## Abstract

Soot aggregates (SAs)–fractal clusters of small, spherical carbonaceous monomers–modulate the incoming visible solar radiation and contribute significantly to climate forcing. Experimentalists and climate modelers typically assume a spherical morphology for SAs when computing their optical properties, causing significant errors. Here, we calculate the optical properties of freshly-generated (fractal dimension *D _{f}* = 1.8) and aged (

*D*= 2.6) SAs at 550 nm wavelength using the numerically-exact superposition

_{f}*T*-Matrix method. These properties were expressed as functions of equivalent aerosol diameters as measured by contemporary aerosol instruments. This work improves upon previous efforts wherein SA optical properties were computed as a function of monomer number, rendering them unusable in practical applications. Future research will address the sensitivity of variation in refractive index, fractal prefactor, and monomer overlap of SAs on the reported empirical relationships.

© 2015 Optical Society of America

## 1. Introduction

Soot is formed from high-temperature, incomplete combustion of hydrocarbon materials. In a typical combustion system, soot particles first get formed as spherical monomers (30-50 nm diameter), after which they undergo Brownian collisions in 3-dimensional (3-d) space to form clusters of monomers (hereafter referred to as “aggregates”) [1]. This aggregate formation mechanism is now well studied as the Diffusion Limited Cluster Aggregation (DLCA) [1, 2]. The resulting non-spherical morphology of the soot aggregates (SAs) can be represented by the fractal scaling law for length-scales much greater than the monomer size [1, 3, 4]:

where*N*is the number of particles enclosed within a radius of gyration

*R*,

_{g}*r*is the average monomer radius,

_{o}*D*is the non-integral fractal dimension, and

_{f}*k*is a constant prefactor. Thus, the structure of fractal aggregates can be completely defined using the parameters

_{o}*D*and

_{f}*k*, which quantify the aggregate geometry and packing factor of monomers, respectively [5]. Both parameters are governed by the aggregate formation mechanism and the aging processes encountered in the atmosphere [6]. Fresh aggregates formed by the DLCA mechanism have been universally observed to yield

_{o}*D*and

_{f}*k*values of ≈1.8 and 1.2, respectively [1, 4]. After release into the atmosphere, SAs could change their morphology owing to atmospheric processing to form more compact, near-spherical structures with a

_{o}*D*≈2.6 [1].

_{f}The significance of SA optical properties lies in the fact that these particles are one of the most important short-lived drivers of climate change. Uncertainty in direct radiative forcing (RF) estimates due to these particles currently plagues climate models [3, 7]. Estimation of RF from aerosols requires knowledge of their intrinsic optical properties–single scattering albedo (*ω*) and asymmetry parameter (*g*) as a function of wavelength [1]. The value of *ω* is a measure of brightness; it is 0 for a perfectly absorbing particle and 1 for a perfectly scattering particle. At a given wavelength, *ω* is defined as the ratio of the scattering cross-section (*C _{scat}*) to extinction cross-section, which is the sum of scattering and absorption (

*C*) cross-sections:

_{abs}The directionality of scattering by a particle is represented by *g*, which is empirically related to the hemispherical backscatter fraction *f*, a parameter experimentally measurable using an integrating nephelometer, as:

*b*is the total volume scattering coefficient at wavelength

_{s,i}*i*and

*b*is the corresponding back scattering coefficient [8]. The intrinsic optical properties of SAs have been shown to be very sensitive to aggregate morphology [9, 10].

_{bs,i}The current uncertainty in estimates of RF due to SAs is largely attributable to lack of accurate parameterization of wavelength-dependent particle optical properties in climate models and uncertainties in spatio-temporal soot mass distribution. The well-known Lorenz-Mie theory, which computes numerically-exact optical properties for homogeneous, spherical particles, cannot be applied to non-spherical SAs. As a convenient way out, current models usually approximate SA as equivalent spheres, and use Lorenz-Mie analytical solutions for estimating their radiative properties [7]. This approach often introduces large errors in RF estimations and gives rise to discrepancies between measurements and model predictions [9–11].

