Capillary forces in suspension rheology

Structure of capillary suspensions and their versatile applications in the creation of smart materials

In this article, we review recent research in the field of capillary suspensions and highlight a variety of applications in the field of smart materials. Capillary suspensions are liquid-liquid-solid ternary systems where one liquid is only present in a few percent and induces a strong, capillary-induced particle network. These suspensions have a large potential for exploitation, particularly in the production of porous materials since the paste itself and the properties of the final material can be adapted. We also discuss the rheological properties of the suspension and network structure to highlight the various ways these systems can be tuned.

Negative normal stress differences N1-N2 in a low concentration capillary suspension

Here, negative normal stress differences are reported in capillary suspensions, i.e. particle suspensions in a two-fluid system that creates strong capillary attractions, at a solid concentration of 25%, and a volume fraction that has heretofore been considered too low to show such normal stress differences. Such capillary suspensions have strong particle networks and are shear thinning for the entire range of shear rates studied. Capillary suspensions exist in two states: a pendular state when the secondary fluid preferentially wets the particles, and a capillary state when the bulk fluid is preferentially wetting. In the pendular state, the system undergoes a transition from a positive normal stress difference at high shear rates to a negative stress difference at low shear rates. These results are an indication of flexible flocs in the pendular state that are able to rotate to reorientate in the vorticity direction under shear. Analogous experiments were also conducted for the capillary state, where only a negative normal stress difference occurs. The capillary state system forms more network contacts due to droplet breakup at higher shear rates, which enhances the importance of hydrodynamic interactions in the non-colloidal suspension.

Capillary assembly as a tool for the heterogeneous integration of micro- and nanoscale objects

During the past decade, capillary assembly in topographical templates has evolved into an efficient method for the heterogeneous integration of micro- and nano-scale objects on a variety of surfaces. This assembly route has been applied to a large spectrum of materials of micrometer to nanometer dimensions, supplied in the form of aqueous colloidal suspensions. Using systems produced via bulk synthesis affords a huge flexibility in the choice of materials, holding promise for the realization of novel superior devices in the fields of optics, electronics and health, if they can be integrated into surface structures in a fast, simple, and reliable way. In this review, the working principles of capillary assembly and its fundamental process parameters are first presented and discussed. We then examine the latest developments in template design and tool optimization to perform capillary assembly in more robust and efficient ways. This is followed by a focus on the broad range of functional materials that have been realized using capillary assembly, from single components to large-scale heterogeneous multi-component assemblies. We then review current applications of capillary assembly, especially in optics, electronics, and in biomaterials. We conclude with a short summary and an outlook for future developments.

Radical polymerization of capillary bridges between micron-sized particles in liquid bulk phase as a low temperature route to produce porous solid materials

We present a generic and versatile low temperature route to produce macro-porous bodies with porosity and pore size distribution that are adjustable in a wide range. Capillary suspensions, where the minor fluid is a monomer, are used as pre-cursors. The monomer is preferentially located between the particles, creating capillary bridges, resulting in a strong, percolating network. Thermally induced polymerization of these bridges at temperatures below 100 °C for less than 5 hours and subsequent removal of the bulk fluid yields macroscopic, self-supporting solid bodies with high porosity. This process is demonstrated using methylmethacrylate and hydroxyethylmethacrlyate with glass particles as a model system. The produced PMMA had a molecular weight of about 500.000 g/mol and dispersity about three. Application specific porous bodies, including PMMA particles connected by PMMA bridges, micron-sized capsules containing phase change material with high inner surface, and porous graphite membranes with high electrical conductivity, are also shown.

Rheology of particle/water/oil three-phase dispersions: Electrostatic vs. capillary bridge forces

HYPOTHESIS

Particle/water/oil three-phase capillary suspensions possess the remarkable property to solidify upon the addition of minimal amount of the second (dispersed) liquid. The hardening of these suspensions is due to capillary bridges, which interconnect the particles (pendular state). Electrostatic repulsion across the oily phase, where Debye screening by electrolyte is missing, could also influence the hardness of these suspensions.

EXPERIMENTS

We present data for oil-continuous suspensions with aqueous capillary bridges between hydrophilic SiO₂ particles at particle volume fractions 35-45%. The hardness is characterized by the yield stress Y for two different oils: mineral (hexadecane) and vegetable (soybean oil).

