Polymer chains, dynamics realistic

The dynamical problem for disordered one-dimensional polymer chains with realistic structures as those of the complex polymers has... 

These models are designed to reproduce the random movement of flexible polymer chains in a solvent or melt in a more or less realistic way. Simulational results which reproduce in simple cases the so-called Rouse [49] or Zimm [50] dynamics, depending on whether hydrodynamic interactions in the system are neglected or not, appear appropriate for studying diffusion, relaxation, and transport properties in general. In all dynamic models the monomers perform small displacements per unit time while the connectivity of the chains is preserved during the simulation. 

Coarse-grained polymer models neglect the chemical detail of a specific polymer chain and include only excluded volume and topology (chain connectivity) as the properties determining universal behavior of polymers. They can be formulated for the continuum (off-lattice) as well as for a lattice. For all coarse-grained models, the repeat unit or monomer unit represents a section of a chemically realistic chain. MD techniques are employed to study dynamics with off-lattice models, whereas MC techniques are used for the lattice models and for efficient equilibration of the continuum models.36 2 A tutorial on coarse-grained modeling can be found in this book series.43... 

Attempts have been made to identify primitive motions from measurements of mechanical and dielectric relaxation (89) and to model the short time end of the relaxation spectrum (90). Methods have been developed recently for calculating the complete dynamical behavior of chains with idealized local structure (91,92). An apparent internal chain viscosity has been observed at high frequencies in dilute polymer solutions which is proportional to solvent viscosity (93) and which presumably appears when the external driving frequency is comparable to the frequency of the primitive rotations (94,95). The beginnings of an analysis of dynamics in the rotational isomeric model have been made (96). However, no general solution applicable for all frequency ranges has been found for chains with realistic local structure. 

The local dynamics is naturally strongly dependent on the exact chemical nature and structure of the polymer one studies. The large scale dynamics, however, is largely universal and is described with the Rouse model whereas for longer chains the tube model and reptation concept is believed to describe the chain dynamics [2]. It is easy to see that no single simulation method can capture the physics of polymer dynamics on all these length and time scales [3]. For situations where we can ignore quantum effects (which can, however, be important in polymer crystals [4]) MD simulations with chemically realistic force fields are the method of choice to study local relaxation. 

Similar restrictive comments need to be made about the existing simulations of spinodal decomposition in polymer blends. First studies addressed the two-dimensional case. In this case, there is no chain interpenetration, and since polymer mixtures in d =2 dimensions are not predicted to become mean-field-like for AT —> oo, one expects the same behavior as in small molecule mixtures, and this is what has been found. Using the slithering snake algorithm, the dynamics of the model is neither realistic at short times (where the chain dynamics should be described by Rouse-model type motions ) nor at late times (where hydrodynamic effects play a role, as discussed above, at least in d=3 dimensions). 

But these models are not very realistic for actual flexible polymer chains, and one cannot expect to understand polymer dynamics without turning to models which take into account the molecular nature of polymers. In polymers, each bond is subjected to particular anisotropic constraints due to neighboring bonds. Rouse (21) proposed to model the chain by a sequence of beads separated by springs. The random forces exerted by the viscous environment are localized on the beads. In spite of its crudeness, this early model contains the two essential features of pol3rmer dynamics, i.e. the connectivity and the flexibility. It leads to a master equation for the orientation probability ... 

The parameter /r tunes the stiffness of the potential. It is chosen such that the repulsive part of the Leimard-Jones potential makes a crossing of bonds highly improbable (e.g., k= 30). This off-lattice model has a rather realistic equation of state and reproduces many experimental features of polymer solutions. Due to the attractive interactions the model exhibits a liquid-vapour coexistence, and an isolated chain undergoes a transition from a self-avoiding walk at high temperatures to a collapsed globule at low temperatures. Since all interactions are continuous, the model is tractable by Monte Carlo simulations as well as by molecular dynamics. Generalizations of the Leimard-Jones potential to anisotropic pair interactions are available e.g., the Gay-Beme potential [29]. This latter potential has been employed to study non-spherical particles that possibly fomi liquid crystalline phases. 

A relaxation spectrum similar to that of Fig. 4.2 is obtained for the diffusional motion of a local-jump stochastic model of IV+ 1 beads joined by N links each of length b, if a weak correlation in the direction of nearest neighbor links is taken into account for the probability of jumps (US). On the other hand, relaxation spectra similar to that of the Rouse theory (27) are obtained for the above mentioned model or for stochastic models of lattice chain type (i 14-116) without the correlation. Iwata examined the Brownian motion of more realistic models for vinyl polymers and obtained detailed spectra of relaxation times of the diffusional motion 117-119). However, this type of theory has not gone so far as to predict stationary values of the dynamic viscosity at high frequencies. 

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