Long-range interactions between soft colloidal particles in slit-pore geometries

Combining theoretical and experimental techniques, we investigate the structure formation of charged colloidal suspensions of silica particles in bulk and in spatial confinement (slit-pore geometry). Our focus is to identify characteristic length scales determining typical quantities, such as the position of the main peak of the bulk structure factor and the period of the oscillatory force profile in the slitpore. We obtain these quantities from integral equations/SANS experiments (bulk) and Monte Carlo simulations/colloidal probe-AFM measurements (confinement), in which the theoretical calculations are based on the Derjaguin-Landau-Verwey-Overbeck (DLVO) potential. Both in bulk and in the slitpore, we find excellent qualitative and quantitative agreement between theory and experiment as long as the ionic strength chosen in the DLVO potential is sufficiently low (implying a relatively long-ranged interaction). In particular, the bulk properties of these systems obey the widely accepted density scaling of xi proportional to phi(-1/3). On the other hand, systems with larger ionic strengths and, consequently, more short-ranged interactions do not obey such power law behavior and rather resemble an uncharged hard-sphere fluid, in which the relevant length scale is the particle diameter.     

Double layer interactions between charge-regulated colloidal surfaces: pair potentials for spherical particles bearing ionogenic surface groups

The variational approach of Reiner and Radke (1990) is employed to investigate the effect of surface charge regulation upon the double layer interaction free energy V_e of pairs of colloidal particles immersed in an electrolyte. A model for dissociating surface groups that permits the consideration of an arbitrary number ofion-complexation reactions is introduced. The variational method is then used to derive (in the Poisson-Boltzmann approximation) the configurational free energy functional Ω of an ensemble of particles bearing such groups. The Debye-Hückel (DH) linearization process is applied to this functional, and ensuing consistency issues are examined.The DH free energy is extremized for a configuration of two interacting flat plates, and Derjaguin's (1934 and 1939) method is used to obtain an approximate analytical form for V_e for two different-sized spherical particles bearing different surface groups. This second problem is next considered from the perspective ofLevine's (1934, 1939b) exact multipole expansion of the electrostatic potential surrounding two axisymmetric particles. It is shown that the linear superposition approximation (LSA) for V_e developed by Levine (1939c) and Verwey and Overbeek (1948) emerges rigorously from this formulation in the limit of large interparticle separations. The interaction free energy from Levine's expansion is calculated te a six digit accuracy for identical spheres over the range of regulated behavior from fixed surface charge density q_s to fixed surface potential ψ_s for surface-surface separation h to Debye length λ ratios from 0 to 2 and ratios of the particle radius a to λ of 0.1, 1, and 10. These results are compared to those obtained from Derjaguin's method and the linear superposition approximation. Derjaguin's method is only quantitatively accurate (in error by less than 10%) for the largest value of a/λ and becomes progressively less so as the boundary is changed from perfectly regulating (constant ψ_s) to unregulated (constant q_s). Agreement of the LSA with the exact V_e is good over a wide range of parameters, but worsens for large a/λ and small h/λ. Appendices present extensions of our approach to surfaces bearing more than one type of complexing group and to the consideration of Stern layer formation at the particle-electrolyte boundary in the context of a standard model for metal oxide-aqueous interfaces.     

Extended DLVO interactions between spherical particles and rough surfaces

An "extended DLVO" approach that includes Lifshitz-van der Waals, Lewis acid-base, and electrostatic double layer interactions is used to describe interaction energies between spherical particles and rough surfaces. Favorable, unfavorable, and intermediate deposition conditions are simulated using surface properties representing common aquatic colloids and polymeric membranes. The surface element integration (SEI) technique and Derjaguin's integration method are employed to calculate interaction energy. Numerical simulations using SEI demonstrate that nanometer scale surface roughness features can produce a distribution of interaction energy profiles. Local interaction energies are statistically analyzed to define representative interaction energy profiles-minimum, average, and maximum-for various combinations of simulated particles and surfaces. In all cases, the magnitude of the average interaction energy profile is reduced, but the reduction of energy depends on particle size, asperity size, and density of asperities. In some cases, a surface that is on average unfavorable for deposition (repulsive) may possess locally favorable (attractive) sites solely due to nanoscale surface roughness. A weighted average of the analytical sphere-sphere and sphere-plate expressions of Derjaguin reasonably approximates the average interaction energy profiles predicted by the SEI model, where the weighting factor is based on the fraction of interactions involving asperities.     

