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Fractal morphology-governed structural build-up in reactive particulate suspensions: insights from tricalcium silicate hydration

This study develops a modeling framework integrating boundary nucleation and growth kinetics with a fractal-based hydrate-bridging model to demonstrate how the evolution of hydrate morphology and internal connectivity, quantified by a fractal parameter $\beta$ , governs the early-age structural build-up and rheological behavior of tricalcium silicate pastes under varying water-to-solid ratios and calcium nitrate dosages.

Original authors: Yanliang Ji, Alexander Mezhov, Hongwei Tian, Tamino Hirsch, Thomas Matschei, Dietmar Stephan

Published 2026-07-31

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Yanliang Ji, Alexander Mezhov, Hongwei Tian, Tamino Hirsch, Thomas Matschei, Dietmar Stephan

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

The Secret Life of Wet Cement

Imagine you are mixing a batch of concrete. At first, it's just a soupy slurry of water and rock dust, easy to pour and spread. But if you leave it alone, something magical happens: it turns into a solid, unbreakable stone. This transformation, called hydration, is one of the most important chemical reactions on Earth, holding up our bridges, skyscrapers, and sidewalks. But how does a liquid turn into a rock? It's not just about the water drying up; it's about tiny, invisible particles inside the mix starting to grow and stick together, forming a microscopic web that locks everything in place.

Scientists have long known that this process depends on two main things: how much water you have and how fast the chemical reaction happens. If you have too much water, the particles are too far apart to grab each other. If the reaction is too slow, the mix stays soupy for too long. But there is a hidden layer to this story: the shape of the tiny crystals growing inside the mix. Do they grow like tight, compact balls, or do they stretch out like fluffy, branching trees? This paper dives deep into that question, using a special type of cement ingredient called tricalcium silicate (C₃S) as a model to understand how these microscopic shapes dictate when and how our concrete hardens.

The Paper's Story: Growing Trees in a Jar

In this study, researchers Yanliang Ji, Alexander Mezhov, and their team from universities in Germany and China decided to play detective with the microscopic world of cement. They wanted to build a "rulebook" that connects the speed of the chemical reaction to the shape of the growing crystals, and finally, to the strength of the wet paste. To do this, they didn't just watch the cement; they built a mathematical model that treats the growing crystals like a game of "connect the dots" played by tiny particles.

The Experiment: Speeding Up the Clock
The team mixed pure C₃S powder with water in different amounts (water-to-solid ratios of 0.32, 0.36, and 0.40). They also added a chemical accelerator called calcium nitrate in doses of 0%, 3%, and 6%. Think of the water ratio as the "roominess" of the party: more water means the particles have more space to dance, while less water forces them to bump into each other sooner. The calcium nitrate acts like a DJ turning up the music, making the particles react much faster.

They watched these mixes using three different "eyes":

The Penetration Test: They stuck a metal probe into the paste to see how hard it was to push down, measuring the "static yield stress" (how much force is needed to make the soup move).
The Oscillating Shear: They gently wiggled the paste to measure its stiffness, like testing how jiggly a gelatin dessert is.
The Microscope and Sensors: They used special tools like Scanning Electron Microscopy (SEM) to take pictures of the crystals, Dynamic Light Scattering (DLS) to guess their size, and Nuclear Magnetic Resonance (NMR) to listen to the water molecules trapped inside the pores.

The Discovery: The "Fractal" Shape Factor
The researchers developed a model that combines the heat released by the chemical reaction with a concept called "fractal growth." In simple terms, a fractal is a shape that looks similar at every scale, like a fern leaf or a lightning bolt. The team introduced a special number, called β (beta), to describe the shape of the growing crystals.

