Mixture factor

The pH of the reaction mixture (Factor C) has the most significant effect on the yield of titanium, while exhibiting abnormal behavior pH the yield of Ti decreases as pH increasing. Wu et al. [214] also observed similar phenomena in their investigation. 

For a mixture with q variables (i.e., q components), the mixture factor space is a subspace of the respective -variables in Euclidean space. In Figure 8.5, Figure 8.6, and Figure 8.7, we see the relationship between the mixture coordinate system and the respective Euclidean space. Figure 8.5 illustrates the case of a binary mixture. The constraint described by Equation 8.11 holds for points A, B, and C however, only point B and all points on the heavy line in Figure 8.5 are points from the mixture space satisfying the conditions described by the constraints in Equation 8.10 as well. 

Figure 8.6 illustrates the relationship between Euclidean and mixture-factor space for three variables. Here, we see that the set of the mixture points lying on... 

The reactivity of a particular diene depends on the concentration of the s-cis conformation in the equilibrium mixture. Factors that increase the concentration of this conformation make the diene more reactive. As an example of this effect, consider 2,3-dimethyl-1,3-butadiene ... 

Recently, Riviere and Brooks (2007) published a method to improve the prediction of dermal absorption of compounds dosed in complex chemical mixtures. The method predicts dermal absorption or penetration of topically applied compounds by developing quantitative structure-property relationship (QSPR) models based on linear free energy relations (LFERs). The QSPR equations are used to describe individual compound penetration based on the molecular descriptors for the compound, and these are modified by a mixture factor (MF), which accounts for the physical-chemical properties of the vehicle and mixture components. Principal components analysis is used to calculate the MF based on percentage composition of the vehicle and mixture components and physical-chemical properties. 

Our laboratory has focused significant research on the effects of chemical mixtures on dermal absorption of penetrant compounds. In order to incorporate mixture effects, our laboratory has been exploring use of an additional term operationally called the mixture factor (MF), yielding... 

Figure 24.2 QSAR plots showing effects of including a mixture factor (MF) [Refractive Index] on prediction of dermal permeability when dosed in complex chemical mixtures. (Top plate) No MF included n = 288, R2 = 0.57, Q2 = 0.56, s = 0.55, F = 77. (Bottom plate) MF included n = 288, R2 = 0.80, Q2 = 0.79, s = 0.37, F = 192.

This is the k dimensional space defined by the k coded variables Z, for the continuous quantitative factors being investigated. If we examine only two factors, keeping all other conditions constant, it can be represented as a (two dimensional) plane. Except for the special case of mixtures, factor space is defined in terms of independent variables. If there are 3 factors, the factor space can be represented diagrammatically in three dimensions. 

Figure 9.1 (a) Relationship between orthogonal axes and mixture factor space, (b) Ternary diagram. 

A typical example is one of a matrix tablet containing a mixture of 2 polymers (Xi, X2) and 2 diluents (Xj, X4), where the total level of polymer is fixed at a fraction P. There are 4 components, and therefore the domain may be defined in 3 dimensional space within the tetrahedron of the 4 variable mixture factor space. There is a second restriction, however, of x, + x = P and so the domain must lie on a plane within that space. For fixed major components, the number of independent variables is equal to the number of minor components, minus the number of major components. In the following discussion we will restrict ourselves to 2 major components. 

To incorporate mixture effects, another term was added, the mixture factor (MF), yielding... 

Figure 14.5 Observed vs. predicted log Kp in PSFT without inclusion of a mixture factor.

THERMODYNAMICS OF POLYMER MIXTURES FACTORS AFFECTING MISCIBILITY IN POLYMER BLENDS... 

In contrast to process factors, mixture factors are sometimes modeled with cubic models. In this case the term in brackets in equation (4) should be added to the model. The -coefficients of equations (2)-(4) are obtained by multiple regression, at least when the number of experiments is higher than the number of coefficients. The knowledge gathered from a multiple regression may be used to verify whether all terms in the model are required, to validate the model, to try to obtain the most precise possible estimate of the coefficients, and to try to obtain the smallest prediction error of the optimum or other regions of interest. 

Usually two-level designs will be applied first to decide which factors are important and only these important factors will be studied with higher-level designs. There are two special cases one is the case of mixture factors, and in the other case process factors are used for which it is not possible to control the levels, because nature does. In these cases, a D-optimal design or a mapping design would be appropriate (see Section 3.4). 

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