2336
Langmuir 1992,8, 2336-2340
Dilatant Flow of Flocculated Suspensions Yasufumi Otsubo Department of Image Science, Faculty of Engineering, Chiba University, Yayoi-cho 1-33, Chiba-shi 260, Japan
Received March 9,1992. I n Final Form:June 29, 1992 The polymer coil whose size is comparable in solution to particle diameter makes a flexible bridge between two particles. The steady-shearviscoeityand dynamicviscoeiaaticitywere measured for suspensions flocculated by the bridging mechanism. When both the particle and polymer concentrations are increased beyond some critical levels, the suspensions show unique flow profiles consisting of Newtonian viscosity in the limit of zero shear rate, dilatant flow at moderate shear rates, and pseudoplastic flow at high shear rates. The existence of Newtonianviscosityindicatesthat polymer bridges are constantlyforming,breaking, and re-forming in a quiescent state. Therefore, the adsorption-desorption procesa is considered to be reversible. When the shear rate is increased, the polymer coils are rapidly extended and the restoring forces cause an increase in resistance to flow. Although the dilatant flow is primarily attributed to the extension of bridge, the network structure of unbounded flocs is essential. The combination effect of extension of flexiblebridges between particles and formation of network structure is discussed on the basis of the percolation theory.
Introduction The conformation of a polymer chain adsorbed at a solidliquid interface is explained in terms of trains of segments which attach to the solid surface and loops of segments which extend into liquid. Since a great variety of conformations can be achieved by flexible polymer chains adsorbed on the fine particles, additions of polymers to colloidal suspensions drastically change the stability according to the interaction energies among the polymer, the liquid, and the particles. A high-molecular-weight polymer can adsorb onto more than two particles and can cause flocculation by a bridging mechanism. In polymer bridging, more than one polymer molecule is required to bridge large particles, whereas for small particles, only a small segment of a polymer chain is required, and thus one molecule can extend through many bridges.' we have reported that silica In previous particles with diameters in the range of 10-20 nm are easily flocculated by addition of polyacrylamide with molecular weight of about 5 X lo6. Since the particle diameter is of the same order as the loop length of adsorbed polymer, one polymer chain is expected to make many bridges in series. When both the particle and polymer concentrations are increased above some critical levels, the suspensions show elastic responses to small strains. The elastic properties have been analyzed on the basis of site-bond per~olation!~~ because the flocs are compared to the clusters consisting of sites (particles) connected by bonds (bridges) in the percolation concept. For suspensions in which the network structure is developed with a constant bridging probability, the elastic properties show a little dependence on the diameter. However, the elasticity is markedly decreased,* when the particle diameter is increased to 45 nm. The adsorption and sedimentation experiments have shown that the bridging distance is decreased to a great extent. The results imply that the attractive force between polymer chain and particle surface (1) Iler, R. K. J. Colloid Interface Sci. 1971, 37, 364. (2) Otaubo, Y.; Umeya, K. J.Rheol. 1984,28,95. (3) Otsubo, Y.; Watanabe, K. J. Non-Newtonian Fluid Mech. 1987, 24, 265. (4) Otaubo, Y.; Watanabe, K. J. ColloidIaterface Sci. 1988,122,346. (5) Otaubo, Y.; Watanabe, K. Colloids Surf.1990, 50, 341. (6) Otaubo, Y.Langmuir 1990, 6, 114. (7) Otaubo, Y.; Nakane, Y. Langmuir 1991, 7, 1118. (8) Otsubo, Y . J. Colloid Interface Sci., in press.
is strong, and hence the polymer chain lies flat on the surface of large particles. In suspensions flocculated by polymer bridging, the rheological properties strongly depend on the conformation of adsorbed polymer chain. In the present study, the suepensions are flocculated by a soluble polymer, the coil size of which is comparable to particle size in the solution phase. Two particles can be bridged by a flexible coil unless the polymer chains have very strong affiiity for the particle surface. The flexible bonds lead to deformable and highly elastic flocs. Their effects on the rheology of suspensions are investigated.
