Does optically effective complex refractive index of internal-mixed aerosols have a physically-based meaning?

Xiaolin Zhang; Huan Jiang; Mao Mao; Yan Yin

doi:10.1364/OE.27.0A1216

1. Introduction

Aerosol particles, emitted from various natural or anthropogenic sources, are ubiquitous in the atmosphere, and can affect the earth’s radiation budget directly by scattering and absorbing solar radiation [1]. Meanwhile, they can impact it indirectly by acting as condensation nuclei in cloud formation, thus affecting the radiative properties and lifetimes of clouds [2]. The large uncertainty in estimating the effects of aerosols on climate is mainly caused by uncertainties in the estimation of their optical properties, which in turn depend on their microphysical properties [3,4]. As one of the crucial microphysical parameters, aerosol complex refractive index (ACRI) governs both particle scattering and absorption, and should be known for modeling their radiative effects. Meanwhile, the magnitude of radiative forcing is sensitive to ACRI, and the dependency of radiative forcing on its imaginary part is even more pronounced than the dependency on real part [5,6]. In particular, the ACRI of black carbon aerosols plays a significant role in the radiative forcing at the top of atmosphere due to their determination of the absorption amount.

The ACRI can be determined approximately from bulk chemical compositions and known refractive indices of pure components based on volume mixing rules [7,8]. This traditional approach is on the basis of the high dependency of ACRI on particle chemical compositions [9]. However, the determination of the refractive index of each individual particle within an ambient aerosol population is infeasible, and an optically effective ACRI is usually applied to represent the whole size distribution [10,11]. In addition, aerosol models tend to predict more and more details of aerosol microphysics (such as size distribution, chemical composition, and mixing state), from which aerosol optical properties can be calculated. Such a modeling approach requires estimating optically effective ACRI from observations, which could serve as a test for the models.

Many studies focus on the determinations of optically effective ACRI from simultaneously measured aerosol optical properties based on the Mie theory, although the application of a spheroidal dust model to deduce ACRI is also performed [12,13]. With the Mie model, Raut and Chazette [5,14] retrieve optically effective ACRI utilizing two constrains of aerosol optical properties with a synergy between lidar, sunphotometer and other in situ measurements. Based on aerosol absorption coefficients measured by a three-wavelength Particle Soot Absorption Photometer (PSAP) [15], or Spectral Optical Absorption Photometer (SOAP) [16], optically effective ACRIs can be derived with a Mie theory based data analysis scheme. With the assumption of spherical particles and chemical homogeneity of aerosol samples, optically effective ACRIs can also be deduced relying on simultaneous measurements of size distributions, scattering and absorption coefficients [17–20]. Nonetheless, as shown in the comparison by Muller et al. [21], retrieved optically effective ACRIs based on various techniques showed partly a reasonable agreement with evident differences for imaginary part. This may be associated with uncertainties in the assumption of particle shape and uncertainties in the measurements of aerosol optical properties by the instruments. Moreover, some studies report some unreasonable low values (real part less than 1.3) of optically effective ACRI retrieved from measured aerosol optical properties [e.g., 22,23], and it is widely known that the real parts close to or even below that of water is unrealistic. Do these extreme low real parts of retrieved optically effective ACRI exist theoretically, or are they just induced by the inaccuracies during retrieval, e.g., uncertainties in aerosol optical measurements or non-spherical particles? Does the optically effective ACRI of internal-mixed particle have a physically-based meaning like real ACRI? Numerical modeling of internal-mixed aerosol particles with accurate light-scattering calculation methods (such as core-shell Mie) could avoid these uncertainties, and become a powerful means to improve our understanding on retrieved optically effective ACRI, which is, nevertheless, generally few in literature.

Here, numerical determinations of optically effective ACRIs retrieved from exactly calculated optical properties of polydisperse core-shell coated BC particles with a Mie theory based data analysis scheme are carried out. The code of core-shell Mie is employed to accurately calculate the scattering and absorption properties of simulated coated BC. The paper is organized as follows. Section 2 introduces retrieval approach of optically effective ACRI of numerically simulated coated BC, and the factors affecting retrieval results are discussed in Section 3. Section 4 concludes the study.

