Langmuir 2008, 24, 1141-1146
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Micromechanics and Contact Forces of Colloidal Aggregates in the Presence of Surfactants† John P. Pantina‡ and Eric M. Furst* UniVersity of Delaware, Department of Chemical Engineering, Newark, Delaware 19716 ReceiVed August 1, 2007. In Final Form: September 19, 2007 We report measurements of the bending mechanics of colloidal aggregates consisting of poly(methyl methacrylate) (PMMA) flocculated with 250 mM MgCl2 in the presence of either pentaethylene glycol monododecyl ether (C12E5), a nonionic surfactant, or sodium dodecyl sulfate (SDS), an anionic surfactant. In the absence of surfactant, singly bonded aggregates exhibit a substantial bond rigidity, κ0, in the linear bending regime. With the addition of surfactant, the tangential restoring force between particles becomes weaker; aggregates exhibit nonlinear mechanics at a lower critical bending moment, Mc, and the bond rigidity decreases. The decrease in κ0 is related to the reduction of the surface energy of adhesion between particles, WSL. We find that WSL decreases with increasing surfactant concentration below the critical micelle concentration (cmc). However, above the cmc, WSL remains constant within experimental error. These results confirm the relation between the bond rigidity and the surface energy of adhesion and clearly demonstrate that, on the basis of this relationship, surface-active agents provide a means of tuning the macroscopic elasticity and yield stress of colloidal gels. Last, the mechanics of the critical moment is consistent with the surfactant lowering the stress at which the contact line between the particles de-pins.
Introduction
Part of the Molecular and Surface Forces special issue. * Corresponding author. E-mail:
[email protected]. Tel: (302) 831-0102. Fax: (302) 831-1048. ‡ Current address: National Starch and Chemical Company, 10 Finderne Avenue, Bridgewater, New Jersey 08807.
presence of MgCl2. Because surface-active agents should decrease the adhesion energy, this enables us to test further the hypothesis that the energetics of adhesion plays the primary role in determining the interactions in gels composed of strongly flocculated latex particles. Furthermore, understanding the influence that additives such as surfactants have on the nearcontact interactions between particles and the micromechanics of the gels that they form enables a rational approach to engineering the rheology of gels for industrial applications. Surfactants are commonly used in industry as dispersing agents to enhance the stability of colloidal suspensions, where they can provide steric stability and additional charge stabilization or even mitigate the effects heterogeneous particle interactions.9-11 The adsorption of surfactants onto particle surfaces has therefore been a topic of intense study, especially with respect to their effect on suspension rheology. Sjo¨berg and co-workers, for example, measured the shear viscosity of concentrated solutions of kaolin particles in the presence of nonphenol poly(ethylene oxide) (NP(EO)10) and sodium dodecylbenzene solphonate (SDBS).9 It was found that anionic surfactant SDBS reduced the shear viscosity by more than 2 orders of magnitude. However, the change in viscosity due to nonionic surfactant NP(EO)10 was negligible. This was attributed to the increased magnitude of the particle zeta potential due to the adsorption of SDBS, whereas the zeta potential was relatively unchanged by the addition of NP(EO)10. Similarly, Panya and co-workers studied the effect of ionic surfactant on the shear viscosity of ceramic glaze suspensions composed of limestone, quartz, feldspar, and kaolin.12 Although some of the particles had a positive zeta potential, the net charge of the mixture was negative. At low concentrations of cationic surfactant cetylpyridinium chloride (CpCl), the viscosity increased to a maximum before dropping below the
(1) Potanin, A. A.; DeRooij, R.; van den Ende, D.; Mellema, D. J. Chem. Phys. 1995, 102, 5845. (2) Potanin, A. A.; Russel, W. B. Phys. ReV. E 1996, 53, 3702. (3) Krall, A. H.; Weitz, D. A. Phys. ReV. Lett. 1998, 80, 778. (4) Pantina, J. P.; Furst, E. M. Langmuir 2004, 20, 3940. (5) Pantina, J. P.; Furst, E. M. Phys. ReV. Lett. 2005, 94, 138301. (6) Pantina, J. P.; Furst, E. M. Langmuir 2006, 22, 5282. (7) Laxton, P. B.; Berg, J. C. Colloids Surf., A 2007, 30, 37. (8) Furst, E. M.; Pantina, J. P. Phys. ReV. E 2007, 75, 050402(R).
