Multi-body coalescence in Pickering emulsions

Wu, Tong; Wang, Haitao; Jing, Benxin; Liu, Fang; Burns, Peter C.; Na, Chongzheng

doi:10.1038/ncomms6929

Article
Published: 12 January 2015

Multi-body coalescence in Pickering emulsions

Tong Wu¹^na1,
Haitao Wang¹^na1,
Benxin Jing²,
Fang Liu³,
Peter C. Burns^1,4 &
…
Chongzheng Na¹

Nature Communications volume 6, Article number: 5929 (2015) Cite this article

6758 Accesses
28 Citations
1 Altmetric
Metrics details

Subjects

Abstract

Particle-stabilized Pickering emulsions have shown unusual behaviours such as the formation of non-spherical droplets and the sudden halt of coalescence between individual droplets. Here we report another unusual behaviour of Pickering emulsions—the simultaneous coalescence of multiple droplets in a single event. Using latex particles, silica particles and carbon nanotubes as model stabilizers, we show that multi-body coalescence can occur in both water-in-oil and oil-in-water emulsions. The number of droplets involved in the nth coalscence event equals four times the corresponding number of the tetrahedral sequence in close packing. Furthermore, coalescence is promoted by repulsive latex and silica particles but inhibited by attractive carbon nanotubes. The revelation of multi-body coalescence is expected to help better understand Pickering emulsions in natural systems and improve their designs in engineering applications.

You have full access to this article via your institution.

Download PDF

Rapid demulsification of pickering emulsions triggered by controllable magnetic field

Article Open access 06 October 2020

Multiple nanoemulsions

Article 28 January 2020

The vortex-driven dynamics of droplets within droplets

Article Open access 04 January 2021

Introduction

Pickering emulsions are made of particle-stabilized droplets suspended in an immiscible continuous liquid phase^1,2. They are important soft matter systems that form naturally in crude oils³ and food products⁴ and have been engineered for drug delivery⁵, water purification⁶ and material processing^7,8,9,10. Compared with ordinary emulsions, Pickering emulsions are distinctively stable because the removal of interfacial particles requires a large amount of energy¹¹. When individual Pickering droplets are forced to coalesce, the extraordinary stability brought about by interfacial particles can lead to the formation of non-spherical droplets^12,13,14 and the arrest of droplet coalescence^15,16. Little is known, however, about the coalescence of a collection of hundreds and thousands of Pickering droplets as in real emulsions. This is particularly important when particle stabilizers are used to produce near-monodispersed droplets¹⁷, during which the distribution of droplet size can be significantly broadened by coalescence under gravity, floatation and shear^18,19.

Here we report for the first time that the presence of stabilizers at the oil–water interface can lead to multi-body coalescence in an ensemble of Pickering droplets—a phenomenon that has not been reported for either Pickering or ordinary emulsions. More interestingly, the number of droplets involved in coalescence equals four times the corresponding number of the tetrahedral sequence, indicating the inclusion of all closely packed nearest neighbours in a single coalescene event. As a result, a magic size distribition is produced with distinctive maxima related to each other through the cubic root of four times the tetrahedral numbers. Futhermore, interactions between stabilizers are found to affect the probability of coalescence by varying interfacial tension. Using model stabilizers including latex particles, silica particles and carbon nanotubes (CNTs), we show that coalescence is promoted by interparticle repulsion but inhibited by interparticle attraction.

Results

Selection of emulsion systems

To investigate coalescence of Pickering droplets in an ensemble, we select three representative emulsion systems, including latex particle-stabilized water droplets in dodecane, silica particle-stabilized 1,2-dichlorobenzene (DCB) droplets in water and CNT-stabilized water droplets in dodecane. Comparisons between the first two systems will show that multi-body coalescence occurs in both water-in-oil and oil-in-water emulsions. Subsequent comparisons with the CNT system will reveal differences between stabilizers lacking and having attractive interactions. Our results are organized in four sections as follows, including emulsion preparation and droplet size analysis, evolution of droplet size through multi-body coalescence, polydispersity and size evolution and coalescence probability and interparticle force.

