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Evaluation of tortuosity based on an experimental study of a solute diffusion in a porous medium

This paper presents experimental findings on the diffusion tortuosity of a porous medium modeled by monosized hard spheres, demonstrating how variations in porosity and skeleton shape influence tortuosity values ranging from 1.20 to 1.46 and comparing these results with theoretical predictions and existing literature.

Original authors: Denis Polezhaev, Anastasia Teryokhina

Published 2026-07-15

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Denis Polezhaev, Anastasia Teryokhina

Original paper licensed under CC BY 4.0 (https://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 crowded hallway where people are trying to walk from one end to the other. If the hallway is empty, they zip straight through. But if you fill the hallway with a giant, chaotic pile of bowling balls, the people have to weave, dodge, and take winding paths just to get past. They still get to the other side, but it takes them much longer.

In the world of science, this "weaving path" is called tortuosity. It's a measure of how twisty and turny a path is inside a porous material (like a sponge, soil, or rock). The more obstacles in the way, the higher the tortuosity, and the slower things move through it.

A team of researchers from Perm State Humanitarian-Pedagogical University decided to play with this idea using a giant, transparent "hallway" filled with steel balls. Here is what they discovered, how they did it, and what it means for our understanding of how things move through messy spaces.

The Great Ball Pit Experiment

To study this, the scientists built a tall, thin, rectangular glass box. Think of it like a giant, vertical fish tank, but very narrow—only about 2 to 4 millimeters thick. They filled this box with thousands of identical steel balls, creating a model of a porous medium.

To see how fast things moved, they used a special trick. They filled the bottom half of the box with water containing a glowing orange dye called Rhodamine B. The top half was filled with plain, clear water. To keep the heavy orange water from just sinking and mixing with the clear water due to gravity, they added a pinch of salt to the orange water to make it slightly denser.

Then, they turned on a green laser sheet that sliced through the box. Because Rhodamine B glows orange when hit by green light, the scientists could watch the orange dye slowly creep upward into the clear water. This creeping is called diffusion.

By taking photos over many hours, they measured how wide the "mixing zone" (the blurry orange line between the two waters) got. They found that the width of this zone grew in a very predictable way: the square of the width grew linearly with time. This allowed them to calculate exactly how fast the dye was moving through the steel ball maze.

The Twisty Truth

Here is the big finding: The dye moved slower through the steel ball maze than it would have in a cup of plain water.

In the free water, the dye molecules could take a straight shot. In the ball maze, they had to dodge the steel spheres, taking a longer, more winding route. The researchers calculated a number called diffusion tortuosity ( $\tau_d$ ) to describe this slowdown. It's simply the ratio of how fast the dye moves in free water versus how fast it moves in the porous maze.

The Result: In their experiments, the tortuosity ranged from 1.20 to 1.46.
What that means: A value of 1.0 would mean a straight path. A value of 1.46 means the dye had to travel a path roughly 46% longer than the straight line just to get to the same spot.

The Porosity Puzzle

The scientists wanted to see how the "tightness" of the ball pile changed the twistiness. They called this tightness porosity ( $P$ ), which is basically the percentage of the box that is empty space (water) versus solid steel.

They created different models by changing two things:

The size of the balls: They used balls with diameters of 1.98 mm, 3.00 mm, and 3.98 mm.
The width of the box: They adjusted the gap between the glass walls.

When the box was just wide enough to fit exactly one layer of balls (a "tight squeeze"), the porosity was low (around 0.40 to 0.43), and the tortuosity was high (1.43 to 1.46). The path was very twisty because the balls were packed tight against the walls and each other.

When they made the box slightly wider, allowing the balls to sit a bit more loosely, the porosity increased to 0.55, and the tortuosity dropped to 1.20. The path became less twisty because there was more open space to flow through.

What They Ruled Out (and What They Didn't)

The researchers were very careful to check if the size of the balls mattered. They compared experiments with small balls (1.98 mm) and larger balls (3.00 mm and 3.98 mm) when the balls were packed in that "tight squeeze" configuration.

They found that the size of the balls didn't matter. Whether the balls were small or large, if the packing was tight, the tortuosity was the same. The slowdown was purely about how the balls were arranged, not how big they were.

However, they did find a snag when the balls were packed in a slightly looser way (where the balls touched opposite walls of the box). In these specific cases, the experimental data didn't perfectly match the famous mathematical formulas that scientists usually use to predict tortuosity.

The paper explicitly argues against the idea that a single, simple formula (like the one proposed by Weissberg: $\tau_d = 1 - 0.5\ln P$ ) works perfectly for every situation. While that formula worked great for the tightly packed balls, it didn't quite fit the looser arrangements the team tested. The authors suggest that in looser packs, the shape of the gaps between the balls and the walls might be creating a different kind of "traffic jam" that the old formulas don't account for.

How Sure Are They?

The authors are very confident in their measurements. They didn't just guess or run a computer simulation; they physically built the models, measured the dye movement with a camera and laser, and calculated the numbers directly.

Proven: They proved that for tightly packed spheres, their results match previous theories and other scientists' data.
Measured: They measured the tortuosity values (1.20 to 1.46) and the diffusion coefficients (like 2.89 $\cdot$ 10 $^{-6}$ cm $^2$ /s for the porous medium).
Suggested: They suggest that the relationship between porosity and tortuosity is more complex than previously thought for looser packs. They propose a new formula ( $\tau_d = 0.83P^{-0.62}$ ) that fits their new data better, but they present this as a description of their specific findings rather than a universal law for all porous materials.