(9) Sjo¨berg, M.; Bergsto¨m, L.; Larsson, A.; Sjo¨stro¨m,. E. Colloids Surf., A 1999, 159, 197. (10) Feick, J. D.; Chukwumah, N.; Noel, A. E.; Velegol, D. Langmuir 2004, 20, 3090. (11) Thwar, P. K.; Velegol, D. Langmuir 2002, 18, 7328. (12) Panya, P.; Wanless, E. J.; Arquero, O.; Franks, G. V. J. Am. Ceram. Soc. 2005, 88, 540. (13) Corrin, M. L.; Harkins, W. D. J. Am. Chem. Soc. 1947, 69, 684.
The rheological properties of colloidal gels arise from the mechanical response of the microstructure to macroscopically imposed stresses and strains. These micromechanics are, in turn, governed by the nanometer-scale interactions between particles.1-3 Because colloidal gels are found in numerous technological applications and processes, understanding the relationships between colloidal forces, microstructural mechanics, and rheology is of critical importance. For this reason, laser tweezers have been used to measure the mechanical properties of particle aggregates that mimic the stress-transmitting microstructure of colloidal gels.4-6 Such experiments connect the elasticity of colloidal “bonds” to the interfacial properties of the particles, such as the work of adhesion. On the basis of this premise, it has been shown that the micromechanical rigidity of aggregates can be tailored by taking advantage of the adsorption of divalent ions, which contribute to the adhesion energy between particles.5 Subsequent bulk rheological measurements have demonstrated that micromechanical experiments provide a quantitative understanding of the elastic modulus of latex particle gels, thus effectively bridging the molecular length scales of the particle interfaces and contact interactions to macroscopic mechanical properties.7 Similarly, a relationship between the yield stress of gels and the nonlinear micromechanics of aggregates has recently been shown.8 Here, we examine the effect of anionic and nonionic surfactants on the bending mechanics of similar aggregated structures in the †
10.1021/la7023617 CCC: $40.75 © 2008 American Chemical Society Published on Web 12/04/2007
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initial value at higher concentrations of CpCl. The concentration at which the viscosity reached a maximum was found to coincide with point at which the net potential of the mixture was zero. We begin by providing an overview of the experimental details, including a brief description of the laser tweezer apparatus used to perform these micromechanical measurements. Next, results are presented describing how the addition of surfactants to the solution alters the mechanical response of colloidal aggregates to an applied bending moment. Measurements of the critical bending moment and single-bond rigidity of chain aggregates are made at various surfactant concentrations. We demonstrate that the observed influence of surfactants on the single-bond rigidity may be understood in terms of our previously developed model that accounts for the surface adhesion energy between particles.5,6 Finally, we discuss the mechanics of the contact region between aggregated particles and its relationship to the nonlinear mechanics of aggregates. Experimental Section Laser Twezeers. The laser tweezer apparatus used in this work is constructed using an inverted microscope (Zeiss Axiovert 200) to enable simultaneous trapping and imaging via video microscopy. A 4-W CW Nd:YAG laser (λ ) 1064 nm, Coherent Compass 1064400M) is used to generate the traps. The trap positions within the microscope focal plane are manipulated by controlling the angle of the beam at the back aperture of the objective using a pair of perpendicular acousto-optic deflectors (AODs, AA Opto-electronics AA.DTS.XY-400). The AOD provides a spatial resolution of 0.2 nm and a rate of trap repositioning on the order of several kilohertz. The beam is then focused onto the sample using a 63× NA 1.2 water immersion microscope objective (Zeiss C-Apochromat). A high numerical objective is necessary to maximize the gradient force generated by the laser, whereas the use of a water immersion objective allows for particle trapping throughout the entire sample, which is approximately 200-300 µm in depth. Our objective allows us to trap over a range of 100 × 100 µm2 in the image plane using the AOD. A more detailed description of the experimental apparatus can be found in ref 4. Materials and Sample Preparation. Experiments are performed on monodisperse spherical latex particles composed of poly(methyl methacrylate) (PMMA). The PMMA particles (Bangs Laboratories, Inc.) have an average diameter of 2a ) 1.47 ( 0.1 µm and are obtained in a stock solution of solids that is 