Formulation Designs
From LabAutopedia
A LabAutopedia invited article
Experimental Strategy: Formulation (Mixture) Designs
Authored by: James N. Cawse, Ph.D.;Cawse and Effect LLC
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Simple Designs
The most common design in combinatorial chemistry is a formulation design. In conventional DOE’s, a formulation or mixture design[1] is straightforward and requires relatively few points, because when the response space is smooth, a small simplex (Figure 1) will allow generation of a complete model. In a combinatorial system, the search is on for new phases with substantial changes in properties over narrow concentration ranges.[2] This requires a far more tightly gridded space with many more runs (Figure 2). Software such as Symyx Library Studio is well optimized for generating 3-component grids of this type. (Figure 3) [Note: Symyx High Productivity Research is now Freeslate.]
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| Figure 1. Simplex mixture design | Figure 2. Grid mixture design (10% steps) | Figure 3. Symyx Library Studio design Image copyright Symyx Corporation; used by permission. |
Constrained Designs
When the ranges of components in a mixture design are constrained, the shape of the experimental space changes from a simplex to a complex polytope (Figure 4). Standard DOE software (e.g. Design-Expert (DX7), JMP) can calculate the shape of the polytope, but selection of appropriate experimental points can be something of an art. JMP locates the vertices, centers of edges, and overall centroid (Figure 4a), a result that gives too much emphasis to the edge of the space. DX7 locates those points plus several others (Figure 4b); it gives a more complete set of points that can then be used as candidates for an optimal design. Finally, it is straightforward to simply select a spacefilling set of points (Figure 4c). These situations are easy to work with in ternary systems but get much more complex in higher dimensions!
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| Figure 4a. Constrained region with JMP points (vertices, centers of edges, overall centroid). | Figure 4b. Constrained region with DX7 points (JMP points + internal check points). | Figure 4c Constrained region with grid points. |
Higher Dimensions
With four components, things become more complicated because a four-component mixture cannot be mapped to a flat surface. Symyx Library Studio, for example, offers four-component gradients (Figures 5a, 5c)that start at either the corners or the edges of a square plate, but neither offers complete coverage of the mixture space (Figures 5b, 5d). Obtaining complete coverage requires pasting an externally generated design (Figure 6a) into Library Studio. Design of such high-dimensional mixture systems can be done using advanced DOE software but the problems of optimally locating points as shown in the 3-component case above become even more significant.
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| Figure 5a. Diagonal Gradient in Symyx Library Studio[3] | Figure 5b. Actual location of points in diagonal gradient. | Figure 5c. Orthogonal Gradient in Symyx Library Studio[4] | Figure 5d. Actual location of points in orthogonal gradient. |
Another concern in higher dimensional mixtures is the preponderance of runs which are on the “edges” of the simplex. When the concentration ranges of the components starts at zero, it is remarkably easy for n-dimensional simplices to have a vanishingly small proportion of runs with all n components present (Figure 6b). Note that only four of the mixtures in this 56-run design are actually quaternary blends.
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| Figure 6a. A complete 4-component simplex in 20% steps. | Figure 6b. The actual mixtures present in the simplex. |
Visualization
Formulation designs are relatively tricky to visualize since they are in the form of simplexes (triangles, tetrahedra…). The triangles do not graph properly on Excel, but they can be graphed in most DOE software such as Design-Expert or JMP. To the best of my knowledge there is no software package that enables proper visualization of a tetrahedral 4-component design, as shown in Figure 6a. Cawse and Effect LLC has designed an Excel spreadsheet that directly visualizes a 3-component design and generates coordinates for properly visualizing a 4-component design if you have 3-D graphing software.
References
- ↑ Cornell, J., Experiments with Mixtures, 3rd Ed., Wiley-Interscience, NY, 2002.
- ↑ Cawse, J.N., Ed., Experimental Design for Combinatorial and High Throughput Materials Development, Wiley-Interscience, NY, 2002.
- ↑ Image copyright Symyx Corporation; used by permission.
- ↑ Image copyright Symyx Corporation; used by permission.