Several studies have been conducted in recent years to take into account the fractal nature of SAs and calculate their numerically-exact optical properties [3, 12, 13]. Without exception, these studies calculated the optical properties as a function of aggregate *D _{f}* and

*N*. These results, although highlight the stark differences in optical properties between spherical and fractal morphology, are ultimately found to be unusable by aerosol experimentalists and climate modelers in their research. Both

*D*and

_{f}*N*are very difficult to measure or infer directly from available aerosol instruments. Aerosol sizing instruments almost always measure size of a non-spherical particle in terms of an equivalent diameter, defined as the diameter of a sphere that exhibits identical property (e.g., aerodynamic, electrical mobility) to that of the investigated non-spherical particle [14].

Common instruments used for measuring equivalent aerosol diameters include the Scanning Mobility Particle Sizer (SMPS), the Single Particle Soot Photometer (SP2), and the Soot Particle - Aerosol Mass Spectrometer (SP-AMS). The SMPS, which is the most widely used aerosol size characterization instrument, measures a number size distribution for aerosols based on their electrical motilities [14]. It measures the particle mobility diameter, *d _{m}*, which is the diameter of a sphere having the same electrical mobility as that of an unknown particle. The SP2, on the other hand, calculates a mass equivalent diameter,

*d*, for SAs by measuring the visible-range radiation emitted from the refractory fraction of carbon in a particle as it is heated to its boiling point [15]. The SP-AMS is a combination of time-of-flight aerosol mass spectrometer (ToF AMS) and the SP2, and measures the vacuum aerodynamic diameter

_{me}*d*of a particle [16]. This diameter is defined as the diameter of a sphere with a density of 1 kg/m

_{Va}^{3}and has the same settling velocity as the investigated non-spherical particle in the free-molecular regime [14, 17].

The motivation behind this work was to provide simple mathematical formulations connecting numerically-exact optical properties of SAs and their measurable equivalent diameters. Computer simulation of a statistically significant number of 3-dimensional (3-d) SAs with *D _{f}* = 1.8 and 2.6, mimicking freshly-emitted and atmospherically aged particles, respectively, were first performed. Next, the resulting values of

*d*,

_{m}*d*and

_{me}*d*and corresponding single wavelength (550 nm) optical properties–

_{Va}*C*,

_{scat}*C*and

_{abs}*g*–were calculated for the aggregates. Optimization algorithms were subsequently employed to correlate the computed optical properties and equivalent diameters of the particles. Finally, simple empirical equations, with highest correlation among the parameters were tabulated.

## 2. Methods

A patented aerosol simulation package, *FracMAP* [18], was used to generate 3-d fractal aggregates using the particle-cluster aggregation technique. Details about the algorithms used in this simulation package can be found in recent publications [18, 19]. In brief, values of *N*, *D _{f}* and

*k*are specified by the user prior to starting the aggregate simulation. The aggregation algorithm initially proceeds by randomly attaching two monomers in point-contact to each other. This is followed by controlled addition of new monomers to the aggregate in point-contact such that resulting

_{o}*R*of the new aggregate satisfies Eq. (1) for the user-specified values of

_{g}*N*,

*D*and

_{f}*k*. After an aggregate is generated, the package identifies all its possible stable resting orientations by randomly rotating the particle in 3-d space and checking whether for each orientation, the aggregate center of mass rests above the area projected by three or more contact points of the fractal aggregate [18, 19]. If the center of mass lies within the area, the particular orientation is deemed as “stable”, and the package proceeds to generate a pixelated 2-dimensional (2-d) projection image of the aggregate in that orientation. The package finally analyzes the aggregate image to determine its various 2-d structural properties. This way,

_{0}*FracMAP*mimics the entire procedure from image generation to 2-d structural characterization of real-world aggregates (see Fig. 1) as performed using an electron microscope and image processing software (e.g. ImageJ) [19].

For this study, 3-d aggregates were generated with *D _{f}* values of 1.8 and 2.6 for representing freshly-emitted and aged states of SAs. The prefactor

*k*in our simulations was fixed at 1.19 per past recommendation [20]. For each

_{o}*D*, at least one hundred aggregates were generated with

_{f}*N*chosen randomly between 10 and 300 for each aggregate. A soot monomer diameter of 50 nm, typically observed in real-world particles [3], was used in our calculations. Each 3-d aggregate was oriented in 25 random orientations, with ~15 orientations, on an average, identified to be stable. For every identified stable orientation, three aggregate equivalent diameters –

*d*,

_{m}*d*and

_{me}*d*– were calculated.