Emulsion preparation and droplet size analysis

The three stabilizers and typical Pickering emulsions made from them are shown in Fig. 1. Latex and silica particles (Fig. 1a,b) are spheres with diameters of 0.8 μm and 1 μm, respectively. CNTs decorated with surface tension-tuning magnetite (Fe₃O₄) nanoparticles (Fig. 1c) are several micrometers long and have a diameter of ca. 15 nm (ref. 6). Pickering emulsions stabilized by latex particles, silica particles and CNTs are prepared following a conventional protocol²⁰, involving two consecutive steps. First, water, oil and stabilizers are mixed and then shaken vigorously by hand for 10 min (Fig. 1d–f), forming Pickering emulsions containing stabilizer-wrapped droplets²¹. Then, emulsions are left standing undisturbed on top of a bench for 10 min, allowing droplets to precipitate (Fig. 1g–i), forming a closely packed ensemble (Fig. 1j–l) to induce coalescence. Pickering droplets prepared following this protocol have low uniformity indices between 0.2 and 0.4 (Supplementary Fig. 1)^22,23,24, suggesting that the emulsions have only experienced limited coalescence²¹.

**Figure 1: Pickering emulsions made of three types of stabilizers.**

After coalescence is complete, diameters of at least 500 droplets are measured using an optical microscope. Histograms, as shown in Fig. 2a–c, are constructed to facilitate detailed analyses of droplet size distribution. For all three types of emulsions, the diameter histogram can be readily deconvoluted into a series of Gaussian functions, indicating that each emulsion consists of several normally distributed populations of droplets. Of note, deconvolution is only possible when histograms are generated using at least 500 measurements. Similar histograms reported in the literature are usually made with significantly less measurements, often in the order of 50–100 (ref. 21). Under such conditions, the combination of multiple normal distributions degenerates to a single log-normal distribution²⁵.

**Figure 2: Size distributions of three types of Pickering droplets.**

The mean of each deconvoluted normal distribution, d_n (n=0, 1, 2, 3), represents the mean diameter of the corresponding droplet population. We find that d_n decreases with increasing stabilizer-to-droplet mass ratio α, as shown in Fig. 2d–f. The inverse dependence of d_n on α can be readily explained by matching the total surface area of droplets and the total cross-section area of interfacial stabilizers²⁵:

where ρ is the specific gravity of stabilizers with respect to the droplet phase, η is the porosity of interfacial packing and τ_n is packing thickness. Conformation of experimental data to equation (1) indicates that fulfiling the interfacial area requirement for accommodating all stabilizer particles is an important determinant of droplet size.

We further compare d_n (n>0) with d₀, revealing a linear relationship between them:

as shown in Fig. 2g–i. The scaling factor k_n is estimated from the slope of linear regression. For n=1, 2 and 3, k₁≈1.6, k₂≈2.5 and k₃≈3.4 for all three emulsion systems. The conservation of k_n's among different emulsion systems suggests the presence of a univeral mechanism that controls the evolution of droplet size.

Evolution of droplet size through multi-body coalescence

To elucidate the mechanism of size evolution for Pickering droplets, we first focus on the mean diameter of deconvoluted droplet population without considering dispersion of the population. As shown in Fig. 3a (see Supplementary Table 1 for data), k_n equals the cubic root of four times the corresponding tetrahedral number, T_n:

**Figure 3: Scaling of Pickering droplet diameters according to the tetrahedral number sequence.**

suggesting that a d_n droplet has the same volume as T_n d₀ droplets and thus is formed by their coalescence. For n=1, 2 and 3, T_n=4, 16 and 40; therefore, the coalescence of these Pickering droplets is multi-body in nature.

