Mechanics and Microstructures of Concentrated Particle Gels

Mechanics and Microstructures of Concentrated Particle Gels

Hans M. Wyss,w,z Elena V. Tervoort, and Ludwig J. Gauckler

Nonmetallic Inorganic Materials, Department of Materials, Wolfgang–Pauli–Strasse 10, ETH Zurich, Zurich CH-8093,Switzerland

It is often assumed that the viscoelastic properties of dense col-loids are determined by the colloid volume fraction, the inter-action potential, as well as the particle size distribution andshape. The dependence of the viscoelastic behavior of particlesuspensions and gels on these parameters has been widely stud-ied, and is well understood in many cases. In contrast, ourknowledge on the influence of microstructure on mechanical andrheological properties, in particular for high solid loading sus-pensions as used in ceramic processing, is much less developed.This aspect has been the focus of recent experiments, whichshow that small changes in microstructure can have dramaticeffects on the mechanics and dynamics of concentrated colloidalgels. In this article, we attempt to give an overview of the influ-ence of microstructure on the mechanical and rheological prop-erties of colloidal systems. Particular attention is given tocolloidal particle gels at high volume fractions.

I. Introduction

COLLOIDAL suspensions show a wide range of viscoelastic,dynamic, and structural behavior. Many industrial appli-

cations make use of colloidal suspensions and of the possibilityto change and control their behavior. In ceramic processing, avariety of different methods are used to exploit the wide-phasebehavior of colloids to produce a dense, homogeneous packingof particles via ‘‘wet processing’’ routes. For good reviews of thedifferent processing methods, see Lewis1 and Sigmund et al.2

It is known that the mechanical behavior of a colloidal geldepends on the size and shape of the particles, the interparticleforces, the volume fraction, and the spatial arrangement of theparticles or, what is called the microstructure of a particle net-work. The structural, mechanical, and dynamic properties of acolloidal suspension can change drastically, when one or severalof these parameters are varied. However, the main focus in thestudy of the mechanical behavior of dense colloidal suspensions

and gels has been limited to a large extent to the influences ofvolume fraction and interparticle forces.

The effect of the interparticle pair potential on the rheologicalbehavior of suspensions was studied by several authors.3–6 Avariety of other contributions focus mainly on the dependenceon the particle volume fraction.7–9

Microstructural characterization is of tremendous impor-tance in ceramic processing and technology. The characteriza-tion of microstructure of sintered ceramic bodies is a standardprocedure that is essential for the optimization of the mechan-ical behavior of ceramic parts. It is, therefore, surprising thatonly a few studies have investigated the microstructures of ce-ramic materials in the ‘‘wet’’ stage—in a concentrated suspen-sion or gel of particles.

Because of the difficulties involved in characterizing andquantifying a microstructure in a colloidal system, it is easierand more straightforward to study the relation of the macro-scopic properties to the other relevant parameters. However, anunderstanding of the mechanics of flocculated suspensions canonly be achieved when the structure of the particle network istaken into account10; the interparticle potential alone is notsufficient.

Even one of the simplest colloidal systems, a suspension ofhard-sphere particles, shows a complex viscoelastic behaviorthat is tightly linked to changes in the microstructure. At highvolume fractions, such a system shows pronounced shear thin-ning, reflecting a shear-induced structural rearrangement (seeSection II). In order to understand the mechanisms that governthe macroscopic properties of a colloidal system, profoundknowledge of the structural properties is thus essential.

Of particular interest is the influence of the microstructure onthe mechanical behavior of particle gels. At low solid loadings,theoretical models based on the fractal description of networkstructures are successful in linking the structural to the mechan-ical behavior (see Section III). These models are in good agree-ment with results from experiments11–13 and from computersimulations.14 However, the fractal description of the networkstructure must break down at some point toward increasingvolume fractions,7,15 and the measurement of fractal dimensionsbecomes increasingly difficult at high volume fractions. Never-theless, the fractal description provides a way to link the micro-structure, or at least some characteristic quantities that describethe microstructure, to the viscoelastic behavior directly.

For highly concentrated gels, however, our knowledge on therelations between structural and mechanical properties is ratherlimited. We identify two main problems that make it difficult to

Journal

J. Am. Ceram. Soc., 88 [9] 2337–2348 (2005)

DOI: 10.1111/j.1551-2916.2005.00622.x

r 2005 The American Ceramic Society

Feature

D. Green—contributing editor

H.W. was supported by the Swiss National Science Foundation under Grant 2002–4 19.wAuthor to whom correspondence should be addressed. e-mail: [email protected].

eduzDivision of Engineering and Applied Sciences, Harvard University, Cambridge, MA

02138

Manuscript No. 20105. Received February 2, 2005; approved June 2, 2005.

establish a relation between network structure and mechanicalbehavior in concentrated colloidal gels.

Firstly, the reproducible formation of a colloidal gel at highvolume fractions cannot be performed as easily as in a dilutedsystem. The direct addition of salt or acid/base to stable suspen-sions will lead to colloidal gels of irreproducible microstructures.Because of the elevated aggregation rates in concentrated sus-pensions, the system will generally aggregate locally before thesalt or acid/base is homogeneously distributed. In rheologicalstudies of concentrated gels, a high shear rate is usually appliedprior to the measurements in order to set reproducible startingconditions.7,10,16 However, in this case the microstructures of theresulting gel networks are determined mainly by the shearingprocess, and not by the initial aggregation process itself. As aconsequence, the formation of different concentrated gels of sig-nificantly different microstructure will be difficult to achieve if ahigh shear rate has to be applied. It is also important to note thatstrong shearing of a particle gel will not always make its struc-ture more homogeneous. By contrast, strong shearing will leadto a highly inhomogeneous structure, as it favors the formationof strong, densely packed aggregates in the system.17,18

Secondly, the characterization of microstructures in concen-trated colloids proves difficult, as the standard methods used forthe characterization of diluted colloidal suspensions cannot beeasily applied to highly concentrated systems. For example,static light scattering, the standard method for the characteri-zation of structure in colloidal gels, fails for systems that exhibitstrong multiple scattering.

