Forces between single pairs of charged colloids in aqueous salt solutions

Formation of Two-Dimensional Magnetically Responsive Clusters Using Hematite Particles Self-Assembled via Particle-Induced Heating at an Interface

Hematite particles, which exhibit a high magnetic moment, are used to apply large forces on physical and biological systems under magnetic fields to investigate various phenomena, such as those of rheology and micromanipulation. However, the magnetic confinement of these particles requires complicated field configurations. On the other hand, laser-assisted optical confinement of single hematite particles results in thermophoresis and subsequent ejection of the particle from the laser spot. Herein, we explore an alternative strategy to induce the self-assembly of hematite. In this strategy, with indirect influence from an optically confined and heated upconverting particle (UCP) at an air-water interface, there is the generation of convection currents that facilitate assembly. We also show that the assembly remains at the interface even after removal of the laser light. The hematite particle assemblies can then be moved using magnetic fields and employed to perform interfacial rheology.

Renormalized charge and dielectric effects in colloidal interactions: a numerical solution of the nonlinear Poisson-Boltzmann equation for unknown boundary conditions

 The Derjaguin-Landau-Verwey-Overbeek (DLVO) theory, introduced more than 70 years ago, is a hallmark of colloidal particle modeling. For highly charged particles in the dilute regime, it is often supplemented by Alexander's prescription (Alexander et al. in J Chem Phys 80:5776, 1984) for using a renormalized charge. Here, we solve the problem of the interaction between two charged colloids at finite ionic strength, including dielectric mismatch effects, using an efficient numerical scheme to solve the nonlinear Poisson-Boltzmann (NPB) equation with unknown boundary conditions. Our results perfectly match the analytical predictions for the renormalized charge by Trizac and coworkers (Aubouy et al. in J Phys A 36:5835, 2003). Moreover, they allow us to reinterpret previous molecular dynamics (MD) simulation results by Kreer et al. (Phys Rev E 74:021401, 2006), rendering them now in agreement with the expected behavior. We furthermore find that the influence of polarization becomes important only when the Debye layers overlap significantly.

Estimating orientation of optically trapped, near vertical, microsphere dimers using central moments and off-focus imaging

Near vertical optically trapped dimers, composed of pairs of microspheres, and constructed in situ, were imaged in bright-field in flow and at rest, and with displacement Δz from the transverse xy imaging plane of an inverted microscope. Image first central moments μ₀₁ were measured, and their dependence on the imposed flow velocity of the surrounding fluid was calculated. This dependence was related to the at-rest restricted diffusion statistics. It was assumed that, for small perturbations, the torque T on the dimer was proportional to the velocity of flow v and resulting angular deflection Δθ so that T∝v∝Δθ. Displacements Δz at which v∝Δμ₀₁∝Δθ, which are typically off focus, were examined in more detail; in this range, Δθ=hΔμ₀₁. The hydrodynamics of the dimer were modeled as that of a prolate ellipsoid, and the constant of proportionality h was determined by comparing the short-time mean-squared variation measured during diffusion to that predicted by the model calculation: h²⟨Δμ012(t)⟩=⟨Δθ²(t)⟩. With h determined, the optical trap stiffness k_θ was determined from the long-time restricted diffusion of the dimer. The measured k_θ and Δθ can then be used compute torque: T=k_θΔθ, potentially enabling the near vertical optically trapped dimer to be used as a torque probe.

Inside the ensemble: unlocking the potential of one-at-a-time experiments with lab-on-a-chip automation