_{Va}The *d _{m}* of an aggregate, in the transition flow regime, is equal to its stably-oriented projected area equivalent diameter as estimated from electron microscopy images [21]. Following this observation,

*d*size distributions for our simulated aggregates were calculated from their stably-oriented projected images. The

_{m}*d*of an aggregate was calculated as:

_{me}*d*( = 50 nm) is the diameter of monomers and

_{mono}*N*is the number of primary particles of an aggregate. The aggregate vacuum aerodynamic diameter

*d*was calculated using the established equation [14]:where ${\rho}_{p}$ is the particle density (1.8 kg/m

_{Va}^{3}for SA [7]:), ${\rho}_{o}$ is the standard density (1 kg/m

^{3}),

*d*is the volume equivalent diameter and $\chi $ is the dynamic shape factor in the free-molecular regime. In this work,

_{ve}*d*was assumed to be equal to

_{ve}*d*, implying that the aggregates do not have any internal voids [14]. This is a reasonable assumption for chain-like freshly-emitted SA, but aged SAs may have up to 10% internal voids [22]. The dynamic shape factor

_{me}*χ*, in a given flow regime, is a measure of the increase in drag force because of the non-spherical shape of the aggregate. The general expression for the shape factor for any flow regime is obtained as [14]:Here C is the Cunningham slip correction factor and $K{n}_{ve}$ and $K{n}_{m}$ are the values of the Knudsen number corresponding to

*d*and

_{ve}*d*, respectively. A mean free path of 4 m, representative of the high vacuum (10

_{m}^{−3}to10

^{−4}Pa) region at the end of the expansion zone in an AMS [14], was used for calculating

*χ*using Eq. (6).

The numerically-exact superposition *T*-matrix method [3, 12] was used to determine the optical properties, *C _{scat}*,

*C*and

_{abs}*g*, of the simulated aggregates as a function of

*N*and

*D*. This method expresses the incident and transmitted ﬁelds as series of vector spherical functions. The relationship between the incident and transmitted electromagnetic field is captured by a 2 x 2 transition super matrix (or the

_{f}*T*-matrix), which is a function of the intrinsic properties of the particle and the coordinate system [12, 23, 24]. Each element of the matrices within this super matrix is calculated by numerically integrating over the particle surface. Past computational investigations conducted using this method have shown that

*D*is an important parameter for accurately estimating the optical properties of an aggregate [25]. Interactions between the monomers are significant for chain-like structures and become even more important as aggregates age and form closely packed compact structures [3]. A complex refractive index of 1.95–0.79i was chosen for the soot monomers in this work following the recommendation of Bond and Bergstrom for atmospherically relevant SAs [22].

_{f}Relationships between optical properties and equivalent aerosol diameters were determined using a non-linear least squares optimization technique known as the Trust-Region-Reflective Least Squares, implemented using the MATLAB Curve Fitting Toolbox. The goodness of fit for these equations is characterized by adjusted R-squared (R^{2}) and Root Mean Square Error (RMSE).

## 3. Results and discussion

The calculated mean values of the three aerosol equivalent diameters are presented in Table 1. While *d _{me}* of an aggregate is independent of its

*D*, the

_{f}*d*values are observed to be decreasing with increasing

_{m}*D*. This decrease becomes significant (as much as 25%) with increasing

_{f}*N*. An explanation of this phenomenon is that

*D*= 1.8 aggregates are open-ended fractals, who yield a larger 2-d projected area and

_{f}*d*compared to the more collapsed and compact structured

_{m}*D*= 2.6 aggregates. An alternate way of stating this is that

_{f}*D*= 1.8 aggregates experience more drag force and therefore have lower mobility, which translates to a larger equivalent sphere diameter. From the perspective of drag force,

_{f}*D*= 1.8 aggregates have smaller terminal velocities, which results in smaller

_{f}*d*than those of

_{Va}*D*= 2.6 aggregates for a fixed

_{f}*N*. This effect of drag force on an aggregate is captured by the shape factor

*χ*(Eq. (6), which is a direct measure of the increase in drag force for a particle with its structural departure from spherical shape.