II. Role of Microstructure in Stable Suspensions

As mentioned before, the microstructure has an important in-fluence on the rheological properties even in a stable colloidalsuspension.19–21 At high concentrations, colloidal suspensionscan show pronounced non-Newtonian effects in their flow, inparticular, shear thinning and shear thickening.19,21–25 The vis-cosity of the suspension is no longer independent of the shearrate, but changes with the shear rate. Both phenomena can beexplained by a change in microstructure in the suspension, in-duced by the shear. The occurrence of structural rearrangementsupon shear has been experimentally confirmed by both staticlight scattering 26 and neutron scattering.8,27 In the following,we will focus on the shear thinning effect to illustrate the im-portance of microstructure on the viscoelastic behavior of sta-ble particle suspensions. For a review of shear thickening(dilatancy) in concentrated suspensions, we refer to Barnes.28

By comparing the typical timescale for Brownian motion andfor the shearing process, one can estimate the typical shear rateat which the structure of the suspension should begin to be per-turbed by the flow. A characteristic time tD for diffusion is givenby the time it takes a sphere to move a distance equal to its ownsize. A characteristic time tshear for the flow is given by the in-verse of the shear rate. The ratio between these two character-istic times is known as the Peclet number Pe and is thus given by:

Pe ! 6pZa3

kBT(1)

where Z is the viscosity of the liquid phase, and a is the particlesize. At Peclet numbers much smaller than unity, Brownianmotion is able to maintain the structure of the suspension in astate essentially indistinguishable from the unsheared state. Atshear rates corresponding to a Peclet number greater than unity,Brownian motion can no longer restore the structure of the sus-pension to its equilibrium state. Experiments on model systemshave shown that flow curves for suspensions of different particlesizes fall onto a single master curve, when plotted as a functionof the dimensionless Peclet number.20,25 The high- and low-shear limits of the viscosity Z are in good agreement with

ZZ0

! 1" ffc

! ""2

(2)

where fc # 0.63 and fc # 0.71 for the low- and the high-shearlimit, respectively. Whereas for the low-shear limit, the criticalvolume fraction corresponds to the volume fraction of randomclose packing, for the high-shear limit it is closer to a crystallinevolume fraction. The value of the critical volume fractionfc # 0.71 reflects the fact that in the high-shear limit, orderingoccurs only in layers and not in a three-dimensional (3D) crys-talline structure. Thus, although Eq. (2) is purely empirical, thevalues for the characteristic volume fractions themselves reflectthe importance of microstructural arrangement. In Fig. 1, weillustrate the shear thinning effect with a plot of viscosity as afunction of shear stress in a model hard sphere system.

As a model system, we use polymethyl methacrylate particlesof radius a5 136 nm dispersed in a mixture of cycloheptylebromide and decalin; the volume fraction is f5 0.41. For detailsregarding this model system, see Prasad et al.,29 Manley et al.,30

Segre et al.,31 Pham et al.,32 Poon et al., 33 and Yethiraj and VanBlaaderen.34 The viscosity Z in the transition region between thelow- and the high-shear region is well described by

Z ! Z1 $ Z0

1

1$ s=sc% &n(3)

A fit to the data in Fig. 1, shown as a dotted line, yields n5 2and sc5 0.83 Pa.

The shear-thinning and shear-thickening effects in a concen-trated suspension illustrate the importance of the microstructurethe viscoelastic properties of colloidal systems. It is obvious thatin a colloidal gel, microstructure could play an even more im-portant role, while a thorough understanding of these effects willbe harder to achieve than for a stable suspension.

III. Fractal Description of Microstructures for Low VolumeFraction Colloidal Gels

If the interaction between colloidal particles is strongly attrac-tive, a colloidal system will form aggregates. At low volumefractions, the structure of such aggregates is accurately describedas fractal.13,35 This means that the number of particles containedin an aggregate scales with its size as a power law:

N / Rdf (4)

where the exponent df, called the fractal dimension, is smallerthan the Euclidean dimension d. This implies, because the den-

Fig. 1. Shear thinning effect for a model hard-sphere system at volumefraction f5 0.41. Normalized viscosity as a function of the shear stresss. The dotted line shows a fit to the data using Eq. (3).

2338 Journal of the American Ceramic Society—Wyss et al. Vol. 88, No. 9

sity of the fractal clusters thus decreases as their size increases,that upon formation of fractal aggregates in a suspension, thesystem will ultimately form a volume-filling gel. The values ofthe fractal dimension df for different aggregation mechanismsare well established from computer simulations as well as fromexperiments at low volume fractions.11–14

Aggregation mechanisms can be divided into two differentcategories: diffusion-limited cluster aggregation (DLCA) andreaction-limited cluster aggregation (RLCA). In DLCA, eachcollision between particles during aggregation leads to an irre-versible bond, whereas in RLCA, particles can escape from abond site and penetrate deeper into the aggregated structures.Therefore, aggregates formed by RLCA are denser and have ahigher fractal dimension than DLCA aggregates.

The size of flocks in a fractal gel network follows directlyfrom the condition that the density fc of the fractal clusters willdecrease during aggregation until, when the volume of the flocksfills all space, it reaches the volume fraction f of particles in theoverall suspension. Thus, the characteristic size Rc of fractalclusters scales as

Rc / f1=%df"d& (5)

As a result of the fractal nature of the microstructure, the me-chanical properties of fractal gels can be directly linked to theparameters that describe their microstructure. The fractal di-mension df describes the overall structure of the fractal clusters.As the elastic constant ke of a cluster depends on the mechanismfor bearing stress, two additional parameters, the parameters eand db, are needed to describe the geometry of the main stress-bearing chain of particles within a cluster. The elastic constant keof a cluster then scales as36–39

ke /1

RdbR2e (6)

where the exponent e, with 0rer1, describes the stress-bearingmechanism of the chain. The case e5 0 applies to a straightchain, where the stress is transmitted by bond stretching.

