There is a large variety of internal structures inside atmospheric dust particles, making them inherently inhomogeneous. Such structures may have a large effect on ground-level and atmospheric radiation. So far, dust particle internal structures and their effect on the light scattering properties have proved to be hard to quantify, in part due to challenges in obtaining information about these structures. Recently, internal structures of individual dust particles were revealed through focused ion beam milling and analyzed. Here, we perform a sensitivity study to evaluate the optical impacts of some of the typical internal structures revealed. To obtain suitable model particles, the first step is to generate inhomogeneous particles with varying internal structures by using an algorithm that is based on three-dimensional Voronoi tessellation. The parameters for the particle generation are obtained from studies of real-world Asian dust particles. The second step is to generate homogeneous versions of the generated particles by using an effective-medium approximation, for comparison. Third, light scattering by both versions of these particles is simulated with discrete dipole approximation code. This allows us to see how different internal structures affect light scattering, and how important it is to account for these structures explicitly. Further, this allows us to estimate the potential inaccuracies caused by using only homogeneous model particles for atmospheric studies and remote-sensing measurements. The results show that the effects vary greatly between different kinds of internal structures and single-scattering quantity considered, but for most structure types the effects are overall notable. Most significantly, hematite inclusions in particles impact light scattering heavily. Furthermore, internal pores and hematite-rich coating both affect some form of light scattering noticeably. Based on this work, it seems that it is exceedingly important that the effects of dust particle internal structures on light scattering are accounted for in a wide variety of applications.

Mineral dust particles are an important part of the atmosphere

The reasons for the large variance of the impact of dust on radiation can be
many, but in addition to the obvious variations in particle size and
concentration, shapes, surface roughness characteristics and internal
structures may play a role. It is known that many dust particles contain
materials with significant dielectric contrast, for example iron oxides or
internal pores. Transmission electron microscopic (TEM) analysis of
cross-sections of single particles showed that internal pores of varying
sizes are common features of Asian dust particles. Additionally, submicron
iron oxides, such as hematite and goethite, are often distributed within clay
medium

While particle size distributions and many other population-level parameters
are possible to measure by remote sensing

Here we carry out a sensitivity study on the impacts of internal structures. Our approach is to create an algorithm that allows us to generate discrete dipole approximation (DDA) models of particles with desired types and amounts of internal structures, for which accurate light scattering simulations can be then easily run, thus allowing taking internal structures into account explicitly and accurately. What we aim to do is to generate a set of particles that are complex-shaped and irregular, as are real dust particles, and possess internal structure characteristics that resemble those observed in real dust. The clear benefit of this pure modeling approach is that it allows us to calculate accurate values for individual optical properties, linked to known individual physical properties. We will be testing how various internal structures change light scattering compared to a baseline version, which is composed only of optically similar minerals. Further, knowing the composition of the inhomogeneous particles completely allows us to calculate homogeneous versions of the same particles with an effective-medium approximation (EMA), and simulate light scattering by both the inhomogeneous and the homogeneous versions of the particles. This, then, allows us to quantify the errors in light scattering caused by using the homogeneous version of the particle instead of the true form of the particle.

The main purpose of this study is to investigate the effects of different
types of internal structures on the single-scattering properties. We will be
examining particles with various internal structures, such as empty cavities
and materials with high real and imaginary refractive index, such as hematite
and other iron oxides. The article is structured as follows:
Sect.

Asian dust is an important mineral dust lifted from arid regions in
northwestern China and southern Mongolia, and transported long range across
East Asia and the North Pacific

Example internal structure and mineral distribution in Asian dust particle.
Panel

The foundation of all radiative effects comes from single-scattering interactions.
The scattering matrix for a particle,

A schematic two-dimensional figure of the particle generation process.
Panel

Here, we study the effects of internal structures primarily in terms of individual scattering matrix elements, from which all other effects can be determined. We also link these effects to relevant radiative transfer and remote-sensing quantities by calculating four scalar scattering quantities, which are described below.

