Langmuir 1991, 7, 1118-1123
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Simulation of Bridging Flocculation and Elastic Percolation in Suspensions Yasufumi Otaubo' and Yoshinori Nakane Department of Image Science and Technology, Faculty of Engineering, Chiba University, 1-33 Yayoi-cho, Chiba-shi 260, Japan Received October 5, 1990. I n Final Form: December 19, 1990 The suspensionsflocculated by bridging show elastic responses at low frequencies when both the particle and polymer concentrationsare increased beyond some criticallevels. Boundaries for elastic percolation were determined in terms of the site-bondpercolation, In bridging flocculation, many bridges are formed by one polymer chain, and a series of bridges cannot be broken down to noninteracting bonds. Therefore, the critical probability for bond percolation, pcB,varies with number of bridges formed by one polymer chain, Nb, whereas the critical site probability, pes, is independent. As a result, scaling analysis is not applicableto the bond process. Flocs in which particles are connected by polymer chains were generated on the computer and paBwas geometrically determined as a function of Nb in a two-dimensional square lattice. For random-bond percolation at Nb = 1, the simulation gives the same value as the theoretical prediction, but pce decreases with increasing Nb. The discrepancy between the experimental results and simulation appears in the Nb dependence of pee when Nb is large. In strongly interacted systems, the formation of infinite flocsin the geometrical sense does not necessarily lead to the appearance of susceptible elasticity. The fractal dimension of percolating floc decreases with increasing Nb. The elastic properties above the percolation threshold may be closely connected with the inherent structure of flocs.
Introduction Flocculated suspensions respond elastically to small deformation when the particle concentration exceeds some critical value. It is accepted that the responsible structure is a three-dimensional network of unbounded flocs because direct transmission of elastic forcesthrough floc-floc bonds is essential for susceptible elasticity. The distribution of particles in the floc and the network formation process are often described through percolation theory. Site percolation deals with the distribution of cluster sizes for particles distributed at random with probability, p S ,in an infinite lattice composed of sites linked together by bonds. The theory predicts that unbounded clusters are built up above the critical site probability, pcs. In recent years, the elastic properties of percolating networks have received increasing attention from both theoretical1* and experimentaFI0 points. In actual suspensions, the critical particle concentration for the appearance of elastic responses depends strongly on the mode of floc formation, the particle size, and the strength of attractive forces. However, based on scaling analyses, the elastic modulus G can be expressed by the following equation near the critical concentration C,
G = k(C - C,)" where C is the particle concentration, k is a constant, and n is the critical exponent. Many authors7-I2have reported (1) De Gennes, P. G. J. Phys., Lett. 1976, 37, L-1. (2) Feng, S.; Sen, P. N. Phys. Rev. Lett. 1984,52, 216. ( 3 ) Kantor, Y.; Webman, 1. Phys. Reu. Lett. 1984,52,1891. (4) Feng, S.; Sen, P. N.; Halperin, B. I.; Lobb, C. J. Phys. Reu. B: Condens. Matter 1984,30,5386. (6) Garcia-Molina, R.; Guinea, F.; Lous, E. Phys. Reu. Lett. 1988,60, 124. (6) Mall,S.; Ruseel, W. B. J. Rheol. 1987,31, 651. (7) h e ] , A. Rheol. Acta 1982,21, 72. (8)Buscall, R.; McGowan, I. J.; Mills, P. D. A.; Stewart, R. F.; Sutton, D.; White, L. F.; Yates, G. E. J. Non-Newtonian Fluid Mech. 1987,24, 183. (9) Sonntag, R. C.; Russel, W. B. J. Colloid Interface Sei. 1987, 116, 485. (10) Heyes, D. M. J. Non-Newtonian Fluid Mech. 1988, 27,47. (11) Russel, W. B. Powder Technol. 1987,51, 15. (12) Tadros, Th. F. Langmuir 1990,6,28.
