Langmuir 2003, 19, 3589-3595
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Viscoelastic Scaling of Colloidal Gels in Polymer Solutions Eugene E. Pashkovski,* James G. Masters, and Ammanuel Mehreteab Colgate Palmolive Company, Research and Development Division, Technology Center, 909 River Road, P.O. Box 1343, Piscataway, New Jersey 08855-1343 Received June 18, 2002. In Final Form: January 29, 2003
The structure and formation of particulate silica networks in biopolymer solutions were studied using rheological measurements and diffusing wave spectroscopy (DWS). For samples with different volume fractions of silica, the frequency-dependent viscoelastic moduli can be scaled onto one master curve by shifting the data along the moduli and frequency axes. Viscoelastic scaling of silica suspended in carboxy methyl cellulose (CMC) solution shows a linear relationship between the scaling factors. For silica suspensions in xanthan solutions, only vertical scaling is needed to obtain the master curve. It was found that the elasticity of the polymer matrix affects the structure of particulate networks as well as the kinetics of structural recovery after preshear. The fractal dimension of the silica network was found to be lower in polymer solutions with high elasticity (xanthan) in comparison to those with low elasticity (CMC). DWS measurements reveal strongly subdiffusive and nearly diffusive motion of particles in xanthan and CMC solutions, respectively. The difference in mobility of particles explains structural differences between silica networks formed in CMC and xanthan solutions as well as different kinetics of viscoelastic recovery after shear.
1. Introduction The rheology of concentrated suspensions and particulate gels is a subject of extensive research due to the industrial importance of these systems. The rheology of colloidal gels was recently studied both theoretically and experimentally.1-3 The main emphasis of these studies was to find the relationship between linear viscoelastic properties and the structure of gels above the critical volume fraction φc that corresponds to the gel point. For interacting particles, the value of φc can be very low compared with the critical packing concentration (φ ∼ 0.63). In these systems, the elastic modulus scales with φ as G′(φ) ∝ (φ/φc - 1)x, where the exponent x depends on the fractal dimension df of the particulate network and φ > φc (typically, 3.7 < x < 4.52). Computer simulations of network structure predict variations of fractal dimension depending on the dimensionality of the particle trajectory dt.4 In Newtonian fluids, dt ) 2, and the mean square displacement (MSD) 〈∆r2(t)〉 ∝ t2/dt; the diffusion coefficient D t 〈∆r2(t)〉/t ∝ t2/dt-1 is time-independent; and D ) kBT/f (f ) 6πη0a for particles of radius a dispersed in the liquid of viscosity η0). In depletion or bridging flocculated colloidal suspensions, a low amount of polymers does not alter significantly the frictional properties of the continuous phase. In this paper, we study the rheology of colloidal networks formed in concentrated biopolymer solutions. In these systems, the role of viscoelasticity of the environment in aggregation becomes dominant. In viscoelastic polymer solutions and gels, the behavior of colloidal particles is subdiffusive;5,6 that is, D(t) t 〈∆r2(t)〉/t ∝ t2/dt-1 decreases with time. This may significantly lower the * E-mail:
[email protected]. (1) Rueb, C. G.; Zukoski, C. F. J. Rheol. 1997, 41, 197-218. (2) Potanin, A. A.; De Rooij, R.; Van den Ende, D.; Mellema, J. J. Chem. Phys. 1995, 102, 5845-5855. (3) De Rooij, R.; Van den Ende, D.; Duits, M. G. H.; Mellema, J. Phys. Rev. E 1994, 49, 3038-3049. (4) Meakin, P. In Kinetics of Aggregation and Gelation; Family, F., Landau, D., Eds.; Elsevier: Amsterdam-New York, 1984; pp 91-99.
