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Shear Thickening and Time-Dependent Rheological Behavior in Aqueous Polyacrylic Ester Dispersions
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
182, 172–178 (1996)
0448
Shear Thickening and Time-Dependent Rheological Behavior in Aqueous Polyacrylic Ester Dispersions J. XU,* A. M. JAMIESON,† S. Q. WANG,† ,1
AND
S. QUTUBUDDIN * ,† ,1
Departments of *Chemical Engineering and †Macromolecular Science, Case Western Reserve University, Cleveland, Ohio 44106 Received November 6, 1995; accepted March 20, 1996
We have observed, in a commercial aqueous polyacrylic ester dispersion, a shear thickening phenomenon that exhibits timedependent rheological behavior. The critical shear rate gg c for the shear thickening transition varies with volume fraction, temperature, pH, and particle size distribution, in a manner which indicates that the phenomenon is associated with a reversible shearinduced colloidal order–disorder transition. However, we find that, in the shear thickening region, the rheology may also evolve with time. Our observations suggest that the time-dependence is caused by the temporary formation of particle clusters at high shear rates. q 1996 Academic Press, Inc. Key Words: shear thickening; order–disorder transition; flowinduced clusters.
INTRODUCTION
Flow instabilities are sometimes observed in coating applications of certain aqueous polymer emulsions. Such effects may be associated with shear thickening behavior, which is commonly observed in concentrated colloidal dispersions of uniform particle size. This type of shear thickening is characterized by an abrupt, sometimes discontinuous, rise in viscosity above a critical shear rate with no apparent particle aggregation. The sudden viscosity increase is generally believed to be due to a shear-induced order– disorder transition. Among the first to describe the phenomenon was Hoffman (1, 2), whose early work established the basis for modeling such flow instabilities in terms of torque balance and energy balance, and who produced a mechanism, based on light scattering evidence, that a transition takes place from a two-dimensional ordered state to a threedimensional disordered state. The transition occurs when the hydrodynamic forces on the particles destabilize the layered structure. Further studies were subsequently carried out to characterize the ordered or layered flow region, and supporting evidence for the order–disorder transition was obtained from small angle neutron scattering (3, 4), light scattering (1, 5), 1
To whom correspondence should be addressed.
JCIS 4326
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6g13$$$341
gh c Å
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e0erc 20 kh , 6ha 2
[1]
where e0 is the permittivity of vacuum, er is the relative dielectric constant of the solvent, c0 is the surface potential, h is the solvent viscosity, a is the particle radius, h is the average interparticle separation, and k is the reciprocal Debye length. This model predicts behavior that is qualitatively similar to the earlier, more elaborate analysis of Hoffman (2) but exhibits significant quantitative differences. For example, as noted by Boersma et al. (12), the analysis of Hoffman (2) includes van der Waals forces which lead to a decrease in gg c for smaller particles. Support for Eq. [1] has been obtained from dynamic simulations (10). In a review paper, Barnes (13) discussed the importance
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and light diffraction (6). Dynamic simulations have reproduced many of the essential features of the shear thickening rheology (7–10). In sterically- or electrostatically-stabilized dispersions of uniform spherical particles, shear flow produces an ordered two-dimensional layered structure. At sufficiently high shear rates, hydrodynamic forces overcome the interparticle repulsion forces so that the ordered state is broken up, resulting in a disordered state with an apparent geometrical jamming leading to a sudden rise in viscosity. The disordered state may involve the transitory formation of large particle aggregates (11, 10) or a random threedimensional network (2). Although there is no general agreement about the exact structural changes that take place, there appears to be consensus on the transition from a freeflowing ordered structure to a less regular, more dissipative state (13). Boersma et al. ( 12 ) recently presented a simple analysis, based on a balance between shear forces and interparticle forces, which is able to predict the dependence of the critical shear rate on the particle volume fraction ( interparticle distance ) , the magnitude of the stabilizing force, the dispersion medium viscosity, and the particle radius. An expression was derived for the dependence of the critical shear rate gg c ( 12 ) ,
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of time effects in the shear-thickening rheology of certain dispersions and stated that, during loop tests, ‘‘given enough time, the structure induced in the shear-thickening region during the high shear rate part of the loop can relax.’’ Timedependent behavior was also studied by Boersma et al. (11), who reported that, at shear rates slightly above the critical shear rate for shear thickening, stress fluctuations in time occur, corresponding to transitions back-and-forth between the ordered and disordered states. These authors also speculated about reversible formation of large clusters in the disordered state. In addition, at even higher shear rates, the dispersion viscosity was found to decrease systematically with time at a fixed shear rate. Boersma et al. (11) attributed this thixotropic behavior as due to a ‘‘reordering of the dispersion.’’ In this paper, we investigate shear-thickening behavior in an aqueous polyacrylic ester dispersion. The experimental results suggest that its origin is associated with a reversible shear-induced order-to-disorder transition. We have further explored time-dependent rheological behavior which occurs in the high shear regime beyond the shear thickening transition. Our observations lead us to suggest that this effect is due to stress-induced particle flocculation leading to temporary particle clustering. Within the lifetime of these particle clusters, the colloidal dispersion effectively exhibits an increase in particle size polydispersity, which, in turn, causes an increase in the critical shear rate for shear-thickening, and hence one observes a hysteresis loop with the descending flow curve falling below the ascending curve. EXPERIMENTAL
The dispersion system under study was a commercial aqueous acrylic ester emulsion with anionic stabilization (HYCAR, B. F. Goodrich Inc.). The solids fraction, as received, was 45.2% by volume. Higher concentrations of the dispersion were obtained by evaporating the aqueous solvent at room temperature, while stirring continuously using a magnetic stirrer. The average particle diameter D, calculated from quasi-elastic light scattering measurements of the translational diffusion coefficient Dt , is 270 nm, and the size polydispersity m2 , defined as the variance in the decay rate distribution, m2 Å [ GV 2 0 ( GV ) 2 ]/( GV ) 2 , is 0.09, where G is the decay rate. The glass transition temperature of the polymer is 067C. Rheological measurements were performed using a CarriMed 50 controlled stress rheometer. The testing geometry used was cone and plate (20 mm diameter with 0.57 cone angle) with a solvent trap to prevent the solvent evaporation in the sample during tests. The rheological measurements were carried out at 257C unless otherwise specified. The maximum shear rate attainable in this cone/plate geometry is 5157 s 01 .
