[Paper Review] The structure of dust aggregates in hierarchical coagulation

C. Dominik, D. Paszun+1 more|arXiv (Cornell University)|Nov 1, 2016

Particle Dynamics in Fluid Flows3 citations

TL;DR

This paper investigates hierarchical dust coagulation in protoplanetary disks, showing that aggregates grow via a modified Particle-Cluster Aggregation (PCA)-like process where smaller projectiles compactly build a dense core, resulting in non-fractal, porous particles with filling factors dominated by the projectile's intrinsic porosity. The key finding is that in turbulent environments with flat mass distributions, hierarchical growth dominates, preventing the formation of ultra-low-density fractal aggregates and implying early structural compactification from void-filling by smaller grains.

ABSTRACT

Dust coagulation in interstellar space and protoplanetary disks is usually treated as one of 2 extreme cases: Particle-Cluster Aggregation and Cluster-Cluster Aggregation. In this paper we study the process of hierarchical growth, where aggregates are built from significantly smaller aggregates (but not monomers). We show that this process can be understood as a modified, PCA-like process that produces porous, but non-fractal particles whose filling factor is chiefly determined by the porosity of the building blocks. We also show that in a coagulation environment where relative velocities are driven by turbulence, a logarithmically flat mass distribution (equal mass per mass decade) as it is typically found in environments where fragmentation replenishes small grains, leads to a situation where small particles and aggregates dominate the growth of large ones. Therefore, in such environments, hierarchical growth should be seen as the norm. Consequently, we predict that the aggregates in such environments are not fractals with extremely low densities as they would result from extrapolation fractal laws to large sizes. The compactification of aggregates does not only result from collisions with enough energy to restructure aggregates - it starts already earlier by filling voids in particles with smaller grains that contribute to the growth.

Motivation & Objective

• To understand the structural evolution of dust aggregates formed through hierarchical coagulation, where large aggregates grow by accreting much smaller projectiles.
• To determine whether such growth leads to fractal-like or compact, non-fractal structures, especially under turbulent conditions.
• To assess the role of mass distribution—particularly logarithmically flat (equal mass per mass decade) distributions—in shaping the dominance of hierarchical growth over CCA or PCA.
• To investigate how early-stage void-filling by small projectiles contributes to aggregate compactification before high-energy restructuring occurs.
• To challenge the assumption that large dust aggregates in protoplanetary disks are fractal with extremely low densities, based on extrapolation of small-scale laws.

Proposed method

• Simulates hierarchical coagulation using a modified PCA model where a large target aggregate grows by successive collisions with smaller, uniformly sized projectiles.
• Models the growth process under turbulent relative velocities, assuming a logarithmically flat mass distribution (equal mass per mass decade) to reflect environments where small particles are replenished by fragmentation.
• Uses a nearest-neighbor search algorithm to determine collision partners and simulate the spatial arrangement of projectiles during accretion.
• Applies a causality condition from Ball & Witten (1984) to constrain the fractal dimension, showing that PCA-like growth leads to non-fractal, constant-density cores.
• Introduces a filling factor model (Eq. 11) that approximates the final core density based on the volume filling factor of the projectiles and the PCA filling factor (0.15).
• Analyzes the threshold energy for restructuring and bouncing, comparing silicate grains and ice-mantled grains to assess how surface energy affects growth limits.

Experimental results

Research questions

• RQ1 Does hierarchical coagulation—where large aggregates grow by accreting much smaller projectiles—produce fractal or non-fractal structures?
• RQ2 How does a logarithmically flat mass distribution (equal mass per mass decade) influence the dominance of hierarchical growth over CCA or PCA?
• RQ3 To what extent does void-filling by smaller projectiles contribute to aggregate compactification before high-energy restructuring occurs?
• RQ4 Can the filling factor of the final aggregate core be predicted from the intrinsic porosity of the projectiles, even when they are themselves porous CCA-like aggregates?
• RQ5 What is the role of surface energy (e.g., ice vs. silicate) in determining the threshold for restructuring during hierarchical growth?

Key findings

• Hierarchical coagulation produces non-fractal, compact cores with constant density, governed by the PCA-like accretion of small projectiles, contradicting the expectation of fractal growth at large sizes.
• The filling factor of the core is well-approximated by the product of the projectile's volume filling factor and the intrinsic PCA filling factor of 0.15, even when projectiles are porous CCA aggregates.
• In environments with a logarithmically flat mass distribution, small particles and aggregates dominate the growth of large ones, making hierarchical growth the norm rather than an exception.
• Aggregates grow several orders of magnitude in mass before restructuring occurs, with larger growth possible under weaker turbulence (e.g., α = 10⁻⁴), leading to very low filling factors.
• Compactification begins early—not only from high-energy collisions but also from the filling of voids by smaller projectiles during accretion, reducing porosity progressively.
• The threshold energy for restructuring is higher by about an order of magnitude for ice-mantled grains (γ_ice/γ_sil = 14.8), delaying structural changes during growth.