The single-scattering albedo,

The asymmetry parameter

In many lidar applications, a quantity called lidar ratio,

Linear depolarization ratio,

As an example, single spherical, isotropic particles have

We performed the light scattering simulations of the generated inhomogeneous
and homogeneous particles with a discrete dipole approximation (DDA)

DDA is generally accurate as long as the target dipole resolution is
sufficient. In this work, the target shapes for all size parameters were
composed of roughly two hundred and twenty thousand dipoles. The value

Test simulations with higher dipole resolution for our inhomogeneous targets
showed that

We used a three-particle ensemble for results, and each internal structure case of each of these three particles was simulated with size parameters

The simulations were run on the Finnish Meteorological Institute Cray XC30 supercomputer Voima.
In the calculations we used 64 computer cores per simulation, and 10 concurrent simulations were run in parallel.
With this setup, the total amount of CPU time used was approximately 110 000

Our method for generating the particle models involves multiple stages that
can be run separately if needed. The overview of each main stage is given
below. The algorithm assumes that the particle can be represented by
a regular three-dimensional lattice of individual volume elements, or
dipoles. With a good enough dipole resolution, the representation can
replicate most large-scale structures of real dust particles with
a sufficient accuracy. This representation also allows us to trivially
convert the algorithm output to a DDA format for light scattering
simulations. The generation process is summarized in
Fig.

To generate computational models of realistic particles with internal
structure, we employ an algorithm with three-dimensional Voronoi tessellation
at its core

The generation of model shapes begins with an enclosed, discretized space
composed of empty volume elements. The first step of the algorithm is to
randomly place a given number of points within this volume. These act as

The formula for weighted distance is:

After the whole volume has been divided into cells, the volume is culled to
extract a model particle from it. Here we have used an ellipsoid with the
same axis proportions as the ellipsoidal grain axis proportions as the
culling shape. The generated particles have aspect ratios close to 1.5. The
culling is done in such a way that each cell with at least one element
outside of the culling shape is removed from the volume, and the remaining
cells form the particle. In this work we used 800 seeds within the original
volume, yielding mean cell size of roughly 1200 elements, which translates to
roughly 0.5

The next step is to separate the cells from each other. This is accomplished by finding the geometrical center of the particle, and forming unit vectors that point from it to the seeds of each cell. The cells are then moved to the direction specified by their corresponding vectors by a user-specified distance, and the final locations are discretized by rounding the cell element to the nearest integer. Therefore, the cells do not change sizes or shapes, but are separated from each other. This step is to allow separating individual cells or crystals from each other, which is often the case also with real dust particles and thus yields more realistic model shapes. The displacement length used in this work was 5, where 1 is the size of one lattice element, or a dipole in the DDA targets.

The cell separation creates gaps between the cells, and the next step is to
fill these gaps, and to soften sharp edges and other roughness
characteristics around the particle. This is accomplished with a method
called concave hull

The fifth step is to coat the particles. The method for coating is simple: a layer of coating is added by setting each empty element that is orthogonally adjacent to a non-empty element as non-empty, assigning these elements to be composed of a coating material. Multi-layer coatings are formed by using the method iteratively. We used a three-layer coating in this work for both the normal coating and the hematite-rich coating cases.

There is an additional optional step, which can be used to add further internal structures in the form of inclusions. To generate inclusions in the original shape we insert nodes inside the particle. Nodes are generated simply by finding a random element in the particle, and growing a sphere of a given radius around the element, replacing parts of any existing cell with the node cell.

Here we have used nodes to generate both hematite inclusions and internal pores. The approach allows us to add these features into generated model particles without introducing any other changes in their shapes. For both the hematite nodes and internal pores, we generated 20 nodes with a radius of eight element lengths, which made the nodes comparable in size to cells.

The above section contains the technical description of the shape-generating
algorithm. The model is given physical relevance by introducing

The end product of this process is a list of volume elements that contain their position and refractive index. This is exactly what is needed for DDA simulations, hence making it straightforward to simulate scattering by these particles.

Once the particles with internal structure have been generated with the
algorithm described above, we will generate their homogenized versions. This
is achieved with a simple effective-medium approximation (EMA). We calculate
the effective refractive index

Different mixing rules are known to perform differently depending on the
particle type

The mineral contents for each of the inhomogeneous cases. CM refers
to clay mixture, Empty refers to internal pores, and HRCM refers to
hematite-rich clay mixture. Volume fraction (VF) columns 1–5 correspond to
Cases 1–5. Percentages may not sum to 100 % due to rounding in the
displayed values. EMA

For the first test particle, there was a very large variability between different EMA's, with the average refractive index and the inverse Maxwell Garnett EMA's performing the best. For the second test particle, all of the mixing rules performed decently at small sizes, but poorly at large sizes, and very similarly to each other at all sizes. Therefore, our conclusions is that out of the five mixing rules tested, not one performed better than the one selected here, and therefore the one selected here is appropriate for more detailed comparisons.