Materials and Methods Materials. The suspensionswere composed of styrene-methyl acrylate copolymer particles, poly(acry1ic acid), Triton X-100, and water. The particles were formed by emulsion copolymerization with a styrenelmethylacrylate monomer ratio of 40160. The pH value was adjustad with hydrochloric acid to pH 2. The diameter and densityof copolymer particles were 80 nm and 1.13 X 103kgm", respectively. The stocksuspensions without soluble polymer were electrostatipllystabilized. Triton X-100surfactant (Union Carbide Co.) is an ethoxylatedoctylphenol. Poly(acry1ic acid) (PAA) with a molecular weight of M , = 4.5 X 105 was obtained from Polysciencea, Inc., and was used as received. The mean size of an isolated polymer chain in solution may be determined from the intrinsic viscosity [q] by the equation.9 where (S*)1/2is the root-mean-squareradius of gyration and @ is the Flory-Fox parameter. The intrinsic viscosity of PAA was 2.5 dL g-1 in aqueous solution at pH 2 and the radius of gyration was about 30 nm. The sample suspensions were prepared at concentrations up to 35% by volume. The PAA concentration of suspensions was in the range of 0-2.0% by weight based on the water. The rheological measurements were carried out for the suspensions stored at 25 O C for 1 day. Methods. Steady-flow and dynamic viscoelastic properties were measured using a Couette geometry on a Rheometrics RFS I1rheometer. The diameters of the bob and cup were 30 and 32 mm, respectively. The immersion length of the bob was 30 mm. The measuring shear rates y were from 2 X to 8 X lo18-1 in steady-flowmeasurements. The frequencies o were from 2 X 10-2to 1 x 102 8-1, and the strain was 0.05 in dynamic measurementa. The dynamic viscoelasticresponses were linear at strains up to 2 for all suspensions. In addition, the suspensions showed (9) Flory, P. J. Rinciple of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953.
0743-7463/92/2408-2336$03.00/0 Q 1992 American Chemical Society
Langmuir, Vol. 8, No. 9,1992 2331
Dilatant Flow of Flocculated Suspensions
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no time dependence and hence the initial conditions before the measurements did not affect the rheological data. The 10vol % suspensions were centrifugedat lOOOg for more than 50 h. The final sedimentation volume gives the concentration of the dispersed phase, from which the bridging distance in the flocs can be determined. The amount of polymer adsorbed on the particles was also calculated from the viscosity of supernatant solution.
Results and Discussion Effects of Particle and Polymer Concentrations. First of all,therheological behavior of suspensionswithout surfactant was measured a t 25 "Cas a function of particle and polymer concentrations. Figure 1 shows the shear rate dependence of viscosity for 15 vol 5% suspensions in solutions containing PAA at different concentrations. At a polymer concentration of 0.4 wt % ,the viscosity is low and the flow is slightly pseudoplastic at high shear rates. With increasing polymer concentration, the viscosity increases over the entire range of shear rates. In contrast to ordinary flocculated suspensions, which show pseudoplastic flow in awide range of shear rates, the flow behavior for the suspensions containing PAA a t concentrations of 0.5 wt % and above is quite different. The viscosity abruptly begins to increase, goes through a maximum, and then markedly decreases as the shear rate is increased. The shear rate at which the dilatant flow begins decreases with increasing polymer concentration. Another interesting point is that the viscosity first increases with polymer concentration and the rate of change in viscosity level decreases above 1.5 w t 5%. The degree of flocculation seems to reach saturation. Figure 2 shows the shear rate dependence of viscosity of 30 vol % suspensions. At low polymer concentrations, the flow is pseudoplastic over the entire range of shear rates. At 0.3 w t % and above, the suspensions show dilatant flow and the maximum viscosity. When the flocculation is induced in suspensions, the viscosity is increased, especially at low shear rates, and the flow becomes pseudoplastic. The flocculated suspensions in which a three-dimensional network structure is formed over the system often show a yield stress with infinite viscosity. The effect of flocculation on the rheology of suspensions is generally striking at low shear rates or low frequencies. However, it looks likely that the pseudoplastic tendency of 30 vol % suspensions disappears at high polymer concentrations. The suspension without polymer
lo-'
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Figure 2. Shear rate dependence of viscosity for 30 vol % suspensions in solutionsof PAA at different concentrations: 0.1 (0); 0.2 (a);0.3 (e);0.4 (0);0.5 (e);0.6 wt % (0).
