rheology of colloidal suspensions flocculated by reversible bridging

* Tel. and fax: #81-43-290-3128.E-mail address: [email protected] (Y. Otsubo).

Chemical Engineering Science 56 (2001) 2939}2946

Rheology of colloidal suspensions #occulated byreversible bridging

Yasufumi Otsubo*Faculty of Engineering, Department of Urban Environment Systems, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba 263-8522, Japan

Received 14 April 2000; received in revised form 27 June 2000; accepted 22 August 2000

Abstract

The steady-shear viscosity, dynamic viscoelasticity, and normal stress behavior were measured for suspensions #occulated byreversible bridging of polymer with weak a$nity for the particle surface. Since the bridges are constantly forming, breaking, andre-forming by thermal energy, the #ow becomes Newtonian at very low shear rates. When subjected to shear "elds, the polymerbridges are highly extended and this produces shear-thickening #ow. Although the extension of #exible coil connecting particles is anintrinsic mechanism, the network structure is essential for shear-thickening #ow of suspensions. The appearance of shear thickening isanalyzed by percolation theory. The shear-thickening suspensions generate a striking normal stress e!ect, which gives a measure ofelasticity. The scaling analysis of elastic behavior near percolation threshold shows that the critical exponent of the reciprocal of thesteady-shear compliance with respect to the di!erence of the particle concentration from the critical value is 1.7. The elastic network of#exible polymer bridges can be modeled by an isotropic force constant. Therefore, the shear-thickening #ow for suspensions#occulated by reversible bridging can be explained by the nonlinear elasticity due to entropy e!ect of extended bridges. � 2001Elsevier Science Ltd. All rights reserved.

Keywords: Bridging #occulation; Shear thickening; Viscoelasticity; Elastic percolation; Critical exponent

1. Introduction

In polymer adsorption at a solid}liquid interface, onlya portion of segments of the chain is in direct contactwith the solid surface. Each polymer chain is attached insequence separated by segments which extend away fromthe surface into the solution. Since the segments extend-ing from one particle can adsorb onto another particle,#occulation of suspensions can be modeled by a bridgingmechanism in which a polymer chain binds two or moreparticles together (Healy, 1961; Healy & La Mer, 1964). Itis generally accepted (Iler, 1971; Fleer & Lyklema, 1974)that bridging #occulation occurs when the polymer chainis long enough and the surface coverage by adsorbedpolymer is low. In ordinary conditions, the polymerchain may not be able to desorb simultaneously from allsites and hence the polymer adsorption is essentiallyirreversible. The polymer bridges are not broken by ther-mal energy. If the three dimensional network of #ocs isdeveloped, the system has a certain shape, the deforma-

tion of which reaches an equilibrium after a long timeunder a constant stress. The highly #occulated suspen-sions behave as solids. However, the bridges are not verystrong. In shear "elds, the structural breakdown is pro-gressively induced with increasing shear rate. The #ow ofsuspensions #occulated by polymer bridging is shear-thinning in a wide range of shear rates.

The irreversibility of polymer adsorption arises fromthe multipoint attachment to the surface. When the poly-mer chains do not have strong a$nity for the surface, thefraction of trains which are in direct contact with thesurface is decreased. Therefore, the weak a$nity betweenpolymer chains and particle surface causes the adsorp-tion}desorption of the whole polymer to reversibly occurby thermal energy. Polymer bridges are constantly form-ing, breaking and re-forming in a quiescent state. Asa result, the suspensions #occulated by reversible bridg-ing are characterized as Newtonian #uids in the limit ofzero shear rate (Otsubo, 1992). In addition, because oflow fraction of train, the adsorbed polymer coil mayremain relatively spherical between particles. Two par-ticles are connected by a #exible bridge, if the coil size inthe solution is comparable to the particle size. The

0009-2509/01/$ - see front matter � 2001 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 0 0 ) 0 0 4 7 8 - 4

Fig. 1. Shear rate dependence of viscosity for 15 vol% suspensionsin solutions of PAA at di!erent concentrations: 0.4 (�); 0.5 (Z); 0.7 (�);1.0 (�); 1.5 (7); 2.0 wt% (�) (adapted from Otsubo, 1992).

