Directional dependence of thermal emission from nonspherical carbon particles

Aerosol Science 40 (2009) 790 -- 801

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Directional dependence of thermal emission from nonspherical carbon particles Nobuhiro Motekia, ∗ , Yutaka Kondoa , Nobuyuki Takegawaa , Shin-ichi Nakamurab a b

Research Center of Advanced Science and Technology, University of Tokyo, Room-414 Bldg.-3, 4-6-1, Komaba, Meguro-ku, Tokyo 153-8904, Japan Center for Instrumental Analysis, College of Science and Engineering, Aoyama Gakuin University, 5-10-1 Fuchinobe, Sagamihara-shi, Kanagawa 229-8558, Japan

A R T I C L E

I N F O

Article history: Received 21 March 2009 Received in revised form 9 May 2009 Accepted 11 May 2009

Keywords: Particle shape Thermal emission Light scattering Laser-induced incandescence Single-particle

A B S T R A C T

The directional intensity distribution of thermal emission from nonspherical particles can be predicted in the geometric optics limit using the Kirchhoff's law, or, in general case, using the Rytov's theory based on the fluctuation–dissipation theorem. This study demonstrates the first experimental evidence of the directional variation of thermal emission from nonspherical particles of a size smaller than the geometric optics limit. We used a method of laser-induced incandescence with multi-angle detectors to observe the directional dependences of thermal emission from individual carbon particles. For laboratory carbon particles with various shapes, the measured directional dependences of thermal emission were consistent with theoretical calculations for model nonspherical particles. This study provides a new physical principle for measuring the shape of aerosols according to the directional dependence of their thermal emission and is especially useful for online, in situ shape classification of carbon particles. © 2009 Elsevier Ltd. All rights reserved.

1. Introduction Online, in situ measurement of aerosol particle shape is important for detecting hazardous particles (e.g., asbestos) in a working environment. In addition, a priori information regarding particle shape is necessary for data interpretation in aerodynamic and mobility-based particle sizing, and in the remote sensing of aerosols. Validation of lidar remote sensing of aerosol properties (e.g., Sassen, 2000, chapter 14) requires in situ, online observation of particle shapes in atmospheric aerosols. The angular dependence of light scattering has been used as a principle for on-line, in situ measurements of the nonsphericity of aerosol particles (Barton, ¨ Hirst, Kaye, & Clark, 2000; Sachweh, Barthel, Polke, Umhauer, & Buttner, 1999). No other physical principle, other than this angular light-scattering method, has been used for online and in situ classification of individual particle shapes. To compensate for the inherent ambiguities of the angular light-scattering method, a new independent physical principle for online, in situ measurement of particle shape is desired. This paper demonstrates evidence of the directional dependence of thermal emission from nonspherical carbon particles with sizes comparable to the wavelength, using a method of single particle measurement of laser-induced incandescence (Moteki & Kondo, 2007; Stephens, Turner, & Sandberg, 2003). A priori information regarding the shape of the carbon particles is obtained based on transmission electron microscope images of bulk samples and the mass-to-mobility relationship of the aerosolized carbon samples. Theoretical predictions of the directional dependence of the thermal emission from model nonspherical carbon particles were found to be consistent with measurements of laboratory carbon particles with various shapes. In addition to the thermal emission, light-scattering properties of the carbon particles were simultaneously measured as independent data supporting the validity of the model calculations. This study demonstrates that the degree of directional dependence of thermal ∗ Corresponding author. Fax: +81 3 5452 5148. E-mail address: [email protected] (N. Moteki). 0021-8502/$ - see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.jaerosci.2009.05.003

