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Langmuir 1999, 15, 1960-1965
A Nonlinear Elastic Model for Shear Thickening of Suspensions Flocculated by Reversible Bridging Yasufumi Otsubo Department of Urban Environment Systems, Faculty of Engineering, Chiba University, Yayoi-cho 1-33, Inage-ku, Chiba-shi 263-8522, Japan Received August 31, 1998. In Final Form: December 22, 1998 The flow of suspensions flocculated by reversible bridging of polymer with weak affinity for the particle surface is Newtonian in the limit of zero shear rate, shear-thickening at moderate shear rates, and plastic with a constant stress independent of shear rate in high shear fields. The Newtonian flow implies that the polymer bridges are constantly forming and breaking in a quiescent state. The stress after cessation of steady shear exponentially relaxes at long times. The relaxation time is not affected by the shear rate. The strain-dependent curve of storage modulus at a constant frequency shows a rapid increase when the strain is increased above a critical level. The critical strain is independent of frequency. Therefore, the shear thickening may be primarily attributed to the elastic effect of extended bridges. Since the polymer coils are forced to desorb at some degree of extension, the suspensions become nearly plastic at high shear rates. The intrinsic mechanism of shear-thickening flow for suspensions flocculated by reversible polymer bridging is the nonlinear elasticity due to entropy effect of extended bridges and forced desorption due to hydrodynamic effect. By combination of the nonlinear elasticity and single relaxation time, a rheological model is derived to quantitatively express the shear-thickening flow. The model prediction and experimental results are in good agreement.
Introduction The most important feature of polymer adsorption is that not all the segments of a chain are in contact with the surface. The conformation of adsorbed polymer chain is explained in terms of trains of segments which attach to the surface, loops of segments which extend into solution between trains, and tails which are the dangling ends of chains.1-3 The train and loop segments reversibly interchange. However, the polymer chain attaches to the surface at many points. The mobility of a polymer chain is markedly reduced by adsorption, and the movement of the whole polymer is very slow. In practice, polymer adsorption is essentially irreversible in many cases because the polymer may not be able to desorb simultaneously from all sites. When the polymer chains do not have very strong affinity for the surface, the fraction of loops increases at the expense of trains. The chain has a lower fraction of adsorbed segments and more conformational freedom. The irreversibility of strong adsorption arises from multipoint attachment to the solid surface. Therefore, the decrease in the fraction of trains due to weak adsorption causes the adsorption-desorption process to occur reversibly by Brownian motion. In colloidal suspensions, the segments extending from one particle can adsorb onto another particle and bind them together. The effect is referred to as polymer bridging.4,5 Flocculation of colloidal suspensions by the bridging mechanism takes place under conditions where the polymer chain is long enough and the surface coverage by adsorbed polymer is low. When the reversible adsorption is present due to polymer chains with weak affinity for the particle surface, the polymer bridges are constantly forming, breaking, and re-forming in zero shear fields. In (1) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1979, 83, 1619. (2) Scheutjens, J. M. H. M.; Fleer, G. J. J. Phys. Chem. 1980, 84, 179. (3) Ploehn, H. J.; Russel, W. B. Adv. Chem. Eng. 1990, 15, 137. (4) Iler, R. K. J. Colloid Interface Sci. 1971, 37, 364. (5) Fleer, G. J.; Lyklema, J. J. Colloid Interface Sci. 1974, 46, 1.
contrast to ordinary flocculated suspensions whose flow profile is plastic at very low shear rates or shear thinning over a wide range of shear rates, the suspensions flocculated by reversible bridging are Newtonian in the limit of zero shear rate. At shear rates where the time scale of coil extension is much longer than that of desorption, the polymer coils have an equilibrium conformation unchanged from zero shear state. The striking contraction of chain on the solid surface is not observed in the case of weak adsorption. Since two particles can be bridged by a flexible polymer coil, elastic and deformable flocs are built up in suspensions flocculated by reversible bridging. In shear fields, the bridges are subjected to rapid extension. The extended bridges can lead to high resistance to flow due to the restoring forces. As a result, the flow becomes shear thickening at intermediate shear rates.6-8 The restoring forces produced by rapid extension of flexible bridges are responsible for the shear thickening. The strain energy stored in extended bridges is released by train desorption in shear filed. Hence, the onset of shear thickening can be analyzed in connection with this relaxation process of polymer chains. The dominant variables affecting the shear thickening are the lifetime and extensibility of bridges which are controlled by changing the adsorption affinity of polymers. In the present paper, shear-thickening behavior is studied for suspensions flocculated by reversible bridging of polymer coils whose size is comparable to the particle diameter. To quantitatively describe the results, a rheological model is developed. The comparison of experimental results and model prediction will be discussed in relation to the nonlinear elasticity of bridges. Materials and Methods Materials. The suspensions were composed of styrene-methyl acrylate copolymer particles, poly(acrylic acid) (PAA), surfactant, and water. The pH value was adjusted with hydrochloric acid to (6) Otsubo, Y. Langmuir 1992, 8, 2336. (7) Otsubo, Y. J. Rheol. 1993, 37, 799. (8) Otsubo, Y. J. Colloid Interface Sci. 1994, 163, 507.
10.1021/la9811362 CCC: $18.00 © 1999 American Chemical Society Published on Web 02/19/1999
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Figure 1. Shear rate dependence of viscosity in a 2.0 wt % PAA solution without surfactant at different temperatures: 15 (O); 25 (y); 35 (k); 45 °C (b). The broken line indicates the transition from Newtonian to shear-thickening flow. pH 2. The stock suspension without PAA and surfactant was electrostatically stabilized. The particles were formed by emulsion copolymerization with a styrene/methyl acrylate monomer ratio of 40/60. The surfactant was an ethoxylated octylphenol (Triton X-100 from Union Carbide Co.). The diameter of the copolymer particles was 80 nm and the density was 1.13 × 103 k gm-3.The poly(acrylic acid) (PAA) with a molecular weight of Mw ) 4.5 × 105 was obtained from Polysciences, Inc., and was used as received. The mean size of an isolated polymer coil calculated from the intrinsic viscosity is 62 nm in aqueous solution at pH 2. The sample suspensions were prepared at a concentration of 15 vol %. This was accomplished by diluting the stock suspension with calculated weights of aqueous solutions which contained surfactant in addition to 10 wt % PAA. The final PAA concentration was 2.0% and the surfactant concentration was in the range of 0-2.0 wt % based on the water. The rheological measurements were carried out after the suspensions were stored at 25 °C under gentle shear on a rolling device for 1 week. Because of initial conditioning prior to measurements, the results were highly reproducible. Aging did not have a significant effect on the rheological behavior unless the period exceeded 1 month. Methods. Steady-shear viscosity, stress relaxation after cessation of steady shear, and dynamic viscoelasticity were measured using a cone-and-plate geometry on a Haake RS100 rheometer. The cone diameter was 60 mm and the gap angle between the cone and plate was 2°. The measuring shear rates were from 2 × 10-2 to 7 × 102 s-1 in steady-flow measurements. In stress relaxation experiments, the stress decay function was measured after cessation of steady shear at different constant shear rates. The frequencies were from 1.4 × 10-1 to 6.3 × 101 s-1 and strain amplitude was from 2 × 10-2 to 3.0 × 101 in dynamic measurements. The rheological measurements were carried out at 15, 25, 35, and 45 °C.
