Non-Monotonic Marangoni Suppression of Hydrodynamic Coarsening in Bicontinuous Liquid-Liquid Phase Separation

This study demonstrates that soluble surfactants suppress hydrodynamic coarsening in bicontinuous liquid-liquid phase separation primarily through Marangoni stresses rather than reduced interfacial tension, with the inhibition peaking at an intermediate Péclet number due to an optimal balance between surfactant replenishment and gradient retention.

The Gist

Imagine you are making a salad dressing. You shake a bottle of oil and vinegar, and for a moment, they mix into a cloudy, swirling mess. But if you stop shaking, they eventually separate back into two distinct layers. In the world of physics, this process is called phase separation.

Now, imagine that while they are separating, the oil and vinegar are trying to merge into big blobs. This merging process is called "coarsening." Usually, the bigger blobs just eat the smaller ones, and the mixture separates faster and faster.

But what happens if you add a little bit of soap (a surfactant) to the mix? You might think the soap just makes the oil and vinegar "slippery" and helps them separate faster. However, this paper reveals a surprising twist: Soap can actually act like a brake, slowing down the separation process.

Here is the story of how and why this happens, explained through simple analogies.

1. The Invisible Traffic Cop: The Marangoni Effect

When oil and vinegar separate, they create a boundary line (an interface). If you add soap, the soap molecules love to sit on this boundary line.

Usually, soap spreads out evenly. But in a moving fluid, the soap gets pushed around. Imagine a crowd of people (soap molecules) trying to walk across a moving walkway (the fluid flow).

• Some areas get crowded with soap.
• Other areas get empty.

This creates a problem: Soap makes the boundary "slippery" (low tension), while bare spots are "sticky" (high tension).

Nature hates this imbalance. The sticky parts pull harder on the slippery parts. This creates a force called the Marangoni stress. Think of it like a traffic cop standing on the boundary line. When the oil and vinegar try to merge (coalesce), this traffic cop pulls them apart, stopping them from joining forces.

2. The Main Discovery: It's Not About the Soap Amount, It's About the Soap Movement

The researchers asked: "Does adding more soap always slow down the separation?"

You might guess:

• No soap? Fast separation.
• A lot of soap? Very slow separation.

But the answer is no. The effect is non-monotonic. This means the "braking" power doesn't just get stronger as you add more soap. Instead, there is a "Goldilocks Zone."

• Too little movement (Low Speed): The soap spreads out too evenly. The traffic cop is everywhere, but because the soap is spread thin and smooth, there's no "pulling" force. The brake doesn't work well.
• Too much movement (High Speed): The fluid moves so fast that it sweeps the soap away faster than new soap can arrive from the bulk liquid. The boundary line becomes "naked" in some spots. The traffic cop is there, but he's running out of fuel. The brake fails.
• Just right (Medium Speed): This is the sweet spot (specifically, a value called Pe = 10). The fluid moves fast enough to create "traffic jams" of soap (creating strong pulling forces), but slow enough that new soap can still arrive to keep the boundary covered. This is when the separation is slowed down the most.

3. The Analogy: The "Replenishment vs. Gradient" Battle

To understand why the "Goldilocks Zone" exists, imagine a leaky bucket being filled by a hose while someone is trying to pour water out of it.

• The Hose (Diffusion): This brings fresh soap from the bulk liquid to the boundary.
• The Pouring (Convection/Flow): This moves the soap around, creating clumps and empty spots.

The Winning Strategy (The Goldilocks Zone):
You need the hose to be strong enough to keep the bucket full (so the soap doesn't run out), but the pouring needs to be strong enough to keep the water uneven (so the "traffic cop" has something to pull on).

• If the hose is too strong (Low Speed): The water stays perfectly flat. No unevenness = no pulling force.
• If the pouring is too strong (High Speed): The bucket empties faster than the hose can fill it. No soap left = no pulling force.
• The Balance: The hose keeps the bucket full, but the pouring keeps the water sloshing. This creates the perfect storm of soap availability and soap unevenness, creating the strongest brake on the separation process.

4. Why Does This Matter?

This isn't just about salad dressing. This process happens in:

• Making plastics: To create materials with specific textures.
• Biological cells: Inside our cells, liquids separate to form organelles (like tiny factories).
• Oil recovery: Extracting oil from rock.

The paper tells us that if we want to control how these materials form, we can't just add "more soap." We have to tune the speed of the flow relative to how fast the soap moves. By finding that "Goldilocks Zone," engineers can design materials that stay mixed longer or separate in very specific, useful shapes.

Summary

• The Problem: Oil and water naturally want to merge into big blobs.
• The Solution: Adding soap creates a "traffic cop" (Marangoni stress) that stops them from merging.
• The Twist: The traffic cop only works best at a medium speed.
  • Too slow? The soap is too smooth to pull.
  • Too fast? The soap runs out.
  • Just right? The soap is both plentiful and uneven, creating the strongest brake.

This paper maps out exactly how to find that "just right" speed to control how complex fluids behave.

Technical Summary

1. Problem Statement

Liquid-liquid phase separation (LLPS) in multicomponent fluids often leads to the formation of bicontinuous domains that undergo hydrodynamic coarsening, where domain sizes grow over time. While the addition of soluble surfactants is known to retard this process, the specific mechanistic role of Marangoni stresses (tangential stresses caused by interfacial tension gradients) versus the simple reduction of mean interfacial tension remains poorly understood.

