Effective capillary interaction of spherical particles at fluid interfaces

Critical Casimir forces in soft matter

We review recent advances in the theoretical, numerical, and experimental studies of critical Casimir forces in soft matter, with particular emphasis on their relevance for the structures of colloidal suspensions and on their dynamics. Distinct from other interactions which act in soft matter, such as electrostatic and van der Waals forces, critical Casimir forces are effective interactions characterised by the possibility to control reversibly their strength via minute temperature changes, while their attractive or repulsive character is conveniently determined via surface treatments or by structuring the involved surfaces. These features make critical Casimir forces excellent candidates for controlling the equilibrium and dynamical properties of individual colloids or colloidal dispersions as well as for possible applications in micro-mechanical systems. In the past 25 years a number of theoretical and experimental studies have been devoted to investigating these forces primarily under thermal equilibrium conditions, while their dynamical and non-equilibrium behaviour is a largely unexplored subject open for future investigations.

Capillary force on an 'inert' colloid: a physical analogy to dielectrophoresis

"Inert" colloids are μm-scale particles that create no distortion when trapped at a planar fluid-fluid interface. When placed in a curved interface, however, such colloids can create interfacial distortions of quadrupolar symmetry - so-called "induced capillary quadrupoles." The present work explores the analogy between capillary quadrupoles and electric dipoles, and the forces exerted on them by a symmetry-breaking gradient. In doing so, we weigh in on an outstanding debate as to whether a curvature gradient can induce a capillary force on an inert colloid. We argue that this force exists, for the opposite would imply that all dielectrophoretic forces vanish in two dimensions (2D). We justify our claim by solving 2D Laplace problems of electrostatics and capillary statics involving a single particle placed within a large circular shell with an imposed gradient. We show that the static boundary condition on the outer shell must be considered when applying the principle of virtual work to compute the force on the particle, as verified by a direct calculation of this force through integration of the particle stresses. Our investigation highlights some of the subtleties that emerge in virtual work calculations of capillary statics and electrostatics, thereby clarifying and extending previous results in the field. The broader implication of our results is that inert particles - including particles with planar, pinned contact lines and equilibrium contact angles - interact through interparticle capillary forces that scale quadratically with the deviatoric curvature of the host interface, contrary to recent claims made in the literature.

Microspheres viscous drag at a deformed fluid interface: particle's weight and electrical charges effects

When a microparticle is trapped at a fluid interface, particle's electrical charge and weight combine to deform the interface. Such deformation is expected to affect the particle diffusion via hydrodynamics boundary conditions. Using available models of particle-induced electrostatic deformation of the interface and particle dynamics at the interface, we are able to analytically predict particle diffusion coefficient values in a large range of particle's contact angle and size. This might offer a solid background of numerical values to compare with for future experimental studies in the field of particle diffusion at a fluid interface.

Janus Particles at Fluid Interfaces: Stability and Interfacial Rheology

The use of the Janus motif in colloidal particles, i.e., anisotropic surface properties on opposite faces, has gained significant attention in the bottom-up assembly of novel functional structures, design of active nanomotors, biological sensing and imaging, and polymer blend compatibilization. This review is focused on the behavior of Janus particles in interfacial systems, such as particle-stabilized (i.e., Pickering) emulsions and foams, where stabilization is achieved through the binding of particles to fluid interfaces. In many such applications, the interface could be subjected to deformations, producing compression and shear stresses. Besides the physicochemical properties of the particle, their behavior under flow will also impact the performance of the resulting system. This review article provides a synopsis of interfacial stability and rheology in particle-laden interfaces to highlight the role of the Janus motif, and how particle anisotropy affects interfacial mechanics.

Capillary interactions between soft capsules protruding through thin fluid films

When a suspension dries, the suspending fluid evaporates, leaving behind a dry film composed of the suspended particles. During the final stages of drying, the height of the fluid film on the substrate drops below the particle size, inducing local interface deformations that lead to strong capillary interactions among the particles. Although capillary interactions between rigid particles are well studied, much is still to be understood about the behaviour of soft particles and the role of their softness during the final stages of film drying. Here, we use our recently-introduced numerical method that couples a fluid described using the lattice Boltzmann approach to a finite element description of deformable objects to investigate the drying process of a film with suspended soft particles. Our measured menisci deformations and lateral capillary forces, which agree well with previous theoretical and experimental works in case of rigid particles, show that the deformations become smaller with increasing particle softness, resulting in weaker lateral interaction forces. At large interparticle distances, the force approaches that of rigid particles. Finally, we investigate the time dependent formation of particle clusters at the late stages of the film drying.

Charging-driven coarsening and melting of a colloidal nanoparticle monolayer at an ionic liquid-vacuum interface

We induce and investigate the coarsening and melting dynamics of an initially static nanoparticle colloidal monolayer at an ionic liquid-vacuum interface, driven by a focused, scanning electron beam. Coarsening occurs through grain interface migration and larger-scale motions such as grain rotations, often facilitated by sliding dislocations. The progressive decrease in area fraction that drives melting of the monolayer is explained using an electrowetting model whereby particles at the interface are solvated once their accumulating charge recruits sufficient counterions to subsume the particle. Subject to stochastic particle removal from the monolayer, melting is recapitulated in simulations with a Lennard-Jones potential. This new driving mechanism for colloidal systems, whose dynamical timescales we show can be controlled with the accelerating voltage, opens the possibility to manipulate particle interactions dynamically without need to vary particle intrinsic properties or surface treatments. Furthermore, the decrease in particle size availed by electron imaging presents opportunities to observe force and time scales in a lesser-explored regime intermediate between typical colloidal and molecular systems.

