Bead-spring friction coefficient

Like the original Rouse model, the GRM involves a bead-spring friction coefficient. The three-dimensional conformation of a polymer chain is decomposed into Fourier modes which relax according to spectrum of relaxation times. The relaxation time corresponding to the Fourier mode with index p is... 

It should be observed that in this equation the vector a = Vln r appears, but the vector bj, = Vln , has dropped out. Equation (13.17) is valid for any bead-spring model for which all beads and friction coefficients are the same also external forces have been neglected in developing this result. 

In an attempt to describe the behavior at large chain deformations, de Gennes [7] incorporated into the dumbbell model the FENE spring law along with a variable bead friction coefficient which increases linearly with the interbead distance ... 

Each submolecule will experience a frictional drag with the solvent represented by the frictional coefficient /0. This drag is related to the frictional coefficient of the monomer unit (0- If there are x monomer units per link then the frictional coefficient of a link is x(0- If we aPply a step strain to the polymer chain it will deform and its entropy will fall. In order to attain its equilibrium conformation and maximum entropy the chain will rearrange itself by diffusion. The instantaneous elastic response can be thought of as being due to an entropic spring . The drag on each submolecule can be treated in terms of the motion of the N+ 1 ends of the submolecules. We can think of these as beads linked... 

Fig. 3.1 Bead-spring-bead model of a Gaussian chain as assumed in tbe Rouse model. Tbe beads are connected by entropic springs and are subject to a frictional force where v is the bead velocity and fo the bead friction coefficient...

A free-draining model with equal bead friction coefficients and W = const, such as a stiff bead-spring polymer, may be efficiently simulated with... 

The Rouse-Bueche model (97,98) replaces the real molecule of n main chain atoms by a mechanical chain of N +1 beads joined in sequence by N linear springs. The frictional interactions with the medium, which are distributed uniformly along the length of the real molecule to give a molecular frictional coefficient n(0, ate concentrated at regular intervals in the beads. The frictional... 

Spring-bead models relate frictional force to the relative velocity of the medium at the point of interaction. The entanglement friction coefficient above is defined in terms of the relative velocity of the passing chain. Since the coupling point lies, on the average, midway between the centers of the two molecules involved, the macroscopic shear rate must be doubled when applying the result to a spring-bead model. Substitution of 2 CE for Con in the Rouse expression for viscosity yields... 

The beads represent entanglement sites which are distributed uniformly along the chain contour the frictional coefficients increase rapidly with distance from the chain ends. The spring constant also depends on contour position, being governed by the mean equilibrium distance of that position from the center of gravity. The resulting spectrum is narrower than the Rouse spectrum, and for E > 1 ... 

Internal viscosity (Section 4) provides another possible source of shear-rate dependence. For sufficiently rapid disturbances, a spring-bead model with internal viscosity acts like a rigid body for sufficiently slow disturbances it is flexible and indefinitely extensible. The analytical difficulties for coupled, non-linear spring-bead systems are equally severe in linear spring-bead systems with internal viscosity. Even the elastic dumbbell with internal viscosity has only been solved exactly in the limit of small e (559), where e is the ratio of internal friction coefficient to molecular (external) friction coefficient Co n. For this case, the viscosity decreases with shear rate. 

The first successful molecular model of polymer dynamics was developed by Rouse. The chain in the Rouse model is represented as N beads connected by springs of root-mean-square size b, as shown in Fig. 8.2. The beads in the Rouse model only interact with each other through the connecting springs. Each bead is characterized by its own independent friction with friction coefficient (. Solvent is assumed to be freely draining through the chain as it moves. 

The Rouse model is the simplest molecular model of polymer dynamics. The chain is mapped onto a system of beads connected by springs. There are no hydrodynamic interactions between beads. The surrounding medium only affects the motion of the chain through the friction coefficient of the beads. In polymer melts, hydrodynamic interactions are screened by the presence of other chains. Unentangled chains in a polymer melt relax by Rouse motion, with monomer friction coefficient C- The friction coefficient of the whole chain is NQ, making tha diffusion coefficient inversely proportional to chain length ... 

A polymer ehain can be approximated by a set of balls connected by springs. The springs aeeount for the elastic behaviour of the chain and the beads are subject to viscous forces. In the Rouse model [35], the elastic force due to a spring connecting two beads is/= bAr, where Ar is the extension of the spring and the spring eonstant is h = T/a ffpis the root-mean-square distance of two successive beads. The viseous foree that acts on a bead is the produet of the bead velocity u and of the friction coefficient of a bead. With these assumptions, one finds for... 

The model molecule consists on n + 1 light-scattering beads each with identical isotropic polarizabilites a. The beads are connected by springs (or segments in the previous description) which provide a restoring force linear in the displacement if some beads stray from their equilibrium separations. Each bead interacts with the surrounding medium through identical frictional coefficients f and, in addition, Brownian forces are exerted on the beads by solvent molecules, (see Fig. 8.8.1). 

This model consists of two identical beads with bead friction coefficient C joined by a Hookean spring with spring constant H. 

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. 

The starting point for molecular models for polymer dynamics based on the ideas introduced in Section 14.2.3 is the Rouse model for an isolated chain in a viscous medium, in which the chain is taken to behave as a sequence of m beads linked by Gaussian springs [Figure 14.9(a)] [13-16]. The chain interacts with the solvent via the beads, and the solvent is assumed to drain freely as the chain moves. Hence, Eq. (22) leads to Eqs. (45), where N is the number of links between adjacent beads, C is a friction coefficient per bead and r is the position of the ith bead. 

The left-hand side represents the viscous force, the parameter Cr designates the friction coefficient per bead. On the right-hand side we have the elastic forces originating from the adjacent beads, which are located at the positions r/ i and r/4.1. The force constant 6r of the springs depends on the mean squared end-to-end distances of the Rouse-sequences, a, and follows from Eq. (6.21) as... 

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. 

The rigid sphere is not an accurate picture of how a polymer molecule affects the flow of the fluid in which it is dissolved, because the fluid can penetrate within the molecule. This recognition led to the development of models in which the molecule is represented as a chain of beads, which contain all the mass of the molecule, connected by springs. In the free-draining model of Rouse [45], there is no effect of one bead on the flow pattern around other beads. This model starts from Stokes law, which gives the drag force f on a sphere in a Newtonian fluid flowing past it at the velocity U as proportional to the radius a of the sphere. In terms of the coefficient of friction f =Fj /U), Stokes law for flow past a sphere is ... 

From the bead movement, it is possible to roughly estimate the spring constant. Notice that the movement isn t a reciprocating oscillation, but an over-damped one (solid line in Fig. 59a). The viscous friction should play an important role. Since the bead velocity is slow (v=1.5 /tm/s at maximum, the dashed line in Fig. 59a), it is reasonable to assume the viscous resistance, a non-conservative force that is always opposed to its direction of motion, is proportional to the speed, in other words /vis=cv, where c is a coefficient. In this case the spring oscillation can be described by the following equation ... 

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