Colloidal Aggregate Micromechanics in the Presence of Divalent Ions

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Colloidal Aggregate Micromechanics in the Presence of Divalent Ions John P. Pantina and Eric M. Furst* UniVersity of Delaware, Department of Chemical Engineering, Newark, Delaware 19716 ReceiVed December 16, 2005. In Final Form: April 7, 2006 Colloidal gels exhibit rheological properties, such as a yield stress and elasticity, which arise from the manner in which stress is transmitted through the microstructure. Insight into the mechanisms of stress transmission is critical in developing a full understanding of the mechanics of these materials. Paramount to this is a thorough knowledge of the interparticle interactions. In this work, we use optical trapping to study interactions between poly(methyl methacrylate) (PMMA) particles in adhesive contact by measuring the bending elasticity of directly assembled colloidal aggregates under various physicochemical conditions. The simplified geometry of the aggregate enables us determine the single-bond rigidity, which can then be related to the work of adhesion, WSL, through the Johnson-KendallRoberts (JKR) theory of adhesion. We find that WSL is independent of ionic strength in flocculating monovalent salt solutions. However, more complex behavior is observed for divalent salts. Using zeta-potential measurements, we show that divalent cations adsorb to the particle surface. This results in the formation of ionic bridges between particles in adhesive contact. A model of the aggregate micromechanics that considers the divalent ion contribution to the surface energy provides a direct link between the interfacial properties of the particles, nanoscale contact interactions between particles, and the bulk gel modulus.

I. Introduction Colloidal dispersions flocculate in the presence of sufficiently strong attractive forces to form fractal aggregates.1-4 The radius of gyration, ξ, of an aggregate of k particles with radii a is given by ξ ) ak1/df, where df is the fractal dimension, which typically ranges from 1.8 for diffusion-limited cluster aggregation (DLCA) to 2.1 for reaction-limited aggregation (RLA).5,6 The suspension is said to have gelled when flocs grow to an average radius, ξc, at which point they pack to form a sample-spanning microstructure. Ultimately, the rheological properties of a colloidal gel depend on the response of the gel backbone, the portion of the microstructure that is active in stress transmission. The micromechanics of the backbone are, in turn, governed by the nanoscale interactions between particles.7-9 The dependence of gel rheology on the microstructural mechanics and interparticle interactions is highlighted by a model in which the gel microstructure can be treated as a length-dependent spring over the lengthscale, s9

κ(s) ) κ0(a/s)2+db

(1)

where κ0 is the “bond” rigidity due to the adhesion between colloidal particles and db is the bond dimension, which describes the fractal geometry of the backbone structure.10 The gel elastic * To whom correpondence should be addressed. E-mail: [email protected]. Phone: (302) 831-0102. Fax: (302) 831-1048. (1) Larson, R. G. The Structure and Rheology of Complex Fluids; Oxford University Press: New York, 1999. (2) Schaefer, D. W.; Martin, J. E.; Wiltzius, P.; Cannell, D. S. Phys. ReV. Lett. 1984, 52, 2371. (3) Dimon, P.; Sinha, S. K.; Weitz, D. A.; Safinya, C. R.; Smith, G. S.; Varady, W. A.; Lindsay, H. M. Phys. ReV. Lett. 1986, 57, 595. (4) Aubert, C.; Cannell, D. S. Phys. ReV. Lett. 1986, 56, 738. (5) Jullien, R. L.; Kolb, M.; Boter, R. J. Phys. Lett. 1984, 45, L211. (6) Russel, W. B.; Saville, D. A.; Schowalter, W. R. Colloidal Dispersions; Cambridge University Press: New York, 1989. (7) Potanin, A. A.; Russel, W. B. Phys. ReV. E. 1996, 53, 3702. (8) Potanin, A. A.; DeRooij, R.; Van de Ende, D.; Mellema, J. J. Chem. Phys. 1995, 102, 5845. (9) Krall, A. H.; Weitz, D. A. Phys. ReV. Lett. 1998, 80, 778. (10) Kantor, Y.; Webman, I. Phys. ReV. Lett. 1984, 52, 1891.

modulus is dominated by the bending rigidity of the largest cluster sizes,

G)

