Viscoelastic analogy

Now we consider roller ball indentations. Let us consider first a rigid ball that does not roll but indents a viscoelastic half-space this analysis will be extended to a rolling ball. This is a typical situation in which the elastic-viscoelastic analogy is, in general, no longer applicable. 

Viscoelasticity. When in addition, the flnid or the material stores some capacitive energy, an elastic property comes in snpplement to the dynamic and kinematic viscosities for featuring the system. The storage and the transport of capacitive energy is then supported by a combined property of viscoelasticity, analogous to a kinetic constant in physical chemistry (see case study All Reactive Chemical Species in Chapter 4 and case stndy 116 Viscoelastic Relaxation in Chapter 11). 

It is useful to note that the dynamic behavior of any system that incorporates both energy storage and energy dissipation must have at least one characteristic time. Another example is an electrical circuit that includes both resistance and capacitance. Furthermore, we note that Eq. 4.15 is the same as Eq. 4.12, with Fq replaced by Gq and % by T. The Maxwell element is thus said to be a mechanical analog of the viscoelastic behavior described by Eq. 4.12. It will often prove useful in our discussion of the linear viscoelastic behavior of polymers to refer to the viscoelastic analog of the Maxwell element. 

While the exponential stress relaxation predicted by the viscoelastic analog of the Mawell element, ie., a single exponential, is qualitatively similar to the relaxation of polymeric liquids, it does not describe the detailed response of real materials. If, however, it is generalized by assembling a number of Maxwell elements in parallel, it is possible to fit the behavior of real materials to a level of accuracy limited only by the precision and time-range of the experimental data. This leads to the generalized, or multi-mode. Maxwell model for linear viscoelastic behavior, which is represented mathematically by a sum of exponentials as shown by Eq. 4.16. 

The Maxwell and Voigt models of the last two sections have been investigated in all sorts of combinations. For our purposes, it is sufficient that they provide us with a way of thinking about relaxation and creep experiments. Probably one of the reasons that the various combinations of springs and dash-pots have been so popular as a way of representing viscoelastic phenomena is the fact that simple and direct comparison is possible between mechanical and electrical networks, as shown in Table 3.3. In this parallel, the compliance of a spring is equivalent to the capacitance of a condenser and the viscosity of a dashpot is equivalent to the resistance of a resistor. The analogy is complete... 

The dry-processed, peel-apart system (Fig. 8b) used for negative surprint apphcations (39,44) is analogous to the peel-apart system described for the oveday proofing apphcation (see Fig. 7) except that the photopolymer layer does not contain added colorant. The same steps ate requited to produce the image. The peel-apart system rehes on the adhesion balance that results after each exposure and coversheet removal of the sequentially laminated layer. Each peel step is followed by the apphcation of the appropriate process-colored toners on a tacky adhesive to produce the image from the negative separations. The mechanism of the peel-apart process has been described in a viscoelastic model (45—51) and is shown in Figure 8c. 

Although the diffusion mechanism can be seen as mechanical but occurring at molecular dimensions, van der Waals intermolecular interactions and conformational entropic energy provide an additional mechanism that increases adhesion [62]. It is interesting to note the analogy that exists between this mechanism at the molecular level with the adherence, adhesion and viscoelastic deformations concept applied for a macroscopic adhesive. 

In polymer electrolytes (even prevailingly crystalline), most of ions are transported via the mobile amorphous regions. The ion conduction should therefore be related to viscoelastic properties of the polymeric host and described by models analogous to that for ion transport in liquids. These include either the free volume model or the configurational entropy model . The former is based on the assumption that thermal fluctuations of the polymer skeleton open occasionally free volumes into which the ionic (or other) species can migrate. For classical liquid electrolytes, the free volume per molecule, vf, is defined as ... 

