TPDSci Effective optical particle shape Sphere

Effective optical particle shape: A sphere

The spherical particle shape has been widely used to model light scattering by nonspherical particles since the emergence of the Mie theory. This trend has intensified in the early 1960's after the advent of computers enabled fast calculation of light scattering properties of homogeneous spheres.

It has frequently been argued that random orientation "averages" effects of the non-spherical particle shape on light scattering and thus justifies the use of a spherical model for randomly-oriented particles. Unfortunately, this argument is rebutted by both computational studies, where the properties of the particles and the dispersion can be easily controlled (for example, Mishchenko MI 1993) as well as by measurements (for example, Muñoz O et al 2004, Volten H et al 1998, Holland AC and Gange 1970).

The spherical model has been found especially problematic in simulating the mid- and large-angle phase function of non-spherical particles, as well as the other elements of the scattering matrix within the entire scattering angle. An interesting exception is the phase function in the vicinity of the scattering angle of 150° (for example, Kokhanovsky AA 1996). Indeed, several computational studies for dispersions of size-distributed nonspherical particles (for example, Mishchenko MI et al 1996c - their Fig. 1, 1996a - their Figs. 10, 11, Mishchenko MI and Travis 1994b - their Fig. 8) suggest that the phase function is little dependent on the particle shape at about that angle. Remarkably, the existence of a crossover point for dispersions of size-distributed non-absorbing particles greater than the wavelength of light is implied by the fact that while (1) the phase function of nonspheres is relatively flat for the scattering angle, θ, in a range of ~90°-180°, the phase function of a dispersion of projection-equivalent spheres dips below the non-spherical phase function at mid scattering angles, say θ ≈ 70°-140°, and climbs above it when the scattering angle approaches 180° (for example, Mishchenko MI et al 1996a, their Fig. 11).

Only in the small-angle region of the scattering angle does the homogeneous sphere model provide a relatively good approximation for the scattering function of non-spherical particles. In particular, Kahnert (2004) has recently shown that the spherical shape model reproduces well the integral optical properties (attenuation coefficient and scattering coefficient) which heavily depend on the small-angle angular pattern, and - as expected - the forward peak of the scattering function of fine dust aerosol [a log-normal PSD with an average radius, r = 0.7 and SD(log r) = 1.9, and refractive index, m, of 1.65 - 0.005i at a wavelength of 550 nm]. Other elements of the scattering matrix were poorly simulated with the spherical model. This is consistent with the results of extensive modeling of the phase function of power-law size-distributed spheroids and cylinders (Mishchenko MI et al 1996a: m = 1.53 - 0.008i, Mishchenko MI et al 1995a) which show that the phase function of these non-spherical particles is essentially similar to that of equal-surface spheres at scattering angles of less than ~20° for particle sizes greater than the wavelength of light. Note that some computational studies on even "moderately" non-spherical particles suggest errors on the order of 10% even in the small-angle region (for example, Wiscombe WJ and Mugnai 1988).

Kahnert and Nousiainen (2006) also approximated the scattering functions of irregular dust particles from a database of experimental scattering matrix elements by scattering functions of spheres. They found that the average cosine, g (i.e. the asymmetry factor of the scattering function) of irregular particles with a refractive index with a real part on the order of 1.5 to 1.7 and an imaginary part between 10^-5 and 10^-3 (mineral aerosols in the visible) can be mis-estimated by ~10% in the case of size distributions with an effective size about 1 µm to as much as about -50% for the effecive size about 10 µm. The approximation error: g_exp - g_sph changed from positive for the smallest particles examined (~ 1 µm) to primarily negative as the effective particle size increased, where g_exp is the experimental value and g_sph is the approximated value, assuming the spherical particle shape. Note that the experimental value, g_exp, involved an extrapolation of the experimental scattering function into a small-angle range (0° to 5°), which itself introduces an uncertainty in that value.

Grenfell and Warren (1999) note that the deficiency of a sphere in representing optical properties of a nonspherical particle, such as a hexagonal ice crystal, is rooted in the fact that the sphere of equal volume has insufficient cross section area to correctly represent light scattering, while the sphere of equal surface has too large a volume to correctly represent light absorption by the nonphserical particle. In a radical departure from a representation a single nonspherical particle by a single equivalent sphere, they proposed to substitute a single nonphserical particle by a collection of independent spheres with the volume-to-area ratio equal to that of the particle. With this approach they approximated reasonably well some integral optical properties (attenuation efficiency and the average cosine of the scattering angle, i.e. asymmetry factor) of a hexagonal ice column in a wide particle size range, although the phase function was not so well approximated. Neshyba SP et al (2003) extended this particle shape model to hexagonal plates, while Grenfell TC et al (2005) used it to model some integral optical properties of hollow hexagonal columns.

CITATION:
Jonasz M. 2006. Effective optical particle shape: A sphere (www.tpdsci.com/tpc/EfOptPtShpSph.php). In: Top. Part. Disp. Sci. (www.tpdsci.com).

HISTORY:
Published: 05-Apr-2006
Modified: 08-Apr-2008
Reviewed: 06-Apr-2006

Topics in Particle and Dispersion Science