Dominique Langevin
email: dominique.langevin@université-paris-saclay.fr

Recent Advances on Emulsion and Foam Stability

In this perspective paper, we highlight the numerous open problems in the topic of stability of emulsions and foams, focusing on the simplest case of dispersions stabilised by surfactants. There are three main destabilization processes, gravity induced evolution, Ostwald ripening and drops or bubble coalescence, which are analysed separately. The discussion is restricted to the case of newtonian fluids, deprived of microstructure, except for the presence of micelles. Thanks to continuing efforts and recent breakthroughs, we show that the understanding of emulsion and foam stability is progressing. Many problems are still open however, and much work remains to be done along the lines outlined in the paper.

Introduction

Emulsions are dispersions of oil in water (O/W emulsion) or of water in oil (W/O), whereas foams are dispersions of gas in a liquid, frequently water [START_REF] Langevin | Microemulsions and Foams[END_REF] . Both emulsions and foams are stabilized by surfactants, proteins or small particles. In some cases, for instance when particles are used, dispersions of water in gas can be obtained [START_REF] Binks | Colloidal Particles at a Range of Fluid-Fluid Interfaces[END_REF] . When the liquid fraction  of the surrounding (continuous) liquid in emulsions and foams is large, the inner fluid is dispersed in the form of spherical drops or bubbles ; in the second case, the dispersion is usually called bubbly liquid instead of foam. When  is close to about 36%, the dispersed spherical objects come in close contact (the volume fraction of a random compact assembly of spheres is 64%). Below  ~ 36%, the spheres are deformed into polyhedra and the dispersions no longer flow, the surface energy cost due to the spheres' deformation being stored in the form of elastic energy. This situation is common to foams, but is also encountered in high internal phase emulsions (HIPE) such as creams. A scheme of the evolution of foam structure with increasing liquid fraction is shown on figure 1. The evolution is similar for O/W emulsions stable enough to form HIPEs. Most emulsions however invert when the drop fraction is increased : O/W emulsions do not really transform into W/O emulsions, but rather form multiple emulsions because a surfactant used to stabilize O/W emulsions cannot stabilise W/O emulsions. This inversion was called catastrophic by Salager and coworkers [START_REF] Salager | Emulsion Formulation Engineering for the Practitioner[END_REF] .

Figure 1. Scheme of the evolution of the structure of an aqueous foam when the water liquid fraction  is increased. Above  ~ 36%, a bubly liquid is obtained. The evolution is the same for the emulsions stable enough to form HIPEs.

The liquid fraction of 36% correspond to a situation where hard spheres are randomly close packed. It signals the transition between fluid and solid systems, currently called jamming transition. There are frequently interactions between the drops/bubbles In emulsions and foams, and the jamming transition can be significantly shifted : for instance, it can increase up to  ~ 70% in adhesive emulsions [START_REF] Datta | Rheology of Attractive Emulsions[END_REF] . Let us finally mention that bicontinuous structures have been discovered in particlestabilized systems. [START_REF] Herzig | Bicontinuous Emulsions Stabilized Solely by Colloidal Particles[END_REF] The stability of emulsions depends on the liquid fraction  as will be discussed in the following. So far, it is still very difficult to predict how stable these dispersions will be, other than empirically. Their stability is determined by an interplay between different phenomena : gravity induced motion of drops/bubbles, Ostwald ripening and drop/bubble coalescence. Gravity decreases the external liquid fraction , hence favours ripening and coalescence; alternatively, ripening and coalescence lead to an increase of drop/bubble size and the decrease of  due to gravity is then more rapid. In the following, we will summarize the recent advances in the understanding of the three different phenomena. The discussion on coalescence will be the longest, because coalescence is the less understood phenomenon.

Gravity effects

Because of the density difference between dispersed and external phases, gravity has an important effect, especially in foams for which the density difference is large. As a consequence, air bubbles rise and the liquid drains between them. The process is usually called drainage, whereas in emulsion, it is rather called sedimentation if the drops are heavier than the external phase, or creaming if they are lighter.

The motion of individual bubbles or drops in the presence of gravity has received a lot of attention. When they are small, their velocity V is given by the Stokes formula

𝑉 𝑆𝑡 = 2|𝜌 𝑐 -𝜌 𝑑 |𝑔𝑅 2 9𝜋𝜂 𝑐 (1)
where R is the radius of the drops/bubbles, c and d are respectively the density of the continuous and disperse phases, g is the accelaration of gravity and c the viscosity of the continuous phase. This expression has been established for solid spheres and is valid when the surface of the drops/bubbles remains immobile during their motion.