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emission varies substantially with particle shape, demonstrating that multi-directional detection of thermal emission is a new physical principle that can be applied to single-particle measurement of particle shape, especially of carbon particles. 2. Physical basis Kirchhoff's law states the equality between radiative emissivity and absorptivity for each direction, wavelength, and polarization. The original form of Kirchhoff's law was derived from the geometric optics limit (i.e., any curvature of the body is much larger than the wavelength), under which any effects of diffraction can be neglected (Planck, 1914). Because of the manifestation of diffraction, the applicability of Kirchhoff's law is nontrivial for bodies of a size comparable to or smaller than the wavelength, as is the case for atmospheric aerosols or any other small particles not much larger than the wavelength under consideration. For spherical particles, Kirchhoff's law has been extended beyond the geometric optics limit. For spherical particles of arbitrary size compared to the wavelength, the equivalence of emissivity to absorption efficiency (i.e., the ratio of absorption cross-section to geometric cross-section) has been predicted from the principle of detailed balance while considering radiative equilibrium in an opaque cavity (Bohren & Huffman, 1983; Landau & Lifshitz, 1980). Laboratory experiments by Egan and Hilgeman (1984) demonstrated the validity of the prediction for di-2-ethylhexyl sebacate droplets. In recent papers using physical modeling of the thermal emission from small particles, spherical particle shape was commonly assumed because of the validated formula of emissivity (e.g., Filippov, Markus, & Roth, 1999; Hansen & Campbell, 1998; Melton, 1984; Moteki & Kondo, 2007). Although past studies on laser-induced incandescence (LII) have used a model of fractal aggregates to calculate emissivity of a volume ¨ u, ¨ 1998; Koyl ¨ u, ¨ 1996; Schulz et al., 2006), these studies did not consider containing many soot particles (Farias, Carvalho, & Koyl the emissivity of single particles. In contrast to spherical particles, it is impossible to deduce the directionality of thermal emission from nonspherical particles of a size comparable to the wavelength by the conventional argument that considers radiative equilibrium in an opaque cavity (Landau & Lifshitz, 1980). Rytov (1953) was the first to theoretically deduce the directionality of thermal emission from a nonspherical body with arbitrary size: he used the fluctuation–dissipation theorem (e.g., Bekefi, 1966; Callen & Welton, 1951; Landau & Lifshitz, 1980) to calculate the thermal radiation originating from micro-scale fluctuating electric currents inside a body under boundary conditions associated with shape. In addition, Tsang (1984) derived succinct formulas for thermal emission from arbitrary particles on the basis of a similar physical idea. Notably, the conventional Kirchhoff's law is correct only under the geometric optics limit, and the law's extension to spherical particles with arbitrary size can be regarded as special cases of Rytov's theory. In spite of these theoretical developments, no experimental evidence has been reported regarding Rytov's theory for nonspherical particles of a size smaller than the range of the geometric optics limit. 3. Methods 3.1. Single-particle detection of thermal emission and light scattering A single-particle soot photometer (Baumgardner, Kok, & Raga, 2004; Gao et al., 2007; Moteki & Kondo, 2007; Moteki & Kondo, 2008; Schwarz et al., 2006; Stephens et al., 2003) was slightly modified to measure the directional dependences of thermal emission and light scattering from individual particles (Fig. 1). Particles are introduced to the laser beam through an aerosol jet directed orthogonal to the plane parallel to the detectors. Individual particles pass though a Gaussian laser beam (i.e., TEM00 mode) of ∼1 mm diameter within ∼30 s, with the transit velocity determined solely by the velocity of the sheath flow of the aerosol jet. Four optical detectors are mounted in the horizontal plane at mutually orthogonal positions. A light-collection lens in front of each detector collects either thermal emission or scattered light within a cone of 30◦ half angle (i.e., solid angle 1 and 2 shown in Fig. 1). Two identical photo-multiplier tubes (PMT model H6779, Hamamatsu Photonics, Inc., Japan) for visible thermal emission ( = 350–550 nm) detection are placed on one side of the intra-cavity laser beam (upper side in Fig. 1), whereas two Si-avalanche photodiodes (Si-APD; Model C30916E and C30927E, Perkin Elmer, Inc., USA) for light-scattering ( = 1064 nm) detection are mounted on the other side of the laser beam (under side in Fig. 1). In this paper, we refer to the differential scattering cross-section of a particle integrated over the solid angle of light collection of the photometer (i.e., 1 or 2 in Fig. 1) as the partial scattering cross-section. Following the notation of Moteki and Kondo (2008), the position of a particle inside the laser beam is measured as the distance from the center of the Gaussian, in units of the standard deviation () of the Gaussian function. The C30927E light-scattering detector is a position-sensitive detector (direction-2 in Fig. 1) that can estimate the position of a particle inside the laser beam from the signal waveform (Gao et al., 2007; Schwarz et al., 2008; Shiraiwa et al., 2008). If the position of the particle inside the laser beam is known, the partial scattering cross-section can be estimated from the scattering waveforms at arbitrary positions in the laser beam (Gao et al., 2007; Moteki & Kondo, 2008). For the thermal emission channels, interference from the 1064-nm intra-cavity Nd:YAG laser or 808-nm pumping diode laser is negligible because the sensitivity of the detector at a wavelength of 808 nm or longer is zero. For each light-scattering channel, a long-pass optical filter RG-850 (Schott, Inc.) is placed in the lens system to block stray light from the 808-nm pumping laser. For carbon particles, only the time domain of the light-scattering signals before onset of visible thermal emission is used for analysis of the scattering properties to avoid interference from thermal emission. The cutoff frequency of the low-pass filter of the electronic amplifiers was adjusted to be 5 MHz, which is equal to the data acquisition frequency (i.e., every 0.2 s) of the analog-to-digital converter to obtain maximum information from the signal waveforms.

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300-550 nm Thermal emission Direction-2

300-550 nm Thermal emission Direction-1

T

PM

PM

T

Aerosol jet

ΔΩ1

ΔΩ2

AP D 1064 nm Light-scattering Direction-2 (Position-sensitive)

ΔΩ1

1064 nm Intra-cavity laser beam

D

Detection volume

AP

ΔΩ2

1064 nm Light-scattering Direction-1

Fig. 1. Schematic diagram of the modified single-particle soot photometer used to measure the directional dependences of thermal emission and light scattering.

3.2. Theoretical response to nonspherical aerosols This section describes the theoretical formulation of the thermal emission and light-scattering signals measured by the photometer that were used for the physical interpretation of the experimental results in the following chapters. For calculating thermal emission, the particle temperature T was assumed to be homogeneous inside the particle. To support this assumption, we made a simple calculation of the thermal diffusion inside a spherical graphite particle. The time for heat to cross a 1.0-mdiameter particle was estimated to be 1 ns, several orders of magnitude shorter than the typical time for notable temperature change of a graphite particle in the laser beam of the photometer (i.e., 0.1–1 s) estimated by model calculations (Moteki & Kondo, 2007). Thermal emission from a nonspherical particle was formulated on the basis of Rytov's theory: the emission cross-section (i.e., the emissivity multiplied by the geometric cross-section) of thermal radiation is equivalent to the absorption cross-section Cabs for radiation propagating in the negative direction for each direction, each wavelength, and each state of polarization (Bekefi, 1966; Rytov, 1953). For a thermal emission detector that collects light over the solid angle , the signal amplitude STE as a function of time t can be expressed as   () Pe (T(t), ) Cabs (, v, shape, , , , , t) d d, (1) STE (t) =  

where v is the particle volume; , , and  the set of Euler's angle of rotation of the particle;  the range of wavelength considered; and  the spectral responsivity of the detector. The Planck function Pe in unit of radiance Pe (T, ) =

