Particle Rafts and Armored Droplets

Particles fl oating at interfaces are commonly observed in nature, as well as in industrial processes.When the particles are non-Brownian particles,large deformations of the interface are created that induce long-ranged capillary interactions and lead to the formation of particle rafts with unique characteristics. In this review we discuss recent efforts in investigating particle raft formation and the role of the rafts’ own weight during dynamic clustering. Under speci fi c conditions, these rafts can ultimately collapse and sink. When subjected to external or internal forces,the raft undergoes large deformations that test the mechanical characteristics of this interfacial composite material. It can behave as a continuous elastic sheet under compression, although its discrete nature can also trigger its fragmentation via interparticle interactions.Finally,armored droplets,drops covered by a protective shell of particles, can lose their integrity when submitted to dynamic deformations, resulting in the ejection of particles or the fracturing of the armor. Open questions to understand the properties of this material are highlighted and future research to understand the fundamental physics of particle rafts, the customization of the cluster formation, or the disassembly of this collective material is suggested.


INTRODUCTION
A raft is known as the most primitive structure used for transportation over water.It usually consists of a at assembly of buoyant materials that keep it a oat.Such oating structures can also self-organize at the interface between two uids of different densities ρ 1 and ρ 2 (with ρ 1 > ρ 2 ) via capillary interactions and are then called particle rafts.The study of such particle rafts seems particularly relevant to take on some of the challenges imposed by the numerous ecological crises we are currently living through.Shade balls have been dispersed at the surface of lakes all over the world to form large rafts protecting lakes from evaporation due to the increasing temperatures (Lehmann et al. 2019).Giant plastic garbage patches have gradually formed as a result of marine pollution in every ocean on the planet.They consist of fragments of all sizes of plastic debris gathered into clusters by ocean currents and trapped in vortices, which thus keep accumulating and damage ecosystems (Cózar et al. 2014).When rafts are formed of dense particles ρ part (with ρ part > ρ 1 > ρ 2 ), they can also be used to clean up oil spills.In this case, the raft undergoes strong deformations and sinks while encapsulating the lighter uid in the process (Abkarian et al. 2013).For industrial purposes, and at a smaller scale, colloidal particles trapped at a liquid interface also self-assemble into a 2D cluster and are commonly used nowadays to stabilize emulsions (Pickering 1907, Binks & Horozov 2006).
The capillary attraction leading to the formation of such rafts occurs when the shape of an interface is perturbed by several objects.To minimize the surface energy, the deformations overlap and create an attractive force between the objects.They then self-assemble into a 2D monolayer at the interface.Understanding the formation of aggregates or clusters has always been a challenge, and the 2D visualization imposed by the liquid interface makes the process easier by allowing researchers to use statistical tools to quantify the formation of aggregates in general, as well as to investigate the role of capillary forces, which is in itself a topic of interest.Moreover, the longranged attraction is linked to the capillary length c = γ 12 / ρg, with γ 12 the interfacial tension and ρ = (ρ 1 − ρ 2 ) the density difference between the two uids.Short-ranged attractive or repulsive forces (Van der Waals or electromagnetic interactions) can also play an important role in the dynamics near the particle clustering (Domìnguez et al. 2010).Interestingly, this longranged capillary clustering via interface deformation has also been compared to 2D gravitational attraction.It was found that a tunable cutoff length related to the value of c in uences the collapse or limits the clustering of the particles (Bleibel et al. 2011).
The interface deformations can be due to the wetting properties or the shape of the particles (Danov et al. 2005, Loudet et al. 2005); however, in this review, we focus on rafts that are composed of large particles that have self-assembled via the deformation of the interface created by their weight.Such particle rafts are also called granular rafts when the particle size R part is comparable to the capillary length c .In this case, the capillary interactions cannot be controlled by the particle's shape alone, and the raft can undergo large deformations in order to sustain its weight.
The properties of particle rafts are controlled by capillary interactions during the aggregation process.The superhydrophobicity of re ants allows them to form strong cohesive rafts that can self-heal to protect the colony (Mlot et al. 2011) (Figure 1a).Mosquito eggs or whirligig beetles form long chains (Figure 1b) when interacting due to their ellipsoidal shape (Loudet & Pouligny 2011, Voise et al. 2011).Particles can be designed in order to create organized structures (Bowden et al. 1997).An external eld can also guide particles during their capillary self-assembly.Liu et al. (2018) discussed the role of surface curvature in order to move particles along curvature gradients: Cylindrical particles migrate along curvature gradient lines formed by a liquid interface pinned to a micropost and orient the colloid's position (Figure 1c).At the millimeter scale, the dynamic selfassembly of objects has also been investigated in a series of experiments involving the formation of patterns of oating magnetic disks subjected to a magnetic eld (Figure 1d) (Grzybowski et   2000, Wang et al. 2017) and more recently with experiments in which the shapes of oating objects change due to temperature changes or light actuation (Hu et al. 2020, Vandewalle et al. 2020).These promising studies show that a raft's overall structure can be customized via a programmable self-assembly of the individual objects.Finally, a particularly interesting feature of particle rafts is that they behave as a single entity but also exhibit granular characteristics due to their discrete assembly.The rheology and mechanics of these interfaces have been the subject of extensive investigations in the past decade.In particular, owing to this dual nature, various interfacial phenomena such as buckling or fracturing can be observed in particle-laden interfaces.