Although this is most often the case, it was found early that when the fluids are carefully cleaned, the velocity V was larger than the Stokes velocity by a factor 3/2, as expected for spheres with mobile surfaces. Surfactants confer to the interface a resistance to deformation, characterized by viscoelastic parameters. Levich has shown that when the resistance to compression is large enough, the Stokes velocity can be used [START_REF] Levich | Physicochemical Hydrodynamics[END_REF] . He introduced a surface compression modulus E defined as :

𝐸 = 1 𝐴 𝑑𝛾 𝑑𝐴 (2)
 being the surface tension and A the surface area.

The transition between the two limits, mobile and rigid surfaces, is observed when the surfactant concentration is increased and has been extensively studied 7 8 . When the concentration is still larger, one expect that the resistance to compression will go through a maximum and then decrease to zero : at large concentration, the exchanges of surfactant between surface and bulk are fast, and the resistance to compression vanishes as also predicted by Levich. This effect has been observed during the motion of drops/bubbles in tubes under the action of a pressure drop [START_REF] Stebe | Remobilizing Surfactant Retarded Fluid Particle Interfaces. II. Controlling the Surface Mobility at Interfaces of Solutions Containing Surface Active Components[END_REF] , but not yet for gravity motion.

When the drops/bubbles are large, their velocity V is large and they can deform during their motion, changing the velocity in a complex manner. When the Reynolds number is large, instabilities can occur. These effects have received a lot of attention [START_REF] Clift | Drops, and Particles[END_REF] and are now reasonably well understood.

The motion of a large number of drops/bubbles is a still more difficult problem and is not fully clarified for the time being [START_REF] Mudde | Gravity-Driven Bubbly Flows[END_REF] . An empirical expression was proposed for solid spheres by Richardson and Zaki [START_REF] Richardson | The Sedimentation of a Suspension of Uniform Spheres under Conditions of Viscous Flow[END_REF] and is accurate up to large drop/bubble volume fractions:

𝑉 = 𝑉 𝑠𝑡 𝜙 𝑛 (3)
The exponent n varies with the Reynolds number Re and at low Re n = 4.5 [START_REF] Chen | The of Bubbles in Shear Thinning Viscoelastic Fluids[END_REF] .

The extreme case of a continuous liquid fraction close to zero is surprisingly better understood, after extensive studies with foams. The liquid drains through the network of interstices between bubbles called Plateau borders. The velocity of drainage is proportional to Vst, but much smaller [START_REF] Saint-Jalmes | Physical Chemistry in Foam Drainage and Coarsening[END_REF] :

𝑉 = 𝐾 𝑑𝑟𝑎𝑖𝑛 𝜌 𝑐 𝑔𝑅 2 𝜂 𝑐 𝜙 𝛼 (4)
where Kdrain is a foam permeability (Kdrain~ 1/300) and  an exponent depending on surface mobility :  = 1 for rigid surfaces, ½ for mobile surfaces. Here the surface mobility is related to the surface shear viscosity. This result only applies to sufficiently dry foams, for which  is equal to a few percent.

A theoretical description of drainage of wetter foams is still awaiting.

When drainage is completed, the liquid fraction vertical profile remains constant, with a dry foam at the top, and a wetter one in contact with the drained liquid. This profile has been determined theoretically and found in agreement not only with experiments with foams, but also with emulsions [START_REF] Maestro | Liquid Dispersions under Gravity: Volume Fraction Profile and Osmotic Pressure[END_REF] .

Ostwald ripening

The internal pressure in drops/bubbles exceeds the external pressure by a quantity equal to the capillary pressure, equal to 2/R for spherical drops/bubbles. Due to this excess pressure, the internal fluid is transferred by diffusion through the continuous phase, from the small drops/bubbles to the larger ones. The problem is the same than in alloys and a theory was established by Lifchitz, Slyozov and Wagner (LSW theory) in the limit  → 1 [START_REF] Taylor | Ostwald Ripening in Emulsions[END_REF] . This theory predicts that the radius increases with time as R ~ t 1/3 with a rate  = dR 3 /dt given by:

Ω 3 = 8 9
𝛾𝐷 𝑚 𝑆 𝑉 𝑚 𝑘 𝐵 𝑇 (5) where Dm is the diffusion coefficient of molecules of the dispersed phase into the continuous phase, Vm their molecular volume, and S the solubility of the dispersed phase in the continuous phase. For bubbly liquids, the solubility S is generally expressed in terms of Henry constant He:

Ω 3 = 8 9 𝛾𝐷 𝑚 𝐻𝑒 𝑉 𝑚 (6)
He being expressed in mole m -3 Pa -1 .

