Copolymer statistical analysis

For a statistical analysis of copolymer sequences, different mathematical techniques are used. For mathematically oriented researchers, a copolymer sequence might be considered as a string of symbols whose correlation structure can be characterized completely by all possible monomer-monomer correlation functions. Since the correlations at long distances are typically small, it is important to use the best possible estimates to measure the correlations, otherwise the error due to a finite sample size can be as large as the correlation value itself. 

Synthesis of block copolymer GPC analysis of the polymerization solution indicated that all the monomer was consumed. The presence of unconsumed initiator was observed. PEG tosylate eluted at 26.2 ml and had UV/ARI of 8000. The chromatogram of a solution of run 1 is given in Figure la. The peak at 24.2 ml corresponds to that of the block copolymer. The peak at 26.2 ml has UV/ARI a. 8000 and should be the unreacted PEG ditosylate. Figure lb shows the elution profile of the of the acetone-washed copolymer and it is free from PEG tosylate. The presence of unconsumed initiator indicated slow initiation. Also, since the two ends of PEG tosylate are identical in reactivity by random statistics (12) we expect the copolymer to be a mixture of ABA type trlblock (initiation at both ends) and AB type dlblock (initiation at one end) copolymers. This was observed, as shown by the solid line of Figure 2 when the copolymer was eluted through a different combination of columns. The peak at 24.0 ml corresponds to the triblock copolymer and the peak at 25.4 ml corresponds to the dlblock copolymer. 

The following section will describe the principal treatment of different types of molecular heterogeneity. Emphasis will be on the analysis of functional homopolymers (telechelics, macromonomers), copolymers (statistical, block, graft), and polymer blends. A detailed description of all experimental steps will be given for one or two representative examples of each group. Information on the application of the described procedures to different heteropolymer structures will be summarized briefly in the Sect. 5.1. 

Statistical analysis of the generated copolymer sequences revealed that one can effectively control the degree of randomness by simply varying the value of feBi7 eAA -... 

There is a large body of published monomer reactivity ratios summarized in reference 5. Values are typically determined from a series of low conversion experiments in which copolymer composition Fp is measured (e.g., by NMR) as a function of monomer composition, and ri and T2 estimated from fitting the data set according to Equation 3.35. The estimation is best accomplished by non-linear techniques, and a statistical analysis... 

The purpose of this chapter, therefore, is to describe the basic concepts of the statistical analysis of copolymer sequence distribution. The necessary relationships between various comonomer sequence abundances are introduced, along with simple statistical models based on monomer addition probabilities. The relationships between the statistical models and propagation models based on reactivity ratios are discussed. The use of these models is then illustrated by means of selected examples. Techniques for extracting sequence information from in situ NMR measurements are also described. Finally, the statistical analysis of chemically modified polymers is introduced with examples. 

The development of statistical analyses of polymers was pioneered by Bovey [1,2] and Price [3] and has also been discussed in detail by Randall [4]. Here, the intention is to discuss only the principal elements of copolymer statistics required for the analysis of the various sequence abundances observed using NMR spectroscopy. For the full derivation of copolymer statistics, the interested reader is referred to the original work of these authors. It should also be noted that most of what follows is limited to A/B copolymer systems. Whilst it is relatively straightforward to extend the analysis to copolymers containing more than two comonomers, the number of equations required to define the system increases dramatically. Furthermore, for such systems, it proves increasingly difficult to assign their NMR spectra in the detail required for complete analysis. 

In order to rank performance of catalysts in copol)unerization with respect to comonomer incorporation and comonomer sequence distribution, copolymerization parameters have proven to be very useful. They are determined by means of nuclear magnetic resonance spectroscopy (NMR copolymer sequence analysis) taking into account the Markovian statistics of chain growth, as reviewed by Randall [4], Galimberti and coworkers [5] described the analysis of EPM prepared by means of... 

Catania (Italy) Author of more than 50 publications and of 18 international invited lectures. He is currently working in the field of characterization of polymers and copolymers. He is an editorial board member of Rapid Communications in Mass Spectrometry. Research interests Structural characterization of polymers by mass-spectrometric techniques MALDI for the analysis of polymers and copolymers chain statistics applied to copolymer sequence analysis MonteCarlo simulations Bivariate distributions of chain size, and composition in high conversion copolymers. 

