Identification and particle sizing of submicron mineral dust by using complex forward-scattering amplitude data

Abstract

An accurate observational dataset of the size-resolved concentration of mineral dust particles in the atmosphere, hydrosphere, and cryosphere is needed for investigating the effect of mineral dust on the climate with earth system models. However, automated measurements of dust particles remain challenging due to the complexities of the physicochemical properties (e.g., shape and mineralogy) of individual dust particles and the difficulties in discriminating dust from other particulate components (e.g., black carbon). Here, we suggest the use of complex forward-scattering amplitude data obtained by the single particle extinction and scattering (SPES) method as a low-cost optical approach for identification and quantification of silicate (aluminosilicates + quartz) particles, the major particulate component of desert dust. We focus here on the submicron particle-size range to challenge the robust identification of wavelength or smaller scale dust solely according to the principle of elastic light scattering. The two-dimensional nature of the complex scattering amplitude data allows us to identify silicate-dominant particles in waterborne dust samples and discriminate them from light-absorbing particulate components (e.g., hematite). We demonstrate that the two-dimensional dataspace of the complex amplitude allows an accurate retrieval of the particle-size distribution of silicate particles thanks to the simultaneous retrieval of the particle’s effective refractive index. We discuss some notable differences in our results from those retrieved from conventional elastic light scattering approaches.

Copyright © 2022 American Association for Aerosol Research

EDITOR:

• Hans Moosmüller

1. Introduction

Mineral dust aerosol emitted from desert and dried soil regions plays a crucial role in the climate system. Dust aerosol affects both shortwave and longwave radiation in the atmosphere (Liao and Seinfeld 1998; Sokolik and Toon 1996). Dust contributes a substantial fraction of global aerosol optical depth (Moosmüller et al. 2012; Satheesh and Moorthy 2005). Radiation perturbation by mineral dust can alter the atmospheric circulation and precipitation (Bangalath and Stenchikov 2015). Mineral dust acts as both cloud condensation nuclei and ice nucleating particles in the atmosphere, affecting the radiative properties and lifetime of clouds (Hoose and Möhler 2012; Murray et al. 2011). The structural iron embedded within the aluminosilicate minerals of dust aerosol is considered to act as a fertilizer for oceanic phytoplankton in high-nutrient, low-chlorophyll regions (Jickells et al. 2005; Journet et al. 2008). The climatically relevant particle size of dust aerosol ranges from ∼0.2 to ∼20 μm (Kok et al. 2017). Accurate observational constraint of the dust particle-size distribution within this size range is needed for evaluating the dust radiative forcing, which is a superposition of the negative forcing of fine dust and potentially positive forcing of coarse dust (Miller et al. 2006). Clay-sized dust (0.2 μm ≤ D ≤ 2 μm), which is usually minor in terms of mass, could be responsible for about half of the global dust aerosol optical depth (Kok et al. 2017). Clay-sized dust accounts for a substantial fraction of the total number concentration of desert dust in remote regions due to its longer lifetime, as is also evidenced from analyses of the Greenland ice core (Nagatsuka et al. 2021). An accurate measurement of particle-size distributions of desert dust has been technically challenging because of the inherent variety of physicochemical properties (e.g., shape, mineralogy) of individual particles and the need for robust discrimination from coexisting non-dust particles. A robust discrimination of desert dust from other particulate components in environmental samples can be even harder in the submicron particle-size range wherein combustion-originated particles (e.g., black carbon) can also be predominant.

Elastic light scattering (Hulst and van de Hulst 1981; Jones 1999; Xu 2015; Romanov and Yurkin 2021) has long been employed as a principle for low cost, field deployable, noninvasive fast measurements of the number concentration of particles as a function of particle size. Here, we focus our review of the conventional methods for dust measurement on those based on the principle of elastic light scattering, which is of direct relevance to our method.

Inline particle sensing based on the extinction of a light beam by single particles has been used for monitoring the size-resolved number concentration of water-insoluble particles in ice core samples (Ruth et al. 2002; Simonsen et al. 2018), and particles detected in the supermicron particle-size range have often been regarded as dust on the basis of tight correlation with non-sea-salt Ca²⁺ (Lambert et al. 2012). The assumption of dust for all the detected particles would be erroneous for submicron particle sizes in which the concentration of black carbon particles originating from natural wildfires and anthropogenic fossil fuel combustion would not be negligible (Kaspari et al. 2011; Zennaro et al. 2014).

