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High Internal-Phase-Ratio Emulsions (HIPEs)
The earliest HIPEs
An emulsion consists of an ensemble of droplets, filled with a liquid and dispersed in another non-miscible liquid. The former is known as the internal phase while the latter is called the continuous phase. The rheological properties of the emulsion are largely determined by the volume fraction of the internal phase, φ, in the system. At low values of φ, emulsions generally behave and flow like the prevailing continuous phase. As φ increases, internal-phase domains get in the way of one another, and the emulsion may acquire a resistance to flow and become increasingly solid-like. Remarkably, this flow behaviour was reported by Pickering as early as 1907, in oil-in-water emulsions stabilised with soap:
“An ordinary emulsion, although containing 70 to 80 per cent of mobile paraffin oil, is as viscid as thick cream, and its viscidity increases with the proportion of oil present… With very high percentages of oil, the emulsion becomes practically solid, resembling a blancmange. Emulsions containing as much as 99 per cent of… oil have been made, the remaining… being a 1 per cent solution of soap.”3
Although casually qualified as “ordinary” in his article, these emulsions that Pickering described were far from it. He had succeeded in formulating high internal-phase-ratio emulsions (HIPEs) without inverting the emulsion topology. Modern convention now defines HIPE as an emulsion in which φ exceeds 0.74, the densest close-packing factor in a monodisperse system of spheres (Ostwald, 1910)4. Keeping in mind that the first emulsifier was produced only two decades later5, Pickering’s success with HIPEs up to φ = 0.99 was all the more impressive. He further noted that “such strong emulsions, however, cannot be obtained directly; they must be made by taking a weaker emulsion, and gradually increasing the [oil] in it, churning it after each addition.” While he made no explicit allusion to it, we now know that this prescribed protocol of slowly adding internal phase under constant mixing is in fact the age-old method (or at least it was formalized in 1820 in a French recipe book “Le Cuisinier Royal”) for making mayonnaise6, an oil-in-water HIPE stabilised by proteins and lecithin in egg yolks. The 21-page paper then went on to explore the nature of oil and of emulsifiers on the formulation’s stability and properties. Experimentation at the time was limited to visual inspection, or at best observation under an optical microscope, and thus explanations were deductive and qualitative. Nonetheless, Pickering noted several more very interesting observations which we shall come back to later:
His emulsions were semi-solid and liquefied quickly when exposed to dry air but not to moist air.
The emulsions contained a transparent jelly, also void of observable structure, but very stable and instantly turned opaque upon contact with water, giving a milky emulsion.
On a side note, Pickering described in this same paper emulsions that would later be named after him: internal-phase droplets stabilised by solid particles at their interfaces.
Between the 1930s and 1950s, surfactant science had considerably progressed and more emulsifiers were being produced. For instance, in the food science industry, synthetic emulsifiers such as mono-and diglycerides began to find their foothold in the production of margarine7. Elsewhere, the role of surfactants in interfacial physics was driven by military applications during the Second World War and its immediate aftermath, notably and unfortunately in the field of chemical warfare agents and their effects on lung alveoli8. This leap forward meant that emulsifiers were more readily available, allowing for the formulation of more metastable emulsions.
The internal structure of a HIPE
How is space filled?
There are two routes that can lead to high internal-phase ratios. The first uses a broad distribution of droplet sizes to fill space, whereas the second takes advantage of the deformability of liquid droplets to achieve the same goal. It would appear that empirical literature published before the 60s, including the earlier-cited paper by Pickering, assumed the first approach: that droplets remained spherical and therefore necessarily had to resort to polydispersity to fill space. Adopting this premise is understandable due to technological limitations at the time; it might be that deforming spherical droplets was unfathomable simply because it had not been observed before. Indeed, electron microscopy images of polyhedral emulsion droplets (Figure 2) were only published in 1965 by Groves and Scarlett in Nature and they cited just two earlier – albeit somewhat obscure – works that observed the same: a German book titled “Emulsionen” by Manegold (1952) and an English PhD thesis by Phipps (1963).
Figure 2: electron microscopy image of polyhedral emulsion droplets (Groves and Scarlett, 1965).9
We note here that Groves and Scarlett were working with emulsions at φ = 0.225 and showed that local clustering of spherical droplets could lead to deformation that gave polyhedral emulsion droplets. They further wrote that Manegold “described the formation of honeycomb or ‘polyhedral drop foams’ in unstable close-packed emulsions with disperse phase ratios in excess of 0.74”9. This would be the first-known instance of a HIPE that exploited the second approach of droplet deformation to fill space.