Empirical equations correlating the numerically-exact aggregate optical properties *C _{scat}*,

*C*and

_{abs}*g*with the three aerosol diameters,

*d*,

_{m}*d*, and

_{Va}*d*are reported for fresh (

_{me}*D*= 1.8) and aged (

_{f}*D*= 2.6) SAs (Tables 2-4). The aggregate optical cross-sections exhibit simple power-law dependencies on the equivalent aerosol diameters. For correlating

_{f}*g*, simple polynomial functional forms were found give the best fits to the data. At minimum, all the equations fits satisfy adjusted R

^{2}≥0.86 and RMSE < 0.03.

Figures 2-4 show the best fit curves for the relationship between aggregate optical properties and equivalent diameters. These plots are typically non-linear, owing to the complex electromagnetic interactions between the point-contacting monomers in an aggregate.

From Fig. 2, it can be seen that that the aggregate optical cross-sections of aged SAs have a steeper variation with mobility diameter compared to those of fresh SAs. Generally speaking, as aggregate *D _{f}* changes from 1.8 to 2.6, the aggregate morphology becomes more compact giving rise to intensification of monomer-monomer electromagnetic wave interactions. This manifests as a non-linear change in the aggregate scattering cross-sections and directionality which in turn affects the

*ω*and

*g*values. It must be noted that for a given value of

*d*, an aged SA would contains a larger number of monomers (see Table 1), or a larger amount of material that can scatter and absorb light, than a fresh SA. This effect is partly responsible for the large gap between the fresh and aged SA curves in Fig. 2. On the other hand,

_{m}*d*is only a function of

_{me}*N*and therefore the fresh and aged SA curves in Fig. 3 only reflect the increase in monomer-monomer interaction with aging. This significantly affects

*C*and

_{scat}*g*, while the change in

*C*due to aging is very small.

_{abs}The variation of optical properties with *d _{Va}* is steeper for fresh SA than for aged SA (Fig. 4), because a given value of

*d*corresponds to a larger number of monomers for a fresh SA than for an aged SA (Table 1). We found that for fresh SA, optical properties are not as strongly correlated with

_{Va}*d*as they are with

_{Va}*d*and

_{m}*d*. For fresh SA,

_{me}*d*shows a non-monotonic variation with

_{Va}*N*, wherein the same value of

*d*could correspond to two different values of

_{Va}*N*. In contrast, the investigated optical properties monotonically increase with

*N*. This leads to a many-to-one relationship between optical properties and

*d*, resulting in relatively lower R

_{Va}^{2}values for fresh SA (Table 4).

## 4. Recommendations and future work

As a first step towards bridging the gap between theoretical and applicable non-spherical aerosol optics, we have developed relationships between numerically-exact calculations of radiative properties of soot aggregates, equivalent aerosol (mobility, mass equivalent and vacuum aerodynamic) diameters, and fractal dimensions. We found that a larger fractal dimension corresponds to more intense interactions between monomers, from a more compact structure, and usually serves to intensify the variation of radiative properties with aggregate size. The empirical equations presented here are strictly valid for values of *D _{f}* close to 1.8 and 2.6, representative of freshly-emitted and aged soot aggregates, respectively. These equations represent the best estimates of radiative properties as a function of aerosol size as measured by commonly occurring aerosol instrumentation such as SMPS, SP2, and the SP-AMS.

It should be noted that variations in refractive index of SAs could alter the reported empirical relationships, especially for mass-equivalent aerosol diameter. Also noteworthy to mention is that freshly-emitted SAs may have variations in their fractal prefactors, while aged SAs may be hydrophilic in nature thereby resulting in different composition and/or surface structures than those simulated in this study. The fractal generation methodology and *T*-matrix calculations used in this work do not account for this effect. A thorough sensitivity analysis of varying fractal parameters and refractive index on the equivalent diameter and optical properties of SAs is needed. Future work is also needed to investigate the influence of monomer overlap and sintering in comparison to point-contacting (as done in this study) on the variation in optical properties.

## Acknowledgments

This material is based upon work supported by the National Science Foundation under Grant No. AGS1455215 and by NASA ROSES under Grant No. NNX15AI66G.