Static and dynamic behavior of magnetic particles at fluid interfaces

This perspective work reviews the current status of research on magnetic particles at fluid interfaces. The article gives both a unified overview of recent experimental advances and theoretical studies centered on very different phenomena that share a common characteristic: they involve adsorbed magnetic particles that range in size from a few nanometers to several millimeters. Because of their capability of being remotely piloted through controllable external fields, magnetic particles have proven essential as building blocks in the design of new techniques, smart materials and micromachines, with new tunable properties and prospective applications in engineering and biotechnology. Once adsorbed at a fluid-fluid interfase, in a process that can be facilitated via the application of magnetic field gradients, these particles often result sorely confined to two dimensions (2D). In this configuration, inter-particle forces directed along the perpendicular to the interface are typically very small compared to the surface forces. Hence, the confinement and symmetry breaking introduced by the presence of the surface play an important role on the response of the system to the application of an external field. In monolayers of particles where the magnetic is predominant interaction, the states reached are strongly determined by the mode and orientation of the applied field, which promote different patterns and processes. Furthermore, they can reproduce some of the dynamic assemblies displayed in bulk or form new ones, that take advantage of the interfacial phenomena or of the symmetry breaking introduce by the confining boundary. Magnetic colloids are also widely used for unraveling the guiding principles of 2D dynamic self-assembly, in designs devised for producing interface transport, as tiny probes for assessing interfacial rheological properties, neglecting the bulk and inertia contributions, as well as actuated stabilizing agents in foams and emulsions.

Dimerization and structure formation of colloids via capillarity at curved fluid interfaces

Capillary interactions are ubiquitous between colloids trapped at fluid interfaces. Generally, colloids in fluid interfaces have pinned, undulated contact lines that distort the interface around them. To minimize the area, and therefore the energy of these distortions, colloids interact and assemble in a manner that depends on the shape of the host interface. On curved interfaces, capillary interactions direct isolated colloid motion along deviatoric curvature gradients. This directed motion relies on the leading order, long-ranged quadrupolar distortions made by the colloids' undulated pinned contact lines. Here we study pair interactions and dimer formation of colloids on non-uniformly curved fluid interfaces. Pair interaction energies are inferred to be order of 104kBT, and interacting forces are of order 10-1 pN for 10 micron particles adsorbed on interfaces formed around a 250 micron micropost. We compare experiments to analysis for the pair interaction energy, and identify criteria for dimers to form. We also study the formation of trapped structures by multiple particles to discern the influence of the underlying interface shape and the contact line undulations. By comparison to Monte Carlo simulations with potentials of interactions based on analysis, we find that higher order terms in the distortion fields generated by the particles play a major role in the structure formation on the curved interface. These interactions are determined by the particle's contact line and the host interface shape, and can be used to assemble particles independent of their material properties.

Particle clusters at fluid-fluid interfaces: equilibrium profiles, structural mechanics and stability against detachment

We investigate clustering of particles at an initially flat fluid-fluid interface of surface tension γ under an external force f directed perpendicular to the interface. We employ analytical theory, numerical energy minimization (Surface Evolver) and computational fluid dynamics (the Lattice-Boltzmann method) to study the equilibrium deformation of the interface and structural mechanics of the clusters, in particular at the onset of instability. In the case of incompressible clusters, we find that the equilibrium 3D interface profiles are uniquely determined by the length scale γ/(fn0), where n0 is the particle surface number density, and a non-dimensional shape parameter f2Nn0/γ2. The scaling remains valid in the whole regime of forces f, i.e., even close to the stability limit fcrit. In the cases with an initial hexagonal arrangement of the particles, upon f approaching fcrit, our simulations additionally reveal the emergence of curvature-induced defects and 2D stress anisotropy. We develop stability diagrams in terms of f, N (we study 7 ≤ N ≤ 61), and the contact angle θp at the particles and identify three unstable regimes corresponding to (i) collective detachment of the whole cluster from the interface, (ii) ejection of individual particles, and (iii) both detachment and ejection. We also discuss possible metastable states. Altogether, our results may help in better understanding and controlling the particle interfacial instabilities with potential uses in synthesis of new materials, environmental sciences and microfluidics.

Capillarity-driven migration of small objects: A critical review

The phenomena on the capillarity-driven migration of small objects are full of interest for both scientific and engineering communities, and a critical review is thereby presented. The small objects mentioned here deal with the non-deformable objects, such as particles, rods, disks and metal sheets; and besides them, the soft objects are considered, such as droplets and bubbles. Two types of interfaces are analyzed, i.e., the solid-fluid interface and the fluid-fluid interface. Due to the easily deformable properties of the soft objects and distorted interfacial shapes induced by small objects, a more convenient way to obtain the driving force is through the potential energy of the system. The asymmetric factors causing the object migration include the asymmetric configuration of the interface, and the difference between the interfacial tensions. Finally, a simple outlook on the potential applications of small object migration is made. These behaviors may cast new light on the design of microfluidics and new devices, environment cleaning, oil and gas displacement and mineral industries.