HIPE droplet geometry
By 1966, thanks to advances in surfactant chemistry, and armed with the idea that deforming spherical droplets into polyhedral ones was possible, Lissant would revisit HIPEs and undertake the task of extensively describing droplet geometry in his formulations10. He focused primarily on the second approach of liquid droplet deformability, beginning with a strictly monodisperse population (in practice, we may qualify a population as “monodisperse” by convention if the largest droplet is no more than ten times bigger than the smallest one). From a purely geometrical standpoint, Lissant argued that droplets remained spherical and may rearrange themselves up to φ = 0.74, the close-packing limit. Beyond 0.74, the spherical droplets undergo deformation and adopt two types of packing possible: rhomboidal dodecahedral (RDH) packing between φ = 0.74 and 0.94 (Figure 3), and tetrakaidecahedral (TKDH) packing for φ > 0.94 (Figure 4) because it requires less angular distortion, giving greater geometrical stability. In both cases, the distorted droplets are effectively gridlocked: they are all in direct contact while the continuous phase exists as flat thin films in-between, meeting at Plateau borders11,12 (Figure 5).
Linking the macroscopic properties of a HIPE with its microscopic structure
Flow behaviour
In the same paper, Lissant described the preparation of oil-in-water HIPE using the mayonnaise method. The continuous phase was a 14% (v/v) aqueous solution of non-ionic surfactant and the internal phase was paraffinic mineral oil (kerosene). He found that the yield stress of his HIPEs increased as φ approaches ever closer to unity10. In fact, Lissant subsequently extended his geometrical musings to emulsions of lower internal-phase ratios, stating that monodisperse emulsions were expected to begin exhibiting non-Newtonian flow at φ = 0.52 due to droplets having to deform when moving past one another, and it was empirically found to be the case13. Qualitatively, we may understand the origin of a HIPE’s gel-like behaviour and resistance to flow as such: displacing a gridlocked droplet requires deforming it further through stretching, and this action is opposed by the tension in the surfactant thin films separating the droplets. Lissant’s comprehensive description on internal-phase droplet geometry paved the way for establishing and quantifying the relationship between macroscopic flow behaviours of HIPEs and their microscopic structures through rheological studies conducted by Princen et al. in the 1980s.
Not only did Princen theoretically formulate equation of states for film structures in HIPEs11,14–17, extensive experimental work spanning nearly the entire decade allowed him and his team to verify these equations and/or draw empirical relationships linking hydrodynamic and thermodynamic parameters with HIPE characteristics, such as φ, droplet size, interfacial tension and contact angle.12,17 Princen began by re-examining the mathematics behind packing structures and supported Lissant’s assertion of RDH packing for φ up to 0.9615. However, in addition to using the mayonnaise method17,19 for their experiments, they also innovatively made use of creaming and sedimentation – either by gravity or by centrifuging14,15 – to further concentrate their HIPEs and attain even higher φ by draining continuous phase from the Plateau borders where they are primarily found.20,21 Creaming was bound to occur in the emulsions due to density differences between the two liquid phases. If the interfaces were well stabilised by surfactant molecules, then the dispersed-phase droplets could be physically concentrated in the cream without undergoing coalescence events. Using this method, it would be possible to attain even φ = 0.9918,22–24. Physically, this translates into the thinning of Plateau borders which eventually lead to their collapse at φ > 0.964 and a measurable contact angle between the surfactant films. Taking into account these additional geometrical constraints, Princen et al. concluded that the regular pentagonal dodecahedron (RPD) (Figure 6) packing was more likely than TKDH at such high values of φ. Incidentally, this RPD geometry was also the model put forward by Manegold in Emulsionen (1952).15 We note here that packing with long-range order has been found to be incompatible with five-fold symmetry.
The group subsequently investigated the rheological properties of HIPEs as a function of HIPE droplet geometry.17,18,20,25 By thinning and draining the Plateau borders where most of the continuous phase was found, their contributions to mechanical properties of HIPEs could be discounted, and a clearer link between the macroscopic measurable properties could be drawn with those of the surfactant thin films separating internal-phase droplets. Key results from the group’s exploits are summarized below (non-exhaustively):
Yield stress and storage modulus G’ increase with internal volume fraction φ, interfacial tension and decrease with droplet size.17,20,25
Yield stress of the HIPE increases with surfactant film thickness.20
Coalescence2.1.3.2.HIPE stability
Several interesting observations were also made during their rheological experiments on the stability of HIPEs. Plastic containers and instruments were reported to incite oil droplet coalescence at walls.20,25 Mechanical strain exerted on the HIPE was also observed to break the emulsion: when subjected to shear or flow, the thin films became depleted of surfactant molecules. Diffusion of surfactant from within the films or from Plateau borders being too slow to repair these flaws26, interfacial tension rose and led to emulsion instability.20 However, this view of rising interfacial Dilution tension resulting in instability rising has since been questioned.
