Liquid Foam films stabilised by particles
The ability of particles to stabilise thin liquid films in emulsions was reviewed by Pickering , building on earlier work by Ramsden . Many of the same fundamental processes occur whether the system is a water-in-oil film (emulsion) or a water-in-air film (foam). Particles attached to thin foam films can both stabilise or destabilise the film and hence, to understand the conditions under which stability is promoted, consideration must also be given to those conditions under which stability is actively reduced. Research into particle stabilised foams has therefore progressed hand-in-hand with research into the use of particles as antifoaming agents.
In a two-phase foam, liquid is drawn from the film into the Plateau borders under the action of capillary pressure, which arises from liquid draining out of the foam. As more liquid drains from the foam, capillary pressure rises, causing the films to thin until they fail. The capillary pressure at which this occurs at is known as the critical capillary pressure, Pcrit. In the case of a particle stabilised froth (three phase foam), the particles attached to the foam films increase Pcrit. As a higher capillary pressure is required to cause film failure, the thin liquid films have a longer lifetime, producing a more stable froth. The value of Pcrit for a particle stabilised thin liquid film is determined by a complex relationship between the particle hydrophobicity, shape and packing density and pattern on the film.
The particle-film system
The interaction between the particles and the thin liquid film is complex and challenging to study both experimentally and theoretically. A useful method for investigating the fundamental behaviour of the system is to reduce it to two dimensions (2D) and examine a single circular particle bridging both sides of the thin liquid film, as shown in Figure 1.
The 2D particle-film system
Figure 1 showing the structure of a particle stabilised film, the geometric dimensions involved and the effect of contact angle on the position of the LV interface.
The contact angle (θ), particle radius (RP), separation distance (SPP) and radius of curvature of the liquid-vapour (LV) interface (RL) and curvature of the film are used to describe the 2D system, with θ defined as the angle that forms between the LV interface and the solid-liquid (SL) interface at the point where the solid, liquid and gas phases meet (Figure 1). This is called the three point contact (TPC) labelled in Figure 1. The Young-Laplace law (equation 1) relates the curvature of the LV interface (r1,2) and surface tension (γ) to the difference in pressure (ΔP) between the two phases. In three dimensions (3D), r1 and r2 are the orthogonal radii of curvature of the film, however in 2D r2 tends to infinity and thus equation 1 simplifies to equation 2 (with r=RL). In the case of a particle stabilised thin liquid film ΔP is the capillary pressure drawing liquid out of the film via the Plateau borders.∆P=γ1r1+1r2 / 1
∆P=γr / 2
Figure 2 The effect of contact angle and increased capillary pressure (P) on the film thickness (h)
In the top set of images in Figure 2 it can be seen that at a capillary pressure of 0 (P=0) the LV interfaces are flat and adopt a position on the particle surface that ensures θ is maintained at the TPC. As contact angle increases to 90°, with P=0, the liquid film thickness (2h) decreases until at 90°, h=0, and the film fails. This film failure at 90° is known as bridging dewetting, first proposed by Garrett , and is widely accepted as the mechanism through which particles with large contact angles (θ>90°) destroy films. In this mechanism, an anti-foaming particle bridges the film, but instead of preventing liquid from draining out of the film by holding the TPCs apart, its high contact angle causes them to be drawn together and the film fails once they meet a the particle surface. A discussion of particles acting as anti-foamers is beyond the scope of this chapter but has been reviewed recently by Garrett  (Upcoming COCIS review 2015, will get details).
If the contact angle remains below 90° the film will remain stable for a range of values for P>0. The bottom row of images in Figure 2 show that as P increases, so does the curvature (1/RL) of the LV interface until h=0 and film failure occurs.