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FIG. 1. Shear rate sweep of a 59.5% (volume fraction) polyacrylic ester dispersion HYCAR, with the ascending sweep (triangles) followed immediately by a descending sweep (circles).
RESULTS AND DISCUSSION
A typical flow response during a shear loop, involving an ascending shear rate sweep followed by a descending one, is shown in Fig. 1 at 59.5% solid volume fraction. The shear thickening transition occurs at a shear rate gg c Å 1800 s 01 and is preceded by a slight decrease in viscosity (shear thinning). Here we refer to the shear rate at which the sudden viscosity increase occurs as the ‘‘critical shear rate.’’ This phenomenon does not seem to involve any particle agglomerates since the shear thickening transition is instantaneously reversible. Thus, as is evident in Fig. 1, the ascending curve and descending curve superimpose on each other, and no hysteresis loop can be detected within experimental error. Note that the shear sweep in Fig. 1 is truncated because, as we discuss further below, the reversibility of the shear-thickening phenomenon exists only over a limited range of shear rates. As discussed earlier, the sudden shear thickening transition is a result of a breakdown of the ordered layer structure, which exists at low shear rates. The transition occurs when there is an imbalance of the hydrodynamic forces and other forces (i.e., the electric double-layer repulsion, the van de Waals attraction, and Brownian forces). Among these, Brownian forces that determine the equilibrium and low shear dispersion microstructure are usually neglected for suspensions of large particles. In the present system, however, due to the relatively small particle size, Brownian forces should still be compared with the hydrodynamic force. To obtain an estimate of the range of shear rates in which Brownian forces are important, we evaluate the ratio of the hydrodynamic force, which can be written as (14) 6ph0gg r a 2r(a/h), to the Brownian forces, which can be expressed as kTra 01r(a/h), where (h / 2a) is the average interparticle distance. This ratio gives a dimensionless parameter Pe Å 6h0gg a 3 /kT, which is the Pe´clet number. In-
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FIG. 3. Critical shear rate gg c as a function of solid volume fraction.
FIG. 2. Shear rate sweep of HYCAR at different solid concentrations.
serting the values h0 Å 0.01 poise, T Å 298.2 K, a Å 2.7 1 10 07 m, k Å 1.38 1 10 023 J/K, we obtain Pe É 0.1gg . Thus, when gg ú 100 s 01 , the Brownian forces are negligibly weak compared to the hydrodynamic forces. Since the critical shear rate gg c is around 1800 s 01 for the order–disorder transition, for which the Brownian forces are clearly infinitesimal, it is the interparticle ordering forces that are overcome by the hydrodynamic force. More discussion on the influence of Brownian motion on shear thickening dispersions can be found in papers by Krieger (15) and Boersma et al. (11). In what follows, we first describe the influence of system parameters (particle concentration, temperature, pH, and particle size polydispersity) on the reversible shear thickening rheology and then present observations on the timedependent behavior at high shear rates.
fraction f in Fig. 3, appears to change inversely with f. This trend is in agreement with the previous experimental data of Boersma et al. (12) and is further consistent with the expectation that at higher particle concentration, the stronger hydrodynamic forces destabilize the layered flow at lower shear rates. Currently no analytical theory is available to correlate the magnitude of the transition to the volume fraction f. Clearly there is a strong similarity in the behavior of our system with those observed in earlier studies (12). b. Effect of Temperature Figure 4 shows the influence of temperature on the shear thickening of dispersion at 58.6% solid volume fraction. Shear sweeps were performed at temperatures from 10 to 307C. At higher temperatures, the critical shear rate is higher.
a. Effect of Particle Concentration The critical shear rate is a strong decreasing function of solid concentration, as shown in Fig. 2. When the solid concentration is raised from 57.6 to 60.5%, the critical shear rate decreases rapidly from about 4000 1/s to 900 1/s. It is known that suspensions of uniformly sized spherical particles typically have a critical volume fraction of about 60% (independent of the actual particle size), beyond which shear thickening takes place theoretically at zero shear rate. For a polydisperse suspension, however, the critical volume can be higher than 60%. The data in Fig. 1 are truncated because the viscous torque acting on the cone and plate exceeds the torque limit of the instrument. This experimental limitation can be avoided by employing smaller flow cells. The critical shear rate, gg c , when plotted as a function of the solid volume
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FIG. 4. Effect of temperature on shear thickening of HYCAR at 58.6% solid volume fraction.
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not produce a temperature dependence as strong as that indicated by our experimental data. In this respect, it should be noted that Eq. [1] neglects van der Waals forces, which, in dynamic simulations, have a strong effect in sharpening and intensifying the shear-thickening transition (10). To resolve these questions, other independent methods must be employed to explore the temperature dependence of interparticle interactions. Such a subject is, however, beyond the scope of the present work. c. Effect of pH
FIG. 5. Comparison of measured and predicted (Ref. (12)) critical shear rates as a function of temperature.
