Dynamic relaxation modulus

The scaling of the relaxation modulus G(t) with time (Eq. 1-1) at the LST was first detected experimentally [5-7]. Subsequently, dynamic scaling based on percolation theory used the relation between diffusion coefficient and longest relaxation time of a single cluster to calculate a relaxation time spectrum for the sum of all clusters [39], This resulted in the same scaling relation for G(t) with an exponent n following Eq. 1-14. 

It is interesting to note here that the cluster mass distribution and the relaxation modulus G(t) at the LST scale with cluster mass and with time, respectively, while all other variables (dynamic and static) scale with the distance from pc in the vicinity of the gel point. 

If the self-similar spectrum extends over a sufficiently wide time window, approximate solutions for the relaxation modulus G(t) and the dynamic moduli G (co), G"(co) might be explored by neglecting the end effects... 

It turns out that stress relaxation following a simple shear deformation is seldom employed experimentally. A more common technique is to measure the steady state response to small sinusoidal deformations as a function of angular frequency to. The dynamic storage modulus G (to) and loss modulus G"(to) in small sinusoidal deformations are related to G(t) ... 

With physical aging at 140 °C in nitrogen/dark atmosphere, the dynamic storage modulus is very sensitive to aging time. The modulus increased from 13 GPa (10 minutes-aged) to 18 GPa for samples aged up to 105 minutes at 140 °C (see Fig. 16). These results agree with observations made in the stress relaxation experi-... 

The extensional dynamic storage modulus E and the loss factor tan 8 for a series of linear polyethylene tapes of different draw ratios are shown in Fig. 30(a) and (b). There are two features worthy of particular note. First, the modulus at low temperatures is about 160 GPa, which is about one half of the theoretical modulus and the maximum value obtained from neutron diffraction and other measurements. Secondly, the a and y relaxations are both dearly visible even in the highest draw ratio material, although the magnitude of tan 5 for the y relaxation reduces with increasing draw ratio. 

When reptation is used to develop a description of the linear viscoelasticity of polymer melts [5, 6], the same underlying hypothesis ismade, and the same phenomenological parameter Ng appears. Basically, to describe the relaxation after a step strain, for example, each chain is assumed to first reorganise inside its deformed tube, with a Rouse-like dynamics, and then to slowly return to isotropy, relaxing the deformed tube by reptation (see the paper by Montfort et al in this book). Along these lines, the plateau relaxation modulus, the steady state compliance and the zero shear viscosity should be respectively ... 

As the Rouse dynamics is assmned to be linear with respect to the MWD and that the A and HF processes are mass independent, we define the relaxation modulus of a polydisperse linear polymer by ... 

As mentioned above, it is very difficult, for experimental reasons, to measure the relaxation modulus or the creep compliance at times below 1 s. In this time scale region, dynamic mechanical viscoelastic functions are widely employed (5,6). However, in these methods the measured forces and displacements are not simply related to the stress and strain in the samples. Moreover, in the case of dynamic experiments, inertial effects are frequently important, and this fact must be taken into account in the theoretical methods developed to calculate complex viscoelastic functions from experimental results. 

Because the relaxation spectra are similar for transient and dynamic relaxation viscoelastic functions, H t) can also be obtained from the storage relaxation modulus. The plot of the kernel of the integral of Eq. (9.8), x /(l + (o x ), versus logcax is a sigmoidal curve that intercepts the ordinate axis at 0.5 and reaches the value of 1 in the limit cox oo (see Fig. 9.5). The kernel can be approximated by the step function... 

Rouse dynamics. After some lengthy mathematical handling, beyond the scope of this book, the Rouse theory predicts that the relaxation modulus is given by... 

The friction coefficient is customarily obtained from either the relaxation or retardation spectrum, H x) or L x), respectively. At short times, i.e., on the transition from the glassy-like to the rubbery plateau, the viscoelastic processes obey Rouse dynamics, and the relaxation modulus is given by Eq. (11.45). Since H x) = —dG/d nx t, one obtains... 

In most of the cases, an ultrasonic wave propagates adiabatically, so the (20) looks more naturally its right-hand side represents the adiabatic (non-relaxed) modulus and non-adiabatic contribution to the dynamic modulus. Recall that the relaxed (or isothermal) modulus should be regarded as quasi-static one. Figure 1 shows the frequency-dependent factor of non-adiabatic contribution as function of cox. One can see that transformation from isothermal-like to adiabatic-like propagation occurs in the vicinity cox = 1. The velocity of ultrasound is increased in this region, while the attenuation reaches its maximum value. 