The Bottom Line

This study is like a masterclass in navigating a maze. It shows us that when you squeeze a bunch of identical balls together, the path for a liquid to flow through them gets significantly longer and twistier. The tighter the squeeze, the harder it is for the liquid to get through.

While the size of the balls doesn't change the difficulty of the maze, the arrangement of the balls does. The researchers showed that while we have good maps for very tight mazes, the maps for slightly looser mazes might need a little updating. Their new measurements provide a clearer picture of how fluids actually move through the tiny, twisting tunnels of the world around us, from soil to oil reservoirs.

Technical Summary: Evaluation of Tortuosity Based on an Experimental Study of Solute Diffusion in a Porous Medium

Problem Statement
Tortuosity is a fundamental parameter characterizing the geometry and transport properties of porous materials, such as sedimentary rocks and soils. While numerous theoretical and semi-empirical models exist to correlate tortuosity with porosity, significant discrepancies remain, particularly regarding the specific influence of pore morphology. A notable gap in the literature is the lack of experimental data investigating the relationship between diffusion tortuosity ( $\tau_d$ ) and porosity ( $P$ ) for porous media with skeletons composed of hard spheres, specifically within the high-porosity range ( $P > 0.4$ ). Previous studies often relied on mixtures of spheres with varying diameters to alter porosity, potentially introducing complexity in pore structure analysis.

Methodology
The authors employed an experimental technique to measure diffusion tortuosity by comparing the molecular diffusion coefficient of a solute in free fluid ( $D_m$ ) against its diffusion coefficient within a fluid-saturated porous medium ( $D_p$ ).

Experimental Setup: The study utilized a vertical rectangular cell (200 mm height, 90 mm width) with variable thickness ( $h$ ) ranging from 2.00 to 4.00 mm. The cell was filled with monosized hard steel spheres of diameters $d = 1.98$ , $3.00$, or $3.98$ mm.
Porosity Control: Porosity was varied between 0.40 and 0.55 by adjusting the cell thickness relative to the sphere diameter. Two packing configurations were tested:
1. Hexagonal Packing: Sphere diameter equaled cell thickness ( $h/d \approx 1$ ), where spheres contacted both lateral walls and neighbors, creating a dense, ordered structure (Models M1, M5, M6).
2. Variable Thickness Packing: Cell thickness exceeded sphere diameter ( $h/d > 1$ ), where spheres contacted only one wall, allowing for looser packing and higher porosity (Models M2–M4).
Solute and Measurement: Water served as the solvent, with Rhodamine B (fluorescent dye) as the solute. Sodium chloride was added to the lower Rhodamine B solution to increase density and prevent convective mixing.
Data Acquisition: A green laser sheet (532 nm) illuminated the cell, exciting the Rhodamine B to emit orange light (610 nm). A camera with an orange filter recorded the fluorescence intensity, which was proportional to solute concentration.
Analysis: The width of the diffusion zone ( $\Delta y$ ) was tracked over time. The diffusion coefficient $D_p$ was derived from the linear relationship between the square of the diffusion zone width and time ( $\Delta y^2 \propto 4\pi D_p t$ ). Tortuosity was calculated as $\tau_d = D_m / D_p$ .

Key Results

Diffusion Coefficients: The molecular diffusion coefficient in free water was established as $D_m = 4.2 \times 10^{-6}$ cm $^2$ /s. The effective diffusion coefficients ( $D_p$ ) in the porous media were consistently lower, ranging from approximately $2.89 \times 10^{-6}$ to $3.5 \times 10^{-6}$ cm $^2$ /s depending on the model.
Tortuosity Values: The measured diffusion tortuosity ( $\tau_d$ $τ_{d}$ ) varied from 1.20 to 1.46 across the tested porosity range (0.40 to 0.55).
- Densely packed models (M1, M5, M6; $P \approx 0.40-0.43$ ) exhibited higher tortuosity ( $\tau_d \approx 1.43-1.46$ ).
- Looser packed models (M2–M4; $P \approx 0.44-0.55$ ) exhibited lower tortuosity ( $\tau_d \approx 1.20-1.37$ ).
Sphere Size Independence: The study confirmed that for a given porosity and packing configuration, the diffusion coefficient is independent of the absolute sphere diameter.
Comparison with Theory and Literature:
- The experimental data for densely packed spheres ( $P \approx 0.4$ ) showed good agreement with theoretical predictions by Weissberg ( $\tau_d = 1 - 0.5 \ln P$ ) and previous experimental data from Currie and Delgado.
- In the looser packing regime ( $P > 0.4$ ), the new data deviated from Weissberg's theoretical curve.
- The authors fitted a new empirical power-law relation to their data: $\tau_d = 0.83 P^{-0.62}$ .

Significance and Claims
The paper claims to provide a new experimental technique for measuring diffusion tortuosity and to fill a specific data gap regarding the $\tau_d - P$ relationship in the high-porosity range ( $P > 0.4$ ) for hard-sphere skeletons. The authors assert that their results:

Validate the use of laser-induced fluorescence for precise measurement of diffusion zones in porous media.
Demonstrate that while theoretical models (like Weissberg's) align well with experimental data in the limit of close packing, they may require adjustment for looser porous media configurations.
Confirm that the geometry of the pore space (specifically the interaction between spheres and cell walls) significantly influences tortuosity, even when porosity is the primary variable.

The study concludes that the obtained data are consistent with known results in the limit of close packing but highlights the complexity of predicting tortuosity in less dense, ordered porous structures, suggesting that pore shape and local geometry play critical roles alongside bulk porosity.