A critical unresolved question is how the transport of soluble surfactants—governed by the balance between diffusion and convection (characterized by the surfactant Péclet number, $Pe_\psi$)—regulates these Marangoni stresses. Specifically, it is unclear why the suppression of coarsening might depend non-monotonically on $Pe_\psi$, rather than simply increasing or decreasing with surfactant mobility.

2. Methodology

The authors employed a validated two-order-parameter phase-field model coupled with the incompressible Navier-Stokes equations to simulate the system.

• Governing Equations:
  • Phase Field ($\phi$) and Surfactant ($\psi$): Described by convective Cahn-Hilliard equations. The chemical potentials are derived from a free-energy functional that includes Ginzburg-Landau terms, Flory-Huggins entropy, and Langmuir-type adsorption/interaction terms.
  • Hydrodynamics: The Navier-Stokes equations include body forces representing the Capillary force (normal to the interface) and the Marangoni force (tangential to the interface, driven by $\nabla \ln(1-\psi)$).
• Numerical Scheme:
  • Finite-volume method on a uniform Marker-and-Cell (MAC) mesh.
  • Advection terms discretized using a fifth-order WENO scheme; time advancement uses explicit Adams-Bashforth (advection) and Euler/Crank-Nicolson (diffusion/viscous) schemes.
  • The projection method solves the coupled momentum and continuity equations.
• Simulation Modes:
  1. Model B: Purely diffusive (no hydrodynamics).
  2. Model H: Hydrodynamic coarsening (no surfactants).
  3. Model HS: Hydrodynamic coarsening with soluble surfactants and full coupling (including Marangoni forces).
• Validation: The model was validated against classical scaling laws ($R(t) \sim t^{1/3}$ for Model B; $t, t^{2/3}, t^{1/2}$ regimes for Model H) and benchmarked against analytical solutions for surfactant-laden droplet deformation in shear flow.

3. Key Contributions

1. Mechanistic Distinction: The study definitively separates the effects of mean interfacial tension reduction from Marangoni stresses. It demonstrates that the suppression of coarsening is driven primarily by Marangoni stresses, not merely by a lower average surface tension.
2. Non-Monotonic Dependence: The paper identifies and explains a non-monotonic relationship between the surfactant Péclet number ($Pe_\psi$) and the suppression of coarsening. The strongest inhibition occurs at an intermediate value ($Pe_\psi = 10$), rather than at low ($Pe_\psi = 1$) or high ($Pe_\psi = 100$) values.
3. Transport-Controlled Mechanism: The authors propose a "transport-controlled" mechanism where the efficacy of Marangoni suppression depends on a competition between surfactant replenishment (diffusion from bulk) and gradient retention (advection preserving heterogeneity).

4. Key Results

A. Marangoni Stresses vs. Mean Tension Reduction

• Simulations comparing clean interfaces, surfactant-laden interfaces with Marangoni forces, and surfactant-laden interfaces without Marangoni forces (artificially deactivated) revealed that:
  • Removing Marangoni forces (while keeping surfactants present) resulted in coarsening dynamics nearly identical to the clean interface case.
  • Including Marangoni forces significantly reduced the growth rate of the characteristic domain size $R(t)$ and altered the phase morphology.
• Physical Mechanism: Marangoni stresses hinder the drainage of thin liquid films between approaching interfaces. As interfaces approach, surfactants are convected to the film edges, creating concentration gradients. These gradients induce Marangoni stresses that oppose the flow, delaying coalescence and reorganizing local vortical flows.

B. Non-Monotonic Dependence on $Pe_\psi$

The study analyzed three regimes of surfactant transport ($Pe_\psi = 1, 10, 100$):

• Low $Pe_\psi$ (Diffusion Dominated, $Pe_\psi = 1$): Diffusion is efficient at replenishing surfactants at the interface, but it also rapidly smooths out interfacial concentration gradients. The lack of persistent gradients results in weak Marangoni stresses and minimal suppression.
• High $Pe_\psi$ (Advection Dominated, $Pe_\psi = 100$): Advection preserves interfacial heterogeneity (gradients), but diffusion is too slow to replenish surfactants from the bulk to the interface. The interface becomes depleted of surfactant, leading to weak Marangoni stresses despite the presence of gradients.
• Intermediate $Pe_\psi$ (Optimal Balance, $Pe_\psi = 10$): This regime achieves the "sweet spot." Diffusion is fast enough to maintain a high mean interfacial surfactant loading, while advection is strong enough to sustain significant concentration gradients. This coexistence yields the strongest Marangoni stresses and the maximum suppression of hydrodynamic coarsening.

C. Morphological Evolution

• Marangoni forces do not just slow down coarsening; they alter the pathway. They reorganize local vortical flows, causing the fragmentation of coherent interfacial protrusions into smaller segments, thereby changing the statistical evolution of the bicontinuous structure.

5. Significance and Implications

• Theoretical Insight: The work provides a fundamental understanding of how soluble surfactants dynamically regulate phase separation, moving beyond static equilibrium views to a dynamic transport-controlled perspective.
• Material Design: The findings offer a strategy for controlling microstructure evolution in complex fluid systems (e.g., emulsions, polymer blends, porous polymers). By tuning the Péclet number (via flow rates or surfactant diffusivity), one can optimize the suppression of coarsening to stabilize specific bicontinuous morphologies.
• Future Directions: The framework sets the stage for investigating more complex scenarios, including interfacial rheology, asymmetric material properties, and multicomponent mixtures, which are relevant to biological organization and advanced material fabrication.

In summary, the paper establishes that soluble surfactants regulate bicontinuous hydrodynamic coarsening not by passively lowering surface tension, but by actively generating Marangoni stresses through a transport-limited competition between replenishment and gradient retention.