Tailoring the interfacial assembly of colloidal particles by engineering the mechanical properties of the interface

Fluid interfaces can be used as a platform for promoting the direct and spontaneous self-assembly of colloidal particles, where the driving force is the reduction in interfacial energy. In addition, fluid interfaces allow fine-tuning of the particles ensemble by an external force, such as the presence of an imposed interfacial flow, or by engineering the interparticle interactions dictated by the interplay of interfacial forces. As a consequence, a wide-ranging set of interfacial structures can be achieved from liquid-like layers, which can flow under stress, to amorphous solids that are able to sustain static stress. Here, far from a comprehensive overview of the interfacial assembly of colloidal particles, different ways of tailoring it by rationally designing the rheological properties of the interface are provided, with a focus on experimental and theoretical methods and model systems that have been recently exploited. In particular, ligand-coated nanoparticles, with a strong emphasis on the effect of the ligands on the interfacial structure and the rheological properties, and soft nanogel particles, in which an environmental factor, such as the temperature, drives to different interfacial structures and mechanical responses will be further discussed.     

An approximate method for calculating depletion and structural interactions between colloidal particles

An approximate method is evaluated for calculating the second virial coefficient in a dilute macromolecular solution bounded by two interfaces. The approximation is essentially the superposition of the coefficients calculated independently for each surface. To test this approach, the depletion interaction between two particles in a solution of nonadsorbing, spherical macromolecules is calculated in systems with either hard-wall or long-range electrostatic interactions. In all systems tested, the interparticle interaction energy calculated with the approximation is found to be in good agreement with that calculated using the exact approach (e.g., error less than 2% at the smallest separations). The primary advantage of this approximation is a significant reduction in the computation time required for calculating the depletion interaction, especially in charged systems. The paper also shows that the expressions for predicting the depletion interaction in purely hard-sphere systems can be used in dilute ionic systems, provided the appropriate effective macromolecule size is used. For the attractive depletion interaction, this effective size is determined by the range of the particle-macromolecule interaction (as opposed to the macromolecule-macromolecule interaction).     

Interactions between spheroidal colloidal particles

Using Derjaguin's approximation, we have evaluated the interaction energy associated with van der Waals, electrostatic, depletion, and capillary forces between colloidal spheroids. If the interaction range between spheroids is distinctly smaller than the lengths of their principal axes, then simple pair potentials that depend on particle distance and orientation can be derived. Attractive interactions between adjacent spheroids favor their parallel alignment. Parallel spheroids can be arranged into a variety of densely packed configurations. All of these configurations turn out to have the same lattice energy. We discuss the implications of this degeneracy with respect to the stability of photonic crystals consisting of spheroids.     

Closure-based density functional theory applied to interfacial colloidal fluids

The behavior of dense colloidal fluids near surfaces can now be probed in great detail with experimental techniques like confocal microscopy. In fact, we are approaching a point where quantitative comparisons of experiment with particle-level theory, such as classical density functional theory (DFT), are appropriate. In a forward sense, we may use a known surface potential to predict a particle density distribution function from DFT; in an inverse sense, we may use an experimentally measured particle density distribution function to predict the underlying surface potential from DFT. In this paper, we tested the ability of the closure-based DFT of Zhou and Ruckenstein (J. Chem. Phys. 2000, 112, 8079-8082) to perform forward and inverse calculations on potential models commonly employed for colloidal particles and surfaces. To reduce sources of uncertainty in this initial study, Monte Carlo simulation results played the role of experimental data. The combination of Rogers-Young and modified-Verlet closures consistently performed well across the different potential models. For a reasonable range of choices of the density, temperature, and potential parameters, the inversion procedure yielded particle-surface potentials to an accuracy on the order of 0.1kT.     

Model colloidal fluid with competing interactions: bulk and interfacial properties

Using a simple mean field density functional theory (DFT), the authors investigate the structure and phase behavior of a model colloidal fluid composed of particles interacting via a pair potential which has a hard core of diameter sigma, is attractive Yukawa at intermediate separations, and is repulsive Yukawa at large separations. The authors analyze the form of the asymptotic decay of the bulk fluid correlation functions, comparing results from DFT with those from the self-consistent Ornstein-Zernike approximation (SCOZA). In both theories the authors find rich crossover behavior, whereby the ultimate decay of correlation functions changes from monotonic to long wavelength damped oscillatory decay on crossing certain lines in the phase diagram or sometimes from oscillatory to oscillatory with a longer wavelength. For some choices of potential parameters the authors find, within the DFT, a lambda line at which the fluid becomes unstable with respect to periodic density fluctuations. SCOZA fails to yield solutions for state points near such a lambda line. The propensity towards clustering of particles, which is reflected by the presence of a long wavelength (>sigma) slowly decaying oscillatory pair correlation function, and a structure factor that exhibits a very sharp maximum at small but nonzero wave numbers, is enhanced in states near the lambda line. The authors present density profiles for the planar liquid-gas interface and for fluids adsorbed at a planar hard wall. The presence of a nearby lambda transition gives rise to pronounced long wavelength oscillations in the one-body density profiles at both types of interface.     