The case e5 1 describes a purely isotropic chain, where stresstransmission through a bond-bending mechanism dominates.These two extreme cases are illustrated in Fig. 2. The geometryof the chain will determine its elastic properties.

The exponent db is called the bond dimension or chemicaldimension and must be at least 1 to provide a closed path. Anupper bound to db is given by min[df, 5/3], where a value of 5/3corresponds to a self-avoiding chain. Computer simulationshave suggested that db # 1.35 for a 3D percolation network.40

The exponents e and db are not independent. For straight chains(bond stretching), the parameters are e5 0 and db5 1. For amore detailed description, see De Rooij et al.37 and Potanin.39

The scaling behavior of the shear modulus G0 follows directlyfrom Eq. (6):

G0 / keRc

/ R"%1$2e$db&

c (7)

For a fractal structure, with Eq. (5), one finally obtains

G0 / f%1$2e$db&=%d"df & ! fm1 (8)

Through similar scaling arguments, de Rooij et al. derive anexpression for the scaling behavior of the yield stress sy of aparticle network as follows 37:

sy #fextR2

c

1R"%2$e&c / f%2$e&=%d"df & ' fm2 (9)

The scaling laws for the storage modulus G0 and the yield stresssy are linked to the stress-bearing mechanism via the para-meter e:

For a bond-stretching mechanism (e5 0):

m1 !1$ dbd " df

and m2 !2

d " df(10)

For the case of bond bending (e5 1):

m1 !3$ dbd " df

and m2 !3

d " df(11)

The values of the exponents m1 and m2 are directly linked tothe fractal dimension df of the aggregates. Low values for theexponents indicate anisotropic aggregate structures, whereashigher values indicate more isotropic aggregate structures orhigher fractal dimension.37,39

Thus, the fractal description of the network structure and theabove scaling concepts provide a clear-cut relation between thestructural and mechanical properties of fractal particle networksat low volume fractions. However, at very high volume frac-tions, the fractal description of network structure is no longerappropriate. Nevertheless, the scaling concepts summarized inthis section might be useful to some extent in describing the be-havior of highly concentrated systems as well.

IV. Microstructural Characterization of Gels

Information on microstructures is one of the most useful andessential measures when it comes to understanding the macro-scopic properties of materials. In Table I, we list some of themethods that can be used to access microstructures in particlesuspensions and gels. For a comprehensive review of aggregatecharacterization techniques, we refer to a recent article byBushell et al.41

In the following, we will briefly discuss the various methods,as well as their advantages and disadvantages with respect to thecharacterization of the microstructure in dense, highly turbidcolloidal suspensions.

The characterization of the structure in dense colloidal sus-pensions and gels is often an experimental challenge. In partic-ular for ceramic systems it is hard to access the microstructureexperimentally. One reason for this is that the refractive indexdifference between the solid and the liquid phases is usually toohigh to enable measurement of their microstructure using stand-ard light scattering techniques (Table I (1)).42 A meaningfulevaluation of standard light scattering experiments relies onsamples that show only negligible multiple scattering. By usingmore advanced light scattering setups such as a multicolor or a3D setup (Table I (2)),43,44 contributions of multiple scatteringto the detected signal can be suppressed. However, thesemethods are still limited to samples of rather low turbidity, assome measurable amount of single scattering intensity alwayshas to be present.

Using diffusing-wave spectroscopy (DWS) (Table I (3)),45,46

the dynamics of highly turbid systems can be analyzed. DWSextends traditional dynamic light scattering to media with strongmultiple scattering by treating the transport of light as a diffu-sion process. No direct structural information is obtained from

Fig. 2. Schematic representation of different chain geometries. (a) Iso-tropic chain: dominated by bond bending. (b) Straight chain: pure bond-stretching mechanism.

September 2005 Concentrated Particle Gels 2339

such a measurement, although measuring the turbidity itselfgives some information on the homogeneity of the microstruc-ture.47 The single-scattering regime can also be accessed by usingX rays instead of visible light. However, accessing large enoughlength scales requires detection at very low angles. Small-angleX-ray scattering (Table I (4))48–50 is suitable for the character-ization of microstructures only if the particle size is sufficientlysmall.

Also, in small-angle neutron scattering (SANS) (TableI(5)),48,51 present detectors set an upper bound to the accessiblelength scales at around 200 nm.

However, a recently developed technique called spin-echoSANS (SESANS) (Table I (6))52–56 allows to extend the acces-sible length scales dramatically. Structural information can beobtained at length scales spanning three orders of magnitude:from 10 nm up to 7 mm.56 The technique is based on the Larmorprecession of polarized neutrons in magnetic fields with inclinedfaces. The precession of the polarization vector of neutrons in amagnetic field is proportional to the wavelength, magnetic field,and path length through this field. It is therefore possible to de-termine the z component of the wave vector momentum transferQz as a function of the precession angle f. The SESANS signalcan be interpreted in real space; it is possible to determine theparticle density autocorrelation function directly from the meas-urement. In the first applications of the SESANS technique tothe characterization of colloidal systems, Krouglov et al.55 showthis directly by measuring the structure of hard-sphere suspen-sions at different concentrations; Bouwman et al.56 show thatthe technique is able to directly identify the phase transition of acolloidal system from a gas via a liquid to a solid. The largerange of accessible length scales combined with the possibility ofmeasuring samples of very high volume fraction, confers thismethod with considerable potential for future studies of thestructure of concentrated colloidal systems.