Results First, the rheology of suspension dispersed in a 2.0 wt % PAA solution without surfactant was examined. Figure 1 shows the shear-rate dependence of viscosity at different temperatures. The viscosity of 15 vol % suspension without PAA is 3.5 × 10-3 Pa s at 25 °C. Because the viscosity of 2.0 wt % PAA solution was 1.25 × 10-2 Pa s at pH 2, the addition of PAA causes an increase in viscosity of suspension. The suspension was centrifuged at 1000g for more than 50 h. The final sedimentation volume gives the concentration of the dispersed phase. The amount of polymer adsorbed on the particles can be calculated from the viscosity of supernatant solution. The particle concentration in the sediment was 26 vol %, and the adsorbance was 50 mg/g particles. Since the suspension studied contains sufficient polymer, the particle surface my be completely covered with polymer. The suspension is highly flocculated by polymer bridging. However, the
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flow behavior of sample suspension is quite different at all temperatures from that of ordinary flocculated suspensions which show shear-thinning profiles over a wide range of shear rates. The flow is Newtonian even at very low shear rates, and the suspension behaves as a fluid. As the shear rate is increased, the viscosity abruptly begins to increase, goes through a maximum, and then markedly decreases. With increasing temperature, the Newtonian viscosity decreases and the critical shear rate at the onset of shearthickening flow increases. The transition boundary from Newtonian to shear-thickening profiles lies on a straight line with a slope of about -1. The critical shear rate is inversely proportional to the Newtonian viscosity. Beyond the peak viscosity, the viscosity rapidly decreases. In this region, the flow is almost plastic, because the shear stress seems independent of shear rate. Concentrated suspensions often cause slip at solid boundaries in rheometers. The errors due to wall slip can be corrected by repeating viscosity measurements twice at different gap distances in a parallel plate geometry.9 Although the reading was slightly scattered at high shear rates, the curves of shear stress versus apparent shear rate for different gaps were almost the same. The wall slip was not observed for the sample suspension. The shear-thickening flow may arise primarily from the extension of flexible bridges. In high shear fields, the extended bridges are forced to desorb. Since the desorption of polymer coils takes place at some constant force, the suspension shows nearly plastic flow at high shear rates. Polymer chains containing a small fraction of strongly associating groups exhibit similar shear-thickening flow.10,11 This can be attributed to the shear-induced formation of interchain association at the expense of intrachain association.12 According to the statistical mechanical model,13 the shear-thickening flow is due to the decrease in entropy of polymer chains in the network during extension by shear. In the suspensions, the extension of bridges may be responsible for the increase in resistance to flow. In fact, shear thickening is observed when both the particle and polymer concentration are increased beyond critical values. The boundary condition for the appearance of shear-thickening flow can be analyzed as a percolation process because the flocs can be compared to clusters consisting of sites (particles) connected by bonds (bridges). From the sedimentation and adsorption experiments, the author has found good correspondence between rheological boundary and percolation threshold for network formation, by using a simplified model in which the particles are arranged in hexagonal packing in the floc.6,7 It must be stressed that the unbounded flocs, that is, the connections of bridges over the system, are essential for shear-thickening flow as overall responses of suspensions. Figure 2 shows the frequency dependence of storage modulus G′ and loss modulus G′′ at temperatures of 15 and 45 °C. Because of low strain amplitude, the viscoelastic responses are regarded as linear. Both moduli rapidly decrease with decreasing frequency, and at low frequencies the loss modulus is predominant. It is well-known that the viscoelastic function (G′, G′′) of ordinary flocculated suspensions shows a plateau at low frequencies. The (9) Yoshimura, A.; Prud’homme, R. K. J. Rheol. 1988, 32, 53. (10) Jenkins, R. D.; Silebi, C. A.; El-Aasser, M. S. ACS Symp. Ser. 1991, No. 462, 222. (11) Lundberg, D. J.; Glass, J. E.; Eley, R. R. J. Rheol. 1991, 35, 1255. (12) Witten, T. A.; Cohen, M. H. Macromolecules 1985, 18, 1915. (13) Vrahopoulou, E. P.; McHugh, A. J. J. Rheol. 1987, 31, 371.
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Figure 2. Frequency dependence of storage modulus (O) and loss modulus (b) at 15 and 45 °C.
plateau has been explained by an additional relaxation process due to network structure of particles. However, this plateau region was not observed for all suspensions, irrespective of temperature. The lack of plateau in frequency-dependent curves indicates the relatively weak interactions between particles with short relaxation times. It looks likely that the relaxation time becomes shorter at higher temperatures. To understand the detailed mechanism of shearthickening flow, the elastic modulus of extended bridges must be analyzed as a function of relaxation time in nonlinear regions. Figure 3 shows the stress relaxation behavior after cessation of steady shear at 25 °C. The ratio of stress to shear rate at the limit of t ) 0 corresponds to the steady-shear viscosity. In high shear fields, the stress relaxation curves are not strongly affected by the shear rate. This may be a manifestation of the plastic nature. At long times the stress decreases linearly with time for all shear rates. Therefore, the stress relaxation is governed by the longest relaxation mechanism and the time-dependent function is expressed by the following equation.
σ ∝ exp(-t/τm)
Figure 3. Stress relaxation after cessation of steady shear at 25 °C: 3 (O); 12 (#); 30 (k); 120 s-1 (b).