FINDINGS AND MODELLING

The comparison of data for the "mirror" systems of water- and oil-continuous capillary suspensions shows that Y is lower for the oil-continuous ones. The theoretical model of yield stress is upgraded by including a contribution from electrostatic repulsion, which partially counterbalances the capillary-bridge attraction and renders the suspensions softer. The particle charge density determined from data fits is close to that obtained in experiments with monolayers from charged colloid particles at oil/water interfaces. The results could contribute for better understanding, quantitative prediction and control of the mechanical properties of solid/liquid/liquid capillary suspensions.

Capillary Structured Suspensions from In Situ Hydrophobized Calcium Carbonate Particles Suspended in a Polar Liquid Media

We demonstrate that capillary suspensions can be formed from hydrophilic calcium carbonate particles suspended in a polar continuous media and connected by capillary bridges formed of minute amounts of an immiscible secondary liquid phase. This was achieved in two different polar continuous phases, water and glycerol, and three different oils, oleic acid, isopropyl myristate, and peppermint oil as a secondary liquid phase. The capillary structuring of the suspension was made possible through local in situ hydrophobization of the calcium carbonate particles dispersed in the polar media by adding very small amounts of oleic acid to the secondary liquid phase. We observed a strong increase in the viscosity of the calcium carbonate suspension by several orders of magnitude upon addition of the secondary oil phase compared with the same suspension without secondary liquid phase or without oleic acid. The stability and the rheological properties of the obtained capillary structured materials were studied in relation to the physical properties of the system such as the particle size, interfacial tension between the primary and secondary liquid phases, as well as the particle contact angle at this liquid-liquid interface. We also determined the minimal concentrations of the secondary liquid phase at fixed particle concentration as well as the minimal particle concentration at fixed secondary phase concentration needed to form a capillary suspension. Capillary suspensions formed by this method can find application in structuring pharmaceutical and food formulations as well as a variety of home and personal care products.

Hardening of particle/oil/water suspensions due to capillary bridges: Experimental yield stress and theoretical interpretation

Suspensions of colloid particles possess the remarkable property to solidify upon the addition of minimal amount of a second liquid that preferentially wets the particles. The hardening is due to the formation of capillary bridges (pendular rings), which connect the particles. Here, we review works on the mechanical properties of such suspensions and related works on the capillary-bridge force, and present new rheological data for the weakly studied concentration range 30-55 vol% particles. The mechanical strength of the solidified capillary suspensions, characterized by the yield stress Y, is measured at the elastic limit for various volume fractions of the particles and the preferentially wetting liquid. A quantitative theoretical model is developed, which relates Y with the maximum of the capillary-bridge force, projected on the shear plane. A semi-empirical expression for the mean number of capillary bridges per particle is proposed. The model agrees very well with the experimental data and gives a quantitative description of the yield stress, which increases with the rise of interfacial tension and with the volume fractions of particles and capillary bridges, but decreases with the rise of particle radius and contact angle. The quantitative description of capillary force is based on the exact theory and numerical calculation of the capillary bridge profile at various bridge volumes and contact angles. An analytical formula for Y is also derived. The comparison of the theoretical and experimental strain at the elastic limit reveals that the fluidization of the capillary suspension takes place only in a deformation zone of thickness up to several hundred particle diameters, which is adjacent to the rheometer's mobile plate. The reported experimental results refer to water-continuous suspension with hydrophobic particles and oily capillary bridges. The comparison of data for bridges from soybean oil and hexadecane surprisingly indicate that the yield strength is greater for the suspension with soybean oil despite its lower interfacial tension against water. The result can be explained with the different contact angles of the two oils in agreement with the theoretical predictions. The results could contribute for a better understanding, quantitative prediction and control of the mechanical properties of three-phase capillary suspensions solid/liquid/liquid.

