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Particle size distribution

The particle size is one of the most important characteristics of particulate materials. It directly affects several properties, from the accessibility of minerals during processing to the mouthfeel of many foods. In industry, the aim of particle size measurement is to first find a correlation between the particle size and the property of interest (e.g. mouthfeel, reactivity, sintering behavior).This information can then be used to modify the properties of the substances via the particle size and also use the particle size as a parameter for quality insurance purposes.

Today, measuring the particle size distribution of a substance is an easy and straightforward task thanks to the modern instrumentation available which often enables measurement in less than one minute. To determine particle size there is a large variety of techniques available, which deliver a similarly large selection of results in the form of means, averages, modes, and other parameters. The key to understanding the results is to first know the meaning of these parameters. In this article we will introduce the basic terms and their use in particle size analysis and discuss the advantages and disadvantages of using single parameters vs. multiple parameters to characterize the particle size distribution.

Describing the particle size distribution by a single parameter

Figure 1: Theoretical particle size distribution of a simple mixture in different weightings

Obtaining a meaningful description of a particulate system with many different particle sizes and shapes using just one or two parameters is challenging. To deliver useful information, the chosen parameter should be easy to calculate, should be specific enough, and should have a direct connection to the physical property of interest.

Different measurement techniques “see” the particles in a different way, which translates to a different weighting as the result. The different weightings are:

Number-weighted distribution (% number)
Intensity-weighted distribution (% in surface)
Volume-weighted distribution (% in volume)

As an example, as a microscope sees the diameter of each particle, using this technique will deliver a number-weighted result. In contrast, the diffraction of light is proportional to the volume of the particle, therefore techniques such as laser diffraction or X-ray diffraction deliver volume-weighted results.

Figure 1 shows an example of how the particle size distribution (PSD) of a mixture containing 1 µm, 2 µm and 3 µm particles would look like in different weightings.

Table 1: Particle size distributions delivered by different measurement techniques
Weighting	Measurement technique
Number-weighted distribution (% number)	Microscopy, dynamic image analysis
Intensity-weighted distribution (% in surface)	Dynamic light scattering (DLS)
Volume-weighted distribution (% in volume)	Sedimentation, laser diffraction, X-ray diffraction

As Figure 1 illustrates, the main difference between these weightings is the representation of the fine fraction compared to the coarse fraction. Number weighting gives equal representation to the fine and the coarse fraction. In contrast, in the volume-weighted results a few large particles might outweigh numerous smaller particles. Both representations have their merits, depending on what is required from a measurement: For someone working in the mining industry, it is important to know the size of the particles in a given volume, while in microbiology, where individual cells are counted, the number-based distribution is preferred.

The remaining sections of this article will look at the selection of options to visualize, analyze and compare particle size distributions, as illustrated by an example of a measurement using the laser diffraction technique.

Using laser diffraction to obtain a single parameter for quality control

Figure 2: Typical results of a particle size analysis by laser diffraction. CC BY license

When using laser diffraction, the two most common visual representations are the density or frequency distribution (q₃), see Figure 2, and the cumulative curve (Q₃), shown in Figure 3. The former displays the probability of finding a particle with diameter d in the population, while the latter shows the percentage of particles which are smaller (undersize) or bigger (oversize) than diameter d. The two are in strong mathematical correlation, as the cumulative curve is a simple integral of the frequency curve:

$$Q_3(r)=\int_{0}^{r} q_3(r) dr$$

Read more about the principles of laser diffraction.

Graphical depiction is practical, as it gives lots of information at a glance. A few examples are the modality, the relative ratios of the particle populations, the broadness, and the general shape of the distribution. Although a visual input is fast and convenient, for documentation and comparison purposes numerical values are indispensable. The three most important options for describing a particle size distribution by a single parameter are:

the mode
the median
the mean

The mode

The mode is defined as the most commonly occurring size. It is the peak position on the frequency distribution curve. This frequency distribution curve is a straightforward figure for strictly monomodal (having only one peak) and narrow samples, such as latex beads, or monodisperse (containing only particles of very similar sizes) nanoparticle populations. However, its limitations become clear once more than one particle population is present in the sample (bimodal or polymodal substances) or when characterizing polydisperse samples. Figure 2 shows an example with three samples, all of which have a very similar mode although the particle distribution of the three samples is clearly different.

Figure 3: Three different samples with the same mode, but clearly different particle size distributions

The median

The median is the value separating the higher half of the data from the lower half. It is the determined particle size from which half of the particles are smaller and half are larger. This is also called D₅₀ (see Figure 1). The median is easy to determine from the cumulative distribution curve (s. Figure 4). However, it has the same problem as the mode: Several distributions may have the same median; therefore it is not reliable as a single figure.

The mean

The mean is the single most valuable measure of a sample. There are many different mean diameters defined, based on the weighting of the mean. The three most important means are:

Number-weighted mean diameter
Surface-weighted mean diameter
Volume-weighted mean diameter

The calculation of these means follows the general formula:

$$D[p,q]=\frac{\sum n*d^p}{\sum n*d^q} $$

These means are equally important because – as already mentioned earlier – depending on the application people are interested in different properties of the sample. Table 2 summarizes the means and gives a few applicative examples.