fractal dimension of particulate gels. The rheology of complex systems exhibiting a dual contribution of polymer and colloidal gels is quite intriguing. In systems studied here, both polymer solutions and particulate gels contribute to elasticity. One should expect that these systems are less fragile than purely colloidal networks and have a wider range of linear viscoelasticity. In the linear regime of deformation, purely elastic behavior of the network is usually discussed in the literature,7 and the dissipation of energy is assumed to be negligible. This may be the case for highly elastic systems probed at low frequencies. For colloidal gels formed by attractive particles, the highfrequency dissipation plays an important role, as was recently found by Trappe and Weitz.8 They have shown that the Newtonian background fluid contributes to the viscous losses of the system. As a result, the elastic and loss moduli cross over when frequency-dependent viscous losses match the frequency-independent elastic modulus, G′ ) G′′ ) ωcrossη0. Moreover, the dependencies of G′ and G′′ on frequency collapse onto one master curve when the curves are shifted along the frequency and the moduli axes for gels with different φ. For polymeric background fluids, viscoelastic scaling may significantly change due to the elastic contribution of polymer solutions. Therefore, a more detailed picture is needed to describe the role of polymeric background fluids in the rheological behavior of complex colloidal gels. In this study, we probe the structure and rheology of polymer solutions using diffusing wave spectroscopy (DWS). From this method, one obtains a comprehensive dynamic picture of the colloidal particles (tracers) dispersed in polymer solutions.5 Measurements of the mean square displacement of tracers reveal the structure and (5) Mason, T. G.; Gisler, T.; Kroy, K.; Frey, E.; Weitz, D. J. Rheol. 2000, 44, 917-928. (6) Nisato, G.; He´braud, P.; Munch, J.-P.; Candau, S. J. Phys. Rev. E 2000, 61, 2879-2887. (7) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: New York, 1999; Chapter 7. (8) Trappe, V.; Weitz, D. A. Phys. Rev. Lett. 2000, 85, 449-452.
10.1021/la026087e CCC: $25.00 © 2003 American Chemical Society Published on Web 03/14/2003
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dynamics of polymer solutions. In our study, DWS is employed to calculate rheological functions in a much wider frequency range compared to that for conventional mechanical measurements. In addition, measurement of particle mobility by DWS may be especially informative for highly structured solutions of natural polysaccharides such as carboxymethyl cellulose (CMC) and xanthan gum utilized in this study. In this article, we make an attempt to unify the rheology of particulate gels with independent information on particle mobility in the viscoelastic environment.
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3. Results and Discussion 3.1. Microrheology of Polymer Solutions. The microrheological properties of viscoelastic fluids can be obtained by monitoring the mobility of well-characterized tracer particles. The mean square displacement of particles 〈∆r2(t)〉 can be calculated from the field correlation function
g1(t) ) xg2(t) - 1 )
∫0∞P(S) exp(- 31k02〈∆r2(t)〉l*S ) dS (1)
2. Experimental Section 2.1. Preparation of Samples. The polymer gums xanthan and CMC (500T), supplied by Kelco and Noviant, respectively, were used as received. The polymers were dissolved in 50/50 mixtures of water/glycerin at 1 wt % to form the continuous phase. Polydisperse silica (ZEO 115, Huber Corp.) was dispersed in the continuous phase using a high shear mixer. The system was further homogenized by gentle mixing under vacuum for several hours. The resulting stock gel with high volume fraction of silica was then diluted with the continuous phase to form dispersions at lower volume fraction. After dilution, samples were gently remixed to form homogeneous dispersions. No separation occurs for these samples during experiments. Dispersions of silica in CMC and xanthan solutions with φ > 0.1 are stable at least for several weeks. 2.2. Methods. An advanced rheological system (ARES, Rheometric Scientific, Inc.), equipped with standard and double wall Couette cells, was utilized to measure linear viscoelastic properties of gels. Circulating water controlled the temperature of the cells. To ensure the unique shear history, all measurements were started after a 1-h equilibration after sample loading. After equilibration, the frequency sweep was performed at a very small strain. Strain sweeps at ω ) 10 rad/s were performed to ensure that the frequency sweeps were carried out in the linear regime. The critical strain values that limit the linear viscoelastic regime for CMC and xanthan-based dispersions with the highest amount of silica were 0.19% and 0.31%, respectively. Therefore, the values of strain (0.05%) employed in frequency sweep experiments were significantly lower. The gels were then subjected to a strong oscillatory deformation (100% at 100 rad/s) for 1 min to break the gel structure. The recovery of the elastic modulus was recorded for 2 h using small amplitude oscillatory shear at 10 rad/s. After the recovery, the first two steps were repeated. A DWS optical system was utilized similar to that described in ref 9. The beam of an Ar+ ion laser of wavelength λ ) 488 nm was focused to the sample cell using the F ) 10 mm lens. The polarizing prism was used to ensure that only multiply scattered light was detected. The scattered light was collected and then equally split by a system of optical fibers (OZ optics). Two identical photomultipliers (Hamamatsu) were used to amplify the optical signal. The signals were cross-correlated with a digital correlator BI-9000 (Brookhaven Instruments, Inc.). The Brookhaven software for dynamic light scattering was employed for data collection. A stepper motor was used to control vertical displacement of nonergodic samples to achieve proper averaging of the scattered light intensity. Monodisperse carboxylated latex particles (Interfacial Dynamics Corporation) were used in DWS experiments. The particles were dispersed in polymer solutions at a fixed volume fraction ∼ 0.003. The dispersions were placed into the rectangular glass cells (Vitrocom, Hellma) with the cell thicknesses L ) 2 and 5 mm. The reference mean free path l/ref was determined by fitting experimental data for suspensions of particles in water (φ ) 0.0029) to calculated correlation functions. Three latex dispersions of particles with d ) 0.45, 1.1, and 1.5 µm were utilized. The mean free paths for the samples were estimated from the intensity of transmitted light to be / lSAM /l/ref ) ISAM/Iref. (9) Weitz D. A.; Pine, D. J. In Dynamic Light Scattering; Brown, W., Ed.; Oxford: 1992; Chapter 16.