Furthermore, the transition becomes less abrupt. This upward shift of the critical shear rate with increasing temperature is again very similar to observations of previous studies by Laun et al. (16) and Boersma et al. (12). Consistent with the conclusions of Laun et al. (16), we find that temperature has a stronger effect on the critical shear rate than on the magnitude of the solvent viscosity. In other words, the increase of gg c with temperature cannot be explained by the temperature dependence of the solvent viscosity. In contrast, Boersma et al. (12) claimed that their experimental data supported the theoretical model described by Eq. [1] in the sense that the temperature dependence of gg c derives solely from that of the solvent viscosity. Thus it appears that the temperature dependence may be quite system specific. In any event, we show in Fig. 5 a log–log plot of the critical shear rate for our system versus the temperature-dependent solvent viscosity. Assuming that only the solvent viscosity hs depends on temperature T, Eq. [1] predicts that gg c } 1/ hs , i.e., the solid line in Fig. 5 with slope 01. However, the gg c values observed in our experimental data, when plotted as a function of hs , exhibit a much larger slope of 04. Clearly we have to conclude that the change in solvent viscosity over the temperature range from 10 to 307C is insufficient to explain the large shift in critical shear rate. This result suggests that the interparticle forces required to maintain the layered structure in the dispersion may become weaker at lower temperatures. The origin of this temperature effect is unclear. According to the model of Boersma et al. (12), increase of temperature T would in principle influence the interparticle forces through changes in dielectric constant and surface potential (see Eq. [1]). Still, such effects would
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By adding acetic acid or ammonium hydroxide, the pH of the dispersion was varied. The influence of pH on the shear thickening transition is demonstrated in Fig. 6. A decrease in pH shifts the critical shear rate to lower values whereas increasing the pH postpones the transition. For pH ú 4.9, the shear thickening phenomenon disappears in the accessible shear rate range. Since the dispersion is anionically stabilized, upon increasing pH, the particle surface potential (z -potential) increases, and hence the dispersion stability is enhanced, as discussed by Laun et al. (17) and Boersma et al. (12). As a result, a higher critical shear rate is required to produce the disordered state. Equation [1] indicates that increasing surface potential will result in higher gg c . Thus the observed pH-dependence appears to be consistent with the theoretical prediction of Boersma et al. (12). Another feature of the data in Fig. 6 is that the low-shear viscosity increases with pH. This is consistent with the earlier results of Laun et al. (17) and with the expected effect of an increase in interparticle repulsion on the viscosity (second electroviscous effect). The observation that a relatively large increase in low-shear viscosity occurs upon increasing pH from 4.5 to 4.9, and that the critical shear rate moves to values higher than those accessible with our rheometer, is presumably related to the fact that the principal ionic species
FIG. 6. Effect of pH on shear thickening of HYCAR at 59.5% solid volume fraction.
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solvent. Laun et al. (17) also found that size polydispersity increases the viscosity level at small stresses, presumably also due to some cluster formation. It is also possible that ordering is disrupted by the SiO2 particles. In summary, it is clear that all the pertinent system variables influence the shear thickening behavior of the polymer latex dispersion in a fashion expected of the phenomenon commonly associated with a flow-induced order-to-disorder transition. We now turn to a presentation and discussion of experimental observations of time-dependent changes in the high-shear rheology of the shear thickening state. e. Time-Dependent Rheology in the Shear Thickening Region FIG. 7. Effect of particle size polydispersity on shear thickening of HYCAR at a total solid volume fraction of 59.5%, with 0, 5, and 10% volume fraction of the polymer dispersion particles replaced by silica spheres 50 nm in diameter.
on the particle surface is carboxylate, whose pKa is about 4.7. Thus near pH 4.7 the emulsion particle rapidly becomes more negatively charged and the interparticle repulsion becomes correspondingly stronger, leading to a larger low-shear viscosity and higher critical shear rate, again in agreement with the studies of Boersma et al. (12). d. Effect of Particle Size Polydispersity Silica particles (provided by Nissan Chemical America Corporation) of uniform diameter around 50 nm (about 1/5 the size of the emulsion particles), were added to the dispersions in different quantities. The total solids content was kept constant, providing a basis for comparison; i.e., the overall effect was to replace some of the original large polymer latex particles with smaller silica particles. Figure 7 shows the results of shear sweep experiments on dispersions with different amounts of silica particles. Consistent with the findings of Boersma et al. (12), we see that the shear thickening phenomenon is sensitive to the particle size polydispersity. A more polydisperse dispersion apparently requires a higher critical shear rate to enter the disordered phase. This is in agreement with reasonable expectations that the dispersion is less prone to mechanical jamming in presence of a higher degree of polydispersity. We note, however, that since the low-shear viscosity also increases on addition of the SiO2 particles, it appears that some particle agglomeration may have occurred. Tsenoglou (18) has discussed phenomenological scaling laws regarding the effect of interparticle aggregation on the viscosity of agglomerated suspensions. Following this analysis, the increase in the lowshear viscosity can be interpreted as due to the fact that formation of aggregates at a fixed solid content will increase the effective hydrodynamic volume due to the entrapped
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In contrast to the reversible flow behavior shown in Fig. 1, upon further increasing the shear rate in a shear rate sweep, we observed that the dispersion first attains a maximum viscosity and then its viscosity starts to decrease, as shown in Fig. 8. Further, in shear loops which are extended to these higher shear rates, the dispersion viscosity in the shear thickening region is no longer immediately reversible. Comparing the two shear loops shown in Figs. 1 and 8, respectively, we see that above a higher shear rate (an upper critical shear rate) than the initial critical shear rate for shear thickening, a new dispersion structure may form temporarily. The descending curve (where the shear rate is continuously decreased) in the hysteresis loop indicates that the shearinduced new structural state responds to flow somewhat differently, as evidenced by the difference in viscosity between the ascending curve and descending curve. It is plausible that temporary clusters form under sufficiently high stresses in the disordered state. These transient clusters increase the effective polydispersity and reduce dispersion viscosity in the disordered state, because mechanical jamming is less severe and less likely to occur. As a result, the descending curve is below the ascending curve.