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can be approximated as Newtonian. It is of interest to note that the highly flocculated suspensions are also Newtonian a t low shear rates. Figure 3 shows the shear rate dependence of viscosity for suspensions in a 1.0 w t % PAA solution. The dilatent flow occurs a t particle concentrations above 10 vol %. The particle concentration also has a critical value for appearance of dilatant flow. The shear rate at which dilatant flow begins decreases with increasing particle concentration. Although an increase in particle concentration causes adrastic increase in the Newtonian viscosity a t low shear rates, the magnitude of viscosity enhancement in dilatant region is smaller for concentrated suspensions. Beyond the maximum viscosity, the flow is almost plastic (because the shear stress is independent of shear rate) and the particle concentration does not have significant effects. The non-Newtonian viscosity profiles typical of the sample suspensions consist of a low-shear-rate Newtonian viscosity, a dilatant region a t moderate shear rates, and a pseudoplastic (nearly plastic) region at high shear rates. The dilatant flow and maximum viscosity are observed when both the particle and polymer concentrations exceed some critical values. Figure 4 shows the boundary for appearance of dilatant flow. Although the suspensions
Otsubo
2338 Langmuir, Vol. 8, No. 9, 1992 *
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Figure 6. Frequency dependence of storage modulus for 30 vol % Suspensions in solutions of PAA at different concentrations: 0.1 (0);0.2 (0);0.3 (e);0.4 (01; 0.6 (e);0.6 wt % (0).
I
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IO2 (s-') Figure 6. Frequency dependence of low modulus for 30 vol % suspensions in solutions of PAA at different concentrations: 0.1 (0); 0.2 (0); 0.3 (e);0.4 (01; 0.6 (e);0.6 wt % (0). lo-2 10-I IO0 Angular frequency
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with compositions above the line are dilatant at high shear rates, the magnitude of viscosity enhancement is large in the vicinity of the boundary. Figures 5 and 6 show the frequency dependence of storage and loss moduli, respectively, for 30 vol % suspensions in solutions of PAA at different concentrations. At low frequencies, the loss modulus ie larger than the storage modulus and both the moduli linearly decrease
with decreasing frequency. It is well-knoWn1*l2 that the viecoelaaticfunction of flocculated suspensionsin polymer solutionsshows a plateau at low frequencies. The plateau has been considered to be a manifestation of network structure of particles. An important feature is that with increasingparticle concentration or degree of flocculation, the height of the plateau increases and the terminal flow region shifts toward the low frequency side. The threedimensional network structure provides an additional relaxation process. However, the plateau region was not observed for all suspensions studied. The floc structure may be constructed because the relative viscosity, defined as the viscosity of suspeneions divided by that of the medium, is considerably large. The lack of plateau in frequency-dependent curve reflects the relatively weak interactions between particles and short relaxation time. In flocculated suspensions, the formation of flocs traps part of the continuous medium and leads to a larger effective volume of dispersed phase than that of the primary particles. To understand the floc structures, the adsorption and sedimentation experiments were carried out for 10 vol % suspensions in polymer solutions at concentrationsof 0.3-1.Owt % . The particle concentration in the sediment was 26 vol 7% and the adsorbance was 60 mg/g particles,irrespectiveof polymer concentration.The average number of polymer coils which adsorb onto one particle can be calculated from the adaorbance and molecular weight, and the value determined is 20. Amsuming that the particles are arranged in hexagonal packing in the sediment, the mean distance between particle surfaces is estimated to be 33 nm. In polymer adsorption on flat surfaces, the thickness of the adsorbed polymer layer is roughly of the same order as the radius of gyration of an isolated polymer coil in solution. The conformation of adsorbed coils on fine particles must undergo changes because the fraction of segments of an adsorbed coil in trains decreases and loop size becomes relatively larger. The resulta show that although the contraction of polymer coil occurs by adsorption, the bridging distance is comparable to the coil size. Therefore, two particles may be connectad by one polymer coil. It can be stressed that the bonds between particles are very flexible. The viscosity behavior observed for the suspensions is similar to that for dilute solutions of 888oci8ting polymers.1sJ4 Polymer chains containiig a small fraction of strongly associating groups exhibit unusual rheological properties in solution due to intrachain and interchain aseociation~.~~J~ Witten and Cohenl' have explained the dilatant flow of associating polymer solutione. Shear flow serves to elongate the polymer chains and break some of the intrachain associations, forming interchain aseociations. The flow may increase the number of interchain associations. However, at low polymer concentration, the association and clustering of chains cannot take place. The most important factor which affecta the association process is the volume concentration of polymer chains. In (10)Onogi, S.;Mataumoto,T.;Warmhina,Y. Tram. SOC.Rhsol. lWS, 17, 175. (11)Mateumoto, T.;Hitomi, C.; Onogi, S . Tram. Soc.Rheol. 1976,19, 541. (12)Umeya, K.;Otaubo, Y. J. Rheol. 1980,24,299. (13)Duvdevaui, I.; Agarwal, P. K.; Lundberg, R. D. Polym. Eng. Sci. 1982,22,499. (14)Jenkins, R.D.; Silebi, C. A.; E l - h r , M.S . ACS Symp. Ser. 1991,No. 4 2 , 222. (15)Lundberg, R. D.;Makowski,H. S. J. Polym. Sci.,Polym. Phy8. Ed. 1980,18,1821. (16)Lundbrg, R.D.;Phillips,R.R.J. Polym. Sci., Polym. Phy8. Ed. 1982,20,1143. (17)Witten, T.A.;Cohen,M. H.Macromolecules L B S , 18,1915.
