Simple bead-spring

Since this behavior is universal, it is obvious that the simplest simulation models which contain the essential aspects of polymers are sufficient to study these phenomena. Two typical examples of such models are the bond fluctuation Monte Carlo model and the simple bead-spring model employed in molecular dynamics simulations. Both models are illustrated in Fig. 6. 

Finally, there is another model commonly used in simulations - a simple bead-spring model for chain molecules. The bead-spring model is often referred to as a meso-scale model because the beads and springs represent the average properties of much larger molecules. In this model, monomers separated by distance r interact through a two -body potential, often of the truncated LJ form ... 

In between the molecular and macroscopic approach several mesoscopic scale techniques are available, like lattice bond fluctuation models [15] and coarse grained techniques, such as smoothed particle hydrody-namics[16], and dissipative particle dynamics (DPD) [17, 18]. The latter is based on the concept of matter particles , representing a large number of atoms or molecules, interacting via soft potentials and subjected to dissipative and fluctuating forces. Polymeric system can be described by simple bead-spring models, with the advantage that a full chain can be represented by just a few particles, typically from 20 to 50 [18, 19]. 

The replicated data approach is readily extendable to macromolecules. Many MD studies of polymers have used the simple bead-spring model... 

Potentials and Geometry. We will describe results for films containing linear short-chain molecules. These are modeled using a simple bead-spring potential that has been used extensively in studies of polymer structure and dynamics (24). Each spherical monomer within the molecule interacts with all other monomers through a Lennard-Jones (LJ) potential Vy that is truncated beyond rc. For monomers separated by a distance r[Pg.92]

Many questions about large scales have already been answered with simple bead-spring models. These models can reproduce scaling behaviors, thereby contributing to our basic understanding of how complex systems behave. However, to obtain numerical results that can be compared directly with experiments, one needs a meso-scale model that does not represent a simple generic polymer but instead represents the identity of the specific polymer being studied. A combination of atomistic and meso-scale models is needed, and the models have to be mapped onto each other as uniquely as possible. 

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,... 

A simple generic bead spring model of chains can be used to study universal polymer properties that do not depend on specific chemical details. Bonds between neighbouring Lennard-Jones particles in a chain can be represented by the finite extension non-linear elastic (FENE) potential. 

The starting point of a molecular constitutive theory is a simple mechanical model for the molecule that captures its most salient traits. Thus, flexible polymer molecules have been represented by elastic dumbbells and bead-spring chains, and rigid polymers by rigid dumbbells and rigid rods. For its simplicity, the evolution of the model molecule is easily described by a convection-diffusion equation. Then a Fokker-Planck equation is written for the probability distribution function of an ensemble of these molecules. Finally, the macroscopic stress tensor is derived in terms of the distribution function. This kinetic theory framework was pioneered by Kirkwood (see, for example, Ref. ). 

The elastic dumbbell model studied iu Chapter 6 is both structurally and djmamicaUy too simple for a poljmier. However, the derivation of its constitutive equation illustrates the main theoretical steps involved. In this chapter we shall apply these theoretical results to a Gaussian chain (or Rouse chain) containing many bead-spring segments (Rouse segments). First we obtain the Smoluchowski equation for the bond vectors. After transforming to the normal coordinates, the Smoluchowski equation for each normal mode is equivalent in form to the equation for the elastic dumbbell. 

Agarwal and Mashelkar first analyzed contradictory reports from Kim [190], Gryte [191], and Singh [192, 193], and proposed a simple mechanistic model [209]. In stark contrast to the concept of preferential scission of side chains, their model reveals decreased shear stability by grafting side chains. They extended Odell and Keller s bead-rod model [27]. The backbone was modeled as a fully extended rod with Ni = 2m+ beads (Fig. 25). p grafted bead-spring chains having g beads with... 

There are other models based on springs and dashpots such as the simple Kelvin-Voigt model for viscoelastic solid and the Burgers model. Reader is referred to Refs. [1-5] for details. Other elementary models are the dumbbell, bead-spring representations, network, and kinetic theories. However, the most notable limitation of all these models is their restriction to small strain and strain rates [2, 3]. 

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