FIG. 8. Hysteresis loop at high shear rates (HYCAR, 59.5% solid volume fraction), with the ascending sweep followed by the descending sweep.
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FIG. 9. Effect of prolonged shear on HYCAR at 59.5% solid volume fraction: (a) Repeated shear rate sweeps. Initial sweep is indicated by the dashed curve and the solid curve represents a second sweep performed immediately after the steady-shear experiment described in (b). (b) Viscosity change during steady shear at 3000 s 01 , which was applied immediately following the initial shear rate sweep (dashed curve in (a)).
A series of rheological tests were performed, combining shear rate sweeps and prolonged steady shear to gain more insight into the time-dependent flow behavior at high shear rates. An initial shear rate sweep produces a response, represented in Fig. 9a by the dashed line, which is very similar to the ascending curve shown in Fig. 8. This measurement was immediately followed by a prolonged shear at 3000 s 01 (slightly above the upper critical shear rate). Figure 9b shows that, under this prolonged shear, the dispersion viscosity decreases significantly in time before finally reaching a steady state 15 min later (at t Å 900 s). We believe this viscosity decrease to be the result of stress-induced particle clustering which produces an effectively higher polydispersity and reduces the severity of viscosity buildup in the disordered state. This conclusion is obviously consistent with that drawn from the data in Figure 8. A subsequent shear rate sweep performed immediately following this prolonged shear described in Fig. 9b shows a larger low-shear viscosity and a considerably higher critical shear rate for shear thickening, as the solid line in Fig. 9a indicates. Thus Fig. 9a resembles the polydispersity effect shown in Fig. 7 and further supports that the high stresses may have produced temporary particle clusters, which result in a higher effective particle polydispersity and postpone the order-to-disorder transition. The comparison between the dashed and solid lines in Fig. 9a indicates that the stress-induced particle clusters may be relatively long-lived. It is interesting to determine the lifetime of the stressinduced particle cluster formation and whether the dispersion ever recovers its initial equilibrium structure. Figure 10 shows that there is a substantial degree of recovery toward the initial equilibrium state after 2.0 h of recovery time. This implies that the flow-induced clusters are not permanent and will break up due to thermal Brownian motion.
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CONCLUSIONS
In conclusion, we have observed interesting rheological behavior of a commercial aqueous polymer latex dispersion. The majority of the flow characteristics are consistent with the well-known phenomenon of a shear-induced order-todisorder transition. Many of the salient features can be explained in terms of a force balance between the interparticle repulsion and hydrodynamic viscous forces acting on each particle. The shear thickening behavior of the present polymer dispersion is found to depend on system parameters such as particle concentration, polydispersity, and pH in a predictable way. The origin of the temperature dependence remains elusive and requires further investigations in the future.
FIG. 10. Structural recovery is apparent in a shear rate sweep performed 2 h after the experiment described in Fig. 9, when compared with the initial shear rate sweep.
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A new flow-induced structural change has been studied at shear rates well above the critical shear rate for the shear thickening transition. The temporary formation of efficiently packed particle clusters is proposed to explain the timedependent rheological response at such high shear rates. It appears that the stress-induced clustering produces a higher effective size polydispersity in the dispersion. The signature of the increased polydispersity is the reduced dispersion viscosity in the shear-thickening state, a higher viscosity in the low-shear region, and the systematic shift of the critical shear rate for the order–disorder transition to higher values. The stress-induced clusters apparently are not permanent and slowly break up over time. Thus, after sufficient time has elapsed, the rheology of the dispersion containing these clusters returns significantly toward that of a virgin sample that is free of any preshear history. ACKNOWLEDGMENTS This work was supported by the Edison Polymer Innovation Corporation (EPIC), Center for Adhesives Sealants and Coatings (ECASC). The authors also thank Professor Kingman Strohl for providing the Carri-Med CS50, B. F. Goodrich Inc. for providing the sample, and Dr. Yuntao Hu for obtaining light scattering results.
