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Investigation of the link between micromechanical interparticle bond rigidity measurements and macroscopic shear moduli of colloidal gels
Colloids and Surfaces A: Physicochem. Eng. Aspects 301 (2007) 137–140
Investigation of the link between micromechanical interparticle bond rigidity measurements and macroscopic shear moduli of colloidal gels Peter B. Laxton, John C. Berg ∗ Department of Chemical Engineering, University of Washington, Box 351750, Seattle, WA 98195, USA Received 2 September 2006; received in revised form 19 November 2006; accepted 7 December 2006 Available online 14 December 2006
Abstract Recent research has been reported on the development of a direct measurement of the particle–particle bond rigidity, κo , between aggregated colloids using a micromechanical three point bending test of assembled linear aggregates employing laser tweezers [J.P. Pantina, E.M. Furst, Colloidal aggregate micromechanics in the presence of divalent ions, Langmuir 22 (2006) 5282–5288]. Current theory suggests how one may use measured κo to estimate bulk colloidal gel shear moduli, G. In cases where the theory is appropriate, direct measurement of κo allows this estimation to be made regardless of the source of interparticle attractions. This work investigates the comparison of G-values computed using the reported κo measurements to values obtained by bulk rheometry. Bulk colloidal gel shear moduli are predicted within a factor of two, and the general trends predicted are observed. © 2006 Elsevier B.V. All rights reserved. Keywords: Colloidal gel; Interparticle bond rigidity; Gel modulus; Scaling law for gel rheology
1. Introduction
dependence takes the form
Colloidal gels are common in industry and everyday experience and are also interesting subjects of fundamental colloid science. The ability to predict the mechanical properties of these gels would provide formulators with a valuable tool as well as providing insight into the behavior of these systems. The recent development of a measurement technique by Pantina and Furst [1,2] provides access to the final piece of information necessary to calculate colloidal gel shear modulus, G, and thus the ability to predict this property. This work was set out to compare measured values of G with values calculated in terms of the particle–particle bond rigidity, κo , reported by Pantina and Furst [1]. The scaling behavior of the gel modulus, viz., G ∼ φμ , where φ is the volume fraction of the particles, has been observed repeatedly over the course of the last few decades [3–11]. The scaling exponent as well as a prefactor to the scaling law have been derived through a spring model. The spring constant, κ, of fractal objects is size dependent [5]. The length scale, s,
κ(s) = κo
∗
Corresponding author. Tel.: +1 206 543 2029; fax: +1 206 543 3778. E-mail address: [email protected] (J.C. Berg).
0927-7757/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2006.12.035
a β s
,
(1)
where a is the particle radius and κo , referred to as the “bond rigidity,” is the spring constant between a pair of interacting particles within the cluster. κo is due to adhesion between particles [1], and one would expect a large value in relatively strong gels where electrostatic repulsions have been effectively screened, while lower κo -values are expected in weaker gels. The elasticity exponent, β = 2 + db , where the bond dimension, db , describes the fractal structure of the gel backbone and must be greater than unity for connectivity [12]. For colloidal suspensions, the critical length scale for gelation, ξ c , representing the average floc size, is exponentially dependent on the floc fractal dimension, df [2,4] in accord with, ξc = aφ1/3−df .
