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Backscatter of agglomerate particles
Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 163 – 171 www.elsevier.com/locate/jqsrt
Backscatter of agglomerate particles Evgenij Zubkoa,∗ , Yuriy Shkuratova , Matthew Hartb , Jay Eversoleb , Gorden Videenc a Institute of Astronomy, Kharkov National University, 35 Sumskaya St., Kharkov, 61022, Ukraine b Naval Research Laboratory, 4555 Overlook Av. SW, Washington, DC 20375-5320, USA c Army Research Laboratory AMSRL-CI-EM, 2800 Powder Mill Road, Adelphi, MD 20783, USA
Received 24 February 2004; accepted 25 March 2004
Abstract We use the discrete dipole approximation (DDA) to study the backscatter of agglomerate particles consisting of oblong monomers. We examine the effects of monomer number and packing structure on the resulting negative polarization branch at small phase angle. We find large a dependence on the orientation of the monomers within the agglomerate and a smaller dependence on the number of monomers, suggesting that the mechanism producing the negative polarization minimum depends strongly on the interactions between the individual monomers. 䉷 2004 Elsevier Ltd. All rights reserved. PACS: 260.5430; 280.0280; 290.1350; 290.3770; 290.4210; 290.5870 Keywords: Light scattering; Bacteria-like particles; Negative polarization; Opposition effect 2
1. Introduction Rapid methods of detecting threat aerosols are necessary for environmental and security applications. Elastic light scattering may prove to be a valuable instrument in such diagnostics because it is rapid and components for building light-scattering detectors are relatively inexpensive, do not require reagents, and may be automated. One particular class of aerosols of interest is a cluster of spores. Spores of interest, like B. subtilis, tend to be approximately pill-shaped, or spheroidal, having aspect ratios of approximately 2, and major ∗ Corresponding author. Tel.: +38-0572-43-24-28; fax: +38-0572-43-24-28
E-mail address: [email protected] (E. Zubko) 0022-4073/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.03.026
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Fig. 1. A cluster of B. subtilis spores.
axes between 0.5 and 1.0 m. Clusters of spores may be formed when aerosolized droplets containing multiple spores evaporate. What remains is a very dense cluster of nearly identical particles (see Fig. 1). These aerosols may contain one to several hundred spores, and vary greatly in size and morphology. In many of the photomicrographs that we have seen, we have noticed that the spores tend to be oriented so that their major axes are approximately perpendicular to the radial axis drawn from the approximate center of the cluster to the approximate center of each spore. Acquiring information about a component of a particle system is not always a trivial task. Light scattering is used most successfully to acquire general information, like particle size through forwardscattering minima locations, and refractive index through pattern visibility. Some shape information may also be obtained [1–3]. In the detector system, we are not interested in acquiring characterizing parameters (like the size) of the total aerosol, but the characterizing parameters of the individual components that make up the aerosol. The cluster itself may be composed of a few to several hundred spores, and there is enormous variability in the scattering from such systems. To illustrate the difficulties of such a problem, we consider the forward scatter that is predominantly determined by the size and shape of the entire particle. For example, the forward-scatter intensity from a cluster of several hundred spores would have a signal that resembles that of a similarly sized dust particle, and would look completely different from that of a cluster of a few spores, since the sizes of these systems differ greatly. Another parameter that is often studied is small-scale roughness. However, for such particles, roughness must be large to perturb the scattering significantly, and even then it tends to wash out the structure that might contain characterizing information [4,5]. The backscatter region tends to be more sensitive to particle irregularities and heterogeneities than the forward-scatter region. We focus on one particular feature in the back-scatter region that has received a great deal of interest especially in the astrophysical community: a negative polarization minimum near the exact back-scatter region (see [6–12], and references therein). The polarization opposition effect (POE) refers to the narrow, asymmetric branch that occurs within a few degrees of the exact back-scatter and is the result of the coherent interference or reciprocal rays. The negative polarization branch (NPB) refers to