Several attempts have been made to identify dust within the framework of elastic light scattering, utilizing the fact that most of the individual dust particles are nonspherical. The degree of depolarization of light scattered to the side or backwards has often been used as an optical proxy for dust identification (Sugimoto et al. 2002; Kobayashi et al. 2014). Inline digital holography, which is also based on elastic light scattering, has been used for coarse dust identification according to the non-circularity of the holographic image (Berg and Videen 2011; Kemppinen et al. 2020; Ravasio et al. 2021). This holographic approach for shape reconstruction becomes difficult for particle sizes comparable to or smaller than the wavelength. Besides, dust aerosol particles collected in the atmosphere have been reported to be less nonspherical in smaller particle-size ranges (Huebert et al. 2000; Koren, Ganor, and Joseph 2001; Okada et al. 2001). Therefore, none of the above dust identification approaches based on elastic light scattering at visible wavelengths would be practical in the submicron particle-size range. Even after successful dust identification, the optical particle sizing would suffer from uncertainties due to unconstrained physicochemical properties (e.g., particle shape, internal porosity, and complex refractive index).

The recently proposed Single-Particle Extinction and Scattering (SPES: Potenza, Sanvito, and Pullia 2015) method that measures the complex forward-scattering amplitude of single particles is a new approach within the framework of elastic light scattering. The two-dimensional dataspace (the real and imaginary parts of the scattering amplitude) of SPES has been shown to be useful for discriminating different types of particles according to differences in either the complex refractive index or particle shape (Villa et al. 2016; Potenza et al. 2017).

In this study, we evaluate the applicability of the SPES method for identification and size-quantification of particulate silicate minerals (aluminosilicates + quartz), the major mineralogical component of desert dust, in the submicron particle-size range (0.2 μm ≤ D ≤ 1 μm). We selected water (instead of air) as a working fluid for future applications of SPES to inline measurements of particles in ice core, rainwater, ocean water, and atmospheric aerosols collected into water.

2. Methods and materials

2.1. Single particle extinction and scattering (SPES) method

The SPES method used here has been described in detail in Moteki (2020); here we focus only on the necessary points. In the SPES, one of the two diagonal elements of the complex forward-scattering amplitude matrix $S_{22}\left(0°\right)$ of each particle in the sample flow is determined from a measurement of the interference of incident and scattered fields in forward direction (Figure 1). We properly take into account the non-negligible contribution of the scattered power in addition to the interference power at the surface of the photodetector (Moteki 2020). The measured $S_{22}$ data are therefore denoted as $\left({\mathrm{ReS}}_{22},{{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}\right),$ in which the theoretical correction to the measured ${\mathrm{Im}S}_{22}$ values according to the scattered power contribution is denoted with the subscript “sc”. In our instrumental setup, a small systematic error probably caused by beam aberration was observed in the original $S_{22}$ data that had been derived according to Eqs. (29) and (36) of Moteki (2020). To correct this error, the original ${\mathrm{Re}S}_{22}$ and ${{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}$ data were multiplied by the tuning coefficients 1.02 and 1.16, respectively. The tuning coefficients were determined so as to minimize the discrepancies between measured and theoretical $\left({\mathrm{Re}S}_{22},{{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}\right)$ values for polystyrene (PS) spheres. Figure 2 shows a scatterplot of the measured $\left({\mathrm{Re}S}_{22},{{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}\right)$ data for PS spheres with six different sizes after this correction. The theoretical $\left({\mathrm{Re}S}_{22},{{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}\right)$ values were computed using Mie theory, assuming the complex refractive indices of PS and water to be 1.585 + 0i and 1.332 + 0i, respectively, at the laser wavelength 632.8 nm. The PS refractive index value was given by linearly interpolating the measured complex refractive index of bulk PS material as a function of wavelength (0.4–2 μm) available from the supplementary material of Zhang et al. (2020). The water refractive index value was given by linearly interpolating the data given by the Table 1 of Hale and Querry (1973), neglecting the imaginary part (∼10⁻⁸). Measured and theoretical results agree within 10% relative error for each particle size.