#exibility and lifetime of bridges determine the dynamicsof #oc structure in shear "elds. In this review, the rheol-ogy of suspensions #occulated by reversible bridging isdiscussed in relation to the mechanical properties ofpolymer bridges and network structures of #ocs.

2. Materials and methods

2.1. Materials

The suspensions were composed of styrene-methylacrylate copolymer particles, poly(acrylic acid), andwater. The pH value was adjusted with hydrochloric acidto pH2. The stock suspension without poly(acrylic acid)was electrostatically stabilized. The particles were formedby emulsion copolymerization with a styrene/methylacrylate monomer ratio of 40/60. The diameter of par-ticles was 80 nm and the density was 1.13�10� kg m��.The poly(acrylic acid) (PAA) with a molecular weight ofM

�"4.5�10� was obtained from Polysciences, Inc.

and was used as received. The mean size of an isolatedpolymer coil calculated from the intrinsic viscosity is62 nm in aqueous solution at pH2. The sample suspen-sions were prepared at concentrations up to 35% byvolume. The PAA concentration was in the range of0}2.0% by weight based on the water. The rheologicalmeasurements were carried out after the suspensionswere stored at 253C under gentle shear on a rolling devicefor one week. Aging did not have a signi"cant e!ect onthe rheological behavior unless the period exceeded byone month.

2.2. Methods

Steady-shear viscosity �, stress relaxation after cessa-tion of steady shear, and dynamic viscoelasticityGH ("G�#iG�) were measured using a cone-and-plategeometry on a Haake RS100 rheometer. The cone dia-meter was 60 mm and the gap angle between the coneand plate was 23. The measuring shear rates �� were from2.0�10�� to 7.0�10� s�� in steady-#ow measurements.In stress relaxation, the stress decay function was mea-sured after cessation of steady shear at di!erent constantshear rates. The frequencies � were from 1.4�10�� to6.3�10� s�� and strain amplitude was from 2.0�10��

to 3.0�10� in dynamic measurements. The "rst normalstress di!erence N

�were measured with a cone-and-plate

geometry on a Haake CV20N rheometer. The cone dia-meter was 40.94 mm and the gap angle between the coneand plate was 43. The measuring shear rates were from2.0�10�� to 3.0�10� s��. All rheological measurementswere carried out at 253C.

Adsorption of polymer on the particle surface wasmeasured with 10 and 15 vol% suspensions. For deter-mination of the concentration of nonadsorbed polymer,

the #ocs were separated by centrifugation at 1000g formore than 50 h, and the viscosity of supernatant solutionwas measured. The amount of polymer adsorbed on theparticles was calculated from the residual polymer con-centration. The "nal sedimentation volume gives theconcentration of the dispersed phase, from which thebridging distance in the #ocs can be determined.

3. Results and discussion

3.1. Nonlinear viscoelasticity

Fig. 1 shows the shear rate dependence of viscosity for15 vol% suspensions in solutions containing PAA atdi!erent concentrations. At a polymer concentration of0.4 wt%, the viscosity is low and the #ow is slightlyshear-thinning at high shear rates. With increasing poly-mer concentration, the viscosity increases over the entirerange of shear rates. In contrast to ordinary #occulatedsuspensions which show shear-thinning #ow, the viscos-ity behavior of the suspensions containing PAA at0.5 wt% and above is quite di!erent. The viscosity ab-ruptly begins to increase, goes through a maximum, andthen markedly decreases as the shear rate is increased(Otsubo, 1992; Otsubo, 1993). The shear rate at which theshear-thickening #ow begins decreases with increasingpolymer concentration.

