🔬 condensed matter

Phase-Field Models for Particle-Stabilised Emulsions

This paper introduces a computationally efficient phase-field model that resolves the coupled dynamics of liquid phase separation and nanoparticle adsorption to simulate large-scale particle-stabilised emulsions, successfully demonstrating how higher nanoparticle concentrations reduce domain sizes in bicontinuous interfacially jammed emulsion gels (bijels) formed via solvent-transfer induced phase separation.

Original authors: Elisabeth C. Eij, Joost de Graaf, Martin F. Haase, Jesse M. Steenhoff

Published 2026-02-19

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CC BY 4.0

Original authors: Elisabeth C. Eij, Joost de Graaf, Martin F. Haase, Jesse M. Steenhoff

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to keep a salad dressing (oil and vinegar) from separating. Usually, you shake it, and it mixes for a while, but eventually, the oil and vinegar split apart again. To stop this, you add an emulsifier, like mustard, which acts like a glue holding the tiny oil droplets together.

In the world of science, researchers use nanoparticles (tiny, tiny specks of solid material) instead of mustard to create super-stable mixtures. These are called particle-stabilised emulsions. They are so strong they can be used for everything from drug delivery to energy storage.

However, there is a big problem for scientists trying to study them on a computer: The "Pixel" Problem.

The Problem: Too Many Tiny Things

To simulate these mixtures on a computer, scientists usually have to draw every single nanoparticle individually. Imagine trying to simulate a beach by drawing every single grain of sand. If you want to see a large beach (or a large drop of emulsion) over a long period of time, your computer would need to track billions of grains of sand. It would take longer than the age of the universe to finish the calculation.

The particles are too small, and the process is too slow for current computer methods.

The Solution: The "Crowd" Analogy

This paper introduces a clever new way to look at the problem. Instead of drawing every single grain of sand, the researchers decided to treat the nanoparticles like a fog or a crowd.

Think of it this way:

Old Way: You try to track every single person in a massive concert crowd to see how they move. (Too hard!)
New Way: You treat the crowd as a single, flowing "density field." You don't care where John is standing; you just care that the crowd is getting thicker in that area.

By using a mathematical approach called Phase-Field Theory, the researchers turned the billions of individual particles into a smooth, continuous "fog" that moves and changes density. This allows them to simulate huge, complex systems on a computer in a reasonable amount of time.

The "Traffic Jam" Discovery

The researchers used this new "fog" model to study a specific type of emulsion called a Bijel (Bicontinuous Interfacially Jammed Emulsion Gel).

Here is how the simulation works in simple terms:

The Separation: Two liquids that hate each other (like oil and water) start to separate.
The Rush: As they separate, they create a boundary line (an interface). The nanoparticles (the "fog") rush to this line because they love to sit there.
The Jam: Once too many nanoparticles pile up on the boundary, they get stuck. It's like a traffic jam on a highway. The cars (particles) are so packed together they can't move anymore.
The Freeze: Because the particles are jammed, they act like a solid wall. They stop the oil and water from merging any further. The mixture is "frozen" in a perfect, stable state.

What They Found

Using this new "fog" model, the scientists discovered two main things:

More Particles = Smaller Droplets: If you have a lot of nanoparticles, they jam the traffic very quickly. This stops the separation early, resulting in tiny, uniform droplets. If you have fewer particles, they take longer to jam, allowing the droplets to grow larger before getting stuck.
The "Gradient" Effect: In a specific manufacturing process called STrIPS (where a solvent is removed to trigger the separation), the researchers saw a fascinating pattern. The nanoparticles jammed first near the top (where the solvent left first), creating tiny droplets. Deeper down, where the solvent left later, the droplets had more time to grow before jamming. This created a "gradient" of sizes, just like what is seen in real-life experiments.

Why This Matters

This paper is a breakthrough because it bridges the gap between the microscopic world (individual particles) and the macroscopic world (large-scale materials).

Before: Scientists could see the particles but couldn't simulate the big picture.
Now: They can simulate the big picture and still understand how the particles control the shape and stability of the material.

The Bottom Line:
The researchers invented a new "mathematical lens" that lets them zoom out and see the whole forest without losing track of the trees. This helps them design better, stronger, and more useful materials for medicine, energy, and industry without needing a supercomputer the size of a city.

1. Problem Statement

Particle-stabilised emulsions, such as Pickering emulsions and bicontinuous interfacially jammed emulsion gels (bijels), are critical in soft matter science. Their formation and stability rely on the interplay between liquid-liquid phase separation and the adsorption of nanoparticles to the interface.

Limitation of Current Methods: Existing simulation techniques (e.g., Molecular Dynamics, Dissipative Particle Dynamics, Lattice-Boltzmann) typically resolve individual nanoparticles. While accurate at micro- and mesoscales, these methods are computationally prohibitive for systems involving macroscopic diffusion and large numbers of particles (e.g., billions of nanoparticles).
Specific Challenge: The formation of bijels via Solvent-Transfer Induced Phase Separation (STrIPS) occurs over seconds and hundreds of micrometers. Resolving individual particles at these spatiotemporal scales is currently unattainable.
Gap in Phase-Field Theory: Previous attempts to apply phase-field models to these systems often still resolved individual particles or suffered from unphysical artifacts (such as negative interfacial tension and interface distortion) when coupling particle density with liquid phase separation.

