Polymers, kinetic modeling methods

This closure property is also inherent to a set of differential equations for arbitrary sequences Uk in macromolecules of linear copolymers as well as for analogous fragments in branched polymers. Hence, in principle, the kinetic method enables the determination of statistical characteristics of the chemical structure of noncyclic polymers, provided the Flory principle holds for all the chemical reactions involved in their synthesis. It is essential here that the Flory principle is meant not in its original version but in the extended one [2]. Hence under mathematical modeling the employment of the kinetic models of macro-molecular reactions where the violation of ideality is connected only with the short-range effects will not create new fundamental problems as compared with ideal models. 

In the past, the equivalence between the size distribution generated by the Smoluchowski equation and simple statistical methods [9, 12, 40-42] was a source of some confusion. The Spouge proof and the numerical results obtained for the kinetics models with more complex aggregation physics, e.g., with a presence of substitution effects [43,44], revealed the non-equivalence of kinetics and statistical models of polymerization processes. More elaborated statistical models, however, with the complete analysis made repeatedly at small time intervals have been shown to produce polymer size distributions equivalent to those generated kinetically [45]. Recently, Faliagas [46] has demonstrated that the kinetics and statistical models which are both the mean-field models can be considered as special cases of a general stochastic Markov process. 

When the statistical moments of the distribution of macromolecules in size and composition (SC distribution) are supposed to be found rather than the distribution itself, the problem is substantially simplified. The fact is that for the processes of synthesis of polymers describable by the ideal kinetic model, the set of the statistical moments is always closed. The same closure property is peculiar to a set of differential equations for the probability of arbitrary sequences t//j in linear copolymers and analogous fragments in branched polymers. Therefore, the kinetic method permits finding any statistical characteristics of loopless polymers, provided the Flory principle works for all chemical reactions of their synthesis. This assertion rests on the fact that linear and branched polymers being formed under the applicability of the ideal kinetic model are Markovian and Gordonian polymers, respectively. 

MSI) that uses the same time-dependent Ginzburg Landau kinetic equation as CDS, but starts from (arbitrary) bead models for polymer chains. The methods have been summarized elsewhere. Examples of recent applications include LB simulations of viscoelastic effects in complex fluids under oscillatory shear,DPD simulations of microphase separation in block copoly-mers ° and mesophase formation in amphiphiles, and cell dynamics simulations applied to block copolymers under shear. - DPD is able to reproduce many features of analytical mean field theory but in addition it is possible to study effects such as hydrodynamic interactions. The use of cell dynamics simulations to model non-linear rheology (especially the effect of large amplitude oscillatory shear) in block copolymer miscrostructures is currently being investigated. ... 

To address polymer network formation from nonlinear chain-growth polymerization (or copolymerization), kinetic methods are more appropriate [23, 75-83], Some of the most successful kinetic models to address this type of system are based on the method of moments [23, 75-77, 79, 80, 82, 84], Some divergence problems at the vicinity of the gelation point are common with the method of moments, although there are practical ways to avoid this situation [80], A more refined kinetic method to address the issue of modeling the dynamics of gelation in... 

Hence, the performed above analysis has shown that different solvents using in low-temperature nonequilibriiun polycondensation process can result not only in symthesized polymer quantitative characteristics change, but also in reaction mechanism and polymer chain structure change. This effect is comparable with the observed one at the same polymer receiving by methods of equilibrium and nonequilibrium polycondensation. Let us note, that the fractal analysis and irreversible aggregation models allow in principle to predict symthesized polymer properties as a function of a solvent, used in synthesis process. The stated above results confirm Al-exandrowicz s conclusion [134] about the fact that kinetics of branched polymers formation effects on their topological structures distribution and macromolecules mean shape. 

The cured and the liquid polymers degrade essentially by the same mechanism (see Equation 6.1). The kinetic analysis of the isothermal and dynamic thermogravimetric data of the liquid polysulfide polymer cured with ammonium dichromate is explained by a kinetic model based on random initiation, followed by rapid termination by disproportionation. The average overall activation energy obtained by different methods for the decomposition is 145.3 kj/mole ... 

The major limitation of kinetic models using the method of moments is that they only track average quantities. More detail is sometimes required (e.g., to examine the combined effect of chain-scission and LCB on polymer architecture). In such cases, mechanistic... 

The deprotection kinetics of alicyclic polymer resist systems designed for 193 nm lithography was examined using JR and fluorescence spectroscopic techniques. A kinetic model was developed that simulates the deprotection of the resists fairly well. A new, simple, and reliable method for monitoring photoinduced acid generation in polymer films and in solutions of the kind used in 193 nm and deep-UV lithography was developed. This technique could find application in the study of diffusional processes in thin polymer films. 

As far as the present work is concerned, the relevance of numerical stochastic methods for polymer dynamics in micro/macro calculations resides in their ability to yield (within error bars) exact numerical solutions to dynamic models which are insoluble in the framework of polymer kinetic theory. In addition, and mainly as a consequence of the correspondence between Fokker Planck and stochastic differential equations, complex polymer dynamics can be mapped onto extremely efficient computational schemes. Another reason for the efficiency of stochastic dynamic models for polymer melts stems from the reduction of a many-chain problem to a single-chain or two-chain representation, i.e., to linear computational complexity in the number of particles. This circumstance permits the treatment of global ensembles consisting of several tens of millions of particles on current hardware, corresponding to local ensemble sizes of O(IO ) particles per element. 

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