Figure 4. Shear rate dependence of viscosity (O) and relaxation time (b) at 25 °C.
(1)
where τm is the longest relaxation time. Figure 4 shows the viscosity and relaxation time plotted against the shear rate at 25 °C. Although both the stress and viscosity at t ) 0 are increased by a factor of about 10 in the shearthickening region, the longest relaxation time from eq 1 is almost constant (about 30 s). The shear-thickening flow is attributed to the extension of flexible bridges. But the stress relaxation experiments clearly indicate that the longest relaxation time is not affected by the degree of extension. Many structuring fluids such as suspensions and polymer solutions show nonlinear viscoelasticity under large deformation. At very low strains, the storage modulus shows very little dependence on the strain. Under large strains, the storage and loss moduli are drastically decreased. The rapid decrease of moduli can be related to the breakdown of internal structures. To examine the nonlinear elastic effect, the strain dependence of the storage modulus was measured at different frequencies. Figure 5 shows the results at 25 °C. At low strains, the storage modulus is constant and the viscoelastic response is linear. When the strain is increased above some critical
Figure 5. Effect of angular frequency on the strain dependence of storage modulus at 25 °C: 0.135 (b); 0.628 (k); 6.28 (y); 62.8 s-1 (O).
level, the storage modulus shows a rapid increase. Except at a frequency of 62.8 s-1, the curve shape is very similar to that of steady-shear viscosity. Presumably the increase in storage modulus reflects the same rheology as shear thickening. In steady shear, the shear rate has a critical value for the appearance of shear thickening. However, the sharp increase in storage modulus occurs when the strain is increased up to 3-5, independent of frequency. In oscillatory shear, there exists a critical strain, above which the storage modulus begins to rapidly increase and
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Figure 6. Effect of temperature on the strain dependence of storage modulus at 6.28 s-1: 15 (O); 25 (y); 35 (k); 45 °C (b).
below which linear responses are observed. The dynamic viscoelasticity under large strains is controlled by the strain, rather than by the shear rate. The mechanical properties of flexible bridges are analogous to the rubbery elasticity, of which the elastic modulus markedly increases in extended states. In previous work,8 the author has reported that the suspensions flocculated by reversible bridging of polymers generated a striking normal stress effect in the shearthickening region. The first normal stress difference provides a measure of elasticity. Since the elasticity is induced by the extension of flexible bridges, the strain energy may be stored as a decrease in entropy of polymer chains. The internal strain to which the flexible bridges is subjected at the maximum extension can be calculated through steady shear compliance, and the value is estimated to be 3-5. The critical strain determined through the normal stress difference and the strain dependence of the storage modulus are almost the same. The nonlinear elasticity of polymer bridges plays an important role in controlling the shear-thickening flow. Figure 6 shows the effect of temperature on the strain dependence of the storage modulus at a frequency of 6.28 s-1. At 45 °C, the storage modulus is of the order of 10-1 Pa. The accurate values of storage moduli at large strains were not determined, because of the limitations of the rheometer. However, the strain-dependent function in the range of 15-35 °C clearly shows the sharp increase at strains of 3-8. Although the increasing region slightly shifts toward the large strain side, the effect of temperature on the critical strain is weak. Therefore, the primary factor dominating the nonlinear rheological behavior is the internal strain in flexible polymer bridges. In suspensions flocculated by reversible bridging, the adsorption-desorption of the polymer coil on the particles reversibly takes place by thermal energy. The lifetime of bridges may be a strong function of temperature. From Figure 1, a linear relation between the Newtonian viscosity and relaxation time is expected. The lifetime of a bridge can be related to the adsorption affinity of the polymer coils. The number of attaching points along the chain is decreased by surfactant adsorption. In the second stage of these experiments, the adsorption affinity was modified by the addition of surfactant. The effect of lifetime of bridges on the rheology was studied at 25 °C to provide more insight into the mechanism. Figure 7 shows the shear rate dependence of viscosity for the suspension in a 2.0 wt % PAA solution containing surfactant at different concentrations. With increasing surfactant concentration, the Newtonian viscosity decreases and the critical shear rate increases. At 2.0 wt %,
Figure 7. Effect of surfactant concentration on the viscosity behavior at 25 °C: 0 (O); 1.0 (y); 1.5 (k); 2.0 wt % (b).