provided that the intensity correlation function g2(t) is measured by DWS.9 In this equation, P(S) represents a distribution function of the light path length S through the sample, l* is the mean free path of photons, and k0 ) (2π/λ) is a constant (λ ) 488 nm). For transmission geometry with a point source employed in this study, one can use an approximation:
g1(t) ) C
∫Q
∞
2[(1 + ξ) - (1 - ξ)e-2ζξ] 2
2 -2ξ
(1 + ξ) - (1 - ξ) e
ξe-(1-ζ)ξ dξ (2)
where Q t (L/l*)k0(〈∆r2(t)〉1/2, t 2l*/(3L), ζ t l*/L, C is the normalization constant chosen that g1(0) ) 1, and L is the thickness of sample cell. By solving this equation numerically, one can obtain the value of 〈∆r2(t)〉 using experimentally measured g1(t). For systems in thermal equilibrium, the generalized Stokes-Einstein relation provides the relationship between the mean square displacement of spheres and the viscoelasticity of the complex fluid. The viscoelastic spectrum G ˜ (s) as a function of Laplace frequency5 s is given by
G ˜ (s) )
kBT πas〈∆r˜ 2(s)〉
(3)
Here, kB is the Boltzmann constant, and the tilde denotes the unilateral Laplace transform. The time-domain creep compliance can be obtained from eq 3 and the simple relationship between the shear modulus and the shear creep compliance, J ˜ ) 1/(sG ˜ ):5
J(t) )
πa 〈∆r2(t)〉 kBT
(4)
Equations 3 and 4 denote that the local viscoelasticity near the spheres is equivalent to the macroscopic viscoelasticity.5 This is the case if the particles are much larger than the typical length of spatial inhomogeneity. For linear polymer solutions in a good solvent, the length ξ ) RF(cpol/c*)-3/4 sets the spatial inhomogeneity of the transient polymer network. Here C* is the critical overlap concentration and RF is Flory’s radius, which is of the order of several hundred angstroms. Thus, microparticles having diameter d ∼ 0.5-1.5 µm are at least 1 order of magnitude larger than the mesh size for polymer concentration cpol ≈ c*. In this case, the micro-viscoelastic environment experienced by the probe particles is similar to the macroscopic viscoelasticity.
