Other mixture designs

When the component proportions must obey lower limits, the experimental region becomes constrained and the design results are much easier to analyze in terms of pseudocomponents, as we have just seen. In 

The goal of the investigation was to study the elongation imtU ruptiue and swelhng, in dioxane, of pol3rmeric films made from polyisobutene (PIB), polyethylene (PE) and paraffin wax (PW). For technical reasons, the proportions of these components in the mixtures were restricted to the following intervals  

Because aU the intervals are different, these inequalities define an irregular hexagon inside the concentration triangle, shown in Fig. 7.8a. The points belonging to this hexagon represent the mixtures that in principle can be studied. With these specifications, the pseudocomponents are defined by the expressions  

To define the design, we need to consider what models could be appropriate to describe the behavior of the two responses — film elongation and swelling. Unless we already know a good deal about the system imder study, this usually cannot be done before the experiments are made. Also, different responses may require different models. Since it is possible that a description of the results will require a special cubic model, we had better choose a design with at least seven different mixtures. 

Examining the (X X) matrix for these two designs, one can conclude that the first design (points 1-7 in Fig. 7.8a) produces coefficient estimates 10-70% more precise than those of the second design, and for this reason it was chosen. When the experiments were performed, mixture number 6 did not produce a viable film and had to be substituted by mixture 6, with composition defined by the midpoint of the 1-6 side. 

---

Using experimental design such as Surface Response Method optimises the product formulation. This method is more satisfactory and effective than other methods such as classical one-at-a-time or mathematical methods because it can study many variables simultaneously with a low number of observations, saving time and costs [6]. Hence in this research, statistical experimental design or mixture design is used in this work in order to optimise the MUF resin formulation. 

For the regression analysis of a mixture design of this type, the NOCONSTANT regression command in MINITAB was used. Because of the constraint that the sum of all components must equal unity, the resultant models are in the form of Scheffe polynomials(13), in which the constant term is included in the other coefficients. However, the calculation of correlation coefficients and F values given by MINITAB are not correct for this situation. Therefore, these values had to be calculated in a separate program. Again, the computer made these repetitive and Involved calculations easily. The correct equations are shown below (13) ... 

The equations developed by Alexander and Klug [1,45] have also been used in the quantitative analysis of polymorphic mixtures. If there is a mixture of two polymorphs, one is considered the unknown component while the other is designated the matrix. The intensity of line i of component 1 (the unknown component), /il( is determined in mixtures containing different weight fractions of components 1 and 2 (the matrix). When dealing with polymorphic mixtures, however, Eq. (15) is much simplified. Since fix = /a2, Eq. (15) reduces to... 

In some mixture design problems (such as formulations), it may not be necessary to consider processing issues and hence we would not have the process model constraints. In this case the problem becomes a simple mixing problem, which would already have been addressed by the miscibility criteria in sub-problem 4M. Hence, for these problems, we will not need sub-problem 5M. Also in some cases we might have to identify a mixture whose constituents perform different functions such as solvents and anti solvents for crystallization. In such cases we would have to formulate and solve more than one single compound design problems to identify the constituents and then solve the final two sub-problems to identify the optimal mixture. In certain cases we may not have process model constraints, however, we may still have to solve an optimization problem with other constraints, in sub-problem 4 and sub-problem 5m respectively. 

Many mixtures exhibit edge effects such that the behavior of the formulation shows drastic changes when one or more of the components is omitted from the mixture [Anderson and McLean (1974)]. Thus, if simple empirical models such as Equations 12.90 and 12.91 are to be used to model the system, it is often best to work in regions that have all components present. Such systems can be prepared with so-called pseudo-components [Cornell (1990)] as shown in the lower two panels of Figure 12.33. The pseudo-components correspond to the vertexes in these designs and are seen to be mixtures that are relatively rich in one of the components. In practice, the pseudo-components can be prepared first, and then the other mixtures in the design can be prepared from these pseudo-components. 

Diethylamino-4-methylcoumarin is used to sensitize a weakly fluorescent second material in a mixture designed for use as an in situ flaw detector in metal surfaces. The energy absorbed by the coumarin is transferred to a second component with little energy loss by non-radiative processes. The blue fluorescence of the coumarin is replaced by the yellow-green of the other component, to which the eye is more sensitive. 

Mixture designs are applied in cases where the levels of individual components in a formulation require optimization, but where the system is constrained by a maximum value for the overall formulation. In other words, a mixture design is often considered at this stage when the quantities of the factors must add to a fixed total. In a mixture experiment, the factors are proportions of different components of a blend. Mixture designs allow for the specification of constraints on each of the factors, such as a maximum and/or minimum value for each component, as well as for the sum and/or ratio of two or more of the factors. These designs are very specific in nature and are tied to the specific constraints that are unique to the particular formulation. However, as with the discussion of the fractional factorial designs, in order to be most efficient, it is important to provide realistic prior expectations on anticipated effects so the smallest design can be set up to fit the simplest realistic model to the data. 

The optimum mixture ratio for a given propellant combination is in turn influenced by the other primary design parameters, chamber pressure and expansion ratio. 

Modem detergent products are complex mixtures of many different ingredients. Typical formulations consist of surfactants, builders, and other additives designed to maximize performance for the consumer while maintaining reasonable raw material and manufacturing costs. Typical detergent formulations contain multiple surfactant types to optimize performance and product stability.21,22 Performance additives such as bleaches, bleach activators,... 

Table 2.42 illustrates two constrained mixture designs, one with six and die other... 

Dimensionless degrees of freedom do not always transfer to other column designs as perfectly as shown in Fig. 7.1. In many cases some deviations in the concentration profiles have to be taken into account. These cases will now be demonstrated using a second exemplary separation problem the chromatographic separation of 1 1 mixture of glucose and fructose on ion-exchange resin Amberlite CR 1320 Ca from Rohm Haas (325 pm particle diameter) (Tab. 7.2). 

The criteria for maximum allowable air voids content in the final pavement was taken at 15% with a unit weight of about 125 lb/cu ft as established by Shell (2). Although many different mixture ratios were tested, the major comparisons were made with the Shell Thermopave mixture design, i.e., 80.5% sand-6% asphalt-13.5 % sulfur by weight. Under different situations other designs may be equally attractive both technically and economically. 

Mixture design was used to help in formulation of a sustained release tablet, based on a hydrophilic cellulose polymer, which swells in the presence of water, and so impedes the release of the soluble active substance. Drug release is by a combination of diffusion and erosion. The formulators wished to examine the effect of changing the proportions of polymer, and of the different diluents (9). The constraints on the formulation are given in table 10.8. Other components (drug substance, lubricant) are to be considered as fixed, at least for the time being. 

Chemometrics has been defined in some texts [155] as the entire process whereby data are transformed into information used for decision-making. It is this definition that is the most applicable to separation sciences, more specifically in method development and optimisation in liquid chromatography. In this example, chemometrics has been used to predict optimum separation conditions based on empirical data and other separation information. Chemometric approaches to method development can be based on either sequential simplex models [156] or simultaneous fixed factorial designs [157] or interactive mixture designs [158] which combine the advantages of simultaneous and simplex models. 

---