Pickering had previously observed that his gel-like oil-in-water HIPE readily dissolved in water to give a milky emulsion. Princen would observe the same. He introduced the notion of “compression” applied to HIPE: if the concentrated emulsion was allowed to be at equilibrium with bulk continuous phase, then it would be considered “uncompressed”. Therefore by his reckoning, the mayonnaise method would give “compressed” emulsions because there was no excess bulk continuous phase. He advised that a practical way to differentiate the two types would be to add some excess continuous phase to a HIPE – if it is absorbed by the HIPE then said HIPE is compressed.15 Clearly then, the ease of dilution that Pickering had observed with his jelly samples indicated that he had compressed HIPEs. Princen explained this affinity towards dilution in terms of osmotic pressure in the HIPE; allowing more continuous phase to enter the HIPE’s interdroplet regions would relieve this osmotic pressure14,20 and counter the Laplace pressure that distorted the droplets24, allowing them to relax and return to their original spherical shape. Princen and Kiss also observed that a HIPE was stable when diluted by the constituent continuous phase, showing no signs of coalescence nor ripening over a period of 4 weeks. Interfacial tension was also found to remain constant.17,20 This was to be expected since the diluting medium contained the same surfactant concentration as the continuous phase and so the surfactant’s chemical potential should remain constant.
The physical chemistry of HIPEs
Clearly, there was a link between the rheological and thermodynamic properties of HIPEs and their microscopic structure. Mason et al. would continue work in this direction with monodisperse HIPEs in the 1990s. Theoretical reasoning had, until then, been resolved largely in a two-dimensional plane. Thanks to numerical simulations, Mason et al. could now develop three-dimensional models that took into account droplet positions and repulsions, and formulate equations of state based on these models. Compared with empirical measurements, their models for osmotic pressure and rheological properties (storage modulus, yield stress) as a function of φ appeared to describe reality fairly well.24,27,28 They used polymer dialysis to impose a very high osmotic pressure on a HIPE by drawing out as much continuous phase as possible, a clever experimental technique invented by Parsegian et al. (1986)29, thereby further concentrating it more than even centrifugation creaming could (Figure 7).
Figure 7: osmotic pressure in a concentrated emulsion obtained with different compression techniques – gravitational creaming (diamonds); centrifugal creaming (solid circles); polymer dialysis (blank circles).24
Mason et al. would also go on to perform neutron scattering on monodisperse concentrated nanoemulsions to gain insight on the in-situ organisation of droplets as a function of φ.30,31 Although the emulsions were prepared only up until φ = 0.72 and therefore not strictly HIPE, evidence suggesting droplet deformation as φ approached the close-packing limit was found. This set of experiments was – to quote the authors – “strategic”, as conventional light scattering methods would require dilution of the HIPEs in order to avoid multiple scattering, whereas here the concentrated emulsions could be studied as is. This would inspire us to use X-ray and neutron scattering to study our HIPEs. We note here that another way to avoid multiple scattering would be to match the refractive index of the emulsion to the solvent32.
Practical uses of a HIPE
Controlling the droplet-size distribution
Fundamental research on HIPE geometry and properties appeared to wane towards the end of 1990s and attention shifted to the conditions surrounding the emulsification process.
Mason and Bibette would discover that a polydisperse concentrated emulsion crudely prepared by the mayonnaise method could be made highly monodisperse through shear rupturing33–36 (Figure 8). The continuous phase had to be viscoelastic enough to delay the onset of capillary instability, allowing the internal liquid stream to be stretched for a longer period before breaking up into droplets of more or less equal sizes. The viscoelastic property of the continuous phase was conferred by solubilizing large amounts of non-ionic surfactant, up to 40%. It was also determined that a high internal volume fraction significantly raised the effective viscosity of the HIPE35. Therefore, shear rupturing behaviour depended on the effective viscosity in the medium, rather than just the rheology of the continuous phase (Figure 9). These results overturned earlier rheological studies conducted by Schwartz and Princen who maintained that the dispersed phase’s viscosity had negligible effect on the overall behaviour21.