κ(ξc) κ0 (3+db)/(3-df) ) φ ξc a

(2)

where ξc ≈ aφ1/df-3. Equation 2 accurately captures the volume fraction dependence of the gel modulus.9 However, until recently, little was known about the bond rigidity and its dependence on physicochemical conditions, such as the ionic species, ionic strength, and surface chemistry of the particles. Because colloidal gels are found in numerous technological applications and processes, understanding the relationship between interparticle interactions, backbone micromechanics, and rheology is of critical importance. Using laser tweezers, we measure the micromechanics of assembled chain aggregates consisting of polymer colloids under conditions of high ionic strength, as shown in Figure 1. These chain aggregates mimic the backbone of colloidal gels; therefore, the experiments enable us to independently investigate the mechanical response of the gel microstructure. More importantly, the simplified geometry provides critical insight into the interparticle interactions between Brownian particles at nanometer separations. The interaction between colloidal particles is traditionally understood in terms of the Derjaguin-Landau-VerweyOverbeek theory (DLVO).11,12 DLVO describes interparticle interactions as a sum of contributions from van der Waals attraction and the double-layer repulsion. However, DLVO is known to give erroneous results at near-contact separations between colloidal particles.13-16 Mechanisms such as the influence of surface asperities17,18 and solvation forces19 have been (11) Verwey, E. G. W.; Overbeek, J. J. G. Theory of the Stability of Lyophobic Colloids; Elsevier: Amsterdam, 1948. (12) Derjaguin, B. V.; Landau, L. Acta Physiochem. 1941, 14, 633. (13) Behrens, S. H.; Borkovec, M.; Schurtenberger, P. Langmuir 1998, 14, 1951. (14) Behrens, S. H.; Christl, D. T.; Emmerzael, R.; Schurtenberger, P.; Borkovec, M. Langmuir 2000, 16, 2566. (15) Velegol, D.; Thwar, P. K. Langmuir 2001, 17, 7687. (16) Pantina, J. P.; Furst, E. M. Langmuir 2004, 20, 3940. (17) Suresh, L.; Walz, J. Y. J. Colloid Interface Sci. 1996, 183, 199.

10.1021/la0534120 CCC: $33.50 © 2006 American Chemical Society Published on Web 05/13/2006

Colloidal Aggregate Micromechanics

Figure 1. Bending a 21-particle chain of 1.47 µm PMMA particles in 100 mM CaCl2 using three time-shared optical traps. The end particles of the aggregate are held in stationary traps, while the center trap is translated perpendicular to the aggregate at a rate of 20 nm/s. The scale bar is 10 µm.

previously described. Recently, we demonstrated that the Johnson-Kendall-Roberts (JKR) theory of adhesion can be used to understand the elasticity of colloidal bonds found in strong gels.20 JKR theory therefore provides a direct link between the surface adhesion energy between particles and the bending mechanics of colloidal aggregates. Here, we extend this model to account for the contribution of ion bridging to the energy of adhesion between particles in solutions of divalent salts. We begin by briefly describing the laser tweezer apparatus used to perform these experiments and the experimental methods used to directly assemble and measure the mechanical properties of model aggregates. Next, we discuss measurements of the elastic response of the chains to an applied bending moment. The aggregate elasticity is related to the elasticity of a single colloidal bond, which is then related to the energy of adhesion between the particles.20 Next, using zeta-potential measurements, we demonstrate that divalent ions adsorb to the particle surfaces, resulting in the formation of ionic cross-links between aggregated particles. The Stern equation is used to model the ion adsorption. Finally, we show that ionic cross-links provide a significant contribution to the total energy of adhesion between colloidal particles. We conclude with final remarks and future directions for this work. II. Experimental Section A. Laser Tweezers. The laser tweezer apparatus used in the experiments employs an inverted microscope (Zeiss Axiovert 200). The traps are generated with a 4-W CW Nd:YAG laser (λ ) 1064 nm, Coherent Compass 1064-400M). Typical laser powers, measured at the objective back aperture, are set between 250 and 300 mW, resulting in a maximum trapping force for the time-shared tweezers of ∼15 pN. Note that lower trapping forces are used to increase the force sensitivity of the traps when measuring the bending rigidity of aggregates. A pair of perpendicular acousto-optic deflectors (AODs, AA Optoelectronics AA.DTS.XY-400) is used to manipulate the angle of the beam at the back aperture of the objective, allowing for control of the trap position in the focal plane of the microscope. The AOD provides a spatial resolution of 0.2 nm and a rate of trap repositioning on the order of several kilohertz. The AOD is imaged (18) Suresh, L.; Walz, J. Y. J. Colloid Interface Sci. 1997, 196, 177. (19) Franks, G. V.; Johnson, S. B.; Scales, P. J.; Boger, D. V.; Healy, T. W. Langmuir 1999, 15, 4411. (20) Pantina, J. P.; Furst, E. M. Phys. ReV. Lett. 2005, 94, 138301.