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REFERENCES 1. Hoffman, R. L., Trans. Soc. Rheol. 16, 155 (1972). 2. Hoffman, R. L., J. Colloid Interface Sci. 46(3), 491 (1974). 3. Ackerson, B. J., Hayter, J. B., Clark, N. A., and Cotter, L., J. Chem. Phys. 84(4), 2344 (1986). 4. Laun, H. M., Bung, R., Hess, S., Loose, W., and Hess, O., J. Rheol. 36(4), 743 (1992). 5. Wagner, N. J., and Russel, W. B., Phys. Fluids A 2(4), 491 (1990). 6. Tomita, M., and Van de Ven, T. G. M., J. Colloid Interface Sci. 99(2), 374 (1984). 7. Woodcock, L., Chem. Phys. Lett. 9, 455 (1984). 8. Barnes, H. A., Edwards, M. F., and Woodcock, L. V., Chem. Eng. Sci. 42, 591 (1987). 9. Bossis, G., and Brady, J. F., Phys. Sci. A 49, 89 (1993). 10. Boersma, W. H., Laven, J., and Stein, H. N., J. Rheol. 39(5), 841 (1995). 11. Boersma, W. H., Baets, P. J. M., Laven, J., and Stein, H. N., J. Rheol. 35, 1093 (1991). 12. Boersma, W. H., Laven, J., and Stein, H. N., AIChE J. 36(3), 321 (1990). 13. Barnes, H. A., J. Rheol. 33(2), 329 (1989). 14. Laven, J., Boersma, W. H., Kaldasch, J., and Stein, H. N., ‘‘Advances in Structured and Heterogeneous Continua,’’ p. 479. Allerton Press, New York, 1994. 15. Krieger, I. M., Adv. Colloid Interface Sci. 3, 111 (1972). 16. Laun, H. M., Bung, R., and Schmidt, F., J. Rheol. 35(6), 999 (1991). 17. Laun, H. M., Die Angew. Makromol. Chem. 123/124, 335 (1984). 18. Tsenoglou, C., J. Rheol. 34(1), 15 (1990).
coida
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182, 172–178 (1996)
0448
Shear Thickening and Time-Dependent Rheological Behavior in Aqueous Polyacrylic Ester Dispersions J. XU,* A. M. JAMIESON,† S. Q. WANG,† ,1
AND
S. QUTUBUDDIN * ,† ,1
Departments of *Chemical Engineering and †Macromolecular Science, Case Western Reserve University, Cleveland, Ohio 44106 Received November 6, 1995; accepted March 20, 1996
We have observed, in a commercial aqueous polyacrylic ester dispersion, a shear thickening phenomenon that exhibits timedependent rheological behavior. The critical shear rate gg c for the shear thickening transition varies with volume fraction, temperature, pH, and particle size distribution, in a manner which indicates that the phenomenon is associated with a reversible shearinduced colloidal order–disorder transition. However, we find that, in the shear thickening region, the rheology may also evolve with time. Our observations suggest that the time-dependence is caused by the temporary formation of particle clusters at high shear rates. q 1996 Academic Press, Inc. Key Words: shear thickening; order–disorder transition; flowinduced clusters.
INTRODUCTION
Flow instabilities are sometimes observed in coating applications of certain aqueous polymer emulsions. Such effects may be associated with shear thickening behavior, which is commonly observed in concentrated colloidal dispersions of uniform particle size. This type of shear thickening is characterized by an abrupt, sometimes discontinuous, rise in viscosity above a critical shear rate with no apparent particle aggregation. The sudden viscosity increase is generally believed to be due to a shear-induced order– disorder transition. Among the first to describe the phenomenon was Hoffman (1, 2), whose early work established the basis for modeling such flow instabilities in terms of torque balance and energy balance, and who produced a mechanism, based on light scattering evidence, that a transition takes place from a two-dimensional ordered state to a threedimensional disordered state. The transition occurs when the hydrodynamic forces on the particles destabilize the layered structure. Further studies were subsequently carried out to characterize the ordered or layered flow region, and supporting evidence for the order–disorder transition was obtained from small angle neutron scattering (3, 4), light scattering (1, 5), 1
To whom correspondence should be addressed.
JCIS 4326
/
6g13$$$341
gh c Å
07-24-96 08:20:05
e0erc 20 kh , 6ha 2
[1]
where e0 is the permittivity of vacuum, er is the relative dielectric constant of the solvent, c0 is the surface potential, h is the solvent viscosity, a is the particle radius, h is the average interparticle separation, and k is the reciprocal Debye length. This model predicts behavior that is qualitatively similar to the earlier, more elaborate analysis of Hoffman (2) but exhibits significant quantitative differences. For example, as noted by Boersma et al. (12), the analysis of Hoffman (2) includes van der Waals forces which lead to a decrease in gg c for smaller particles. Support for Eq. [1] has been obtained from dynamic simulations (10). In a review paper, Barnes (13) discussed the importance
172
0021-9797/96 $18.00 Copyright q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
AID
and light diffraction (6). Dynamic simulations have reproduced many of the essential features of the shear thickening rheology (7–10). In sterically- or electrostatically-stabilized dispersions of uniform spherical particles, shear flow produces an ordered two-dimensional layered structure. At sufficiently high shear rates, hydrodynamic forces overcome the interparticle repulsion forces so that the ordered state is broken up, resulting in a disordered state with an apparent geometrical jamming leading to a sudden rise in viscosity. The disordered state may involve the transitory formation of large particle aggregates (11, 10) or a random threedimensional network (2). Although there is no general agreement about the exact structural changes that take place, there appears to be consensus on the transition from a freeflowing ordered structure to a less regular, more dissipative state (13). Boersma et al. ( 12 ) recently presented a simple analysis, based on a balance between shear forces and interparticle forces, which is able to predict the dependence of the critical shear rate on the particle volume fraction ( interparticle distance ) , the magnitude of the stabilizing force, the dispersion medium viscosity, and the particle radius. An expression was derived for the dependence of the critical shear rate gg c ( 12 ) ,
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of time effects in the shear-thickening rheology of certain dispersions and stated that, during loop tests, ‘‘given enough time, the structure induced in the shear-thickening region during the high shear rate part of the loop can relax.’’ Timedependent behavior was also studied by Boersma et al. (11), who reported that, at shear rates slightly above the critical shear rate for shear thickening, stress fluctuations in time occur, corresponding to transitions back-and-forth between the ordered and disordered states. These authors also speculated about reversible formation of large clusters in the disordered state. In addition, at even higher shear rates, the dispersion viscosity was found to decrease systematically with time at a fixed shear rate. Boersma et al. (11) attributed this thixotropic behavior as due to a ‘‘reordering of the dispersion.’’ In this paper, we investigate shear-thickening behavior in an aqueous polyacrylic ester dispersion. The experimental results suggest that its origin is associated with a reversible shear-induced order-to-disorder transition. We have further explored time-dependent rheological behavior which occurs in the high shear regime beyond the shear thickening transition. Our observations lead us to suggest that this effect is due to stress-induced particle flocculation leading to temporary particle clustering. Within the lifetime of these particle clusters, the colloidal dispersion effectively exhibits an increase in particle size polydispersity, which, in turn, causes an increase in the critical shear rate for shear-thickening, and hence one observes a hysteresis loop with the descending flow curve falling below the ascending curve. EXPERIMENTAL
