Adhesion of an epoxy primer on steel.

1. Runs used in total

21 runs: 17 in the screening design and 4 in the confirmation. None failed.

2. Suggested optimum

About 132 N/cm² pull-off adhesion. The two confirmation runs at these settings gave 134.2 and 131.5 N/cm².

3. Recommended settings

X02, X04, X05 and X06 have no effect we could measure, so you can set them to whatever is cheapest.

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

6. Summary

I started with a definitive screening design, because 17 runs cover all eight factors. It finds main effects without mixing them up with two-factor interactions, and it can also show curvature. A follow-up optimization design was unlikely to be needed, because the best settings would probably sit at the edges of the ranges.

Two factors matter most. More silane (X01) and a rougher surface (X03) both raise adhesion, and they work together: each one helps most when the other is high (interaction +7 N/cm²). Thinner films (X07) also help, by about 4 N/cm² per coded unit. Cure temperature (X08) had a small effect, and the other four factors had none. With five terms, the model explained 98 % of the variation (adjusted R²). It predicted about 131 N/cm² at the best corner.

The confirmation runs tested that corner at both cure temperatures, with two repeats each. They gave 132.9 N/cm² on average at 20 °C and 131.8 N/cm² at 60 °C. The difference is within the noise, so cure temperature does not matter here, and 20 °C saves energy. The result matches the prediction, so I stopped there.

1. Runs used in total

21 runs: 17 screening runs and 4 confirmation runs. No runs failed.

2. Suggested optimum

About 132 N/cm² pull-off adhesion. The two confirmation runs at these settings gave 132.3 and 134.0 N/cm².

3. Recommended settings

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

6. Summary

I started with a definitive screening design because it tests all 8 factors in 17 runs. It also separates main effects from two-factor interactions and can show curvature. If only a few factors matter, it often gives the whole answer in one step.

Three factors matter: silane (+12 N/cm² per coded unit), surface profile (+6.6) and film thickness (−5.9). Silane and surface profile also interact (+7.5), so silane works best on a rough surface. With these four terms the model fits well (adjusted R² 0.95, residual SD 3.6 N/cm²). The epoxy:amine index, degrease time, dilution, flash-off time and cure temperature showed no effect, so you can set them for cost or speed.

The model put the optimum at a corner: high silane, rough profile, thin film. A weak, non-significant curvature term suggested the best film thickness might be a little above 40 µm. I tested this with two repeat runs at 40 µm and two at 50 µm. The 40 µm runs averaged 133.2 N/cm² and the 50 µm runs 129.0 N/cm². This confirms the prediction and the thin film. The optimum lies at the edge of the allowed ranges, so more silane or a rougher profile might raise adhesion further if you can widen those ranges.

1. Runs used in total

21 runs: 17 screening runs and 4 confirmation runs. No runs failed.

2. Suggested optimum

About 131 N/cm² pull-off adhesion. The model predicts 131.7, and the four confirmation runs averaged 130.6.

3. Recommended settings

The factors marked “no effect” can go wherever is cheapest or most practical.

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

6. Summary

I started with a definitive screening design because it handles 8 factors in 17 runs. It estimates every main effect without bias from two-factor interactions, and it also shows curvature. Few factors were expected to matter, so a single screening design could be enough.

Three factors drive adhesion. More silane helps most (+11.9 N/cm² per coded unit), a rougher surface helps (+6.6), and a thicker film hurts (−5.7). Silane and roughness also work together (+7.2): silane does much more on a rough surface. With these four terms the model explains the data with a residual error of about 1.7 N/cm². That left no sign of curvature or other effects. Epoxy:amine index, degrease time, dilution, flash-off time and cure temperature showed no effect inside their ranges.

The best point lies at a corner of the region, so a response-surface design was not needed. Four repeat runs at the recommended settings gave 129.4 to 131.8 N/cm², which agrees with the prediction.

1. Runs used in total

21 runs. None failed.

2. Suggested optimum

About 132 N/cm² pull-off adhesion. The four confirmation runs gave 130.3 to 132.4 N/cm², with a mean of 131.7.

3. Recommended settings

X02, X04, X05, X06 and X08 had no clear effect. You can set them to whatever is cheapest or easiest.

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

6. Summary

Adhesion depends on three of the eight factors: silane, surface profile and film thickness. More silane and a rougher surface raise adhesion. A thinner film raises it too. Silane and roughness work best together: more silane helps much more on a rough surface than on a smooth one.

I started with a definitive screening design, built from a conference matrix of order 8. With 17 runs it estimates all eight main effects, and these estimates are not biased by two-factor interactions or curvature. This makes it the smallest design that can screen eight factors and still find curvature. The main effects (in coded units) were silane +8.5, surface profile +6.5 and film thickness -5.1. All the others were under 2.

The main-effect model still left a large residual error of 11.6 N/cm². Effect heredity pointed to an interaction among the three active factors. Adding the silane × surface profile term (+9.2) cut the residual to 3.2 N/cm². No other interaction or squared term helped. Since the model is linear plus one interaction, the best point is at a corner: silane high, rough surface, thin film.

The model predicted about 128 N/cm² there. Four replicate confirmation runs gave a mean of 131.7 N/cm² with little scatter, which supports the model. The best screening run had reached 127.8 N/cm², so I needed no further optimization design.

1. Runs used in total

22 runs. None failed.

2. Suggested optimum

About 131 N/cm² pull-off adhesion. The model predicts 130; the four confirmation runs averaged 132.

3. Recommended settings

X02, X04, X05, X06 and X08 had no effect we could measure, so choose them for cost or processing.

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

6. Summary

I started with a definitive screening design: 17 runs plus one extra centre point. With eight factors, it estimates every main effect free of two-factor interactions, and it tests for curvature, all in one step. If only a few factors matter, the same data also fit a model with their interactions, so no second stage may be needed.

Three factors matter. More silane raises adhesion by about 10 N/cm² per coded unit. A rougher surface raises it by about 7. A thicker film lowers it by about 5. Silane and surface profile also work together (interaction +7): silane helps most on a rough surface. The other five factors had no clear effect. We found no curvature, so the best point sits at a corner of the region. The model with these four terms explains 95 % of the variation, with a residual noise of about 3 N/cm².

To check the model, I ran four repeats at the best corner: 2 % silane, Ra 2.5 µm, 40 µm film, and the other factors at their midpoints. They gave 131 to 134 N/cm², in line with the prediction of 130. More adhesion would need settings outside the allowed ranges, such as a rougher profile or more silane.

1. Runs used in total

23 runs. None failed.

2. Suggested optimum

About 131 N/cm² pull-off adhesion. The model predicts 131.5, and three confirmation runs gave a mean of 130.8.

3. Recommended settings

• X01 Silane adhesion promoter: 2 % (+1)
• X02 Epoxy:amine index: 1.0 (0)
• X03 Surface profile (Ra): 2.5 µm (+1)
• X04 Degrease time: 5.5 min (0)
• X05 Primer dilution: 7.5 % (0)
• X06 Flash-off time: 10 min (0)
• X07 Dry film thickness: 40 µm (-1)
• X08 Cure temperature: 40 °C (0)

X02, X04, X05, X06 and X08 had no clear effect. You can set them to whatever is cheapest or easiest.

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

6. Summary

I started with a definitive screening design (DSD): 17 runs for 8 factors. A DSD estimates every main effect without bias from two-factor interactions or curvature, so one design could screen the factors and point to the optimum.

Three factors matter. Silane (X01, +10.4) and surface profile (X03, +6.7) raise adhesion, and film thickness (X07, −5.9) lowers it. The silane × profile interaction is large (+8.7): silane helps most on a rough surface. A model with these four terms cut the residual error from 11 to 4 N/cm². The other five factors showed nothing above the noise.

The model puts the optimum at high silane, rough profile and thin film. Design 2 tested this with three repeat runs each at 40 µm and 60 µm film thickness. The thin film gave 130.8 N/cm² and the medium film gave 126.6. Both values agree with the model, so I stopped there.

1. Runs used in total

25 runs. None failed.