Measurements of the Ostwald ripening rate in alloys with  between 90 and 20%, showed that the experimental rates were only at most three times larger than those of the LSW theory [START_REF] Baldan | Review Progress in Ostwald Ripening Theories and Their Applications to Nickel-Base Superalloys -Part I: Ostwald Ripening Theories[END_REF] .

Experiments with emulsions however rarely fitted with the theory, and deviations by factors of order 10 are currently observed. For instance, the ripening rate depends on the amount of surfactant in excess in the continuous phase, but no rational explanation was yet found [START_REF] Ariyaprakai | Influence of Surfactant Structure on the Contribution of Micelles to Ostwald Ripening in Oil-in-Water Emulsions[END_REF] . A mechanism specific to emulsions was recently identified: exchange of disperse phase during contact between drops [START_REF] Roger | Emulsion Ripening through Molecular Exchange at Droplet Contacts[END_REF] . The exchange is facilitated when the surfactant molecules are small. However and although the coarsening rate can be reasonably estimated by the LSW theory, its exact value cannot yet be predicted.

In dry foams, the gas transfer occurs mainly through the thin liquid films formed between bubbles, instead of the larger spaces between bubbles, that scale with the bubble radius. As a result, the bubbles radius increases more rapidly: R ~ t 1/2 . The ripening rate  such as dR 2 /dt =  f (), f () being the fraction of bubble surfaces covered by films, is given by:

Ω 2 = 𝑘 𝛾 𝐷 𝑚 𝐻𝑒 𝑉 𝑚 ℎ (7)
where k is a geometrical coefficient related to bubble shape ( k ~ 1.37 for Kelvin tetradecahedra) and h the film thickness [START_REF] Hilgenfeldt | Dynamics of Coarsening Foams: Accelerated and Self-Limiting Drainage[END_REF] .

As in Ostwald ripening of emulsions, the experimental determinations of the rate 3 differ from the predictions by factors of order 10 21 . This is probably because drainage produces rapid changes of liquid fraction, difficult to control. Indeed, a good agreement was obtained in experiments in which the coupling of drainage and coarsening has been modeled [START_REF] Hilgenfeldt | Dynamics of Coarsening Foams: Accelerated and Self-Limiting Drainage[END_REF] . However, experiments with relatively dry foams  < 15%, where the liquid fraction was maintained constant thanks to a rotation of the foam samples led to rates significantly different from the predicted ones [START_REF] Saint-Jalmes | Differences between Protein and Surfactant Foams: Microscopic Properties, Stability and Coarsening[END_REF] . It is possible that residual drainage affected these measurements.

Recent experiments performed in microgravity conditions in the International Space Station [START_REF] Cohen-Addad | Coarsening of Wet Foams. A Microgravity Study[END_REF] were found in good agreement with Equation 7and allowed to determine the function f(). In turn the rates measured in bubbly liquids are much larger than those predicted, possibly because of gas exchanges through transiently formed films as found in emulsions [START_REF] Roger | Emulsion Ripening through Molecular Exchange at Droplet Contacts[END_REF] .

Coalescence

Coalescence is the rupture of the liquid films formed between drops/bubbles, either during collisions, or at rest in HIPEs or foams. Coalescence is the less well understood destabilization process in emulsions and foams. One of the reasons is the large number of different coalescence scenarios.

For instance, hydrophobic particles can bridge aqueous films and rupture them through a dewetting process. This is one of the mechanism of action of antifoams 24 . In the following, we will focus on dispersions containing only surfactants and distinguish between dispersions with high and low stability, which destabilize in a very different manner [START_REF] Kabalnov | Macroemulsion Stability: The Oriented Wedge Theory Revisited[END_REF] .