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a wider, symmetric branch located approximately within a few tens of degrees from the exact back-scatter [13–17]. The mechanism for this effect has not been established, but it may be the result of near-field effects, single-particle scattering, or some combination of mechanisms [12,18]. Single-particle scattering tends to contain this particular feature and its dependence on particle parameters is not yet known. Some natural complex media are known to exhibit both branches of negative polarization [6], and it may be significant in spore-detector development that the only single-component laboratory sample that displays both of these features is a substrate composed of lycopodium spores [11,19]. The primary reason for pursuing the POE and NPB for spore detection, rather than another scattering feature, is that it might be possible to retrieve information about the individual components that make up an aerosol particle using the POE or NPB. It is known, for instance that the position of the POE is determined by path differences of reciprocal rays. This is determined by the spatial frequencies of the particle. For an agglomerate of spores, these frequencies correspond to the sizes of the individual spores; hence, information on the minima position might correspond directly with monomer size. Furthermore, because the spores that make up an aerosol cluster all are approximately the same dimensions, we would expect that the minima from such clusters would be especially pronounced in comparison with the minima from other background aerosols. In a previous manuscript in which we examined the negative polarization minima from aerosol clusters of monomers having different aspect ratios, we found negative polarization minima amplitudes between 5% and 10%, and this depth increases with aspect ratio [20]. In considering polarization effects, there currently are two classes of scattering systems: (1) single particles that exhibit a NPB, and (2) complex extended media that may exhibit both a NPB and a POE. In the former, the light is considered to be single-scattered; whereas, in the latter, the light may be considered both single-scattered and multiple-scattered by the many components making up the system. Agglomerate aerosol particles can be considered to lie somewhere between these two classes of particles. While the particle system is finite, with size on the order of wavelength, there are multiple components that may give rise to multiple scattering effects. How these manifest themselves in this system is unknown and it is hoped that their study may yield insights into the mechanism producing the NPB.
2. Model Because we are interested in agglomerates of non-spherical particles, we use a numerical technique with a discretized particle space. In our simulations, the advantage of being able to include particle asymmetry and orientation as parameters appears to outweigh the significant cost savings of using a more rapid semi-analytical technique [10,12,21–23]. This advantage is illustrated in a previous publication in which we showed that the introduction of monomer asphericity significantly increased the magnitude of the polarization in particle agglomerates [20]. The DDA method is described in many works, e.g., [24–26]. Thus we present here only the model of the irregular particles. We use our DDA code that exploits the fast Fourier transformation and conjugate gradient method [26]. Our calculations have been made with seven PCs working in parallel. The calculations were performed by ensemble averaging over 100 particle realizations including orientation averaging. We construct the monomers that form the agglomerate particles from a finite cylinder with semi-spherical butt ends. The aspect ratio and size of the monomer can be varied by changing the length and width of the finite cylinder. In these studies we keep the aspect ratio fixed to match approximately that of B. subtilis spores, 2.5.
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Fig. 2. Monomers represented by dipoles: (a) spherical; (b) spherically capped finite cylinder; and (c–f) various debris particles.
3. Results In this paper we study the backscatter from various non-spherical particle systems, specifically, agglomerates of non-spherical particles. Some non-spherical particles of interest that we have studied are shown in Fig. 2. These include (a) spherical particles, (b) pill-shaped particles (spherically capped finite cylinders) and (c–f) various debris aggregates. Particles of the last type are clusters consisting of monomers with very irregular shape. To generate these particles we begin with a lattice 64 × 64 × 64. By the Monte Carlo method we choose a small number of knots, seed centers of matter and of vacuum. Each center from the rest quantity is then marked as the closest center of the matter or vacuum. Each cluster is generated with 4 seeds of the matter and 12 seeds of vacuum. The clusters have packing density = 0.25. The size of the particle corresponds to x = 10. Fig. 3 shows the intensities and polarizations of scattered light for the spore simulate shown in Fig. 2b in different particle orientations. We see that the intensities tend to be higher in the mid- to backscatter regions for broadside incidence in the plane perpendicular to the major axis. This is simply a validation of the Fourier relationship between the particle and its light scatter.