Dilatant Flow of Flocculated Suspensions
order for interchain association to occur, the chains in solution must overlap. Therefore, at concentrations slightly above the overlap thresholds, the delicate balance between intrachain and interchain associations results in dilatant flow. However, in the suspensions studied, it is difficult to expect the shear-induced formation of interchain associations at the expense of intrachain associations. The statistical mechanical network modells has shown that the dilatant flow is induced by the decrease in entropy of polymer chain in the network during extension by shear. In suspensions, the extension of chains bridging particles can provide an additional energy dissipation. Although the flexible bridge between two particles is primarily responsible for resistance to flow, the three-dimensional network structure of bridges is required for dilatant behavior of suspensions. The sediment of suspensions which contain sufficient adsorbing polymer gives the limiting structure in which the polymer bridging is fully developed and the particles are arranged at the maximum concentration. At particle concentrations above 26 vol % ,all polymer chains adsorb onto particles and efficiently make bridges when the polymer concentration is low. Therefore, the effect of polymer concentration on the floc structure can be analyzed as the bond percolation process. In the bond percolation, the network of unbounded floc is constructed when the occupancy of bonds in a lattice whose sites are fully occupied exceeds a critical value, pcB.According to the empirical equationlg for the bond problem, the percolation threshold for various three-dimensional lattices is zpCBE 3/2, where z is the coordination number. In concentrated suspensions, the average number of bridges for each particle is estimated from the critical polymer concentration for appearance of dilatant flow on the assumption that one polymer coil makes one bridge. The obtained value is 1.5 for 30 vol % suspension and in good agreement with the percolation threshold for network formation. On the other hand, it is considered that two adjacent particles are bridged in dilute suspensions which contain sufficient polymer. Presumably this occurs at relatively high polymer concentrations, because all adsorbed coils do not always act as effective bridges. The effect of particle concentration on the floc structure is described through the site percolation process. In the site percolation, the sites are distributed at random in a lattice and two adjacent sites are necessarily connectedby a bond. The network structure is also developed above the critical site probability, pcs. The 7.5 vol % suspension showed dilatant behavior at polymer concentrations above 1.4 wt % . However, the dilatant behavior was not observed for 5.0 vol % and more diluted suspensions even though the polymer concentration is increased beyond 2.0 w t % .The critical particle concentration seems to be 5.0 vol % . The rheology of suspensions is often discussed in relation to a normalized concentration defined as the ratio of volume concentration of primary particles to the maximum packing concentration. The normalized concentration may correspond to the site probability. The critical site probability is determined as 0.2. For the site percolation of hexagonal close-packed lattice, the critical probability pcs has been estimated to be 0.204 by the Monte Carlo method.20 The theoretical prediction for percolation and the experimental value for appearance of dilatant flow are in good agreement. Therefore, it is concluded that the (18)Vrahopoulou, E.P.;McHugh, A. J. J.Rheol. 1987,31, 371. (19)Ziman, J. M. J.Phys. C 1968,1 , 1532. (20)Frish, H.L.;Sonnenblick, E.; Vyssotaky, V. A.; Hammersley, J. M.Phys. Rev. 1961,124,1021.
