Polymers shear modulus

Figure 3. Emulsion copolymer of chloro-prene/cyanoprene (14% cyanoprene polymer) shear modulus plotted over temperature.

R, Lucklum, P, Hauptmann Determination of polymer shear modulus with quartz crystal resonators, Pp, 123-40. 

Polymers Shear modulus G, GPa Young s modulus E, GPa Tensile stress, CTuT, MPa Tensile strain, 6ut,% Specific Poisson s heat ratio, p. C, J/(kg Thermal conductivity K) X at 2 K, mW/(m-K) Coefficient of thermal expansion AL/L,%... 

We observed above that the Rouse expression for the shear modulus is the same function as that written for a set of Maxwell elements, except that the summations are over all modes of vibration and the parameters are characteristic of the polymers and not springs and dashpots. Table 3.5 shows that this parallel extends throughout the moduli and compliances that we have discussed in this chapter. In Table 3.5 we observe the following ... 

Superabsorbents. Water-sweUable polymers are used extensively in consumer articles and for industrial appUcations. Most of these polymers are cross-linked acryUc copolymers of metal salts of acryUc acid and acrylamide or other monomers such as 2-acrylamido-2-methylpropanesulfonic acid. These hydrogel forming systems can have high gel strength as measured by the shear modulus (134). Sometimes inorganic water-insoluble powder is blended with the polymer to increase gel strength (135). Patents describe processes for making cross-linked polyurethane foams which contain superabsorbent polymers (136,137). 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

The factor 3 appears because the viscosity is defined for shear deformation - as is the shear modulus G. For tensile deformation we want the viscous equivalent of Young s modulus . The answer is 3ri, for much the same reason that = (8/3)G 3G - see Chapter 3.) Data giving C and Q for polymers are available from suppliers. Then... 

These two moduli are not material constants and typical variations are shown in Fig. 5.3. As with the viscous components, the tensile modulus tends to be about three times the shear modulus at low stresses. Fig. 5.3 has been included here as an introduction to the type of behaviour which can be expected from a polymer melt as it flows. The methods used to obtain this data will be described later, when the effects of temperature and pressure will also be discussed. 

The solidity of gel electrolytes results from chain entanglements. At high temperatures they flow like liquids, but on cooling they show a small increase in the shear modulus at temperatures well above T. This is the liquid-to-rubber transition. The values of shear modulus and viscosity for rubbery solids are considerably lower than those for glass forming liquids at an equivalent structural relaxation time. The local or microscopic viscosity relaxation time of the rubbery material, which is reflected in the 7], obeys a VTF equation with a pre-exponential factor equivalent to that for small-molecule liquids. Above the liquid-to-rubber transition, the VTF equation is also obeyed but the pre-exponential term for viscosity is much larger than is typical for small-molecule liquids and is dependent on the polymer molecular weight. 

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

Shear modulus, polyamide, 138 Sheet molding compounds (SMCs), 30 Shoe sole products, 205 Shore hardness gauge, 243 Side-chain liquid crystalline polymers, 49 Side reactions, in transition metal coupling, 477... 

In case of copper some rheological experiments carried out at a given polymer concentration and increasing amoimt of cations indicates that copper/pectin systems in the one-phase domain behave as a viscoelastic liquid rather than a viscoelastic solid referred to as true gel (G (co) = G, when to—>0 with Gg the equilibrium shear modulus)[35]. Despite the lack of experimental data the range in cation and polymer concentration in which true gels may be observed seemed very limited. These results corroborate the strength of the binding of copper by pectins evidenced by the properties of the phase separation curves. 

A general characteristic of polymers is that their hardnesses tend to be proportional to their elastic moduli, particularly their shear moduli (Flores et al., 2000). However, the shear modulus is often anisotropic so an average value may not be an appropriate measure of hardness. The modulus for the plane of shear should be a better indicator. 

The viscoelastic behavior of concentrated (20% w/w)aqueous polystryene latex dispersions (particle radius 92nm), in the presence of physically adsorbed poly(vinyl alcohol), has been investigated as a function of surface coverage by the polymer using creep measurements. From the creep curves both the instantaneous shear modulus, G0, and residual viscosity, nQ, were calculated. 