Figure 8: (a) the polydispersity of a concentrated emulsion prepared by the mayonnaise method is greatly refined into a monodisperse population (b) through shear rupturing. The periodicity (or space invariance) of the resulting HIPE is demonstrated by the small-angle light scattering spectra displaying six bright Bragg spots as shown in the inset.33
Figure 9: shear rupturing of a single droplet in a viscoelastic continuous phase. The viscoelastic character is obtained either by solubilizing large amounts of surfactant, or by considering the effective viscosity of other droplets, especially in a HIPE.37
As previously mentioned, modern HIPE research was focused on the monodisperse variant and obtaining emulsions of controlled sizes was especially enticing especially for industrial applications. This penchant was fulfilled by the invention of the Couette Injection Mixer38 (Figure 10) which enabled a high-throughput means of production.
Figure 10: the Couette injection mixer for producing highly monodisperse emulsions, based on the shear rupturing principle as described by Mason and Bibette.35
Emulsifying viscous oils
Repeated observations in the domain of colloidal sciences holds that efficient emulsification cannot take place if the viscosity of the internal phase exceeds 3.5 times that of the external phase due to poor mixing (Grace, 1982).39 To overcome this problem, we may exploit the increased effective viscosity by working at higher internal-phase ratios. In the 2000s, making HIPEs was shown to be a practical way of emulsifying viscous silicone oils36,40, with the most extreme example sitting pretty at 300 Pa.s37. All the HIPEs thus made were of the monodisperse variant. Tcholakova et al. (2011) also found that the oil’s viscosity did not have much influence on the droplet-size distribution in HIPEs beyond φ > 0.840.
Making a HIPE: choosing a surfactant
We have already noted that HIPEs could be fabricated either by the mayonnaise method3,10,17,19,33,35 or by creaming under gravity or centrifugation15,22–24,27,30,34 an already concentrated emulsion, or even a combination of both41. Regardless of protocols, it was frequently emphasized in literature that the choice of surfactant was crucial.
The surfactants commonly used were either non-ionic10 (alcohol ethoxylates19, sorbitan esters41, fatty acid glycerides41, polyoxyethylene alkyl/phenyl ether33,35–37,40,41, polyoxyethylene fatty alcohol41, polyvinyl alcohol40), solubilised at around 7-30% in the continuous phase, or anionic (sodium dodecyl sulfate15,17,22,24,27,30,34, fatty acid salt or soap3,15, ammonium alkyl ether sulfate19, sodium lauryl ether sulfate37) typically solubilised at 0.2-2% in the continuous phase. The nature of surfactant necessary depends on the HIPE topology desired – direct (oil-in-water) or inverse (water-in-oil). Lissant stressed the importance of beginning with the right internal phase10, and to ensure that this liquid stayed the internal phase even as more is gradually added, the Bancroft rule needs to be followed: the continuous phase of the HIPE will be the one favoured by the surfactant26. Kabalnov and Wennerström (1996) would later interpret and explain Bancroft’s rule through physical processes.
Non-ionic surfactants
Temperature is an additional consideration when working with non-ionic polyoxyethylene surfactants for there is a Phase Inversion Temperature (PIT), intrinsic to each surfactant, at which the surfactant attains its spontaneous curvature, , where is the radius of a micelle. PIT 42 At temperatures below the PIT, the curvature of increases as the hydrophilic chain length increases.= 1⁄ the surfactant, is positive and . Then, should a pore spontaneously nucleate between two oil droplets, the surfactant monolayer situated at the pore acquires a high curvature energy and ≫ linear tension (Kabalnov and Wennerström, 1996)26. The nucleated pore is consequently energetically unstable and closes up (Figure 11), ensuring that the emulsions obtained are oil-in-water.
Figure 11: depiction of surfactant monolayer frustration should a pore open between two oil droplets. The curvature of the surfactant at the pore is very high and therefore the pore is energetically unstable and closes back up. Figure adapted from Kabalnov and Wennerström (1996)26.
The amount of surfactant to use depends then on the nature of the surfactant. Non-ionic surfactants form a protective monolayer around dispersed-phase droplets, preventing sterically any coalescence events when two droplets meet.26,43,44 If we consider that a non-ionic surfactant molecule occupies about 50Å2 in area45, to fully saturate every dispersed-phase droplet surface at just φ = 0.74, we would require a minimum surfactant concentration of 1-10% in the continuous phase (for dispersed droplets 1µm and 100nm in size respectively, supposing they are monodisperse: the smaller the dispersed droplets the more surface area there is to cover).