E. Zubko et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 163 – 171
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100 1 - End-on 10 I(θ) I(180˚)
3 - Short-axis plane
1 2 - Long-axis plane 100
3 - Short-axis plane
P(θ),%
50
1 - End-on
0
-50
2 -Long-axis plane
-100 0
60
120
180
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Fig. 3. Intensity and polarization corresponding to different illumination/observation geometries: (1) major axis is parallel with incident field direction; (2) major axis is perpendicular to incident beam and scattering plane includes major axis and incident beam; (3) major axis is perpendicular to incident beam and scattering plane is perpendicular to major axis.
Thus we would expect monomer–monomer interactions to be maximized when their major axes are aligned in parallel, so that particles lie in the path of maximal scattering intensity of the other particles. In ray-tracing algorithms, this increases the probability of backscattering. There is greatest likelihood of such alignment if the particles are oriented with their major axes perpendicular to the radial vector (Fig. 4a) rather than completely random orientation (Fig. 4b), or with the major axes oriented in the radial direction (Fig. 4c). Fig. 5 shows orientation-averaged back-scattering intensity and polarization of the agglomerates shown in Fig. 4. The NPB of such agglomerates shows a very strong dependence on orientation. The NPB minimum is deepest when the monomers are aligned as in Fig. 4a and decreases as their alignment in the radial direction becomes more pronounced. When they are aligned as in Fig. 4c, the NPB minimum does not even exist. This simulation suggests that the NPB is the result of interactions occurring between monomers. Another numerical experiment that we can perform with the DDA is to examine the effect of the number of monomers on the back-scatter response. By increasing the number of equal-volume monomers, we increase the total size parameter of the agglomerate, while keeping the monomer parameters and packing densities the same in all cases. In Fig. 6 we present phase curves of the normalized intensity and degree of linear polarization of agglomerate particles. The average backscatter polarization for the single monomer shows a positive response. As the monomers are brought together to form agglomerates, the polarization response displays the negative feature. For the x = 10 agglomerate, the minimum is located
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Fig. 4. Agglomerate particles whose constituent monomers have different orientations: (a) monomers whose long axes are oriented perpendicular to the radial vector drawn from the particle center; (b) randomly oriented monomers; and (c) monomers whose long axes are oriented along the radial vector drawn from the particle center. For all particles, m = 1.59 + 0i and xeq ≈ 10.
approximately 13◦ from the exact backscatter and decreases to 10◦ for the x = 20 agglomerate; i.e., we find that when the size of the agglomerate increases, the polarization minimum position decreases to smaller phase angles. Typically we expect the amount of oscillatory structure in the scattering response to be approximately proportional to the size parameter; hence, we might expect the angular minimum position to be inversely proportional to the size parameter. This is not the case in Fig. 6, since the
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1 c
I (θ) I(180)
b 0.8
a
10
P(θ),%
5 c 0 b -5
180
a
170
160
150
θ,°
Fig. 5. Intensity and polarization as functions of scattering angle for agglomerate particles shown in Fig. 4.
difference between the minima position for the x = 10 and 20 is only a few degrees, and not a factor of two. If we consider minima produced via the coherent-backscatter mechanism, the position depends on the pathlength difference of reciprocal rays. For the agglomerate particles in our study, the statistics remain largely the same; however, for the larger particles, there is the possibility of larger pathlength differences being present, corresponding to the size of the agglomerate. The inclusion of these pathlength differences would tend to move the minimum closer to the exact backscatter direction. Whether or not this is the mechanism responsible for the minima, it is consistent with what we see in our modeling results. For detector applications, it would be ideal for the minimum position to remain constant for agglomerates of different number monomers. Unfortunately, this does not appear to be the case; however, in the spore agglomerates under study, the minimum position does not vary as rapidly as inversely proportional to the total agglomerate size. This small dependence may not prove fatal for its use in detector applications. Furthermore, the simulations point to a dependence on internal structure that may prove useful for characterizing particle heterogeneity.