The advent of technologies that allow the interactions of individual microscopic particles to be probed "one-at-a-time" has paved the way for new experimental avenues of enquiry in colloidal systems. For example, investigating whether a particular pair of colloidal particles isolated from a macroscopic sample might adhere to each other when brought into close proximity is certainly possible. However, given the probabilistic nature of the process (different particles within the ensemble may have slightly different surface charge distributions and asperities, and interaction energies involved can be close to thermal values), it is important that many hundreds or thousands of pairs of particles are tested under each set of experimental conditions of interest. Currently it is still an arduous task to perform such an experiment a sufficient number of times in order to acquire a data-set that truly represents the ensemble. Herein an automated particle collider for measuring particle-particle interactions has been realized by combining elements of microfluidics, holographic optical tweezers and image processing. Each individual measurement consists of confining two particles within a predetermined chemical micro-environment, and observing whether their interactions lead to aggregation. To automate the measurements, computer software consisting of LabVIEW and Red Tweezers with a custom plugin was used. Preliminary experiments carried out using 1 μm diameter polystyrene particles demonstrated that many hundreds of pairwise-interaction measurements could be carried out autonomously within a matter of hours. Further exemplar real-world experiments, designed to examine the stickiness of emulsion drops as a function of bulk measurements of the ζ-potential (zeta potential) of the sample, were then performed. It is envisaged that such robust approaches to the automation of "one-at-a-time" experiments will find applications in a large number of areas, and enable previously unthinkable experiments to be carried out in a timely fashion, thus allowing the focus to shift away from tedious experimental frustrations to more profound scientific questions.

Patchy Colloidal Clusters with Broken Symmetry

Colloidal clusters are prepared by assembling positively charged cross-linked polystyrene (PS) particles onto negatively charged liquid cores of swollen polymer particles. PS particles at the interface of the liquid core are closely packed around the core due to interfacial wetting. Then, by evaporating solvent in the liquid cores, polymers in the cores are solidified and the clusters are cemented. As the swelling ratio of PS cores increases, cores at the center of colloidal clusters are exposed, forming patchy colloidal clusters. Finally, by density gradient centrifugation, high-purity symmetric colloidal clusters are obtained. When silica-PS core-shell particles are swollen and serve as the liquid cores, hybrid colloidal clusters are obtained in which each silica nanoparticle is relocated to the liquid core interface during the swelling-deswelling process breaking symmetry in colloidal clusters as the silica nanoparticle in the core is comparable in size with the PS particle in the shell. The configuration of colloidal clusters is determined once the number of particles around the liquid core is given, which depends on the size ratio of the liquid core and shell particle. Since hybrid clusters are heavier than PS particles, they can be purified using centrifugation.

A self-consistent Ornstein-Zernike jellium for highly charged colloids (microgels) in suspensions with added salt

This work discusses a jellium scheme, built within the framework of the multicomponent Ornstein-Zernike (OZ) equation, which is capable of describing the collective structure of suspensions of highly charged colloids with added salt, even in the presence of finite-size multivalent microions. This approach uses a suitable approximation to decouple the microion-microion correlations from the macroion-microion profiles, which in combination with the methodology from the dressed ion theory (DIT) gives a full account of the electrostatic effective potential among the colloids. The main advantages of the present contribution reside in its ability to manage the short-range potentials and non-linear correlations among the microions, as well as its realistic characterization of the ionic clouds surrounding each macroion. The structure factors predicted by this jellium scheme are contrasted with previously reported experimental results for microgel suspensions with monovalent salts (2019Phys. Rev. E100032602), thus validating its high accuracy in these situations. The present theoretical analysis is then extended to microgel suspensions with multivalent salts, which reveals the prominent influence of the counterion valence on the makeup of the effective potentials. Although the induced differences may be difficult to identify through the mesoscopic structure, our results suggest that the microgel collapsing transition may be used to enhance these distinct effects, thus giving a feasible experimental probe for these phenomena.

The Casimir Interaction between Spheres Immersed in Electrolytes

We investigate the Casimir interaction between two dielectric spheres immersed in an electrolyte solution. Since ionized solutions typically correspond to a plasma frequency much smaller than kBT/ħ at room temperature, only the contribution of the zeroth Matsubara frequency is affected by ionic screening. We follow the electrostatic fluctuational approach and derive the zero-frequency contribution from the linear Poisson-Boltzmann (Debye-Hückel) equation for the geometry of two spherical surfaces of arbitrary radii. We show that a contribution from monopole fluctuations, which is reminiscent of the Kirkwood-Shumaker interaction, arises from the exclusion of ionic charge in the volume occupied by the spheres. Alongside the contribution from dipole fluctuations, such monopolar term provides the leading-order Casimir energy for very small spheres. Finally, we also investigate the large sphere limit and the conditions for validity of the proximity force (Derjaguin) approximation. Altogether, our results represent the first step towards a full scattering approach to the screening of the Casimir interaction between spheres that takes into account the nonlocal response of the electrolyte solution.