Multi-body coalescence requires droplets to be closely packed, which is facilitated by the density difference between water and oil in our experimental systems (cf. Fig. 1g–i)²⁶. As illustrated in Fig. 3b and Table 1, four nearest-neighbouring d₀ droplets form a tetrahedron in a face-centred close (FCC) packed ensemble. When all four droplets coalesce simultaneously, the new droplet has a diameter of , where T₁=4. The d₁ droplet has 12 nearest d₀ neighbours in FCC, yielding a d₂ droplet after coalescing with the d₁ droplet: , where T₂=4+12=16. Similarly, a d₃ droplet is formed by the coalescence of the d₂ droplet with its 24 nearest neighbours: , where T₃=16+24=40.

Table 1 Progression of coalescence in FCC packing*.

Full size table

We further examine multi-body coalescence by considering the material conservation of interfacial stabilizers before and after coalescence, which requires:

By combining this equation with equation (2), we obtain:

for constants ρ and η. Indeed, equation (5) holds for all three emulsion systems as illustrated in Fig. 3c (see Supplementary Table 2 for data).

Polydispersity and size evolution

In the analyses described above, we have assumed that each deconvoluted droplet population has a single diameter, d_n (n=0, 1, 2, 3), equal to the mean of the Gaussian fit of experimental data. For the 0th droplet population, this assumption can be validated by considering that d₀ droplets are formed under vigorous shaking—an independent and identical process with a finite variance²⁷. However, will the coalescence of normally distributed d₀ droplets produce normally distributed d_n (n>0) droplets?

Coalescence progresses through the conservation of volume:

According to equation (6), we can prove that the probability density function of d_n is the T_n-fold convolution power of the probability density function of (see Supplementary Note 1 for derivation), which cannot be evaluated analytically. To obtain the distribution of the nth droplet population, we resort to the Monte Carlo method, which computes one million d_n values from randomly selected d₀'s using equation (6).

The histograms of simulated d_n's (n>0) are shown in Fig. 4a, along with the normally distributed 0th population. The histograms can be well-approximated by normal distributions (similarity to normality >99.8%, as measured by Kolmogorov–Smirnov statistic)^28,29, confirming that the normal distribution is conserved through coalescence. The means of simulated d_n's are compared with those estimated from experimental data in Fig. 4b. The two data sets exhibit an excellent linear correlation with a near-unity slope of 1.01(±0.01) (R²=0.96), validating the use of Gaussian fits to estimate d_n's.

**Figure 4: Evolution of diameter distribution through coalescence examined by Monte Carlo simulation.**

Coalescence probability and interparticle force

Although Pickering droplets prepared with different stabilizers coalesce following the same tetrahedral sequence, the selection of stabilizer can, however, affect coalescence probability. This is revealed by examining the variation of relative abundance N_n/N_T (n=0, 1, 2, 3) of each droplet population with α, as shown in Fig. 5. Here N_n is the number of d_n droplets estimated by integrating the nth Gaussian fit and . As α increases, N_n/N_T (n>0) increases at the expense of N₀/N_T for latex and silica-stabilized droplets (Fig. 5a,b), indicating that the addition of stabilizers promotes coalescence. For CNT-stabilized droplets (Fig. 5c), the opposite is observed, revealing improved stability of d₀ droplets and suppressed coalescence with the addition of CNTs.

**Figure 5: Relative abundances of Pickering droplets from different normal populations.**

To understand why coalescence is promoted by latex and silica particles but suppressed by CNTs, we divide coalescence into two consecutive processes: packing and fusion, as illustrated in Fig. 6. Packing presses d₀ droplets together and transforms them from spheres to rounded polyhedrons³⁰. To pack droplets sufficiently close for coalescence, Laplace pressure p₀ must be overcome by the external pressure provided by the droplets’ weight, P:

**Figure 6: Two consecutive processes of coalescence.**

where γ is interfacial tension. Fusion between droplets then happens with the rupture of the separating liquid film, which requires the internal pressure of polyhedral droplets, p_p, to exceed the disjoining pressure of the film, Π (a property of the continuous phase)^31,32:

where C is a constant related to droplet packing fraction.