Langmuir, Vol. 22, No. 12, 2006 5283 onto the back aperture of a 63× NA 1.2 water immersion microscope objective (Zeiss C-Apochromat). The high numerical aperture objective is necessary to maximize the gradient force generated by the laser, while overfilling the back aperture ensures the maximum trapping efficiency. The water immersion objective minimizes the effect of spherical aberrations in aqueous samples, while providing a larger working distance (∼200 µm) as compared to oil immersion objectives, which have trapping depths of about 10 µm from the coverslip. This enables us to conduct experiments far from the sample interfaces. A more detailed description of the experimental apparatus can be found in ref 16. B. 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 10% w/w solids in water. The particles are triple washed in MilliQ water before being diluted to a solids volume fraction of φ ≈ 10-4. The exact solids volume fraction is insignificant to our measurements since each aggregate is directly assembled from individual particles. However, we find that φ ≈ 10-4 provides the optimal number of particles in the suspension, such that it is not difficult to locate and trap a number of particles while simultaneously keeping the concentration low enough that stray particles do not interfere with the assembly process or micromechanical measurements. Samples for each experiment are made using standard glass microscope slides (Fisher, size 25 × 75 × 1 mm3). An adhesive spacer creates a gap between the microscope slide and the coverslip (no. 1.5, Corning Labware and Equipment) with a depth of ∼200300 µm. A drop of the colloidal solution is then placed next to the gap between the coverslip and the slide and is pulled into the cell by capillary forces. Once the cell is completely filled with solution, the gaps are sealed using a fast UV-curing epoxy (Norland Products, NOA 81) to prevent drying and convective flow in the sample. C. Aggregate Assembly. Aggregates are assembled using the discrete trapping method described in ref 16. This technique is based on time-shared optical trapping. After an array of time-shared traps is created, all of the traps are filled with particles. Due to 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. As a result, 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 separation between the particles is decreased until van der Waals interactions induce aggregation. The advantages of this method over the other techniques described in ref 16 include having exact knowledge and direct control over each particle position. This allows us to precisely position the traps over the particle centers in threepoint bending experiments, ensuring that a tensile or compressive force is not being applied to the chain. Measurements of the bending elasticity were performed on aggregates that ranged in size from 7 to 25 particles in length. D. Measurements of Aggreage Elasticity. The elasticity of the chain aggregates is measured using a three-point bending geometry, as shown in Figure 1. The two end particles are held in stationary traps, and a bending moment is applied by translating a trap positioned on the center particle perpendicular to the aggregate at a velocity slow enough to create negligible hydrodynamic 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 while the particle is held. 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.21 Finally, the chain elasticity is defined as κ ) Fbend/δ, where δ is the transverse deflection of the center particle relative to the two end points. All (21) Crocker, J. C.; Grier, D. G. J. Colloid Interface Sci. 1996, 179, 298.

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measurements reported here are within the linear elastic regime of colloidal aggregates.20 E. Zeta-Potential Measurements. The electrophoretic mobility of dilute solutions of PMMA particles (φ ≈ 10-4) is measured as a function of salt concentration using a commercial phase analysis light scattering device (Brookhaven Instruments Corp, Zeta-PALS). Three measurements at 25 °C are taken at each salt concentration, with individual measurements consisting of 10 cycles. The zetapotential, ζ, of the particles is calculated using the Smoluchowski equation, which is valid for κa . 1, where κ-1 is the Debye length.6

III. Results and Discussion A. Aggregate Bending Rigidity. At all salt concentrations examined, particles aggregate to form rigid microstructures. The bonds formed between particles resist tensile forces of up to 15 pN, the maximum trapping force of our laser tweezers. Figure 1 shows the shape progression of a typical chain aggregate in response to a bending moment. Earlier, we demonstrated that the particle positions yi(xi) in such aggregates are in good agreement with the shape expected from a thin rigid rod centered at x ) 0 under similar load conditions,

y(x) )