2. Methodology: retrieval approach

Our retrieval approach is generally similar to corresponding methods described in other studies [e.g., 23–25], with the differences being that aerosol inherent optical properties are exactly calculated rather than measured, and simulated particle shapes are not altered during retrieval. To avoid uncertainties induced by different numerical methods, the core-shell Mie is applied to calculate inherent optical properties of modeled coated BC particles, while the Mie is employed to retrieve their optically effective ACRIs. We consider BC coated by non-absorbing sulfate with a core-shell structure as a case study, which may represent some aged BC particles under polluted urban environments [26]. Among all aerosol optical properties, particle scattering and absorption are selected for retrieval, since both are governed by the real and imaginary part of ACRI, respectively. For ambient aerosols, the bulk optical properties averaged over a certain size distribution are meaningful, and an ensemble of coated BC with different sizes but same shell/core ratio ( $D_{p} / D_{c}$ , particle diameter divided by BC core diameter) is considered. The internal-mixed BC-sulfate particles are assumed to follow a lognormal size distribution as:

n (r) = \frac{1}{\sqrt{2 π} r \ln (σ_{g})} \exp [- {(\frac{\ln (r) - \ln (r_{g})}{\sqrt{2} \ln (σ_{g})})}^{2}],

where

r_{g}

is the geometric mean radius, and

σ_{g}

is the geometric standard deviation [e.g., 27,28]. For Aitken, accumulation and coarse modes, the radius ranges are set as 0.005–0.05 μm, 0.05–0.5 μm and 0.5–5.0 μm, respectively. To better understand the behaviors of retrieved optically effective ACRI at different particle size mode, we use the size distributions in Aitken, accumulation and coarse modes separately, which are similar to those utilized in the aerosol-climate models [29]. In this study, the values of

r_{g}

are assumed as 0.03 μm, 0.075 μm and 0.75 μm, while

σ_{g}

values of 1.59, 1.59 and 2.0 are considered for Aitken, accumulation and coarse coated BC, respectively [29,30]. As particle size distribution is given, the bulk scattering and absorption cross sections can be easily obtained following:

〈 C_{s c a} 〉 = \int_{r_{\min}}^{r_{\max}} C_{s c a} (r) n (r) d (r),

〈 C_{a b s} 〉 = \int_{r_{\min}}^{r_{\max}} C_{a b s} (r) n (r) d (r) .

Given that the ensemble-averaged scattering and absorption cross sections are obtained, the optically effective ACRI is determined by a Mie based iterative scheme using the same size distributions. Exploiting all calculations, the designed inversion scheme for retrieving the optically effective ACRI follows. On the Basis of a guess for a real, n, and imaginary part, k, of the ACRI at a given wavelength, two look-up tables are built from the database with known size distribution. One look-up table encompasses the scattering cross sections whilst the other contains the absorption cross sections. Considering all possible aerosol compositions in ambient atmosphere, in the calculation, the real part of refractive index varies from 1.1 to 2.5 with an equidistant space of 0.001 and the imaginary part changed from 10⁻⁹ to 1.0 with a logarithmic interval of 0.005. Then the retrieval scheme simultaneously alters n and k, and scans through all possible ACRIs within an expected resolution until it minimizes $χ^{2}$ :

χ^{2} (n, k) = \frac{1}{N} \sum_{i = 1}^{N_{\}} [{(\frac{C_{s c a}_{, c a l c u l a t e d} (n, k) - C_{s c a, i n h e r e n t}}{C_{s c a}_{, i n h e r e n t}})}^{2}_{i} + {(\frac{C_{a b s,}_{c a l c u l a t e d} (n, k) - C_{a b s,}_{i n h e r e n t}}{C_{a b s,}_{i n h e r e n t}})}^{2}_{i}],

where

C_{s c a}_{, i n h e r e n t}

and

C_{a b s,}_{i n h e r e n t}

are inherent scattering and absorption cross sections of simulated coated BC particles,

χ^{2} (n, k)

the fractional difference of calculated scattering and absorption cross sections relative to the inherent properties, and N the number of calculation considered in the retrieval. The

χ^{2} (n, k)

for scattering plus absorption is minimized by optimizing initial guess ACRI values, and an optically effective ACRI is yielded at this wavelength. As the absorption is chiefly controlled by the magnitude of k whereas the scattering is primarily governed by the magnitude of n, the minimization of

χ^{2} (n, k)

should derive a unique retrieval result of optically effective ACRI.

