Viscoelastic scaling

The PS-gas systems studied by Kwag (1998) follow the same viscoelastic scaling principle as the PDMS-CQ2 system. Figure 11.6 shows a master... 

Variation of Viscoelastic Scaling Factor ac with Composition... 

We have seen in the previous sections that viscoelastic scaling, employing the scaling factor ac, produces master viscosity curves for polymer-gas solutions that are identical to the master curve for the pure polymer. This means that the effect of dissolved gas on the rheology of polymer melts can be described entirely by the variation of ac with gas content. We have not, of course, demonstrated that all polymer-gas systems follow this scaling beha-... 

The viscoelastic scaling factor ac is displayed as a function of C02 content for PDMS-C02 systems and PS-C02 systems in Figure 11.7. Clearly, ac varies much more sharply with C02 content for the PS-C02 systems, which are 45-70 °C above Tg of pure PS, than for the PDMS-C02 systems, which are 178-208 °C above Tg of pure PDMS. Moreover, the fine details of the data show that the slope of the ac versus wco, curves decrease with increasing temperature for both the PDMS-C02 and PS-C02 data. Thus, temperature exerts a strong influence on ac. At temperatures within 75 °C of Tg of the pure polymer, a few percent dissolved gas dramatically reduces the viscosity of the melt, reflected by the ac values that are on the order of 10-2 to 10 3. 

Prediction of Viscoelastic Scaling Factor from Free-Volume Theory... 

Simple free-volume theories such as Doolittle s equation (Doolittle, 1951) suggest that the viscosity of liquids varies with the exponential of the fractional free volume. Viscoelastic scaling theories based on the free-volume... 

Figure I 1.7. Variation of viscoelastic scaling factors with gas content for PS-C02 and PDMS-C02 systems. Lower scaling factor values for PS-C02 system, compared with PDMS-C02 system, are due to the closer proximity of the experimental temperatures to Tg of the pure polymer. The top curve displaying results for iso-free volume dilution of high-Mw polystyrene by low-Af polystyrene represents the effect on viscosity of volumetric dilution of high-Mw chains. Viscosity reductions for polymer-gas systems are significantly lower than the iso-free volume dilution curve, indicating that viscosity reduction is primarily due to free volume contributed by dissolved gas.

Figure 11.8. Viscoelastic scaling factor for the PDMS-C02 system predicted by eq. 11.1, with the Sanchez-Lacombe equation of state employed to evaluate specific volume, as calculated by Gerhardt et al. (1998). The solid line is the prediction at 50 °C, and the broken line is the prediction for 80 °C. The predicted curves are compared to the data of Gerhardt et al. (1997) A data for 50 °C data for 80 °C.

All polymer-gas systems studied here exhibit ideal viscoelastic scaling, whereby viscosity measurements taken at different gas compositions can be unified to a master curve of reduced viscosity r (c, y)/a versus reduced... 

A modified version of the free-volume theory is used to calculate the viscoelastic scaling factor or the Newtonian viscosity reduction where the fractional free volumes of pure polymer and polymer-SCF mixtures are determined from thermodynamic data and equation-of-state models. The significance of the combined EOS and free-volume theory is that the viscoelastic scaling factor can be predicted accurately without requiring any mixture rheological data. 

Gerhardt, L.J., et al., Concentration-deperulent viscoelastic scaling models for polydimethysiloxane melts with dissolved carbon dioxide. Journal of Polymer Science Part B-Polymer Physics, 1998. 36(11) p. 1911-1918. 

Royer, J. R., DeSimone, J. M. Khan, S. A. (2001). High-pressure rheology and viscoelastic scaling predictions of polymer melts containing liquid and supercritical carbon dioxide. Journal ( Polymer Science Part B-Polymer Physics 39(23) 3055-3066. 

The relaxation and creep experiments that were described in the preceding sections are known as transient experiments. They begin, run their course, and end. A different experimental approach, called a dynamic experiment, involves stresses and strains that vary periodically. Our concern will be with sinusoidal oscillations of frequency v in cycles per second (Hz) or co in radians per second. Remember that there are 2ir radians in a full cycle, so co = 2nv. The reciprocal of CO gives the period of the oscillation and defines the time scale of the experiment. In connection with the relaxation and creep experiments, we observed that the maximum viscoelastic effect was observed when the time scale of the experiment is close to r. At a fixed temperature and for a specific sample, r or the spectrum of r values is fixed. If it does not correspond to the time scale of a transient experiment, we will lose a considerable amount of information about the viscoelastic response of the system. In a dynamic experiment it may... 