For aqueous systems, freeze-fracture techniques can be usedto access the microstructure (Table I (7))57,58 This can be per-formed by applying a coating to the surface of the frozen frac-ture surface and then examining this templated layer usingtransmission electron microscopy. Alternatively, one can exam-ine the fracture surface directly by using a scanning electron

microscope (SEM) equipped with a Cryo stage. However, thestandard freezing techniques are likely to produce large ice crys-tals in the liquid phase of the samples and thereby distort themicrostructure. The best method to preserve the structure ofaqueous samples during freezing is high-pressure freezing (TableI (8)).59 As the crystallization of water requires more energy athigh pressures than at ambient pressure, the formation andgrowth of ice crystals in the liquid phase can be adequately sup-pressed. This technique has recently been applied to ceramicsystems.60,61

The most direct way of accessing 3D microstructure is con-focal microscopy (Table I (9)).62–64 However, this technique alsorequires a low refractive index mismatch between the colloidsand the solvent. It is therefore not suitable for ceramic systemsor for attractive systems that rely on van der Waals forces. Thistechnique has recently been used to study weakly attractive gelsand colloidal glasses in model systems.62–64

V. Structure and Mechanical Properties of Concentrated Gels

Linking microstructure to macroscopic behavior is of centralinterest in materials science and technology. Nevertheless, rela-tively little is known about the relation between microstructuresand mechanical properties of concentrated colloidal suspensionsand gels. The main difficulties in this endeavor lie in the exper-imental limitations when it comes to studying and quantifyingthe microstructure of these systems.

Channell et al.10 report one of the few studies that elucidatethis relation in a systematic way. The study focuses on thecharacterization of the mechanical behavior of gels. The com-pressive yield stress sy as a function of volume fraction f is ob-tained by analyzing the density profiles of samples aftersubjecting them to a compressive stress in a centrifuge. A de-tailed discussion of this technique is given elsewhere.65,66 Theadvantage of the technique is that a whole curve of yield stressversus volume fraction is obtained from a single measurement.The stress at a given point in the sample is given by the load of allthe material on top of this point. Thus, the equilibrium volumefraction for a whole spectrum of stresses can be obtained by

Table I. Different Techniques Applied in Characterizing the Microstructures of Particle Gels

Method and references Capabilities Limitations

(1) Standard static anddynamic lightscattering42

Allows the characterization of structure anddynamics in colloidal systems of low volumefraction

Limited to samples of very low turbidity,where multiple scattering can be neglected

(2) Multicolor or 3D setupfor static and dynamiclight scattering43,44

Allows suppression of contributions of multiplescattering in the signal

Still limited to rather low turbidity, as somemeasurable amount of single scattering alwayshas to be present

(3) Diffusing wavespectroscopy45,46

Allows to extend dynamic light scattering tohighly turbid suspensions and gels. Givesinformation on the average dynamics of particlesin the system

Only limited structural information47

(4) Small-angle X-rayscattering48–50

Gives structural information in terms of the staticstructure factor S(q)

Upper limit to length scales that can beaccessed because of small wavelength

(5) Small-angle neutronscattering

Gives structural information in terms of the staticstructure factor S(q)

Present detectors set an upper bound to theaccessible length scales at around 200 nm

(6) Spin-echo small-angleneutron scattering52,56

Gives structural information in terms of the staticstructure factor S(q) on length scales between 10nm and 7 mm

Facilities not readily available

(7) Freeze-fracturetechniques57,58

Gives real space structural information Crystal formation in the liquid phase duringfreezing is likely to distort microstructure

(8) High-pressure freezingand fracture49,61

Gives real space structural informationCrystal formation in aqueous systems can beadequately suppressed

Limited to aqueous systems

(9) Confocalmicroscopy62,64

Gives detailed 3D structural information in realspace

Limited to low-turbidity samples or tomeasurements near the cover slip (walleffects). Fluorescent labeling necessary

3D, three-dimensional.


analyzing a single sample. In order to achieve different micro-structures, the authors use a very simple, yet efficient, technique.Different levels of homogeneity are achieved by preparing thesamples in different ways, but with identical chemical com-position. Homogeneous samples are obtained by preparingsuspensions at a given volume fraction f0 and subsequent ho-mogenization by high-shear mixing. Lower levels of homogene-ity are achieved by carefully diluting samples gelled at a highervolume fraction f4f0, and subsequent shearing by either handmixing or by ultrasonication. During the measurement of thecompressive yield stress, this low volume fraction structure isthen consolidated in the centrifuge, which leads to a volumefraction gradient in the sample. The lowest volume fraction isfound at the upper surface of the gel, where the volume fractionafter centrifugation should still be the same as the overall volumefraction f0 before centrifugation. Figure 3 shows a schematicillustration of a microstructure obtained from diluting a moreconcentrated gel under gentle stirring. The new gel network isbuilt up from individual clusters that are connected only througha few weak bonds to form the new volume-spanning network.

The experimental data indicate that lower mixing intensitieslead to lower compressive yield stresses. Figure 4 shows thecompressive yield stress as a function of volume fraction for gelsthat were obtained from the different homogenization methods.If high-shear mixing is used for homogenization of these sam-ples, the compressive yield stress is independent of the volumefraction where the gels were initially prepared. Thus, the appliedhigh-shear mixing seems to be sufficient to achieve a reproduc-ible microstructure, dependent only on the sample composition,but not the sample’s preparation history. The lowest compress-ive yield stresses are observed for samples subjected to the lowestmixing intensity (hand mixing). Note also that above the volumefraction where the samples were originally prepared, the yieldstresses are independent of the homogenization used in the di-lution process.

By using different mixing intensities and starting volume frac-tions f4f0, samples of different microstructures are obtained.Even though the authors do not characterize microstructuresexperimentally, it is reasonable to assume that the different lev-els of homogenization lead to different aggregate sizes. The low-er the mixing intensity, the larger the aggregates and the lowerthe compressive yield stresses.