(2)
The colloidal gel shear modulus is directly related to the floc spring constant as follows [2], G=
κ(ξc ) . ξc
(3)
138
P.B. Laxton, J.C. Berg / Colloids and Surfaces A: Physicochem. Eng. Aspects 301 (2007) 137–140
By substitution, the equation for the modulus then becomes, G=
κ0 (3+db /3−df ) . φ a
(4)
The form of the exponential dependence of the modulus on fractal dimension has been corroborated through experiment [3–5]. The bond rigidity was not a measurable quantity until Pantina and Furst developed a micromechanical three point bending test applied to particle chains employing laser tweezers. Eq. (4) can then be fully realized in terms of measurable quantities. Particle radius is routinely measured using many techniques, including light scattering and microscopy. Aggregate fractal dimension is commonly measured with static light scattering, SLS [13]. The bond dimension, db , has been shown through simulation to be 1.1 for gels made up of diffusion-limited clusters [12]. Now with the ability to measure κo it is possible to calculate G using Eq. (4). Predicting rheological parameters of colloidal gels in terms of number and strength of particle contacts has been done previously with some success [14–18]. However, these methods depend on the ability to estimate the strength of interparticle attractions, typically according to DLVO theory. As long as aggregates form rigid interparticle bonds, the direct measurement of κo allows for the calculation of bulk properties regardless of the applicability of DLVO theory. It is important to note that this may not be the case in depletion gels, where a lubricating layer of polymer may preclude interparticle bonds from having the necessary restoring force. Pantina and Furst report κo data for a series of aqueous linear aggregates with various counterion types and concentrations. They formed linear aggregates of 1.47 m diameter poly(methyl methacrylate) (PMMA) spheres by optical trapping. The aggregates were created in aqueous solutions of CaCl2 , MgCl2 , and NaCl at various concentrations. This allowed them to observe different bond strengths due to varied physicochemical conditions, including the non-DLVO behavior resulting from divalent ion effects. Micromechanical testing of assembled aggregates is a powerful technique allowing for direct access to the strength of interparticle bonding. The present work focuses on one aspect of the application of data measured in this way, viz., predicting bulk mechanical properties of colloidal gels. The objective of this work is to explore applicability of micromechanically measured values of κo to estimating macroscopic gel properties by calculating G and comparing this value with G as measured by bulk rheology. 2. Materials and methods 2.1. Materials Materials were chosen to match the systems used in Pantina and Furst [2] as closely as possible. Gels were prepared by aggregating 1.50 m average diameter spherical PMMA latex particles (Bangs Laboratories, Inc., Fishers, IN) which have a reported specific gravity of 1.19. These particles were washed
by dilution in deionized water and subsequent centrifuging of the dispersion at 3500 rpm for 15 min in an International Micro Centrifuge (International Equipment Co., Needham Hts., MA). The wash was repeated until the supernatant was found to have a surface tension equal to that of the deionized water. Three washes proved sufficient. Since these particles are too large to measure aggregate fractal dimension by SLS, it was assumed that df developed by aggregation of smaller particles of the same type under identical conditions as the gels studied would be representative of gel df . Therefore, 60 nm average diameter PMMA spheres (Bangs Laboratories, Inc.) were used for df determination. The salts used to induce aggregation and gelation, CaCl2 , MgCl2 , and NaCl, were all Baker Analyzed A.C.S. Reagent grade (J.T. Baker, Inc., Phillipsburg, NJ). 2.2. Sample preparation Gel preparation was complicated by the formation of ‘armored bubbles’ [19], i.e. air bubbles stabilized by the preferential adsorption of particles to the air–liquid interface. Particle stabilized bubbles readily develop upon mixing of these aqueous PMMA suspensions. These bubbles lead to irreproducible gel rheology, and it was therefore necessary to avoid their presence in the gel samples. This need was met by the following sample preparation procedure. The washed particle centrifuge cake was redispersed by physical mixing accompanied by sonication in an Ultrasonic Cleaner (Cole Parmer, Vernon Hills, IL) bath sonicator for 5 min in enough deionized water so that φ < 0.1. Particle stabilized bubbles did occur in low φ dispersions; however, foaming was greatly reduced. Concentrated stock salt solution was then added to these dilute dispersions, and gelation and sedimentation were allowed to occur over night. The supernatant of these sedimented gels was poured off and the sediment was centrifuged at 3500 rpm for 30 min. The supernatant was again poured off leaving a cake with φ ≈ 0.45. This cake was used as the gel sample for shear modulus determination. Exact volume fractions were determined by measuring weight fraction using a simple thermal gravimetric technique, and then calculating φ from the known particle and solvent densities. 2.3. Fractal dimension measurement Salt solutions of various concentrations were prepared and filtered through 0.22 m syringe filters (Whatman, Trenton, NJ) into SLS sample cells. The glass cells had been soaked in concentrated sulfuric acid and rinsed with filtered water. Drops of concentrated 60 nm diameter PMMA spheres were added to the salt solutions and subsequently mixed by gently shaking the container by hand. These systems were then allowed to sit overnight. Scattering intensity, I, at various scattering angles, θ, was measured for each sample using a BI-200SM goniometer and BI-DS photomultiplier (Brookhaven Instrument Corp., Holtsville, NY) together with a 633 nm, 15 mW HeNe laser (Melles Girot, Irvine, CA). Prior to measurement, the index matching fluid, decalin (J.T. Baker, Inc.), was filtered by pumping it through a 0.2 m filter in order to remove dust.
