Chain length distribution function

Fig.8 Polymer chain length distribution function P(N) for various polymer concentrations. P(N) has been norma-lized(EP(N) = 1) (DNP)...

It is challenging to derive chain length distribution functions for branching and aoss-linking systems. In such cases, we often resort to the use of the method of moments.The ith moments of radical and dead polymer chains are defined as... 

In the two-site model, it is assumed that the catalytic sites have the same polymerization activity (propagation activity) but they differ in their chain transfer capabilities. The two-site model is the simplest of the multisite model and its main advantage is that the number of adjustable parameters is minimal. The weight chain length distribution function of the two-site model takes the following form ... 

From expression (2.24) it follows that even at asymptotically large chain length, distribution function is not Gaussian, which in Fourier presentation looks as follows ... 

For the transformation of the macrocomposite model to a molecular composite model for the ultimate strength of the fibre the following assumptions are made (1) the rods in the macrocomposite are replaced by the parallel-oriented polymer chains or by larger entities like bundles of chains forming fibrils and (2) the function of the matrix in the composite, in particular the rod-matrix interface, is taken over by the intermolecular bonds between the chains or fibrils. In order to evaluate the effect of the chain length distribution on the ultimate strength the monodisperse distribution, the Flory distribution, the half-Gauss and the uniform distribution are considered. 

In the previous sections the theoretical relations describing the strength of a polymer fibre as a function of the intrinsic parameters, such as the chain modulus, the modulus for shear between adjacent chains, the orientation distribution and the chain length distribution, have been discussed. In this section the dependence of the strength on the time and the temperature will be investigated. 

The length and the distribution of chain lengths are functions of the temperature, pressure, residence time, catalyst characteristics, and the proportion of ethylene present in the reaction, A measure of this is the mole ratio of ethylene, which measures the weight of ethylene compared to the weight of triethyl aluminum in scales related to their atomic weights. As an example, Table 15-2 shows how the distribution of chain lengths can vary, using different mole ratios of ethylene to triethyl aluminum. 

In PB networks, the variation of the line shape as a function of the applied stress was interpreted in terms of a chain length distribution. Shorter chains may be more oriented than longer ones, at a given elongation [18], which may lead to a non-affine behaviour at the chain scale. The question of the spatial scale to which the deformation is affinely transmitted, has been investigated intensively by small angle neutron scattering [64]. However, it may happen as well that the chain portions close to junction points are more oriented (have a more restricted mobility) than those in the middle of the chains [19]. 

For a nonodisperse polymer. My = M = M for a polydisperse polymer the two kinds of averages are not equal but are related by a constant factor that depends on the distribution function F M) and the parameter a. A commonly encountered distribution function, and one that is likely to be valid for PVOH, is one that arises if the probability of a chain-termination reaction during the polymerization is constant with time and independent of the chain length already achieved. It is also the most likely function for the product resulting from cleavage with periodate if the head-to-head structures can be assumed to be randomly distributed along the PVOH chains. This distribution function is... 

The number of monomeric units in one average macromolecule of a polymer sample in which the chain length distribution is represented by the function F(n), is expressed by... 

The reactions within the DR and AC series are given by the probabilities (1 — x ) and (1 — y ), respectively. Because the chemical character of the intermediates is not a function of chain length the probabilities are assumed to be independent of n, therefore x = xandy = y. In this model, due to the branching of the photochemical reactions, the intermediates are not expected to become infinitely long. Thus, a distribution of chain lengths will be obtained for the DR intermediates, different from those of the AC and SO intermediates. The different chain length distributions — obtained under stationary conditions — are dependent on the probabilities x and y and on the different rate constants k and They follow from the kinetic equations... 

The main feature of polymers is their MMD, which is well known and understood today. However, several other properties in which the breadth of distribution are important and influence polymer behavior (see Figure 1) include physical, the classical chain-length distribution chemical, two or more comonomers are incorporated in different fractions topological, polymer architecture may differ (e.g., linear, branched, grafted, cyclic, star or comb-like, and dendritic) structural, comonomer placement may be random, block, alternating, and so on and functional, distribution of chain functions (e.g., all chain ends or only some carry specific groups). Other properties the polymers may disperse (tacticity and crystallite dimensions) are not of the same general interest or cannot be characterized by solution methods. 

Henry constant for absorption of gas in liquid Free energy change Heat of reaction Initiator for polymerization, modified Bessel functions, electric current Electric current density Adsorption constant Chemical equilibrium constant Specific rate constant of reaction, mass-transfer coefficient Length of path in reactor Lack of fit sum of squares Average molecular weight in polymers, dead polymer species, monomer Number of moles in electrochemical reaction Molar flow rate, molar flux Number chain length distribution Number molecular weight distribution... 

Abstract This article summarizes recent developments in the synthesis of polypeptides and hybrid peptide copolymers. Traditional methods used to polymerize -amino acid-N-carboxyanhydrides (NCAs) are described, and limitations in the utility of these systems for the preparation of polypeptides are discussed. Recently developed initiators and methods are also discussed that allow polypeptide synthesis with good control over chain length, chain length distribution, and chain-end functionality. The latter feature is particularly useful for the preparation of polypeptide hybrid copolymers. The methods and strategies for the preparation of such hybrid copolymers are described, as well as analysis of the synthetic scope of the different methods. Finally, issues relating to obtaining these highly functional copolymers in pure form are detailed. 

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