In the Laboratory
Mixture Design Experiments Applied to the Formulation of Colorant Solutions
W
J. M. Gozálvez* Department of Chemical and Nuclear Engineering, Universidad Politécnica de Valencia, C/ Camino de Vera s/n, 46022 Valencia, Spain; *
[email protected] J. C. García-Díaz Department of Applied Statistics, Operations Research and Quality, Universidad Politécnica de Valencia, C/ Camino de Vera s/n, 46022 Valencia, Spain
The “design of experiments” (DOE) is a set of statistical techniques that allows the experimenter to select the most influential factors on an experimental response and to obtain their optimum values (1). The DOE will provide the appropriate set of experiments to perform in the laboratory to obtain the maximum information with the minimum number of experiments. Many real problems in the chemical industry can be solved satisfactorily with the application of the most common DOE techniques: factorial design and central composite design. Factorial design is used to identify the most significant factors and central composite design, also known as response surface methodology, and expresses the dependence of one response in terms of independent factors in order to optimize it. Nevertheless, there is a problem of great interest for the chemical industry in which these techniques are not applicable: the optimization of the mixture characteristics. In this case, the intensive properties of the mixture depend on the proportions of the components and not on the quantity of the mixture. As the factors to be considered are the fractions of every component and their levels are not independent, the central composite designs cannot be employed. This special case must be addressed with a special kind of DOE known as mixture design. The introduction of this topic should be a part of undergraduate chemical education (2). Students must know that the experimental results are nonrepetitive and subjected to natural deviations. They must also learn how to plan their experiments in the optimization procedure. With the adequate use of statistical software in the laboratory it is possible to understand complicated concepts and perform an experimental optimization. Mixture Design The starting point of the theory of DOE with mixtures is that the sum of the fractions of the components must be equal to unity and their proportions must be non-negative (3). In the case of three components, the factorial space constituted by all of the possible fractions of the components is a triangle whose vertices correspond to pure components. In mixture problems, the objective is to find a mathematical model, EQ, useful for forecasting the values of the response variable Q in term of its components xi. If f is the true value of Q and e is the experimental error, then Q = f ( x1, x 2, x 3 ( + e
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Similar to other DOE methodologies, the mixture design models are typically first- or second-degree polynomials. However, due to the restrictions implicit in fractions, the models do not include constant terms. The most typical models to represent Q are linear: E (Q ) = β1 x1 + β2 x 2 + β3 x 3
quadratic: E (Q ) = β1 x1 + β2 x 2 + β3 x 3 + β1,2 x1 x 2 + β1, 3 x1 x 3 + β2, 3 x 2 x 3 The linear model is used when the effects of the components in the mixture are additive and the response variable can be defined as a linear combination of their fractions. The quadratic model considers antagonistic or synergic interactions between pairs of components of the mixture. There is also a model called special cubic to consider interactions among the three components. The meaning of the terms is relatively simple. The linear coefficient βi in the equations represents the expected average response for the pure components (xi = 1, xj≠i = 0). The binary coefficients βi,j take into account the curvature of the response owing to nonlinear effects between pairs of components and represent synergic mixture effects when positive or antagonistic when negative. To obtain the coefficients it is necessary to carry out a set of experiments in the laboratory. Each experiment consists of a combination of factors for which the characteristic Q is measured. The different combinations are planned to optimize the information that can be obtained and constitute what is known as an experimental design. The most common types of mixture experimental designs are simplex-lattice, simplex-centroid, and augmented simplex design. The simplex-lattice design consists of a set of experimental tests uniformly spaced in the triangle. This set of experiments is obtained by the combination of m + 1 (where m is the number of levels) component fractions, that is, xi = 0, 1兾m, 2兾m, …, 1 where i = 1, 2, 3. For m = 1 we have the simplex-lattice for linear models that consists of one observation of the characteristic of the pure components that corresponds to the triangle vertices (Table 1 and Figure 1). Since three parameters in the model are considered, this design is not able to estimate the experimental error or to prove the validity of the adjustment. This limitation can be resolved using a simplex-centroid design that adds a central point of
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In the Laboratory
unknown. The researcher is interested in the simplest model that describes the results with an acceptable error level. The use of statistical software allows selecting the most appropriate model and analyzing the results. Colorimetric Coordinates in the Color Space CIELab
Figure 1. Combinations for the simplex designs: simplex-lattice– points 1–6; simplex-centroid–points 1–7; and augmented simplex– points 1–10.