Electrostatic interaction of particles trapped at fluid interfaces: effects of geometry and wetting properties

The electrostatic interaction between pairs of spherical or macroscopically long, parallel cylindrical colloids trapped at fluid interfaces is studied theoretically for the case of small inter-particle separations. Starting from the effective interaction between two planar walls and by using the Derjaguin approximation, we address the issue of how the electrostatic interaction between such particles is influenced by their curvatures and by the wetting contact angle at their surfaces. Regarding the influence of curvature, our findings suggest that the discrepancies between linear and nonlinear Poisson-Boltzmann theory, which have been noticed before for planar walls, also occur for spheres and macroscopically long, parallel cylinders, though their magnitude depends on the wetting contact angle. Concerning the influence of the wetting contact angle θ simple relations are obtained for equally sized particles which indicate that the inter-particle force varies significantly with θ only within an interval around 90°. This interval depends on the Debye length of the fluids and on the size of the particles but not on their shape. For unequally sized particles, a more complicated relation is obtained for the variation of the inter-particle force with the wetting contact angle.

In situ X-ray scattering observation of two-dimensional interfacial colloidal crystallization

Charged colloids at interfaces hold such a simple configuration that their interactions are supposed to be fully elucidated in the framework of classical electrostatics, yet the mysterious existence of attractive forces between these like-charged particles has puzzled the scientific community for decades. Here, we perform the in situ grazing-incidence small-angle X-ray scattering study of the dynamic self-assembling process of two-dimensional interfacial colloids. This approach allows simultaneous monitoring of the in-plane structure and ordering and the out-of-plane immersion depth variation. Upon compression, the system undergoes multiple metastable intermediate states before the stable hexagonal close-packed monolayer forms under van der Waals attraction. Remarkably, the immersion depth of colloidal particles is found to increase as the interparticle distance decreases. Numerical simulations demonstrate the interface around a colloid is deformed by the electrostatic force from its neighboring particles, which induces the long-range capillary attraction.

Colloids adsorbed at fluid interfaces can self-assemble into crystal, but the detail remains largely unknown due to experimental challenges. Using in situ X-ray scattering, Wu et al. show that out-of-plane electrostatic force induces in-plane capillary attraction between like-charged particles.

2D Stokesian Approach to Modeling Flow Induced Deformation of Particle Laden Interfaces

A Stokesian dynamics simulation of the effect of surface Couette flow on the microstructure of particles irreversibly adsorbed to an interface is presented. Rather than modeling both bulk phases, the interface, and particles in a full 3D simulation, known interfacial interactions between adsorbed particles are used to create a 2D model from a top down perspective. This novel methodology is easy to implement and computationally inexpensive, which makes it favorable to simulate behavior of particles under applied flow at fluid-fluid interfaces. The methodology is used to examine microstructure deformation of monodisperse, rigid spherical colloids with repulsive interactions when a surface Couette flow is imposed. Simulation results compare favorably to experimental results taken from literature, showing that interparticle forces must be 1 order of magnitude greater than viscous drag for microstructure to transition from aligned particle strings to rotation of local hexagonal domains. Additionally, it is demonstrated that hydrodynamic interactions between particles play a significant role in the magnitude of these microstructure deformations.

Self-assembly of cubic colloidal particles at fluid-fluid interfaces by hexapolar capillary interactions

Colloidal particles adsorbed at fluid-fluid interfaces can self-assemble, thanks to capillary interactions, into 2D ordered structures. Recently, it has been predicted by theoretical and numerical calculations [G. Soligno et al., Phys. Rev. Lett., 2016, 116, 258001] that cubes with smooth edges adsorbed at a flat fluid-fluid interface generate hexapolar capillary deformations that cause the particles to self-assemble into honeycomb and hexagonal lattices, at equilibrium and for Young's contact angle π/2. Here we extend these results. Firstly, we show that capillary interactions induced by hexapolar deformations can drive the particles at the interface to form also thermodynamically-stable square lattices, in addition to honeycomb and hexagonal lattices. Then, we study the effects of tuning the particle shape on the particle self-assembly at the interface, considering, respectively, smooth-edge cubes, sharp-edge cubes, slightly truncated-edge cubes, and highly truncated-edge cubes. In our calculations, both capillary and hard-particle interactions are taken into account. We show that such variations in the particle shape significantly affect both qualitatively and quantitatively the self-assembly of the particles at the interface, and we sum up our results in the form of temperature-density phase diagrams. For example, using typical experimental parameters, our results show that only 4-to-5 nm sized sharp-edge and smooth-edge cubes can self-assemble into a honeycomb lattice, while slightly and highly truncated-edge cubes can form a honeycomb lattice only if they have a 8-to-12 and 10-to-16 nm size, respectively, for the same experimental parameters. Also, our results show that the capillarity-induced square lattice phase is stable only for the smooth-edge and truncated-edge cubes, but not for the sharp-edge cubes.

