Rouse model spring constant

A polymer chain can be approximated by a set of balls connected by springs. The springs account 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 f= bAr, where Ar is the extension of the spring and the spring constant is ii = rtRis the root-mean-square distance of two successive beads. The viscous force that acts on a bead is... 

In view of the comparative intensitivity of JeR to the properties of B, it seems unlikely that terminal spectra which are narrow enough to agree with experimental compliance data (that is, to make JeR inversely proportional to E) can be produced merely by introducing a distribution of frictional coefficients within the molecule. The same holds true if the spring constants of the Rouse model are allowed to vary with position (241). 

In 1953 Rouse published a paper to describe theoretically the flow of polymers in dilute solutions. The polymer molecule is assumed to exist as a statistical coil and is subdivided into N submolecules. Each submolecule is thought of a solid bead. The beads behave as Gaussian chains and their entropy-elastic recovery can be described by a spring with a spring constant hkT/cP-, where a is the average end-to-end distance of a submolecule and k is the Boltzmann constant. The model is shown in Figure 8.9. 

Comparison of the forms of equations 58 to 61 with equations 21 to 23 of Chapter 9 and equations 23 and 24 of Chapter 3 shows that the time and frequency dependence correspond to a generalized Maxwell model as in the Rouse theory and its various modifications, but here the spring constants (or discrete contributions to the relaxation spectrum) are not necessarily all equal they are proportional to the concentrations of the various types of strands, v e. The molecular weight does not enter explicitly, but it may be expected that the higher the molecular weight the greater the concentrations of strands which find it difficult to leave the network and hence have large values of the time parameter... 

A more realistic representation of polymer chains is the Rouse model [59], which considers a polymer molecule to be a linear chain of N free-draining beads interconnected by springs (each of time constant Xu = il4H), and predicts Eq. (16) to apply. 

Indeed, starting from this picture, one can verify the stress-optical rule and set it on a microscopic basis. We describe a chain in the spirit of the Rouse-model, as sketched in Fig. 7.24. Each polymer is subdivided in sequences of equal size, long enough to ensure that they behave like elastic springs, with a force constant 6r given by Eq. (6.18)... 

The hydrodynamic scaling model is an extension of the Kirkwood-Riseman model for polymer dynamics(l). The original model considered a single polymer molecule. It effectively treats a polymer coil as a bag of beads. For their collective coordinates, the beads have three center-of-mass translations, three rotations around the center of mass, and unspecified other coordinates. The use of rotation coordinates causes the Kirkwood-Riseman model to differ from the Rouse and Zimm models(2,3). The other collective coordinates of the Kirkwood-Riseman model are lumped as internal coordinates whose fluctuations are in first approximation ignored. The beads are linked end-to-end, the links serving to estabhsh and maintain the coil s bead density and radius of gyration. However, the spring constant of the finks only affects the time evolution of the internal coordinates it has no effect on translation or rotation of the coil as a whole. 

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... 

In the theories for dilute solutions of flexible molecules based on the bead-spring model, the contribution of the solute to the storage shear modulus, loss modulus, or relaxation modulus is given by a series of terms the magnitude of each of which is proportional to nkT, i.e., to cRTjM, as in equation 18 of Chapter 9 alternatively, the definition of [C ]y as the zero-concentration limit of G M/cRT (equations 1 and 6 of Chapter 9) implies that all contributions are proportional to nkT. Each contribution is associated with a relaxation time which is proportional to [ri Ti)sM/RT-, the proportionality constant (= for r i) depends on which theory applies (Rouse, Zimm, etc.) but is independent of temperature, as is evident, for example, in equation 27 of Chapter 9. Thus the temperature dependence of viscoelastic properties enters in four variables [r ], t/j, T explicitly, and c (which decreases slightly with increasing temperature because of thermal expansion). 

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