P.B. Laxton, J.C. Berg / Colloids and Surfaces A: Physicochem. Eng. Aspects 301 (2007) 137–140
Software automatically collected I versus θ data and generated plots of log I versus log q. Where q, the scattering vector, is a function of system properties and θ. The negative of the slope of log I versus log q plots were then taken to be df [13]. Each measurement was repeated five times. The fractal dimension measured in this way is an estimate of the true fractal dimension of the rheology samples. If some deviation does exist between the estimate and true fractal dimensions, it is likely that the actual gels, as prepared by the method described in Section 2.2, would tend to be more consolidated than the measured aggregates. This would result in a reduced value for the calculated G, where an increase of 0.1 in df would result in about a 35% reduction in G. 2.4. Shear modulus measurement Gel shear moduli were measured at a Peltier-system controlled temperature of 25.0 ± 0.1 ◦ C with a Modular Compact Rheometer (MCR) 300 (Anton Paar, Ashland, VA). Gel samples were held between 50-mm diameter sand blasted plate and a parallel smooth plate. The surface roughness of the sand blasted plate ensures that slip due to depletion at the tool-sample interface does not affect the measurement. At high frequencies, ω, the storage modulus, G , of gels tends to asymptote to some value, and Glim ω→∞ = G [20]. Therefore, in order to most closely approximate the shear modulus, the highest available frequency (100 Hz) allowed by the instrument was used to measure storage modulus. This is a good approximation as long as the characteristic relaxation time of the gel is much longer than the period of the applied strain; which, at 100 Hz, is generally the case for weak gels [20]. Each run consisted of an increase in oscillation amplitude from 0.01 to 100% at a constant frequency of 100 Hz. The resulting torque on the sand blasted plate was recorded, and from this value the storage modulus was calculated by the instrument automatically. The storage modulus at the lowest measurable amplitude was taken to be the best approximation of the shear modulus. Low amplitude values were used in order to ensure that the linear viscoelastic region was probed.
139
Fig. 1. Log I vs. log q plot from SLS of 60 nm PMMA spheres aggregated in 50 mM CaCl2 solution. The line is a least squares fit to the data with a slope of −2.08, giving df = 2.08. Table 1 Summary of aqueous salt media used, together with values of the bond rigidity, κo , and fractal dimension, df , used for calculation of the gel modulus, G Salt concentration (mM)
κ (N/m)
df
CaCl2
10 50 100 200 400
0.02 0.04 0.28 0.70 0.21
2.07 2.08 1.80 2.08 2.00
MgCl2
50 250 375 500
0.06 0.20 0.31 0.61
2.03 2.00 1.95 2.05
NaCl
100 500
0.02 0.04
2.00 1.91
3. Results and discussion Gels prepared as described in Section 2.2 resulted in φ ranging from 0.37–0.47. Measured values of df used in calculating G were between 1.8 and 2.1 ± 0.1, where the uncertainty represents one standard deviation calculated from the five individual measurements. Fig. 1 shows a typical example of the determination of the fractal dimension, and Table 1 lists df values used in calculating G. Figs. 2–4 show gel modulus as calculated according to Eq. (4) using Pantina and Furst values of κo , listed in Table 1, compared with gel modulus measured by conventional oscillatory rheometry, as described in Section 2.4. Error bars on the calculated points are due to the reported uncertainties in κo as well as measured uncertainties in df . In some cases the error bars are large; therefore it does not make sense to attempt to verify the exact values of κo . Despite this limitation it is valuable to compare the data within the given level of certainty. Although some
Fig. 2. Log–log plot of calculated and measured G vs. CaCl2 concentration.