Acknowledgements This work was supported by the TechBase Program on Chemical and Biological Defense and by the Battlefield Environment Directorate under the auspices of the U.S. Army Research Office
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E. Zubko et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 163 – 171 1 1
I(θ) I(180˚)
0.9 0.8 3 0.7
2 4
0.6 1
3
P(θ),%
0
-5 4
2
-10 180
170
θ,°
160
150
Fig. 6. Average normalized intensity and degree of linear polarization of agglomerates of different size. In all cases, the packing density = 0.25, monomer aspect ratio equals 2.5, refractive index m = 1.59 + 0i, and size parameter of an equivalent-volume sphere for the monomers xeq =1.86. Curve 1 corresponds to one monomer. Curves 2 corresponds to 40 monomers (size parameter of aggregate x = 10), 69 monomers (x = 12), and 316 monomers (x = 20).
Scientific Services Program administered by Battelle (Delivery Order 291, Contract. No. DAAD19-02D-0001). References [1] Holler S, Auger J-C, Stout B, Pan Y, Bottiger J, Chang R, Videen G. Observations and calculations of light scattering from clusters of spheres. Appl Opt 2000;39:6873–87. [2] Videen G, Sun W, Fu Q, Secker DR, Greenaway R, Kaye PH, Hirst E, Bartley D. Light scattering from deformed droplets and droplets with inclusions: II. Theoretical treatment. Appl Opt 2000;39:5031–9. [3] Secker DR, Greenaway R, Kaye PH, Hirst E, Bartley D, Videen G. Light scattering from deformed droplets and droplets with inclusions: I. Experimental results. Appl Opt 2000;39:5023–30. [4] Sun W, Nousiainen T, Muinonen K, Fu Q, Loeb NG, Videen G. Light scattering by Gaussian particles: A solution with finite-difference time domain technique. JQSRT 2003;79-80:1083–90. [5] Mishchenko MI, Lacis AA. Morphology-dependent resonances of nearly spherical particles in random orientation. Appl Opt 2003;42:5551–6. [6] Rosenbush V, Kiselev N, Avramchuk V, Mishchenko M. Photometric and polarimetric opposition phenomena exhibited by solar system bodies. in: Videen G, Kocifaj M., (editors.) Optics of cosmic dust. Dordrecht: Kluwer Academic Publishers; 2002. pp. 191–224. [7] Mishchenko M, Tishkovets V, Litvinov P. Exact results of the vector theory of coherent backscattering from discrete random media: an overview. in: Videen G, Kocifaj M., (editors.) Optics of cosmic dust. Dordrecht: Kluwer Academic Publishers; 2002. pp. 239–60.
