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For a long time the measuring of the particle size distribution has been an utilized technology, based on the for a 100 years well understood scattering of light on small particles. The commercial use of this technology though, only became possible with the availability of effective and inexpensive laser systems and computers.
Here shall the basic information about the measuring of the particle size distribution with the aid of the static light scattering be conveyed. Additionally you can find all important terms explained in our well-arranged encyclopedia.
In the twenties of the past century the Swiss astronomer R.J. Tümpler discovered, that far away star clusters appeared darker as it was based on their distance to be expected. He figured that part of the starlight on its way to earth is lost. Few years later the American astronomer E.P. Hubble observed that the median number of visible galaxies in the direction of the centre of our milky way in the constellation Sagittarius, is clearly less as when for example viewing towards the direction of the Big Dipper. Besides brightly lit clouds of gas, predominantly consisting of hydrogen one finds in the so called galactic plane also numerous dark areas, which almost completely swallow the light of objects situated behind. The interstellar dust was discovered.
The interstellar dusts consists of mostly very small particles - their typical diameter is between 0.1 and 1 µm - which scatter and absorb the star light. Since these particles cannot be examined with alternative procedures like for example the electron microscopy, was and still is the application of the light scattering theory for astronomers during the research of interstellar or interplanetary dust of great significance.
In a laboratory, quasi for earthly applications the circumstances are simpler or one should even say the challenges are different ones. Here of course the optical composition of the entire system can be adjusted to the demands and most of the time more is known about the examined sample material as in the case of the particles of the universe. Starlight with its wide wave length spectrum can be replaced with monochromatic laser light and the chemical composition of the sample material is often very well known. Instead new difficulties occur, especially during the suitable preparation of the to be measured particle collective.
Basically the design is always the same: A light beam, mostly supplied by a laser, shines through the sample to be measured and behind it, the intensity distribution caused by the scattering is picked up with a detector. Here already, it shall be mentioned, that the particle collective to be measured should be available in a sufficient dilution and should not form clusters or even better said should not create agglomerates. Then the measured intensity distribution shows a system of numerous more or less concentric rings, which their spacing correlates with the particle size. Large particles create close neighbouring rings, small particles create rings further apart. When now determining the distance of each ring, the particle size can be calculated from the results.
Now before looking at the technical implementation of this simple principle, it makes sense to view the main features of the relevant physical processes.
When illuminating a particle with light, different effects occur, which together lead to a reduction or extinction of the incident light. The extinction is basically the sum of absorption and scattering.
Let’s initially view the absorption. Here part of the electromagnetic energy of the incidence light is absorbed by the particle and changed into another energy form, mostly in thermal energy. This energy, is either through infrared radiance (thermal radiation) or through convection of the surrounding medium emitted, an effect which is without significance with dynamic laser scattering. The size of the absorption is for sufficiently large, non-transparent particles merely obtained from their geometric transverse section. “Sufficiently large” means in this context, their diameter is clearly above the wave length of the used light. With marginal particle sizes and with opaque particles, the circumstances are more complicated, the absorption co-efficient of the material must be known in order to link the absorption and particle size. In the Mie-Theory, the absorption plays a major part. But more about this topic that later.
Now let’s address the scattering. Here, initially a distinction is drawn between basically two different forms of scattering: The inelastic scattering whereas the energy and therefore the wavelength of the light changes and the elastic scattering, where the wave length remains. For us, the latter is solely of interest, and the inelastic scattering shall not be discussed and with “scattering” it is always referred to the “elastic scattering”.
Scattering identifies everything deflecting the incident light of its original direction. It can be divided in three parts, first: the reflection, second the refraction and third the diffraction.
The Reflection occurs mostly on the surface of the particles and along with the geometric optic and is described according to the law “angle of incidence equals angle of reflection” When viewing the angle dependent complete course, of an intensity distribution caused by scattering, the reflexion on the surface of a sphere provides a very smooth share. Basically, reflexion with transparent materials can also occur on inner border-areas, which is especially important in connection with the refraction.
