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On the impact of non-sphericity and small-scale surface roughness on the optical properties of hematite aerosols
Journal of Quantitative Spectroscopy & Radiative Transfer 112 (2011) 1815–1824
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On the impact of non-sphericity and small-scale surface roughness on the optical properties of hematite aerosols ¨ Michael Kahnert a,, Timo Nousiainen b, Paivi Mauno b a b
Swedish Meteorological and Hydrological Institute, Folkborgsv¨ agen 1, S-601 76 Norrk¨ oping, Sweden Department of Physics, P.O. Box 48, FI-00014, University of Helsinki, Finland
a r t i c l e i n f o
abstract
Available online 26 January 2011
We perform a comparative modelling study to investigate how different morphological features influence the optical properties of hematite aerosols. We consider high-order Chebyshev particles as a proxy for aerosol with a small-scale surface roughness, and spheroids as a model for nonspherical aerosols with a smooth boundary surface. The modelling results are compared to those obtained for homogeneous spherical particles. It is found that for hematite particles with an absorption efficiency of order unity the difference in optical properties between spheres and spheroids disappears. For optically softer particles, such as ice particles at far-infrared wavelengths, this effect can be observed for absorption efficiencies lower than unity. The convergence of the optical properties of spheres and spheroids is caused by absorption and quenching of internal resonances inside the particles, which depend both on the imaginary part of the refractive index and on the size parameter, and to some extent on the real part of the refractive index. By contrast, small-scale surface roughness becomes the dominant morphological feature for large particles. This effect is likely to depend on the amplitude of the surface roughness, the relative significance of internal resonances, and possibly on the real part of the refractive index. The extinction cross section is rather insensitive to surface roughness, while the single-scattering albedo, asymmetry parameter, and the Mueller matrix are strongly influenced. Small-scale surface roughness reduces the backscattering cross section by up to a factor of 2–3 as compared to size-equivalent particles with a smooth boundary surface. This can have important implications for the interpretation of lidar backscattering observations. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Scattering Aerosols T-matrix Surface roughness Hematite Lidar
1. Introduction Iron oxide mineral aerosols, such as hematite (Fe2O3), are found in both the terrestrial and the Martian atmosphere. In comparison to many other types of mineral dust aerosols, they can have large absorption cross sections and low single scattering albedos at visible wavelengths. Scanning electron microscope images show that hematite particles display departures from spherical shape on various size-scales [1].
Corresponding author.
E-mail address: [email protected] (M. Kahnert). 0022-4073/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2011.01.022
Modelling the optical properties of such morphologically complex particles is a challenging task with high relevance to remote sensing of planetary atmospheres as well as in climate modelling. The optical properties of weakly absorbing spherical particles as a function of particle size, refractive index, or scattering angle display characteristic resonance features that are rather atypical for non-spherical particles. Optical properties of non-spherical particles have been modelled by a variety of different model shapes, each of them emphasising different morphological aspects. In this article we investigate morphological features on different size scales, and we focus on the optical properties of
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hematite aerosols. In particular, we investigate the effect of small-scale surface roughness. By this we mean a perturbation of the particle surface with a (mean) perturbation wavelength L and a (mean) perturbation amplitude A that are small compared to the characteristic size r0 of the particle and small compared to the wavelength l of incident light. Non-spherical particles with smooth surfaces are often modelled by homogeneous spheroids. Their geometry is characterised by the size and the aspect ratio, thus introducing only one additional parameter as compared to homogeneous spheres. Averaging the optical properties of spheroids over not only sizes, but also orientations and aspect ratios strongly reduces the resonance features observed for a size distribution of spheres [2]. However, these model particles neglect the effects of surface roughness and geometric irregularities. A highly flexible model particle that can account for such morphological features is the Gaussian random particle model [3]. These model particles have boundary surfaces that are randomly deformed on variable scales. Such model particles can be employed for not only mimicking non-spherical shapes, but also for reproducing the shape statistics of an ensemble of realistic nonspherical aerosols (e.g. [4,5]). Also, these particles can be employed for mimicking small-scale surface-roughness. Computations of optical properties of irregular particles, such as Gaussian random particles, are considerably more demanding than computations for highly symmetric particles, such as spheroids. The reason for this can be explained by the use of group theory [6,7]. Chebyshev particles offer a computationally more expedient way of mimicking the effects of surface roughness, at the price of sacrificing the capability of modelling shape statistics and the effects of shape irregularity. A Chebyshev particle is constructed by rotating the curve rðyÞ ¼ r0 ½1 þ eT‘ ðyÞ
ð1Þ
about the vertical axis [8]. In this equation, T‘ ðyÞ ¼ cosð‘yÞ denotes a Chebyshev polynomial of order ‘, r0 is the radius of the unperturbed sphere, and e is the deformation parameter, where 1 r e o 1. Thus the perturbation amplitude is given by A ¼ er0 , and the wavelength of the perturbation on the particle surface is L ¼ 2pr0 =‘ (since there are ‘ oscillations on the circumference 2pr0 of the unperturbed sphere). Fig. 1 shows, as an example, the geometry of a Chebyshev particle with ‘ ¼ 45 and e ¼ 0:05. For low orders of ‘, Chebyshev particles can be used as a simple proxy for particles with a ‘‘bumpy’’ surface. For sufficiently high orders of ‘ and for small e, the amplitude A and the length scale L of the surface perturbations become much smaller than the particle size and the wavelength of light. Thus high-order Chebyshev particles are a simple model for particles with small-scale surface roughness. Recently, computations of the optical properties of Chebyshev particles of orders up to ‘ ¼ 45 have been reported [9]. It was found that the optical properties initially vary with ‘ at low orders, and then converge at high orders. So the optical properties of high-order Chebyshev particles differ, in general, from those of smooth
Fig. 1. Chebyshev particle with polynomial order ‘ ¼ 45 and deformation parameter e ¼ 0:05.
spheres, but they do not depend on the length scale L, if L is small. Following these results, small-scale surface roughness means, in our terminology:
The roughness length scale L is sufficiently small (i.e. ‘
is sufficiently large) so that any further decrease in L (increase in ‘) does not alter the optical properties. For Chebyshev particles of radius r0 and order ‘, L ¼ 2pr0 =‘. L 5 2pr0 , i.e., L is much smaller than the circumference of the particle. L 5 l, where l is the wavelength of light. A 5 r0 , where A is the roughness amplitude. For Chebyshev particles with a deformation parameter e, A ¼ r0 e. A 5 l. A is sufficiently large so that the optical properties of a particle with a perturbed boundary surface differ from those of the corresponding unperturbed geometry.
Note that for particles smaller than the wavelength, there may exist no particles with a ‘‘small-scale’’ surface roughness in this sense, since it may not be possible to simultaneously satisfy the last three conditions. An important observation in [9] was that the phase functions of highly absorbing, high-order Chebyshev particles and homogeneous spheres of comparable size display clear differences. This result is even more striking in view of the fact that differences between the phase functions of spheroids and spheres disappear with increasing imaginary part of the refractive index [10,11]. A spheroid can be obtained by a homeomorphic deformation of a sphere with a perturbation ‘‘wavelength’’ L equal to half of the circumference 2pr0 of the sphere, i.e. L ¼ pr0 . (Think of r0 as a mean size a or0 o b, where a and b are the semiminor and semimajor axis lengths of the spheroid. The spheroidal geometry can then be thought of as the result of a perturbation of a sphere of radius r0, where the spatial period of that perturbation is L ¼ pr0 , and the perturbation amplitude is A=ba.) So, departures from spherical shape that correspond to a perturbation with L pr0 have a
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minor effect on the optical properties of highly absorbing particles, while small-scale surface roughness ðL 52pr0 Þ can have a strong impact. To understand the optical properties of aerosols composed of moderately and highly absorbing material, it is therefore essential to investigate the importance of smallscale surface roughness. In our study we employ highorder Chebyshev particles as a proxy for particles with small-scale surface roughness. Further, we employ spheroids as a model geometry for smooth nonspherical particles with a deformation length scale L pr0 . The main focus of our study is hematite, a representative for moderately absorbing minerals. For comparison, we also study ice at far-infrared wavelength, where ice becomes strongly absorbing. We emphasise that in our terminology we discriminate between weakly or strongly absorbing materials and weakly or strongly absorbing particles. The former term refers to the material’s absorption properties as expressed by the imaginary part k of the refractive index m ¼ n þ ik. The absorption properties of particles are expressed by the absorption efficiency Qabs or absorption cross section Cabs. In addition to k, the former also depends on the size parameter x ¼ 2pr0 =l, and the latter depends on the size r0 of the particle. In Section 2 we briefly outline the methodology. Computational results are shown in Section 3. The results are discussed in Section 4.
