Computer - Aided Tools for Optimal Mixture Design
Powerful desktop computer tools now make it easy to optimize paint formulations. Aided by the computer, statistically based design of experiments (DOE) — a proven method for making breakthrough improvements in cost and performance — can be applied.1 The latest versions of dedicated DOE software2 exhibit more versatility than ever to create optimal designs that handle any combination of mixture components, processing factors (such as time or temperature) and categorical variables (such as supplier and material type). These computer programs easily manipulate almost any number of responses in powerful optimization routines that reveal “sweet spots” — the operating windows that meet all specifications at minimal cost.
This article reviews the basic principles of mixture design, and applies state-of-the-art tools for optimal design to the formulation of a coating.
Laying the Groundwork — A Simple Mixture ProblemLet’s begin with a simple mixture design that allows three components to vary from 0–100%. Other ingredients may be present in the final formulation, but by holding these at fixed proportions, the experimenter can set them aside. The resulting experimental region forms a simplex, a geometric figure with one more vertex than the number of dimensions. In two dimensions, a simplex is a triangle. In three dimensions, a simplex is a tetrahedron, and so on.
Figure 1 shows the experimental region, or “mixture space,” for the three unconstrained components. The three corners, or vertices, of the resulting triangle (a simplex) represent the maximum allowed for a specific component. For example, at the top of the triangle, Component A reaches its upper limit of 100%, with B and C at 0%. At any point in the triangular region, the ingredients total to 100%. This is called the “total constraint.”
The points shown on Figure 1 represent a reasonable candidate set of points for a mixture design. It includes the pure mixtures of A, B and C shown at the vertices, plus the three binary blends (A-B, A-C, and B-C), which are represented by the mid-points at the edges of the triangle. These points surround the three-part blend (one-third each of A, B and C) at the “centroid” of the triangle. The other six points in Figure 1, called “check blends,” fill the remaining gaps in the mixture space. They form a small triangle within the bigger triangle. If time and materials run short, the check blends can be skipped.
A simplex will always result from designs where all components can vary from 0–100%. This geometry also occurs in special cases where components are constrained to equal ranges, such as those given for the following case study.
Adding Categorical Factors – A Case StudyFormulators of an automotive clearcoat explored the impact of three components (monomer, crosslinker and resin) on two key responses: hardness and solids. Based on knowledge of the chemistry, they restricted the component levels as shown in Table 1.
Note the categorical choices for supplier of the monomer and crosslinker. Suppliers must not be mixed. For example, monomer M1 cannot be blended with M2. The same rule applies to the three possible crosslinker suppliers (X1, X2 and X3). The formulators decided not to vary supplier of the resin.
The constraints shown in Table 1 form a simplex, so the pattern of points shown in Figure 1 could be used for the experimental design. However, in this case, each formula must be re-applied as many as six times to cover all combinations of the two monomers and three crosslinkers. Multiplying these six categorical combinations times the 13 points for the proposed mixture design produces a total of 78 blends, which far exceeds the limitations of time and material. A smaller number of blends can provide enough information to get a decent fit of the data.
Optimal Design Tools Pare Down the Number of Required BlendsThe invention of computers made it feasible to apply matrix-based algorithms that select an ideal subset of experiments. The two most popular algorithmic methods are called “a-optimal” and “d-optimal.” These are just two of the so-called “alphabetic optimality criteria” developed by statisticians.3 The general procedure for optimal design is:
1 Specify the order of polynomial: first, second, third or beyond, that may be needed to model the response.
2 Generate a “candidate set” with more than enough points to fit the specified model.
3 From the candidate set, select the minimum number of points needed to fit the model.
First-order polynomials will only model linear behavior. A second order polynomial, called a “quadratic,” will reveal two-component interactions. More complex interactions can be modeled by third-order polynomials. Design of experiments can be likened to design of bridges: when in doubt, build it stout. In other words, if you’re not sure what to do, invest in a higher-order polynomial. The added blends needed to fit the bigger models may reap great benefits through discovery of unexpected interactions.
The actual selection procedure involves computer-intensive matrix calculations designed to produce model coefficients of maximum possible precision. Years ago, an experimenter who wanted to apply optimal design would have to describe the problem to a statistician, who then would go to a mainframe computer and, perhaps days later, come back with the experimental plan. Now, anyone equipped with a personal computer and the proper software can generate an optimal design. Let’s see how this was done for the automotive coatings case.
