Phase-Field Models for Particle-Stabilised Emulsions

TL;DR

The paper tackles the challenge of modeling particle-stabilised emulsions at scales where particle-resolved simulations are impractical. It develops a phase-field framework with a free-energy formulation and coupled dynamics to describe liquid phase separation and nanoparticle adsorption, addressing early instabilities in standard Cahn-Hilliard approaches by decoupling the liquid and particle dynamics and implementing density-dependent mobilities that produce jammed interfacial layers. The approach yields stable Pickering emulsions and bijels, and applied to STrIPS bijels it reveals a domain-size gradient and an inverse relation between overall domain size and nanoparticle concentration, consistent with experimental observations. Limitations include the absence of hydrodynamics and droplet coalescence, but the framework offers a versatile platform for exploring morphology and can be extended to incorporate more physics and wetting effects, with open data provided for replication and extension.

Abstract

Particle-stabilised emulsions are a cornerstone of soft matter science due to their broad application and fundamental relevance. Computer simulations provide key insights into the formation and behaviour of these emulsions, yet current methods are limited by the spatiotemporal scales accessible for study. The principal issue is that particles are resolved individually. In this work, an alternative strategy is introduced based on phase-field theory, for which we establish the framework. By evolving continuous fields, large-scale dynamics can be simulated in a computationally efficient manner. Our approach is then applied to model the complex formation of a bicontinuous interfacially jammed emulsion gel (bijel) via solvent-transfer induced phase separation (STrIPS). By resolving the coupled dynamics of liquid phase separation and nanoparticle adsorption, the model allows for the characterisation of the influence of nanoparticles on the morphology. Higher concentrations of nanoparticles are found to reduce the average domain size of STrIPS bijels, in line with previous experimental evidence. The presented phase-field model thus represents a promising approach for the morphological investigation of complex particle-stabilised emulsions.

Figures (6)

• Figure 1: Schematic 2D representations of the different morphologies of particle-stabilised emulsions. The Pickering emulsion (left) is made up of dispersed droplets in a continuous medium, with nanoparticles occupying the surface of the droplets. In contrast, the bijel (right) is a continuous structure of interwoven liquid channels, separated by a percolating sheet of nanoparticles.
• Figure 2: (A) Equilibrium solutions of governing Eqs. (\ref{['ScaledCH1']}) and (\ref{['ScaledCH2']}) for a 1D liquid interface, using different values of the attachment parameter $\tilde{\alpha}$. In both panels, the dashed grey line indicates the centre of the interface. The top and bottom panels show the density profiles of the liquid $\phi$ and nanoparticles $\psi$, respectively. For the sake of illustrating the interface, space is discretised with $\Delta\tilde{x}=0.5$ rather than the usual $\Delta\tilde{x}=1$. (B) Evolution of the interfacial tension (solid lines) and interfacial excess (dashed lines) while forming the interfaces in (A). The interfacial tension is calculated via Eq. (\ref{['IFT']}), while the interfacial excess here is defined as the difference in liquid composition between the left bulk phase and the interface at $\tilde{x}=4.5$.
• Figure 3: The top and middle panel show the density profiles emerging from the dynamic Eqs. (\ref{['DynamicNoJam1']}) and (\ref{['DynamicNoJam2']}) for a 1D liquid interface with increasing values of the attachment parameter $\tilde{\alpha}$. The liquid ($\phi$) density profiles in the top panel completely overlap, showing no significant distortion. The density profiles for non-jamming and jamming nanoparticles ($\psi$) are shown in the middle and bottom panel, respectively. The latter profiles follow from Eq. (\ref{['DynamicJam2']}), where the nanoparticle-dependent mobility $\tilde{M}_\psi(\psi)$ is calculated via the general mobility function (\ref{['MobilityFunction']}) for $\psi_c=0.95$. The liquid profiles from the coupled Eq. (\ref{['DynamicJam1']}) are identical to those shown in the top panel and therefore omitted here.
• Figure 4: Phase-field simulations showing the formation of the two types of particle-stabilised emulsion: the bijel and the Pickering emulsion. From a homogeneous mixture of immiscible liquids and nanoparticles, with either a critical ($\phi=0.50$) or off-critical ($\phi=0.25$) initial composition of the liquids, phase separation is initiated. The resulting morphologies represent the bijel and the Pickering emulsion, respectively. The state of the liquid $\phi$, nanoparticles $\psi$ and general mobility $\tilde{M}_i /\tilde{M}_i^0$ are presented at different stages of the simulation, expressed in percentages of its total duration. This total duration is $\tilde{t}=100$ for the bijel and $\tilde{t}=500$ for the Pickering emulsion, corresponding to their different dynamics of phase separation. Here, the general mobility $\tilde{M}_i /\tilde{M}_i^0$ is calculated in accordance with Eq. (\ref{['MobilityFunction']}) for $\psi_c=0.60$, while the length of each simulated domain is $\tilde{L}=128$ and the initial concentration of nanoparticles is $\psi_0=0.40$ with $\tilde{\alpha}=60$.
• Figure 5: (A) Histograms showing the time-dependent distributions of the droplet diameter $\tilde{D}$ in 2D Pickering emulsions, both with and without nanoparticles. For the Pickering emulsions with nanoparticles, the initial particle density is set to $\psi_0=0.40$ with $\tilde{\alpha}=60$ and $\psi_c=0.60$. In the histograms, filled and empty bars represent the presence and absence of nanoparticles, respectively. While the distributions of Pickering emulsions with nanoparticles show no significant change over time, the absence of nanoparticles results in considerable structural coarsening due to Ostwald ripening. (B) Profiles of the droplet number $N_d$ and the average droplet diameter $\tilde{D}_\mathrm{av}$ during the formation of 2D Pickering emulsions, shown for different values of the initial nanoparticle density $\psi_0$. (C) 2D morphologies representing Pickering emulsions with different initial nanoparticle density $\psi_0$, taken at $\tilde{t}=1000$. Higher nanoparticle densities lead to phase separation being arrested at earlier stages, resulting in Pickering emulsions with smaller, more monodisperse droplets. Note that the simulations in (C) use a system length of $\tilde{L}=256$ for ease of visualisation, while those in (A) and (B) employ a larger value of $\tilde{L}=512$ to improve statistical robustness.