10% w/w solids in water. The particles are triply washed in Milli-Q water before being diluted to a solids volume fraction of φ ≈ 10-4. The double-layer repulsion is screened through the addition of magnesium chloride (MgCl2, Sigma-Aldrich) to a final concentration of 250 mM. In this work, we study the interactions of particles in the presences of both nonionic and anionic surfactants. The anionic surfactant used was sodium dodecyl sulfate (SDS, EM Science). Bulk solutions of 400 mM SDS were prepared and diluted to the desired final concentration. The critical micelle concentration (cmc) of SDS in the absence of added electrolyte is 8 mM. However, the addition of salt to an ionic surfactant solution is known to depress the cmc.12,14 Corrin and Harkins found that the cmc is affected only by the concentration of the ion opposite in charge to the micelle.13 They provide an empirical relationship for calculating the cmc of SDS in the presence of salt, log c* ) -0.46 log m+ - 3.25
(1)
where c* is the cmc in the presence of an added salt and m+ is the concentration of counterions in solution. Both terms have units of mol/L. Thus, for an SDS in solution with 250 mM MgCl2, the cmc is c* ≈ 1 mM. (14) Rosen, M. J. Surfactants and Interfacial Phenomena, 3rd ed.; Wiley: Hoboken, NJ, 2004.
Figure 1. (A) Schematic drawing of the three-point bending experiment. Stationary traps are positioned on the two end particles, which serve as forces sensors. A bending moment is applied to the aggregate using a third trap positioned on the center particle, translating in the y direction at a rate of 20 nm/s. (B) Micrograph of an aggregate deforming. The scale bar is 5 µm. (C) Diagram of the experimental apparatus. Multiple optical traps are generated by time sharing a single beam using an acousto-optic deflector (AOD). The nonionic surfactant is pentaethylene glycol monododecyl ether (C12E5, Sigma-Aldrich). The cmc of C12E5 in the absence of an added electrolyte is 85 µM. Unlike ionic surfactants, the cmc’s of nonionic surfactants are not significantly affected by the presence of salts in solution (i.e., c* ≈ cmc).14 Bulk solutions of 2 mM C12E5 were prepared and diluted to the final desired concentration. We note that we do not expect the surface adsorption of surfactants to shift the bulk concentration significantly relative to the cmc for either surfactant because the concentration of particles used in our samples is extremely low. Measurements of Aggregate Elasticity. Samples for each experiment are made using standard glass microscope slides (Fisher, size 25 × 75 × 1 mm3). An adhesive spacer is used to form a gap between the glass coverslip (no. 1.5, Corning Labware & Equipment) and the microscope slide. A solution consisting of the particles, salt, and surfactant at the final dilution concentration is introduced such that it fills the gap, and the ends are sealed with a fast-curing U.V. epoxy (Norland Products, NOA 81) to prevent drying and convective flow within the sample. Particle aggregates are directly assembled using the discrete trapping method described previously.4 This method is based on the principles of time-shared optical trapping, in which multiple traps are created through the use of high-frequency, high-precision steering of a single laser beam.15-17 After an array of time-shared traps is created, all of the traps are filled with particles. Because of the low particle volume fractions used in our samples and the relatively large size of the particles, the time required to fill as few as three traps via particle diffusion is untenable. Therefore, we actively seek particles to fill the array of traps by translating the microscope stage to scan through the sample. After all traps in the array are filled, the gap between the particles is decreased until van der Waals interactions induce aggregation. The advantage of this method over the other techniques is the direct control over each particle position.4 This assures that tensile or compressive forces are not being applied to the chain during micromechanical experiments, such as chain bending. Using time-shared optical traps, a bending moment is applied to the chain using a three-point trap configuration, shown in Figure 1. The two end particles are held in stationary traps as a third trap, positioned on the center particle, translates perpendicular to the aggregate at a velocity slow enough to create negligible hydrodynamic (15) Misawa, H.; Sasaki, K.; Koshioka, M.; Kitamura, N.; Masuhara, H. Appl. Phys. Lett. 1992, 60, 310. (16) Visscher, K.; Brakenhoff, G. J.; Krol, J. J. Cytometry 1993, 14, 105. (17) Mio, C.; Marr, D. W. M. Langmuir 1999, 15, 8565.
