Particle size distribution

The particle size distribution [Jillavenkatesa A, Dapkunas S J, Lin-Sien Lum, "Particle Size Characterization", NIST Special Publication 960-1, 2001] (PSD) of a powder, or granular material, or particles dispersed in fluid, is a list of values or a mathematical function that defines the relative amounts of particles present, sorted according to size. PSD is also known as "grain size distribution" [Sivakugan N, "Soil Classification", James Cook University Geoengineering lecture handout, 2000] .The method used to determine PSD is called particle size analysis, and the apparatus a particle size analyzer [cite book|author=James P M Syvitski (editor)|title=Principles, Methods and Application of Particle Size Analysis|publisher=Cambridge University Press|year=2007|id=ISBN-13: 9780521044615]

ignificance of PSD

The PSD of a material can be important in understanding its physical and chemical properties. It affects the strength and load-bearing properties of rocks and soils. It affects the reactivity of solids participating in chemical reactions, and needs to be tightly controlled in many industrial products such as the manufacture of printer toner, cosmetics, etc.

Types of PSD

The way PSD is expressed is usually defined by the method by which it is determined. The easiest-understood method of determination is sieve analysis, where powder is separated on sieves of different sizes. Thus, the PSD is defined in terms of discrete size ranges: e.g. "% of sample between 45 μm and 53 μm", when sieves of these sizes are used. The PSD is usually determined over a list of size ranges that covers nearly all the sizes present in the sample. Some methods of determination allow much narrower size ranges to be defined than can be obtained by use of sieves, and are applicable to particle sizes outside the range available in sieves. However, the idea of the notional "sieve", that "retains" particles above a certain size, and "passes" particles below that size, is universally used in presenting PSD data of all kinds.

The PSD may be expressed as a "range" analysis, in which the amount in each size range is listed in order. It may also be presented in "cumulative" form, in which the total of all sizes "retained" or "passed" by a single notional "sieve" is given for a range of sizes. Range analysis is suitable when a particular ideal mid-range particle size is being sought, while cumulative analysis is used where the amount of "under-size" or "over-size" must be controlled.

The way in which "size" is expressed is open to a wide range of interpretations. A simple treatment assumes the particles are spheres that will just pass through a square hole in a "sieve". In practice, particles are irregular - often extremely so, for example in the case of fibrous materials - and the way in which such particles are characterized during analysis is very dependent on the method of measurement used.

Graphical representation of PSD

ampling

Before PSD can be determined, it is vital that a precisely representative sample is obtained. The material to be analyzed must be carefully blended, and the sample withdrawn using techniques that avoid size segregation. Particular attention must be paid to avoidance of loss of fines during maniputation of the sample.

PSD measurement techniques

ieve analysis

This continues to be used for many measurements because of its simplicity, cheapness, and ease of interpretation. Methods may be simple shaking of the sample in sieves until the amount retained becomes more or less constant. Alternatively, the sample may be washed through with a non-reacting liquid (usually water) or blown through with an air current. The most obvious disadvantage is that the smallest practical sieve size is 20-40 µm, and many PSDs are concerned with much smaller sizes than this. A 20 μm sieve is exceedingly fragile, and it is very difficult to get material to pass through it. Another disadvantage is that the amount of energy used to sieve the sample is arbitrarily determined. Over-energetic sieving causes attrition of the particles and thus changes the PSD, while insufficient energy fails to break down loose agglomerates.

Optical counting methods

PSDs can be measured microscopically by sizing against a graticule and counting, but for a statistically valid analysis, millions of particles must be measured. This is impossibly arduous when done manually, but automated analysis of electron micrographs is now commercially available.

Electroresistance counting methods

An example of this is the Coulter counter, which measures the momentary changes in the conductivity of a liquid passing through an orifice that take place when individual non-conducting particles pass through. The particle count is obtained by counting pulses, and the size is dependent on the size of each pulse.

edimentation techniques

These are based upon study of the terminal velocity acquired by particles suspended in a viscous liquid. Sedimentation time is longest for the finest particles, so this technique is useful for sizes below 10 μm, but sub-micrometer particles can't be reliably measured due to the effects of Brownian motion. Typical apparatus diperses the sample in liquid, then measures the optical density of successive layers using visible light or x-rays.

Laser diffraction methods

These depend upon analysis of the "halo" of diffracted light produced when a laser beam passes through a dispersion of particles in air or in a liquid. The angle of diffraction increases as particle size decreases, so that this method is particularly good for measuring sizes below 1 μm. Advances in sophisticated data processing and automation have allowed this to become the dominant method used in industrial PSD determination. A particular advantage is that the technique can generate a continuous measurement for analyzing process streams.

Acoustic spectroscopy

Instead of light, this method employs ultrasound for collecting information on the particles that are dispersed in fluid. Alternative term is ultrasound attenuation spectroscopy. :Dispersed particles absorb and scatter ultrasound similarly to light. This has been known since Lord Rayleigh developed the first theory of "ultrasound scattering" and published a book "The Theory of Sound" in 1878 [Lord Rayleigh, "The Theory of Sound", vol.2, Macmillan and Co, NY, second edition, 1896, first edition, 1878.] . There have been hundreds of papers studying ultrasound propagation through fluid particulates in the 20th century [Dukhin, A.S. and Goetz P.J. "Ultrasound for characterizing colloids", Elsevier, 2002] . It turns out that instead of measuring "scattered energy versus angle", as in case of light, in the case of ultrasound, measuring of "transmitted energy versus frequency" is a better choice. The resulting ultrasound attenuation frequency spectra are the raw data for calculating particle size distribution. It can be measured for any fluid system with no sample preparation, no dilution. This is a big advantage of this method over others. Calculation of particle size distribution is based on theoretical models that are well verified up to 50% by volume of dispersed particles. See the Dukhin and Goetz reference, where there are many examples of successful particle size distribution measurements.

Mathematical models for PSD

Probability distributions

* The log-normal distribution is often used to approximate the particle size distribution of aerosols, aquatic particles and pulverized material.
* The Weibull distribution or Rosin Rammler distribution is a useful distribution for representing particle size distributions generated by grinding, milling and crushing operations.

References