2. Methods Laboratory measurements of the Mueller matrix of hematite aerosols at l ¼ 632:8 nm have been reported in [1]. Hematite is a birefringent material with a real part of n= 2.9 for the extraordinary axis and 3.1 for the ordinary axis. Following [1], we neglect birefringence in our calculations and assume m =3.0 + 0.1i. It has been shown in [12] for mineral dust flakes composed of weakly absorbing material that the effects of birefringence are small and mostly limited to polarisation properties. For more strongly absorbing material, such as hematite, the effect of birefringence is expected to be even smaller. As a proxy for non-spherical particles with smooth surfaces we use a size-shape distribution of homogeneous spheroids. Computations have been performed by use of the T-matrix program described in [13]. The volumeequivalent particle radii covered the range between 20 nm and 1:4 mm, with a step-size of 20 nm. This corresponds to a size-parameter resolution of 0.2, which agrees with the recommendations given in [14] for homogeneous spheroids. The shape distribution contains aspect ratios of a/b=1.2, 1.4, y,2.6 for oblate spheroids, and a/b=1/1.2, 1/1.4, y,1/2.6 for prolate spheroids, where a and b denote the size of the spheroid perpendicular to and along the main rotational symmetry axis of the spheroid, respectively. It is sometimes more convenient to characterise the shape of a spheroid by the so-called shape parameter x, which was introduced in [15] according to ( 1b=a : a o b ðprolateÞ x¼ ð2Þ a=b1 : a 4 b ðoblateÞ
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In terms of this parameter, our shape distribution consists of particles with x ¼ 1:6,1:4, . . . ,0:2 (prolate) and x ¼ 0:2,0:4, . . . ,1:6 (oblate). This linearised scale facilitates the parameterisation of the shape distribution. Attempts to fit the Mueller matrix elements of nonspherical particles with spheroidal model particles have consistently shown that a best-fit Mueller matrix is generally obtained by putting more weight on those spheroids that deviate strongly from spherical shape (e.g. [15–18]). Based on this observation, it has been proposed to parameterise the shape distribution of spheroids by a simple power law jxjq . Comparisons with laboratory measurements of the scattering phase function of feldspar aerosols yielded very good agreement between measurements and modelling studies for a power law jxj3 [17]. Following these results, we assume a jxj3 shape distribution for our hematite spheroids; all results presented for spheroids are thus averaged over this shape distribution. As a reference, we also perform Mie calculations for a size distribution of homogeneous spheres. Computations have been performed by using a Mie program based on the T-matrix program described in [13]. Computations were performed for sizes from 5 nm up to 1:4 mm, with a step-size of 2.5 nm. This corresponds to a size-parameter resolution of 0.025. Again, this resolution agrees with the recommendations given in [14] for homogeneous spheres. Note that those recommendations apply to weakly absorbing particles. For moderately and highly absorbing particles, such as hematite, quenching of internal resonances may have allowed for the reduction of the size resolution to a certain degree. Our choice of size resolution is thus rather conservative. Numerical computations of the optical properties of Chebyshev particles by use of T-matrix methods are notoriously difficult because of numerical ill-conditioning problems. We use the T-matrix code ‘‘mieschka’’ [19] for this task. This code achieves a high numerical stability owing to its strict automated convergence tests. The same code has been used in [9] for computing the optical properties of high-order Chebyshev particles. As mentioned earlier, the optical properties of particles with small-scale surface roughness are independent of L if L is sufficiently small. We tested this convergence of the optical properties with respect to L. For instance, we found that choosing ‘ ¼ 45 for a particle size r0 ¼ 1:4 mm yields a Mueller matrix that does not change upon further increasing ‘. We repeated this test for various particle sizes and found, in general, that by choosing the Chebyshev order ‘ such that L ¼ 2pr0 =‘ t l=4 we are well within the small-scale surface roughness regime throughout the entire size range, so that the optical properties do not change upon further increasing ‘. Thus, in our computations we use a size-dependent Chebyshev order that increases up to ‘ ¼ 53 for the largest particles. (Incidentally, we repeated all calculations for a fixed Chebyshev order of ‘ ¼ 45 for all particle sizes and obtained, as expected, identical optical properties as in the case with L ¼ 2pr0 =‘ t l=4.) We use two different deformation parameters e ¼ 0:03 and 0:05, and we consider particle sizes in the range 5 nm21:4 mm, with a step-size of 5 nm,
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which corresponds to a size-parameter resolution of 0.05. Orientational averaging has been performed analytically. 3. Results We start our investigation by comparing the computed Mueller matrices obtained with spherical, spheroidal, and Chebyshev model particles to laboratory measurements [1] from the Amsterdam Light Scattering Database (http://www.iaa.es/scattering/). Fig. 2 shows the Mueller matrix elements computed for a size distribution of Chebyshev particles with e ¼ 0:05 (thick solid line), spheres (thin solid line), and a jxj3 size-shape distribution of spheroids (dashed line). The normalised laboratory observations based on [1] are represented by symbols with error bars. The measurements only cover the angular range from 51 to 1731. To normalise the phase function, the forward-scattering peak (0–51) has been supplemented by Lorenz–Mie computations based on the measured size distribution. Since the forward-scattering peak is mainly sensitive to size rather than to particle shape, Lorenz–Mie theory works well for this purpose. In the backscattering direction, the phase function has been extrapolated by simply extending the value at 173–1801. Since the phase function has very low values in the backscattering direction, the normalisation integral is
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insensitive to the extrapolation in the backscattering direction. The agreement of the measured and modelled phase functions is rather poor. There are several factors that contribute to this. The deviations in the forward-scattering direction strongly suggest uncertainties in the size distribution. The measured size distribution is shown in Fig. 3; it has been measured for particle sizes of 0:0824:7 mm. Due to computational constraints we only performed computations up to a cut-off size of 1:4 mm. However, we did perform sensitivity tests with Mie computations up to sizes of 4:7 mm (not shown). It was found that cutting off the size distribution at 1:4 mm has a minor effect on the phase function. On the other hand, the size distribution measurements extend only down to particle sizes of 80 nm, which is still well above the Rayleigh regime. In our computations we set the particle number densities equal to zero for sizes below 80 nm. Sensitivity studies with different approaches to extrapolate the size measurements to smaller sizes revealed, somewhat surprisingly, a high sensitivity of the phase function to the assumed number distribution of particles with sizes below 80 nm. In particular, non-zero number densities of small particles can broaden the forwardscattering peak, thus improving the agreement with the observations. Another potential source of error is the
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Fig. 2. Mueller matrix elements computed for an ensemble of Chebyshev particles, spheres, and spheroids. The laboratory observations are based on [1], which have been normalised as explained in the text.
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uncertainty in the refractive index. Third, hematite particles are neither spheres, spheroids, nor Chebyshev particles in their shapes, so all the model shapes are approximations. The third factor is likely to be the most important. The uncertainties in the size distribution and refractive index complicate the comparison of modelled and measured Mueller matrix elements. However, comparison of the three different models leads to a number of interesting observations. All three model particles yield rather similar phase functions P11. So, in comparison to spheres, neither a shape distribution of smooth spheroids nor Chebyshev particles with a small-scale surface roughness
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r [μm] Fig. 3. Measured size distribution of the hematite sample, taken from [1].
display significant differences in the angular distribution of the scattered intensity. The Mueller matrix elements computed with spheres and Chebyshev particles are very similar, while results obtained for spheroids display clear differences. This indicates that small-scale surface roughness plays a minor role for this particular sample of hematite. We attribute this observation mainly to the fact that the hematite particles are rather small with an effective radius of only 0:4 mm. An interesting feature is the bimodality of the P12/P11 element. This feature was also noted in [20] for small, monodisperse agglomerated debris particles with k Z 0:5. Our simulations show that the feature persists for spheres and Chebyshev particles when size-integrated, and it is also apparent in the measurements. Simulations in [20] suggest that the feature disappears if particle size increases or k decreases, so it may be indicative of small, absorbing particles. Overall, the spheroids match the measurements better than the spheres or the Chebyshev shapes. However, even the spheroids perform poorly. We are not aware of other successful model fits to the hematite sample, so the hematite sample remains a challenge for our modelling capabilities. To attain a better understanding of the significance of surface roughness effects we consider the optical properties as a function of particle size. Fig. 4 shows the extinction efficiency Qext, single-scattering albedo o, asymmetry parameter g, and backscattering cross section Cbak for different model particles. The size of the Chebyshev particles is the radius of the unperturbed sphere r0, while the size of the spheroids is understood as the volume-equivalent radius. The extinction efficiencies Qext of Chebyshev particles and spheres are indistinguishable over the entire size range. They converge to an asymptotic value slightly different from
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Fig. 4. Qext (upper left), o (upper right), g (lower left), and Cbak (lower right) for hematite Chebyshev particles with e ¼ 0:05 (thick solid line), e ¼ 0:03 3 (dash-dotted line), for spheres (thin solid line), and spheroids with a jx j shape distribution (dashed line).
