🔬 condensed matter

Ultra-slow capillary rise on hydrogel surfaces

This paper reports the discovery of ultra-slow capillary rise on agarose hydrogels that defies classical models, leading to the development of a new porous-network-based model that enables non-invasive, high-resolution estimation of hydrogel permeability for biomedical applications.

Original authors: Anagha Datar, Joonas Ryssy, Aku O. Toivonen, Matilda Backholm

Published 2026-04-20

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Anagha Datar, Joonas Ryssy, Aku O. Toivonen, Matilda Backholm

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

The Big Idea: A "Sponge" That Drinks Too Slowly

Imagine you have a glass straw and you dip the tip into a glass of water. What happens? The water shoots up the straw almost instantly, racing against gravity until it finds its balance. This is called capillary rise, and it's a well-known trick of physics that happens in plants, paper towels, and sponges.

Now, imagine dipping that same straw into a jiggly gelatin dessert (specifically, an agarose hydrogel, which is a type of jelly used in labs). You might expect the water inside the gel to rush up the straw just like it does in a cup of water.

But it doesn't.

Instead of a race, the liquid moves at a snail's pace. It's so slow that it takes minutes to climb just a tiny bit. The researchers in this paper discovered this "ultra-slow" rise and realized that the old rules of physics (which work for water in a glass tube) completely fail to explain what's happening here.

The Problem: Why the Old Rules Don't Work

In the old model, the "brakes" on the liquid are the friction inside the glass tube itself. Think of it like running down a hallway; the friction of your shoes against the floor slows you down.

But in this experiment, the liquid isn't just moving through the glass tube; it has to squeeze out of the gel first. The gel is like a microscopic maze made of a tangled net of polymers.

The Analogy: Imagine trying to drink a milkshake through a straw.
- Normal Liquid (Water): The milkshake is thin. The only resistance is the straw itself. Whoosh! It goes up fast.
- Hydrogel: The milkshake is actually a giant, thick sponge. To get the liquid into your straw, you first have to squeeze the liquid out of the sponge's tiny holes. The sponge is so dense with tiny holes that the liquid gets stuck and drags its feet.

The researchers found that the "brakes" aren't the glass tube anymore; the brakes are the tiny pores inside the gel.

The Solution: A New Rulebook

Since the old math didn't work, the team (led by Matilda Backholm at Aalto University) wrote a new equation. They used a concept called Darcy's Law, which is usually used to describe how water moves through soil or sand.

They realized that the speed of the liquid climbing the straw depends on:

How "squeezable" the gel is (its permeability).
How sticky the liquid is (viscosity).
How tight the squeeze is (the size of the straw).

When they plugged their new math into the data, it matched perfectly. The liquid wasn't moving slowly because it was lazy; it was moving slowly because it was fighting its way through a microscopic forest of polymer strands.

Why Does This Matter? (The "Superpower" of the Experiment)

This isn't just a cool physics trick; it's a new tool for scientists.

1. It's a Non-Invasive "X-Ray" for Gels
Usually, to measure how porous a gel is (how easy it is for water to flow through), scientists have to do messy things:

Freeze the gel (which can break the structure).
Squeeze it (which changes its shape).
Dry it out for days (which ruins it).

This new method is like listening to a heartbeat. You just touch the gel with a tiny straw, watch the liquid creep up, and voila—you know exactly how porous the gel is. You don't have to cut, freeze, or crush the sample. It's gentle and fast.

2. It Can "See" Tiny Details
Because the straw is so thin (microscopic), the researchers can measure the porosity of the gel at a very specific spot.

The Analogy: Imagine a field of grass where some patches are thick and some are thin. Old methods would take a big shovel, dig up a huge chunk of the field, and give you an average of the whole thing.
This new method is like using a needle to poke just one blade of grass to see how thick that specific spot is.

3. It Helps with Medicine and Biology
Hydrogels are used in everything from contact lenses to artificial skin and drug delivery. Cells (like skin cells or bacteria) move through these gels.

If the gel is too "tight" (low permeability), cells can't move.
If it's too "loose," it might not hold together.
This new technique allows scientists to map out exactly how easy it is for cells to move across a gel surface, which is crucial for designing better artificial tissues or understanding how plant roots grow in soil.

The Takeaway

The researchers found that when you try to pull liquid out of a jelly-like gel, it moves incredibly slowly because the gel acts like a dense, microscopic maze. By understanding this "traffic jam," they created a new, gentle, and highly precise way to measure how "open" or "closed" these gels are. This helps scientists build better materials for medicine, agriculture, and biology without damaging the delicate samples they are studying.

1. Problem Statement

Hydrogels are soft, crosslinked polymer networks widely used in biomedical applications (e.g., drug delivery, tissue engineering, bioadhesion) and agricultural research (mimicking soil). A critical property governing their performance is permeability (fluid transport through the network).