2. Suggested optimum

About 132 N/cm² pull-off adhesion after seven days. Four confirmation runs averaged 131.7 N/cm², against 98 N/cm² at the centre of the ranges.

3. Recommended settings

X02, X04, X05, X06 and X08 had no effect we could measure. Set them for cost or speed instead.

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

6. Summary

I started with a definitive screening design, built from a 10 × 10 conference matrix with two columns dropped. That gives 21 runs for 8 factors. The two spare columns give it more power against the noise. A DSD keeps main effects free of two-factor interactions and also shows curvature. That let me screen and model in one step, so I might not need a second design.

Only three factors mattered. More silane (+9 N/cm² per coded unit) and a rougher surface (+7) raised adhesion. A thicker film lowered it (−7). Silane and surface profile also interact strongly (+7): silane helps far more on a rough surface. A model with these terms explained 96 % of the variation. No other factor or curvature was significant.

The model put the optimum at a corner: most silane, roughest surface, thinnest film. I ran four repeats there to confirm it. They averaged 131.7 N/cm², within the model’s prediction. The optimum sits on the edge of the allowed ranges, so a further gain would need more silane, a rougher profile or a thinner film than the ranges allow.

1. Runs used in total

27 runs. None failed.

2. Suggested optimum

About 130 N/cm² pull-off adhesion. The model predicts 131 N/cm². Four confirmation runs averaged 129.5 N/cm². The start point, the center of all ranges, gave about 101 N/cm².

3. Recommended settings

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

6. Summary

Three of the eight factors control adhesion: silane, surface profile and film thickness. More silane and a rougher surface each raise adhesion. Silane and roughness also work together: each helps much more when the other is high, and this joint effect adds about 8 N/cm². A thinner film raises adhesion as well. The other five factors had no effect we could measure, so set them for cost or ease of work.

I started with a definitive screening design because it screens all eight factors in few runs. It keeps main effects free of two-factor interactions and also shows curvature. Three center points measured the noise, which was about 3 N/cm².

The screen found the three factors and the interaction clearly, with no real curvature. A model with these terms fit well (adjusted R² 0.97), so no further optimization design was needed. The best point is at the corner of the range, so the four confirmation runs went there. They averaged 129.5 N/cm², close to the 131 N/cm² the model predicted.

1. Runs used in total

29 runs, none failed.

2. Suggested optimum

About 131 N/cm² pull-off adhesion. The four confirmation runs averaged 131.4 N/cm².

3. Recommended settings

• X01 Silane adhesion promoter: 2 % (+1)
• X02 Epoxy:amine index: 1.0 (0)
• X03 Surface profile (Ra): 2.5 µm (+1)
• X04 Degrease time: 5.5 min (0)
• X05 Primer dilution: 0 % (-1)
• X06 Flash-off time: 5 min (-1)
• X07 Dry film thickness: 40 µm (-1)
• X08 Cure temperature: 40 °C (0)

4. Designs in order

5. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-DSD.csv, design-2-DSD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

Three factors drive adhesion: silane level, surface profile and film thickness. Silane and surface profile also work together. Each helps more when the other is high, so the best primer has both at maximum. A thinner film gives better adhesion. The other five factors had no clear effect inside their ranges.

I started with a 12-run Plackett-Burman design to screen all eight factors in as few runs as possible. It picked out silane and surface profile. Degrease time and the remaining factors showed smaller effects that were hard to tell apart from noise.

Next I ran a 13-run definitive screening design on silane, epoxy:amine index, surface profile, degrease time and cure temperature, with the other three factors fixed at low. A DSD can find curvature and keeps main effects clear of two-factor interactions. Fitting a model to both designs together gave a strong silane × surface profile interaction (t ≈ 10). That interaction also explains the apparent degrease-time effect in the first design, since the two are confounded there. Film thickness had a real negative effect, about −5.5 N/cm² per coded unit. I found no curvature in the epoxy:amine index or cure temperature, so the model points to a corner of the range rather than a peak in the middle.

Four confirmation runs at the recommended settings gave 129 to 133 N/cm². The model predicted about 136 N/cm², which is near enough given a run-to-run noise of about 3 N/cm². The factors that showed no effect can be set wherever is cheapest or quickest. The values above are only suggestions.

1. Runs used in total

38 runs. None failed.

2. Suggested optimum

About 129 N/cm² pull-off adhesion (model prediction). The four confirmation runs averaged 131 N/cm².

3. Recommended settings

• X01 Silane adhesion promoter: 2 % (+1)
• X02 Epoxy:amine index: 1.0 (0)
• X03 Surface profile (Ra): 2.5 µm (+1)
• X04 Degrease time: 5.5 min (0)
• X05 Primer dilution: 7.5 % (0)
• X06 Flash-off time: 10 min (0)
• X07 Dry film thickness: 40 µm (-1)
• X08 Cure temperature: 40 °C (0)

X02, X04, X05, X06 and X08 had no effect we could detect. You can set them to whatever is cheapest or most practical.

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

Three of the eight factors control adhesion: silane, surface profile and film thickness. The best primer has the most silane and the roughest surface, applied as a thin film.

I started with a definitive screening design, which tests all eight factors in 17 runs. It can pick out the strong main effects and also shows curvature. Silane and surface profile raised adhesion, and a thicker film lowered it. The other five factors showed nothing.

Next I ran a face-centred central composite design on those three factors, holding the other five at their centre settings. Together with the screening data, it fits a full quadratic model, and the residual noise was about 2.4 N/cm². The model added one big finding: silane and surface profile work together (interaction +8.3). Silane helps most on a rough surface. Thickness interacted only slightly with surface profile, and none of the curvature terms were real. So the best point sits in a corner of the region.

Four repeat runs at that corner gave 127 to 134 N/cm², with a mean of 131. That agrees with the predicted 129, so the model holds.

1. Runs used in total

20 runs (17 + 3). No run failed.

2. Suggested optimum

About 129 N/cm². The model predicts 131. The three confirmation runs gave 130.6, 127.8 and 126.5 (mean 128.3).

3. Recommended settings

X02, X04, X05, X06 and X08 had no effect. Set them wherever is cheapest.

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

6. Summary

Eight factors and no prior knowledge called for a screening design. I chose a 17-run definitive screening design. It estimates all main effects and also shows curvature and interactions, so one design could both screen and model.

Three factors mattered: silane (+11 per coded unit), surface profile (+6) and film thickness (-7). Silane and surface profile also interact (+9): the promoter helps most on a rough surface. Squared terms were not significant, so the response is close to linear and the best point is a corner of the region. The other five factors had effects below the noise (about 2 N/cm²).

I then ran three repeats at the predicted best corner. They averaged 128.3, close to the 131 predicted, so the model holds and no further runs were needed.

1. Runs used in total

21 runs sent to the lab (17 + 4). None failed.

2. Suggested optimum

About 132 N/cm² pull-off adhesion. The model predicted 131.7 and the four confirmation runs averaged 131.6.

3. Recommended settings

X02, X04, X05, X06 and X08 did not matter, so set them to whatever is cheapest or easiest inside their ranges.

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

6. Summary

I used a 17-run definitive screening design (DSD) for the eight factors. It screens main effects and still shows curvature and some interactions, so one step could both screen and point to an optimum. A one-step fractional design would have confused the interactions with the main effects.

The DSD showed three factors that matter: more silane (X01), a rougher surface (X03) and a thinner film (X07) all raise adhesion. Silane and roughness also work together: each helps more when the other is high (X01×X03). Noise was about 4 N/cm² after fitting. The other five factors showed no effect. Cure temperature (X08) may interact with film thickness, but the evidence was weak.

The fitted model put the best corner at X01 +1, X03 +1, X07 -1. I tested it with four runs, two of them repeats. They matched the prediction. Cure temperature made no clear difference, so I recommend the mid-point.

The study is small. The optimum sits at a corner of the region, so higher silane or a thinner film may do even better, but those settings are outside the allowed ranges.

1. Runs used in total

27 runs. No run failed.

2. Suggested optimum

About 130 N/cm² (standard error about 1.2). The three confirmation runs at the optimum measured 131, 126 and 133.