Fig. 2 shows the frequency dependence of storagemodulus for 30 vol% suspensions in solutions of PAA atdi!erent concentrations. At low frequencies, the storagemodulus linearly decreases with decreasing frequency. Itis known that the viscoelastic function of #occulatedsuspensions shows a plateau at low frequencies. The

2940 Y. Otsubo / Chemical Engineering Science 56 (2001) 2939}2946

Fig. 2. Frequency dependence of storage modulus for 30 vol% suspen-sions in solutions of PAA at di!erent concentrations: 0.1 (�); 0.2 (Z); 0.3(�); 0.4 (�); 0.5 (7); 0.6 wt% (�) (adapted from Otsubo, 1992).

Fig. 3. Stress relaxation after cessation of steady shear for 15 vol%suspension in a 2.0 wt% PAA solution: 3 (�); 12 (Z); 30 (7); 120 s�� (�)(adapted from Otsubo, 1999).plateau has been considered to be a manifestation of

network structure which is developed by rigid bonds withlong relaxation times (Otsubo, 1990). However, the pla-teau region was not observed for all suspensions studied.Cawdery and Vincent (1995) reported that the PAA caninduce bridging #occulation below pH3 because the pKaof carboxyl groups is about 4. The high relative viscosityof the sample suspensions indicates the existence of #ocstructure. An important feature is that with increasingpolymer concentration, the terminal #ow region shiftstoward the low frequency side. The three-dimensionalnetwork structure provides an additional relaxation pro-cess. The lack of plateau in frequency-dependent curveimplies that the bonds between particles relax by thermalenergy. The suspensions are #occulated by reversiblebridging, in which the particle bonds are constantlyforming and breaking. The lifetime of bridges determinesthe relaxation time of suspensions. In shear "elds, wherethe time scale of adsorption}desorption is shorter thanthat of coil deformation, the #ow becomes Newtonian.

When subjected to high shear "elds, the polymerbridges are extended to a great extent and this produceshigh resistance to #ow. The shear thickening can beattributed to the extension of #exible bridges. The exten-sion of polymer chain causes a decrease in entropy,resulting in the accumulation of energy. The shear thick-ening can be associated with elastic responses, althoughthe strain energy stored in extended bridges relaxes bydesorption. To understand this point, the elastic proper-ties of shear-thickening suspensions must be analyzed asa function of relaxation time in nonlinear regions. Fig. 3shows the stress relaxation behavior after cessation ofsteady shear for 15 vol% suspension in a 2.0 wt% PAA

solution. In high shear "elds where the suspension be-comes shear-thinning, the stress relaxation curves are notstrongly a!ected by the shear rate. This may be a re#ec-tion of the plastic nature. At long times, the stress � de-creases linearly with time for all shear rates. The stressrelaxation is governed by the longest relaxation mecha-nism and the time-dependent function is expressed by thefollowing equation:

�Jexp(!t/��

), (1)

where ��

is the longest relaxation time. Fig. 4 shows theviscosity and relaxation time plotted against the shearrate. Although both the stress and viscosity at t"0 areincreased by a factor of about 10 in the shear-thickeningregion, the longest relaxation time is almost constant(about 30 s). The stress relaxation experiments clearlyindicate that the longest relaxation time is not a!ected bythe viscosity pro"le.

Many structuring #uids such as suspensions and poly-mer solutions show nonlinear viscoelasticity under largedeformation. At very low strains, the storage and lossmoduli show very little dependence on the strain. Underlarge strains, the moduli are drastically decreased. Therapid decrease of moduli can be related to the breakdownof internal structures. To examine the nonlinear elastice!ect, the strain dependence of the storage modulus wasmeasured at di!erent frequencies. Fig. 5 shows the resultsfor 15 vol% suspension in a 2.0 wt% PAA solution. Atlow strains, the storage modulus is constant and theviscoelastic response is linear. As the strain is increased

Y. Otsubo / Chemical Engineering Science 56 (2001) 2939}2946 2941

Fig. 4. Shear rate dependence of viscosity (�) and relaxation time (�)for 15 vol% suspension in a 2.0 wt% PAA solution (adapted fromOtsubo, 1999).