The coatings formulators faced a difficult modeling task — combining mixture design with categorical factors. The resulting first order (categorical) model contains four terms: one to estimate the overall average effect, one for the effect of changing from one monomer supplier to the other, and two for the effect of changing from one crosslinker to either of the two alternatives.
For the mixture side of the problem, the formulators expected some complex chemistry, so they chose a third order mixture model called a “special cubic”.
Given these design criteria, the software2 generated a candidate set comprised of the six possible combinations of the two sets of suppliers for monomer and crosslinker, crossed with the 13 blends shown on Figure 1, for a total of 78 points. It then used d-optimal criteria to select the 28 points needed for the desired model. To add statistical power, the software replicated the five points with highest leverage, a statistical measure of relative influence. It also added five additional points to fill in some of the gaps and test for lack of fit. The resulting 38-point design, and associated response values for hardness and solids, is shown in Table 2. The blends are listed by an identification scheme that puts replicates in sequential order. However, the actual run order was randomized to ensure against biases caused by lurking factors such as material degradation, machine wear and drifting ambient conditions. The order of experiments must be randomized, or invalidation of the entire design may occur.
Response Surface Graphics Tell the StoryUsing least-squares regression, the DOE software fitted the response data to the 28-term equation for the coatings system. Two statistically significant models were produced — one for each response. These models then became the basis for response surface graphics, which greatly simplify the interpretation of results. Figure 2 shows the effect of the categorical factors on hardness.
The M2 monomer, labeled D+ with triangular symbols, came out on top of M1 (labeled D-). The crosslinker supplier, shown on the X-axis, produced a much bigger effect on hardness. The customer required a level of 10 or higher for hardness. This could be achieved only by the X1 supplier of crosslinker.
Figure 3 shows the predicted hardness results as a function of the component levels for monomer M2, crosslinker X1 and the resin.
The contour labeled 10 represents a crucial boundary. To the right of this contour, hardness falls below specification. The most crucial variable is amount of resin. As resin concentration approaches the high level (70%) at the lower right, the predicted hardness falls well below the target. The maximum hardness occurs somewhere along the opposite edge where resin amount is at its minimum level. The flag, which displays a predicted hardness value, stems from one of the actual points in the d-optimal design at concentration of monomer, crosslinker and resin of 12.5%, 32.5% and 55%, respectively.
Figure 4 shows a similar plot for the second response, solids, which must exceed 50%. The monomer and crosslinker remain fixed at M2 and X1, respectively, which were chosen to achieve maximum hardness.
The area where solids exceed 50% extends from the centroid down to the corner of maximum resin (70%). This region did not exhibit outstanding hardness. Obviously a tradeoff must be made between maximizing hardness or maximizing solids. Ideally, there will be an overlap of the in-specification areas on Figures 3 and 4.
Multiple Response Optimization Finds the “Sweet Spot”By shading out areas outside of specification (hardness less than 10 and solids less than 50), the software created the graph shown in Figure 5. It reveals a sweet spot for formulation of the automotive clearcoat.
The flag is arbitrarily set near the middle of the operating window. It provides a reasonable choice for a good formulation: 11.28% of M2 monomer, 28.40% of X1 crosslinker, and the remainder (60.32%) as resin. The software offers more sophisticated numerical search algorithms that will find the most desirable mixture within the operating window. Cost can be taken into account before making the final decision. For example, in this case, if the M2 monomer is relatively expensive, an alternate solution can be found by switching to the M1 supplier.
It is important to note that any recommended formulations must be verified by way of confirmation runs. Remember that the recommended optimum is only a prediction based on a very select (but optimal) set of sample blends.
Benefits of DOEThe case study on the automotive clearcoat shows how advanced tools of DOE can be applied to optimize a formulation. The optimal design approach, so handy in this case for dealing with categorical factors, more commonly finds application for formulations that don’t fit a neat simplex space. It also helps in situations where process factors are desired. Now you can mix your cake and bake it too! Yet another type of optimal design incorporates the amount of the formulation. For example, component levels and the thickness of a coating can be varied simultaneously. In all these cases, specialized DOE software can provide optimal designs that explore critical variables in coatings systems. The ultimate benefit comes from discovery of operating windows that satisfy all customer specifications most economically. If you accomplish this, you and your company will gain a competitive advantage and generate big profits.
For more information on design-of-experiment software, contact Mark J. Anderson, Stat-Ease Inc., 2021 East Hennepin Ave, #191, Minneapolis, MN 55413; phone 612/378.9449, ext. 13; fax 612/378.2152; e-mail: Mark@StatEase.com.