From bijels to Pickering emulsions: A lattice Boltzmann study

Particle stabilized emulsions are ubiquitous in the food and cosmetics industry, but our understanding of the influence of microscopic fluid-particle and particle-particle interactions on the macroscopic rheology is still limited. In this paper we present a simulation algorithm based on a multicomponent lattice Boltzmann model to describe the solvents combined with a molecular dynamics solver for the description of the solved particles. It is shown that the model allows a wide variation of fluid properties and arbitrary contact angles on the particle surfaces. We demonstrate its applicability by studying the transition from a “bicontinuous interfacially jammed emulsion gel” (bijel) to a “Pickering emulsion” in dependence on the contact angle, the particle concentration, and the ratio of the solvents.

Figure 1

${\sum }_c{\rho }^c$ after 2000 timesteps in the presence of a particle with $r_{\mathrm{par}}=10$ at an interface at the center of the shown area. The cut on the left (a) shows a particle being filled with virtual fluid, while on the right (b) the particle is empty as in the original algorithm. Without the virtual fluid, the halo of increased density can clearly be seen, while adding the virtual fluid successfully allows to correct for this inconsistency. The parameters of the simulation were ${\rho }_0=1$ and $\tau =1$ for both species, the system of size ${48}^3$ lattice sites was initially divided into two lamellae of width 24 lattice sites with density ${\rho }^{\mathrm{r}}={\rho }^{\mathrm{b}}=0.7$, respectively. All units are given in lattice units throughout this paper.

Figure 2

Definition of the contact angle $\theta$ for a particle of radius $r_{\mathrm{par}}$ at the interface between the two fluid components. $\Delta h$ is the distance from the particle center to the interface.

Figure 3

Contact angle versus time for a $48\times 48\times 256$ system with particle color 0.02, $r_{\mathrm{par}}=10$, $g_{\mathrm{br}}=0.08$, and fluid density ${\rho }_{\mathrm{init}}=0.6$. After a rapid change from about ${93}^{°}$ to about ${107}^{°}$ the contact angle stays at a constant value and only shows small oscillations with an amplitude of about $1^{°}$.

Figure 4

(a) Contact angle versus particle size. The system size is $48\times 48\times 256$, the particle color 0.02, $g_{\mathrm{br}}=0.08$, and ${\rho }_{\mathrm{init}}=0.6$. The average measured angle and errors given by the standard deviation over timesteps from $6\times {10}^5$ to $9\times {10}^5$ are shown. The contact angle for $r_{\mathrm{par}}=2$ is $(120.9\pm 6.0){}^{°}$, for particles larger than $r_{\mathrm{par}}=5$ the error stays below $1.5^{°}$ and the angles are in the range of ${106}^{°}$ to ${110}^{°}$. Particles with a noninteger radius show larger contact angles, which can be adhered to a discretization effect. (b) Contact angle versus particle color for ${\rho }_{\mathrm{init}}=0.7$. The data can be fitted with the equation $\theta =442\times \Delta \rho +90$ (dashed line).

Figure 5

(a) Contact angle versus $g_{\mathrm{br}}$. For $g_{\mathrm{br}}⩾0.1$ a contact angle of $93.0^{°}$ with an error smaller than $0.1^{°}$ is measured. The contact angle increases from this value at $g_{\mathrm{br}}=0.1$ to $98.0^{°}$ at $g_{\mathrm{br}}=0.08$ with an error below one degree. $g_{\mathrm{br}}=0.075$ results in a contact angle of $(113.6\pm 0.3){}^{°}$ and for smaller $g_{\mathrm{br}}$ the particle detaches from the interface. (b) Contact angle versus the initial fluid density ${\rho }_{\mathrm{init}}$. ${\rho }_{\mathrm{init}}=0.9$ results in a contact angle of $92.5^{°}$. Reducing ${\rho }_{\mathrm{init}}$ causes the contact angle to decrease to $108.4^{°}$ at ${\rho }_{\mathrm{init}}=0.55$.

Figure 6

Average domain size for a system with $g_{\mathrm{br}}=0.08$, ${\rho }_{\mathrm{init}}=0.7$, and a particle concentration of 20%. During the first $2.5\times {10}^4$ steps of the simulation the average domain size grows from below 5 to its final value of about 31 lattice units.

Figure 7

2D cut of the order parameter $\varphi$ at $z=0$ through the system studied in Fig. 6. The spots where $\varphi =0$ correspond to the regions occupied by particles. $\varphi$ shows pronounced differences between $t=5000$ and $t=10\hspace{0.5em}000$, while at timesteps $2.5\times {10}^5$ and $2.8\times {10}^5$ only minor rearrangements can be observed.

Figure 8

3D visualization of the system presented in Figs. 6 and 7. Shown are the particles [in green (light gray)] and the two fluids [in red (medium gray) and blue (dark gray), respectively]. The visualizations for $t=2.5\times {10}^5$ and $t=2.8\times {10}^5$ nicely depict the bicontinuouity of the fluids and the attachment of the particles to the interface.

Figure 9

(a) Average domain size versus $g_{\mathrm{br}}$ at $t=2.8\times {10}^5$. With increasing $g_{\mathrm{br}}$ $L$ decreases from 33.7 lattice units at $g_{\mathrm{br}}=0.07$ to 28 lattice units at $g_{\mathrm{br}}=0.125$. (b) Average domain size versus ${\rho }_{\mathrm{init}}$. ${\rho }_{\mathrm{init}}=0.6$ leads to a $L=32$ lattice units. A larger value of ${\rho }_{\mathrm{init}}=0.9$ reduces $L$ to a value slightly below 30 lattice units. Error bars are given by the maximum deviation of $L_x$, $L_y$, $L_z$ from the mean.

Figure 10

Average domain size versus particle concentration $\alpha$. Decreasing $\alpha$ from 0.35 to 0.15 leads to an increase of $L$ from 21.5 lattice units to 36 lattice units. If the concentration is further reduced to 0.05, finite size effects start to occur. Also shown is $L=\frac{3.86}{\alpha }+10.85$, the result of a fit to the concentration values between 0.15 and 0.35. Error bars are given by the maximum deviation of $L_x$, $L_y$, $L_z$ from the mean.

Figure 11

Average domain size over time for a system of size ${256}^3$, $g_{\mathrm{br}}=0.09$, ${\rho }_{\mathrm{init}}=0.66$, fluid ratio 1:3, particle color $-0.01$, and particle concentration $\alpha =0.15$. After a rapid growth of the domain size to over 25 lattice units during the first 25 000 timesteps the domain size increases only slowly to slightly over 27 lattice units at timestep $3.0\times {10}^5$.

Figure 12

3D visualization of the system described in Fig. 11. The particles [green (light gray)] and the fluids [red (medium gray) and blue (dark gray)] are shown. The particles are attached to the interface between the fluid components and the red fluid forms spherical droplets inside the continuous blue fluid. While the change from timesteps 5000 to 10 000 is significant the droplet growth is almost at rest between $t=2.5\times {10}^5$ and $t=3.0\times {10}^5$.

Figure 13

2D cut at $z=0$ through the system described in Fig. 11 at different times. Shown is the order parameter $\varphi$.

Figure 14

Average domain size after $1\times {10}^6$ timesteps versus particle concentration $\alpha$. System size ${256}^3$, $g_{\mathrm{br}}=0.08$, ${\rho }_{\mathrm{init}}=0.7$, fluid ratio 1:3, and particle color $-0.01$. Increasing $\alpha$ leads to a decreasing average domain size. Also shown is a fit with equation $\frac{2.12}{\alpha }+18.91$. Error bars are given by the maximum deviation of $L_x$, $L_y$, $L_z$ from the mean.

Figure 15

Average domain size after $1.5\times {10}^5$ timesteps versus contact angle $\theta$. System size ${256}^3$, $g_{\mathrm{br}}=0.08$, ${\rho }_{\mathrm{init}}=0.7$, fluid ratio 5:9, and particle concentration $\alpha =0.15$. Neutrally wetting particles lead to a small $L$ with a large deviation between the spatial directions while strongly colored particles lead to a larger average domain size with almost no deviations. Error bars are given by the maximum deviation of $L_x$, $L_y$, $L_z$ from the mean.

Figure 16

System state in dependence on the contact angle and concentration after $3\times {10}^4$ timesteps. The system size is ${128}^3$, ${\rho }_{\mathrm{init}}=0.7$, and the ratio of the two fluid species is kept fixed at 5:9. For contact angles larger than 90${}^{°}$ the system always relaxes toward a bijel, while for strongly negative coloring a Pickering emulsion is obtained. The line is a guide to the eye.

Figure 17

System state in dependence on the contact angle and the fluid ratio after $3\times {10}^4$ timesteps. The system size is ${128}^3$, ${\rho }_{\mathrm{init}}=0.7$, and the particle concentration is kept fixed at 0.2. For fluid ratios between 1:1 and 3:4 also for contact angles larger than ${90}^{°}$ a bijel is obtained. For larger concentration ratios it depends on the particle color or contact angle if a bijel or a Pickering emulsion is produced. The line is a guide to the eye.