Mie’s theory

For particles below 200 nm, Rayleigh s theory holds, which considers the scattering intensity to be proportional to the 6th potency of the particle diameter. Both Fraunhofer s and Rayleigh s theories are only approximations of Mie s theory which claims that the scattering intensity depends on the scattering angle, the absorption, the size of the particles as well as on the refractive indices of both the particles and the dispersion medium. 

In Mie s theory, the scattering diameter Qs and the absorption diameter QA are related to the particle size D, the wavelength A, and the optical constants of the material (refractive index n and absorption index k). 

Mie s Theory. Mie applied the Maxwell equations to a model in which a plane wave front meets an optically isotropic sphere with refractive index n and absorption index k [1.26]. Integration gives the values of the absorption cross section QA and the scattering cross section Qs these dimensionless numbers relate the proportion of absorption and scattering to the geometric diameter of the particle. The theory has provided useful insights into the effect of particle size on the color properties of pigments. 

The consequences of Mie s theory for absorption (i.e., for tinting strength) are now considered. Calculations from Mie s theory, using the relative refractive index n and the absorption index k, are given in Figure 8 [1.30]. The parameter a on the abscissa can once more be taken as a relative measure of the particle size. The following conclusions may be drawn ... 

Although Mie s theory was first published in 1908, computations of scattering coefficients were not tabulated to any extent until the 1940s (Lowan, 1948), and then the available tables were quite limited. Nevertheless, considering that each data point represented many hours of error-free calculation with a desk calculator, the accuracy of these early tables is indeed remarkable. 

More exact computations of this relationship, based on Mie s theory, are shown in Figs. 17.8 and 17.9. A device known as the aerosol owl can be used to make these measurements. It consists of a viewing chamber on which a telescope is mounted that can be rotated through roughly 160°. A protractor is attached to the telescope so that the angle at which a red is observed can be noted. The number of reds found... 

Eor a sphere, the conditions of the RDGA might not be matched because the refractive index of the sphere could be substantially different from that of the suspension medium. If so, P( ) should be calculated by Mie s theory. This is the case of PS (polystyrene) latex spheres in water. 

The main premises of this technique are given in the monograph (Klenin et al., 1977a) with calibrations of all the characteristic functions for inonodisperse, spherical particles within wide ranges of a Euid m computed with the use of Mie s theory. Since then, new results have been obtained concerning the problem of phase sepsu ation in polymer systems. 

Heller ct al. (1962), Klenin et al. (1977a) have tabulated n from Mie s theory in wide ranges of a and m (Figure 2.14). 

One can see that the exponent vtiriea from 2 to 4 and docs not depend on m (cf. the results of Mie s theory within this range of n in Figure 2.14). 

Thus, metal nanocrystals of various sizes exhibit characteristic colors depending on their diameters and the dielectric constant of the surrounding medium. Typical size-dependent changes in the optical spectra are shown in Fig. 1.16. Mathematical methods have been developed to accurately compute the higher order terms in Mie s theory [71]. Several off-the-shelf programs are available to apply Mie s theory and its extensions to a variety of nanoparticles [72]. One such code called BHCOAT, presented in the book by Bohren and Huffman [73] is very popular. 

The remission spectra (Figure 19.4) can be elegantly interpreted by means of the optical constants, on the basis of Mie s theory. Small remission values for blue light, due to the solid-state absorption band (40-460 nm) are followed by a steep incline at 500 nm. At longer wavelengths the optical window is open to green and red light, which can be scattered effectively by the TiOj particles of the white reduction. 

More specifically at ITIES, a lot of effort has been dedicated to nucleation and growth of nanoparticles for electrocatalytic studies. In 1998, Schiffrin deposited gold particles at an ITIES by electrochemical reduction of tetraoctylammonium tetrachloroaurate in 1,2-DCE using ferrocyanide in water as the electron donor. Their growth was monitored in situ by transmission UV-VIS spectroscopy, and the spectra have been qualitatively analyzed using Mie s theory [331]. The nucleation mechanism was later addressed by Johans et al. [334,335] using dibutyl-... 

---