**A qualitative description of the shape of particles often involves mere visual inspection –in the case of fine samples usually with the aid of a microscope. Characterizations like "needle-shaped crystals," "compact granulate," "splintered grain," or "rounded sand"; quite easily evoke a corresponding picture in the mind of every reader. However, such descriptions are insufficient when evaluating a material’s suitability for a specific application.**

For instance, if sand is to be used as a construction material, the grains must not be too round. In such a scenario, it becomes important to precisely quantify the degree of roundness, in order to determine the suitability of the raw material. Here, quantitative particle shape analysis comes into play.

The Dynamic Image Analysis (DIA)’s measurement technique is particularly suitable for the purpose of accurately determining a range of shape characteristics of individual particles. By analyzing large quantities of material, it provides meaningful results in a short span of time.

Besides introducing the shape parameters characterized particularly well by DIA, this article also illustrates shape analysis with the CAMSIZER systems, citing practical examples.

## Describing Particle Shape

“Shape” as a term encompasses different geometric properties of the particles studied. Some of the parameters describe the grains’ outer shape, including, for instance, the aspect ratio (width divided by length, macroshape). On the other hand, meso- or microshape refers to the analysis of finer structures, such as curvature of the corners or surface roughness. ISO 9276-6 describes the shape analysis.

The user must select the type of description(s) from the multitude of ways of describing the particle shape that best reflects the sample material. Naturally, this will depend on the robustness, uniqueness, and manageability of the shape parameter as well as type of application and accessibility for the measuring system. In fact, different definitions are found in the literature for some parameters. These definitions and the characteristics of the measuring technique used should be taken into consideration.

## Determination of Particle Shape by Dynamic Image Analysis

In DIA a camera system captures a stream of moving particles . The particle images are analyzed in real-time by being recorded as shadow projections (Fig. 1). Thus, in just a few minutes of measurement time, a high-resolution size and shape distribution is available. This measurement result is based on the evaluation of several thousand, hundreds of thousands, or sometimes even millions of individual particles – depending on the sample and measuring system.

**Figure 1.** Two state-of-the-art DIA instruments - CAMSIZER P4 (left) and CAMSIZER X2 (right) from Retch Technology. The P4 model is suitable for the rapid analysis of free-flowing bulk materials in a size range of 20 μm to 30 mm, the X2 model is optimized for fine, powdery sample materials in a size range from 0.8 μm to 8 mm.

Particle images: Activated carbon (left), sugar crystals (center), expandable polystyrene, EPS (right).

## Particle Shape Definitions

Usually, values for shape parameters range between 0 and 1. Values equal to or close to 1 characterize a nearly spherical, regular, or isometric particle. The following section introduces the most commonly used shape parameters in DIA.

**Aspect ratio (width / length)**

The aspect ratio is the quotient of particle width and particle length (Fig. 2). Thus, aspect ratio functions as a description of the outer form or geometry (macroshape) of the particle. Different widths or length definitions may be used for the determination of the aspect ratio – for instance, minimum and maximum Feret diameter, chord measurements, or only ratios are considered in which width and length are perpendicular to each other.

Thus, the numbers may differ slightly depending on the method chosen. Nevertheless, aspect ratio is an extremely easy to determine, stable and robust shape parameter allowing for largely reliable measurement, irrespective of the absolute particle size.

**Circularity**

Fig. 2. shows the formula for calculating circularity – i.e. the ratio of the area (A) of the particle projection to its perimeter (P). Circularity, in other words, can be considered as a measure of the similarity of the particle to a sphere. Thus, a rough particle surface implies low numerical values of the parameter circularity.

Accurate measurement of the circumference is required to use this parameter – i.e. circularity – in a meaningful manner. However, this can only be achieved sufficiently well by aid of adequate magnification. Thus, circularity measurement results depend largely on the particle size. While this can predominantly be compensated by suitable correction factors, it cannot completely be prevented.

What’s more, it is difficult to compare circularity results by different optical systems with optical design and different magnification. Thus, circularity is a somewhat less robust shape parameter than aspect ratio.

**Figure 2.** Frequently used shape parameters in dynamic image analysis.

**Symmetry **

As the name suggests, symmetry is a measure of the eccentricity of the particle image. To establish symmetry, first the centroid of the particle projection is determined, followed by the calculation of the minimum ratio of two opposing semi axes through this point (Fig. 2).

Symmetry is then determined from this ratio, such that a perfectly symmetrical particle is given the value 1 (in other words, the distance from centroid to perimeter is the same in each direction). It is important to note that this holds true for a circle as well as for many other polygons, for instance, rectangles.

**Convexity**

Convexity is known by many names, such as concavity, or "solidity." However, these terms all describe the ratio of the real area of a particle projection to its convex hull. To determine the convex hull, an envelope is required to compensate all the roughness (Fig.2) which can best be imagined like a rubber band stretched around the particle image.

**Compactness**

Similar to circularity, compactness can also be interpreted as a similarity of the particle projection with a circle (e.g., for circles, compactness = circularity = 1). Compactness is calculated from the ratio of the area of the particle image to the particle length.

However, the compactness formula does not require the perimeter – rather the length of the particle, which is metrologically more clearly defined and therefore more accessible to the analyzers (Fig. 2). Thus, compactness is the more robust parameter and a perfect substitute for circularity.

**Roundness**

ISO 9276-6 explains roundness based on the description of the curvature of the particle images. Thus, it is a measure of the particle’s corners’ smoothness. While there are various definitions presented in the literature; this article follows those proposed by Wadell (1932, 1935), Krumbein (1941) and Krumbein & Sloss (1963). These definitions have thus far proven useful in sedimentology and for the analysis of raw sand and quartz grains; however, they are also equally applicable to other areas.

