Influence of the Processes Parameters on the Properties of The ...

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Chapter 3. Analytical Methods and Designs <strong>of</strong> Experiments obtain <strong>the</strong> particle size distribution. Three <strong>the</strong>ories may be used for that: Rayleigh’ <strong>the</strong>ory, Lorenz- Mie’ <strong>the</strong>ory and Fraunh<strong>of</strong>er’ <strong>the</strong>ory; 3.2.1 Rayleigh’ Theory Figure 3.7: Scheme <strong>of</strong> laser diffraction <strong>of</strong> a spherical particle. Rayleigh’ <strong>the</strong>ory demands that <strong>the</strong> particle size is much smaller than <strong>the</strong> wavelength <strong>of</strong> incident light. In that case, <strong>the</strong> whole particle behaves similarly in a homogeneous electric field. The incident light penetrates <strong>the</strong> particle due to <strong>the</strong> polarizability <strong>of</strong> <strong>the</strong> particle. The penetration time is short compared to <strong>the</strong> period <strong>of</strong> incident light. Induced dipole moment is formed when electric charges <strong>of</strong> non polar particle are forced apart by subjecting <strong>the</strong> particle to electromagnetic wave. Like so, <strong>the</strong> polarized particle is created. The electric field and <strong>the</strong> dipole moment oscillate synchronously and <strong>the</strong> axis <strong>of</strong> <strong>the</strong> dipole moment is downright to <strong>the</strong> incident light as in Figure 3.8, which also describes <strong>the</strong> intensity <strong>of</strong> scattering to different directions [Xu, 2000]. According to this <strong>the</strong>ory, <strong>the</strong> laser beam is assumed to not only be diffracted by <strong>the</strong> particles, but is also reflected and diffused. The light will spread until <strong>the</strong>re is a variation in <strong>the</strong> refraction index <strong>of</strong> <strong>the</strong> propagation environment. This index variation will create a refraction <strong>of</strong> <strong>the</strong> monochromatic beam; <strong>the</strong> laser will reach <strong>the</strong> detector having been subjected to several variations in its propagation direction. Figure 3.8: Three dimensional model <strong>of</strong> scattering from a dipole. [Xu, 2000] 3.2.2 Lorenz-Mie’ Theory Lorenz-Mie’ <strong>the</strong>ory (or Mie’ <strong>the</strong>ory) is more detailed and wider <strong>the</strong>ory <strong>of</strong> <strong>the</strong> light scattering than Rayleigh’ <strong>the</strong>ory. It can be used for spherical particles which can be small, large, transparent or opaque. According to Lorenz-Mie <strong>the</strong>ory, <strong>the</strong> intensity <strong>of</strong> scattering from <strong>the</strong> surface <strong>of</strong> <strong>the</strong> particle (primary scattering) can be predicted with <strong>the</strong> refractive indexes <strong>of</strong> <strong>the</strong> particle and <strong>the</strong> medium. Lorenz-Mie’ <strong>the</strong>ory - 68 -
Chapter 3. Analytical Methods and Designs <strong>of</strong> Experiments also caters <strong>the</strong> light refraction with <strong>the</strong> particle (secondary scattering) which is very important when particle diameter is below 50 μm. This is also mentioned in <strong>the</strong> standard for laser diffraction measurements (ISO 133201) [Kippax, 2005]. According to Lorenz-Mie’ <strong>the</strong>ory <strong>the</strong> scattering patterns <strong>of</strong> spheres are symmetric with axis <strong>of</strong> <strong>the</strong> incident light. Scattering minima and maxima are in different angles, if <strong>the</strong> properties <strong>of</strong> <strong>the</strong> particles vary. Figure 3.9, shows <strong>the</strong> diffraction patterns <strong>of</strong> two particles with different sizes [Xu, 2000]. With large particle (solid line) <strong>the</strong> peak <strong>of</strong> intensity is stronger than with small particle (dashed line) and <strong>the</strong> minimum intensity is much closer to axis <strong>of</strong> <strong>the</strong> incident light. The intensity peaks are in <strong>the</strong> same locations in <strong>the</strong> both positive and negative angles because <strong>the</strong> symmetrical nature <strong>of</strong> scattering. Very illustrative way <strong>of</strong> displaying <strong>the</strong> intensity distribution is also a radial graph as in Figure 3.9. The bold trace describes <strong>the</strong> intensity <strong>of</strong> scattered light in different angles [Xu, 2000]. Figure 3.9: Scattering patterns <strong>of</strong> two particles <strong>of</strong> a different size. [Xu, 2000] 3.2.3 Fraunh<strong>of</strong>er’ Theory Fraunh<strong>of</strong>er’ <strong>the</strong>ory covers <strong>the</strong> light diffraction in <strong>the</strong> aperture which is described in Fresnel- Kirch<strong>of</strong>f’ diffraction integral [Brittain, 2003]. In Fresnel diffraction (Figure 3.10-A), distances from <strong>the</strong> point source and <strong>the</strong> screen to <strong>the</strong> obstacle forming <strong>the</strong> diffraction pattern are relatively short. In Fraunh<strong>of</strong>er’ diffraction (Figure 3.10-B), <strong>the</strong> distances are much longer and all lines from <strong>the</strong> source to <strong>the</strong> obstacle and forward to <strong>the</strong> screen can be considered parallel. In Figure 3.10-C, <strong>the</strong> lens forms smaller image <strong>of</strong> <strong>the</strong> same diffraction pattern which would be formed on <strong>the</strong> screen extremely far without <strong>the</strong> lens [Young and Freedman, 2000]. (A) (B) (C) Figure 3.10: Principles <strong>of</strong> Fresnel’ diffraction (A) and Fraunh<strong>of</strong>er’ diffraction (B and C). [Young and Freedman, 2000] - 69 -
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