Now we will discuss a procedure of reconstruction the temperature dependence of the relaxed and unrelaxed elastic moduli. We proposed before that the unrelaxed modulus, which describes the Jahn-Teller contribution, vanishes. Actually, the dynamic modulus measured in an experiment is the total one containing the contribution of the Jahn-Teller system as a summand. So, even the dynamic modulus which contains the unrelaxed Jahn-Teller contribution should be non-zero and can have a certain temperature dependence that is not associated with the Janh-Teller impurities. As well, the relaxed modulus for this reason can differ from one described with the expression (45). To deal with the impurity s contribution only, we can measure the temperature dependence of the dynamic modulus for an un-doped crystal and subtract it from one obtained for the the doped crystal. But it requires two specimens (doped and un-doped) and two experiments. More easy is to reconstruct the relaxed and unrelaxed moduli with the help of the data relating to the doped crystal. To derive the necessary expressions we will use the (20) and (21) and... 

If the setup makes it possible to measure the ultrasonic velocity only (or the dynamic elastic modulus), a similar technique can be developed for reconstruction the relaxation time and all other parameters characterizing the Jahn-Teller system. In this case we need two temperature dependences vg (T) (or cg (T)) the first one (denoted without superscript) obtained on the doped specimen and the second (superscript (2)) - on the un-doped. At high enough temperatures these dependences should be identical, while at low temperatures they should differ due to the Jahn-Teller effect. So, contribution of the Jahn-Teller system to the total dynamic modulus Acg may be written as... 

Fig. 8 Elastic moduli ci = (cim + C1122 + 2ci3i3)/2 vs. inverse temperature obtained for 54.4 MHz in ZnSe Cr with concentration of the dopand ncr = 10 ° cm. Filled circles represent the real part of the dynamic modulus (q — co)/cq, open circles represent the relaxed modulus (c —co)/cq, and open triangles represent the unrelaxed modulus (c —cq )/cq. The initial reference modulus Co was taken as an extrapolation of ci(T) to F = 0 K. After Fig. 6 in [17]...

The relaxation time r is a fundamental dynamic property of all viscoelastic liquids. Polymer liquids have multiple relaxation modes, each with its own relaxation time. Any stress relaxation modulus can be described by a series combination of Maxwell elements. 

The time dependence of the stress relaxation modulus in semidilute unentangled solution is sketched in Fig. 8.10. Experimental verification of Rouse dynamics for frequencies smaller than 1/r was shown in Fig. 8.5, for a semidilute unentangled polyelectrolyte solution. 

The full-time dependence of the stress relaxation modulus ot randomly branched unentangled polymers is best derived from the fractal dynamics of Section 8.8 using the relaxation rate spectrum P( ) ... 

There are three different regimes of polymer dynamics on three different length and time scales for an entangled polymer solution in an athermal solvent. The stress relaxation modulus of such a solution is shown in Fig. 9.9. Two of the regimes are identical to those discussed in Section 9.2.2 and the other regime was discussed in Section 8.5. 

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 ... 

Show that a stress relaxation modulus of an entangled but non-concate-nated melt of rings on the basis of the single chain dynamic modes described in Problem 9.31 is... 

Winter (Winter and Chambon, 1986, Winter, 1987) also described an isothermal dynamic relaxation test to measure the gel point. He noted that the gel point coincides with a power-law relationship between the relaxation modulus (G) and relaxation time (t) G = St ", where S and n are constants. An isothermal step strain test measures the relaxation modulus as a function of time after an instantaneously applied strain. The gel time can be measured as the point at which the profile of the relaxation modulus can be expressed by this power law. This model is equivalent to tan <5 being independent of frequency. 

Still another relationship between experimental parameters is a direct consequence of the Boltzmann superposition principle. We will derive the equations relating the shear stress relaxation modulus G(t) to the in-phase and out-of-phase dynamic shear moduli G oi) and G"(co) starting from equation (2-46)... 

The buffering action of a coating in this situation is determined by the relaxation modulus of the coating material. The relaxation modulus may be measured on a film cast from the material by carrying out tensile-stress relaxation measurements with a suitable apparatus such as a Rheovibron dynamic viscoelastometer operated in a static mode. Figure 13 (inset) displays such measurements for the four coating materials used on the fibers measured in Figure 12. The measurements were carried out at 23 °C at small tensile strains, where the materials exhibit linear viscoelastic behavior. 

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