Polymer bead-spring model

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. 

Concluding this section, one should mention also the method of molecular dynamics (MD) in which one employs again a bead-spring model [33,70,71] of a polymer chain where each monomer is coupled to a heat bath. Monomers which are connected along the backbone of a chain interact via Eq. (8) whereas non-bonded monomers are assumed usually to exert Lennard-Jones forces on each other. Then the time evolution of the system is obtained by integrating numerically the equation of motion for each monomer i... 

Again, the OLMC bead-spring model (Sec. IIB 2) is used, with a host matrix of an equilibrated dense solution of polymer chains quenched at different concentrations Cots. Eq. (7) for the probability IF of a random monomer displacement in direction Ax, Ay, Az is given by... 

A. Milchev, K. Binder. Osmotic pressure, atomic pressure and the virial equation of state of polymer solutions Monte Carlo simulations of a bead-spring model. Macromol Theory Simul 5 915-929, 1994. 

First approaches at modeling the viscoelasticity of polymer solutions on the basis of a molecular theory can be traced back to Rouse [33], who derived the so-called bead-spring model for flexible coiled polymers. It is assumed that the macromolecules can be treated as threads consisting of N beads freely jointed by (N-l) springs. Furthermore, it is considered that the solution is ideally dilute, so that intermolecular interactions can be neglected. 

We now turn to a characterization of the dynamics in a polymer melt where, as it is supercooled, it approaches its glass transition temperature. We begin by looking at the translational dynamics in a bead-spring model and consider its analysis in terms of MCT. 

Figure 4.52 Single-molecule bead spring models for (a) dilute polymer solution and (b) polymer melt. From R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena, 2nd ed. Copyright 2002 by John Wiley Sons, Inc. This material is used by permission of John Wiley Sons, Inc.

Lodge,A.S., Wu,Y.-J. Constitutive equations for polymer solutions derived from the bead/spring model of Rouse and Zimm. Rheol. Acta 10,539-553 (1971). 

Single-molecule theories originated in early polymer physics work (45) to describe the flow behavior of very dilute polymer solutions, which are free of interpolymer chain effects. Most commonly, the macromolecular chain, capable of viscoelastic response, is represented by the well-known bead-spring model or cartoon, shown in Fig. 3.8(a), which consists of a series of small spheres connected to elastic springs. 

Comparison with experimental data demonstrates that the bead-spring model allows one to describe correctly linear viscoelastic behaviour of dilute polymer solutions in wide range of frequencies (see Section 6.2.2), if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. The validity of the theory for non-linear region is restricted by the terms of the second power with respect to velocity gradient for non-steady-state flow and by the terms of the third order for steady-state flow due to approximations taken in Chapter 2, when relaxation modes of macromolecule were being determined. 

Brunn PO (1984) Polymer migration phenomena based on the general bead-spring model for flexible polymers. J Chem Phys 80 5821-5826 Bueche F (1956) Viscosity of polymers in concentrated solutions. J Chem Phys 25 599-600 Chandrasekhar S (1943) Stochastic problems in physics and astronomy. Rev Modern Phys 15(l) l-89... 

Calculations with molecular structure models, as performed by Bueche, Rouse and others, in general based on bead spring models, predict that for monodisperse polymers the largest relaxation time is equal to... 

One model which has been extensively used to model polymers in the continuum is the bead-spring model. In this model a polymer chain consists of Nbeads (mers) connected by a spring. The easiest way to include excluded volume interactions is to represent the beads as spheres centered at each connection point on the chain. The spheres can either be hard or soft. For soft spheres, a Lennard-Jones interaction is often used, where the interaction between monomers is... 

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,... 

Time constants based on molecular theories have been derived for rod hke and bead-spring models (Bird et al., 1977b Ferry, 1980). For example, Whitcomb and Macosko (1978) showed that the conformation of xanthan gum in solution is rod like with some flexibility. The bead-spring model has found extensive use in the literature on polymers. The development of molecular theory for dilute solutions of linear... 

They have been developed based on either molecular structure or continuum mechanics where the molecular structure is not considered explicitly and the response of a material is independent of the coordinate system (principle of material indifference). In the former, the polymer molecules are represented by mechanical models and a probability distribution of the molecules, and relationships between macroscopic quantities of interest are derived. Three models have found extensive use in rheology the bead-spring model for dilute polymer solutions, and the transient net work and the reptation models for concentrated polymer solutions and polymer melts. 

In Fig. 3-5a, the polymer coil is modeled as a series of beads equally spaced along the polymer backbone and connected to each other by springs. The beads account for the viscous forces and the springs the elastic forces in the molecule the portion of the chain represented by a single spring is called a submolecule. The bead-spring model is... 

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. 

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