(

)

-Fbend Lx2 |x|3 EI 4 6

(3)

In eq 3, E is the Young’s modulus, I is the area moment of inertia, and L is the length of the aggregate. The curvature of the chains demonstrates the existence of tangential forces between particles that results in single aggregate bonds that are capable of supporting torques. Prior to this work, the presence of tangential forces on the order of 0.1 pN had been inferred from observations of particle deposition22 and differential electrophoresis23 for colloidal particles separated by tens of nanometers, yet many microrheological models of gels assume that interparticle interactions are centrosymmetric.7,8 Therefore, particles connected to a network by a single bond would exhibit free rotation, thus requiring multiply bonded structures to support a stress. If particles did undergo free rotation, we would expect the chain aggregates to respond to the bending moment by forming a triangular structure with a pivot point at the center particle. Previously, we verified that the chain elasticity scales as κ ∼ s-3, as expected for db ≈ 1.20 Using eq 1, we calculate κ0 from the measured elasticity of each aggregate. Figure 2 shows the single-bond rigidity as a function of salt concentration for three inorganic salts, NaCl, MgCl2, and CaCl2. For the monovalent salt NaCl, κ0 exhibits a constant value of ∼0.2 N/m, with a slight increase at the highest salt concentration (500 mM). The divalent salts MgCl2 and CaCl2 exhibit more complex behavior. The singlebond elasticity increases by more than an order of magnitude for both salts between 10 and 200 mM, with CaCl2 demonstrating a greater initial increase than MgCl2. This is in agreement with the known increase of adhesion forces between surfaces with acidic residues.24 Surprisingly, at the highest concentration of CaCl2, κ0 decreases.

Figure 2. Measurements of the particle bond rigidity, κ0, for solutions of (b) MgCl2, (4) CaCl2, and (9) NaCl. The solid lines are calculated values of κ0 based on the JKR adhesion between particles. The energy of adhesion between particles is calculated using the YoungDupre´ equation with additional terms that account for the crosslinking energy due to adsorbed divalent ions and Coulombic repulsion.

calculated using the JKR theory of adhesion.25 For equal-size elastic spheres in the absence of an applied load,

(

)

3πa2WSL ac ) 2K

1/3

(4)

where WSL is the adhesion energy per unit area, and the bulk modulus is K ) 2E/3(1 - ν2). The Young’s modulus of PMMA is E ) 3100 MPa and Poisson ratio is ν ) 0.4.26 Using the area moment of inertia of a cylinder for the neck region between particles, we find that the single-bond elasticity is

κ0 )

3πa4c E 4a3

(5)

The single-bond bending rigidity is determined by the nature and mechanical properties of the contact region between particles. Assuming a circular region of contact, the contact radius, ac, is

Combining eqs 4 and 5 results in a direct relationship between the energy of adhesion between colloidal particles and the elasticity of the bonds that they form. This establishes the critical link between the mechanical properties of a strong gel and the interactions between particles, which are ultimately controllable by changing the physicochemical conditions or the interfacial properties of the particles. Next, we calculate the energy of adhesion between the PMMA surfaces at vanishing divalent salt concentrations, W0SL. The energy of adhesion is written in terms of the energy of cohesion between the PMMA solid and liquid half surfaces, WSS and WLL, respectively, and the interfacial energy of the solid-liquid surface, γSL,27

(22) Dabros, T.; van de Ven, T. G. M. Colloid Polym. Sci. 1983, 261, 694. (23) Velegol, D.; Catana, S.; Anderson, J.; Garoff, S. Phys. ReV. Lett. 1999, 83, 1243. (24) Ederth, T.; Claesson, P. M. J. Colloid Interface Sci. 2000, 229, 123.

(25) Johnson, K. L.; Kendall, K.; Roberts, A. D. Proc. R. Soc. London A 1971, 324, 301. (26) Schreyer, G. Konstruieren mit Kunststoffen; Carl Hanser: Mu¨nchen, 1972. (27) Israelchvili, J. N. Intermolecular and Surface Forces; Academic Press: London, 1992.