Fig. 3. Log–log plot of calculated and measured G vs. MgCl2 concentration.
140
P.B. Laxton, J.C. Berg / Colloids and Surfaces A: Physicochem. Eng. Aspects 301 (2007) 137–140
References
Fig. 4. Log–log plot of calculated and measured G vs. NaCl concentration.
deviation between the calculated and measured values exists, Eq. (4), with the reported values of κo can predict G to within a factor of 2. It is also important to note that both calculated and measured G follow similar trends with salt concentration. Pantina and Furst discuss in particular the effects of divalent ion adsorption, specifically noting that charge reversal occurs above 200 mM CaCl2 . This phenomenon causes the observed maximum of Fig. 2. Data for κo above the concentration required for charge reversal in MgCl2 systems was not available; therefore, a monotonic trend was expected and is observed in Fig. 3. Divalent ion absorption effects make the usefulness of the direct measurement of κo clear. 4. Summary Bulk rheological measurements of colloid gels created from particles and media nearly identical to those used by Pantina and Furst to investigate κo have been used to explore the link between the microscopic κo measurement and macroscopic properties. The data presented attest to the applicability of Pantina and Furst micromechanical determinations of κo to the calculation of bulk modulus, G. Direct measurement of κo thus allows bulk colloidal gel rheology to be predicted within a reasonable degree of certainty, even in cases where DLVO theory breaks down. Acknowledgment This work was supported by the U.S. Department of Energy (Contract DE-FG02-04ER-63796).
[1] J.P. Pantina, E.M. Furst, Elasticity and critical bending moment of model colloidal aggregates, Phys. Rev. Lett. 94 (2005) 138301-1–138301-4. [2] J.P. Pantina, E.M. Furst, Colloidal aggregate micromechanics in the presence of divalent ions, Langmuir 22 (2006) 5282–5288. [3] R. Buscall, P.D.A. Mills, J.W. Goodwin, D.W. Lawson, Scaling behaviour of the rheology of aggregate networks formed from colloidal particles, J. Chem. Soc., Faraday Trans. 1 84 (1988) 4249–4260. [4] W.-H. Shih, W.Y. Shih, S.-I. Kim, J. Liu, I.A. Aksay, Scaling behavior of the elastic properties of colloidal gels, Phys. Rev. A 47 (1990) 4772–4780. [5] A.H. Krall, D.A. Weitz, Internal dynamics and elasticity of fractal colloidal gels, Phys. Rev. Lett. 80 (1998) 778–781. [6] M. Chen, W.B. Russel, Characteristics of flocculated silica dispersions, J. Colloid Interface Sci. 141 (1991) 564–577. [7] R. de Rooij, D. van den Ende, M.H.G. Duits, J. Mellema, Elasticity of weakly aggregating polystyrene latex dispersions, Phys. Rev. E 49 (1994) 3038–3049. [8] J.W. Goodwin, R.W. Hughes, S.J. Partidge, C.F. Zukoski, The elasticity of weakly flocculated suspensions, J. Chem. Phys. 85 (1986) 559–567. [9] M.C. Grant, W.B. Russel, Volume-fraction dependence of elastic-moduli and transition-temperatures for colloidal silica-gels, Phys. Rev. E 47 (1993) 2606–2614. [10] J.A. Yanez, E. Laarz, L. Bergstrom, Viscoelastic properties of particle gels, J. Colloid Interface Sci. 209 (1999) 162–172. [11] S.A. Shah, Y.L. Chen, K.S. Schweizer, C.F. Zukoski, Viscoelasticity and rheology of depletion flocculated gels and fluids, J. Chem. Phys. 119 (2003) 8747–8760. [12] P. Meakin, Formation of fractal clusters and networks by irreversible diffusion-limited aggregation, Phys. Rev. Lett. 51 (1983) 1119– 1122. [13] D.W. Schaefer, J.E. Martin, P. Wiltzius, D.S. Cannell, Fractal geometry of colloidal aggregates, Phys. Rev. Lett. 52 (1984) 2371–2374. [14] R.G. Larson, The Structure and Rheology of Complex Fluids, Oxford University Press, New York, 1999. [15] Y.K. Leong, B.C. Ong, Critical zeta potential and the Hamaker constant of oxides of water, Powder Technol. 