Calculations. The statistical analysis of the copolymer sequencing was performed in terms of Markov chain theory, using the 2nd order Markov chain model. The equations were numerically solved with respect to the conditional probabilities by least squares minimization of the deviations of calculated and measured pentad contributions. 

I can thus conclude by recalling what I said at the beginning of my talk, namely that the future of these studies, if any, lies in the possibility of developing more and more refined experimental techniques able to give a detailed description of the sequence distribution in copolymers and at present only NMR and to a minor extent IR spectroscopy fulfil these requirements then I hope that the new concepts of copolymer statistics discussed here could represent, for the scientists dealing with structural analysis of chemical and steric copolymers by NMR, a wider framework for representing and correlating their data. 

Table 21.4 Quantification of elementary reaction processes through statistical analysis of AFM images of isolated 4-arm polystyrene comb stars (Reprinted with permission from M. Schappacher and A. Deffieux, AFM image analysis applied to the investigation of elementary reactions in the synthesis of comb star copolymers, Macromolecules, 38, 4942—4946, 2005 2005 American Chemical Society.)...

Several workers have discnssed different aspects of the technique [1-6] covering design of apparatns [1, 4], robotic interfaces [2], polymer degradation, statistical analysis of data [4] and end-gronp analysis in copolymers [5]. 

Precision—The precision of this test method was determined by statistical analysis of inteilaboratory results. In this study, dilution solvents were limited to xylene or kerosine. Some laboraUnies chose to use Babington-type nebulizers, peristaltic pumps, and background correction. Fourteen laboratories analyz twelve specimens in duplicate. The samples were one gas turbine used oil, four gasoline engine used oils, two truck diesel engine used oils, two marine ei e used oils, SRM 1085 diluted in SRM 1083 (base oil) to contain approximately 40 mg/kg of eleven different metals (this oil also contained 8 mass % of an ethylene-propylene copolymer viscosity index improver), SRM 1085 diluted in SRM 1083 to contain approximately 40 mg/kg of twelve different metals, SRM 1085 diluted in SRM 1083 to contain approximately 2 mg/kg of 12 different metals. 

For a complete description of the monomer sequence distribution and relative stereochemical configuration of monomer sequences, at least in terms of N- and M-centred triads, it is necessary to take into consideration as many as 10 different triads with a central N unit, which may be magnetically distinguishable as is shown in the scheme of Figure 7.22. Since both monomeric repeating units have a quaternary pseudoasymmetric carbon, from a stereochemical point of view 10 triads with a central M unit sensitive to tacticity must also be considered. The statistical analysis of the monomer sequence distribution and of the stereochemical configuration of the copolymer sequences was carried under the following assumptions with respect to the chemical composition of copolymer sequences, it was assumed that... 

A general purpose program has been developed for the analysis of NMR spectra of polymers. A database contains the peak assignments, stereosequence names for homopolymers or monomer sequence names for copolymers, and intensities are analyzed automatically in terms of Bernoullian or Markov statistical propagation models. A calculated spectrum is compared with the experimental spectrum until optimized probabilities, for addition of the next polymer unit, that are associated with the statistical model are produced. 

Statistical and block copolymers based on ethylene oxide (EO) and propylene oxide (PO) are important precursors of polyurethanes. Their detailed chemical structure, that is, the chemical composition, block length, and molar mass of the individual blocks may be decisive for the properties of the final product. For triblock copolymers HO (EO) (PO)m(EO) OH, the detailed analysis relates to the determination of the total molar mass and the degrees of polymerization of the inner PPO block (m) and the outer PEO blocks (n). 

The copolymerization of lactones took place through enzyme catalysis [92]. The copolymerization of e-CL with d-VL catalyzed by lipase PF affords the corresponding copolymer having a molecular weight of several thousand. From 13C NMR analysis, the copolymer was found to be of random structure having both units, suggesting the frequent occurrence of transesterifications between the polyesters. In the copolymerization of 8-OL with e-CL or DDL, random copolyesters were also formed [84], whereas the copolymer from e-CL and PDL was not statistically random [88]. 

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