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Figure 3. Critical bending moment of PMMA chains in 250 mM MgCl2 at various surfactant concentrations. The arrows indicate 1 standard deviation of the critical bending moment in the absence of surfactant. The critical micelle concentration for SDS is c* ) 1 mM. For C12E5, c* ) 85 µM.
Figure 2. (A) Particle positions for an 11-particle chain during a bending experiment in the presence of 250 mM MgCl2 and 0.1 mM SDS. Open circles show the particle positions for M < Mc. Closed circles show the particle positions immediately prior to the center trap release. (B) Particle positions for a 9-particle chain during a bending experiment in the presence of 250 mM MgCl2 and 100 mM SDS. drag (20 nm/s). The force applied to the chain, Fbend, is calculated by measuring the displacement of the end particles from their equilibrium positions within the stationary traps. Force versus displacement calibrations for the optical traps are made prior to each experiment by translating the microscope stage at various velocities as individual particles are held in the trap. The force exerted on the particle due to the translating fluid is calculated using Stokes’ law. Particle positions are measured with an accuracy of (33 nm using a weighted centroid method.18
Results Figure 2a,b shows the shape progression of a typical aggregate in response to a bending moment when 250 mM MgCl2 and 0.1 mM SDS are present in solution. The particle positions in Figure 2a demonstrate a time point in the experiment prior to reaching the critical bending moment, Mc.5 Also shown in Figure 2a are the positions of the particles within the chain at a time after Mc had been exceeded and prior to the release of the center trap from the chain. (The maximum trapping force in this experiment was 10.5 pN.) In both instances, the particle positions are in good agreement with the shape expected from bending a thin rigid rod,
y(x) )
(
)
-Fbend Lx2 |x|3 EI 4 6
(2)
where E is Young’s modulus, I is the area moment of inertia, and L is the length of the aggregate. This is similar to the behavior observed for chains aggregated with only inorganic salts.5,6 The curvature of these chains demonstrates that the tangential restoring (18) Crocker, J. C.; Grier, D. G. J. Colloid Interface Sci. 1996, 179, 298.
force between the particles is significant, and the particles do not undergo rotation. In contrast, Figure 2b shows the shape progression of a typical aggregate in response to a bending moment in 250 mM MgCl2 and 100 mM SDS. The chain shape is measured prior to Mc being reached and immediately prior to the center particle’s release from the translating trap. The maximum trapping force in this experiment was 3.6 pN. We observe that for M < Mc the shape is in good agreement with eq 2. However, when the bending moment exceeds the critical moment, M > Mc, the particles begin to rotate or slide around each other, and the aggregate shape is more triangular than curved, with a pivot point at the center. This is similar to the stress chain deformation assumed in microrheological models of colloidal gels for singly connected chains.1,2 However, such models assume that there is no energy barrier to particle rotation, whereas here rotation does not occur until Mc has been exceeded. Nevertheless, we can conclude that the tangential interaction between the PMMA particles has been significantly reduced by the presence of SDS. We use the critical bending moment to quantify the reduction of the tangential interactions between particles due to the presence of surfactant. It is calculated by identifying the position of the particle that undergoes a rearrangement, x*. Then,
Mc ) F/bend
(L2 - |x - L2|) *
(3)
where F/bend is the maximum force measured immediately prior to the rearrangement. Figure 3 shows Mc for chains in 250 mM MgCl2 and either SDS or C12E5 at various surfactant concentrations. Three features of the data are apparent in Figure 3. First, we find that Mc decreases with increasing surfactant concentration for both SDS and C12E5. This is consistent with our earlier observation of the effect of surfactant on the shape of the chains. Second, Mc is within experimental error of the value in the absence of surfactant for c/c* < 0.1 for both SDS and C12E5. Finally, at a similar normalized surfactant concentration in the range of 0.1 e c/c* e 10, the measured values of Mc for both SDS and C12E5 are within the experimental error. Therefore, the charge on the SDS does not affect the critical bending moment. Below the critical bending moment, the aggregates exhibit an elastic response. The bending elasticity of a chain is defined as κ ) Fbend/δ, where Fbend is the applied force and δ is the transverse deflection of the center particle relative to the two end particles.