134 (2003) 249–254. [16] N. Sakairi, M. Kobayashi, Y. Adachi, Effects of salt concentration on the yield stress of sodium montmorillonite suspension, J. Colloid Interface Sci. 283 (2005) 245–250. [17] P.B. Laxton, J.C. Berg, Relating clay yield stress to colloidal properties, J. Colloid Interface Sci. 296 (2006) 749–755. [18] P.J. Scales, P.C. Kapur, S.B. Johnson, T.W. Healy, Shear yield stress of partially flocculated colloidal suspensions, AIChE J. 44 (1998) 538–544. [19] A.B. Subramaniam, C. Mejean, M. Abkarian, H.A. Stone, Microstructure, morphology, and lifetime of armored bubbles exposed to surfactants, Langmuir 22 (2006) 5986–5990. [20] J.W. Goodwin, Some uses of rheology in colloid science, in: J.W. Goodwin (Ed.), Colloidal Dispersions, Dorset Press, Dorchester, 1982, pp. 165–195.
Investigation of the link between micromechanical interparticle bond rigidity measurements and macroscopic shear moduli of colloidal gels Peter B. Laxton, John C. Berg ∗ Department of Chemical Engineering, University of Washington, Box 351750, Seattle, WA 98195, USA Received 2 September 2006; received in revised form 19 November 2006; accepted 7 December 2006 Available online 14 December 2006
Abstract Recent research has been reported on the development of a direct measurement of the particle–particle bond rigidity, κo , between aggregated colloids using a micromechanical three point bending test of assembled linear aggregates employing laser tweezers [J.P. Pantina, E.M. Furst, Colloidal aggregate micromechanics in the presence of divalent ions, Langmuir 22 (2006) 5282–5288]. Current theory suggests how one may use measured κo to estimate bulk colloidal gel shear moduli, G. In cases where the theory is appropriate, direct measurement of κo allows this estimation to be made regardless of the source of interparticle attractions. This work investigates the comparison of G-values computed using the reported κo measurements to values obtained by bulk rheometry. Bulk colloidal gel shear moduli are predicted within a factor of two, and the general trends predicted are observed. © 2006 Elsevier B.V. All rights reserved. Keywords: Colloidal gel; Interparticle bond rigidity; Gel modulus; Scaling law for gel rheology
1. Introduction
dependence takes the form
Colloidal gels are common in industry and everyday experience and are also interesting subjects of fundamental colloid science. The ability to predict the mechanical properties of these gels would provide formulators with a valuable tool as well as providing insight into the behavior of these systems. The recent development of a measurement technique by Pantina and Furst [1,2] provides access to the final piece of information necessary to calculate colloidal gel shear modulus, G, and thus the ability to predict this property. This work was set out to compare measured values of G with values calculated in terms of the particle–particle bond rigidity, κo , reported by Pantina and Furst [1]. The scaling behavior of the gel modulus, viz., G ∼ φμ , where φ is the volume fraction of the particles, has been observed repeatedly over the course of the last few decades [3–11]. The scaling exponent as well as a prefactor to the scaling law have been derived through a spring model. The spring constant, κ, of fractal objects is size dependent [5]. The length scale, s,
κ(s) = κo
∗
Corresponding author. Tel.: +1 206 543 2029; fax: +1 206 543 3778. E-mail address: [email protected] (J.C. Berg).