coordinates (1兾3, 1兾3, 1兾3) to determine whether the model is appropriate. The simplex-lattice design for quadratic models (m = 2) includes binary mixtures corresponding to the medium point in the lines that connect the vertices in order to estimate nonlinear effects. A more complete study can be performed augmenting the simplex-centroid with additional points known as axial runs (Figure 1). These points include the three components and are placed at a distance d = 2兾3 from the centroid in the radii that connect it with the vertices. Once the type of experimental design is chosen, the experimental tests will be carried out in random order to obtain the values of the response variable. The next step is the analysis of the experimental results and the creation of models by means of polynomial regression. Many times, regression models are used as approximation functions when the relation between the response variable Q and the factors xi is
The formulation of a mixture of dyes to obtain a specific color is an example of an industrial problem where the design of mixture can be employed. The perception of color has a subjective interpretation; however, numerical values are needed to employ DOE techniques. The CIELab system was created as an objective tool to standardize the color measurement (4). This system is based on the theory of opposite colors that establish that a color cannot be green and red or blue and yellow at the same time and that it is possible to represent all the colors in a system of coordinates L, a*, and b*, that remain inside a solid sphere. The vertical axis is the L coordinate that represents luminosity and ranges from 0% (black) to 100% (white). The coordinates a* and b* form a horizontal plane that represents the transitions green–red (a* = ᎑60 to a* = +60) and blue–yellow (b* = ᎑60 to b* = +60). The CIELab coordinates for a solution can be obtained from the tri-stimulus values XYZ. These can be calculated from the convolution products of the color spectra of the samples corresponding to a standard illuminant of xenon D65, which is referred to a standard observer, CIE 64, with an angle of observation of 10⬚. Experiment Employing Mixture Design with Three Colorant Solutions To illustrate the technique of mixture design and the color characterization we designed a set of laboratory experiments to relate the proportions in which three colorant solutions are mixed with the color obtained. A complete description of this procedure is included in the Supplemental Material.W An example of a practical session and typical results follows. It is worth mentioning that the time of the experiment is short enough to allow the students to complete a whole set of tests in one session of approximately three hours. The main sections of this procedure are the selection
Table 1. Fractions for Quadratic Simplex Designs for Three Components Exp
648
Simplex-Lattice
Simplex-Centroid
Augmented Simplex
x1
x2
x3
x1
x2
x3
x1
x2
x3
1
1
0
0
1
0
0
1
0
0
2
0
1
0
0
1
0
0
1
0
3
0
0
1
0
0
1
0
0
1
4
1/2
1/2
0
1/2
1/2
0
1/2
1/2
0
5
1/2
0
1/2
1/2
0
1/2
1/2
0
1/2
6
0
1/2
1/2
0
1/2
1/2
0
1/2
1/2
7
---
---
---
1/3
1/3
1/3
1/3
1/3
1/3
8
---
---
---
---
---
---
1/6
1/6
2/3
9
---
---
---
---
---
---
1/6
2/3
1/6
10
---
---
---
---
---
---
2/3
1/6
1/6
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In the Laboratory
of the appropriate combinations using DOE, the carryingout of the experiments in the laboratory, and the subsequent analysis of the experimental results.
Materials Three food colorants with sufficiently different spectra were used to produce the solutions in order to obtain a range of colors as wide as possible: tartrazine (yellow), Ponceau 4R (red), and indigo carmine (blue) designated by the letters Y, R, and B. Tartrazine and Ponceau 4R are synthetic azoic dyes. Indigo carmine is a synthetic coal tar dye, normally produced by a synthesis of indoxyl. To avoid one colorant masking the others owing to different molar absorptivities, the concentrations of the colorants were adjusted to have solutions of similar absorbance (Table 2). Planning the Experiments After the theoretical explanation, the lab started with the planning of the experiments using the software Statgraphics 5. It was necessary to define the name of the factors and response variables and to select the type of design. The factors were the different dyes. The levels were the volumetric fractions of each individual solution (xB, xY, xR) and the response variables to be determined were the color coordinates L, a*, and b*. The type of mixture design can be selected in the computer or can be a user-specified design. In our case the design chosen was an augmented simplex-centroid design with axial runs and two repetitions of the centroid, obtaining the factor combinations shown in Table 3. The program provided a datasheet to be filled out in the laboratory.