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1 1 WSL ) WSS + WLL - γSL 2 2

(6)

This is, in turn, related to the surface energy of the PMMA, γS, and the surface tension of the aqueous phase, γL,

WSL ) γS + γL - γSL

(7)

which, when combined with the Young equation,

γS ) γSL + γL(cos θ)

(8)

yields the Young-Dupre´ equation,

W0SL ) γL(1 + cos θ)

(9)

where θ is the solid-liquid contact angle. The PMMA-water contact angle is θ ) 73.7°,28 and the surface tension of water, which is constant for the range of salt concentrations we investigate, is γL ) 72 mN/m.29 Note that the PMMA surface energy can also be approximated by

γS )

AH

(10)

12πD20

where D0 ) 0.165 nm and AH is the nonretarded Hamaker constant of PMMA.27 The value calculated from eq 10, γS ) 10 mJ/m2, is in reasonable agreement with those inferred from contact angle measurements on PMMA surfaces used above (25-45 mJ/m2).28 From eqs 4, 5, and 9, we calculate a single-bond elasticity that is independent of salt concentration, κ0 ) 0.8 N/m. Qualitatively, this agrees with the results for NaCl, although it is larger than the measured value by approximately a factor of 4. The quantitative difference is likely to be due to the contribution of repulsive electrostatic interactions between the charged surface groups, which are not accounted for by the adhesion energy above.30,31 In this work, the experiments are conducted at high ionic strengths where we expect the double-layer repulsions to be significantly screened and the aggregated particles to be in primary minima. However, Coulombic interactions between charged surface groups are likely to occur, reducing the overall surface adhesion energy. This Coulombic repulsion is expressed as an additional term in the adhesion energy

(σe) A

WES SL ) wES

2

c

(11)

where Ac is the area of the contact region, and wES is the energy per interacting charge pair. For the NaCl measurements, we find 2 WES SL ) -61 ( 3 mJ/m . Returning to the strong dependence of κ0 on the divalent ion concentration, the above calculation suggests that divalent ions interact strongly with the PMMA particle surfaces and contribute significantly to the total energy of adhesion. In the following sections, we discuss the adsorption of divalent ions to the PMMA particle surface and the ion-mediated adhesion between neighboring particles. It is known that divalent ions are capable of enhancing the attraction between colloidal particles. For example, theoretical calculations based on the primitive model have shown that correlations between divalent ions induce attractive interac(28) Kwok, D. Y.; Leung, A.; Lam, C. N. C.; Li, A.; Wu, R.; Neumann, W. J. Colloid Interface Sci. 1998, 206, 44. (29) Jarvis, N. L.; Scheiman, M. A. J. Phys. Chem. 1968, 72, 74. (30) Go¨tzinger, M.; Peukert, W. Powder Technol. 2003, 130, 102. (31) Hayes, D. A. J. Adhes. Sci. Technol. 1995, 9, 1063.

Figure 3. Salt concentration dependence of the particle zeta-potential for (a) CaCl2 and (b) MgCl2. The solid line is a fit of the linearized Stern equation to the CaCl2 data. The Stern model does not describe the adsorption isotherm of MgCl2.

tions in excess of van der Waals contributions, a phenomenon that the Poisson-Boltzmann theory does not capture.32,33 B. Divalent Ion Adsorption on PMMA. There is a long history of interest in the adsorption of divalent ions to surfaces.27,34 Ion adsorption has strong implications for the adhesion between cell membranes, where Ca2+ ions and other divalent species are known to adsorb to lipid bilayers.35 In studies of thiohexadecanoic acid monolayers in aqueous solution, Ca2+ and Mg2+ both were shown to adsorb strongly to carboxylic acid groups compared to Na+, resulting in a decrease of the surface potential and an increase in the adhesion force between the monolayers.24 The adsorption of poly(acrylic acid) (PAA) onto mica surfaces in the presence of CaCl2 has also been attributed to the adsorption of Ca2+ ion onto the mica, which forms an ionic bridge between the mica and the PAA.36 Figure 3 shows the measurements of the zeta-potential at various concentrations of CaCl2 and MgCl2. At 10 mM NaCl, the zeta-potential of the PMMA particles is -40 mV. However, at 10 mM CaCl2 and MgCl2, the zeta-potential is -19 and -9 mV, respectively. At concentrations above ∼10 mM, the zetapotential for CaCl2 is less than MgCl2. A crossover to a positive surface potential is observed between 200 and 400 mM CaCl2, while a similar crossover is observed for MgCl2 at concentrations between 400 and 600 mM. Furthermore, an immediate connection between the zeta-potetial and bending mechanics can be made. Plotting the single-bond bending rigidity versus the zeta-potential (Figure 4) collapses the MgCl2 and CaCl2 data onto a single (32) Kjellander, R.; Mare`elja, S.; Pashley, R. M.; Quirk, J. P. J. Phys. Chem. 1988, 92, 6489. (33) Kjellander, R.; Mare`elja, S.; Pashley, R. M.; Quirk, J. P. J. Chem. Phys. 1990, 92, 4399. (34) Healy, T. W.; White, L. R. AdV. Colloid Interface Sci. 1978, 9, 303. (35) McLaughlin, S.; Mulrine, N.; Gresalfi, T.; Vaio, G.; McLaughlin, A. J. Gen. Physiol. 1981, 77, 445. (36) Berg, J. M.; Claesson, P. M.; Neuman, R. D. J. Colloid Interface Sci. 1993, 161, 182.