The dispersion system under study was a commercial aqueous acrylic ester emulsion with anionic stabilization (HYCAR, B. F. Goodrich Inc.). The solids fraction, as received, was 45.2% by volume. Higher concentrations of the dispersion were obtained by evaporating the aqueous solvent at room temperature, while stirring continuously using a magnetic stirrer. The average particle diameter D, calculated from quasi-elastic light scattering measurements of the translational diffusion coefficient Dt , is 270 nm, and the size polydispersity m2 , defined as the variance in the decay rate distribution, m2 Å [ GV 2 0 ( GV ) 2 ]/( GV ) 2 , is 0.09, where G is the decay rate. The glass transition temperature of the polymer is 067C. Rheological measurements were performed using a CarriMed 50 controlled stress rheometer. The testing geometry used was cone and plate (20 mm diameter with 0.57 cone angle) with a solvent trap to prevent the solvent evaporation in the sample during tests. The rheological measurements were carried out at 257C unless otherwise specified. The maximum shear rate attainable in this cone/plate geometry is 5157 s 01 .
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FIG. 1. Shear rate sweep of a 59.5% (volume fraction) polyacrylic ester dispersion HYCAR, with the ascending sweep (triangles) followed immediately by a descending sweep (circles).
RESULTS AND DISCUSSION
A typical flow response during a shear loop, involving an ascending shear rate sweep followed by a descending one, is shown in Fig. 1 at 59.5% solid volume fraction. The shear thickening transition occurs at a shear rate gg c Å 1800 s 01 and is preceded by a slight decrease in viscosity (shear thinning). Here we refer to the shear rate at which the sudden viscosity increase occurs as the ‘‘critical shear rate.’’ This phenomenon does not seem to involve any particle agglomerates since the shear thickening transition is instantaneously reversible. Thus, as is evident in Fig. 1, the ascending curve and descending curve superimpose on each other, and no hysteresis loop can be detected within experimental error. Note that the shear sweep in Fig. 1 is truncated because, as we discuss further below, the reversibility of the shear-thickening phenomenon exists only over a limited range of shear rates. As discussed earlier, the sudden shear thickening transition is a result of a breakdown of the ordered layer structure, which exists at low shear rates. The transition occurs when there is an imbalance of the hydrodynamic forces and other forces (i.e., the electric double-layer repulsion, the van de Waals attraction, and Brownian forces). Among these, Brownian forces that determine the equilibrium and low shear dispersion microstructure are usually neglected for suspensions of large particles. In the present system, however, due to the relatively small particle size, Brownian forces should still be compared with the hydrodynamic force. To obtain an estimate of the range of shear rates in which Brownian forces are important, we evaluate the ratio of the hydrodynamic force, which can be written as (14) 6ph0gg r a 2r(a/h), to the Brownian forces, which can be expressed as kTra 01r(a/h), where (h / 2a) is the average interparticle distance. This ratio gives a dimensionless parameter Pe Å 6h0gg a 3 /kT, which is the Pe´clet number. In-
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FIG. 3. Critical shear rate gg c as a function of solid volume fraction.
FIG. 2. Shear rate sweep of HYCAR at different solid concentrations.
serting the values h0 Å 0.01 poise, T Å 298.2 K, a Å 2.7 1 10 07 m, k Å 1.38 1 10 023 J/K, we obtain Pe É 0.1gg . Thus, when gg ú 100 s 01 , the Brownian forces are negligibly weak compared to the hydrodynamic forces. Since the critical shear rate gg c is around 1800 s 01 for the order–disorder transition, for which the Brownian forces are clearly infinitesimal, it is the interparticle ordering forces that are overcome by the hydrodynamic force. More discussion on the influence of Brownian motion on shear thickening dispersions can be found in papers by Krieger (15) and Boersma et al. (11). In what follows, we first describe the influence of system parameters (particle concentration, temperature, pH, and particle size polydispersity) on the reversible shear thickening rheology and then present observations on the timedependent behavior at high shear rates.
fraction f in Fig. 3, appears to change inversely with f. This trend is in agreement with the previous experimental data of Boersma et al. (12) and is further consistent with the expectation that at higher particle concentration, the stronger hydrodynamic forces destabilize the layered flow at lower shear rates. Currently no analytical theory is available to correlate the magnitude of the transition to the volume fraction f. Clearly there is a strong similarity in the behavior of our system with those observed in earlier studies (12). b. Effect of Temperature Figure 4 shows the influence of temperature on the shear thickening of dispersion at 58.6% solid volume fraction. Shear sweeps were performed at temperatures from 10 to 307C. At higher temperatures, the critical shear rate is higher.
a. Effect of Particle Concentration The critical shear rate is a strong decreasing function of solid concentration, as shown in Fig. 2. When the solid concentration is raised from 57.6 to 60.5%, the critical shear rate decreases rapidly from about 4000 1/s to 900 1/s. It is known that suspensions of uniformly sized spherical particles typically have a critical volume fraction of about 60% (independent of the actual particle size), beyond which shear thickening takes place theoretically at zero shear rate. For a polydisperse suspension, however, the critical volume can be higher than 60%. The data in Fig. 1 are truncated because the viscous torque acting on the cone and plate exceeds the torque limit of the instrument. This experimental limitation can be avoided by employing smaller flow cells. The critical shear rate, gg c , when plotted as a function of the solid volume
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FIG. 4. Effect of temperature on shear thickening of HYCAR at 58.6% solid volume fraction.