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concentrations by a factor of about 1/2 (to compensate for the over-prediction of Cbak by smooth model particles). Note that the phase functions of Chebyshev shapes are very similar to those of spheres and spheroids at the backscattering direction in Fig. 2. This indicates that particles with sizes r0 4 500 nm do not contribute significantly to the size-integrated results. Why do the optical properties of spheres and spheroids converge with increasing particle size? In [9] it has been noted that the optical properties of spheres and spheroids converge with increasing imaginary part k of the refractive index. This indicates that differences in the optical properties between spheres and spheroids are mainly caused by internal optical resonances inside the particle. When the particle becomes highly absorbing the optical resonances are quenched. As a result, the differences in optical properties between spheres and spheroids disappear. Recall that a ‘‘highly absorbing particle’’ is a particle with a high absorption efficiency Qabs. The latter depends on both k and the size parameter x. This leads us to hypothesise that the convergence of the optical properties of spheres and spheroids apparent in Fig. 4 are due to quenching of internal resonances caused by an increase of Qabs with increasing size parameter. The critical transition size of 0:5 mm corresponds to a size parameter of x 5, and to Qabs 1. To further investigate the impact of surface roughness we now take a little side-track and consider the optical properties of ice at a far-infrared wavelength of 47 mm, using a refractive index of 1.34+0.83i. Results for ice particle sizes up to 100 mm are shown in Fig. 5. We see that o and g computed with spheres and spheroids are nearly indistinguishable over virtually the entire size range. Cbak computed with spheres displays small oscillations, but is otherwise
that of spheroids. The extinction cross section Cext at large sizes (not shown) is about 7% larger for spheroids than for spheres. This can be explained by the notorious difficulties of comparing sizes of spherical and non-spherical particles. Although the volume-equivalent radius is a reasonable measure for size comparisons, it does not necessarily yield identical extinction cross sections for spheres and spheroids. The single-scattering albedo o ¼ Csca =Cext (where Csca denotes the scattering cross section) is a relative measure that is often less sensitive to how we define size-equivalence. As can be seen in Fig. 4 (upper left), o at particle radii r0 o 500 nm is different for spheres and spheroids and comparable for spheres and Chebyshev particles. For increasing sizes r0 4 500 nm, o converges for spheres and spheroids, while it diverges for spheres and Chebyshev particles. The same behaviour can be observed for g (lower left panel). Equally remarkable is the behaviour of Cbak as a function of size for the three model particles (lower right). For spheres one observes conspicuous resonance features, which are absent for spheroids (owing to the averaging over orientations and shape parameters). However, for particle sizes r0 4500 nm Cbak computed for spheres seems to oscillate, to a good approximation, about that computed for spheroids. So Cbak averaged over any size distribution dominated by particles with sizes r0 4 500 nm is expected to give similar results for spheres and spheroids. By contrast, Chebyshev particles predict significantly lower values of Cbak for r0 4 500 nm (roughly half as much). This could have important implications for the interpretation of observed lidar backscattering coefficients. If one observes backscattering signals from moderately or highly absorbing particles with surface roughness, an inversion algorithm that neglects the effects of surface roughness could under-predict aerosol 3
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Fig. 5. Qext (upper left), o (upper right), g (lower left), and Cbak (lower right) for ice Chebyshev particles with e ¼ 0:05 (thick solid line), spheres (thin solid 3 line), and spheroids with a jx j shape distribution (dashed line).
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very similar to that computed with spheroids over the entire size range. Remarkably enough, Cbak at r0 ¼ 100 mm computed for spheres or spheroids is larger than that computed for Chebyshev particles by almost a factor of 3. Again, this indicates that neglecting the effect of small-scale surface roughness is potentially an important source of error in inverse modelling of lidar backscattering observations. For the hematite particles the convergence of the optical properties of spheres and spheroids occurred at an absorption efficiency of Qabs 1 and a size parameter of x 5. For our ice particles, Qabs 1 corresponds to a particle size of r0 5 mm, while x 5 corresponds to r0 37 mm. Clearly, the convergence of the optical properties of spheres and spheroids seems to correlate to Qabs rather than to x. This strongly supports our hypothesis that this convergence effect is caused by quenching of internal resonances due to absorption inside the particle. However, a close analysis of our results reveals that the optical properties of ice spheres and spheroids are very close even for sizes r0 o 5 mm. A possible explanation is that the ice particles are optically softer than the hematite aerosols (i.e. the real part n of their refractive index is smaller). Therefore, less of the incoming radiation is scattered at the particle surface, and more penetrates into the particle, where it can be absorbed, whence the quenching of internal resonances becomes significant even for Qabs t1. Next, we take a closer look at the diverging optical properties of spheres and Chebyshev particles. For hematite, the onset of the divergence is at r0 0:5 mm. It is obviously a coincidence that convergence of spheres and spheroids occurs at the same particle size. This becomes apparent in Fig. 5. For ice, the divergence of spheres and Chebyshev particles occurs at r0 30 mm, while spheres and spheroids give almost identical optical properties over the entire size range. This indicates that the convergence of spheres and spheroids on one hand, and the divergence of spheres and Chebyshev particles on the other must be related to different physical processes. A plausible explanation for the divergence of spheres and Chebyshev particles may be that surface roughness effects first become apparent when the perturbation amplitude on the particle surface in comparison to the wavelength, A=l ¼ er0 =l, exceeds a certain threshold value. For hematite, Chebyshev particles with e ¼ 0:05 and r0 0:5 mm have A=l 0:04. For ice, Chebyshev particles with e ¼ 0:05 and r0 30 mm give A=l 0:03, which is of the same order. However, we would expect that a decrease in e would result in a corresponding increase in the critical particle size at which the divergence effect occurs. We can see in Fig. 4 that this is not the case. The optical properties of Chebyshev particles with e ¼ 0:03 (dash-dotted line) start to diverge from those of spheres at about the same size ð0:5 mmÞ as for e ¼ 0:05 (thick solid line). It is quite possible that the significance of small-scale surface roughness depends on a complex interplay of several factors. The relative perturbation amplitude A=l is certainly one important factor. Quenching of internal resonances may also contribute, because in weakly absorbing wavelength-scale particles the effect of small-scale surface roughness may be completely overshadowed by the
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effect of internal resonances. Another important factor may be the real part n of the refractive index, which determines how much radiation is actually scattered at the particle surface. We now understand why we did not see any significant differences between the Mueller matrix elements of Chebyshev particles and spheres for hematite in Fig. 2. It is because the size distribution of this sample is dominated by particles below the critical size of 0:5 mm, at which the transition to the small-scale surface roughness regime occurs. To attain a better understanding of the impact of small-scale surface roughness on the differential scattering behaviour of hematite we consider a lognormal size distribution with r0 =500 nm and s ¼ 0:8. This size distribution contains a large fraction of particles above the critical size. We average the Mueller matrix elements over this size distribution for each of the three model particles. The result is shown in Fig. 6. For most of the Mueller matrix elements, spheres and spheroids yield similar results, while Chebyshev particles display a different angular dependence. This clearly shows that the effect of small-scale surface roughness ðL 5 2pr0 Þ dominates over the effect of large-scale nonsphericity ðL pr0 Þ, even though, on average, the perturbation amplitude A=b a of the ensemble of spheroids is much larger than that of the Chebyshev particles! The phase functions P11 of spheres and spheroids are nearly indistinguishable, while that of Chebyshev particles is characterised by less side- and backscattering. This explains the high asymmetry parameter and the small Cbak observed for large Chebyshev particles in Fig. 4. The element P22, which enters into the linear depolarisation ratio dL ¼ ðP11 P22 Þ=ðP11 þP22 Þ, is equal to unity for both spheres and Chebyshev particles. The corresponding results obtained for spheroids do not significantly deviate from unity either. The deviation of P22 from unity is usually considered to be a good indication for nonsphericity. However, for particles with sufficiently high absorption efficiencies, the sensitivity of P22 to particle shape is strongly reduced. The elements P33, P44, and, most importantly, P12 are distinctly different for Chebyshev particles, and rather similar for spheres and spheroids, thus underlining the importance of surface roughness for the polarisation properties. The element P34 is rather similar for all three particle models, except near the backscattering direction, where Chebyshev particles display a local minimum. To get some idea of what happens at larger size parameters, we investigate the Mueller matrix at a fixed size of r ¼ 1:4 mm, which corresponds to a size parameter of x =14. This is the largest size parameter considered in our computations. Fig. 7 reveals that the phase function of Chebyshev particles displays considerably less side-scattering than that of either spheres or spheroids. Interestingly enough, the element P22 deviates from unity not only for spheroids but also for Chebyshev particles. For spheroids, the departure from unity is strongest in the forward-scattering hemisphere, while Chebyshev particles depolarise mainly in the backscattering hemisphere. However, the effect is very weak. This indicates that depolarisation measurements can, in principle, easily confuse spherical particles and strongly absorbing
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Fig. 6. Mueller matrix computed for an ensemble of Chebyshev particles (thin solid line), spheres (dashed line), and spheroids (thick solid line), assuming a lognormal size distribution with r0 = 500 nm and s ¼ 0:8.
