- Email: [email protected]
Determination of refractive index for absorbing spheres
Optik 124 (2013) 5254–5258
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Determination of refractive index for absorbing spheres Chang Gyu Woo a , Sukbeom You b , Jeonghoon Lee c,∗ a
Samsung Electronics, 416 Maetan 3-dong, Yeongtong-gu, Suwon 443-742, Republic of Korea Samsung Display, #95 Samsung-ro, Giheung-gu, Yongin 446-711, Republic of Korea c School of Mechanical Engineering, Korea University of Technology and Education, Cheonan 330-708, Republic of Korea b
a r t i c l e
i n f o
Article history: Received 15 October 2012 Accepted 16 March 2013
Keywords: Refractive index Absorption MAAP
a b s t r a c t Complex refractive index for submicron absorbing spheres was determined by simultaneously measuring absorption coefficients and number concentrations with a multiangle absorption photometer and a condensation particle counter, respectively. The absorption cross section calculated from the absorption coefficient and the number concentration was fitted to the absorption cross section calculated using Mie theory. For homogeneously dyed spheres, the absorption cross section depended on the size of sphere. For non-homogeneously dyed spheres, however, the absorption cross section had nothing to do with the size of sphere. It is concluded that the ability to measure refractive index by our method held good as long as the absorption phenomenon takes place in the near field surrounding particles surface. To the best of our knowledge, it is for the first time that the refractive index was determined by only absorption measurements and theory. This method is believed to help understand the fundamental of the optical properties for other absorbing particles. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction The optical light absorption is one of the most important properties in various fields such as bioengineering research and climate research. In the bioengineering field, light-absorbing particles are used to stain bio organism. In the sustainable energy community, to improve the absorption in solar cell device, considerable attention has been paid to the reduction in the physical thickness of the solar photovoltaic absorber layers [1] and unique properties of nanomaterials for applications in dye-sensitized solar cells have been extensively investigated [2]. In the climate change research field, the light absorption by the particles of the sunlight is considered important because it heats and warms up the boundary layer of the atmosphere so that impacts the regional weather [3]. Information about the refractive index (RI) of absorbing particles is more or less difficult to find though the RI is an important optical property. The absorbing particles include soot and black carbon (BC). For various carbons, usually referred in atmospheric sciences literature, the real part of the RI ranges from 1.34 to 2.5, and the imaginary part of the RI describing the optical light absorption ranges from 0.07 to 1.46 [3]. This wide range of RI for carbon particles results in the large uncertainty in estimating the optical properties. Several studies have been conducted to retrieve the refractive index by performing the extinction experiments [4,5]. Light refraction was
∗ Corresponding author. Tel.: +82 41 560 1151; fax: +82 41 560 1253. E-mail address: [email protected] (J. Lee). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.03.083
also used to measure effective refractive index of a turbid colloidal suspension [6]. It is noted that the imaginary part of the refractive index is related to the light absorption. The previous studies involved extinction experiments rather than absorption measurements when measuring absorption properties [7]. It is believed to be appropriate that an experimental method incorporating only the absorption measurements without performing refraction experiments or extinction experiments can estimate the imaginary part of the refractive index for absorbing spheres within a relatively reasonable precision. In the present study, we solve Mie theory in order to estimate the complex RI for the absorbing spheres. 2. Apparatus and method Fig. 1 shows the apparatus of the experimental setup for the present study. We produced two kinds of light absorbing particles. Nigrosin and black polystyrene latex were generated using a constant output atomizer (Dong Sung Industry). Nigrosin is a mixture of synthetic black dyes made by heating a mixture of nitrobenzene, aniline and aniline hydrochloride in the presence of a copper or iron catalyst. In biology, nigrosin is used for negative staining of bacteria. Its main industrial uses are as a colorant for lacquers and varnishes and in marker-pen inks [8]. On the other hand, the black polystyrene latex particles are spheres on the surface of which blackish pigment is dyed. Polystyrene latex particles are suspended in water without being agglomerated [9]. Distilled water-based aqueous solutions of the polystyrene latex particles were prepared for the production of particles of interest. Particle-free nitrogen was
C.G. Woo et al. / Optik 124 (2013) 5254–5258
5255
is expressed as the sum of the real part and the imaginary part as below. m = n + ik
Fig. 1. Schematics of the experimental apparatus used in the present study.