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not produce a temperature dependence as strong as that indicated by our experimental data. In this respect, it should be noted that Eq. [1] neglects van der Waals forces, which, in dynamic simulations, have a strong effect in sharpening and intensifying the shear-thickening transition (10). To resolve these questions, other independent methods must be employed to explore the temperature dependence of interparticle interactions. Such a subject is, however, beyond the scope of the present work. c. Effect of pH
FIG. 5. Comparison of measured and predicted (Ref. (12)) critical shear rates as a function of temperature.
Furthermore, the transition becomes less abrupt. This upward shift of the critical shear rate with increasing temperature is again very similar to observations of previous studies by Laun et al. (16) and Boersma et al. (12). Consistent with the conclusions of Laun et al. (16), we find that temperature has a stronger effect on the critical shear rate than on the magnitude of the solvent viscosity. In other words, the increase of gg c with temperature cannot be explained by the temperature dependence of the solvent viscosity. In contrast, Boersma et al. (12) claimed that their experimental data supported the theoretical model described by Eq. [1] in the sense that the temperature dependence of gg c derives solely from that of the solvent viscosity. Thus it appears that the temperature dependence may be quite system specific. In any event, we show in Fig. 5 a log–log plot of the critical shear rate for our system versus the temperature-dependent solvent viscosity. Assuming that only the solvent viscosity hs depends on temperature T, Eq. [1] predicts that gg c } 1/ hs , i.e., the solid line in Fig. 5 with slope 01. However, the gg c values observed in our experimental data, when plotted as a function of hs , exhibit a much larger slope of 04. Clearly we have to conclude that the change in solvent viscosity over the temperature range from 10 to 307C is insufficient to explain the large shift in critical shear rate. This result suggests that the interparticle forces required to maintain the layered structure in the dispersion may become weaker at lower temperatures. The origin of this temperature effect is unclear. According to the model of Boersma et al. (12), increase of temperature T would in principle influence the interparticle forces through changes in dielectric constant and surface potential (see Eq. [1]). Still, such effects would
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By adding acetic acid or ammonium hydroxide, the pH of the dispersion was varied. The influence of pH on the shear thickening transition is demonstrated in Fig. 6. A decrease in pH shifts the critical shear rate to lower values whereas increasing the pH postpones the transition. For pH ú 4.9, the shear thickening phenomenon disappears in the accessible shear rate range. Since the dispersion is anionically stabilized, upon increasing pH, the particle surface potential (z -potential) increases, and hence the dispersion stability is enhanced, as discussed by Laun et al. (17) and Boersma et al. (12). As a result, a higher critical shear rate is required to produce the disordered state. Equation [1] indicates that increasing surface potential will result in higher gg c . Thus the observed pH-dependence appears to be consistent with the theoretical prediction of Boersma et al. (12). Another feature of the data in Fig. 6 is that the low-shear viscosity increases with pH. This is consistent with the earlier results of Laun et al. (17) and with the expected effect of an increase in interparticle repulsion on the viscosity (second electroviscous effect). The observation that a relatively large increase in low-shear viscosity occurs upon increasing pH from 4.5 to 4.9, and that the critical shear rate moves to values higher than those accessible with our rheometer, is presumably related to the fact that the principal ionic species
FIG. 6. Effect of pH on shear thickening of HYCAR at 59.5% solid volume fraction.
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solvent. Laun et al. (17) also found that size polydispersity increases the viscosity level at small stresses, presumably also due to some cluster formation. It is also possible that ordering is disrupted by the SiO2 particles. In summary, it is clear that all the pertinent system variables influence the shear thickening behavior of the polymer latex dispersion in a fashion expected of the phenomenon commonly associated with a flow-induced order-to-disorder transition. We now turn to a presentation and discussion of experimental observations of time-dependent changes in the high-shear rheology of the shear thickening state. e. Time-Dependent Rheology in the Shear Thickening Region FIG. 7. Effect of particle size polydispersity on shear thickening of HYCAR at a total solid volume fraction of 59.5%, with 0, 5, and 10% volume fraction of the polymer dispersion particles replaced by silica spheres 50 nm in diameter.