0927-7757/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.colsurfa.2006.12.035
a β s
,
(1)
where a is the particle radius and κo , referred to as the “bond rigidity,” is the spring constant between a pair of interacting particles within the cluster. κo is due to adhesion between particles [1], and one would expect a large value in relatively strong gels where electrostatic repulsions have been effectively screened, while lower κo -values are expected in weaker gels. The elasticity exponent, β = 2 + db , where the bond dimension, db , describes the fractal structure of the gel backbone and must be greater than unity for connectivity [12]. For colloidal suspensions, the critical length scale for gelation, ξ c , representing the average floc size, is exponentially dependent on the floc fractal dimension, df [2,4] in accord with, ξc = aφ1/3−df .
(2)
The colloidal gel shear modulus is directly related to the floc spring constant as follows [2], G=
κ(ξc ) . ξc
(3)
138
P.B. Laxton, J.C. Berg / Colloids and Surfaces A: Physicochem. Eng. Aspects 301 (2007) 137–140
By substitution, the equation for the modulus then becomes, G=
κ0 (3+db /3−df ) . φ a
(4)
The form of the exponential dependence of the modulus on fractal dimension has been corroborated through experiment [3–5]. The bond rigidity was not a measurable quantity until Pantina and Furst developed a micromechanical three point bending test applied to particle chains employing laser tweezers. Eq. (4) can then be fully realized in terms of measurable quantities. Particle radius is routinely measured using many techniques, including light scattering and microscopy. Aggregate fractal dimension is commonly measured with static light scattering, SLS [13]. The bond dimension, db , has been shown through simulation to be 1.1 for gels made up of diffusion-limited clusters [12]. Now with the ability to measure κo it is possible to calculate G using Eq. (4). Predicting rheological parameters of colloidal gels in terms of number and strength of particle contacts has been done previously with some success [14–18]. However, these methods depend on the ability to estimate the strength of interparticle attractions, typically according to DLVO theory. As long as aggregates form rigid interparticle bonds, the direct measurement of κo allows for the calculation of bulk properties regardless of the applicability of DLVO theory. It is important to note that this may not be the case in depletion gels, where a lubricating layer of polymer may preclude interparticle bonds from having the necessary restoring force. Pantina and Furst report κo data for a series of aqueous linear aggregates with various counterion types and concentrations. They formed linear aggregates of 1.47 m diameter poly(methyl methacrylate) (PMMA) spheres by optical trapping. The aggregates were created in aqueous solutions of CaCl2 , MgCl2 , and NaCl at various concentrations. This allowed them to observe different bond strengths due to varied physicochemical conditions, including the non-DLVO behavior resulting from divalent ion effects. Micromechanical testing of assembled aggregates is a powerful technique allowing for direct access to the strength of interparticle bonding. The present work focuses on one aspect of the application of data measured in this way, viz., predicting bulk mechanical properties of colloidal gels. The objective of this work is to explore applicability of micromechanically measured values of κo to estimating macroscopic gel properties by calculating G and comparing this value with G as measured by bulk rheology. 2. Materials and methods 2.1. Materials Materials were chosen to match the systems used in Pantina and Furst [2] as closely as possible. Gels were prepared by aggregating 1.50 m average diameter spherical PMMA latex particles (Bangs Laboratories, Inc., Fishers, IN) which have a reported specific gravity of 1.19. These particles were washed
by dilution in deionized water and subsequent centrifuging of the dispersion at 3500 rpm for 15 min in an International Micro Centrifuge (International Equipment Co., Needham Hts., MA). The wash was repeated until the supernatant was found to have a surface tension equal to that of the deionized water. Three washes proved sufficient. Since these particles are too large to measure aggregate fractal dimension by SLS, it was assumed that df developed by aggregation of smaller particles of the same type under identical conditions as the gels studied would be representative of gel df . Therefore, 60 nm average diameter PMMA spheres (Bangs Laboratories, Inc.) were used for df determination. The salts used to induce aggregation and gelation, CaCl2 , MgCl2 , and NaCl, were all Baker Analyzed A.C.S. Reagent grade (J.T. Baker, Inc., Phillipsburg, NJ). 