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Equation 3 can be rewritten in terms of the complex modulus:10
G(s) )
kBT 2
|
πa〈∆r (t)〉Γ[1 + R(t)]
(5)
s)1/t
where the argument of the Γ-function (1 + R(t)) is defined by the logarithmic slope of the mean square displacement:
R(t) )
d ln 〈∆r2(t)〉 d ln t
(6)
We used the following approximation11 to decompose the complex modulus to the elastic and the loss components:
{[ (π2 - 1) + π2]R(t)}[1 - β(t)] π π G′′(ω) ) G(ω) cos{[β(t)( - 1) + ][1 - R(t)]}[1 - β(t)], 2 2 G′(ω) ) G(ω) cos β(t)
t ) 1/ω (7)
with β(t) ) d ln[a(t)]/d ln t. Equations 4-7 give complete rheological information under the assumption that the probe particles “see” the polymer solution as a continuum. In natural polymers such us polysaccharides, the probe particles may “see” structural inhomogeneities on the length scale of several microns. Furthermore, the introduction of spherical probe particles perturbs the medium by reducing the network density, as noted by Levine and Lubensky.12 They show that mutual fluctuations of two separated particles provide a more accurate determination of the complex shear modulus. For inhomogeneous materials, the use of singleparticle microrheology (DWS) for calculating viscoelastic functions is problematic; nevertheless, it may reveal qualitative information on local structure and mobility. The frequency range at which the microrheological calculations are valid depends on the material parameters of viscoelastic medium and particle size, ω g ωB ) (G/ η0)(ξ/a)2,13 where G is the modulus, η0 is the solvent viscosity, and ξ is the mesh size of the polymer network. Below the threshold frequency ωB, the longitudinal compression modes of the network are excited and there is no direct comparison to the shear flow, which is the case for mechanical measurements. This frequency is ωB ∼ 10 rad/s for G ) 10 dyn/cm2, η0 ) 1 cP, and ξ/a ) 0.1. The upper frequency limit depends on the inertial effects that define the size of particles to be a < (Fω22G*)-1/2.13 Here F is the density of the fluid and G* is the complex modulus; at a ) 20 µm the upper frequency limit is still very high, ω ∼ 6 × 104 rad/s. To characterize CMC and xanthan solutions, we used polystyrene particles of different size. The correlation functions for 0.4% CMC solution with microparticles with d ) 0.45, 1.1, and 1.5 µm are shown in Figure 1a. At this concentration, correlation functions decay at t e ls, and the samples are ergodic. The values of 〈∆r2(t)〉 shown in Figure 1b were calculated from eq 2. For comparison, the values of 〈∆r2(t)〉 for particles dispersed in a 50/50 water/ (10) Mason, T. G.; Gang, Hu; Weitz, D. A. J. Opt. Soc. Am. A 1997, 14, 139-149. (11) Dasgupta, B. R.; Tee, S.-Y.; Crocker, J. C.; Frisken, B. J.; Weitz, D. A. Phys. Rev. E 2002, 65, 051505. (12) Levine, A. J.; Lubensky, T. C. Phys. Rev. Lett. 2000, 85, 17741777. (13) Schnurr, B.; Gittes, F.; MacKintosh, F. C.; Schmidt, C. F. Macromolecules 1997, 30, 7781-7792.
Figure 1. Intensity autocorrelation functions (a), mean square displacements (MSDs) (b), and creep compliance (c) for carboxylated latex particles dispersed in 0.4% CMC solution: d ) 0.45 µm (circles), 1.1 µm (triangles), and 1.5 µm (squares). For comparison, the MSD of 0.45 µm particles in a 50/50 water/ glycerol mixture is presented (diamonds).
glycerin mixture are given on the same graph. The logarithmic slope R(t) t d ln 〈∆r2(t)〉/d ln t slightly increases with time; the value of R is very close to 1 for 1.1 and 0.45 µm particles and R ∼ 0.95 for 1.5 µm particles. Slightly subdiffusive motion of 1.5-µm particles shows that the dimensionality of the particle trajectory dt ) 2/R(t) ≈ 2.1 is slightly higher than that of the Brownian trajectory dt ) 2. For comparison, Brownian diffusion of 0.45-µm particles in a 1:1 water/glycerin mixture is shown in Figure 1b. The values of creep compliance J(t) estimated from eq 2 for solutions with 1.1 and 1.5 µm microparticles almost collapse onto one curve (Figure 1c), whereas 0.45-µm particles have lower values of the creep compliance. A possible explanation is that the small particles introduce less disturbances of the local network structure. In contrast to CMC, xanthan solutions are highly elastic even at relatively low concentrations. For xanthan solution of 0.4%, the smaller the particles, the faster the decay (Figure 2a). The mean square displacement of particles with d ) 0.45 and 1.5 µm differs 2 orders of magnitude even at short times (Figure 2b). The mobility of particles in xanthan solution dramatically decreases with their size. In the homogeneous medium, the value of 〈∆r2(t)〉 would change with the particle size as 〈∆r2(t)〉 ∝ 1/d, according to the Stokes law. In contrast to that, we found that
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Figure 3. Values of G′ and G′′ (filled and open symbols, respectively) measured as a function of frequency for xanthan solutions: c ) 0.3 (circles), 0.5 (triangles), 0.8 (squares), and 1.1 (diamonds). Inset: Dependence of G′ vs concentration (circles); the solid line represents a fit to the equation G′ ∝ (c - c0) with c0 ) 0.08 ( 0.025%.
