🔬 condensed matter

Kinetic theory of emulsions with matter supply

This contribution extends the Lifshitz-Slyozov-Wagner theory to model emulsions with continuous material supply, revealing distinct coagulation kinetics and droplet size distribution behaviors under sustained supersaturation compared to conditions of constant supply, which is particularly significant for non-conserved biological systems such as biomolecular condensates.

Original authors: Jacqueline Janssen, Frank Jülicher, Christoph A. Weber

Published 2026-04-28

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jacqueline Janssen, Frank Jülicher, Christoph A. Weber

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 a bowl of soup filled with tiny oil droplets floating in water. In a normal, passive situation (like a bowl of soup sitting on a table), these droplets play a game of "survival of the fittest." Large droplets grow larger, small ones shrink and eventually disappear. This is called Ostwald ripening. It happens because large droplets hoard available oil more efficiently, causing the small ones to starve. Eventually, only one giant droplet remains.

This article poses a new question: What happens if we keep pouring oil into the soup while this game is ongoing?

The authors, Jacqueline Janssen, Frank Jülicher, and Christoph Weber, have developed a new mathematical theory to predict how these droplets behave when constantly supplied with fresh material. They examined two specific ways this "feeding" can occur and found that the rules of the game change completely depending on how the oil reaches the droplets.

Here is a breakdown of their findings using simple analogies:

The two ways droplets grow

The article identifies two "bottlenecks" that control how fast a droplet can grow:

The Diffusion Bottleneck: Imagine the oil must swim through the water to reach the droplet. If the water is thick or the distance is large, the oil gets tired before it arrives. This is diffusion-limited.
The Door Bottleneck: Imagine the oil swims quickly, but the droplet has a very narrow, sticky door. The oil cannot get in fast enough, even if enough of it is waiting outside. This is interface-resistance-limited.

Scenario 1: Maintaining constant "supersaturation"

The Analogy: Imagine a magical faucet that perfectly adjusts its flow. As soon as a droplet swallows a drop of oil, the faucet immediately adds more oil to the soup to keep the total amount of oil in the water exactly the same. The system's "hunger" never changes.

The Result:

No Competition: Since the oil supply is infinite and perfectly balanced, the droplets stop fighting each other. They grow independently, like individual plants in a garden receiving exactly the same amount of water.
The Outcome:
- If the bottleneck is swimming (diffusion), the droplets will eventually all become the same size, and the diversity of sizes disappears (the distribution "narrows").
- If the bottleneck is the door (interface), all droplets grow at a constant, uniform speed, maintaining their original shape and diversity, simply shifting toward ever-larger sizes.

Scenario 2: A constant "drip" of supply

The Analogy: Imagine a faucet dripping oil at a constant, slow rhythm, regardless of how many droplets are in the soup. It is a fixed amount of new oil per minute.

The Result:

The Diffusion Bottleneck (Swimming is slow):
- If the drip is slow, the large droplets still eat the small ones (as in the passive soup).
- If the drip is fast, something strange happens: The small droplets receive enough food to grow, and the large ones do not grow much faster. The droplets stop competing and begin growing independently. The soup is full of droplets that are all roughly the same size (a "narrowed" distribution).
- Key Insight: The final outcome depends entirely on how fast you drip the oil. There is no single "universal" rule here.
The Door Bottleneck (The door is sticky):
- This is the article's biggest discovery. Although oil is added at a constant rate, the system finds a universal rhythm.
- Regardless of how fast you drip the oil (as long as it is constant), the droplets eventually settle into a specific pattern. They grow at a predictable speed, and the distribution of their sizes follows a specific, unchangeable form.
- The Surprise: The speed at which the average droplet grows turns out to be independent of how fast you feed the system. It is as if the "sticky door" sets a speed limit that the system follows, regardless of how much food you stuff into it.

Why this matters (according to the article)

The authors suggest this theory is crucial for understanding biomolecular condensates in living cells. These are tiny, liquid-like droplets in our cells (such as stress granules or the nucleolus) that organize our biology.

Unlike a bowl of soup, cells are "active" systems. They constantly produce new proteins and RNA (the "matter supply").
The article suggests that in these cells, the "door" (the interface) is often the bottleneck.
Therefore, the behavior of these cellular droplets might follow the "universal law" discovered by the authors: They grow in a predictable, steady manner determined by the cell's internal chemistry, not just by the amount of material floating around.

Summary

In short, this article updates the classic rules of droplet growth for a world where droplets are constantly fed.

If you keep the "hunger" constant, the droplets grow independently.
If you give a constant "drip" of food, the outcome depends on the bottleneck. If the bottleneck is the "door," the system finds a universal, predictable rhythm that ignores the specific amount of added food. This helps explain how droplets in living cells can grow and organize themselves efficiently.

1. Problem Statement

The article addresses the kinetics of emulsion coarsening (Ostwald ripening) in non-conserved systems, where a continuous external supply of matter occurs.

Context: Classical coarsening theories (Lifshitz-Slyozov-Wagner or LSW) assume a conserved total amount of droplet material, leading to a broadening of the droplet size distribution, where larger droplets grow at the expense of smaller ones until a single droplet remains.
The Gap: Many biological systems (e.g., biomolecular condensates like nucleoli or stress granules) and chemically driven synthetic emulsions are active. They are maintained far from thermodynamic equilibrium by the continuous synthesis of droplet components (matter supply).
The Challenge: Existing theories for active emulsions often focus on diffusion-limited transport. It remains unclear how different types of matter supply (constant supersaturation vs. constant flux) and different transport regimes (diffusion-limited vs. interfacial-resistance-limited) influence the universality of coarsening kinetics and the resulting droplet size distributions.