FORMATION OF A PARTICLE RAFT
A uid-uid interface is created when a junction is formed between two immiscible uids: a lower phase denoted as phase 1 and an upper phase denoted as phase 2 throughout this review.When objects are sprinkled at such an interface, they start to interact via capillary forces.These forces have been studied for over 70 years for two identical objects.In this section, we investigate the aggregation of multiple objects at a liquid interface.We rst brie y introduce the different types of capillary forces and in particular the interaction of noncolloidal particles, which is the focus this review.We then brie y recall the dynamics of interaction between two identical objects at a liquid interface and detail the forces at play.Finally we describe how two deformable nonidentical bodies interact, which then enables us to understand the global dynamics of particles interacting on a liquid interface.

Capillary Forces Between Two Particles: From Colloids to Large Objects
Particles trapped at a liquid interface interact with each other via capillary forces (Figure 2).These forces can be divided into two categories (Kralchevsky & Denkov 2001): normal forces and lateral capillary forces.Normal forces bind objects together by a capillary bridge: either a liquid bridge in a gas or a gas bridge in a liquid.This attractive force is produced by a pressure drop across the curved interface, as well as by the surface tension exerted along the contact line.It is directed normally to the plane of the contact line (Figure 2a).This force is at play in sandcastle-building (Hornbaker et al. 1997), the dispersion of pigments, or the adhesion of powders on surfaces, for a Normal forces b Flotation forces c Immersion forces

Protière
Bond number: dimensionless number comparing capillary and gravity forces, Particles interacting with a liquid interface, with ρ 1 ≤ ρ part .(a) For Bond number Bo 1, perfectly smooth spheres will get trapped at the interface without creating any distortion.(b) For Bo ∼ 1, the particle's weight creates a deformation of the interface.Two such particles will undergo an attractive capillary force and move toward each other in order to minimize surface energy.Red arrows indicate the particles' capillary attraction at the interface.(c) For Bo 1, the particles will sink.
example (Kralchevsky & Nagayama 2001).The objects can also distort the liquid interface once trapped and are then set in motion so as to minimize the interfacial area.This creates a second category of capillary forces called lateral forces.
When the deformation of the interface is due to a perturbation of the contact line, we call the forces of interaction lateral immersion forces (Kralchevsky & Nagayama 1994) (Figure 2c).This phenomenon is mostly observed at the colloidal scale where Brownian motion is overcome by capillary forces, which can appear when the particles are con ned inside a liquid layer that decays either at in nity or at a nite distance.When they oat at the surface of a liquid, the force is produced by an irregular contact line due to the surface roughness or chemical inhomogeneities (Stamou et al. 2000).For perfectly smooth colloidal spheres at equilibrium wetting conditions, they will leave the interface around them unperturbed (Figure 3a).In this situation, the Bond number Bo = (R part / c ) 2 , the dimensionless number comparing capillary and gravity forces, is much smaller than 1.Moreover, nonspherical shapes can also create distortion elds around them which play a central role in particle interactions that we will not detail here.
In this review, we focus on lateral capillary forces observed when the shape of an interface is perturbed by the weight of the objects (Figure 3b).The interface is then pulled down locally.In this case, the Bond number Bo is close to 1.We call this force a lateral otation force, and it creates an interaction between objects at a uid interface (Figure 2b).At an air-water interface, the deformation can be directed downward if the particle is denser than water or upward if the object is considered lighter (a bubble, for instance).For spherical particles, the deformation is axisymmetric (the bead acts as a monopole).Of course, if the particle's weight exceeds its buoyancy, it will not remain at the interface as it will sink (Figure 3c).
We wish to draw the reader's attention to the fact that the wettability of the particles (and thus the contact angle) can also play an important role [reported at length by Kralchevsky & Nagayama (1994)], but since we are discussing particle rafts, where the weight of the particles is primarily responsible for the clustering of the particles, we have decided, for clarity reasons, not to discuss it further in this review.

Dynamics of Two Particles
In this section we discuss the capillary interaction of two identical objects at a liquid interface.We rst describe the forces at play and then detail the hydrodynamic interaction.
2.2.1.The Cheerios effect.We focus here on the interaction of objects at a liquid interface for which gravity plays an important role.This interaction is also commonly called the Cheerios effect (Vella & Mahadevan 2005), after the clustering of cereal observed when one pours cereal into a bowl of milk.The particle's radius R part is larger than 10 µm and its weight or buoyancy cannot be neglected (Figure 3b).The attractive force between two bubbles oating on a uid was rst studied by Nicolson (1949), who proposed a model using a linear superposition approximation of the interface deformations to describe the force of interaction between the bubbles.Since then, this approximation has been used in a variety of works: for the interaction between parallel in nite cylinders (Gifford & Scriven 1971, Allain & Cloitre 1993) or particles oating on a liquid (Chan et al. 1981, Kralchevsky et al. 1992, Paunov et al. 1993, Fournier & Galatola 2002, Vassileva et al. 2005).These works have led to an analytical solution for the capillary force between two particles.However, this expression cannot be used directly, as it depends on the position of the contact line around each particle, a physical quantity unknown a priori.The calculation by Vella & Mahadevan (2005) considers the vertical equilibrium of a single particle at an interface (Binks & Horozov 2006) and then adds the second particle to calculate an interface energy.
The shape of the interface between two objects results from how the particles deform the interface: their weight, their wettability, and the induced buoyancy.These parameters can be taken into account with dimensionless numbers: the Bond number Bo, the particle-liquid density ratio In the case of small spheres (R part â c ) where the shape of the interface is isotropic around each bead and its deformations are small, the calculation leads to an energy of interaction equal to the product between the effective weight of one particle, F g + F buoy = 2πγ 12 R part Bo , and the vertical deformation created by the other: with K i the modi ed Bessel function of the second kind of order i.By deriving Equation 1 with respect to r (the radial coordinate), we obtain the nal expression for the horizontal force between two identical particles at a uid-air interface:

The hydrodynamics of two identical spheres at a uid-liquid interface.
At this scale, this capillary force F cap is counterbalanced by a drag around each particle (Figure 4a).We assume that the particles can be slowed down by the upper phase of density ρ 2 .The particle's inertia can also be neglected: The particle moves in a creeping ow once attached to the interface.We thus have the Reynolds number Re = ρV R part /µ 2 1, with µ 2 the characteristic viscosity of the upper uid and V the characteristic particle velocity.The viscous stresses compared with surface tension are negligible, which gives the capillary number Ca = µ 2 V/γ 12 1.The drag force acting on a particle F drag can be expressed as a Stokes drag corrected by a mobility function G for two spheres in a single phase, in order to take into account the drainage of the liquid between the particles as they get closer: where L is the distance between the centers of the two spheres.The term k accounts for the fact that the particles are immersed in two phases (Dani et al. 2015) and depend on the contact angle of the spheres and the phases' viscosities (Danov et al. 1995(Danov et al. , 2000;;Pozrikidis 2007).The mobility function G was tabulated by Batchelor (1976) and Batchelor & Green (1972) and can be approximated by various interpolation formulas (Vassileva et al. 2006, Boneva et al. 2007)  Re 1, there is no inertia and the equilibrium of forces (Equations 2 and 3) gives an expression for the velocity of one particle separated by a distance L from a second identical particle: This model is then veri ed experimentally by recording the aggregation dynamics of two identical spheres and measuring their velocities.We rst observe an increase in velocity as the particles get closer until a maximum is reached.From this point onward, the drainage of the liquid separating the two spheres will make the velocity decrease rapidly until contact is reached (Figure 4b).The tting parameter k has been discussed in different experimental cases by varying the wettability and the viscosity of the two uids (Vassileva et al. 2006, Dalbe et al. 2011, Dani et al. 2015).

Clustering Dynamics of Large Objects
The models and experiments we have discussed so far concern the interactions of two identical particles.To understand the formation of particle rafts, we now detail the interactions of larger clusters.

Interaction of two particle rafts.
Once two spherical particles are in contact they form a new aggregate at the interface.These 2D structures interact with other large clusters.These assemblies of particles have been considered as single entities in order to investigate the role of cluster dimensions on their interactions.Indeed, the capillary force experienced by each raft can strongly depend on its size and on the external interfacial deformation.Lagarde et al. (2019) focused on the interaction of such large objects, where dense millimetersized beads act as monopoles.Such 2D structures are exible and exhibit high deformations due to both their weight and their geometrical extension.They describe the capillary forces generated by each raft, as well as the drag due to their individual motion (Figure 4c).Because the deformation around a particle raft strongly depends on its size, the capillary interaction can exceed by several orders of magnitude the forces created by individual beads, with a very strong dependence on the number of particles (Figure 4d).The model for a two-particle interaction can be generalized to the interaction of two particle rafts.The dependence of the drag and the capillary force experienced by a raft is then quanti ed depending on both its size (number of particles n) and the size of the other interacting raft: 5 .
with n A and n B the number of particles in rafts A and B, respectively.We must note that the model used here is based in part on the assumption that the deformations due to the weight remain small when compared to the capillary length c .However, this is not always the case, as a granular raft's vertical deformation can reach c /2 and its lateral extension can be close to c .This could explain the disagreement between the experimental and theoretical speeds measured for large clusters (yellow curves in Figure 4d).

2.3.2.
Clustering.The next step is to explore the aggregation of a random number of particle rafts to understand the clustering dynamics and the formation of such monolayers of grains.This clustering has been studied extensively for colloidal particles, where the motion is completely controlled by Brownian thermal agitation.Universal laws that characterize aggregating systems have been theorized (Leyvraz 2003), from the fragmentation or merging of asteroids due to the collisions of planets (Lee 2000) to the formation of a fog resulting from the suspension of micrometer-sized water droplets that coalesce and break apart (Friedlander et al. 2000).In all these physical systems, statistical tools are used to describe the ordering arising from the interaction of these individual objects (Weitz & Lin 1986, Lin et al. 1989).

THE STABILITY OF A PARTICLE RAFT
We have discussed how particles, once placed at a liquid interface, attract each other, via a capillary force F cap , to minimize the deformation of the interface.These interactions lead to the formation of a monolayer of grains.We now wish to focus on the stable equilibrium state of this new entity.At equilibrium, one object is straddling the interface between the two uids.When an object is destabilized, it becomes out of equilibrium and sinks.First, we discuss the stability of one object, before turning to how the interaction of two or several objects at a uid interface can lead to a collective sinking instability of the raft.Finally, we discuss how this new structure consisting of an assembly of distinct objects becomes out of equilibrium and describe various collapsing processes.

Individual Particles
Schoolchildren may recall that when an object is placed at the surface of a liquid, Archimedes' principle determines whether the object will oat or sink: The upward force on a body immersed or oating in a liquid is equal to the weight of liquid displaced by that body.Surprisingly, only recently has this principle been generalized to incorporate surface tension (Keller 1998), which plays an important role when the object's size is comparable to the length of the deformed interface, c .In this case, one has to take into account the liquid displaced not only by the object's weight but also by its meniscus.Objects with densities ρ part signi cantly larger than that of a liquid will only oat for R part â c (Figure 3b).In the previous section, we introduced the parameter D, which corresponds to a density ratio and is used, along with Bo, to characterize the deformation of a uid interface.In his review, Vella (2015) suggested considering this same parameter D when evaluating whether an object can oat in equilibrium at an interface and found that, to determine an object's ability to oat, one has to consider a variety of its characteristics: its density and size, which directly affect its oatability, but also its shape and surface properties, which impact the length of the contact line and thus play a role in the equilibrium state of these small objects at a uid interface.