3. Recommended settings

X02, X04, X05, X06 and X08 had no clear effect. Set them where it is cheapest or easiest.

4. Designs in order

5. Run sheets

• design-1-DSD.csv, design-1-DSD-results.csv
• design-2-confirmation.csv, design-2-confirmation-results.csv

6. Summary

I used a definitive screening design (DSD) first, because eight factors are many and I did not know if any effect curves. Twenty-one runs gave main effects, curvature and some interaction information. I built it from a 10-column conference matrix and dropped two columns.

The DSD showed three active factors: silane (up), surface profile (up) and film thickness (down). One interaction stood out: silane with surface profile. Silane helps more on a rougher surface. With it in the model, the residual noise fell from about 8 to about 3 N/cm². The other five factors, and all squared terms, were not clearly different from noise.

The best corner of the three active factors is at the edge of the allowed range, so I used six confirmation runs: three at the predicted optimum and three at neighbouring corners. All gave the predicted pattern. The optimum is about 130 N/cm², and the other corners were lower (98 to 123).

The model is adhesion = 99.4 + 9.4·X01 + 6.7·X03 - 6.3·X07 + 8.3·X01·X03, fitted to all 27 runs. A weak X07 by X08 interaction (cure temperature) was hinted at but I did not chase it.

1. Runs used in total

28 runs sent to the lab. None failed.

2. Suggested optimum

About 130 N/cm². The two confirmation runs gave 129.8 and 131.3.

3. Recommended settings

X02, X04, X05, X06 and X08 had no clear effect, so any value in range works. I left them at the centre.

4. Designs in order

5. Run sheets

• design-1-fractional-factorial.csv, design-1-fractional-factorial-results.csv
• design-2-full-factorial.csv, design-2-full-factorial-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

I used a screen-then-model-then-confirm procedure. Design 1 was a 16-run resolution IV fraction with two centre runs. It was the cheapest way to look at all 8 factors at once. Three factors stood out: silane (+10 per coded unit), surface profile (+6.6) and film thickness (-4.9). The other five had effects of 1.5 or less. The centre runs sat close to the factorial mean, so curvature looked small.

The fraction also showed one large two-factor signal, aliased with three other pairs. By effect heredity I read it as silane × surface profile, since both parents are active.

Design 2 tested that reading. It was a full 2^3 on the three active factors, with the rest at centre. The fit was adhesion = 99.2 + 9.0·X01 + 6.7·X03 − 6.1·X07 + 9.4·X01·X03. The other terms were near zero. Silane and a rough surface help most when used together. Thin films help.

Design 3 ran two repeats at the predicted corner (X01 +1, X03 +1, X07 −1). They gave 129.8 and 131.3, in line with the predicted 130.4. The optimum sits at the edge of the allowed ranges, so a higher value may exist outside them. I could not test that.

1. Runs used in total

29 runs. None failed.

2. Suggested optimum

About 131 N/cm². The three confirmation runs gave 130.3, 132.9 and 130.9 (mean 131.3).

3. Recommended settings

X02, X04, X05, X06 and X08 have no effect, so any setting in range works. They sit at the centre.

4. Designs in order

5. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-PB-foldover.csv, design-2-PB-foldover-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

I first screened all 8 factors with a 12-run Plackett-Burman design and two centre points. Only X01 and X03 stood out clearly. The rest were close to the noise (about 3 N/cm²), so I could not tell them apart from it.

I then ran the fold-over. With the 24 factorial runs combined, main effects are free of two-factor aliasing. Only X01 (+10.3), X03 (+6.5) and X07 (-6.5) were real. The five others were under 1.

The model fitted the data well: adhesion = 100 + 10.3·X01 + 6.5·X03 + 8.3·X01·X03 − 6.5·X07. The X01·X03 interaction is large: silane works best on a rough surface. The centre points showed no curvature. Every effect is linear, so the best point is a corner: X01 high, X03 high, X07 low.

The confirmation runs at that corner matched the prediction of 131.6 N/cm², so I did not run a response surface design.

1. Runs used in total

32 runs sent to the lab (12 + 17 + 3). None failed.

2. Suggested optimum

About 129 N/cm² pull-off adhesion. The model predicted about 131, and three confirmation runs gave 131.1, 129.6 and 126.8 (mean 129.1).

3. Recommended settings

X04 to X08 were set by the sign of their small screening effects. They were not tested again on their own.

4. Designs in order

5. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

I screened all 8 factors with a 12-run Plackett-Burman design, the smallest sheet that covers 8 factors. Silane (X01) had the largest effect, then surface profile (X03) and the epoxy:amine index (X02). The other five effects were small, and the noise was large (residual SD about 8.6 on 3 degrees of freedom), so I kept only the top three.

Next I ran a 17-run face-centred CCD on X01, X02 and X03, with the other five fixed at their favourable sign. It gave a much lower noise level (SD about 1.7). X02 had no effect. Silane and surface profile both help, and they work together: the silane gain is large only on the rough surface (interaction of about 8.6). Curvature was negligible, so the best point is the corner at high silane and high roughness.

Three runs at that corner confirmed the prediction. Because the CCD fixed X04 to X08 at one setting, I can say nothing firm about them beyond the weak screening hints.

1. Runs used in total

34 runs, none failed (14 + 16 + 4).

2. Suggested optimum

About 133 N/cm². Four confirmation runs at these settings averaged 134.3 N/cm².

3. Recommended settings

The last five factors do not matter. Pick whatever is cheapest inside the range.

4. Designs in order

5. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-fractional-factorial.csv, design-2-fractional-factorial-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

I screened all eight factors with a 12-run Plackett-Burman design plus two centre points. Silane (X01) raised adhesion, and surface profile (X03) probably did too. Film thickness (X07) looked harmful. The centre points matched the average of the corners, so there was no sign of curvature.

To pin this down I ran a 16-run fractional factorial on the six leading factors and held the other two at centre. It confirmed X01, X03 and X07, and showed that X02, X04 and X06 have no effect. The pooled model of 30 runs is: adhesion = 100.8 + 10.3·X01 + 6.4·X03 - 7.4·X07 + 7.8·X01·X03, with noise near 2.5 N/cm². Silane and a rough surface work better together.

Every effect is monotone, so the best point is a corner: high silane, high roughness, thin film. Four runs there gave 130 to 138 N/cm², in line with the model’s 133. I did not test X05 and X08 beyond the screening design; their effects there were small and not consistent, so I left them at centre.

1. Runs used in total

34 runs. None failed.

2. Suggested optimum

About 130 N/cm². The three confirmation runs gave 124.9, 132.9 and 131.1 (mean 129.6).

3. Recommended settings

X02, X04, X06 had no clear effect, so mid-range is fine. X05 and X08 were set low because their screening effects were slightly negative, but they were never tested separately at that corner.

4. Designs in order

5. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-fractional-factorial.csv, design-2-fractional-factorial-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

I started with a 12-run Plackett-Burman screen of all eight factors. It is the cheapest way to test eight main effects. Noise was large (about 11 N/cm²), but the silane promoter (X01) and surface profile (X03) stood out.

Design 2 was a 16-run 2^(6-2) fractional factorial plus 3 centre points. It covered the six most promising factors (X01, X02, X03, X04, X06, X07), with X05 and X08 held low. The centre points sat at the factorial mean, so there is no sign of curvature. Pooling both designs showed that X01, X03 and a lower film thickness (X07) help, and that X01 and X03 interact strongly: the promoter works much better on a rough surface. The other factors did nothing measurable.

Because the model is linear plus interactions and the optimum sits at a corner, I skipped a response surface design. Three replicates at the predicted best corner (design 3) matched the prediction of about 132, so I stopped.

1. Runs used in total

34 runs (12 + 18 + 4). None failed.

2. Suggested optimum

About 132 N/cm² pull-off adhesion. The model predicted 132.2, and four confirmation runs averaged 132.7 (range 129 to 138).

3. Recommended settings

4. Designs in order

5. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-fractional-factorial.csv, design-2-fractional-factorial-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

I used three steps: screen all eight factors, study the likely ones in more detail, then confirm. The aim was to reach the optimum with few runs.

