Gaussian spring-bead chain

For an unperturbed (Gaussian) spring-bead chain the mean square and mean reciprocal rij ) of the distance rij between its beads i and j are equal to a i — j and 6/na Y i —These are well-known facts and are cited here without proof (see Ref. [2]). How are these averages expressed when the chain is perturbed by intrachain excluded-volume interactions This problem was first brought us by Peterlin [56] in 1955. Solution of it needs information about Wij(R), but, as mentioned above, work on this function is as yet in the process of development. 

Under the 9 condition the intrachain interference factor Hi k) is not affected by the presence of other chains, so that its value Hie k) at any concentration is equal to that at c = 0. For a Gaussian spring-bead chain Pij(r) at this limit is given exactly by [15]... 

We consider a spring-bead chain consisting of N Gaussian springs. Its contour length L is expressed by... 

Now we join the N scatterers by identical Gaussian springs to form a spring-bead chain. Then our object becomes a fiexible linear polymer in dilute solution. We introduce a quantity Q, k) defined by... 

The primitive chain in the Doi-Edwards theory is a smooth wire of constant length. Less coarse-grained and thus closer to actual polymer molecules is the spring-bead chain model. In this model, the beads wriggle individually under the constraint that they are connected by Gaussian springs. Their motions differ depending on the timescale on which we look at the chain. To discuss this problem we consider the discrete chain version of g t) defined by eq 2.16, i.e.. 

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...

Zimm s model (1956) is also a chain of beads connected by ideal springs. The chain consists of N identical segments joining + 1 identical beads with complete flexibility at each bead. Each segment, which is similar to a submolecule, is supposed to have a Gaussian probability function. The major difference between the two models lies in the interaction between the individual beads. In the Rouse model, such interaction is ignored in Zimm s model, such interaction is taken into consideration. 

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 Rouse model starts from such a Gaussian chain representing a coarsegrained polymer model, where springs represent the entropic forces between hypothetic beads [6] (Fig. 3.1). 

For the bead-spring model, a/b has to be smaller than 1/2 to avoid the interpenetration of the neighboring spheres. The value of ft described by Eq. (3.1) satisfies this criterion and so is consistent with the model on which the theory is based However, it should be noted that this favorable result is obtained within the framework of the Zimm theory. The value of ft at the non-free draining limit is 1/4 for Gaussian chains but it is different from 1/4 for chains of other distribution. Moreover,... 

As discussed in Chapter 1, a Gaussian chain is physically equivalent to a string of beads connected by harmonic springs with the elastic constant ikT/lP (Eq. (1.47) with 6 given by Eq. (1.44)). Here each bead is regarded as a Brownian particle in modeling the chain d3mamics. Such a model was first proposed by Rouse and has been the basis of molecular theories for the dynamics of polymeric liquids. 34... 

The elastic dumbbell model studied iu Chapter 6 is both structurally and djmamicaUy too simple for a poljmier. However, the derivation of its constitutive equation illustrates the main theoretical steps involved. In this chapter we shall apply these theoretical results to a Gaussian chain (or Rouse chain) containing many bead-spring segments (Rouse segments). First we obtain the Smoluchowski equation for the bond vectors. After transforming to the normal coordinates, the Smoluchowski equation for each normal mode is equivalent in form to the equation for the elastic dumbbell. 

The Gaussian chain is often represented by a mechanical model (see Fig. 2.4) N + 1) beads are considered to be connected by a harmonic spring whose potential energy is given by... 

We consider a long Gaussian chain made up of n + 1 beads connected by n harmonic springs, each of elastic constant k = 3 (this is actually K = iksT/f, but takes the simpler form because of our choice of the energy and length units). Its Hamiltonian is ... 

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. 

It is convenient to consider the simplest Gaussian chain modd point-like beads (units) coimected by elastic springs (bonds), figure 12(b). To calculate the free energy F = F[c] (such energy is known as effertive Hanriltoiuan in the condensed matter physics) it is useful to adopt the following scheme (1) to divide the system in cells of size 2,2 1... 

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