Figure 8. Effect of surfactant concentration on the strain dependence of storage modulus at 6.28 s-1: 0 (O); 1.0 (y); 1.5 (k); 2.0 wt % (b).
the increment of viscosity in the shear-thickening region is very small. From Figure 1, the addition of surfactant has almost the same effect on the shear-thickening behavior as an increase in temperature. The coverage of particle surface with surfactant causes a decrease in the number of adsorption sites per chain. Since the chance for desorption of chains from the surface determines the lifetime of bridges, the relaxation time is decreased with increasing surfactant adsorption. Figure 8 shows the effect of surfactant concentration on the nonlinear behavior of the storage modulus at a frequency of 6.28 s-1. At surfactant concentrations below 1.5 wt %, a sharp increase in the storage modulus can be observed as the strain is increased up to 3. The critical strain depends only weakly on the surfactant concentration. An interesting finding with respect to nonlinear viscoelastic behavior is that the increase in temperature and surfactant concentration gives rise to the reduction of relaxation time, while the critical strain at the onset of rapid increase in storage modulus is not strongly affected. Discussion The shear-thickening flow of suspensions flocculated by reversible bridging can be attributed to the nonlinear elasticity of flexible bridges and forced desorption at large extension. Although the relaxation time varies with
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temperature and surfactant concentration, the critical strain for the appearance of the nonlinear effect is approximately constant. Hence, the nonlinear relaxation modulus G(t,γ) of reversible bridges can be given by the following expression14,15
G(t,γ) ) h(γ) exp(-t/τ)
(2)
where h(γ) is the nonlinear damping function (this function decreases with increasing strain for most polymer solutions and is generally called damping function) and τ is the relaxation time. From the strain dependence of storage modulus, h(γ) may increase with strain at the beginning of the nonlinear effect. As a simple approach, an exponential function is adopted
h(γ) ) G0 exp(Rγ)
(3)
where R is a constant. The steady-shear viscosity is expressed by the integral of the relaxation modulus.
η)
∫0 h(γ) exp(-t/τ) dt ∞
(4)
The final formula of viscosity at shear rates below 1/Rτ is
η)
G0τ (1 - Rγ˘ τ)2
γ˘ < 1/Rτ
(5)
The viscosity is constant (Newtonian viscosity ) G0τ) in the low shear fields and rises sharply with increasing shear rate. The model predicts that η f ∞ at γ˘ f 1/Rτ. But, the actual suspensions are almost plastic at high shear rates. This implies that the structural rupture occurs at a constant shear stress. The origin of the rupture of the bridge is the forced desorption of polymer chain from the particle surface. The required force for desorption or strength of bridge can be coupled with the number of attaching points per chain on the particle. In the elastic model proposed, the constant shear stress at high shear rates, σ0, is assumed to be proportional to the number of attaching points per chain. In addition, the relaxation time can be approximated as a quadratic function of the number of attaching points on the basis of the Rouse theory,16 which is derived to explain the viscoelastic relaxation of an isolated polymer coil. The relaxation of polymer bridges is governed by the adsorption-desorption process, which varies with the adsorbed conformation. Although the Rouse theory predicts that the number of segments constructing the polymer coil shows the quadratic effect on the relaxation time, the number of attaching points is adopted to express the relaxation of reversible polymer bridges in the present model. Therefore, the relation between the relaxation time and constant shear stress at high shear rates is given by the following equation
σ02 ) Aτ
(6)
where A is a constant depending on the adsorption affinity. Let us now compare the model to the experimental data. At 25 °C the Newtonian viscosity is 1.5 Pa s from Figure 1, the longest relaxation time is 30 s from Figure 3, and the storage modulus in the linear region is 20 Pa at 62.8 s-1 from Figure 5. These values cannot satisfy eq 5 in the (14) Doi, M.; Edwards, S. F. J. Chem. Soc., Faraday Trans. 2 1978, 74, 1789. (15) Osaki, K.; Bessho, N.; Kojimoto, T,; Kurata, M. J. Rheol. 1980, 24, 125, (16) Rouse, P. E. J. Chem. Phys. 1953, 21, 1272.