Fractal approaches to characterize the structure of capillary suspensions using rheology and confocal microscopy

The rheological properties of a particle suspension can be substantially altered by adding a small amount of a secondary fluid that is immiscible with the bulk phase. The drastic change in the strength of these capillary suspensions arises due to the capillary forces, induced by the added liquid, leading to a percolating particle network. Using rheological scaling models, fractal dimensions are deduced from the yield stress and from oscillatory strain amplitude sweep data as function of the solid volume fraction. Exponents obtained using aluminum-oxide-based capillary suspensions, with a preferentially wetting secondary fluid, indicate an increase in the particle gel's fractal dimension with increasing particle size. This may be explained by a corresponding relative reduction in the capillary force compared to other forces. Confocal images using a glass model system show the microstructure to consist of compact particle flocs interconnected by a sparse backbone. Thus, using the rheological models two different fractal dimensionalities are distinguished - a lower network backbone dimension (D = 1.86-2.05) and an intrafloc dimension (D = 2.57-2.74). The latter is higher due to the higher local solid volume fraction inside of the flocs compared to the sparse backbone. Both of these dimensions are compared with values obtained by analysis of spatial particle positions from 3D confocal microscopy images, where dimensions between 2.43 and 2.63 are computed, lying between the two dimension ranges obtained from rheology. The fractal dimensions determined via this method corroborate the increase in structural compactness with increasing particle size.

Thermally Responsive Capillary Suspensions

We demonstrate that stimulus-responsive capillary-structured materials can be formed from hydrophobized calcium carbonate particles suspended in a non-polar phase (silicone oil) and bridged by very small amounts of a hydrogel as the secondary aqueous phase. Inclusion of thermally responsive polymers into the aqueous phase yielded a capillary-structured suspension whose rheology is controlled by a change in temperature and can increase its complex modulus by several orders of magnitude because of the gelation of the capillary bridges between the solid particles. We demonstrate that the rheology of the capillary suspension and its response upon temperature changes can be controlled by the gelling properties as little as 0.1 w/w % of the secondary aqueous phase containing 2 wt % of the gelling carbohydrate. Doping the secondary (aqueous) phase with methyl cellulose, which gels at elevated temperatures, gave capillary-structured materials whose viscosity and structural strength can increase by several orders of magnitude as the temperature is increased past the gelling temperature of the methyl cellulose solution. Increasing the methyl cellulose concentration from 0 to 2 w/w % in the secondary (aqueous) phase increases the complex modulus and the yield stress of the capillary suspension of 10 w/w % hydrophobized calcium carbonate in silicone oil by 2 orders of magnitude at a fixed temperature. By using an aqueous solution of a low melting point agarose as a secondary liquid phase, which melts as the temperature is raised, we produced capillary-structured materials whose viscosity and structural strength can decrease by several orders of magnitude as the temperature is increased past the melting temperature of the agarose solution. The development of thermally responsive capillary suspensions can find potential applications in structuring of smart home and personal care products as well as in temperature-triggered change in rheology and release of flavors in foods and actives in pharmaceutical formulations.

A microstructure-composition map of a ternary liquid/liquid/particle system with partially-wetting particles

We examine the effect of composition on the morphology of a ternary mixture comprising two molten polymeric liquid phases (polyisobutylene and polyethylene oxide) and micron-scale spherical silica particles. The silica particles were treated with silanes to make them partially wetted by both polymers. Particle loadings up to 30 vol% are examined while varying the fluid phase ratios across a wide range. Numerous effects of particle addition are catalogued, stabilization of Pickering emulsions and of interfacially-jammed co-continuous microstructures, meniscus-bridging of particles, particle-induced coalescence of the dispersed phase, and significant shifts in the phase inversion composition. Many of the effects are asymmetric, for example particle-induced coalescence is more severe and drop sizes are larger when polyisobutylene is the continuous phase, and particles promote phase continuity of the polyethylene oxide. These asymmetries are likely attributable to a slight preferential wettability of the particles towards the polyethylene oxide. A state map is constructed which classifies the various microstructures within a triangular composition diagram. Comparisons are made between this diagram vs. a previous one constructed for the case when particles are fully-wetted by polyethylene oxide.

Colloidal suspensions in one-phase mixed solvents under shear flow

We numerically studied the behaviour of colloidal suspensions in one-phase binary liquid mixtures under shear flows. Far from the phase-separation point, the colloidal particles are well dispersed and the suspension exhibits a Newtonian viscosity. When the mixture is close to the coexistence curve, the colloidal particles aggregate by attractive interactions due to the concentration heterogeneity caused by surface wetting, and the viscosity of the suspension increases. Near the phase-separation point, the viscosity increases when the fraction of species favoured by the surface of a colloid particle is small. The mixture also exhibits shear thinning behaviour, since the aggregated structure is rearranged into small clusters due to the shear flow. Our simulations indicate that the concentration profile around each particle is not significantly disturbed by the shear flow at the onset of the structural rearrangements. The effective interaction is independent of the shear flow and remains isotropic.

