Copolymers chain statistics

Hence, the stated above results have shown, that conversion degree and the reduced viscosity, obtained in PUAr synthesis process, are a funetion of copolymer chain statistical flexibility the more rigid chain is, the higher Q and tired are. The fractal analysis methods allow to make this correlation quantitative treatment. From the ehemieal point of view the values Q and tired depend on eomonomers functional groups activity % The higher % is, the larger the values Q and tired are. The value also defines a synthesized copolymer type. 

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. 

Functional and end-functional polymers are precursors to block and graft copolymers and, in some cases, polymer networks. Copolymers with in-chain functionality may be simply prepared in copolymerizations by using a functional monomer. However, obtaining a desired distribution requires consideration of the chain statistics and, for low molecular weight polymers, the specificity of the initiation and termination processes, l hese issues are discussed in Section 7.5.6... 

In terms of a statistical approach to every macromolecule of a copolymer specimen there can be put in correspondence a certain realization of a stochastic process of conventional movement along a copolymer chain. This movement can be conveniently thought of as a sequence of random transitions from a unit of the chain to the neighboring one. Here the probability of finding a monomeric unit of a particular type at every step is predetermined by the stochastic process which describes the particular polymer specimen. To consider the set of trajec-... 

Statistical characteristics of the second type define the microstructure of copolymer chains. The best known characteristics in this category are the fractions P [/k) (probabilities) of sequences Uk involving k monomeric units. The simplest among them are the dyads U2, the complete set of which, for example, for a binary copolymer is composed of four pairs of monomeric units M2M, M2M2. The number of the types of k-ad in chains of m-component copolymers grows exponentially as mk so that with practical purposes in mind it is generally enough to restrict the consideration to sequences Uk] with moderate values of k. Their calculation turns out to be rather useful... 

Monomer concentrations Ma a=, ...,m) in a reaction system have no time to alter during the period of formation of every macromolecule so that the propagation of any copolymer chain occurs under fixed external conditions. This permits one to calculate the statistical characteristics of the products of copolymerization under specified values Ma and then to average all these instantaneous characteristics with allowance for the drift of monomer concentrations during the synthesis. Such a two-stage procedure of calculation, where first statistical problems are solved before dealing with dynamic ones, is exclusively predetermined by the very specificity of free-radical copolymerization and does not depend on the kinetic model chosen. The latter gives the explicit dependencies of the instantaneous statistical characteristics on monomers concentrations and the rate constants of the elementary reactions. 

Statistical co-crystallization of different constitutional repeating units, which may either belong to the same copolymer chains (copolymer isomorphism) or originate from different homopolymer chains (homopolymer isomorphism). 

For the statistical copolymer the distribution may follow different statistical laws, for example, Bemoullian (zero-order Markov), first- or second-order Markov, depending on the specific reactants and the method of synthesis. This is discussed further in Secs. 6-2 and 6-5. Many statistical copolymers are produced via Bemoullian processes wherein the various groups are randomly distributed along the copolymer chain such copolymers are random copolymers. The terminology used in this book is that recommended by IUPAC [Ring et al., 1985]. However, most literature references use the term random copolymer independent of the type of statistical distribution (which seldom is known). 

The copolymer described by Eq. 6-1, referred to as a statistical copolymer, has a distribution of the two monomer units along the copolymer chain that follows some statistical law, for example, Bemoullian (zero-order Markov) or first- or second-order Markov. Copolymers formed via Bemoullian processes have the two monomer units distributed randomly and are referred to as random copolymers. The reader is cautioned that the distinction between the terms statistical and random, recommended by IUPAC [IUPAC, 1991, in press], has often not been followed in the literature. Most references use the term random copolymer independent of the type of statistical process involved in synthesizing the copolymer. There are three other types of copolymer structures—alternating, block, and graft. The alternating copolymer contains the two monomer units in equimolar amounts in a regular alternating distribution ... 

This section deals with the effect of monomeric sequences in copolymer chains upon TLC separation. A possibility of separating copolymers by the difference in their chain architectures was first demonstrated by Kamiyama et al. S9 For the preliminary TLC experiment they used copolymers composed of styrene and methyl methacrylate, for the reason that this comonomer pair is endowed with the possibility of being polymerized to three different chain architectures, namely, alternating61 , statistical, and block. 

