Statistical polymers

Kleinert H (2004) Path integrals in quantum mechanics, statistics, polymer physics, and financial markets. 3rd edition. World Scientific Singapore River Edge, NJ, p xxvi, 1468 p. For the quantum mechanical integral equation, see Section 1.9 For the variational perturbation theory, see Chapters... 

Kleinert, H., Path Integrals in Quantum Mechanics, Statistics, Polymer Physics and Financial Markets, (3rd edition), World Scientific, Singapore... 

Smirnova and coworkers studied the influence of various types of ionizing radiations on the physiomechanical characteristics of a statistical polymer of butadiene and acrylonitrile137. Although the polymer is a statistical polymer, the nature of its thermo-mechanical curve indicates a block nature of the polymeric basis of the rubber there is a... 

Shunmugam R, Tew GN. Efficient route to well-characterized homo, block, and statistical polymers containing terpyridine in the side chain. J Polym Sci Part A Polym Chem 2005 43 5831-5843. 

The ratio (Re/R)3 = (I)3 is thus implicit in the value of in the Flory-Fox equation and has a value of 0.49, corresponding to the Flory-Fox value of 2.1 X 1023. It is clear from Equations 1, 2, and 3 that [77] M cannot be related to the statistical polymer dimensions h and R without a knowledge of , i.e., < , which varies with solvent for a given polymer. It follows, that if all species having the same [77] M elute together from the GPC columns, then only Re can be the universal parameter, since will not be the same for all solute-solvent pairs and h and R will not be equally correct for universal calibration. 

In spite of the academic significance of this synthesis, the industrial value of 8-lactone as a versatile starting material has remained low because of the few applications reported to date. Dinjus and coworkers [74] described the synthesis of polymers via the photoinitiated polyaddition of 8-lactone with various dithiols which displayed intact lactone fragments within the polymer backbone. However, the complicated synthesis and the formation of rather statistic polymers with hard-to-tune characteristics hindered their potential industrial applications. 

IV. POLYMER MODELS OF MICROPOROUS MATERIALS LIKE SILICA/ALUMINA GELS. STATISTICAL POLYMER METHOD... 

The statistical polymer method proposed recently by the author [6] considers polymeric systems as sets of assemblages possessing structures averaged over all these of polymers containing the same numbers of monomeric units, i.e., statistical polymers. For the case when one is interested in the evaluation of the weight distribution and/or additive (extensive) parameters like energy, entropy, etc., one can consider statistical polymers instead branched cross-linked ones. 

The problem of the evaluation of the weight distribution of polymers produced by a reaction of polymerization has the general solution only for the irreversible polymerization, whereas gel systems are more convenient when considering the reversible case. The methods still used for the description of irreversible polymerization do not satisfy the general reversible case. Nevertheless, the statistical polymer method allows the treatment of polymeric systems produced by reversible polymerization [6]. 

Let us consider the statistical polymer method in the following order ... 

Statistical polymer is defined as the averaged structure exhibiting all possible structures of polymers containing the same number of monomeric units. 

Processes of polymerization-destruction in a polymeric system are described as reactions of statistical polymers ... 

We note that the one-component version of the statistical polymer method is applicable also to some multicomponent systems. For instance, if we consider a multicomponent system in which only for one (first) component m > 1 and the values of m for other components are equal one, the branch and cross-link formation are determined only by the parameters of the first component. If no specific interaction appears, such system is correctly described by Eqs. (65)-(76), while one has to take into account that in this description the formal monomeric unit which appears in the equations contains one monomeric unit of first component and the proportional numbers of other ones. 

For some problems like building theoretical isotherm curves of adsorption, the evaluation of the energetic distribution of pores in the system is very important. The statistical polymer method allows us to solve such problems [6]. 

Let us denote the number of monomers possessing (each) 5-vacancies on Ath level as Bsk(N). Of course, Bm (N) = U (N). The total number of units possessing 5-vacancies in the statistical polymer is Bsz(N Bmz = U-z(N). The change of the values of B because of reaction of polymerization when a monomer is captured by the statistical polymer is given by ... 