The 12-run Plackett-Burman design covered all eight factors. Silane (X01) and surface profile (X03) stood out, and X02, X05 and X07 were borderline. Noise was large (about 10 N/cm²), so I did not drop the borderline factors yet.

The second design studied those five factors in 16 runs plus 2 center runs. A half fraction of five factors can separate main effects from two-factor interactions. It showed that X01, X03 and X07 matter, and that X01 and X03 work together: adhesion jumps to about 125 only when both are high. X02 and X05 had no effect. The two center runs agreed closely (102 and 104), and the center sat only a little above the corner average, so I kept a linear model with the X01 by X03 interaction.

The model has silane high, roughness high and thin film. Both X01 and X03 are at their limits, so I stopped there. Four runs at that point matched the prediction, which confirmed the optimum. X04, X06 and X08 showed only small effects in the screen, within the noise, and I left them at the center. The study only tested their effects in the screening design.

1. Runs used in total

35 runs sent to the lab. None failed.

2. Suggested optimum

About 130 N/cm². The three confirmation runs gave 130.9, 133.1 and 124.7 (mean 129.5).

3. Recommended settings

Only X01, X03 and X07 matter. The other settings are my choice, not a result: X04, X05, X06 and X08 showed no clear effect, and X02 showed none.

4. Designs in order

5. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-factorial.csv, design-2-factorial-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

I first had to sort 8 factors into important and unimportant ones. A 12-run Plackett-Burman design does that with few runs, and two center points gave a first look at noise and curvature. X01 (silane), X03 (surface profile) and X07 (film thickness) stood out, with X02 close behind. The noise was too large to be sure of any of them.

To settle it, I ran a full 2^4 factorial on X01, X02, X03 and X07, with the weak factors held at their favourable ends. This design can show interactions. It confirmed X01 (+9.0), X03 (+6.9) and X07 (-6.2), and found a strong X01×X03 interaction (+7.8): silane helps most on a rough surface. X02 had no effect. The center points matched the factorial mean (100.1 vs 100.3), so there is no curvature, and a response-surface design was not needed.

The best corner is silane high, rough surface, thin film. A model gives about 130 N/cm² there. Three confirmation runs at that point averaged 129.5, so the model holds, including the settings I had fixed for X04, X05, X06 and X08.

1. Runs used in total

31 runs, none failed: 15 in the screen, 11 in the follow-up, 5 in confirmation.

2. Suggested optimum

132 N/cm² pull-off adhesion after seven days, measured directly at the recommended settings (five replicate runs, mean 132.0, standard deviation 2.6).

3. Recommended settings

Silane and surface profile drive adhesion and set the recommendation. The other six factors showed no reliable effect; each is set to its most favourable direction from the screen, at no extra cost in runs.

4. Designs in order

5. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

The first design screened all eight factors in 12 runs, using a Plackett-Burman design: it assumes interactions are small next to main effects, a fair bet this early. Three centre points checked for curvature. Two factors stood out: the silane adhesion promoter and the surface profile, both with positive effects and both far more significant than the rest. The centre points showed no curvature, so a straight line fit the data well.

The second design zoomed in on those two factors with a face-centred central composite design, holding the other six at their favourable settings from the screen. This design can catch curvature and interactions, not just straight lines. Both main effects came back strong, joined by a clear positive interaction: pushing both factors to their high settings together gained more than pushing either one alone. Still no curvature, so the response kept climbing to the edge of the allowed range. The optimum sits at a corner of the design box, both factors high.

The last step sent five replicate runs at the recommended settings, to get a trustworthy estimate of the adhesion and confirm the model. The measured mean, 132.0 N/cm², landed within half a unit of the model’s prediction of 132.1.

1. Runs used in total

39 runs, all measured. None failed.

2. Suggested optimum

129 N/cm² pull-off adhesion after seven days.

3. Recommended settings

4. Designs in order

5. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-fractional-factorial.csv, design-2-fractional-factorial-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

Testing all eight factors at full resolution would take too many runs, so the study went in three steps: screen, confirm, then verify.

Step one screened all eight factors in 12 Plackett-Burman runs plus three centre points. Three factors stood out: the silane adhesion promoter, the surface profile, and the dry film thickness. Cure temperature and flash-off time showed smaller effects, and epoxy:amine index, degrease time and primer dilution were weaker still. The centre points showed almost no curvature, so the response behaves close to a flat plane over the tested region, which rules out the need for a response-surface design later.

Step two checked this with a resolution V half-fraction (16 runs) on the five factors that looked active, plus three more centre points. This resolution keeps main effects and two-factor interactions free of each other, which the screening design could not do. The three weakest factors were fixed at the setting step one had favoured rather than tested again. Only the silane promoter, the surface profile, and the dry film thickness mattered, and the silane promoter and surface profile reinforced each other through a strong positive interaction. Cure temperature and flash-off time, which had looked mildly important in screening, turned out to be noise. Curvature stayed negligible, so the optimum sits at a boundary rather than an interior peak.

With two positive main effects reinforcing each other and no curvature, the optimum is a corner of the tested region: silane promoter high, surface profile high, dry film thickness low, with the other five factors fixed as found in step one. The model from step two predicted about 128 N/cm² there. Step three ran five replicate confirmation runs at these settings, which averaged 129 N/cm², matching the prediction and giving a direct read on measurement noise at the recommended point.

1. Runs used in total

42 runs, all measured. None failed.

2. Suggested optimum

131 N/cm² pull-off adhesion after seven days, averaged across five confirmation runs at the recommended settings (individual results: 131.9, 130.3, 128.2, 128.1, 131.2).

3. Recommended settings

Only silane, surface profile, and dry film thickness move adhesion. The other four factors showed no effect, so they’re set at the middle of their range.

4. Designs in order

5. Run sheets

• design-1-PB.csv / design-1-PB-results.csv
• design-2-PB-foldover.csv / design-2-PB-foldover-results.csv
• design-3-full-factorial.csv / design-3-full-factorial-results.csv
• design-4-confirmation.csv / design-4-confirmation-results.csv

6. Summary

The study ran in three steps: screen all eight factors, test the ones that mattered properly, then confirm the result.

A 12-run Plackett-Burman design with two centre points checked all eight factors at once. Silane and surface profile came out significant. Degrease time, dilution, and cure temperature showed small trends that didn’t clear the noise. Plackett-Burman designs mix main effects with interactions, so a 12-run foldover (every sign flipped) came next. Across the combined 24 runs, only silane, surface profile, and dry film thickness held up. The early trends in degrease time and cure temperature turned out to be noise.

A full 2³ factorial plus three centre points (11 runs) then tested these three factors properly, interactions included. It found no curvature, so the response is flat enough that the best setting sits at a corner of the range rather than somewhere in the middle. It also found a strong interaction: silane and surface profile boost each other, so raising both together beats raising either one alone. A thinner film still helped on its own, but the silane-roughness pairing drives most of the gain.

Five confirmation runs at the recommended corner (high silane, rough surface, thin film, other factors at mid-range) landed tightly around 130 N/cm², matching the factorial model’s prediction.

1. Runs used in total

42 runs: 17 in the screening design, 20 in the optimization design, and 5 confirmation runs. None failed.

2. Suggested optimum

About 130 N/cm² pull-off adhesion after seven days, measured at the recommended settings (mean of 5 confirmation runs: 130.3 N/cm², range 125.7 to 135.2).

3. Recommended settings

Four factors (silane, surface profile, dry film thickness, cure temperature) drive adhesion and sit at the edge of their allowed range. The other four have no measurable effect, so hold them at a convenient mid-point.

4. Designs in order

5. Run sheets

• design-1-DSD.csv / design-1-DSD-results.csv
• design-2-CCD.csv / design-2-CCD-results.csv
• design-3-confirmation.csv / design-3-confirmation-results.csv

6. Summary

We started with a 17-run definitive screening design across all eight factors. It needed only 2k+1 runs, yet still let us estimate each factor’s own curvature, so we could tell early whether the optimum sits inside the range or at an edge. The main effects showed four factors with a clear, large effect: silane promoter and surface profile raised adhesion, dry film thickness and cure temperature lowered it. The other four (epoxy:amine index, degrease time, primer dilution, flash-off time) had effects too small to separate from noise. We dropped them from further testing and fixed them at a mid-point setting.