A recent study by Franks et al.67 focuses on the relationshipbetween aggregate size and the rheological properties of apacked sediment of aggregates. The authors induced the forma-tion aggregates by flocculation of alumina particles in a dilutesuspension, varying for instance the salt concentration toachieve different aggregate sizes. The aggregates formed subse-quently settle to form a dense sediment at the bottom of thecontainer. By adjusting the salt concentration of the sediment,the interparticle forces for different samples can be equilibratedbetween samples of originally differing salt concentration. Thus,the presented method enables the separation of the effects ofaggregate size (i.e. microstructure) from those of interparticleforces. Aggregate sizes can be measured in the dilute suspension,before the aggregates settle to form a dense sediment.

They find that when other aggregate properties are held con-stant, the shear and compressive yield stresses of sedimentsformed from larger aggregates are generally higher at all vol-ume fractions. Also, the larger aggregates produce a sediment oflower maximum packing fractions than smaller aggregates.

Using a different approach to control microstructure, anotherrecent study 47,60,61,68,69 has focused on both the characterizationof the microstructure as well as the mechanical properties ofconcentrated gels. Control over the microstructure is achievedby using different chemical pathways of coagulating electrostat-ically stabilized gels, using a destabilization method originallydeveloped for the processing of high-performance ceramics, di-rect coagulation casting (DCC).70–72 The method is based onenzyme-catalyzed chemical reactions, which can be controlledby both enzyme concentration and temperature. It allows theformation of gels of highly concentrated particles without per-turbing the microstructures that develop during the gelationprocess. The gels can be produced by two different destabilizat-ion mechanisms that lead to significantly different microstruc-tures: either the pH of the suspensions is shifted toward theirisoelectric point (DpH method), or the ionic strength of the sus-pensions is increased at a constant pH (DI method) (DCC tech-nique (see Box 1)).

Fig. 3. Schematic illustration of the microstructure obtained by dilut-ing a more concentrated particle gel with only gentle stirring. The newgel network is built up from individual clusters that are connected inweak bonds to form a volume-spanning network (after).10

Fig. 4. Compressive yield stress versus volume fraction for particle gelsformed by centrifugation of a low volume fraction gel. The differentvolume fractions correspond to different layers within the centrifugedsample. The initial low volume fraction gels are obtained by diluting agel of volume fraction of f5 0.45 using different homogenization meth-ods: hand mixing (open diamonds), ultrasonication (full diamonds), andhigh shear mixing (circles). Also shown are results for a gel that wasdirectly prepared at low volume fraction and homogenized using highshear mixing (squares) (after).10


The authors use these reactions and apply them to concen-trated particle gels to investigate the relations between the ki-netics of the destabilization process, the dynamics of particles,the evolving microstructure, and the resulting mechanical prop-erties. The dynamics of particles during the aggregation processis followed by using DWS. This light scattering technique allowsderiving quantitative information about the dynamics of parti-cles in very turbid suspensions. While in conventional light scat-tering experiments the samples have to be highly diluted, DWSextends traditional dynamic light scattering to media with strongmultiple scattering by treating the transport of light as a diffu-sion process. We derive quantitative information about the sol–gel transition and the viscoelastic properties of the gels, as wellas a characterization of changes in the microstructure. DI gelsshow stronger elastic moduli and exhibit a less homogeneousmicrostructure than DpH gels.47

To further characterize the dynamics of the destabiliza-tion process as well as the evolving microstructures, the authorsperformed static light scattering measurements on suspen-sions of concentrated silica particles.68,74 Here, problems be-cause of multiple scattering can be avoided by using a very thinsample thickness. In addition, the refractive index difference be-tween the aqueous phase and the particles can be reduced byadding sucrose to the liquid phase of the suspensions. Whileduring the DpH destabilization, the scattering curve shows sig-nificant changes only after some characteristic delay time, itchanges continuously during the DI destabilization. This be-havior is attributed to the formation of a weak pre-gel structurein the suspensions, as a shallow secondary minimum appears inthe interparticle potential.

The real-space structure of concentrated gels is directly ac-cessed by combining high-pressure freezing with Cryo-SEM. Byusing the high-pressure freezing technique, the formation andgrowth of the ice crystals are adequately reduced.59

Microstructures can thus be examined on fracture surfaces ofhigh-pressure frozen particle gels. In Fig. 5(a), we show the ob-tained SEM images of systems at a volume fraction of f5 0.4,for a DpH-destabilized suspension and a DI-destabilized suspen-sion. Structural differences are evident and in good agree-ment with a recent Brownian dynamics simulation study.75,76

A visualization of the structures obtained in this simulation isshown in Fig. 5(b). Results show a less homogeneous structurefor DI gels, with inhomogeneities on the length scale of a fewparticle diameters.

The viscoelastic behavior of concentrated DpH and DI gels ischaracterized by using standard macroscopic rheological meas-urements and, for samples that can support their own weight,compressive tests. In Fig. 6, we show these results for the twodifferent destabilization methods as a function of volume frac-tion. Notable differences in the mechanical behavior of the twosystems are observed. For the inhomogeneous DI gels, signifi-cantly higher yield stresses are observed. In oscillative rheolog-ical measurements, the same behavior is observed; both the yieldstresses and the storage moduli G0 are significantly higher for theless homogeneous DI systems.61,69,77

An analysis of the scaling behavior of yield strains suggests afundamental difference in the stress-bearing mechanism of thetwo systems, with stresses supported by a bond-bending mech-anism for DpH gels and a bond-stretching mechanism for DIgels.69

All results combined strongly indicate that microstructure hasa tremendous influence on the mechanical behavior of thesesystems. The two different variants of the in situ destabilizationprocess naturally lead to gels of significantly different micro-structures.