Dispersions with low stability

Some emulsions and foams destabilize a few minutes after being formed. This happens in general when the surfactant concentration is low, below the critical micellar concentration (cmc) for nonionic surfactants and about two orders of magnitude below for ionic surfactants [START_REF] Petkova | Foamability of Aqueous Solutions: Role of Surfactant Type and Concentration[END_REF] . Exerowa proposed that these concentrations correspond to the onset of formation of stable liquid films between drops and bubbles [START_REF] Exerowa | Foam and Foam Films: Theory, Experiment, Application[END_REF] . Such films are in equilibrium under the action of forces between film surfaces, attractive van der Waals forces and repulsive forces, electrostatic for ionic surfactants, and steric for nonionic ones. Figure 2 shows the schematic variation with film thickness of the force per unit area, called disjoining pressure d (figure 2). In unstable emulsion and foams, only van der Waals forces are present, and the films never reach an equilibrium thickness.

When two drops or bubbles approach at a velocity V and when the distance h between their surfaces becomes small, these surfaces may deform and flatten out. Flattening occurs when the hydrodynamic pressure Phyd at the film center exceeds the capillary pressure Pcap in the drops/bubbles. It is only then that films form strictly speaking. For drops/bubbles of radius R with immobile surface boundary conditions, when P hyd > P cap and h < h flat , the surfaces flatten:

ℎ 𝑓𝑙𝑎𝑡 = 𝑅 2 √ 3𝜂 𝑐 𝑉 𝛾 (8)

Film thinning and rupture

Shortly after film formation, the flattened regions deforms and dimples form [START_REF] Chan | Film Drainage and Coalescence between Deformable Drops and Bubbles[END_REF] (see Figure 3) . This occurs when the film thickness is about twice smaller than hflat. After the formation of the dimple, the surfaces continue approaching until coalescence occurs.

The problem of film rupture was extensively investigated. A model was proposed early by Sheludko and Vrij, in which rupture is due to the amplification of capillary waves at the film surfaces when the film thickness becomes small enough 29 30 . The Vrij's model predicts a critical film thickness for rupture of a few tens of nm, although recent experiments showed that the films studied rupture at much smaller thicknesses, a few angströms [START_REF] Vakarelski | Dynamic Interactions between Microbubbles in Water[END_REF] . A hydrodynamic model, with no adjustable parameters accounted very well for the rupture times. Although the fluids used were surfactantfree, the surface was assumed to be immobile. As discussed earlier, this can arise from residual contamination. It should be noted that the effect of residual contamination depends on the type of experiment performed : the same interface can behave as mobile or rigid, depending of the type of applied constraint.

Evidence of coalescence induced by fluctuations were reported in solutions containing particles and polymers [START_REF] Aarts | Direct Visual Observation of Thermal Capillary Waves[END_REF] . However, in these systems, the surface tension was ultralow and the surface fluctuations were present all along the drop surfaces, not only in the regions where two drops were approaching as assumed in the Vrij-Sheludko model. Studies of films with mobile surfaces were performed recently and evidence of surface fluctuations was found when the film thinning velocity was small. However, although the films ruptured when they were 30 nm thick, the observations were not consistent with the Vrij's model. [START_REF] Chatzigiannakis | Breakup of Thin Liquid Films: From Stochastic to Deterministic[END_REF] Figure 3. Vertical profile of a silicone oil film between glycerol drops at various times after dimple formation: experiments (symbols) and theory (lines). The inset shows interference fringes observed when illuminating the film perpendicularly to its surfaces. Reproduced with permission from reference [START_REF] Chan | Film Drainage and Coalescence between Deformable Drops and Bubbles[END_REF] . Copyright 2011, Royal Society of Chemistry.

In the experiments of ref [START_REF] Vakarelski | Dynamic Interactions between Microbubbles in Water[END_REF] the films remained centrosymmetric during thinning. In ref [START_REF] Chatzigiannakis | Breakup of Thin Liquid Films: From Stochastic to Deterministic[END_REF] , an asymmetry developed when the fluctuations appreared. Asymmetrical film thinning was reported early, but attributed to a hydrodynamic instability occuring when the surface rheological parameters are small [START_REF] Joye | Numerical Simulation of Instability Causing Asymmetric Drainage in Foam Films[END_REF] . This instability causes the films to rupture when they are still quite thick.