particles with non-spherical or rough boundary surfaces. Another conspicuous feature of the Mueller matrix elements of large Chebyshev particles is the high oscillation amplitude (even though the results are orientation averaged). Such high resonances are seen also for rough spheres by [21] and dusted spheres by [22], indicating that small deformations of spheres are not sufficient to eliminate the strong resonances associated to spherical scatterers. A recent literature review [23] suggests that smallscale surface roughness would show some impact to individual scattering matrix elements already at size parameters near unity. The reviewed studies had focused on weakly absorbing particles, so we analysed our results to see whether the finding also applies here. We found that the matrix elements start to deviate at size parameters of about x= 1.5 for hematite and x =2 for ice. Ice at l ¼ 47 mm has a lower value of n and a higher value of k, which means that more radiative energy penetrates into the particle, and more of the penetrated radiation is absorbed. This may be the reason why the effect of small-scale surface roughness becomes more pronounced for smaller size parameters in the case of hematite. The data are currently very limited, however, and based on several different models for roughness. Therefore, this finding should be considered preliminary. Note that the
critical size parameter at which surface roughness effects become important also depends on the metric used for defining the onset of divergence, which in turn depends on the accuracy required in any given application.
4. Conclusions and outlook The focus of our study was to investigate the impact of departures from spherical shape on different size-scales L on the optical properties of hematite aerosols. High-order Chebyshev particles were used as a proxy for particles with small-scale surface roughness, for which L 5 2pr0 and L 5 l. A shape distribution of spheroids was employed as a proxy for smooth nonspherical particles, i.e., particles that can be obtained by a homeomorphic deformation of a sphere with a deformation ‘‘wavelength’’ L pr0 . A main conclusion of this study is that the morphological properties of spheroids ðL pr0 Þ are mainly important for particles with low absorption efficiencies. The morphology of high-order Chebyshev particles ðL 5 2pr0 Þ becomes significant at larger size parameters. For instance, it was found that the backscattering cross section of spheres and spheroids can be as much as 2–3 times as high as that of Chebyshev particles. Thus, neglecting surface roughness
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Fig. 7. Mueller matrix computed for Chebyshev particles (thin solid line), spheres (dashed line), and spheroids (thick solid line) with a fixed particle size r ¼ 1:4 mm.
effects can introduce large biases in the interpretation of lidar backscattering measurements. The difference between spherical particles and smooth nonspherical spheroids are prominent for low absorption efficiencies Qabs, and disappear for larger values of order Qabs 1. The observed differences at low values of Qabs are caused by internal optical resonances inside the particles, which become quenched as the absorption efficiency Qabs of the particles increases. This increase in Qabs can either be achieved by increasing the particle size parameter, as in our study, or by increasing the imaginary part k of the refractive index, as in [9]. Comparison with corresponding results for ice at a far-infrared wavelength confirm that the convergence of the optical properties of spheres and spheroids mainly depends on Qabs rather than on the size parameter x. Further, the quenching of resonance effects seems to be more efficient for optically soft particles (i.e. particles with low values of the real part n of the refractive index). This is due to less scattering by the particle surface, which allows more radiative energy to penetrate into the absorbing material. By contrast, the optical properties of ice spheres and high-order Chebyshev particles start diverging at larger particle sizes. It is difficult to identify a single cause for this effect. For instance, the amplitude A of the surface
roughness is likely to be important for the divergence of the optical properties of particles with smooth and rough surfaces. However, comparison of hematite Chebyshev particles with relative amplitudes e ¼ A=r0 ¼ 0:03 and e ¼ 0:05 did not provide any clear evidence that the onset of divergence decreases in proportion to an increase in e. Comparison of results of hematite and ice indicates that the onset of divergence does not correlate to Qabs, so it is probably not directly related to the quenching of internal resonances. However, it is conceivable that for particles composed of weakly absorbing material the effect of surface roughness will be obscured by the effect of internal resonances. So Qabs may, at least, have an indirect significance for the observability of surface roughness effects. The real part n of the refractive index determines how optically hard the material is. This can also be expected to contribute to the importance of small-scale surface roughness. The results of this study allow us to clearly formulate future research required to better understand the importance of surface roughness. First, we need to study a variety of different materials and wavelengths to better understand the conditions under which small-scale surface roughness becomes the dominant morphological property. It will be particularly interesting to study aerosols composed of
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weakly absorbing material, for which surface-roughness effects can be expected to compete with effects related to internal resonances. To this end we need, second, to significantly extent our current capabilities of existing computational methods. Approximate methods, such as the geometric optics approximation, are not valid for particles with small-scale surface roughness. On the other hand, the applicability of currently available T-matrix programs to high-order Chebyshev particles is rather limited in terms of accessible size parameters. At larger size parameters computations can become unstable and inaccurate due to numerical ill-conditioning problems. We therefore need to improve our current models with special focus on particles with rough surfaces, such as high-order Chebyshev particles. Third, we will need more studies for different shape models, as our conclusions are based on the use of spherical, spheroidal, and Chebyshev model particles only. Fourth, it would be helpful to obtain additional measurements for hematite aerosols with larger effective radii. At a wavelength of l ¼ 632:8 nm, hematite samples that are dominated by particles larger than 500 nm are expected to be most relevant. Fifth, there may be cases for which the optical properties are simultaneously sensitive to large-scale departures from spherical shape ðL pr0 Þ and small-scale surface roughness ðL 5 2pr0 Þ. Such cases could, e.g., be modelled by spheroids perturbed by a Chebyshev polynomial [9]. Again, one needs to improve currently available T-matrix programs to be able to use such more advanced model particles for sufficiently large size parameters. Finally, it would be interesting to study the significance of 3D surface roughness effects. Electromagnetic scattering computations for nonaxisymmetric particles tend to be highly demanding. To keep such computations manageable, this will likely involve the exploitation of discrete particle symmetries [6].
Acknowledgements ˜ oz for making her The authors are grateful to Olga Mun measurement data publicly available. We are further grateful to Michael Mishchenko and Tom Rother for making their respective T-matrix programs available. M. Kahnert acknowledges funding from the Swedish Research Council under contract 80438701. T. Nousiainen and P. Mauno acknowledge the funding by the Academy of Finland (contracts 125180 and 12148). M. Thomas and two anonymous reviewers are acknowledged for helpful comments and suggestions. References ˜ oz O, Volten H, Hovenier JW, Min M, Shkuratov YG, Jalava JP, [1] Mun et al. Experimental and computational study of light scattering by irregular particles with extreme refractive indices: hematite and rutile. Astron Astrophys 2006;446:525–35.
