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Particle Size and Particle size distribution, short tutorial
Particle size is a geometric characteristic that is usually assigned to material objects with sizes ranging from nanometers to millimeters. There is a wide variety of real systems that contain particles within this size range. These systems are generally polydisperse, meaning that the particles in a particular sample vary in size. Therefore, in order to adequately characterize such systems, it is not sufficient to simply determine a particle size. We must instead determine the system’s particle size distribution. There is a section below which presents the definition of this notion.
Measurement of the particle size and particle size distribution is an extremely important tool for research and development, as well as quality control, in many industries. These include pharmaceuticals, cements (see Newsletter #20), ceramics (see Newsletter #3), paints, emulsions (see Newsletters #4,10,17), etc. Particles in such system vary not only in size, but in some other properties, such as rheological, shapes, colors, densities, chemical etc. That is why many different methods have been developed for their characterization. In general, particle size analyzers could be roughly organized in three groups: 1) counters; 2) fractionation methods; 3) macroscopic fitting methods. Below we present links to several International Standards that provide descriptions of some of these methods.
Methods classified as macroscopic fitting include light scattering, x-ray scattering, neutron scattering, and acoustic spectroscopy. All of these methods rely on measuring a certain property of a sample that contains particles and then calculating the particle size distribution from this measured raw data using an appropriate theory. For instance, in the case of static light scattering instruments, the raw data is the intensity of scattered light versus angle. Calculation of the particle size distribution requires a theory which relates this information to particle size (this is sometimes called a “solution of an ill-defined inverse problem”). These theoretical aspects of the light scattering are well known and described in details in the ISO standards referenced below.
Acoustic spectroscopy also belongs to the group of macroscopic fitting methods, but it stands alone from other macroscopic methods due to its capability of characterizing concentrated systems without the need for dilution. This method is based on measuring the attenuation of ultrasound at the set of frequencies in the MHz range. This attenuation spectrum is the raw data used for calculating a particle size distribution. There is a well-developed theory that takes into account particles’ interactions, which is imperative for extracting particle size distributions from attenuation spectra in concentrated systems. Details of this method are described in the International Standard ISO 20998-1:2006 “Measurement and characterization of particles by acoustic methods”, see reference below.
Verification of the theory for acoustic spectroscopy has been conducted by many different groups with wide variety of samples. Some results of such verification are presented in the book [3]. This method yields particle size distributions that are very close to the other characterization techniques, if proper care is given to transition from concentrated sample for acoustics to diluted sample for other methods. Here we present the result of such a verification test that compares DLS, Centrifugation, ELS, X-ray scattering, and acoustic spectroscopy. All measurements were performed on suspensions of silica Koestrosol. This silica serves as a basis for an EU certified reference material. Details of these experimental tests are described in reference 5.
Table. Particle size of silica Koestrosol, according to different measuring methods.
Method |
Particle size value [nm]. |
Dynamic light scattering, intensity massed harmonic mean diameter by cumulants method. |
19.0 ± 0.6 |
Centrifuge liquid sedimentation, intensity based modal diameter. |
20.1 ± 1.3 |
Electron microscopy, number based mean diameter. |
19.4 ± 1.3 |
Small X-rays scattering, number based mean diameter. |
21.8 ± 0.7 |
Acoustics, mass based median diameter. |
22.4 ± 0.5 |
Acoustic spectroscopy makes possible particle size measurement in concentrated dispersions and emulsions with no dilution and no sample preparation. This is critical in many cases when dilution affects particle size distribution. Also, this method can be applied for structured dispersions. Structure, usually in the form of polymer networks, contributes to ultrasound attenuation by itself, but this contribution can be subtracted using appropriate existing theory (see Newsletter #8). In addition, acoustic spectroscopy can resolve particle size distributions of different species of particles in mixed dispersions, assuming that they have different densities (see Newsletter #6).
This method can be used for continuous monitoring of particle size during milling (see Newsletter #18), crystallization, and other industrial processes. One of the most important applications is particle sizing of nano-dispersions and other nano-particulates. Acoustics can monitor the presence of nano-particles with a precision of 1%, see Newsletter #21. Nano-particles can be masked by larger particles when other methods are used.
PARTICLE SIZE DISTRIBUTION.