Nonaffine deformation and tunable yielding of colloidal assemblies at the air-water interface

Silica nanoparticles trapped at the air-water interface form a 2D solid state with amorphous order. We propose a theoretical model to describe how this solid-like state deforms under a shear strain ramp up to and beyond a yielding point which leads to plastic flow. The model accounts for all the particle-level and many-body physics of the system: nonaffine displacements, local connectivity and its evolution in terms of cage-breaking, and interparticle interactions mediated by the particle chemistry and colloidal forces. The model is able to reproduce experimental data with only two non-trivial fitting parameters: the relaxation time of the cage and the viscous relaxation time. The interparticle spring constant contains information about the strength of interparticle bonding which is tuned by the amount of surfactant that renders the particles hydrophobic and mutually attractive. This framework opens up the possibility of quantitatively tuning and rationally designing the mechanical response of colloidal assemblies at the air-water interface. Also, it provides a mechanistic explanation for the observed non-monotonic dependence of yield strain on surfactant concentration.

Dynamic self-assembly and self-organized transport of magnetic micro-swimmers

We demonstrate experimentally and in computer simulations that magnetic microfloaters can self-organize into various functional structures while energized by an external alternating (ac) magnetic field. The structures exhibit self-propelled motion and an ability to carry a cargo along a pre-defined path. The morphology of the self-assembled swimmers is controlled by the frequency and amplitude of the magnetic field.

Nano- and microparticles at fluid and biological interfaces

Systems with interfaces are abundant in both technological applications and biology. While a fluid interface separates two fluids, membranes separate the inside of vesicles from the outside, the interior of biological cells from the environment, and compartmentalize cells into organelles. The physical properties of interfaces are characterized by interface tension, those of membranes are characterized by bending and stretching elasticity. Amphiphilic molecules like surfactants that are added to a system with two immiscible fluids decrease the interface tension and induce a bending rigidity. Lipid bilayer membranes of vesicles can be stretched or compressed by osmotic pressure; in biological cells, also the presence of a cytoskeleton can induce membrane tension. If the thickness of the interface or the membrane is small compared with its lateral extension, both can be described using two-dimensional mathematical surfaces embedded in three-dimensional space. We review recent work on the interaction of particles with interfaces and membranes. This can be micrometer-sized particles at interfaces that stabilise emulsions or form colloidosomes, as well as typically nanometer-sized particles at membranes, such as viruses, parasites, and engineered drug delivery systems. In both cases, we first discuss the interaction of single particles with interfaces and membranes, e.g. particles in external fields, non-spherical particles, and particles at curved interfaces, followed by interface-mediated interaction between two particles, many-particle interactions, interface and membrane curvature-induced phenomena, and applications.

Collective colloid diffusion under soft two-dimensional confinement

This work presents a numerical and theoretical investigation of the collective dynamics of colloids in an unbounded solution but trapped in a harmonic potential. Under strict two-dimensional confinement (infinitely stiff trap) the collective colloidal diffusion is enhanced and diverges at zero wave number (like k^{-1}), due to the hydrodynamic propagation of the confining force across the layer. The analytic solution for the collective diffusion of colloids under a Gaussian trap of width δ still shows enhanced diffusion for large wavelengths kδ<1, while a gradual transition to normal diffusion for kδ>1. At intermediate and short wavelengths, we illustrate to what extent the hydrodynamic enhancement of diffusion is masked by the conservative forces between colloids. At very large wavelengths, the collective diffusion becomes faster than the solvent momentum transport and a transition from Stokesian dynamics to inertial dynamics takes place. Using our inertial coupling method code (resolving fluid inertia), we study this transition by performing simulations at small Schmidt number. Simulations confirm theoretical predictions for the k→0 limit [Phys. Rev. E 90, 062314 (2014)PLEEE81539-375510.1103/PhysRevE.90.062314] showing negative density-density time correlations. However, at finite k simulations show deviations from the theory.

Tuning the Structure and Rheology of Polystyrene Particles at the Air-Water Interface by Varying the pH

We form films of carboxylated polystyrene particles (C-PS) at the air-water interface and investigate the effect of subphase pH on their structure and rheology by using a suite of complementary experimental techniques. Our results suggest that electrostatic interactions drive the stability and the structural order of the films. In particular, we show that by increasing the pH of the subphase from 9 up to 13, the films exhibit a gradual transition from solid to liquidlike, which is accompanied by a loss of the long-range order (that characterizes them at lower values of pH). Direct optical visualization of the layers, scanning electron microscopy, and surface pressure isotherms indicate that the particles deposited at the interface form three-dimensional structures involving clusters, with the latter being suppressed and a quasi-2D particle configuration eventually reached at the highest pH values. Evidently, the properties of colloidal films can be tailored significantly by altering the pH of the subphase.