that the critical exponent for flocculated suspensions is in the range of 2.0-6.0. Colloidal suspensions flocculated by polymer bridging also show elastic responses when both the particle and polymer concentrationsare increased beyond some critical levels. In such suspensions, the flocs are considered to consist of sites (particles) connected by bonds (bridges). Therefore, the elastic properties can be analyzed in terms of site-bond percolation.I3-l5 In the site-bond percolation, for any site probability above pcs,there exists a percolation threshold for the bond probability pB. In a previous paper,I6 a pS-pB diagram for elastic percolation was determined by measuring the dynamic viscoelasticity of silica suspensions as a function of particle and polymer concentrations and the effect of particle size on the percolation boundary was examined. The most significant result is that the critical bond probability depends on the particle diameter. In bridging flocculation, a polymer chain adsorbs onto two or more particles whereas each bond is regarded as distributed at random in theoretical calculation. The particle diameter causes changes in the bridging distance and, in turn, the number of bridges formed by one polymer chain. This may be responsible for the dependence of pcBon the particle diameter. In this study, to provide more definitive data, the viscoelastic measurements have been carried out using polymers with different molecular weights. In addition, large flocs in which many particles are connected by a series of bridges have been grown on the computer. The elastic percolation in suspensions flocculated by polymer bridging will be discussed in relation to the geometric properties of floc structures.
Viscoelasticity and Percolation Boundary Materials and Methods. The suspensions were composed of silica, polyacrylamide (PAAm), glycerin, and water. The media were solutions of PAAm at concen(13) Otaubo, Y.; Watanabe, K.J. Colloid Interface Sci. 1989,127,214. (14) Otaubo, Y.; Watanabe, K.J. Colloid Interface Sci. 1989,133,491. (15) Otaubo, Y.; Watanabe, K.Colloids Surf. 1989, 41, 303. (16) Otaubo, Y. Langmuir 1990, 6, 114.
0743-7463/91/2407-ll18$02.50/0 0 1991 American Chemical Society
Langmuir, Vol. 7, NO.6, 1991 1119
Bridging Flocculation in Suspensions 1
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Figure 1. Frequency dependence of storage modulus for suspensions of 20-nm silica in a 0.5 wt % solution of PAAm with molecular weight Mw = 5 X 108. trations from 0.05to 0.5% by weight in a mixture of glycerin and water with 50/50mixing ratio. Two PAAm samples having molecular weights of 7 X lo6 (Nitto Chemical Industry Co.) and 5 X lo6 (Aldrich Chemical Co.)were used as received. The silica samples were Aerosil 130, 200, and 300 from Degussa Co., the particle diameters of which were 20, 15, and 10 nm, respectively. Silica suspensions were prepared at concentrations up to 18% by weight. The suspensions were ultrasonicated for 10 min to eliminate entrapped air and aged for more than 10 days until they gave their equilibrium responses. Dynamic viscoelasticitywas measured in the frequency range of 1.5 X to 6.3 X 10' s-l using a parallel disk geometry on a Rheometrics System IV rheometer. Measurements were carried out at low strain amplitude where the storage modulus G' and loss modulus G"showed linear viscoelastic responses. When the concentrationsof particle and polymer are low, both moduli have comparable values. For highly flocculated suspensions the elastic responses are predominant. Sothe results for storage modulus G' only will be presented. The 3 wt % suspensions in a 0.5 wt % PAAm solution were centrifuged to evaluate the floc concentration and bridging distance between particles in the flocs. From the viscosity of supernatant solutions, the amount of polymer adsorbed on the particles was calculated. Results. Figure 1shows the frequency dependence of storage modulus for suspensions of 20-nm silica in a 0.5 w t % solution of PAAm with a molecular weight M, = 5 X 106. At particle concentrations of 5 wt 7% and above, the viscoelasticfunction has a plateau, the height of which increases with particle concentration. At 4 wt %, the plateau does not appear, but the storage modulus rapidly decreases with decreasing frequency. The silica particles have no detectable effect on the storage modulus for more dilute suspensions. One can find a critical concentration for appearance of plateau or elastic responses at low frequencies. In the present study, it is assumed that the elastic percolation occurs at a particle concentration where the storage modulus exceeds 5 Pa in the frequency range of 10-"10-1 s-1.
