**Size and shape are measures that greatly impact particulate behavior such as flowability, compaction, and mixing. Spherical particles are adequately characterized by size-only techniques however, irregular particles require additional shape measurements to best characterize them.**

More irregular shapes are harder to characterize, but for powders of use in manufacturing, it is required to determine a more significant amount of parameters that describe the sample in its entirety, and that can predict performance. The irregularity of particles and their use in final products will indicate the shape measurements needed to properly gain a better understanding and control to optimize the final product.

Dynamic image analysis benefits from the use of a large sample population. As such, statistical histograms are the optimal method to demonstrate statistical distributions of a measurement parameter of a particle.

## Distribution Histograms

To illustrate the statistical results from sample analysis, results are divided into small classes or “bins,” and the number of particles in each size bin is reported. Size information can be displayed in volume, number, and surface-area weighted histograms, each offering valuable information about the analyzed sample. In some cases, shape measurements are size-independent. In such cases, these fraction measures range from a value of zero to one. The following graph is a size histogram, and it shows a good view of the actual distribution. Size data is usually depicted graphically on a log scale axis. Size and shape histograms are commonly used to compare differences in lots of samples.

*Typical Bounding Rectangle width size histogram*

*Image Credit: Vision Analytical *

Other shape statistics (non-size data) such as smoothness, circularity, opacity or aspect ratio is reported on a linear scale. Each bin reports its recorded quantity of particles:

*Typical linear axis histogram for smoothness*

*Image Credit: Vision Analytical *

As shown below, a volume-weighted histogram plot will emphasize the presence of larger particles over the smaller ones, given the more significant volume larger particles have compared to a higher number of smaller ones. Below are the volume-weighted results of the same sample shown in a number-weighted histogram above. Volume-weighted histograms are helpful to identify small amounts of agglomerates in samples. Number-weighted histograms are helpful to identify large quantities of fine particles in samples, resulting in clogging or filtration issues. Of course, using dynamic image analysis, objective evidence of these particles are given in the form of thumbnail images of every measured particle:

*A volume-weighted Bounding Rectangle width histogram*

*Image Credit: Vision Analytical *

The following chart explains why one sample can have different number-weighted and volume-weighted histograms. This also shows the benefit of always studying both weighted statistics and using both to compare and contrast. The number and volume weighting can offer value in observing small quantities of agglomerates (larger particles) by reporting by volume and a higher amount of fines (smaller particles) reporting by number.

*Image Credit: Vision Analytical *

## Characterization of Distributions

### Means

A distribution is characterized by a mean as a single number. A mean value allows some information of the sizes present but does not issue any indication about the shape of the distribution or how broad or narrow it is. The number or arithmetic mean is purely the average value. It is frequently denoted as D_{1,0}. Other means take into account volume weighting and area.

Assuming sphericity of particles, the generic definition of a weighted D_{p,q} mean diameter is

Definition of D_{p,q} means

The diameter means that are most often of use in characterizing a particle sample are:

In these definitions, n_{i} is the count in size bin number i, and d_{i} is the representative diameter of that size bin. The sums are over all particles, and N is the particle count total.

Descriptions of these means in words are as follows:

Arithmetic mean diameter, D[1,0] : the average of the diameters of all the particles in the sample.

Surface mean diameter, D[2,0] : the diameter of a particle whose surface area, if multiplied by the total number of droplets, will equal the total surface area of the sample.

Volume mean diameter, D[3,0] : the diameter of a particle whose volume, if multiplied by the total number of particles, will equate all of the sample’s volume.

Surface moment mean diameter, D[3,2] (“Sauter mean”): the diameter of a particle whose ratio of volume to surface area is the same as that of the complete sample. Mathematically, if V is the total volume and A is the total surface area of the sample, D3,2 = 6 * (V / A).

Volume moment mean diameter, D[4,3]. This value is a superior indicator than other means of which particle sizes contain the majority of the volume.

The Mode is the most frequent size present.

The Harmonic mean is **N / Σ (n _{i} / d_{i})**

### Geometric Means

Geometric means reflect the visual weighting of a log-size axis. The geometric mean diameter will appear as the center of a distribution on a log scale, while the usual arithmetic mean may sit a lot lower on the size scale (as smaller sizes are a lot more numerous than larger sizes).

*Arithmetic and geometric means*

*Image Credit: Vision Analytical *

To compute the geometric mean, use the logs of the x-axis values ( log (d_{i}) in place of d_{i} ) :

The Geometric mean is **Σ [n _{i} log(d_{i})] / N **

With the use of the same manner, the geometric versions of the other means and standard deviation may be calculated.

### Measures of Spread

Standard deviation measures how wide the distribution is:

where μ = mean diameter (D_{10}). It has units of microns (for a size distribution).

The Coefficient of Variance is the ratio of the standard deviation to the mean: CV = σ / μ . To express as a percent multiply by 100. Being a ratio, this statistic does not have units.

### Percentiles

Percentiles are a way of conducting size information as one or more numbers. The Median size splits the particles into two parts containing equal counts. Strictly speaking, this is the Number Median. It is also known as the 50^{th} or the 50% percentile.

The 10^{th} percentile defines the size with the property that 10% of the particle count is less than that size. Any other percentile can be defined similarly. Commonly, three percentiles (10%, 50% and 90%) are used as a characterization that is uncomplicated but provides additional information to a single mean. Additional percentiles are available and customizable in the Insight software to meet the needs of any specific industry requirement.

The Volume Median, or 50^{th} percentile by volume, splits the volume of the sample into two equal pieces. The two classes will hold equal volume but not an equal count of particles. Other volume percentiles are defined similarly; as an example, the 25^{th} volume percentile is the size such that particles smaller than that size are representative of a quarter of the volume.

### Other Characterizations

*Image Credit: Vision Analytical *

Skewness is an indicator of how asymmetrical the shape of the distribution is, about the center. A positive value will mean there are further counts on the right side of center, normally in the appearance of a tail. A negative value will mean it tails to the left.

Skewness = **Σ n _{i} (d_{i} – μ)^{3} / (σ^{3} N)** (σ = std. dev., μ = mean)

Kurtosis is an indicator of how much the shape differs from the typical bell curve in a vertical sense.

Kurtosis = **[Σ n _{i} (d_{i} – μ)^{4} / (σ^{4} N)] / - 3** (σ = std. dev., μ = mean)

This information has been sourced, reviewed and adapted from materials provided by Vision Analytical Inc.

For more information on this source, please visit Vision Analytical Inc.