In this introduction, the viscoelastic properties of polymers are represented as the summation of mechanical analog responses to applied stress. This discussion is thus only intended to be very introductory. Any in-depth discussion of polymer viscoelasticity involves the use of tensors, and this high-level mathematics topic is beyond the scope of what will be presented in this book. Earlier in the chapter the concept of elastic and viscous properties of polymers was briefly introduced. A purely viscous response can be represented by a mechanical dash pot, as shown in Fig. 3.10(a). This purely viscous response is normally the response of interest in routine extruder calculations. For those familiar with the suspension of an automobile, this would represent the shock absorber in the front suspension. If a stress is applied to this element it will continue to elongate as long as the stress is applied. When the stress is removed there will be no recovery in the strain that has occurred. The next mechanical element is the spring (Fig. 3.10[b]), and it represents a purely elastic response of the polymer. If a stress is applied to this element, the element will elongate until the strain and the force are in equilibrium with the stress, and then the element will remain at that strain until the stress is removed. The strain is inversely proportional to the spring modulus. The initial strain and the total strain recovery upon removal of the stress are considered to be instantaneous. 

Other types of linear viscoelastic experiments may be used. Dynamic shear compliance measurements provide the storage and loss compliances J (co) and J"(co). An equation analogous to Eq.(3.12) is available for determining the initial modulus from J"(co) ... 

A summary of analytic expressions obtained in this manner for all the viscoelastic functions is presented in Table 4 and 5 for the linear and cubic arrays. The well-known phenomenological analogy (8) between dynamic compliance and dielectric permittivity allows the formal use of Eqs. (T 5), (T 6), and (T 11), (T 12) for the dielectric constant, e (co), and loss, e"(co), of the linear and cubic arrays, respectively (see Table 6). The derivations of these equations are elaborated in the next section and certain molecular weight trends are discussed. 

Up to this point concern has been with viscoelastic functions. In view of the phenomenological analogy between the dielectric permittivity and... 

The calculation of residual stresses in the polymerization process during the formation of an amorphous material was formulated earlier.12 The theory was based on a model of a linear viscoelastic material with properties dependent on temperature T and the degree of conversion p. In this model the effect of the degree of conversion was treated by a new "polymerization-time" superposition method, which is analogous to the temperature-time superposition discussed earlier. 

The rates of relaxation and retardation processes above the glass temperature are strongly dependent on the viscosity and thus on the fraction of free volume present. Because the viscosity not only depends on temperature but also on static pressure (the glass transition temperature increases approximately 1 °C per 20 bar of pressure) it is not surprising that pressure also affects the viscoelastic processes. A qualitatively relation analogous to Eq. (13.121) can be readily derived (Ferry, 1980) ... 

Viscoelastic properties of polymer solutions may be of practical importance, e.g. in the flow of these solutions through technical equipment. For concentrated polymer solutions the viscoelastic properties show great analogy with those of polymer melts. For dilute solutions (c < ccr) the analogy decreases with decreasing concentration. 

This expression is of the same shape as that of stress relaxation of viscoelastic materials (Chap. 13). By analogy 1/k is called the "relaxation time" (t). Since chemical reactions normally satisfy an Arrhenius type of equation in their temperature dependence, the variation of relaxation time with temperature may be expressed as follows ... 

On a global scale, the linear viscoelastic behavior of the polymer chains in the nanocomposites, as detected by conventional rheometry, is dramatically altered when the chains are tethered to the surface of the silicate or are in close proximity to the silicate layers as in intercalated nanocomposites. Some of these systems show close analogies to other intrinsically anisotropic materials such as block copolymers and smectic liquid crystalline polymers and provide model systems to understand the dynamics of polymer brushes. Finally, the polymer melt-brushes exhibit intriguing non-linear viscoelastic behavior, which shows strainhardening with a characteric critical strain amplitude that is only a function of the interlayer distance. These results provide complementary information to that obtained for solution brushes using the SFA, and are attributed to chain stretching associated with the space-filling requirements of a melt brush. 

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