The interfacial tension includes contributions from both stabilizer-wrapped droplets, γ_d, and interactions between interfacial stabilizers, γ_s:

For stabilized droplets²,

where γ_ow is the oil–water interfacial tension and θ is the contact angle formed by the continuous phase, the stabilizer surface and the droplet phase. According to equation (10), γ_d is constant for a given emulsion system; therefore, γ varies with γ_s.

γ_s can arise from the electrostatic repulsion between interfacial stabilizers. Latex particles, silica particles and CNTs are all negatively charged, as confirmed by their negative zeta potentials in water (latex, −18(±7) mV; silica, −21(±7) mV; CNTs, −13(±1) mV). Charge-induced repulsion, γ_cp, pushes stabilizer particles away from one another, reducing interfacial tension that pulls stabilizers together (that is, γ_s=−γ_cp<0):

An indication of interparticle repulsion is the random close packing³³ patterns formed by latex and silica particles at the oil–water interface and revealed by confocal laser scanning microscopy, as shown in Fig. 7a–c and d–f, respectively. With low γ, equation (7) is readily fulfiled. The probability of coalescence is thus controlled by the difference between p_p and Π according to equation (8). As α increases, d₀ decreases according to equation (1). This leads to an increase of p_p, improving the chances of overcoming Π to coalesce and produce more d_n (n>0) droplets at greater α (cf. Fig. 5a,b).

**Figure 7: False-colour confocal laser scanning micrographs of interfacial stabilizers.**

Different from latex and silica particles, CNTs form an extended network at the oil–water interface, as revealed by the confocal micrographs shown in Fig. 7g–i. Formation of the network can be attributed to strong π–π attractions between individual nanotubes, which overtake electrostatic repulsions between them (that is, γ_s=γ_π–π−γ_cp>0)^6,34:

With high γ, equation (8) is readily fulfiled, leaving the control of coalescence probability to equation (7). As α increases, d₀ decreases and p₀ increases, resulting in a decrease of coalescence and minimal amounts of d_n (n>0) droplets with large α (cf. Fig. 5c).

Discussion

We have shown that closely packed Pickering droplets can coalesce through a multi-body mechanism. We hypothesize that the determining factor of multi-body coalescence is the presence of stabilizers at the oil–water interface, which slows down coalescence. In ordinary emulsions where droplets are stabilized by surfactant molecules or ions, coalescence happens rapidly between two droplets^35,36. Recent measurements have, however, shown that coalescence between two particle-stabilized droplets is orders of magnitude slower³⁷. The extended transition time provides an opportunity for all of the nearest neighbours to be involved in a single coalescence event once coalescence is initiated between two droplets.

We formulate the multi-body coalescence theory in the FCC configuration. If Pickering droplets are packed in the hexagonal close packing (HCP) configuration, the number of droplets involved in the first coalescence event is the same as in FCC but decreases gradually for the second and third events, as illustrated in Table 2. The k_n values for HCP are 1.6, 2.4 and 2.5 compared with 1.6, 2.5 and 3.4 for FCC. According to experimentally determined k_n's, the packing of Pickering droplets is better represented by FCC. Nonetheless, multi-body coalescence requires only short-range ordering because significant coalescence occurs in the first few coordination shells surrounding an interstitial void. In the longer range, the lack of organization, such as that in random close packing³³, should not affect the outcome of multi-body coalescence in Pickering emulsions.

Table 2 Progression of coalescence in HCP*.

Full size table

Methods

Reagents

Reagent-grade chemicals were purchased from Sigma-Aldrich and Fisher Scientific except where otherwise stated. Deionized (DI) water (18.2 MΩ cm⁻¹) used in solution making, washing and rinsing was generated using a Millipore system (Billerica, MA, USA) on site. We prepared Pickering emulsions by shaking and standing (see below)²⁰.

Latex particle-stabilized water droplets in dodecane

Latex particles were obtained by drying an aqueous solution in vacuum overnight. The particles were then dispersed in dodecane (99%, TCI America) at various concentrations. To make Pickering emulsions, 50 μl DI water was added to 1 ml dodecane. The mixture was shaken by hand vigorously for 10 min and then left standing on bench for 10 min.