Ionic surfactants
On the other hand, ionic surfactants do not have to saturate a droplet’s surface in order to protect it. Electrostatic repulsion22,34,46 from the adsorbed ionic surfactant molecules prevent droplets from coming into close contact with each other, and the repulsion acts over a distance known as the Debye length, typically around 10nm47. The Debye length is inversely proportional to the square-root of ionic strength of the medium. Given that the ionic strength depends on the concentration of all charged species present, the Debye length is shorter if the concentration of charges is higher. Hence, we want to put as little ionic surfactant as possible in the system, but not too little: in the event where droplets do collide with each other, either due to Brownian motion or mechanical agitation, surfactant molecules can be knocked off and we want to have just enough surfactant in the system so that these molecules can quickly diffuse to readsorb onto the droplet surface. Surfaces of freshly created droplets may also be protected by a Marangoni effect as described by Taisne, Walstra and Cabane (1996)48. The CMCs of common anionic surfactants are usually in the range of 0.1-1% (w/w)49,50, with that of SDS at 0.23%22.
Current applications
Safety fuel
Since HIPEs pack an exceedingly high amount of one phase into another immiscible phase, they have found their use in applications that require maximum active ingredient efficiency per unit volume of formulation. A notable example is a gel-like safety fuel, where jet engine fuel was the dispersed phase and formamide – a hydrophilic organic liquid – the continuous, stabilised with a mixture of non-ionic surfactants. This was first published in 1968 by Nixon and Beerbower.51 The HIPE was observed under microscope to have polyhedral deformed droplets.
Oil recovery
HIPEs have also been used as fracturing fluids in oil recovery, as described in the presently-active patents.52–54 Fracturing fluid are typically highly viscous, either oil-based or water-based thickened with cross-linking polymers. The fluid is pumped underground into rock formations, delivering large suspended solid particles into cracks that would then keep the fractures open. However, subterranean heat causes the fluids to lose their viscosity, thereby reducing their rock-fracturing efficiency. Polymer thickeners may also be left behind in the cracks. The first HIPEs proposed for this application were oil-in-water, and exploited a HIPE’s high viscosity as a result of the gridlocking of deformed oil droplets up to φ = 0.8.52 The fracturing fluid was then removed by breaking the emulsion; either by adding a demulsifier or by stabilising the HIPE fracturing fluid with cationic surfactant that would adsorb onto the rock wall underground. Non-ionic surfactants (1-6% in the continuous phase) were later proposed as an alternative: by choosing surfactants with PITs close to the ambient temperature in the rock formation, the HIPE would invert when thermally equilibrated with its surroundings, and turn into a fluid easy to remove.53 A later innovation would propose using n inverse HIPE in order to reduce costs associated with the large amount of oil necessary to formulate a direct HIPE. Using an inverse HIPE was also purportedly more eco-friendly as it did not leave behind polymer residues in the rock formation.54
Emulsion templating
More recent applications of HIPEs are in polymer synthesis, specifically in a process known as emulsion templating.55 If the polymerisable species is present only in the continuous phase, then after polymerisation and removal of the internal phase, only the interconnected network of continuous films and Plateau borders would be left (Figure 12). The result is a highly porous material known as a polyHIPE.51 It was also found that the amount of surfactant stabilising the interfaces played a role in the final structure of the polyHIPE. Similarly, high density, low-porosity materials such as latex can also be obtained if the polymerisable species is found instead in the dispersed phase (Figure 13).56 Finally, if both phases are polymerised, a composite material would be the final product.57 The advantage of this process is that nanosized porosities can be achieved, so long as the HIPE droplet sizes are well-controlled.