2. Methodology

The authors develop a theoretical framework that extends the classical LSW theory to include matter supply.

Model Setup:
- The system consists of $N$ spherical droplets in a volume $V_{sys}$ , coupled to a reservoir.
- Dynamics: The evolution is determined by the coarse-grained background concentration $\bar{c}(t)$ and the individual droplet radii $R_i(t)$ .
- Fluxes: The growth rate of a droplet is determined by the flux across its interface, accounting for both bulk diffusion ( $D_{out}$ ) and interfacial resistance ( $k$ ). The transition between regimes is controlled by the dimensionless parameter $\beta = kR/D_{out}$ .
- Supply Cases: Two different supply scenarios are analyzed:
  1. Constant Supersaturation: The background concentration $\bar{c}$ is held constant ( $\epsilon = \text{const}$ ), requiring the supply rate $J(t)$ to increase with droplet growth.
  2. Constant Matter Supply: The flux density $J$ is constant ( $J = \text{const}$ ).
Analytical Approach:
- Single and Two-Droplet: The authors first analyze the dynamics of single and two competing droplets to understand coupling mechanisms and fixed points in phase space.
- Continuum Limit: For large $N$ , they derive a continuity equation for the droplet size distribution function $N(R, t)$ .
- Scaling Analysis: They use separation of variables ansätze ( $N(R,t) = g(t)h(\rho)\rho$ ) to find asymptotic solutions for the distribution shape and scaling laws for the mean radius $\langle R \rangle$ .

3. Main Contributions

The article provides a unified theory distinguishing four different universality classes based on the transport regime and the type of matter supply:

A. Constant Supersaturation ( $\epsilon = \text{const}$ )

Decoupling: Droplets grow independently because the background concentration is fixed; there is no competition for material.
Diffusion-Limited Regime ( $\beta \gg 1$ ): The distribution narrows over time and approaches a delta function. The mean radius scales as $\langle R \rangle \propto t^{1/2}$ .
Interfacial-Resistance-Limited Regime ( $\beta \ll 1$ ): The distribution maintains a fixed shape but drifts with increasing velocity toward larger radii. The mean radius scales linearly, $\langle R \rangle \propto t$ .

B. Constant Matter Supply ( $J = \text{const}$ )

Diffusion-Limited Regime ( $\beta \gg 1$ ):
- The behavior is non-universal.
- There is a transition between broadening and narrowing distributions, depending on the supply strength relative to the droplet number density.
- If the supply is strong enough, the distribution narrows to a monodisperse state (delta peak), but the scaling factor depends on initial conditions and supply strength.
Interfacial-Resistance-Limited Regime ( $\beta \ll 1$ ):
- New Universal Law: The authors discover a universal coarsening kinetics independent of the supply strength $J$ .
- Distribution: The droplet size distribution broadens and follows a closed analytical expression (Equation 48), which differs from the passive LSW solution.
- Scaling: The mean radius follows a power law $\langle R \rangle \propto t^{1/2}$ , identical to the passive interfacial-resistance-limited case, but with a different prefactor.
- Mechanism: Initially, the droplets appear decoupled (constant standard deviation) but re-couple at late times via the background concentration, leading to the universal broadening distribution.

4. Main Results

Regime	Supply Type	Distribution Behavior	Scaling of Mean Radius $\langle R \rangle$	Universality
Diffusion-Limited	Passive ( $J=0$ )	Broadening	$t^{1/3}$	Universal (LSW)
Diffusion-Limited	Const. Supersaturation	Narrowing	$t^{1/2}$	Non-universal
Diffusion-Limited	Const. Supply	Narrowing or Broadening	$t^{1/3}$ (Prefactor depends on $J$ )	Non-universal
Interfacial-Resistance-Limited	Passive ( $J=0$ )	Broadening	$t^{1/2}$	Universal (Wagner)
Interfacial-Resistance-Limited	Const. Supersaturation	Drifting (Fixed Shape)	$t$	Non-universal
Interfacial-Resistance-Limited	Const. Supply	Broadening	$t^{1/2}$	Universal (New Law)

Convergence of Critical Radius: In the interfacial-resistance-limited regime with constant supply, the critical radius $R_c(t)$ converges to the same asymptotic behavior as the passive system ( $R_c \propto t^{1/2}$ ) despite the continuous matter supply.
Closed Solution: The article derives an exact analytical expression for the universal droplet size distribution in the interfacial-resistance-limited regime with constant supply:
$h(\rho)\rho = \rho \left(\frac{2}{2-\rho}\right)^3 \exp\left(-\frac{\rho}{2-\rho}\right)$
where $\rho = R/R_c$ .

5. Significance and Implications

Biological Relevance: The theory is highly relevant for biomolecular condensates in living cells. The article suggests that the linear growth of nucleoli observed in C. elegans ( $\langle R \rangle \propto t$ ) corresponds to the constant supersaturation regime, driven by continuous rRNA transcription.
Physics of Active Matter: It generalizes the understanding of active emulsions and shows that matter supply can either decouple droplets (leading to narrowing) or maintain coupling (leading to universal broadening), depending on the rate-limiting step.
Experimental Guidance: Distinguishing between diffusion-limited and interfacial-resistance-limited regimes provides a framework for interpreting experimental data in chemically driven emulsions and cellular condensates. The finding that interfacial-resistance-limited active systems exhibit universal kinetics despite non-conserved matter is a counterintuitive and significant result.

In summary, this work extends the fundamental LSW theory to active, non-conserved systems and shows that while diffusion-limited active emulsions exhibit complex, non-universal behavior, interfacial-resistance-limited active emulsions with constant supply converge to a new, robust universal law.