Table 2. Concentration and Absorbance of Pure Solutions Soln
Colorant
Conc/ (mg L᎑1)
Peak Wavelength/ nm
Abs
Y
Tartrazine
08.0
429
0.342
R
Ponceau 4R
10.0
508
0.318
B
Indigo carmine
07.5
610
0.328
Table 3. Color Results from the Different Experiments Exp
Factors
Responses
xY
xR
xB
L
a*
b*
1
1
0
0
98.847
᎑11.467
30.602
2
0
1
0
89.499
3
0
0
1
87.821
24.363 ᎑11.215
4.330 ᎑16.926
4
1/2
1/2
0
94.129
1/2
0
1/2
93.208
7.278 ᎑13.939
17.766
5 6
0
1/2
1/2
88.014
7.192
6.965 ᎑7.606
7
1/3
1/3
1/3
91.596
8
1/6
1/6
2/3
89.421
0.015 ᎑6.160
5.812 ᎑5.898
11.335 ᎑6.461
17.867
9
1/6
2/3
1/6
91.070
10
2/3
1/6
1/6
95.590
11
1/3
1/3
1/3
91.545
12
1/3
1/3
1/3
91.213
0.081 ᎑6.707
4.442 5.657 2.277
Laboratory Experiments Once the combinations to be performed were stated, the students started to prepare the different mixtures using the pure colorant solutions. They previously checked the absorbance of these solutions using Table 2. A spectrophotometer was employed to obtain the tri-stimulus coordinates from the transmittance measurements of the samples in the interval 400–700 nm. The spectra were transformed into L, a*, and b* coordinates (see the Supplemental MaterialW). A typical result is shown in Table 3. Analysis of the Results The statistical analysis of the results (Figure 2) was performed after having entered the laboratory results in the Statgraphics design file. Students were asked to select the model order with the highest R 2 value. This parameter indicates the percentage of the variation of the response variable that is explained by the factors. In this case, quadratic models were adequate to represent all the color coordinates. The second step was to simplify the models discarding the nonsignificant interactions between factors. L = 98.816 x Y + 89.653 xR + 87.678 xB − 2.766 xR xB a * = −11.213 x Y + 24.542 xR − 10.9732 xB − 9.970 xY xB b * = 30.673 x Y + 4.373 xR − 16.845 xB − 4.920 xR xB
For example, the interpretation of the model for L is the fol-
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Figure 2. Statistical results for the response variable L.
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mately ∆C = 1, no color difference could be distinguished between these solutions. Hazards In the preparation of pure solutions, the contact of the colorant with the skin or dust inhalation should be avoided as these substances can be irritating or cause damage to the eyes. If pure colorant solutions are previously prepared the experiment does not present any chemical hazards as the colorants used are in the same concentration as in food. Nevertheless, we recommend that students always wear gloves and safety goggles in the laboratory. Conclusions
Figure 3. Estimated response surface for variable L.
lowing: The most influential factor is xY owing to the high value of this coefficient. The factors xR and xB present an antagonistic effect because the coefficient of their product is negative. The model allowed the representation of the response surface (Figure 3) using the software.
Use of the Models To Obtain a Solution of a Specified Color The second objective was the estimation of the fractions that produce a color as similar as possible to an arbitrary solution. As an example, a mixture of fractions not present in the design (xY1 = 0.167, xR1 = 0.500, xB1 = 0.333) was made by the instructor. The students measured the following color coordinates for the sample: L1 = 89.957, a1* = 6.250, and b1* = 0.892. The solution of the system equation constituted by every model equation equal to this color coordinates gave the combination: xY2 = 0.166, xR2 = 0.501, and xB2 = 0.331. This composition was prepared and gave L2 = 89.887, a2* = 6.662, and b2* = 0.921. For these values the normalized color difference, ∆C, was calculated in CIELab units as the Euclidean distance in the tridimensional space: ∆C = L1* − L*2
2
+ a1* − a *2
2
+ b1* − b2*
2
= 0.51
As the discrimination limit for the human eye is approxi-
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The simple laboratory proposed allows teaching the students a very useful technique for the optimization of mixture fraction factors commonly used in the chemical industry. Using mixture design the students reinforce their knowledge in design of experiments and statistics and they see these techniques as useful tools in chemistry. From an educational point of view it is important that the experiments are sufficiently short so that each group of students can complete an experimental set. This lab meets this condition and, additionally, introduces color theory and advance spectrometry concepts. The colorants employed can be changed. An interesting variation of this laboratory would consist of the study of the color of a dyed surface, employing a spectrophotometer for the characterization of the color of solids. On the other hand, mixture design can be extended to a wide variety of problems in chemistry and chemical engineering where the results depend on the proportions of the components used. WSupplemental
Material
A students handout, details of preparation, calculations, and notes for the instructors are available in this issue of JCE Online. Literature Cited 1. Montgomery, D. C. Design and Analysis of Experiments, 5th ed.; John Wiley & Sons: New York, 2001. 2. Stolzberg, R. J. J. Chem. Educ. 1997, 74, 216–220. 3. Cornell, J. A. Experiments with Mixtures. Designs, Models, and the Analysis of Mixture Data; John Wiley & Sons: New York, 1990. 4. CIE Publication 15.2. Colorimetry, 2nd ed.; CIE Publications Offices: Salem, MA, 1986.
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