Zheligovskaya et al. [55] have simulated the adsorption of quasirandom adsorption-tuned copolymers (ATC). The critical adsorption energy as well as some characteristics of the adsorbed single chains (statistics of trains, loops, and tails) were studied. All these properties were compared with those... 

Fig. 22 a Measured force curves of linear segmented poly(N-isopropylacrylamide-seg-styrene) (PNIPAM-seg-St) copolymer chains adsorbed on a hydrophobic PS substrate in water, b Statistics of the distance between two adjacent peaks in the measured force curves [97]... 

Metallocene catalysts show low r values, which allows easy incorporation of bulky cycloolefins into the growing copolymer chain. Surprisingly, the ethylene reactivity ratio in copolymerisation with cyclopentene in the presence of a (ThindCH2)2ZrCl2-based catalyst (r = 2.2) and in copolymerisation with norbornene in the presence of catalysts characterised by Cs and Ci symmetry (ri 3.4 and 3.1 respectively) is considerably lower than that for the copolymerisation of ethylene with propylene (r = 6.6 at 37 °C). Various catalysts produce copolymers of structures that are between statistical and alternating [468]. 

All the statistical characteristics of copolymer chain structure and composition inhomogeneity, (including the ones reported in the above papers) can be easily calculated by means of the Markov chain formalism for any of kinetic models presented in Sect. 2. Then it does not seem advisable for the solution of such problems to apply the Monte-Carlo method with which the simulation of the copolymer chain growth was carried out [83-93]. 

Figure 1 Macromolecular architectures linear macromolecular chains (homopolymer, block-copolymer and statistical copolymer [14]), brushed-polymer (= linear chains attached to a polymer-chain brush-polymer, brush-copolymers [14]), star polymer [4], mikto-star-polymer [16], arborescent graft polymer (=repeated grafting of linear chains on a macromolecule [17,18]), dendrimer (= maximally branched, regular polymer [15])...

A mixture of two monomers that can be homopo-lymerized by a metal catalyst can be copolymerized as in conventional radical systems. In fact, various pairs of methacrylates, acrylates, and styrenes have been copolymerized by the metal catalysts in random or statistical fashion, and the copolymerizations appear to also have the characteristics of a living process. The monomer reactivity ratio and sequence distributions of the comonomer units, as discussed already, seem very similar to those in the conventional free radical systems, although the detailed analysis should be awaited as described above. Apart from the mechanistic study (section II.F.3), the metal-catalyzed systems afford random or statistical copolymers of controlled molecular weights and sharp MWDs, where, because of the living nature, there are almost no differences in composition distribution in each copolymer chain in a single sample, in sharp contrast to conventional random copolymers, in which there is a considerable compositional distribution from chain to chain. Figure 26 shows the random copolymers thus prepared by the metal-catalyzed living radical polymerizations. 

The relative NMR intensities of triad sequences are a function of the run number and the mole fraction of the A or B siloxane units. Appropriate combinations of different NMR intensities provided expressions that lead to the run number. If the A and B units are statistically distributed along the copolymer chain, the run number will be... 

The simplest, from the viewpoint of topological structure, are the linear polymers. Depending on the number m of the types of monomeric units they differentiate homopolymers (ra=1) and copolymers (m>2). In the most trivial case molecules in a homopolymer are merely identified by the number Z of monomeric units involved, whereas the composition of a copolymer macromolecule is defined by vector 1 with components lly..., Za,..., Zm equal to the numbers of monomeric units of each type. At identical composition these molecules can vary in microstructure which is characterized by the manner of alternation of different units in a copolymer chain. Because the values of the average degree of polymerization l=lx+... +Zm in synthetic copolymers normally constitute 102-104 it becomes clear that the number of conceivable types of isomers with different microstructure turns out to be practically infinite. Naturally, a quantitative description of any polymer specimen comprising macromolecules with such an impressive number of configurations can be performed exclusively by statistical methods. 

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