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[8] Muinonen K, Videen G, Zubko E, Shkuratov Yu Numerical techniques for backscattering by random media. in: Videen G, Kocifaj M., (editors.) Optics of cosmic dust. Dordrecht: Kluwer Academic Publishers; 2002. pp. 261–82. [9] Shkuratov YG, Ovcharenko A. Experimental modeling of opposition effect and negative polarization of regolith-like samples. in: Videen G, Kocifaj M., (editors.) Optics of cosmic dust. Dordrecht: Kluwer Academic Publishers; 2002. [10] Litvinov PV, Tishkovets VP, Muinonen K, Videen G. Coherent opposition effect for discrete random media. in: van Tiggelen B, Skipetrov SE., (editors.) Wave scattering in complex media. Dordrecht: Kluwer; 2003. pp. 567–81. [11] Shkuratov Y, Videen G, Kreslavsky M, Belskaya I, Kaydash V, Ovcharenko A, Omelchenko V, Opanasenko N, Zubko E. Scattering properties of planetary regoliths near opposition. in: Videen G, Yatskiv Y, Mishchenko M., (editors.) Photopolarimetry in remote sensing. Dordrecht: Kluwer; 2004. [12] TishkovetsV, Litvinov P, Petrova E, Jockers K, Mishchenko M. Backscattering effects for discrete random media: theoretical results. in: Videen G, Yatskiv Y, Mishchenko M., (editors.) Photopolarimetry in remote sensing. Dordrecht: Kluwer; 2004. [13] Levasseur-Regourd AC, Hadamcik E, Renard JB. Evidence for two classes of comets from their polarimetric properties at large phase angles. Astron Astrophys 1996;313:327–33. [14] Volten H, Muñoz O, Rol E, de Haan JF, Vassen W, Hoovenier JW. Scattering matrices of mineral aerosol particles at 441.6 and 632.8 nm. J Geophys Res 2000;106 (D15):17,375–401. [15] Muñoz O, Volten H, de Haan JF, Vassen W, Hovenier JW. Experimental determination of scattering matrices of randomly oriented fly ash and clay particles at 442 and 633 nm. J Geophys Res 2001;106 D19::22,833–44. [16] Levasseur-Regourd AC. in: Videen G, Yatskiv Y, Mishchenko M., (editors.) Polarimetry of dust in the solar system: observations and measurements. Dordrecht: Kluwer; 2004. [17] Muñoz O, Volten H, Hovenier JW. Experimental light scattering matrices relevant to cosmic dust. in: Videen G, Kocifaj M., (editors.) Optics of cosmic dust. Dordrecht: Kluwer Academic Publishers; 2002. pp. 57–70. [18] Shkuratov Yu, Ovcharenko A, Zubko E, Volten H, Muñoz O, Videen G. The negative polarization of light scattered from particulate surfaces and of independently scattering particles, JQSRT, this issue. [19] Shkuratov Yu, Muinonen K, Bowell E, Lumme K, Peltoniemi J, Kreslavsky M, Stankevich D, Tishkovetz V, Opanasenko N, Melkumova L. A critical review of theoretical models of negatively polarized light scattered by atmosphereless solar system bodies. Earth, Moon and Planets 1994;65:201. [20] Zubko E, Shkuratov Yu, Hart M, Eversole J, Videen G. Backscattering and negative Polarization of agglomerate particles. Opt Lett 2003;28:1504–6. [21] Tishkovets VP. Multiple scattering of light by a layer of discrete random medium: backscattering. JQSRT 2002;72:123. [22] Tishkovets VP, Litvinov PV, Lyubchenko MV. Coherent opposition effects for semi-infinite discrete random medium in the double-scattering approximation. JQSRT 2002;72:803. [23] Mishchenko MI, Travis LD, Lacis AA. Scattering absorption and emission of light by small particles. Cambridge: Cambridge University Press; 2002 . [24] Purcell EM, Pennypacker CR. Scattering and absorption of light by nonspherical dielectric grains. Astrophys J 1973;186: 705–14. [25] Draine BT. The discrete-dipole approximation and its application to the interstellar graphite grains. Astrophys J 1988;333:848–72. [26] Draine BT, Flatau PJ. Discrete-dipole approximation for scattering calculations. J Opt Soc Am 1994;A11:1491–8.
