Infinite polymer chains approach

Two theoretical approaches for calculating NMR chemical shift of polymers and its application to structural characterization have been described. One is that model molecules such as dimer, trimer, etc., as a local structure of polymer chains, are in the calculation by combining quantum chemistry and statistical mechanics. This approach has been applied to polymer systems in the solution, amorphous and solid states. Another approach is to employ the tight-binding molecular orbital theory to describe the NMR chemical shift and electronic structure of infinite polymer chains with periodic structure. This approach has been applied to polymer systems in the solid state. These approaches have been successfully applied to structural characterization of polymers... 

M is a complex function of composition, molecular weight and temperature. Two analytical approaches have been directed toward obtaining expressions for M. The simplest one goes back to Levinson who treated a one-dimensional random walk on an infinite lattice with randomly distributed traps. [19] Although this is clearly unrealistic for real, finite length polymer chains, it does help to set certain critical concepts. Thus, we spend some time on it before considering the results applicable to real polymers. 

Using the critical conversion and Equation 10, values for the parameter s have been calculated and are listed In Table II. This value of s Is a cumulative average value over the entire conversion Interval from zero to gelation. At the very beginning of the reaction, the polymer chains are infinitely diluted by monomer which causes the probability of a radical on one chain attacking a pendant vinyl on another to approach zero. Therefore, In the limit of zero conversion, the Instantaneous value of s will be zero. As the reaction progresses, the polymer concentration Increases and with It the probability of Intermolecular contact and reaction. The Instantaneous value of the parameter s will therefore rise and continue to rise up to the gel point qualitatively as shown In Figure 3. 

Kinetic approaches represent realistic and comprehensive description of the mechanism of network formation. Under this approach, reaction rates are proportional to the concentration of unreacted functional groups involved in a specific reaction times an associated proportionality constant (the kinetic rate constant). This method can be applied to the examination of different reactor types. It is based on population balances derived from a reaction scheme. An infinite set of mass balance equations will result, one for each polymer chain length present in the reaction system. This leads to ordinary differential or algebraic equations, depending on the reactor type under consideration. This set of equations must be solved to obtain the desired information on polymer distribution, and thus instantaneous and accumulated chain polymer properties can be calculated. In the introductory paragraphs of Section... 

Manning proposed a linear counterion condensation theory to account for the low activity of counterions in polyelectrolyte solutions 14). The basic idea of the theo is that there is a critical charge density on a polymer chain beyond which some counterions will condense to the polymer chain to lower the charge density, otherwise the ener of the system would approach infinite. The concept of this theory has been widely accepted. The shortcoming of the linear countmon condensation is that it predicts that counterion condensation is independent of ionic strength in the solution, which is not in agreement with experimental observations. Counterion condensation can be obtained duectly by solving the nonlinear Poisson-Boltzmann equation. 

Flory PJ (1953) Principles of polymer chemistry. Cornell University Press, Ithaca Freed KF (1987) Renormalization group theory of macromolecules. Wiley, New York Freed KF, Dudowicz J, Stukalin EB, Douglas JF (2010) General approach to polymer chains confined by interacting boundaries. J Chem Phys 133 094901 Fuoss RM, Katchalsky A, Lifson S (1951) The potential of an infinite rod-like molecule and the distribution of the counter ions. Proc Natl Acad Sci USA 37 579-589 Golestanian R, Kardar M, Liverpool TB (1999) Collapse of stiff polyelectrolytes due to counterion fluctuations. Phys Rev Lett 82 4456-4459 Guggenheim FA (1952) Mixtures. The Clarendon Press, Oxford... 

The second repulsive force arises from the loss of configurational entropy when the chains overlap [20]. This is schematically illustrated in Figure 14.7, whereby the polymer chain is represented by a simple rod with one attachment point to the surface. When the two surfaces are separated at infinite distance, each chain will have a number of configurations, that are determined by the volume of the hemisphere swept by the rod. When the two surfaces approach to a distance h that is smaller then the radius of the hemisphere swept by the rod, the volume available to the chains... 

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