Much more information can be obtained by examining the mechanical properties of a viscoelastic material over an extensive temperature range. A convenient nondestmctive method is the measurement of torsional modulus. A number of instmments are available (13—18). More details on use and interpretation of these measurements may be found in references 8 and 19—25. An increase in modulus value means an increase in polymer hardness or stiffness. The various regions of elastic behavior are shown in Figure 1. Curve A of Figure 1 is that of a soft polymer, curve B of a hard polymer. To a close approximation both are transpositions of each other on the temperature scale. A copolymer curve would fall between those of the homopolymers, with the displacement depending on the amount of hard monomer in the copolymer (26—28). 

International Rubber Hardness. The International mbber hardness test (ASTM D1415) (2) for elastomers is similar to the Rockwell test ia that the measured property is the difference ia penetration of a standard steel ball between minor and major loads. The viscoelastic properties of elastomers require that a load appHcation time, usually 30 seconds, be a part of the test procedure. The hardness number is read directly on a scale of 0 to 100 upon return to the minor load. International mbber hardness numbers are often considered equivalent to Durometer hardness numbers but differences ia iadenters, loads, and test time preclude such a relationship. 

Whether a viscoelastic material behaves as a viscous Hquid or an elastic soHd depends on the relation between the time scale of the experiment and the time required for the system to respond to stress or deformation. Although the concept of a single relaxation time is generally inappHcable to real materials, a mean characteristic time can be defined as the time required for a stress to decay to 1/ of its elastic response to a step change in strain. The... 

A parameter indicating whether viscoelastic effects are important is the Deborah number, which is the ratio of the characteristic relaxation time of the fluid to the characteristic time scale of the flow. For small Deborah numbers, the relaxation is fast compared to the characteristic time of the flow, and the fluid behavior is purely viscous. For veiy large Deborah numbers, the behavior closely resembles that of an elastic solid. 

The premise of the above analysis is the fact that it has treated the interfacial and bulk viscoelasticity equally (linearly viscoelastic experiencing similar time scales of relaxation). Falsafi et al. make an assumption that the adhesion energy G is constant in the course of loading experiments and its value corresponds to the thermodynamic work of adhesion W. By incorporating the time-dependent part of K t) into the left-hand side (LHS) of Eq. 61 and convoluting it with the evolution of the cube of the contact radius in the entire course of the contact, one can generate a set of [LHS(t), P(0J data. By applying the same procedure described for the elastic case, now the set of [LHS(t), / (Ol points can be fitted to the Eq. 61 for the best values of A"(I) and W. 

Wahl, K.J., Stepnowski, S.V. and Unertl, W.N., Viscoelastic effects in nanometer-scale contacts under shear. Tribal. Lett., 5, 103-107 (1998). 

Micro-mechanical processes that control the adhesion and fracture of elastomeric polymers occur at two different size scales. On the size scale of the chain the failure is by breakage of Van der Waals attraction, chain pull-out or by chain scission. The viscoelastic deformation in which most of the energy is dissipated occurs at a larger size scale but is controlled by the processes that occur on the scale of a chain. The situation is, in principle, very similar to that of glassy polymers except that crack growth rate and temperature dependence of the micromechanical processes are very important. 

Fig. 7 gives an example of such a comparison between a number of different polymer simulations and an experiment. The data contain a variety of Monte Carlo simulations employing different models, molecular dynamics simulations, as well as experimental results for polyethylene. Within the error bars this universal analysis of the diffusion constant is independent of the chemical species, be they simple computer models or real chemical materials. Thus, on this level, the simplified models are the most suitable models for investigating polymer materials. (For polymers with side branches or more complicated monomers, the situation is not that clear cut.) It also shows that the so-called entanglement length or entanglement molecular mass Mg is the universal scaling variable which allows one to compare different polymeric melts in order to interpret their viscoelastic behavior. 

P. E. Rouse. The theory of nonlinear viscoelastic properties of dilute solutions of scaling polymers. J Chem Phys 27 1273-1280, 1953. 

Adsorption of rubber over the nanosilica particles alters the viscoelastic responses. Analysis of dynamic mechanical properties therefore provides a direct clue of the mbber-silica interaction. Figure 3.22 shows the variation in storage modulus (log scale) and tan 8 against temperature for ACM-silica, ENR-silica, and in situ acrylic copolymer and terpolymer-silica hybrid nanocomposites. 

---