Scattering Regime Analysis

Independent vs. Dependent Scattering Criteria

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.

Physical Parameters and Calculations

To determine the scattering regime, the simulator calculates several geometric and electromagnetic parameters. For polydisperse systems (where nRadius > 1), MieSimulatorGUI calculates an effective radius based on the volume-weighted average of the particle population.

1. Volume Fraction ($f_v$)

The volume fraction represents the ratio of the volume occupied by particles to $1 mm^3$ volume of the medium.

• Monodisperse:

$$f_v = \frac{V_{sphere} \cdot N}{10^9}$$

• Polydisperse:

$$f_v = \frac{\sum_{i=1}^{n} V_i \cdot N_i}{10^9}$$

2. Total Number Density ($N_{total}$)

• Monodisperse:

$$N_{total} = N$$

• Polydisperse:

$$N_{total} = \sum_{i=1}^{n} N_i$$

3. Effective Radius ($r_{eff}$)

• Monodisperse:

$$r_{eff} = r_{sphere}$$

• Polydisperse:

  First, compute average volume per particle ($V_{avg}$) and then derive $r_{eff}$.

$$V_{avg} = \frac{\sum_{i=1}^{n} V_i \cdot N_i}{N_{total}}$$ $$r_{eff} = \sqrt[3]{\frac{3 \cdot V_{avg}}{4\pi}}$$

4. Size Parameter ($\chi$)

Size Parameter ($\chi$) is then calculated using this effective radius and the wavelength ($\lambda_{med} = \lambda /n_{med} $). If a wavelength range is provided, use the longest wavelength.

$$\chi = \frac{2\pi \cdot r_{eff}}{\lambda_{med}}$$

5. Inter-particle Distance ($L$)

This $L$ represents the average distance between the centers of two adjacent particles. Imagine each particle sitting in the center of its own cube, where $L$ is the side length of that cube. It is derived by taking the cube root of the reciprocal of $N_{total}$. The factor of $1000$ serves as a scale conversion to ensure $L$ is in $\mu m$.

$$L = \sqrt[3]{\frac{1}{N_{total}}} \times 1000 $$

6. Clearance-to-Wavelength Ratio ($c/\lambda$)

The clearance ($c$) is the physical gap between the surfaces of two spheres. Even if $L$ is large, a large particle radius can result in a very small clearance, leading to near-field interactions.

$$c = L - (2 \cdot r_{eff})$$

The Clearance-to-Wavelength ratio is

$$\frac{c}{\lambda} = \frac{L - (2 \cdot r_{eff})}{\lambda_{med}}$$

Regime Classification Logic

The code evaluates the scattering regime based on the volume fraction and geometric thresholds:

High Concentration Regime ($f_v > 0.1$)

Near-field interactions and coherent scattering effects dominate regardless of the size parameter (Tien and Drolen, 1987). In concentrated suspensions reaching $f_v$ values of $0.2$ to $0.3$, dependent effects are the defining characteristic (Yalcin et al., 2022).

Transitional Regime ($0.006 < f_v \leq 0.1$)

Dependent effects may occur based on the ratio of interparticle distance and wavelength.

• Per Galy et al. (2020), the threshold for independent behavior is:

  • If $\chi \leq 2$: Requires $c/\lambda > 2$ for independent scattering.
  • If $\chi > 2$: Requires $c/\lambda > 5$ for independent scattering.

• Per Tien and Drolen (1987), the threshold for independent behavior is:

  • $c/\lambda > 0.5$ for independent scattering.

Low Concentration Regime ($f_v \leq 0.006$)

Interactions between particles are generally negligible (Tien and Drolen, 1987). However, if the Size Parameter $\chi > 0.388$, the tool still verifies the Clearance-to-Wavelength ratio to ensure far-field interference is not present.

User Alerts

Whenever input parameters deviate from the criteria for independent scattering, MieSimulatorGUI displays a warning message similar to the one shown below:

References

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

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

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

Last edited: Feb 6, 2026