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{C++S-} ) K1[C++]0{S-}

(16)

where K1 is the intrinsic association constant, [C++]0 is the bulk concentration of the cation at the particle surface, and {S-} is the surface density of free charge sites. The total number of negative adsorption sites, {S}tot, is the sum of the unoccupied and associated sites

{S}tot ) {S-} + {C++S-}

Figure 4. Bond elasticity for the PMMA particles in divalent salt solutions collapse onto a single curve when plotted against the particle zeta-potential. The decrease in κ0 between CaCl2 concentrations 200 and 400 mM corresponds to a charge reversal of the particles.

curve, and the downturn in κ0 for CaCl2 corresponds to a charge reversal of the particles. The change in zeta-potential of the particles in solutions of divalent salts is related to the adsorption of ions to the PMMA surfaces. In the remainder of this section, we derive the Stern equation35 that describes the surface concentration of ions, which is then used to calculate the divalent ion contribution to the surface adhesion energy. First, we note that the surface charge density, σ, of the particles is related to the zeta-potential via the Grahame equation27

σ2 ) 20kBT(

∑F0i - ∑F∞i)

(12)

where 0 is the permittivity of free space,  is the dielectric constant of water, F∞i is the concentration of ion species i in the bulk, and F0i is the concentration of ion species i at the particle surface. The Boltzman distribution is used to the relate the ion concentration Csalt at the particle surface to the bulk concentration and the zeta-potential,

F0i ) F∞ie-zieζ/kBT

(13)

In the Boltzman equation, zi is the valency of ionic species i, and e is the fundamental unit of charge. Substituting eq 13 into the Grahame equation and writing in terms of divalent salt concentration Csalt in units of mol/L yields

{ ( )

( ) }

2eψ eψ + 2 exp -3 kBT kBT (14)

σ2 ) 2000NA0kBTCsalt exp -

where NA is Avogrado’s constant. Since, |ζ| < |e/kBT| ≈ 25 mV for the divalent salt concentrations investigated, we use the linearized form of eq 14 for low potentials,

σ)

( )x eζ kBT

6000NA0kBT[Csalt]

(15)

Typical sources of charge on PMMA particles are due to sulfonate ester groups from the polymerization initiator and acidic residues from the subsequent hydrolysis of these sulfonate groups and methyl methacrylate.37 Following the derivation of the Stern equation,35 we assume a Langmuir adsorption isotherm, which gives a surface concentration of the 1:1 complex of the cationbinding site, {C++S-}, as (37) Shim, S. E.; Kim, K.; Oh, S.; Choe, S. Macromol. Res. 2004, 12, 240.

(17)

Again, we linearize the Boltzman distribution of the cation concentration at the particle surface. Equations 16 and 17, in combination with a charge balance where the C++S- complex contributes a net +1 charge and the free site contributes a -1 charge, yields for a divalent salt,

σ)

{S}tote 1 + K1Csalt

(

2eζ 1kBT

){

(

K1Csalt 1 -

) }

2eζ -1 kBT

(18)