Early rupture was also observed in emulsions with large concentrations of surfactant. This happens when the partition coefficient of the surfactant between oil and water is close to one. When the surface is compressed, the surfactant can dissolve very rapidly in both oil and in water, and the resistance to compression vanishes, leading to very rapid film thinning: the surface tension gradients produced during film thinning cannot be sustained and no Marangoni force is present to slow down film thinning, the film surfaces are mobile [START_REF] Marquez | Instability of Emulsions Made with Surfactant-Oil-Water Systems at Optimum Formulation with Ultralow Interfacial Tension[END_REF] . This is likely also the mecanism behind the action of demulsifiers which are surfactants equally partitioned between oil and water [START_REF] Marquez | Breaking of Water-In-Crude Oil Emulsions. Part 9. New Interfacial Rheology Characteristics Measured Using a Spinning Drop Rheometer at Optimum Formulation[END_REF] .

Coalescence without film formation

When the velocity V is very small, and/or if the drops/ bubbles are very small, no films are formed, Phyd remaining smaller than Pcap. At small gap thickness and under the influence of van der Waals attraction, the surfaces bend outwards (protrude toward each other), and two pimples are created [START_REF] Chan | Film Drainage and Coalescence between Deformable Drops and Bubbles[END_REF] .

Pimples form because the van der Waals disjoining pressure increases faster than the hydrodynamic pressure when the gap thickness decreases. Pimples spontaneously grow until the surfaces touch each other and drops/bubbles coalesce. In the opposite case of very high velocities, drops or bubbles approaching an interface may rebound after the collision.

Experiments were performed recently by Politova et al who measured the lifetimes of water drops of dilute aqueous surfactant solutions pressed by gravity onto a flat oil-water interface [START_REF] Politova | Coalescence Stability of Water-in-Oil Drops: Effects of Drop Size and Surfactant Concentration[END_REF] . The data for small drops has been analyzed assuming that the lifetime was controlled by the velocity of approach, and that coalescence occurs when h reaches the value at which the van der Waals disjoining pressure equals the capillary pressure. The agreement between calculated lifetimes and measured ones was rather good for small drops (for which films are not expected to form)

Politova et al also made experiments with larger drops, for which flat films should form between the drops and the oil-water interface. Coalescence was assumed to occur when the film thickness reaches the critical thickness in Vrij's model. The agreement with the experiments is not as good as for the small drops, probable because of the inadequacy of Vrij's model in this case [START_REF] Politova | Coalescence Stability of Water-in-Oil Drops: Effects of Drop Size and Surfactant Concentration[END_REF] .

In summary, when the surfactant concentration is not sufficient for equilibrium films to form, emulsions and foams are very unstable and destabilize after only a few minutes. Only attractive van der Waals forces are present between drops/bubbles surfaces and accelerate the drops/bubbles approach when they are close enough. Instabilities may lead to asymmetric deformations, rapidly followed by rupture when the distance between drops/bubbles is still large. The surface fluctuations recently observed might be precursors of this instability, explaining why the Vrij-Sheludko model that assumes centro-symmetry is not appropriate [START_REF] Chatzigiannakis | Breakup of Thin Liquid Films: From Stochastic to Deterministic[END_REF] .

The lifetime of drops/bubbles is controlled by their velocity of approach, the coalescence itself being very rapid (typically of the order of milliseconds). It should be noted that the lifetime then depends of the type of experiment performed: it is different for instance when coalescence is observed during the creaming of an emulsion or by looking at the collision between a single drop and a microscopic oil-water interface. [START_REF] Plassard | Impact of Electrolytes on Produced Water Destabilization[END_REF] The trends can even be opposite, when for instance a polymer is added: on one hand, the polymer can lead to flocculation and accelerate coalescence during emulsion creaming, and on another hand increase the viscosity of the continuous phase, slowing down film thinning in single drop experiments, thus slowing down coalescence [START_REF] Plassard | Impact of Polymers and Surfactants on the stability of emulsions produced after water injection into oil wells[END_REF] .

Dispersions with high stability

When the surfactant concentration is large enough, stable equilibrium films can form. In the case of emulsions, the surfactant should be present in the continuous phase of the emulsion (Bancroft rule), in order for the surface tension gradients produced during film thinning to remain active and slow down thinning. There is indeed not enough surfactant in the thin films to replenish the depleted surfaces 40 . The lifetime of the films is then larger than the film thinning time, because the time of film rupture becomes important and dominates in the very stable films. Figure 4 shows the evolution of film lifetime with surfactant concentration before and after the onset of stable film formation (black spot in the figures on the right) [START_REF] Politova | Factors Affecting the Stability of Water-Oil-Water Emulsion Films[END_REF] .