A logical step in these experiments is to determine what hap-pens if the structure formed in situ is perturbed by applying alarge shear rate (as done in many rheological experiments to setreproducible starting conditions). The result69,78 is surprisingand shows again the strong dependence of mechanical behavioron the structural arrangement of particles in a gel network. Re-sults again suggest an increase in mechanical properties of con-

centrated particle gels with decreasing homogeneity of theirmicrostructures. Shearing a homogeneous DpH gel leads to astrong increase of the shear modulus and yield stress, whileshearing the more inhomogeneous DI gel instead leads to astrong decrease of the elastic and yield properties. The influenceof the microstructure is very evident here, as all the otherparameters (including the interparticle potential) remain un-changed during the shearing process.

Another very direct and simple way to manipulate the micro-structures of gelling colloids is to introduce voids into the struc-ture intentionally. This can be achieved by adding alkalineswellable thickeners (ASTs) to the liquid phase of a colloidalalumina suspension. Hesselbarth et al.79 use polyacrylic acids,such as Acusol 820 or Acusol 830, that are slightly internallycross-linked. These are introduced at low pH (pH 4.5) into highvolume fraction ceramic suspensions. At this pH, these ASTpolymers are small (diameter # 80 nm) insoluble polymer par-ticles that have only a minor influence on the viscosity of thesuspensions.

In Fig. 7, we plot the diameter of these polymers, as measuredby dynamic light scattering in dilute aqueous suspension, as afunction of pH. The diameter of the AST polymers increases byapproximately a factor of 10 between pH 4 and pH 9, the iso-electric point of Al2O3. Thus, increasing the pH in such an alu-mina suspension from pH 4 to pH 9 by using the DCCtechnique (see Box 1) then leads not only to destabilization ofthe colloids but also to a drastic increase in the size of these ASTpolymers. The AST polymers thus produce inhomogeneities inthe evolving microstructure, which can be controlled by the sizeand the concentration of the added AST polymer.

The presence of small amounts of AST polymers (0.4 wt%with respect to alumina) results in a more than 10-fold increasein the yield strength and elastic modulus of the wet coagulatedgels. Figure 8 shows the compressive yield stress sy as a functionof the AST polymer content. This drastic change in the me-chanical behavior is a result of the formation of a less homoge-neous microstructure, caused by the presence of the swollenAST polymers. In case higher urea concentrations are used thereaction first produce a coagulated gel with swollen polymersand high yield strength but further ions created at the buffer pH(5 9) of the urea hydrolysis reaction lead to shielding of thedeprotonated sidegroups of the polymers. This causes the pol-ymer particles to shrink again and the gel relaxes to a more ho-mogenous microstructure with a lower yield strength (arrows inFig. 8).

Thus, in contrast to the study of Channell et al.,10 the resultsof Wyss et al.47,60,61,68,69 and Hesselbarth et al.79 show a signif-icant increase of the elastic and yield properties of concentratedparticle gels with decreasing homogeneity of their microstruc-tures.

In the following, we discuss this apparent contradiction andattempt to provide a qualitative explanation. Figure 9 shows aschematic representation of the different kinds of microstruc-ture. Figure 9(a) shows a microstructure obtained by diluting amore concentrated, homogeneous gel under slow stirring. Theresulting structure consists of dense aggregates that are looselypacked with only a few bonds between aggregates.

Although the mechanical behavior of the aggregates them-selves will be much stronger than that of a homogenous network(Fig. 9(b)), the overall gel structure will be weaker because of thesparse and weak bonds between the aggregates. Channell et al.10

present a model that takes into account both the inter-aggregateand intra-aggregate bonds and agrees well with their experimen-tal data. It is also evident from this model that the overallmechanical behavior for a packing of large aggregates will bedominated by the weakness of the inter-aggregate bonds. Alter-natively, one could view the aggregates as stiff particles andconsider the scaling of the expected mechanical behavior withthe size of these ‘‘aggregate-particles.’’ For packed particles withfixed coordination number and fixed interaction strength, oneexpects the mechanical properties to scale with the number ofbonds per volume. This implies that the mechanical strength, for


Box 1. DCC

DCC70–72 is one of the few methods73 that allow a true in situ destabilization of colloidal suspensions by a direct change of theinterparticle potential. Electrostatically stabilized suspensions are destabilized by shifting the pH of the suspensions toward theirIEP or by increasing the ionic strength in the system. The DCC technique allows for in situ destabilization of the suspensions,without perturbing the gel formation. The onset and speed of the reaction are controlled by both temperature and enzymeconcentration. This is achieved by the use of time-delayed internal chemical reactions, such as enzyme-catalyzed hydrolysisreactions, or hydrolysis reactions induced by elevated temperatures. For the destabilization of alumina suspensions, thehydrolysis of urea,

NH2CONH2 $H2O ! CO2 $ 2NH3 ! HCO"3 $ 2NH$

4 %B1&catalyzed by the enzyme urease can be used. The pH of the suspension is shifted because of the products of this reaction to abuffer pH of 9 in the presence of the enzyme urease. For other oxides with IEPs different from that of alumina, many otherenzymatic destabilization systems are available.54

Furthermore, the ionic strength in a suspension is increased homogeneously and continuously at the buffer pH of 9 by theproducts of Eq. (B1).

DpH and DI methods At low ionic strength and at pH values far from the IEP, an oxide particle suspension is electrostaticallystabilized. This is shown in the schematic stability diagram in Fig. B1. Thus, the suspension is destabilized by either shifting the pHtoward the IEP or by increasing the ionic strength to high values. The two different destabilization pathways are referred to asDpH destabilization and DI destabilization, respectively.

Although in the final stage of destabilization, the interparticle potentials for the two routes might be comparable, the pathway ofdestabilization—the way the interparticle potentials change during the continuous destabilization process—is completely different.

For the two different methods, the evolution of DLVO potentials shows characteristic differences. Interaction potentials, aspredicted bythe DLVO theory using parameters typical for alumina suspensions, are plotted in Fig. B2. A secondary minimum inthe DLVO potential appears during the DDI destabilization, whereas for the case of DpH destabilization, no secondary minimum isobserved.