Figure 4. Master curves obtained using time-temperature superposition for xanthan solutions: c ) 0.6 and 1% (open and filled symbols, respectively). The measurements of G′ (circles) and G′′ (triangles) were performed at T ) 15-45 °C.
Figure 2. Intensity autocorrelation functions (a), mean square displacements (b), and creep compliance (c) for carboxylated latex particles dispersed in 0.4% xanthan solution: d ) 0.45 µm (circles), 1.1 µm (triangles), and 1.5 µm (squares). For comparison, the MSD of 0.45 µm particles in a 50/50 water/ glycerol mixture is presented (diamonds).
〈∆r2(t)〉 ∝ e-d/ζ with ζ ) 0.24 µm for c ) 0.4%. The parameter ζ sets the length scale of spatial heterogeneity in xanthan solutions. This result indicates that the particles are caged between the xanthan microgel blobs, creating local distortion and some excessive stress; this explains the decrease of apparent creep compliance with particle size (Figure 2c). Thus, the elasticity of xanthan paste is controlled by repulsive interactions between the blobs (see next section). This explains very low values of the logarithmic slope for xanthan solutions (R(t) ∼ 0.4-0.5) that indicate strongly subdiffusive motion of particles (Figure 2b). Such a low value of R(t) in combination with small displacement 〈∆r2(t)〉1/2 ≈ 1 nm (t < 1 s) indicates very slow collective mobility of xanthan paste. In the time window of our experiment, this system displays rubberlike behavior, whereas CMC solution is liquidlike. One can conclude that xanthan and CMC solutions have qualitatively
different structures. In both cases, the size of probe particles is in the range of spatial inhomogeneity of solutions, indicating that application of eqs 5-7 for calculating viscoelastic moduli is problematic. Nevertheless, we perform such calculations in order to gather qualitative information on the high-frequency dependencies of elastic moduli. 3.2. Rheology of Xanthan and CMC Solutions. Xanthan solutions are highly elastic; the storage modulus of 0.3-1.1% solutions measured at 0.1-100 rad/s exceeds significantly the loss modulus (Figure 3). Furthermore, the storage modulus scales with concentration as G′ ∝ (c - c0) with c0 ) 0.08 ( 0.025% (see inset in Figure 3). Linear scaling of G′ observed for xanthan paste is similar to that for compressed emulsions14 and microgel suspensions;15 for entangled polymer solutions in a good solvent, this dependence is much steeper, G′ ∝ c9/4.16 At c . c0, the elastic microgel particles of xanthan become compressed, and the storage and loss moduli are almost frequencyindependent. At high frequencies, the slopes for G′ and G′′ are similar; for instance, G′(ω) ∝ ω0.41 and G′′(ω) ∝ ω0.38 for c ) 0.3%, and no high-frequency crossover is observed at ω < 100 rad/s. To expand the frequency window, we used time-temperature superposition (TTS) for 0.6 and 1% xanthan solutions by measuring G′(ω) and G′′(ω) at T ) 15-45 °C (Figure 4). Remarkably, the curves G′(ω) and G′′(ω) are almost parallel in the frequency range covering more than four decades (up to ∼2000 rad/s). Values of G′′(ω) and G′′(ω) for ω g 2000 rad/s were (14) Prinsen, H. M.; Kiss, A. D. J. Colloid. Interface Sci. 1986, 112, 427-437. (15) Berli, C. L. A.; Quemada, D. Langmuir 2000, 16, 7968-7974. (16) Rubinstein, M. Theoretical Challenges in Polymer Dynamics. In Theoretical Challenges in the Dynamics of Complex Fluids; McLeish, T., Ed.; NATO ASI Series; Dordrecht, 1997; Chapter 3.
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Figure 5. Storage moduli (solid symbols) and loss moduli (open symbols) for 0.4% xanthan solution. The circles and triangles correspond to the rheological and DWS measurements, respectively.