2.2. Sample preparation Gel preparation was complicated by the formation of ‘armored bubbles’ [19], i.e. air bubbles stabilized by the preferential adsorption of particles to the air–liquid interface. Particle stabilized bubbles readily develop upon mixing of these aqueous PMMA suspensions. These bubbles lead to irreproducible gel rheology, and it was therefore necessary to avoid their presence in the gel samples. This need was met by the following sample preparation procedure. The washed particle centrifuge cake was redispersed by physical mixing accompanied by sonication in an Ultrasonic Cleaner (Cole Parmer, Vernon Hills, IL) bath sonicator for 5 min in enough deionized water so that φ < 0.1. Particle stabilized bubbles did occur in low φ dispersions; however, foaming was greatly reduced. Concentrated stock salt solution was then added to these dilute dispersions, and gelation and sedimentation were allowed to occur over night. The supernatant of these sedimented gels was poured off and the sediment was centrifuged at 3500 rpm for 30 min. The supernatant was again poured off leaving a cake with φ ≈ 0.45. This cake was used as the gel sample for shear modulus determination. Exact volume fractions were determined by measuring weight fraction using a simple thermal gravimetric technique, and then calculating φ from the known particle and solvent densities. 2.3. Fractal dimension measurement Salt solutions of various concentrations were prepared and filtered through 0.22 m syringe filters (Whatman, Trenton, NJ) into SLS sample cells. The glass cells had been soaked in concentrated sulfuric acid and rinsed with filtered water. Drops of concentrated 60 nm diameter PMMA spheres were added to the salt solutions and subsequently mixed by gently shaking the container by hand. These systems were then allowed to sit overnight. Scattering intensity, I, at various scattering angles, θ, was measured for each sample using a BI-200SM goniometer and BI-DS photomultiplier (Brookhaven Instrument Corp., Holtsville, NY) together with a 633 nm, 15 mW HeNe laser (Melles Girot, Irvine, CA). Prior to measurement, the index matching fluid, decalin (J.T. Baker, Inc.), was filtered by pumping it through a 0.2 m filter in order to remove dust.
P.B. Laxton, J.C. Berg / Colloids and Surfaces A: Physicochem. Eng. Aspects 301 (2007) 137–140
Software automatically collected I versus θ data and generated plots of log I versus log q. Where q, the scattering vector, is a function of system properties and θ. The negative of the slope of log I versus log q plots were then taken to be df [13]. Each measurement was repeated five times. The fractal dimension measured in this way is an estimate of the true fractal dimension of the rheology samples. If some deviation does exist between the estimate and true fractal dimensions, it is likely that the actual gels, as prepared by the method described in Section 2.2, would tend to be more consolidated than the measured aggregates. This would result in a reduced value for the calculated G, where an increase of 0.1 in df would result in about a 35% reduction in G. 2.4. Shear modulus measurement Gel shear moduli were measured at a Peltier-system controlled temperature of 25.0 ± 0.1 ◦ C with a Modular Compact Rheometer (MCR) 300 (Anton Paar, Ashland, VA). Gel samples were held between 50-mm diameter sand blasted plate and a parallel smooth plate. The surface roughness of the sand blasted plate ensures that slip due to depletion at the tool-sample interface does not affect the measurement. At high frequencies, ω, the storage modulus, G , of gels tends to asymptote to some value, and Glim ω→∞ = G [20]. Therefore, in order to most closely approximate the shear modulus, the highest available frequency (100 Hz) allowed by the instrument was used to measure storage modulus. This is a good approximation as long as the characteristic relaxation time of the gel is much longer than the period of the applied strain; which, at 100 Hz, is generally the case for weak gels [20]. Each run consisted of an increase in oscillation amplitude from 0.01 to 100% at a constant frequency of 100 Hz. The resulting torque on the sand blasted plate was recorded, and from this value the storage modulus was calculated by the instrument automatically. The storage modulus at the lowest measurable amplitude was taken to be the best approximation of the shear modulus. Low amplitude values were used in order to ensure that the linear viscoelastic region was probed.
