Independent Scattering Approximation

The independent scattering approximation assumes that the total scattering from a group of particles is simply the linear sum of the scattering from each individual particle. This assumption fails in densely packed media, where near-field interactions cause "dependent" scattering effects as particles influence one another’s electromagnetic fields.

Based on established literature, the transition between regimes is determined by the volume fraction ($f_v$)

• $f_v \le 0.006$: Independent Scattering – Interactions between particles are negligible (Tien and Drolen (1987)). However, for particles with a size parameter ($\chi = \pi D/\lambda$) $\chi > 0.388$, the tool continues to verify the clearance ($c$) to wavelength ($\lambda$) ratio to ensure farfield interference is not present.
• $0.006 < f_v \le 0.1$: Transitional Regime – Dependent effects may occur depending on the ratio of interparticle distance and wavelength. According to Galy et al. (2020), the threshold for independent behavior is defined by the following geometric criteria:
  • For $\chi \leq 2$, independent scattering requires $c/\lambda > 2$.
  • For $\chi > 2$, independent scattering requires $c/\lambda > 5$.
• $f_v > 0.1$: Dependent Scattering Near-field interactions and coherent scattering effects dominate (Tien and Drolen, 1987). Furthermore, Yalcin et al. (2022) reinforces that in thick and concentrated suspensions reaching $f_v$ values of $0.2$ to $0.3$, dependent effects are the defining characteristic of the medium.

MieSimulatorGUI provides the following warning message whenever the input parameters enter the Transitional or Dependent regimes based on the criteria above.

High Volume Fraction Alert: The current volume fraction (or concentration) exceeds the recommended range for independent scattering. Since MieSimulatorGUI is best suited for dilute mixtures, results at this concentration should be interpreted with caution.

References:

Tien, C. L., & Drolen, B. L. (1987). Thermal radiation in particulate media with dependent and independent scattering. Annual Review of Numerical Fluid Mechanics and Heat Transfer, 1, 1–32. https://doi.org/10.1615/AnnualRevHeatTransfer.v1.30

Galy, T., Huang, D., & Pilon, L. (2020). Revisiting independent versus dependent scattering regimes in suspensions or aggregates of spherical particles. Journal of Quantitative Spectroscopy and Radiative Transfer, 246, 106924. https://doi.org/10.1016/j.jqsrt.2020.106924

Yalcin, R. A., Lee, T., Kashanchi, G. N., Markkanen, J., Martinez, R., Tolbert, S. H., & Pilon, L. (2022). Dependent scattering in thick and concentrated colloidal suspensions. ACS Photonics, 9(10), 3318–3332. https://doi.org/10.1021/acsphotonics.2c00664