injected into the atomizer at about 1.5 bar to produce the particle stream, which may contain the droplet of water, so the relative humidity of the particle stream is higher than that of the indoor air stream. A diffusion drier (Dong Sung Industry) was used to dry out the moisture which was included in the particle stream. The number concentration of the particle depends on the concentration of the solution and the injection pressure of the nitrogen. The number concentration was controlled using a dilution system consisting of a ball valve and a high efficiency particulate air filter (Pall Life Science, 12144). The dried particle stream passed through a soft X-ray neutralizer (HCT, 4530). Then, the particle size was classified using a differential mobility analyzer equipped with a long concentric column (Dong Sung Industry). The number concentrations of the size selected particles were measured by a condensation particle counter (TSI 3775). Simultaneously, absorption coefficients of the size selected monodisperse particles were measured by a multiangle absorption photometer (Thermo Scientific 5012). The wavelength of the light used in the present study is 670 nm. Absorption coefficient is expressed as a number concentration multiplied by an absorption cross section. Therefore, it is needed to acquire the absorption coefficient data as a function of the number concentration to obtain the absorption cross section. The slope of the graph of absorption coefficient versus number concentration allows us to estimate the absorption cross section. Absorption efficiencies of the sphere are defined as a ratio of the absorption cross section to the geometric cross section. The absorption efficiencies can be plotted as a function of size parameter which is defined as the ratio of the circumference of one spherical particle to the wavelength of the light source as below [10]. x = ˛d =
d
(1)
The x is the size parameter and d is the diameter of a spherical particle and is the wavelength of the light. From the Mie theory, absorption cross section is expressed as below 2 [(2n + 1){Re{an + bn } − (|an |2 + |bn |2 )}] ˛2 ∞
Cabs =
(2)
n=1
where an and bn are scattering coefficients and ˛ is / [10]. The scattering coefficients are expressed as below an =
bn =
m2 jn (mx)[xjn (x)] − 1 jn (x)[mxjn (mx)] m2 jn (mx)[xhn (x)] − 1 hn (x)[mxjn (mx)] (1)
(1)
1 jn (mx)[xjn (x)] − jn (x)[mxjn (mx)] 1 jn (mx)[xhn (x)] − hn (x)[mxjn (mx)] (1)
(1)
(3)
(4) (1)
where jn is the spherical Bessel function of the first kind and hn is the spherical Bessel function of the third kind. In Eqs. (3) and (4), m denotes the refractive index of the sphere [10]. The refractive index
(5)
The real part of the refractive index is related to the light scattering and the imaginary part of the refractive index is related to the light absorption. Therefore, for the absorbing particles, the real part of the refractive index is not associated to the light absorption. Moreover, it is reasonable to extract the absorptive refractive index, that is, the imaginary part of the refractive index, by the ‘absorption’ measurements rather than extinction measurements. Some studies on the determination of the refractive index from the extinction experiments have been reported [4,5]. Their approaches were based on spectral measurements of the ‘extinction’ of light in order to retrieve the refractive index. The extinction of light happens by the combined action of scattering and absorption. It is noted that the refractive index includes the information about both the scattering and the absorption, not the extinction. We determine the imaginary part of the refractive index by analyzing the measured absorption coefficient data. Usually, the scattering properties and extinction properties are sensitive to the particle shape. Especially, the optical properties of the soil-derived mineral particles are reported to be sensitive to particle shape [11]. Hence, we imagine that the absorption properties are also sensitive to the particle shape. In this situation, we adopt a spherical Mie model to simplify the complexity of particle shape. The measured absorption efficiencies of spheres were fitted to the absorption efficiencies calculated from Mie theory on condition that the chi-square of the difference between the measured absorption efficiencies and the calculated absorption efficiencies are floated. Here, the curve fitting depends only on the refractive index of the particle. By minimizing the chi-square, we attempted to obtain the complex refractive index of the particles. 3. Results and discussion Comparison with the previous study in the determination of the complex refractive index for nigrosin was performed. Our MATLAB (MathWorks) code compares the measured absorption efficiency with the absorption efficiency calculated from the Mie code [12] for the set of the real part and the imaginary part of the refractive index. The chi-squares were saved for every set of the real part of the RI and the imaginary part of the RI. Then, the minimum of the chisquare was searched for by using the built-in function, ‘fminsearch’. The resolution was increased step by step until our code found the minimum within the desired precision. Finally, the fitted curve was plotted using the refractive index that minimized the chi-square of the data set. Previously, the RI of nigrosin particles was measured and it was suggested that the value was 1.7 + 0.31i [7]. However, the refractive index estimated in our study was 1.55 + 0.38i. Fig. 2 shows that our code estimated the imaginary part of the refractive index slightly different from previous results. We believe that the discrepancy between our results and the previous study is due to the fact that the previous study fitted the ‘extinction’ efficiency, not absorption efficiency. It is noted that our experiment is based on absorption measurements. It is worthwhile to investigate the effect of the number of data points on the result. In the previous study done by Lack et al. [7], the number of data participated in the curve fitting was 10. We chose 9 data points by a jackknifing method and curve-fitted to the Mie absorption efficiency using the selected 9 data points. Then, 10 sets of 9 data points allowed us to estimate 10 refractive indexes. In the same way, we selected 8 data points among the 10 data points
5256
C.G. Woo et al. / Optik 124 (2013) 5254–5258
60 50
Frequency (%)
n=9 40 30 20 10 0 0.20 Fig. 2. Comparison of our code by curve fitting the absorption efficiencies data from Lack et al. [7]. Absorption efficiencies were also superimposed with the absorption efficiencies.
0.25
0.30 Imaginary part of RI
0.35
0.40
0.30 Imaginary part of RI
0.35
0.40
60 50 n=8
Frequency (%)
and then estimated the refractive index. The results are listed in Table 1. The average value for the case of n = 9 is very similar to that of n = 8. However, the range of the refractive index is quite different. Fig. 3 shows the histogram of the imaginary part of the refractive index estimated for both cases of n = 9 and n = 8. In both cases, the imaginary refractive index ranges from approximately 0.2 to 0.39. Of course, the uncertainty is reduced as the number of data points increases. The estimated refractive index becomes deviated from the 1.55 + 0.38i as the number of data points is reduced.