Self-Assembly of Cubes into 2D Hexagonal and Honeycomb Lattices by Hexapolar Capillary Interactions

Particles adsorbed at a fluid-fluid interface induce capillary deformations that determine their orientations and generate mutual capillary interactions which drive them to assemble into 2D ordered structures. We numerically calculate, by energy minimization, the capillary deformations induced by adsorbed cubes for various Young's contact angles. First, we show that capillarity is crucial not only for quantitative, but also for qualitative predictions of equilibrium configurations of a single cube. For a Young's contact angle close to 90°, we show that a single-adsorbed cube generates a hexapolar interface deformation with three rises and three depressions. Thanks to the threefold symmetry of this hexapole, strongly directional capillary interactions drive the cubes to self-assemble into hexagonal or graphenelike honeycomb lattices. By a simple free-energy model, we predict a density-temperature phase diagram in which both the honeycomb and hexagonal lattice phases are present as stable states.

Active colloids at fluid interfaces

If an active Janus particle is trapped at the interface between a liquid and a fluid, its self-propelled motion along the interface is affected by a net torque on the particle due to the viscosity contrast between the two adjacent fluid phases. For a simple model of an active, spherical Janus colloid we analyze the conditions under which translation occurs along the interface and we provide estimates of the corresponding persistence length. We show that under certain conditions the persistence length of such a particle is significantly larger than the corresponding one in the bulk liquid, which is in line with the trends observed in recent experimental studies.

Hydrodynamics of Particles at an Oil-Water Interface

This study is a theoretical and experimental investigation of the hydrodynamics of the mutual approach of two floating spherical particles moving along an oil-water interface. An analytical expression is obtained for the (inertialess) Stokes drag for an isolated particle translating on a flat interface as a function of the immersion depth into the water phase for the case in which the viscosity of the oil is much larger than that of the water. An approximation for the viscous drag due to the mutual approach of identical spheres is formulated as the product of the isolated drag multiplied by the resistance of approaching spheres in an infinite medium. Experiments are undertaken on the capillary attraction of large, millimeter-sized Teflon spheres floating at the interface between a very viscous oil and water. With the use of image visualization and particle tracking, the separation distance as a function of time [[Formula: see text](t)] is measured along with the immersion depth and predicted by setting the capillary attraction force equal to the viscous drag resistance. The excellent agreement validates the approximating formula.

Lateral capillary interactions between colloids beneath an oil-water interface that are driven by out-of-plane electrostatic double-layer interactions

We study the lateral capillary interactions between colloids beneath an oil-water interface that lead to closely packed two-dimensional self-assembled colloidal crystals. These capillary forces are caused by the overlap of deformed interfaces above colloids on a solid substrate. The interface deformation is due to the electrostatic disjoining pressure between the charged particles and the charged oil-water interface. It is notable that the short-range (i.e., on the nanometer scale) and out-of-plane electrostatic double-layer interactions, which occur through an aqueous phase, can generate the long-range lateral capillary attraction (i.e., on the micrometer scale).

Interfacial Assembly of Surfactant-Decorated Nanoparticles: On the Rheological Description of a Colloidal 2D Glass

We address the rheology of assemblies of surfactant-decorated silica nanoparticles irreversibly adsorbed at the gas/liquid interface. Positively charged surfactant molecules (such as CTAB) bind to silica nanoparticle surfaces, and the resulting particle-surfactant complexes adsorb at gas/liquid interfaces. The surfactant molecules control the wettability of such decorated nanoparticles and their adsorption. The interparticle forces can be tuned by changing the surfactant concentration Cs. Increasing Cs, in addition to a decrease of the particles wettability, leads to an increase of the area fraction of particles at the interface. Oscillatory shear measurements (strain- and frequency-sweep) have been performed. Here, we explore the effect of the surfactant concentration Cs. At high enough Cs, the interface is highly packed, and an overall solidlike response is observed, with 2D glass properties.

Interaction between colloidal particles on an oil-water interface in dilute and dense phases

The interaction between micron-sized charged colloidal particles at polar/non-polar liquid interfaces remains surprisingly poorly understood for a relatively simple physical chemistry system. By measuring the pair correlation function g(r) for different densities of polystyrene particles at the decane-water interface, and using a powerful predictor-corrector inversion scheme, effective pair-interaction potentials can be obtained up to fairly high densities, and these reproduce the experimental g(r) in forward simulations, so are self consistent. While at low densities these potentials agree with published dipole-dipole repulsion, measured by various methods, an apparent density dependence and long range attraction are obtained when the density is higher. This condition is thus explored in an alternative fashion, measuring the local mobility of colloids when confined by their neighbors. This method of extracting interaction potentials gives results that are consistent with dipolar repulsion throughout the concentration range, with the same magnitude as in the dilute limit. We are unable to rule out the density dependence based on the experimental accuracy of our data, but we show that incomplete equilibration of the experimental system, which would be possible despite long waiting times due to the very strong repulsions, is a possible cause of artefacts in the inverted potentials. We conclude that to within the precision of these measurements, the dilute pair potential remains valid at high density in this system.

