🔬 condensed matter

Spontaneous currents determine capillary rise in active matter

This paper derives an active form of Jurin's law demonstrating that spontaneous particle currents, rather than surface tension alone, drive capillary rise in active fluids, leading to shape-dependent phenomena that defy equilibrium predictions.

Original authors: Xinyi Dong, Yongfeng Zhao

Published 2026-08-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Xinyi Dong, Yongfeng Zhao

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

In the quiet physics of everyday liquids, a simple rule has long governed how water climbs a narrow tube. When a glass straw is dipped into a glass of water, the liquid rises slightly above the surrounding level, defying gravity for a short distance. This phenomenon, known as capillary action, is a delicate balance between the pull of gravity and the surface tension of the liquid, which acts like a stretched elastic skin trying to minimize its area. For centuries, scientists have understood that the height this liquid reaches depends only on how wide the tube is, how heavy the liquid is, and how strongly the liquid clings to the tube's walls. This relationship, a staple of fluid mechanics, suggests that the behavior of the liquid is a local event, determined entirely by the immediate conditions inside the tube.

However, the world of "active matter" turns these familiar rules upside down. Active matter consists of materials made of tiny, self-propelled units—like bacteria or synthetic micro-robots—that constantly consume energy to move on their own. Unlike a passive drop of water that sits still unless pushed, these active particles are always in motion, jostling and swimming even when the system appears calm. When researchers began studying how these active fluids interact with surfaces, they encountered a puzzling contradiction. Theory suggested that the surface tension of such active fluids should be negative, a state that should theoretically cause the liquid to be pushed down rather than pulled up. Yet, experiments and computer models showed that these active fluids still climbed up tubes, often reaching heights that defied the predictions of the old laws. The question remained: what invisible force was overcoming the negative surface tension to drive the liquid upward?

A team of researchers at Soochow University in China has now solved this mystery by looking at the hidden currents flowing within the active fluid. By simulating the behavior of thousands of self-propelled particles moving under gravity, they discovered that the rise of the liquid is not driven by surface tension alone, but by a persistent, self-organized flow of particles that acts like a mechanical pump. In their simulations, the active particles do not simply sit in the tube; they circulate in complex patterns, constantly bumping against the walls and pushing each other upward. The researchers found that this internal drag, generated by the movement of the particles themselves, is strong enough to reverse the effect of the negative surface tension and lift the liquid against gravity. This discovery rewrites the fundamental law of capillary action for active materials, replacing a simple balance of forces with a more complex equation that includes the momentum of these internal currents.

The most surprising aspect of this new understanding is that the height the liquid reaches is no longer determined solely by the width of the tube. In the world of passive fluids, the shape of the tube's exterior or the length of the tube beyond the liquid level makes no difference to how high the water climbs. For active fluids, however, the entire global configuration of the system matters. The researchers demonstrated that the shape of the tube's tip plays a critical role. When they simulated a tube with a sharp, pointed tip, the liquid rose to a certain height. When they changed the tip to a smooth, rounded shape, the liquid climbed significantly higher, even though the width of the tube and the properties of the liquid remained exactly the same. This happens because the sharp tip creates a specific kind of disturbance in the particle flow, generating a vortex that acts as a ratchet, injecting particles into the tube. A rounded tip, lacking this sharp geometric discontinuity, generates a different, often stronger, flow pattern that pushes more particles upward.

This sensitivity to global shape extends beyond just the tip. The researchers found that the number of tubes in a cluster also changes the outcome. When multiple tubes are placed close together, the liquid level in each one drops compared to when a single tube stands alone. This is because the circulating currents that drive the liquid up are generated not just inside the tubes, but also in the space around them. As more tubes are added, the available space for these external currents to form is reduced, and the particles are distributed across more openings, lowering the rise in each individual tube. Furthermore, the stiffness of the outer walls of the tube influences the result. If the outer wall is made softer, the liquid wets it differently, creating additional currents that force more particles into the tube and increase the height of the rise. Even the length of the tube matters, a factor that is irrelevant for passive water. In tubes with rounded tips, making the tube longer actually causes the liquid to climb lower, whereas in tubes with pointed tips, a longer tube can lead to a higher rise.

The researchers arrived at these conclusions by running detailed computer simulations of active particles moving in two dimensions. They modeled the particles as tiny spheres that repel each other when they get too close and move with a constant speed in a specific direction, occasionally changing their orientation due to random noise. They placed these particles in a gravitational field and inserted a virtual tube into the mixture. By tracking the position and movement of every particle, they were able to calculate the forces at play. They confirmed that the upward force generated by the particle currents, which they termed "drag," perfectly balanced the downward pull of gravity and the negative surface tension. This balance allowed them to derive a new version of the classic law of capillary action, one that explicitly includes the contribution of these spontaneous currents.

The findings suggest that the behavior of active fluids is inherently non-local, meaning that what happens in one part of the system is deeply connected to the shape and configuration of the entire system. The liquid does not just react to the immediate walls of the tube; it responds to the global flow patterns established by the entire container. This challenges the intuition built from centuries of studying passive liquids, where local conditions are usually sufficient to predict the outcome. The study indicates that to fully understand wetting and capillary action in active systems, one must account for these long-range, self-organized currents. While the research was conducted through simulations, the principles uncovered offer a framework for understanding real-world active fluids, such as bacterial suspensions or synthetic micro-swimmers, where similar currents are known to exist. The work highlights that in the realm of active matter, the whole is truly greater than the sum of its parts, and the shape of the container is just as important as the liquid it holds.