This study considers an incident wavelength of 550 nm, and relevant refractive indices of BC and sulfate are $1.85 - 0.71 i$ [31] and $1.52 - 5.0 \times 10^{- 4} i$ [32], respectively. The $D_{p} / D_{c}$ ranges are assumed to be 1.1–2.7 for coated BC particles based on the measurements in London [33] and Beijing [34]. With exampled coated BC model defined and the accurate Mie methods (including core-shell Mie), it is possible to study retrieved optically effective ACRIs with more details.

3. Results and discussion

3.1 Optically effective ACRIs based on different look-up tables

Figure 1 illustrates the optically effective ACRIs retrieved from scattering and absorption properties of polydisperse coated BC particles with the aforementioned size distributions based on the look-up tables of all possible refractive indices. The differences, which are also shown in Fig. 1, are relative errors of coated BC scattering and absorption cross sections induced by retrieved optically effective ACRIs compared to their inherent properties. The retrieved ACRIs of coated BC aerosols are optically effective, since the relative errors for both scattering and absorption cross sections are within 1%, 1%, and 5%, in Aitken, accumulation and coarse modes, respectively (see Fig. 1, bottom row). Figure 1 clearly shows that retrieved optically effective ACRI of coated BC is quite sensitive to its shell/core ratio and size mode. It is expected that, both n and k of optically effective ACRI are decreasing with increasing $D_{p} / D_{c}$ in Aitken and accumulation modes. As $D_{p} / D_{c}$ augments from 1.1 to 2.7, retrieved optically effective ACRIs lessen from $1.795 - 5.370 \times 10^{- 1} i$ to $1.543 - 3.548 \times 10^{- 2} i$ , and from $1.741 - 5.128 \times 10^{-1} i$ to $1.527 - 3.802 \times 10^{- 2} i$ at 550 nm for Aitken and accumulation modes, respectively. Nevertheless, in coarse mode, the n of retrieved optically effective ACRI is complicated as a function of $D_{p} / D_{c}$ , although its k decreases with the increase of $D_{p} / D_{c}$ . It is profound to note that retrieved optically effective ACRI of coarse coated BC with some shell/core ratios values show extreme real parts, i.e., $n < 1.52$ or $n > 1.85$ . These extreme real parts seem to be unreasonable and have no physically-based sense, since the lowest n of the composition of our modeled internal-mixed particle is sulfate with n of 1.52 and the highest is BC with $n = 1.85$ . Virkkula et al. [22] report some unreasonable low values of retrieved effective real refractive index ( $n < 1.3$ ) of dry aerosol in the Antarctic boundary layer, and give possible explanations, such as non-analyzed biogenic particles, or non-spherical particles. However, our exact numerical investigation indicates that these unreasonable real parts of optically effective ACRI exist indeed in theory for coarse coated BC if unlimited look-up tables, considering all possible ACRIs, are used for retrieval.

Fig. 1 Retrieved optically effective complex refractive indices of polydisperse coated BC with a core/shell structure based on different look-up tables and their induced differences of scattering and absorption cross sections as a function of shell/core in nuclei (a, d, g), accumulation (b, e, h), and coarse (c, f, i) modes, respectively. The solid interiors denote the results based on the look-up table with a range of all possible refractive indices, while the open ones indicate those from the look-up table with refractive indices within a limited range of only known compositions. For the complex refractive indices, the black squares denote the real part whist the red circles indicate the imaginary part. For the differences, the up triangles and down triangles indicate the relative errors of scattering and absorption cross sections, respectively.

