1
Particle Analysis

Particle size and size distribution are fundamentally important in determining how a powder will behave in its bulk form. Measuring the particle size distribution, describing it in graphical and mathematical form and comparing it with other distributions are therefore important tasks for the particle technologist and are introduced in this chapter. Particle “size” can be described unambiguously by a single number only for a distribution of monosized spheres, but real particles are neither spherical nor monosized. We therefore need to understand how to describe particle shape and what effects this has on measurement and calculation. In many industrial applications it is common to represent an entire distribution by some sort of averaged single number. Calculation methods and choices for this are shown. Finally, we also introduce methods for measurement of the surface area of particle distributions.

1.1 PARTICLE SIZE

Size is the most fundamental of particle properties. We will see throughout this book that the size of a particle affects all of its properties. For example, larger particles usually flow freely whereas smaller particles do not. Larger particles dissolve slowly and smaller ones more quickly, resulting in different pharmaceutical effectiveness. Light is scattered strongly from small particles but much less so from larger ones, resulting in different atmospheric effects and a different appearance of painted surfaces, for example.

The objects we describe as particles can cover a wide range of sizes, as shown in Figure 1.1, from large molecules (of order 0.01 μm or 10−8 m) to bricks (of order 10 cm or 10−1 m). Particles almost always come in large numbers (there are hundreds of millions of salt particles on your dinner table right now!) and as a distribution of sizes. Mono‐sized distributions, containing only one particle size or a very narrow distribution of sizes, are very rare (pollen is an example in nature). Distributions can be wide, often very wide. For example, your container of salt will include not only salt crystals of around 0.3 mm or 300 μm but also broken salt dust particles down to 1 μm in size. In many processes and products the entire distribution is important; those very fine particles can have a big effect on how the whole distribution behaves.

Figure 1.1 Ranges of particle size

What do we mean by particle size? This might seem like a simple question but in general it is not. If the particle is a sphere, the obvious answer is that its size is the same as its diameter. What if the particle is a cube? (Crystalline particles usually have an angular shape and crystals of common salt – sodium chloride – are roughly cubic.) In this case, it might seem logical to choose the side length of the cube to represent its size but as shown in Figure 1.2, there are other choices and the maximum dimension is actually times the side length a.

Another way of looking at the problem of selection of a representative or equivalent diameter is to calculate the size of a sphere which has the same property as the non‐spherical particle we are interested in. Two widely used possibilities are:

Figure 1.2 Sizes of spheres and cubes

the diameter of a sphere of equivalent projected area A

or the diameter of a sphere of equivalent volume V

For the cube considered earlier, Figure 1.3 shows how the projected area may take the value a2 or a2 or a2 according to the orientation of the particle, or, of course, values in between these for other orientations. The equivalent projected area diameter will then depend upon the orientation and in these three cases will take values of and .

Note that the equivalent volume diameter xV, does not depend on the orientation.

The example of a cube is relatively easy to deal with. Real particles seldom have a regular shape but approximation to a spheroid, plate or rod may sometimes be useful.

For irregular particles, the projected image can be used to obtain a number of types of shape factors, as illustrated in Figure 1.4.

Roundness SR, is a measure of how closely the particle outline resembles a circle:

where P is the perimeter length and A is the area. Roundness is defined such that the value for a circle is 1. The roundness of other particles is then less than 1.

The Feret diameter is obtained by taking the distance between two parallel lines on either side of the particle perimeter, as shown. By varying the angle φ it is possible to find the minimum and maximum Feret diameters; a shape factor can be simply obtained from the ratio between them.

Many other measures of shape can be defined.

Figure 1.3 Equivalent areas for a cube

Figure 1.4 Shape and the Feret diameter

1.2 DESCRIPTION OF POPULATIONS OF PARTICLES

A population of particles is described by a particle size distribution. Particle size distributions may be expressed as frequency distributions or cumulative distributions. These are illustrated in Figure 1.5. The two are related mathematically in that the cumulative distribution is the integral of the frequency distribution; i.e. if the cumulative distribution is denoted as F, then the frequency distribution is dF/dx. For simplicity, dF/dx is often written as f(x). The distributions can be by number, surface, mass or volume (where particle density does not vary with size, the mass distribution is the same as the volume distribution). Incorporating this information into the notation, fN(x) is the frequency distribution by number, fS(x) is the frequency distribution by surface, FS is the cumulative distribution by surface and FM is the cumulative distribution by mass. In reality, for many particles these distributions are smooth continuous curves. However, size measurement methods usually divide the size spectrum into size ranges or classes and the size distribution becomes a histogram.

For a given population of particles, the distributions by mass, number and surface can differ dramatically, as can be seen in Figure 1.6.

1.3 CONVERSION BETWEEN DISTRIBUTIONS

Many modern size analysis instruments measure particles individually and therefore produce a number distribution, which is rarely the one which is of most practical use. These instruments include software to convert the measured distribution into more practical distributions by mass, surface, etc.

Relating the size distributions by number fN(x), and by surface fS(x) for a population of particles having the same geometric shape but different size:

Figure 1.5 Typical differential and cumulative frequency distributions

Figure 1.6 Comparison between distributions

If N is the total number of particles in the population, the number of particles in the size range x to x + dx = NfN(x)dx and the surface area of these particles = (x2αS)NfN(x)dx, where αS is the factor relating the linear dimension of the particle to its surface area.

Therefore, the fraction of the total surface area contributed by these particles [fS(x)dx] is:

where S is the total surface area of the population of particles.

For a given population, the total number of particles, N, and the total surface area, S are constant. Also, assuming particle shape is independent of size, i.e. all particles have the same shape, αS is constant, and so

where

Similarly, for the distribution by volume

where

where V is the total volume of the population of particles and αV is the factor relating the linear dimension of the particle to its volume.

And for the distribution by mass

where

assuming particle density ρp is independent of size, i.e. all the particles have the same density.

The constants kS, kV and km may be found by using the fact that

Thus, when we convert between distributions it is necessary to make assumptions about the constancy of shape and density with size. If these assumptions are not valid, the conversions are likely to be in error. For example, this approach would not be valid in the case where the population consists of whole spheres and broken pieces of spheres, because shape will then vary with size. Also, calculation errors are introduced into the conversions. For example, imagine that we used an electron microscope to produce a number distribution of size with a measurement error of ±2%. Converting the number distribution to a mass distribution we triple the error involved (i.e. the error becomes ±6%). For these reasons, conversions between distributions are to be avoided...