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Mixture design in epoxy formulations?

robhanssen

Does anyone have experience with using the mixture design in epoxy formulations? The difficulty is that there is a linear constraint on the formulation that I can't enter.

The constraint is (amine 1)/AHEW1 + (amine 2)/AHEW2 = (epoxy 1)/EEW1 + (epoxy 2)/EEW2, where AHEW and EEW are known properties of the amines and epoxies.

The constraints only allow me to enter a greater than/smaller constraint, not an equivalence. If the amine/epoxy balance is not at equivalence, the material will not cure and therefore not give useful information.

As background: we make epoxy and amine-modified materials that go into coatings. These materials influence the coatings in certain ways (viscosity, hardness, flexibility, water-resistance, scratch-resistance, etc.). We would like to map performance of our materials in complex mixtures.

Accepted answers

louv

I like the new space filling mixture design in the Mixture DOE platform as well.

DOE(

Mixture Design,

{Add Response( Maximize, "Y", ., ., . ), Change Factor Settings( 1, 0, 1, "X1" ),

Change Factor Settings( 2, 0, 1, "X2" ), Change Factor Settings( 3, 0, 1, "X3" ),

Add Factor( Mixture, 0, 1, "X4", 0 ), Set Random Seed( 2130893620 ),

Add Constraint( [-0.025 -0.005 0.006 0.004 0, 0.025 0.005 -0.006 -0.004 0] ),

Mixture Design Type( Space Filling, 20 ), Make Table}

)

All comments

louv

Rob,

Looks like you are dealing with a Mixture of Mixtures scenario which is documented in the online DOE guide.

http://www.jmp.com/support/downloads/pdf/jmp1001/doe_guide.pdf

Just type "mixture of mixtures" in the search.

You have to add two linear constraints. One where X1+X2 ≤0.5 and the other where X1+X2≥0.5.

The document details adjusting the model as well in the Custom Design platform.

robhanssen

Thanks, Lou.

The issue is that no combination of any of the components is going to be constant. The material properties (AHEW and EEW) can be very different in magnitutude, e.g. AHEW1=40, AHEW2=200, EEW1=170, EEW2=250. A valid formulation for this would be (am1,am2,ep1,ep2) = (0.106, 0.213, 0.532, 0.149). Another valid formulation for this would be (am1,am2,ep1,ep2) = (0.139, 0.069, 0.347, 0.444).

The only real constraint is the amine/epoxy balance ( sum( am_i/AHEW_i) = sum( ep_j/EEW_j) ). I've tried adding the two-constraints method(1) into the custom designer, but it gave a warning and an error message(2,3).

(1) two-contraints, actual number used were the ones mentioned above.

(1/AHEW1)*AM1 + (1/AHEW2)*AM2 + (-1/EEW1)*EP1 + (-1/EEW2)*EP2 >= 0

(1/AHEW1)*AM1 + (1/AHEW2)*AM2 + (-1/EEW1)*EP1 + (-1/EEW2)*EP2 <= 0

(2) A subset of the factor constraints is equivalent to an equality constraint. The resulting design may be singular if the model effects are not chosen carefully.

(3) Optimal designer failed to converge.

louv

Rob,

If AHEW1=40 and AHEW2=200, EEW1=170 and EEW2=250 could your units of X1, X2, X3, and X4 be normalized to accommodate the constants? Then using the two constraint method of X1+X2-X3-X4 ≤0 and X1+X2-X3-X4≥0 along with removal of the X4 term in the model to avoid singularity (see DOE guide on mixture of mixtures) might get the job done?

If X1+X2 = X3+X4 as you mentioned originally then the constraint should be X1+X2-X3-X4≤0 and X1+X2-X3-X4≥0 not X1+X2+X3+X4≤0 and X1+X2+X3+X4≥0?

You might also need 4 total constraints to also make X1+X2 = X3+X4

robhanssen

I don't know if I fully understand your answer. The constraint for the total mixture should be that X1+X2+X3+X4=1. If I would scale the units of X_i to normalize for the AHEW or EEW, the sum of normalized components would not necessarily be 1 (or any constant value) anymore.

Ryan_Lekivetz

When you tried the above and the optimal designer failed to converge, did you remove an additional effect for the second equality constraint? Running the script below and clicking the "Make Design" button (notice only 3 effects in the model), I am getting a design that looks to meet the constraints in JMP10.0.2 and JMP11.2. You may want to refine the 1/170 more than I have. If the AHEWi & EEWi's are also varying, it becomes a much more difficult problem. DOE( Custom Design, {Add Response( Maximize, "Y", ., ., . ), Add Factor( Mixture, 0, 1, "AM1", 0 ), Add Factor( Mixture, 0, 1, "AM2", 0 ), Add Factor( Mixture, 0, 1, "EP1", 0 ), Add Factor( Mixture, 0, 1, "EP2", 0 ), Set Random Seed( 1485792608 ), Add Constraint( [0.025 0.005 -0.00588 -0.004 0, -0.025 -0.005 0.00588 0.004 0] ), Add Term( {2, 1} ), Add Term( {3, 1} ), Add Term( {4, 1} ), Add Alias Term( {1, 1}, {2, 1} ), Add Alias Term( {1, 1}, {3, 1} ), Add Alias Term( {1, 1}, {4, 1} ), Add Alias Term( {2, 1}, {3, 1} ), Add Alias Term( {2, 1}, {4, 1} ), Add Alias Term( {3, 1}, {4, 1} ), Set Sample Size( 10 )} );

louv

I like the new space filling mixture design in the Mixture DOE platform as well.

DOE(

Mixture Design,

{Add Response( Maximize, "Y", ., ., . ), Change Factor Settings( 1, 0, 1, "X1" ),

Change Factor Settings( 2, 0, 1, "X2" ), Change Factor Settings( 3, 0, 1, "X3" ),

Add Factor( Mixture, 0, 1, "X4", 0 ), Set Random Seed( 2130893620 ),

Add Constraint( [-0.025 -0.005 0.006 0.004 0, 0.025 0.005 -0.006 -0.004 0] ),

Mixture Design Type( Space Filling, 20 ), Make Table}

)

robhanssen

As an answer to both @LouV and @rlek2:

Removing a factor from the model (e.g. X4 a.k.a. EP2) makes JMP setup a model that is valid according to the formulation constraint that was set.

The issues with that option is that changing that factor over a large enough range will have a strong influence on the coating properties. If the factor is not captured in the model, the model is probably not useful.

If you go by the number examples in one of my examples above: if the formulation changes from (stiff component EP1 ,flexible component EP2) = (0.532, 0.149) to (0.347, 0.444), you could imagine a big change in coating properties - caused by both EP1 and EP2 changes.