Films form even between small drops when repulsive interactions are present: when the distance h between drops/bubbles becomes comparable to the range of surface forces and when these forces are repulsive, the capillary pressure is opposed by the disjoining pressure and drops/bubbles surfaces flatten. It was observed that the films form only close to their equilibrium thickness when they are very small (diameters of the order of 10 m) [START_REF] Velev | Investigation of Thin Liquid Films of Small Diameters and High Capillary Pressures by a Miniaturized Cell[END_REF] .

For fully immobile surfaces, the velocity of film thinning is given by the Reynolds expression :

𝑉 Re = - dℎ dt = 2ℎ 3 3𝜂 𝑐 𝑅 2 {Δ𝑃 -𝜋 𝑑 (ℎ)} ( 9
)
where P is the pressure difference between film centre and border. This pressure contains contributions of the capillary pressure (always present and due to film border curvature), gravity when the film is not horizontal and eventual externally applied forces. In general, the surface tension gradients are not sufficient to ensure that the surfaces are immobile, and the thinning velocity is larger than VRe. It is sometimes assumed that equation 9 can be used by introducing a smaller viscosity eff = n 2 c , the index n being a mobility factor: 0 < n <1. This is not correct, because n is not constant: the thinning velocity decreases faster that h 3 when h decreases 40 and tends toward the Reynolds velocity when the film thickness approaches it equilibrium value [START_REF] Ivanov | Thin Liquid Films[END_REF] . This value depends on the applied pressure and is such that P = d(h) (VRe = 0).

When repulsive forces are present, the equilibrium films should never rupture. Models such as the Vrij model are inappropriate, because it assumes that the force is attractive (van der Waals). However, the film surfaces being partly mobile, surface concentration fluctuations are possible and if the local concentration is small enough, the repulsive forces may disappear, allowing the film to rupture. De Gennes estimated the corresponding activation energy [START_REF] De Gennes | Some Remarks on Coalescence in Emulsions or Foams[END_REF] : U* = kBT  2 /< 2 > where kB is the Boltzmann constant, T the absolute temperature,  the surface concentration,  its variation and < 2 > the mean square amplitude of surface concentration fluctuations [START_REF] Langevin | Adsorbed Monolayers[END_REF] :

〈𝛿Γ 2 〉 = 𝑘 𝐵 𝑇 Γ 2 𝐸 𝐴 ℎ𝑜𝑙𝑒 , ( 10
)
where E is the surface compression elastic modulus (see equation 2) and A hole the area of the region depleted in surfactant. When a film ruptures, the critical hole radius is half the film thickness [START_REF] De Vries | Foam Stability: Part V. Mechanism of Film Rupture[END_REF] .

Taking  ~  E ~ 40 mN/m, h ~ 10 nm, U* is about a few hundred kBT. De Gennes then quoted that intrinsic film rupture is extremely difficult and can only occur with external agents such as antifoam particles are present. However, for a film to rupture,  only needs to be about /10, concentration at which films are no longer stable [START_REF] Exerowa | Foam and Foam Films: Theory, Experiment, Application[END_REF] .

De Gennes related the activation energy U* to the compression elastic energy and estimated the coalescence frequency:

1 τ = 1 τ 0 A A hole exp (- U* k B T )= 1 τ 0 A A hole exp (- δΓ 2 𝐸 A hole Γ 2 k B T ) (11)
where  is a molecular time, equal to the hole dimension divided by the velocity of compression waves and A is the contact area of two emulsion drops/bubbles [de Gennes, unpublished, see ref [START_REF] Langevin | Microemulsions and Foams[END_REF] page 113, for details]. This relation shows that the stability of emulsions and foams frequently appears correlated with the surface compression modulus E, at least for surfactants irreversibly adsorbed, for which the modulus is not affected by exchanges between surface and bulk.