Particles as surfactants
Surfactants and particles both stabilise foams, but, arguably, they do so through different mechanisms and on different scales. They do however also exhibit similarities, as discussed by Binks . It was proposed that just as a surfactant's properties can be described in terms of the hydrophile-lipophile balance (HLB), those of spherical particles can be described in terms of contact angle and size. The HLB is an important parameter used to characterise the relative efficiency of a surfactant’s hydrophobic and hydrophilic parts whereas the hydrophobicity of a particle affects its readiness to adsorb at an interface. Once adsorbed a particle is held strongly at the two-phase interface whereas surfactant molecules are adsorbing and desorbing dynamically from the interface over very short time scales.
Binks  also reported that as nano- or micro-particles are not dissolved in the liquid phase there are no solubility based phenomena such as the formation of micelles, as found with surfactants. It is, however, possible for the particles to form loose aggregates in the film and become trapped between the opposite interfaces. Here they hold the opposite sides apart and prolong film lifetime [8, 9].
Binks  used equation 3 to calculate the energy required to remove a particle from an air-water or oil-water interface, γαβ is the surface tension of the interface the particle is attached at and E is the energy.E=πRP2γαβ(1±cosθ)2 / 3
The plus or minus signs represent the energy required to remove the particle from the interface into one of the fluid phases. E is at a maximum when θ=90° and decreases rapidly either side of this. E also decreases with the square of RP and particles of a size comparable to a surfactant molecule (5nm) are very easily detached, suggesting that there is a lower limit to the size of particle that can stabilise a film effectively.
2D models of particles in thin liquid films
In the case of the 2D model described in Figure 1, as the capillary pressure increases and the liquid drains from the film, the LV interfaces distort around the particles to maintain the contact angle at the particle’s surface. As the LV interfaces thin between the particles, they maintain the radius of curvature defined by the Young-Laplace law. Eventually the capillary pressure will reach Pcrit, the opposite sides of the film will touch, and the film fails (Figure 2). Both Denkov et al.  and Ali et al.  investigated the 2D case, but used different approaches.
Denkov et al.  approached the problem from the perspective of Pickering emulsions; the theory is applicable to particle stabilised foam films as well. Here the spherical particles are assumed to occupy a circular cell whose area corresponds to the packing density of the particles on the film. The distortion of the meniscus surrounding the particle is assumed axi-symmetric, centred on the particle, so that the 2D case can be evaluated.
Ali et al.  also used a similar approach to investigate the 2D case albeit approaching the problem from a more geometric perspective. They concluded, much like Denkov et al. , that the problem of particle stabilisation can be treated elegantly by supposing the particles are evenly distributed throughout the film. In reality, small perturbations will cause unbalanced forces to act on each particle, causing them to clump together in the film and opening up areas of empty film surface that reduces film stability.
Spherical particles in thin liquid films
The analytical model derived by Ali et al. , returns a value of Pcrit from the contact angle and particle spacing (equation 4), where Spp is the distance from the centre of the particle to the centre of the film (Figure 1). These 2D results are plotted in Figure 3 along with results from 3D numerical simulations.Pcrit=2γcosθSpp2-Rp2 / 4
From equation 4, Pcrit tends to infinity for closely packed particles when Spp=Rp. The densest possible packing of spheres on a film is close packed hexagonal, when the spaces correspond to Spp=1.155. However, even using Spp=1.155 as the minimum value for the separation in equation 4, the 2D case is still unable to take into account the complex topology that the LV interface adopts in 3D. It is clear that this requires a 3D approach to fully understand the film stability.
Figure 3 Showing a comparison of analytical and numerical models of Pcrit for single layer of particles stabilising the liquid film. The analytical model shown for single layers is from , the two vertical lines represent the minimum packing area for hexagonal (solid) and square (dashed) packing.
The 2D analysis shows that circular particles can both stabilise or destroy films. The shortcoming of the 2D analysis has been touched upon; particles are rarely smooth, uniform spheres and do not distribute evenly in a film. In reality, the LV interface surrounding the particles has a complex topology that affects the film stability. With access to more powerful computing power and numerical modelling tools it has become possible to expand the 2D models into 3D.