For the second stage we built a 20-run response surface design on the four active factors: a half-fraction factorial, face-centered axial points, and four center points. The half-fraction confounds some two-factor interactions with each other, but it cut the run count by eight against a full design, and we cared more about the four main factors than about telling interactions apart. The fitted model explained 97% of the variation in adhesion. None of the quadratic terms were significant, so the response keeps rising or falling in a straight line across the tested range, with no peak inside it. That pointed the optimum to a corner of the design: high silane, high surface profile, low dry film thickness, low cure temperature.

Five confirmation trials at that corner averaged 130.3 N/cm², close to the model’s prediction of 133.2 N/cm² and well above anything seen in the first two designs. Because the response surface is flat rather than curved, the true optimum may sit past the edge of the tested range for these four factors. A follow-up study that widens the range for silane, surface profile, dry film thickness, and cure temperature could push adhesion higher still.

1. Runs used in total

42 runs, none failed: 19 in the screening design, 18 in the optimization design, 5 in the confirmation run.

2. Suggested optimum

131 N/cm² pull-off adhesion after seven days, at the settings below.

3. Recommended settings

4. Designs in order

5. Run sheets

• design-1-fractional-factorial.csv / design-1-fractional-factorial-results.csv
• design-2-CCD.csv / design-2-CCD-results.csv
• design-3-confirmation.csv / design-3-confirmation-results.csv

6. Summary

With 8 factors, testing every combination would take too many runs, so we started with a resolution IV fractional factorial. This design still keeps every main effect clear of two-factor interactions, and three center points let us check for curvature at no extra cost.

Three factors came out significant: silane adhesion promoter (X01), surface profile (X03), and dry film thickness (X07). The other five, including epoxy:amine index and cure temperature, had no measurable effect. The center points showed no curvature, but a screening design this flat can still hide interactions or a bend in the response, so we moved to a response surface design before trusting that reading.

For the second design, we built a face-centered CCD on the three active factors and held the rest at their mid-level. This let us fit curvature and interactions directly, and both turned out to matter: silane and surface profile interact, each one boosts adhesion more when the other is also high, and dry film thickness has a real optimum partway between its low and high setting, not at either edge. The fitted model pointed to the best combination: silane and surface profile both at their high end, dry film thickness pulled down from the mid-point, for a predicted 131 N/cm².

Five repeats at those settings averaged 130 N/cm², close enough to the prediction and in line with the noise seen throughout the study. The settings in Section 3 stand.

1. Runs used in total

43 runs, none failed: 19 for screening, 19 for optimization, 5 for confirmation.

2. Suggested optimum

The model predicts 137 N/cm² at the recommended settings. Five confirmation runs there measured 133 N/cm² on average (std 3.8), close to the prediction.

3. Recommended settings

X02, X04, X05, X06 and X08 had no measurable effect on adhesion, so their settings can move freely within range without hurting the result. We’ve set them to their mid-point.

4. Designs in order

5. Run sheets

• design-1-fractional-factorial.csv / design-1-fractional-factorial-results.csv
• design-2-CCD.csv / design-2-CCD-results.csv
• design-3-confirmation.csv / design-3-confirmation-results.csv

6. Summary

We started with a resolution IV fractional factorial (16 runs) plus three center points, to screen all eight factors at once in few runs. Three factors stood out above the noise: silane adhesion promoter and surface profile both raised adhesion, and dry film thickness lowered it. The other five factors stayed inside the noise band set by the center points, so we dropped them. The screening data also showed a strong interaction: silane only helps when the surface profile is rough, and does little on a smooth one.

That interaction meant we couldn’t just push each good factor to its best level and call it done. We needed the shape of the response surface, so we ran a face-centered central composite design on the three active factors, holding the other five at mid-point. A quadratic model fit the data well (R² = 0.97) and placed the optimum at the edge of the allowed range: full silane, the roughest surface profile, and the thinnest film.

Five confirmation runs at those settings averaged 133 N/cm², close to the model’s prediction of 137 N/cm². That agreement supports both the model and the recommended settings above.

1. Runs used in total

47 runs, all measured. None failed.

2. Suggested optimum

We expect pull-off adhesion of about 130 N/cm² after seven days at the recommended settings. That is up from about 99 N/cm² at the centre of the original ranges, the baseline our first design measured.

3. Recommended settings

4. Designs in order

5. Run sheets

• design-1-PB.csv and design-1-PB-results.csv
• design-2-BBD.csv and design-2-BBD-results.csv
• design-3-confirmation.csv and design-3-confirmation-results.csv

6. Summary

We started with a Plackett-Burman screen of all eight factors, plus three centre points to check for curvature. Every factor showed an effect bigger than the noise in the centre points, except the epoxy:amine index, which was small. The centre points also sat above the average of the cube runs, a sign of curvature, so a simple linear model would not be enough to find the optimum.

Running full response-surface designs on all eight factors would have cost far more runs than the study needed. Four factors stood out with the largest effects and one representative from each factor group: the silane level and surface profile (the two biggest effects), flash-off time, and cure temperature. We fixed the other four factors at the level the screen favoured and built a Box-Behnken design around the remaining four. This design estimates curvature and two-factor interactions with a moderate number of runs.

The Box-Behnken results showed that silane level and surface profile both help adhesion on their own, and reinforce each other: raising both together works better than raising either alone. Flash-off time and cure temperature barely matter on their own, but they interact: a long flash-off pairs best with a low cure temperature. None of the four factors showed real curvature inside this design, so the fitted model points to a corner of the range rather than a point in the middle.

Five confirmation runs at the recommended settings measured adhesion between 128 and 132 N/cm², averaging about 130 N/cm². That is close to the 137 N/cm² the model predicted and well above the roughly 99 N/cm² baseline from the first design, so we recommend these settings.

1. Runs used in total

49 runs, none failed.

2. Suggested optimum

About 129 N/cm² pull-off adhesion after seven days, measured directly at the recommended settings (four confirmation runs: 129.7, 124.7, 130.0, 130.2).

3. Recommended settings

4. Designs in order

5. Run sheets

• design-1-PB.csv, design-1-PB-results.csv
• design-2-CCD.csv, design-2-CCD-results.csv
• design-3-confirmation.csv, design-3-confirmation-results.csv

6. Summary

We began with a 12-run Plackett-Burman design plus two centre points, to screen all eight factors at once. Five stood out: the silane promoter and surface profile helped adhesion, while the epoxy:amine index, primer dilution and dry film thickness hurt it. Degrease time, flash-off time and cure temperature had effects too small to trust, so we set those aside.

Next we built a face-centred central composite design around the five active factors, large enough to fit curves and interactions, not just straight lines. We held the three set-aside factors at the levels the screening favoured. This design showed that silane and surface profile reinforce each other: together they explain most of the gain in adhesion. Primer dilution helped at both ends of its range rather than in the middle, so we pushed it to its highest level. Dry film thickness kept the negative effect seen in the screening stage, with no curve or interaction attached.

We used the fitted model to search the whole allowed range for the best combination, then ran it four times to confirm. Measured adhesion landed close to the model’s prediction, with run-to-run spread typical of the lab’s noise. The resulting settings raise adhesion to about 129 N/cm², up from the roughly 95 to 100 N/cm² seen at the centre of the original ranges.

1. Runs used in total

49 runs sent to the lab. All 49 were measured; none failed.

2. Suggested optimum

131 N/cm² pull-off adhesion after seven days, at the recommended settings below. Five confirmation runs at these settings measured 127.3, 129.1, 130.1, 134.8 and 134.9 N/cm², for a mean of 131.2 N/cm², close to the model’s prediction.

3. Recommended settings

X02, X04, X05, X06 and X08 did not move adhesion enough to matter. Their settings above are a reasonable, low-risk choice, not ones the data demand.

