsmall as long as their major axis resides in quadrants Ior III. Hence, flocs with larger aspect ratios have theirbackbones oriented more preferentially in the quadrantsII and IV, whereas on average smaller flocs are morecommon with their orientation in the quadrants I and III.This leads to an anisotropy of the type seen in Figure 2.It is important to point out that the major axis of theanisotropy of the FFT is oriented in the flow direction(0); i.e., the corresponding structure has longer correlationlengths perpendicular to the flow direction in real space.The anisotropy cannot be a consequence of stretching andbreakup of individual flocs. These would on average leadto an orientation along the extension axis, i.e., with anangle of 45 with respect to the flow direction.25-27 Also,floc structures will only be able to take up a limiteddeformation, typically on the order of 10% or less,28 beforethey break up. Consequently, much smaller degrees ofanisotropy would be expected if this were the dominatingmechanism. Finally, it is important to note that flocs thatinterlock obtain a common rotation speed and that theflocs after breakup are different from the ones thatinterlocked along the compressional axis. Hence, theanisotropy observed is truly a result of direction-dependentaggregation and breakup processes and not of hydro-dynamic clustering.
It is possible to test the hypothesis that the differencein floc size and aspect ratio in the different sectors of theshear flow field is responsible for the anisotropy. Exten-sional flows are irrotational; hence, the rotational mech-anism suggested above cannot play a role there, as onlyextension and compression occur. Therefore, a four-rollmill apparatus was inserted into the interface, with theaxes of the flow field as defined in Figure 3b. The evolutionof the structure close to the stagnation point was measuredmicroscopically. A typical micrograph during an exten-sional flow, for a suspension with salt in the aqueous phaseand at an average surface coverage of 0.35, is shown inFigure 3a. The overall stretching rate is 0.2 s-1. To evaluatethe anisotropy, the FFT was calculated in the same wayas in the case of shear flow. The isocontour plot displaysno pronounced anisotropy. Moreover, it can be concludedfrom the microscopic observations that the small aniso-tropy is due to reorientation of the nonspherical flocs bythe elongational flow rather than the stretching of theflocs. As in the case of shear flow, the particles within theflocs did not change their relative positions, consistentwith the strong attractive interactions. Comparing theresults of Figures 3c and 2b, it can be concluded that theanisotropy in extensional flow is less pronounced thanthe anisotropy in the shear flow, confirming our hypothesisthat the rotation plays an important role in the develop-ment of the anisotropy in shear.
To evaluate the effect of the strength of the interactionpotential on structural anisotropy during flow, dispersionswith a small amount of SDS were studied. The micrographof Figure 4 shows a structure observed during shear flowat a shear rate of 0.033 s-1 for this sample. The system
has less compact flocs and smaller voids than a similarsurfactant-free dispersion when the images are comparedat the indicated shear rate and surface coverage (compareFigures 2a and 4a). This results in a finer structure witha more frequent occurrence of breakup and aggregationevents. The snapshot of the structure during flow in Figure4a was chosen explicitly to show long flocs that have justbeen formed and consequently still have their axis orientedalong the compression axis of the flow field (-45). AnFFT, averaged over 25 images, displays a pronouncedanisotropy as shown in Figure 4b. The major axis of thepattern is again oriented along the velocity direction. Inaddition to the changes in the dispersion microstructureby breakup and aggregation, rearrangements of particlesinside the flocs were also observed. Along the compres-sional axis these consist of particles sliding past each other,resulting in densification of the flocs. When reaching theextensional axis the opposite occurs, here the particlessometimes can be observed to slide past each other,rendering the local structure more open. Combined withbreakupandaggregation theserearrangements contributequalitatively in the same way to the anisotropy of themicrostructure as the breakup and aggregation phenom-ena themselves. Flocs with their backbone in the quad-
(25) Jeffery, G. B. Proc. R. Soc. London, Ser. A 1922, 102, 161.(26) Torres, F. E.; Russel, W. B.; Schowalter, W. R. J. Colloid Interface
Sci. 1991, 145, 51.(27) Van de Ven, T. G. M. Colloidal Hydrodynamics; Academic
Press: London, 1989; p 532.(28) Ourieva, G. Instability in sterically stabilized suspensions. Ph.D.
Thesis, KU Leuven, 1999.
Figure 3. (a) Microscopy image (snapshot), (b) four-roll mill,and (c) isocontour plot of a 2D-FFT (averaged over 25 microscopicimages) as observed during steady-state extensional flow at asurface coverage of 0.35, a extensional rate of 0.2 s-1, and a saltconcentration of 0.4 M.
