Particle characterization is an important aspect of quality control of chemical and biological suspensions, slurries and dispersions due to their economic impact in industries [1]. Apart from concentration and particles size, the optical constants i.e. the complex refractive index () may provide significant information which is important to control processes for the production of particulate systems such as latexes, suspensions and dispersions. Further, in order to estimate particle size and size distribution using methods based on light scattering theories, accurate values of the optical constants are required [2].
Ma et al [3] used Monte Carlo simulations in combination with Mie theory and experimental measurements of diffuse reflectance and transmittance to determine the complex refractive index of polystyrene microspheres. They reported values for the range 370 – 1600 nm using highly diluted monodisperse suspensions of polystyrene. In many practical situations, it is necessary to deal with higher concentrations and polydisperse systems. The effect of particle size, size distribution, shape and concentration on the uncertainties in the estimated and
, and on the particle size analysis using these estimates is not known.
This paper presents a general method for the estimation of optical constants in the UV-Vis-NIR region, from a polydisperse (broad and/or multimodal) suspension. The optical constants and
are obtained by an inversion technique using the adding-doubling method to solve the radiative transfer equation [4] in combination with theories of light scattering by small particles such as the Mie theory [5] and measurements of total diffuse reflectance and transmittance using an integrating sphere setup.
The inversion method compares values of total diffuse reflectance, Rtdm, and transmittance, Ttdm, measured using a single integrating sphere method against the values of simulated total diffuse reflectance, Rtds, and transmittance, Ttds, calculated using the radiative transfer equation to model multiple scattering in turbid media [6,7]. The equation of radiative transfer is given by:
(1)
where is the specific intensity at a point r with radiation along the direction
,
is the bulk extinction coefficient,
is the bulk scattering coefficient and
is the bulk absorption coefficient.
is the phase function which is a measure of the angular distribution of the scattered light.
The bulk scattering and absorption coefficients are functions of the particle concentration and the scattering and absorption cross-sections of the species at wavelength of the incident beam, and are written as:
(2)
where c is the particle volume fraction, is the absorption cross-section of the particles, and kw is the imaginary part of the complex refractive index of water.
is the scattering cross section. The scattering cross section for polydisperse suspension is given by:
(3)
where r is the number density of the particles, f(D) is the fraction of particles of diameter D. For monodisperse systems this parameter is set to 1 while for polydisperse dispersions the appropriate size distribution function should be used. The differential cross-section F is function of D, the relative complex refractive index mr, which is the ratio of the particle refractive index and the refractive index of the medium (in this case water), is the scattered angle and
is the azimuth angle. The differential cross-section is computed using Mie theory [5]. The static structure factor S, is a function of the inter-particle interaction energy VT and represents the microstructure of the suspension. For dilute suspensions, where inter-particle interactions are negligible, the value of S is set to 1. In this study, it is assumed that the inter-particle interactions are negligible since moderately low concentrations are used.
The phase function is required to solve the radiative transfer equation. Two different phase functions are implemented, the exact Mie phase function for spherical particles and the Henyey-Greenstein function, which is a simple approximation of the real phase function [4]. The Mie phase function is given by:
(4)
Where S1(cos) and S2(cos
) are the scattering amplitudes computed using the Mie theory. The Henyey-Greenstein function is given by:
(5)
where g is the anisotropic factor and is the average cosine of the phase function:
(6)
Once that the coefficients ,
and the phase function are determined the radiative transfer equation, RTE, is solved. The adding-doubling method, ADD, is used to solve the RTE for plane–parallel geometries [7,8]. This method is an accurate numerical technique based on matrices for solving the radiative transfer equation that considers multiple scattering, and takes into consideration anisotropic scattering, Fresnel boundary conditions and inhomogeneous or homogeneous layers, for arbitrarily thick samples with relatively fast computations.
Total diffuse reflectance, Rtdm, and transmittance, Ttdm, are measured on the range 450 to 1800 nm of the spectra by the comparison method using a single integrating sphere attached to a Cary 5000 UV-Vis-NIR spectrophotometer for polydisperse and monodisperse polystyrene microspheres suspensions. Two different multimodal polydisperse polystyrene suspensions of different concentrations in which the inter-particle interactions can be neglected were prepared by mixing different proportions of monomodal polystyrene suspensions of different diameters. Monodisperse polystyrene suspensions from Duke Scientific Co of 10% by weight of solids of different diameters were used to prepare multimodal polydisperse suspensions.
The optical constants are estimated by minimising the objective function given by the absolute deviation of simulated total diffuse reflectance and transmittance from the experimental values:
(7)
The inversion method using the exact Mie calculations is very computationally intensive and time consuming. For this reason usually the phase function is modelled using the simple approximation of Henyey-Greenstein function. Comparative study on the estimation of the optical constants and
using the two different phase functions, Mie and Henyey-Greenstein implemented in the adding-doubling method is presented.
Sensitivity analysis on the estimation of and
using monodisperse and polydisperse (narrowly and broadly distributed) suspensions to study the effect of the polydispersity on the estimated values of the optical constants are included. An error analysis is conducted using replicates of total diffuse reflectance and transmittance measurements for a fixed value of concentration and particle size distribution to establish the error bounds of the refractive index. The effect of the particle concentration on the estimated values of
and
are also discussed.
Keywords: Optical constants, particle size distribution, a single integrating sphere, polydisperse, diffuse reflectance.
References
[1] SCOTT, D. M., Characterizing Particle Characterization, Particle & Particle Systems Characterization, 2003, 20, pp. 305-310.
[2] BARTLETT, M A. and JIANG H., Effect of refractive index on the measurement of optical properties in turbid media, APPLIED OPTICS, 2001, 40, 10, pp 1735- 1741.
[3] MA, X., LU Q. L., BROCK, S. R., JACOBS, K. M., YANG, P. and HU, X-H., Determination of Complex Refractive Index of Polystyrene Microspheres from 370 to 1610 nm, Physics in Medicine and Biology, 2003 48, pp 4165-4172.
[4] PRAHL, S.A., The Adding Doubling Method. In Optical Thermal Response of Laser Irradiated Tissue (Eds., Welch, A. J. and van Gemert, M. J. C.) Plenum Press, New York, 1995 pp. 101-129.
[5] BOHREN, C. F. and HUFFMAN, D.R., Absorption and Scattering of Light by Small Particles, John Wiley & Sons, 1983 New York.
[6] ISHIMARU, A., Wave Propagation and Scattering in Random Media. IEEE/OUP series on electromagnetic wave theory. 1997, New York: IEEE Press. xxv, 574.
[7] HULST, V.d., Light Scattering by Small Particles. First ed. Structure of matter series. 1964, USA: John Wiley & Sons, Inc. p.p. 470.
[8] BREWSTER, M. Q., Thermal Radiative Transfer and Properties. John Wiley & Sons, 1992 USA. p.p 188.
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