Laser Induced Aggregation of Light Absorbing Particles by Marangoni Convection

Laser induced Marangoni convection can be used to accumulate micro-particles. In this paper, a method is developed to control and accumulate the light absorbing particles dispersed in a thin solution layer. The particles are irradiated by a focused laser beam. Due to the photothermal effect of the particles, the laser heating generates a thermal gradient and induces a convective flow around the laser’s heating center. The convective flow drives the particles to accumulate and form a particle aggregate close to the laser’s heating center. The motion of particles is dominated by the Marangoni convection. When the laser power is high, the vapor bubbles generated by laser heating on particles strengthen the convection, which accelerates the particles’ aggregation.

Synthetic non-classical luminescence generation by enhanced silica nanophotonics based on nano-bio-FRET

Fluorescent silica nanoparticles (NPs–(SiO₂–Fluo)) were synthesized based on the classical Störber method for cyanobacteria labelling. Modified mono-coloured SiO₂ NPs with fluorescein (Fl) and rhodamine B (RhB) were obtained (NPs–(SiO₂–Fl) and NPs–(SiO₂–RhB)). Moreover, multi-coloured SiO₂ NPs, via the incorporation of both emitters (NPs–(SiO₂–RhB–Fl)), were tuned for optimal emissions and the biodetection of cyanobacteria. NPs–(SiO₂–Fl) and NPs–(SiO₂–RhB–Fl) were optimized for detection via laser fluorescence microscopy and in-flow cytometry with laser excitation and fluorescence detection. By TEM, homogeneous SiO₂ NPs of 180.0 nm in diameter were recorded. These sizes were slightly increased due to the covalent linking incorporation of fluorescent dye emitters to 210.0 nm with mono-coloured fluorescent modified amine-organosilanes, and to 340.0 nm in diameter with multi-coloured dye incorporation. NPs–(SiO₂–Fluo) showed variable emission depending on the dye emitter concentration, quantum yield and applied luminescent pathway. Thus, mono-coloured NPs–(SiO₂–Fl) and NPs–(SiO₂–RhB) showed diminished emissions in comparison to multi-coloured NPs–(SiO₂–RhB–Fl). This enhancement was explained by fluorescence resonance energy transfer (FRET) between Fl as a fluorescent energy donor and RhB as an energy acceptor produced within the nanoarchitecture, produced only in the presence of both fluorophores with the appropriate laser excitation of the energy donor. The depositions of the nano-emitters on cyanobacteria by non-covalent interactions were observed by TEM and laser fluorescence microscopy. For multi-coloured NPs–(SiO₂–RhB–Fl) labelling, bio-FRET was observed between the emission of the nano-labellers and the natural fluorophores from the cyanobacteria that quenched the emission of the whole nano-biostructure in comparison to mono-coloured NPs–(SiO₂–Fl) labelling. This fact was explained and discussed in terms of different fluorescence energy transfer from the nanolabellers towards different natural chromophore coupling. In the presence of NPs–(SiO₂–RhB–Fl) and NPs–(SiO₂–RhB), the emission was coupled with lower quantum yield chromophores; while upon the application of NPs–(SiO₂–Fl), it was coupled with higher quantum yield chromophores. In this manner, for enhanced luminescent nanoplatform tracking, the multi-coloured NPs–(SiO₂–RhB–Fl) showed improved properties; but more highly luminescent bio-surfaces were generated with mono-coloured NPs–(SiO₂–Fl) that permitted faster cyanobacteria detection and counting by laser fluorescence microscopy, and by in-flow cytometry with laser excitation and fluorescence detection.

Fluorescent silica nanophotonics for cyanobacteria labelling.