During the refraction, according to the refraction law of Snellius, the direction of a light beam changes directions during the transition between two materials with different indexes of refraction. For example, hits a light beam a drop of rain, so it is refracted towards the middle of the droplet, and in the further course, on the outer edge of the raindrop, it is continuously reflected back inside the raindrop again and again. A part of the irradiation leaves with each reflexion the droplet. With this image for example, the creation of a rainbow can be explained, but also numerous structural details of particle intensity distributions are observed during the laser scattering measurement.
In order to understand diffraction, one has to imagine the light beam as a wide wave front, which hits a particle and partially encircles it, similar to a water wave, which meets a pole or even a larger obstacle. Through the superimposition of various parts of the broken wave front (interference) behind the particle, the characteristic diffraction patters evolves, which its direct course is described with the Fraunhofer-Theory and is clearly defined by the diameter of the particles. The here shown illustration is the graphic depiction of the scattering amplitude of a sphere shaped particle which can be exactly described with the so called Bessel Function. Here the central diffraction maximum for only very small scattering angles is clearly recognizable, which show the highest intensity for the scattered light. Towards larger scattering angles - and therefore to larger distances of the detector centre of the corresponding measuring instrument – then follow alternatively dark and light rings, which their distance, like already mentioned above, directly co-relate with the particle diameter: The tighter the rings the larger the particles and vice versa.
Strictly speaking, the above mentioned only applies to sufficiently large enough particles, whereas „sufficiently large“ again means here just like with the absorption that their diameter is clearly above the wavelength of the used light. For particle diameters in the scale of light wave length, the already briefly mentioned above Mie theory is applied. The Mie theory is the complete solution of the Maxwell equations for the scattering of electromagnetic waves on spheric particles. Now what does this mean? Well, one can imagine that electromagnetic light waves couple onto the atoms and molecules in a particle so to speak and get these oscillating. These oscillations generate then in turn electromagnetic waves, light waves of the same wave length to be exact (like already mentioned, we are only discussing elastic scattering here), which can be radiated in all possible directions. The overlapping of the individual waves from the various areas of the particle causes then a formation of a characteristic intensity distribution, which different from the Fraunhofer- diffraction can be observed not only in the forward direction, but also in scattering angles larger than 90 degrees.
Based on the Maxwell equations, which describe the distribution of electromagnetic waves in general terms, examined Gustav Mie at the beginning of the 20th century, effects during the light scattering in colloidal metal solutions, especially the scattering of light of fine gold particles and developed for this as one of the first ones, a complete theory, which later was named after him.
Now, after the important processes during the light scattering - at least suggestively - have been described, now the exact execution of the optical design of a laser particle sizer shall be described. The already above illustrated basic alignment, can be achieved with two different concepts. Besides the mentioned components (laser – measurement cell – detector) inside the optical path, a converging lens must be integrated, which focuses the scattered light onto the detector. Since the converging lens from the spatial distribution of the scattered light at the location of the particle (in the measuring cell) with its Fourier transformed (on the detector) creates, is the converging lens also regarded as the Fourier lens. The formation of this Fourier-lens makes the decisive difference between so called conventional design and the inverse Fourier design.
Let’s start with the conventional design: Here a sufficiently broad enough, parallel laser beam is generated, in which then the measuring cell with the scattering particles is inserted. Between the measurement cell and the detector the Fourier lens is positioned. With this alignment the focal distance of the Fourier lens is determined, and a variation of it makes the exchange of the lens necessary. This must be adjusted with a high degree of accuracy, since especially with larger particles very small angles are measured and a tipping of the Fourier lens, directly has a very large influence on the result of the measurement. An additional disadvantage of this set up, is the limited possibility, to measure large scattering angles. And these large scattering angles are – which we know – necessary for the measuring of really small particles.