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To avoid these unreasonable extreme real parts and make retrieved optically effective ACRIs be able to show a physically-based sense, the results based on limited look-up tables with guessed real and imaginary parts within refractive index ranges of known compositions, are also portrayed in Fig. 1 for comparison. For the limited ACRI look-up tables in our retrieval, guessed n varies from 1.52 to 1.85 with an equidistant space of 0.001 while k alters from $5.0 \times 10^{- 4}$ to 0.71 with a logarithmic interval of 0.005 (see Table 1). It is expected that, with the limited look-up tables considering known aerosol compositions, retrieved optically effective ACRI of coated BC in coarse mode appears reasonable, and its k of becomes smooth as a function of shell/core ratio, although relative errors of scattering and absorption cross sections are within 10%. This indicates that, knowing particle chemical compositions and then employing limited ACRI look-up tables, will benefit the retrieval of reasonable optically effective ACRIs of coarse particles. Nevertheless, in Aitken and accumulation modes, the same retrieval results of optically effective ACRIs are obtained based on different ACRI look-up tables, which are different from the pattern in coarse mode. In general, the optically effective ACRIs of internal-mixed particles in Aitken and accumulation modes have physically-based meanings like real ACRI, and can produce effective optical properties. However, it should be noticed that the notion of refractive index becomes somewhat ill-defined if applied to coarse internal-mixed particles, since it is called the optically effective refractive index. Meanwhile, to avoid the lack of physically-based meaning for retrieved optically effective ACRI, limited ACRI look-up tables with the consideration of the ACRIs of known aerosol compositions may be needed.

Table 1. Different Look-up Tables for Retrieval

View Table

3.2 Comparisons with the volume weighted average and effective medium theory

In addition to the optically effective ACRI, some other approaches are also applied to approximately determine the effective refractive index of internal-mixed particles, including the volume weighted average (VWA) method and effective medium theory. The VWV method approximates the effective ACRI with volume fractions of each chemical composition and its pure ACRI, which is popularly employed in the state-of-the-art aerosol-climate models [e.g., 35–37]. The effective medium theory approximates the optical properties of the heterogeneous scatterer by those of a homogeneous particle of the same shape, and the effective dielectric constant is derived by requiring the forward-scattering amplitude to be zero [e.g., 38]. The Bruggeman effective medium (BEM) theory [39], one of the most commonly used effective medium theory, is considered to compare with the optically effective ACRI. With known volume fractions and refractive indices of aerosol compositions, it is straight to calculate the effective ACRIs based on the VWA and BEM.

Figure 2 illustrates the effective ACRIs of polydisperse coated BC derived from the VWA and BEM methods compared to the optically effective ACRIs retrieved with limited look-up tables considering known aerosol compositions. The effective ACRIs based on the BEM and VWA are similar, and for a fixed shell/core ratio, derived ACRI results in Aitken, accumulation and coarse modes are expressed identically. With coated BC becomes larger (i.e., from Aitken mode to coarse mode), induced errors of both particle scattering and absorption by the BEM and VWA are generally increasing. For Aitken, accumulation and coarse coated BC particles, induced errors of scattering cross sections by the VWA are within 6%, 15% and 40% while those of absorption cross sections are less than 3%, 4% and 116%, respectively, which are higher than corresponding errors caused by the optically effective ACRI (see Fig. 2g, 2h and 2i). The effective ACRIs given by the VWA are close to retrieved optically effective ACRI with differences of both n and k within 4% in Aitken and accumulation modes. Nonetheless, for coarse coated BC, prominent differences are found between the optically effective ACRIs based on limited look-up tables considering aerosol compositions and the ACRIs given by the VWA. As $D_{p} / D_{c}$ increases in the range of 1.1–2.7, retrieved imaginary parts of optically effective ACRIs of coarse coated BC decrease smoothly from $1.549 \times 10^{- 1}$ to $1.035 \times 10^{-2}$ , which are significantly lower than those of effective ACRIs based on the VWA by a factor of nearly 3. This indicates that the VWA can result in large uncertainties in coated BC absorption in coarse mode, and it overestimate the absorption of coarse coated BC by a factor of ~2 for large $D_{p} / D_{c}$ . This may be one of the reasons why modelled aerosol optical depth is 20% larger than observed [40], as the VWA is widely applied in the state-of-the-art aerosol-climate models. Overall, the VWA and EMT produce acceptable results for coated BC in Aitken and accumulation modes, whereas the optically effective ACRI, rather than the effective ACRI given by the VWA, may be considered for coarse internal-mixed particles in aerosol-climate models.