Forces between solid surfaces in aqueous electrolyte solutions

This review addresses experimental findings obtained with direct force measurements between two similar or dissimilar solid surfaces in aqueous electrolyte solutions. Interpretation of these measurements is mainly put forward in terms of the classical theory of Derjaguin, Landau, Verwey, and Overbeek (DLVO). This theory invokes a superposition of attractive van der Waals forces and repulsive double layer forces. DLVO theory is shown to be extremely reliable, even in the case of multivalent ions. However, such a description is only successful, when appropriate surface charge densities, charge regulation characteristics, and ion pairing or complexation equilibria in solution are considered. Deviations from DLVO theory only manifest themselves at distances of typically below few nm. More long-ranged non-DLVO forces can be observed in some situations, particularly, in concentrated electrolyte solutions, in the presence of strongly adsorbed layers, or for hydrophobic surfaces. The latter forces probably originate from patch-charge surface heterogeneities, which can be induced by ion-ion correlation effects, charge fluctuations, or other types of surface heterogeneities.

In Situ Measurement of Depletion Caused by SDBS Micelles on the Surface of Silica Particles Using Optical Tweezers

Dual-trap optical tweezers have been used to directly measure the interaction forces between two silica particles upon controlling the concentration of the ionic surfactant sodium dodecylbenzenesulfonate (SDBS). By capturing two silica particles in one spot optical trap and one linear optical trap and controlling the linear trap to bring one particle to approach another sufficiently closer, the interaction forces between these two particles can be measured as the separation distance changes. Results showed that with increasing concentrations of SDBS, the interaction force between the two silica particles emerges at closer surface distance between two silica particles. Only repulsive force exists between silica particles below the critical micelle concentration (cmc) of SDBS and it could be well-fitted using the classical Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. However, the depletion attraction force appears above the cmc of SDBS which is induced by the generation of SDBS micelles. By in situ measurement of the interaction force between two silica particles in the presence of different concentrations of SDBS, the depletion force can be quantitatively calculated.

Decoupling conservative forces and hydrodynamic interactions between optically trapped spheres

Characterizing the interactions between colloidal particles is important, both from a fundamental perspective as well as due to its technological importance. However, current methods to measure the interaction forces between two colloids have significant limitations. Here we describe a method that exploits the fluctuation spectra of two optically trapped microspheres in order to extract, and decouple, the conservative forces acting between them and their hydrodynamic coupling. We demonstrate the proposed method with two silica microspheres, and find good agreement between our results and previous predictions for the hydrodynamic and electrostatic interactions between the spheres.

Measuring Inner Layer Capacitance with the Colloidal Probe Technique

The colloidal probe technique was used to measure the inner layer capacitance of an electrical double layer. In particular, the forces were measured between silica surfaces and sulfate latex surfaces in solutions of monovalent salts of different alkali metals. The force profiles were interpreted with Poisson-Boltzmann theory with charge regulation, whereby the diffuse layer potential and the regulation properties of the interface were obtained. While the diffuse layer potential was measured in this fashion in the past, we are able to extract the regulation properties of the inner layer, in particular, its capacitance. We find systematic trends with the type of alkali metal ion and the salt concentration. The observed trends could be caused by difference in ion hydration, variation of the binding capacitance, and changes of the effective dielectric constant within the Stern layer. Our results are in agreement with recent experiments involving the water-silica interface based on a completely independent method using X-ray photoelectron spectroscopy in a liquid microjet. This agreement confirms the validity of our approach, which further provides a means to probe other types of interfaces than silica.

Experimental characterization and modeling of optical tweezer particle handling dynamics

We report a new framework for a quantitative understanding of optical trapping (OT) particle handling dynamics. We present a novel three-dimensional particle-based model that includes optical, hydrodynamic, and inter-particle forces. This semi-empirical colloid model is based on an open-source simulation code known as LAMMPS (large-scale atomic/molecular massively parallel simulator) and properly recapitulates the full OT force profile beyond the typical linear approximations valid near the trap center. Simulations are carried out with typical system parameters relevant for our experimental holographic optical trapping (HOT) system, including varied particle sizes, trap movement speeds, and beam powers. Furthermore, we present a new experimental method for measuring both the stable and metastable boundaries of the optical force profile to inform or validate the model's underlying force profile. We show that our framework is a powerful tool for accurately predicting particle behavior in a practical experimental OT setup and can be used to characterize and predict particle handling dynamics within any arbitrary OT force profile.

Electrostatic interactions in soft particle systems: mesoscale simulations of ionic liquids

Computer simulations provide a unique insight into the microscopic details, molecular interactions and dynamic behavior responsible for many distinct physicochemical properties of ionic liquids. Due to the sluggish and heterogeneous dynamics and the long-ranged nanostructured nature of ionic liquids, coarse-grained meso-scale simulations provide an indispensable complement to detailed first-principles calculations and atomistic simulations allowing studies over extended length and time scales with a modest computational cost. Here, we present extensive coarse-grained simulations on a series of ionic liquids of the 1-alkyl-3-methylimidazolium (alkyl = butyl, heptyl-, and decyl-) family with Cl, [BF4], and [PF6] counterions. Liquid densities, microstructures, translational diffusion coefficients, and re-orientational motion of these model ionic liquid systems have been systematically studied over a wide temperature range. The addition of neutral beads in cationic models leads to a transition of liquid morphologies from dispersed apolar beads in a polar framework to that characterized by bi-continuous sponge-like interpenetrating networks in liquid matrices. Translational diffusion coefficients of both cations and anions decrease upon lengthening of the neutral chains in the cationic models and by enlarging molecular sizes of the anionic groups. Similar features are observed in re-orientational motion and time scales of different cationic models within the studied temperature range. The comparison of the liquid properties of the ionic systems with their neutral counterparts indicates that the distinctive microstructures and dynamical quantities of the model ionic liquid systems are intrinsically related to Coulombic interactions. Finally, we compared the computational efficiencies of three linearly scaling O(N log N) Ewald summation methods, the particle-particle particle-mesh method, the particle-mesh Ewald summation method, and the Ewald summation method based on a non-uniform fast Fourier transform technique, to calculate electrostatic interactions. Coarse-grained simulations were performed using the GALAMOST and the GROMACS packages and hardware efficiently utilizing graphics processing units on a set of extended [1-decyl-3-methylimidazolium][BF4] ionic liquid systems of up to 131 072 ion pairs.

Interactions between charged particles with bathing multivalent counterions: experiments vs. dressed ion theory

We compare the recent experimentally measured forces between charged colloidal particles, as well as their effective surface potentials (surface charge) in the presence of multivalent counterions in a bathing monovalent salt solution, with the predictions of the dressed ion theory of strongly charged colloidal systems. The benchmark for comparison is provided by the DLVO theory and the deviations from its predictions at small separations are taken as an indication of the additional non-DLVO attractions that can be fitted by an additional phenomenological exponential term. The parameters characterizing this non-DLVO exponential term as well as the dependencies of the effective potential on the counterion concentration and valency predicted by the dressed ion theory are well within the experimental values. This suggests that the deviations from the DLVO theory are probably caused by ion correlations as formalized within the dressed ion theory.

Interactions between similar and dissimilar charged interfaces in the presence of multivalent anions

With direct force measurements, we identify a short-ranged attraction, which acts not only between similar interfaces, but also between dissimilar ones.

Attractive non-DLVO forces induced by adsorption of monovalent organic ions

Direct force measurements between negatively charged colloidal particles were carried out using an atomic force microscope (AFM) in aqueous solutions containing monovalent organic cations, namely tetraphenylarsonium (Ph₄As⁺), 1-hexyl-3-methylimidazolium (HMIM⁺), and 1-octyl-3-methylimidazolium (OMIM⁺).

Interactions between silica particles in the presence of multivalent coions

Forces between charged silica particles in solutions of multivalent coions are measured with colloidal probe technique based on atomic force microscopy. The concentration of 1 : z electrolytes is systematically varied to understand the behavior of electrostatic interactions and double-layer properties in these systems. Although the coions are multivalent the Derjaguin, Landau, Verwey, and Overbeek (DLVO) theory perfectly describes the measured force profiles. The diffuse-layer potentials and regulation properties are extracted from the forces profiles by using the DLVO theory. The dependencies of the diffuse-layer potential and regulation parameter shift to lower concentration with increasing coion valence when plotted as a function of concentration of 1 : z salt. Interestingly, these profiles collapse to a master curve if plotted as a function of monovalent counterion concentration.

