## Particle size distribution

Powders consist of particles with varying particle sizes, so in many cases, the particle sizes of individual particles are managed collectively as a distribution. This distribution is referred to as “particle size distribution” or “granulometry distribution.” Particle size distribution is represented using acquired data through “frequency distribution (histogram)” and “cumulative distribution.”

The distribution of particle sizes in powders is presented in terms of how often each size occurs (frequency distribution) and the accumulation of sizes up to a certain point (cumulative distribution).

## Frequency Distribution (Histogram)

A frequency distribution, often referred to as a histogram, displays the proportion of particles for each class (size range). Let’s explain this using the example of measuring particle sizes through sieving. A sample of 200g is sieved using 10 different sieves with varying mesh sizes. The table summarizes the amount of powder remaining on each sieve. For instance, 2g of powder could not pass through a sieve with an opening of 1000μm, indicating that these particles have a size larger than 1000μm. Since the total weight is 200g, particles with a size greater than 1000μm constitute 1% of the total.

Moving on, when powder passes through the 1000μm sieve but remains on the 900μm sieve with 6g, it means these particles have sizes larger than 900μm and equal to or smaller than 1000μm. The particles within this size range contribute 3% to the total weight.

In this manner, graphing the proportion of particles remaining on each sieve provides the frequency distribution, which is also known as a histogram.

In a frequency distribution (histogram), it’s important to note that there are ranges like “greater than 1000μm” or “greater than 900μm and equal to or less than 1000μm” for each data point. In other words, it doesn’t indicate the proportion of specific particle sizes.

A frequency distribution (histogram) allows you to quickly see the range of the most frequent particle sizes and the spread (variability) of particle sizes. However, these values depend on how the intervals are set, so caution is necessary when interpreting them. For instance, if we change the interval of the previous data to 100-250, the result would appear as shown in the figure below. In this result, it might seem like there’s a peak around 475μm (the midpoint of the interval [350,600]), but in reality, the peak is in the interval [500,600]. Thus, the information interpreted can vary based on the chosen interval settings.

## Cumulative Distribution:

A cumulative distribution represents the proportion of particles with particle sizes below (or above) a certain threshold. When aggregating below the threshold, it’s referred to as the “undersize cumulative distribution,” and when aggregating above the threshold, it’s called the “oversize cumulative distribution.” We’ll focus on the undersize cumulative distribution for now.

When the threshold is infinitely small, particles with sizes below that threshold don’t exist, resulting in 0%. On the other hand, when the threshold is infinitely large, all particles are included, leading to 100%.

Using the example from the frequency distribution (histogram), the cumulative distribution would appear as follows.

The characteristic of a cumulative distribution is that, unlike a frequency distribution (histogram), it is independent of interval settings. Therefore, regardless of who aggregates the data, the same distribution can be obtained. On the other hand, it becomes more challenging to directly read the range of the most frequent particle sizes, as is possible with a histogram. In a cumulative distribution, the proportion of particle sizes is represented by the slope of the graph.

In the example above, you can observe that the steepest slope is around 500μm, indicating the highest concentration of particles. The particle size at which the cumulative distribution reaches 50% is known as D50 (median diameter). Additionally, based on the values of the cumulative distribution, particle sizes corresponding to 10% and 90% are represented as D10 and D90, respectively.