Problem Statement
Capillary rise in passive fluids is traditionally governed by Jurin's Law, which describes a balance between surface tension and gravity, predicting that the rise height depends solely on tube width, surface tension, and gravity. However, recent observations of active fluids (composed of self-propelled particles) exhibit capillary rise despite possessing a negative mechanical liquid-gas surface tension ( $\gamma_{LG} < 0$ ). In equilibrium, negative surface tension would predict a capillary drop, not a rise. Furthermore, experimental and numerical studies have shown that the capillary rise in active matter depends on the global shape of the tube (e.g., pointed vs. round tips) and other non-local factors, contradicting the local nature of Jurin's Law in passive systems. The mechanism driving this counterintuitive rise and the origin of its non-local dependencies remained unexplained.

Methodology
The authors investigate this phenomenon using a microscopic model of overdamped active Brownian particles in two dimensions. The particles interact via short-range repulsive forces and are subjected to uniform gravity. The system operates in a regime of strong self-propulsion and repulsion, leading to motility-induced phase separation (MIPS) with a negative surface tension.

Simulation Setup: A tube of width $w$ is inserted into the active liquid. The tube walls are modeled using a repulsive harmonic potential. The study compares tubes with pointed tips ( $d=0$ ) and round tips ( $d>0$ ), as well as systems with multiple tubes and varying outer wall stiffness.
Theoretical Derivation: Starting from the Fokker-Planck equation for the particle dynamics, the authors integrate out angular dependencies to derive a steady-state force balance equation. This equation includes the divergence of the active stress tensor, gravitational forces, wall interactions, and an emergent particle current term.
Force Balance Analysis: By integrating the force balance equation along the tube axis and comparing it with the bulk behavior far from the tube, the authors derive a modified force balance equation that explicitly accounts for the drag exerted by particle currents.

Key Contributions and Results

Derivation of Active Jurin's Law: The paper establishes a new form of Jurin's Law derived from microscopic dynamics. The total capillary rise height $H$ is expressed as the sum of two terms:
$H = H_\gamma + H_J$
where $H_\gamma$ is the contribution from surface tension (which is negative in this active regime) and $H_J$ is the contribution from particle currents.
Role of Spontaneous Currents: The authors identify self-organized particle currents near surfaces as the missing key factor. These currents generate a drag force ( $f_d$ ) that opposes gravity. Crucially, this drag is large enough to overcome the negative surface tension, resulting in a net capillary rise.
Non-Locality and Global Dependence: Unlike passive Jurin's Law, the active capillary rise is a non-local phenomenon. The magnitude of the current drag ( $f_d$ $f_{d}$ ) depends on the global configuration of the system, including:
- Tube Tip Geometry: Pointed tips introduce a geometric discontinuity that excites complex vortex currents, whereas round tips mitigate this flow field. However, in the specific configurations compared in the paper, the net drag of the currents created by the pointed tips is lower than that created by the round tips. Consequently, the capillary rise is systematically lower in tubes with pointed tips and higher in tubes with round tips, which generate stronger drag.
- Tube Number: In a "comb-like" arrangement of multiple tubes, the capillary rise per tube decreases as the number of tubes increases because the external currents are distributed among them.
- Outer Wall Stiffness: The stiffness of the tube's outer wall influences the wetting angle and the formation of capillary bridges, which in turn modulates the circulating currents and the rise height.
- Tube Length: The rise height scales differently with tube length $L$ depending on the tip geometry. For round tubes, $H$ decreases with $L$ (dominated by external currents decaying as $L^{-1}$ ). For pointed tips, the scaling is more complex; while $H$ can increase with $L$ if the drag of currents above the tube is kept constant, the general behavior depends on the interplay between tip-induced currents and external sources.
Scaling Exponents: The study notes that the scaling of capillary rise with gravity ( $H \sim g^{-\alpha}$ ) deviates from the passive expectation ( $\alpha=1$ ). The authors attribute the previously reported anomalous exponents (e.g., $\approx -0.77$ ) to finite-size effects and the complex interplay of current drag rather than a fundamental breakdown of the force balance.

Significance and Claims
The paper claims to provide a comprehensive microscopic explanation for capillary rise in active fluids with negative surface tension. By identifying spontaneous particle currents as the driving mechanism, the authors resolve the paradox of why active fluids climb against gravity despite negative surface tension.
The work highlights that capillary action in active matter is fundamentally a non-equilibrium, non-local phenomenon. The rise height is not determined solely by local properties (tube width, contact angle) but is sensitive to the global system configuration, including the shape of the tube, the number of tubes, and the stiffness of external boundaries.
The authors conclude that spontaneous currents are a defining feature of non-equilibrium systems that break the universality expectations derived from equilibrium intuition. They suggest that a complete theoretical account of wetting phenomena in active systems must incorporate these non-local effects, potentially through explicit dynamics of orientation fields or convolution terms in hydrodynamic descriptions. The theory is presented for dry active systems with momentum-preserving interactions, with the authors noting that extensions to momentum-conserving fluids and partial wetting scenarios are promising directions for future research.

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