Backscatter of agglomerate particles Evgenij Zubkoa,∗ , Yuriy Shkuratova , Matthew Hartb , Jay Eversoleb , Gorden Videenc a Institute of Astronomy, Kharkov National University, 35 Sumskaya St., Kharkov, 61022, Ukraine b Naval Research Laboratory, 4555 Overlook Av. SW, Washington, DC 20375-5320, USA c Army Research Laboratory AMSRL-CI-EM, 2800 Powder Mill Road, Adelphi, MD 20783, USA
Received 24 February 2004; accepted 25 March 2004
Abstract We use the discrete dipole approximation (DDA) to study the backscatter of agglomerate particles consisting of oblong monomers. We examine the effects of monomer number and packing structure on the resulting negative polarization branch at small phase angle. We find large a dependence on the orientation of the monomers within the agglomerate and a smaller dependence on the number of monomers, suggesting that the mechanism producing the negative polarization minimum depends strongly on the interactions between the individual monomers. 䉷 2004 Elsevier Ltd. All rights reserved. PACS: 260.5430; 280.0280; 290.1350; 290.3770; 290.4210; 290.5870 Keywords: Light scattering; Bacteria-like particles; Negative polarization; Opposition effect 2
1. Introduction Rapid methods of detecting threat aerosols are necessary for environmental and security applications. Elastic light scattering may prove to be a valuable instrument in such diagnostics because it is rapid and components for building light-scattering detectors are relatively inexpensive, do not require reagents, and may be automated. One particular class of aerosols of interest is a cluster of spores. Spores of interest, like B. subtilis, tend to be approximately pill-shaped, or spheroidal, having aspect ratios of approximately 2, and major ∗ Corresponding author. Tel.: +38-0572-43-24-28; fax: +38-0572-43-24-28
E-mail address: [email protected] (E. Zubko) 0022-4073/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2004.03.026
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Fig. 1. A cluster of B. subtilis spores.
axes between 0.5 and 1.0 m. Clusters of spores may be formed when aerosolized droplets containing multiple spores evaporate. What remains is a very dense cluster of nearly identical particles (see Fig. 1). These aerosols may contain one to several hundred spores, and vary greatly in size and morphology. In many of the photomicrographs that we have seen, we have noticed that the spores tend to be oriented so that their major axes are approximately perpendicular to the radial axis drawn from the approximate center of the cluster to the approximate center of each spore. Acquiring information about a component of a particle system is not always a trivial task. Light scattering is used most successfully to acquire general information, like particle size through forwardscattering minima locations, and refractive index through pattern visibility. Some shape information may also be obtained [1–3]. In the detector system, we are not interested in acquiring characterizing parameters (like the size) of the total aerosol, but the characterizing parameters of the individual components that make up the aerosol. The cluster itself may be composed of a few to several hundred spores, and there is enormous variability in the scattering from such systems. To illustrate the difficulties of such a problem, we consider the forward scatter that is predominantly determined by the size and shape of the entire particle. For example, the forward-scatter intensity from a cluster of several hundred spores would have a signal that resembles that of a similarly sized dust particle, and would look completely different from that of a cluster of a few spores, since the sizes of these systems differ greatly. Another parameter that is often studied is small-scale roughness. However, for such particles, roughness must be large to perturb the scattering significantly, and even then it tends to wash out the structure that might contain characterizing information [4,5]. The backscatter region tends to be more sensitive to particle irregularities and heterogeneities than the forward-scatter region. We focus on one particular feature in the back-scatter region that has received a great deal of interest especially in the astrophysical community: a negative polarization minimum near the exact back-scatter region (see [6–12], and references therein). The polarization opposition effect (POE) refers to the narrow, asymmetric branch that occurs within a few degrees of the exact back-scatter and is the result of the coherent interference or reciprocal rays. The negative polarization branch (NPB) refers to
E. Zubko et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 163 – 171
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a wider, symmetric branch located approximately within a few tens of degrees from the exact back-scatter [13–17]. The mechanism for this effect has not been established, but it may be the result of near-field effects, single-particle scattering, or some combination of mechanisms [12,18]. Single-particle scattering tends to contain this particular feature and its dependence on particle parameters is not yet known. Some natural complex media are known to exhibit both branches of negative polarization [6], and it may be significant in spore-detector development that the only single-component laboratory sample that displays both of these features is a substrate composed of lycopodium spores [11,19]. The primary reason for pursuing the POE and NPB for spore detection, rather than another scattering feature, is that it might be possible to retrieve information about the individual components that make up an aerosol particle using the POE or NPB. It is known, for instance that the position of the POE is determined by path differences of reciprocal rays. This is determined by the spatial frequencies of the particle. For an agglomerate of spores, these frequencies correspond to the sizes of the individual spores; hence, information on the minima position might correspond directly with monomer size. Furthermore, because the spores that make up an aerosol cluster all are approximately the same dimensions, we would expect that the minima from such clusters would be especially pronounced in comparison with the minima from other background aerosols. In a previous manuscript in which we examined the negative polarization minima from aerosol clusters of monomers having different aspect ratios, we found negative polarization minima amplitudes between 5% and 10%, and this depth increases with aspect ratio [20]. In considering polarization effects, there currently are two classes of scattering systems: (1) single particles that exhibit a NPB, and (2) complex extended media that may exhibit both a NPB and a POE. In the former, the light is considered to be single-scattered; whereas, in the latter, the light may be considered both single-scattered and multiple-scattered by the many components making up the system. Agglomerate aerosol particles can be considered to lie somewhere between these two classes of particles. While the particle system is finite, with size on the order of wavelength, there are multiple components that may give rise to multiple scattering effects. How these manifest themselves in this system is unknown and it is hoped that their study may yield insights into the mechanism producing the NPB.
