Domain and droplet sizes in emulsions stabilized by colloidal particles

Particle-stabilized emulsions are commonly used in various industrial applications. These emulsions can present in different forms, such as Pickering emulsions or bijels, which can be distinguished by their different topologies and rheology. We numerically investigate the effect of the volume fraction and the uniform wettability of the stabilizing spherical particles in mixtures of two fluids. For this, we use the well-established three-dimensional lattice Boltzmann method, extended to allow for the added colloidal particles with non-neutral wetting properties. We obtain data on the domain sizes in the emulsions by using both structure functions and the Hoshen-Kopelman (HK) algorithm, and we demonstrate that both methods have their own (dis)advantages. We confirm an inverse dependence between the concentration of particles and the average radius of the stabilized droplets. Furthermore, we demonstrate the effect of particles detaching from interfaces on the emulsion properties and domain-size measurements.

Figure 1

Examples of emulsion states stabilized by spherical particles: The left picture showcases a Pickering emulsion (hydrophilic particles; ${\theta }_p={77}^{\circ })$: discrete droplets (red) suspended in the medium (blue) and stabilized by particles (green). The center picture shows an intermediate state between a Pickering emulsion and a bijel (neutrally wetting particles ${\theta }_p={90}^{\circ })$: a large continuous red domain has formed, but some discrete droplets remain. Finally, the right picture depicts a bijel (hydrophobic particles; ${\theta }_p={103}^{\circ })$: two intertwined continuous domains.

Figure 2

Time evolution of the surface weighted average droplet radius $\overline{R_d}$ of a Pickering emulsion, following Arditty et al. [32]. We use the cluster sizes obtained through the Hoshen-Kopelman algorithm to calculate the radii $R_d$ of the droplets. The same qualitative behavior as in Arditty et al. (in their Figs. 4 and 5) is observed: the droplet growth is rapid at first, but slows down until $\overline{R_d}$ converges to an asymptotic value.

Figure 3

Dependence of the surface weighted average droplet radius $\overline{R_d}$ of a Pickering emulsion on the particle volume fraction $\Xi$, following Arditty et al. [32]. We observe a linear relation with an offset; the offset can be explained by the facts that the system is finite, and that the interfaces are diffuse.

Figure 4

Averaged lateral domain size $\Lambda$ as a function of particle volume fraction $\Xi$ for various particle contact angles ${\theta }_p$. These values are then fit using an inverse relation with an offset, following Arditty et al. [32] as in Jansen and Harting [29] resulting in $\Lambda \left(\Xi \right)=7.11+2.36/\Xi$ for ${\theta }_p={72}^{\circ }$ (solid line), $\Lambda \left(\Xi \right)=6.61+2.80/\Xi$ for ${\theta }_p={90}^{\circ }$ (dashed line), and $\Lambda \left(\Xi \right)=6.14+2.83/\Xi$ for ${\theta }_p={107}^{\circ }$ (dotted line). The domain sizes are largest in the case of neutrally wetting particles.

Figure 5

Average lateral domain size $\Lambda$ as a function of particle contact angle ${\theta }_p$ for various particle volume fractions $\Xi$. The domain size is largest for neutrally wetting particles. The interfacial curvature induced by the particles leads to smaller droplet sizes for Pickering emulsions. The reduction of $\Lambda$ as the contact angle deviates far from ${90}^{\circ }$ can be explained by the fact that particles have detached from the interface and moved into the bulk. They are then detected as domains of size $2r_p=10.0$, which depresses the average domain size toward that value.

Figure 6

Examples of degenerate emulsions: When the particles are strongly hydrophilic (here, ${\theta }_p={50}^{\circ })$ or hydrophobic (here, ${\theta }_p={130}^{\circ })$, they lose their ability to stabilize the system as their detachment energy decreases. In extreme cases, most particles will have moved into their preferred fluid, leaving a very small effective number of stabilizing particles at the interfaces. This strongly increases the average domain sizes, and the resulting state can hardly be called an emulsion at all.

Figure 7

Reduced average cluster size $I_{\mathrm{av}}$ after equilibration of $t={10}^5$ time steps as a function of the contact angle ${\theta }_p$ of the stabilizing particles. After subtracting the effect of the largest cluster, the bijel has zero average cluster size, while the Pickering emulsions have nonzero sizes, with a dependence on the particle volume fraction $\Xi$. For all data points, the system size is $V_{\mathrm{sim}}={256}^3$ lattice sites, the fluid-fluid ratio $\chi =0.56$ $({\rho }_{\mathrm{init}}^r=0.5$ and ${\rho }_{\mathrm{init}}^b=0.9)$, and the surface tension is $\sigma =0.014$. The error bars are calculated through the standard deviation of the droplet sizes.

Figure 8

We compare the values of the reduced average cluster size (shown as radius $R_d$, open symbols) to those of the averaged lateral domain size $\Lambda$ (filled symbols) for the same datasets. Very large differences exist: in the Pickering regime, the value of $\Lambda$ is depressed because the method detects the particle size and interparticle distances as characteristic length scales, while the cluster sizes are calculated based on the number of lattice sites of the fluid domains. In the bijel regime, the cluster size of the single domain in a bijel will not provide meaningful information on the size of local structures.

Figure 9

Following Arditty et al. [32], we present our data on a logarithmic scale for the droplet sizes. It is evident that after the early time evolution, when the droplets are small but numerous and the resulting distribution is smooth, the statistics of even five combined Pickering emulsions of a ${256}^3$ system are inadequate (note the change in range of the vertical axes). Acquiring the necessary statistics to smooth out the latter distributions is computationally prohibitive.