## References and Links

**1. **R. K. Chakrabarty, N. D. Beres, H. Moosmüller, S. China, C. Mazzoleni, M. K. Dubey, L. Liu, and M. I. Mishchenko, “Soot superaggregates from flaming wildfires and their direct radiative forcing,” Sci. Rep. **4**, 5508 (2014). [CrossRef] [PubMed]

**2. **C. Sorensen, “Light scattering by fractal aggregates: a review,” Aerosol Sci. Technol. **35**(2), 648–687 (2001). [CrossRef]

**3. **L. Liu, M. I. Mishchenko, and W. P. Arnott, “A study of radiative properties of fractal soot aggregates using the superposition T-matrix method,” J. Quant. Spectrosc. Radiat. Transf. **109**(15), 2656–2663 (2008). [CrossRef]

**4. **C. M. Sorensen, J. Cai, and N. Lu, “Light-scattering measurements of monomer size, monomers per aggregate, and fractal dimension for soot aggregates in flames,” Appl. Opt. **31**(30), 6547–6557 (1992). [CrossRef] [PubMed]

**5. **M. K. Wu and S. K. Friedlander, “Note on the power law equation for fractal-like aerosol agglomerates,” J. Colloid Interface Sci. **159**(1), 246–248 (1993). [CrossRef]

**6. **R. Zhang, A. F. Khalizov, J. Pagels, D. Zhang, H. Xue, and P. H. McMurry, “Variability in morphology, hygroscopicity, and optical properties of soot aerosols during atmospheric processing,” Proc. Natl. Acad. Sci. U.S.A. **105**(30), 10291–10296 (2008). [CrossRef] [PubMed]

**7. **T. C. Bond, S. J. Doherty, D. Fahey, P. Forster, T. Berntsen, B. DeAngelo, M. Flanner, S. Ghan, B. Kärcher, D. Koch, S. Kinne, Y. Kondo, P. K. Quinn, M. C. Sarofim, M. G. Schultz, M. Schulz, C. Venkataraman, H. Zhang, S. Zhang, N. Bellouin, S. K. Guttikunda, P. K. Hopke, M. Z. Jacobson, J. W. Kaiser, Z. Klimont, U. Lohmann, J. P. Schwarz, D. Shindell, T. Storelvmo, S. G. Warren, and C. S. Zender, “Bounding the role of black carbon in the climate system: A scientific assessment,” J. Geophys. Res. Atmos. **118**(11), 5380–5552 (2013). [CrossRef]

**8. **E. Andrews, P. Sheridan, M. Fiebig, A. McComiskey, J. Ogren, P. Arnott, D. Covert, R. Elleman, R. Gasparini, D. Collins, H. Jonsson, B. Schmid, and J. Wang, “Comparison of methods for deriving aerosol asymmetry parameter,” J. Geophys. Res. Atmos. **111**(D5), D05S04 (2006). [CrossRef]

**9. **C. D. Cappa, T. B. Onasch, P. Massoli, D. R. Worsnop, T. S. Bates, E. S. Cross, P. Davidovits, J. Hakala, K. L. Hayden, B. T. Jobson, K. R. Kolesar, D. A. Lack, B. M. Lerner, S. M. Li, D. Mellon, I. Nuaaman, J. S. Olfert, T. Petäjä, P. K. Quinn, C. Song, R. Subramanian, E. J. Williams, and R. A. Zaveri, “Radiative absorption enhancements due to the mixing state of atmospheric black carbon,” Science **337**(6098), 1078–1081 (2012). [CrossRef] [PubMed]

**10. **H. Li, C. Liu, L. Bi, P. Yang, and G. W. Kattawar, “Numerical accuracy of “equivalent” spherical approximations for computing ensemble-averaged scattering properties of fractal soot aggregates,” J. Quant. Spectrosc. Radiat. Transf. **111**(14), 2127–2132 (2010). [CrossRef]

**11. **R. K. Chakrabarty, H. Moosmüller, W. P. Arnott, M. A. Garro, J. G. Slowik, E. S. Cross, J.-H. Han, P. Davidovits, T. B. Onasch, and D. R. Worsnop, “Light scattering and absorption by fractal-like carbonaceous chain aggregates: comparison of theories and experiment,” Appl. Opt. **46**(28), 6990–7006 (2007). [CrossRef] [PubMed]

