Bead spring model of Rouse

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). 

Stockmayer and Kennedy (1975) conducted a seminal smdy on the chain dynamics of Rouse chains of AB-type diblock or ABA-type triblock copolymers by modifying the bead—spring model of Rouse for linear flexible homopofymers (see Chapter 4). They calculated the spectrum of relaxation times ftp biodc) block copolymer in terms of the terminal relaxation times for the Rouse chains for the A and B blocks. Once the values of are determined, one can calculate linear dynamic viscoelastic... 

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. 

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. 

FIG. 16.9 Logarithmic plots of [G ]R and [G"]R vs. cot, for bead-spring models. (A) Rouse free-draining (negligible hydrodynamic interaction) (B) Zimm non-draining (dominant hydrodynamic interaction). After Ferry, General References, 1980. 

Fig. 5.2 An illustration of the bead-spring model of the Rouse chain...

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... 

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. 

Zimm [34] extended the bead-spring model by additionally taking hydrodynamic interactions into account. These interactions lead to changes in the medium velocity in the surroundings of each bead, by beads of the same chain. It is worth noting that neither the Rouse nor the Zimm model predicts a shear rate dependency of rj. Moreover, it is assumed that the beads are jointed by an ideally Hookean spring, i.e. they obey a strictly linear force law. 

Figure 13 Temperature dependence of the time scales for the first five Rouse modes in the bead-spring model in the vicinity of the MCT Tc. 

Figure 14 Master curve generated from mean-square displacements at different temperatures, plotting them against the diffusion coefficient at that temperature times time. Shown are only the envelopes of this procedure for the monomer displacement in the bead-spring model and for the atom displacement in a binary Lennard-Jones mixture. Also indicated are the long-time Fickian diffusion limit, the Rouse-like subdiffusive regime for the bead-spring model ( ° 63), the MCT von Schweidler description of the plateau regime, and typical length scales R2 and R2e of the bead-spring model.

It is interesting to examine the bead-spring models to see what flow-induced configurational changes would be required in order to develop N2 values of the proper magnitude and sign. In the Rouse model, the components of the stress tensor are related directly to averages of the internal coordinates of the beads. For the simplest case of the elastic dumbbell ... 

In the Rouse model, a chain of N monomers is mapped onto a bead spring chain of N beads connected by springs. 

This latter model was employed by Rouse (27) and by Bueche (28) in the calculation of viscoelasticity and is sometimes called the Rouse model. It was used later by Zimm (29) in a more general calculation which may be regarded as an application of the Kirkwood theory. As illustrated in Fig. 2.1, the Rouse model is composed of N + 1 frictional elements represented by beads connected in a linear array with N elastic elements or springs, hence the bead-spring model designation. The frictional element is assumed to represent the translational friction... 

Just as the Gaussian chain is the basic paradigm of the statistics of polymer solutions, so is its extension to the bead-spring model still basic to current work in the held of polymer dynamics. The two limiting cases of free draining (no hydrodynamic interaction between beads, characterized by the draining parameter A = 0) and non-free draining (dominant hydrodynamic interaction, A= CO, due to Rouse and Zimm, respectively, are sufficiently familiar that the approach is often known as the Rouse-Zimm model. ... 

The Rouse modeP is the simplest version of the bead-spring model that can treat the chain dynamics. The model assumes that the beads have no excluded volume (they are essentially a point) and that there are no hydrodynamic interactions... 

The explicit expressions for [17] for the three cases of the bead-spring model are 1. Rouse model. With Eq. 3.136,... 

Another very important analytically solvable case is the harmonic oscillator. This term is used for a mechanical system in which potential energy depends quadratically on displacement from the equilibrium position. The harmonic oscillator is very important, as it is an interacting system (i.e., a system with nonzero potential energy), which admits an analytical solution. A diatomic molecule, linked by a chemical bond with potential energy described by Eq. (2), is a typical example that is reasonably well described by the harmonic oscillator model. A chain with harmonic potentials along its bonds (bead-spring model), often invoked in polymer theories such as the Rouse theory of viscoelasticity, can be described as a set of coupled harmonic oscillators. 

The segmental motion of a polymer chain was successfully described by a bead-spring model, discussed by Rouse [17] in the so-called free-draining limit and by Zimm [18] in the hydrodynamic limit, de Gennes [19,20] calculated the coherent and incoherent intermediate scattering functions for both the Rouse and Zimm models. In the low Q and long time limit, the time decay of the intermediate scattering function depends on and and the Q dependence of the... 

Two general classical bead-spring models have been developed for the description and analysis of the motions of flexible chains (see chapter Conformational and Dynamic Behavior of Polymer and Polyelectrolyte Chains in Dilute Solutions ). The Rouse model [54] is simpler (it does not take into account hydrodynamic correlations). The more advanced Zimm model accounts for hydrodynamic correlations and provides better description of the behavior [55]. In both cases, solution of the derived equations provides the so-called normal modes (relaxation times of different types of motions). The first mode describes the slowest motion of the... 

In the theories for dilute solutions of flexible molecules based on the bead-spring model, the contribution of the solute to the storage shear modulus, loss modulus, or relaxation modulus is given by a series of terms the magnitude of each of which is proportional to nkT, i.e., to cRTjM, as in equation 18 of Chapter 9 alternatively, the definition of [C ]y as the zero-concentration limit of G M/cRT (equations 1 and 6 of Chapter 9) implies that all contributions are proportional to nkT. Each contribution is associated with a relaxation time which is proportional to [ri Ti)sM/RT-, the proportionality constant (= for r i) depends on which theory applies (Rouse, Zimm, etc.) but is independent of temperature, as is evident, for example, in equation 27 of Chapter 9. Thus the temperature dependence of viscoelastic properties enters in four variables [r ], t/j, T explicitly, and c (which decreases slightly with increasing temperature because of thermal expansion). 

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-... 

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