Mixture, statistically uniform

Danckwerts expresses the goodness of mixing by two statistically defined quantities, the scale and the intensity of segregation. He states that his treatment is suitable chiefly for mixtures where the smallest particles capable of independent movement are very small compared to the size of the portions which will normally be taken for use or analysis. For his analysis, he assumes that the mixture is uniform in texture that is, it cannot be divided into two parts of equal size in which the mean concentration or the scale or intensity of segregation differ significantly. He further states This is the most important limitation on the practical value of the definitions and tests which will be proposed. He emphasizes the fact that large scale segregation, caused for example by... 

Most of the assumptions are based on idealized models, indicating the limitations of the mathematical methods employed and the quantity and type of experimental data available. For example, the details of the combinatorial entropy of a binary mixture may be well understood, but modeling requires, in large measure, uniformity so the statistical relationships can be determined. This uniformity is manifested in mixing rules and a minimum number of adjustable parameters so as to avoid problems related to the mathematics, eg, local minima and multiple solutions. 

If a relatively small amount of a black (additive) component, q, must be mixed uniformly into a white matrix, p, such that a certain small volume element of the mixture contains the same amount of the black component, a maximum particle size of the black component exists at or below which a uniform mixture can statistically be guaranteed. Figure 136 illustrates this principle it is assumed that the white matrix is a continuum and will not hinder the relative movement of the black component particles during mixing. 

Such uniform fats do not occur to any extent in nature. Natural fats are statistical mixtures of glycerol triesters with three different fatty acids usually containing 12-20 carbon atoms and 0-4 cw-configured double bonds. A rule that seems often to be followed is that the few percent of unsaturated acids in animal fats are located on the central carbon atom of glycerol, whereas in plant fats their position is rather on Cl or C3. 

Molecular weight of a polymer and a certain concentration of the plasticizer determine the uniformity blend. If the molecular weight of PVC used in CPE-PVC-DOP blend is doubled, the concentration of DOP needed to obtain a homogeneous mixture is increased from 55 wt% to 70 wt% of DOP. A statistical thermodynamic approach was developed for mixing of polymers, small molecules (plasticizers), and holes which are different in sizes. Plasticizer efficiency is determined by polymer-plasticizer interaction and plasticizer segment size. 

The degree of uniformity of a mixed product may be measured by analysis of a number of spot samples. Powder mixers act on two or more separate materials to intermingle them. Once a material is randomly distributed through another, mixing may be considered to be complete. Based on these concepts, the well-known statistical parameters mean and standard deviation of component concentration can be used to characterize the state of a mixture. If spot samples are taken at random from a mixture and analyzed, the standard deviation of the analyses s about the average value of the fraction of a specific powder x is estimated by the following relation ... 

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