[2] Mishchenko MI, Travis LD, Kahn RA, West RA. Modeling phase functions for dustlike tropospheric aerosols using a shape mixture of randomly oriented polydisperse spheroids. J Geophys Res 1997;102:16,831–47. [3] Muinonen K, Nousiainen T, Fast P, Lumme K, Peltoniemi JI. Light scattering by gaussian random particles: ray optics approximation. J Quant Spectrosc Radiat Transfer 1996;55:577–601. ¨ anen ¨ [4] Nousiainen T, Muinonen K, Rais P. Scattering of light by saharan particles in a modified ray-optics approximation. J Geophys Res 200310.1029/2001JD001277. [5] Veihelmann B, Nousiainen T, Kahnert M, van der Zande WJ. Light scattering by small feldspar particles simulated using the gaussian random sphere geometry. J Quant Spectrosc Radiat Transfer 2006;100(1–3):393–405. [6] Kahnert M. Irreducible representations of finite groups in the T matrix formulation of the electromagnetic scattering problem. J Opt Soc Am A 2005;22:1187–99. [7] Kahnert M. Boundary symmetries in linear differential and integral equation problems applied to the self-consistent Green’s function formalism of acoustic and electromagnetic scattering. Opt Commun 2006;265:383–93. [8] Mugnai A, Wiscombe WJ. Scattering from nonspherical Chebyshev particles. I: cross sections, single scattering albedo, asymmetry factor, and backscattering fraction. Appl Opt 1986;25:1235–44. [9] Rother T, Schmidt K, Wauer J, Shcherbakov V, Gaeyt J-F. Light scattering on Chebyshev particles of higher order. Appl Opt 2006;45:6030–7. [10] van de Hulst HC. Light scattering by small particles. New York: John Wiley & Sons, Inc.; 1957. [11] Mishchenko MI, Travis LD, Lacis AA. Scattering, absorption, and emission of light by small particles. Cambridge: Cambridge University Press; 2002. [12] Nousiainen T, Zubko E, Niemi JV, Kupiainen K, Lehtinen M, Muinonen K, et al. Single-scattering modeling of thin, birefringent mineral-dust flakes using the discrete-dipole approximation. J Geophys Res 2009;114:D07207, doi:10.1029/2008JD011564. [13] Mishchenko MI. Light scattering by randomly oriented axially symmetric particles. J Opt Soc Am A 1991;8:871–82. [14] Schmidt K, Wauer J, Rother T, Trautmann T. Scattering database for spheroidal particles. Appl Opt 2009;48:2154–64. [15] Kahnert FM, Stamnes JJ, Stamnes K. Using simple particle shapes to model the stokes scattering matrix of ensembles of wavelengthsized particles with complex shapes: possibilities and limitations. J Quant Spectrosc Radiat Transfer 2002;74:167–82. [16] Kahnert FM. Reproducing the optical properties of fine desert dust aerosols using ensembles of simple model particles. J Quant Spectrosc Radiat Transfer 2004;85:231–49. [17] Nousiainen T, Kahnert M, Veihelmann B. Light scattering modeling of small feldspar aerosol particles using polyhedral prisms and spheroids. J Quant Spectrosc Radiat Transfer 2006;101:471–87. [18] Dubovik O, Sinyuk A, Lapyonok T, Holben BN, Mishchenko MI, Yang P, et al. Application of spheroid models to account for aerosol particle nonsphericity in remote sensing of desert dust. J Geophys Res 2006;111:D11208. [19] Rother T. Electromagnetic wave scattering on nonspherical particles. Berlin: Springer; 2009. [20] Zubko E, Kimura H, Shkuratov Y, Muinonen K, Yamamoto T, Okamoto H, et al. Effect of absorption on light scattering by agglomerated debris particles. J Quant Spectrosc Radiat Transfer 2009;110:1741–9, doi:10.1016/j.jqsrt.2008.12.006. [21] Li C, Kattawar GW, Yang P. Effects of surface roughness on light scattering by small particles. J Quant Spectrosc Radiat Transfer 2004;89:123–31. [22] Mishchenko MI, Dlugach JM, Mackowski DW. Light scattering by wavelength-sized particles ‘‘dusted’’ with subwavelength-sized grains. Opt Lett 2011;36:337–9. [23] Nousiainen T. Optical modeling of mineral dust particles: a review. J Quant Spectrosc Radiat Transfer 2009;110:1261–79, doi: 10.1016/j.jqsrt2009.03.002.
Contents lists available at ScienceDirect
Journal of Quantitative Spectroscopy & Radiative Transfer journal homepage: www.elsevier.com/locate/jqsrt
On the impact of non-sphericity and small-scale surface roughness on the optical properties of hematite aerosols ¨ Michael Kahnert a,, Timo Nousiainen b, Paivi Mauno b a b
Swedish Meteorological and Hydrological Institute, Folkborgsv¨ agen 1, S-601 76 Norrk¨ oping, Sweden Department of Physics, P.O. Box 48, FI-00014, University of Helsinki, Finland
a r t i c l e i n f o
abstract
Available online 26 January 2011
We perform a comparative modelling study to investigate how different morphological features influence the optical properties of hematite aerosols. We consider high-order Chebyshev particles as a proxy for aerosol with a small-scale surface roughness, and spheroids as a model for nonspherical aerosols with a smooth boundary surface. The modelling results are compared to those obtained for homogeneous spherical particles. It is found that for hematite particles with an absorption efficiency of order unity the difference in optical properties between spheres and spheroids disappears. For optically softer particles, such as ice particles at far-infrared wavelengths, this effect can be observed for absorption efficiencies lower than unity. The convergence of the optical properties of spheres and spheroids is caused by absorption and quenching of internal resonances inside the particles, which depend both on the imaginary part of the refractive index and on the size parameter, and to some extent on the real part of the refractive index. By contrast, small-scale surface roughness becomes the dominant morphological feature for large particles. This effect is likely to depend on the amplitude of the surface roughness, the relative significance of internal resonances, and possibly on the real part of the refractive index. The extinction cross section is rather insensitive to surface roughness, while the single-scattering albedo, asymmetry parameter, and the Mueller matrix are strongly influenced. Small-scale surface roughness reduces the backscattering cross section by up to a factor of 2–3 as compared to size-equivalent particles with a smooth boundary surface. This can have important implications for the interpretation of lidar backscattering observations. & 2011 Elsevier Ltd. All rights reserved.
Keywords: Scattering Aerosols T-matrix Surface roughness Hematite Lidar
1. Introduction Iron oxide mineral aerosols, such as hematite (Fe2O3), are found in both the terrestrial and the Martian atmosphere. In comparison to many other types of mineral dust aerosols, they can have large absorption cross sections and low single scattering albedos at visible wavelengths. Scanning electron microscope images show that hematite particles display departures from spherical shape on various size-scales [1].
Corresponding author.
E-mail address: [email protected] (M. Kahnert). 0022-4073/$ - see front matter & 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.jqsrt.2011.01.022
Modelling the optical properties of such morphologically complex particles is a challenging task with high relevance to remote sensing of planetary atmospheres as well as in climate modelling. The optical properties of weakly absorbing spherical particles as a function of particle size, refractive index, or scattering angle display characteristic resonance features that are rather atypical for non-spherical particles. Optical properties of non-spherical particles have been modelled by a variety of different model shapes, each of them emphasising different morphological aspects. In this article we investigate morphological features on different size scales, and we focus on the optical properties of
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hematite aerosols. In particular, we investigate the effect of small-scale surface roughness. By this we mean a perturbation of the particle surface with a (mean) perturbation wavelength L and a (mean) perturbation amplitude A that are small compared to the characteristic size r0 of the particle and small compared to the wavelength l of incident light. Non-spherical particles with smooth surfaces are often modelled by homogeneous spheroids. Their geometry is characterised by the size and the aspect ratio, thus introducing only one additional parameter as compared to homogeneous spheres. Averaging the optical properties of spheroids over not only sizes, but also orientations and aspect ratios strongly reduces the resonance features observed for a size distribution of spheres [2]. However, these model particles neglect the effects of surface roughness and geometric irregularities. A highly flexible model particle that can account for such morphological features is the Gaussian random particle model [3]. These model particles have boundary surfaces that are randomly deformed on variable scales. Such model particles can be employed for not only mimicking non-spherical shapes, but also for reproducing the shape statistics of an ensemble of realistic nonspherical aerosols (e.g. [4,5]). Also, these particles can be employed for mimicking small-scale surface-roughness. Computations of optical properties of irregular particles, such as Gaussian random particles, are considerably more demanding than computations for highly symmetric particles, such as spheroids. The reason for this can be explained by the use of group theory [6,7]. Chebyshev particles offer a computationally more expedient way of mimicking the effects of surface roughness, at the price of sacrificing the capability of modelling shape statistics and the effects of shape irregularity. A Chebyshev particle is constructed by rotating the curve rðyÞ ¼ r0 ½1 þ eT‘ ðyÞ
ð1Þ
about the vertical axis [8]. In this equation, T‘ ðyÞ ¼ cosð‘yÞ denotes a Chebyshev polynomial of order ‘, r0 is the radius of the unperturbed sphere, and e is the deformation parameter, where 1 r e o 1. Thus the perturbation amplitude is given by A ¼ er0 , and the wavelength of the perturbation on the particle surface is L ¼ 2pr0 =‘ (since there are ‘ oscillations on the circumference 2pr0 of the unperturbed sphere). Fig. 1 shows, as an example, the geometry of a Chebyshev particle with ‘ ¼ 45 and e ¼ 0:05. For low orders of ‘, Chebyshev particles can be used as a simple proxy for particles with a ‘‘bumpy’’ surface. For sufficiently high orders of ‘ and for small e, the amplitude A and the length scale L of the surface perturbations become much smaller than the particle size and the wavelength of light. Thus high-order Chebyshev particles are a simple model for particles with small-scale surface roughness. Recently, computations of the optical properties of Chebyshev particles of orders up to ‘ ¼ 45 have been reported [9]. It was found that the optical properties initially vary with ‘ at low orders, and then converge at high orders. So the optical properties of high-order Chebyshev particles differ, in general, from those of smooth
Fig. 1. Chebyshev particle with polynomial order ‘ ¼ 45 and deformation parameter e ¼ 0:05.
spheres, but they do not depend on the length scale L, if L is small. Following these results, small-scale surface roughness means, in our terminology:
The roughness length scale L is sufficiently small (i.e. ‘
is sufficiently large) so that any further decrease in L (increase in ‘) does not alter the optical properties. For Chebyshev particles of radius r0 and order ‘, L ¼ 2pr0 =‘. L 5 2pr0 , i.e., L is much smaller than the circumference of the particle. L 5 l, where l is the wavelength of light. A 5 r0 , where A is the roughness amplitude. For Chebyshev particles with a deformation parameter e, A ¼ r0 e. A 5 l. A is sufficiently large so that the optical properties of a particle with a perturbed boundary surface differ from those of the corresponding unperturbed geometry.