Two Particles
The equilibrium conditions under which an object oats at an interface have been the topic of studies since Archimedes' principle.However, when several objects interact at the surface of a liquid, this compromises their equilibrium and the newly formed object may eventually sink.This topic has been investigated only recently.Indeed, since the oatability of an object is related to the shape of its contact line, when objects interact at a uid surface during the formation of a particle raft, the shape of the contact line may vary and change the objects' load-bearing capacity (Vella et al. 2006b).In particular, Cooray et al. (2017) have found that otation capacities can be enhanced or reduced depending on the size of and the distance between particles, as the hydrostatic pressure is in competition with the surface tension when the meniscus changes shape at the contact line during the interaction of the two objects.This may have some impact on the raft's formation.

Assembly of Particles at an Interface: Collective Stability
Once the stability of a couple of particles is understood, the next step consists in adding progressively more particles to increase the size of the cluster and see if the newly formed particle raft will stay a oat.It is dif cult to predict the general conditions of stability of a raft: Although individual grains may produce deformations of the interface, when these deformations overlap and form this structure, they produce a global deformation that may not be predicted only by summing up the local capillary interactions.We therefore consider the raft as a new composite material made of the particles and the liquid, thus obtaining a scaled density ratio D, where the load ρ part at the scale of the raft thickness R part is compared to the buoyancy ρ at the scale of the capillary length c : With this parameter in mind, we can test the stability of a raft for different particle and liquid characteristics.Thus far, we have considered the case where the upper uid is a gas of negligible density, but Protière et al. (2017) studied the stability of an axisymmetric raft formed of dense particles at an oil-water interface.In this case, the capillary attraction is long ranged, c ≈ 10 mm, and more importantly, the raft can undergo strong deformations before sinking.As particles are added, the raft grows, becomes denser, and lowers signi cantly, displacing a large amount of liquid from under it.Similarly to Vella et al. (2006b), the authors reported that for D < 3 a raft can grow in nitely large without sinking, as the hydrostatic pressure will always support its weight.However, for D > 3, there always is a critical size at which the raft will collapse, as it would need to lower deeper than the capillary length in order to oat, which surface tension does not permit; thus, the raft will sink (Figure 5d).We must note that the way the particles are deposited impacts the self-assembly of the objects at the liquid interface and plays a signi cant role in the limits of the overall system's stability.In particular, when particles are poured (Abkarian et al. 2013, Ong et al. 2019, Cervantes-Álvarez Protière et al. 2020) or stacked ( Jones et al. 2015), they sink for a smaller number of particles than they do when they self-assemble in a monolayer (Figure 5f ).The speci c con guration of the raft increases its potential load, as surface tension effects can increase with the raft size.For the same reason, a linear or axisymmetric raft will not sink for the same number of particles (Figure 5d).

Beyond the Stability of a Particle Raft
Particle rafts are out of equilibrium once capillary forces cannot sustain their weight.They then start to collapse and sink.This destabilization process can also be triggered by applying a force on the raft or increasing its load.In particular, Larmour et al. (2008) studied how large oating rafts made of superhydrophobic copper particles can support objects that would normally sink (Figure 5a-c).These sheets are exible since they can deform when an object is placed on them, but they are robust enough that they do not tear or crack under loading.The sheet can then wrap heavy objects and the cohesion between the copper particles prohibits the penetration of liquid inside.Similarly, Jambon-Puillet et al. ( 2018) studied how centimetric drops can be placed on a particle raft and oat on a bath of the same liquid.The raft is deformed by the load of the oating drop and the particles' weight, but it also prevents the coalescence of the drop with the liquid, as it acts as an armor.The raft can also be pushed down from above with a cylinder (Abkarian et al. 2013) to initiate its collapse in a controlled way: The particles slide down along the cylinder at a constant speed and thus sink continuously when reaching its tip.
When out of equilibrium, the raft then sinks below the interface.The dynamics of the sinking process is again linked to the dimensionless parameter D. For large or dense particles (D > 3 in Figure 5d), the raft bows in the middle and encapsulates the upper liquid phase into a single large droplet as it sinks.For small or less dense particles (D < 3 in Figure 5e), as explained in the previous paragraph, the raft could be in nitely long and never sink.However, we have found that in this case, when rafts become several times longer than the capillary length, they can become metastable, as they are compressed by their own weight and wrinkles appear along the raft that grow and lead to total collapse (Protière et al. 2017, Guzowski & Gim 2019).The sinking process here is quite different, as one observes an interfacial granular jet that destabilizes due to a Rayleigh-Plateau-like instability and thus breaks down into millimeter-sized armored droplets, which we discuss in the last section of this review.

MECHANICAL PROPERTIES OF A PARTICLE RAFT
Several works have focused on investigating the mechanical properties of particle rafts.Contrary to suspensions, the particles form a 2D aggregate that is easy to visualize and probe.When subjected to various external stresses, it has been found that these aggregates undergo different types of structural changes (Figure 6), such as (a) rearrangements of the particles within the raft due to shear ows or low stresses that drive the particles to slide along one another; (b) free or controlled compression of the raft, which creates deformations that lead to its buckling; and (c) external ows that may force the particles to detach from the raft's rim and lead to its erosion or the breaking up of the raft into large pieces.