40 30 20 10 0 0.20
4. Nigrosin: homogeneously dyed spheres The refractive index for various sized nigrosin spherical particles generated in the present study was estimated using our code. Fig. 4 shows the measured absorption cross section for nigrosin particles. Each data point was an average of about 30 data measured at every minute for 30 min. Measured absorption cross section increases as the diameter of the nigrosin particle increases. The error bars are one standard deviations of the mean values. It is certain that the absorption efficiency should be larger than 1 since the measured absorption cross section is larger than the geometric cross section for the entire range of measurements. Absorption processes include geometrical blocking, internal reflection, internal refraction, large angle diffraction and wave resonance [13]. Incident beam may be blocked similar to the Beer’s absorption and the photons beyond the physical cross section are changed into surface wave to be damped as they propagate. Beam internally penetrating particle is absorbed through refraction and reflection and fades away similar to the surface wave. Therefore, the absorption efficiency larger than 1 implies that all photons incident on the particles are absorbed, and plus,
0.25
Fig. 3. The effect of the number of data points. Uncertainty of the refractive index depending on the number of data points.
some photons beyond the geometric cross section of the particles are also absorbed. Fig. 5 shows absorption efficiency for the nigrosin particles generated in the present study. The line depicts a fit to the measurement in which both the real part and the imaginary part of the complex refractive index are the free parameters. We varied the real part of the RI from 0 to 2.000 and the imaginary part from 0 to 0.500. The refractive index of the nigrosin particles determined at the wavelength of 670 nm is approximately 1.70 + 0.24i, which imaginary refractive index is comparable to the literature value, 1.67 + 0.26i [14] if we consider that the refractive index is inversely proportional to the wavelength.
# of data
n = 10 n=9 Avg Std Max Min n=8 Avg Std Max Min
3
2
Table 1 Refractive index estimated from the selected data.
Absorption cross section (m /part.)
4
Refractive index n
k
1.544
0.377
1.572 0.094 1.837 1.519
0.364 0.045 0.383 0.236
1.576 0.095 1.885 1.498
0.361 0.046 0.388 0.205
2
-13
10
7 6 5 4
Measured absortpion cross section Geometric cross section
3 2
400
600 800 Diameter (nm)
1000
Fig. 4. Absorption cross section of nigrosin for various sizes.
1200
C.G. Woo et al. / Optik 124 (2013) 5254–5258
Fig. 5. Absorption efficiencies versus size parameter for nigrosin particles. Shown is refractive index estimated from curve fitting of the absorption efficiency using Mie theory.
5. Black polystyrene latex: non-homogeneously dyed spheres According to the manufacturer’s technical note [15], the black polystyrene latex particles are made by dye diffusion-entrapment method, which involves the swelling of the polystyrene latex in a dye solvent. The dye penetrates into the polystyrene latex matrix and is entrapped when the solvent is removed through evaporation or transfer to an aqueous phase. The caveat in this manufacturing process is that the penetration depth cannot be controlled. In the case of the black spheres such as nigrosin particles, refractive index does not change along the size of particles. However, the refractive index of the larger black polystyrene latex spheres can be different from that of the smaller ones because the dye pigment may be non-uniformly penetrated. Therefore, it is impossible to perform curve fitting as a function of size for the refractive index. Instead, it is worthwhile to solve the absorption efficiency equation with which the measured absorption efficiency becomes equal to the calculated absorption efficiency. We attempted to assess the refractive index in two different ways: (i) bulk analysis and (ii) Mie calculation. We calculated the refractive index by iterating the Mie code until the relative error became less than 0.01%. We believe that the black dye differently penetrates the surface of the latex spheres without regard to the particle size. The measured absorption cross sections for the two black polystyrene latexes were 1.49 × 10−14 and 1.37 × 10−14 m2 /part for 233 nm and 980 nm, respectively [16]. It is interesting that the absorption cross section decreased as the black polystyrene latex became larger. Since the volume of the 233 nm black polystyrene latex particle is the smaller one, it is not unreasonable to assume that the dye was fully penetrated for the 233 nm black polystyrene latex particles to the core. For the homogeneous bulk materials, the absorption coefficient, Babs , is related to the refractive index as follows [10]: Babs =
4k
(6)
Here, the absorption coefficient can be thought of as the absorption cross section for one individual sphere, resulting in Babs = abs /(/6d3 ). Therefore, the imaginary part of the refractive index is expressed as k=
abs (d3 /6) 4
(7)
Using Eq. (7), the imaginary part of the refractive index was calculated to be 0.120 ± 0.027. The uncertainty for the imaginary refractive index was calculated from the uncertainties of the
5257
absorption cross section ( abs ) and the diameter (d) in Eq. (7) using the same procedure of uncertainty propagation as shown in elsewhere [17]. In another way, the imaginary part of the refractive index was calculated to be 0.112i from the Mie calculation on the assumption of 1.66 for the real part of the refractive index. It is interesting that two techniques, which depend on completely different physical principles for refractive index measurement, gave similar values when considering that the theory applied assumes that the dye penetrated completely and homogenously. For the 980 nm black polystyrene latex, however, it is probable that the black dye did not penetrate completely. We imagine that the black dye penetrates the surface of the latex spheres shallowly, so that the latex sphere is incompletely dyed. The imaginary part of the RI measured by the Mie calculation was 0.00049 ± 0.0001. However, the imaginary part of the RI estimated from the bulk method was 0.001482, which does not exist in the range of the imaginary refractive index calculated by Mie code. Obviously, 980 nm black polystyrene latex is even less blackish than 233 nm one. The reduced absorption cross section for the larger particle demonstrates that the light absorption for the dyed polystyrene materials has nothing to do with the particle size. In addition, the different refractive indexes measured by different principles are the evidence that the dye did not penetrate homogeneously for the 980 nm black latex sphere as opposed to the 233 nm one. Photobleaching effect could explain the low absorption cross section for the larger spheres. The black dye insufficiently adsorbed on the surface of the polystyrene sphere due likely to low surface area compared to the smaller one could be bleached by the light. Therefore, it is probable that the absorption cross section for 980 nm one is smaller than that for 233 nm one because of the photobleaching effect even though the 980 nm one is geometrically larger than 233 nm one. Not only the optical artifacts but also geometrical issue can affect the absorption cross section. Spheres can collide with each others to form large agglomerate. In this case, Mie theory should be replaced by other optics theory associated with the shape of the particles [18]. More detailed discussion needs further study by measuring the absorption spectrum in visible range. This is beyond the scope of the present study.
