Analysis of Particle Size Distribution

The particle size distribution of a given material is an important analysis parameter in quality control processes and research applications, because many other product properties are directly related to it. Particle size distribution influences material properties like flow and conveying behavior (for bulk materials), reactivity, abrasiveness, solubility, extraction and reaction behavior, taste, compressibility, and many more.

The analysis of particle size distribution is an established procedure in many laboratories. Depending on the sample material and the scope of the examination, various methods are used for this purpose. These include Laser Diffraction (LD), Dynamic Light Scattering (DLS), Dynamic Image Analysis (DIA) or Sieve Analysis. Typically, suspensions, emulsions, and bulk materials are analyzed, in exceptional cases also aerosols (sprays).

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Methods to determine the particle size distribution

Most samples are so-called polydisperse systems, which means that the particles are not of the same size, but of different sizes. A particle size distribution indicates the percentage of particles of a certain size (or in a certain size interval). These intervals are also called size classes or fractions.

A simple example is shown below. Here, a mixture of grinding balls has been separated by size: 5 mm, 10 mm, 15 mm and 40 mm:

Quantification can now be performed in several ways:

1. weighing: Each fraction contains 190 g of sample, or 25% of the total amount or weight. These values also correspond to the fraction of the total volume, since mass and volume can be treated equivalently, provided that the density does not change with particle size. 
2. counting: In total, the sample consists of 573 objects distributed into the four fractions. Since there is only one 40 mm sphere, this now accounts for only 0.2% of the total number, rather than 25% as in the mass-based distribution. On the other hand, the 490 spheres with a diameter of 5 mm have a share of 85.5%.

Thus, depending on the type of evaluation (number or mass/volume), one obtains a very different particle size distribution for the same sample.

Some particle size analyzers provide number-based distributions (Dynamic Image Analysis), others mass-based (Sieve Analysis) or volume-based particle size distributions (Laser Diffraction). With a suitable model, the distributions can be converted into each other. One special case is Dynamic Light Scattering, in which very often intensity-based particle size distributions are reported. This means that the different sizes are represented according to their contribution to the overall scattering intensity. This leads to a strong representation of large particles since the scattering intensity is decreasing with size by a factor of 106.

Parameters generated from a particle size distribution analysis

Many statistical parameters can be derived from a particle size distribution. The cumulative distribution is particularly suitable for this purpose. Among the most important parameters are certainly the percentiles. These indicate in each case the size x below which a certain quantity of the sample lies. Percentiles thus answer, for example, the questions "Below which size are the 10% smallest particles?" or "Above which size are the 5% largest particles?" Percentiles can be read directly from the Q or 1-Q curve.

Percentiles are denoted by the letter d followed by the % value. Thus, d10 = 83 µm, d50 = 330 µm, and d90 = 1600 µm means that 10% of the sample is smaller than 83 µm, 50% is smaller than 330 µm, and 90% is smaller than 1600 µm. Alternative notations are x10/50/90 or D 0.1/0.5/0.9 The d50 value is also called "median" and it divides the particle size distribution into equal amounts of “smaller” and “larger” particles. Usually d10, d50 and d90 are reported for a particle size distribution.

This makes it easy to characterize the middle or central point of the distribution, as well as the upper and lower ends with three values. This specification is not always useful, but it usually gives a good overview. Any number of percentile values can be defined, e.g. d16, d84, d95, d99 etc. However, attention must also be paid to whether the sensitivity of the measurement method is sufficient to reliably detect percentiles close to 0% or close to 100%. A d100 value is not clearly defined and therefore meaningless. If 100% of the particles are < 2mm, then this is also true for all larger x-values, which are also d100 values.

This figure shows how percentiles can be read directly from the cumulative curve. 

Percentiles like d10, d50, d90 can be obtained directly from the cumulative curve

Mean values (or mean particle size) can also be calculated from the tabulated values. This is done by multiplying the quantity in each measurement class by the mean size measurement class and summing the individual values. Various methods exist to calculate a mean, some are described in ISO 9276-2. To also characterize the distribution width, the standard deviation around the mean value can be used, or the span value. This is calculated as (d90 - d10) / d50. The wider the distribution, the larger the standard deviation and span.

The x-value at which the density distribution reaches a maximum (or the most frequently occupied measurement class) is called the mode size. Particle size distributions with multiple maximum values in the density distribution are referred to as multimodal (or bimodal, trimodal, etc.).

A special issue in the analysis of particle size distributions is the determination of oversize and undersize particles. These are small portions of particles that are significantly larger or significantly smaller than the bulk of the sample. In the cumulative curve, the presence of oversize or undersize is manifested by a step, in the density distribution by a small second peak (second maximum) outside the actual distribution. This oversize or undersize is best characterized by Q or 1-Q values at a suitable size x.

The example below shows a particle size distribution with 5% oversize. Here, 95 % of the particles are below 1 mm, the oversize has a size of 1 - 1.25 mm. This can be quantified by Q3(1 mm) = 95% or 1-Q3(1 mm) = 5%. This example also shows that the addition of oversize increases the mean particle size, while the median remains unchanged. Alternatively, the presence of oversize can also be described by the increased d95.

Particle size distribution of a monomodal material (red) as Q3 and q3 curve. If 5 % of particles 1 – 1.25 mm is added, this results in a bimodal distribution. The 10 % and 50 % percentiles remain unchanged, mean and standards deviation increase. The oversize is best characterized by d95 or Q3 at 1 mm