Rearrangements within a Particle Raft
At rest, a raft consisting of monodisperse particles will form a hexagonal lattice.Camoin & Blanc (1985) were the rst to report an experimental study on 2D aggregates subjected to a shear, which they compared to the attractive capillary forces, and observed the formation of fractal structures.Whether in shear or extensional ow (Stancik et al. 2002(Stancik et al. , 2003) )  surface area at the interface impacts the aggregate's structure (Berhanu & Kudrolli 2010).At low concentration (or high shear rates), particles may align in the ow direction and slide against each other, thus provoking a stretching of the aggregate.At high concentrations (or low shear rates), grain-to-grain interactions are strong and the particles are blocked in their lattice positions.In this case, domains within the cluster may rotate to accommodate the ow's direction.The force linked to rearrangements can be measured for a compressed 2D crystal made of monodisperse droplets (Ono-dit-Biot et al. 2020).It would be interesting to study the role of friction during rearrangements and compare those measurements to the controllable adhesion within 2D droplet crystals.

Buckling of a Particle Raft
A particle raft can also be compressed so as to increase its packing fraction in a controlled way (Figure 7a).There comes a critical packing area where the particles cannot rearrange anymore: They are jammed.This creates a sudden global change of behavior where instead of observing in-plane motion within the raft such as small-or large-scale arrangements, we have a sudden out-of-plane deformation of the entire cluster and the raft starts to buckle.Similarly to surfactants at a liquid interface, this behavior can be quanti ed using a Langmuir trough equipped with Wilhelmy plates to measure the surface tension of the interface while the quantity of particles remains constant (Aveyard et al. 2000).

Elastic behavior.
Once the interface has buckled, the particles are jammed and the raft reaches a transition from liquid-like to solid-like.Several works have focused on understanding this instability by considering the raft as a continuous elastic sheet oating at the surface of a uid.The raft can then be characterized by its bending stiffness per unit width, B = ER 3 part 12(1−ν 2 ) , with R part the sheet thickness, ν the Poisson ratio, and a Young's modulus E, which depends on the characteristics of the interface (Vella et al. 2004).For large compressions, wrinkles are observed along the surface of the raft (Figure 7d, f ) at a characteristic wavelength λ.For an elastic sheet, we have  After measuring λ we can deduce an effective bending rigidity of the particle raft.With this 2D approach, Vella et al. (2004) considered the mean stress and strain within the raft and hypothesized that the strain only depends on the liquid surface area.They obtained an effective Young's modulus that depends on the 2D packing fraction φ: which we can substitute in Equation 8to nd Instead of compressing the raft, Planchette et al. (2012b) oscillated a plate orthogonally to its interface to excite surface waves on the particle-laden surface (Figure 7b).They used a modi ed dispersion relation to include a bending term and the particles' inertia to characterize the raft's wave propagation.The authors proposed a model that takes into account the wettability of the particles via the contact angle θ (Kralchevsky et al. 2005); since the particles are adsorbed at the interface they have to move upward or downward to accommodate the pinning of the contact line.
www.annualreviews.org• Particle Rafts and Armored Droplets Janssen effect: the weight of grains in a silo is distributed both vertically and laterally due to the transmission of forces between grains in contact; the latter is responsible for the collapse of the silo's walls, which are not built to resist such loads This induces a variation of the free energy as a function of the interface's curvature.This model leads to the same relation for the bending rigidity that was deduced from the con guration of the particles within the raft under compression, λ ∼ R part , with a prefactor that depends on the wetting properties of the particles.The term θ can be varied, but it does not modify the bending rigidity signi cantly enough to rule out the elastic sheet analogy.
Using both experimental techniques (compression or wave propagation), Petit et al. (2016) showed that the buckling of bidisperse particle rafts highlights the dual nature of the raft: Its elasticity is con rmed via collective effects and its granular character is also observed via the propagation of stresses through force chains.The limits of the analogy of a particle raft with a continuous elastic sheet have been tested with various experimental setups.

Granular characteristics. When further investigating the compression of rafts in a
Langmuir trough, Cicuta & Vella (2009) have shown that by orienting the Wilhelmy plates parallel or orthogonal to the moving barriers, the critical buckling pressure will vary.The exponential decay for the surface pressure is then explained by arguing that the stress propagates through force chains and the walls absorb part of the stress, similarly to the Janssen effect observed in 3D granular silos ( Janssen 1895).The role of friction is also discussed by Saavedra et al. (2018), who distributed exible pressure sensors along the raft and measured the gradual development of internal stresses within the raft.They found that the screening of the pressure stress is much stronger and may be due to localized compaction effects.Similarly, local plastic instabilities are linked to a weakening of the shear response during the jamming transition (Varshney et al. 2012) and attributed to a local depinning of the contact line.
Most studies have focused on small deformations, in which the onset of buckling of the raft is linked to frictional forces once the particles have reached the jamming concentration.This elastic behavior can be studied in a controlled way by compressing these rafts beyond wrinkling.Jambon-Puillet et al. (2017) pushed this analogy even further by checking experimentally whether it is still valid for large deformations.They investigated the buckling instability in the compression of granular rafts in the linear and nonlinear regimes.In this case, a second wavelength of wrinkles was measured [which was also observed by Kassuga & Rothstein (2015)], and at larger compression, they observed the localization of the wrinkles into a single fold that grows until the entire raft sinks (Figure 7d,e).This is reminiscent of the reversible wrinkle-to-fold transition observed for a oating elastic sheet.After further probing of the limits of a continuum elastic model, it was found that the granular raft exhibits speci c mechanical properties, in particular, an irreversible plastic transition, which is observed in hysteretic behavior during compression cycles.When the particles are stirred, forcing new random particle arrangements, networks of intense force chains may be broken and the raft can recover its elastic properties.The raft's dual nature-granular and elastic-is thus highlighted during this process.
By locally indenting the raft (Figure 7c), researchers can probe the elastic response of a particle raft to normal forces as well (Zuo et al. 2017, He et al. 2020).For thin elastic sheets, wrinkles extend radially outward from the indentation, but for rafts, particles can rearrange, and their motions away from the indentation allow the lattice to accommodate the azimuthal compression without leading to a buckling instability, which shows that the raft behaves as a uid interface (Figure 7g).The raft can also withstand large indentation depths, act as a mechanical protection to the interface, and enable the production of armored droplets of controlled volume and particle coverage (Abkarian et al. 2013).A body force such as a magnetic eld can also be used to induce an upward bending of the raft, and the forces needed to create a deformation can then be quantitatively measured from the magnetic forces (Zuo et al. 2021). Protière