Figure 6. Storage moduli (solid symbols) and loss moduli (open symbols) for 0.4% CMC solution. The triangles and the circles correspond to time-temperature superposition of the rheological data and DWS measurements, respectively.
recovered from DWS data for 0.27% xanthan solution with 0.45 µm latex particles (see Figure 5). For ω > ωB ≈ 10 rad/s, a good agreement for this particular system was found, indicating that 0.45 µm particles do not change the local elasticity of the paste, which has a correlation length of the order of ζ ) 0.24 µm. Bigger particles would create excessive local stresses around them, causing the viscoelastic moduli to be overestimated. The discrepancy between mechanical and DWS data for ω < 10 rad/s may be due to the effect of longitudinal compression modes, as discussed in the previous section. In the range 102 < ω < 104 rad/s, the moduli scale as G′ = G′′ ∝ ω0.45. Unlike xanthan, 0.4% CMC solution displays liquidlike behavior with G′(ω) < G′′(ω) (Figure 6). A master curve shown in Figure 6 was obtained by time-temperature superposition; this procedure allowed for increasing the upper frequency limit only to 200 rad/s. The values of G′(ω) and G′′(ω) calculated from DWS cover the highfrequency range; the loss modulus is significantly higher while the storage modulus is similar to that measured in the rheometer. This discrepancy may be due to the exclusion of microparticles from the regions of high polymer concentration; this would indicate a strongly aggregated state of CMC molecules. These aggregates, called spherical or fringed micelles in ref 17, may contribute to the slightly decreasing slope of G′ at low frequency (Figure 6). The absence of the terminal relaxation time and the plateau regime for CMC solutions up to c ) 1.0% indicate that the CMC micelles do not overlap. The frequency range available in DWS for CMC solutions is limited by the fast decay of the intensity correlation function g2(t); therefore, the mechanical and DWS data
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shown in Figure 6 do not overlap. For xanthan gel, the DWS and mechanical frequency ranges overlap due to low mobility of tracers and longer decay time of g2(t). The high-frequency regime that is observed for CMC (G′ ∝ ω3/4) was recently described by Morse18 in terms of the chain-bending modes. No contribution of the bending modes was found for 0.4% xanthan solution; the moduli behaved as predicted by the Mooney-Rivlin model for the rubberlike systems.19 Microrheological analysis helps us to understand complex behavior of colloidal particles dispersed in polymer solutions. Nevertheless, some limitations of this analysis should be mentioned. The experimental scheme we utilized is based on time-averaging of g2(t), which is valid for thermal equilibrium systems. Nonequilibrium colloidal systems display changes in the course of measurements, which is the case for concentrated xanthan solutions. Therefore, in this study we used solutions of relatively low concentrations, which were found to be ergodic and display no aging effects. The main findings, however, can be used for more concentrated systems. Using vertical displacement of a sample cell, we performed measurements for more concentrated solutions and found that the slopes R(t) t d ln 〈∆r2(t)〉/d ln t did not significantly change for c < 0.8% for xanthan and for c < 1.9% for CMC. We found, however, significant change in the slow dynamics of xanthan solutions observed for times similar to the sample age; this effect will be described in detail elsewhere. However, if the observation time is much shorter than the sample age, DWS measurements give a good estimate of the high-frequency slopes of G′ and G′′. 3.3. Rheology of Colloidal Gels. The rheology of particulate gels depends on contributions of both the background fluid and the network formed by the particles. For particulate gels in Newtonian background fluids (carbon black particles in oil), Trappe and Weitz8 suggested that the fluid contributes exclusively to the viscous part of the complex modulus while the particulate network is purely elastic. The elastic modulus of the network is almost frequency-independent whereas the loss modulus increases as G′′(ω) ∝ ω. Thus, the viscous and elastic contributions are equal at the crossover frequency. The existence of high-frequency crossover suggests that the rheological responses of the background fluid and the particulate network are effectively independent of each other. Therefore, the viscoelastic moduli measured for samples with different volume fractions of solid particles can be normalized by the plateau value of the elastic network and by the crossover frequency. For a given sample, the crossover occurs when the viscous and elastic contributions are equal, G′(ωc) ) G′′(ωc) ) η0ωc (with η0 being the viscosity of the fluid). Then, the shift of G′(φ,ω) and G′′(φ,ω) curves collected for samples with varying volume fraction φ can be used to obtain the master curve. The vertical and horizontal shift factors are defined as b(φ) ) G′(φ,ω)/G′(φmin,ω) and a(φ) ) ωc(φ)/ωc(φmin), respectively. For Newtonian background fluids, the relationship between b(φ) and a(φ) must be linear. The difference between the system described in ref 8 and our systems is that the background fluids are viscoelastic. CMC solution slightly deviates from Newtonian fluids, and G′′ > G′. In contrast to CMC, xanthan microgel suspensions are highly elastic. The relationship between the scaling factors may significantly change for dispersions of particles in poly(17) Zugenmeier, P. Supramolecular Structure of Polysaccharides. In Polysaccharides. Structural Diversity and Functional Versatility; Dumitriu, S., Ed.; Marcel Dekker, Inc.: New York, 1998; Chapter 2. (18) Morse, D. C. Macromolecules 1998, 31, 7044-7067. (19) Ferry, J. D. Viscoelastic Properties of Polymers, 3rd ed.; John Wiley and Sons: New York, 1980; Chapter 10.