3D Printing by Multiphase Silicone/Water Capillary Inks

3D printing of polymers is accomplished easily with thermoplastics as the extruded hot melt solidifies rapidly during the printing process. Printing with liquid polymer precursors is more challenging due to their longer curing times. One curable liquid polymer of specific interest is polydimethylsiloxane (PDMS). This study demonstrates a new efficient technique for 3D printing with PDMS by using a capillary suspension ink containing PDMS in the form of both precured microbeads and uncured liquid precursor, dispersed in water as continuous medium. The PDMS microbeads are held together in thixotropic granular paste by capillary attraction induced by the liquid precursor. These capillary suspensions possess high storage moduli and yield stresses that are needed for direct ink writing. They could be 3D printed and cured both in air and under water. The resulting PDMS structures are remarkably elastic, flexible, and extensible. As the ink is made of porous, biocompatible silicone that can be printed directly inside aqueous medium, it can be used in 3D printed biomedical products, or in applications such as direct printing of bioscaffolds on live tissue. This study demonstrates a number of examples using the high softness, elasticity, and resilience of these 3D printed structures.

Influence of mixing conditions on the rheological properties and structure of capillary suspensions

The rheological properties of a suspension can be dramatically altered by adding a small amount of a secondary fluid that is immiscible with the bulk liquid. These capillary suspensions exist either in the pendular state where the secondary fluid preferentially wets the particles or the capillary state where the bulk fluid is preferentially wetting. The yield stress, as well as storage and loss moduli, depends on the size and distribution of secondary phase droplets created during sample preparation. Enhanced droplet breakup leads to stronger sample structures. In capillary state systems, this can be achieved by increasing the mixing speed and time of turbulent mixing using a dissolver stirrer. In the pendular state, increased mixing speed also leads to better droplet breakup, but spherical agglomeration is favored at longer times decreasing the yield stress. Additional mixing with a ball mill is shown to be beneficial to sample strength. The influence of viscosity variance between the bulk and second fluid on the droplet breakup is excluded by performing experiments with viscosity-matched fluids. These experiments show that the capillary state competes with the formation of Pickering emulsion droplets and is often more difficult to achieve than the pendular state.

Suppressing Crack Formation in Particulate Systems by Utilizing Capillary Forces

Cracks, formed during the drying of particulate films, can reduce the effectiveness or even render products useless. We present a novel, generic approach to suppress crack formation in thin films made from hard particle suspensions, which are otherwise highly susceptible to cracking, using the capillary force between particles present when a trace amount of an immiscible liquid is added to a suspension. This secondary liquid preserves the particle cohesion, modifying the structure and increasing the drying rate. Crack-free films can be produced at thicknesses much greater than the critical cracking thickness for a suspension without capillary interactions, and even persists after sintering. This capillary suspension strategy is applicable to a broad range of materials, including suspensions of metals, semiconductive and ceramic oxides, or glassy polymeric particles, and can be easily implemented in many industrial processes since it is based on well-established unit operations. Promising fields of application include ceramic foils and printed electronic devices.

Dependence of the Internal Structure on Water/Particle Volume Ratio in an Amphiphilic Janus Particle-Water-Oil Ternary System: From Micelle-like Clusters to Emulsions of Spherical Droplets

Amphiphilic Janus particles (AJP), composed of hydrophilic and hydrophobic hemispheres, are one of the simplest anisotropic colloids, and they exhibit higher surface activities than particles with homogeneous surface properties. Consequently, a ternary system of AJP, water, and oil can form extremely stable Pickering emulsions, with internal structures that depend on the Janus structure of the particles and the system composition. However, the detail of these structures has not been fully explored, especially for the composition range where the amount of the minority liquid phase and AJP are comparable, where one would expect the Janus characteristics to be directly reflected. In this study, we varied the volume ratio of the particles and the minority liquid phase, water, by 2 orders of magnitude around the comparable composition range, and observed the resultant structures at the resolution of the individual particle dimensions by optical microscopy. When the volume ratio of water is smaller than that of the Janus particles, capillary interactions between the hydrophilic hemispheres of the particles induce micelle-like clusters in which the hydrophilic sides of the particles face inward. With increasing water content, these clusters grow into a rodlike morphology. When the water volume exceeds that of the particles, the structure transforms into an emulsion state composed of spherical droplets, colloidosomes, because of the surface activity of particles at the liquid-liquid interface. Thus, we found that a change in volume fraction alters the mechanism of structure formation in the ternary system, and large resulting morphological changes in the self-assembled structures reflect the anisotropy of the particles. The self-assembly shows essential commonalities with that in microemulsions of surfactant molecules, however the AJP system is stabilized only kinetically. Analysis of the dependence of the emulsion droplet size on composition shows that almost all the particles are adsorbed at the water-oil interface; i.e., the particles show ideal surface activity.