Testing the Cohesion of a Particle Raft: Erosion, Fractures, and Fragmentation
When an external ow is applied to a capillary raft, single particles can detach from the parent raft's edges.We call this process surface erosion.The raft remains mostly intact except for a few particles lost at its edges (Lagarde & Protière 2020).When the raft breaks up into larger pieces, we have a fragmentation of the raft.Studying how and where aggregates break allows researchers to test the cohesion of a raft and investigate the interparticle interactions within it.Vassileva et al. (2007) discussed the critical shear rate for breakup by erosion or fragmentation.In particular, they found that the two processes occur at different shear rates, as the normal forces in the center of the aggregate are larger than those near its rim.Moreover, the critical shear rate depends weakly on the aggregate size of the raft and the particles' radii.Since friction forces are larger for large particles (due to a larger grain-to-grain contact), large particle aggregates will hardly rearrange, while a small defect may induce crack growth and further fracturing; in contrast, for smaller particles, erosion is possible.One model based on the balance between the viscous drag, which creates deformations that facilitate the breaking up of the raft, and the attractive capillary forces that bind the particles into an aggregate predicts the cluster size distribution in a shear ow (Huang et al. 2012).
Another way to test the cohesion of a raft is to apply a radial stress.Kim et al. (2019) proposed an experimental method where a densely packed raft is placed in a funnel and the water level is progressively raised.In this case, depending on the intensity of the attractive force between the particles (which they controlled by varying their density), the raft will fracture or the particles will move and rearrange to heal small fractures.By performing quasi-static tensile tests on a particle raft, Xiao et al. (2020) observed similar plastic rearrangements and failures.The ductility of the material can be tuned by varying the interaction range for the capillary attractions.Large particles interact at shorter distances; for example, the rafts then fracture at small tensile deformations, leading to brittle behavior.In contrast, rafts made of small particles have long-ranged interactions and can rearrange more easily; by preventing any eventual voids from growing, they show less brittle behavior.Fractures are also observed when locally adding a surfactant at the center of a raft (Vella et al. 2006a).In this case, the crack propagation is mostly due to the advection of surfactant toward the crack tip, but its branching seems mostly due to the raft's solid-like response.When impacting a droplet on the surface of the particle raft, Planchette et al. (2018) observed that the mechanisms leading to its rupture and robustness are based on the size distribution of its particles as collective effects become apparent.These results con rm the importance of the steric repulsion between the particles in the solid-like response of particle rafts.

ARMORED DROPLETS
In this section, we describe a few results detailing the mechanical properties of armored droplets (i.e., liquid droplets adorned with a monolayer of particles adsorbed at their curved surfaces).With such systems, the mechanical properties of particle-laden drop surfaces can be investigated via various experimental con gurations.
As we have detailed in the previous section, the jamming of a particle-laden interface occurs when the particle concentration at the interface increases (or, conversely, when the surface area decreases).Shrinking the volume of armored droplets is a common method to reduce their interfacial area and force the compaction of the particles at their surfaces.This phenomenon leads to a similar solid-like behavior discussed above for rafts, with wrinkles appearing at the surface of the droplet (Monteux et al. 2007).With this protective shell, coalescence, which would increase surface area and therefore lead to less stable droplets, is also inhibited.To modify the armored droplet's volume, one can progressively remove the bulk liquid and simultaneously measure its internal pressure P. Alternatively, one can quasi-statically compress the shell by slowly evaporating the droplet, thereby keeping the boundary conditions intact, and image the droplet shape and granular interface to measure the variation of the solid fraction and the drop's effective surface tension (Lagubeau et al. 2014).At rst, the armored droplet's pressure corresponds to the Laplace pressure 2γ 12 /R (with R the droplet radius and γ 12 the interfacial tension), but then it decreases drastically to 0 due to the collapse of the droplet as jamming occurs, the particles' interactions make the pressure drop, and the surface starts to wrinkle.This critical collapse pressure P c can be compared to the behavior of the collapse of a solid spherical shell using various physical models (Pitois et al. 2015, Taccoen et al. 2016, Pitois & Rouyer 2019).Similar observations have been made for armored bubbles where, interestingly, jamming particles at their liquid-gas interface can extend their lifetimes inde nitely (Pitois & Rouyer 2019).
Armored droplets with jammed particles at their surfaces can be squeezed mechanically (Rendos et al. 2017) or can rupture upon impact (Aussillous & Quéré 2006, Planchette et al. 2012a).These two methods probe the droplets' robustness directly or indirectly ( Ji et al. 2020).These large deformations lead to fracturing of the shell.Indeed, upon impact at a critical velocity, an armored droplet spreads on the solid substrate and thus increases its surface area.This creates a hole in the armor, which can then wet the solid surface in this region.The armored droplet will adhere to the surface at this point, and its integrity is lost.The rupture mechanism is still unknown but seems to depend on the particle size: A single crack is observed for small particles and multiple holes are observed for larger particles, which shows that interparticle forces play an important role here (Figure 8a).