Figure 4. (a) Microscopy image (snapshot) and (b) isocontourplot of a 2D-FFT (averaged over 25 microscopic images) asobserved during steady-state flow at a surface coverage of 0.33,a shear rate of 0.098 s-1, and an SDS concentration of 1 mM.
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rants following the compression axis will be denser dueto rearrangements and larger due to aggregation. Whenthey pass the extension axis, flocs become somewhat lessdense and break up.
To investigate anisotropy at smaller length scales, thepair distribution function g(r) is used. It describes thelocal particle density n(r), relative to the average particledensity n, as a function of the position r with respect tothe center of a reference particle.
Figure 5 shows the two-dimensional pair distributionfunctions during shear flow at a shear rate of 0.098 s-1computed for both the surfactant-free and SDS-containingsuspensions. The single images that were used in thisanalysis contained about 2700 particles. For each of theparticles g(r) was calculated, and this was averaged over25 images. The combination of the optical hardware andthe used averaging procedure resulted in a spatialresolution of 0.25 m. The quadrants indicated in Figure5 correspond to those of Figure 1a. A distinct anisotropycan be observed between the quadrants following thecompressional axis (I and III) and those the extensionalaxis (II and IV). Hence, we can define an orientationaveraged scalar pair distribution function:
The brackets ... denote a spatial average over a specifiedarc-segment region. With the image processing proceduresdescribed in Materials and Methods, a distributionfunction g1(r) was generated by spatially averaging overthe quadrants following the compressional axis, i.e.,quadrants II and IV. Similarly, a function g2(r) wasobtained by averaging over the quadrants following theextensional axis, i.e., quadrants I and III.
Figure 6 gives the results for a shear rate of 0.098 s-1,computed as an average over 25 single images for bothtypes of suspension used here. The differences betweenthe two directions are small but significant, consideringthat the error bars are smaller than the symbols in Figure6. The functions g1(r) have a higher first peak than g2(r),indicating that more particles are close together inquadrants II and IV. It can be concluded that theanisotropy induced by the shear flow persists even at thesmall length scales probed by the correlation functions.In the range of shear rates covered here, i.e., 0.033-0.33s-1, the results remain very similar to those of Figure 6.It should be noted that in Figure 6 the pair distributionfunction displays a small nonzero tail below the hard-sphere limit of r ) 2a. Some uncertainty in the determi-nation of the exact particle positions combined withpolydispersity in particle size is responsible for thisdeviation.
When the functions g1(r) and g2(r) are calculated for thequadrants formed by the velocity and velocity-gradientaxes, i.e., rotated over 45 with respect to the orientationof Figure 6, no anisotropy is observed and the functionsare identical. This indicates that these directions aresymmetry axes for the structure at small length scales.As the anisotropy is not oriented along the compressionand extension axes, stretching of flocs cannot explain thephenomena. In the case of floc stretching, the anisotropyon a local scale is not expected to be very pronounced, asthe change of interparticle spacing within a floc is nearzero for the rigid flocs and quite limited for the systemswith surfactant. Aggregation events, taking place alongthe compressional axis, however, cause the averageparticle spacing to decrease. The particles residing at theedges of flocs see their local neighborhood changed, which
results in a decrease of the position of the maximum ing1(r). Likewise, along the extensional axis the averageparticle distance slightly increases when breakup eventstake place: g2(r) shows slightly larger separation dis-tances. When comparing Figure 6a with Figure 6b thedifference in the interparticle distance is more pronouncedfor the system containing SDS. This is consistent withthe fact that in these flocs rearrangements are possible.This enhances the anisotropy already caused by thedirectional dependence of aggregation and breakup.
An alternative method to characterize the anisotropyof g(r) has been used to evaluate the effect of shear on thepair correlation function for the case of liquidlike struc-tures. It has been shown that g(r) can be expanded asseries of harmonics, perturbing the radial average.29 Thismethod has been used to quantify the effect of shear onthe pair distribution function for stable colloidal disper-sions.30 For the two-dimensional case the expansionbecomes
where gs(r) is the isotropic part of the pair distributionfunction and x and y are the unit vectors in respectively
Figure 5. Two-dimensional pair distribution functions at ashear rate of 0.098 s-1: (a) surfactant-free dispersion, (b)dispersion with 1 mM SDS added.
g(r) ) gs(r) + g+(r)xy + g-(r)x2 - y2
2+ ... (2)
9138 Langmuir, Vol. 19, No. 22, 2003 Hoekstra et al.
the flow and gradient directions. Although the truncationof the expansion does not necessarily give sufficientinformation to fully describe aggregated colloids undershear, it can be used as an additional tool to quantify theanisotropy in g(r). Figure 7 shows the amplitudes of theharmonic terms, g+(r) and g-(r), as a function of the radialdistance. They were calculated by multiplying the patternsof Figure 5 with the appropriate harmonics (cos sinin the case of g+ and 0.5(cos2 - sin2) in the case ofg-, with ) 0 in the flow direction). The isotropic part isnot shown.
