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Vibration-Induced Densification, Microstructural Reconstruction, and Force Chain Topology of Polydisperse Hollow Glass Microspheres via Discrete Element Method

This study utilizes a calibrated Discrete Element Method to reveal that polydisperse hollow glass microspheres undergo a three-stage vibration-induced densification process, identifying optimal "energy sweet spots" for maximum packing density and establishing a linear scaling law between local coordination number and particle diameter.

Original authors: Haozhe Jin, Tianhao Ma, Chao Wang, Xiaofei Liu, Jie Luo, Xu Xu, Kang Xu, Pengqing Huang

Published 2026-10-07

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CC BY 4.0

Original authors: Haozhe Jin, Tianhao Ma, Chao Wang, Xiaofei Liu, Jie Luo, Xu Xu, Kang Xu, Pengqing Huang

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

In the world of energy storage, keeping things cold is often just as critical as keeping them hot. For the massive spherical tanks that hold liquid hydrogen, a fuel that powers everything from rockets to heavy trucks, the challenge is to prevent heat from leaking in and turning the precious fuel back into gas. These tanks rely on a thick layer of insulation, much like the foam in a thermos, but on a scale that can fill a room. The material of choice for this job is often hollow glass microspheres, tiny spheres of glass with a vacuum or thin gas inside. These spheres are incredibly efficient at blocking heat, but their performance depends entirely on how they are packed together. If they are loosely arranged, heat can sneak through the gaps; if they are packed too tightly or unevenly, the structure might fail under pressure. The way these microscopic spheres settle, touch, and support one another determines the tank's ability to keep its contents frozen for years.

Researchers at Zhejiang Sci-Tech University set out to understand exactly how these tiny glass balls behave when they are shaken, a scenario that happens naturally during the transport and operation of large storage tanks. Using a powerful computer simulation technique that tracks the movement of individual particles, they recreated a container filled with millions of these microspheres. Unlike previous studies that assumed all the spheres were the same size, this team used real-world data to create a mix of spheres ranging from very small to quite large, mimicking the actual material used in industry. They then subjected this digital pile to a wide variety of vibrations, changing the speed and the strength of the shaking to see how the arrangement of the spheres changed.

The study revealed that simply shaking the container does not always make the packing tighter. In fact, the relationship between the shaking speed and the resulting density is surprisingly complex. The researchers found that there are specific combinations of vibration speed and shaking distance where the spheres settle into their most compact state, reaching a density where the empty space between them is minimized. They identified two such "sweet spots": one where the container vibrates at 40 times per second with a stroke of 1.25 millimeters, and another at 60 times per second with a stroke of 1.50 millimeters. At these precise settings, the spheres rearrange themselves so that the smaller ones nestle perfectly into the gaps between the larger ones, creating a stable, dense structure.

However, the team also discovered that shaking too hard or at the wrong speed can have the opposite effect. If the vibration is too intense or the frequency is mismatched, the spheres begin to bounce and float, a state known as fluidization, which actually makes the pile looser and less effective as an insulator. This happens because the energy of the shaking tears apart the delicate network of contacts holding the spheres together, preventing them from settling into a tight formation. The simulations showed that the process of densification happens in distinct stages. Initially, the shaking breaks apart unstable, temporary connections between spheres that formed when they were first poured in. As the vibration continues, the spheres collide and slide past one another, searching for better positions. Finally, when the shaking stops, the spheres settle into a new, more stable arrangement where they touch each other more effectively, creating a stronger framework that can bear weight without collapsing.

A key finding of the research is how the size of the spheres influences their role in this packed structure. The computer model showed that larger spheres act as the main skeleton of the pile, supporting the weight and forming the primary load-bearing paths. These large spheres tend to have many neighbors touching them. In contrast, the tiny spheres act as fillers, slipping into the voids left by the larger ones. Because they are trapped in these small spaces, they have fewer neighbors touching them. The researchers established a clear mathematical link between the size of a sphere and the number of neighbors it touches, confirming that larger spheres consistently have more contact points than smaller ones. This difference in how spheres connect is crucial for understanding how forces travel through the insulation layer.

The study also looked at the invisible forces that hold the pile together. By analyzing the pressure at the points where spheres touch, the researchers saw that the vibration helps to redistribute these forces. In a loose pile, some spheres might carry too much weight while others carry almost none, creating weak spots. The optimal vibration settings help to even out these forces, ensuring that the load is shared more evenly across the entire structure. This redistribution allows the insulation layer to become more robust and less likely to settle unevenly over time. The work provides a clear guide for engineers on how to fill these massive tanks. By applying the right amount of vibration during the filling process, they can ensure the insulation is packed as tightly as possible, maximizing the tank's efficiency and safety without the risk of the material shifting later during transport. This research turns a complex physical problem into a set of practical instructions, showing that the secret to better insulation lies not just in the material itself, but in the precise way it is settled into place.