6. Conclusion Real part and imaginary part of refractive index for absorbing spheres were determined using a multiangle absorption photometer together with Mie theory. For the absorbing spheres generated using a constant output atomizer, absorption coefficient and the number concentration were measured through a multiangle absorption photometer and a condensation particle counter, respectively. The absorption cross section for nigrosin was estimated from the slope of the plot of the absorption coefficients as a function of the particle number concentrations. The measured absorption cross sections were fitted to the absorption cross sections calculated from the Mie theory. For the nigrosin, the refractive index obtained by the best fit corresponded to the 1.70 + 0.24i, which was obtained by minimizing the chi-square of the data. The refractive index measured by our method holds valid as long as the absorption phenomenon takes place in the near field surrounding particles surface. This method was applied to the measurement of the optical properties for non-homogeneously dyed absorbing particle, black polystyrene latex. It was observed that the larger particle was less black than the smaller one. In this case, the absorption cross section was independent on the particle size. As far as we know, it is unprecedented that the refractive index was estimated from the absorption experiments and the refractive index for the non-homogeneous absorbing particles was investigated from the refractive index point of view.
5258
C.G. Woo et al. / Optik 124 (2013) 5254–5258
Acknowledgements This work was supported by Basic Science Research Program (No. 2012R1A1B4002700) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST). We give special thanks to Mr. S Han for collecting some of absorption data. References [1] H. Atwater, A. Polman, Plasmonics for improved photovoltaic devices, Nat. Mater. 9 (2010) 205–213. [2] M. Grätzel, Photoelectrochemical cells, Nature 414 (2001) 338–344. [3] N.A. Marley, J.S. Gaffney, C. Baird, C.A. Blazer, P.J. Drayton, J.E. Frederick, An empirical method for the determination of the complex refractive index of sizefractionated atmospheric aerosols for radiative transfer calculations, Aerosol Sci. Technol. 34 (2001) 535–549. [4] G.E. Thomas, S.F. Bass, R.G. Grainger, A. Lambert, Retrieval of aerosol refractive index from extinction spectra with a damped harmonic-oscillator band model, Appl. Opt. 44 (2005) 1332–1341. [5] M. Wendisch, W. von Hoyningen-Huene, Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements, Atmos. Environ. 28 (1994) 785–792. [6] A. Reyes-Coronado, A. Garcia-Valenzuela, C. Sánchez-Pérez, R.G. Barrera, Measurement of the effective refractive index of a turbid colloidal suspension using light refraction, New J. Phys. 7 (2005) 89–90.
[7] D.A. Lack, E.R. Lovejoy, T. Baynard, A. Pettersson, A.R. Ravishankara, Aerosol absorption measurement using photoacoustic spectroscopy: sensitivity, calibration, and uncertainty developments, Aerosol Sci. Technol. 40 (2006) 697–708. [8] F.J. Green, The Sigma–Aldrich Handbook of Dyes Stains and Indicators, Aldrich, Milwaukee, 1990. [9] S. Deguchi, K. Tsujii, Supercritical water: a fascinating medium for soft matter, Soft Matter 3 (2007) 797–803. [10] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, Inc., New York, 1983. [11] O.V. Kalashnikova, I.N. Sokolik, Modeling the radiative properties of nonspherical soil-derived mineral aerosols, J. Quant. Spectrosc. Radiat. Transfer 87 (2004) 137–166. [12] C. Mätzler, MATLAB functions for Mie scattering and absorption version 2, Institut für Angewandte Physik, Research Report No. 2002-11, 2002. [13] D.L. Mitchell, Parameterization of the Mie extinction and absorption coefficients for water clouds, J. Atmos. Sci. 57 (2000) 1311–1326. [14] D.M. Garvey, R.G. Pinnick, Response characteristics of the particle measuring systems active scattering aerosol spectrometer probe (ASASP-X), Aerosol Sci. Technol. 2 (1983) 477. [15] Bangs Lab, http://www.bangslabs.com/sites/default/files/bangs/docs/pdf/103. pdf [16] Y.-K. Jeong, J. Lee, Optical properties of polystyrene black dyed microspheres, Curr. Appl. Phys. 12 (2012) S76–S79. [17] J. Lee, Estimation of emission properties for silica particles using thermal radiation spectroscopy, Appl. Opt. 50 (2011) 4262–4267. [18] J. Lee, Effect of structure factor on aggregate number concentration estimated using Rayleigh–Debye–Gans scattering theory, Opt. Appl. 41 (2011) 519–525.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Determination of refractive index for absorbing spheres Chang Gyu Woo a , Sukbeom You b , Jeonghoon Lee c,∗ a
Samsung Electronics, 416 Maetan 3-dong, Yeongtong-gu, Suwon 443-742, Republic of Korea Samsung Display, #95 Samsung-ro, Giheung-gu, Yongin 446-711, Republic of Korea c School of Mechanical Engineering, Korea University of Technology and Education, Cheonan 330-708, Republic of Korea b
