Polymers statistical problem solving

For this reason, computer simulation methods (Monte Carlo and molecular and Brownian dynamic methods) have been developed not only for solving the problems of polymer statistics, but also for the investigation of the dynamic properties and the intramolecular mobility of polymers. 

Another kind of situation arises when it is necessary to take into account the long-range effects. Here, as a rule, attempts to obtain analytical results have not met with success. Unlike the case of the ideal model the equations for statistical moments of distribution of polymers for size and composition as well as for the fractions of the fragments of macromolecules turn out normally to be unclosed. Consequently, to determine the above statistical characteristics, the necessity arises for a numerical solution to the material balance equations for the concentration of molecules with a fixed number of monomeric units and reactive centers. The difficulties in solving the infinite set of ordinary differential equations emerging here can be obviated by switching from discrete variables, characterizing macromolecule size and composition, to continuous ones. In this case the mathematical problem may be reduced to the solution of one or several partial differential equations. 

Although recent years have witnessed an impressive confluence of experiments and statistical theories, presently there is no comprehensive understanding of the interrelation between chemical sequences in synthetic copolymers and the conditions of synthesis. One has merely to glance at recent literature in polymer science and biophysics to realize that the problem of sequence-property relationship is by no means entirely solved. As always, in these circumstances, an alternative to analytical theories is computer simulations, which are designed to obtain a numerical answer without knowledge of an analytical solution. 

Both statistical and kinetic methods can be used, in principle, for the description of destruction, the process inverse to polymerization. However, the main problem limiting their applicability consists in difficulties of taking into account of all opportunities for breaking bonds inside the polymeric molecules. This problem is solved for chain polymers and some of the simplest kinds of branched polymers, for instance, polymers containing a limited number of units and those with a specified structure, but not for the general case of macromolecules. 

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]. 

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. 

The parameters a = l/rij5 the number of which equals m(m — IX are reciprocal reactivity ratios (2.8) of binary copolymers. Markov chain theory allows one, without any trouble, to calculate at any m, all the necessary statistical characteristics of the copolymers, which are formed at given composition x of the monomer feed mixture. For instance, the instantaneous composition of the multicomponent copolymer is still determined by means of formulae (3.7) and (3.8), the sums which now contain m items. In the general case the problems of the calculation of the instantaneous values of sequence distribution and composition distribution of the Markov multicomponent copolymers were also solved [53, 6]. The availability of the simple algebraic expressions puts in question the expediency of the application of the Monte-Carlo method, which was used in the case of terpolymerization [85,99-103], for the calculations of the above statistical characteristics. Actually, the probability of any sequence MjMjWk. .. Mrl 4s of consecutive monomer units, selected randomly from a polymer chain is calculated by means of the elementary formula ... 

Many technical problems that may be encountered, say, with a new thermoplastic, will already have been met and solved with polymers, like rubber, that have been in the marketplace for a comparatively long time. It is not often possible to recognize and use such parallels, however, if the parameters of the molecular weight distributions in the different cases are not measured in the same units. This results in much unnecessary rediscovery of old answers, and the engineer or scientist who can interpret both Mooney and melt index values in terms of statistical parameters of the molecular weight distributions of the respective rubber and thermoplastic may save considerable time and efl ort. 

From a conceptual viewpoint the primary theoretiotl problem yet to be solved is the stress transfer mechanism in polymer solids. As noted earlier, polymers have statistical structures v4ien in the glassy state and a rather broad spectrum of order-disorder when in the crystalline state. Detailed analysis of stress transfer throi a glassy structure requires comprehensive analysis of chain conformation in the (nonequilibrium) glass which in turn requires an imderstanding of both the intramolecular and intermolecular energetics. 

Despite writing the chapters quite independently, the authors wanted to give a true unity to the book. Thus, throughout the work, they aimed at using coherent notation and reasonable designations. Consequently, logic sometimes forced them to distance themselves somewhat from awkward traditions. Nevertheless, this problem of notation has not always been easy to solve, due to the large number of disciplines concerned by the study of polymers namely, computer simulation, statistical mechanics and theory of liquids, description of the... 

Hence, the suggested approaches do not allow solving the problem of the internal links distribution for a polymer chain completely. In the present work we propose its analytic solution in terms of SARWstrict statistics, that is, without taking into account of the so-called volume interaction. 

It has been noticed several times [3, 4] that the configuration of soft polymer chains differs from the RW trajectory in one important aspect it must not intersect. This limitation, known as long-range ordering effect or excluded volume effect, requires new statistics, i.e., statistics of self-avoiding walks (SAW). The attempts made so far [3] have not succeeded in solving this problem completely. 

Monte Carlo simulation is a technique for solving stochastic problems that is widely used in the area of polymer science and engineering. Specifically for studies of polymer microstructures, it has been used to predict the CCD of copolymers and the distribution of stereoregularity [82,83]. One of the advantages of this technique is that one can obtain detailed statistical information of chain structures simply from relatively easy to measure polymer properties such as the CC and the molecular weight. 

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