Compositional data, modeling

Evaluation of reactivity ratios from the copolymer composition equation requires only composition data—that is, analytical chemistry-and has been the method most widely used to evaluate rj and t2. As noted in the last section, this method assumes terminal control and seeks the best fit of the data to that model. It offers no means for testing the model and, as we shall see, is subject to enough uncertainty to make even self-consistency difficult to achieve. 

It has been argued that for a majority of copolymerizations, composition data can be adequately predicted by the terminal model copolymer composition equation (eqs. 5-9). However, in that composition data are not particularly good for model discrimination, any conclusion regarding the widespread applicability of the implicit penultimate model on this basis is premature. 

Mechanisms for copolymerization involving complexes between the monomers were first proposed to explain the high degree of alternation observed in some copolymerizations. They have also been put forward, usually as alternatives to the penultimate model, to explain anomalous (not consistent with the terminal model) composition data in certain copolymerizations.65"74... 

It is also possible to process copolymer composition data to obtain reactivity ratios for higher order models (e.g. penultimate model or complex participation, etc.). However, composition data have low power in model discrimination (Sections 7.3.1.2 and 7.3.1.3). There has been much published on the subject of the design of experiments for reactivity ratio determination and model discrimination.49 "8 136 137 Attention must be paid to the information that is required the optimal design for obtaining terminal model reactivity ratios may not be ideal for model discrimination.49... 

NMR spectroscopy has made possible the characterization of copolymers in terms of their monomer sequence distribution. The area has been reviewed by Randall,100 Bovey,139 Tonelli,101 Hatada140 and others. Information on monomer sequence distribution is substantially more powerful than simple composition data with respect to model discrimination,25,49 Although many authors have used the distribution of triad fractions to confirm the adequacy or otherwise of various models, only a few25 58,141 have used dyad or triad fractions to calculate reactivity ratios directly. 

Mayo-Lewis Binary Copolymeriration Model. In this exeimple we consider the Mayo-Lewis model for describing binary copolymerization. The procedure for estimating the kinetic parameters expressed as reactivity ratios from composition data is discussed in detail in our earlier paper (1 ). Here diad fractions, which are the relative numbers of MjMj, MiMj, M Mj and MjMj sequences as measured by NMR are used. NMR, while extremely useful, cannot distinguish between MiM and M Mi sequences and... 

In their paper Hill and coworkers discriminate between alternative copolymerization models by fitting the models to composition data and then predicting sequence distributions based on the fitted models. Measured and fitted sequence distributions are then compared. A better approach taken here is to fit the models to the sequence distribution data directly. 

Figure 1 Time-dependent composition data is shown for the hydrogenation of aqueous 3-buten-2-ol for both (a) ultrasound irradiated and (b) magnetically stirred systems. The symbols correspond to experimental measurements (3-buten-2-ol 3BEN20L-solid circles 3-buten-2-one 3BEN20NE-open hourglass 2-butanone 2BONE-open triangles 2-butanol 2BOL-crossed squares). The lines in the ultrasound experiment simply connect the data points, whereas for the stirred experiment the lines correspond to a modeled fit (see text).

The detailed composition, referring to classes of compounds, is shown for C6 in Figure 9.3 with and without precolumn hydrogenation. In addition to paraffins, there are olefins—mainly with terminal double bond—and small amounts of alcohols (and aldehydes). The low detection limit of gas chromatography (GC) analysis allows precise determination even of minor compounds and provides exhaustive composition data also for use in kinetic modeling. Because of the short sampling duration of ca. 0.1 s,8 time-resolved selectivity data are obtained. 

The data modeled are from gas chromatograms obtained for Aroclors 1242, 1248, 1254 and 1260. The unknown samples are from the anaysis of used transformer oil obtained from a waste dump in New Jersey. The concentration of individual isomers in selected Aroclor and transformer oil samples are given in Appendix I. The data are organized in a matrix in which the first four data entries for each sample in row 1 of the data array (Table 2, Apendix I) designate the composition of the sample. For standards, these four variables represent the fractional parts of Aroclor 1242, 1248, 1254, or 1260, respectively, that were combined. Results from the analysis of transformer oil (samples 21-23) are of unknown fractional composition and variables 1 through 4 are null entries. In the examples that follow data from samples analyzed (Table 1, Appendix I) were used in part or in total to illustrate the PLS method. 

So from electrical data, it is possible to get information on partial thermodynamic functions of the salt and then develop thermodynamic models for quantitative interpretation of the conductivity variation with composition. These models are not very different from those already developed for molten salt mixtures or metallic alloys. 

Very few of the references in Tables 1-3 attempt any quantitative modelling of their NMR data in terms of cell microstructure or composition. Such models would be extremely useful in choosing the optimum acquisition pulse sequences and for rationalising differences between sample batches, varieties and the effects of harvesting times and storage conditions. The Numerical Cell Model referred to earlier is a first step in this direction but more realistic cell morphologies could be tackled with finite element and Monte Carlo numerical methods. 

Fig. 3.46 (a) The temperature of melting and decomposition peaks of NaBH for all composites from Figs. 3.44 and 3.45 as a function of the MgH content. Numbers in parenthesis show the number of multiple data points for this specific composition. Data for the (NaBH + Xwt%Mg) mixtures where Xwt% is an equivalent amount of Mg corresponding to the Mg decomposed from Xwt%MgHj are also included, (b) Analysis of the yield of MgB based on the Vajo et al. [196-198] model adopted for the (NaBH + MgH ) system... 

These types of models, while incomplete, are steps toward the formulation of composite models, which depend on future availability of compositional data. Moreover, these structural models are an important aid in understanding the interactions between anthropogenic chemicals and terrestrial organic matter. However, due to the heterogeneity of humic substances in the environment, provision of an exact, general structure does not seem feasible. 

Yeo, et al. [23,24] went on to make more complete studies of modulus-composition data using cross-poly(n-butyl acrylate)-Inter-cross-polystyrene, PnBA/PS, see Figure 6. Both the Davies and the Budlansky models fit reasonably well over wide ranges of composition, especially the Budlansky model. Other models, which in one form or another assume one continuous and one disperse phase, fit much less well. 

Figures 6-12 and 6-13 shows plots of copolymer composition and propagation rate constant, respectively, versus comonomer feed composition for styrene-diethyl fumarate copolymerization at 40°C with AIBN [Ma et al., 2001]. The system follows well the implicit penultimate model. The copolymer composition data follow the terminal model within experimental error, which is less than 2% in this system. The propagation rate constant shows a penultimate effect, and the results conform well to the implicit penultimate model with si = 0.055, S2 — 0.32.

The ability to determine which copolymerization model best describes the behavior of a particular comonomer pair depends on the quality of the experimental data. There are many reports in the literature where different workers conclude that a different model describes the same comonomer pair. This occurs when the accuracy and precision of the composition data are insufficient to easily discriminate between the different models or composition data are not obtained over a wide range of experimental conditions (feed composition, monomer concentration, temperature). There are comonomer pairs where the behavior is not sufficiently extreme in terms of depropagation or complex participation or penultimate effect such that even with the best composition data it may not be possible to conclude that only one model fits the composition data [Hill et al., 1985 Moad et al., 1989]. 

The sequence distributions expected for the different models have been described [Hill et al., 1982, 1983 Howell et al., 1970 Tirrell, 1986] (Sec. 6-5a). Sequence distributions obtained by 13C NMR are sometimes more useful than composition data for discriminating between different copolymerization models. For example, while composition data for the radical copolymerization of styrene-acrylonitrile are consistent with either the penultimate or complex participation model, sequence distributions show the penultimate model to give the best fit. 

Among the multivariate statistical techniques that have been used as source-receptor models, factor analysis is the most widely employed. The basic objective of factor analysis is to allow the variation within a set of data to determine the number of independent causalities, i.e. sources of particles. It also permits the combination of the measured variables into new axes for the system that can be related to specific particle sources. The principles of factor analysis are reviewed and the principal components method is illustrated by the reanalysis of aerosol composition results from Charleston, West Virginia. An alternative approach to factor analysis. Target Transformation Factor Analysis, is introduced and its application to a subset of particle composition data from the Regional Air Pollution Study (RAPS) of St. Louis, Missouri is presented. 

Derived from CMC and compositional data. l Model calculation from Equation 1. Regular solution model. 

The release kinetics of polyelectrolyte-containing controlled release compositions were modeled by Ozturk et al. [331]. According to this analysis the drug release rate depends on intrinsic solubilities as well as pKa values of the drug and polymer. Explicit relationships between release rates and these factors were derived, resulting in successful predictions of experimental data. 

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