Fig. 2 Retrieved complex refractive indices of polydisperse coated BC with a core/shell structure based on different retrieval approaches and their induced differences of scattering and absorption cross sections as a function of shell/core in nuclei (a, d, g), accumulation (b, e, h), and coarse (c, f, i) modes, respectively. The solid, open and half-right interiors denote the results on the basis of the optically effective method, Bruggeman effective medium theory, and volume weighted average method, respectively. For the complex refractive indices, the black squares denote the real part while the red circles indicate the imaginary part. For the differences, the up triangles and down triangles denote the relative errors of scattering and absorption cross sections, respectively.

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4. Conclusions

The study explores numerical determinations of optically effective ACRIs from exactly calculated scattering and absorption properties of polydisperse coated BC particles, which may represent some internal-mixed aged BC particles under polluted urban environments. The core-shell Mie is employed to accurately calculate inherent optical properties of coated BC particles, whereas the Mie is utilized to retrieve their optically effective ACRIs without any particle shapes altered during retrieval.

Our exact results indicate that the optically effective ACRI of internal-mixed BC particles does have a physically-based meaning like real ACRI, unless limited ACRI look-up tables considering aerosol compositions are applied for retrieval. The look-up tables without limiting ACRI ranges based on known particle compositions could theoretically cause extreme real parts of retrieved optically effective ACRI of coarse coated BC with some shell/core ratios, which appear unreasonable. Nevertheless, if coated BC particles are in Aitken or accumulation modes, their optically effective ACRIs are not affect by the ACRI looking-up tables and have the physically-based sense.

Compared with retrieved optically effective ACRIs of coarse coated BC based on limited look-up tables with aerosol compositions considered, the imaginary parts approximated by the VWA are significantly higher by a factor of nearly 3. The VWA shows acceptable performance for coated BC in Aitken and accumulation modes. However, it could overestimate the absorption of coarse coated BC by a factor of ~2 for large $D_{p} / D_{c}$ , and this may be one of the reasons why modelled aerosol optical depth is 20% larger than observed [40]. Note that some other approaches are also applied to approximately determine the effective refractive index of internal-mixed particles, including the volume weighted average method and effective medium theory, in addition to the optically effective ACRI [e.g., 41,42]. Meanwhile, the relationship between the effective refractive index and effective mass density of aerosol particles has been explored empirically, and the consistency of the mixing rules with the index-density relationship is also examined [43]. Overall, this study suggests the optically effective ACRI, rather than the effective ACRI given by the VWA, should be considered for coarse internal-mixed particles in the state-of-the-art aerosol-climate models.

Funding

National Natural Science Foundation of China (NSFC) (91644224, 41505127, 21406189); Natural Science Foundation of Jiangsu Province (BK20150901); Natural Science Foundation of the Jiangsu Higher Education Institutions of China (15KJB170009); Open Foundation of Key Laboratory of Meteorological Disaster, Ministry of Education (KLME201810); Startup Foundation for introducing Talent of NUIST (2015r002, 2014r011); China Postdoctoral Science Foundation Funded Project (2016M591883)

Acknowledgments

We also gratefully appreciate the supports from Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase).

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Does optically effective complex refractive index of internal-mixed aerosols have a physically-based meaning?

Abstract

1. Introduction

2. Methodology: retrieval approach

3. Results and discussion

3.1 Optically effective ACRIs based on different look-up tables

3.2 Comparisons with the volume weighted average and effective medium theory

4. Conclusions

Funding

Acknowledgments

References

Cited By

Figures (2)

Tables (1)

Equations (4)

Optics Express

Refractive	Look-up table		Limited look-up table considering compositions
index	minimum	maximum	minimum	maximum
n	1.1	2.5	1.52	1.85
k	10⁻⁹	1.0	5*10⁻⁴	0.71