Silica particle-stabilized DCB droplets in water

Silica particles were first coated with (3-aminopropyl)trimethoxysilane (APTMS, 97%) to modify their surface wettability³⁸. To do so, 0.1 ml particle solution (10 wt%) was dried in an oven at 120 °C overnight and was then mixed with 10 ml toluene (≥99.8%) and 100 μl APTMS. The mixture was shaken for 2 h. The particles were washed with toluene five times and with ethanol three times. The particles were then dried in an oven overnight to remove residual ethanol and immersed in water before use. To make a Pickering emulsion, 50 μl 1,2-DCB (99%, Alfa Aesar) was added to 1 ml DI water with different concentrations of silica particles. The mixture was shaken by hand vigorously and left standing undisturbed following the same procedure for making latex-stabilized emulsions.

CNT-stabilized water droplets in dodecane

Magnetite-decorated CNTs were prepared using multi-walled CNTs synthesized by chemical vapour deposition⁶. Catalysts were removed by washing with nitric acid. CNTs were then decorated with 10-nm magnetite nanoparticles using the polyol reduction method³⁹. To make a CNT-stabilized Pickering emulsion, CNTs were dispersed in 10 ml water by sonication for 10 min, followed by an addition of 0.5 ml dodecane. The mixture was shaken and left standing quiescently following the same protocol for making latex and silica-stabilized emulsions.

Optical microscopy

Diameters of particle-stabilized droplets were measured using images taken by an optical microscope (Motic BA300POL). To do so, emulsions were poured on either glass (for silica-stabilized droplets) or plastic (for latex and CNT-stabilized droplets) Petri dishes. The emulsions were then diluted with the corresponding continuous phases to minimize droplet overlapping in the imaging field. For each sample, ~30 images were taken randomly with a × 10 objective lens (resolution: 1.25 μm per pixel). Diameters were measured using software ImageJ⁴⁰. A few droplets stabilized by silica particles (<5%) were found at the arrested coalescence state with non-spherical shapes (Supplementary Fig. 2). They were excluded in subsequent diameter analyses.

Confocal laser scanning microscopy

Interfacial stabilizers were visualized using a confocal laser scanning microscope (Nikon A1R) equipped with a × 100 Plan Apo total internal reflection fluorescence objective lens. The oil phases were illuminated using oil-soluble Nile red. Water was illuminated using Alex Fluor 488. Concentrations of the fluorescent dyes were: latex particle-stabilized water droplets in dodecane, 0.001 mM Alex Fluor 488 in water and 0.1 mM Nile red in dodecane; silica particle-stabilized DCB droplets in water, 0.03 mM Nile red in DCB and 0.01 mM Alex Fluor 488 in water; CNT-stabilized water droplets in dodecane, 0.01 mM Alex Fluor 488 in water and 0.03 mM Nile red in dodecane. The oil-in-water emulsion stabilized by silica particles was imaged using a custom-made hydrophilic glass reservoir. To image water-in-oil emulsions stabilized by latex particles and CNTs, the reservoir was treated with a 1:100 octadecyltrichlorosilane toluene solution to create a hydrophobic coating before use.

Measurement of particle surface charge

Zeta potentials of latex particles, silica particles and CNTs were measured using a ZetaPlus analyzer (Brookhaven Instruments) in water at concentrations of 0.2 mg l⁻¹, 0.2 mg l⁻¹ and 0.12 mg l⁻¹, respectively. Water pH was adjusted according to the conditions in corresponding emulsions. Latex particles and CNTs were dispersed in water in equilibrium with atmospheric carbon dioxide at pH 5.6. Silica particles were dispersed in dilute sodium hydroxide solution at pH 7.0. Before measurements, stabilizer suspensions were sonicated for 30 min. For each stabilizer, five measurements were made.

