🔬 condensed matter

Jamming states in random sequential adsorption of diffusion-limited aggregates

This study reveals that the irreversible adsorption of diffusion-limited aggregation clusters on a square lattice exhibits jamming densities that decrease as a power law with cluster size, where increased shape diversity promotes denser packing and distinct scaling behaviors in density fluctuations depending on whether the cluster shape pool is fixed or refreshed across realizations.

Original authors: Fahad Puthalath, Dipanjan Mandal, Sumanta Kundu

Published 2026-08-26

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CC BY 4.0

Original authors: Fahad Puthalath, Dipanjan Mandal, Sumanta Kundu

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

Nature is full of patterns that look like they were drawn by a frantic, branching hand. From the jagged veins of a lightning strike to the intricate, feathery edges of a snowflake, these shapes appear in everything from mineral deposits in rocks to the way bacteria spread across a petri dish. Scientists call these ramified, or branching, structures fractals, and they are a common sight in both the natural world and in engineered materials like thin films used in technology. A fundamental question in physics is how these irregular, tree-like objects pack together when they are dropped onto a surface one by one. Unlike smooth spheres or perfect squares, which fit together in predictable ways, these branching clusters tangle and interlock, creating a messy, crowded layer that eventually stops growing. This final, stuck state is known as a jamming state, and understanding how tightly these shapes can pack is crucial for designing everything from better batteries to more efficient filters.

To explore this, a team of researchers set out to simulate the process of dropping these branching clusters onto a flat, square grid. They used a method called random sequential adsorption, which mimics a game of chance where objects are placed one at a time in random spots. If a new object fits without overlapping anything already there, it stays. If it hits an existing object, it is rejected, and the process tries again with a new spot. This continues until no more objects can fit, leaving the surface in a jammed, packed state. The researchers were particularly interested in how the shape of the object and the variety of shapes used affected the final density of the packed layer. They focused on clusters that grow in a specific way known as diffusion-limited aggregation, a process where particles wander randomly until they stick to a growing cluster, creating the characteristic branching, fractal shapes seen in nature.

The team ran massive computer simulations to see what happened when they dropped these clusters onto a grid. They tested two main scenarios. In the first, they used a single, specific branching shape for the entire experiment, dropping that exact same shape over and over again. In the second, they generated a brand new, unique branching shape for every single attempt, ensuring that the surface was covered with a constantly changing variety of forms. They also tested a middle ground where they used a fixed collection of different shapes, varying the size of that collection to see if having more variety changed the outcome. The simulations covered a vast range of cluster sizes, from tiny groups of just two particles up to massive clusters containing over four thousand particles.

The results revealed a clear pattern: as the clusters grew larger, the final packed density of the layer decreased. The researchers found that this drop in density followed a predictable mathematical trend, where larger clusters left more empty space between them. However, the specific rate at which this happened depended on how much variety was in the mix. When the researchers used a single, fixed shape, the packing was less dense than when they introduced a variety of shapes. In fact, increasing the diversity of shapes allowed the clusters to pack more tightly together. This suggests that having a mix of different branching forms helps the objects find better ways to nestle into the gaps left by their neighbors, much like how a pile of mixed-sized rocks might settle more tightly than a pile of identical-sized stones.

Another surprising discovery concerned the stability of the results. When the researchers used a single, fixed shape, the final density varied slightly from one simulation run to another, but these variations became smaller and smaller as the grid size increased, following a standard rule of physics. However, when they used a constantly changing variety of shapes, the variations in density did not shrink as the grid got larger; they remained constant. This indicates that the randomness introduced by constantly changing the shape of the objects creates a different kind of uncertainty that does not disappear even in very large systems.

Despite these differences, the researchers found that as the clusters became extremely large, the distinction between using a single shape and using a variety of shapes began to fade. Because these branching clusters are statistically self-similar—meaning their overall pattern looks roughly the same whether you look at a small part or the whole thing—the specific details of any single shape matter less as the object grows. In the limit of infinitely large clusters, the results from using a single shape and using a variety of shapes are expected to become identical. The study also confirmed that introducing a pool of many different shapes consistently led to denser packing than using just one, with the effect becoming more pronounced as the number of available shapes increased.

These findings offer a clearer picture of how irregular, fractal-like objects behave when they crowd together. While the specific shape of a cluster matters for smaller sizes, the underlying statistical nature of these branching forms eventually dominates the outcome. The work highlights that diversity in shape promotes tighter packing, a principle that could be useful for engineers designing materials where space efficiency is key. By understanding how these complex, natural-looking shapes interact, scientists can better predict how they will settle and pack in real-world applications, from the formation of mineral deposits to the fabrication of advanced nanomaterials. The study concludes that while the path to a jammed state is influenced by the variety of shapes available, the fundamental geometry of these fractal clusters ensures that, given enough size, they will all behave in a remarkably similar way.

