To discuss the principles of the mixing process, a situation will be considered where there are equal quantities of two powdered components of the same size, shape and density that are required to be mixed, the only difference between them being their colour. This situation will not, of course, occur practically but it will serve to simplify the discussion of the mixing process and allow some important considerations to be illustrated with the help of statistical analysis.

The Unmixed, Perfect mix And Random Mix In Mixing Process

If the components are represented by coloured cubes, then a two-dimensional representation of the initial unmixed or completely segregated state can be shown as Fig. 1a.

From the definition of mixing, the ideal situation or perfect mix in this case would be produced when each particle lies adjacent to a particle of the other component (i.e. each particle lies as closely as possible in contact with a particle of the other component). This is shown in Fig. 1b, where it can be seen that the components are as evenly distributed as possible. If this mix was viewed in three dimensions, then behind and in front of each coloured particle would be a white particle and vice versa. Powder mixing, however, is a ‘chance’ process, and while the situation shown in Fig. 1b could arise, the odds against it are so great that for practical purposes it can be considered impossible. For example, if there are only 200 particles present, the chance of a perfect mix occurring is approximately 1 in 10⁶⁰ and is similar to the chance of the situation in Fig. 1a occurring after prolonged mixing. In practice, the best type of mix likely to be obtained will have the components under consideration distributed as indicated in Fig. 1c. This is referred to as a random mix, which can be defined as a mix where the probability of selecting a particular type of particle is the same at all positions in the mix and is equal to the proportion of such particles in the total mix.

If any two adjacent particles are selected from the random mix shown:

• the chance of picking two coloured particles = 1 in 4 (25%)
• the chance of picking two white particles = 1 in 4 (25%)
• and the chance of picking one of each = 2 in 4 (50%).

If any two adjacent particles are selected from the perfect mix shown in Fig. 1b, there will always be one coloured particle and one white particle.

Thus if the samples taken from a random mix contain only two particles, then in 25% of cases the sample will contain no white particles and in 25% of cases it will contain no coloured particles. It may help in this and subsequent discussions to imagine the coloured particles as being the active drug and the white particles the inert excipient.

It can be seen that, in practice, the components will not be perfectly evenly distributed, i.e. there will not be full mixing. But if an overall view is taken, the components can be described as being mixed as in the total sample (Fig. 1c) the amount of each component is approximately similar (48.8% coloured and 51.2% white). If, however, Fig. 1c is considered as 16 different blocks of 25 particles, then it can be seen that the number of coloured particles in the blocks ranges from 6 to 19 (24% to 76% of the total number of particles in each block). Careful examination of Fig. 1c shows that as the number of particles in the sample increases, then the closer will be the proportion of each component to that which would occur with a perfect mix. This is a very important consideration in powder mixing, and is discussed in more detail in the following sections.

Reference:

• Aulton, M. E., & Taylor, K. (2018). Aulton’s Pharmaceutics: The Design and Manufacture of Medicines, 5th ed. Elsevier.