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Figure 7. (a) Values of G′ and G′′ (filled and open symbols, respectively) measured as a function of frequency for silica suspensions in 1% CMC solution: φ ) 0.101 (circles), 0.127 (triangles), 0.160 (squares), and 0.195 (diamonds). (b) Master curve obtained by shifting the data along the moduli and frequency axes using two-dimensional minimization. The inset shows the relationship between the vertical (b) and the horizontal (a) shift factors. Note that, at high frequencies, G′′ ∝ ω0.56; this indicates the non-Newtonian character of losses for the polymeric background fluid.
meric fluids. Let us examine this effect first for the CMC/ silica systems. Figure 7a shows frequency dependences of viscoelastic moduli for suspensions of silica in CMC solutions. The crossover point is observed only for gels with relatively small volume fraction φ; for higher values of φ, the crossover frequency exceeds the maximum frequency of the rheometer (100 rad/s). Therefore, the shift factors were determined by two-dimensional minimization (Orchestrator software). The master curve in Figure 7b shows perfect scaling of viscoelastic moduli similar to that found by Trappe and Weitz8 but with one significant difference. At high frequencies, the slope of G′′(ω) is approximately 0.56, showing a significant decrease in viscous losses compared with the case of Newtonian fluids, for which the slope is 1. However, a linear relationship between the shift factors still holds for CMC (see inset in Figure 7b). This simple relationship may not be valid for the systems where microgel particles form the high modulus network (xanthan in our case, see Figure 4). In this case, solid particles are caged between soft microgels; therefore, the behavior of polymeric and solid particles becomes strongly coupled. Figure 8a shows the frequency dependences of viscoelastic moduli for suspensions of silica in xanthan matrix. No crossover is observed, even at the lowest values of φ. The elastic and loss moduli can be scaled using only the vertical shifts (Figure 8b). The vertical shift factors can be used to determine the structure of particulate networks, assuming their self-similarity, which implies the power law scaling of b(φ).
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Figure 8. (a) Values of G′ and G′′ (filled and open symbols, respectively) measured as a function of frequency for silica suspensions in 1% xanthan solution: φ ) 0.114 (circles), 0.146 (triangles), 0.181 (squares), and 0.209 (diamonds). (b) Master curve obtained by vertically shifting the data along the moduli axis. Note that G′′ is almost frequency-independent, indicating a very low level of viscous losses in the system.
Figure 9. Vertical shift factors as a function of silica volume fraction for CMC (filled symbols) and xanthan (open symbols) systems. Circles and triangles correspond to measurements before and after preshear, respectively.
Figure 9 shows the log-log plots of b(φ) ∝ φx with the exponents x ) 6.71 ( 0.2 and x ) 3.15 ( 0.2 for CMC and xanthan-based gels, respectively. For xanthan systems, this behavior occurs for φ > 0.135, suggesting that for φ < 0.135 the particles do not form the fractal network. The exponent x can be associated with the fractal dimension of the gel df, x ) (3 + db)/(3 - df),20 where db is the fractal dimension of the backbone that supports the stress. As 1 e db e df, the values of fractal dimensions may be found in the range 2.2 e df e 2.5 for CMC and 1.7 e df e 2.0 for xanthan-based systems. Thus, the particles of silica form denser clusters in CMC than in xanthan solutions. Note that the scaling factors may depend on the age of gels. To take into account the time factor, we performed preshear of samples and measured the moduli after a 2-h (20) Shish, W.-H.; Shish, W. Y.; Kim, S.-I.; et al. Rhys. Rev. A 1990, 42, 4774-4779.
