Rouse-model stress relaxation modulus

The value of the stress relaxation modulus at the relaxation time G(x) is of the order of kT per chain in either the Rouse or Zimm models, just as the strands of a network in Chapter 7 stored of order kT of elastic energy ... 

Combining Eqs (8.45) and (8.46) approximates the stress relaxation modulus for the Rouse model at intermediate time scales ... 

This expression effectively interpolates between a modulus level of order kT per monomer at the shortest Rouse mode t ro) to a modulus level of order kT per chain at the longest Rouse mode (/ = tr xqN ) using a power law. We already know that the stress relaxation modulus has an exponential decay beyond its longest relaxation time [Eq. (7.112)]. Therefore, an approximate description of the stress relaxation modulus of the Rouse model is the product of [Eq. (8.47)] and an exponential cutoff ... 

For high frequencies uj > 1 /tq, there are no relaxation modes in the Rouse model. The storage modulus becomes independent of frequency, and equal to the short time stress relaxation modulus, which is kT per monomer G uj) (pkTjb. This high-frequency saturation is not included in Eqs (8.49) and (8.50). At low frequencies a < 1/tr, the storage modulus is proportional to the square of frequency and the loss modulus is pro-portional to frequency, as is the case for the terminal response of any viscoelastic liquid. 

Stress relaxation modulus predicted by the Rouse model for a melt of unentangled chains with jV= 10 . The solid curve is the exact Rouse result [Eq. (8.55)] and the dotted curve is the approximate Rouse result [Eq. (8.48)]. 

In 0-solvents 1/2), the stress relaxation modulus decays as the - 2/3 power of time, while in good solvents (i/ 0.588) G(t) decays approximately as the - 0.57 power of time. Like the stress relaxation modulus of the Rouse model [Eq. (8.47)], Eq. (8.63) crosses over from kT per monomer at the monomer relaxation time tq to kT per chain at the relaxation time of the chain tz TqN [Eq. (8.25)]. Once again, an excellent approximation to the stress relaxation modulus predicted by the Zimm... 

The time-dependent viscoelastic response of polymers is broken down into individual modes that relax on the scale of subsections of the chain with Njp monomers. The Rouse and Zimm models have different structure of their mode spectra, which translates into different power law exponents for the stress relaxation modulus G t) ... 

Calculate the stress relaxation modulus G(t), valid for all times longer than the relaxation time of a monomer, for a monodisperse three-dimensional melt of unentangled flexible fractal polymers that have fractal dimension V <1. Assume complete hydrodynamic screening. Hint Keep the fractal dimension general and make sure your result coincides with the Rouse model for V — 2. 

Calculate the stress relaxation modulus of the Rouse model (Eq. 8.55) by showing that after a small step shear strain 7 at time t 0 the correlation function of normal modes decays as Xpx t)X y(t)) — i kTjkp) exp (- tjxp). 

The Rouse model predicts that the stress relaxation modulus on these short time scales decays inversely proportional to the square root of time [Eq. (8.47)] ... 

Tnteractions are not important. The dynamics on these intermediate scales (for r < t< Te) are described by the Rouse model with stress relaxation modulus similar to the Rouse result for unentangled solutions [Eq. (8.90) with the long time limit the Rouse time of an entanglement strand Tg]. At Te, the stress relaxation modulus has decayed to the plateau modulus Gg[kT per entanglement strand, Eq. [(9.37), see Fig. 9.9)]. The ratio of osmotic pressure and plateau modulus at any concentration in semidilute solution -in athermal solvents is proportional to the number of Kuhn monomers in ... 

Constraint release has a limited effect on the diffusion coefficient it is important only for the diffusion of very long chains in a matrix of much shorter chains and can be neglected in monodisperse solutions and melts. The effect of constraint release on stress relaxation is much more important than on the diffusion and cannot be neglected even for monodisperse systems. Constraint release can be described by Rouse motion of the tube. The stress relaxation modulus for the Rouse model decays as the reciprocal square root of time [Eq. (8.47)] ... 

This square-root dependence on tw is a fundamental featme of linear chains in the Rouse model. The shear modulus at intermediate frequencies is a signature of the internal, intra-chain dynamics, which is determined by the topology of the GGS. As stressed before, the viscoelastic relaxation forms can be expressed through the relaxation spectrum H r), see Eq. 27. Here one finds [3] ... 

We now consider predictions of the viscoelastic properties of unentangled polymer melts based on the Rouse model. Consider the situation where a sudden strain is imposed on a polymer. Then the stress remaining in the specimen at time t can be determined from a material property referred to as the stress relaxation modulus G(t). The G(t) for the Rouse model is given by (Doi and Edwards 1986 Rouse 1953)... 

The shear relaxation modulus Gs t) and the first normal-stress difference function G i(t), both normalized on a per-segment basis and with kT set to 1, are obtained from the constitutive equation of the Rouse model (Eq. (7.55) with Sp replaced by Tp) as... 

Let us add here some remarks on the normal stress difference. According to the Rouse-Zimm model [132,133] the first normal stress difference may be related to the storage modulus G. Taking into account only the longest relaxation time x, one gets... 

---