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Figure 2 shows the frequency dependence of storage modulus for 8wt % suspensionsof 20-nmsilica in solutions containing PAAm with M, = 5 X 106 at different concentrations. The frequency-dependentcurve of storage modulus gives the plateau when the polymer concentration is increased to 0.15 wt %. The polymer concentration also has a critical value for elastic percolation. For percolating systems, the storage modulus slowly increases with polymer concentration in contrast to its abrupt increase near the threshold and then approaches a constant value above 0.4 wt 9%. At high polymer concentrations, the degree of bridging is expected to reach saturation. By repeating the dynamic viscoelastic measurements, the storagemodulus can be determined as a function of particle and polymer concentrations. The site-bond percolation process has thresholds which are given by a set of p s and pB. To construct the ps-pB diagram for elastic percolation, we must translate the particle concentration into the site probability and the polymer concentration into the bond probability. In the site percolation process, two adjacent occupied sites are necessarily connected; that is, pB = 1.0. The complete bridging implies that two adjacent particles are also inevitably connected. The elasticproperties of suspensions which contain sufficient adsorbing polymer would be explained by the site percolation. Therefore, it is considered that the sediment of suspension with fully developed polymer bridging has the limiting structure with p s = 1.0 andpB = 1.0. To understand the floc structures, the adsorption and sedimentation experiments were carried out. The results are summarized in Table I. In the sediments of flocculated suspensions, the volume fraction must be lower than the maximum random packing, 0.64,which ordinary takes place in noninteracting suspensions. In addition, if all bonds are fully occupied by the adsorbed polymer chains, the coordination number is
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1120 Langmuir, Vol. 7, No. 6, 1991 I o4 h
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Figure 4. Effect of Nb on the percolation boundary. estimated as about 6. Hence, the silica particles are assumed to be arranged in a simple cubic lattice in the sediment. The average number of bridges, Nb, which are formed by one polymer chain can be calculated from the adsorbance and molecular weights and is also cited for each system in Table I. Since the limiting floc structure is known, the percolation boundary is determined through the procedure described in the previous paper.16 For instance, the pS-pB diagram for determination of the percolation boundary is shown in Figure 3 using the suspensions of 20-nm silica in solutions of PAAm with M, = 5 X 106, As mentioned above, the percolating system has a storage modulus larger than 5 Pa over the entire range of frequency. Figure 4 shows the effect of Nb on the percolation boundary in the pS-pB diagram. The critical site probability, pcS, is about 0.35 and is in good agreement with the theoretical predictions for the simple cubic lattice.17-19 However, the critical bond probability, pcB,varies with Nb. In bridging flocculation, one polymer chain can make many bridges in series, although the existing percolation theories ignore correlations between sites or bonds. The dependence of pcB on Nb may be due to long-range interactions of bonds. The relation betweenPCBand bond distribution in the flocs will be discussed later. (17) Frish,H. L.;Sonnenblick, E.; Vyssotsky, V. A.; Hammersley, J. M. Phye. Rev. 1961,124, 1021. (18) Sykes, M.F.;Emam, J. W . J. Math. Phys. 1964,5, 1117. (19) Sykes, M.F.;Essam, J. W . Phys. Reo. 1964, 133, A310.
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Figure 6. Critical behavior of storage modulus near pee for the bond percolation process. Scaling arguments enable us to show a power law dependence of elasticity on the difference of site probability from the critical value. For flocculated suspensions, the dependence of storage modulus on the particle concentration is generally analyzed as the site percolation process. The storage modulus has been determined as a function of p s and pB. We have used the scaling analysis to estimate two critical exponents for the site percolation with pB = 1.0 and the bond percolation with p s = 1.0. Figures 5 and 6 show the critical behavior of the storage modulus near pcs for the site process and near pcBfor the bond process, respectively. Irrespective Of Nb,pCsis about 0.35, but pcBvaries from 0.1 to 0.25 depending on Nb. For the site process, the plots are closely related by a straight line with a slope of 4.0. In the elastic network the bonds transmit central and transverse forces,whereas in electrical networks conductors carry a scalar quantity. Mewis et aL20 have reported that the elasticity and conductivity of carbon black suspensions have the same percolation thresholds and increase exponentially with particle concentration. Their results indicate that the elastic network of carbon black can be modeled by an isotropic force (20) Mewis, J.; De Groot, L. M.; Helsen, J. A. Colloids Surf. 1987,22,
271.
Langmuir, Vol. 7, No. 6, 1991 1121
Bridging Flocculation in Suspensions constant. However, it has been generally shown that the elasticity and conductivity problems show different scaling behavior near threshold. According to recent s t ~ d i e s ~ ~ ~ l - ~ ~ the critical exponent for the elastic network may be in the range of 3.64.0. The suspensions studied give a reasonable value. On the other hand, the linear dependence is not seen for the bond process. Although the plots are highly scattered, the storage modulus at pB- PcB= 0.04 is less than 10 Pa for all suspensions. It seems likely that the critical exponent is very high near the threshold and decreases with increasing bond probability. Whether or not the scaling on pB- pcBis applicable to the bond process for elastic behavior of suspensions flocculated by polymer bridging is suspect. From a statistical point, the density of bonds which belong to the same polymer chain may decrease with increasing radius. Besides the percolation threshold, the storage modulus as a function of pB- pcB is considered to depend on the floc structure or Nb.
