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# Field Scanning Methods of particle size measurement

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out to monitor the wet grinding of submicron color pigment using diluted, extracted samples. Further work was proposed to investigate the effect of particle shape. A redesign of the FBR-sensor was performed to reduce the measurement volume and increase the upper volume concentration level to 1%. The improvements also permitted a low number concentration to be monitored (<10^ particles cm" )^ so that the instrument could be used in a stream-scanning mode and as an in-line counter to monitor particles in gases. 10.3.7 Optical back-scattering The mixing process of pesticide dispersion in a spray tank mounted on a tractor was monitored using optical back-scattering probes mounted in the tank [89]. A granulated powder was fully dispersed in 20 s whereas a powder product took 4 min. 10.3.8 Transmission fluctuation spectroscopy A transmission signal measured on a flowing suspension of particles shows significant fluctuations which contain complete information on particle size and concentration. Details have been published in three parts [90-92]. In parts one and two the basic properties were studied for a beam of uniform intensity. The theory was extended in part three to a Gaussian beam with experimental tests to follow. 10. 4 Light scattering theory 10.4.1 The Rayleigh region (d« X) In the Rayleigh region the intensity of the scattered radiation in a direction making 6 with an incident beam of unit intensity is given by [45]: Io^\a f i2 2^Y(l + cos^ )^ ^^ I ^ ^ ( 1 0 . 1 4 ) \^m J Ir" where a is the particle polarizability, r is the distance from the particle to the point of observation and X^ is the wavelength of the incident radiation in the medium surrounding the particle. 540 Powder sampling and particle size determination The intensity is the sum of two terms: L, = \cc\ 2n \ 4 2r' and h=M 2n \^mj cos^ <9 1? which refer respectively to the intensities of the vertically and horizontally polarized components. For spheres: a-\a 2 3(m-\)V (10.15) Equation (10.15) has been applied to very small particles but is more relevant to the size determination of molecules. For m-\->Q a = ( » , ^ - l ) ^ . 2 ( \u2122 - , ) ^ (10.16) where m is the refractive index of the particle relative to that of the surrounding medium and Fis the volume of the particle. 10.4.2 The Rayleigh-Gans region (D < A) The Rayleigh-Gans equation for the angular dependence of the intensity of the scattered light is given for spherical particles of low refractive index by the equation [45]: j0=h \u2014{smu - ucosu) (l + cos^^) (10.17) Again, the two terms in the fmal brackets refer respectively to the intensities of the vertically and horizontally polarized components, /Q is the intensity of the incident beam, D is particle diameter and: 2TID . e u = sm \u2014 A^ 1 2 and k = 2n_ (10.18) Equation (10.17) reduces to equation (10.14) when the middle term is equal to one. The scattering pattern is however modified by this term, thus Field scann ing m ethods 541 enabling a size determination to be carried out in the Rayleigh-Gans region. Differentiating equation (10.17) with respect to u and putting d/^dw = 0, for minimum intensity gives: sin u-ucos u = 0 (10.19) and, for maximum intensity: 3wcos u - u^sin u -3sin u =0 (10.20) The first minimum is at w = 4.4934 radians corresponding to: - ^ s i n 4 = i : ^ = 0.715 (10.21) X^ 2 2n Similarly the first maximum occurs at: \u2014 s i n ^ = 0.916 (10.22) A graphical solution for all maxima and minima has been determined by Pierce and Maron [93] who, together with Elder [94,95] extended equations (10.21) and (10.22) beyond the Rayleigh-Gans region, {m-l^' 0) to 1.00 < w < 1.55, deriving the following formulae: \u2014 s i n ^ = 1.062 - 0.347m (10.23) \u2014sin ^ = 1.379 - 0.463m (10.24) These equations give the positions of the first intensity minimum and maximum respectively. The range of validity of these equations has been investigated by Kerker et.al. [96]. The angular positions determined experimentally were found to depend on concentration hence it is necessary to take reading at several concentrations and extrapolate to zero concentration. This concentration dependency has also been noticed with depolarization and dissymetry methods becoming negligible only when particle separation exceeds 200 radii [97]. 542 Powder sampling and particle size determination 10.4.3 High order Tyndall spectra (HOTS) When a dilute suspension of sufficiently large, mono-disperse, spherical particles is irradiated with white light, vivid colors appear at various angles to the incident beam. The angular positions of the spectra depend on m and A hence they may be used to determine particle size in colloidal suspensions. Since the red and green bands predominate it is usual to observe the ratios of the intensities of the vertical component of the red and green light in the scattered radiation as a function of 9. When these ratios, R = r^ed^ C^Teen ^^^ plotted against d, curves showing maxima and minima appear, the maxima being the red order, the minima the green. Smaller particles yield only one order but the number of orders increases with increasing particle size. HOTS have been studied extensively in monodisperse sulfur solutions by La Mer et.al. [98-101], by Kenyon [102], in aerosols by Sinclair and La Mer [103], in polystyrene latices by La Mer and Plessner [105] and in butadiene latices by Maron and Elder [106]. The following equations derived by Maron and Elder for the angular positions of the first red and green order, 0^^ and 0^^ are particular examples of equation (10.22). Dsin Dsin ^ 1 = 0.2300 (10.25) ^.1 -0.3120 (10.26) Pierce and Maron [107] show that the angular positions of the red orders are identical with the angles at which minima occur in the intensity of the scattered light when the incident light has a wavelength of y .^ Similarly, the angular positions of the green orders coincide with the angles at which minima occur with incident light of wavelength A/. Consequently, for the same effective incident wavelengths, minimum intensity and HOTS represent equivalent measurements. For equations (10.25) and (10.26), solving with equation (10.23) with m= 1.17 gives: 1.062 - 0.347(1.17) = 0.2300/^ .*. /^= 0.3506 jLim (0.4673 jum in vaccuo) Field scanning methods 543 1.062-0.347(1.17) = 0.3120/A; .'. A'^=0. 4756 |am (0.6340 jum in vaccuo) This method is qualitative unless A^ and /^ are known. The above equations yield weight average diameters. With increasing m the evaluation of D becomes more difficult due to reduction in the intensity of the maximum and broadening of the peak. The method has been used for the size range 0.26 to 1.01 |Lim [108]. 10.4.4 Light diffraction The far-field diffraction pattern of an assembly of particles yields information concerning their size distribution. For particles dispersed on a transparent slide the geometrically scattered part of the incident beam may be eliminated by coating the slide with aluminum and then removing the particles. The particle size distribution of the particles can then be determined from the resulting far-field diffraction pattern [109-112]. This effect was utilized in the Talbot diffraction size frequency analyzer (DISA) [113] that used a series of interchangeable filters to determine the number size distribution of the resulting apertures. Talbot later described a filter that transmitted amounts of light proportional to particle volume [114]. He also incorporated the principle in a slurry sizer, the Talbot Spacial Period Spectrometer [115,116]. Gucker et. al. [117] developed equations from the Lorentz-Mie theory relating the size of the Airy points to particle size. Davidson and Haller [118] applied these equations to 0.07 to 0.50 \\m latices deposited on microscope slides and obtained poor agreement that they attributed to strong particle-slide interaction.