DoE: Mixture designs

When it comes to working with mixtures, such as detergent formulations, beverages, copolymers, processed foods and so on, Mixture Designs are a great tool for finding out the optimal composition. The mathematics of Mixture Designs were developed mainly by Scheffé and Cox, but current users of Mixture DoE won't have to care about the theory behind these models. With the existing DoE software, running a Mixture Design will just require deciding on the components of our mixture, knowing the basic preconditions of these designs, and deciding on the type of fitting model we want to apply to our system (eg, a liner, quadratic or cubic fit).


The precondition in a Mixture Design is that the proportions of all components must add up to 1 (100%). Mathematically, the design region for a mixture of k components is a simplex, a regular polymer of dimension k-1, and k vertices. For instances, with a two component system, the simplex is the line segment from (1,0) to (0,1). For three components, the simplex is a triangle with vertices (1,0,0), (0,1,0) and (0,0,1). For four components, we would obtain a regular tetrahedron, and so on.

The figure shows a mixture design with three components (A,B,C). The vertices in the simplex region represent a 100% composition of each individual components. The mid points on each side correspond to 50% mixtures on the components of each connecting vertex, and the central point represents a mixture with equal amounts of each of them.

The figure below shows some of the most common mixture designs for a three component system. Black dots indicate the experimental points used in the model, and white dots are experiments carried out to check the proper fit. Notice that when we are uncertain about the behaviour of our system, moving from a linear to a quadratic or cubic model simply represents adding more experimental points to our design, so that we can start from the simplest, least time consuming model, and scale-up as needed.

 

In many occasions, the researcher does not want to explore the full experimental simplex region covering all possible combinations. By setting restrictions, it is possible to focus on a small portion of the experimental domain. The figure below shows the experimental sub-region for a 3 component mixture in which some restrictions have been applied.

After completing the experiments, the results of our mixture design can be easily visualized with the existing DoE software in the form of level curves in a 2D plot, or as response surfaces in 3D plots. The figures below are examples o 2D and 3D outputs that can be obtained with commercial software such as Design Expert. The example shows the viscosity response of a solution containing three different surfactants.