2hc2 5

 (ehc/kT − 1)

(2)

should be used in Eq. (1), where h is the Planck constant; c the light velocity; k the Boltzmann constant. For the optical system of the single-particle soot photometer (Fig. 1), the STE ratio for two directions 1 and 2 (i.e., STE−1 (t) and STE−2 (t)) is   STE−1 (t)   () Pe (T(t), ) Cabs (, v, shape, , , , , t) d d   RTE (t) = . (3) =  1 STE−2 (t) 2  () Pe (T(t), ) Cabs , v, shape, , , , , t d d The value of RTE (t) deviates from unity only for nonspherical particles because of the different absorption cross-sections between directions 1 and 2 . Within a time domain of constant temperature and constant particle volume, RTE (t) is a function only of

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Table 1 Properties of the laboratory carbon particles used in this study. Sample name

Supplier and product information

Selection mobility diameter (nm)

Mass for this mobility diameter (fg)

Product of shape factor and void factor k0 and its uncertainty (%)

Shape model consistent with TEM images and mass-to-mobility relationship

Shape model consistent with (s.d. of RSC , s.d. of RTE ) value

Glassy carbon

Alfa Aesar, USA, stock# 38 001 Acheson, USA, ultra-fine graphite Alfa Aesar, USA, 300 mesh, stock# 10 130 Alfa Aesar, USA, stock# 41 773 Alfa Aesar, USA, stock# 40 971

600

210

1.0 (4%)

Sphere

Sphere

750

110

1.6 (10%)

700

370

1.0 (4%)

750

310

1.1 (4%)

900

190

1.6 (10%)

Spheroid of 0.2 < q < 0.3 Spheroid of 0.4 < q < 0.6 Spheroid of 2 < q < 2.5 Fractal aggregate with 2.3 < Df < 2.5

900

410

1.2 (5%)

900

310

1.3 (6%)

Spheroid of 0.1 < q < 10 Spheroid of 0.5 < q < 2 Spheroid of 0.3 < q < 3 Fractal aggregate with dpp = 50 nm, Df = 2.65 Fractal aggregate with dpp = 20 nm, Df = 3.0 Fractal aggregate with dpp = 20 nm, Df = 3.0

Aquadag Microcrystalline graphite Colloidal graphite Fullerene soot

Aqua-Black 001 Aqua-Black 162

Tokai carbon, Japan, carbon black ink Tokai carbon, Japan, carbon black ink

Fractal aggregate with 2.8 < Df < 3.0 Fractal aggregate with 2.8 < Df < 3.0

the shape and orientation of the particle. Within this time domain, RTE (t) is a useful measure for observing correlations between the particle shape and the directional dependence of thermal emission. For a light-scattering detector that collects light over the solid angle , the signal amplitude SSC as a function of time t can be expressed as SSC (t) = I(t) 



dCSC

 d

(v, shape, , , , , t) d,

(4)

where I is the irradiance of incident laser beam,  the responsivity of the detector, and dCSC /d the differential scattering crosssection. For optical system of the single-particle soot photometer (Fig. 1), the ratio of SSC (t) for two directions 1 and 2 (i.e., SSC−1 (t) and SSC−2 (t)) is  SSC−1 (t)  RSC (t) = = 1 SSC−2 (t) 2

dCSC d dCSC d

(v, shape, , , , , t) d (v, shape, , , , , t) d

.

(5)

Because of the equivalence of two incident waves traveling in opposite directions, the partial scattering cross-sections for the directions 1 and 2 are the same for spherical particles (see Fig. 1). Therefore, the value of RSC (t) deviates from unity only for nonspherical particles because of the different partial scattering cross-sections between directions 1 and 2 . Within a time domain of constant particle volume, RSC (t) is a function only of shape and orientation of the particle. 3.3. Laboratory particles Seven commercially available carbon particles (Table 1) were used in experiments to investigate the effect of particle shape on the directional dependence of thermal emission and light scattering as described in the following sections. Transmission electron microscope (TEM; Model JEM-4010, JEOL, Inc., Japan) images of bulk samples were taken to investigate the morphologies of structural elements which consist of the aerosolized samples. Glassy carbon is a collection of spherical carbon particles with diameters ranging from about 0.1 to 10 m (Fig. 2a). Aquadag is a collection of small fragments of thin plates of crystalline graphite (Fig. 2b). Unfortunately, TEM images of microcrystalline graphite could not be taken for this study because of a restriction in the experimental schedule, but many microscope images of microcrystalline graphite are available in the literature (e.g., Wang, Gai, Yang, & Shen, 2008). According to the literature, the typical shape of microcrystalline graphite powder is that of a nonspherical solid with a stout aspect ratio. A TEM image of colloidal graphite appears in Moteki and Kondo (2007); that image shows a collection of irregular fragments of graphite. Aqua-Black 001 is a collection of aggregates of small primary particles with diameters of approximately 20 nm (Fig. 2c). TEM images of Aqua-Black 162, not shown in this paper, were very similar to those of Aqua-Black 001. Fullerene soot is a collection of aggregates of small carbon spherules of approximately 50 nm diameter (Fig. 2d). Primary particles of fullerene soot tend to conglutinate with each other. The bulk carbon samples were aerosolized by atomizing their pure water suspensions and drying them in a diffusion dryer with silica gel. The dried aerosol particles were electrically neutralized and size-selected by a differential mobility analyzer (DMA; Model3081, TSI, Inc., USA) before being introduced into the single-particle soot photometer. Aerosolized glassy carbon aerosols downstream of the DMA were isolated single spheres, and other aerosolized samples were likely aggregates consisting of small spherical or nonspherical elements according to their mass-to-mobility relationships (Appendix A). Temporal changes of the partial scattering cross-section (Moteki & Kondo, 2008) of individual carbon particles inside the laser beam of the photometer were measured to estimate the amount of volatile coating on them. The selection mobility diameters