In total, we studied five distinct internal structure cases. The five cases
are:

For each case, the particle resembles the “baseline” Case 1 in other respects than the added features. Therefore, for instance, apart from the added hematite, the mineral volume fractions in Case 2 resemble those in Case 1. While they are not identical due to the stochastic nature of inclusion locations, they are close enough to plausibly assume that the changes seen are not caused by the differing non-hematite mineral content but instead are caused by hematite. In fact, in the text below we compare the other cases to Case 1 for specifically this reason.

For all of these cases, we use a three-particle ensemble for all of the results. We decided to use an ensemble to average out oscillations by single particles, and to see if the effects of different internal structures are consistent across all three different generated particles. Therefore, the particle generator is run three times with the same input parameters, the DDA simulations are run for all three versions of the five cases, and for each case we calculate the average of the three results. Because the generator is stochastic in nature, the three individual particles differ from each other despite having identical input parameters; however, all of the results are qualitatively similar for each individual particle in such a way that inter-particle variability is smaller than the difference between inhomogeneous and homogenized cases, or between inhomogeneous case and the baseline, for all scattering matrix elements and at most scattering angles.

Table

Example cross-sections of Case 1 versions of all three of the ensemble
constituent particles (Panels

Due to the very large variability of the types and structures of real-world
dust particles and the lack of information of the three-dimensional
structures of the particles, quantitative validation of the generated shapes
is challenging. Instead, we can compare the particle compositions to those of
real particles, and compare the cross-sections visually. Looking at
Fig.

In addition to the grains and the coating, the model particles contain
inclusions and pores, as described above. The nodes are generated to be
comparable to grains in size, that is, diameters of 0.5–1

Here we show the effect of taking inhomogeneity into account in light
scattering simulations for several different internal structure scenarios.
The scattering matrix elements will be analyzed as a function of the
scattering angle after integrating the values over a size distribution. As
the size distribution we use a lognormal distribution with the geometric mean
radius

As the simulations are carried out separately for each particle size and only then averaged over the size distribution, we can easily estimate how a different choice of size distribution would impact the results. For example, had a wider size distribution been chosen, the results would have changed to some degree due to assigning a larger weight to larger particles compared to the current size distribution. Exact changes would depend on the inhomogeneity case and scattering matrix element in question, but based on the results for individual sizes, for example Case 5 EMA errors would have been increased. We speculate this is due to larger interaction between the radiation and the thin hematite-rich coating, causing the EMA to perform worse for large size parameters than for small size parameters.

It should be reiterated that the primary purpose here is not to study the single-scattering properties themselves, but how they differ when the internal structure is accounted for either explicitly or through an effective medium approximation. In particular, we are interested in establishing which types of internal structures have large effects on scattering. For each of the five cases, we show light scattering by the inhomogeneous particle (called IHG), light scattering by IHG Case 1 (called the baseline), as well as the homogeneous version of the particle (called EMA), for straightforward comparison of the effects of inhomogeneity. As a reminder, the baseline and the EMA versions of the particles are identical in size and shape to the corresponding IHG versions, and the only difference is in the local refractive indices within the lattice elements.

Below, we study each of the six independent scattering matrix elements
separately. For each matrix element, we show all of the five cases, comparing
the IHG version of the particle with the baseline and the EMA versions. Case
1, the baseline, is not discussed separately along the other cases because
EMA values for it are virtually identical to the IHG values for all of the
scattering matrix elements.

Added contrast shows a clear effect on

For

Figure

It is clear from Fig.

Lastly, the effects of the added forms of internal structure on

As a practical consideration of identifying particle internal structures from
measurements, we recommend polarization measurements. While producing an
identification algorithm would require a very large amount of additional
work, it seems that, for example, positive degree of linear polarization
(

In addition to the effect of internal structures on the scattering matrix
elements, we also explore the impacts of the same types of internal
structures on four scalar quantities that are often used in climate or
remote-sensing applications. These quantities, namely co-albedo, asymmetry
parameter, linear depolarization ratio and lidar ratio, are shown below as
a function of the particle size parameter. The format is similar to that used
for the scattering matrix elements, where we show results separately for the
IHG, the baseline, and the EMA particles, and compare them to see how added
internal structures affect the values. In addition to the
size-parameter-dependent figures, we also show the values of the
size-distribution-averaged results in Table

Co-albedo for the inhomogeneous (IHG), the baseline, and the homogeneous (EMA) versions of all five particle cases as a function of the size parameter of the particle.