4. Designs in order

5. Run sheets

• design-1-PB.csv, results in design-1-PB-results.csv
• design-2-CCD.csv, results in design-2-CCD-results.csv
• design-3-confirmation.csv, results in design-3-confirmation-results.csv

6. Summary

The study ran in three stages: screen all eight factors, build a response surface around the ones that mattered, then confirm the optimum the model pointed to.

The first stage used a 12-run Plackett-Burman design, the standard choice for screening eight factors in as few runs as possible. Three centre points were added to check for curvature. Silane (X01) and surface profile (X03) had large positive effects, dry film thickness (X07) had a large negative effect, and primer dilution (X05) and flash-off time (X06) showed smaller effects worth a closer look. Epoxy:amine index (X02), degrease time (X04) and cure temperature (X08) looked unimportant, so the next stage fixed each at its better level and spent the remaining runs on the factors that mattered.

The second stage built a face-centred central composite design around the five candidate factors, with sixteen factorial runs (a resolution-V half-fraction, so no two-factor interaction is hidden by another), ten axial points and three centre points. This design supports a full quadratic model, which is what finding an optimum, rather than just a direction, requires. The fit confirmed X01, X03 and X07 as the real drivers, with a strong positive interaction between X01 and X03: silane and surface profile reinforce each other, so pushing both up at once gains more than pushing either one alone. X05 and X06 turned out not to matter after all, and none of the curvature terms were significant, which means the response keeps climbing or falling across the tested range rather than peaking inside it. The optimum therefore sits at a corner of the design: silane and surface profile at their high settings, dry film thickness at its low setting.

The third stage ran five confirmation trials at that corner. The measured mean, 131.2 N/cm², sat close to the model’s prediction of 131.4 N/cm², which supports both the model and the recommended settings.

1. Runs used in total

62 runs, all measured. None failed.

2. Suggested optimum

About 131 N/cm², with noise of a few N/cm² between repeats at this point.

3. Recommended settings

Epoxy:amine index, degrease time, primer dilution, flash-off time and cure temperature did not move adhesion. Pick values inside their ranges for whatever suits production; the levels above are the ranges’ midpoints.

4. Designs in order

5. Run sheets

• design-1-fractional-factorial.csv / design-1-fractional-factorial-results.csv
• design-2-D-optimal.csv / design-2-D-optimal-results.csv
• design-3-confirmation.csv / design-3-confirmation-results.csv

6. Summary

The study ran in three steps: screen all eight factors, model the ones that mattered, then confirm the predicted optimum.

Design 1 was a resolution IV fractional factorial, 16 runs plus three centre points. This keeps main effects clear of two-factor interactions, which is what a screening step needs. Silane adhesion promoter, surface profile, and dry film thickness stood out, with primer dilution and cure temperature as smaller effects. Degrease time and flash-off time did nothing. The centre points showed no curvature.

Design 2 followed up on the six factors that looked active, with the other two fixed at their mid-range values. A full factorial on six factors would have cost 64 runs, and resolution IV designs alias interactions, so a response surface step still needed to check for curvature and interactions. A D-optimal design covers both at lower cost: 34 points chosen by coordinate exchange to fit a full quadratic model, plus four centre points, for 38 runs. The fit kept silane adhesion promoter, surface profile, and dry film thickness as the only factors that mattered, found a strong interaction between silane and surface profile, and again found no curvature. Epoxy:amine index, primer dilution, and cure temperature dropped out once the model included that interaction, so design 1 had picked up their effects as noise or confounding, not a real signal. With no curvature and the model down to straight lines and one cross term, the best setting sits at a corner of the allowed ranges, not somewhere in the middle.

Design 3 checked that corner directly: five repeats with silane adhesion promoter and surface profile at their high end, dry film thickness at its low end, and the rest at mid-range. The five measurements averaged 131 N/cm², matching the model’s prediction.

1. Runs used in total

37 runs. None failed.

2. Suggested optimum

About 131 N/cm², from seven measurements at the recommended settings (range 127 to 134). The starting centre of the ranges gives about 102, so this is a gain of roughly 30 %.

3. Recommended settings

Only X01, X03 and X07 matter. The other five can be set wherever production prefers.

4. Designs in order

5. Run sheets

6. Summary

Eight factors, a noisy response and no idea which factors matter: that calls for screening first. A 16-run resolution IV fraction covers all eight and keeps main effects clear of two-factor interactions. Three centre points came along to check for curvature.

The screen was clear. Three factors moved the response: silane promoter (+21), surface profile (+13) and film thickness (-12). The other five stayed inside the noise. One two-factor interaction showed up at +14, but a resolution IV design cannot say which one it is: X01X03, X02X07, X04X06 and X05X08 all share the same column.

Eight runs settled it. A full 2^3 in silane, profile and film thickness, with the other five held at mid-level, estimates every interaction among the three cleanly. Silane x profile came back at +18; the other two interactions were near zero. Since X02, X04, X05, X06 and X08 sat at mid-level in these runs, their interactions could not contribute, so the alias chain was broken.

That leaves a simple picture. More silane helps, and it helps much more on a rough surface. Thin films beat thick ones. The best corner is silane high, profile high, film thin.

Design 3 asked whether the real optimum sits inside the ranges rather than at that corner. Pulling each of the three factors back to mid-level dropped the response by 10 to 20 N/cm² each time, so the surface rises all the way to the edge and there is nothing to find in the middle. No response surface design was needed. One run in the same set put all five inert factors at their low level and measured the same value as the corner, which confirmed they are inert at the optimum too.

Four confirmation runs at the recommended settings averaged 130 N/cm², in line with the earlier measurements there.

1. Runs used in total

39 runs. None failed.

2. Suggested optimum

About 133 N/cm² pull-off adhesion. Four confirmation runs at the recommended settings gave 132.2, 133.9, 130.8 and 128.8 N/cm², a mean of 131.4. The centre of the design sits near 101, so this is a gain of about 30 %.

3. Recommended settings

Only X01, X03 and X07 matter. The other five can be set wherever cost or line speed suits you.

4. Designs in order

5. Run sheets

6. Summary

Eight factors and no prior knowledge call for a screen first, so design 1 was a 16-run resolution IV fraction with three centre points. Resolution IV keeps main effects clear of two-factor interactions, which is what you need when you do not yet know which factors are live.

The centre points put the noise at about 3 N/cm². Against that, three main effects stood out: silane promoter (+11.9 per coded unit), surface profile (+6.5) and film thickness (-6.2). The other five factors were all under 1.3, well inside noise. One interaction chain also came through at 6.2. Its four members are X01xX03, X02xX07, X04xX06 and X05xX08; only X01xX03 joins two active factors, so that is the one. The centre points matched the factorial mean almost exactly, so nothing suggested curvature yet.

Design 2 tested that reading and looked for a peak inside the range. A face-centred central composite on the three live factors, with everything else held at mid-level, costs 16 runs and estimates all squared terms. The three main effects and the X01xX03 interaction came back strongly; every squared term was inside noise. So the response is a tilted plane, not a dome, and the best point has to lie on a corner.

The model points at high silane, rough profile and thin film. Silane and profile help each other: rough steel gives the promoter more surface to bond to, which is why the interaction is positive and why pushing both to their limits pays more than either alone. Thin films adhere better here, most likely because a thick layer builds more shrinkage stress at the interface.

Design 3 confirmed it. Four replicates at that corner averaged 131.4, within noise of the predicted 135. If you want more adhesion than this, the ranges themselves are the limit: more silane, a rougher blast profile and a thinner film all sit outside the window you gave.

1. Runs used in total

41 runs. None failed.

2. Suggested optimum

About 131 N/cm². That is the mean of three confirmation runs at the recommended settings (129.2, 131.4, 133.4). The starting level in the middle of all ranges was about 102 N/cm², so adhesion goes up by roughly 30 per cent.

3. Recommended settings

Three factors drive the result: silane, surface profile and film thickness. X02, X05 and X08 move adhesion by about 1 to 2 N/cm², which is inside the noise, so set them where they help and do not pay for them. X04 and X06 did nothing, so pick whatever suits production.

