Copolymers descriptions

Since PEC is a copolymer, description of its interaction with PS is more complex than if it were simply a homopolymer. Binary interaction models have been presented which suggest that copolymer miscibility with a homopolymer can be enhanced by endothermic interactions between the unlike repeat units of the copolymer (11-13). In its simplest form (11). the binary interaction model for the heat of mixing, AHaix, of a copolymer. A, containing repeat units 1 and 2, with a homopolymer, B, containing repeat units 3, is given by... 

Figure 11. Isotachopherograms of poly(aeryleunide-co-acrylate) of different compositions euid injected at identical acrylate concentration of 0.02 tiq/tii. acrylate. Copolymer descriptions given in Tcd)le V. Leading electrolyte (L) 0.01 H Cl , pH 8.2...

In the discussion of these combined topics, we use statistics extensively because the description of microstructure requires this kind of approach. This is the basis for merging a discussion of copolymers and stereoregular polymers into a single chapter. In other respects these two classes of materials and the processes which produce them are very different and their description leads us into some rather diverse areas. 

Instead of devoting more space to copolymers, we turn next to stereoregular polymers, in which many of the descriptions of microstructure developed in Sec. 7.6 can also find application. 

A description of modified ethylene—tetrafluoroethylene copolymers and their classification is given by the American Society for Testing and Materials under the designation D3159-83 (36). A comprehensive listing of industrial and military specifications is avaHable (37). 

Olig omerization and Polymerization. Siace an aHyl radical is stable, linear a-olefins are not readily polymerized by free-radical processes such as those employed ia the polymerization of styrene. However, ia the presence of Ziegler-Natta catalysts, these a-olefins can be smoothly converted to copolymers of various descriptions. Addition of higher olefins during polymerization of ethylene is commonly practiced to yield finished polymers with improved physical characteristics. 

Poly(ethyl methacrylate) (PEMA) yields truly compatible blends with poly(vinyl acetate) up to 20% PEMA concentration (133). Synergistic improvement in material properties was observed. Poly(ethylene oxide) forms compatible homogeneous blends with poly(vinyl acetate) (134). The T of the blends and the crystaUizabiUty of the PEO depend on the composition. The miscibility window of poly(vinyl acetate) and its copolymers with alkyl acrylates can be broadened through the incorporation of acryUc acid as a third component (135). A description of compatible and incompatible blends of poly(vinyl acetate) and other copolymers has been compiled (136). Blends of poly(vinyl acetate) copolymers with urethanes can provide improved heat resistance to the product providing reduced creep rates in adhesives used for vinyl laminating (137). 

Copolymers of acrylonitrile [107-13-1] are used in extmsion and molding appHcations. Commercially important comonomers for barrier appHcations include styrene and methyl acrylate. As the comonomer content is increased, the permeabiUties increase as shown in Figure 3. These copolymers are not moisture-sensitive. Table 7 contains descriptions of three high nitrile barrier polymers. Barex and Cycopac resins are mbber-modified to improve the mechanical properties. 

Yet another variant of self-assembly relies on the repulsion between blocks of suitably constituted block copolymers, leading to fine-scale patterns of organisation. One very recent description of this approach is by de Rosa et al. (2000). Details of this kind of approach as cultivated at Oak Ridge National Laboratory can also be found on the internet (ORNL 2000). 

Sedimentation systems with the particle size of 0.1-1.0 pm were obtained by mechanical dispersion of the copolymers with subsequent washing, fractionation, and separation of fractions. Micrograin forms (gr.) were synthesized by suspension copolymerization and fractionated. For the description of properties of weakly swelling and weakly dissociating gels, Katchalsky [36] has proposed an equation which contains the electrostatic potential eip... 

Tire simplest model for describing binary copolyinerization of two monomers, Ma and Mr, is the terminal model. The model has been applied to a vast number of systems and, in most cases, appears to give an adequate description of the overall copolymer composition at least for low conversions. The limitations of the terminal model generally only become obvious when attempting to describe the monomer sequence distribution or the polymerization kinetics. Even though the terminal model does not always provide an accurate description of the copolymerization process, it remains useful for making qualitative predictions, as a starting point for parameter estimation and it is simple to apply. 