Therefore, approximately 25 years ago, as an alternative the inverse Fourier design was introduced. The FRITSCH GmbH was the first company, which with the first model of the ANALYSETTE 22 type series, used an inverse Fourier optic for the particle size determination. Despite a lot of scepticism in the beginning, the principle in the meantime was adopted by many manufacturers. Now how does it look? The difference to the conventional design is, that here the Fourier lens is situated in front of the measurement cell, so it is not traversed by a parallel, but instead by a convergent laser beam. The scattered light is therefore without additional optical elements focussed directly onto the detector. Despite this basically same set up of the individual components (laser – Fourier lens – measuring cell – detector) the designs of the inverse Fourier optic of the various models greatly differ in significant details.
With a very common approach small scattering angles – namely large particles –are covered with a main detector, whereas for the large scattering angles of the small particles, a lateral detector system is used. For very large scattering angles close to 180°, a second system must be integrated, which often consists of a blue light source – usually a LED – with a lens and a detector. The main disadvantage of this design is, that during each measurement the entire measuring range of the instrument will be covered (only the area of very small diameters can by switching on and off of the second light source be specifically measured respectively omitted). Why is this a disadvantage? The vast majority of the samples to be measured show a size distribution which only partially covers the entire measurement area of the used instrument. A large measurement area is therefore essentially interesting in order to analyse as many sample systems as possible. With the design of the Fourier lens, in many or even in the most cases, an unnecessary measurement area is covered, which comes at a high cost: Reduced measuring accuracy, lesser particle size dissolution and reduced sensitivity. And the larger the entire measuring range of an instrument is the more drastic is this effect. How come?
In the most basic case, a sample consists of a strict mono disperse material, i.e. the intensity distribution shows a simple ring structure, whereas the particle size can be directly determined. The more exact that this intensity distribution can be measured, the more exact then is the recoverable result. This means, the exactness of the measurement is directly dependent on the amount of the measuring channels, which are available for the current measuring interval. Now, during a measuring, if the maximum usable measuring area is always covered, the diffraction rings with sufficient intensity are - for example for a sample with large particles - always confined to the central area of the detector. The number of detector elements in this central area is naturally low in comparison, whereas the outer channels for such material are quasi unused.
To apply a comparative example: It is almost like measuring the voltage of a 1.5 V battery with a measuring instrument, adjusted to a measurement range from 0 to 1000 V.
For the separation of two closely neighbouring particle sizes, results from an analogue argumentation, also the dependence of the dissolution of the effective amount of the utilized detector elements: In order to exactly measure fine differences in the intensity distribution and - if possible - large number of elements is absolutely necessary.
Now in order to avoid the disadvantage of an unnecessary large measuring area, the patented by FRITSCH principal of the movable measuring cell position is used with the ANALYSETTE 22 line. Here the position of the measuring cell between the Fourier lens and the detector is varied, whereas the covered measuring area is adjusted to the respective demands. And it works like this:
The left illustration shows the situation for large particles. With a measuring cell far away from the detector, the only slightly scattered light waves cover the entire detector and all channels are utilized during the measurement.
Is alternatively the measurement cell positioned closely to detector, so now the heavily scattered light waves of the small particles are measured with the full resolution of the detector.
If necessary both cell positions are combined further, so that during a measurement the entire possible measuring range of an instrument is covered, but now with double the amount of effective detector elements.
Laser scattering determines Volume! The result tells you how many percent of the complete sample volume is filled with particles smaller than x µm. This is often called Q3(x).
And you also can get the answer to the question: "How many percent of the complete sample volume is filled with particles with a diameter between x and y µm?" This is then called dQ3(x).
Illustration: Particle size distribution of fly ash measured with ANALYSETTE 22 MicroTec plus. The continuous line is the so called sum curve Q3(x), the bars represent the value of dQ3(x).