Here is what they found:

More Water, Bigger Branches: When they used more water (higher water-to-solid ratios), the β value went up. This means the crystals grew into larger, more open, and "fluffier" structures, like a sprawling tree with wide branches. Because these branches are loose, they take up more space but aren't as tightly packed.
The Accelerator Effect: When they added calcium nitrate, the reaction sped up dramatically. The heat data showed the peak reaction happening much earlier (dropping from 5 hours 10 minutes down to just 2 hours 53 minutes for the 6% dose). Interestingly, the accelerator made the crystals grow smaller but more numerous. It's like having a huge crowd of people trying to build a wall; if everyone starts building at once, they end up with many small, tight clusters rather than a few big ones.
The Shape-Strength Link: The team found a direct link between the shape factor β and how the paste hardened. A higher β (fluffier, more open crystals) meant the paste filled the empty spaces (pores) differently. They discovered that as the crystals grew, the "β" value didn't just stay the same; it changed over time. Initially, the structures simplified (became more open), but later, they started to get complex again, re-connecting to form a stronger network.

What the Model Tells Us
The paper suggests that the way cement hardens isn't just about how much crystal is formed, but how it is arranged.

The "Bridge" Theory: The model assumes that the strength comes from tiny "bridges" of crystal forming between the solid particles. The researchers found that the number of these bridges and their shape (described by β) could accurately predict the stiffness of the paste.
The NMR Connection: They also found that the β value correlated with the NMR results. When the crystals had a higher β (more open), the water trapped inside the pores had a different "relaxation time," suggesting the crystals were filling the gaps in a specific way.
The Simulation: To back this up, they ran computer simulations (Monte Carlo) where virtual particles stuck together. These simulations showed that when particles have more space or stick together in certain ways, they naturally form these open, tree-like fractal shapes, just like the real cement did.

The Bottom Line
This paper doesn't claim to have solved all the mysteries of concrete, but it offers a powerful new way to look at it. It suggests that we can understand the "stiffening" of wet cement by looking at the fractal geometry of the crystals growing inside it. By tracking a single number (β), scientists can now better predict how fast a mix will harden and how strong it will become, simply by knowing how much water is in the mix and what chemicals are added. It turns the messy, chaotic growth of cement into a predictable pattern of branching trees, helping engineers design better, stronger, and faster-setting materials for the future.

Technical Summary: Fractal Morphology-Governed Structural Build-up in Reactive Particulate Suspensions

Problem Statement
The structural build-up of reactive particulate suspensions, such as cementitious pastes during early hydration, is a complex multiscale process. It involves the simultaneous evolution of chemical reactions, phase transformations, and microstructural changes. While existing models (e.g., Yodel, thixotropy models) link microstructural evolution to macroscopic rheological properties like yield stress, many rely on idealized assumptions. They often fail to adequately account for the evolving quantity, morphology, and complexity of hydration products across multiple scales. Specifically, there is a need for a physically informed framework that quantitatively links the disordered, fractal-like growth of hydration products (specifically Calcium Silicate Hydrate, C-S-H) to the macroscopic structural stiffening observed in fresh pastes.

Methodology
This study employs tricalcium silicate (C₃S) pastes as a model system to investigate the relationship between hydrate morphology and structural build-up. The research integrates experimental characterization with a novel modeling framework:

Experimental Characterization:
- Rheology: Structural build-up was measured via slow penetration tests (to determine apparent static yield stress, $\Delta\tau_0$ ) and small-amplitude oscillatory shear (SAOS) tests (to determine relative storage modulus, $\Delta G'$ ).
- Hydration Kinetics: Isothermal calorimetry was used to monitor heat flow.
- Microstructure & Pore Structure: Low-field $^1$ H Nuclear Magnetic Resonance (NMR) monitored transverse relaxation times ( $T_2$ ) to assess pore filling and water confinement. Dynamic Light Scattering (DLS) measured hydrate-related nanoparticle sizes, and Scanning Electron Microscopy (SEM) provided visual evidence of hydrate morphology.
- Variables: Experiments were conducted across varying water-to-solid ratios (0.32–0.40) and calcium nitrate dosages (0–6.0 wt.%).
Modeling Framework:
- Kinetics: The Boundary Nucleation and Growth (BNG) model was fitted to calorimetric data to quantify the transformed volume fraction of hydration products, $X_T(t)$ .
- Hydrate-Bridging Model: A fractal-based model was developed where structural response $S(t)$ scales with the probability of interparticle bridge formation. This probability depends on the characteristic size of bridging hydrates ( $\varepsilon_L$ ) and interparticle spacing ( $h$ ).
- Fractal Scaling: The model assumes hydrate growth follows a reaction-limited aggregation (RLA) mechanism. The characteristic size $\varepsilon_L$ is linked to the number of hydration units via a fractal dimension. A key fitted parameter, $\beta$ , serves as a morphology-related scaling exponent derived from the fractal dimension.
- Simulation: Monte Carlo simulations were used to analogize hydrate growth along particle boundaries, validating the relationship between particle number, sticking probability, and cluster morphology.

Key Results

Model Performance: The integrated framework successfully reproduced the time-dependent evolution of structural build-up ( $\Delta\tau_0$ and $\Delta G'$ ) across all tested mix designs with high correlation coefficients ( $R^2 > 0.99$ for penetration tests).
The Parameter $\beta$ : The fitted parameter $\beta$ $β$ acts as a descriptor of hydrate morphology and internal connectivity.
- Water-to-Solid Ratio: Higher water-to-solid ratios resulted in higher $\beta$ values, corresponding to larger, more open, and less compact hydrate structures (confirmed by DLS and SEM).
- Calcium Nitrate: In calcium nitrate-modified systems, the addition of accelerators led to a non-linear response. While higher dosages generally increased $\beta$ (indicating more open structures over time), the 3 wt.% dosage showed the smallest apparent hydrate clusters. This aligns with classical dissolution-precipitation theory, where increased supersaturation enhances nucleation rates, partitioning solutes among more nuclei and restricting individual crystal growth.
Morphology and Pore Filling: A positive correlation was found between $\beta$ and the change in $T_2$ relaxation time ( $\Delta T_2$ ) at comparable hydration levels. Higher $\beta$ values (simpler, more open morphologies) were associated with greater changes in $T_2$ , suggesting that hydrate morphology significantly influences pore filling and water confinement.
Temporal Evolution: The temporal evolution of $\beta$ revealed a non-monotonic trend: an initial phase of structural simplification (increasing $\beta$ ) followed by partial re-complexification (decreasing $\beta$ ) at later stages. This dynamic transition was consistent with Monte Carlo simulation results showing a shift from compact boundary clusters to open, tree-like structures, and eventually to more connected networks.

Significance and Claims
The paper claims to provide a "physically informed interpretation" of the link between hydrate-scale structural evolution and early-age rheological behavior. Its primary contributions are:

Bridging Scales: It establishes a quantitative link between the fractal morphology of hydration products and macroscopic structural build-up, moving beyond idealized assumptions to account for the disordered nature of early-age microstructures.
Descriptor $\beta$ : It introduces $\beta$ as a model-derived descriptor that captures the evolving internal connectivity and spatial arrangement of hydration products, validated against SEM, DLS, and NMR data.
Mechanistic Insight: The study demonstrates that structural build-up is governed not only by the amount of hydration products but critically by their spatial arrangement and morphology.
Validation: The consistency between the model parameters, experimental observations (SEM/DLS/NMR), and classical nucleation theory supports the proposed framework's ability to interpret microstructural refinement and pore-filling behavior.

The authors conclude that this framework offers a route for understanding structure formation in reactive particulate systems governed by interfacial growth and disordered aggregation, though they note limitations regarding its direct applicability to complex multi-phase cement systems with overlapping reactions (e.g., C3A hydration) and the simplifications inherent in 2D Monte Carlo simulations.

The Secret Life of Wet Cement

The Paper's Story: Growing Trees in a Jar

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