Highly conductive, printable pastes from capillary suspensions

We have used the capillary suspension phenomenon to design conductive pastes for printed electronic applications, such as front side metallization of solar cells, without non-volatile, organic additives that often deteriorate electrical properties. Adding a small amount of a second, immiscible fluid to a suspension creates a network of liquid bridges between the particles. This capillary force-controlled microstructure allows for tuning the flow behavior in a wide range. Yield stress and low-shear viscosity can be adjusted such that long-term stability is provided by inhibiting sedimentation, and, even more importantly, narrow line widths and high aspect ratios are accessible. These ternary mixtures, called capillary suspensions, exhibit a strong degree of shear thinning that allows for conventional coating or printing equipment to be used. Finally, the secondary fluid, beneficial for stability and processing of the wet paste, completely evaporates during drying and sintering. Thus, we obtained high purity silver and nickel layers with a conductivity two times greater than could be obtained with state-of-the-art, commercial materials. This revolutionary concept can be easily applied to other systems using inorganic or even organic conductive particles and represents a fundamental paradigm change to the formulation of pastes for printed electronics.

Tunable Design of Structural Colors Produced by Pseudo-1D Photonic Crystals of Graphene Oxide

It is broadly observed that graphene oxide (GO) films appear transparent with a thickness of about several nanometers, whereas they appear dark brown or almost black with thickness of more than 1 μm. The basic color mechanism of GO film on a sub-micrometer scale, however, is not well understood. This study reports on GO pseudo-1D photonic crystals (p1D-PhCs) exhibiting tunable structural colors in the visible wavelength range owing to its 1D Bragg nanostructures. Striking structural colors of GO p1D-PhCs could be tuned by simply changing either the volume or concentration of the aqueous GO dispersion during vacuum filtration. Moreover, the quantitative relationship between thickness and reflection wavelength of GO p1D-PhCs has been revealed, thereby providing a theoretical basis to rationally design structural colors of GO p1D-PhCs. The spectral response of GO p1D-PhCs to humidity is also obtained clearly showing the wavelength shift of GO p1D-PhCs at differently relative humidity values and thus encouraging the integration of structural color printing and the humidity-responsive property of GO p1D-PhCs to develop a visible and fast-responsive anti-counterfeiting label. The results pave the way for a variety of potential applications of GO in optics, structural color printing, sensing, and anti-counterfeiting.

Low internal pressure in femtoliter water capillary bridges reduces evaporation rates

Capillary bridges are usually formed by a small liquid volume in a confined space between two solid surfaces. They can have a lower internal pressure than the surrounding pressure for volumes of the order of femtoliters. Femtoliter capillary bridges with relatively rapid evaporation rates are difficult to explore experimentally. To understand in detail the evaporation of femtoliter capillary bridges, we present a feasible experimental method to directly visualize how water bridges evaporate between a microsphere and a flat substrate in still air using transmission X-ray microscopy. Precise measurements of evaporation rates for water bridges show that lower water pressure than surrounding pressure can significantly decrease evaporation through the suppression of vapor diffusion. This finding provides insight into the evaporation of ultrasmall capillary bridges.

Highly Porous Materials with Unique Mechanical Properties from Smart Capillary Suspensions

Smart capillary suspensions are used to fabricate macroporous solids with unique features regarding porosity and mechanical strength from a wide range of materials, including carbon layers and polyethylene membranes, even if sintering or high-temperature treatment is not feasible. High-strength porous ceramics are obtained, tailoring neck and pore shape via controlled deposition of fine particles at the sintering necks.