Langmuir, Vol. 8, No. 9,1992 2339
cc 10) ,” .-
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dilatant flow of suspensions results from the network structure consistingof particles bridged by flexible polymer chains. Effects of Temperature and Surfactant. Polymer adsorption is considered to be irreversible because polymer may attach to the surface at several points along ita chain and not be able to desorb simultaneously from all sites. In general, the bridging flocculation is essentially irreversible and the bond is not broken down in a quiescent state. Since the structural breakdown is progressively induced with increasing shear rate, the flow of flocculated suspensions is plastic with infinite viscosity or pseudoplastic. However, the suspensions studied show Newtonian flow at low shear rates. This indicates that the polymer bridges which make up the network are constantly forming, breaking, and re-forming by Brownian motion. The asdorption-desorption of polymer coil on particles reversibly takes place. The lifetime of a bridge may be a strong function of temperature. Figure 7 shows the shear rate dependence of viscosity of 15 vol % suspension in a 2.0 wt % PAA solution at different temperatures. With increasing temperature, the Newtonian viscosity decreases as might be expected and the critical shear rate at the onset of dilatant flow drastically increases. In addition, one can find that the critical shear rate is inversely proportional to the Newtonian viscosity at low temperatures because the transition boundary is approximated by a straight line with a slope of about -1. Although the network of unbounded flocs is required, the dilatant flow arises primarily from the extension of bridge which is formed by a polymer coil. The coil connecting two particles is extended under shear fields and contributes to energy storage. The force that determines the steady-flow properties is the restoring force. The desorption of coils from the particles plays a role in dissipating the energy stored in extension. Under conditions where the time scale of coil extension is much longer than that of desorption, the polymer coils have the equilibrium conformation and the flow becomes Newtonian. As the shear rate is increased, the coils are rapidly extended before they have a chance to dissipate the stored energy; this results in dilatant behavior. However, the coils are forced to desorb at some degree of extension. Since the force required for desorption of each coil and the total number of bridges in the system are constant, the suspensions show nearly plastic flow at high shear rates. The effect of temperature on the onset of dilatant flow can be discussed in connection with the relaxation process
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2340 Langmuir, Vol. 8,No.9,1992
of polymer chains. Most theories on polymer dynamics imply that the viscosity-shear rate curve is a universal function when plotted as 7/70 versus yA/l, where 70 is the Newtonian viscosity in the limit of zero shear rate and A1 is a characteristicrelaxation time. For melts and solutions of linear polymers, the A1 is proportional to ~ o l Twhere , T is the temperature. The zero shear viscosity is the dominant variable affecting the relaxation time in polymer liquid. From the fact that the transition. behavior from Newtonian to dilatant flow is scaled on 770, the results obtained show the same rheology as the polymer liquids. The network theories predict that the dilatant flow occurs when the Weissenbergnumber 7x1 approaches unity. The A1 can be determined as qdG,, where G , is the pseudoequilibrium modulus. In analogy to rubber elasticity, the G, is calculated by integration of Gr' over the appropriate range if the maximum of G" appears.21 Since the maximum was not observed in Figure 6, the upper limit of integration was adjusted to o = lo2 s-l as a first approximation. The obtained value of A1 is about 0.3 s for 30 vol % suspension in a 0.6 wt % polymer solution and the Weissenberg number at the maximum viscosity is 0.9. The Weissenberg number at the maximum viscosity has a tendency to increase with decreasing particle concentration. The importance of the Weissenbergnumber shows that the intrinsic mechanism of dilatant and pseudoplastic flow is the relaxation of extended bridges due to the desorption in shear fields. When the particles are covered with surfactant molecules, the polymer bridges hardly occur. The addition of surfactant causes a decrease in the number of bridges. Figure 8 shows the effect of surfactant on the viscosity behavior of a 15 vol % suspension in a 2.0 w t 95 PAA solution at 25 O C . With increasing surfactant concentration, the Newtonian viscosity decreases and the critical shear rate increases. Above 2.0wt % ,the dilatant behavior disappears. Referring back to Figure 1, the increase in surfactant concentration has almost the same effect on the dilatant behavior as the decrease in polymer concentration. Especially for the suspensions with fully developed bridging, the Newtonian viscosity is not sensitive to the addition of polymer or surfactant. The addition of surfactant causes a decrease in the number of bridges and in turn an increase in the concentration of nonadsorbed polymer. Therefore, in spite of high Newtonian viscosity, the suspensions containing surfactant have relatively high critical shear rates. The results about surfactant effects
Conclusions A polymer coil makes a flexible bridge to bind two particles whose diameter is comparable to the coil size. The suspensions flocculated by this bridging mechanism show unique viscosity profiles consisting of a low-shearrate Newtonian viscosity, a dilatant region at moderate shear rates, and a pseudoplastic region at high shear rates. The Newtonian flow implies that the polymer bridges are constantly forming,breaking, and re-forming by Brownian motion. Therefore, the adsorption-desorption process reversibly occurs, whereas under ordinary conditions it is considered to be irreversible. When the network of unbounded flocs connected by flexiblebridges is developed over the system, the suspensions show dilatant flow. This is due to the extension of bridges. Since the coils are rapidly extended before the desorption, the increase in restoring force causes dilatant flow. However, the high extension causes the desorption of coils from the particles. Because of breakdown of bridge at constant force, the flow drastically changes from dilatant to nearly plastic when the shear rate is increased.
(21) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; Wiley: New York, 1980.
Registry No. (&)(MA)(copolymer),25036-19-5;PAA, 900301-4.
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indicate that the number of bridges in the system must exceed some critical value and a network of unbounded floc must be developed for dilatant flow.