Additional information

How to cite this article: Wu, T. et al. Multi-body coalescence in Pickering emulsions. Nat. Commun. 6:5929 doi: 10.1038/ncomms6929 (2015).

References

Pickering, S. U. Emulsions. J. Chem. Soc. 91, 2001–2021 (1907).
Article Google Scholar
Binks, B. P. Particles as surfactants-Similarities and differences. Curr. Opin. Colloid Interface Sci. 7, 21–41 (2002).
Article CAS Google Scholar
McLean, J. D. & Kilpatrick, P. K. Effects of asphaltene solvency on stability of water-in-crude-oil emulsions. J. Colloid Interface Sci. 189, 242–253 (1997).
Article ADS CAS Google Scholar
Dickinson, E. Food emulsions and foams: stabilization by particles. Curr. Opin. Colloid Interface Sci. 15, 40–49 (2010).
Article CAS Google Scholar
Dinsmore, A. D. et al. Colloidosomes: selectively permeable capsules composed of colloidal particles. Science 298, 1006–1009 (2002).
Article ADS CAS Google Scholar
Wang, H. et al. Removal of oil droplets from contaminated water using magnetic carbon nanotubes. Water Res. 47, 4198–4205 (2013).
Article CAS Google Scholar
Koos, E. & Willenbacher, N. Capillary forces in suspension rheology. Science 331, 897–900 (2011).
Article ADS CAS Google Scholar
Stratford, K., Adhikari, R., Pagonabarraga, I., Desplat, J. C. & Cates, M. E. Colloidal jamming at interfaces: a route to fluid-bicontinuous gels. Science 309, 2198–2201 (2005).
Article ADS CAS Google Scholar
Crossley, S., Faria, J., Shen, M. & Resasco, D. E. Solid nanoparticles that catalyze biofuel upgrade reactions at the water/oil interface. Science 327, 68–72 (2010).
Article ADS CAS Google Scholar
Velev, O. D., Lenhoff, A. M. & Kaler, E. W. A class of microstructured particles through colloidal crystallization. Science 287, 2240–2243 (2000).
Article ADS CAS Google Scholar
Aussillous, P. & Quere, D. Liquid marbles. Nature 411, 924–927 (2001).
Article ADS CAS Google Scholar
Subramaniam, A. B., Abkarian, M., Mahadevan, L. & Stone, H. A. Non-spherical bubbles. Nature 438, 930–930 (2005).
Article ADS Google Scholar
Cui, M., Emrick, T. & Russell, T. P. Stabilizing liquid drops in nonequilibrium shapes by the interfacial jamming of nanoparticles. Science 342, 460–463 (2013).
Article ADS CAS Google Scholar
Abkarian, M. et al. Dissolution arest and stability of particle-covered bubbles. Phys. Rev. Lett. 99, 188301 (2007).
Article ADS Google Scholar
Pawar, A. B., Caggioni, M., Ergun, R., Hartel, R. W. & Spicer, P. T. Arrested coalescence in Pickering emulsions. Soft Matter 7, 7710–7716 (2011).
Article ADS CAS Google Scholar
Whitby, C. P. & Krebsz, M. Coalescence in concentrated Pickering emulsions under shear. Soft Matter 10, 4848–4854 (2014).
Article ADS CAS Google Scholar
Nie, Z., Park, J. I., Li, W., Bon, S. A. F. & Kumacheva, E. An “inside-out” microfluidic approach to monodisperse emulsions stabilized by solid particles. J. Am. Chem. Soc. 130, 16508–16509 (2008).
Article CAS Google Scholar
Hasmy, A., Paredes, R., Sonneville-Aubrun, O., Cabane, B. & Botet, R. Dynamical transition in a model for dry foams. Phys. Rev. Lett. 82, 3368–3371 (1999).
Article ADS CAS Google Scholar
Whitby, C. P., Fischer, F. E., Fornasiero, D. & Ralston, J. Shear-induced coalescence of oil-in-water Pickering emulsions. J. Colloid Interface Sci. 361, 170–177 (2011).
Article ADS CAS Google Scholar
Wiley, R. M. Limited coalescence of oil droplets in coarse oil-in-water emulsions. J. Colloid Sci. 9, 427–437 (1954).
Article CAS Google Scholar
