Statistical models copolymers

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. 

In the past three decades, industrial polymerization research and development aimed at controlling average polymer properties such as molecular weight averages, melt flow index and copolymer composition. These properties were modeled using either first principle models or empirical models represented by differential equations or statistical model equations. However, recent advances in polymerization chemistry, polymerization catalysis, polymer characterization techniques, and computational tools are making the molecular level design and control of polymer microstructure a reality. 

In their statistical model for microphase separation of block copolymers, Leary and Williams (43) proposed the concept of a separation temperature Ts. It is defined as the temperature at which a first-order transition occurs when the domain structure is at equilibrium with a homogeneous melt, i.e.,... 

Fig. 21.1. Plots of melting point (TM) vs. 3HV mole fraction (Fv) of bacterially synthesized random P(3HB-co-3HV)s. The circles indicate T values for P(3HB-co-3HV) samples which show a single DSC melting peak. The T values indicated by triangles are those for P(3HB-co-3HV) samples which were bacterially synthesized as mixtures composed of two main copolymer components with different 3HV mole fractions, and so show well-resolved two DSC melting peaks. For these mixed samples, the 3HV contents of two components were determined by analyzing the solution H NMR spectra based on the statistical model (for details of spectral analysis, see Ref. [28]). Hence, for these samples, the Tm values are plotted against the 3HV fractions of the component copolymers. (Reproduced from Ref. [28] with permission.)...

Montaudo, M.S., Ballistreri, A., and Montaudo, G., Determination of Microstructure in Copolymers. Statistical Modeling and Computer Simulation of Mass Spectra, Macromolecules, 24, 5051 (1991). 

Chain statistics modeling of the mass spectral intensities of copolymers has been used to derive information on the distribution of monomers along the copolymer chain (Chapter 2), and an automated procedure to find the composition and the sequence of copolymers has been developed. 

The MS peak intensities of the PET/PEA samples reacted from 10 up to 270 min. were used to estimate the copolymer composition, the extent of exchange and the number average block lengths by the chain statistics modeling of the mass spectra of copolymers (Chapter 2). ... 

Montaudo, M. S. and Montaudo G., Further Studies on the Composition and Microstructure of Copolymers by Statistical Modeling of Their Mass Sp>ectra, Macromolecules, 25, 4264, 1992. 

In this work, coupled SEC-NMR analysis has been demonstrated for three samples of alginates. The NMR data have been treated with two-component 1 order Markov statistical models. The first component reflects a mostly G homopolymer, and the second con onent is a MG copolymer with an almost random or alternating sequence distribution. The relevance of the two components to the epimerization reactions has been noted. 

As an extension to this type of sequence information, it has become common practice to check for conformity of the observed sequence distribution to an appropriate statistical model. The rationale for this is that, as well as providing a more complete description of copolymer microstructure, conformity to a particular model can impart important information concerning the mechanism of polymer propagation. In addition, in favourable cases, the relative reactivities of the comonomers can also be established. The more traditional methods for assessing comonomer reactivities are significantly more laborious than NMR methods and often relatively crude. 

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. 

Other statistical models. The second-order Markov model is sometimes also applied to copolymers. Here, the probability of addition of a given monomer depends not only on the identity of the chain end monomer, but also on the nature of the preceding or penultimate monomer unit. As there are then four possible types of chain end to consider (namely, -AA, -AB, -BA, and -BB), there are eight addition probabilities which describe addition of the A and B monomers (e.g. Pg g represents the probability of B adding to a -BA chain end). As with the first-order Markov case, only half of these are independent because (Paaa + aab) = ( aba + abb) = Equations for... 

In the next section, examples are given of how these statistical models have been used to examine NMR-derived sequence information so revealing the details of copolymer microstructure and propagation. 

This section is not intended as an exhaustive review of the literature concerned with NMR studies of copolymers. Instead, examples have been chosen to reflect and reinforce the principles described in the preceding section. It seems logical here to treat these examples approximately in the order in which the statistical models were introduced. 

The ability of NMR spectroscopy to distinguish between dyads, triads, and higher n-ad sequences in copolymers makes it an especially powerful tool for the polymer chemist interested in the fine details of molecular structure. In this chapter, we have seen that simple expressions derived from the relative abundances of various sequence types allow characterisation of the copolymer microstructure in terms of number-average sequence lengths, without any recourse to a statistical model. A more detailed examination of copolymer... 

