Paint & Coatings Industry - March 2008 - (Page 56)

deSigns of the Times Slacking Off! Edito s ote: his r icle Editor’s Note: This article is Part Editor’s Note: This article is Part 14 in a series; see PCI, art in series ; see PCI, series ries PCI, March 2007, March 2007 for the previ s installment. 0 evio tallment March 2007, for the previous installment. previous nstallmen n previous articles1,2 factorial designs were described as being especially well suited to studying continuous variables, like pressure, temperature, time and speed, and categorical variables, like chemical functionality type, lot number and position. Mixture designs were described as being especially suited to studying variables that add to 1 or 100%, like blends, formulations, reactions, oligomers and polymers. I have a colleague who has a Ph.D. in applied statistics and champions experimental design wherever he can, but has never used mixture statistics. When a mixture design is called for, he refers his clients to me. Before he knew me (and even now when he thinks he can get away with it), whenever he had a mixture to study, he used a slack-variable, factorial design. This article discusses the differences between a slack-variable, factorial design and a mixture design. I foun that each f the ingredients sh u be limited found that each of the ingredients should be limited und un ac he ngredien s should red ents ho houl m ted found to the range between 0.3 and 0.4 in order t achieve to the range between 0.3 and 0.4 in order to achieve h an e etwe n .3 nd 4 rde chie useab formulat ns. This can be describ by using useable formulations This can be described by using ble bl ormulatio lt scribe useable formulations. h described the following notation: 0.30 ≤ A ≤ 0.40 0.30 ≤ B ≤ 0.40 0.30 ≤ C ≤ 0.40 A Factorial Design The formulation to be studied contains three ingredients. Through initial evaluations the researcher has Table 1 | Studying a mixture with a three-factor, factorial design. Experiment Factorial Design Standard Order A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 –1 +1 –1 +1 –1 +1 –1 +1 0 +1.4 –1.4 0 0 0 0 Factorial Design With Levels C A 0.30 0.40 0.30 0.40 0.30 0.40 0.30 0.40 0.34 0.42 0.28 0.33 0.33 0.33 0.33 Factorial Design With Levels Normalized to a Total of One C 0.30 0.30 0.30 0.30 0.40 0.40 0.40 0.40 0.33 0.33 0.33 0.33 0.33 0.42 0.28 B –1 –1 +1 +1 –1 –1 +1 +1 0 0 0 +1.4 –1.4 0 0 B 0.30 0.30 0.40 0.40 0.30 0.30 0.40 0.40 0.33 0.33 0.33 0.42 0.28 0.33 0.33 A 0.34 0.40 0.30 0.36 0.30 0.36 0.28 0.34 0.34 0.39 0.30 0.30 0.35 0.31 0.35 B 0.33 0.30 0.40 0.36 0.30 0.28 0.36 0.33 0.33 0.31 0.35 0.39 0.30 0.30 0.35 C 0.33 0.30 0.30 0.28 0.40 0.36 0.36 0.33 0.33 0.30 0.35 0.31 0.35 0.39 0.30 –1 –1 –1 –1 +1 +1 +1 +1 0 0 0 0 0 +1.4 –1.4 If a standard factorial design was to be used, a low level, coded as -1, would be set for each variable at 0.30 and the high level, coded as +1, at 0.40. A three-factor, factorial design of this type is given in standard order in Table 1. A center point experiment is included, and experiments for estimating the quadratic coefficients are also included. The first columns for A, B and C are shown in the coded form; the second columns for A, B and C are shown with the actual levels of ingredients; and the third columns for A, B and C are shown with normalized levels of ingredients. The levels have to be normalized because several of the sums of ingredients in several of the experiments do not total to 1. The sum of all ingredients has to total 1. Experiments 1, 11, 13 and 15 have ingredients that total less than 1. Experiments 4, 6, 7, 8, 10, 12 and 14 have ingredients that total more than 1. Now new problems arise. After the experiments were normalized to 1, the first problem is that several experiments have normalized levels of ingredients below the desired 0.3. Another problem is that Experiment 1 with the –1, –1, –1 coding has the same normalized levels as Experiment 8, with the +1, +1, +1 coding and Experiment 9, with the 0, 0, 0 coding. This prevents doing any meaningful mathematical analysis. A Slack-Variable, Factorial Design One way out of this is to use only two of the ingredients in a two-factor, factorial design. The third factor is called the slack variable. In the example, A and B are the factorial variables and C is the slack variable. First, one sets the levels of A and B and then sets C to whatever is needed to get A + B + C = 1. This is laid out in Table 2. One issue is that the C variable falls outside of the desired experimental range. This is an artifact of the By Richard R. Roesler, Senior Principal Scientist | Bayer MaterialScience LLC, Pittsburgh, PA 56 MARCH 2008 | W W W . P C I M A G . C O M http://WWW.PCIMAG.COM

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Viewpoint Industry News Calendar of Events Company News Mergers and Acquisitions Names in the News Marketplace Quarterly Stock Watch Reaching Strict VOC Requirements With Outstanding Durability for Industrial Maintenance and Marine Coatings A Unique Fungicide for Wood-Filled Plastic deSigns of the Times: Slacking Off! From Rocks to Nanos Shear-Free Dispersion Process Fluorinated Oxetane Oligomers as Versatile Intermediates for Polymer Modification The Effects of TPO Composition on Adhesion and Proposed Chemical Mechanism Supplier Showcase Color Management Materials Watch Products Classifieds Advertiser Index

Paint & Coatings Industry - March 2008

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