Size-dependent self-organization of colloidal particles on chemically patterned surfaces

A study of the self-organization of colloidal particles during the evaporation of particle solutions on chemically patterned surfaces is presented. On a surface with hydrophilic and hydrophobic regions, colloidal particles form compact structures on the hydrophilic sites. When a colloidal solution containing a mixture of particles with a variation in size is used, the number density of each type of particle deposited on the hydrophilic islands after evaporation decreases with increasing particle size. This makes it possible to produce a concentration gradient of the particles on islands of different sizes. It is shown that this technique could allow for particle separation.     

Small-angle neutron scattering study of concentrated colloidal dispersions: the electrostatic/steric composite interactions between colloidal particles

Small-angle neutron scattering was used to investigate the interactions in concentrated colloidal dispersions containing silica or polystyrene latex with adsorbed polyethyleneoxide (PEO). In these dispersions of charged particles, both electrostatic and steric repulsions are present. The PEO layer was made invisible to neutrons through contrast matching. The effect of the interparticle repulsion was clearly shown in the scattering spectra by the appearance of a peak at low Q. The effective potentials can be well described by the Hayter-Penfold/Yukawa (HPY) potential. In the silica dispersions studied, the layer thickness is small, hence the electrostatic potential dominates and the potential has a lower concentration dependence. In the dispersions of polystyrene latex, the adsorbed layer is thicker; consequently, the electrostatic potential dominates at low volume fraction (the potential has a lower concentration dependence), and the steric potential dominates at higher volume fraction (the potential has a higher concentration dependence). This study also suggests that when more than one potential is present the stronger one has a dominant influence in determining the structure factor. This finding makes it possible to describe the multicomponential interactions by a single function.     

Interactions between colloidal particles in polymer solutions: A density functional theory study

We present a density functional theory study of colloidal interactions in a concentrated polymer solution. The colloids are modeled as hard spheres and polymers are modeled as freely jointed tangent hard sphere chains. Our theoretical results for the polymer-mediated mean force between two dilute colloids are compared with recent simulation data for this model. Theory is shown to be in good agreement with simulation. We compute the colloid-colloid potential of mean force and the second virial coefficient, and analyze the behavior of these quantities as a function of the polymer solution density, the polymer chain length, and the colloid/polymer bead size ratio.     

Interactions between colloidal particles in amphiphilic mixtures: a density functional theory study

We present a density functional theory study of interactions between spherical colloidal particles in amphiphile solutions. Theory is found to be in good agreement with previously published molecular dynamics simulations. It is used to analyze the effect of the amphiphile solution bulk density, the chain length, and the solvent mole fraction on the potential of mean force between the particles. The general features of the potential of mean force are rationalized in terms of formation of layers and bilayers of amphiphilic molecules in the intercolloidal gap. Theory yields the same general trends as observed in simulations and in experiments. In particular, the computed mean force changes its character from repulsive to attractive and back to repulsive as the solvent mole fraction is gradually increased.     

Depletion force between anisometric colloidal particles

A simple mathematical model for the depletion force between two arbitrarily shaped large convex colloidal particles immersed in a suspension of small spherical particles is proposed. Using differential geometry, the interaction potential is expressed in terms of the mean and Gaussian curvature of the particle surfaces. The accuracy of theoretical results is tested by Monte Carlo simulations for parallel and nonparallel circular cylinders. The agreement between theoretical results and simulated data is very good if the density of the depletion agent is not too high.     

Polarization and interactions of colloidal particles in ac electric fields

Micrometer-sized polystyrene particles form two-dimensional crystals in alternating current (ac) electric fields. The induced dipole-dipole interaction is the dominant force that drives this assembly. We report measurements of forces between colloidal particles in ac electric fields using optical tweezers and find good agreement with the point dipole model. The magnitude of the pair interaction forces depends strongly on the bulk solution conductivity and decreases as the ionic strength increases. The forces also decrease with increasing field frequency. The salt and frequency dependences are consistent with double layer polarization with a characteristic relaxation frequency omega(CD) approximately a(2)/D, where a is the particle radius and D is the ion diffusivity. This enables us to reinterpret the order-disorder transition reported for micrometer-sized polystyrene particles [Lumsdon et al., Langmuir 20, 2108 (2004)], including the dependence on particle size, frequency, and ionic strength. These results provide a rational framework for identifying assembly conditions of colloidal particles in ac fields over a wide range of parameters.     