2. Methodology

The authors developed a novel phase-field framework that treats nanoparticles as a continuous field rather than discrete objects, enabling large-scale simulations.

Free-Energy Functional: The model is based on a free-energy functional ( $F$ $F$ ) comprising three parts:
1. Bulk free energy of the immiscible liquids (Flory-Huggins).
2. Bulk free energy of the nanoparticles (ideal mixture approximation).
3. Interfacial energy coupling the liquid gradient ( $\nabla\phi$ ) and nanoparticle density ( $\psi$ ).
Derivation of Dynamic Equations:
- Initial attempts using standard multi-component Cahn-Hilliard equations revealed a critical flaw: high nanoparticle accumulation at the interface caused severe distortion of the liquid interface and unphysical negative interfacial tensions due to a feedback loop in the chemical potential.
- Proposed Solution: The authors introduced a partially decoupled set of dynamic equations. They modified the chemical potential of the liquid so that it is not directly influenced by the nanoparticle density term, while the nanoparticle chemical potential remains coupled to the liquid interface gradient. This prevents interface distortion while maintaining the physical mechanism of particle adsorption.
Jamming Mechanism: To model the stabilization of the emulsion, the authors introduced density-dependent mobilities. When the local nanoparticle density ( $\psi$ ) exceeds a critical threshold ( $\psi_c$ ), the mobility of both the liquid and particles drops sharply. This simulates the "jamming" of the interfacial layer, which pins the interface and arrests further phase separation.
Application to STrIPS: The model was extended to simulate STrIPS by adding a solvent field. The solvent diffuses out of the system, increasing the interaction parameter ( $\chi$ ) between liquids, triggering spinodal decomposition, and allowing nanoparticles to adsorb and jam the resulting interfaces.

3. Key Contributions

Novel Phase-Field Formalism: Established a computationally efficient framework for particle-stabilised emulsions that avoids resolving individual particles, allowing simulation of macroscopic scales (micrometers to hundreds of micrometers).
Resolution of Numerical Instabilities: Solved the issue of interface distortion and negative interfacial tension found in previous coupled Cahn-Hilliard approaches by decoupling the liquid chemical potential from the nanoparticle density.
Mechanism of Arrest: Demonstrated that phase separation arrest is driven by the kinetic jamming of nanoparticles at the interface, which suppresses Ostwald ripening.
Integration with STrIPS: Successfully merged the particle-stabilised emulsion model with solvent-transfer dynamics to simulate the complex formation of bijels.

4. Key Results

Formation of Emulsions: The model successfully reproduced the formation of both Pickering emulsions (off-critical composition, droplet morphology) and bijels (critical composition, bicontinuous network).
Stability and Coarsening:
- Simulations without nanoparticles showed continuous coarsening via Ostwald ripening (droplet size increased, polydispersity increased).
- With nanoparticles, coarsening was arrested once the interfacial density exceeded the jamming threshold. The system reached a stable state with constant droplet/domain size.
- Higher initial nanoparticle concentrations ( $\psi_0$ ) led to faster jamming, resulting in smaller, more monodisperse droplets.
STrIPS Bijel Morphology:
- The model reproduced the characteristic domain size gradient observed in experimental STrIPS bijels: domains are smallest near the solvent-removal interface and grow larger deeper into the substrate.
- This gradient arises because solvent depletion is faster near the boundary, causing rapid phase separation and immediate nanoparticle jamming (small domains). Deeper in the system, slower solvent removal allows domains to coarsen before jamming occurs.
- Inverse Relationship: Higher nanoparticle concentrations resulted in smaller average domain sizes across the entire bijel structure, consistent with experimental trends.

5. Significance and Limitations

Significance: This work provides a powerful tool for investigating the morphological evolution of complex particle-stabilised emulsions at scales previously inaccessible to particle-resolved simulations. It bridges the gap between microscopic particle interactions and macroscopic material properties, offering insights into the design of bijels for applications in catalysis, energy storage, and membrane separation.
Limitations & Future Work:
- Thermodynamic Rigor: The dynamic equations are "non-integrable" (not strictly derivable from a single free-energy functional), a common approximation in active matter but less thermodynamically rigorous than full particle-resolved models.
- Hydrodynamics: The current model neglects hydrodynamics (convection), which may play a role in later stages of coarsening and STrIPS processes.
- Coalescence: The model suppresses droplet coalescence entirely due to the lack of Brownian motion and hydrodynamic interactions, which are present in real Pickering emulsions.
- Future Directions: The authors suggest incorporating hydrodynamics, thermal fluctuations (to allow coalescence), and more detailed particle properties (shape, contact angle) to further refine the model.