The distinctive properties of densely tethered chains were first noted by Alexander [7] in 1977. His theoretical analysis concerned the end-adsorption of terminally functionalized polymers on a flat surface. Further elaboration by de Gennes [8] and by Cantor [9] stressed the utility of tethered chains to the description of self-assembled block copolymers. The next important step was taken by Daoud and Cotton [10] in 1982 in a model for star polymers. This model generalizes the... 

Polymer products synthesized in laboratories and in industry represent a set of individual chemical compounds whose number is practically infinite. Macro-molecules of such products can differ in their degree of polymerization, tactici-ty, number of branchings and the lengths that connect their polymer chains, as well as in other characteristics which describe the configuration of the macromolecule. In the case of copolymers their macromolecules are known to also vary in composition and the character of the alternation of monomeric units of different types. As a rule, it is impossible to provide an exhaustive quantitative description of such a polymer system, i.e. to indicate concentrations of all individual compounds with a particular chemical (primary) structure. However, for many practical purposes it is often enough to define a polymer specimen only in terms of partial distributions of molecules for some of their main characteristics (such as, for instance, molecular weight or composition) avoiding completely a... 

The microheterogeneity coefficient was introduced only for the description of the microstructure of binary copolymers with symmetric units. At increased number of unit types and/or when account is taken of structural isomerism, the role of Km will be performed by other parameters analogous to it. A general strategy for the choice of these latter has been elaborated in detail [12], while their values have been measured via NMR spectroscopic techniques for a variety of polycondensation polymers [13]. 

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 exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

Somewhat more complicated is the Markov chain describing the products of polycondensation with participation of asymmetric monomers. Any of them, AjSaAj, comprises a tail-to-head oriented monomeric unit Sa. It has been demonstrated [55,56] that the description of molecules of polycondensation copolymers can be performed using the Markov chain whose transient states correspond to the oriented units. A transient state of this chain ij corresponds to a monomeric unit at the left and right edge of which the groups A, and A are positioned, respectively. A state ji corresponds here to the same unit but is oriented in the opposite direction. However, a drawback of this Markov chain worthy of mention is the excessive number of its states. 

In what follows, we use simple mean-field theories to predict polymer phase diagrams and then use numerical simulations to study the kinetics of polymer crystallization behaviors and the morphologies of the resulting polymer crystals. More specifically, in the molecular driving forces for the crystallization of statistical copolymers, the distinction of comonomer sequences from monomer sequences can be represented by the absence (presence) of parallel attractions. We also devote considerable attention to the study of the free-energy landscape of single-chain homopolymer crystallites. For readers interested in the computational techniques that we used, we provide a detailed description in the Appendix. ... 

At the mesoscopic level of description the Landau-Ginzburg model of the phase transitions in diblock copolymer system was formulated by Leibler [36]... 

Thus, as can be inferred from the foregoing, the calculation of any statistical characteristics of the chemical structure of Markovian copolymers is rather easy to perform. The methods of statistical chemistry [1,3] can reveal the conditions for obtaining a copolymer under which the sequence distribution in macromolecules will be describable by a Markov chain as well as to establish the dependence of elements vap of transition matrix Q of this chain on the kinetic and stoichiometric parameters of a reaction system. It has been rigorously proved [ 1,3] that Markovian copolymers are formed in such reaction systems where the Flory principle can be applied for the description of macromolecular reactions. According to this fundamental principle, the reactivity of a reactive center in a polymer molecule is believed to be independent of its configuration as well as of the location of this center inside a macromolecule. 

For many synthetic copolymers, it becomes possible to calculate all desired statistical characteristics of their primary structure, provided the sequence is described by a Markov chain. Although stochastic process 31 in the case of proteinlike copolymers is not a Markov chain, an exhaustive statistic description of their chemical structure can be performed by means of an auxiliary stochastic process 3iib whose states correspond to labeled monomeric units. As a label for unit M , it was suggested [23] to use its distance r from the center of the globule. The state of this stationary stochastic process 31 is a pair of numbers, (a, r), the first of which belongs to a discrete set while the second one corresponds to a continuous set. Stochastic process ib is remarkable for being stationary and Markovian. The probability of the transition from state a, r ) to state (/i, r") for the process of conventional movement along a heteropolymer macromolecule is described by the matrix-function of transition intensities... 

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