Kabalnov and Wennerstrom proposed an alternative model to relate emulsion stability to the surfactant layer curvature energy and applied the model to nonionic surfactant near the temperature at which the tension is minimum [START_REF] Kabalnov | Macroemulsion Stability: The Oriented Wedge Theory Revisited[END_REF] . They found an activation energy given by: U*/kBT = 0.43 + 30.9 T, T being the temperature difference with respect to the temperature at which the tension is minimum. The change of the activation energy with temperature is thus drastic, and compatible with experimental observations. There are however many drawbacks in the model. Kabalnov and Wennerstrom used an expression of the curvature energy that is not valid for the very thin equilibrium films formed by nonionic surfactants: the radius of curvature of the hole is comparable to the surfactant molecular length. In addition, they used a surface tension expression that neglect the variation of the elastic constants with temperature and the contribution of dispersion entropy. They also used a positive gaussian bending elastic modulus, whereas a careful analysis of surface tension data led later to a negative modulus [START_REF] Sottmann | Ultralow Interfacial Tensions in Water-n-Alkane-Surfactant Systems[END_REF] . The activation energy calculated with the data of ref [START_REF] Sottmann | Ultralow Interfacial Tensions in Water-n-Alkane-Surfactant Systems[END_REF] becomes U*/kBT= 11.5 + 66.6 T, and the predicted change in activation energy with temperature is no longer drastic as observed in the experiments. The model could apply to ionic surfactants, for which the equilibrium films are thicker and the expression of the curvature energy more appropriate. However, this energy becomes negligible compared to the other surface energy contributions in systems with larger surface tension. Equation 11 predicts that the film lifetime is inversely proportional to film area. It has been early recognized that coalescence begins when emulsion drops reach a critical size, for instance during their growth due to Ostwald ripening [START_REF] Georgieva | On the Possible Role of Surface Elasticity in Emulsion Stability[END_REF] . In other studies, it was claimed that coalescence occurs above a critical capillary pressure 50 51 . In these experiments however, both the size and the capillary pressure vary, and it is not easy to disentangle their role. An experiment with 2D foams was designed to control both parameters independently. It was then shown that at constant capillary pressure, the film lifetime is proportional to film area as predicted by Equation 11. [START_REF] Forel | Coalescence in Two-Dimensional Foams: A Purely Statistical Process Dependent on Film Area[END_REF] The role of the capillary pressure was recently clarified using simple bottle tests in which the creaming of emulsions is monitored. Changing the oil fraction allows to obtain a larger heigth of the creamed emulsion H, and to vary the pressure (Figure 5). The results are in excellent agreement with a model based on the fact that the film lifetime is proportional to film area : increasing H increases the contact area between drops [START_REF] Dinh | Coalescence in Concentrated Emulsions: Theoretical Predictions and Comparison with Experimental Bottle Test Behaviour[END_REF] . The results have been used to compare the coalescence probability with the model of Wennerstrom and Kabalnov, despite the drawbacks of this model 53 54 .

Conclusions and perspectives

The discussion presented highlights the numerous open problems in the topic of stability of emulsions and foams stabilised by surfactants. The discussion was restricted to surfactant solutions without microstructures, except for the presence of micelles. In practice, these dispersions can be stabilised by polymers, particles or their mixtures including with surfactants. The constitutive fluids can also be viscoelastic, which bring additional complexity.

We have tried to list the recent advances, together with the still open problems and considered separately the processes involved in the destabilization : gravity effects, ripening and coalescence.

Because none of them is yet fully described, we did not attempt to discuss the interplay between them Gravity effects are perhaps the better understood ones, the phenomena at play being well identified.

It is now recognized that the question of interface mobility is specific to the type of hydrodynamic flow considered and that the same interface could well behave as mobile and immobile in different situations. The rheological parameters characterizing this mobility can be different, for instance the compression modulus in the motion of isolated drops/bubbles and the surface shear viscosity in foam drainage. The description of the gravity effects is now satisfactory for very dilute dispersions, even when the drops or bubbles are large and deformed. The gravity drainage of dry foams is also well described. The intermediate cases where the continous or the disperse liquid fractions are larger than a few % remains to be improved.

Ostwald ripening is less well understood although the ripening rates are seldom larger or smaller than the theoretical predictions by a factor 10. The role of free surfactant is still not clear, despite numerous studies. The exchange of dispersed phase during collisions recently evidenced might help to clarify the issue. Studies of wet foams (or HIPEs) are in progress and will fill the missing liquid fraction gap.

