# Geometrical structure and thermal conductivity of dust aggregates formed via ballistic cluster-cluster aggregation

## TL;DR

This paper provides a theoretical analysis of the geometrical structure of dust aggregates formed via ballistic cluster-cluster aggregation and investigates how their thermal conductivity depends on their filling factor and structure.

## Contribution

It introduces new fractal parameters and formulas linking aggregate structure to thermal conductivity, advancing understanding of dust aggregate properties.

## Key findings

- Fractal dimension Df approximately 1.88 for BCCA clusters.
- Geodesic radius scales with N as N^{0.710}.
- Thermal conductivity scales with filling factor as φ^{(1+α)/(3−Df)}.

## Abstract

We herein report a theoretical study of the geometrical structure of porous dust aggregates formed via ballistic cluster-cluster aggregation (BCCA). We calculated the gyration radius $R_{\rm gyr}$ and the graph-based geodesic radius $R_{\rm geo}$ as a function of the number of constituent particles $N$. We found that $R_{\rm gyr} / r_{0} \sim N^{0.531 \pm 0.011}$ and $R_{\rm geo} / r_{0} \sim N^{0.710 \pm 0.013}$, where $r_{0}$ is the radius of constituent particles. Furthermore, we defined two constants that characterize the geometrical structure of fractal aggregates: $D_{\rm f}$ and $\alpha$. The definition of $D_{\rm f}$ and $\alpha$ are $N \sim {( R_{\rm gyr} / r_{0} )}^{D_{\rm f}}$ and ${R_{\rm geo}} / {r_{0}} \sim {\left( {R_{\rm gyr}} / {r_{0}} \right)}^{\alpha}$, respectively. Our study revealed that $D_{\rm f} \simeq 1.88$ and $\alpha \simeq 1.34$ for the clusters of the BCCA. In addition, we also studied the filling factor dependence of thermal conductivity of statically compressed fractal aggregates. From this study, we reveal that the thermal conductivity of statically compressed aggregates $k$ is given by $k \sim 2 k_{\rm mat} {( r_{\rm c} / r_{0} )} \phi^{(1 + \alpha) / (3 - D_{\rm f})}$, where $k_{\rm mat}$ is the material thermal conductivity, $r_{\rm c}$ is the contact radius of constituent particles, and $\phi$ is the filling factor of dust aggregates.