From the relative magnitude of g+ and g- in Figure 7it can be concluded that the contribution to the anisotropyis determined by the harmonic with amplitude g-, whichindicates that the anisotropy has the flow direction as itsprimary axis. The other harmonic, which has the com-pression axis as its primary axis, does not contribute tog(r). The orientation of the anisotropy is consistent withthe proposed mechanism for anisotropy development asdiscussed above. Furthermore, Figure 7b indicates thatthe anisotropy is more pronounced for the surfactant-containing dispersion, in which the flocs can internallyrearrange. As in Figure 6, nonzero values are observedfor r < 2a, which is unrealistic for a system containingmonodisperse hard-core particles. The discrepancy hasthe same origin in both cases. The effect, however, is morepronounced here due to the weighing with harmonicfunctions. At small values of r, the azimuthal integrationthat is required to determine values of the amplitudessuffers from the limited azimuthal information. At largerseparation distances this error is greatly reduced.
The mechanism for anisotropy development in thesetwo-dimensional flocculated suspensions in shear flow isrelated to the directional dependence of breakup andaggregation. The latter are closely linked to the nature ofa shear flow. This might explain the very generic natureof the anisotropy, having been observed here in 2Daggregated systems, as well as in a wide variety offlocculated systems in three dimensions.2-5 A crucialexperiment to be performed to prove that the proposedmechanism plays a role in 3D systems is probing scatteringfrom the flow-vorticity plane.
Evolution of the Coordination Number. Withparticle positions identified in the same manner as forthe determination of the radial distribution functions, thedistribution of the number of nearest neighbors can becalculated. Small variations in focus and variations andillumination now have a pronounced effect on the imageanalysis. In an ideal case the coordination number isobtained by counting the number of particle centers thatreside in a shell around a single particle with a radiusequal to the particle diameter. All particles with center-to-center separations smaller than or equal to this cutoffradius are identified as nearest neighbors. Because of someuncertainties of the exact location of the particle centerson the pixel map, some extra thickness has to be addedto the shell to identify all neighbors. The optimum shelldiameter used to identify particles as neighbors had to bedetermined by a calibration procedure based on trial anderror. The number of neighbors obtained on a single imagewere compared to the results obtained using manualcounting for a number of test images. An added shellthickness of 3 pixels wide gave the best result for thesalt-based system. This corresponds to a cutoff radiusequal to the minimum in the pair distribution function.31With the SDS-based system, 2 pixels had to be used forthe reasons mentioned above, resulting in a somewhat
(29) Hess, S.; Hanley, H. M. J. Phys. Lett. 1983, 98A, 35.(30) Mitchell, P. D.; Heyes, D. M.; Melrose, J. R. J. Chem. Soc.,
Faraday Trans. 1995, 91, 1975.
Figure 6. Direction-dependent pair distribution functions g1-(r) and g2(r) at a shear rate of 0.098 s-1: (a) surfactant-freedispersion; (b) dispersion with 1 mM of SDS added to thesubphase.
Figure 7. Amplitudes of the harmonic expansion of the pairdistribution function at a shear rate of 0.098 s-1: (a) oddharmonic; (b) even harmonic.
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smaller cutoff radius. The limited degree of arbitrarinessin the choice of the shell width, combined with the discretenature of the pixel maps, implies that the absolute valuesof the coordination number should be handled withcaution. It should be stressed, however, that the trendsobserved when varying shear rate were independent ofthe choice of the shell width. Hence, the conclusions arenot affected by it.
Applying the image analysis algorithm to 25 images ateach shear rate, distributions of the coordination numbercould be obtained. An example is shown in Figure 8 forthe system containing salt, for shear rates ranging from0.033 to 0.33 s-1. The lower limit for the shear rates wasimposed by the resolution of the stepper motor. Beyond0.33 s-1 mechanical vibrations caused an unstable surfaceflow. An average coordination number was subsequentlycalculated from the distributions. The evolution of thisaverage coordination number is given in Figure 9.