2. Model Because we are interested in agglomerates of non-spherical particles, we use a numerical technique with a discretized particle space. In our simulations, the advantage of being able to include particle asymmetry and orientation as parameters appears to outweigh the significant cost savings of using a more rapid semi-analytical technique [10,12,21–23]. This advantage is illustrated in a previous publication in which we showed that the introduction of monomer asphericity significantly increased the magnitude of the polarization in particle agglomerates [20]. The DDA method is described in many works, e.g., [24–26]. Thus we present here only the model of the irregular particles. We use our DDA code that exploits the fast Fourier transformation and conjugate gradient method [26]. Our calculations have been made with seven PCs working in parallel. The calculations were performed by ensemble averaging over 100 particle realizations including orientation averaging. We construct the monomers that form the agglomerate particles from a finite cylinder with semi-spherical butt ends. The aspect ratio and size of the monomer can be varied by changing the length and width of the finite cylinder. In these studies we keep the aspect ratio fixed to match approximately that of B. subtilis spores, 2.5.
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Fig. 2. Monomers represented by dipoles: (a) spherical; (b) spherically capped finite cylinder; and (c–f) various debris particles.
3. Results In this paper we study the backscatter from various non-spherical particle systems, specifically, agglomerates of non-spherical particles. Some non-spherical particles of interest that we have studied are shown in Fig. 2. These include (a) spherical particles, (b) pill-shaped particles (spherically capped finite cylinders) and (c–f) various debris aggregates. Particles of the last type are clusters consisting of monomers with very irregular shape. To generate these particles we begin with a lattice 64 × 64 × 64. By the Monte Carlo method we choose a small number of knots, seed centers of matter and of vacuum. Each center from the rest quantity is then marked as the closest center of the matter or vacuum. Each cluster is generated with 4 seeds of the matter and 12 seeds of vacuum. The clusters have packing density = 0.25. The size of the particle corresponds to x = 10. Fig. 3 shows the intensities and polarizations of scattered light for the spore simulate shown in Fig. 2b in different particle orientations. We see that the intensities tend to be higher in the mid- to backscatter regions for broadside incidence in the plane perpendicular to the major axis. This is simply a validation of the Fourier relationship between the particle and its light scatter.
E. Zubko et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 163 – 171
167
100 1 - End-on 10 I(θ) I(180˚)
3 - Short-axis plane
1 2 - Long-axis plane 100
3 - Short-axis plane
P(θ),%
50
1 - End-on
0
-50
2 -Long-axis plane
-100 0
60
120
180
θ,°
Fig. 3. Intensity and polarization corresponding to different illumination/observation geometries: (1) major axis is parallel with incident field direction; (2) major axis is perpendicular to incident beam and scattering plane includes major axis and incident beam; (3) major axis is perpendicular to incident beam and scattering plane is perpendicular to major axis.
