Statistics of macromolecules

The first of these assumptions, generally accepted in macromolecular chemistry [1,3], is correct enough when considering the propagation reaction under copolymerization of the majority of monomers. Simple estimates reported in paper [74] support the correctness of the second assumption. As for the third one, it is true, strictly speaking, only under 0-conditions. The conformational statistics of macromolecules in a thermodynamically good solvent is known [30] to differ from the Gaussian one. Nevertheless, this distinction may hardly influence the qualitative conclusions of the simplest theory of interphase copolymerization. To which extent the account of the excluded volume of macromolecules will affect quantitative results of this theory, may be revealed exclusively by computer simulations. 

The statistical basis of diffusion requires arguments that may be familiar from kinetic molecular theory. Elementary concepts from the theory of random walks and its relation to diffusion form the third topic, which is covered in Section 2.6. As is well known, the random walk statistics can also be used for describing configurational statistics of macromolecules under some simplifying assumptions this is outlined in Section 2.7. 

Helfand and Tagami model is based on self-consistent field that determines the configurational statistics of macromolecules in the interfacial region. At the interface, the interactions between statistic segments of polymers A and B are determined by the thermodynamic binary interaction parameter, Since the polymers are immiscible, there are repulsive enthalpic effects that must be balanced by the entropic ones that cause chains A and B to intermingle. 

The addition of nanoparticles into the polymer matrix is a way to affect the chain conformation structure as well as conformation dynamics and, therefore, attention to the conformation statistics of macromolecules will be paid in this chapter at the expense of constitution and configuration structure. 

Grosberg A Y and Khokhlov A R 1994 Statistical Physics of Macromolecules (AlP Series in Polymers and Complex Materials) (New York AlP)... 

Another important sub-case, of disorder in macromolecular crystals, corresponds to the statistical occurrence of two specific orientations only, at well defined positions in a 3-D lattice, of a group of macromolecules or of each single macromolecule. 

Hashimoto T., Shibayama M., Kawai H., and Meier D. J. Confined chain statistics of block polymers and estimation of optical anisotropy and domain size. Macromolecules, 18, 1855, 1985. 

A polymer coil does not only possess a structure on the atomistic scale of a few A, corresponding to the length of covalent bonds and interatomic distances characteristic of macromolecules are coils that more or less, obey Gaussian statistics and have a diameter of the order of hundreds of A (Fig. 1.2) [17]. Structures of intermediate length scales also occur e. g., characterized by the persistence length. For a simulation of a polymer melt, one should consider a box that contains many such chains that interpenetrate each other, i. e., a box with a linear dimension of several hundred A or more, in order to ensure that no artefacts occur attributable to the finite size of the simulation box or the periodic boundary conditions at the surfaces of the box. This ne-... 

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. 

For a number of copolymers, whose kinetics of formation is described by nonideal models, the statistics of alternation of monomeric units in macromolecules cannot be characterized by a Markov chain however, it may be reduced to the extended Markov chain provided that units apart from their chemical nature... 

Under current treatment of statistical method a set of the states of the Markovian stochastic process describing the ensemble of macromolecules with labeled units can be not only discrete but also continuous. So, for instance, when the description of the products of living anionic copolymerization is performed within the framework of a terminal model the role of the label characterizing the state of a monomeric unit is played by the moment when this unit forms in the course of a macroradical growth [25]. 

Upon expressing from the equilibrium condition the complex concentration M12 through the concentrations of monomers, and substituting the expression found into relationship (21) we obtain, invoking the formalism of the Markov chains, final formulas enabling us to calculate instantaneous statistical characteristics of the ensemble of macromolecules with colored units. A subsequent color erasing procedure is carried out in the manner described above. For example, when calculating instantaneous copolymer composition, this procedure corresponds to the summation of the appropriate components of the stationary vector jt of the extended Markov chain ... 

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 ... 

Knowing the functions (26) and(27) it is possible by means of the formalism of the theory of Markovian processes [53] to find any statistical characteristic in an ensemble of macromolecules with labeled units. A subsequent label erasing procedure is carried out by integration of the obtained expressions over time of the formation of monomeric units. Examples of the application of this algorithm are reported elsewhere [25]. 

Once the particular branching process that specifies the probability measure on the set of macromolecules of a polymer specimen has been identified, the statistical method provides the possibility to determine any statistical characteristic of the chemical structure of this specimen. In particular, the dependence of the weight fraction of a sol on conversion can be calculated by formulas [extending those (55)] which are obtainable from (61) provided the value of dummy variable s is put unity ... 

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