Bead-spring model of polymer

Fong, J.T. Peterlin, A. Matrix algebra and eigen values for the bead/spring model of polymer solutions. J. Res. 

In the Rouse-Zimm bead spring model of polymer solution dynamics, the long-range global motions are associated with a broad spectrum of relaxation times given by equation (10) and where tj is the relaxation time of the h such normal mode of the chain... 

The bead-spring model of polymer chains utilized for molecular dynamics calculations in Ref. 19 consists of beads interacting through a truncated and shifted 12-6 Lennard-Jones potential... 

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). 

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. 

In the simulation study of fluorescence anisotropy decay, a generic bead-spring model of the polymer was used. It is schematically shown in Fig. 19. Each bead can represent one or several monomer units in a real polymer. The degree of dissociation, a, is defined as the fraction of monomer units carrying electric charges. The interaction between monomer units of the polymer is modeled by the Lennard-Jones potential and the solvent quality is controlled by the depth of this potential, e. As shown by Micka, Holm and Kremer, 0.34 corresponds to the theta state [146]. The simulation study was performed for several values of > 0.33, i.e., under poor solvent conditions. The simulation technique used was MD coupled to a Langevin thermostat, i.e., the polymer was simulated in an implicit solvent. The counterions were simulated explicitly. A more detailed description of the polymer model can be found in the original paper [87]. 

To take a polymer example, consider simulations of the bead-spring model of the linear polymer chain from Fig. 2, at temperature T. The probability density to find it in a partieular eonformation F, specified by the set of coordinates of its N monomers, r = ri,r2,...,rjv, is a product of probability densities for the bonds [compare Eq. (12)] it ean be considered as being a canonical distribution of the form of Eq. (46) (here we use F to denote a point in configuration space, not in phase spaee). 

As a preliminary step, we consider the dynamics of the macromolecular coil moving in the flow of a viscous liquid. The bead-spring model of a macromolecule is usually used to investigate large-scale or low-frequency dynamics of a macromolecular coil, while molecules of solvent are considered to constitute a continuum - viscous liquid. This is a mesoscopic approach to the dynamics of dilute solutions of polymers. There is no intention to collect all the available results and methods concerning the dynamics of a macromolecule in viscous liquid in this section. They can be found elsewhere [9,29]. We need to consider the results for dilute solutions mainly as a background to the discussion of the dynamics of a macromolecule in very concentrated solutions and melts of polymers. 

These bead-spring models of Rouse and Kirkwood-Riseman-Zimm suffer from the artificiality of the beads and springs. The bead friction coefficient is an ad hoc phenomenological coefficient. This should arise naturally from the frictional forces coupling the polymer and solvent directly with the continuous version of the chain without beads and springs. 

We employ the freely-jointed bead-spring model of a polymer as it has been successfully used for simulations of neutral polymers . Each polymer chain consists of... 

Although the bead-spring models of the previous section can be considered representative of the behavior of dilute polymer solutions, they cannot be used to predict the behavior of concentrated solutions or melts due to the formation of entanglements. Here, we turn instead to network models. The motivation for using these models is the success of the theory of rubber elasticity in explaining the stress-strain behavior of a network of polymer molecules linked together by... 

Lodge, A. S., and Y. J. Wu, Constitutive Equations for Polymer Solutions Derived from the Bead/Spring Model of Rouse and Zimm, Rheol. Acta, 10, 539-553, 1971. Tam, K. C., and C. Tiu, Steady and Dynamic Shear Properties of Aqueous Polymer Solutions, J. Rheol, 33, 257-280, 1989. 

Gordon, R. J., and A. E. Everage, Jr., Bead-Spring Model of Dilute Polymer Solutions Continuum Modifications and an Explicit Constitutive Equation, J. AppL Polym. Set, 15, 1903-1909, 1971. 

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. 

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