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Preparation history can outweigh descriptor definition in frictionless bidisperse packings below jamming

In frictionless bidisperse packings below jamming, the assembly route (preparation history) exerts a stronger influence on fine-network connectivity descriptors than the definition of the contact gap threshold, challenging the assumption that morphology is solely a function of state variables.

Original authors: Quoc-Bao Nguyen, Hoang-Quan Nguyen, Thai-Son Vu

Published 2026-10-28

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CC BY 4.0

Original authors: Quoc-Bao Nguyen, Hoang-Quan Nguyen, Thai-Son Vu

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 pile of sand. To a geologist or an engineer, the behavior of that pile—how it holds a road, how it flows through a hopper, or how it supports a building—depends on how the grains are arranged. For decades, scientists have operated on a comforting assumption: if you know the size of the grains, how many of them there are, and how tightly they are packed, you know the structure of the pile. It was believed that these numbers alone dictated the shape of the microscopic web of contacts between the grains. If two piles had the same ingredients and the same density, they were assumed to be structurally identical, regardless of how they got there. This idea underpins much of our understanding of soil mechanics and the design of composite materials. But what if the history of the pile matters more than its current ingredients? What if the way the sand was poured, shaken, or settled leaves a permanent fingerprint that changes how the material behaves, even when the numbers look exactly the same?

This question sits at the heart of a new study by researchers at Hanoi University of Civil Engineering and the University of Transport and Communications. They set out to test whether the "recipe" used to build a granular packing is more important than the standard measurements used to describe it. To do this, they turned to a computer simulation, creating a virtual world filled with millions of smooth, frictionless spheres of two different sizes. In the real world, creating a perfect, repeatable pile of sand is nearly impossible; in the computer, the researchers could control every variable with absolute precision. They built two types of piles that were identical in every measurable way: the same ratio of large to small balls, the same number of each, and the same overall density. The only difference was the method used to assemble them. One group of piles was built by growing the spheres from tiny seeds until they filled the space, a process that leaves many particles touching or nearly touching. The other group started with the first type of pile and was then subjected to a virtual "shaking" and relaxing process, designed to jostle the particles into a more settled state without changing the overall density.

The researchers then examined a specific feature of these piles: the connectivity of the smaller spheres. They asked a simple question: at what point do the small balls form a continuous, system-spanning network? In the world of granular materials, this network is crucial because it often determines how forces travel through the material or how fluids might flow through it. To find this "tipping point," they defined a rule: two small balls are considered connected if the gap between their surfaces is smaller than a tiny fraction of their diameter. They then varied this gap rule, making it stricter or more lenient, to see how much the tipping point moved. They expected that changing this rule would shift the result, as the definition of "connected" is somewhat arbitrary.

However, the results revealed a startling reality. When the researchers compared the two piles that were built differently but were otherwise identical, the difference in their connectivity was massive. The pile built by the "growth" method formed a connected network of small balls at a much lower concentration of those balls than the pile that had been shaken and settled. In fact, the shift caused by simply changing the assembly method was far larger than the shift caused by changing the definition of a connection by a factor of ten. At one specific density, changing the assembly route moved the tipping point by a significant margin, while making the connection rule ten times more generous only moved it by a fraction of that amount. This means that the history of how the material was prepared can outweigh the very definition used to measure its structure.

The researchers dug deeper to understand where this difference lived. They looked at the tiny gaps between the particles. They found that the "growth" piles were full of pairs of small balls that were almost touching, sitting in a narrow window of space just a few percent of a ball's diameter apart. The shaking process effectively drained these near-touching pairs away, leaving the shaken piles with far fewer of these critical contacts. This difference was so localized and so pronounced that it completely altered the behavior of the network. The study showed that this effect was not a fluke of a small sample size; it held true even when the number of particles was increased significantly. Furthermore, the researchers developed a way to measure this "history" as a continuous number, rather than just a label. They found that this single number could predict the connectivity of the pile with high accuracy, but only if the prediction was made for piles with the same overall density. If the density changed, the relationship broke down, proving that the history and the density are intertwined in a complex way that cannot be ignored.

The implications of this finding extend beyond a virtual pile of balls. In the real world, materials like soil, concrete, or pharmaceutical powders are often described by their composition and density alone. This study suggests that such descriptions are incomplete. If two samples of soil have the same grain sizes and the same density, but one was deposited by a river and the other was compacted by a machine, they may behave very differently. The study demonstrated that this difference in preparation leads to a measurable difference in mechanical stiffness; the "grown" virtual material was nearly seven percent stiffer than its "shaken" counterpart, despite having the exact same ingredients. This challenges the standard models used by engineers to predict how materials will respond to stress. It suggests that to truly understand a granular material, one must account for how it was made, not just what it is made of. The history of the pile is not just a footnote; it is a fundamental part of the material's identity, capable of shifting its properties more than the arbitrary rules we use to measure it.

Technical Summary: Preparation History in Frictionless Bidisperse Packings

Problem Statement
Structural and constitutive descriptions of granular packings typically rely on the assumption that fixing macroscopic state variables (size ratio, fine fraction, and crowding) uniquely determines the packing morphology and any descriptor derived from it. While this assumption is well-established for jammed states where preparation history is known to matter, its validity for unjammed, frictionless bidisperse packings far below the jamming point remains untested. Specifically, it is unclear whether the assembly route (preparation protocol) alters contact-based descriptors more significantly than the arbitrary choice of the descriptor's own parameters (e.g., the gap threshold used to define connectivity). This study investigates whether "preparation history can outweigh descriptor definition" in the context of fine-network connectivity in sub-jamming packings.