Combining eqs 18 and 15 results in a linearized Stern equation, which is solved numerically to predict the zeta-potential of the PMMA particles as a function of the divalent salt concentration. The line in Figure 3a is a fit of the Stern equation using the fitting parameters K1 ) 2.82 ( 0.14 M-1 and {S}tot ) 5.56 × 1016 ( 2.24 × 1016 m-2 where the errors represent the 95% confidence interval of the parameter fit. As mentioned earlier, the zeta-potential of the particles at 10 mM NaCl is -40 mV, corresponding to a charge density of 6.28 × 1016 m-2. This is in good agreement with the fitted value for {S}tot, suggesting that ions adsorb to the charged sites, as assumed. Furthermore, the zeta-potential crossover from negative to positive values between 200 and 400 mM CaCl2 suggests that the assumption of 1:1 complex formation in eq 16 is correct. If the Ca2+ ions formed a 2:1 complex with the binding sites, C++S2-, we would expect the zeta-potential to asymptote to zero. In contrast to CaCl2, MgCl2 exhibits a significantly stronger initial adsorption. The Stern equation does not fit the zeta-potential measured for the salt concentrations we examine. Notice, however, that a point of zero charge is still observed between 400 and 600 mM MgCl2. Although the Stern equation does not describe the concentration dependence of the zeta-potential for MgCl2 solutions, the surface charge density derived from the zetapotential still provides an accurate measure of the number of adsorption sites occupied by the divalent ion. This is reinforced by the collapse of the data shown in Figure 4. C. Salt Dependent Contributions to WSL. To calculate the bending rigidity of aggregates formed in divalent salt solutions, we consider the contribution of ion association with the PMMA surfaces to the adhesion energy. As divalent ions adsorb to the surface of particles, they increase the association through the formation of ionic bridges and decrease the Coulombic repulsion due to the decrease in total surface charge density. The total adhesion energy is expressed as

WSL ) W0SL + WSSL + WES SL

(19)

where WSSL is the contribution of the adsorbed divalent ions forming cross-links to charged sites on the adjacent particle and WES SL is the energy due to a Coulombic repulsion between the charged surfaces. These two additional contributions to WSL will provide a dependence on the divalent salt concentration that is observed in experiments.

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Figure 5. Total energy of adhesion of PMMA particles as a function of the particle zeta-potential (bottom axis) and fractional coverage of the adsorbed divalent ions (top axis). A maximum in WSL occurs at θ ) 0.5, which corresponds to the point of zero net charge.

To calculate WSSL, we use the parameters obtained from the Stern equation fit to the CaCl2 zeta-potential data to determine the fractional coverage of the adsorbed divalent species onto the binding sites

K1[Ca+2]0 {CaS+1} θˆ ) ) {S}tot 1 + K1[Ca+2]0

(20)

When two particles come into contact, the number of ionic crosslinks formed by adsorbed divalent ions is proportional to the number of occupied sites on a particle and open sites on the adjacent particle, n ∝ θˆ (1 - θˆ ). Note that a maximum occurs at θˆ ) 0.5, corresponding to the point of zero charge on the particles, at which point ζ ) 0. Physically, this means that an equal number of charged surface sites and Ca2+S- complex sites are present, resulting a net zero charge of the particles. Assuming WSSL is a sum of the total number the cross-links between two adjacent particles, each contributing an equal energy per ionic cross-link, wS, the energy per unit area is

WSSL ) 4{S}totwSθˆ (1 - θˆ )

(21)

Divalent ion adsorption also reduces the Coulombic repulsion between charged surface sites. By performing a charge balance similar to the one used to derive eq 18, we rewrite the particle surface charge density as σ/e ) {S}tot - 2θˆ {S}tot. Assuming that the contact region is circular, we substitute Ac ) πa2c into eq 11 to find 2 ˆ )2πa2c WES SL ) wES{S}tot(1 - 2θ

(22)

Combining eqs 9, 19, 21, and 22 yields an expression for the total energy of adhesion of the particles; when substituted into eq 4, this results in a cubic equation for the contact area radius ac,

2K 3 ac - wES{S}2tot(1 - 2θˆ )2πa2c - 4{S}totwSθˆ (1 - θˆ ) 3πa2 γL(1 + cos θ) ) 0 (23) Using eq 23 as the contact radii calculated in eq 5, we fit κ0 for the CaCl2 data. The adhesion energy model is in excellent agreement with measurements of the single-bond elasticity as a function of the zeta-potential of the particles, as shown in Figure 4. The corresponding adhesion energies are shown in Figure 5