Note that for particles smaller than the wavelength, there may exist no particles with a ‘‘small-scale’’ surface roughness in this sense, since it may not be possible to simultaneously satisfy the last three conditions. An important observation in [9] was that the phase functions of highly absorbing, high-order Chebyshev particles and homogeneous spheres of comparable size display clear differences. This result is even more striking in view of the fact that differences between the phase functions of spheroids and spheres disappear with increasing imaginary part of the refractive index [10,11]. A spheroid can be obtained by a homeomorphic deformation of a sphere with a perturbation ‘‘wavelength’’ L equal to half of the circumference 2pr0 of the sphere, i.e. L ¼ pr0 . (Think of r0 as a mean size a or0 o b, where a and b are the semiminor and semimajor axis lengths of the spheroid. The spheroidal geometry can then be thought of as the result of a perturbation of a sphere of radius r0, where the spatial period of that perturbation is L ¼ pr0 , and the perturbation amplitude is A=ba.) So, departures from spherical shape that correspond to a perturbation with L pr0 have a
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minor effect on the optical properties of highly absorbing particles, while small-scale surface roughness ðL 52pr0 Þ can have a strong impact. To understand the optical properties of aerosols composed of moderately and highly absorbing material, it is therefore essential to investigate the importance of smallscale surface roughness. In our study we employ highorder Chebyshev particles as a proxy for particles with small-scale surface roughness. Further, we employ spheroids as a model geometry for smooth nonspherical particles with a deformation length scale L pr0 . The main focus of our study is hematite, a representative for moderately absorbing minerals. For comparison, we also study ice at far-infrared wavelength, where ice becomes strongly absorbing. We emphasise that in our terminology we discriminate between weakly or strongly absorbing materials and weakly or strongly absorbing particles. The former term refers to the material’s absorption properties as expressed by the imaginary part k of the refractive index m ¼ n þ ik. The absorption properties of particles are expressed by the absorption efficiency Qabs or absorption cross section Cabs. In addition to k, the former also depends on the size parameter x ¼ 2pr0 =l, and the latter depends on the size r0 of the particle. In Section 2 we briefly outline the methodology. Computational results are shown in Section 3. The results are discussed in Section 4.
2. Methods Laboratory measurements of the Mueller matrix of hematite aerosols at l ¼ 632:8 nm have been reported in [1]. Hematite is a birefringent material with a real part of n= 2.9 for the extraordinary axis and 3.1 for the ordinary axis. Following [1], we neglect birefringence in our calculations and assume m =3.0 + 0.1i. It has been shown in [12] for mineral dust flakes composed of weakly absorbing material that the effects of birefringence are small and mostly limited to polarisation properties. For more strongly absorbing material, such as hematite, the effect of birefringence is expected to be even smaller. As a proxy for non-spherical particles with smooth surfaces we use a size-shape distribution of homogeneous spheroids. Computations have been performed by use of the T-matrix program described in [13]. The volumeequivalent particle radii covered the range between 20 nm and 1:4 mm, with a step-size of 20 nm. This corresponds to a size-parameter resolution of 0.2, which agrees with the recommendations given in [14] for homogeneous spheroids. The shape distribution contains aspect ratios of a/b=1.2, 1.4, y,2.6 for oblate spheroids, and a/b=1/1.2, 1/1.4, y,1/2.6 for prolate spheroids, where a and b denote the size of the spheroid perpendicular to and along the main rotational symmetry axis of the spheroid, respectively. It is sometimes more convenient to characterise the shape of a spheroid by the so-called shape parameter x, which was introduced in [15] according to ( 1b=a : a o b ðprolateÞ x¼ ð2Þ a=b1 : a 4 b ðoblateÞ
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In terms of this parameter, our shape distribution consists of particles with x ¼ 1:6,1:4, . . . ,0:2 (prolate) and x ¼ 0:2,0:4, . . . ,1:6 (oblate). This linearised scale facilitates the parameterisation of the shape distribution. Attempts to fit the Mueller matrix elements of nonspherical particles with spheroidal model particles have consistently shown that a best-fit Mueller matrix is generally obtained by putting more weight on those spheroids that deviate strongly from spherical shape (e.g. [15–18]). Based on this observation, it has been proposed to parameterise the shape distribution of spheroids by a simple power law jxjq . Comparisons with laboratory measurements of the scattering phase function of feldspar aerosols yielded very good agreement between measurements and modelling studies for a power law jxj3 [17]. Following these results, we assume a jxj3 shape distribution for our hematite spheroids; all results presented for spheroids are thus averaged over this shape distribution. As a reference, we also perform Mie calculations for a size distribution of homogeneous spheres. Computations have been performed by using a Mie program based on the T-matrix program described in [13]. Computations were performed for sizes from 5 nm up to 1:4 mm, with a step-size of 2.5 nm. This corresponds to a size-parameter resolution of 0.025. Again, this resolution agrees with the recommendations given in [14] for homogeneous spheres. Note that those recommendations apply to weakly absorbing particles. For moderately and highly absorbing particles, such as hematite, quenching of internal resonances may have allowed for the reduction of the size resolution to a certain degree. Our choice of size resolution is thus rather conservative. Numerical computations of the optical properties of Chebyshev particles by use of T-matrix methods are notoriously difficult because of numerical ill-conditioning problems. We use the T-matrix code ‘‘mieschka’’ [19] for this task. This code achieves a high numerical stability owing to its strict automated convergence tests. The same code has been used in [9] for computing the optical properties of high-order Chebyshev particles. As mentioned earlier, the optical properties of particles with small-scale surface roughness are independent of L if L is sufficiently small. We tested this convergence of the optical properties with respect to L. For instance, we found that choosing ‘ ¼ 45 for a particle size r0 ¼ 1:4 mm yields a Mueller matrix that does not change upon further increasing ‘. We repeated this test for various particle sizes and found, in general, that by choosing the Chebyshev order ‘ such that L ¼ 2pr0 =‘ t l=4 we are well within the small-scale surface roughness regime throughout the entire size range, so that the optical properties do not change upon further increasing ‘. Thus, in our computations we use a size-dependent Chebyshev order that increases up to ‘ ¼ 53 for the largest particles. (Incidentally, we repeated all calculations for a fixed Chebyshev order of ‘ ¼ 45 for all particle sizes and obtained, as expected, identical optical properties as in the case with L ¼ 2pr0 =‘ t l=4.) We use two different deformation parameters e ¼ 0:03 and 0:05, and we consider particle sizes in the range 5 nm21:4 mm, with a step-size of 5 nm,
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which corresponds to a size-parameter resolution of 0.05. Orientational averaging has been performed analytically. 3. Results We start our investigation by comparing the computed Mueller matrices obtained with spherical, spheroidal, and Chebyshev model particles to laboratory measurements [1] from the Amsterdam Light Scattering Database (http://www.iaa.es/scattering/). Fig. 2 shows the Mueller matrix elements computed for a size distribution of Chebyshev particles with e ¼ 0:05 (thick solid line), spheres (thin solid line), and a jxj3 size-shape distribution of spheroids (dashed line). The normalised laboratory observations based on [1] are represented by symbols with error bars. The measurements only cover the angular range from 51 to 1731. To normalise the phase function, the forward-scattering peak (0–51) has been supplemented by Lorenz–Mie computations based on the measured size distribution. Since the forward-scattering peak is mainly sensitive to size rather than to particle shape, Lorenz–Mie theory works well for this purpose. In the backscattering direction, the phase function has been extrapolated by simply extending the value at 173–1801. Since the phase function has very low values in the backscattering direction, the normalisation integral is
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insensitive to the extrapolation in the backscattering direction. The agreement of the measured and modelled phase functions is rather poor. There are several factors that contribute to this. The deviations in the forward-scattering direction strongly suggest uncertainties in the size distribution. The measured size distribution is shown in Fig. 3; it has been measured for particle sizes of 0:0824:7 mm. Due to computational constraints we only performed computations up to a cut-off size of 1:4 mm. However, we did perform sensitivity tests with Mie computations up to sizes of 4:7 mm (not shown). It was found that cutting off the size distribution at 1:4 mm has a minor effect on the phase function. On the other hand, the size distribution measurements extend only down to particle sizes of 80 nm, which is still well above the Rayleigh regime. In our computations we set the particle number densities equal to zero for sizes below 80 nm. Sensitivity studies with different approaches to extrapolate the size measurements to smaller sizes revealed, somewhat surprisingly, a high sensitivity of the phase function to the assumed number distribution of particles with sizes below 80 nm. In particular, non-zero number densities of small particles can broaden the forwardscattering peak, thus improving the agreement with the observations. Another potential source of error is the
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Fig. 2. Mueller matrix elements computed for an ensemble of Chebyshev particles, spheres, and spheroids. The laboratory observations are based on [1], which have been normalised as explained in the text.