In the surfactant-free dispersion, a densification of theflocs with increasing shear rate can be observed (Figure9), at least in the shear rate range under investigation.Shear densification has been observed in 3D aggregatedsuspensions as well.34,35 Breakup of flocs typically occursat their weakest link, which is usually a single contactbetween two particles. Hence, breakup is almost neutralwith respect to the average coordination number as thenewly formed edge particles already had a low coordinationnumber inside the original floc. Aggregation does not haveto occur between single particles. Whole clusters caninterlock, resulting in several particle bonds that are being
created. These multiple particle bonds cause stronger flocsthat remain stable. The result is an increased averagecoordination number for the clusters. The mechanism ofshear densification is consistent with experimental andsimulation results for rapid coagulation in shear flows byTorres et al.18,26 in which cluster-cluster aggregationdominates. Erosion of the flocs by the hydrodynamicstresses of the surrounding fluid was not observed, aswas the case for the experiments on single flocs reportedabove.
The systems containing SDS in the aqueous phase (i.e.,with internalparticlemobility) showtheoppositebehavior,in line with the predictions of recent simulations for 2Dsupensions.32,33 In the system with surfactant the mech-anism for shear densification competes with flow-inducedchanges in the floc size. Breaking up of flocs in smallerentities creates an increasing number particles residingat the edge of the flocs. These particles have a lowercoordination number than those in the internal regionsof the flocs. As a result, the average coordination numbershould start to decrease when flocs become sufficientlysmall. Accurate measurements of the actual floc size werenot possible with the present optical system. It can beanticipated that in the salt-based system a decrease ofthe average coordination number should occur at highershear rates. This higher shear rate range could not beinvestigated with the experimental setup used due tomechanical instabilities.
Considering the uncertainties in the absolute value ofthe coordination number, it was attempted to determinecontour lengths as a function of shear rate. Again anabsolute comparison turned out to be difficult. The relativechanges with shear rate are, however, in reasonableagreement with those based on average coordinationnumber. For the system with salt the contour lengthdecreases with increasing shear rate. In the case of thesystem with surfactant the change with shear rate issmaller than the scatter on the data.
The effect of shear flow on reversibly aggregated 2-Dsuspensions has been studied. It has been shown thatshear flow induces anisotropy in flocculated 2D systems.Systems where the flocs are built up from rigidly bondedparticles, and those where the particles in the flocs arelinked by mobile bonds, display an anisotropic structure.The anisotropy has been characterized by means of anonspherical FFT, a direction-dependent pair distributionfunction, and an expansion of g(r) using harmonic func-tions. The FFT patterns are oriented along the flowdirections, indicating a structure with the longest cor-relation length perpendicular to the flow direction. Theresults for segments centered around 45 and 135 areidentical, indicating that the anisotropy cannot be at-tributed to a deformation of the flocs. Systems with rigidflocs and systems with flocs that can be deformed showthe same type of anisotropy. A mechanism based on adirectional dependence of breakup and aggregation phe-nomena is proposed to explain this anisotropy. It iscorroborated by the differences observed between exten-sional and shear flow. Anisotropy persists at small lengthscales, as is demonstrated by means of the orientationaldependence of radial pair distribution functions and evenmore clearly by the expansion of g(r) in harmonics.Flexibility of the bonds between particles causes acharacteristic change in the anisotropy of these functions.
It is also shown that shear flow can densify the flocs insystems composed of rigidly bonded particles. This can be
(31) Dinsmore, A. D.; Weeks, E. R.; Prasad, V.; Levitt, A. C.; Weitz,D. A. Appl. Opt. 2001, 40, 4152.
(32) Doi, M.; Chen, D. J. Chem. Phys. 1989, 90, 5271.(33) Chen, D.; Doi, M. J. Colloid Interface Sci. 1999, 212, 286.(34) Rueb, C. J.; Zukoski, C. F. J. Rheol. 1997, 41 197.(35) Wolthers, W.; Duits, H. M. G.; van den Ende, D.; Mellema, J.
J. Rheol. 1996, 40, 799.
Figure 8. Distribution of the coordination number as a functionof shear rate for the system with salt added.
Figure 9. Average coordination number as a function of shearrate for both dispersions.
9140 Langmuir, Vol. 19, No. 22, 2003 Hoekstra et al.
explained by the fact that breakup is almost neutral withrespect to the average coordination number whereasaggregation is not. The latter does not have to occur at aparticle-particle level whereas the first preferentiallyoccurs at single particle contacts. When the particlesbecome less tightly bound, as in the systems containingSDS, a decrease of the coordination number with shearrate is observed. This might be attributed to a decrease
of the average floc size, a phenomenon that dominates atsmall floc sizes.
Acknowledgment. Some of the authors acknowledgethe Fund for Scientific Research-Flanders (FWO-Vlaan-deren) for funding through FWO-G.0208.00.NLOT (J.V.and J.M.), NATO for a NATO-CLG (J.V. and G.G.F.), andDuPont for a Young Faculty Grant (J.V.).LA034582K
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