Polymer dynamic moduli

The dynamic mechanical properties of VDC—VC copolymers have been studied in detail. The incorporation of VC units in the polymer results in a drop in dynamic modulus because of the reduction in crystallinity. However, the glass-transition temperature is raised therefore, the softening effect observed at room temperature is accompanied by increased brittleness at lower temperatures. These copolymers are normally plasticized in order to avoid this. Small amounts of plasticizer (2—10 wt %) depress T significantly without loss of strength at room temperature. At higher levels of VC, the T of the copolymer is above room temperature and the modulus rises again. A minimum in modulus or maximum in softness is usually observed in copolymers in which T is above room temperature. A thermomechanical analysis of VDC—AN (acrylonitrile) and VDC—MMA (methyl methacrylate) copolymer systems shows a minimum in softening point at 79.4 and 68.1 mol % VDC, respectively (86). 

Butyl-type polymers exhibit high damping, and the viscous part of the dynamic modulus is uniquely broad as a function of frequency or temperature. Molded mbber parts for damping and shock absorption find wide appHcation in automotive suspension bumpers, auto exhaust hangers, and body mounts. 

The discussion of the results of measuring the dynamic properties of filled polymers is based very often on the idea of correlation of the G" and t functions, which is not always expressed directly. However, due to a very sharp dependence of a dynamic modulus on the amplitude, it is not clear how to understand this correlation. 

Moreover, if for pure polymer melts the correlation of the behavior of the functions ri (co) andrify) under the condition of comparing as y takes place, as a general rule, but for filled polymers such correlation vanishes. Therefore the results of measuring frequency dependences of a dynamic modulus or dynamic viscosity should not be compared with the behavior of the material during steady flow. 

As regards a qualitative pattern of influence of the filler on dynamic properties of melts of filled polymers, the situation in many respects is the similar described above for yield stress and viscosity. Indeed, the interpretation of the field of yield stress, estimated by a dynamic modulus, was given in an appropriate section. 

Polymer Dynamic shear modulus (frequency > 1 Hz) S/MPa Quasi-static Young s modulus (frequency 0.01 Hz) E/MPa Ratio 3S/E... 

Experimentally DMTA is carried out on a small specimen of polymer held in a temperature-controlled chamber. The specimen is subjected to a sinusoidal mechanical loading (stress), which induces a corresponding extension (strain) in the material. The technique of DMTA essentially uses these measurements to evaluate a property known as the complex dynamic modulus, , which is resolved into two component parts, the storage modulus, E and the loss modulus, E . Mathematically these moduli are out of phase by an angle 5, the ratio of these moduli being defined as tan 5, Le. 

Fig. 1.2 Richness of dynamic modulus in a bulk polymer and its molecular origin. The associated length scales vary from the typical bond length ( A) at low temperatures to interchain distances ( 10 A) around the glass transition. In the plateau regime of the modulus typical scales involve distances between entanglements of the order of 50-100 A. In the flow regime the relevant length scale is determined by the proper chain dimensions...

At low temperature the material is in the glassy state and only small ampU-tude motions hke vibrations, short range rotations or secondary relaxations are possible. Below the glass transition temperature Tg the secondary /J-re-laxation as observed by dielectric spectroscopy and the methyl group rotations maybe observed. In addition, at high frequencies the vibrational dynamics, in particular the so called Boson peak, characterizes the dynamic behaviour of amorphous polyisoprene. The secondary relaxations cause the first small step in the dynamic modulus of such a polymer system. 

Rheological properties of filled polymers can be characterised by the same parameters as any fluid medium, including shear viscosity and its interdependence with applied shear stress and shear rate elongational viscosity under conditions of uniaxial extension and real and imaginary components of a complex dynamic modulus which depend on applied frequency [1]. The presence of fillers in viscoelastic polymers is generally considered to reduce melt elasticity and hence influence dependent phenomena such as die swell [2]. 

Here three constants appear Go is the equilibrium modulus of elasticity 0p is the characteristic relaxation time, and AG is the relaxation part of elastic modulus. There are six measured quantities (components of the dynamic modulus for three frequencies) for any curing time. It is essential that the relaxation characteristics are related to actual physical mechanisms the Go value reflects the existence of a three-dimensional network of permanent (chemical) bonds 0p and AG are related to the relaxation process due to the segmental flexibility of the polymer chains. According to the model, in-termolecular interactions are modelled by assuming the existence of a network of temporary bonds, which are sometimes interpreted as physical (or geometrical) long-chain entanglements. 

Thus, one may conclude that, in the region of comparatively low frequencies, the schematic representation of the macromolecule by a subchain, taking into account intramolecular friction, the volume effects, and the hydrodynamic interaction, make it possible to explain the dependence of the viscoelastic behaviour of dilute polymer solutions on the molecular weight, temperature, and frequency. At low frequencies, the description becomes universal. In order to describe the frequency dependence of the dynamic modulus at higher frequencies, internal relaxation process has to be considered as was shown in Section 6.2.4. 

There are plenty of measurements of dynamic modulus of nearly monodisperse polymers starting with pioneering works of Onogi et al. (1970) and Vinogradov et al. (1972a). The more recent examples of the similar dependencies can be found in papers by Baumgaertel et al. (1990, 1992) for polybutadiene and for polystyrene and in paper by Pakula et al. (1996) for polyisoprene. 

Experiments reveal that the dynamic modulus and the characteristic quantities (6.58) depend on the polymer concentration c and length M of the macromolecule (Ferry 1980), and these dependencies are implied through the parameters of the theory. 

The considered system contains no macromolecules of the matrix and n macromolecules of the additive per unit volume and can be characterised by dynamic modulus G(u). The medium, in which the macromolecules of the additive move, is a system consisting of a linear polymer of molecular weight Mo, which is characterised by the modulus Gq(u) = — iurjo(u). The change of dynamic modulus, taking into account the fact that some of the macromolecules of the matrix have been replaced by impurity macromolecules, can be written as... 

For the weakly entangled system, the steady-state modulus depends on the molecular weight of polymer as M 1, while for strongly entangled system, the steady-state modulus does not depend on the molecular weight of polymer, which is consistent with typical experimental data for concentrated polymer systems (Graessley 1974). The expression for the modulus is exactly the same as for the plateau value of the dynamic modulus (equations (6.52) and (6.58)) Expressions (9.42) lead to the following relation for the ratio of the normal stresses differences... 

Notwithstanding the simplifying assumptions in the dynamics of macromolecules, the sets of constitutive relations derived in Section 9.2.1 for polymer systems, are rather cumbersome. Now, it is expedient to employ additional assumptions to obtain reasonable approximations to many-mode constitutive relations. It can be seen that the constitutive equations are valid for the small mode numbers a, in fact, the first few modes determines main contribution to viscoelasticity. The very form of dependence of the dynamical modulus in Fig. 17 in Chapter 6 suggests to try to use the first modes to describe low-frequency viscoelastic behaviour. So, one can reduce the number of modes to minimum, while two cases have to be considered separately. 

Pokrovskii VN, Volkov VS (1978b) The calculation of relaxation time and dynamical modulus of linear polymers in one-molecular approximation with self-consistency. (A new approach to the theory of viscoelasticity of linear polymers). Polym Sci USSR 20 3029-3037... 

In some cases, network structure is modified by aminolysis reactions25. An example is the polymer formed from diglycidylic ester of o-phthalic acid and diaminodiphenilmethane. Aminolysis makes the chain between crosslinks shorter and influences the properties of the polymer (dynamic shear modulus in a rubbery... 

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