Table of contents :
1. Introduction
2. Literature review
2.1 High Internal-Phase-Ratio Emulsions (HIPEs)
2.1.1 The earliest HIPEs
2.1.2 The internal structure of a HIPE
2.1.2.1 How is space filled?
2.1.2.2 HIPE droplet geometry
2.1.3 Linking the macroscopic properties of a HIPE with its microscopic structure
2.1.3.1. Flow behaviour
2.1.3.2. HIPE stability
Coalescence
Dilution
2.1.4 The physical chemistry of HIPEs
2.1.5 Practical uses of a HIPE
2.1.5.1 Controlling the droplet-size distribution
2.1.5.2 Emulsifying viscous oils
2.1.6. Making a HIPE: choosing a surfactant
2.1.6.1 Non-ionic surfactants
2.1.6.2 Ionic surfactants
2.1.7 Current applications
2.1.7.1 Safety fuel
2.1.7.2 Oil recovery
2.1.7.3 Emulsion templating
2.1.8 Future explorations
Filling space with scale invariance
2.2 Apollonian packing of spheres
2.2.1 A timeless problem
2.2.2 Present-day research
2.2.3 Fractals
3. Materials and methods for HIPEs
3.1 Preparing HIPEs
3.1.1 Tools required
3.1.2 Protocol
3.1.3 Reproducibility of the protocol
3.1.3.1 Chemical properties of oil
Incompatibility with alkanes
Compatibility with silicone oil
3.1.3.2 Physical properties of oil
Viscosity
3.2 Characterizing HIPEs
3.2.1 Rheology
3.2.1.1 Selecting a shearing geometry
Shape
Surface roughness
3.2.1.2 Protocol
3.2.2 Measuring droplet-size distribution by Light Scattering
3.2.2.1 Protocol
3.2.2.2 Interpreting the results
Number density or volume density?
Revealing a power-law behaviour
3.2.3 Small-Angle X-ray Scattering (SAXS)
3.2.3.1 Protocol
3.2.3.2 Data treatment
Background subtraction
Calculating the experimental structure factor, ()
Spectra fitting
3.2.4 Numerical simulations
3.2.4.1 Osculatory Random Apollonian Packing
3.2.4.2 Calculating the Apollonian structure factor, ()
3.2.4.3 Smoluchowski coalescence algorithm
Coalescence in a monodisperse population
Coalescence in a polydisperse population
Protocol
4. Results and Discussion
4.1 Macroscopic behaviour of a HIPE vs. its composition
4.1.1 Increasing gel-like character with increasing surfactant concentration
4.1.2 Rheology
4.1.3 Different droplet geometries corresponding to each regime
4.1.4 Size distribution of each type of HIPE
4.1.4.1 How to judge polydispersity in power-law distributions
4.1.4.2 Coalescence and fragmentation in HIPEs
Coalescence provokes polydispersity
Fragmentation causes power-law size distributions in all HIPEs
A novel approach: simultaneous fragmentation and coalescence in liquid HIPEs
4.1.5 Link between a HIPE’s microstructure and its rheological behaviour
4.1.5.1 Rheological properties of solid and hybrid HIPEs
4.1.5.2 Measure of surfactant film thickness by SAXS
4.1.5.3 The Farris effect
4.1.5.4 Newtonian flow behaviour in liquid HIPE
4.2 Evolution of liquid HIPEs
4.2.1 Evolution towards an Apollonian exponent
4.2.2 Persistence of the Apollonian exponent
4.2.3 Proof of Apollonian droplet packing through SAXS
4.2.4 Confirming Light Scattering observations by SAXS
4.2.4.1 Evolving towards RAP after 1 week
4.2.4.2 Increasing surface area with time
4.2.5 Existence of swollen micelles observed by SAXS
4.2.5.1 Two populations of swollen micelles
4.2.5.2 Making swollen micelles by the PIT method
4.3 Discussions on Apollonian HIPEs
4.3.1 Creation of smaller droplets in a HIPE
4.3.1.1 How are smaller droplets created during coalescence?
4.3.1.2 Why are smaller droplets created during coalescence in a liquid HIPE?
Evicted surfactants cannot be evacuated by the continuous phase
Evicted surfactant molecules are confined locally
Free energy is reduced by creating small droplets rather than maintaining flat films
4.3.1.3 Summary on the coalescence-fragmentation mechanism
4.3.2 How is an Apollonian HIPE so metastable?
4.3.2.1 Measure of coalescence rate
4.3.2.2 The consequence of coalescence-fragmentation
4.3.2.3 Thermodynamic considerations
Δ in an Apollonian structure
in an Apollonian structure
Elastic energy
Surface energy
5. Suggestions for applications of Apollonian HIPEs
5.1 Solid material with space-filling structure and ultra-low porosity
5.2 Photon or electron trap
5.3 Controlled release of drugs
5.3.1 Drug release from a spherical carrier by diffusion
5.3.2 Drug release by the dissolution of a solid sphere
6. Conclusion
Résumé en français
