Attractive Forces between Charged Colloidal Particles Induced by

Letter pubs.acs.org/JPCL

Attractive Forces between Charged Colloidal Particles Induced by Multivalent Ions Revealed by Confronting Aggregation and Direct Force Measurements Prashant Sinha, Istvan Szilagyi, F. Javier Montes Ruiz-Cabello, Plinio Maroni, and Michal Borkovec* Department of Inorganic and Analytical Chemistry, University of Geneva, Sciences II, Quai Ernest-Ansermet 30, 1205 Geneva, Switzerland S Supporting Information *

ABSTRACT: Interactions involving charged particles in the presence of multivalent ions are relevant in wide-range of phenomena, including condensation of nucleic acids, cement hardening, or water treatment. Here, we study such interactions by combining direct force measurements with atomic force microscopy (AFM) and aggregation studies with time-resolved light scattering for particles originating from the same colloidal suspension for the first time. Classical DLVO theory is found to be only applicable for monovalent and divalent ions. For ions of higher valence, charge inversion and additional non-DLVO attractive forces are observed. These attractive forces can be attributed to surface charge heterogeneities, which leads to stability ratios that are calculated from direct force measurements to be higher than the experimental ones. Ion−ion correlations are equally important as they induce the charge inversion in the presence of trivalent or tetravalent ions, and they enhance the surface charge heterogeneities. Such heterogeneities therefore play an essential role in controlling interactions in particle suspensions containing multivalent ions. SECTION: Glasses, Colloids, Polymers, and Soft Matter

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either.2,11,16−19 A clearer picture would emerge if such measurements could be complemented by direct force measurements involving the same particles, but this task is out of reach for many techniques available. The surface forces apparatus (SFA) can only measure forces between macroscopic mica sheets,8,20 total internal reflection microscopy is restricted to the sphere-plane geometry,21 and video-microscopy techniques require particles that are too large for reliable aggregation measurements.15,22,23 The colloidal probe technique based on the atomic force microscope (AFM) is normally used to probe forces in the sphere-plane geometry,24,25 but has been extended to measure forces between an AFM-tip and deposited particles26 as well as between pairs of individual particles.27,28 Here we measure forces and aggregation rates involving particles from the same colloidal suspension for the first time. These measurements became possible since forces between pairs of particles of 1.0 μm in diameter could be measured by the multiparticle colloidal probe technique (Figure 1a). These particles are sufficiently small to be used for aggregation studies whereby sedimentation and multiple-scattering effects become negligible. By combining these techniques, new information

nteraction forces involving macromolecules, nanoparticles, or water−solid interfaces are strongly modified by multivalent ions, and such effects are essential in the condensation of nucleic acids, crystal growth, cement hardening, or water treatment.1−3 The recent focus on multivalent ions was triggered by the discovery that the Poisson−Boltzmann (PB) approximation describing electrical double-layers may fail in their presence due to neglect of ion−ion correlations.3−7 As a consequence, one ought to question the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory that postulates that such forces are dominated by a superposition of attractive van der Waals forces and repulsive double-layer forces, the latter being precisely obtained from the doubtful PB approximation.8,9 Still, the DLVO theory triumphed with the derivation of the classical Schulze−Hardy rule, which captures the influence of multivalent ions on the stability of colloidal suspensions.9−11 This rule states that the critical coagulation concentration (CCC) defining the onset of fast aggregation scales as z−6 where z is the counterion valence. While this relation has been confirmed experimentally, eventual importance of surface charge heterogeneities was equally put forward.12−15 It is probably fair to say that no consensus concerning the forces acting between colloidal particles in the presence of multivalent ions has been reached so far. Precise measurements of aggregation rate constants between colloidal particles, which now can be routinely achieved with time-resolved light scattering or single particle counting techniques, do not help to disentangle these questions © 2013 American Chemical Society

Received: January 9, 2013 Accepted: February 5, 2013 Published: February 5, 2013 648

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Figure 1. Forces measured by the colloidal probe technique. (A) Scheme of the experimental setup. (B) Forces in the presence of a divalent anion SO2− 4 , where repulsive forces weaken with increasing salt concentration and finally become attractive. Solid lines are best fits with DLVO theory. Reentrant forces in the presence of a tetravalent anion [Fe(CN)6]4− at (C) low and (D) high concentrations.

Figure 2. Parameters extracted from measured force profiles. Solid lines are empirical functions used for calculation of the aggregation rate constants. (A) Electric surface potentials whose sign has been inferred from electrophoretic mobility. (B) Amplitude of the attractive exponential non-DLVO contribution.