Figure 1. Schematic diagram of our SPES instrument with an illustration of the interferometric detection of scattered wave from each particle in forward direction.

Figure 2. Scatterplot of ${{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}$ versus ${\mathrm{Re}S}_{22}$ for polystyrene spheres measured by the SPES instruments. The black dots indicate single-particle data. The red-filled circles with error bars indicate the median and 10th–90th percentile ranges of single-particle data for each particle size. The blue-filled circles with error bars indicate the theoretical values and their uncertainty ranges for each particle size. The uncertainty ranges of the theoretical values were computed from the expanded uncertainty (k = 2) of each particle size specified in the sample datasheet.

Table 1. List of samples.

Sample type	Name	Production information
Synthetic	Fullerene soot	Alfa Aesar, Inc., Stock#40971, Lot#G25N20
	Magnetite (Fe3O4)	Kojundo Chemical Laboratory Co., Ltd., FEO17PB
	Goethite (FeOOH)	Kojundo Chemical Laboratory Co., Ltd., FEI16PB
	Hematite (Fe2O3)	Kojundo Chemical Laboratory Co., Ltd., FEO14PB
	Ferrihydrite (5Fe2O3•9H2O)	Synthesized by Dr. Takahashi (Univ. of Tokyo)
	Quartz	Kojundo Chemical Laboratory Co., Ltd., SIO08PB
	Snomax®	Snomax® international
Natural	NX illite	Arginotec (NX Nanopowder, B + M Nottenkamper, Munich, Germany)
	KGa-1b	The Clay Mineral Society
	Arizona test dust (ATD)	Powder Technologies, Inc., ISOTEST DUST 12103-1, A2 fine
	Gobi dust	Environmental Certified Reference Materials No.30, National Institute for Environmental Studies, Japan.

2.2. Samples

Table 1 lists the synthetic and natural samples used in our experiment. Transmission electron microscopy (TEM) images of individual particles of these samples are shown in Figure 3. We used seven different synthetic pure powder materials as proxies for several ubiquitous compositions of water-insoluble aerosols. Fullerene soot powder (Moteki et al. 2009; Moteki and Kondo 2010; Gysel et al. 2011) was used as a proxy for refractory black carbon. Hematite, magnetite, goethite, and ferrihydrite powders were used as proxies for light-absorbing iron oxides/hydroxides. Quartz powder (>99.9% purity) was used as a proxy for particulate silicates in desert dust. Snomax® powder made from the bacterium Pseudomonas syringae (Wex et al. 2015) was used as a proxy for bio-aerosols. For laboratory proxies of desert dust aerosols, we used four reference materials. The KGa-1b powder is dominated by kaolinite (∼96% purity) (Murray et al. 2011). The NX illite powder consists of illite (∼60%), interstratified illite-smectite (∼14%), feldspar (∼10%), kaolinite (∼7%), quartz (∼7%), and carbonate (∼2%) (Broadley et al. 2012). Kaolinite and illite are known as major mineralogical components of clay-sized soil particles (Ito and Wagai 2017). Arizona test dust (ATD) consists of feldspar (∼33%), quartz (∼17%), interstratified illite-smectite (∼10%), illite (∼8%), carbonate (6%), kaolinite (2%), and others (∼24%) (Broadley et al. 2012). The ATD was reported to be similar in composition to the desert dust aerosols near the source regions (Broadley et al. 2012). Gobi Kosa is a reference material of Asian desert dust aerosols that was prepared through soil sampling at the Gobi Desert followed by aerodynamic particle-size fractionation (Nishikawa et al. 2013). Its X-ray diffraction pattern implies that Gobi Kosa contains quartz, feldspar, calcite, and clay minerals (group of aluminum phyllosilicates) (Nishikawa et al. 2013). Each sample was dispersed in pure water for the SPES analyses. To avoid clogging the flow cell, we removed most of the supermicron-sized waterborne particles through gravitational particle-size fractionation (24–30 h at ∼5 °C). Therefore, the measured particle-size distributions reported here should be different from the original distributions of the powder sample.

Figure 3. Representative TEM images for each of the (a) synthetic and (b) natural samples. In (b), four elemental mappings are shown for each TEM image. The TEM images were obtained using a 120-kV TEM (JEM-1400, JEOL) and the elemental mappings were measured using an energy-dispersive X-ray spectrometer (EDS, Oxford Instruments) with scanning TEM mode (STEM-EDS).