3D models of particles in thin liquid films
Morris et al.  investigated the effect of particle packing in 3D using the Surface Evolver  programme. Here, regular hexagonal and square packing arrangements were simulated. Both models used idealised uniform spheres, akin to the 2D models, and were able to determine the topology of the film surrounding the particles.
Particle packing density on the film (APP) is calculated using equation 5 (where np is the number of particles and AC is the area of the periodic cell). Although both packing arrangements yield similar film stabilities for a given θ and APP, hexagonal packing can achieve a higher packing density and so the greatest film stability. Comparing 2D and 3D predictions of Pcrit (Figure 3), it can be seen that the 2D model over-predicts.App=Ac-npRp2πnp / 5
When a thin liquid film is heavily loaded with ideal spherical particles they can pack with square and hexagonal regularity [12, 13]. For partial loading, the capillary forces will force the particles to draw together into irregular agglomerates . This requires a statistical analysis of repeated model results from random particle distributions in the film.
Morris et al.  performed repeated (more than a thousand times) simulations of periodic cells containing up to 20 particles randomly placed in a thin liquid film. The simulations used the same packing densities but different packing arrangements, to allow statistical analysis of the film stability. They produced a relationship between packing density, particle contact angle and Pcrit (equation 6); K is a fitted constant (2.31) and App the area of empty film per particle (defined in equation 5).Pcrit=KγcosθApp / 6
Double layers of particles in thin liquid films
Thus far the analysis has considered only single layers of particles in the film. However, for particle stabilised foams that are heavily laden with particles, it is not uncommon to see double layers of particles in the film.
The stability and energy considerations here are slightly different [8, 13, 16, 17, 18]. Under Certain conditions a double layer of particles can stabilise a film, even when the particles have a contact angle greater than 90°, up to a theoretical limit of 129°. The case of double (or more) layers of particles stabilising a thin liquid film has similarities with work done on porous flow and the capillary bridging of a pore. For example, Hilden and Trumble  and Cox et al  both used the Surface Evolver to investigate the complex shape of the LV interface as it travels through a pore.
.A similar approach was used by Morris et al.  to investigate the effect of contact angle and particle packing on the stability of double layers in 3D, which built on previous studies [13, 18] with the different predictions for Pcrit compared in Figure 4. The equation from  used to predict Pcrit for a close packed double layer of particles is given in equation 7; where γ is the surface tension at the oil-water interface and αmax is a geometric parameter αmax =θ+arccos(√3 sinθ/2).Pcrit=2γ1-3sin2θ4Rp23-sinαmax / 7
Figure 4 A comparison of analytical and numerical models of Pcrit for single and double layers of particles stabilising the liquid film. The analytical results shown for a single layer is from , that shown for double layers from .
Morris et al. assume a static, uniformly packed, double layer of spherical particles  and identify three possible film failure modes; particle bridging, capillary pressure driven failure and film inversion. In all three cases as the capillary pressure increases, so too does the curvature of the LV interfaces. This draws the film surface towards the particles and interface on the opposite side of the film. The first case, particle bridging (Figure 5, top) occurs when the particles have a contact angle greater than 90°. Once the curvature of the LV interface brings it into contact with particles in the opposite layer it bridges them and the film fails immediately as the particles cause bridging-dewetting of the film. The second mode, capillary pressure driven failure (Figure 5, middle) is analogous to the standard film failure mode and occurs when θ<90°; the LV interface touches the particle in the opposite layer and attaches to them. As the pressure continues to rise the curvature of the LV interfaces eventually brings them into contact with each other and failure occurs. Finally, film inversion (Figure 5, bottom) is analogous to the bridging of a pore. In this case the particles are closely packed together and the LV interfaces cannot sustain the curvature required to bridge the opposite side of the film. Instead, once the maximum sustainable capillary pressure is reached, the LV interface inverts itself, bridging the film and causing immediate failure. Films stabilised by double layer systems are much more computationally intensive to model than single layers. As such both dynamic and non-spherical particle systems are yet to be investigated theoretically in great detail.