Figure 9. Comparison between the experimental results (circles) and model prediction (solid lines) for shear-thickening flow of suspensions without surfactant at different temperatures. The temperature is 15, 25, 35, and 45 °C from top to bottom.
zero shear limit. However, the onset of shear thickening can be discussed in relation to the relaxation process of polymer chains. From the prediction through molecular network theories,13 the shear-thickening flow appears when the Weissenberg number γ˘ τ, which represents the relative strength of shear field, approaches unity. The characteristic relaxation time, τ, can be determined as η0/Ge, where η0 is the zero-shear viscosity and Ge the pseudoequilibrium modulus. From the frequency dependence of G′, the Ge is expected to be larger than 20 Pa. By assuming η0 ) 1.5 Pa and Ge ) 20 Pa, the characteristic relaxation time can be estimated as a first approximation and the obtained value is 0.075 s. For further calculation, the following values are used; η0 ) 1.5 Pa, G0 ) 20 Pa, and τ ) 0.075 s. Also the R is determined to be 0.5 from the strain dependence of storage modulus. The viscosity curve becomes plastic at high shear rates, yielding a constant stress of σ0 ) 200 Pa at 25 °C. Therefore, the value of A in eq 6 is 5.33 × 105 Pa2 s-1. Referring back to Figure 1, which shows that the transition from Newtonian to shearthickening profiles is scaled on γ˘ η0, the Newtonian viscosity is inversely proportional to the characteristic relaxation time and the G0 is independent of temperature. Finally, the viscosity behavior at different temperatures can be predicted by combining these parameters, eq 5, and eq 6. Figure 9 shows the comparison between the experimental results and model prediction for shearthickening flow for suspensions without surfactant at different temperatures. The viscosity increase is rather gentle in the predicted curves, whereas the experimental data show a very rapid increase in a very narrow range of shear rates. Although a slight discrepancy can be seen in the viscosity jump region, the overall flow profiles are somewhat in agreement. At shear rates where the shear-thickening flow appears, the internal strain to which the flexible bridge is subjected reaches the maximum. The maximum internal strain being taken as 6, the energy stored in extended bridge is about 17 kT, on the assumption that the mechanical properties of a flexible bridge are analogous to rubbery elasticity. The energy stored at the maximum extension may be equated to the amount of energy required to separate the bridged particles. It is possible to relate the constant stress at high shear rates, σ0, to the energy
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required for rupture of the bridge, Esep, by expansion of the work of Tadros17,18
σ0 ) (3ΦzEsep)/πd3
(7)
where Φ is the volume fraction of particles, z the coordination number, and d the particle diameter. From the sedimentation experiments, the limiting structure in which the particles are fully bridged with a coordination number of 12 may be achieved in 26 vol % suspension. If z is 6 in 15 vol % suspension, σ0 is estimated to be 115 Pa at 25 °C. Although the estimated value is smaller than the experimental results (200 Pa), it serves for an initial estimation. Therefore, it can be concluded that the shear thickening of suspensions flocculated by reversible bridging occurs when the flexible bridges with nonlinear elasticity are highly extended within the lifetime. Conclusions The suspensions flocculated by reversible polymer bridging show shear-thickening flow in a narrow range of (17) Tadros, Th. F. Langmuir 1990, 6, 28. (18) Tadros, Th. F. Zsednai, A. Colloids Surf. 1990, 49, 103.
shear rates. The relaxation time in the stress relaxation process after cessation of steady shear is constant, irrespective of shear rate. The storage modulus at a constant frequency shows a rapid increase when the strain is increased above a critical level. The critical strain is independent of frequency. The shear thickening may arise from the nonlinear elasticity of flexible bridges. In high shear fields, the suspensions become nearly plastic, because the polymer coils are forced to desorb at some degree of extension. The nonlinear viscoelastic model with a single relaxation time is proposed to quantitatively describe the shear-thickening flow. The model prediction reasonably agrees with the experimental data. The intrinsic mechanism of shear thickening and subsequent plastic flow profiles for suspensions flocculated by reversible polymer bridging is the nonlinear elasticity due to the entropy effect of extended bridges and the forced desorption due to the hydrodynamic effect. Acknowledgment. This work was supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science, and Culture, Japan, for which the author is grateful. LA9811362