Screened Coulomb interactions with non-uniform surface charge

The screened Coulomb interaction between a pair of infinite parallel planes with spatially varying surface charge is considered in the limit of small electrical potentials for arbitrary Debye lengths. A simple expression for the disjoining pressure is derived in terms of a two-dimensional integral in Fourier space. The integral is evaluated for periodic and random charge distributions and the disjoining pressure is expressed as a sum over Fourier–Bloch reciprocal lattice vectors or in terms of an integral involving the autocorrelation function, respectively. The force between planes with a finite area of uniform charge, a model for the DLVO interaction between finite surfaces, is also calculated. It is shown that the overspill of the charge cloud beyond the region immediately between the charged areas results in a reduction of the disjoining pressure, as reported by us recently in the long Debye length limit for planes of finite width.

Repulsion Between Finite Charged Plates with Strongly Overlapped Electric Double Layers

Screened Coulomb interactions between uniformly charged flat plates are considered at very small plate separations for which the Debye layers are strongly overlapped, in the limit of small electrical potentials. If the plates are of infinite length, the disjoining pressure between the plates decays as an inverse power of the plate separation. If the plates are of finite length, we show that screening Debye layer charges close to the edge of the plates are no longer constrained to stay between the plates, but instead spill out into the surrounding electrolyte. The resulting change in the disjoining pressure is calculated analytically: the force between the plates is reduced by this edge correction when the charge density is uniform over the surface of the plates, and is increased when the surface is at constant potential. A similar change in disjoining pressure due to loss of lateral confinement of the Debye layer charges should occur whenever the sizes of the interacting charged objects become small enough to approach the Debye scale. We investigate the effect here in the context of a two-dimensional model problem that is sufficiently simple to yield analytical results.

Effects of non-Gaussian Brownian motion on direct force optical tweezers measurements of the electrostatic forces between pairs of colloidal particles

Measurements of the electrostatic force with separation between a fixed and an optically trapped colloidal particle are examined with experiment, simulation and analytical calculation. Non-Gaussian Brownian motion is observed in the position of the optically trapped particle when particles are close and traps weak. As a consequence of this motion, a simple least squares parameterization of direct force measurements, in which force is inferred from the displacement of an optically trapped particle as separation is gradually decreased, contains forces generated by the rectification of thermal fluctuations in addition to those originating directly from the electrostatic interaction between the particles. Thus, when particles are close and traps weak, simply fitting the measured direct force measurement to DLVO theory extracts parameters with modified meanings when compared to the original formulation. In such cases, however, physically meaningful DLVO parameters can be recovered by comparing the measured non-Gaussian statistics to those predicted by solutions to Smoluchowski's equation for diffusion in a potential.

Charge Regulation in the Electrical Double Layer: Ion Adsorption and Surface Interactions

Charge regulation in the electrical double layer has important implications for ion adsorption, interparticle forces, colloidal stability, and deposition phenomena. Although charge regulation generally receives little attention, its consequences can be major, especially when considering interactions between unequally charged surfaces. The present article discusses common approaches to quantify such phenomena, especially within classical Poisson-Boltzmann theory, and pinpoints numerous situations where a consideration of charge regulation is essential. For the interpretation of interaction energy profiles, we advocate the use of the constant regulation approximation, which summarizes the surface properties in terms of two quantities, namely, the diffuse layer potential and the regulation parameter. This description also captures some pronounced regulation effects observed in the presence of multivalent ions.

Measuring the interaction between a pair of emulsion droplets using dual-trap optical tweezers

Force–separation curves measured from a singe pair of emulsion droplets as a function of salt concentration (fits are DLVO theory).

Nanometer-ranged attraction induced by multivalent ions between similar and dissimilar surfaces probed using an atomic force microscope (AFM)

Forces between similar and dissimilar surfaces are quantified and a short-ranged attraction can be identified.

Surface adsorption of sulfonated poly(phenylene sulfone)/C14TAB mixtures and its correlation with foam film stability

Polyelectrolyte/surfactant mixtures of rigid monosulfonated poly(phenylene sulfone) (sPSO₂-220) and tetradecyl trimethylammonium bromide (C₁₄TAB) were investigated by surface tension, surface elasticity and foam film stability measurements.

Developing a video tracking method to study interactions between close pairs of optically trapped particles in three dimensions

We develop a video tracking method that utilizes an interpolation-based normalized cross-correlation approach to track the position of microscopic spherical particles in three dimensions. Subnanometer resolution is demonstrated. The method does not assume that the particle's image is radially symmetric, making it useful for determining the position when particles are close and their images overlap. This is demonstrated in a study of the electrostatic and hydrodynamic interactions between a pair of beads in dual laser tweezers traps.