**12. **H. Moosmüller, R. Chakrabarty, and W. Arnott, “Aerosol light absorption and its measurement: A review,” J. Quant. Spectrosc. Radiat. Transf. **110**(11), 844–878 (2009). [CrossRef]

**13. **M. I. Mishchenko, N. T. Zakharova, G. Videen, N. G. Khlebtsov, and T. Wriedt, “Comprehensive T-matrix reference database: A 2007-2009 update,” J. Quant. Spectrosc. Radiat. Transf. **111**(4), 650–658 (2010). [CrossRef]

**14. **P. F. DeCarlo, J. G. Slowik, D. R. Worsnop, P. Davidovits, and J. L. Jimenez, “Particle morphology and density characterization by combined mobility and aerodynamic diameter measurements. Part 1: Theory,” Aerosol Sci. Technol. **38**(12), 1185–1205 (2004). [CrossRef]

**15. **M. Laborde, M. Schnaiter, C. Linke, H. Saathoff, K. Naumann, O. Möhler, S. Berlenz, U. Wagner, J. Taylor, D. Liu, M. Flynn, J. D. Allan, H. Coe, K. Heimerl, F. Dahlkötter, B. Weinzierl, A. G. Wollny, M. Zanatta, J. Cozic, P. Laj, R. Hitzenberger, J. P. Schwarz, and M. Gysel, “Single Particle Soot Photometer intercomparison at the AIDA chamber,” Atmos. Meas. Tech. **5**(12), 3077–3097 (2012). [CrossRef]

**16. **T. Onasch, A. Trimborn, E. Fortner, J. Jayne, G. Kok, L. Williams, P. Davidovits, and D. Worsnop, “Soot particle aerosol mass spectrometer: development, validation, and initial application,” Aerosol Sci. Technol. **46**(7), 804–817 (2012). [CrossRef]

**17. **M. Lindskog and E. Nordin, *Ageing of Diesel Aerosols: Design and Implementation of a Teflon Simulation Chamber* (Lund University, 2009).

**18. **R. K. Chakrabarty, M. A. Garro, S. Chancellor, C. M. Herald, and H. Moosmüller, “FracMAP: A user-interactive package for performing simulation and orientation-specific morphology analysis of fractal-like solid nano-agglomerates,” Comput. Phys. Commun. **180**, 1376–1381 (2009). [CrossRef]

**19. **R. K. Chakrabarty, M. A. Garro, B. A. Garro, S. Chancellor, H. Moosmüller, and C. M. Herald, “Simulation of aggregates with point-contacting monomers in the cluster–dilute regime. Part 1: Determining the most reliable technique for obtaining three-dimensional fractal dimension from two-dimensional images,” Aerosol Sci. Technol. **45**(1), 75–80 (2011). [CrossRef]

**20. **C. M. Sorensen and G. C. Roberts, “The prefactor of fractal aggregates,” J. Colloid Interface Sci. **186**(2), 447–452 (1997). [CrossRef] [PubMed]

**21. **K. Park, D. B. Kittelson, and P. H. McMurry, “Structural properties of diesel exhaust particles measured by transmission electron microscopy (TEM): Relationships to particle mass and mobility,” Aerosol Sci. Technol. **38**(9), 881–889 (2004). [CrossRef]

**22. **T. C. Bond and R. W. Bergstrom, “Light absorption by carbonaceous particles: An investigative review,” Aerosol Sci. Technol. **40**(1), 27–67 (2006). [CrossRef]

**23. **D. W. Mackowski and M. I. Mishchenko, “Calculation of the T-matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A **13**(11), 2266–2278 (1996). [CrossRef]

**24. **M. I. Mishchenko, L. D. Travis, and A. A. Lacis, *Scattering, Absorption, and Emission of Light by Small Particles* (Cambridge University Press, 2002).

**25. **L. Liu and M. I. Mishchenko, “Effects of aggregation on scattering and radiative properties of soot aerosols,” J. Geophys. Res. Atmos. **110**(D11), D11211 (2005). [CrossRef]