Arditty, S., Whitby, C. P., Binks, B. P., Schmitt, V. & Leal-Calderon, F. Some general features of limited coalescence in solid-stabilized emulsions. Eur. Phys. J. E Soft Matter 11, 273–281 (2003).
Article CAS Google Scholar
Binks, B. P. & Whitby, C. P. Silica particle-stabilized emulsions of silicone oil and water: aspects of emulsification. Langmuir 20, 1130–1137 (2004).
Article CAS Google Scholar
Romero, A., Cordobes, F., Puppo, M. C., Guerreroa, A. & Bengoechea, C. Rheology and droplet size distribution of emulsions stabilized by crayfish flour. Food Hydrocolloids 22, 1033–1043 (2008).
Article CAS Google Scholar
Schmitt, V., Cattelet, C. & Leal-Calderon, F. Coarsening of alkane-in-water emulsions stabilized by nonionic poly(oxyethylene) surfactants: The role of molecular permeation and coalescence. Langmuir 20, 46–52 (2004).
Article CAS Google Scholar
Wang, H. & Hobbie, E. K. Amphiphobic carbon nanotubes as macroemulsion surfactants. Langmuir 19, 3091–3093 (2003).
Article CAS Google Scholar
Whitby, C. P., Lotte, L. & Lang, C. Structure of concentrated oil-in-water Pickering emulsions. Soft Matter 8, 3784–3789 (2012).
Google Scholar
Irani, R. R. Particle Size: Measurement, Interpretation, and Application John Wiley & Sons (1963).
Kolmogorov, A. N. Sulla determinazione empirica di una legge di distribuzione. G. Ist. Ital. Attuari 4, 83–91 (1933).
MATH Google Scholar
Smirnov, N. Table for estimating the goodness of fit of empirical distributions. Ann. Math. Stat. 19, 279–281 (1948).
Article MathSciNet Google Scholar
Bibette, J., Morse, D. C., Witten, T. A. & Weitz, D. A. Stability criteria for emulsions. Phys. Rev. Lett. 69, 2439–2442 (1992).
Article ADS CAS Google Scholar
Princen, H. M. Highly concentrated emulsions I. cylindrical systems. J. Colloid Interface Sci. 71, 55–66 (1979).
Article ADS Google Scholar
Princen, H. M., Aronson, M. P. & Moser, J. C. Highly concentrated emulsions II. Real systems. The effect of film thickness and contact angle on the volume fraction in creamed emulsions. J. Colloid Interface Sci. 75, 246–270 (1980).
Article ADS CAS Google Scholar
Clusel, M., Corwin, E. I., Siemens, A. O. N. & Brujic, J. A. 'Granocentric' model for random packing of jammed emulsions. Nature 460, 611–615 (2009).
Article ADS CAS Google Scholar
Feng, T., Hoagland, D. A. & Russell, T. P. Assembly of acid-functionalized single-walled carbon nanotubes at oil/water interfaces. Langmuir 30, 1072–1079 (2014).
Article CAS Google Scholar
von Smoluchowski, M. Experiments on a mathematical theory of kinetic coagulation of coloid solutions. Z. Phys. Chem. Stoechiom. Verwandtschafts. 92, 129–168 (1917).
CAS Google Scholar
Aldous, D. J. Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists. Bernoulli 5, 3–48 (1999).
Article MathSciNet Google Scholar
Chen, G. et al. Coalescence of Pickering emulsion droplets induced by an electric field. Phys. Rev. Lett. 110, 064502 (2013).
Article ADS Google Scholar
Krysztafkiewicz, A., Binkowski, S. & Wysocka, W. Pigments on amorphous silica carriers. Powder Technol. 132, 190–195 (2003).
Article CAS Google Scholar
Wang, H., Cao, L., Yan, S., Huang, N. & Xiao, Z. An efficient method for decoration of the multiwalled carbon nanotubes with nearly monodispersed magnetite nanoparticles. Mater. Sci. Eng. B 164, 191–194 (2009).
Article CAS Google Scholar
Schneider, C. A., Rasband, W. S. & Eliceiri, K. W. NIH Image to ImageJ: 25 years of image analysis. Nat. Methods 9, 671–675 (2012).
Article CAS Google Scholar