a r t i c l e
i n f o
Article history: Received 15 October 2012 Accepted 16 March 2013
Keywords: Refractive index Absorption MAAP
a b s t r a c t Complex refractive index for submicron absorbing spheres was determined by simultaneously measuring absorption coefficients and number concentrations with a multiangle absorption photometer and a condensation particle counter, respectively. The absorption cross section calculated from the absorption coefficient and the number concentration was fitted to the absorption cross section calculated using Mie theory. For homogeneously dyed spheres, the absorption cross section depended on the size of sphere. For non-homogeneously dyed spheres, however, the absorption cross section had nothing to do with the size of sphere. It is concluded that the ability to measure refractive index by our method held good as long as the absorption phenomenon takes place in the near field surrounding particles surface. To the best of our knowledge, it is for the first time that the refractive index was determined by only absorption measurements and theory. This method is believed to help understand the fundamental of the optical properties for other absorbing particles. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction The optical light absorption is one of the most important properties in various fields such as bioengineering research and climate research. In the bioengineering field, light-absorbing particles are used to stain bio organism. In the sustainable energy community, to improve the absorption in solar cell device, considerable attention has been paid to the reduction in the physical thickness of the solar photovoltaic absorber layers [1] and unique properties of nanomaterials for applications in dye-sensitized solar cells have been extensively investigated [2]. In the climate change research field, the light absorption by the particles of the sunlight is considered important because it heats and warms up the boundary layer of the atmosphere so that impacts the regional weather [3]. Information about the refractive index (RI) of absorbing particles is more or less difficult to find though the RI is an important optical property. The absorbing particles include soot and black carbon (BC). For various carbons, usually referred in atmospheric sciences literature, the real part of the RI ranges from 1.34 to 2.5, and the imaginary part of the RI describing the optical light absorption ranges from 0.07 to 1.46 [3]. This wide range of RI for carbon particles results in the large uncertainty in estimating the optical properties. Several studies have been conducted to retrieve the refractive index by performing the extinction experiments [4,5]. Light refraction was
∗ Corresponding author. Tel.: +82 41 560 1151; fax: +82 41 560 1253. E-mail address: [email protected] (J. Lee). 0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.03.083
also used to measure effective refractive index of a turbid colloidal suspension [6]. It is noted that the imaginary part of the refractive index is related to the light absorption. The previous studies involved extinction experiments rather than absorption measurements when measuring absorption properties [7]. It is believed to be appropriate that an experimental method incorporating only the absorption measurements without performing refraction experiments or extinction experiments can estimate the imaginary part of the refractive index for absorbing spheres within a relatively reasonable precision. In the present study, we solve Mie theory in order to estimate the complex RI for the absorbing spheres. 2. Apparatus and method Fig. 1 shows the apparatus of the experimental setup for the present study. We produced two kinds of light absorbing particles. Nigrosin and black polystyrene latex were generated using a constant output atomizer (Dong Sung Industry). Nigrosin is a mixture of synthetic black dyes made by heating a mixture of nitrobenzene, aniline and aniline hydrochloride in the presence of a copper or iron catalyst. In biology, nigrosin is used for negative staining of bacteria. Its main industrial uses are as a colorant for lacquers and varnishes and in marker-pen inks [8]. On the other hand, the black polystyrene latex particles are spheres on the surface of which blackish pigment is dyed. Polystyrene latex particles are suspended in water without being agglomerated [9]. Distilled water-based aqueous solutions of the polystyrene latex particles were prepared for the production of particles of interest. Particle-free nitrogen was
C.G. Woo et al. / Optik 124 (2013) 5254–5258
5255
is expressed as the sum of the real part and the imaginary part as below. m = n + ik
Fig. 1. Schematics of the experimental apparatus used in the present study.