Fig. 5. E!ect of angular frequency on the strain dependence of storagemodulus for 15 vol% suspension in a 2.0 wt% PAA solution: 0.135 (�);0.628 (7); 6.28 (�); 62.8 s�� (�) (adapted from Otsubo, 1999).

Fig. 6. E!ect of particle concentration on the viscosity behavior forsuspensions in a 1.0 wt% PAA solution: 7.5 (�); 10 (Z); 15 (�); 20 (�);25 vol% (7) (adapted from Otsubo, 1992).

above some critical level, the storage modulus showsa rapid increase. The curve shape is very similar to that ofsteady-shear viscosity. Presumably the increase in stor-age modulus implies the same rheology as shear thicken-ing. In steady shear, the shear rate has a critical value forthe appearance of shear thickening. If the shear rate is theprimary factor for sharp increase in storage modulus, thecritical strain would linearly increase with decreasingfrequency. However, a sharp increase occurs when thestrain is increased up to 3}5, independent of frequency. Inoscillatory shear, the strain has a critical value, abovewhich the storage modulus begins to rapidly increase andbelow which linear responses are observed. The dynamicviscoelasticity under large strains is controlled by thestrain, rather than the shear rate. The mechanical proper-ties of #exible bridges are analogous to the rubberyelasticity. The sudden onset of shear thickening can betheoretically derived by a nonlinear elastic model inwhich the relaxation modulus exponentially increaseswith strain (Otsubo, 1999).

3.2. Percolation analysis

Fig. 6 shows the shear rate dependence of viscosity forsuspensions in a 1.0 wt% PAA solution. The #ow ofdilute suspensions is Newtonian. A shear thickening oc-curs at particle concentrations above 10 vol%. The par-ticle concentration also has a critical value for shearthickening. The shear rate at the onset of shear thicken-ing decreases with increasing particle concentration. Al-though an increase in particle concentration causesa drastic increase in the Newtonian viscosity at low shearrates, the magnitude of viscosity enhancement in shear-thickening region is smaller for concentrated suspen-sions. Beyond the maximum viscosity, the #ow is almostplastic and the particle concentration does not havesigni"cant e!ects. The non-Newtonian viscosity pro"lestypical of the sample suspensions consist of a low-shear-rate Newtonian viscosity, a shear-thickening region atmoderate shear rates, and a shear-thinning (nearly plas-tic) region at high shear rates.

The shear-thickening #ow and maximum viscosity areobserved when both the particle and polymer concentra-tions exceed some critical values. Fig. 7 shows the bound-ary for shear-thickening #ow. Although the suspensionswith compositions above the line are shear-thickening athigh shear rates, the magnitude of viscosity enhancementis large in the vicinity of the boundary. The boundarycondition can be discussed in terms of percolationprocess.

Dilute suspensions of #occulated particles consist ofa collection of discrete #ocs. In more concentratedsuspensions, the #ocs cease to be discrete and anetwork structure is formed. The distribution of particlesin the media and the network formation process can be


Fig. 7. Transition boundary from shear-thinning (�) to shear-thicken-ing (�) suspensions (adapted from Otsubo, 1992).

described through percolation theory. Site percolationdeals with the distribution of cluster sizes for particlesdistributed in an in"nite lattice composed of sites linkedtogether by bonds. When sites are occupied at randomwith probability p� and adjacent occupied sites are con-nected, the process has a critical probability p�

�above

which unbounded clusters of connected sites are con-structed. In a similar manner, unbounded clusters areexpected when the occupancy of bonds p� in a latticewhose sites are fully occupied exceeds a critical value p�

�.

The process is called bond percolation. Many authorshave discussed the critical percolation probability forvarious lattices (Sykes & Essam, 1964; Vyssotsky, Gor-don, & Frisch, 1961). In addition, the percolation processfor p�(1.0 and p�(1.0 is referred to as site-bondpercolation. The site-bond percolation process hasthresholds which are given as a set of p� and p�. Insuspensions #occulated by polymer bridging, the #ocsare considered to consist of sites (particles) connected bybonds (bridges).