Colloidal Gels in Polymer Solutions
Langmuir, Vol. 19, No. 9, 2003 3595
In comparison, the evolution of G′R(t) for xanthan-based gels is quite different; the reduced elastic modulus increases at short times and plateaus at longer times. All dependencies collapse onto one curve (with the exception of that for φ ) 0.185, Figure 10b). Figure 10c shows the time evolution of the elastic modulus G′(t)/G′(0) for φ ) 0.180 dispersions of silica in CMC and xanthan solutions; qualitatively different behavior of CMC and xanthan systems may be a result of differences in particle mobility that were revealed by DWS. The preshear breaks the colloidal network into fragments (flocs), and the increase in elasticity after preshear is due to formation of chains between the flocs. The gel stiffness is approximately proportional to the number of particles that form the chains connecting the flocs.2 The kinetics of chain formation is much slower for xanthan-based systems due to strongly subdiffusive mobility of particles. However, to further investigate the relationship between particle mobility and elasticity of particulate networks, it is necessary to obtain information on the mobility of microparticles on larger time and length scales than are available in DWS.
Figure 10. Relative changes of the elastic modulus vs time after preshear for φ ) 0.18 silica in CMC (a) and xanthan (b).
recovery. The vertical shift factors estimated before and after recovery were apparently identical (Figure 9). The exponent x is a measure of the density of the fractal network. A strong difference in density of particulate networks formed in CMC and xanthan solutions may be the result of different mobility of particles, as shown by DWS. Strongly subdiffusive mobility of particles in xanthan significantly slows down the aggregation of the colloidal network. Aggregation of colloidal gels can be monitored rheologically. The elastic modulus G′(t) was measured after preshear as a function of time at ω ) 10 rad/s (see Experimental Section). To compare behavior of gels with different volume fractions of silica, the time evolution of the modulus is presented as G′R(t) ) [G′(t) - G′(0)]/[G′(2 h) - G′(0)] with the elastic moduli at t ) 0 and t ) 2 h being G′(0) and G′(2 h), respectively. For CMC-based systems, the value of G′R(t) increases monotonically, and all curves collapse onto one master curve for 0.095 e φ e 0.185 (Figure 10a).
4. Conclusion In this paper, rheology and DWS were utilized to study the structure and mechanism of colloidal gel formation in viscoelastic polymer solutions. We have found that the elasticity of a polymer matrix affects the structure of colloidal gels. The procedure of viscoelastic scaling developed for colloidal gels formed in Newtonian fluids8 has been applied for systems with non-Newtonian (polymeric) background fluids. According to this procedure, the frequency dependencies of G′(ω) and G′′(ω) for particulate gels with systematically varied volume fraction φ can be scaled onto one master curve. For Newtonian fluids, viscoelastic scaling is based on the idea that the frequency-independent elastic modulus of the colloidal network is matched by the viscous losses at the crossover point; as a result, a(φ) ) b(φ). For polymeric background fluids, this relationship changes due to the elastic contribution of the fluid. The elastic contribution of polymeric liquids is frequency-dependent; to estimate high-frequency behavior of polymer solutions in a much wider frequency range than that available using regular rheometry, we use DWS. For CMC solutions at ω ) 102-104 rad/s, the viscous losses dominate over the elastic contribution, whereas for xanthan pastes G′(ω) ≈ G′′(ω) ∝ ω0.45. This rubberlike behavior of xanthan paste strongly affects the observed scaling of viscoelastic moduli, indicating that viscous losses play an insignificant role in the rheology of these systems. Further analysis suggests that xanthan particles form an elastic paste with the storage modulus increasing as a linear function of polymer concentration. Strongly subdiffusive motion of the tracer particles dispersed in xanthan paste further supports this result. In contrast to this, tracers in CMC solutions display nearly diffusive behavior. Acknowledgment. The authors thank D. Weitz, V. Trappe, and B. Dasgupta for many stimulating discussions and J. Rouse for his assistance in setting up the DWS apparatus. We also thank L. Szeles for preparation of samples. LA026087E