DCC, direct coagulation casting; IEP, isoelectric point; DLVO, Derjaguin–Landau–Verwey–Overbeek.

Fig.B2. Evolution of Derjaguin–Landau–Verwey–Overbeek-potentials for the two destabilization mechanisms. A Hamaker constant ofAH5 5( 10"20 J and a particle radius of 200 nm is used. (a) Decrease of surface charge at a constant ionic strength of I5 0.001 mol/L (DpHdestabilization). z potential values in the curves are z5 60 mV (i), 30 mV (ii), 20 mV (iii), 10 mV (iv), and 0 mV (v). (b) DI destabilization: Increase ofionic strength at constant surface charge (The z potential is kept constant at z5 60 mV). Ionic strength values in the curves are I5 0.001 mol/L (i),0.02 mol/L (ii), 0.1 mol/L (iii), 0.2 mol/L (iv), 1.0 mol/L (v).

Fig.B1. Schematic stability diagram for a colloidal suspension. At low ionicstrength and at pH values far from the isoelectric point (IEP) the suspensionis electrostatically stabilized. Moving the pH toward the IEP (pathway (a),DpH destabilization) or increasing the ionic strength to high values (pathway(b), DI destabilization) leads to aggregation of the suspension as the surfacecharge or the Debye screening length, respectively, is reduced.


example, the compressive yield stress sy, decreases with the sizeRaggr of the aggregate as:

sy / R"3aggr (12)

The structures shown schematically in Figs. 9(c) and 9(d) arecompletely different. They were formed by an in situ process,yielding structural properties that cannot be achieved by mixingor shearing. In a Gedanken Experiment, we view this kind ofstructure as a two-component system, with one phase consistingof voids (filled with water) and the other phase consisting of amore concentrated, homogeneous gel phase. The gel phase ispercolating the whole macroscopic volume, while the otherphase consists of isolated pores. This is in contrast to the caseof Fig. 9(a), where the liquid phase between aggregates spans theoverall volume.

VI. Summary and Conclusions

The microstructure of concentrated particle gels strongly influ-ences their rheological properties. A significant number ofstudies have focused on this issue in recent years, aiming atdescribing relations between the structural and rheological andmechanical properties of colloidal gel networks. Nevertheless,our understanding of the role of microstructure in concentratedcolloids is still rather limited. More work remains to be carriedout to gain a more thorough understanding.

The reasons for this lack of understanding are twofold:A first important issue is the way microstructure is quantified.

Even if detailed knowledge on the microstructure of a particlegel is experimentally accessible, for instance as a full set of par-ticle positions in the network, this information has to be trans-lated into a useful measure that describes the macroscopicstructure and allows a link to, for instance, the rheological

Box 2. DWS

DWS extends DLS to the highly multiple scattering regime.Traditional DLS relies on samples that show only negligible multiple scattering. In this case, the length scale 1/q that one

observes in an experiment is related to the scattering angle y via

q ! 4pnl

sin%y=2& %B2&

where k is the wavelength of the laser, and n is the refractive index of the sample. Observations of the fluctuations in the detectedintensity then yields information on the dynamics of the system on the length scale 1/q. The fluctuations in intensity arecharacterized by the temporal intensity autocorrelation function g2(q, t):

g2%t& !I%t&I%t$ t&h i

I%t&2D E %B3&

The intensities at time t and at time (t1s) are correlated as long as during the lag time s the particles have moved only over adistance much shorter than the length scale 1/q. When the particle displacements become much larger than this length scale, thetwo intensities will no longer be correlated. For a stable suspension of monodisperse particles, the electric field autocorrelationfunction

g1%t& !###################g2%t& " 1

p%B4&

will decay as

g1%t& ! e"Dq2t %B5&

where D is the diffusion constant of the particles. The size of the particles can thus be derived from such a measurement.In the case of a highly multiple scattering sample, the scenario is different. The detection angle h no longer yields any useful

information about length scales, as each photon passing through the sample would have been scattered by many particles beforeit reaches the detector. The movement of each of these particles will contribute to the total loss of temporal correlation in thedetected signal. In DWS, the transport of photons through a highly multiple scattering sample is treated as a diffusionprocess.45,46 This allows the theoretical prediction of the distribution of the path length of photons through the sample fordifferent sample geometries. An important parameter is the transport mean free path l). It is the distance traveled in the sampleafter which the direction of the photon is completely randomized. The electric field autocorrelation function is then given by

g1%t& !Z 1

0P%s& e"*%s=l)&k2 Dt%t&h i+ds %B6&

where P(s) is the distribution of photon path lengths, L is the cell thickness, and /Dr2S is the mean square displacement of theparticles. Thus, DWS allows measurement of particle sizes in stable suspensions. The analysis of the particle dynamics can alsobe used to characterize the viscoelastic behavior of colloidal gels.84

The main drawback of the technique is its inability to access different length scales. As many particles contribute to the loss oftemporal correlation for each photon path, all measurements represent average properties of the system. On the other hand, thisfact also implies that the technique allows measurement of the dynamics of the system at extremely short length scales. Eachparticle only has to move a very short distance (typically sub-nanometers) in order for a photon path length to change by awavelength of light.However, the best advantage of DWS is the possibility to characterize highly turbid colloidal systems that cannot be studied

by means of conventional light scattering. DWS is thus ideally suited for the characterization of ceramic suspensions, whosestrong turbidity excludes them from access by other techniques.

DLS, dynamic light scattering.


behavior of the system. It is unclear whether quantities like theradial pair correlation function g(r), or the structure factor S(q)are able to capture the essential properties of microstructuresthat can be related to the properties of a macroscopic gel net-work in a meaningful way. Alternative ways of quantifyingmicrostructures 76,80–82 should be considered in the future. Theymight provide us with a much better measure of the essentialstructural properties of particle networks.