Simulation of Bond Percolation Procedure. The flocs were grown by using a 200 X 200 square lattice. The simulation was started by placing a seed bond a t the center of lattice and generating a polymer chain of specified length with self-avoiding random configuration. The polymer chain contains a number of sites that are connected by a series of bonds and represents a floc. Another seed bond was randomly placed and the self-avoidingrandom walk was executed to form the second floc. Similarly, polymer chains or flocs with self-avoiding random configuration and specified length were grown until a network structure of unbounded flocs was formed. Newly introduced chains were permitted to occupy the vacant bonds. In the process of self-avoiding random walks, the end of a given polymer chain was often trapped in the flocs. In this case, the random walk was stopped. Therefore, although the maximum length is limited, the polymer chains do not necessarily have the same length. The average length of polymer chain, Nb,is defined as the ratio of the total number of occupied bonds to that of polymer chains. Of course, for Nb = 1, the bonds are random and independently distributed. This process is referred to as random-bond percolation. When the steps of random walk stuck out of the boundary of lattice, the walk was continued until a specified number of bonds was incorporated into the floc or the end of chain was trapped. Only the occupied bonds in the field of 200 X 200 lattice were taken into account for the determination of percolation threshold. In this simulation, it is assumed that the percolation occurs when a large floc extending to two opposing boundary lines is formed. The cluster analysis was made by means of the algorithm of Hoshen and Kopelman.24 Results. Figure 7 shows the network structure at the critical bond probability, pcB= 0.5, for Nb = 1. Each bond has been formed is shown as a short line connecting two sites; the sites are not shown. For the random-bond percolation in a square lattice, the critical bond probability is theoretically shown to be 0.5.25 Our simulation gives the same critical value. The system consists of an infinitely large floc which extends from one end of the lattice to the other and many discrete flocs with different sizes. The (21) Woignier, T.; Phalippou, J.; Sempere, R.; Pelous, J. J . Phys. Fr. 1988, 49, 289. (22) Deptuck, D.; Harrison, J. P.; Zawadzki, P. Phys. Rev. Lett. 1985, 54, 913. (23) Benguigui, L. Phys. Reu. Lett. 1984,53, 2028. (24) Hoshen, J.; Kopelman, R. Phys. Reu. B: Solid State 1976, 14, 3428. (25) Shank, V. K. S.;Kirkpatrick, S. Adu. Phys. 1971, 20, 325.
Figure 7. Network structureat the critical bond probability for = 1. Holes (isolated flocs) are contoured.
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Figure 8. Network structureat the critical bond probability for = 800. The percolating floc is contoured. size distribution of finite flocs is known to be fractal. Figure 8 shows the network structure a t the critical bond probability, pcB= 0.22,for Nb = 800. The simulations were repeated several times under the same condition. However, the critical bond probability was not constant, especially a t large values of Nb. In the case of Nb = 800, pcBvariesfrom0.18to0.25. Itcan beseenthatthenetwork consists of flocs with a peculiar size. A collection of sites connected in series by one polymer chain may behave as a unit floc, while each bond is regarded as a unit in the random-bond percolation. It must be stressed that the size of unit floc varies with Nb. Figure 9 shows the critical bond probability,pcB,plotted against the length of polymer chain, Nb. Two important points are noted with regard to the critical behavior. First, pcBdecreases with increasing Nb. The random-bond percolation theory assumes that each bond between two neighboring sites on an infinite periodic lattice is formed Nb
1122 Langmuir, Vol. 7, No. 6, 1991
Otsubo and Nakane
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randomly with probability pB. The probability is the only parameter. In percolation theory, the most important concept is recognition of geometrical connections. Through the observation of geometrical connections, we can know the cluster size and percolation threshold. Recently other types of percolation in which the bonds, instead of being randomly distributed, are correlateddue to the interactions between them have been extensively studied. In the simulation, we introduce very strong interactions that many bonds are connected in series. Therefore, this may lead to a decrease in pcB.Secondly,the plots of pcBscatter as Nb is increased. It seems impossible to absolutely determine pcB as a singular point even under constant length of polymer chain. A large number of configurations can be achieved by flexible polymer chains. Since the unit flocs have a great variety of configurations and sizes, pcBstatistically scatters in a certain range depending on Nb. The scattering of the storage modulus shown in Figure 6 can be connected with the uncertainty of pcB. This implies that for suspensions flocculated by strong interactions, difficulties arise in the determination of percolation threshold and scaling analysis.