injected into the atomizer at about 1.5 bar to produce the particle stream, which may contain the droplet of water, so the relative humidity of the particle stream is higher than that of the indoor air stream. A diffusion drier (Dong Sung Industry) was used to dry out the moisture which was included in the particle stream. The number concentration of the particle depends on the concentration of the solution and the injection pressure of the nitrogen. The number concentration was controlled using a dilution system consisting of a ball valve and a high efficiency particulate air filter (Pall Life Science, 12144). The dried particle stream passed through a soft X-ray neutralizer (HCT, 4530). Then, the particle size was classified using a differential mobility analyzer equipped with a long concentric column (Dong Sung Industry). The number concentrations of the size selected particles were measured by a condensation particle counter (TSI 3775). Simultaneously, absorption coefficients of the size selected monodisperse particles were measured by a multiangle absorption photometer (Thermo Scientific 5012). The wavelength of the light used in the present study is 670 nm. Absorption coefficient is expressed as a number concentration multiplied by an absorption cross section. Therefore, it is needed to acquire the absorption coefficient data as a function of the number concentration to obtain the absorption cross section. The slope of the graph of absorption coefficient versus number concentration allows us to estimate the absorption cross section. Absorption efficiencies of the sphere are defined as a ratio of the absorption cross section to the geometric cross section. The absorption efficiencies can be plotted as a function of size parameter which is defined as the ratio of the circumference of one spherical particle to the wavelength of the light source as below [10]. x = ˛d =
d
(1)
The x is the size parameter and d is the diameter of a spherical particle and is the wavelength of the light. From the Mie theory, absorption cross section is expressed as below 2 [(2n + 1){Re{an + bn } − (|an |2 + |bn |2 )}] ˛2 ∞
Cabs =
(2)
n=1
where an and bn are scattering coefficients and ˛ is / [10]. The scattering coefficients are expressed as below an =
bn =
m2 jn (mx)[xjn (x)] − 1 jn (x)[mxjn (mx)] m2 jn (mx)[xhn (x)] − 1 hn (x)[mxjn (mx)] (1)
(1)
1 jn (mx)[xjn (x)] − jn (x)[mxjn (mx)] 1 jn (mx)[xhn (x)] − hn (x)[mxjn (mx)] (1)
(1)
(3)
(4) (1)
where jn is the spherical Bessel function of the first kind and hn is the spherical Bessel function of the third kind. In Eqs. (3) and (4), m denotes the refractive index of the sphere [10]. The refractive index
(5)
The real part of the refractive index is related to the light scattering and the imaginary part of the refractive index is related to the light absorption. Therefore, for the absorbing particles, the real part of the refractive index is not associated to the light absorption. Moreover, it is reasonable to extract the absorptive refractive index, that is, the imaginary part of the refractive index, by the ‘absorption’ measurements rather than extinction measurements. Some studies on the determination of the refractive index from the extinction experiments have been reported [4,5]. Their approaches were based on spectral measurements of the ‘extinction’ of light in order to retrieve the refractive index. The extinction of light happens by the combined action of scattering and absorption. It is noted that the refractive index includes the information about both the scattering and the absorption, not the extinction. We determine the imaginary part of the refractive index by analyzing the measured absorption coefficient data. Usually, the scattering properties and extinction properties are sensitive to the particle shape. Especially, the optical properties of the soil-derived mineral particles are reported to be sensitive to particle shape [11]. Hence, we imagine that the absorption properties are also sensitive to the particle shape. In this situation, we adopt a spherical Mie model to simplify the complexity of particle shape. The measured absorption efficiencies of spheres were fitted to the absorption efficiencies calculated from Mie theory on condition that the chi-square of the difference between the measured absorption efficiencies and the calculated absorption efficiencies are floated. Here, the curve fitting depends only on the refractive index of the particle. By minimizing the chi-square, we attempted to obtain the complex refractive index of the particles. 3. Results and discussion Comparison with the previous study in the determination of the complex refractive index for nigrosin was performed. Our MATLAB (MathWorks) code compares the measured absorption efficiency with the absorption efficiency calculated from the Mie code [12] for the set of the real part and the imaginary part of the refractive index. The chi-squares were saved for every set of the real part of the RI and the imaginary part of the RI. Then, the minimum of the chisquare was searched for by using the built-in function, ‘fminsearch’. The resolution was increased step by step until our code found the minimum within the desired precision. Finally, the fitted curve was plotted using the refractive index that minimized the chi-square of the data set. Previously, the RI of nigrosin particles was measured and it was suggested that the value was 1.7 + 0.31i [7]. However, the refractive index estimated in our study was 1.55 + 0.38i. Fig. 2 shows that our code estimated the imaginary part of the refractive index slightly different from previous results. We believe that the discrepancy between our results and the previous study is due to the fact that the previous study fitted the ‘extinction’ efficiency, not absorption efficiency. It is noted that our experiment is based on absorption measurements. It is worthwhile to investigate the effect of the number of data points on the result. In the previous study done by Lack et al. [7], the number of data participated in the curve fitting was 10. We chose 9 data points by a jackknifing method and curve-fitted to the Mie absorption efficiency using the selected 9 data points. Then, 10 sets of 9 data points allowed us to estimate 10 refractive indexes. In the same way, we selected 8 data points among the 10 data points
5256
C.G. Woo et al. / Optik 124 (2013) 5254–5258
60 50
Frequency (%)
n=9 40 30 20 10 0 0.20 Fig. 2. Comparison of our code by curve fitting the absorption efficiencies data from Lack et al. [7]. Absorption efficiencies were also superimposed with the absorption efficiencies.
0.25
0.30 Imaginary part of RI
0.35
0.40
0.30 Imaginary part of RI
0.35
0.40
60 50 n=8
Frequency (%)
and then estimated the refractive index. The results are listed in Table 1. The average value for the case of n = 9 is very similar to that of n = 8. However, the range of the refractive index is quite different. Fig. 3 shows the histogram of the imaginary part of the refractive index estimated for both cases of n = 9 and n = 8. In both cases, the imaginary refractive index ranges from approximately 0.2 to 0.39. Of course, the uncertainty is reduced as the number of data points increases. The estimated refractive index becomes deviated from the 1.55 + 0.38i as the number of data points is reduced.