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uncertainty in the refractive index. Third, hematite particles are neither spheres, spheroids, nor Chebyshev particles in their shapes, so all the model shapes are approximations. The third factor is likely to be the most important. The uncertainties in the size distribution and refractive index complicate the comparison of modelled and measured Mueller matrix elements. However, comparison of the three different models leads to a number of interesting observations. All three model particles yield rather similar phase functions P11. So, in comparison to spheres, neither a shape distribution of smooth spheroids nor Chebyshev particles with a small-scale surface roughness
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r [μm] Fig. 3. Measured size distribution of the hematite sample, taken from [1].
display significant differences in the angular distribution of the scattered intensity. The Mueller matrix elements computed with spheres and Chebyshev particles are very similar, while results obtained for spheroids display clear differences. This indicates that small-scale surface roughness plays a minor role for this particular sample of hematite. We attribute this observation mainly to the fact that the hematite particles are rather small with an effective radius of only 0:4 mm. An interesting feature is the bimodality of the P12/P11 element. This feature was also noted in [20] for small, monodisperse agglomerated debris particles with k Z 0:5. Our simulations show that the feature persists for spheres and Chebyshev particles when size-integrated, and it is also apparent in the measurements. Simulations in [20] suggest that the feature disappears if particle size increases or k decreases, so it may be indicative of small, absorbing particles. Overall, the spheroids match the measurements better than the spheres or the Chebyshev shapes. However, even the spheroids perform poorly. We are not aware of other successful model fits to the hematite sample, so the hematite sample remains a challenge for our modelling capabilities. To attain a better understanding of the significance of surface roughness effects we consider the optical properties as a function of particle size. Fig. 4 shows the extinction efficiency Qext, single-scattering albedo o, asymmetry parameter g, and backscattering cross section Cbak for different model particles. The size of the Chebyshev particles is the radius of the unperturbed sphere r0, while the size of the spheroids is understood as the volume-equivalent radius. The extinction efficiencies Qext of Chebyshev particles and spheres are indistinguishable over the entire size range. They converge to an asymptotic value slightly different from
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Fig. 4. Qext (upper left), o (upper right), g (lower left), and Cbak (lower right) for hematite Chebyshev particles with e ¼ 0:05 (thick solid line), e ¼ 0:03 3 (dash-dotted line), for spheres (thin solid line), and spheroids with a jx j shape distribution (dashed line).
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concentrations by a factor of about 1/2 (to compensate for the over-prediction of Cbak by smooth model particles). Note that the phase functions of Chebyshev shapes are very similar to those of spheres and spheroids at the backscattering direction in Fig. 2. This indicates that particles with sizes r0 4 500 nm do not contribute significantly to the size-integrated results. Why do the optical properties of spheres and spheroids converge with increasing particle size? In [9] it has been noted that the optical properties of spheres and spheroids converge with increasing imaginary part k of the refractive index. This indicates that differences in the optical properties between spheres and spheroids are mainly caused by internal optical resonances inside the particle. When the particle becomes highly absorbing the optical resonances are quenched. As a result, the differences in optical properties between spheres and spheroids disappear. Recall that a ‘‘highly absorbing particle’’ is a particle with a high absorption efficiency Qabs. The latter depends on both k and the size parameter x. This leads us to hypothesise that the convergence of the optical properties of spheres and spheroids apparent in Fig. 4 are due to quenching of internal resonances caused by an increase of Qabs with increasing size parameter. The critical transition size of 0:5 mm corresponds to a size parameter of x 5, and to Qabs 1. To further investigate the impact of surface roughness we now take a little side-track and consider the optical properties of ice at a far-infrared wavelength of 47 mm, using a refractive index of 1.34+0.83i. Results for ice particle sizes up to 100 mm are shown in Fig. 5. We see that o and g computed with spheres and spheroids are nearly indistinguishable over virtually the entire size range. Cbak computed with spheres displays small oscillations, but is otherwise
that of spheroids. The extinction cross section Cext at large sizes (not shown) is about 7% larger for spheroids than for spheres. This can be explained by the notorious difficulties of comparing sizes of spherical and non-spherical particles. Although the volume-equivalent radius is a reasonable measure for size comparisons, it does not necessarily yield identical extinction cross sections for spheres and spheroids. The single-scattering albedo o ¼ Csca =Cext (where Csca denotes the scattering cross section) is a relative measure that is often less sensitive to how we define size-equivalence. As can be seen in Fig. 4 (upper left), o at particle radii r0 o 500 nm is different for spheres and spheroids and comparable for spheres and Chebyshev particles. For increasing sizes r0 4 500 nm, o converges for spheres and spheroids, while it diverges for spheres and Chebyshev particles. The same behaviour can be observed for g (lower left panel). Equally remarkable is the behaviour of Cbak as a function of size for the three model particles (lower right). For spheres one observes conspicuous resonance features, which are absent for spheroids (owing to the averaging over orientations and shape parameters). However, for particle sizes r0 4500 nm Cbak computed for spheres seems to oscillate, to a good approximation, about that computed for spheroids. So Cbak averaged over any size distribution dominated by particles with sizes r0 4 500 nm is expected to give similar results for spheres and spheroids. By contrast, Chebyshev particles predict significantly lower values of Cbak for r0 4 500 nm (roughly half as much). This could have important implications for the interpretation of observed lidar backscattering coefficients. If one observes backscattering signals from moderately or highly absorbing particles with surface roughness, an inversion algorithm that neglects the effects of surface roughness could under-predict aerosol 3
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very similar to that computed with spheroids over the entire size range. Remarkably enough, Cbak at r0 ¼ 100 mm computed for spheres or spheroids is larger than that computed for Chebyshev particles by almost a factor of 3. Again, this indicates that neglecting the effect of small-scale surface roughness is potentially an important source of error in inverse modelling of lidar backscattering observations. For the hematite particles the convergence of the optical properties of spheres and spheroids occurred at an absorption efficiency of Qabs 1 and a size parameter of x 5. For our ice particles, Qabs 1 corresponds to a particle size of r0 5 mm, while x 5 corresponds to r0 37 mm. Clearly, the convergence of the optical properties of spheres and spheroids seems to correlate to Qabs rather than to x. This strongly supports our hypothesis that this convergence effect is caused by quenching of internal resonances due to absorption inside the particle. However, a close analysis of our results reveals that the optical properties of ice spheres and spheroids are very close even for sizes r0 o 5 mm. A possible explanation is that the ice particles are optically softer than the hematite aerosols (i.e. the real part n of their refractive index is smaller). Therefore, less of the incoming radiation is scattered at the particle surface, and more penetrates into the particle, where it can be absorbed, whence the quenching of internal resonances becomes significant even for Qabs t1. Next, we take a closer look at the diverging optical properties of spheres and Chebyshev particles. For hematite, the onset of the divergence is at r0 0:5 mm. It is obviously a coincidence that convergence of spheres and spheroids occurs at the same particle size. This becomes apparent in Fig. 5. For ice, the divergence of spheres and Chebyshev particles occurs at r0 30 mm, while spheres and spheroids give almost identical optical properties over the entire size range. This indicates that the convergence of spheres and spheroids on one hand, and the divergence of spheres and Chebyshev particles on the other must be related to different physical processes. A plausible explanation for the divergence of spheres and Chebyshev particles may be that surface roughness effects first become apparent when the perturbation amplitude on the particle surface in comparison to the wavelength, A=l ¼ er0 =l, exceeds a certain threshold value. For hematite, Chebyshev particles with e ¼ 0:05 and r0 0:5 mm have A=l 0:04. For ice, Chebyshev particles with e ¼ 0:05 and r0 30 mm give A=l 0:03, which is of the same order. However, we would expect that a decrease in e would result in a corresponding increase in the critical particle size at which the divergence effect occurs. We can see in Fig. 4 that this is not the case. The optical properties of Chebyshev particles with e ¼ 0:03 (dash-dotted line) start to diverge from those of spheres at about the same size ð0:5 mmÞ as for e ¼ 0:05 (thick solid line). It is quite possible that the significance of small-scale surface roughness depends on a complex interplay of several factors. The relative perturbation amplitude A=l is certainly one important factor. Quenching of internal resonances may also contribute, because in weakly absorbing wavelength-scale particles the effect of small-scale surface roughness may be completely overshadowed by the