Discussions The flocs generated on the computer are two-dimensional. Strictly speaking, the simulation is not valid for the three-dimensional flocs that exist in the solutions. However, some investigations have reported2&29that the clustering dynamics, nucleation, ordering, and growth of three-dimensional systems can be described in terms of hard-sphere models in two dimensions. The simulation of bond percolation can predict the relation between interaction strength and percolation threshold. The comparison between the experimental results and simulation discloses a different dependence of pcBon Nb. In Figure 4, pcBreaches a minimum around Nb N 100 in contrast to the simulation showing that pcBdecreaseswith increasing Nb. A criterion used in deciding whether the system is percolating is the geometrical connections. However, it has been pointed out2I6 that since bonds transmit current and a central force but not transverse forces, parts of an infinite network would conduct but not support a shear stress. The conductivity threshold can be lower than the elasticity threshold for purely central forces. For the same reason, the threshold geometrically determined does not necessarilygive the Nb dependence similar to the elasticity threshold. In addition, the unit flocs are (26) Takano, K.; Hachisu, S . J. Chem. Phys. 1977,67,2604. (27) Richter, R.; Sander, L. M.; Chena, -. Z. J. Colloid Interface Sci. 1984,100, 203. (28) Onoda, G. Y. Phys. Reu. Lett. 1986,55, 226. (29) Keefer, K. D.; Schaefer, D. W. Phys. Reu. Lett. 1986, 56, 2376.
connected by a few bonds in Figure 8. The narrowest channel consists of a single chain. Even though this single chain transmits forces, it may be difficult to detect the elasticity as overall responses of network. The detectable elasticity is essential for the determination of percolation threshold. Actually, the sensitivity of a rheometer gives another factor in determining the threshold. In the random-bond percolation, as expected from Figure 7,it is reasonable to accept that the drastic change in transport properties occurs at the criticalthreshold in the geometrical sense. To analyze the elastic properties, theories must account for the inherent structure of percolating network. Next, the fractal dimension was examined for the percolating floc (infinite floc) produced by the simulation. In dilute suspensions, the particles form discrete flocs, the size of which far exceeds the range of forces holding them together. The short-range structure in the floc depends on the strength and nature of the interactions between two particles prior to flocculation. However, the interesting phenomena in irreversiblerandom flocculation is that systems with only short-range forces and no longrange order can form closely resembling flocs whose longrange structures are distinctive and statistically scaleinvariant. It has been ~ h 0 ~ that n ~large ~ flocs 1 ~ ~ ~ produced by irreversible process are of fractal geometry. The flocculated structures are characterized by the fractal dimension D which is defined as
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N ( R ) RD (2) where N ( R ) is the number of particles in a single floc as a function of radius R. Since the average density of a fractal floc decreases with increasing radius, the fractal dimension is always less than the Euclidean dimension. For the two-dimensional random percolation in a square lattice, the percolating floc has been found to be D = 1.9 at the critical probability.% Although there is at present relatively little quantitative information about the effects of long-range interactions on the fractal geometries in three-dimensions,the fractal analysis of a two-dimensional percolating floc serves to understand the morphology of network structure. Figure 10 shows the fractal dimension plotted against Nb. The radius R has been chosen at a distance from the center of lattice within the limits of 2oLb-63oLb,where Lb is the bond length. At Nb C 10, D is about 1.9 and in good agreement with the theoretical value. However, D decreases with increasing&. When we attempt to substitute a long chain for parts of continuous bonds in Figure 7, a very branched polymer is required. In a sense, the fractal ~~
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(30) Weitz, D. A.; Oliveria, M. Phys. Rev. Lett. 1984,52,1433. (31) Aubert, C.; Cannell, D. S. Phys. Reu. Lett. 1986,56, 738. (32) Witten, T. A.; Sander, L. M. Phys. Reu. Lett. 1981, 47, 1400. (33) Nienhuis, B. Phys. Rev. Lett. 1982, 49, 1062.
Bridging Flocculation in Suspensions
dimension is a measure to characterize the branching structures. Hence, the low fractal dimension at Nb = 800 may be attributed to the linear conformation of polymer chains. In ordinary suspensions, the particles are held together by attractive forces such as the London-van der Waals forces. The strong attractive forces do not directly
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result in the promotion of fibrous structure. It is interestingtostudy the relation between the intarparticle forces and the fractal dimension of flocs. Also,for further study, the elastic properties of isolated fractal flocs would be a key to a better understanding of elastic percolation in flocculated suspensions.