40 30 20 10 0 0.20
4. Nigrosin: homogeneously dyed spheres The refractive index for various sized nigrosin spherical particles generated in the present study was estimated using our code. Fig. 4 shows the measured absorption cross section for nigrosin particles. Each data point was an average of about 30 data measured at every minute for 30 min. Measured absorption cross section increases as the diameter of the nigrosin particle increases. The error bars are one standard deviations of the mean values. It is certain that the absorption efficiency should be larger than 1 since the measured absorption cross section is larger than the geometric cross section for the entire range of measurements. Absorption processes include geometrical blocking, internal reflection, internal refraction, large angle diffraction and wave resonance [13]. Incident beam may be blocked similar to the Beer’s absorption and the photons beyond the physical cross section are changed into surface wave to be damped as they propagate. Beam internally penetrating particle is absorbed through refraction and reflection and fades away similar to the surface wave. Therefore, the absorption efficiency larger than 1 implies that all photons incident on the particles are absorbed, and plus,
0.25
Fig. 3. The effect of the number of data points. Uncertainty of the refractive index depending on the number of data points.
some photons beyond the geometric cross section of the particles are also absorbed. Fig. 5 shows absorption efficiency for the nigrosin particles generated in the present study. The line depicts a fit to the measurement in which both the real part and the imaginary part of the complex refractive index are the free parameters. We varied the real part of the RI from 0 to 2.000 and the imaginary part from 0 to 0.500. The refractive index of the nigrosin particles determined at the wavelength of 670 nm is approximately 1.70 + 0.24i, which imaginary refractive index is comparable to the literature value, 1.67 + 0.26i [14] if we consider that the refractive index is inversely proportional to the wavelength.
# of data
n = 10 n=9 Avg Std Max Min n=8 Avg Std Max Min
3
2
Table 1 Refractive index estimated from the selected data.
Absorption cross section (m /part.)
4
Refractive index n
k
1.544
0.377
1.572 0.094 1.837 1.519
0.364 0.045 0.383 0.236
1.576 0.095 1.885 1.498
0.361 0.046 0.388 0.205
2
-13
10
7 6 5 4
Measured absortpion cross section Geometric cross section
3 2
400
600 800 Diameter (nm)
1000
Fig. 4. Absorption cross section of nigrosin for various sizes.
1200
C.G. Woo et al. / Optik 124 (2013) 5254–5258
Fig. 5. Absorption efficiencies versus size parameter for nigrosin particles. Shown is refractive index estimated from curve fitting of the absorption efficiency using Mie theory.
5. Black polystyrene latex: non-homogeneously dyed spheres According to the manufacturer’s technical note [15], the black polystyrene latex particles are made by dye diffusion-entrapment method, which involves the swelling of the polystyrene latex in a dye solvent. The dye penetrates into the polystyrene latex matrix and is entrapped when the solvent is removed through evaporation or transfer to an aqueous phase. The caveat in this manufacturing process is that the penetration depth cannot be controlled. In the case of the black spheres such as nigrosin particles, refractive index does not change along the size of particles. However, the refractive index of the larger black polystyrene latex spheres can be different from that of the smaller ones because the dye pigment may be non-uniformly penetrated. Therefore, it is impossible to perform curve fitting as a function of size for the refractive index. Instead, it is worthwhile to solve the absorption efficiency equation with which the measured absorption efficiency becomes equal to the calculated absorption efficiency. We attempted to assess the refractive index in two different ways: (i) bulk analysis and (ii) Mie calculation. We calculated the refractive index by iterating the Mie code until the relative error became less than 0.01%. We believe that the black dye differently penetrates the surface of the latex spheres without regard to the particle size. The measured absorption cross sections for the two black polystyrene latexes were 1.49 × 10−14 and 1.37 × 10−14 m2 /part for 233 nm and 980 nm, respectively [16]. It is interesting that the absorption cross section decreased as the black polystyrene latex became larger. Since the volume of the 233 nm black polystyrene latex particle is the smaller one, it is not unreasonable to assume that the dye was fully penetrated for the 233 nm black polystyrene latex particles to the core. For the homogeneous bulk materials, the absorption coefficient, Babs , is related to the refractive index as follows [10]: Babs =
4k
(6)
Here, the absorption coefficient can be thought of as the absorption cross section for one individual sphere, resulting in Babs = abs /(/6d3 ). Therefore, the imaginary part of the refractive index is expressed as k=
abs (d3 /6) 4
(7)
Using Eq. (7), the imaginary part of the refractive index was calculated to be 0.120 ± 0.027. The uncertainty for the imaginary refractive index was calculated from the uncertainties of the
5257
absorption cross section ( abs ) and the diameter (d) in Eq. (7) using the same procedure of uncertainty propagation as shown in elsewhere [17]. In another way, the imaginary part of the refractive index was calculated to be 0.112i from the Mie calculation on the assumption of 1.66 for the real part of the refractive index. It is interesting that two techniques, which depend on completely different physical principles for refractive index measurement, gave similar values when considering that the theory applied assumes that the dye penetrated completely and homogenously. For the 980 nm black polystyrene latex, however, it is probable that the black dye did not penetrate completely. We imagine that the black dye penetrates the surface of the latex spheres shallowly, so that the latex sphere is incompletely dyed. The imaginary part of the RI measured by the Mie calculation was 0.00049 ± 0.0001. However, the imaginary part of the RI estimated from the bulk method was 0.001482, which does not exist in the range of the imaginary refractive index calculated by Mie code. Obviously, 980 nm black polystyrene latex is even less blackish than 233 nm one. The reduced absorption cross section for the larger particle demonstrates that the light absorption for the dyed polystyrene materials has nothing to do with the particle size. In addition, the different refractive indexes measured by different principles are the evidence that the dye did not penetrate homogeneously for the 980 nm black latex sphere as opposed to the 233 nm one. Photobleaching effect could explain the low absorption cross section for the larger spheres. The black dye insufficiently adsorbed on the surface of the polystyrene sphere due likely to low surface area compared to the smaller one could be bleached by the light. Therefore, it is probable that the absorption cross section for 980 nm one is smaller than that for 233 nm one because of the photobleaching effect even though the 980 nm one is geometrically larger than 233 nm one. Not only the optical artifacts but also geometrical issue can affect the absorption cross section. Spheres can collide with each others to form large agglomerate. In this case, Mie theory should be replaced by other optics theory associated with the shape of the particles [18]. More detailed discussion needs further study by measuring the absorption spectrum in visible range. This is beyond the scope of the present study.
