Bead-and-spring chain

The equilibrium features of the unperturbed bead-and-spring chain are summarized by the relationships... 

When the bead-and-spring chain is not in the ideal state, the intramolecular force is given in Eqn. (3.1.3). As it may be seen, in general, the force is not simply transmitted by first-neighboring atoms, but it has a long-range character. The relaxation times are given by Eqn. (3.1.11) after they are known, the dynamic viscosity i (cu) and the atomic correlation function B(k, t) are obtained from Eqs. (3.1.15) and (3.1.18) (for the periodic chain), and the complex modulus and dynamic structure factors are easily constructed. 

Let us consider a solution of long polymer chains (N—> ) adopting the simplest beads-and-springs chain model (Figure 12(b)). Two beads located at r, and q are interaaing with the effective potential energy (r,-r,) which includes... 

The first property can be deduced from the results of Section 1.02.5.2. Concentration fluctuations are weak since the melt compression modulus is high cv tl, where c is the concentration of repeat rmits. The correlation length of density fluctuations f is defined in eqn [96] which is qualitatively applicable for cv 1 giving that is, f is comparable with the statistical segment b for the standard beads-and-springs chain model. (More generally f in a polymer melt is comparable with the chain persistence length I)... 

The bead and spring model is clearly based on mechanical elements just as the Maxwell and Voigt models were. There is a difference, however. The latter merely describe a mechanical system which behaves the same as a polymer sample, while the former relates these elements to actual polymer chains. As a mechanical system, the differential equations represented by Eq. (3.89) have been thoroughly investigated. The results are somewhat complicated, so we shall not go into the method of solution, except for the following observations ... 

Usually, MD methods are applied to polymer systems in order to obtain short-time properties corresponding to problems where the influence of solvent molecules has to be explicitly included. Then the models are usually atomic representations of both chain and solvent molecules. Realistic potentials for non-bonded interactions between non-bonded atoms should be incorporated. Appropriate methods can be employed to maintain constraints corresponding to fixed bond lengths, bond angles and restricted torsional barriers in the molecules [117]. For atomic models, the simulation time steps are typically of the order of femtoseconds (10 s). However, some simulations have been performed with idealized polymer representations [118], such as Bead and Spring or Bead and Rod models whose units interact through parametric attractive-repulsive potentials. 

Figure 7.5 Bead-and-spring model of a polymer chain.

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 ... 

The Bead-and-Spring Model in Bad and Good Solvents Dynamics of the Collapsed Chain... 

In the next section we shall consider the equilibrium properties of some typical models of unperturbed chains with an increasing degree of complexity. They are (i) the bead-and-spring phantom chain (ii) the phantom chain with nearest-neighbor correlation and (iii) the unperturbed real chain... 

Is such a deformable chain model inconsistent with the non-Newtonian intrinsic viscosity Finding an answer to this question is the goal of this paper. To this end, the viscosity of xanthan solutions was measured over a broad range of shear stress, including especially the low-shear Newtonian limit which has not been measured by Whitcomb and Macosko. The intrinsic viscosity at various shear stresses was then determined and the resultant experimental curve was compared to theoretical expectations for a flexible chain (bead-and-spring) model. 

Only recently has the theory of chain dynamics been extended by Peterlin (J [) and by Fixman (12) to encompass the known non-Newtonian intrinsic viscosity ofTlexible polymers. This theory, which is an extension of the Rouse-Zimm bead-and-spring model but which includes excluded volume effects, is much more complex than that for undeformable ellipsoids, and approximations are needed to make the problem tractable. Nevertheless, this theory agrees remarkably well (J2) with observations on polystyrene, which is surely a flexible chain. In particular, the theory predicts quite well the characteristic shear stress at which the intrinsic viscosity of polystyrene begins to drop from its low-shear Newtonian plateau. 

The chain is modeled as a system of beads and springs undergoing Brownian motion in a viscous medium. The other polymer chains provide the viscous medium for any individual chain. The inherent dynamics can be represented in terms of N relaxation modes, where N is the number of statistical subunits in the chain. The shear relaxation modulus G(f) is given by ... 

Figure 12 (a) A coarse-grained chain of Mg=6 superunits (groups of g=5 consecutive repeat units) with grouped bond vectors (b) The beads-and-springs model of a Gaussian chain h.rz. are the position vectors of the beads. 

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