Technical Summary: Vibration-Induced Densification, Microstructural Reconstruction, and Force Chain Topology of Polydisperse Hollow Glass Microspheres via Discrete Element Method

Problem Statement
The efficiency of thermal insulation in large-scale liquid hydrogen storage tanks relies heavily on the packing density and topological structure of the insulating layer, typically composed of hollow glass microspheres (HGMs). While HGMs offer superior thermal and mechanical properties compared to traditional perlite, their macroscopic thermal conductivity is not an intrinsic material property but is dictated by the microscopic packing topology (porosity, coordination number, and contact network). Current theoretical models often rely on idealized, regular packing assumptions (e.g., FCC, BCC) or monodisperse particle distributions, which fail to capture the complex, random packing of real-world polydisperse HGMs. Furthermore, the dynamic response of these granular beds to external excitations—such as mechanical vibrations during transport or operation—remains insufficiently characterized. Existing research often overlooks the coupled nonlinear effects of vibration frequency, amplitude, and continuous particle size distribution, leading to gaps in predicting how dynamic restructuring alters porosity and force chain networks.

Methodology
This study employs a multi-scale approach combining experimental characterization and Discrete Element Method (DEM) simulations to investigate polydisperse HGM systems.

Experimental Characterization: T15 hollow glass microspheres were analyzed using Scanning Electron Microscopy (SEM). Image processing algorithms (FFT filtering, Otsu binarization, Watershed segmentation) were used to extract geometric parameters from over 3,800 particles. The particle size distribution was statistically fitted to a normal distribution (mean $\mu = 40.35$ $\mu$ m, standard deviation $\sigma = 26.78$ $\mu$ m).
DEM Simulation Framework: A coarse-grained DEM model was developed using EDEM software to overcome the prohibitive computational cost of simulating micron-scale particles directly. Based on the Parcel concept, real particles were aggregated into statistically equivalent coarse grains with a scaling factor of $f_{cg}=100$ . The Hertz-Mindlin no-slip contact model, augmented with rolling friction, was utilized to simulate elastic and frictional interactions.
Simulation Protocol: A 5×5 parameter matrix was established, varying vibration frequency ( $f = 20–100$ Hz) and amplitude ( $A = 0.5–1.5$ mm). The simulation followed a three-stage timeline: initial free-fall settling, vertical sinusoidal vibration excitation, and static stabilization. Key metrics included global porosity, local coordination number (CN), and the probability density functions (PDF) of dimensionless normal contact forces.

Key Contributions and Results

Static Topological Characterization:
- The study established that polydisperse HGMs exhibit a "broad-peaked" coordination number distribution, contrasting with the "narrow-peaked" distribution of monodisperse systems.
- A quantitative linear scaling law was derived linking the local coordination number ( $CN$ ) to the particle diameter ( $d_g$ ): $CN = 0.886d_g + 2.390$ . This reveals that large particles act as high-coordination "supernodes" (local CN reaching 10–15), while small particles trapped in voids serve as low-coordination fillers (CN < 4).
- Static polydisperse packing reduced initial porosity to 42.86% (compared to 43.67% for monodisperse) but remained in a metastable state with significant pseudo-contacts.
Dynamic Densification Mechanisms:
- The research identified a three-stage densification process: "pseudo-contact breakage," "skeletal restructuring," and "steady compaction."
- Nonlinear Coupling: Porosity evolution is not monotonic with vibration intensity. Instead, it depends on the specific coupling of frequency and amplitude.
- Energy Sweet Spots: Two optimal parameter regimes were identified where compaction is maximized without inducing fluidization: 40 Hz / 1.25 mm and 60 Hz / 1.50 mm. Under these conditions, the global porosity reached a minimum of 37.3%.
- Fluidization Thresholds: Excessive vibration energy (e.g., high frequency at low amplitude or high amplitude at low frequency) triggered particle suspension, shear expansion, and fluidization, causing porosity to rebound to levels exceeding 43%.
Force Chain and Contact Network Reconstruction:
- The study demonstrated that densification is driven by the elimination of unstable "pseudo-contacts" (metastable arches) rather than a simple increase in contact count. The average coordination number initially drops sharply as weak contacts break, then stabilizes at a lower but more effective value.
- Analysis of force chain PDFs showed that optimal vibration redistributes loads, eliminating weak contacts and strengthening the remaining force chains. This creates a robust skeletal network capable of withstanding higher stresses, evidenced by an extension of the force distribution tail.

Significance and Claims
The authors claim that this work provides a fundamental framework for understanding the micro-mechanical topology of polydisperse granular media under dynamic excitation. By rejecting idealized packing assumptions and utilizing a coarse-grained DEM model calibrated with real SEM data, the study offers a more accurate representation of engineering realities.

The primary significance lies in the identification of specific "Energy Sweet Spots" that allow for the optimization of packing processes. These findings provide theoretical guidelines for:

Predicting the performance of thermal insulation layers in liquid hydrogen storage tanks under dynamic service conditions.
Optimizing pre-compaction filling procedures to mitigate secondary settlement caused by transportation vibrations.
Establishing quantitative scaling laws between particle size and coordination number, which are critical for modeling heat transfer pathways in granular assemblies.

The paper concludes that precise control of vibration parameters is essential to achieve deep compaction and stable force chain networks, directly addressing the engineering challenge of maintaining insulation efficiency in cryogenic storage systems.

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