Structure of Particle Networks in Capillary Suspensions with Wetting and Nonwetting Fluids

The mechanical properties of a suspension can be dramatically altered by adding a small amount of a secondary fluid that is immiscible with the bulk phase. The substantial changes in the strength of these capillary suspensions arise due to the capillary force inducing a percolating particle network. Spatial information on the structure of the particle networks is obtained using confocal microscopy. It is possible, for the first time, to visualize the different types of percolating structures of capillary suspensions in situ. These capillary networks are unique from other types of particulate networks due to the nature of the capillary attraction. We investigate the influence of the three-phase contact angle on the structure of an oil-based capillary suspension with silica microspheres. Contact angles smaller than 90° lead to pendular networks of particles connected with single capillary bridges or clusters comparable to the funicular state in wet granular matter, whereas a different clustered structure, the capillary state, forms for angles larger than 90°. Particle pair distribution functions are obtained by image analysis, which demonstrate differences in the network microstructures. When porous particles are used, the pendular conformation also appears for apparent contact angles larger than 90°. The complex shear modulus can be correlated to these microstructural changes. When the percolating structure is formed, the complex shear modulus increases by nearly three decades. Pendular bridges lead to stronger networks than the capillary state network conformations, but the capillary state clusters are nevertheless much stronger than pure suspensions without the added liquid.

Interaction between Polymeric Additives and Secondary Fluids in Capillary Suspensions

Capillary suspensions are ternary systems including a solid and two liquid phases representing a novel formulation platform for pastes with unique processing and end-use properties. Here we have investigated aqueous suspensions of non-Brownian graphite particles including different polymers commonly used as thickening agents or binders in paste formulations. We have studied the interaction between these additives and organic solvents in order to elucidate its effect on the characteristic formation of a particle network structure in corresponding ternary capillary suspension systems. Organic solvents with different polarity have been employed, and in the presence of nonadsorbing poly(ethylene oxide), all of them, whether they preferentially wet the graphite surface or not, induce the formation of a network structure within the suspension as indicated by a strong change in rheological properties. However, when the adsorbing polymers carboxymethylcellulose and poly(vinylpyrrolidone) are included, the drastic change in rheological behavior occurs only when polar organic solvents are used as secondary liquids. Obviously, these solvents can form pendular bridges, finally resulting in a sample-spanning particle network. Vibrational sum frequency spectroscopy provides evidence that these polar liquids remove the adsorbed polymer from the graphite particles. In contrast, nonpolar and nonwetting solvents do not force polymer desorption. In these cases, the formation of a percolating network structure within the suspensions is presumably prevented by the strong steric repulsion among graphite particles, not allowing for the formation of particle clusters encapsulating the secondary liquid. Accordingly, polymeric additives and secondary fluids have to be carefully selected in capillary suspension formulations, then offering a new pathway to customize paste formulations. The polymer may serve to adjust an appropriate viscosity level, and the capillary bridging induces the desired degree of shear thinning. Alternatively, the polymer may be selected with respect to its binding properties in the final dry product, and capillary bridging may be used to control the flow and processing behavior of the wet paste.

Three distinct open-pore morphologies from a single particle-filled polymer blend

Ternary mixtures composed of polyisobutylene (PIB), polyethylene oxide (PEO), and silica particles yield three distinct open-pore morphologies depending on the mixture composition: (1) pendular network (particles bonded together by menisci of PEO); (2) capillary aggregate network (particles and PEO form a combined phase with strongly solid-like properties which forms a percolating network); (3) cocontinuous morphology (silica and the PEO form a highly viscous combined phase which retards interfacial tension-driven coarsening). Remarkably, interfacial tension plays altogether different roles in stabilizing these three morphologies: stabilizing the first, not affecting the second, and destabilizing the last. The first two of these morphologies appear to be generalizable to other systems, e.g. to oil/water/particle mixtures. In all three cases, the pores do not collapse even after flow, i.e. all three porous morphologies are amenable to processing.