Protière
Thus far, we have discussed the particular case of liquid marbles or armored bubbles that are formed at an air-water interface.However, the stability of emulsions is known to be enhanced by adding hydrophobic solid particles at the oil-water interface since the pioneering work of Pickering (1907).Since then, so-called Pickering emulsions have been investigated thoroughly for many applications (Binks & Lumsdon 2001, Binks & Horozov 2006).These particles can be coupled to external elds in order to control their stability and capture or release the liquid inside the capsule.Interestingly, when the density of the particles is larger than the density of the outer liquid, ρ part > ρ 1 , oil-in-water armored droplets can be destabilized by gravity (Tavacoli et al. 2012).This process is entirely governed by a modi ed Bond number, which here compares the external force on one particle, namely gravity, to the maximal capillary force that keeps it adsorbed to the interface: Bo = (R part / c ) 2 .The limiting droplet size for these buoyant armored droplets under gravity is therefore D max = c , or the maximum number of particles N m that can be trapped at the surface of the droplet: 8b).Above this threshold, particles are ejected.
This limit takes into account the characteristics of the two liquid phases but does not account for variations in the particles' densities.Abkarian et al. (2013) highlighted the role of the particles by studying highly dense particles adsorbed at the oil-water interface of centimetric droplets.In this case, by adding oil progressively to such armored droplets, their buoyancy increases and it is the particles' weight that contains the oil and prevents it from escaping the droplet.For a critical oil volume, destabilization occurs via two scenarios depending on the parameters of the experiments: The entire droplet can become buoyant and oat to the surface, or the uncovered portion of the droplet can pinch off, leaving behind a small armored droplet.These armored droplets have been further investigated beyond buckling when subjected to large deformations.In particular, curvature effects can lead to particle desorption instead of buckling when the surface pressure increases for large values of R part /R (Gu & Botto 2018, Guzowski & Gim 2019).Armored droplets undergo strong deformations when placed in an extensional ow or a constriction, or during extremely fast events such as their coalescence or the oscillations due to an acoustic eld (Garbin 2019, de Soëte 2021).These dynamic phenomena also lead to the ejection of particles (via a rapid expulsion or a slow shedding), which is a topic that has only recently been explored (Figure 8c).

SUMMARY POINTS
1. Non-Brownian particles present at a liquid-liquid interface create local deformations that induce long-ranged attractive capillary interactions among objects that selfassemble into a capillary raft: a monolayer of grains.The clustering dynamics depends on the number of particles within the clusters.
2. Although individual particles within a raft oat independently thanks to Archimedes' Principle, the entire particle assembly can become unstable and sink in the case of large otation forces.The critical shape of the capillary raft depends on a modi ed Bond number, which takes into account the characteristics of the particles and the liquid.
3. The mechanical properties of a particle raft have been probed by subjecting it to various external stresses.It behaves as a single entity and can be modeled as a continuous elastic sheet, as wrinkles and folds can be observed when it is compressed, but its discrete nature is also responsible for restructuring within the raft or plastic deformations due to interparticle interactions.
www.annualreviews.org• Particle Rafts and Armored Droplets 4. The integrity of droplets coated by a protective armor of jammed particles, or socalled armored droplets, has been examined via various methods.In particular, armored droplets are found to fracture or eject particles when subjected to large deformations.The stability of the armor, as well as the cohesion between the individual particles, is then lost.

FUTURE ISSUES
1. We still lack a complete model for the clustering dynamics of dispersed particles into a single granular raft where the role of gravity and the additivity are fully described.In this case, objects form clusters with their closest neighbors but will eventually merge with the largest raft due to its strong attractive potential.The clustering process therefore depends on the rate of creation of a given raft compared to its aggregation with other clusters, which is uncommon in the clustering dynamics of passive systems.
2. A particle raft's cohesion is due to the attractive capillary interactions between the grains.The plastic behavior of a granular raft, however, occurs when the particles come into contact.This implies that friction between the grains plays an important role beyond the jamming transition.Bridging the liquid and granular aspects of particle rafts by controlling the friction coef cient, for example, would allow us to describe the plastic transition observed beyond the elastic analogy and fully describe the composite nature of this material.We are still lacking a full granular and elastic model that would include the wrinkling and bending of the raft under compression.
3. Large deformations of armored droplets or granular rafts have been observed when subjected to external body forces, external elds, or various types of ows.They lead to new types of behavior, such as the expulsion of particles or the dynamic collapse of the raft.Investigating these dynamic effects will improve our current knowledge of the mechanics of these composite materials in realistic conditions.
4. Similarly, measurements of the forces needed to break up or fragment a particle raft are still scarce and would offer a different insight into the cohesion of its structure.Moreover, the role of defects during the fragmentation process is still unclear.Once we understand a raft's disassembly we could then create or break large structures whose microstructures could rearrange at interfaces.