Statistical design of experiment (DOE) is an efficient procedure for finding the optimum molar ratio for copolymers having the best property profile. Based on the concepts of response-surface (RS) methodology, developed by Box and Wilson [11], there are four models or polynominals (Table III) useful in our study. For three components, in general, if there are seven to nine experimental data points, the linear, quadratic and special cubic will be applicable for use in predictions. If there are ten or more data points, the full cubic model will also be applicable. At the start of the effort, one prepares a fair number of copolymers with different AA IA NVP ratios and tests for a property one wishes to optimize, with the data fit to the statistical models. Based on the models, new copolymers, with different ratios, are prepared and tested for the desired property improvement. This type procedure significantly lowers the number of copolymers that needs to be prepared and evaluated, in order to identify the ratio needed to give the best mechanical property. 

Matrix-assisted laser desorption ionization MS is well represented in this area [56, 85-94]. Cox et al. [88] used MALDI-MS to characterize low molecular weight polyolefin copolymers of isobutylene and paramethylstyrene and to extract composition information from spectra. Comparison of experimental oligomer distributions to a Bemoullian statistical model revealed severe overrepresentation of oligomers with higher relative amounts of paramethylstyrene. However, good agreement between the model and experimental data was obtained by the introduction of an ionization efficiency term in the model used. The differences between average composition determined by MALDI and NMR were discussed. 

Figure 11.2 Degree of crystallinity (Wc) of statistical ethylene copolymers containing n-propyl or ethyl branches as a function of temperature. Solid lines represent copolymer crystallinity calculated based on measured specific volume 1 = 1.8 n-propyl branches/100 C 2 = 2.0 n-propyl branches/100 C 3 = 4.2 n-pro-pyl branches/100 C 4 = 4.6 n-propyl branches/100 C 6 = 6.8 n-propyl branches/100 C 7 = 7.7 n-propyl branches/100 C 8 = 7.3 ethyl branches/100 C. Dotted lines represent Flory s equilibrium model for copolymer crystallinity, as given in Equation (11.3). Reprinted from Reference [11]. Copyright 1963, with permission from Elsevier.

Statistics of Copolymer Chains. A triad distribution is capable of giving even more microstructural information after comparing an experimental distribution to a calculated distribution having specifically defined chain statistics (3,6,15,16). The calculated statistical distributions can be random or some higher level of a so-called Markovian chain. These models are constructed on the premise of a conditional probability of finding any particular unit or sequence in a copolymer chain. In a perfectly random statistical model, there are no previous conditions required for one unit to follow any other particular unit. These types of statistical models are called Bernoullian or zero-order Markovian. For these statistical models, only mole fraction information is required to determine what the corresponding triad distribution would look like for a perfectly random copolymer, that is. 

In the case of chiral homopolymers, the meso and racemic diads are treated statistically as if they were different comonomers. Some care has to be taken when calculating statistical models for chiral homopolymers because two contiguous units are required to identify meso or racemic configurations. There are examples in the literature where the order of a particular statistical model has been confused by the simple treatment of a chiral homopolymer as an m, r copolymer (6). 

Once the copolymerization parameters of a catalyst system with a given ratio of monomers are known, the composition of the copolymer and all its sequence distributions result, e.g., the complete microstructure of the copolymer is revealed. These parameters characterize any catalyst system and enable different systems to be compared. The major practical problem is the choice of the adequate statistical model. 

The experimental values of V- and M-centred triads from [ H] triads were compared with the theoretical values from Harwood s [171] statistical model using copolymerisation reactivity ratios. The reactivity ratios for free-radical solution copolymerisation of V with M were calculated using the Kelen-Tudos (KT) [172] and the nonlinear error in variables (EVM) [173] methods using the RREVM [174] program. Homonuclear H-2D-COSY and 2D-NOESY NMR of the copolymer sample were recorded for determining the interactions between different protons in the copolymer chain. 

Triad fractions determined from C-NMR spectra of copolymers. Triad fraction calculated using rV = 0.04 and rM = 7.28 in Harwood s statistical model... 

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