6. Conclusion Real part and imaginary part of refractive index for absorbing spheres were determined using a multiangle absorption photometer together with Mie theory. For the absorbing spheres generated using a constant output atomizer, absorption coefficient and the number concentration were measured through a multiangle absorption photometer and a condensation particle counter, respectively. The absorption cross section for nigrosin was estimated from the slope of the plot of the absorption coefficients as a function of the particle number concentrations. The measured absorption cross sections were fitted to the absorption cross sections calculated from the Mie theory. For the nigrosin, the refractive index obtained by the best fit corresponded to the 1.70 + 0.24i, which was obtained by minimizing the chi-square of the data. The refractive index measured by our method holds valid as long as the absorption phenomenon takes place in the near field surrounding particles surface. This method was applied to the measurement of the optical properties for non-homogeneously dyed absorbing particle, black polystyrene latex. It was observed that the larger particle was less black than the smaller one. In this case, the absorption cross section was independent on the particle size. As far as we know, it is unprecedented that the refractive index was estimated from the absorption experiments and the refractive index for the non-homogeneous absorbing particles was investigated from the refractive index point of view.
5258
C.G. Woo et al. / Optik 124 (2013) 5254–5258
Acknowledgements This work was supported by Basic Science Research Program (No. 2012R1A1B4002700) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (MEST). We give special thanks to Mr. S Han for collecting some of absorption data. References [1] H. Atwater, A. Polman, Plasmonics for improved photovoltaic devices, Nat. Mater. 9 (2010) 205–213. [2] M. Grätzel, Photoelectrochemical cells, Nature 414 (2001) 338–344. [3] N.A. Marley, J.S. Gaffney, C. Baird, C.A. Blazer, P.J. Drayton, J.E. Frederick, An empirical method for the determination of the complex refractive index of sizefractionated atmospheric aerosols for radiative transfer calculations, Aerosol Sci. Technol. 34 (2001) 535–549. [4] G.E. Thomas, S.F. Bass, R.G. Grainger, A. Lambert, Retrieval of aerosol refractive index from extinction spectra with a damped harmonic-oscillator band model, Appl. Opt. 44 (2005) 1332–1341. [5] M. Wendisch, W. von Hoyningen-Huene, Possibility of refractive index determination of atmospheric aerosol particles by ground-based solar extinction and scattering measurements, Atmos. Environ. 28 (1994) 785–792. [6] A. Reyes-Coronado, A. Garcia-Valenzuela, C. Sánchez-Pérez, R.G. Barrera, Measurement of the effective refractive index of a turbid colloidal suspension using light refraction, New J. Phys. 7 (2005) 89–90.
[7] D.A. Lack, E.R. Lovejoy, T. Baynard, A. Pettersson, A.R. Ravishankara, Aerosol absorption measurement using photoacoustic spectroscopy: sensitivity, calibration, and uncertainty developments, Aerosol Sci. Technol. 40 (2006) 697–708. [8] F.J. Green, The Sigma–Aldrich Handbook of Dyes Stains and Indicators, Aldrich, Milwaukee, 1990. [9] S. Deguchi, K. Tsujii, Supercritical water: a fascinating medium for soft matter, Soft Matter 3 (2007) 797–803. [10] C.F. Bohren, D.R. Huffman, Absorption and Scattering of Light by Small Particles, John Wiley & Sons, Inc., New York, 1983. [11] O.V. Kalashnikova, I.N. Sokolik, Modeling the radiative properties of nonspherical soil-derived mineral aerosols, J. Quant. Spectrosc. Radiat. Transfer 87 (2004) 137–166. [12] C. Mätzler, MATLAB functions for Mie scattering and absorption version 2, Institut für Angewandte Physik, Research Report No. 2002-11, 2002. [13] D.L. Mitchell, Parameterization of the Mie extinction and absorption coefficients for water clouds, J. Atmos. Sci. 57 (2000) 1311–1326. [14] D.M. Garvey, R.G. Pinnick, Response characteristics of the particle measuring systems active scattering aerosol spectrometer probe (ASASP-X), Aerosol Sci. Technol. 2 (1983) 477. [15] Bangs Lab, http://www.bangslabs.com/sites/default/files/bangs/docs/pdf/103. pdf [16] Y.-K. Jeong, J. Lee, Optical properties of polystyrene black dyed microspheres, Curr. Appl. Phys. 12 (2012) S76–S79. [17] J. Lee, Estimation of emission properties for silica particles using thermal radiation spectroscopy, Appl. Opt. 50 (2011) 4262–4267. [18] J. Lee, Effect of structure factor on aggregate number concentration estimated using Rayleigh–Debye–Gans scattering theory, Opt. Appl. 41 (2011) 519–525.