Long-ranged and soft interactions between charged colloidal particles induced by multivalent coions

Forces between charged particles in aqueous solutions containing multivalent coions and monovalent counterions are studied by the colloidal probe technique. Here, the multivalent ions have the same charge as the particles, which must be contrasted to the frequently studied case where multivalent ions have the opposite sign as the substrate. In the present case, the forces remain repulsive and are dominated by the interactions of the double layers. The valence of the multivalent coion is found to have a profound influence on the shape of the force curve. While for monovalent coions the force profile is exponential down to separations of a few nanometers, the interaction is much softer and longer-ranged in the presence of multivalent coions. The force profiles in the presence of multivalent coions and in the mixtures of monovalent and multivalent coions can be accurately described by Poisson-Boltzmann theory. These results are accurate for different surfaces and even in the case of highly charged particles. This behavior can be explained by the fact that the force profile follows the near-field limit to much larger distances for multivalent coions than for monovalent ones. This limit corresponds to the conditions with no salt, where the coions are expelled between the two surfaces.

Toward optical-tweezers-based force microscopy for airborne microparticles

Optical tweezers have found widespread application in biological and colloidal physics for the measurement of pN forces over nanometer to micrometer length scales. Similar aerosol-phase measurements of interparticle force have not been reported in spite of the potential to better resolve particle coagulation kinetics. Various refractive index mismatches in the beam path as well as the need to explicitly account for gravity and inertial particle motion provide a number of challenges that must be overcome to make such measurements tractable. In this regard, we demonstrate schemes by which the particle position and trap stiffness may be unambiguously measured using bright-field microscopy with resolution comparable with analogous condensed-phase measurements. Moreover, some of the challenges of working with highly dynamic aqueous particles are introduced and exploited to observe size-dependent phenomena in aerosol optical tweezers. Notably, when combined with cavity-enhanced Raman spectroscopy, this provides a unique opportunity to explore trapping forces over a continuum of particle size and refractive index. It is expected that the methods developed will provide a basis for the measurement of pairwise interaction forces in aerosol optical tweezers while providing a probe of fundamental airborne particle trapping dynamics.

Electric double-layer potentials and surface regulation properties measured by colloidal-probe atomic force microscopy

We show how the colloidal-probe technique, which is based on force measurements made with the atomic force microscope, can be used to accurately determine the charging parameters of water-solid interfaces. Besides yielding accurate values of the double-layer or diffuse-layer potential, the method also allows reliable determination of the charge regulation properties of the surfaces. The latter can be quantified with a regulation parameter, which is essential to properly describe forces between interfaces, especially in asymmetric situations when one of the interfaces is charged and the other one is close to neutral. The technique relies on a highly charged probe particle, for which the charging properties are accurately determined by interpreting the double-layer contribution of the measured force profiles in the symmetric sphere-sphere geometry with Poisson-Boltzmann (PB) theory. Once the probe particle is calibrated, this particle is used to measure the force profile between an unknown substrate in the asymmetric sphere-sphere or sphere-plane geometry. From this profile, the diffuse-layer potential and regulation parameter of the substrate can be again determined with PB theory. The technique is highly versatile, as it can be used for a wide variety of substrates, including colloidal particles and planar substrates. The technique is also applicable in salt solutions containing multivalent ions. The current drawbacks of the technique are that it can only be applied up to moderately high salt levels, typically to 10 mM, and only for relatively large particles, typically down to about 1 μm in diameter. How the technique could be extended to higher salt levels and smaller particle size is also briefly discussed.

