# Computational assessment of an effective-sphere model for characterizing colloidal fractal aggregates with holographic microscopy

## TL;DR

This study evaluates the accuracy of an effective-sphere model combined with holographic microscopy for characterizing colloidal fractal aggregates, demonstrating its effectiveness under certain conditions.

## Contribution

It provides a simulation-based validation of the effective-sphere model's ability to measure fractal dimensions of colloidal aggregates using holography.

## Key findings

- Effective-sphere model accurately estimates fractal dimension within 10.5% when multiple scattering is negligible.
- Sensitivity of effective parameters decreases with increasing fractal dimension.
- Maxwell Garnett theory effectively relates effective refractive index and radius for fractal aggregates.

## Abstract

We perform simulations to evaluate a recent experimental technique for using in-line holographic microscopy and an effective-sphere model to measure the population-averaged fractal dimension $D_f$ of an ensemble of colloidal fractal aggregates. In this technique, models based on Lorenz-Mie scattering by a uniform sphere are fit to digital holograms of a population of fractal aggregates to determine the effective refractive indices $n_{eff}$ and effective radii $a_{eff}$ of the aggregates. A scaling relationship between $n_{eff}$ and $a_{eff}$ based on the Maxwell Garnett effective-medium theory then determines $D_f$. Here we use a multisphere superposition code to calculate the exact holograms produced by aggregates with tunable fractal dimensions $D_f$. We show that $n_{eff}$ and $a_{eff}$ become less sensitive to the aggregate orientation as $D_f$ increases. We also show that the Maxwell Garnett scaling relationship correctly determines $D_f$ to within 10.5\% when multiple scattering is negligible and the population-averaged coefficient of determination $\langle R^2\rangle_p > 0.6$, indicating that the holograms are well-described by the effective-sphere model.