Download references

Acknowledgements

C.N. thanks the Department of Energy Office of Nuclear Energy’s Nuclear Energy University Programmes, the National Science Foundation Environmental Engineering Programme and the Notre Dame Sustainable Energy Initiative for financial support. P.C.B.’s contribution was supported by the Energy Frontier Research Center Materials Science of Actinides. We thank Kun-Yi Lin for performing preliminary experiments on carbon nanotubes. We thank Antonio Simonetti, Diogo Bolster and Kapil Khandelwal for their inspiring comments and suggestions.

Author information

Tong Wu and Haitao Wang: These authors contributed equally to this work

Authors and Affiliations

Department of Civil and Environmental Engineering and Earth Sciences, University of Notre Dame, 156 Fitzpatrick Hall, Notre Dame, 46556, Indiana, USA
Tong Wu, Haitao Wang, Peter C. Burns & Chongzheng Na
Department of Chemical and Biomolecular Engineering, University of Notre Dame, 182 Fitzpatrick Hall, Notre Dame, 46556, Indiana, USA
Benxin Jing
Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, 153 Hurley Hall, Notre Dame, 46556, Indiana, USA
Fang Liu
Department of Chemistry and Biochemistry, University of Notre Dame, 251 Nieuwland Science Hall, Notre Dame, Indiana 46556, USA,
Peter C. Burns

Authors

Tong Wu
View author publications
You can also search for this author in PubMed Google Scholar
Haitao Wang
View author publications
You can also search for this author in PubMed Google Scholar
Benxin Jing
View author publications
You can also search for this author in PubMed Google Scholar
Fang Liu
View author publications
You can also search for this author in PubMed Google Scholar
Peter C. Burns
View author publications
You can also search for this author in PubMed Google Scholar
Chongzheng Na
View author publications
You can also search for this author in PubMed Google Scholar

Contributions

C.N., H.W. and T.W. designed the study and analyzed data. T.W. and H.W. performed the experiments. B.J. contributed to the design of confocal measurements. F.L. performed Monte Carlo simulation and contributed to the statistical analysis of experimental data. P.C.B. contributed to the interpretation of droplet packing. All authors contributed to the writing of the paper.

Corresponding author

Correspondence to Chongzheng Na.

Ethics declarations

Competing interests

The authors declare no competing financial interests.

Supplementary information

Supplementary Information

Supplementary Figures 1-2, Supplementary Tables 1-2, Supplementary Note 1 and Supplementary References (PDF 458 kb)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wu, T., Wang, H., Jing, B. et al. Multi-body coalescence in Pickering emulsions. Nat Commun 6, 5929 (2015). https://doi.org/10.1038/ncomms6929

Download citation

Received: 10 June 2014
Accepted: 23 November 2014
Published: 12 January 2015
DOI: https://doi.org/10.1038/ncomms6929

This article is cited by

General approach to surface-accessible plasmonic Pickering emulsions for SERS sensing and interfacial catalysis
- Yingrui Zhang
- Ziwei Ye
- Yikai Xu
Nature Communications (2023)
Enhanced coalescence of two oil droplets with clay particles
- Joung Sook Hong
Korea-Australia Rheology Journal (2019)
Designer liquid-liquid interfaces made from transient double emulsions
- Greet Dockx
- Steffen Geisel
- Jan Vermant
Nature Communications (2018)
Higher-order assembly of crystalline cylindrical micelles into membrane-extendable colloidosomes
- Hongjing Dou
- Mei Li
- Ian Manners
Nature Communications (2017)

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.