4. Designs in order

5. Run sheets

6. Summary

Eight factors and no idea which ones matter, so the first job was screening. I used a 16-run fractional factorial of resolution IV, which keeps every main effect clear of any two-factor interaction. A 12-run Plackett-Burman would not have done that. Four centre points came along to measure the noise and to check for curvature.

The screening gave a clean answer. Silane (+10.4), surface profile (+7.2) and film thickness (-6.5) stood well clear of the repeat noise, which had a standard deviation of 4.6 N/cm². Everything else sat under 2. One interaction chain also stood out at about 9, and by effect heredity it belongs to silane times surface profile: both parents are active, the other candidates in the chain are not. Silane and a rough surface work together, and neither is worth much without the other. The centre points matched the average of the corners, so there was no curvature to chase, and the best point had to sit on a corner: silane high, profile high, film thin.

The second design went to that corner and asked two questions. Do any of the five quiet factors wake up there? An 8-run fractional factorial in them said no. None moved adhesion more than about 2 N/cm², and X04 and X06 even flipped sign between the two designs, which marks them as noise. Is the corner really the top? Four probe runs answered that. Pulling silane and profile back to half strength dropped adhesion from 129 to 119, and raising film thickness to the middle dropped it to 120. The response climbs all the way to the edge of the allowed range, so a response surface design would have spent runs mapping a hill with no peak inside it. I skipped it.

The third design was a straight confirmation. Three candidate settings, three repeats each, differing only in the small factors. They finished at 131.3, 130.0 and 128.5, which is about one noise width apart, so the choice between them hardly matters. I recommend the best of the three.

Worth knowing: silane, surface profile and film thickness all sit against their limits. If the process allows more silane, a rougher blast profile or a thinner film than the ranges here, there is more adhesion to be had, and a short series outside the current range would pay for itself.

Runs used in total

44 runs. None failed.

Suggested optimum

About 138 N/cm² pull-off adhesion, up from about 104 N/cm² at the centre of the ranges.

Recommended settings

The first four factors and X07 drive the result. X05, X06 and X08 barely move it, so set them to whatever suits production.

Designs in order

Run sheets

Summary

I screened all eight factors, then zoomed in on the ones that mattered, then cleaned up the rest.

Design 1 was a 16-run resolution IV fractional factorial with three centre points. Resolution IV keeps the main effects clear of two-factor interactions, which matters here because binder, curing agent and cure conditions often interact in this kind of system. Three factors stood out: silane (+11), surface profile (+7) and film thickness (-6). One alias chain of interactions came back large as well, at +9. The centre points put the noise at about 2.6 N/cm² and showed almost no curvature.

Design 2 tested those three factors on their own, with the other five held at mid-level. Holding the five constant broke the alias chain apart, and the large term turned out to be silane times surface profile, as effect heredity suggested. A face-centred CCD fitted the full quadratic model at the same time. No quadratic term was significant, so the surface is a plane with a twist in it and the best point has to sit on a corner of the range: silane high, profile high, film thickness low. The fitted value there was 129 N/cm².

Design 3 took the five weak factors back to that corner and tested them in eight runs. A screening design is enough for effects this small, and the runs double as a confirmation of the corner. They measured 127 to 137 N/cm², which confirms it. Epoxy:amine index (+2) and degrease time (+2) gave a small extra push, and the other three stayed inside the noise. Adding those two to the corner settings gives the recommended run and about 138 N/cm².

The optimum sits on the edge of the allowed range for five factors, so the true optimum may lie outside it. More silane, a rougher profile or a thinner film would be worth a look if the ranges can be opened up.

Runs used in total

45 runs. None failed.

Suggested optimum

About 131 N/cm² pull-off adhesion. Four confirmation runs at the settings below gave 124.5, 131.9, 132.8 and 134.5 N/cm², a mean of 130.9. The middle of the factor ranges gives about 100 N/cm², so this is a gain of roughly 30 percent.

Recommended settings

Only X01, X03 and X07 matter. X06 helps a little. The other four do nothing I could measure, so I set them where they are cheapest or safest: a stoichiometric index, the shortest degrease, no thinning and an ambient cure.

Designs in order

Run sheets

Summary

Eight factors, an unknown response and noisy measurements. I used the usual three steps: screen to find the factors that matter, map the surface around the winners, then confirm the answer.

Design 1 screened all eight factors in 16 runs. A resolution IV fractional factorial keeps main effects clear of two-factor interactions, which is what you need when you do not yet know what is active. Three centre points gave an estimate of the noise and a test for curvature. The noise came out at about 2.2 N/cm². Three factors stood well clear of it: silane (+9.8), surface profile (+7.3) and film thickness (-3.4). One chain of aliased interactions was large too (+6.7). Effect heredity points at X01 by X03, since both are strong on their own and the other members of the chain involve dead factors. The centre points sat on the plane through the corners, so there was no curvature to chase.

Design 2 mapped the active region. A face-centred central composite on X01, X03 and X07 split the interaction from its aliases and tested for curvature in the factors that matter. It confirmed X01 by X03 at +7.0 and found no curvature: none of the three squared terms reached its own standard error. So the surface is a tilted plane. The response rises steadily towards high silane, a rough profile and a thin film, and the two formulation factors reinforce each other, so the best point has to sit in a corner. Three extra runs checked the two borderline factors from screening at that corner. Dilution did nothing. Flash-off time gave about +2.5, small but in the same direction as in screening, so I kept it long.

Design 3 confirmed the corner. Four replicates at the recommended settings averaged 130.9 N/cm², a little above the 127.6 the model predicted. Two further runs moved the supposedly dead factors one at a time: a long degrease gave 134.0 and a hot cure gave 125.3, both inside the spread of the replicates. That backs leaving those factors free.

One caveat. The optimum sits on the edge of the allowed range for silane, surface profile and film thickness, and the response was still climbing there. More silane, a rougher profile or a thinner film would probably help again. Widening those three ranges is the obvious next experiment.

1. Runs used in total

47 runs. None failed.

2. Suggested optimum

About 131 N/cm² pull-off adhesion. Four confirmation runs at the settings below gave 124.5, 130.6, 132.8 and 133.9 N/cm², a mean of 130.4. The centre of the factor ranges gives about 99, so this is a gain of roughly 30%.

3. Recommended settings

Only the first three matter. The other five had no measurable effect, so I set them to the cheapest level: short degrease, high dilution, short flash-off, and room-temperature cure.

4. Designs in order

5. Run sheets

6. Summary

Eight factors and a noisy response, so I screened first and built a model only once I knew what was worth modelling. The repeated centre points in design 1 put the measurement noise at about 2 N/cm². That set the bar for every effect that followed.

Design 1 was a 16-run resolution IV fraction with three centre points. Three main effects stood far above the noise: silane promoter (+19), surface profile (+12.5) and film thickness (-12.7). The other five came in at 2 or below. One two-factor interaction chain was large (18.3) and held four candidates. Effect heredity pointed at silane x surface profile, because both of those are active and the partners in the other three candidates are not.

Design 2 settled that question and looked for curvature at the same time. I ran a face-centred central composite on the three active factors and held the five spare ones at zero. Holding them at zero removes the three rival interactions from the data altogether, so the silane x profile term is estimated on its own. It came out at +7.0 with a t of 8.4, which confirms the reading. No quadratic term was significant, so the surface is a tilted plane and the best point sits on a corner: silane high, profile high, film thickness low.

Design 3 checked the five spare factors again, this time inside that corner rather than at the centre. An interaction with one of the active factors would show up there as a main effect. All five stayed under 1.8 N/cm², below the noise, so the choice between their levels is a cost question and not a performance one.

Design 4 confirmed the prediction. Four replicates averaged 130.4 against a predicted 130, within the spread of the measurement.

Use the full 2% silane, blast to a rough 2.5 µm profile, and keep the film thin at 40 µm. The silane and the profile reinforce each other, so doing only one of the two gives you less than half the gain. Everything else can be set for cost.

1. Runs used in total

47 runs. None failed.

2. Suggested optimum

About 132 N/cm², measured as the mean of five repeat runs at the settings below. The baseline at mid-level settings was about 96 N/cm², so this is a gain of roughly 37%.

3. Recommended settings

The five factors marked “free” had no measurable effect. Pick them on cost or convenience.

