Computational Studies of Polymer Kinetics

In most situations when, for example, such reactions as chain transfer or termination take place, obtaining analytical expressions for the MWD is rather complicated or even impossible. However, it is often sufficient to know only average degrees of polymerization (DP) 

The most common way to calculate average DPs is to use the well-known mathematical method of statistical moments. By definition, the MWD moment of the ith order is given by p = X] The use of this method decreases the number of equations to solve from almost infinite set of equations for R(l) to several differential equations for p . Therefore, in order to calculate Pn, Pw, and Pz, one has to solve only four equations since Pn = Pi/Po, Pw = P hl h- 

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Studies of polymerization kinetics have a long history. Nevertheless, many problems in this field remain unresolved. Using computational methods, several of these problems are clarified in the chapter hy Litvinenko ( Computational Studies of Polymer Kinetics ). Special attention is paid to the effect of chain transfer reactions on polymer molecular weight and apphcations to different types of polymerization methods. 

In our computer studies of the conformational behavior of the shell-forming chains, we used MC simulations [91, 95] on a simple cubic lattice and studied the shell behavior of a single micelle only. Because we modeled the behavior of shells of kinetically frozen micelles, we simulated a spherical polymer brush tethered to the surface of a hydrophobic spherical core. The association number was taken from the experiment. The size of the core, lattice constant (i.e., the size of the lattice Kuhn segment ) and the effective chain length were recalculated from experimental values on the basis of the coarse graining parameterization [95]. 

This volume consists of four parts. The first part is devoted to theoretical studies and computer simulations. These studies deal with the structure and dynamics of polymers adsorbed at interfaces, equations of state for particles in polymer solutions, interactions in diblock copolymer micelles, and partitioning of biocolloidal particles in biphasic polymer solutions. The second part discusses experimental studies of polymers adsorbed at colloidal surfaces. These studies serve to elucidate the kinetics of polymer adsorption, the hydrodynamic properties of polymer-covered particles, and the configuration of the adsorbed chains. The third part deals with flocculation and stabilization of particles in adsorbing and nonadsorbing polymer solutions. Particular focus is placed on polyelectrolytes in adsorbing solutions, and on nonionic polymers in nonadsorbing solutions. In the final section of the book, the interactions of macromolecules with complex colloidal particles such as micelles, liposomes, and proteins are considered. 

Finally, we have addressed only the phase behavior of solutions, and have not addressed eiflier the structural properties (e.g., as described by the various pair distribution functicms or the size of polymer coils) or the interfacial structure in phase-separated solutions (though we did pay attention to prediction of the interfacial tension between coexisting phases). Also, the kinetics of phase separation (via nucleation and growth or spinodal decomposition) has not been discussed. Thus, we emphasize that, although a few first and promising steps towards the computational modeling of polymer solutions via computer simulations have been taken, many further studies are still necessary to obtain a more complete theoretical understanding of polymer solutions and their properties. 

The main advantage of the simulations is that they provide the exact solution of the model system. The main disadvantage is that they are very demanding computationally and therefore in order to make systematic studies a compromise must be reached between the molecular detail in the model system and the number of studies that can be carried out. Furthermore, in some applications, for example protein adsorption on tethered polymer layers, the time scale for the process may be too long to be able to be simulated even with a simple model. However, combined effective potentials obtained from other theoretical approaches with, for example, BD simulations may lead to studies of the kinetic prop>erties of protein adsorption in tethered protein layers. As computational capabilities increase we may be able to reach more complex and detailed systems by the use of straightforward simulations. 

In principle, all three issues mentioned above could be tackled with fully atomistic MD simulations. Apart from the astronomical resources in computer time that an accurate determination of the phase diagram and a study of the kinetics of phase separation in a fully atomistic model would require, the lack of accurate interaction potentials between the constituents of the polymer + solvent mixture might seriously upset the predictive power of such a brute-force approach. This is particularly relevant for the specific system - hexadecane and carbon dioxide - because a small change of the mixing rule for interactions between unlike segments can alter the qualitative type of phase diagram. 

In what follows, we use simple mean-field theories to predict polymer phase diagrams and then use numerical simulations to study the kinetics of polymer crystallization behaviors and the morphologies of the resulting polymer crystals. More specifically, in the molecular driving forces for the crystallization of statistical copolymers, the distinction of comonomer sequences from monomer sequences can be represented by the absence (presence) of parallel attractions. We also devote considerable attention to the study of the free-energy landscape of single-chain homopolymer crystallites. For readers interested in the computational techniques that we used, we provide a detailed description in the Appendix. ... 

The field of chemical kinetics and reaction engineering has grown over the years. New experimental techniques have been developed to follow the progress of chemical reactions and these have aided study of the fundamentals and mechanisms of chemical reactions. The availability of personal computers has enhanced the simulation of complex chemical reactions and reactor stability analysis. These activities have resulted in improved designs of industrial reactors. An increased number of industrial patents now relate to new catalysts and catalytic processes, synthetic polymers, and novel reactor designs. Lin [1] has given a comprehensive review of chemical reactions involving kinetics and mechanisms. 

The contents of the review are as follows. The dynamics of rodlike polymers are reviewed in Section 2 followed by a review of previous experimental results of the polymerization kinetics of rodlike molecules in Section 3. Theoretical analyses of the problem following Smoluchowski s approach are discussed next (Section 4), and this is followed by a review of computational studies based on multiparticle Brownian dynamics in Section 5. The pairwise Brownian dynamics method is discussed in some detail in Section 6, and the conclusions of the review are given in Section 7. 

The application of computer calculations to DTA studies of the crystallization kinetics of polymers was described by Gornick (51). Calculations were made of the temperature of a polymeric sample during the cooling process using a Burroughs Mode 5500 computer. Morie et al. (52) used an IBM Model 1130 computer to prepare standard vapor-pressure plots of In P versus 1/T the vapor-pressure data being obtained from DTA or DSC curves. The heat of vaporization was calculated by the Haggenmacher method as modified by Fishtine. 

Curro et al. [1981] followed a similar procedure in studies of the aging kinetics of poly(methyl methacrylate) (PMMA). For predicting the shift factors of aging experiments at 23°C, the authors computed from PVT the free-volume function, h = h P, V), and then substituted these into Doolittle s equation (6.63). The resulting prediction agreed with the experimental values, contrasting with the inadequacy of the WLF relation. Next, the polymer aging process was modeled as a diffusion of free volume [Curro et al., 1982]. 

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