on the particle surface is carboxylate, whose pKa is about 4.7. Thus near pH 4.7 the emulsion particle rapidly becomes more negatively charged and the interparticle repulsion becomes correspondingly stronger, leading to a larger low-shear viscosity and higher critical shear rate, again in agreement with the studies of Boersma et al. (12). d. Effect of Particle Size Polydispersity Silica particles (provided by Nissan Chemical America Corporation) of uniform diameter around 50 nm (about 1/5 the size of the emulsion particles), were added to the dispersions in different quantities. The total solids content was kept constant, providing a basis for comparison; i.e., the overall effect was to replace some of the original large polymer latex particles with smaller silica particles. Figure 7 shows the results of shear sweep experiments on dispersions with different amounts of silica particles. Consistent with the findings of Boersma et al. (12), we see that the shear thickening phenomenon is sensitive to the particle size polydispersity. A more polydisperse dispersion apparently requires a higher critical shear rate to enter the disordered phase. This is in agreement with reasonable expectations that the dispersion is less prone to mechanical jamming in presence of a higher degree of polydispersity. We note, however, that since the low-shear viscosity also increases on addition of the SiO2 particles, it appears that some particle agglomeration may have occurred. Tsenoglou (18) has discussed phenomenological scaling laws regarding the effect of interparticle aggregation on the viscosity of agglomerated suspensions. Following this analysis, the increase in the lowshear viscosity can be interpreted as due to the fact that formation of aggregates at a fixed solid content will increase the effective hydrodynamic volume due to the entrapped
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In contrast to the reversible flow behavior shown in Fig. 1, upon further increasing the shear rate in a shear rate sweep, we observed that the dispersion first attains a maximum viscosity and then its viscosity starts to decrease, as shown in Fig. 8. Further, in shear loops which are extended to these higher shear rates, the dispersion viscosity in the shear thickening region is no longer immediately reversible. Comparing the two shear loops shown in Figs. 1 and 8, respectively, we see that above a higher shear rate (an upper critical shear rate) than the initial critical shear rate for shear thickening, a new dispersion structure may form temporarily. The descending curve (where the shear rate is continuously decreased) in the hysteresis loop indicates that the shearinduced new structural state responds to flow somewhat differently, as evidenced by the difference in viscosity between the ascending curve and descending curve. It is plausible that temporary clusters form under sufficiently high stresses in the disordered state. These transient clusters increase the effective polydispersity and reduce dispersion viscosity in the disordered state, because mechanical jamming is less severe and less likely to occur. As a result, the descending curve is below the ascending curve.
FIG. 8. Hysteresis loop at high shear rates (HYCAR, 59.5% solid volume fraction), with the ascending sweep followed by the descending sweep.
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FIG. 9. Effect of prolonged shear on HYCAR at 59.5% solid volume fraction: (a) Repeated shear rate sweeps. Initial sweep is indicated by the dashed curve and the solid curve represents a second sweep performed immediately after the steady-shear experiment described in (b). (b) Viscosity change during steady shear at 3000 s 01 , which was applied immediately following the initial shear rate sweep (dashed curve in (a)).
A series of rheological tests were performed, combining shear rate sweeps and prolonged steady shear to gain more insight into the time-dependent flow behavior at high shear rates. An initial shear rate sweep produces a response, represented in Fig. 9a by the dashed line, which is very similar to the ascending curve shown in Fig. 8. This measurement was immediately followed by a prolonged shear at 3000 s 01 (slightly above the upper critical shear rate). Figure 9b shows that, under this prolonged shear, the dispersion viscosity decreases significantly in time before finally reaching a steady state 15 min later (at t Å 900 s). We believe this viscosity decrease to be the result of stress-induced particle clustering which produces an effectively higher polydispersity and reduces the severity of viscosity buildup in the disordered state. This conclusion is obviously consistent with that drawn from the data in Figure 8. A subsequent shear rate sweep performed immediately following this prolonged shear described in Fig. 9b shows a larger low-shear viscosity and a considerably higher critical shear rate for shear thickening, as the solid line in Fig. 9a indicates. Thus Fig. 9a resembles the polydispersity effect shown in Fig. 7 and further supports that the high stresses may have produced temporary particle clusters, which result in a higher effective particle polydispersity and postpone the order-to-disorder transition. The comparison between the dashed and solid lines in Fig. 9a indicates that the stress-induced particle clusters may be relatively long-lived. It is interesting to determine the lifetime of the stressinduced particle cluster formation and whether the dispersion ever recovers its initial equilibrium structure. Figure 10 shows that there is a substantial degree of recovery toward the initial equilibrium state after 2.0 h of recovery time. This implies that the flow-induced clusters are not permanent and will break up due to thermal Brownian motion.
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CONCLUSIONS
In conclusion, we have observed interesting rheological behavior of a commercial aqueous polymer latex dispersion. The majority of the flow characteristics are consistent with the well-known phenomenon of a shear-induced order-todisorder transition. Many of the salient features can be explained in terms of a force balance between the interparticle repulsion and hydrodynamic viscous forces acting on each particle. The shear thickening behavior of the present polymer dispersion is found to depend on system parameters such as particle concentration, polydispersity, and pH in a predictable way. The origin of the temperature dependence remains elusive and requires further investigations in the future.
FIG. 10. Structural recovery is apparent in a shear rate sweep performed 2 h after the experiment described in Fig. 9, when compared with the initial shear rate sweep.
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A new flow-induced structural change has been studied at shear rates well above the critical shear rate for the shear thickening transition. The temporary formation of efficiently packed particle clusters is proposed to explain the timedependent rheological response at such high shear rates. It appears that the stress-induced clustering produces a higher effective size polydispersity in the dispersion. The signature of the increased polydispersity is the reduced dispersion viscosity in the shear-thickening state, a higher viscosity in the low-shear region, and the systematic shift of the critical shear rate for the order–disorder transition to higher values. The stress-induced clusters apparently are not permanent and slowly break up over time. Thus, after sufficient time has elapsed, the rheology of the dispersion containing these clusters returns significantly toward that of a virgin sample that is free of any preshear history. ACKNOWLEDGMENTS This work was supported by the Edison Polymer Innovation Corporation (EPIC), Center for Adhesives Sealants and Coatings (ECASC). The authors also thank Professor Kingman Strohl for providing the Carri-Med CS50, B. F. Goodrich Inc. for providing the sample, and Dr. Yuntao Hu for obtaining light scattering results.
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