The capillary bridge between two spheres: New closed-form equations in a two century old problem

We discuss progress in obtaining explicit equations for the capillary force between nano and micron sized solid spheres. Early approaches to this two-century old problem adopted approximations to the geometry. With the toroidal approximation, the meridian profile is approximated by an arc, and the approach leads to the capillary force being dependent on the location at which the force is evaluated. The Derjaguin approximation further assumes that the meridian radius is orders of magnitude smaller than the azimuth radius. An explicit expression for the capillary force is obtained, but the equation is limited to sufficiently small liquid volumes and separation distances. Significant progress has been made in recent years in using numerical solutions to derive analytical expressions for capillary bridges. Early numerical investigation established that the maximum separation for stable capillary bridges before rupture scales to the cubic root of the liquid volume. We report new progress in using numerical solutions to obtain more accurate and more general closed-form expressions for capillary bridges. Simple explicit algebraic equations have been observed to fit the numerical results well, leading to a closed-form solution applicable to capillary bridges between equal and unequal spheres and with zero or finite solid-liquid contact angles. The newly derived closed-form equation is more accurate and reduces to the Derjaguin equation when the liquid volume (or half-filling angle) and separation distance are both sufficiently small.

A non-equilibrium state diagram for liquid/fluid/particle mixtures

The equilibrium structures of ternary oil/water/surfactant systems are often represented within a triangular composition diagram with various regions of the triangle corresponding to different equilibrium states. We transplant this idea to ternary liquid/fluid/particle systems that are far from equilibrium. Liquid/liquid/particle mixtures or liquid/gas/particle mixtures yield a wide diversity of morphologies including Pickering emulsions, bijels, pendular aggregates, spherical agglomerates, capillary suspensions, liquid marbles, powdered liquids, and particle-stabilized foams. This paper argues that such ternary liquid/fluid/particle mixtures can be unified into a non-equilibrium state diagram. What is common among all these systems is that the morphology results from an interplay between the preferential wettability of the particles, capillarity, and viscous forces encountered during mixing. Therefore all such systems share certain universal features, regardless of the details of the particles or fluids used. These features guide the construction of a non-equilibrium state diagram which takes the form of a triangular prism, where each triangular cross-section of the prism corresponds to a different relative affinity of the particles towards the two fluids. We classify the prism into regions in which the various morphologies appear and also emphasize the major difference between systems in which the particles are fully-wetted by one of the fluids vs. partially-wetted by both fluids. We also discuss how the state diagram may change with mixing intensity or with interparticle attractions.

Multiple Effects of the Second Fluid on Suspension Viscosity

Previous research has shown that adding a small amount of a second immiscible fluid to particulate suspension can result in a significant influence on viscosity. In this study, the effects of the second fluid addition over a small dosage range on the rheological properties of particle suspension were investigated. As the dosage of the second fluid was increased, the viscosity and yield stress initially decreased then increased and finally decreased again. The microstructure of the suspension was observed using a confocal laser scanning microscope (CLSM) and showed three different states with the increasing dosage of the second fluid: a dispersive sate, cluster state and cell state in sequence. The presence of these states interpreted the non-monotonic trend of viscosity and yield stress in the suspensions.

Nanocapillarity-mediated magnetic assembly of nanoparticles into ultraflexible filaments and reconfigurable networks

The fabrication of multifunctional materials with tunable structure and properties requires programmed binding of their building blocks. For example, particles organized in long-ranged structures by external fields can be bound permanently into stiff chains through electrostatic or van der Waals attraction, or into flexible chains through soft molecular linkers such as surface-grafted DNA or polymers. Here, we show that capillarity-mediated binding between magnetic nanoparticles coated with a liquid lipid shell can be used for the assembly of ultraflexible microfilaments and network structures. These filaments can be magnetically regenerated on mechanical damage, owing to the fluidity of the capillary bridges between nanoparticles and their reversible binding on contact. Nanocapillary forces offer opportunities for assembling dynamically reconfigurable multifunctional materials that could find applications as micromanipulators, microbots with ultrasoft joints, or magnetically self-repairing gels.

Centrifugation-assisted Assembly of Colloidal Silica into Crack-Free and Transferrable Films with Tunable Crystalline Structures

Self-assembly of colloidal particles into colloidal films has many actual and potential applications. While various strategies have been developed to direct the assembly of colloidal particles, fabrication of crack-free and transferrable colloidal film with controllable crystal structures still remains a major challenge. Here we show a centrifugation-assisted assembly of colloidal silica spheres into free-standing colloidal film by using the liquid/liquid interfaces of three immiscible phases. Through independent control of centrifugal force and interparticle electrostatic repulsion, polycrystalline, single-crystalline and quasi-amorphous structures can be readily obtained. More importantly, by dehydration of silica particles during centrifugation, the spontaneous formation of capillary water bridges between particles enables the binding and pre-shrinkage of the assembled array at the fluid interface. Thus the assembled colloidal films are not only crack-free, but also robust and flexible enough to be easily transferred on various planar and curved substrates.