Thus we would expect monomer–monomer interactions to be maximized when their major axes are aligned in parallel, so that particles lie in the path of maximal scattering intensity of the other particles. In ray-tracing algorithms, this increases the probability of backscattering. There is greatest likelihood of such alignment if the particles are oriented with their major axes perpendicular to the radial vector (Fig. 4a) rather than completely random orientation (Fig. 4b), or with the major axes oriented in the radial direction (Fig. 4c). Fig. 5 shows orientation-averaged back-scattering intensity and polarization of the agglomerates shown in Fig. 4. The NPB of such agglomerates shows a very strong dependence on orientation. The NPB minimum is deepest when the monomers are aligned as in Fig. 4a and decreases as their alignment in the radial direction becomes more pronounced. When they are aligned as in Fig. 4c, the NPB minimum does not even exist. This simulation suggests that the NPB is the result of interactions occurring between monomers. Another numerical experiment that we can perform with the DDA is to examine the effect of the number of monomers on the back-scatter response. By increasing the number of equal-volume monomers, we increase the total size parameter of the agglomerate, while keeping the monomer parameters and packing densities the same in all cases. In Fig. 6 we present phase curves of the normalized intensity and degree of linear polarization of agglomerate particles. The average backscatter polarization for the single monomer shows a positive response. As the monomers are brought together to form agglomerates, the polarization response displays the negative feature. For the x = 10 agglomerate, the minimum is located
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E. Zubko et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 88 (2004) 163 – 171
Fig. 4. Agglomerate particles whose constituent monomers have different orientations: (a) monomers whose long axes are oriented perpendicular to the radial vector drawn from the particle center; (b) randomly oriented monomers; and (c) monomers whose long axes are oriented along the radial vector drawn from the particle center. For all particles, m = 1.59 + 0i and xeq ≈ 10.
approximately 13◦ from the exact backscatter and decreases to 10◦ for the x = 20 agglomerate; i.e., we find that when the size of the agglomerate increases, the polarization minimum position decreases to smaller phase angles. Typically we expect the amount of oscillatory structure in the scattering response to be approximately proportional to the size parameter; hence, we might expect the angular minimum position to be inversely proportional to the size parameter. This is not the case in Fig. 6, since the
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Fig. 5. Intensity and polarization as functions of scattering angle for agglomerate particles shown in Fig. 4.
difference between the minima position for the x = 10 and 20 is only a few degrees, and not a factor of two. If we consider minima produced via the coherent-backscatter mechanism, the position depends on the pathlength difference of reciprocal rays. For the agglomerate particles in our study, the statistics remain largely the same; however, for the larger particles, there is the possibility of larger pathlength differences being present, corresponding to the size of the agglomerate. The inclusion of these pathlength differences would tend to move the minimum closer to the exact backscatter direction. Whether or not this is the mechanism responsible for the minima, it is consistent with what we see in our modeling results. For detector applications, it would be ideal for the minimum position to remain constant for agglomerates of different number monomers. Unfortunately, this does not appear to be the case; however, in the spore agglomerates under study, the minimum position does not vary as rapidly as inversely proportional to the total agglomerate size. This small dependence may not prove fatal for its use in detector applications. Furthermore, the simulations point to a dependence on internal structure that may prove useful for characterizing particle heterogeneity.
Acknowledgements This work was supported by the TechBase Program on Chemical and Biological Defense and by the Battlefield Environment Directorate under the auspices of the U.S. Army Research Office
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Fig. 6. Average normalized intensity and degree of linear polarization of agglomerates of different size. In all cases, the packing density = 0.25, monomer aspect ratio equals 2.5, refractive index m = 1.59 + 0i, and size parameter of an equivalent-volume sphere for the monomers xeq =1.86. Curve 1 corresponds to one monomer. Curves 2 corresponds to 40 monomers (size parameter of aggregate x = 10), 69 monomers (x = 12), and 316 monomers (x = 20).
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