Chain stretching is governed by the covalent bonds in the chain and is therefore considered a purely elastic deformation, whereas the intermolecular secondary bonds govern the shear deformation. Hence, the time or frequency dependency of the tensile properties of a polymer fibre can be represented by introducing the time- or frequency-dependent internal shear modulus g(t) or g(v). According to the continuous chain model the fibre modulus is given by the formula... 

As the chain modulus of a polymer cannot be altered in a spinning process, a larger fibre modulus can only be obtained by improving the orientation of the chains and by an increase of the shear modulus g. However, there is one exception. After dissolving native cellulose fibres with the cellulose I conformation and a chain modulus of 138 GPa into a solution, the regenerated fibres obtained by spinning of this solution and subsequent coagulation always have the cellulose II chain conformation with a chain modulus of 88 GPa [26]. 

Estimates of the ultimate shear strength r0 can be obtained from molecular mechanics calculations that are applied to perfect polymer crystals, employing accurate force fields for the secondary bonds between the chains. When the crystal structure of the polymer is known, the increase in the energy can be calculated as a function of the shear displacement of a chain. The derivative of this function is the attracting force between the chains. Its maximum value represents the breaking force, and the corresponding displacement allows the calculation of the maximum allowable shear strain. In Sect. 4 we will present a model for the dependence of the strength on time and temperature. In this model a constant shear modulus g is used, thus r0=gyb. 

Table 8 presents a survey of the basic elastic constants of a series of polymer fibres and the relation with the various kinds of interchain bonds. As shown by this table, the interchain forces not only determine the elastic shear modulus gy but also the creep rate of the fibre. 

Shear modulus, 13 498, 26 777 of dry foams, 12 16 of silicon carbide, 22 526t of vitreous silica, 22 430 of wet foams, 12 17-18 Shear plane, polymer colloid, 20 383, 384 Shear pulverization, of polymer blends, 20 326... 

Figure 2.14 The relative shear modulus, G> = G/GN, of an HEUR gel filled with polyethylmethacrylate particles. GN = 0.4 kPa. Experimental points are shown with the curves calculated for a non-interactive and an interactive filler. The amount of adsorbed polymer, T = 78g/kg offiller, gave a good fit to the experimental data...

O is the stress per unit unstrained area, G the shear modulus, A the deformation ratio, p the density of the dry network. iJ>2 volume fraction of polymer present in the network, V the volume at formation. A=1 for affine behaviour (expected) and 1-2/f for phantom behaviour(1,3). is the molar mass for the perfect network, essentially the molar mass of a chain of v bonds, the number which can form the smallest loop (5-7) see Figure 2. is equal to the... 

When there is no volume change, as when an elastomer is stretched, Poisson s ratio is 0.5. This value decreases as the Tg of the polymer increases and approaches 0.3 for rigid solids such as PVC and ebonite. For simplicity, the polymers dealt with here will be considered to be isotropic viscoelastic solids with a Poisson s ratio of 0.5, and only deformations in tension and shear will be considered. Thus, a shear modulus (G) will usually be used in place of Young s modulus of elasticity E Equation 14.2) where E is about 2.6G at temperatures below Tg. 

Most polymers are applied either as elastomers or as solids. Here, their mechanical properties are the predominant characteristics quantities like the elasticity modulus (Young modulus) E, the shear modulus G, and the temperature-and frequency dependences thereof are of special interest when a material is selected for an application. The mechanical properties of polymers sometimes follow rules which are quite different from those of non-polymeric materials. For example, most polymers do not follow a sudden mechanical load immediately but rather yield slowly, i.e., the deformation increases with time ( retardation ). If the shape of a polymeric item is changed suddenly, the initially high internal stress decreases slowly ( relaxation ). Finally, when an external force (an enforced deformation) is applied to a polymeric material which changes over time with constant (sinus-like) frequency, a phase shift is observed between the force (deformation) and the deformation (internal stress). Therefore, mechanic modules of polymers have to be expressed as complex quantities (see Sect. 2.3.5). 

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