4. Designs in order

5. Run sheets

6. Summary

I ran the study in three steps: screen all eight factors, map the ones that matter, then confirm the answer with repeats.

Design 1 was a resolution IV fractional factorial, 16 corner runs plus 4 centre points. With eight factors and nothing known in advance, this is the cheapest way to see which ones move the response. Resolution IV keeps the main effects clear of two-factor interactions, which I needed because the noise is about 3 N/cm². Three factors stood out: silane promoter (+11), surface profile (+8) and dry film thickness (-5). The other five came in at 1 N/cm² or less, inside the noise. One interaction chain was large (+7.4). Since silane and surface profile were both strong on their own, the interaction almost certainly belongs to that pair. The centre points sat about 4 N/cm² below the corner average, a hint of curvature worth checking.

Design 2 was a face-centred central composite design on the three active factors, with the other five held at mid-level. It separates the interaction from its aliases and measures curvature directly. It did both. The silane by surface profile interaction came back at +8.1 with t = 5.4, which confirms the guess from design 1. All three quadratic terms were small and not significant, so the response surface is a tilted plane with a twist, not a dome. The best point is therefore a corner: silane high, surface profile high, film thickness low.

Design 3 confirmed that corner with five repeats. Five more runs tested whether the weak factors were worth setting, each at the level its screening estimate favoured. Those five runs gave 130.6 against 132.5 for the same corner with the weak factors at mid-level, a difference of -1.9 against a standard error of about 2.0. No gain. The plant can set those five wherever it likes.

What drives adhesion here is the bond to the steel, not the cure or the dilution. Raising the silane promoter to 2% and blasting to a 2.5 µm profile each help on their own, and they help more together. Dropping the film to 40 µm adds about 7 N/cm² more. Thicker films shrink more as they cure, which loads the interface, so this fits what we already know.

1. Runs used in total

52 runs. None failed.

2. Suggested optimum

About 132 N/cm². The model predicted 130 N/cm², and four confirmation runs measured 134.2, 134.0, 131.5 and 129.0, a mean of 132.2.

The starting point was around 100 N/cm², so this is a gain of roughly a third.

3. Recommended settings

Three factors matter. The other five can be set wherever they are cheapest; the values below are their midpoints.

4. Designs in order

5. Run sheets

6. Summary

Eight factors, a noisy measurement and no idea which factors matter. The plan was to screen wide and cheap first, then spend the runs on the few factors that survived.

The first design was a 12-run Plackett-Burman, the smallest orthogonal design that holds 8 factors. Three centre points came along to measure the noise and check for curvature. The noise turned out to be about 2.4 N/cm², and the centre points sat on the average of the corners, so nothing was bending. Silane (X01) and surface profile (X03) stood out. Five more factors looked mildly active, which is what you see when a real interaction leaks into the main effects. A Plackett-Burman design cannot tell those apart.

The second design sorted that out. A 16-run resolution IV fraction gives main effects that are clean of two-factor interactions, and its aliasing pattern differs from the Plackett-Burman one, so the two sets together pull the interactions apart. Fitting all 31 runs at once left four terms: silane, surface profile, their interaction, and film thickness with a negative sign. Everything else fell away. The residual spread was 2.9 N/cm², close to the pure noise, so the model had nothing left to explain.

The interaction between silane and surface profile is large. On a smooth surface, going from 0 % to 2 % silane buys about 4 N/cm². On a rough surface it buys about 35. Roughness on its own, without silane, buys nothing. The two only pay off together, which fits the chemistry: the profile gives the silane the surface area and the anchoring sites it needs.

The third design asked whether the best point lies inside the box or at a corner. A face-centred central composite on the three live factors, with the other five held at their midpoints, gives a full quadratic in 17 runs. All three squared terms came back near zero, and so did the two remaining interactions. The surface is a tilted plane with one twist in it, and its highest point is the corner: most silane, roughest profile, thinnest film.

Four confirmation runs at that corner averaged 132.2 N/cm² against a prediction of 130. That is within the noise, so the model holds.

Two practical notes. All three winning settings sit on the edge of the allowed range, and none of them had flattened out, so stretching the limits on silane, profile or film thickness would probably buy more. Cure temperature, epoxy:amine index, dilution, degrease time and flash-off time had no measurable effect over the ranges tested, so set them for cost and throughput.

1. Runs used in total

54 runs. None failed.

2. Suggested optimum

About 132 N/cm², measured. Three confirmation runs at the recommended settings gave 133.3, 132.3 and 130.2 N/cm².

3. Recommended settings

Only X01, X03 and X07 matter. The other five had no measurable effect, so set them wherever production prefers; the mid-levels above are a safe choice.

4. Designs in order

5. Run sheets

6. Summary

With eight factors and a noisy response, the cheapest route is to screen first and then look closely at the few factors that survive. Four designs, 54 runs.

Design 1 was a 16-run resolution IV fractional factorial with four centre points. Resolution IV keeps main effects clear of two-factor interactions, which matters when you expect interactions. The centre points put the noise at about 1.4 N/cm² and showed almost no curvature. Three main effects stood out: silane promoter (+18), surface profile (+12) and film thickness (-12). One interaction chain came back at +19, far bigger than the rest. Under effect heredity, the only member of that chain with two active parents is silane times surface profile, so that is what I took it to be.

Design 2 tested that reading. A face-centred central composite on the three active factors, with the other five at mid-level, confirmed the interaction (+10.6 per coded unit) and found the other two interactions and all three quadratic terms to be nothing. A surface without curvature peaks at a corner, and the fit put that corner at silane high, profile high, film thickness low, predicting 136 N/cm².

Design 3 went back to the five quiet factors. One screen at the centre is not enough, because a factor can wake up elsewhere in the space. So I screened them again at the new optimum with a 12-run Plackett-Burman. Every effect came in under 2.5 N/cm², inside the noise.

Design 4 confirmed the settings with six runs: three with the inert factors at mid-level, three with them low. The two groups agreed (131.2 against 131.3 N/cm²) and both landed close to the prediction.

So: use the full 2 % silane, blast to the rough 2.5 µm profile, and keep the film thin at 40 µm. Silane and profile work together. Either one alone buys you little, both together buy you a lot. Nothing else on the list is worth controlling for adhesion.

1. Runs used in total

57 runs. None failed.

2. Suggested optimum

About 131 N/cm² pull-off adhesion, against roughly 100 N/cm² at the centre of the factor ranges.

3. Recommended settings

Only X01, X03 and X07 matter. The other five had no measurable effect at this corner, so pick them on cost and convenience. The values above are the cheapest ones we tested.

4. Designs in order

5. Run sheets

6. Summary

Eight factors, a noisy response and no idea which ones matter. The plan was to screen all eight cheaply, map the few that survive, check the rest once more at the good settings, then confirm.

Design 1 screened all eight factors in 16 runs with a resolution IV fractional factorial. Three centre points measured the noise: a standard deviation of about 4.5 N/cm², which makes any effect under about 4.5 N/cm² hard to trust. Three factors cleared that bar by a wide margin: silane promoter (+21), surface profile (+14) and dry film thickness (-13). One pair of factors interacted strongly too, but a resolution IV design cannot say which pair. The centre points sat on the plane through the corners, so the response had no curvature worth chasing.

Design 2 mapped those three factors with a face-centred central composite and held the other five at mid-range. This separated the interaction and tested for curvature. The silane and the surface profile help each other: rough steel only pays off when the promoter is there. None of the three quadratic terms was significant, so the response is a plane in this region and the best point has to be a corner. It was: high silane, high profile, low film thickness, at about 133 N/cm².

That left the unresolved interaction from design 1. If one of the five spare factors mattered at the new corner, holding it at mid-range would have hidden it. Design 3 therefore ran a resolution V fractional factorial on all five at that corner, which estimates every main effect and every two-factor interaction among them. Nothing came through above the noise, so those five can be set for cost.

Design 4 repeated the recommended setting four times and the closest runner-up twice. The six runs averaged 129.7 N/cm². Pooling every run at the best corner gives 130.6 N/cm², in line with the model.