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ac )
(
)
3πa2WSL 2K
1/3
(6)
where WSL is the adhesion energy per unit area and the particle elastic modulus is K ) 2E/3(1 - ν2). Combining eqs 5 and 6 yields
κ0 )
Figure 4. Single-bond rigidty for PMMA aggregates in 250 mM MgCl2 at various surfactant concentrations. The critical micelle concentration for SDS is c* ) 1 mM. For C12E5, c* ) 85 µM. The inset shows the data on the scale of the bending rigidity in the absence of surfactant, indicated by the arrow and box on the ordinate.
Previously, it was demonstrated that the length-dependent spring model,19
κ(s) ) κ0
(as)
2+db
(4)
can be used to calculate the single-bond rigidity, κ0.5 In eq 4, s is the contour length of the chain, and db is its fractal dimension. The chain aggregates in the experiments reported here are assembled to be nearly perfectly straight; therefore, db ) 1. Figure 4 shows a semilog plot of κ0 as a function of surfactant concentration in 250 mM MgCl2. The bending rigidity is very sensitive to the presence of surfactant in solution. At all surfactant concentrations examined, we find that κ0 is significantly lower than the value measured ) 0.21 ( 0.01 N/m, as illustrated in for the bare particles, κbare 0 the inset to Figure 4. Furthermore, κ0 decreases as much as an order of magnitude with increasing surfactant concentration for both SDS and C12E5. However, the most striking feature of Figure 4 is that the slope of the data changes at c ≈ c* for both the SDS and C12E5 solutions, although the effect is more pronounced for SDS. For c/c* > 1, the measured values of κ0 in both SDS and C12E5 solution remain within experimental error of each other. In the following section, we examine the dependence of κ0 on surfactant concentration using a previously developed model that relates the bending mechanics of aggregates to the surface adhesion energy between particles.
Discussion Bending Mechanics and the Energetics of Adhesion. It has been shown earlier that by equating eqs 2 and 4 and using the area moment of inertia of a cylinder of radius ac,5,6
κ0 )
3πac4E 4a3
(5)
For equally sized adhesive, elastic spheres in the absence of an applied mechanical load, the Johnson-Kendall-Roberts (JKR) theory of adhesion predicts a contact radius of20 (19) Kantor, Y.; Webman, I. Phys. ReV. Lett. 1984, 52, 1891. (20) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London, Sect. A 1971, 324, 301. (21) Israelchvili, J. N. Intermolecular and Surface Forces, 2nd ed.; Academic Press: London, 1992.