Numerous recent coalescence studies improved the understanding of this process, the most complex in emulsion and foam destabilization. It is now clear that coalescence mechanisms in unstable emulsions and foams are completely different than the mechanisms at play in more stable dispersions. The lifetime of unstable dispersions is fully determined by the time of approach of drops/bubbles. When the drops/bubbles are sufficiently large, films may form between them. In some cases, the film surfaces do not remain flat and dimples are formed ; when the dimples remain centrosymmetric, coalescence was reported to occur only when the surfaces are very close (a few Å).

In other cases, an instability occurs, the dimple loses its axisymmetry, and the film ruptures when it is still thick (a few tens of nanometers). Asymmetrical thickness fluctuations have also been observed when the approach is sufficiently slow. When the drops are very small, films do not form between them and coalescence possibly occurs via pimple formation. The question of what drives the rupture and of the critical distance at which coalescence occurs are therefore still unclear.

The stable dispersions destabilize after a time that can be long, hours for foams, months for emulsions. Destabilization occurs once the drop/bubble size has reached a critical value, for instance during the ripening process. The probability of film rupture is indeed proportional to the contact area between drops/bubbles if one assumes that coalescence is caused by thermally activated holes in the surfactant layer. With this assumption, coalescence is predicted to be stochastic, as observed in practice. Recent bottle tests experiments showed that it is possible to measure the coalescence probability, and comparisons with models are in progress.

To summarize, thanks to continuing efforts and recent breakthroughs, our understanding of emulsion and foam stability is progressing. Many problems remain open however, and much work is still to be done along the lines outlined in this paper.

Figure 2 .

 2 Figure 2. Schematic representation of the variation of the disjoining pressure  d with film thickness h, including contributions of van der Waals attraction and electrostatic and short-range repulsion. The horizontal dashed lines correspond to different applied pressures P and the vertical ones show the pressures at which the film has reached its equilibrium thickness.

Figure 4 .

 4 Figure 4. Measured lifetimes of films formed by water drops pressed by gravity onto a flat oil-water interface (time required for 50% of films to break after their formation) versus surfactant concentration together with film images. Hexadecane films, stabilized by Span 80, in contact with an aqueous phase, 150 mM NaCl. The distance between the dark vertical bars is 50 m. Reproduced with permission from reference 41 . Copyright 2017, Elsevier.

Figure 5 .

 5 Figure 5. Left : Definition of interface positions measured in the bottle test; before mixing, the water and oil heights are equal. Right : position of the oil-emulsion interface (red circles), and of the water-emulsion interface (blue circles).The first regime corresponds to the creaming of the emulsion. After a quasi-plateau, coalescence starts in the second regime. Reproduced with permission from reference 48 . Copyright 2020, Royal Society of Chemistry.

  at Ecole Normale Supérieure Paris and became afterwards CNRS research scientist, being presently directeur de recherche emeritus. She began her research at the Physics laboratory of Ecole Normale Supérieure, she was post-doc at P.G.de Gennes' laboratory, Collège de France, she directed later the Centre de Recherche Paul Pascal, Bordeaux and she works presently at the Physique des Solides laboratory, University Paris-Saclay. Her research is focused on the dynamic behavior of interfaces between complex fluids (surfactants, polymers, liquid crystals, nanoparticles). She is specialist of interfacial rheology and its relation with foam, emulsion and microemulsion properties. Changes I inserted page numbers to ref 1, 6, 10, 27 and 43. I replaced ref 39 and 46 by publications in journals. Ref 23 and 39 are still in preparation, but they will be submitted shortly and I hope to be able to provide a DOI before the publication of the present article. I added a link to Figure 5 in the text, and I included a better version of this figure, together with a new one of Figure 3. It is no longer possible to download high resolution images from the Soft Matter website, so I took these figures from pdf versions of Google Scholar.

Acknowledgements

I am indebted to many colleagues for fruitful collaborations and discussions : Denis Weaire, Stefan Hutzler, Howard Stone, Annie Colin, Vance Bergeron, Arnaud Saint Jalmes, Cosima Stubenrauch, Wiebke Drenckan, Emmanuelle Rio, Anniina Salonen, Nicolai Denkov, Véronique Schmitt, Fernando Leal-Calderon, François Lequeux, Jean-Louis Salager, Clarence Miller, George Hirasaki among many others, as well as to many doctoral and post-doctoral students.