3. S₂₂ data clusters

Figure 4 shows scatterplots of $\left({\mathrm{Re}S}_{22},{{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}\right)$ data points for the synthetic and natural samples. Figure S1 summarizes these scatterplots in a single panel for ease of comparison between samples. The quartz sample exhibited a unique cluster of $S_{22}$ data points densely distributed along a particular curve (Figure 4a). Each of the iron oxide/hydroxide samples (hematite, magnetite, goethite, and ferrihydrite) exhibited clusters of $S_{22}$ data points with a higher ${{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}/{\mathrm{Re}S}_{22}$ ratio than the quartz sample (Figures 4b–e) due to their higher real and imaginary parts of the refractive index (Huffman and Stapp 1973; Querry 1985; Longtin et al. 1988; Bedidi and Cervelle 1993). The synthetic hematite and magnetite samples each exhibited two distinct clusters of $S_{22}$ data points near the origin (Figures 4b and c) for unknown reasons.

Figure 4. Scatter plots of S₂₂ data points for the individual samples listed in Table 1.

Every natural dust sample exhibited a dense cluster of $S_{22}$ data points with a ${{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}/{\mathrm{Re}S}_{22}$ ratio similar to the quartz sample that accounted for more than 85% of the data points (Figures 4f–i). In addition, each of the natural dust samples exhibited less dense clusters with higher ${{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}/{\mathrm{Re}S}_{22}$ values than that of the major cluster. The major dense cluster is attributable to particles mostly consisting of aluminosilicates, which are major mineralogical components (e.g., interstratified illite-smectite, illite, kaolinite, feldspar, and quartz). TEM analyses of the natural dust samples confirmed the presence of iron oxide particles with smaller number fraction than the aluminosilicate particles (cf. Figure 3b). Such iron oxide particles isolated from the aluminosilicate particles would account for the minor clusters of $S_{22}$ data points with high ${{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}/{\mathrm{Re}S}_{22}$ ratios. The data points with medium ${{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}/{\mathrm{Re}S}_{22}$ ratios between the major and minor clusters can be attributed to particles consisting of various fractions of aluminosilicate and iron oxides.

Figure 5 shows density plots for natural dust samples with a magnified color scale. The ridge curve of the density of $S_{22}$ data points for each iron oxide/hydroxide standard sample is also shown for reference. For convenience, we call the major cluster the “silicate cluster” and the minor cluster including the outlier data points the “light-absorbing cluster”.

Figure 5. Density plots of $S_{22}$ data points for natural dust samples. The solid lines indicate the traces of median values of ${{[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}$ as a function of ${\mathrm{Re}S}_{22}$ for each synthetic sample. Because the $S_{22}$ data points for hematite show two distinct clusters (Figure 4b), we separated the hematite data into two clusters using a boundary line (y = 0.95x – 0.02) and then computed the trace for each cluster.

Conventional elastic light scattering approaches for optical particle sizing have not been able to resolve these two optically distinct components in natural dust samples. Therefore, we have been forced to assume a single physicochemical property (i.e., complex refractive index, particle shape, and mixing state) to estimate the optical particle size for every detected particle. The most notable advantage of the SPES method is an ability to discriminate optically distinct types of particles coexisting in a sample prior to the particle-size estimation. In the following sections, we illustrate a procedure for SPES data analyses to measure the particle-size distribution of the submicron silicate particles in natural dust samples and then discuss the accuracy of results in comparison with the conventional optical particle sizing approaches.

4. Data analyses for the silicate particles

4.1. Clustering

We compared two approaches for discriminating the silicate cluster from other data points. The first appoach (Boundary method) uses an empirical boundary curve to discriminate data points of the silicate cluster from those of light-absorbing cluster. The second approach (HDBSCAN method) is applying Density-Based Clustering Based on Hierarchical Density Estimates (HDBSCAN; Campello, Moulavi, and Sander 2013) to all the data points and assigning the largest identified cluster to the silicate cluster). Appendix A describes the details of these methods. For each sample, the number of $S_{22}$ data points used for the following analysis was fixed at 15,000.

