Modulus bead-spring model

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 ... 

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). 

At low frequencies the loss modulus is linear in frequency and the storage modulus is quadratic for both models. As the frequency exceeds the reciprocal of the relaxation time ii the Rouse model approaches a square root dependence on frequency. The Zimm model varies as the 2/3rd power in frequency. At high frequencies there is some experimental evidence that suggests the storage modulus reaches a plateau value. The loss modulus has a linear dependence on frequency with a slope controlled by the solvent viscosity. Hearst and Tschoegl32 have both illustrated how a parameter h can be introduced into a bead spring... 

The chain is modeled as a system of beads and springs undergoing Brownian motion in a viscous medium. The other polymer chains provide the viscous medium for any individual chain. The inherent dynamics can be represented in terms of N relaxation modes, where N is the number of statistical subunits in the chain. The shear relaxation modulus G(f) is given by ... 

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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