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1 µm

200 nm

200 nm

200 nm

Fig. 2. Transmission electron microscope images of laboratory carbon particles: (a) glassy carbon (Alfa Aesar Inc.); (b) Aquadag (Acheson Inc.); (c) Aqua-Black 001 (Tokai Carbon Inc.); and (d) Fullerene soot (Alfa Aesar).

for this test were set to be 200, 300, and 500 nm. In this procedure, the laser power was reduced accordingly (approximately half of the ordinary condition) to delay the onset of thermal emission. Delaying the onset of thermal emission increases the time domain in which the light-scattering signal is available for analysis. For all samples and selection mobility diameters, the average reduction of the partial scattering cross-section at the point of onset of thermal emission (i.e., at about the −1.3 position) relative to that at the leading edge (i.e., before the −2.5 position) was < 30%. The average reduction of the partial scattering cross-sections was not altered by the insertion of a 400 ◦ C thermo-denuder upstream of the photometer. Therefore, reduction of the partial scattering cross-section in the laser beam before the onset of thermal emission is mainly due to evaporation of some of the coating materials that are not vaporized under high temperatures up to 400 ◦ C. Considering that the scattering cross-section for an arbitrary particle, which is small compared to the wavelength, is proportional to square of its volume, the aerosolized carbon samples might have coating materials up to ∼15% by volume. A 15% change in volume corresponds to a 5% change in size, which will not significantly alter the shape of a particle. Therefore, we neglect any possible effects of coatings on the interpretation of experimental results in the following sections. 4. Results and discussion 4.1. Nonsphericity of laboratory particles In the experiment to investigate the directional dependence of the thermal emission and light scattering from individual carbon particles, aerosolized particles were mobility-selected upstream of the photometer. These mobility diameters (Table 1) were chosen by the following three criteria: (1) the intensities of the thermal emission signals become comparable (i.e., agree to within a factor of 1.5) among different samples; (2) the absolute intensity of thermal emission and light scattering are high enough

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to neglect any effects of background noise; and (3) particle sizes should not be much smaller than the wavelength of thermal emission measured ( = 350–550 nm), otherwise any effects of particle shape on thermal emission diminish, as explained in Section 4.4. To obtain information regarding the shape of these particles with known mobility diameter, the mass distribution of the mobility-selected particles was investigated by a system connecting the DMA and an aerosol particle mass analyzer (APM; Model-302, KANOMAX, Inc., Japan) (Ehara, Hagwood, & Coakley, 1996) in tandem. For all samples, the mass distributions showed a single mode (the mode masses are listed in Table 1), indicating that the morphology of the mobility-selected particles was uniform for each sample. As an indicator of particle nonsphericity, values of the product of the dynamic shape factor k0 and void factor , which approximately equals the ratio of the mobility diameter to the mass equivalent diameter (see Appendix A for details), are summarized in Table 1. As discussed in detail in Appendix A, the value of k0 for a spheroidal particle is approximately equal to that of other plate- or needle-like particles with the same aspect ratio. For nearly spherical and plate- or needle-like particles, as determined from their TEM images (i.e., glassy carbon, Aquadag, microcrystalline graphite, and colloidal graphite), the possible ranges of the aspect ratio q (q < 1 for oblate and q > 1 for prolate) of the spheroids for the given k0 values are shown as an indicator of the particles' nonsphericity. In contrast, for aggregate particles consisting of very small primary particles (i.e., fullerene soot and Aqua-Black 001 and 162), the typical primary particle diameter dpp and fractal dimension Df estimated from their TEM images and mass-to-mobility relationships (see Appendix A) are listed in Table 1. 4.2. Waveforms of STE and SSC for spherical and nonspherical particles Measured waveforms of thermal emission and light scattering from individual particle are shown in Fig. 3 for (a) Glassy carbon with dmob = 600 nm and (b) Aquadag with dmob = 750 nm. According to the discussions in the previous sections, glassy carbon and Aquadag particles are likely to be spherical and highly nonspherical, respectively. For both particles, the position of the onset of thermal emission was estimated to be about −2 from the center of the laser beam by waveform analysis of the positionsensitive detector. The relationship between the physical parameters of a particle (e.g., changes of temperature or diameter) and waveforms of thermal emission and light scattering can be interpreted as follows based on the simulation of Moteki and Kondo (2007). The steep increases in thermal emissions STE (t) during t = 22–23 s are due to heating up to ∼4500 K. The duration of thermal emission signals was ∼10 s, which corresponds to a ∼1.5 distance in the Gaussian laser beam. Therefore, the thermal emission decreases to a negligible level due to evaporation before the particle reaches the center of the laser beam. The increase in light scattering SSC (t) during t = 16–22 s corresponds to the increase of incident irradiance I(t) as the particle transits toward the center of the laser beam. Gradual reductions of STE (t) after the maximum point are due to evaporation of the carbonaceous core. For quantitative observation of the directional dependences of thermal emission and light scattering, the temporal fluctuations of RTE (t) and RSC (t) were analyzed. Changes in particle volume v(t) and temperature T(t) should be as small as possible in the time domains for this analysis, because the shape dependences of RTE and RSC change with these parameters according to Eqs. (3) and (5). According to the model simulations of Moteki and Kondo (2007), the changes in particle volume and temperature were relatively slow in the time domain of intense thermal emission around the STE (t) peak. For these reasons, a 4-s domain starting from the point 1 s after onset of thermal emission was used for observing the fluctuations of RTE (t). In contrast, a 4-s domain that ends at the point of onset of thermal emission was used for observing the fluctuations of RSC (t). In this time domain, the

