Lightscattering Methods

Young's observation that the maxima and minima in a shadow behind an obstacle were caused by interference waves began the interest in the phenomenon of light scattering early in the nineteenth century. In the latter part of the nineteenth century, Maxwell developed the electromagnetic theory of light that linked electrical and optical phenomena. Lord Rayleigh used the new theory of electromagnetic radiation to investigate the scattering of white light by small particles. He formulated an approximation applying to very small particles whose refractive index was small. In order for Rayleigh scattering to apply, the particles also must be small in comparison to the wavelength of the radiation. Rayleigh's approximation is valid for particles of diameter less than one-tenth the wavelength of the incident light. This restriction assumes that the particle has a uniform internal field when the wave passes through, resulting in only one scattering center.

Light scattering was further investigated by Mie, who published his complete theory in 1908. Unlike Rayleigh scattering, Mie accounted for the complete light-scattering pattern by taking into account the electric field inside and outside the particle. As particles increase from 0.10 to 10 times the wavelength of the incident radiation, scattering occurs from more than one point in a single particle and is out of phase. This results in interference and reduced light intensity. The scattering of light was calculated as a power series that takes into consideration the different angles of scattering, wavelength of incident light, differences in refractive indices, and the diameter of a sphere. The Mie theory, unlike the Rayleigh approximation and Debye approximations, may be applied to absorbing and nonabsorbing particles. Although the Mie theory could be used to determine particle size distributions, the calculations are so complex that it was not feasible until the advent of the computer.

Debye's approximation was published in 1909, and it applies to slightly larger particles than the Rayleigh approximation but to a narrower range of refractive indices. As the particle size approaches the wavelength of the incident radiation, different regions of the same particle will behave as scattering centers. There will be interference between the waves of light scattered from the same particle. Therefore, the Rayleigh approximation must be multiplied by a correction factor to account for the interference. This approach accounts for particles with more than two scattering centers and recognizes the tendency for the number of centers to increase with particle size. It also considers that coordinates must be described by a radial distance and two angles.

Another approximation that can be applied to particles larger than the Rayleigh region is the Rayleigh-Gans approximation. For Rayleigh scattering, the vertical components of scattered light remain constant. However, as the particle size increases, the light intensity is inversely proportional to the angle to observation and passes through a minimum. This requires radiation to undergo a small phase shift when passing through the particle. The addition of the phase shift term results in an asymmetric scattering pattern about 90°.

For particles approaching the wavelength of light, the complex Mie theory is required. However, for much larger particles, the contribution of the radiation refracted within the particle diminishes in relation to the radiation diffracted external to the particle. Furthermore, the amount diffracted is independent of the particle's refractive index because it involves rays external to the particle. When particles are four or five times greater than the wavelength of the incident radiation, Mie theory can be reduced to the simpler Fraunhofer diffraction theory. This theory explains that the intensity of light scattered by a particle is proportional to the particle size, whereas the size of the diffraction pattern is inversely proportional to the particle size. Fraunhofer determined that a central stop blocked light proportional to the fourth power of the particle diameter. The transmission filter permitted scattering light proportional to the volume, or third power, of the particle diameter to be measured.

Rayleigh scattering, the Rayleigh-Gans and Debye theories, and diffraction are all well-known limiting cases to the complex Mie theory of light scattering. Two inventions contributed significantly to the use of light scattering for particle analysis. These were the laser, in 1961, and the computer, in the late 1960s. The computer was necessary to perform the manipulations required by Mie theory. The laser was important because it provided coherent monochromatic light of high intensity that allowed the development of dynamic-scattering techniques. Pusey was responsible for the development of the scattering theory concerning lasers. Coherent light means that the phase relationships in a beam are maintained, and random diffraction patterns are formed after striking the particles. The particles in the array are undergoing Brownian motion, and the fluctuations in the scattered light give information about the particle size. The particles are diffusing around their equilibrium positions, resulting in a fluctuation in the number of scattering centers seen by the photodetector. The spectrometer analyzes these fluctuations and obtains the diffusion coefficient, which is related to the particle diameter by the Stokes-Einstein equation.

Calibration

Calibration of optical devices is most frequently performed using polystyrene latex spheres that can be generated from dilute aqueous suspensions and dried before measurement. The ASTM developed a standard for the use of reticles, or disks, that can be placed in the path of the beam of a laser diffraction device for the purpose of calibration. Some manufacturers maintain that their instruments do not require calibration. The "calibration" step may then be regarded as a verification of the alignment of the optics of the instrument.

Forward Light-Scattering Particle Counters

Hodgkinson's [123] exposition on the optical measurement of aerosols, though dated with respect to instrumentation, contains an excellent review of the theoretical aspects of this subject. Fig. 17 illustrates a classic forward light-scattering arrangement.

Optical particle counters have been used for a number of years. The forward scattering of light minimizes the effects of shape and refractive index. The Royco 4100 series (4101 and 4102, Royco Instruments, Inc., Menlo Park, CA) use the near forward light-scatter region (7-17°) and, thus, have the previously mentioned advantage [124,125]. The Royco 4130 and Climet 7000 series (Climet instrument Co., Redlands, CA) minimize the effects of shape and refractive index by using an ellipsoid mirror to collect light scattered from the forward direction [126]. Older models of the Royco and Climet devices, the Royco 220 and Climet 208, are described in detail by Allen [4]. The PMS ASASP-X laser aerosol spectrometer system (Particle Measuring Systems, Inc., Boulder, CO) uses a parabolic mirror to collect all light scattered by a particle over an angular region (35-120°) [127,128]. The particle diameter obtained by these techniques is a projected area diameter. Each of these devices has a small sensing volume, for example, 2.63 mm3 for the Royco 220 [4] and 0.004 mm3 for the PMS ASASP-X [127,128]. Each of these manufacturers markets a number of devices, each targeted at the measurement of airborne particles in a specific size range. Table 2 lists some of the instruments that may be of interest for pharmaceutical purposes. Additional instruments measure particles up to 5.0 mm, for example, Models Turbo-110 and Micro LPCA (PMS), 5120 (Hiac/Royco), CI-7400 (Climet), and 3755 laser particle counter (TSI, St. Paul, MN). However, this does not cover the entire size range of interest for therapeutic purposes. Of note, many of these instruments are designed for use in cleanrooms where monitoring and environmental control of submicrometer particles are the greatest concerns.

Laser Diffraction

As indicated, Fraunhofer diffraction is a special case of Mie theory that can be used to obtain the volume of particles. The Malvern Instruments Model 2600c series of sizers (Malvern Instruments, Southborough, MA) use a laser diffraction method. The instruments are equipped with an IBM-compatible computer that controls the collection, manipulation, and presentation of data. The principle of operation is illustrated in Fig. 18. The instruments consist of a low-power laser transmitter and receiver detector units mounted 50 cm apart. Particles or spray

Figure 17 Classic forward light-scattering instruments. (With permission of Particle Measuring Systems, Inc.)

Table 2 Manufacturers and Specifications of Some Light-Scattering Instruments

Manufacturer and model

Table 2 Manufacturers and Specifications of Some Light-Scattering Instruments

Manufacturer and model

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