139
Fig. 1. Log I vs. log q plot from SLS of 60 nm PMMA spheres aggregated in 50 mM CaCl2 solution. The line is a least squares fit to the data with a slope of −2.08, giving df = 2.08. Table 1 Summary of aqueous salt media used, together with values of the bond rigidity, κo , and fractal dimension, df , used for calculation of the gel modulus, G Salt concentration (mM)
κ (N/m)
df
CaCl2
10 50 100 200 400
0.02 0.04 0.28 0.70 0.21
2.07 2.08 1.80 2.08 2.00
MgCl2
50 250 375 500
0.06 0.20 0.31 0.61
2.03 2.00 1.95 2.05
NaCl
100 500
0.02 0.04
2.00 1.91
3. Results and discussion Gels prepared as described in Section 2.2 resulted in φ ranging from 0.37–0.47. Measured values of df used in calculating G were between 1.8 and 2.1 ± 0.1, where the uncertainty represents one standard deviation calculated from the five individual measurements. Fig. 1 shows a typical example of the determination of the fractal dimension, and Table 1 lists df values used in calculating G. Figs. 2–4 show gel modulus as calculated according to Eq. (4) using Pantina and Furst values of κo , listed in Table 1, compared with gel modulus measured by conventional oscillatory rheometry, as described in Section 2.4. Error bars on the calculated points are due to the reported uncertainties in κo as well as measured uncertainties in df . In some cases the error bars are large; therefore it does not make sense to attempt to verify the exact values of κo . Despite this limitation it is valuable to compare the data within the given level of certainty. Although some
Fig. 2. Log–log plot of calculated and measured G vs. CaCl2 concentration.
Fig. 3. Log–log plot of calculated and measured G vs. MgCl2 concentration.
140
P.B. Laxton, J.C. Berg / Colloids and Surfaces A: Physicochem. Eng. Aspects 301 (2007) 137–140
References
Fig. 4. Log–log plot of calculated and measured G vs. NaCl concentration.
deviation between the calculated and measured values exists, Eq. (4), with the reported values of κo can predict G to within a factor of 2. It is also important to note that both calculated and measured G follow similar trends with salt concentration. Pantina and Furst discuss in particular the effects of divalent ion adsorption, specifically noting that charge reversal occurs above 200 mM CaCl2 . This phenomenon causes the observed maximum of Fig. 2. Data for κo above the concentration required for charge reversal in MgCl2 systems was not available; therefore, a monotonic trend was expected and is observed in Fig. 3. Divalent ion absorption effects make the usefulness of the direct measurement of κo clear. 4. Summary Bulk rheological measurements of colloid gels created from particles and media nearly identical to those used by Pantina and Furst to investigate κo have been used to explore the link between the microscopic κo measurement and macroscopic properties. The data presented attest to the applicability of Pantina and Furst micromechanical determinations of κo to the calculation of bulk modulus, G. Direct measurement of κo thus allows bulk colloidal gel rheology to be predicted within a reasonable degree of certainty, even in cases where DLVO theory breaks down. Acknowledgment This work was supported by the U.S. Department of Energy (Contract DE-FG02-04ER-63796).
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