The Gap: Existing methods to measure hydrogel permeability are often invasive, low-resolution, or destructive. Techniques like diffusion experiments, pressure-driven flow, or mercury intrusion often require freezing, using non-aqueous fluids, or compressing the gel, which alters the pore structure.
The Specific Challenge: There is a lack of non-invasive techniques capable of measuring interfacial fluid transport with high spatial resolution (down to the scale of single cells, ~10 µm). This limitation hinders the study of how local permeability gradients affect 2D cell migration and bioadhesion.

2. Methodology

The authors developed a novel experimental approach using microcapillary rise on solid-like hydrogel surfaces, contrasting it with classical capillary rise on liquids.

Materials:
- Hydrogels: Agarose hydrogels prepared in two formulations: pure agarose and "agarose++" (containing MS media and sucrose, mimicking artificial soil). Concentrations ranged from 0.3% to 2% (w/v), including a sample with a concentration gradient.
- Fluids: The fluid leaking from the gels was characterized for density ( $\rho$ ) and viscosity ( $\eta$ ) using rheometry and gravimetric analysis.
Experimental Setup:
- Microcapillaries: Glass microcapillaries with tip radii ( $R$ ) between 16–32 µm were fabricated.
- Procedure: A dry microcapillary was brought into contact with the hydrogel surface. The rise of the fluid meniscus inside the capillary was tracked via high-speed video microscopy.
- Control: Classical capillary rise experiments were also performed on liquid puddles leaked from the gels to measure surface tension ( $\gamma$ ) and contact angle ( $\theta$ ) using Jurin's law.
Theoretical Modeling:
- The authors rejected classical models (Lucas-Washburn equation), which predict $z(t) \sim t^{1/2}$ (viscous regime) or $z(t) \sim t$ (inertial regime) based on damping within the capillary.
- New Model: They proposed a model where the dominant damping occurs within the porous hydrogel network, not inside the glass capillary. Using Darcy's Law ( $Q = \kappa A \Delta p / \eta L$ ) combined with capillary pressure ( $\Delta p = 2\gamma \cos\theta / R$ ), they derived a linear rise equation:
  $z(t) \approx U t = \left( \frac{2\kappa \gamma \cos\theta}{\eta R^2} \right) t$
  Where $\kappa$ is the Darcy permeability.

3. Key Results

Observation of Ultra-Slow Rise: Upon contacting the hydrogel, the fluid rose in the microcapillary at a constant, ultra-slow velocity (linear scaling $z \sim t$ ), distinct from the rapid rise seen in liquid puddles.
Failure of Classical Models: Fitting the classical Lucas-Washburn equation to the data yielded unphysical surface tension values ( $\sim 10^{-11}$ N/m), proving that the standard viscous damping model (inside the tube) does not apply.
Validation of the New Model:
- The derived linear model based on Darcy flow through the gel pores perfectly matched the experimental data across five different agarose concentrations and two fluid viscosities.
- The measured rise velocity ( $U$ ) decreased significantly as agarose concentration increased (due to reduced permeability).
Quantitative Permeability: The technique allowed for the direct calculation of Darcy permeability ( $\kappa$ ) using the rearranged equation:
$\kappa = \frac{U \eta R^2}{2\gamma \cos\theta}$
The calculated permeabilities agreed with values derived from empirical equations based on water content, validating the method.
Spatial Resolution: The method successfully mapped permeability across a concentration gradient sample. The spatial resolution is determined by the microcapillary tip size (~10–30 µm), which is comparable to the size of single cells.

4. Key Contributions

Discovery of a New Regime: Identified and characterized "ultra-slow capillary rise" on solid-like porous materials, where the dissipation mechanism shifts from the capillary wall to the porous medium itself.
Novel Measurement Technique: Developed a non-invasive, fast, and simple method to measure hydrogel permeability. Unlike traditional methods, it does not require freezing, compression, or invasive fluid injection.
High Spatial Resolution: Demonstrated the ability to probe interfacial permeability at the cellular length scale (~10 µm), a capability previously unavailable with standard bulk measurement techniques.
Theoretical Framework: Provided a modified capillary rise model that accurately describes fluid transport in hydrogel interfaces, bridging the gap between Darcy flow and capillary dynamics.

5. Significance

Biomedical Applications: This technique is crucial for optimizing hydrogel interfaces for bioadhesion and tissue engineering, where local fluid transport properties dictate cell behavior.
Cell Migration Studies: It enables researchers to investigate how spatial gradients in permeability (independent of stiffness) influence 2D cell migration, addressing a major bottleneck in mechanobiology.
Agricultural Science: The method offers a way to study fluid transport in artificial soil models (agarose++), relevant for plant root growth research.
Material Characterization: It provides a reliable way to estimate effective pore size ( $r_p \sim \sqrt{\kappa}$ ) non-invasively, offering an alternative to expensive and destructive imaging techniques like CRYO-SEM.

In summary, the paper presents a paradigm shift in measuring hydrogel properties, moving from bulk, invasive measurements to localized, non-invasive probing of interfacial fluid dynamics, with direct implications for advanced biomedical and biological research.