To understand the #oc structures, the adsorption andsedimentation experiments were carried out for 10 vol%suspensions in polymer solutions at concentrations of0.3}1.0 wt%. The particle concentration in the sedimentwas 26 vol% and the adsorbance was 50 mg/g-particles.Assuming that the particles are arranged in hexagonalpacking in the sediment, the mean distance betweenparticle surfaces is estimated to be 33 nm. Referring backto the experimental section, the diameters of an isolatedcoil and particle are 62 and 80 nm, respectively. Althoughthe contraction of polymer coil occurs by adsorption, thebridging distance is comparable to the coil size and thepolymer bridges may remain spherical. It can be stressedthat the bonds between particles are very #exible.

At particle concentrations above 26 vol%, all polymerchains adsorb onto particles and e!ectively make bridges

when the polymer concentration is low. The e!ect ofpolymer concentration on the #oc structure can be ana-lyzed as the bond percolation. According to the empiricalequation (Ziman, 1968), the percolation threshold forvarious three-dimensional lattices is zp�

�&3/2, where z is

the coordination number. In concentrated suspensions,the average number of bridges for each particle is esti-mated from the critical polymer concentration for ap-pearance of shear thickening on the assumption that onepolymer coil makes one bridge. The value is 1.5 for30 vol% suspension and reasonably acceptable as thepercolation threshold for network formation.

On the other hand, it would be reasonable that twoadjacent particles are bridged in dilute suspensions whichcontain su$cient amount of polymer. The e!ect of par-ticle concentration on the #oc structure is describedthrough the site percolation. The network structure isalso developed above the critical site probability p�

�. The

7.5 vol% suspension showed shear thickening at polymerconcentrations above 1.4 wt%. However, shear thicken-ing was not observed for 5.0 vol% and more dilutesuspensions even though polymer concentration was in-creased beyond 2.0 wt%. The critical particle concentra-tion seems to be 5.0 vol%. The rheology of suspensions isgenerally discussed in relation to a normalized concen-tration de"ned as the ratio of volume concentration ofprimary particles to the maximum packing concentra-tion. The normalized concentration may correspond tothe site probability. The critical site probability is deter-mined as 0.2. For the site percolation of hexagonal close-packed lattice, the critical probability p�

�has been

estimated to be 0.204 (Frisch, Sonnenblick, Vyssotsky,& Hammersley, 1961). The theoretical prediction forpercolation and experimental value for shear-thickening#ow are in good agreement. Therefore, it is concludedthat the shear thickening of suspensions #occulated byreversible bridging results from the network structureconsisting of particles bridged by #exible polymer chains.

3.3. Normal stress behavior and scaling analysis

A comparison between the viscometric and dynamicfunctions will provide useful information on the relax-ation mechanisms. The most popular relationship is theCox}Merz rule (Cox & Merz, 1958) which proposes thatthe steady-shear viscosity � should be the same functionof shear rate �� as the magnitude of complex viscosity �His of frequency �, where �H is given by

�H"[ (��)�#(G�/�)�]��. (2)

This correspondence has been con"rmed experimentallyfor most polymeric liquids. In addition, both the storagemodulus G� and "rst normal stress di!erence N

�are

monotonic increasing functions of � and �� , respectively.It is considered that a relationship analogous to the


Fig. 8. Viscosity (�) and "rst normal stress di!erence (7) in steadyshear and magnitude of complex viscosity (Z) and storage modulus (�)at a strain of 0.05 in oscillatory shear for 5 vol% suspension ina 1.5 wt% PAA solution (adapted from Otsubo, 1994).

Fig. 9. Viscosity (�) and "rst normal stress di!erence (7) in steadyshear and magnitude of complex viscosity (Z) and storage modulus (�)at a strain of 0.05 in oscillatory shear for 20 vol% suspension ina 1.5 wt% PAA solution (adapted from Otsubo, 1994).