Scalar scattering quantities for the size distribution averaged three-particle ensembles. Inhomogeneous (IHG), Case 1 (baseline) and homogeneous (EMA) values are shown separately, as well as their differences as percentages of the IHG value.

Single-scattering co-albedo for the original (IHG), Case 1 (baseline), and
homogenized (EMA) versions of the three-particle ensembles of the five
internal structure cases are shown in Fig.

Asymmetry parameter for the original and homogenized versions of the five
internal structure cases is shown in Fig.

Asymmetry parameter for the inhomogeneous (IHG), the baseline, and the homogeneous (EMA) versions of all five particle cases as a function of the size parameter of the particle.

The linear depolarization ratio for the original and homogenized versions of
the five internal structure cases are shown in
Fig.

Linear depolarization ratio for the inhomogeneous (IHG), the baseline, and the homogeneous (EMA) versions of all five particle cases as a function of the size parameter of the particle.

Lidar ratio for the original and homogenized versions of the five internal
structure cases are shown in Fig.

Lidar ratio for the inhomogeneous (IHG), the baseline, and the homogeneous (EMA) versions of all five particle cases as a function of the size parameter of the particle.

In this work, we studied the effects of dust particle internal structure in
a computational way based on real internal structures revealed by

Five distinct internal structure cases were studied, and for each case we used a three-particle ensemble. First, we studied particles whose composite minerals had similar refractive indices. This was considered our baseline, to which the other cases were compared to. Second, we added hematite inclusions to the baseline particles. Third, we added internal pores to the baseline. Fourth, both hematite inclusions and internal pores were added to the baseline. Fifth, the coating material of the baseline was replaced with a hematite-containing material, but no nodes or pores were added.

These models of internal structures were selected by their common occurrence
in the Asian dust particles on the basis of systematic TEM data provided in

For each of these cases, we studied light scattering by both the
inhomogeneous and homogenized versions of the particles and compared them
against the baseline. The results show that most types of internal structure
have clear effects on light scattering, and that many of those effects are
not properly accounted for by the effective-medium approximation (EMA) that
we used. Our findings are consistent with those of

All four of the scalar variables studied, the single-scattering co-albedo,
asymmetry parameter, linear depolarization ratio and lidar ratio, were
affected noticeably by some forms of internal structure. For co-albedo,
adding hematite content increased the values significantly. For asymmetry
parameter, hematite nodes, but not hematite-rich coating, increased the
values clearly. For linear depolarization ratio, all forms of hematite
lowered the values greatly. Finally, for lidar ratio, added hematite nodes
increased the values two- or three-fold. Interestingly, we can compare the
differences to those caused by adding surface roughness, as done by

Based on these results, it appears that the internal structure of real dust particles needs to be accounted for in single-scattering simulations to obtain accurate results. Not only is it common in real dust particles, it also has major effects on scattering matrix elements and many scalar scattering quantities. Furthermore, the form of the internal structure matters. For example, light scattering changes considerably depending on whether hematite is present as inclusions, or mixed in the coating material. Additionally, at least for the simple mixing rule tested here, a homogenized particle created with an effective-medium approximation is unable to well mimic scattering by the original inhomogeneous version of the particle in most cases. In fact, in some cases using an effective-medium approximation causes results to be more wrong than ignoring the internal structure altogether. Therefore, if accuracy is desired in the results, accounting for internal structure should be done explicitly.

Following up on these results, there are several directions to consider. As a
practical concern, one might try to find an EMA that works very well for some
or all of the inhomogeneity types here. Additionally, replicating the
scattering by the inhomogeneous particles by using detailed modeling results
to fine-tune shape and composition ensembles of simple model shapes, such as
ellipsoids, might lead to much better results in applications. However, as
shown by

The authors wish to thank Maxim Yurkin, Bastiaan van Diedenhoven and an
anonymous referee for their helpful comments in improving the manuscript.
This research has been funded by the Academy of Finland (grant 255718), the
Finnish Funding Agency for Technology and Innovation (Tekes; grant
3155/31/2009), the Magnus Ehrnrooth Foundation, and the National Research
Foundation of Korea grant NRF-2011-0028597. Maxim Yurkin is acknowledged for
making his ADDA code publicly available
(