Capillary assembly of microscale ellipsoidal, cuboidal, and spherical particles at interfaces

Micron-sized anisotropic particles with homogeneous surface properties at a fluid interface can deform the interface due to their shape. The particles thereby create excess interfacial area and interact in order to minimize this area, which lowers the total interfacial energy. We present a systematic investigation of the interface deformations around single ellipsoidal particles and cuboidal particles with rounded edges in the near field for various contact angles and particle aspect ratios. The correlation of these deformations with capillary bond energies-the interaction energies of two particles at contact-quantifies the relation between the interactions and the near-field deformations. We characterize the interactions using effective power laws and investigate how anisotropic particles self-assemble by capillary forces. Interface deformations and particle interactions for cuboidal particles are weaker compared with those for ellipsoidal particles with the same aspect ratios. For both particle shapes, the bound state in side-by-side orientation is most stable, while the interaction in tip-to-side orientation is repulsive. Furthermore, we find capillary attraction between spherical and ellipsoidal particles. Our calculations therefore suggest cluster formation of spherical and ellipsoidal particles, which elucidates the role of spherical particles as stoppers for the growth of worm-like chains of ellipsoidal particles. The interaction between spherical and ellipsoidal particles might also explain the suppression of the "coffee-ring effect" that has been observed for evaporating droplets with mixtures of spherical and ellipsoidal particles. In general, our calculations of the near-field interactions complement previous calculations in the far field and help to predict colloidal assembly and rheological properties of particle-laden interfaces.

Capillary attraction induced collapse of colloidal monolayers at fluid interfaces

We investigate the evolution of a system of colloidal particles, trapped at a fluid interface and interacting via capillary attraction, as a function of the range of capillary interactions and temperature. We address the collapse of an initially homogeneous particle distribution and of a radially symmetric (disk-shaped) distribution of finite size, both theoretically by using a perturbative approach inspired by cosmological models and numerically by means of Brownian dynamics (BD) and dynamical density functional theory (DDFT). The results are summarized in a "dynamical phase diagram", describing a smooth crossover from a collective (gravitational-like) collapse to local (spinodal-like) clustering. In this crossover region, the evolution exhibits a peculiar shock wave behavior at the outer rim of the contracting, disk-shaped distribution.

Capillary force acting on a colloidal particle floating on a deformed interface

We analytically determine the lateral capillary force acting on a spherical colloid trapped at an interface of arbitrary shape. Our calculations, which are valid for colloids that are small with respect to the capillary length, take into account surface tension, pressure and gravity. We relate the force acting on the colloid to the shape of the liquid interface prior to colloid deposition. Our approach is a generalization of a previous study of ours [Ch. Blanc et al., Phys. Rev. Lett., 2013, 111, 058302] to the case in which gravity is present. Our findings are in agreement with the so-called Nicolson superposition approximation [D. Y. C. Chan, J. D. J. Henry and L. R. White, J. Colloid Interface Sci., 1981, 79, 410] and with the curvature-dependent capillary force predicted by Würger [A. Würger, Phys. Rev. E, 2006, 74, 041402], and extend these results by including higher-order terms in the ratio between the size of the colloid and the capillary length. We thoroughly validate our theoretical expressions by means of an exact nonlinear numerical calculation.

Role of geometry and amphiphilicity on capillary-induced interactions between anisotropic Janus particles

We study the capillary interactions between ellipsoidal Janus particles adsorbed at flat liquid-fluid interfaces. In contrast to spherical particles, Janus ellipsoids with a large aspect ratio or a small difference in the wettability of the two regions tend to tilt at equilibrium. The interface deforms around ellipsoids with tilted orientations and thus results in energetic interactions between neighboring particles. We quantify these interactions through evaluation of capillary energy variation as a function of the spacing and angle between the particles. The complex meniscus shape results in a pair interaction potential which cannot be expressed in terms of capillary quadrupoles as in homogeneous ellipsoids. Moreover, Janus ellipsoids in contact exhibit a larger capillary force at side-by-side alignment compared to the tip-to-tip configuration, while these two are of comparable magnitude for their homogeneous counterparts. We evaluate the role of particles aspect ratio and the degree of amphiphilicity on the interparticle force and the capillary torque. The energy landscapes enable prediction of micromechanics of particle chains, which has implications in predicting the interfacial rheology of such particles at fluid interfaces.

Capillary force on a micrometric sphere trapped at a fluid interface exhibiting arbitrary curvature gradients

We report theoretical predictions and measurements of the capillary force acting on a spherical colloid smaller than the capillary length that is placed on a curved fluid interface of arbitrary shape. By coupling direct imaging and interferometry, we are able to measure the in situ colloid contact angle and to correlate its position with respect to the interface curvature. Extremely tiny capillary forces down to femtonewtons can be measured with this method. Measurements agree well with a theory relating the capillary force to the gradient of Gaussian curvature and to the mean curvature of the interface prior to colloidal deposition. Numerical calculations corroborate these results.

Collective dynamics of colloids at fluid interfaces

The evolution of an initially prepared distribution of micron-sized colloidal particles, trapped at a fluid interface and under the action of their mutual capillary attraction, is analyzed by using Brownian dynamics simulations. At a separation λ given by the capillary length of typically 1mm, the distance dependence of this attraction exhibits a crossover from a logarithmic decay, formally analogous to two-dimensional gravity, to an exponential decay. We discuss in detail the adaptation of a particle-mesh algorithm, as used in cosmological simulations to study structure formation due to gravitational collapse, to the present colloidal problem. These simulations confirm the predictions, as far as available, of a mean-field theory developed previously for this problem. The evolution is monitored by quantitative characteristics which are particularly sensitive to the formation of highly inhomogeneous structures. Upon increasing λ the dynamics shows a smooth transition from the spinodal decomposition expected for a simple fluid with short-ranged attraction to the self-gravitational collapse scenario.