A second problem is the lack of control on the possible micro-structures that can be experimentally realized in a reproducibleway. In concentrated and strongly attractive colloidal suspen-

sions, the aggregation times are too fast to allow the formationof a homogeneous and reproducible microstructure by directaddition of salt or acid/base. In order to reach reproducibleconditions, the aggregated gels have to be strongly sheared priorto measurement. However, this implies that the microstructurewill be governed mainly by the pre-shearing process and not bythe particle aggregation kinetics themselves. In situ destabilizat-ion methods offer the possibility of inducing formation gels in areproducible way without shearing or mixing.7,9,71 Even differ-ent kinds of aggregation kinetics, leading to significantly dif-ferent microstructures, can be realized.47,61,71,83 This allows todirectly study the effect of different kinds of microstructures onthe bulk properties of the formed colloidal gels. Another way torealize different microstructures is to concentrate dilute suspen-sions of aggregated clusters into dense gels by centrifugation orpressure filtration. The morphology and size of particle aggre-gates in dilute suspension can be controlled by applying differentshear rates to the suspension. A gel of the compacted clusterswill then have properties that will depend on the structure and

Fig. 5. Top: Scanning electron microscopy images of high-pressurefrozen samples at a volume fraction of f5 0.4. (a) After DpH des-tabilization; (b) after DI destabilization. Bottom: Visualizations ofBrownian dynamics simulations74,75 for fast coagulation (c) and slowcoagulation (d).

Fig. 6. Compressive yield stress sy as a function of volume fraction ffor DpH- versus DI-destabilized gels.

Fig. 7. Diameter of the alkaline swellable thickener (AST) particles inH2O, as measured via dynamic light scattering, as a function of pH. Theconcentration of the AST was 0.05 wt% of dry polymer, relative to H2O.

Fig. 8. Compressive yield strength of wet green bodies with 57 vol%alumina, as a function of polymer content and urea concentration (UC).The yield strength is around 10 kPa for coagulated suspensions withoutpolymers. High yield strength results for coagulated gels with swollenpolymers inducing inhomogeneities in the gel microstructures uponchanging pH from 4 to 9. In case of much higher urea concentrations(1.96 mol/L), after shifting pH from 4 to 9, the salt concentration is in-creased and the swollen polymers shrink as a result of shielding the neg-ative charges of the deprotonated polymer side groups. Thereby, the gelmicrostructures relax, leading again to more homogenous microstruc-tures and to lower yield strengths (indicated by the arrows in the figure).


size of the individual building blocks that constitute the net-work.10

Future studies are necessary to provide us with a deeperunderstanding of the complicated interplays between structur-al and mechanical properties. The challenge will be to com-bine new ways of producing different kinds of microstructureand new, adequate measures to quantify microstructuraldifferences.

Acknowledgments

The authors thank Theo Tervoort for valuable discussions.

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Hans Wyss studied experimentalphysics at the Swiss Federal Insti-tute of Technology (ETH) in Zurich,Switzerland. For his graduate stud-ies, he joined Prof. Ludwig Gauck-ler’s group in the Department ofMaterials at ETH, where he startedto work on ceramics processing andcolloid chemistry. His PhD thesis fo-cused on the experimental character-ization of microstructure in denseceramic suspensions and gels, as

well as on the relation between the microstructure and the me-chanical behavior of these systems.

He is currently working as a post-doctoral fellow at HarvardUniversity in the lab of Prof. David Weitz. His general researchinterest lies on the physics of soft materials such as colloids,emulsions, or polymers. He is particularly interested in under-standing the relations between the dynamics, structure, andmacroscopic properties of these systems.

Elena V. Tervoort received her ac-ademic degrees at the Moscow In-stitute of Chemical Technologyand at the Department of Poly-mers and Composites in the Rus-sian Academy of Science inMoscow, Russia, where sheworked as a research associatefrom 1985 until 1992. Her PhDthesis was on the modification ofpolyolefins with siloxane additivesand anorganic fillers. After a com-

bined post-doctoral stay at DSM research in Geleen, and thePolymer Technology group of the Eindhoven University ofTechnology, both in the Netherlands, she accepted a positionas Research Associate in Prof. Gauckler’s Nonmetallic Inor-ganic Materials group at the Department of Material Science at


the ETH-Zurich in Switzerland. Her research interests includecolloidal aspects of ceramic processing and ceramic foams.

Ludwig J. Gauckler received his de-gree in physics at the University ofStuttgart and his PhD in materialsscience in 1977 with a Thesis onSiAlON that he completed at theMax Planck Institute for Metalsand Materials Research in Stutt-gart. He then carried out researchin the area of high-perform-ance structural and functional ce-ramics as research associate at theUniversity of Michigan, Ann Arbor

in 1977. From 1979 to 1988, he was responsible for the inorganicnon-metallic materials development in the central laboratoriesof Alusuisse-Lonza AG. Since 1988, he is Professor for ‘‘Non-metallic Inorganic Materials’’ in the Department of Material

Science at the ETH-Zurich. He served as head of the Depart-ment from 1991 to 1993. He is guest professor at the TsinghuaUniversity in Beijing since 1993 and a guest professor at theMIT, Boston in 2001.

Ludwig Gauckler and his co-workers received several nation-al and international awards for their work on colloid chemistryfor ceramic processing and high-temperature solid oxide fuelcells, rapid prototyping of ceramics (Cercon), among them, theAward for Real Advances in Materials from NASTS (NationalAssociation for Science, Technology, and Society) and the Fed-eration of Materials Societies in 1994. He is a Fellow ofthe American Ceramic Society and served as President of thescientific advisory board of the Swiss Academy of TechnicalSciences, and is a member of the Academy of Ceramics. His re-search interests include structural ceramics, high Tc-supercon-ductors, computer-assisted modeling of thermodynamics, mixedionic–electronic conducting oxides and colloidal chemistry forceramic processing. Prof. Gauckler has published more than 180scientific papers, and holds 15 patents.


Mechanics and Microstructures of Concentrated Particle Gels

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