Macromolecule bead-spring

A. Milchev, K. Binder. Static and dynamic properties of adsorbed chains at surfaces Monte Carlo simulations of a bead-spring model. Macromolecules 29 343-354, 1996. 

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. 

It is understandable that the resistance coefficient decreases as the hydro-dynamic interaction increases. However, if one uses the bead-spring model of a macromolecule, the resistance coefficient of the whole macromolecule cannot depend on the arbitrary number of subchains N.5 To ensure this, one has to consider that the product hN1/2 does not depend on N which implies that the coefficient of hydrodynamic interaction changes with N as h N A which means, in this situation, that coefficient of resistance of a particle always remains to be proportional to the length of the subchain. All this is valid,... 

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. 

Fig. 8.8.1. Bead-spring model of a flexible macromolecule. All interactions of the macromolecule with light and the surrounding medium occurs through the beads. The springs (segments) merely provide an entropic restoring force to return the beads to their equilibrium separations whenever a shape fluctuation occurs.

The theoretical treatment of such non-Newtonian behavior has been through the development of constitutive equations to replace the Newtonian relationship between stress and strain rate, and to be used in continuum mechanical treatments of real flows. The constitutive equation has often been largely empirical in origin (like, for example, power law fluids), but increasingly has been derived from molecular theories, where macromolecules are modeled as bead-spring, bead-rod, or finitely extendable dumbbells. 

The most studied relaxation processes from the point of view of molecular theories are those governing relaxation function, G,(t), in equation [7.2.4]. According to the Rouse theory, a macromolecule is modeled by a bead-spring chain. The beads are the centers of hydrodynamic interaction of a molecule with a solvent while the springs model elastic linkage between the beads. The polymer macromolecule is subdivided into a number of equal segments (submolecules or subchains) within which the equilibrium is supposed to be achieved thus the model does not permit to describe small-scale motions that are smaller in size than the statistical segment. Maximal relaxation time in a spectrum is expressed in terms of macroscopic parameters of the system, which can be easily measured ... 

In recent years, my group in Durham has been looking at the use of simple coarse-grained models for the simulation of liquid crystalline macromolecules. These models combine anisotropic sites, such as the Gay-Berne potential (discussed above) and either bead-spring or simple united atom models for the polymer chains. A typical potential for such a system is represented by... 

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

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. 

Equations (25) - (29) determine the simplest approach to the dynamics of a macromolecule, even so, it appears to be rather complex if the effects of excluded volume, hydrodynamic interaction, and internal viscosity are taken into account. Due to these effects, all the beads in the chain ought to be considered to interact with each other in a non-linear way. To tackle with the problem, this set of coupled non-linear equations is usually simplified. There exist the different simpler approaches originating in works of Kirkwood and Riseman [46], Rouse [2], Zimm [5], Cerf [4], Peterlin [6] to the dynamics of a bead-spring chain in the flow of viscous liquid. The linearization is usually achieved by using preliminary-averaged forms of the matrix of hydrodynamic resistance (hydrodynamic interaction) [5] and the matrix of the internal viscosity [4]. In the last case, to ensure the proper covariance properties when the coil is rotated as a whole, Eq. (29) must be modified and written thus... 

Since molecular theories of viscoelasticity are available only to describe the behavior of isolated polymer molecules at infinite dilution, efforts have been made over the years for measurements at progressively lower concentrations and it has been finally possible to extrapolate data to zero concentration. The behavior of linear flexible macromolecules is well described by the Rouse-Zimm theory based on a bead-spring model, except at high frequencies . Effects of branching can be taken into account, at least for starshaped molecules. At low and intermediate frequencies, the molec-... 

The most studied relaxation processes from the point of view of molecular theories are those governing relaxation function, Gi(t), in equation [7.2.4]. According to the Rouse theoiy, a macromolecule is modeled by a bead-spring chain. The beads are the centers of... 

In the Rouse model approach, the macromolecule is considered as a bead spring free of any hydrodynamic interaction with the solvent, which then can be of the same nature as the polymer. This approach can be applied to the case of a... 

Model simulating the hydrodynamic properties of a chain macromolecule consisting of a sequence of beads, each of which offers hydrodynamic resistance to the flow of the surrounding medium and is connected to the next bead by a spring which does not contribute to the frictional interaction but which is responsible for the elastic and deformational properties of the chain. The mutual orientation of the springs is random. 

As suggested by Fig. 1, the interbead forces within a single molecule are described by springs. However, if it is desired to take into account the excluded volume effect (the fact that the various segments of the macromolecule have finite volume and hence cannot overlap), then one can in addition include a Lennard-Jones type of interaction between those beads that are not connected by springs. Here again, the interbead force vector is taken to be collinear with the interbead vector. 

Tandon GP, Weng GJ (1984) The effect of aspect ratio of inclusion on elastic properties of unidirectionally aligned composites. Polym Compos 5 327-333 Tanner RI (1960) Full-film lubrication theory for a Maxwell liquid. Int J Mech Sci 1 206-215 Tanner RI (1975) Stresses in dilute solutions of bead-nonlinear-spring macromolecules, ii. Unsteady flows and approximate constitutive relations, constitutive relations. J Rheol 19 37-65... 

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