Dynamic viscoelastic spectra

Paramyosin behaves as an extremely asymmetric a-helical rigid rod in solution, as shown by its hydrodynamic and light-scattering properties 22, 36), its dynamic viscoelastic behavior 1,2), the hypochromicity of its far-ultraviolet absorption spectrum 50), and its optical rotatory properties 12, 56). 

As discussed in this section, the tube-dilation effect, i.e. M J/Me > 1, mainly occurs in the terminal-relaxation region of component two in a binary blend. This effect means that the basic mean-field assumption of the Doi-Edwards theory (Eq. (8.3)) has a dynamic aspect when the molecular-weight distribution of the polymer sample is not narrow. This additional dynamic effect causes the viscoelastic spectrum of a broadly polydisperse sample to be much more complicated to analyze in terms of the tube model, and is the main factor which prevents Eq. (9.19) from being applied... 

Analysis of the dynamical viscoelastic quantities shows that the relaxation spectrum H r) of the two-dimensional network goes as H(t) 1/r [65,68-70]. Hence 2-D networks do indeed show dynamical behavior intermediate between that of linear chains and that of 3-D networks. Moreover, in a fractal picture, square networks may be viewed as being fractals and as having a spectral dimension of 2. Now H(r) 1/t leads to an -behavior for the storage modulus G (a>), see Eig. 4, and to G(f) 1/t. 

Stockmayer and Kennedy (1975) conducted a seminal smdy on the chain dynamics of Rouse chains of AB-type diblock or ABA-type triblock copolymers by modifying the bead—spring model of Rouse for linear flexible homopofymers (see Chapter 4). They calculated the spectrum of relaxation times ftp biodc) block copolymer in terms of the terminal relaxation times for the Rouse chains for the A and B blocks. Once the values of are determined, one can calculate linear dynamic viscoelastic... 

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... 

Distributions of relaxation or retardation times are useful and important both theoretically and practicably, because // can be calculated from /.. (and vice versa) and because from such distributions other types of viscoelastic properties can be calculated. For example, dynamic modulus data can be calculated from experimentally measured stress relaxation data via the resulting // spectrum, or H can be inverted to L, from which creep can be calculated. Alternatively, rather than going from one measured property function to the spectrum to a desired property function [e.g., Eft) — // In Schwarzl has presented a series of easy-to-use approximate equations, including estimated error limits, for converting from one property function to another (11). 

The relaxation spectrum H(0) completely characterizes the viscoelastic properties of a material. H(0) can be found from the measured frequency dependence of the dynamic modulus of elasticity G (co) by means of the following integral equation ... 

The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics variational formulations computational mechanics statics, kinematics and dynamics of rigid and elastic bodies vibrations of solids and structures dynamical systems and chaos the theories of elasticity, plasticity and viscoelasticity composite materials rods, beams, shells and membranes structural control and stability soils, rocks and geomechanics fracture tribology experimental mechanics biomechanics and machine design. 

A powerful technique for the study of orientation and dynamics in viscoelastic media is line shape analysis in deuteron NMR spectroscopy [1]. For example, the average orientation of chain segments in elastomer networks upon macroscopic strain can be determined by this technique [22-31]. For a non-deformed rubber, a single resonance line in the deuterium NMR spectrum is observed [26] while the spectrum splits into a well-defined doublet structure under uniaxial deformation. It was shown that the usual network constraint on the end-to-end vector determines the deuterium line shape under deformation, while the interchain (excluded volume) interactions lead to splitting [26-31]. Deuterium NMR is thus able to monitor the average segmental orientation due to the crosslinks and mean field separately [31]. 

From a fundamental point of view, none of the theoretical results necessary to justify rigorously bifurcation or stability studies have been fully established so far for the equations governing viscoelastic flows (contrary to the Newtonian case). Such results concern, for instance, the relations between linear stability and the spectrum of the associated operator, between linear and nonlinear stability, or the reduction of the dynamics to a centre manifold. 

From the concept of separability, the memory function of the linear viscoelasticity is required. This memory function can be related to a discrete relaxation time spectrum, available firom dynamic experiments, given by ... 

Figure 3.10 Predictions of the temporary network model [Eq. (3-24)] (lines) compared to experimental data (symbols) for start-up of uniaxial extension of Melt 1, a long-chain branched polyethylene, using a relaxation spectrum fit to linear viscoelastic data for this melt. (From Bird et al. Dynamics of Polymeric Liquids. Vol. 1 Fluid Mechanics, Copyright 1987. Reprinted by permission of John Wiley Sons, Inc.)...

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... 

The crystalline phase affects the viscoelastic dynamic functions describing the glass-rubber relaxation. For example, the location of this absorption in the relaxation spectrum is displaced with respect to that of the amorphous polymer and greatly broadened. Consequently, the perturbing effects of crystal entities in dynamic experiments propagate throughout the amorphous fraction. The empirical Boyer-Beaman law (32)... 

The proposed method of data treatment has two advantages (1) It allows assessment of the status of blend miscibility In the melt, and (11) It permits computation of any linear viscoelastic function from a single frequency scan. Once the numerical values of Equation 20 or Equation 21 parameters are established Che relaxation spectrum as well as all linear viscoelastic functions of the material are known. Since there Is a direct relation between the relaxation and Che retardation time spectra, one can compute from Hq(o)) the stress growth function, creep compliance, complex dynamic compliances, etc. 

Thus, once the four parameters of Eq 7.42 are known, the relaxation spectrum, and then any linear viscoelastic function can be calculated. For example, the experimental data of the dynamic storage and loss shear moduli, respectively G and G , or the linear viscoelastic stress growth function in shear or uniaxial elongation can be computed from the dependencies [Utracki and Schlund, 1987] ... 

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