A circle is described in each corner of the particle projection, the mean radius of all corner circles is calculated and divided by the radius of the inscribed circle. Historically, when these parameters were presented, there was no image analysis software, therefore, time-consuming, laborious manual evaluation was inevitable.

Model particles were printed on charts (Fig. 3) to reduce workload. These were then used to classify the particles during evaluation under the microscope. As a consequence, this method is highly susceptible to the user’s individual perception. What’s more, these charts became altered and deteriorated by repeated copying, thus making them hardly comparable.

In contrast, Dynamic image analysis (DIA) provides user-independent, reliable measurement data. However, the roundness determination requires a high degree of magnification and is thus useful only for a particle size of more than 25 pixels.ls.

**Figure 3. **Determination of roundness according to Wadell or Krumbein & Sloss. The chart on the right shows model particles that should allow the classification by roundness and sphericity in the microscopic evaluation. Note that "sphericity" after Krumbein means aspect ratio. A good example of the sometimes confusing nomenclature when describing particle shape.

## Examples of Particle Shape Analysis

**Broken spheres**

In the chemical industry, catalyst materials – in a variety of geometries such as rods, tori, or spherical particles – are often applied to ceramic substrates. Often, however, these carriers may break due to mechanical stress, thus reducing the efficiency of the catalyst and producing an undesired fine fraction. DIA enables the precise determination of the percentage of broken particles, thereby allowing for continuous monitoring of the product quality (Fig. 4).

**Figure 4.** Five analyses of a spherical catalyst carriers with different percentages of broken particles. Broken particles are easily identified as these have an aspect ratio smaller than 0.95. The percentage of these defective particles lies between 5% (red curve) and 32% (turquoise curve). The picture shows two complete and one broken sphere.

**Metal powders **

Usually, metal powders are produced by atomizing a melt. As a result, a range of different particle shapes emerges, depending on whether the cooling process happens in air or in liquid. In fact, the shape of the particle has a significant impact on the powder's flowability and bulk density.

A round particle shape is desirable for additive manufacturing techniques like selective laser sintering. This is why Dynamic Image Analysis is particularly apt for predicting the suitability of a metal powder for different processes based on particle shape analysis (Fig. 5).

**Figure 5. **Comparison of the aspect ratio for 10 different metal powders. Curves plotting on the left side of the diagram indicate irregular shape, curves plotting on the right side indicate regular particle shape. SPHT (sphericity) is circularity squared. The Ti powder has the highest aspect ratio and is well suited for additive manufacturing. By means of image analysis, fused spheres can be identified very well.

**Mixtures **

Shape analysis is often the only suitable method to determine the proportion of the components, especially if a sample has two components that are very similar in size. For instance, a mixture of activated carbon and ion exchanger in water filters or a mixture of glass beads and grip agent, which is used for road markings.

In both cases, the size of both components must be similar to avoid segregation or separation. However, since the particle shape clearly differs, the proportions can be exactly quantified using Dynamic Image Analysis (DIA) (Fig. 6).

**Figure 6.** Mixture of activated carbon (angular particles) and ion exchanger (round particles) from a water filter. The ion exchanger has shape values close to 1, whereas the activated carbon is much smaller than 1. In the mixture, a saddle point forms in the distribution where the mixing ratio can be read directly.

**Roundness**** of sand grains **

Sand is one of the most widely used natural resources, with an annual consumption of more than 40 billion tons. For example, sand is predominantly found in building materials. However, not every type of sand is suitable. For example, well-rounded sand grains, typically found in desert sand, are inappropriate for the construction industry.

However, for use as a proppant material in hydraulic fracking, round sand grains are better suited than angular particles. To reliably characterize the sand, exact, user-independent measurement of the roundness by Dynamic Image Analysis is a method of choice. High sample throughput and complete quality control are ensured, owing to short measurement times ranging between 2 - 5 minutes. The comparability of the roundness values obtained by dynamic image analysis with the microscopic evaluation has been demonstrated (Vos, 2018).

**Figure 7.** Comparison of the roundness of two sand samples: Sample 1 (blue curve) consists of angular grains and has low roundness values. Sample 2 (red curve) consists of wellrounded grains with high roundness values. Particle images can be saved during analysis and then displayed. On the left some of the edgy grains, on the right some rounded grains are shown.

## Conclusion

Dynamic Image Analysis, as demonstrated in this article, is a highly accurate and reliable method for characterizing the particle size and shape of bulk solids and suspensions. Unlike alternative methods, DIA alone delivers reliable and user-independent information about the particle shape.

Since many properties of the sample material are influenced by the particle shape, image analysis provides valuable information to assess the product quality.

## References

- ISO 9276-6:2008 - Representation of results of particle size analysis - Part 6: Descriptive and quantitative representation of particle shape and morphology (ISO 9276-6:2008)
- Krumbein, W.C. (1941). Measurement and geological significance of shape and roundness of sedimentary particles. J. Sed. Petrol., 11, 64-72.
- Krumbein, W.C., Sloss, L.L., (1963). Stratigraphy and sedimentation. Second Edition. W.H. Freeman and Company, San Francisco, 660 pp.
- Vos, K.: A generic prospection strategy for industrial sands with high resistance to compressive stress, unpub. Diss, KU Leuven, 2018
- Wadell, H. (1932). Volume, shape and roundness of rock particles; Journal of Geology. 40, 443-451

Wadell, H. (1935). Volume, shape, and roundness of quartz particles. Journal of Geology., 43, 250-280.

This information has been sourced, reviewed and adapted from materials provided by RETSCH Technology GmbH.

For more information on this source, please visit RETSCH Technology GmbH.