Equilibrium shear modulus

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. 

According to the rubber elasticity theory ( 1, 2), the equilibrium shear modulus, Ge, is proportional to the concentration of EANC s and an additional contribution due to trapped entanglements may also be considered ... 

The equilibrium shear modulus of two similar polyurethane elastomers is shown to depend on both the concentration of elastically active chains, vc, and topological interactions between such chains (trapped entanglements). The elastomers were carefully prepared in different ways from the same amounts of toluene-2,4-diisocyanate, a polypropylene oxide) (PPO) triol, a dihydroxy-terminated PPO, and a monohydroxy PPO in small amount. Provided the network junctions do not fluctuate significantly, the modulus of both elastomers can be expressed as c( 1 + ve/vc)RT, the average value of vth>c being 0.61. The quantity vc equals TeG ax/RT, where TeG ax is the contribution of the topological interactions to the modulus. Both vc and Te were calculated from the sol fraction and the initial formulation. Discussed briefly is the dependence of the ultimate tensile properties on extension rate. 

Figure 5.11 Dependence of the reduced equilibrium shear modulus, Ge/wg// 7" on the molar ratio of [OH]/[NCO] groups, ah, for poly(oxypropylene)triol (Niax LG 56)-4,4 -diisocyanatodiphenylmethane system (—-) limits of the Flory-Erman junction fluctuation rubber elasticity theory. The dependence has been reconstructed from data of ref. [78]...

Deformational measurements were carried out in a uniaxial compression of cylindrical samples and the equilibrium shear modulus G was determined from [11]... 

In Eq. (4.13) NT is the total number of internal degrees of freedom per unit volume which relax by simple diffusion (NT — 3vN for dilute solutions), and t, is the relaxation time of the ith normal mode (/ = 1,2,3NT) for small disturbances. Equation (4.13), together with a stipulation that all relaxation times have the same temperature coefficient, provides, in fact, the molecular basis of time-temperature superposition in linear viscoelasticity. It also reduces to the expression for the equilibrium shear modulus in the kinetic theory of rubber elasticity when tj = oo for some of the modes. 

The proportionality factor G is the equilibrium shear modulus of the network, given by... 

For a viscoelastic solid (like an organogel), any rheological description should give a constant finite elastic modulus and infinite viscosity at zero frequency or long times. The situation is somewhat comparable to that of a cross-linked network [2. The equilibrium shear modulus for small deformations is proportional... 

Experimental detection of the gel point is not always easy since the equilibrium shear modulus is technically zero at the gel point and any applied stress will eventually relax, but only at infinite time. From the classical theory, the attributes of the gel point are an infinite steady-shear viscosity and a zero equilibrium modulus at zero frequency limit (Figure 6-3) (Flory, 1953). These criteria have been widely employed to detect the gel point of chemical gels. However, because continuous shearing affects gel formation, accurate information from viscosity measurement is not possible in the close vicinity of the gel point. Further, information regarding the transition itself could only be obtained by extrapolation, thereby introducing uncertainties in the determination of the gelation moment. 

Table Ila. Values for the Equilibrium Shear Modulus Gq for Different Aqueous Surfactant Solutions...

Notice that the above equation is simply a time-dependent generalization of Hooke s law [Eq. (7.98)]. For viscoelastic solids, G(t) relaxes to a finite "value, called the equilibrium shear modulus G q (see Fig. 7.22, top curve) —... 

An interesting, indirect application of the statistical theory to filled rubbers has been published by Wolff (/33). In recent years increasing use has been made of oscillating disc rheometers to follow the course of vulcanization. These instruments operate at low frequencies and at vulcanization temperatures, so that torque recorded at full cure is a good measure of an equilibrium shear modulus and hence proportional to v. Wolff finds for polymers filled with furnace blacks and cross-linked with dieumyl peroxide ... 

Ge° Equilibrium shear modulus for T>Tg in a polymer crosslinked beyond its gel point. Gic Fracture energy of a brittle material. 

In their work on a specific class of amine-crosslinked epoxy thermosets, Lesser and Crawford [140,141] found that Tg was affected strongly by Mc, by the average crosslink functionality fav (because of the effects of crosslink mobility, as will be discussed in Section 1 l.B.3.a for the equilibrium shear modulus Ge°), and by the chain stiffness. Their data were best represented by Equation 6.18, where Tg(° ) and C, were treated as fitting parameters. 

If the rubbery equilibrium shear modulus does not show evidence of crosslink mobility for some other family of thermosets, then the factor (fav-2)/fav would drop out of Equation 6.18, so that the dependence of Tg on network architecture would be expressed more simply, just in tenns of Mc. It could, in that case, be expressed equivalently as in Equations 6.16 and 6.17. For thermosets known to or expected to manifest crosslink mobility, it should, then, generally be possible to combine the functional form of the dependence on fav shown in Equation 6.18 with Equation 6.16, to obtain Equation 6.19 which is an alternative form for the relationship for the Tg of thermosets manifesting crosslink mobility. 

Figure 13.12. Schematic illustration of the shear viscosity r and the equilibrium shear modulus G as functions of the particle volume fraction <1>. Note that r —as <1> approaches the viscosity percolation threshold from below, while the onset of a nonzero G (indicative of the onset of true solid-like rigidity) occurs immediately above d>. ...

A fifth method to determine the gel point follows from the relationships between the equilibrium shear modulus and the network strudure, as discussed... 

With the aid of this result Morton and Ferry [38] derived a relationship between the equilibrium shear modulus, G, and the sol fraction ... 

If Mw and are known, then a can be calculated with the aid of Eq. (13 thereupon from the equilibrium shear modulus, Ge, the sol fraction, w, can be calculated with the aid of Eqs. (14)-(17) provide the crosslink functionality, f, is known. Subsequently the we t average crosslinking index in the system as a whole, and the number average crosslinking index in the networic fraction,... 

It should be emphasized that just after the gel point the equilibrium shear modulus scales to the third power of Ay = Yw — Yw,g. independent of the molecular weight distribution of the primary polymen... 

Te Nijenhuis [127] showed that the crosslink frmctionality of tlte system mentioned in Fig. 30 increased from 16 to 36 over a period from 0.1 to 1000 h. During this period the equilibrium shear modulus inciea from 500 to 2800 N/m. ... 

Schultz and Myers [185] measured the equilibrium shear modulus as a function of concentration of added borate. For borate concentrations of 2.5-10 mol/m , the modulus is approximately linear with the coircentration of... 

Fig. 94. Equilibrium shear modulus of 20% PAN-DMF gels plotted vs agdng time during ageing at various temperatures. Reproduced from J Polym Sci [Ret 6] by the of John Wiley...

F%.95. Equilibrium shear modulus at 2S°C (right) after preoeeding gelation at — 9.S°C (I ), dotted vs ageing time. Reproduced from J Polym Sd [Ref. 296] by the courtesy of John W3ey Sons, Inc... 

F. 128w Concentration dependence the equilibrium shear modulus of n-tetradecane stdutions SBS Kraton 1101 at 23 °C (constructed from data presented in [369])... 

Near the critical concentration, c , below which no gelation occurs, the power law dependence of the equilibrium shear modulus, Ge, on the concentration, c, is high and variable. However, in many, rather historical, pubUcations [461,463, 464, 466, 4 )-489] it was demonstrated that the quotient G /c or GJc has a constant value in a certain concentration region, or that G or Ge are linear functions of c. For the value of G, its equilibrium value (whatever it may be) should be used. In the previous section it was shown that even after 150 h of ageing the storage modulus still linearly increases with log a ing time. Hence, it... 

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