• Insights

Published on January 27, 2021 by Jenny Maat

Many common ingredients in products made by a variety of industries actually consist of suspension of microparticles. Whether it’s milk powder in food, pigments in paint, or titanium dioxide in sunscreen, these types of particles are not simple spheres. In fact, they come in all shapes and sizes. These characteristics play a large role in the rheological properties of the products they are used in. For instance, think about how sunscreen spreads as it’s applied to the skin. To make an accurate virtual equivalent of real products, the particle shape and size (and the variation therein) needs to be taken into account.

In RheoCube, the particle shape model can recreate all the intricate details of particle shape and roughness. RheoCube can create a wide range of virtual particle shapes using a set of simple parameters. This is done by following a few basic steps:

Step 1. A basic shape is chosen, one that best represents the ingredient particles. In RheoCube, you can choose between a sphere, cube or tetrahedron. 

Step 2. The base shape can be stretched or squeezed (or even both) to match the desired overall particle shape to the ingredient of choice. This is done by altering two aspect ratios of the particle: the Elongation Index (a.k.a. stretching – the ratio between the middle and largest dimensions), and the Flatness Index (a.k.a. squeezing – the ratio between the smallest and middle dimensions) Examples of these transformations for a spherical base shape are shown below. 

Combining the three different base shapes with these shape transformations already creates a wealth of possibilities. This functionality is ideal for new particle design, or getting insight into the effect of particle shape change on a product’s rheological properties. For existing ingredients, these ratios can be obtained from microscopic analysis of the particles. Both the average ratios and the variation in these shape parameters can be used directly as input in RheoCube. 

Step 3. The amount of detail can be fine-tuned even further. This can be done by adding surface roughness to the particle by using spherical harmonics [1]. Generally speaking, spherical harmonics are a collection of spherical functions. An example of such functions are the ones describing the electron orbitals around atoms. They can be used to mathematically describe any particle shape in the same manner as sine and/or cosine functions can be used to mathematically describe any measured function of real values. This last procedure is done with Fourier analysis: using a linear combination of sine and/or cosine functions (or any other standard functions) you can describe any type of real “signal” or measurement of a real variable. The same can be done in three dimensions by using combinations of spherical harmonics.

Another way of fine-tuning the particle shape is by smoothing out the edges. Both actions can also be combined to create a more accurate representation of the ingredient particles. In this way, a close match to the real system can be obtained. To demonstrate this, we made virtual equivalents of several common particulate ingredients across a variety of industries. 

Spray dried skim milk powder as imaged by means of electron microscopy was recreated using a spherical base shape with moderate roughness added. The actual particle is of the order of ~ 15 𝝁m in diameter.

Take an electron microscopy image of titanium dioxide, often used as a food additive and in personal care products (E171) from reference [3]. The particles have a mean size in the order of 200-300 nm in diameter. RheoCube can easily create virtual equivalents of the cubic particles, including the rounded edges and the natural variation in particle length. The RheoCube images show a collection of cubic particles with characteristics matching the titanium dioxide, as well as a close up of a particle with an exact cubic shape and one slightly more stretched taken from the generated collection.

As a final example, the maximum roughness settings are explored in the recreation of dry soft wheat flour particles as imaged in [4] with scanning electron microscopy. The particle highlighted is roughly 50 𝝁m long. The example created with RheoCube is made using a spherical base shape which is slightly stretched and squeezed. On top of that, roughness is added using the maximum roughness settings possible in RheoCube. In this way, very irregular shapes can be obtained.

The particle model can be used not only to create virtual equivalents of existing particles, but also to explore new ideas for particle shapes. With the step-by-step process described above, new forms of particulate matter can be virtually synthesized and tested quickly and simply.