In order to characterize the polydispersity of real dispersions and emulsions we use the well-known concept of the “particle size distribution” (PSD). A detailed description of this concept can be found in the publications of Leschonski [7] and Irani [8]. According to their fundamental works, there are multiple ways to represent PSD depending on the principles of the measurement and particles’ properties. Usually the independent variable (abscissa) describes the physical property chosen to characterize the size, whereas the dependent variable (ordinate) characterizes the type and measure of the quantity – either number, area, volume, or weight. It is assumed that a unique relationship exists between the physical property and a one-dimensional property unequivocally defining “size”. This can be achieved only approximately. For irregular particles, the concept of an “equivalent diameter” allows one to characterize irregular particles. This is the diameter of a sphere that yields the same value of any given physical property when analyzed under the same conditions as the irregularly shaped particle.
The different relative amount of particles, measured in certain size intervals, form a so-called density distribution that represents the first derivative of the cumulative distribution. The cumulative distribution is usually normalized. This distribution determines the amount of particles (in number, length, area, weight or volume depending on r) that are smaller than the equivalent diameter X.
The figure below shows the normalized discrete density distribution or histogram q_{r}(X_{i},X_{i+1}). This distribution specifies the amount of particles having diameters larger than x_{i} and smaller than x_{i+1} and is given by:
The shaded area represents the relative amount of the particles. The histogram transforms to a continuous density distribution when the thickness of the histogram column limits to zero. A histogram is suitable for presentation of the PSD when the value of each particle size fraction is known. It is the so-called “full particle size distribution”. It can be measured using either counting or fractionation techniques. However, in many cases, full information about the PSD is either not available or not even required. That is why in many cases histograms are replaced with various analytical particle size distributions. There are several analytical particle size distributions that approximately describe empirically determined particle size distributions. One of the most useful, and widely used distributions, is the log-normal distribution. Log-normal distributions require that we extract only two parameters from the experimental data – median size and standard deviation. This is a big advantage in many cases when amount of the experimental data is restricted.
International Standards for particle sizing:
ISO 9276-1:1998 Representation of results of particle size analysis -- Part 1: Graphical representation:http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=25860
ISO 9276-5:2005: Representation of results of particle size analysis -- Part 5: Methods of calculation relating to particle size analyses using logarithmic normal probability distribution:http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=39388
ISO 22412:2008 Particle size analysis -- Dynamic light scattering (DLS)http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=40942
ISO 13320:2009 Particle size analysis -- Laser diffraction methodshttp://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=44929
ISO 13317-1:2001 Determination of particle size distribution by gravitational liquid sedimentation methods -- Part 1: General principles and guidelines: http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=21703
ISO 20998-1:2006 Measurement and characterization of particles by acoustic methods -- Part 1: Concepts and procedures in ultrasonic attenuation spectroscopy:http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=39869
ISO 13322-1:2004 Particle size analysis -- Image analysis methods -- Part 1: Static image analysis methods:http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=38664
ISO 13317-1:2001 Determination of particle size distribution by gravitational liquid sedimentation methods -- Part 1: General principles and guidelines: http://www.iso.org/iso/iso_catalogue/catalogue_tc/catalogue_detail.htm?csnumber=21703
REFERENCES.
1. http://en.wikipedia.org/wiki/Particle_size
2. http://en.wikipedia.org/wiki/Particle-size_distribution
3. Dukhin, A.S. and Goetz, J.P. “Characterization of Liquids, Nano- and Microparticulates, and Porous Bodies using Ultrasound”, Elsevier, 2010, 503 pages, 653 references.
4. Dukhin, A.S., Fluck, D., Goetz, P.J., Shilov, V.N., Dukhin, S.S. “Characterization of fractal particles using Acoustics, Electro-acoustics, Light scattering, Image analysis , and Conductivity”, Langmuir, 23 (10), pp. 5338-5351 (2007)http://www.ncbi.nlm.nih.gov/pubmed/17428071
5. J.Lyklema, “Fundamentals on Colloid and Interface Science, Elsevier, 2005,https://www.elsevier.com/books/fundamentals-of-interface-and-colloid-science/lyklema/978-0-12-460529-9
6. Dukhin, A.S., Parlia, S., Klank, D., Lesti, M. “Particle sizing and zeta potential of silica Koestrol (basis for certified reference material ERM-FD100 for Nanoparticles) by Acoustics and Electro-acoustics”,Part.Part.Syst.Charact.,27,pp.165-171, (2010) http://onlinelibrary.wiley.com/doi/10.1002/ppsc.201100038/abstract
7. Leschonski, K. “Representation and Evaluation of Particle Size Analysis data”, Part. Charact., 1, 89-95 (1984)
8. Irani, R.R. and Callis, C.F., “Particle Size: Measurement, Interpretation and Application”, John Wiley & Sons, NY-London, (1971)