Stability Limits of Capillary Bridges: How Contact Angle Hysteresis Affects Morphology Transitions of Liquid Microstructures

The equilibrium shape of a drop in contact with solid surfaces can undergo continuous or discontinuous transitions upon changes in either drop volume or surface energies. In many instances, such transitions involve the motion of the three-phase contact line and are thus sensitive to contact angle hysteresis. Using a combination of electrowetting-based experiments and numerical calculations, we demonstrate for a generic sphere-plate confinement geometry how contact angle hysteresis affects the mechanical stability of competing axisymmetric and nonaxisymmetric drop conformations and qualitatively changes the character of transitions between them.

Capillary foams: stabilization and functionalization of porous liquids and solids

Liquid foams are two-phase systems in which a large volume of gas is dispersed as bubbles in a continuous liquid phase. These foams are ubiquitous in nature. In addition, they are found in industrial applications, such as pharmaceutical formulation, food processing, wastewater treatment, construction, and cosmetics. Recently, we reported a new type of foam material, capillary foam, which is stabilized by the synergistic action of particles and a small amount of an immiscible secondary liquid. In this study, we explore in more detail the foam preparation routes. To illustrate some of the potential applications, we create vividly colored wet and dried foams, which are difficult to prepare using traditional methods, and load-bearing porous solids. The combined action of particles and immiscible secondary fluid confers exceptional stability to capillary foams and many options for functionalization, suggesting a wide range of possible applications.

On the rheology of pendular gels and morphological developments in paste-like ternary systems based on capillary attraction

We investigate capillary bridging-induced gelation phenomena in silica particle suspensions and pastes, where a particle-wetting fluid is added as the third component. Increasing the wetting fluid loading in the ternary system induces a morphological transition from a pendular network to compact capillary aggregates network, with an intermediate funicular state. To our knowledge, the formation of percolated structures from compact capillary aggregates when the volume fraction of a wetting fluid approaches that of the particles is unprecedented. Such structures appear to result from the arrested coalescence of compact capillary aggregates due to the balance between the Laplace pressure and solid-like properties (yield stress, elasticity) of the aggregates. Shear-induced yielding of the ternary systems, linked to their percolating nature, is strongly influenced by the amount of wetting fluid phase. A non-monotonic dependence of the yield stress on the amount of wetting fluid is found, with the maximum yield stress obtained for a wetting fluid-to-particle volume fraction ratio of 0.2-0.3. For pendular systems, linear viscoelastic properties display a soft glassy rheological behavior above the percolation threshold (around 4 vol% particles), and complex viscosity data can be scaled using the high frequency plateau value, as well as a single characteristic relaxation time, which decreases when the particle concentration is increased. In addition, the particle concentration dependence of the yielding transition in the pendular regime appears to be efficiently described by two parameters extracted from the steady state flow curves: the yield stress and the limiting viscosity at a high shear rate. Although these non-colloidal networks result from flow-driven assembly, the scaling laws for our pendular gels are reminiscent of colloidal gels with a fractal geometry. Our studies pinpoint new pathways to create physical gels where the interparticle attraction strength is determined by capillary interactions.

Multi-body coalescence in Pickering emulsions

Particle-stabilized Pickering emulsions have shown unusual behaviours such as the formation of non-spherical droplets and the sudden halt of coalescence between individual droplets. Here we report another unusual behaviour of Pickering emulsions-the simultaneous coalescence of multiple droplets in a single event. Using latex particles, silica particles and carbon nanotubes as model stabilizers, we show that multi-body coalescence can occur in both water-in-oil and oil-in-water emulsions. The number of droplets involved in the nth coalscence event equals four times the corresponding number of the tetrahedral sequence in close packing. Furthermore, coalescence is promoted by repulsive latex and silica particles but inhibited by attractive carbon nanotubes. The revelation of multi-body coalescence is expected to help better understand Pickering emulsions in natural systems and improve their designs in engineering applications.