Supporting Information). The fitted surface potentials are shown in Figure 2a, and they decrease with increasing salt concentration as electrostatic screening suggests. The force profiles for tetravalent ions cannot be reconciled with the simple DLVO picture due to their reentrant nature and presence of additional attractions (Figure 1c,d). The reentrance can be explained by charge inversion as confirmed by

concerning the interactions between charged colloidal particles in the presence of multivalent ions is obtained. Measured force profiles between positively charged amidine polystyrene latex particles in the presence of divalent and tetravalent anions are shown in Figure 1. Forces involving divalent ions could be fitted with the classical DLVO theory with a Hamaker constant of 4.0 × 10−21 J remarkably well (see 649

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Figure 3. Experimental stability ratios of amidine latex particles (points) versus the counterion concentration compared with calculations based on 4− direct force measurements (solid lines) for (A) SO2− 4 and (B) [Fe(CN)6] . Schemes in A show the elementary aggregation process of the doublet formation and the light scattering setup used to measure the aggregation rate in the early stages. Solid lines in B include the non-DLVO contribution, while the dashed lines shows contribution of DLVO forces only. (C) The stability map compares the valence of the counterion with the location of the CCC obtained from light scattering experiments (LS, points) and from estimates based on direct force measurements (AFM, lines). The thick line indicates the Schulze−Hardy rule. (D) Surface separation for the energy barrier at CCC for different valence. The insets in D provide a pictorial representation of the enhancement of surface charge heterogeneities by adsorption of multivalent counterions.

electrophoresis. Similar overcharging phenomena are documented in other systems.11,18 With increasing concentration, the potential passes through the isoelectric point (IEP) and becomes negative. After going through a minimum, it finally decreases in magnitude due to screening. The attractive forces close to IEP are substantially stronger than for divalent counterions, and they are inconsistent with the corresponding Hamaker constant. We interpret this additional non-DLVO attraction as originating from surface charge heterogeneities.12 When such heterogeneities are arranged in a square lattice, the attraction is known to be exponential, and its inverse decay length is given by (κ2 + (2π/a)2)1/2 where κ−1 is the Debye length and a is the lattice spacing (see Supporting Information). When this additional force is added to the DLVO profiles, the observed forces can be reconciled with a = 15 ± 5 nm. The fitted surface potentials and amplitudes of the non-DLVO force are shown in Figure 2. Early stage aggregation rates were measured in suspensions of the same particles as used for the force measurements by time-resolved light scattering.11,19 Figure 3 shows these results as stability ratios, which is defined as W = kfast/k, where kfast is the fast aggregation rate coefficient measured at high salt concentrations and k is the aggregation rate coefficient at the

conditions investigated. Divalent counterions induce the classical salt dependence featuring the slow aggregation regime at low salt concentrations and fast aggregation at high concentrations. For tetravalent counterions, the stability ratio features an intermediate maximum due to charge inversion. The CCCs separating these regimes are summarized in a stability map for all counterions (Figure 3c). Since force profiles and aggregation rates are available for exactly the same particles, the experimental aggregation rates can be compared with the ones calculated from the forces (Figure 3). These calculations involve no adjustable parameters. The map indicates that the CCCs are reproduced quite well for all types of ions, and that the Schulze−Hardy rule indeed reflects the shift of the lowest CCC. For divalent counterions, the experimental stability ratios are predicted well. The experimental fast aggregation rate kfast = 3.8 × 10−18 m3/s agrees with DLVO theory, which yields 6.5 × 10−18 m3/s. However, the agreement is only semiquantitative for tetravalent counterions, and the disagreement becomes especially important near the intermediate maximum. Including attractive non-DLVO forces improves the agreement with experiment somewhat, but substantial discrepancies remain. These discrepancies originate most likely from lateral charge heterogeneities. Relevant interactions occur at distances of 650

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as they induce charge inversion in the presence of ions of higher valence and enhance surface charge heterogeneities. This approach involving the multiparticle colloidal probe technique could also be confronted with other bulk characteristics (e.g., structure factor, osmotic pressure), and thus has the potential to become indispensable tool in our technical repertoire to study colloidal interactions.