Cox}Merz rule will hold between G� and N�:

N�"2G�. (3)

In fact, we can easily deduce that Eq. (3) is the exactrelation in the lower limits of frequency and shear rate.

Fig. 8 shows the viscosity and "rst normal stress di!er-ence in steady shear and the magnitude of complex vis-cosity and storage modulus in oscillatory shear for5 vol% suspension in a 1.5 wt% PAA solution. Appar-ently, the Cox}Merz rule is not applicable at high shearrates. However, the shapes of � and �H curves are similarand the di!erence is not large. Moreover, the plots ofN

�and 2G� approximately lie on a single curve. It can be

seen that the above empirical rules are acceptable forshear-thinning suspensions.

Fig. 9 shows a comparison between the viscometricand dynamic functions for 20 vol% suspensions ina 1.5 wt% PAA solution. The steady-shear viscosity co-incides with the magnitude of complex viscosity in theNewtonian region, while signi"cant discrepancy appearsat high shear rates. The "rst normal stress di!erencerapidly increases and becomes constant in the shear-thickening region. The di!erence between N

�and 2G� is

also very large at high shear rates. The steady-shearviscosity shows the maximum at shear rate where the "rstnormal stress di!erence reaches the equilibrium. FromFigs. 8 and 9, it must be noted that Eqs. (2) and (3) are notapplicable to the rheology of suspensions in the shear-thickening region (Otsubo, 1994).

When the time scale of coil extension is much longerthan that of desorption, the polymer coils have the equi-librium conformation and the #ow becomes Newtonian.In high shear "elds, the coils are rapidly extended before

they dissipate the stored energy; this results in shearthickening. However, the coils are forced to desorb atsome degree of extension. Since the shear forces cause therupture of bridges by desorption, the suspensions arenearly plastic at high shear rates. It is interesting that inshear-thinning region where the breakdown of bridges byforced desorption takes place, the "rst normal stressdi!erence is almost independent of shear rate. The mech-anism of change in viscosity pro"le from shear-thicken-ing to shear-thinning #ow is the relaxation of extendedbridges due to desorption in shear "elds. The dominantvariables are the lifetime and strength of bridges.

Although the shear thickening arises from the nonlin-ear elasticity of #exible bridges due to the extension inhigh shear "elds, the three-dimensional network of un-bounded #ocs is essential. The transition from discrete toin"nite #ocs in #occulated suspensions is accompaniedby extreme changes of material transport properties.Because the particle}particle bonds transmit theelastic forces, the network of unbounded #ocs respondselastically to small deformation. The elastic properties asoverall responses are closely related to the mechanicalproperties of each bond between primary particles andnetwork structure of #ocs.

Scaling analysis enables us to show a power lawdependence of elasticity on the di!erence of particle con-centration from the critical value C

�:

G"k(C!C�)�, (4)

where G is the static elastic modulus, k is a constant, andn is the critical exponent. In ordinary #occulated suspen-sions, the #oc-#oc bonds are not broken down byBrownian motion. Since the frequency-dependent curveof storage modulus shows a plateau at low frequencies,


Fig. 10. Critical behavior of J��

near percolation threshold (adaptedfrom Otsubo, 1994).

the plateau value can be used as the elastic modulus tocharacterize the network structure. However, the re-search interest of this study is focused on the elasticitydue to the deformation of network constructed bya series of temporary bridges.

The "rst normal stress di!erence is generated by elastice!ects. To understand the energy storage and internalstrain �

�of viscoelastic liquids in shear "elds, the steady-

shear compliance J�

is often used:

J�"N

�/2(�� )�, (5)

��"�� J�

. (6)

The reciprocal of the steady-shear compliance, J��

, givesa measure of elasticity. For polymer solutions, the J��

�is

proportional to the number of molecules or to the num-ber of entanglement points in the unit volume. It isreasonable to discuss the network structure in connec-tion with J��

�.