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effect of internal resonances. Another important factor may be the real part n of the refractive index, which determines how much radiation is actually scattered at the particle surface. We now understand why we did not see any significant differences between the Mueller matrix elements of Chebyshev particles and spheres for hematite in Fig. 2. It is because the size distribution of this sample is dominated by particles below the critical size of 0:5 mm, at which the transition to the small-scale surface roughness regime occurs. To attain a better understanding of the impact of small-scale surface roughness on the differential scattering behaviour of hematite we consider a lognormal size distribution with r0 =500 nm and s ¼ 0:8. This size distribution contains a large fraction of particles above the critical size. We average the Mueller matrix elements over this size distribution for each of the three model particles. The result is shown in Fig. 6. For most of the Mueller matrix elements, spheres and spheroids yield similar results, while Chebyshev particles display a different angular dependence. This clearly shows that the effect of small-scale surface roughness ðL 5 2pr0 Þ dominates over the effect of large-scale nonsphericity ðL pr0 Þ, even though, on average, the perturbation amplitude A=b a of the ensemble of spheroids is much larger than that of the Chebyshev particles! The phase functions P11 of spheres and spheroids are nearly indistinguishable, while that of Chebyshev particles is characterised by less side- and backscattering. This explains the high asymmetry parameter and the small Cbak observed for large Chebyshev particles in Fig. 4. The element P22, which enters into the linear depolarisation ratio dL ¼ ðP11 P22 Þ=ðP11 þP22 Þ, is equal to unity for both spheres and Chebyshev particles. The corresponding results obtained for spheroids do not significantly deviate from unity either. The deviation of P22 from unity is usually considered to be a good indication for nonsphericity. However, for particles with sufficiently high absorption efficiencies, the sensitivity of P22 to particle shape is strongly reduced. The elements P33, P44, and, most importantly, P12 are distinctly different for Chebyshev particles, and rather similar for spheres and spheroids, thus underlining the importance of surface roughness for the polarisation properties. The element P34 is rather similar for all three particle models, except near the backscattering direction, where Chebyshev particles display a local minimum. To get some idea of what happens at larger size parameters, we investigate the Mueller matrix at a fixed size of r ¼ 1:4 mm, which corresponds to a size parameter of x =14. This is the largest size parameter considered in our computations. Fig. 7 reveals that the phase function of Chebyshev particles displays considerably less side-scattering than that of either spheres or spheroids. Interestingly enough, the element P22 deviates from unity not only for spheroids but also for Chebyshev particles. For spheroids, the departure from unity is strongest in the forward-scattering hemisphere, while Chebyshev particles depolarise mainly in the backscattering hemisphere. However, the effect is very weak. This indicates that depolarisation measurements can, in principle, easily confuse spherical particles and strongly absorbing
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particles with non-spherical or rough boundary surfaces. Another conspicuous feature of the Mueller matrix elements of large Chebyshev particles is the high oscillation amplitude (even though the results are orientation averaged). Such high resonances are seen also for rough spheres by [21] and dusted spheres by [22], indicating that small deformations of spheres are not sufficient to eliminate the strong resonances associated to spherical scatterers. A recent literature review [23] suggests that smallscale surface roughness would show some impact to individual scattering matrix elements already at size parameters near unity. The reviewed studies had focused on weakly absorbing particles, so we analysed our results to see whether the finding also applies here. We found that the matrix elements start to deviate at size parameters of about x= 1.5 for hematite and x =2 for ice. Ice at l ¼ 47 mm has a lower value of n and a higher value of k, which means that more radiative energy penetrates into the particle, and more of the penetrated radiation is absorbed. This may be the reason why the effect of small-scale surface roughness becomes more pronounced for smaller size parameters in the case of hematite. The data are currently very limited, however, and based on several different models for roughness. Therefore, this finding should be considered preliminary. Note that the
critical size parameter at which surface roughness effects become important also depends on the metric used for defining the onset of divergence, which in turn depends on the accuracy required in any given application.
4. Conclusions and outlook The focus of our study was to investigate the impact of departures from spherical shape on different size-scales L on the optical properties of hematite aerosols. High-order Chebyshev particles were used as a proxy for particles with small-scale surface roughness, for which L 5 2pr0 and L 5 l. A shape distribution of spheroids was employed as a proxy for smooth nonspherical particles, i.e., particles that can be obtained by a homeomorphic deformation of a sphere with a deformation ‘‘wavelength’’ L pr0 . A main conclusion of this study is that the morphological properties of spheroids ðL pr0 Þ are mainly important for particles with low absorption efficiencies. The morphology of high-order Chebyshev particles ðL 5 2pr0 Þ becomes significant at larger size parameters. For instance, it was found that the backscattering cross section of spheres and spheroids can be as much as 2–3 times as high as that of Chebyshev particles. Thus, neglecting surface roughness
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Fig. 7. Mueller matrix computed for Chebyshev particles (thin solid line), spheres (dashed line), and spheroids (thick solid line) with a fixed particle size r ¼ 1:4 mm.
effects can introduce large biases in the interpretation of lidar backscattering measurements. The difference between spherical particles and smooth nonspherical spheroids are prominent for low absorption efficiencies Qabs, and disappear for larger values of order Qabs 1. The observed differences at low values of Qabs are caused by internal optical resonances inside the particles, which become quenched as the absorption efficiency Qabs of the particles increases. This increase in Qabs can either be achieved by increasing the particle size parameter, as in our study, or by increasing the imaginary part k of the refractive index, as in [9]. Comparison with corresponding results for ice at a far-infrared wavelength confirm that the convergence of the optical properties of spheres and spheroids mainly depends on Qabs rather than on the size parameter x. Further, the quenching of resonance effects seems to be more efficient for optically soft particles (i.e. particles with low values of the real part n of the refractive index). This is due to less scattering by the particle surface, which allows more radiative energy to penetrate into the absorbing material. By contrast, the optical properties of ice spheres and high-order Chebyshev particles start diverging at larger particle sizes. It is difficult to identify a single cause for this effect. For instance, the amplitude A of the surface
roughness is likely to be important for the divergence of the optical properties of particles with smooth and rough surfaces. However, comparison of hematite Chebyshev particles with relative amplitudes e ¼ A=r0 ¼ 0:03 and e ¼ 0:05 did not provide any clear evidence that the onset of divergence decreases in proportion to an increase in e. Comparison of results of hematite and ice indicates that the onset of divergence does not correlate to Qabs, so it is probably not directly related to the quenching of internal resonances. However, it is conceivable that for particles composed of weakly absorbing material the effect of surface roughness will be obscured by the effect of internal resonances. So Qabs may, at least, have an indirect significance for the observability of surface roughness effects. The real part n of the refractive index determines how optically hard the material is. This can also be expected to contribute to the importance of small-scale surface roughness. The results of this study allow us to clearly formulate future research required to better understand the importance of surface roughness. First, we need to study a variety of different materials and wavelengths to better understand the conditions under which small-scale surface roughness becomes the dominant morphological property. It will be particularly interesting to study aerosols composed of
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weakly absorbing material, for which surface-roughness effects can be expected to compete with effects related to internal resonances. To this end we need, second, to significantly extent our current capabilities of existing computational methods. Approximate methods, such as the geometric optics approximation, are not valid for particles with small-scale surface roughness. On the other hand, the applicability of currently available T-matrix programs to high-order Chebyshev particles is rather limited in terms of accessible size parameters. At larger size parameters computations can become unstable and inaccurate due to numerical ill-conditioning problems. We therefore need to improve our current models with special focus on particles with rough surfaces, such as high-order Chebyshev particles. Third, we will need more studies for different shape models, as our conclusions are based on the use of spherical, spheroidal, and Chebyshev model particles only. Fourth, it would be helpful to obtain additional measurements for hematite aerosols with larger effective radii. At a wavelength of l ¼ 632:8 nm, hematite samples that are dominated by particles larger than 500 nm are expected to be most relevant. Fifth, there may be cases for which the optical properties are simultaneously sensitive to large-scale departures from spherical shape ðL pr0 Þ and small-scale surface roughness ðL 5 2pr0 Þ. Such cases could, e.g., be modelled by spheroids perturbed by a Chebyshev polynomial [9]. Again, one needs to improve currently available T-matrix programs to be able to use such more advanced model particles for sufficiently large size parameters. Finally, it would be interesting to study the significance of 3D surface roughness effects. Electromagnetic scattering computations for nonaxisymmetric particles tend to be highly demanding. To keep such computations manageable, this will likely involve the exploitation of discrete particle symmetries [6].
Acknowledgements ˜ oz for making her The authors are grateful to Olga Mun measurement data publicly available. We are further grateful to Michael Mishchenko and Tom Rother for making their respective T-matrix programs available. M. Kahnert acknowledges funding from the Swedish Research Council under contract 80438701. T. Nousiainen and P. Mauno acknowledge the funding by the Academy of Finland (contracts 125180 and 12148). M. Thomas and two anonymous reviewers are acknowledged for helpful comments and suggestions. References ˜ oz O, Volten H, Hovenier JW, Min M, Shkuratov YG, Jalava JP, [1] Mun et al. Experimental and computational study of light scattering by irregular particles with extreme refractive indices: hematite and rutile. Astron Astrophys 2006;446:525–35.
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