Methodology
The authors employed numerical simulations of frictionless bidisperse sphere packings in a periodic cubic cell. The study controlled all state variables—size ratio ( $\lambda$ ), fine volume fraction ( $\xi$ ), solid volume fraction ( $\phi$ ), and crowding ( $s = \phi/\phi_m$ )—while varying only the assembly protocol. Two distinct preparation families were compared:

Growth-Prepared (G): Packings generated by seeding spheres uniformly at random and growing them to nominal size, followed by a collective rearrangement scheme to remove overlaps.
Shake-Equilibrated (S): Packings derived from the G states by applying "shake–relax" cycles. In these cycles, particle positions receive Gaussian perturbations (simulating agitation) followed by relaxation, strictly conserving the solid volume fraction. This process drains near-contact pairs left by the growth protocol without compacting the system.

The primary descriptor analyzed was fine-network connectivity, defined via continuum percolation: two fine spheres are linked if their surface gap falls below a threshold $t$ (a fraction of the fine diameter). The connectivity metric was the largest-cluster fraction ( $P_\infty^f$ ). The study systematically compared:

Threshold Sensitivity ( $S_t$ ): The shift in the percolation onset ( $\xi_g$ ) caused by varying the gap threshold $t$ over a tenfold range (2% to 20%).
Preparation Sensitivity ( $S_P$ ): The shift in $\xi_g$ between the G and S families at a fixed reference threshold ( $t=5\%$ ).
History Coordinate ( $\chi_h$ ): A continuous variable normalizing the contact fraction of near-pairs (gaps $< 2\%$ of fine diameter) between the G and S references, used to predict connectivity within a trajectory.

Experiments were conducted at size ratios $\lambda = 4, 6, 8$ and crowding levels $s = 0.60, 0.65, 0.70$ .

Key Results

Preparation Dominates Descriptor Definition (Growth-Prepared): At all tested size ratios, the shift in the percolation onset caused by changing the assembly route ( $S_P$ ) exceeded the shift caused by a tenfold change in the gap threshold ( $S_t$ ) when the threshold sensitivity was measured on the growth-prepared family.
- At $\lambda = 4$ and $s=0.70$ , the preparation shift was $\Delta \xi_g = 0.334$ , whereas a tenfold threshold change shifted the growth-prepared onset by only $0.086$.
- The ratio of preparation sensitivity to threshold sensitivity ( $R = S_P/S_t$ ) was approximately 8.9 per e-fold of the threshold at $\lambda = 4$ (using the growth-prepared denominator).
- At $\lambda = 6$ and $\lambda = 8$ , $R$ remained significantly greater than 1 (lower bounds of 3.3 and 9, respectively), indicating preparation history is the dominant factor relative to the growth-prepared threshold sensitivity.
- Nuance: However, when comparing the preparation shift to the threshold sensitivity of the shake-equilibrated family specifically over the 5–10% threshold band at $\lambda = 4$ , the ratio $R$ fell below one ( $R = 0.83$ per e-fold). This indicates that the threshold sensitivity itself depends on the preparation route, and both sensitivities must be checked in every preparation family.
Localization of Structural Differences: The structural differences between G and S packings are highly localized. The largest discrepancy lies in the population of pairs with gaps smaller than 2% of the fine diameter. The growth protocol leaves a significant fraction of these pairs in contact (or near-contact), while the shake-relax process drains this population. Beyond this narrow window (gaps $> 2\%$ ), the pair distributions of the two families converge.
History Coordinate as a Predictor: The categorical protocol label (G vs. S) can be replaced by the continuous history coordinate $\chi_h$ . Within a fixed condition (fixed $\lambda, \xi, s$ ), $\chi_h$ strongly predicts the largest-cluster fraction ( $P_\infty^f$ ), reducing out-of-sample prediction error by 78% compared to a state-only baseline. However, this relationship is not universal across different crowding levels; a model fitted at $s=0.70$ fails to predict behavior at $s=0.60$ .
Mechanical Consequence: For a matched condition ( $\lambda=4, \xi=0.35, s=0.70$ ), the growth-prepared state exhibited a shear stiffness approximately 6.83% higher than its shake-equilibrated counterpart, despite identical state variables and particle counts. This confirms that the history-dependent morphology directly influences effective mechanical properties.

Significance and Claims
The paper claims that for contact-based descriptors in unjammed granular packings, the assembly route is a more significant determinant of the descriptor's value than the arbitrary choice of the descriptor's internal parameters (gap threshold), provided the comparison is made against the growth-prepared threshold sensitivity.

Challenge to State-Variable Models: The authors argue that morphology-informed effective-property closures (e.g., relating stiffness to state variables and a descriptor $M$ ) cannot be uniquely specified by composition and crowding alone. The descriptor $M$ itself depends on history, necessitating a fourth argument in constitutive models: a history coordinate (analogous to the fabric tensor in soil mechanics).
Methodological Test: The study proposes a direct comparison of onset shifts (preparation vs. threshold) as a necessary test for any contact descriptor before it is incorporated into state-variable models. Crucially, because the threshold sensitivity varies by preparation route (as seen in the shake-equilibrated family at $\lambda=4$ ), both sensitivities must be checked in every preparation family.
Scope: The findings are specific to frictionless, quiescent packings far below jamming. The authors note that while near-contact pair correlations are known to be universal at the jamming limit, the amplitude differences observed here (which drive the connectivity shift) occur at crowding levels ( $s \le 0.70$ ) where such asymptotic universality is not expected to hold.

The work concludes that both sensitivities (preparation and threshold) must be checked in every preparation family before a contact descriptor is used to define the state of a granular system.

Technical Summary: Preparation History in Frictionless Bidisperse Packings

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