Table 2: Weighted mean diameters and their applications
Mean diameter		Importance
Number-weighted mean	D[1,0]	Biological applications, viruses, blood cell counting
Surface-weighted (Sauter) mean	D[3,2]	Fluid dynamics, catalysis [Catalyst AR], fuel combustion
Volume-weighted (De Brouckere) mean	D[4,3]	Mining, comminution, mineral processing [Building Mat AR]

Other methods

Additionally, several methods were developed in the 20^th century in order to provide a single measure for the quality check of products in the mining, milling, and grinding industries. The two most prominent examples are the Rosin-Rammler (RR) model and the Gates-Gaudin-Schuchmann (GGS) model. Both of these provide a linearization of the distribution through a logarithmic calculation and provide a more sensitive median (D_63.5).

Describing the particle size distribution by multiple parameters

A single number is a valuable piece of information for quality control purposes, e.g. when checking for differences in production batches. However, it does not deliver information about the source of the deviation, such as the shape, broadness, modality, etc. A deeper investigation into, and understanding of, the source of differences call for a multiparameter description of the particle size distribution. The following parameters are used in particle sizing techniques.

D-values

The most common values used in several particle sizing techniques are the D-values. These are related to the cumulative distribution and are only meaningful followed by a number, e.g. D_x. Both undersize and oversize D-values can be defined (just like the cumulative curve can be undersize or oversize). If not stated otherwise, the standard is the undersize D-value which means that in general D_x gives the diameter of which x percentage of the particles are smaller. The three most often used D-values are D₁₀, D₅₀, and D₉₀ (see Figure 2), but custom D-values can be found for particular applications (e.g. IV Insulin).

Even though D-values are the most often used results due to their convenience and easy to understand definition, they are typically meant for quality control purposes as an alternative to the single parameter but more calculation-heavy RR (Rosin-Rammler) or GGS (Gates-Gaudin-Schuchmann) method.

Span

In most cases a useful description of a particle size distribution also calls for a way to describe the shape of the distribution. An important measure of any statistical distribution is the width or broadness. In laser diffraction an often-used measure is the span. It is calculated in general by the following formula:

$$ Span = \frac{D_{90} - D_{10}}{D_{50}} $$

However, some custom-defined spans exist for specific applications.

Folk & Ward parameters

Some samples, such as soil and geological samples, have a large variety of grain sizes and are classified accordingly. Their particle size distribution is often complex with very broad polymodal distribution. For the analysis of such samples Folk and Ward (F&W) proposed a statistical grain-size analysis method [Ref: (R.L. Folk, 1957), AR: Soil].

At the core of the method is the conversion of the particle size distribution from millimeters to the phi scale by the following formula:

$$ \phi = -\log_{2}d $$

Followed by the calculation of a set of parameters:

Median: see above. Measured in phi units
Mean: definition as above, however Folk proposed his own formula (R.L. Folk, 1957):
$$ M_z = \frac{\phi_{16} + \phi_{50} + \phi_{84}}{3} $$ The mean is also measured in phi units and is the most widely used of all the parameters.
Skewness: Measure of the asymmetry of the distribution around its peak. It is determined by the equation:
$$ sk_1 = \frac{\phi_{16} + \phi_{84} - 2\phi_{50}}{2(\phi_{84} - \phi_{16})} + \frac{\phi_{5} + \phi_{95}-2\phi_{50}}{2(\phi_{95}-\phi_{5})} $$ The skewness is also given in phi units. There are other means of calculating the skewness. However, the formula proposed by Folk also takes the “tails” into consideration. Perfectly symmetrical distributions have a skewness of 0.00. A positive skew represents an increased fine portion (“tail” on the left), while a negative skew indicates a shift to the coarse materials (“tail” on the right). Folk also suggested a classification:
1. ±0.1 is nearly symmetrical
2. -0.10 to -0.30 is coarse-skewed and +0.1 to +0.3 is fine-skewed
3. -0.30 to -1.00 is strongly coarse skewed and +0.3 to +1.0 is strongly fine-skewed.
Kurtosis: Measure of the deviation from the normal (Gaussian) distribution or in other words the “peakedness”. The formula proposed by Folk is:
$$ k_g=\frac{\phi_{95} - \phi_{5}}{2.44(\phi_{75}-\phi_{25})} $$ Kurtosis is measured in phi units as well. It can be used to determine the number of outliers in the sample. A perfectly Gaussian curve has a kurtosis of 1.00 and is also called mesokurtic. If the sample is more sharp-peaked, hence less outliers are present, then the kurtosis is higher than 1.00 and it is called leptokurtic. If the sample curve is more flat-peaked, meaning there are a large number of outliers in the sample, the value is smaller than 1.00. Such a distribution is called platykurtic.

Phi units furthermore have the advantage of providing an easy-to-read scale of size classes for the classification of soil types starting from boulders (ϕ > -8) down to clay (ϕ < 8).

Conclusion

Particle size analysis is a convenient tool for quality control as well as research and development. Through the careful and mindful selection of the right weighting model, mode, and further parameters, it is possible to extract crucial information also in the research and development phase of new products.

Deeper discussions of special sample groups are given here:

Read more about laser diffraction for particle sizing

References

R.L. Folk, W. W. (1957). Brazos River Bar: A Study in the Significance of Grain Size Parameters. Journal of Sedimentary Petrology, Vol 27, No. 1, pp 3-36.

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