Figure 6 shows the results of applying each clustering method to each natural dust sample. For the KGa-1b sample, the number of data points in the silicate cluster identified by the curve and HDBSCAN methods were 11,567 and 10,988, respectively. In the HDBSCAN method, the number of data points in the silicate cluster was smaller than that determined by the Boundary method because the data points with relatively large $\left|S_{22}\right|$ were classified as noise. This potential underestimation of the silicate cluster by the HDBSCAN method could also be observed regardless of the total number of data points, as illustrated in Figure S2. The appropriate procedure for the clustering can change depending on the coexisting compositions in samples. In the following analyses, we used the silicate cluster determined by the Boundary method. The principal curve, an orthogonal projection of data points into one-dimensional space (Hastie and Stuetzle 1989), was computed for each silicate cluster and is also shown in Figure 6a.

Figure 6. Clustering results of $S_{22}$ data points for natural dust samples. For each sample, the (a) Boundary and (b) HDBSCAN methods were applied to the 15,000 raw data points. The black solid lines in (a) indicate the boundary curve (y = 0.65x^1.4 + 0.008). The blue solid lines with open circles indicate the principal curves for the silicate clusters, wherein the open circles indicate the 0th, 10th, …, 90th, 98th, and 100th percentiles of the arc-length coordinate of the $S_{22}$ data points. The data points with the largest 2% $\left|S_{22}\right|$ values were removed before the principal curve fitting to avoid an overfitting issue. The data number of each cluster is denoted in the legend. The total number of data points shown in each figure is less than 15,000 because of trimming of the raw data points near the origin.

4.2. Bayesian inversion

The Bayesian inversion method of Moteki (2020) was designed to automatically select the most plausible model of particle type among the $L$ candidate models $M_l\mathrm{}(l=1,\dots ,L).$ It provides estimates of the complex refractive index and volume-equivalent size distribution for the selected model, using a tuple of data points within the cluster of each particulate compound. Here, we used six candidate models defined in Table 1 of Moteki (2020): NONABSORBING SPHERE, NONABSORBING OBLATE_SPHEROID, NONABSORBING AGGREGATE, GENERAL SPHERE, GENERAL OBLATE_SPHEROID, and GENERAL AGGREGATE. The GENERAL models allow the imaginary part of the refractive index to be ${0\le m}_i\le 1.5,$ whereas the NONABSORBING models restrict its range to near zero (${0\le m}_i\le 0.0001$). Each of the two nonspherical shape model considered here was made based on T-matrix theory and has a shape parameter that is continuously varied to find the best fit of the simulated data to the measured data. The OBLATE_SPHEROID model allows the horizontal-to-rotational axis ratio between 1.0 and 1.66. The AGGREGATE model, that mimics the fractal-like aggregate of nanospheres, allows the fractal dimension between 1.7 to 2.5.

An application of Bayesian inversion to the tuple of median values of PS spheres data (6 red-filled circles in Figure 1) resulted in the selection of the NONSABSORBING SPHERE model with a MAP complex refractive index of 1.57 + 0.00i, which is in fair agreement with the true properties of the PS spheres.

For each of the four natural dust samples and the synthesized quartz sample, we executed the Bayesian inversion method using a tuple of (0th, 10th, …, 90th, 98th, and 100th) percentile points of the arclength coordinate of the principal curve as the input data vector $\mathit{D}.$ Figure 7 shows the marginal likelihood $m(\mathit{D}|M_l)$ and the maximum a posteriori (MAP) values of the complex refractive index m^MAP for each sample. The model with the largest $m(\mathit{D}|M_l)$ value (i.e., smallest ${-\log }_{10}m(\mathit{D}|M_l)$ value) was selected as the best model because it explained the data with the smallest redundancy in model parameter space. Regardless of shape, the ${-\log }_{10}m(\mathit{D}|M_l)$ value was always smaller in the NONABSORBING model than in the GENERAL model. Moreover, the m_i values even in the GENERAL model were predicted to be very close to zero. These results suggest that the dust particles generating the $S_{22}$ data points in the “silicate cluster” predominantly consist of non-light-absorbing materials.

Figure 7. Bar plots of ${-\log }_{10}m(\mathit{D}|M_l)$ values for the silicate cluster. The MAP estimate of the complex refractive index m^MAP = m_r + im_i is also shown on the top of each bar plot.

In the NONABSORBING constraint, we compare the three different shape models. For each sample, the lowest ${-\log }_{10}m(\mathit{D}|M_l)$ was attained by either the SPHERE or AGGREGATE model. The m_r^MAP values for the AGGREGATE model were 1.87–2.27. This m_r^MAP value range is much higher than the reported m_r range for single crystals of silicate minerals (cf. Table 2) and thus we regard the results of the AGGREGATE model for any of our dust samples to be unphysical. Besides, the m_r^MAP values for the SPHERE model were 1.46–1.50, slightly less than the reported m_r range for single crystals of silicate minerals (cf. Table 2). The smaller m_r^MAP values in the SPHERE model for dust samples could be attained by assuming a presence of the microscopic pores and cracks that shift the effective refractive index of the particle closer to that of the surrounding medium.

Table 2. The effective refractive indices of the silicate clusters estimated according to the Bayesian inversion method.

Sample	Refractive index	Wavelength (nm)	Method
KGa-1b	1.484 (1.461–1.497)^a	632.8	Bayesian inversion of SPES data	This study
NX illite	1.468 (1.450–1.489)^a
ATD	1.476 (1.459–1.493)^a
Gobi dust	1.463 (1.439–1.469)^a
Quartz	1.501 (1.478–1.521)^a
Kaolinite	1.556 and 1.565^b	589.6	Refractive index matching method	Weidler and Friedrich (2007)
	1.566	640	Transmission measurements	Arakawa et al. (1997)
NX Illite	1.586 and 1.588^c	589.6	Refractive index matching method	Friedrich, Steudel, and Weidler (2008)
Quartz	1.544	589.6	Refractive index matching method	Weidler and Friedrich (2007)

Note. Reported refractive index values for pure clay minerals are also shown for reference.

^a The effective refractive index is expressed as m_r^MAP along with the 10th–90th percentile range of the posterior samples of m_r.

^b Kaolinite-PW (Polwhite-B) and kaolinite-GA (Grade-A) produced by IMERYS, respectively.

^c Two test results.

The TEM images (Figure 3b) imply that the particles’ envelopes are mostly nonspherical, apparently in contradiction to the selection of the SPHERE model. However, irregularity of a particle’s envelope may result in the same S₂₂ shift as a certain porosity of a spherical particle’s volume, considering the fact that the S₂₂ value of a constant-volume particle remains unchanged through any shape deformation as long as the internal field integrated over the particle’s volume, that is, the complex structure factor defined by Eq. (10) of Moteki (2020), remains unchanged. Therefore, the observed lower shift of m_r^MAP in the SPHERE model from the refractive index of pure silicate minerals could be caused by any of the presence of microscopic pores and cracks, and nonspherical envelope. A clearer physical interpretation for each sample would be attainable, for example, by combining the TEM data and additional dataset of effective density (Lee et al. 2009) as a function of particle size. In our Bayesian inversion algorithm, any nonspherical shape model that has the larger freedom in model parameter space should always shows lower marginal likelihood than the SPHERE model given that it exhibits the same goodness of fitting the measured S₂₂ data. From these considerations, we suggest that the NONASORBING SPHERE model does not represent the actual shape of silicate particles but a preferred mathematical model with regard to the Bayesian criterion for the particles in the silicate cluster. We use the MAP parameters provided by this model for analyses hereafter. A particle’s size and complex refractive index derived in our Bayesian inversion method may therefore be interpreted respectively as the “envelope size” and “effective complex refractive index” of a porous sphere that has a value of $S_{22}$ equal to that of a real particle. It should be noted that the SPHERE model might not be able to fit the $S_{22}$ data of dust particles in a larger size range (> ∼1μm), wherein the shape effects on the scattering properties will be more pronounced (e.g., Kalashnikova and Sokolik 2004).

For each sample, the particle-size distribution for the silicate cluster was derived by the following procedure. The arclength coordinate value was previously determined for every $S_{22}$ data point in the silicate cluster through principal curve projection. The particle volume for each $S_{22}$ data point was estimated by linearly interpolating the MAP particle volumes with respect to the arclength coordinate. The particle size was defined according to the estimated volume as the diameter of an equal-volume homogeneous sphere.

