Bead/spring chain model

Bird, R. B., Dotson, P. J., and Johnson, N. L., Polymer solution rheology based on a finitely extensible bead-spring chain model, J. Non-Newtonian Fluid Mech., 7, 213-235 (1980). 

Underhill PT and Doyle PS. (2006) Alternative spring force law for bead-spring chain models of the worm-hke chain. Journal of Rheology, 50, pp. 513-529. 

In order to describe rheological properties that depend on a spectrum of relaxation times the dumbbell models are inadequate, and it is necessary to use mechanical models with many beads (ref. 66, chapter 15). The bead-spring chain model of Figure 12 has been the subject of much study " this model poses no particular problems, since the chain geometry is well understood and the matrices needed to facilitate derivations have been presented systematically (ref. 66, tables 15.1-1 and 15.4-1). 

Since the relationship between the nonequilibrium and equilibrium motion cannot be uniquely specified, the validity of the Cox-Merz mle is not a trivial result. In fact, the Cox-Merz mle does not hold for nonentangled polymers The rjly) data of these polymers are insensitive to y and very close to rjo (no thinning), while their r/ co) data are well described by the bead-spring chain model and exhibit a power-law decrease with inaeasing cu > Thus, the validity of the Cox-Merz rule for entangled polymers reflects some characteristic aspect of the nonequilibrium djmamics of these polymers. 

FIG. 6 Illustration of the bond fluctuation Monte Carlo model and the standard bead-spring chain (see, e.g. [4]). 

The behavior of a bead-spring chain immersed in a flowing solvent could be envisioned as the following under the influence of hydrodynamic drag forces (fH), each bead tends to move differently and to distort the equilibrium distance. It is pulled back, however, by the entropic need of the molecule to retain its coiled shape, represented by the restoring forces (fs) and materialized by the spring in the model. The random bombardment of the solvent molecules on the polymer beads is taken into account by time smoothed Brownian forces (fB). Finally inertial forces (f1) are introduced into the forces balance equation by the bead mass (m) times the acceleration ( ) of one bead relative to the others ... 

The system used in the simulations usually consists of solid walls and lubricant molecules, but the specific arrangement of the system depends on the problem under investigation. In early studies, hard spherical molecules, interacting with each other through the Lennard-Jones (L-J) potential, were adopted to model the lubricant [27], but recently we tend to take more realistic models for describing the lubricant molecules. The alkane molecules with flexible linear chains [28,29] and bead-spring chains [7,30] are the examples for the most commonly used molecular architectures. The inter- and intra-molecular potentials, as well as the interactions between the lubricant molecule and solid wall, have to be properly defined in order to get reliable results. Readers who intend to learn more about the specific techniques of the simulations are referred to Refs. [27-29]. 

Fig. 1 a Model bead-spring chain interacting through bond potential Dj, bond angle potential Uq, and van der Waals potential C7v(jw> and b the form of the bond angle potential Ug... 

The inclusion of chain connectivity prevents polymer strands from crossing one another in the course of a computer simulation. In bead-spring polymer models, this typically means that one has to limit the maximal (or typical) extension of a spring connecting the beads that represent the monomers along the chain. This process is most often performed using the so-called finitely extensible, nonlinear elastic (FENE) type potentials44 of Eq. [17]... 

Our model of a polyelectrolyte solution consists of Np flexible bead-spring-chains which are located in a simulation box of length L with periodic boundary conditions. For each chain, a fraction / of the N monomers is monovalently charged (v=l), and fN oppositely charged monovalent counterions are added to obtain an electrically neutral system. In some cases Ns pairs of salt ions were added. The density is given in form of the charged... 

Fig. 12. Equation of state (a) and phase diagram (b) of a bead-spring polymer model. Monomers interact via a truncated and shifted Lennard-Jones potential as in Fig. 6 and neighboring monomers along a molecule are bonded together via a finitely extensible non-linear elastic potential of the form iJpENE(r) = — 15e(iJo/

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

The starting point of a molecular constitutive theory is a simple mechanical model for the molecule that captures its most salient traits. Thus, flexible polymer molecules have been represented by elastic dumbbells and bead-spring chains, and rigid polymers by rigid dumbbells and rigid rods. For its simplicity, the evolution of the model molecule is easily described by a convection-diffusion equation. Then a Fokker-Planck equation is written for the probability distribution function of an ensemble of these molecules. Finally, the macroscopic stress tensor is derived in terms of the distribution function. This kinetic theory framework was pioneered by Kirkwood (see, for example, Ref. ). 

Agarwal and Mashelkar first analyzed contradictory reports from Kim [190], Gryte [191], and Singh [192, 193], and proposed a simple mechanistic model [209]. In stark contrast to the concept of preferential scission of side chains, their model reveals decreased shear stability by grafting side chains. They extended Odell and Keller s bead-rod model [27]. The backbone was modeled as a fully extended rod with Ni = 2m+ beads (Fig. 25). p grafted bead-spring chains having g beads with... 

In 1944 Kramers [1] published a phase-space kinetic theory for the steady-state potential flow of monodisperse dilute polymer systems in which the polymer molecule is modeled as a freely jointed bead-rod chain. Subsequent scholars developed kinetic theories for shearing flows of monodisperse dilute polymer solutions Kirkwood [2] for freely rotating bead-rod chains with equilibnum-averaged hydrodynamic interaction. Rouse [3] and Zimm [4] for freely jointed bead-spring chains, and others. These theories were all formulated m the configuration space of a single polymer chain. 

Once the singlet distribution function has been found, we are in a position to evaluate the various contributions to the fluxes that depend on (see Table 1). In this section we discuss the contnbutions to the stress tensor, and in the next two sections the contnbutions to the mass and heat flux vectors. In these sections, for illustrative purposes, we restrict ourselves to the Rouse bead-spring chain and the Hookean dumbbell models, for which we can use the singlet distribution functions , given in Eqs. (13.5) and (13.8). 

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 ... 

For a dilute solution of polymer Aina low molecular weight solvent B, the polymer molecules are modeled as bead-spring chains. Resistance in the motion of beads is characterized by a friction coefficient As the number of beads is proportional to the polymer molecular weight M, we have Dab 1 / Vm. Table 2.8 shows some values of diffusion coefficients in polymers. In a flow of dilute solution of polymers, the diffiisivity tensor is anisotropic and depends on the velocity gradient. The Maxwell-Stefan equation may predict the diffusion in multicomponent mixtures of polymers. 

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)... 

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