3000

Light-scattering SSC-1 SSC-2

400

2000

200

1000

0

0 16

18

20

22

24 t (µs)

26

28

30

32

Aquadag 800

4000

Thermal emission STE-1 STE-2

600

3000

Light-scattering SSC-1 SSC-2

400

2000

1000

200

Light-scattering signal (2.44 mV)

600

Thermal emission signal (2.44 mV)

4000 Thermal emission STE-1 STE-2

Light-scattering signal (2.44 mV)

Thermal emission signal (2.44 mV)

Glassy carbon 800

5000

1000

5000

1000

0

0 16

18

20

22

24

26

28

30

32

t (µs)

Fig. 3. Measured waveforms of thermal emission and light-scattering signals for observing directions-1 and -2: (a) spherical particles (glassy carbon) and (b) highly nonspherical particles (Aquadag).

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Light-scattering ratio

2.0 Glassy carbon Aquadag 1.5 1.0

2.0

1.0 0.5

0.0

0.0 19.0

20.0 t (µs)

21.0

Glassy carbon Aquadag

1.5

0.5

18.0

Thermal emission ratio

2.5 RTE (= STE-1/STE-2)

RSC (= SSC-1/SSC-2)

2.5

23.0

24.0

25.0 t (µs)

26.0

27.0

Fig. 4. Time-dependent signal ratios of direction-1 to -2 for the waveforms shown in Fig. 3: (a) light scattering and (b) thermal emission. The time domains shown in (a) and (b) correspond to 4-s periods before the onset of thermal emission and around the peak of thermal emission, respectively.

change in particle volume (i.e., evaporation) was small and the waveforms of SSC (t) were well above their background noise level without any interference from thermal emission. The time-series of RSC (t) and RTE (t) for the data shown in Fig. 3 in each specified time domain are shown in Fig. 4. The ratios of light-scattering waveforms RSC (t) shown in Fig. 4a clearly show that the temporal fluctuation of RSC (t) is much stronger for nonspherical Aquadag than for spherical glassy carbon. The vigorous fluctuation of RSC (t) for nonspherical Aquadag is evidence of free rotation of the particles in a laser beam because the partial scattering cross-section of nonspherical particles depends on the orientation (, , ) relative to the coordinate system fixed on the detectors. The time-cycle of the particle rotation certainly has Fourier components of between about 0.1 and 10 s; otherwise the fluctuations of RSC would not be observable in the 4-s time domain of Fig. 4a. Many possible mechanisms might induce the particle rotation, for example, Brownian motion, propulsions by particle evaporation, and angular momentum transfers between radiation and particle. Because of the complexity, any theoretical explanations of observed rotation speed are beyond the scope of this paper. As in the case of light scattering, only the thermal emission ratio RTE (t) for Aquadag fluctuated with time (Fig. 4b), with a typical time-cycle similar to the fluctuation of RSC (t), demonstrating the directional dependence of the thermal emission for nonspherical particles. Interpretation of this directional dependence on the basis of Eq. (3) reveals that changes in the emission cross-section ( = absorption cross-section Cabs ) with particle orientation (i.e., , , ) result in differences in the intensity of thermal emission toward each detector. In the next section, the observed standard deviations of RTE and RSC are compared to their model calculations to validate this theoretical interpretation of directional dependence of thermal emission. 4.3. Model calculations of RTE and RSC For the accurate interpretation of the experimental results that are shown later, we calculated RTE and RSC for various model nonspherical particles using Eqs. (3) and (5), respectively. Spheroids and cuboids with various aspect ratios and fractal aggregates of various fractal dimensions Df were used for the model nonspherical particles. Independent of the particle shape, the mass equivalent diameter of the model nonspherical particles was fixed at dm = 500 nm, which is comparable to the range of dm = 470–750 nm in the experiments. The aspect ratio q of spheroids and cuboids was varied over the range 0.1 < q < 10. A computer program with tunable particle–cluster aggregation has been developed using the algorithm of Filippov, Zurita, and Rosner (2000) to generate fractal aggregates of various Df values ranging from 2.2 to 3.0. The diameter of the primary particle in these aggregates was fixed at 50 nm, comparable to the value for fullerene soot and larger than that for Aqua-Black 001. The fractal prefactor (e.g., Filippov et al., 2000), a parameter associated with the stoutness of branches of aggregates, was fixed at 1.0. The model fractal aggregates with dm = 500 nm consist of 1000 primary particles. Calculations of RTE (Eq. (3)) and RSC (Eq. (5)) require values of the absorption cross-section and differential scattering crosssection, respectively. These optical cross-sections were calculated by the discrete-dipole approximation (Draine & Flatau, 1994; Purcell & Pennypacker, 1973) code DDSCAT 7.0.7 (Draine & Flatau, 2008). For these calculations, the refractive indices of particles were assumed to be n = 2.00–0.80i or 1.75–0.63i, independent of the wavelengths considered here (i.e., 300–1064 nm). These values of n are close to the upper and lower limits of the reported range of refractive indices of light-absorbing carbon with little or no internal voids at a wavelength of 550 nm (Bond & Bergstrom, 2006). The refractive index n, wavenumber k, and dipole separation d should satisfy the criterion |n|kd < 0.5 for accurate calculations of the absorption cross-sections and differential scattering cross-sections (Draine & Flatau, 2008). For calculations in this study, ∼104 dipoles were used for each particle to satisfy the criterion. For calculation of RTE , the particle temperature was assumed to be T = 4500 K. For all model nonspherical