Capillary interactions in Pickering emulsions

The effective capillary interaction potentials for small colloidal particles trapped at the surface of liquid droplets are calculated analytically. Pair potentials between capillary monopoles and dipoles, corresponding to particles floating on a droplet with a fixed center of mass and subjected to external forces and torques, respectively, exhibit a repulsion at large angular separations and an attraction at smaller separations, with the latter resembling the typical behavior for flat interfaces. This change of character is not observed for quadrupoles, corresponding to free particles on a mechanically isolated droplet. The analytical results are compared with the numerical minimization of the surface free energy of the droplet in the presence of spherical or ellipsoidal particles.

How do mosquito eggs self-assemble on the water surface?

This work reports a detailed numerical study of the behavior of ellipsoid-shaped particles adsorbed at fluid interfaces. Former experiments have shown that micrometer-sized prolate ellipsoids aggregate under the action of strong and long-ranged capillary interactions. The latter are due to nonplanar contact lines and to the resulting deformations of the interface in the vicinity of the trapped objects. We first consider the case of a single ellipsoid and examine in detail the influence of contact angle and ellipsoid aspect ratio on interfacial distortions. We then focus on two contacting ellipsoids and study the optimum packing configuration depending on their size and/or aspect ratio mismatch. We thoroughly explore the variety of contact configurations between both ellipsoids and provide corresponding energy maps. Whereas the side-by-side configuration is the most stable state for identical ellipsoids, we find that the mismatched pair adopts an "arrow" configuration in which a finite angle exists between the particles long axes. Such arrows are actually seen in experiments with micron-sized ellipsoids and similarly with millimeter-sized mosquito eggs. These results complement our previous work (J.C. Loudet, B. Pouligny, EPL 85, 28003 (2009)) and highlight the importance of geometrical factors to explain the morphology of aggregated structures at fluid interfaces.

Electric field-induced self-assembly of micro- and nanoparticles of various shapes at two-fluid interfaces

Particle lithography which explores the capability of particles to self-assemble offers an attractive means to manufacture nanostructured materials. Although traditional techniques typically lead to the formation of dense crystals, adjustable non-close-packed crystals are crucial in a number of applications. We have recently proposed a novel method to assemble spherical micro- and nanoparticles into monolayers. The technique consists of trapping particles at a liquid-fluid interface and applying an electric field normal to the interface. Particles rearrange themselves under the influence of interfacial and electrostatic forces to form 2-D hexagonal arrays of long-range order and whose lattice constant depends on the electric field strength and frequency. Furthermore, the existence of an electric field-induced capillary force makes the technique applicable to submicron and nanosized particles. Although spherical particles are often used, non-spherical particles can be beneficial in practice. Here, we review the method, discuss its applicability to particles of various shapes, and present results for particles self-assembly on air-liquid and liquid-liquid interfaces. In the case of non-spherical particles, the self-assembly process, while still taking place, is more complex as particles experience a torque which causes them to rotate relative to one another. This leads to a final arrangement displaying either a dominant orientation or no well-defined orientation. We also discuss the possibility of dislodging the particles from the interface by applying a strong electric field such that the Weber number is of order 1 or larger, a phenomenon which can be utilized to clean particles from liquid-fluid surfaces.

Freezing transition and interaction potential in monolayers of microparticles at fluid interfaces

The structure and the interaction potential of monolayers of charged polystyrene microparticles at fluid interfaces have been studied by optical microscopy. Microparticles of different sizes have been studied over a broad range of surface particle densities. The structural characterization is based on the analysis of images obtained by digital optical microscopy. From the experimental images, radial distribution functions, hexagonal bond order correlation functions, and temporal orientational correlation functions have been calculated for different monolayer states at both the air/water and oil/water interfaces. The interaction potential has been calculated from the structure factor using integral equations within the hypernetted chain closure relationship. For particles trapped at the oil-water interface, it was found that, upon increasing the surface coverage, a freezing transition occurs, that leads to the formation of a 2D crystalline structure. We have studied the freezing densities of particle monolayers at the oil/water interface and compared them with Monte Carlo simulation results reported by H. Löwen. In contrast, at the air-water interface, freezing is inhibited due to the formation of particle aggregates.

Free energy of colloidal particles at the surface of sessile drops

The influence of finite system size on the free energy of a spherical particle floating at the surface of a sessile droplet is studied both analytically and numerically. In the special case that the contact angle at the substrate equals π/2 , a capillary analogue of the method of images is applied in order to calculate small deformations of the droplet shape if an external force is applied to the particle. The type of boundary conditions for the droplet shape at the substrate determines the sign of the capillary monopole associated with the image particle. Therefore, the free energy of the particle, which is proportional to the interaction energy of the original particle with its image, can be of either sign, too. The analytic solutions, given by the Green's function of the capillary equation, are constructed such that the condition of the forces acting on the droplet being balanced and of the volume constraint are fulfilled. Besides the known phenomena of attraction of a particle to a free contact line and repulsion from a pinned one, we observe a local free-energy minimum for the particle being located at the drop apex or at an intermediate angle, respectively. This peculiarity can be traced back to a non-monotonic behavior of the Green's function, which reflects the interplay between the deformations of the droplet shape and the volume constraint.