Figure 8 shows the derived particle-size distributions of the silicate cluster of each dust sample as well as that for the synthetic quartz sample. For comparison, we also show in Figure 8 the two simulated particle-size distributions derived from either the absolute scattering amplitude ($\left|S_{22}\left(0°\right)\right|$) or the imaginary scattering amplitude (${\mathrm{Im}S}_{22}\left(0°\right)$) for all the data points (silicate cluster + light-absorbing cluster). In the simulations, every particle is assumed to be a single homogeneous sphere with a refractive index of 1.56 + 0i, a typical value for single crystals of clay minerals. The $\left|S_{22}\left(0°\right)\right|$-based simulation result mimics the particle-size distribution that would be obtained by using a low-cost optical particle sensor that measures only the angle-integrated power of scattered field by individual particles. Besides, the ${\mathrm{Im}S}_{22}\left(0°\right)$-based simulation mimics the observation by an occultation particle sensor that measures only the extinguished power of the incident field by individual particles. The difference between our results (bars) and the simulations (lines) in Figure 8 implies that an unambiguous physical interpretation of the particle-size distribution obtained through conventional elastic light scattering approaches would not be attainable even for an unmixed dust material.

Figure 8. Retrieved particle-size distributions for the silicate cluster of each dust sample. For comparison, also shown by solid lines are the retrieved particle-size distributions by elastic light scattering approaches that measure only either $\left|S_{22}\left(0°\right)\right|$ or ${\mathrm{Im}S}_{22}\left(0°\right),$ assuming m = 1.56 + 0i to be a typical refractive index value for single crystals of clay minerals.

5. Discussion

Conventional elastic light scattering approaches for particle measurements have mostly been designed to have a one-dimensional dataspace for the signal of either angular-integrated scattering or beam extinction. In a one-dimensional dataspace, assumptions for the complex refractive index and shape are always necessary for particle-size estimation, leading to errors for multi-component environmental samples. A multidimensional dataspace provided by an angle-resolved scattering detector enables determinations of both size and complex refractive index for spherical particles (Chernyshev et al. 1995). However, even the dataspace of an angle-resolved scattering detector loses resolution for the complex refractive index as the particle size gets smaller and approaches the Rayleigh limit.

By contrast, the two-dimensional dataspace provided by complex forward-scattering amplitude sensing has a superior resolution for a particle’s complex refractive index in the particle-size range comparable to or smaller than the wavelength (cf. Figure 1 of Moteki 2020), even at the Rayleigh limit (Taubenblatt and Batchelder 1991). Our approach was therefore able to discriminate silicate-dominant particles from light-absorbing iron-oxide particles coexisting in each desert dust sample in the submicron particle-size range. The discrimination of such optically distinct mineralogical components helps in the acquisition of less ambiguous data of the particle-size distribution of desert dust. The complex forward-scattering amplitude distribution data of dust samples also provides an experimental constraint on the optical properties, in addition to the constraints from the bulk absorption and scattering coefficients (Moosmüller et al. 2012) and theoretical calculations on the basis of individual particle analyses (Kalashnikova and Sokolik 2004; Kemppinen et al. 2015). It should also be mentioned that quality-controlled measurements of the complex forward-scattering amplitude can be performed over a wider particle-size range (Moteki 2021). In future, more realistic shape models, such as the Gaussian ellipsoid (Muinonen and Pieniluoma 2011), should also be considered in the inversion algorithm for an improved estimation of refractive index and distributions in shape and particle’s volume for silicate particles. Comprehensive light-scattering databases for nonspherical particles have been developed for application to remote sensing and radiative transfer simulations (Meng et al. 2010; Saito et al. 2021). A “complex amplitude counterpart” of such databases should also be developed for an improvement of the SPES approach for dust measurement.

Here, we contrast our approach with previously proposed approaches for estimating physical properties of particles from the SPES data. Potenza, Sanvito, and Pullia (2015) used a look up table according to the Mie theory for unique determinations of both particle-size and refractive index for non-absorbing spherical scatterers such as polystyrene and Polymethyl Methacrylate (PMMA). For particles with unknown shape, Villa et al. (2016) propose a use of anomalous diffraction theory, that is applicable to any scatterer with large size parameter and small refractive index contrast (Hulst and van de Hulst 1981), to estimate some geometric properties of waterborne dust particles. Villa et al. (2016) also used a look up table for spheroidal shape model prepared through an exact numerical method to estimate the two-dimensional distribution of concentration with respect to the aspect ratio and particle-volume, assuming the complex refractive index a priori.

