Viscoelasticity logarithmical plots

For a given polymer, the viscoelastic curves (either moduli or compliances) obtained at different temperatures in the plateau and terminal regions are simply affine in the frequency (or time) scale, in a double logarithmic plot. 

A real example of the effect of temperature on the viscoelastic functions at T > Tg is shown in Figure 8.2. Here double logarithmic plots of the compliance function J t) versus time are shown at several temperatures for a solution of polystyrene My — 860,000) in tri-ra-tolyl phosphate (1) in which the weight fraction of polymer is 0.70. Because the glass transition temperature of the solution is 15°C, the isotherms were registered at... 

Double logarithmic plots of the storage relaxation modulus versus frequency for a viscoelastic material are shown in Figure 8.12 (9). By taking into account Eq. (6.3), the correspondence between the results at tempera-... 

The shift factors are usually obtained by empirical methods that involve the horizontal translation of the isotherm representing the reduced viscoelastic functions in the time or frequency domains, in double logarithmic plots with respect to the reference isotherm. However, analysis of the components of the complex relaxation moduli in the terminal region ( 0) permits... 

Intermediate Response. Figure 6 is a double logarithmic plot of o/e vs. time in seconds at three different strain rates for the samples as a function of H O content. To extend the time scale and to correlate results at various , we have used the reduced-variables procedure shown to be applicable in describing the viscoelastic response of rubbery materials (8) as well as of several glassy polymers (6). (To compensate for the effect of different e we plot a/e vs. e/e the latter is simply the time, t.) Superposition over the entire time scale for 0% H2O (upper curve) is excellent except for times close to the fracture times of the materials tested e higher strain rates. For example, a deviat ipn occurs at 10 sec for the material at e = 3.3 x 10 sec... 

Viscoelastic functions depend on both temperature and time. For many polymers, the logarithmic plot of a viscoelastic function at the temperature T may be obtained from that at the temperature Tq by shifting the curve along the logarithmic time axis by the amount of log (T)- This procedure is called time-temperature superposition. The ability to superpose viscoelastic data is known as thermorheological simplicity. Thermorheological simplicity demands that all the molecular mechanisms involved in the relaxation process have the same temperature dependencies. 

Time-temperature equivalence in its simplest form implies that the viscoelastic behaviour at one temperature can be related to that at another temperature by a change in the time-scale only. Consider the idealized double logarithmic plots of creep compliance versus time shown in Figure 6.7(a). The compliances at temperatures T and T2 can be superimposed exactly by a horizontal displacement log at, where at is called the shift factor. Similarly (Figure 6.7(b)), in dynamic... 

The system can be further characterised by measurement of the mechanical spectrum at a strain within the linear viscoelastic region defined by the strain sweep. Here the storage (G ) and loss (G") modulus, and complex viscosity (r) ) are measured as a function of frequency (u)) and plotted on double logarithmic plots. Typical mechanical spectrum of entanglement solutions are shown in Figure 2.9. 

Fig. 53 Logarithmic plots of the height of the second plateaus of the nonlinear viscoelastic functions and the yield stress, Sy against the particle content. (From Ref. 7.) L Polystyrene copolymer particle in polystyrene solution and B Carbon black in polystyrene solution.

At very low frequencies, G" for a viscoelastic liquid should be directly proportional to CO, with a slope of 1 on a logarithmic plot. This is evident in Examples I, II, and III. The proportionality constant is the Newtonian steady-flow viscosity Tjo. as shown below. For a simple Newtonian liquid, G" = cot o over the entire frequency range this would be represented by a straight line with unit slope for the solvent of the dilute solution. Example I. 

The steepness of the viscoelastic functions in the transition zone is governed primarily by the parameter or S in these formulations. Another index of steepness, used by Aklonis, is simply the absolute value of maximum slope of a logarithmic plot of Git) or Jit) in the transition zone. 

With increasing proportion of diluent, the monomeric friction coefficient fo is normally diminished, as evidenced by displacement of logarithmic plots of viscoelastic functions in the transition zone to higher frequencies or shorter times with relatively little change in shape. Examples are shown in Fig. 17-2 for the relaxation spectrum of poly( -bulyl methacrylate), and in Fig. 17-3 for the creep compliance of poly(vinyl acetate), both diluted to varying extents with diethyl phthalate. (In the latter figure, we focus attention now on the transition zone, where log J t) < -6.5 the other zones will be discussed later.) Introduction of diluent displaces the time scale by many orders of magnitude. Similar results were obtained in an ex-... 

Dynamic frequency tests are used to explore the microstructure and network formation of the nanocomposites in presence of multiple fillers and their chemical modifications. The storage modulus (G ) of neat PU, PU/MWCNT, PU/functionalized MWCNT/CB nanocomposites measured at 150 °C is logarithmically plotted as a function of angular frequency (m) in Fig. 13 [ 117]. Incorporation of unmodified MWCNT causes dramatic changes in viscoelasticity of polymer... 

Figure 4.10 Storage and loss moduli divided by Gg versus <0for a viscoelastic fluid as modeled by a single-Maxwell element.The slopes approach two and unity respectively as the frequency approaches zero on this double-logarithmic plot.

Isotherms at several temperatures showing the frequency dependence of the real component / (co) of the complex compliance / (coi) of a viscoelastic material are plotted on a double logarithmic scale in Figure 8.9 (7). At high temperatures and low frequencies, J (co) decreases slightly with increas-... 

As discussed in Sect. 4, in the fluid, MCT-ITT flnds a linear or Newtonian regime in the limit y 0, where it recovers the standard MCT approximation for Newtonian viscosity rio of a viscoelastic fluid [2, 38]. Hence a yrio holds for Pe 1, as shown in Fig. 13, where Pe calculated with the structural relaxation time T is included. As discussed, the growth of T (asymptotically) dominates all transport coefficients of the colloidal suspension and causes a proportional increase in the viscosity j]. For Pe > 1, the non-linear viscosity shear thins, and a increases sublin-early with y. The stress vs strain rate plot in Fig. 13 clearly exhibits a broad crossover between the linear Newtonian and a much weaker (asymptotically) y-independent variation of the stress. In the fluid, the flow curve takes a S-shape in double logarithmic representation, while in the glass it is bent upward only. 

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