Design and fabrication of functional hydrogels through interfacial engineering

Hydrogels have drawn considerable attention in the past two decades due to their excellent biocompatibility and multi-stimuli responsiveness. They have a wide range of applications in the fields related to tissue engineering, sensors and biomedicine. Their applications are strongly influenced by the surface properties of hydrogels and the interfacial interactions between hydrogels and other substrates. In particular, the surface wettability and adhesion of hydrogels decide their applications as drug carriers and wound dressing materials. Nevertheless, there is a lack of systematic discussion on the surface functionalization strategies of hydrogels. Therefore, this review aims at summarizing the strategies of functionalizing the surfaces of hydrogels and bonding hydrogels with other solid substrates. It also explores the challenges and future perspectives of interfacial engineering of hydrogels.     

Electrostatic interactions of colloidal particles at vanishing ionic strength

Electrostatic interactions of colloidal particles are typically screened by mobile ions in the solvent. We measure the forces between isolated pairs of colloidal polymer microspheres as the density of bulk ions vanishes. The ionic strength is controlled by varying the concentration of surfactant (NaAOT) in a nonpolar solvent (hexadecane). While interactions are well-described by the familiar screened-Coulomb form at high surfactant concentrations, they are experimentally indistinguishable from bare Coulomb interactions at low surfactant concentration. Interactions are strongest just above the critical micelle concentration, where particles can obtain high surface potentials without significant screening, kappaa < 1. Exploiting the absence of significant charge renormalization, we are able to construct a simple thermodynamic model capturing the role of reverse micelles in charging the particle surface. These measurements provide novel access to electrostatic forces in the limit where the particle size is much less than the screening length, which is relevant not just to the nonpolar suspensions described here, but also to aqueous suspensions of nanoparticles.     

DLVO interaction of colloidal particles with topographically and chemically heterogeneous surfaces

The DLVO force and potential energy of interaction between microspheres and topographically and chemically heterogeneous surfaces in aqueous solution are computed using a modification of the surface element integration approach. The heterogeneous surface has an array of cylindrical pillars of varying height, diameter, and arrangement to model different nano-topographies. In agreement with previous studies, the nano-topography decreases the size of the potential energy barrier for unfavorable surfaces because the pillars limit the minimum separation distance. The influence of topography is significant even for pillars several nanometers high and is more pronounced if the surface potential of the pillar tops differs from that of the underlying surface. A new force- and energy-averaging model is introduced as a simple method to compute the mean interaction energy or force between the particle and a heterogeneous surface, which differs significantly from a mean-field approach based on the average or nominal surface potential. Small variations in topography are found to remove large energy barriers to colloidal deposition. These results help explain the increased attraction of patchy surfaces towards particles relative to expectations based on typical DLVO calculations, which is particularly significant for surfaces with adsorbed polyelectrolytes.     

Electrostatic interaction between two spherical colloidal particles

Explicit exact analytic expressions are obtained in the form of infinite series for the potential distribution and the potential energy of the electrostatic interaction for the system of two dissimilar spheres in an electrolyte solution on the basis of the linearized Poisson—Boltzmann equation without recourse to Derjaguin's approximation. The leading term of the expression for the interaction energy (the zeroth order approximation) corresponds to the interaction energy that would be obtained if both spheres were ion-penetrable spheres (“soft” spheres). This term is a screened Coulomb interaction due to a simple linear superposition of the unperturbed potentials of the respective spheres, which is proportional to the product of their unperturbed surface potentials. The first-order approximation corresponds to the interaction energy that would be obtained if either sphere were a soft particle (the other being hard). The first-order correction term consists of two sub-terms, each of which is proportional to the square of the unperturbed surface potential of either sphere and does not depend on the unperturbed surface potential of the other sphere, can be interpreted as the interaction between the soft sphere and its image with respect to the hard sphere. This image interaction is attractive if the surface potential of the hard sphere is constant and repulsive if the surface charge density of the sphere is constant. It is shown that Derjaguin's method as well as its extension to the interaction of unequal spheres by Hogg, Healy and Fuerstenau (HHF) is quite a good approximation.     

Photophysical effects between spirobenzopyran-methyl methacrylate-functionalized colloidal particles

Colloidal particles were derivatized with end-grafted polymethylmethacrylate polymer brushes containing varying concentrations of spirobenzopyran photochromic molecules. The polymers were grown from initiator-functionalized silica particles by an atom-transfer radical polymerization (ATRP). These core-shell colloids formed stable suspensions in toluene with the spirobenzopyran in its closed, nonpolar form. However, UV-induced photoswitching of the photochrome to its open, polar merocyanine isomer caused rapid aggregation. The nature of this colloidal stability transition was examined with respect to the spirobenzopyran content in the polymeric brush and solvent polarity. Turbidimetry, wettability studies, UV-vis spectroscopy, suspension rheology, SEM, and visual inspection were utilized to characterize the system photoswitchability. It was found that the system exhibiting the greatest transition in toluene was the copolymer brush composed of 20% spirobenzopyran and 80% methyl methacrylate.