16 Rubber & Plastics News • October 21, 2002 www.rubbernews.com

Technical

Mixture DOE uncovers formulations quickerExecutive summary

This is the second article of a series on design of experiments (DOE). It showshow to uncover “sweet spots” where multiple product specifications can be metin a most desirable way. The first publication provided tools for process break-throughs via two-level factorial designs.1 This follow-up article offers a simplecase study that illustrates how to put rubber or plastics formulations to the testby using powerful statistical methods for mixture design and analysis.

By Mark J. Anderson and Patrick J. Whitcomb

Stat-Ease Inc.

Design of experiments (DOE) tech-niques provide an efficient means foryou to optimize your process. But youshouldn’t restrict your studies only toprocess factors. Adjustments in the for-mulation may prove to be beneficial aswell. A simple, but effective, strategy ofexperimentation involves:

1. finding an ideal formulation viamixture design; and

2. optimizing the process with factori-al design and response surface methods.

This article shows you how to applydesign of experiment methods to yourformulation. Two case studies give you

templates for action.

Why factorial methodsdon’t work well for mixtures

Industrial experimenters typicallyturn to two-level factorials as their firstattempt at DOE. These designs consistof all combinations of each factor at itshigh and low levels. With large numbersof factors, only a fraction of the runsneeds to be completed to produce esti-mates of main effects and simple inter-actions. However, when the response de-pends on proportions of ingredients,such as in chemical or food formula-tions, factorial designs may not makesense. For example, look at what hap-

gular layout, the apexes (point 1, 2 and3) represent pure component blends. Bi-nary blends, which provide estimates ofsecond order effects, occur at the mid-points of the sides on the triangle(points 4, 5 and 6). The centroid (point 7)contains equal amounts of all three in-gredients. The individual proportions gofrom zero to one from base to apex ineach of the three axes. The pattern ofthe points 1 through 7 in the mixture“space” (shown by Fig. 1) forms a text-book design called an augmented sim-plex-centroid.3 The term “simplex” re-lates to the geometry—the simplestfigure with one more vertex than thenumber of dimensions. In this case onlytwo dimensions are needed to graph thethree components on to an equilateraltriangle. A four-component mixture ex-periment requires another dimension insimplex geometry—a tetrahedron (like apyramid, but with three sides, not four).Let’s keep things really simple by only

pens with experiments on lemonade(Table I) that vary the number oflemons vs. cups of sugar water. Stan-dard orders 1 (both factors low) and 4(both factors high) taste the same. Itmakes more sense to look at taste as afunction of the proportion of lemons towater, not the amount. Mixture designaccounts for the dependence of responseon proportionality of ingredients. If youexperiment on formulations where pro-portions matter, not the amount, factori-als won’t work. Use a mixture design.

Case study illustrates asimple mixture design

To illustrate how to apply mixture de-sign, we present a relatively simplestudy that involves three solvents.2 Theexperimenters measured the effects ofthree solvents known to dissolve a par-ticular family of complex organic chemi-cals. They previously had discovered anew compound in this family. It neededto be dissolved for purification purposes,so they needed to find the optimal blendof solvents.

Table II shows the experimental de-

sign in a convenient layout that identi-fies the blends by type. The actual runorder for experiments like this alwaysshould be randomized to counteract anytime-related effects due to aging of ma-terial, etc. Also, we recommend that youalways replicate at least four blends toget a measure of error. In this case, itwould have been helpful to do two eachof the pure materials and also replicatethe three-part blend, called the “cen-troid.”

The geometry of the experimental re-gion can be seen in Fig. 1. In this trian-

TECHNICAL NOTEBOOKEdited by Harold Herzlichh

Table I. Misleading factorial design for lemonade.

Table II. Design matrix and data for solvent study.

Fig. 1. Simplex-centroid mixture design augmented with check blends.

See Mixture, page 18

Fig. 2. Response surface graph of solubility.

18 Rubber & Plastics News • October 21, 2002 www.rubbernews.com

discussing three-component problems.The points in the interior of Fig. 1 be-

tween the centroid and each apex (8, 9and 10) do not come from the simplex-centroid design. The formulators addedthese three-part blends, made up of two-thirds of each respective component andone-sixth each of the other two compo-nents, to provide better coverage of theexperimental region. The interiorpoints, which augment the textbook de-sign to make it more practical, are called“axial check blends.”

Creating a mathematicalmodel

As shown, the solubility response datawere fitted via least squares regressionto a special form of polynomial equationdeveloped by Scheffe.4

Y = 122A + 165B + 178C - 6AB +141AC + 35BC + 799ABC

We call this simply a “mixture model.”Notice that this equation, unlike onesused to graph responses from a process,contains no intercept term, thus ac-counting for the overall constraint thatall mixture components must sum toone. The “Y” (referred to by statisticiansas “Y-hat”) represents the predicted re-sponse. It’s the dependent variable. Theindependent variables (A, B, C), some-times represented mathematically byXs, have been transformed from theiroriginal metric of 0 to 100 percent to acoded format from 0 to 1, thus facilitat-ing interpretation of the resulting coeffi-cients (rounded).