The statistical polymer method presented above still is convenient to the equilibrium model only. However, since that allows estimation of all additive parameters of branched polymers, we can evaluate thermodynamic functions which characterize not only equilibrium but also nonequilibrium situation. 

If the rate of reaction is given by a sum of additive parameters (that is correct in the linear situation), it is additive too, and the statistical polymer method [6] is applicable. In the nonlinear situation, the application of the statistical polymer method is just approximate. The free energy can be evaluated by the negentropy balance method (see Sec. III). 

The statistical polymer method is correctly applicable, first of all, to the evaluation of additive parameters of polymeric systems, that means characteristics satisfying the following equation ... 

There are two principal ways to obtain experimental test results from the statistical polymer method the comparison of the predicted weight distribution and other parameters of polymeric systems with these measured for classical polymers (direct test) and the comparison of predicted structural characteristics with these found from adsorption/desorption and other measurements (indirect test). For gels, the indirect test has many advantages, while the interpretation of experimental data is not always easy, e.g., chemisorption is sometimes mixed to physisorption. 

As the indirect test, the author applied the statistical polymer method to the treatment of data on adsorption on silica/alumina gels the correlation was good for most of treated systems [6]. 

On the other hand, Eq. (136) can be considered as the simple form of the Trommsdorff effect. It is very interesting, because the Trommsdorff effect is not derived from any statistical models of polymerization (as is the statistical polymer). 

FIG. 4 A typical correlation between theoretical (statistical polymer method applied) and empirical (autocatalytic-like) curves of degree of polymerization in time. Degree of polymerization theoretical (Q) empirical (—). 

V. DIRECT MODELING OF MICROPORES IN POLYMERIC (GEL) MATERIALS BY THE COMBINATION OF THE STATISTICAL POLYMER AND FRACTAL METHOD... 

As we noted in Sec. II. B. the use of the statistical polymer method alone does not allow the complete description of microporous systems and could be completed by fractal method. We have noted in Sec. IV that the statistical polymer method, though seems very effective for the estimation of various (first of all additive) parameters of macromolecules (even branched cross-linked ones), does not allow the obtainment of enough information about the empty space between them, i.e., micropores. Sometimes that is not important (e.g., if one is interested mainly in the energy distribution of micropores which is directly determined by that of macromolecules), but sometimes the direct information about micropores (especially their size distribution) is indispensable, and then the combined use of the statistical polymer and fractal... 

To solve the problem of direct modeling of micropores in gel-like polymeric system, let us assume the validity of the fractal approach (Sec. II. Q. If both statistical polymer and fractal methods are applied, macromolecules are considered as random fractals and described by relevant equations, while also terms of the statistical polymer method (Sec. IV) stay valid. 

Let us consider statistical TV-mer (TV —> oo) as the random fractal with dimensionality Df. Such an approach can be compared to the accepted practice of the construction of fractal clusters by the Monte Carlo method of random addition of new units the only difference is the statistical polymer is automatically random and contains all possible structures of randomly constructed clusters (of course, if they contain the same number TV of units). We may assume that the statistical TV-mer can be considered as the averaged structure obtained after the infinite number of operations of constructions of TV-meric clusters. 

We note some obvious advantages of the statistical polymer approach over the Monte Carlo method of the construction of cluster ... 

The application of the Monte Carlo method to the three-dimensional systems is too difficult, whereas the statistical polymer method is applicable in all situations, that offers much more freedom to the researcher. 

The Monte Carlo method furnishes numerical results, whereas the statistical polymer method allows the obtainment of analytical ones (at least, for all additive parameters of macromolecules and polymeric systems). 

To obtain the same result, the statistical polymer method requires much less calculation. 

Since we consider statistical TV-mers (at very large TV) as random fractal-like objects, we need to define the characteristic dimension (size) of the fractal statistical polymer. Let us define the characteristic size of fractal statistical TV-mer (at very large TV) as follows ... 

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