versus the zeta-potential and surface fractional coverage of divalent ion adsorption sites. By fitting eqs 5 and 23 to the measured values of κ0, we find a value for the Coulombic contribution, wES ) 2.59 × 10-20 ( 0.13 × 10-20 J. This corresponds to a particle-surface separation of ∼0.1 nm, which is physically reasonable. However, the fitted value of the energy per divalent cross-link, ws ) 4630 ( 445 kJ/mol, is larger than the energy of a covalent bond, which is typically on the order of 200 kJ/mol. This unphysical value is likely due to surface charge heterogeneity on the PMMA surfaces. Implicitly, our model assumes that the adsorption sites, and thus the surface charge sites, are uniformly distributed across the particle surfaces. Recently, however, it has been shown by rotational differential electrophoresis and AFM imaging that colloidal particles often possess significant surface charge heterogeneity.38,39 The size of the surface “patches” are reported to be ∼50-100 nm,39 which is similar to the values of ac calculated from eq 4. Therefore, if the contact area between particles is a local region of high surface charge, the number of ionic crosslinking sites increases and the energy per cross-link becomes more reasonable. If we assume that the surface charges are hexagonally packed, with a lateral separation between charges close to the Bjerrum length, lb ) 0.7 nm, the resulting local charge density is σ/e ) 3.14 × 1018 m-2. Substituting this value for {S}tot, we find an energy of ionic cross-links, ws ) 86 ( 13 kJ/mol, which is within a reasonable range for ionic interactions. The fit in Figure 4 for the CaCl2 data shows excellent agreement with the MgCl2 data, despite the previous result that the Stern model does not fit the zeta-potential for this salt. These predictions have been mapped onto Figure 2, based on the measured zetapotentials. Our model of particle bond elasticity has progressed from being salt-concentration independent to capturing the complex behavior observed experimentally for PMMA particles in divalent salt solutions. For CaCl2, we capture this behavior on the basis of the knowledge of only the Ca2+ ion concentration, since this species obeys a Langmuir-like adsorption isotherm. Although the adsorption of Mg2+ ions onto PMMA could not be described using the Stern model, we are still able to predict the observed behavior κ0 from the measured zeta-potential of the particles, which provides a measure of the fractional surface coverage of divalent ions. Notice that, for a monovalent salt, we do not expect ion adsorption, and thus independence of salt concentration is maintained

IV. Conclusions We have performed experiments on micromechanics of single colloidal aggregates. These experiments enable us to probe the interactions between colloidal particles under conditions where DLVO theory does not apply. We find that, for strongly aggregated systems, particle bonds are capable of supporting torques, demonstrating the presence of tangential restoring forces. This allows aggregates to be approximated as bending microbeams. Comparing the elasticity expected from a beam to the lengthdependent spring model of a randomly percolated system provides a framework to understanding the particle bond elasticity. The JKR model of adhesion is used to calculate the area of contact between the particles, which provides a direct link between energy of adhesion between the particles and their bond elasticity. As expected from the Young-Dupre´ equation, the energy of adhesion is independent of the salt concentration for monovalent salts. However, divalent salts enhance the attractive energy between particles. We model this as an adsorption of the divalent cation (38) Feick, J. D.; Velegol, D. Langmuir 2002, 18, 3454. (39) Tan, S.; Sherman, R. L.; Qin, D.; Ford, W. T. Langmuir 2005, 21, 43.

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onto the negatively charged particle surface, which results in the formation of ionic bridges between adjacent particles. The adsorption isotherm is found to be ion specific. Thus, the work of adhesion between colloidal particles depends on the ion type used to induce aggregation. The formation of ionic cross-links between particles provides a method of tuning the strength and aging properties of colloidal gels by manipulating the energy of adhesion between particles. For example, the yield stress of a gel is proportional to the number of interparticle bonds per unit area and the strength of the particle bonds1. Therefore, control of the yield stress of a gel may be obtained by understanding how specific electrolytes used as the flocculant, as well as the electrolyte concentration, affect the

Pantina and Furst

energy of adhesion between the particles. This will in turn determine the bond strength, and thus the yield stress of the resulting gel. Alternatively, one can decrease the strength of the gel by reducing the adhesion energy through the addition surfactants. Finally, divalent ions may be an attractive and relatively simple method for controlling interactions between particles with distinct chemical groups. Acknowledgment. The authors gratefully acknowledge the technical skills of Andrew Marshall and helpful discussions with Babatunde Ogunnaike. This work was supported by the National Science Foundation (CAREER CTS-0238689 and CTS-0209936). LA0534120