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0.35 Glassy carbon Aquadag Microcrystalline graphite Colloidal graphite Aqua-Black 001 Aqua-Black 162 Fullerene soot Atmospheric soot (Tokyo, Japan)

Standard deviation of RSC

0.30

0.25

0.20

0.15

0.10

0.05

0.00 0.00

0.04

0.08

0.12

0.16

Standard deviation of RTE Fig. 5. Measured standard deviation (s.d.) of RSC (t) vs. that of RTE (t) for various particle samples with selected mobility diameters indicated in Table 1. For individual particles, the time domains for calculating the s.d. of RTE (t) and s.d. of RSC (t) were selected according to the explanation in the text. Each data point and error bars indicate median value and 25–75 percentile ranges for ∼5000 particles.

particles, ± 500 K changes in the assumed temperature results in up to ± 4% systematic shifts of the calculated RTE values. Such ± 4% changes in RTE never alter the interpretations of the results given in the following sections. 4.4. Fluctuations in RSC and RTE for various particle shapes The relationship observed between the standard deviation (s.d.) of RSC and s.d. of RTE is shown in Fig. 5 for the seven laboratory carbon particles and for soot in the atmosphere. After calculation of the s.d. of RSC and s.d. of RTE for individual particles from their waveforms, data of about 5000 particles were collected to calculate their median, 25 and 75 percentile values shown in this figure. To interpret the observed relationship between the s.d. of RSC and s.d. of RTE , the corresponding theoretical values for model nonspherical particles are shown in Fig. 6. These theoretical values are the s.d. of RSC and s.d. of RTE calculated with the assumption of statistically random particle orientations over 544 different combinations of the Euler angles (, , ) relative to the coordinate system fixed with the detectors. Direct comparison of the measurements in a specific time domain to theories with statistically random orientations is reasonable if the time-cycle of the particle rotation is longer than the sampling interval of the waveform (0.2 s) and shorter than the width of the time domain of analysis (i.e., 4 s). In this study, we assume this condition because the fluctuations of RSC (t) shown in Fig. 4a obviously contain Fourier components of this time-cycle range. According to the theoretical calculations, (s.d. of RTE , s.d. of RSC ) = (0, 0) for spherical particles because they do not have any directional dependence of the partial scattering and absorption cross-sections. Fig. 6 shows that the s.d. of RSC monotonically increases with spherical shape deformation, whereas the s.d. of RTE saturates at some point depending on the model shape. This difference in behavior of the s.d. of RSC and s.d. of RTE is caused by the fact that the directional dependence of scattering is always sensitive to its shape and orientation, but the absorption is not necessarily so. The absorption cross-section becomes independent of the particle shape and depends only on the particle volume under the conditions of the Rayleigh–Gans (RG) approximation (Bohren & Huffman, 1983). The applicability of the RG approximation, namely, the theoretical treatment of the incident wave unaltered by multiple scattering or absorption inside a particle, is better for particles that are optically thin, such as particles that are small compared to the wavelength, particles with a refractive index close to that of the surrounding medium, fluffy aggregates that consist of small primary particles (Sorensen, 2001), and particles with a high aspect ratio so that their thicknesses are small compared to the wavelength. For particles with a high aspect ratio, the RG approximation is better for cases in which the angle of incidence is broadside of the particle because the changes of phase and amplitude inside the particle are smaller for this orientation (Barber & Wang, 1978). For a prolate spheroid, the probability of broadside incidence is greater by a factor of two than for an oblate spheroid for statistically random orientations. Therefore, the directional dependence of the absorption cross-section, namely, the deviation from the RG approximation, is greater for oblate than prolate spheroids.

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0.35 Df = 2.2 1.0

1.0

1.5

2.3 0.20

2.0 0.15 n=2.00-0.80i Oblate spheroid Prolate spheroid Fractal aggregates

2.5

0.10 2.8

2.5

3.0

n=1.75-0.63i Oblate spheroid Prolate spheroid Fractal aggregates

0.05 3.0

0.8

0.6

0.4

0.2

Aspect ratio q of oblate spheroid

0.25

Aspect ratio q of prolate spheroid

Standard deviation of RSC

0.30

0.00 0.00

0.04

0.08 0.12 Standard deviation of RTE

0.16

Fig. 6. Calculated standard deviation (s.d.) of RSC vs. that of RTE for various model nonspherical particles, under the assumption of statistically random orientation over 544 distinct sets of (, , ). Aspect ratios q for the oblate spheroid (q < 1) and prolate spheroid (q > 1) are indicated by the color scales. For fractal aggregates, the fractal dimensions Df for each data points are also shown (2.2–3.0). For each Df value, the average ± s.d. for 10 different aggregates is shown as a data point with error bars. For each model particle, calculations were made with two different refractive indices (n).