(
)
2 3πE 3πa WSL 2K 4a3
4/3
(7)
providing a direct relationship between the energy of adhesion between colloidal particles and the rigidity of the bonds that they form. Figure 6 shows the energy of adhesion calculated from the single-bond elasticity with eq 7 for aggregates at different surfactant concentrations. The adhesion energy is written in terms of the surface energies of the liquid, solid, and liquid-solid interface,
WSL ) γS + γL - γSL
(8)
At vanishing divalent ion concentrations, WSL can then be calculated using the Young-Dupre´ equation
W0SL ) γL(1 + cos θ)
(9)
where γL is the liquid surface tension and θ is the PMMAliquid contact angle.20 Studies by Bargeman and van Voorst Vader on the effect of surfactants on the wettability of hydrophobic solids have shown that there is a linear relationship between the adhesion tension and the liquid surface tension22
γL cos θ ) aγL + b
(10)
In addition, Szymczyk and co-workers recently suggested that empirical parameters a and b are material properties and are independent of surfactant type.23 Using aqueous solutions of cetyltrimethylammonium bromide (CTAB), cetylpyridinium bromide (CPyBr), and mixtures of the two, they found that for PMMA
γL cos θ ) -0.34γL + 44.39
(11)
where γL is given in units of mN/m. Combining eqs 9 and 11 yields for PMMA
W0SL ) 0.66γL + 44.39
(12)
The significance of eq 12 is that the energy of adhesion between particles should be proportional to the liquid surface tension. Thus, a decrease in γL, which is expected with the addition of a surfactant, will reduce the particle adhesion and bond rigidity. The liquid surface tension decreases as the surface excess concentration of surfactant ΓL increases, typically following the classical relationship
ΓL ) -
( )
1 dγL RT d ln c
T
(13)
At c ) c*, the monomeric surfactant molecules begin to form micelles. Because only the monomeric species contributes to the surface excess concentration, a discontinuity in the semilog plot (22) Bargeman, D.; van Voorst Vader, F. J. Colloid Interface Sci. 1973, 42, 467. (23) Szymczyk, K.; Zdziennicka, A.; Jan´czuk, B.; Wo´jcik, W. J. Colloid Interface Sci. 2006, 293, 172.
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region (which gives rise to the linear bending mechanics) versus the mechanics of the solid-solid contact line. Mechanics of the Contact Region. The linear and nonlinear bending mechanics of colloidal aggregates provide insight into the properties of the contact region between particles. First, we note that the deflection of the aggregate (eq 2) is a solution to the elastica, the differential equation describing the bending of a prismatic beam. For small deflections, the elastica can be written as
[
]
d2 d2y EI(x) ) -w dx2 dx2 Figure 5. Energy of adhesion between PMMA particles in 250 mM MgCl2 and surfactant calculated using eq 7. The solid and dashed lines indicate the discontinuity in the surface energy corresponding to the cmc’s for SDS and C12E5, respectively.
Figure 6. Critical bending moment as a function of the contact area radius between particles using eq 5. The solid line indicates a slope of 3, which is consistent with rolling caused by the depinning of the particle-particle contact line.
of γL verses c appears at c ) c*, and γL remains essentially constant at higher surfactant concentrations.14 Indeed, this serves as the basis of surface tension measurements of the critical micelle concentration. It is notable, then, that in Figure 6 the energy of adhesion between particles calculated from the bending mechanics demonstrates a similar dependence on surfactant concentration with respect to the behavior expected from γL. This suggests that that WSL and γL are indeed proportional. We also find that the discontinuity of the slope at c ) c* is less pronounced for C12E5 than for SDS. This effect has been observed in tensiometry measurements and is attributed to impurities in the surfactant.14 In our experiments, C12E5 is used as received from the manufacturer without further purification. Therefore, it is reasonable to believe that impurities may be present, which explains the lack of a sharp transition to a nearly constant value at c ) c*. Finally, we consider the differences between Mc and κ0. Although the critical bending moment decreases with increased surfactant concentration, similar to the bending rigidity, qualitatively it has a different dependence; Mc does not exhibit the same initial rapid decrease as κ0 with the addition of surfactant. Moreover, Mc decreases more gradually as the surfactant concentration increases. This suggests that the interfacial origin of the critical bending moment differs from that of the singlebond elasticity. As we discuss below, this difference may originate in the distinct roles of the overall adhesion across the contact
(14)