Technical Summary: Jamming states in random sequential adsorption of diffusion-limited aggregates

Problem Statement
While the jamming properties of regular geometric objects (e.g., disks, squares, ellipses) in Random Sequential Adsorption (RSA) are well-characterized, the behavior of irregular, fractal-like objects remains largely unexplored. Naturally occurring and engineered deposits, such as mineral formations in fractured rocks, sedimentary transport, and thin-film growth, often exhibit ramified, diffusion-limited aggregation (DLA) morphologies. A fundamental gap exists in understanding how the fractal nature and shape diversity of these DLA clusters influence jamming density, fluctuations, and scaling behavior. Previous studies on self-avoiding walk (SAW) chains were limited to short chain lengths where fractal characteristics were not fully developed. This paper addresses the RSA of large DLA clusters ( $2 \le k \le 4096$ ) on a square lattice to determine how cluster size, shape diversity, and polydispersity affect the jamming state.

Methodology
The authors employ large-scale Monte Carlo simulations on an $L \times L$ square lattice with periodic boundary conditions. The study utilizes the standard RSA protocol where DLA clusters are deposited sequentially at random positions; deposition is accepted only if the cluster does not overlap with previously adsorbed objects. The process terminates when no further clusters can be accommodated, defining the "jamming state."

The study investigates four distinct model variants to control shape diversity:

Monodisperse Fixed-Shape: A single, specific DLA cluster shape is used for all deposition attempts within a run and across all realizations.
Monodisperse Refreshing-Shape: A new, independently generated DLA cluster shape is created for each realization (run) and used throughout that specific run.
Polydisperse Fixed-Pool: A pool of $S$ pre-generated DLA shapes is used across runs.
Polydisperse Refreshing-Pool: A new pool of $S$ shapes is generated for each run.

DLA clusters are generated using the Witten-Sander algorithm with optimized parameters to ensure computational efficiency and verify a fractal dimension $D_f \approx 1.67$ . The simulations utilize a rejection-free algorithm for site selection to track the jamming density $p_j = Nk/L^2$ , where $N$ is the number of adsorbed clusters. Simulations were performed on high-performance computing systems with system sizes up to $L=2048$ and cluster sizes up to $k=4096$ .

Key Results

Jamming Density Scaling: For both monodisperse variants, the average jamming density $p_j(k)$ decreases monotonically with increasing cluster size $k$ . The decay follows a power-law behavior:
$p_j(k) - p_j^\infty \sim k^{-\alpha}$
For the monodisperse refreshing-shape variant, the exponent is found to be $\alpha = 0.374(2)$ with an asymptotic density $p_j^\infty = 0.051(2)$ .
Fluctuation Scaling: The behavior of jamming density fluctuations, $\sigma(L)$ , differs significantly between model variants:
- Fixed-Shape: Fluctuations scale as $\sigma(L) \sim 1/L$ , consistent with universal scaling laws for RSA in 2D.
- Refreshing-Shape: Fluctuations are independent of system size $L$ . The authors attribute this to the ensemble averaging over distinct fixed-shapes; the variance is dominated by the differences in mean jamming densities between different shapes rather than finite-size effects.
- Convergence: As $k \to \infty$ , the difference in variance between the fixed-shape and refreshing-shape variants vanishes as a power law ( $\Delta \sigma^2 \sim k^{-0.871}$ ). The authors argue this convergence arises from the statistical self-similarity of DLA clusters at large scales, causing different shapes to become statistically indistinguishable in terms of jamming properties.
Effect of Polydispersity: Introducing a pool of $S$ distinct shapes (polydispersity) promotes denser packing. The jamming density increases systematically with the pool size $S$ . The power-law exponent $\alpha$ also varies with $S$ , increasing from $0.374(2)$ (monodisperse, $S=1$ ) to $0.417(1)$ for $S=10^6$ .
- For small $k$ , a threshold pool size $S_c(k)$ exists beyond which $p_j$ saturates.
- For large $k$ , $p_j$ varies only weakly with $S$ due to the self-similar nature of the clusters.
- The study notes that even with infinite pool size, the RSA of DLA clusters does not exactly reduce to the RSA of SAW chains due to non-uniform occurrence probabilities of specific shapes in the DLA process (unlike the uniform probability in SAW).

Significance and Claims
The paper claims to provide the first systematic investigation of RSA for large fractal-like DLA clusters, extending the range of cluster sizes far beyond previous studies on SAW chains. The primary contributions are:

Establishing that jamming density for DLA clusters follows a power-law decay with size, with parameters dependent on shape diversity.
Demonstrating that the scaling of jamming density fluctuations is fundamentally altered by the method of shape selection (fixed vs. refreshed), with the latter exhibiting $L$ -independent fluctuations due to shape-to-shape variability.
Showing that shape polydispersity leads to more efficient packing (higher jamming density) and systematically alters the scaling exponent $\alpha$ .
Proposing that the differences between model variants vanish in the thermodynamic limit ( $k \to \infty$ ) due to the statistical self-similarity of DLA clusters.

The authors suggest these findings offer insights into the design of nanostructured materials with tunable properties via fractal cluster deposition. They identify future work in studying transport properties through the disordered media generated by these jamming states but do not propose specific experimental protocols or applications beyond this general statement.

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