The large deviation from the RG approximation for end-on incident directions compensates for the better RG approximation for broadside incident directions. This canceling-out effect causes saturation of the directional dependence of absorption (i.e., the s.d. of RTE ) with increasing aspect ratio (Fig. 6). In addition, the systematic decrease in the s.d. of RTE for lower refractive index values (Fig. 6) is consistent with the physical interpretation of RG approximation theory. In contrast to absorption, the differential scattering cross-section is dependent on the relative position of a coherent array of scattering dipoles inside the particle even under the conditions of the RG approximation. Therefore, the directional dependences of the partial scattering cross-section, the s.d. of the RSC values, increase without any saturation as the nonsphericity (e.g., aspect ratio) of the particle increases. The calculated correlation of (s.d. of RTE , s.d. of RSC ) for fractal aggregates with various fractal dimensions was similar to prolate spheroids with various aspect ratios. Calculated (s.d. of RTE , s.d., of RSC ) values for cuboids (not shown in Fig. 6) were found to be similar to those for spheroids with the same aspect ratio within a difference of ( ± 0.01, ± 0.02). Because of this similarity, it is likely that a spheroid model is a candidate for other needle/plate-like particles for the change of (s.d. of RTE , s.d., of RSC ) according to the change of their aspect ratios. For Aquadag, the median value of the observed (s.d. of RTE , s.d. of RSC ) shown in Fig. 5 was similar to the theoretical values of a spheroid with an aspect ratio range of 0.2 < q < 0.3 as shown in Fig. 6. This range of q is within the possible range estimated from the relationship between mobility and mass (0.1 < q < 10). The results for other nonspherical particles are summarized in Table 1. For microcrystalline graphite and colloidal graphite, the range of q of a spheroid that was inferred based on (s.d. of RSC , s.d. of RTE ) was also included in the range estimated from the relationship between mobility and mass. For Aqua-Black 001 and 162, the fractal dimensions corresponding to the observed (s.d. of RTE , s.d. of RSC ), namely 2.8 < Df < 3.0, were similar to the fractal dimensions estimated from the relationship between mobility and mass. For fullerene soot, the observed s.d. of RTE was similar to the theoretical values of a fractal aggregate with Df = 2.5, whereas the observed s.d. of RSC was similar to the theoretical values for a fractal aggregate with Df = 2.3. This Df value is somewhat different from the Df = 2.65 for fullerene soot estimated from the relationship between mobility and mass. The uncertainties in the assumed values of the refractive index and shape parameters of the fractal aggregate (e.g., primary particle diameter) in the model calculation might cause the discrepancy of the theoretical value from the measurement. Lastly, the (s.d. of RTE , s.d. of RSC ) of ambient soot particles (sampled at the University of Tokyo, Meguro-ku, Tokyo, Japan) was analyzed and compared to the laboratory carbon particles. The mass of individual soot particles was estimated from the peak amplitude of the thermal emission STE (t) assuming linear proportionality between the peak amplitude and particle mass. The linear proportionality between the peak amplitude and particle mass was observed by Slowik et al. (2007) for frame generated soot. If particle temperature at the peak of thermal emission were independent of particle mass, the linear proportionality corresponds to the validity of RG approximation over the considered range of particle mass. However, to our knowledge, no studies on size dependence of incandescent temperature for individual soot particles in a continuum laser beam have been

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reported so far. Therefore, without physical reasoning, we just use the linear proportionality as an empirical function to estimate particle mass of ambient soot. The mass range of ambient soot was selected for this analysis to be about 100–200 fg, which is comparable to the mass range of the laboratory samples in this study. The main emission source of soot in the Tokyo urban area is likely diesel vehicles as discussed in Kondo et al. (2006). The (s.d. of RTE , s.d. of RSC ) of ambient soot particles was found to be similar to the value for fullerene soot (Fig. 5). The observed similarity in (s.d. of RTE , s.d. of RSC ) is consistent with the structural similarities between diesel soot (e.g., Park, Kittelson, & McMurry, 2004) and fullerene soot (Fig. 2d). The series of calculated s.d. of RTE for model fractal aggregates were systematically larger than the observed value for atmospheric soot. Because the RG approximation improves for optically soft particles, the calculated s.d. of RTE will be closer to the observed values if we assume a smaller primary particle diameter. Results and discussions so far have been limited for particles with single size, satisfying the three criteria for size selections given in Section 4.1. Now we show how the results changes with particle size (mass), especially for Aquadag particle which shows most prominent directional dependence of thermal emission (Fig. 5). Although the Aquadag with dmob = 750 nm (110 fg) shows s.d. of RTE = 0.12, large directional dependence of thermal emission compared to other samples, it was observed to diminish down to s.d. of RTE = 0.035 for dmob = 300 nm (10 fg). The s.d. of RTE = 0.035 is comparable to the value for spherical particles which has no directional dependence of thermal emission (see Fig. 5), though the Aquadag particles with dmob = 300 nm has nonspherical envelope (k0 = 1.4) according to the mass-to-mobility relationship. This behavior of s.d. of RTE for nonspherical particle in smaller size is consistent with an interpretation of their thermal emission (absorption) cross-section by the RG approximation, under which any shape effects on thermal emission diminish. 5. Conclusions We have demonstrated the first experimental evidence of the directional dependence of thermal emission from nonspherical particles of a size smaller than the regime of the geometric optics limit, by using the technique of single-particle laser-induced incandescence (i.e., single-particle soot photometer). The signal ratio of thermal emission (RTE ) measured by detectors in two orthogonal directions was used to monitor the directional dependences of thermal emission. Simultaneously, the signal ratio of light scattering (RSC ) measured by detectors placed in two orthogonal directions provided information on the nonsphericity of the particles and their rotation state. The observed correlations between the fluctuations of RTE and RSC for laboratory carbon particles were found to be consistent with calculations using Rytov's theory for randomly oriented model nonspherical carbon particles. This study shows that the directional dependence of thermal emission can be a new physical principle for the detection of particle shape, which is especially useful for carbon particles. The single-particle soot photometer originally has been used to measure carbon mass content inside individual particles from intensity of thermal emission. For this original purpose, it is advisable to use longer wavelength (e.g., red to near-IR wavelength) for thermal emission detection, to increase the penetration depths of thermal emission radiation inside the carbon particles. Under this condition, any shape effects on thermal emission diminish over wider range of carbon mass content. In contrast, for the purpose to measure shape of carbon particles, using shorter wavelength (e.g., blue wavelength) is advisable because the shape effects on directional dependence thermal emission become prominent. Acknowledgments This work was supported by the Grant-in-Aid for Young Scientists (Start-up) Fund of the Japan Society for the Promotion of Science (JSPS), the Ministry of Education, Culture, Sports, Science, and Technology (MEXT), the global environment research fund of the Japanese Ministry of the Environment (B-083), and the Japanese Science and Technology Agency (JST) (SENTAN Program). Appendix A. A.1. Mass-to-mobility relationships used to derive particle shape information In this section, we explain the method used to derive particle shape information from the relationship between mobility and mass. As briefly explained in Section 4.1, mass distributions of mobility-selected particles were measured by the Differential Mobility Analyzer–Aerosol Particle Mass Analyzer (DMA–APM) system (e.g., McMurry, Wang, Park, & Ehara, 2002; Moteki & Kondo, 2007). The experimental details of the DMA–APM system used in this study are explained in Moteki and Kondo (2007). The mass equivalent diameter (dm ) of the particles was calculated for further analysis by assuming a true density of  = 2 g cm−3 . The true densities of several graphite powders were reported to be in the range 1.97–2.26 g cm−3 (Wissler, 2006). In contrast, the true density of diesel soot particles was reported to be 1.77 ± 0.07 g cm−3 (Park, Kittelson, Zachariah, & McMurry, 2004). Considering this reported range of true densities of carbon particles, the assumption of  = 2 g cm−3 causes a ± 4% systematic error in dm . The resistance force FD acting on a particle moving with constant velocity V in air is expressed in terms of the mobility diameter dmob or envelope equivalent diameter de as FD =

