Definition of Particle Size

Particle size and particle size distribution are usually reported in terms of diameter irrespective of the actual particle shape; commonly this diameter is the equivalent sphere diameter, defined by ISO 9276-1 as the diameter of a sphere having the same physical attributes as the particle. In many cases the equivalent sphere is one with the same volume as the particle, but the method of measurement and the property of interest in the powder can lead to the use of other diameters defined by, for example, the surface area, or based on a statistical measurement such as the Feret diameters measured by image analysis. A detailed list of the possible options for description of particle diameters is given in the Recommended Practice Guide 960 published by NIST. Clearly, irregularly shaped particles can lead to very different diameter measurements depending on the definition chosen.

This range of possible descriptions of size gives one indication that different techniques or even different equipment using the same basic technique are likely to produce different size measurements from the same powder sample. Different methods, even different equipment suppliers employing the same basic method, will use different algorithms to convert the effect of an irregularly shaped particle (e.g. on rate of sedimentation or scattering of light) into the equivalent effect that would be produced by an idealised sphere. The effects of the approximations that these algorithms use often become more pronounced at the extremes of a distribution and thus can have a significant effect on values calculated to describe the distribution.

Analysis by laser diffraction is measured in volume, with transformations to number, length and area possible. However caution should be taken as the errors associated with the analysis are cubed through the transformation. The chance of incurring errors is increased in the sub micron region and can be greater than 15%. If a distribution is whole, i.e. there is no missing component then the errors associated with the transformation are minimised. If there is a portion missing then the errors are significant and the transformation should not be relied upon. In the schematic Figure 1 below distribution A is suitable for transformation whilst distribution B has been truncated (e.g. because below the lower limit of measurement) and therefore there is no data for small particles to include in a transformation.

Schematic showing a truncated distribution B which is not suitable for conversion between volume and, for example, area distributions, compared with the narrower but more complete distribution A, which could be converted.

Particle Size Distributions

The distribution can be reported in graphical format, or for the purposes of specifications and quality control measurements more commonly as one or more diameters that describe the size at particular fractions or statistical descriptions of the distribution. 

Graphically the distribution may be shown either as a density (or differential) distribution or as a cumulative plot, generally with increasing particle diameter. Each type has its own advantages. The density distribution presents a clear description of the distribution spread and the peak (mode) and whether the peak is skewed from the centre of the distribution. It will also show if the distribution is multi-modal with more than one peak.

In a cumulative plot small multi-modal peaks may not be easily observed, but this form of graphical output enables simple identification of the fractional distribution of sizes d_nn where the subscript ‘nn’ is the percentage (by volume, area, etc. depending on the definition of d) of particles with dimensions less than d. The values most frequently measured in this form are d10, d50, and d90 which give an indication of size of the fine (d10) and coarse (d90) fractions, and of the median particle size (d50).

While the median particle size indicates the centre of the size distribution, this is unlikely to be the same as the mean particle size. The mean particle size can be defined in many ways; ISO 9276-2 describes methods for calculating this, but many other texts give good descriptions of their derivation.

Number means are familiar to most users, being given simply by the sum of the diameters (or cube root of the sum of the diameters cubed, if volume is considered) divided by the number of particles. This has however a number of disadvantages including the need to count a large number of particles, and the way in which a large number of very small particles can weight the result even though they comprise a very small proportion of the total volume.

Because of these disadvantages, ‘moment means’ are frequently quoted, and in this Guide the de Brouckere mean or volume (equivalent to mass, for a fixed density) moment mean is used, analogous to the centre of gravity of the distribution. Generally this mean is shown by the subscripts ‘4,3’, so using the terminology described earlier, this mean would be shown by x_4,3

or d_4,3. Calculations of moment means are given in the standard ISO 9276-2, although confusingly here the notation used gives this mean as x_1,3 .

Other derivations maybe found elsewhere, which show the calculation simply as: