Average composition distribution

When deriving this expression for the average composition distribution, authors of paper [74] entirely neglected its instantaneous constituent, having taken (as is customary in the quantitative theory of radical copolymerization [3,84]) the Dirac delta-function < ( -X) as the instantaneous composition distribution. Its averaging over conversions, denoted hereinafter by angular brackets, leads to formula (Eq. 101). Note, this formula describes the composition distribution only provided copolymer composition falls in the interval between X(0) and X(p). Otherwise, this distribution function vanishes at all values of composition lying outside the above-mentioned interval. 

The copolymer composition equation only provides the average composition. Not all chains have the same composition. There is a statistical distribution of monomers determined by the reactivity ratios. When chains are short, compositional heterogeneity can mean that not all chains will contain all monomers. 

Gel Permeation Chromatography (CPC) is often the source of molecular wei t averages used in polymerization kinetic modelling Q.,2). Kinetic models also r uire measurement of molecular weight distribution, conversion to polymer, composition of monomers in a copolymerization rea tion mixture, copolymer composition distribution, and sequence length distribution. The GPC chromatogram often reflects these properties (3,. ... 

It should be emphasized that for Markovian copolymers a knowledge of the values of structural parameters of such a kind will suffice to find the probability of any sequence Uk, i.e. for an exhaustive description of the microstructure of the chains of these copolymers with a given average composition. As for the composition distribution of Markovian copolymers, this obeys for any fraction of Z-mers the Gaussian formula whose covariance matrix elements are Dap/l where Dap depend solely on the values of structural parameters [2]. The calculation of their dependence on time, and the stoichiometric and kinetic parameters of the reaction system permits a complete statistical description of the chemical structure of Markovian copolymers to be accomplished. The above reasoning reveals to which extent the mathematical modeling of the processes of the copolymer synthesis is easier to perform provided the alternation of units in macromolecules is known to obey Markovian statistics. 

An average compositional spectrum as well as two-dimensional element distribution maps were acquired simultaneously from each scanned area. Line spectra could also be obtained by allowing the microbeam to scan along a line across the crystal face. The authors examined the base crystal, the smaller crystal as well as the interface region. 

The effects of composition distribution on the morphology of PS-fc-P2VP diblock copolymers were investigated by Matsushita et al. [160]. They produced PS- -P2VP samples with various composition distributions but with constant average composition by blending. If the polydispersity indices of each block were lower than 1.7, the expected lamellar domains were detected (Fig. 49). 

At low-conversion copolymerization in classical systems, the composition of macromolecules X whose value enters in expression (Eq. 69) does not depend on their length l, and thus the weight composition distribution / ( ) (Eq. 1) equals 5(f -X°) where X° = jt(x°). Hence, according to the theory, copolymers prepared in classical systems will be in asymptotic limit (/) -> oo monodisperse in composition. In the next approximation in small parameter 1/(1), where (/) denotes the average chemical size of macromolecules, the weight composition distribution will have a finite width. However, its dispersion specified by formula (Eq. 13) upon the replacement in it of l by (l) will be substantially less than the dispersion of distribution (Eq. 69)... 

Fig. 8 Dependencies on conversion p of monomer mixture composition x (a), instantaneous X (b) and average X) (c) copolymer composition as well as dispersion a2 (d) of the composition distribution calculated at different values of the initial compositions of monomers x°. The calculations have been carried out at values of parameters a and a2 = 1 - fli (Eq. 100) equal to 0.3 and 0.7, respectively... 

The diode array UV/vis spectrophotometer was used to both Identify the polymer exiting and to obtain a quantitative analysis of the copolymer composition distribution. Figure 9 (6) shows the result of summing many individual fraction analyses to see the total copolymer composition distribution. The result had the correct average composition but not the skewed shape expected from theory. Part of the difficulty was the relatively small number of cross fractionations done. 

Fig. 14. Ternary diagram similar to that in Figure 4 showing the distribution of compositions of corroded metallic particles from an LWR fuel sample. Corrosive loss of Mo makes the average composition more Pd-rich.

Tagata and Homma47 analyzed in the aforementioned manner the compositional heterogeneity of two typical commercial SBR samples, E-SBR and S-SBR, which had 23.5 and 20.0 average styrene wt.%, and were produced by an emulsion polymerization and by a solution polymerization with an organometallic catalyst, respectively. The result was that the former gave a distinctly narrower compositional distribution than the latter. GPC experiments on these samples were also carried out, which indicated that the above situation was just the opposite for the molecular-weight distributions. 

The equivalence ratio can be calculated from the Mark-Houwink coefficient, K, of component homopolymers. The composition distribution in the chromatogram of a block copolymer is negligible. The peak point of a block copolymer chromatogram corresponds to the average structure of the polymer. Thus, analysis of block copolymers is reduced to analysis of the peak point. Analyses of anionic block copolymer structures have been successfully accomplished by this peak analysis technique with the aid of equivalence ratio. 

The reason for this discrepancy can be deduced from Fig. 30 all the constituents differing in chemical composition had approximately equal SEC distributions. Dual detection SEC can provide only information on the average composition of the eluted portions. The broad but symmetric CCD of the sample yield roughly constant values for the average at any point of the SEC curve for the whole sample. The investigation is a fine demonstration for the advantage of cross-fractionation over dual-detection... 

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