Fig. 10 shows the critical behavior of J��

near thepercolation threshold. The values of J��

�were deter-

mined using the data at the maximum viscosity, becausethe energy stored in network would reach the maximum.The plots are closely related by a straight line. Therefore,one can "nd that the J��

�is scaled on C}C

�and the

critical exponent (slope of line) is 1.7.de Gennes (1979) pointed out that the elastic modulus

of a polymer gel, modeled by an isotropic force constant,is analogous to the electrical conductivity. For electricalnetwork in which conductors carry a scalar quantity, thecritical exponent is estimated to be 1.8. It has beenreported (Tokita, Niki, & Hikichi, 1984; Allain & Salome,1987) that the value for polymer gels is about 1.9 andagrees with that from scalar elasticity percolation. Thecritical exponent obtained here suggests that the elasticnetwork of particles connected by #exible polymerbridges can be modeled by an isotropic force constant.

For suspensions #occulated by irreversible bridging, thecritical exponent was found to be about 4.0 (Otsubo& Nakane, 1991). The di!erence in critical exponentsshows that two elastic properties belong to di!erentuniversality classes. In the elastic network the bondstransmit central and transverse forces. According to the-oretical studies on percolation of elastic network (Feng& Sen, 1984; Kantor & Webman, 1984), the criticalexponent depends on the local elastic constant which isgiven by a set of central and transverse elastic constants.Therefore, the factor controlling the critical behavior ofelastic network is the vector nature of bond.

In irreversible bridging, one bridge to bind two largeparticles consists of many loops extending to both surfa-ces. Considering that the polymer coils form rigidbridges, the strain required to break the network is verylow. On the other hand, as mentioned above, the suspen-sions #occulated by reversible bridging show linear vis-coelastic responses at strains up to 2 in oscillatory shear.The internal strain to which the network is subjected canbe calculated by Eq. (6) and the value is in the range of3}5 for all shear-thickening suspensions. The networkcan be highly extended in shear "elds and the bond willbe relatively soft. Polymer chains containing a smallfraction of strongly associating groups exhibit similarshear-thickening #ow (Jenkins, Silebi, & El-Aasser, 1991;Lundberg, Glass, & Eley, 1991). This can be attributed tothe shear-induced formation of interchain association atthe expense of intrachain association (Witten & Cohen,1985). Through the statistical mechanical model(Vrahopoulou & McHugh, 1987), the shear thickening isexplained by the decrease in entropy of polymer chains inthe network during extension by shear. In suspensions, itis expected that the elasticity is induced by the extensionof #exible bridges and the strain energy may be stored asa decrease in entropy of polymer chains. The intrinsicmechanism of shear thickening is the entropy elasticity of#exible polymer bridges.

4. Conclusions

(1) The suspensions #occulated by reversible bridgingshow shear-thickening #ow in a narrow range of shearrates.

(2) The shear thickening occurs, when both the particleand polymer concentrations exceed the critical values.From the percolation analysis, the three-dimensionalnetwork of particles is essential for appearance of shearthickening.

(3) The storage modulus at a constant frequency showsa rapid increase under large deformation. The shearthickening may arise from the nonlinear elasticity of#exible bridges.

(4) The shear-thickening suspensions show strikingnormal stress e!ect. Based on the scaling analysis, the


elastic network is modeled by an isotropic force constant.The intrinsic mechanism of shear thickening is the en-tropy elasticity of extended bridges.

Notation

C particle concentration, vol%C

�critical particle concentration, vol%

G static elastic modulus, PaGH complex shear modulus, PaG� storage modulus, PaG� loss modulus, PaJ�

steady-shear compliance, Pa��

n critical exponentN

�"rst normal stress di!erence, Pa

p� bond probabilityp��

critical bond probabilityp� site probabilityp��

critical site probabilityt time, sz coordination number

Greek letters� strain��

internal strain�� shear rate, s��

� viscosity, Pas�H complex viscosity, Pas�� dynamic viscosity, Pas� shear stress, Pa��

longest relaxation time, s� angular frequency, s��

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rheology of colloidal suspensions flocculated by reversible bridging

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