Polymers - Brownian dynamics

Subsequent work by Johansson and Lofroth [183] compared this result with those obtained from Brownian dynamics simulation of hard-sphere diffusion in polymer networks of wormlike chains. They concluded that their theory gave excellent agreement for small particles. For larger particles, the theory predicted a faster diffusion than was observed. They have also compared the diffusion coefficients from Eq. (73) to the experimental values [182] for diffusion of poly(ethylene glycol) in k-carrageenan gels and solutions. It was found that their theory can successfully predict the diffusion of solutes in both flexible and stiff polymer systems. Equation (73) is an example of the so-called stretched exponential function discussed further later. 

Ediger, M. D. and Adolf, D, B. Brownian Dynamics Simulations of Local Polymer Dynamics. VoU 16, pp. 73-110. 

The dynamics of a generic linear, ideal Gaussian chain - as described in the Rouse model [38] - is the starting point and standard description for the Brownian dynamics in polymer melts. In this model the conformational entropy of a chain acts as a resource for restoring forces for chain conformations deviating from thermal equilibrium. First, we attempt to exemphfy the mathematical treatment of chain dynamics problems. Therefore, we have detailed the description such that it may be followed in all steps. In the discussion of further models we have given references to the relevant literature. 

Figure 8.8 Nucleation and growth in a polymer chain (numerical simulation with Brownian dynamics Sakaue et al., unpublished). After staying for a long time in an elongated state, a nucleation centre appears spontaneously on a chain. Then, the densely packed region grows quickly to form a toroid. Essentially the same process has been observed in an experiment with single DNA observations (Yoshikawa and Matsuzawa, 1995,1996).

Nucleation and growth in a polymer chain (numerical simulation with Brownian dynamics 135 Sakaue et al.)... 

Leonov AI (1994) On a self-consistent molecular modelling of linear relaxation phenomena in polymer melts and concentrated solutions. J Rheol 38( 1) 1—11 Liu B, Diinweg B (2003) Translational diffusion of polymer chains with excluded volume and hydrodynamic interactions by Brownian dynamics simulation. J Chem Phys 118(17) 8061-8072... 

In order to determine a system thermodynamically, one has to specify some independent parameters (e.g. N, T, P or V) besides the composition of the system. The most common choice in MC simulation is to specify N, V and T resulting in the canonical ensemble, where the Helmholtz free energy A is the natural thermodynamical potential. However, MC calculations can be performed in any ensemble, where the suitable choice depends on the application. It is straightforward to apply the Metropolis MC algorithm to a simple electric double layer in the iVFT ensemble. It is however, not so efficient for polymers composed of more than a few tens of monomers. For long polymers other algorithms should be considered and the Pivot algorithm [21] offers an efficient alternative. MC simulations provide thermodynamic and structural information, but time-dependent properties are not accessible. If kinetic or time-dependent properties are of interest one has to use molecular dynamic or brownian dynamic simulations. 

Now that we have settled on a model, one needs to choose the appropriate algorithm. Three methods have been used to study polymers in the continuum Monte Carlo, molecular dynamics, and Brownian dynamics. Because the distance between beads is not fixed in the bead-spring model, one can use a very simple set of moves in a Monte Carlo simulation, namely choose a monomer at random and attempt to displace it a random amount in a random direction. The move is then accepted or rejected based on a Boltzmann weight. Although this method works very well for static and dynamic properties in equilibrium, it is not appropriate for studying polymers in a shear flow. This is because the method is purely stochastic and the velocity of a mer is undefined. In a molecular dynamics simulation one can follow the dynamics of each mer since one simply solves Newton s equations of motion for mer i,... 

Fig. 3.9. Dynamical process of the folding of a semiflexible polymer with contour length L/a = 512 and Kuhn length l/a cs 20. (Top) Snapshots obtained through Brownian dynamics simulations, and (bottom) the fluorescence intensity profile of the T4 DNA during the folding and corresponding schematic pictures (see [26] and [34] for more details)...

M. Bishop and J. H. R. Clarke, /. Chem. Phys., 90, 6647 (1989). Brownian Dynamics Study of the Shape of Star and Linear Polymers in Different Regimes. 

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