Cubic models for three-component mixtures

Besides the terms of the additive model, the quadratic model of Eq. (7.12) contains cross terms that describe interactions between two components, and for this reason it is usually able to reproduce the response values at the vertices and on the sides of the concentration triangle, since they represent pure and binary mixtures, respectively. It should not surprise us, however, that non-additive effects involving the simultaneous presence of three components will be important to describe 

The full cubic model for a mixture of three components is given by the equation 

Introducing the identity 1 =xi +X2 +X3 and making the appropriate substitutions, with a little handiwork we can arrive at the expression 

Since this equation has 10 terms, we will have to perform at least 10 distinct runs to determine the values of all the coefficients. Often, however, adding a single cubic term is sufficient to transform the model into a satisfactory description of the experimental region. Eliminating the terms in Eq. (7.18), we arrive at the expression for the special cubic model, which contains only one more term than the quadratic model, and therefore requires only one more run  

The experimental design normally used to determine the values of the coefficients of the special cubic model is called the simplex centroid, which we obtain by simply adding a center point, corresponding to a 1 1 1 ternary mixture, (xi,X2,X3) = (5,5,5) to the simplex lattice design. The coefficient of the cubic term is given by 

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The special cubic model for four-component mixtures has 14 terms, and its coefficients can be estimated from the design shown in Fig. 7.10b. The points on each face now reproduce the arrangement corresponding to the simplex centroid design, which we used to fit the special cubic model for three-component mixtures. 

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