The Bead-Spring Model

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 T. Moreover, it is assumed that the beads are jointed by an ideally Hookean spring, i.e. they obey a strictly linear force law. 

In most cases polymer solutions are not ideally dilute. In fact they exhibit pronounced intermolecular interactions. First approaches dealing with this phenomenon date back to Bueche [35]. Proceeding from the fundamental work of Debye [36] he was able to show that below a critical molar mass Mw the zero-shear viscosity is directly proportional to whereas above this critical value T 0 is found to be proportional to (Mw3,4) [37,38]. This enhanced drag has been attributed to intermolecular couplings. Ferry and co-workers [39] reported that the dynamic behaviour of polymeric liquids is strongly influenced by coupling points. 

For polymer melts or solutions, Graessley [40-42] has shown that for a random coil molecule with a Gaussian segment distribution and a uniform number of segments per unit volume, a shear rate dependent viscosity arises. This effect is attributed to shear-induced entanglement scission. 

Introduction of the reptation concept by De Gennes [43] led to further essential progress. Proceeding from the notion of a reptile-like motion of the polymer chains within a tube of fixed obstacles, De Gennes [43-45], Doi [46, 47] and Edwards [48] were able to confirm Bueche s 3.4-power-law for polymer melts and concentrated polymer solution. This concept has the disadvantage that it is valid only for homogeneous solutions and no statements about flow behaviour at finite shear rates are analysed. 

Non-linear viscoelastic flow phenomena are one of the most characteristic features of polymeric liquids. A matter of very emphasised interest is the first normal stress difference. It is a well-accepted fact that the first normal stress difference Nx is similar to G, a measure of the amount of energy which can be stored reversibly in a viscoelastic fluid, whereas t12 is considered as the portion that is dissipated as viscous flow [49-51]. For concentrated solutions Lodge s theory [52] of an elastic network also predicts normal stresses, which should be associated with the entanglement density. 

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We refer to this model as the bead-spring model and to its theoretical development as the Rouse theory, although Rouse, Bueche, and Zimm have all been associated with its development. 

Figure 9 Chain center of mass self-diffusion coefficient for the bead-spring model as a function of temperature (open circles). The full line is a fit with the Vogel-Fulcher law in Eq. [3]. The dashed and dotted lines are two fits with a power-law divergence at the mode-coupling critical temperature. 

No crystalline order is visible for the bead-spring model upon cooling to the frozen-in phase at T = 0.3. The break in the volume-temperature curve (described in the section on thermodynamic information) occurring between T = 0.4 and T = 0.45 leads us to expect that the two-step decay described by MCT should be observable at simulation temperatures above (and close to) this region. This expectation is borne out in Figure 10, which shows the... 

Figure 10 Intermediate incoherent scattering function for the bead-spring model at T = 0.48 for different values of momentum transfer given in the legend.

Figure 12 Test of the factorization theorem of MCT for the intermediate coherent scattering function for the bead-spring model and a range of -values indicated in the Figure. Data taken from Ref. 132 with permission.

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.

In the discussion on the dynamics in the bead-spring model, we have observed that the position of the amorphous halo marks the relevant local length scale in the melt structure, and it is also central to the MCT treatment of the dynamics. The structural relaxation time in the super-cooled melt is best defined as the time it takes density correlations of this wave number (i.e., the coherent intermediate scattering function) to decay. In simulations one typically uses the time it takes S(q, t) to decay to a value of 0.3 (or 0.1 for larger (/-values). The temperature dependence of this relaxation time scale, which is shown in Figure 20, provides us with a first assessment of the glass transition... 

The bead-spring models are devices to circumvent the complications of the local motion problem and still obtain information on the large-scale configurational relaxations which control viscoelastic behavior. Their utility lies in the... 

Mz = Q2/Q1, and Mz+1 = Q3/Q2- The steady state values of //0 and Je° are independent of the number of elements in the bead-spring model as long as N is sufficiently large. However, the initial modulus depends explicitly on N ... 

The relaxation time distribution of the bead-spring models is discrete. The spectrum is... 

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 ... 

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). 

To represent the molecular structure with reasonable accuracy as well as to reduce computational time, the coarse-grained, bead-spring model [Fig. 1.28(b)] was employed to approximate a PFPE molecule. This simplifies the detailed atomistic information while preserving the essence of the molecular internal structure [167]. The off-lattice MC technique with the bead-spring model was used to examine nanoscale PFPE lubricant film structures and stability with internal degrees of freedom [168],... 

Figure 1.41. Potential energies for the bead-spring model LJ1—Lennard-Jones potential LJ2—van der Waals potential EXP1, EXP2—short-range polar potential FENE—finitely extensible nonlinear elastic potential.

So far, we have demonstrated that the MC simulation (lattice-based SRS model and off-lattice bead-spring model) results are in qualitative agreement with the experiments. A complementary approach is molecular dynamics (MD) simulation using the bead-spring model. Since MD study for PFPE is still the infant stage, we will discuss it only briefly. The equation of motion can be expressed in... 

Figure 1.53. The bead-spring model may be used to simulate PFPE molecules in head-disk interface (HDI), which couple with slider dynamics via steep pressure and temperature gradients, which appears in HAMR technology [35],...

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. 

One model which has been extensively used to model polymers in the continuum is the bead-spring model. In this model a polymer chain consists of Nbeads (mers) connected by a spring. The easiest way to include excluded volume interactions is to represent the beads as spheres centered at each connection point on the chain. The spheres can either be hard or soft. For soft spheres, a Lennard-Jones interaction is often used, where the interaction between monomers is... 

Now that we have settled on a model, one needs to choose the appropriate algorithm. Three methods have been used to study polymers in the continuum Monte Carlo, molecular dynamics, and Brownian dynamics. Because the distance between beads is not fixed in the bead-spring model, one can use a very simple set of moves in a Monte Carlo simulation, namely choose a monomer at random and attempt to displace it a random amount in a random direction. The move is then accepted or rejected based on a Boltzmann weight. Although this method works very well for static and dynamic properties in equilibrium, it is not appropriate for studying polymers in a shear flow. This is because the method is purely stochastic and the velocity of a mer is undefined. In a molecular dynamics simulation one can follow the dynamics of each mer since one simply solves Newton s equations of motion for mer i,... 

The bead—spring model allows the introduction of finite extensibility of the (harmonic springlike) bonds by the introduction of a form of a harmonic-type potential... 

Time constants based on molecular theories have been derived for rod hke and bead-spring models (Bird et al., 1977b Ferry, 1980). For example, Whitcomb and Macosko (1978) showed that the conformation of xanthan gum in solution is rod like with some flexibility. The bead-spring model has found extensive use in the literature on polymers. The development of molecular theory for dilute solutions of linear... 

They have been developed based on either molecular structure or continuum mechanics where the molecular structure is not considered explicitly and the response of a material is independent of the coordinate system (principle of material indifference). In the former, the polymer molecules are represented by mechanical models and a probability distribution of the molecules, and relationships between macroscopic quantities of interest are derived. Three models have found extensive use in rheology the bead-spring model for dilute polymer solutions, and the transient net work and the reptation models for concentrated polymer solutions and polymer melts. 

While empirically useful, the temporary network model gives no indiction of the relationship between the relaxation spectrum and the molecular relaxation processes. In the next sections, this deficiency is addressed by returning to the bead-spring models of Fig. 3-5. 

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