5
. Until now, we have only discussed the self-assembly of passive particles at interfaces.The study of how clusters of active particles behave when con ned to interfaces is now an emerging eld.Active particles driven by an external eld (oriented by surface curvature or another body force, for example) or with their own inertia will potentially affect their collective behavior and directed self-assembly.
6.The two uids creating the interface at which the particles self-assemble into a particle raft are always Newtonian, with properties that do not vary depending on the dynamics.Particle rafts can undergo fast deformations or rapid collapses.At complex uid interfaces, one could expect these processes, as well as the mechanical properties of the particle raft, to be greatly affected and thus completely change their overall behavior. al.
Figure 1 Examples of particle rafts found in nature or formed via capillary-driven self-assembly.(a) Top and side views of the construction of a raft consisting of ∼3,000 ants.The ants redistribute themselves until reaching equilibrium.(b) Mosquito egg rafts oating on water (size: 1 mm).The eggs aggregate at the interface similarly to ellipsoidal micrometer-sized particles.(c) Cylindrical particles' orientation is controlled by a curved interface formed by its pinning to a micropost.(d) Spinning of four circular magnetic objects that self-assemble at the air-water interface.Depending on the spinning speed , the particles will assemble, disassemble, or recon gure.Panels adapted with permission from (a) Mlot et al. (2011), copyright 2011 National Academy of Sciences; (b) Loudet & Pouligny (2011), copyright 2011 EDP Sciences, SIF, and Springer-Verlag; (c) Cavallaro et al. (2011), copyright 2011 National Academy of Sciences; and (d) Wang et al. (2017) (CC BY-NC 4.0).

Figure 2
Figure 2Capillary forces between two objects.(a) Normal capillary forces.The liquid is represented in blue, while the surrounding uid is white.A capillary bridge connects the two interacting particles, leading to an attractive normal force.(b) Flotation forces are caused by the weight of the object, leading to a lateral force.If the two objects de ect the interface in the same direction, the force is attractive; otherwise it is repulsive.(c) Immersion forces can arise in different situations: due to the con nement of colloidal particles inside a liquid layer that decays at in nity (top) or to the undulation of the contact line induced by the particles' surface properties (bottom).Figure adapted from Lagarde (2020) with permission; copyright 2020 the author.
Figure adapted from Lagarde (2020) with permission; copyright 2020 the author.

Figure 4
Figure 4The kinematics of two identical spheres and two particle rafts.(a) The interaction between two identical spheres, both subjected to a capillary attraction and a viscous drag.(b) Speed V of the particle as a function of the distance between the two particles, L/R part .The dashed line corresponds to Equation 4. (c) The interaction between two rafts A and B made of n A and n B particles, respectively.(d) Speed of the raft of increasing size (n A = 1, 5, 10, 20, 40) attracted to a raft of xed size (n B = 60) as a function of the nondimensional distance L/R part .Dashed lines correspond to the velocity found using Equations 5 and 6.Panels adapted with permission from (a,c) Lagarde (2020), copyright 2020 the author; (b) Dalbe et al. (2011), copyright 2011 American Physical Society; and (d) Lagarde et al. (2019), copyright 2019 Royal Society of Chemistry.
Figure 5Sinking instability of a particle raft.(a) Surface of a particle raft made of superhydrophobic copper particles.(b) A water drop can create a large deformation in a sheet without sinking.(c) 1-mL drops on a superhydrophobic particle raft.(d) The transition between oating and sinking for axisymmetric particle rafts: the maximum arclength L of the raft before sinking as a function of the density ratio D. The dashed line corresponds to the analytical model of the oating-sinking transition detailed byProtière et al. (2017).For small values of D, a raft would never sink.The open red circle corresponds to a linear particle raft, whereas the other points correspond to axisymmetric rafts.(e) A raft at an oil-water interface; once unstable, it collapses by forming an oil jet with particles at its interface, which breaks further down into small armored droplets (droplet size ≈5 mm).( f ) A sphere descending at the liquid-liquid interface, pulled toward an existing particle raft by capillary forces to nally self-assemble with the raft (top).A sphere descends from on top of the raft to rearrange within the raft and nally form a stack of particles that is out of equilibrium (bottom).Panels adapted with permission from (a-c)Larmour et al. (2008), copyright 2008 Wiley-VCH; (d) Protière et al. (2017), copyright 2017 American Physical Society; (e) Abkarian et al. (2013), copyright 2013 Nature; and ( f ) Jones et al. (2015), copyright 2015 AIP.
Figure 6 Possible changes in a particle raft subjected to an external force.(a) Reorganization of the internal structure.(b) Buckling of the raft, which induces the formation of cracks, wrinkles, or folds within the monolayer.(c) Erosion, in which a few particles can be pulled from the raft's rim.(d) Fragmentation, in which the raft breaks into various pieces of comparable sizes.Figure adapted with permission from Lagarde (2020), copyright 2020 the author.
Figure 7 Various experimental setups to probe the mechanical properties of a particle raft, such as (a) compression, (b) oscillations to create surface waves, and (c) indentation.(d) By compressing a raft at an oil-water interface we observe wrinkles (view from below), which (e) localize into a single fold upon further compression (particle radius R part = 75 µm).The arrows indicate the direction of compression.( f ) Bottom view of a raft once it has buckled, which can be used to measure the wavelength λ of the wrinkles (R part = 250 µm).(g) By poking a raft from above we observe particle rearrangements within the raft to accommodate the deformation.Panels adapted with permission from (d-f ) Jambon-Puillet (2016), copyright 2016 the author, and (g) He et al. (2020), copyright 2020 Royal Society of Chemistry.
Figure 8 Particle-laden interfaces can lose their integrity.(a) When subjected to a large deformation such as an impact, an armored droplet will fracture its shell.Here we see the top view of the impacts, with particle radius R part = 16 m (left) and R part = 67 µm (right).(b) An armored droplet destabilized by gravity sheds particles until reaching an equilibrium between buoyancy and weight.(c) An armored droplet owing through a constriction will eject particles at large ow rates.Panels adapted with permission from (a) Planchette et al. (2012a), copyright 2012 EPLA; (b) Tavacoli et al. (2012), copyright 2012 American Physical Society; and (c) de Soëte (2021), copyright 2021 the author.