several nanometers (Figure 3d). The energy barrier at CCC moves to larger distances with increasing valence, which excludes eventual contributions from ion−ion correlation forces, hydration forces, or roughness.5,18,19 In particular, forces induced by ion−ion correlations have a subnanometer range,6,7 which is substantially smaller than the one relevant here. We have already argued that for trivalent and tetravalent counterions, the additional non-DLVO attractive forces are consistent with lateral surface charge heterogeneities. Two additional observations support this picture further. (i) The rate calculations assume a radially symmetric force field, which is incorrect for heterogeneous surfaces. Aggregating particles will explore their rotational degrees of freedom to find the lowest free energy path, which results in faster aggregation. In the force measurements, however, particles have little freedom to adjust their mutual orientations. (ii) Variation between the measured force profiles involving different pairs of particles could be sometimes observed (see Supporting Information). Such differences are expected for heterogeneous surfaces since different particles will expose different patches. On the basis of these arguments, lateral charge heterogeneities emerge as the most likely explanation for the observed discrepancies between calculated and experimental stability ratios for tetravalent counterions. While the direct force measurements suggest a characteristic size of these heterogeneities of about 15 nm, their distribution could be wide and include substantially larger patches (i.e., charge polydispersity). Trivalent counterions resemble tetravalent ones, even though the deviations from DLVO theory are less pronounced than in the tetravalent case. The results for monovalent counterions are similar to the divalent ones. For monovalent counterions, however, the energy barrier at CCC is located at subnanometer distances, which is comparable to the distance resolution in the force measurements (Figure 3d). In this case, the force profiles are probably no longer accurate in the relevant distance regime, and contributions from additional short-range interactions cannot be excluded. Effects of ion−ion correlations seem evidenced in the progressive importance of the charge reversal observed in the presence of multivalent ions. However, the resulting surface potentials of these adsorbed layers remain small, and DLVO theory can be still used to describe the force profiles at larger distances. While ion−ion correlations induce attractive forces with a substantially smaller range than the one observed,6,7 these forces could be modified by surface roughness, lateral charge heterogeneities, counterion bridging, or superstructures formed by multivalent ions.5 While the observed attractive nonDLVO forces are likely induced by surface charge heterogeneities, the fact that these forces become more relevant in the presence of multivalent ions suggests the importance of ion− ion correlations. While the importance of such heterogeneities has also been suggested earlier,13,15 the present results indicate that multivalent ions magnify these heterogeneities. For the first time, we were able to combine direct force measurements and aggregation studies involving particles from the same suspension. These techniques reveal that DLVO theory is only applicable in the presence of monovalent and divalent counterions. For ions of higher valence, charge inversion and additional non-DLVO attractive forces are observed. These attractive forces can be attributed to surface charge heterogeneities, which lead to stability ratios calculated from direct force measurements to exceed the experimental ones. However, ion−ion correlation effects are equally relevant

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MATERIALS AND METHODS Positively charged amidine polystyrene latex particles of 1.0 μm in diameter suspended in ultrapure water adjusted to pH 4.0 were mixed with solutions of potassium salts of Cl−, SO2− 4 , [Fe(CN)6]3−, and [Fe(CN)6]4−. Forces between particles were measured with an AFM mounted on an inverse optical microscope. The particle suspension was injected into the AFM fluid cell sealed with a silanized glass plate to which the particles attach spontaneously. A particle was picked up with a silanized tip-less cantilever in the cell and centered above another particle on the substrate. This multiparticle colloidal probe technique features a large internal surface area and thus allows the control of the amount of multivalent ions added precisely. The technique is also less sensitive to impurities. About 100 approach−retraction cycles were recorded per pair of particles, and the noise was reduced by averaging all forces curves and by down sampling to 3 pN. Force profiles between at least three pairs of particles were averaged. Absolute particle aggregation rates were determined in the same suspensions by time-resolved simultaneous static and dynamic light scattering. Stability ratios were obtained by dynamic light scattering at a scattering angle of 90°. The steady-state solution of the diffusion equation was used to calculate the aggregation rate from the interaction potential and the mutual diffusion coefficient. The interaction potential is obtained by integrating the fitted force profiles, whereby the concentration dependence of the parameters is quantified with empirical fits shown in Figure 2. The mutual diffusion coefficient was calculated from the diffusion coefficient of the individual particles measured by dynamic light scattering and by including hydrodynamic interactions between the particles. Further details are given in the Supporting Information.

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ASSOCIATED CONTENT

S Supporting Information *

Additional information is provided concerning materials, methods, experimental data on monovalent and trivalent ions, and references. This material is available free of charge via the Internet http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: + 41 22 379 6405; E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the Swiss National Science Foundation and the University of Geneva. REFERENCES

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