where I(x) is the area moment of inertia, which in this case varies along the length of the particle aggregate, and w is the load distribution. In the aggregate bending experiments, the load is a constant force F applied to the ends. Because the shear force V(x) is related to the load as dV/dx ) -w and the bending moment is related to the shear force as dM/dx ) V, integrating eq 14 yields the result that the shear force in the aggregate is constant, V ) -F, and the bending moment is distributed linearly over the aggregate, M(x) ) F(L/2 - x), even with a nonuniform crosssectional area along its length. Two mechanisms could underlie the critical moment that defines the onset of nonlinear mechanics of an aggregate. First, as suggested in previous work, the critical moment could arise from sliding between particles when the shear force F exceeds the static friction between the polymer particle surfaces.8 In this case, the critical force is F* ) µsL0, where µs is the static coefficient of friction and L0 is the adhesive load between particles. Although static friction is independent of the contact area between bodies, from eq 6, the adhesive load in the case of elastic spheres, L0 ) πaWSL/2, scales with the contact radius as L0 ≈ a3c . Thus, the critical force and critical moment should scale as F* ≈ Mc ≈ a3c . The second mechanism by which colloidal aggregates could fail is due to de-pinning of the solid-solid contact line and subsequent rolling of the adhesive particles. By assuming the stress distribution to be nearly linear across the majority of the contact region, the maximum tensile (normal) stress in the contact region during bending will be
σxx )
Mac I
(15)
at the contact line. If σ/xx represents the critical stress at which the contact line between the particles de-pins, then rearranging and substituting for the area moment of inertia for a circular cross section yields
Mc )
π / 3 σ a 4 xx c
(16)
Using eq 5, we calculate the radius of the contact region from the bond rigidity measured in the linear bending regime. Figure 6 shows that the critical bending moment for both SDS and C12E5 scales as Mc ≈ a3c for all measurements except at the highest concentration of SDS, consistent with both sliding and rolling models of the critical moment. However, a distinguishing trait of the mechanical experiments does support the rolling mechanism over sliding. Because the shear force is expected to be constant over the entire aggregate, all “bonds” are expected to be equally likely to fail in the sliding case. However, the bonds that do fail during experiments are not distributed evenly but tend to occur toward the center of the aggregate, where the bending moment is the greatest.5 Thus, the micromechanical
1146 Langmuir, Vol. 24, No. 4, 2008
experiments support a rolling mechanism caused by the de-pinning of the solid-solid contact line. Interestingly, the nonlinear mechanics of aggregates in the presence of surfactant contrast with those of aggregates in which the surface adhesion energy is controlled by the concentration of divalent ions, which enhances the adhesion energy.8 Under such conditions, it was found that the critical moment scales as Mc ≈ a2c . The origin of this weaker dependence on contact radius is unexplained at this time. Nonetheless, it appears that the presence of surfactant not only decreases the overall adhesion energy, resulting in weaker aggregate mechanics, but may also play an important role in the mechanics of the contact line between the adhesive particles.
Conclusions We have performed experiments on bending mechanics of colloidal aggregates flocculated using a combination of an inorganic salt and either a nonionic or anionic surfactant. At low surfactant concentrations, we find that single-bonded aggregates are capable of supporting torques, reflecting a substantial tangential interaction between the adjacent particles. As the surfactant concentration is increased, however, the tangential restoring force becomes weaker, and the particles begin to roll or slide around each other. To assess this weakening of the tangential interaction quantitatively, we identify and measure both the critical bending moment Mc and single-bond rigidity κ0. Surfactants were found to reduce both κ0 and Mc, although at the lowest surfactant concentrations Mc was found to be similar
Pantina and Furst
to that of the bare particles and κ0 was significantly smaller in the presence of even trace amounts of surfactant. We understand these results in terms of the bond elasticity model presented in our earlier work, which relates κ0 to the surface adhesion energy between particles WSL. We find that WSL decreases for c < c*. However, a discontinuity in the slope is observed in the semilog plot of WSL verses the surfactant concentration at c ≈ c*, above which WSL remains constant within experimental error. This is identical to the behavior of the liquid surface tension around the critical micelle concentration. We therefore conclude that the surfactants decrease κ0 by reducing the adhesion between the elastic particles. The mechanics of the critical moment are consistent with the surfactant lowering the stress at which the contact line between the particles de-pins. These results confirm earlier work relating the bond rigidity to the surface energy of adhesion and clearly demonstrate that, on the basis of this relationship, surface-active agents provide a means of tuning the strength of colloidal gels composed of strongly flocculated latex particles. Finally, we note that the microscopic mechanism responsible for the apparent contact line pinning between particles that gives rise to the bending rigidity and critical moment remains to be understood. Acknowledgment. We gratefully acknowledge helpful discussions with N. Wagner. This work was supported by the National Science Foundation (CBET-0238689 and CBET0500321). LA7023617