Our method presented here is applicable to any waterborne particles, such as particles in rainwater, ocean water, and ice core meltwater. Application to atmospheric aerosols would also be straightforward by resuspending particles collected on a filter substrate into water (e.g., DeMott et al. 2017) or collecting aerosol particles into water (e.g., Weber et al. 2001; Cho et al. 2019). For untreated environmental samples, it might not always be possible to discriminate desert dust silicates from other particulate compounds solely based on the complex forward-scattering amplitude. Appendix B provides a discussion of the possibility of clustering using synthetic data. Additional techniques (e.g., pretreatment of samples) might be necessary for environmental samples to reduce misidentification.

Acknowledgments

We thank Y. Tobo for providing the KGa-1b, NX-illite, and Snomax® particles and for useful comments. We thank Y. Takahashi for providing the ferrihydrite particles. We thank M. Koike, S. Ohata, and Y. Ushikubo for discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the Environment Research and Technology Development Fund (JPMEERF20172003 and JPMEERF20202003) of the Environmental Restoration and Conservation Agency, the Japan Society for the Promotion of Science (JSPS) KAKENHI program (JP19H04236, JP19H04259, and JP19H05699), and the ArCS Project (Arctic Challenge for Sustainability Project, JPMXD1300000000) supported by the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) in Japan.

Appendix A.

Clustering method

We removed data points from around the origin (${\left({\mathrm{Re}{S}_{22}}^2+{\left[\mathrm{Im}{S}_{22}\right]}_{\mathrm{sc}}^2\right)}^{1/2}<0.05\mathrm{\mu }\mathrm{m}$) where the data points of even optically distinct components were severely overlapping. In the Boundary method, we defined two boundary curves: $y=0.65x^{1.4}+0.008$ and $y=0.42x^{1.3}-0.02,$ where $x\equiv {\mathrm{Re}S}_{22}$ and ${y\equiv {[\mathrm{Im}S}_{22}]}_{\mathrm{sc}}.$ The former was used to descriminate the silicate cluster from the light-absorbing cluster. The latter was used only for the procedure described in Appendix B to discriminate the silicate cluster from the cluster of Snomax. In the HDBSCAN method, we transformed the coodinates of S₂₂ dataspace according to the method of Moteki (2020) before application of HDBSCAN clustering to the data. Two hyperparameters of HDBSCAN, “min_cluster_size” and “min_samples,” were tuned for each sample to obtain reasonable clustering results. For all samples except for Gobi dust, we set (min_cluster_size, min_samples) to (20, 30) when N $\le$ 25,000 and (40, 45) when N > 25,000. For the Gobi dust sample, we set (min_cluster_size, min_samples) to (40, 30) for any N.

Appendix B.

Clustering tests using synthetic data

We briefly evaluated the potential errors of our clustering methods for multicomponent samples by using synthetic data. The synthetic data consisted of 45,000 data points prepared by merging 15,000 data points of either of the four dusts (Figures 4f–i), 15,000 data points of fullerene soot (Figure 4k), and 15,000 data points of Snomax (Figure 4l). We designate the clustered data points to one of three distinct clusters: “light-absorbing cluster” (fullerene soot + iron oxides), “silicate cluster” (aluminosilicates + quartz), and “biogenic cluster” (Snomax). Figure B1 shows scatterplots of the clustered data points. Figure B2 shows the number of data points in the silicate cluster extracted from the synthetic data for individual cases. In each case, the number of data points in the silicate cluster determined for pure dust data using the Boundary method is also shown. For both the Boundary and HDBSCAN methods, contamination of the silicate cluster by other components (fullerene soot + Snomax) was less than 5%.

Figure B1. Scatter plots of the synthetic data that were classified by the (a) Boundary and (b) HDBSCAN methods. The black solid and black dashed lines in (a) represent the boundary curves for the curve method. Each of the blue solid lines with open circles indicates the principal curve for each silicate cluster.

Figure B2. Number of data points in the silicate cluster for the synthetic data. Contributions to the number of data points in the silicate cluster from different components are shown. The red-filled circles indicate the original number of data points in the silicate cluster determined for each dust sample.