For experiments like this one, whereeach ingredient can be put in at 100 per-cent, the first-order mixture-model coef-ficients predict the response from thepure components. For example, compo-nent A (methyl-ethyl-ketone or MEK) isthe poorest solvent of the three tested—only 122 grams per liter (g/l) of the newsubstance went into solution vs. 165 and178 for components B (toluene) and C(hexane), respectively.

The second order terms, such as AB,reveal interactions. For responses suchas solubility, where higher is better, pos-itive interaction coefficients indicatesynergism. In this case the combinationof A (MEK) and C (hexane) proved to bemost synergistic according to the bigpositive coefficient (141) for AC. Togeth-er these two solvents work better thaneither one alone. On the other hand,negative interaction coefficients showantagonism between ingredients. Forexample, if the coefficient of -6 for theAB interaction was statistically signifi-cant (it’s not), one could conclude thatingredients A and B work against eachother to make the substance less solu-ble.

In this case, by augmenting their de-sign with check blends, the experi-menters made a sufficient number ofunique formulations to allow estimationof a third-order term: ABC. This term,called a “special cubic,” reveals thethree-component interaction, if any. Thecoefficient of 799 does achieve statisticalsignificance, thus providing solid evi-dence that all three solvents work to-gether to be most efficacious. When youwork with chemical formulations, beprepared for complex interactions of thisdegree.

Response surface graphstell the story

The mixture models become the basis

for response surface graphs, which canbe generated from specialized softwarefor mixture DOE.5 Don’t get boggeddown in the mathematics—let the pic-tures tell the story. The graphs providevaluable insights about your formula-tion. Fig. 2 shows a 3-D representationof the solubility response, with 2-D con-tours projected below it, as function ofthe three solvents. As you might expectfrom the data and discussion so far, thepeak solubility is predicted when allthree solvents are mixed together. Acomputer-aided search reveals an opti-mum composition of 27.58 percent MEK,25.56 percent toluene and 46.85 percenthexane producing a predicted solubilityof nearly 208 g/l. Subsequent confirma-tion blends performed within the nor-mal range of this predicted response, sothis mixture experiment proved to besuccessful.

What if you can’t alloweach ingredient to go in at100 percent?

In many cases it will be unreasonableto vary each ingredient over a range of 0to 100 percent. You must impose con-straints on one or more of the ingredi-ents, or on some combination of ingredi-ents. Your constraints may formcomplex regions that cannot be coveredby the standard mixture designs such asthe simplex centroid. However, a num-ber of statistical software packages cangenerate optimal designs that fit what-ever degree of polynomial, such as theone used for the solubility case, that youthink you need to adequately modelyour response. For example, considermaking play putty as a kitchen chem-istry experiment.6 In this formulation, achemical reaction occurs between a poly-mer (polyvinyl acetate in white glue)and a crosslinker (borax). Water partici-pates as a solvent and modifies thephysical properties (rebound and de-formability) of the resulting dilatantmaterial. Obviously the borax should beconstrained to a narrow range of compo-sition (1-3 percent) relative to the glue

(40-59 percent) and water (40-59 per-cent). Table III shows a design of exper-iments to study the behavior of play put-ty as these ingredients vary. It wasconstructed by filling the mixture spacewith blends spaced at relatively even in-tervals (see Fig. 3), with enough of themto fit a special cubic model such as theone used in the previous example. No-tice that some points are labeled withthe number “2.” These represent blendsto be replicated (in random run order)for estimation of pure error.

ConclusionDesign of experiment methods can be

applied to formulations if you accountfor the unique aspects of mixtures. Byusing appropriate designs, you greatlyaccelerate your exploration of alterna-tive blends. Then with the aid of re-sponse surface graphics based on mix-ture models, you will discover thewinning component combination.

References1. Anderson, M.J., Whitcomb, P.J., BreakthroughImprovements with Experiment Design, Rubberand Plastics News, June 16, 1997, pp. 14-15.2. Del Vecchio, R.J., Design of Experiments,Hanser/Gardner Pub., Cincinnati, 1997, pp. 100-101.3. Cornell, J.A., Experiments with Mixtures, 3rd ed.,John Wiley & Sons Inc., New York, 2002, pp. 60-66.4. Scheffe, H., Experiments with Mixtures, Journalof the Royal Statistical Society, B, Vol. 20, 1958, pp.344-360.5. Helseth, T.J., et al, Design-Expert, Version 6Software for Windows, Stat-Ease Inc., Minneapolis.6. Anderson, M.J., Play Putty Problem, 3 pages ofinstruction and background, Stat-Ease Inc., 2002.7. Anderson, M.J., Whitcomb, P.J., DOE Simplified:Practical Tools for Effective Experimentation, Pro-ductivity Inc., Portland, Ore., (2000).

Technical

MixtureContinued from page 16

Table III. Design of experiments for play putty.

Fig. 3. Optimal design for a complex mixture (play putty).

The authorsMark J. Anderson and Patrick J. Whit-

comb are principals of Stat-Ease Inc. Thechemical engineers co-authored “DOESimplified: Practical Tools for EffectiveExperimentation.”7 Anderson and Whit-comb also have collaborated on numerousarticles on design of experiments, manyof which can be seen or ordered asreprints from www.StatEase.com.

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