Dynamic relaxation spectra

Figure 1. Dynamic relaxation spectra (torsion pendulum, 1 Hz) of polyurethanes based on polymyrcene and polybutadiene polyols. Typical relaxation peaks are shown at the temperatures designated T/j, T and. ...

Next we note that there are two physieally different sources of temperature and pressure dependence of the elastic constants of polymers. One, in common with that exhibited by all inorganic crystals, arises from anharmonic effects in the interatomic or intermolecular interactions. The second is due to the temperature-assisted reversible shear and volumetric relaxations under stress that are particularly prominent in glassy polymers or in the amorphous components of semi-crystalline polymers. The latter are characterized by dynamic relaxation spectra incorporating specific features for different polymers that play a central role in their linear viscoelastic response, which we discuss in more detail in Chapter 5. 

This theory was able to account for both the molecular-weight scaling of the dynamic quantities Dg, r, and x as well as for the shape of the relaxation spectrum (see Fig. 5) apart from one important feature - the constant v in the leading exponential behaviour that multiplies the dimensionless arm molecular weight needed to be adjusted. This can be understood as follows. The prediction of the tube model for the plateau modulus from the stress Eq. (7) is... 

The simplest case of comb polymer is the H-shaped structure in which two side arms of equal length are grafted onto each end of a linear cross-bar [6]. In this case the backbones may reptate, but the reptation time is proportional to the square of Mj, rather than the cube, because the drag is dominated by the dumb-bell-like frictional branch points at the chain ends [45,46]. In this case the dependence on is not a signature of Rouse motion - the relaxation spectrum itself exhibits a characteristic reptation form. The dynamic structure factor would also point to entangled rather than free motion. 

Attempts have been made to identify primitive motions from measurements of mechanical and dielectric relaxation (89) and to model the short time end of the relaxation spectrum (90). Methods have been developed recently for calculating the complete dynamical behavior of chains with idealized local structure (91,92). An apparent internal chain viscosity has been observed at high frequencies in dilute polymer solutions which is proportional to solvent viscosity (93) and which presumably appears when the external driving frequency is comparable to the frequency of the primitive rotations (94,95). The beginnings of an analysis of dynamics in the rotational isomeric model have been made (96). However, no general solution applicable for all frequency ranges has been found for chains with realistic local structure. 

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 ... 

In the example given, the constitutive equation used is a multimode Phan Tien Tanner (PTT). It requires the evaluation of both linear and nonlinear material-response parameters. The linear parameters are a sufficient number of the discrete relaxation spectrum 2, and r]i pairs, which are evaluated from small-strain dynamic experiments. The values of the two nonlinear material-response parameters are evaluated as follows. Three semiarbitrary initial values of the two nonlinear parameters are chosen and the principal normal stress difference field is calculated for each of them using the equation of motion and the multimode PTT. They are compared at each field point (i, j) to the experimentally obtained normal stress difference and used in the following cost function F... 

So far, we have fairly extensively discussed the general aspects of static and dynamic relaxation of core holes. We have also discussed in detail methods for calculating the selfenergy (E). Knowing the self-energy, we know the spectral density of states function A (E) (Eq. (10)) which describes the X-ray photoelectron spectrum (XPS) in the sudden limit of very high photoelectron kinetic energy (Eq. (6)). We will now present numerical results for i(E) and Aj(E) and compare these with experimental XPS spectra and we will find many situations where atomic core holes behave in very unconventional ways. 

The fractal dynamics of holes are diffusive, and the diffusivity depends strongly on the tenuous structure in fractal lattices. The fractal dimension defines the self-similar connectivity of hole motions, the relaxation spectrum, and stretched exponential... 

This chapter discusses the dynamic mechanical properties of polystyrene, styrene copolymers, rubber-modified polystyrene and rubber-modified styrene copolymers. In polystyrene, the experimental relaxation spectrum and its probable molecular origins are reviewed further the effects on the relaxations caused by polymer structure (e.g. tacticity, molecular weight, substituents and crosslinking) and additives (e.g. plasticizers, antioxidants, UV stabilizers, flame retardants and colorants) are assessed. The main relaxation behaviour of styrene copolymers is presented and some of the effects of random copolymerization on secondary mechanical relaxation processes are illustrated on styrene-co-acrylonitrile and styrene-co-methacrylic acid. Finally, in rubber-modified polystyrene and styrene copolymers, it is shown how dynamic mechanical spectroscopy can help in the characterization of rubber phase morphology through the analysis of its main relaxation loss peak. 

The characteristics of the dynamic mechanical spectrum of SMAA show drastic changes compared with those of the aPS homopolymer even at very low molar fractions of the added comonomer. All the changes observed reflect the ionic interactions. The a relaxation temperature increases with increasing methacrylic acid content as a consequence of a stable network of chemical crosslinks due to anhydride bridge formation. The y relaxation could be related to local motion of methacrylic acid due to the breakdown of the weakest hydrogen bonds. The 3 relaxation could be attributed to local motion of the backbone chain induced by the breakdown of stronger hydrogen bonds than those invoked for the y relaxation. 

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 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)... 

Recent numerical experiments by the method of molecular dynamics have shown that, for a chain model consisting of particles joined by ideally rigid bonds, the Van der Waals interactions of chain units cause only a little change in the dependence of relaxation times on the wave vector of normal modes of motions, i.e. in the character and shape of the relaxation spectrum. It was found that for the model chain the important relationship... 

It might be argued that this agreement reflects the method of data treatment rather than a fundamental confirmation of Eq. (2.1) and certain other analogous relations for dynamic behavior. That is, both experiments are treated with relations deduced from a single normal mode calculation used to calculate both r] and the relaxation spectrum H, and both yield the product Thus, rj may be obtained from H (t) as... 

However, with dynamic mechanical relaxation domains of about 100 A can be detected. The dynamic mechanical spectrum of a polymer prepared by system 3 ("Figure 3 ) indeed displays two separate loss peaks one at about -62 °C owing to the glass transition of polyol... 

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. 

The n vs. n dependence is often used while discussing the dynamic data of polymer blends. Since for simple liquids the plot has the form of a semi-circular arc, deviation from a semi-circle is sometimes Identified with Immlsclbillty. However, as frequently demonstrated for homopolymers (16) and homologous polymer blends ( ) the form of the n" vs. n plot is determined by the shape of the relaxation spectrum the molecular polydlsperslty can modify the Cole-Cole plot as much as the presence of the Interphase. As a rule, the presence of (unsubtracted) yield stress appears in the plot as a sudden departure from a semi-circular pattern at higher n level. 

Figure L The low-temperature dynamic mechanical spectrum of Halthane 73-14 is typical of the 73-series polyurethane adhesives. Two secondary relaxations, Tp and Ty, are shown as peaks in the loss modulus at —100° and —150°C. The soft segment glass transition, Tg(SS), occurs at about —50°C. The frequency of oscillation was held constant during the measurement at 0.1 Hz.

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