Polyelectrolyte adsorption, interparticle forces, and colloidal aggregation

This review summarizes the current understanding of adsorption of polyelectrolytes to oppositely charged solid substrates, the resulting interaction forces between such substrates, and consequences for colloidal particle aggregation. The following conclusions can be reached based on experimental findings. Polyelectrolytes adsorb to oppositely charged solid substrates irreversibly up to saturation, whereby loose and thin monolayers are formed. The adsorbed polyelectrolytes normally carry a substantial amount of charge, which leads to a charge reversal. Frequently, the adsorbed films are laterally heterogeneous. With increasing salt levels, the adsorbed mass increases leading to thicker and more homogeneous films. Interaction forces between surfaces coated with saturated polyelectrolyte layers are governed at low salt levels by repulsive electric double layer interactions, and particle suspensions are stable under these conditions. At appropriately high salt levels, the forces become attractive, principally due to van der Waals interactions, but eventually also through other forces, and suspensions become unstable. This situation can be rationalized with the classical theory of Derjaguin, Landau, Verwey, and Overbeek (DLVO). Due to the irreversible nature of the adsorption process, stable unsaturated layers form in colloidal particle suspensions at lower polyelectrolyte doses. An unsaturated polyelectrolyte layer can neutralize the overall particle surface charge. Away from the charge reversal point, electric double layer forces are dominant and particle suspensions are stable. As the charge reversal point is approached, attractive van der Waals forces become important, and particle suspensions become unstable. This behaviour is again in line with the DLVO theory, which may even apply quantitatively, provided the polyelectrolyte films are sufficiently laterally homogeneous. For heterogeneous films, additional attractive patch-charge interactions may become important. Depletion interactions may also lead to attractive forces and suspension destabilization, but such interactions become important only at high polyelectrolyte concentrations.

Large counterions boost the solubility and renormalized charge of suspended nanoparticles

Colloidal particles are ubiquitous in biology and in everyday products such as milk, cosmetics, lubricants, paints, or drugs. The stability and aggregation of colloidal suspensions are of paramount importance in nature and in diverse nanotechnological applications, including the fabrication of photonic materials and scaffolds for biological assemblies, gene therapy, diagnostics, targeted drug delivery, and molecular labeling. Electrolyte solutions have been extensively used to stabilize and direct the assembly of colloidal particles. In electrolytes, the effective electrostatic interactions among the suspended colloids can be changed over various length scales by tuning the ionic concentration. However, a major limitation is gelation or flocculation at high salt concentrations. This is explained by classical theories, which show that the electrostatic repulsion among charged colloids is significantly reduced at high electrolyte concentrations. As a result, these screened colloidal particles are expected to aggregate due to short-range attractive interactions or dispersion forces as the salt concentration increases. We discuss here a robust, tunable mechanism for colloidal stability by which large counterions prevent highly charged nanoparticles from aggregating in salt solutions with concentrations up to 1 M. Large counterions are shown to generate a thicker ionic cloud in the proximity of each charged colloid, which strengthens short-range repulsions among colloidal particles and also increases the corresponding renormalized colloidal charge perceived at larger separation distances. These effects thus provide a reliable stabilization mechanism in a broad range of biological and synthetic colloidal suspensions.

Poisson-Boltzmann description of interaction forces and aggregation rates involving charged colloidal particles in asymmetric electrolytes

Forces and aggregation rates involving spherical particles are studied numerically within the theory of Derjaguin, Landau, Verwey, and Overbeek (DLVO) for asymmetric and mixed electrolytes. Thereby, the double layer interactions are treated at the Debye-Hückel (DH) and Poisson-Boltzmann (PB) levels. The DH model is applicable for weakly charged systems, and effects of ion valence enter only implicitly through the ionic strength. The PB model is necessary for more highly charged systems, and depends on the actual ionic composition. One finds that forces in asymmetric electrolytes at fixed ionic strength weaken when the valence of the counterions is increased or when the valence of the coions is decreased. In symmetric electrolytes, the effect of counterions is more important than the one of the coions. For weakly charged systems, the critical coagulation concentration (CCC) decreases with the square of the valence in symmetric electrolytes, while this decrease is weaker in asymmetric ones. With increasing charge density, the dependence of the CCC on the valence becomes stronger, but the classical Schulze-Hardy decrease with the sixths power of the valence is only recovered for unrealistically high charge densities. Mixtures of electrolytes are treated within the same framework, and one observes that already small amounts of multivalent ions affect the system considerably. An empirical mixing rule is proposed to describe the calculated CCCs.

Attractive Forces between Charged Colloidal Particles Induced by Multivalent Ions Revealed by Confronting Aggregation and Direct Force Measurements

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.