3 de Vk0 3 dmob V = , C(dmob ) C(dadj )

(A.1)

800

N. Moteki et al. / Aerosol Science 40 (2009) 790 -- 801

where  is the viscosity of air, k0 the shape factor of the particle in a continuum regime, and C the Cunningham correction factor (Cheng, Yeh, & Allen, 1988). The adjusted sphere diameter dadj may differ from the envelope equivalent diameter de . Calculation of the Cunningham correction factor by using dadj instead of de is necessary to use the shape factor in a continuum regime k0 (Cheng et al., 1988; Dahneke, 1973). The deviation of C(dadj ) from C(de ) increases with the nonsphericity (e.g., aspect ratio) of the particle (Dahneke, 1973). In the Knudsen number regime in this study ( < 0.3), the difference between C(dadj ) and C(de ) is less than a few percent for needle-like particles with an aspect ratio of 20 (Dahneke, 1973). As shown in Table 1, the nonsphericities (e.g., aspect ratios) of the envelopes of the particles used in this study are likely not as high as those observed for needle-like particles with an aspect ratio of 20. Therefore, we can replace C(dadj ) in Eq. (A.1) by C(de ) within a few percent systematic error, which is small compared to the other experimental uncertainties. After the replacement of C(dadj ) by C(de ) in Eq. (A.1), we have 3 de Vk0 3 dmob V ≈ . C(dmob ) C(de )

(A.2)

The envelope equivalent diameter de differs from the mass equivalent diameter dm by a void factor of de = dm .

(A.3)

The void factor is a parameter related to the amount of void inside the particle envelope. The volume fraction of internal void of a particle can be defined as 1−1/ 3 (Brockmann & Rader, 1990). After replacing de in Eq. (A.2) by Eq. (A.3), we have dmob C( dm ) . dm C(dmob )

k0 =

(A.4)

Under the approximation C(dmob ) ≈ C( dm ), we can express the product of the shape factor k0 and void factor , a parameter having particle shape information, as dmob , dm

k0 ≈

(A.5)

where the right-hand side of Eq. (A.5) can be experimentally determined by the DMA–APM system by assuming a value for the true density. Evaluated values of k0 are listed in Table 1, with their maximum systematic error in parentheses. For each carbon particle, the maximum error due to the approximation C(dmob ) ≈ C( dm ) was calculated using the size dependence of the Cunningham correction factor, and was taken into account when calculating the uncertainty of the k0 value. In the case of a nonspherical envelope, we need to assume a shape model to relate the shape factor k0 to the degree of nonsphericity. In this study, we used a spheroid with various aspect ratios for the shape model. The shape factor of a spheroid approximates well the shape factors of other plate-like and needle-like nonspherical particles (e.g., double cone, cuboid, cylinder, and chain of spheres) with the same aspect ratio (McNown & Malaika, 1950). In addition, the shape factor of a spheroid with an arbitrary aspect ratio can be calculated by an analytical formula (Fuchs, 1964; Kasper, 1982; McNown & Malaika, 1950). For aggregates consisting of small particles, the fractal dimension is a useful parameter for the shape of nonspherical particles (Forrest & Witten, 1979). Assuming a constant primary particle diameter, the particle mass m is proportional to the mobility diameter dmob with the power of the fractal dimension Df under the condition of Df > 2 (Park, Cao, Kittelson, & McMurry, 2003; Schmidt-Ott, Baltensperger, Ga¨ ggeler, & Jost, 1990): D

f . m ∝ dmob

(A.6)

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