Dynamics of colloidal particles with capillary interactions

We investigate the dynamics of colloids at a fluid interface driven by attractive capillary interactions. At submillimeter length scales, the capillary attraction is formally analogous to two-dimensional gravity. In particular it is a nonintegrable interaction and it can be actually relevant for collective phenomena in spite of its weakness at the level of the pair potential. We introduce a mean-field model for the dynamical evolution of the particle number density at the interface. For generic values of the physical parameters the homogeneous distribution is found to be unstable against large-scale clustering driven by the capillary attraction. We also show that for the instability to be observable, the appropriate values for the relevant parameters (colloid radius, surface charge, external electric field, etc.) are experimentally well accessible. Our analysis contributes to current studies of the structure and dynamics of systems governed by long-ranged interactions and points toward their experimental realizations via colloidal suspensions.

Charge interaction between particle-laden fluid interfaces

Experiments are described where two oil/water interfaces laden with charged particles move at close proximity relative to one another. The particles on one of the interfaces were observed to be attracted toward the point of closest approach, forming a denser particle monolayer, while the particles on the opposite interface were repelled away from this point, forming a particle depletion zone. Such particle attraction/repulsion was observed even if one of the interfaces was free of particles. This phenomenon can be explained by the electrostatic interaction between the two interfaces, which causes surface charges (charged particles and ions) to redistribute in order to satisfy surface electric equipotential at each interface. In a forced particle oscillation experiment, we demonstrated the control of charged particle positions on the interface by manipulating charge interaction between interfaces.

Capillary forces between particles at a liquid interface: general theoretical approach and interactions between capillary multipoles

The liquid interface around an adsorbed colloidal particle can be undulated because of roughness or heterogeneity of the particle surface, or due to the fact that the particle has non-spherical (e.g. ellipsoidal or polyhedral) shape. In such case, the meniscus around the particle can be expanded in Fourier series, which is equivalent to a superposition of capillary multipoles, viz. capillary charges, dipoles, quadrupoles, etc. The capillary multipoles attract a growing interest because their interactions have been found to influence the self-assembly of particles at liquid interfaces, as well as the interfacial rheology and the properties of particle-stabilized emulsions and foams. As a rule, the interfacial deformation in the middle between two adsorbed colloidal particles is small. This fact is utilized for derivation of accurate asymptotic expressions for calculating the capillary forces by integration in the midplane, where the Young-Laplace equation can be linearized and the superposition approximation can be applied. Thus, we derived a general integral expression for the capillary force, which was further applied to obtain convenient asymptotic formulas for the force and energy of interaction between capillary multipoles of arbitrary orders. The new analytical expressions have a wider range of validity in comparison with the previously published ones. They are applicable not only for interparticle distances that are much smaller than the capillary length, but also for distances that are comparable or greater than the capillary length.

Anisotropies in thermal Casimir interactions: ellipsoidal colloids trapped at a fluid interface

We study the effective interaction between two ellipsoidal particles at the interface of two fluid phases which are mediated by thermal fluctuations of the interface. In this system the restriction of the long-ranged interface fluctuations by particles gives rise to fluctuation-induced forces which are equivalent to interactions of Casimir type and which are anisotropic in the interface plane. Since the position and the orientation of the colloids with respect to the interface normal may also fluctuate, this system is an example of the Casimir effect with fluctuating boundary conditions. In the approach taken here, the Casimir interaction is rewritten as the interaction between fluctuating multipole moments of an auxiliary charge-density-like field defined on the area enclosed by the contact lines. These fluctuations are coupled to fluctuations of multipole moments of the contact line position (due to the possible position and orientational fluctuations of the colloids). We obtain explicit expressions for the behavior of the Casimir interaction at large distances for arbitrary ellipsoid aspect ratios. If colloid fluctuations are suppressed, the Casimir interaction at large distances is isotropic, attractive, and long ranged (double logarithmic in the distance). If, however, colloid fluctuations are included, the Casimir interaction at large distances changes to a power law in the inverse distance and becomes anisotropic. The leading power is 4 if only vertical fluctuations of the colloid center are allowed, and it becomes 8 if also orientational fluctuations are included.

Attraction between weakly charged silica spheres at a water-air interface induced by surface-charge heterogeneity

We report an optical and atomic force microscopic (AFM) study of interactions between weakly charged silica spheres at a water-air interface. Attractive interactions are observed at intermediate interparticle distances and the amplitude of the attraction increases with the amount of salt (NaCl) added into the water phase. AFM images obtained in the salty water show the formation of patchy charge domains of size approximately 100 nm on the silica surface. The experiment suggests that surface heterogeneity produced during ionization plays an important role in the generation of attractions between like-charged particles at the interface.