Bead-and-spring

We begin the mathematical analysis of the model, by considering the forces acting on one of the beads. If the sample is subject to stress in only one direction, it is sufficient to set up a one-dimensional problem and examine the components of force, velocity, and displacement in the direction of the stress. We assume this to be the z direction. The subchains and their associated beads and springs are indexed from 1 to N we focus attention on the ith. The absolute coordinates of the beads do not concern us, only their displacements. 

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 ... 

Fig. 8a,b. Off-lattice representations of a three-functional star a Bead and Rod model b Bead and Spring model... 

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 5.1 Schematic illustration of bead-and-spring model of atomic force between atoms. Reprinted, by permission, from M. F. Ashby and D. R. H. Jones, Engineering Materials 1, 2nd ed., p. 44. Copyright 1996 by Michael F. Ashby and David R. H. Jones.

Figure 17 A sketch of the rigid units of an oligomeric PFPE molecule (a) the flexible bonds with freely jointed beads and springs for coarse-grained bead-spring model and (b) SRS model with polarity (red arrow).

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

Figure 3.5 (a, b) Illustration of beads-and-springs models. Part (b) shows a two-bead flexible dumbbell. (From Larson 1988.)... 

The derivation of the constitutive equation for the dumbbell model can also be extended to allow multiple beads and springs. From the force-balance equations for such a model, one obtains (Bird et al. 1987b Larson 1988)... 

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... 

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