Modulus equilibrium

Beyond the LST, p > pc, the material is a solid. The solid state manifests itself in a finite value of the relaxation modulus at long times, the so-called equilibrium modulus... 

Steady shear flow properties are sensitive indicators of the approaching gel point for the liquid near LST, p < pc. The zero shear viscosity rj0 and equilibrium modulus Ge grow with power laws [16]... 

Fig. 5. Schematic of the divergence of zero-shear viscosity, rj0, and equilibrium modulus, Ge. The LST is marked by pc...

For the relaxation of the solid near the gel point, the critical gel may serve as a reference state. The long time asymptote of G(t) of the nearly critical gel, the equilibrium modulus Ge, intersects the G(t) = St n of the critical gel at a characteristic time (Fig. 6) which we will define as the longest relaxation time of the nearly critical gel [18]... 

The exponents a and a+ depend not only on the relaxation exponent n, but also on the dynamic exponents s and z for the steady shear viscosity of the sol and the equilibrium modulus of the gel. 

Besides the static scaling relations, scaling of dynamic properties such as viscosity rj and equilibrium modulus Ge [16,34], see Eqs. 1-7 and 1-8, is also predicted. The equilibrium modulus can be extrapolated from dynamic experiments, but it actually is a static property [38]. 

Fig. 10. Experimental values of the gel stiffness S plotted against the relaxation exponent n for crosslinked polycaprolactone at different stoichiometric ratios [59]. The dashed line connects the equilibrium modulus of the fully crosslinked material (on left axis) and the zero shear viscosity of the precursor (on right axis)...

The above equations are generally valid for any isotropic material, including critical gels, as long as the strain amplitude y0 is sufficiently small. The material is completely characterized by the relaxation function G(t) and, in case of a solid, an additional equilibrium modulus Ge. 

Measurement of the equilibrium properties near the LST is difficult because long relaxation times make it impossible to reach equilibrium flow conditions without disruption of the network structure. The fact that some of those properties diverge (e.g. zero-shear viscosity or equilibrium compliance) or equal zero (equilibrium modulus) complicates their determination even more. More promising are time-cure superposition techniques [15] which determine the exponents from the entire relaxation spectrum and not only from the diverging longest mode. 

Adolf and Martin [15] postulated, since the near critical gels are self-similar, that a change in the extent of cure results in a mere change in scale, but the functional form of the relaxation modulus remains the same. They accounted for this change in scale by redefinition of time and by a suitable redefinition of the equilibrium modulus. The data were rescaled as G /Ge(p) and G"/Ge(p) over (oimax(p). The result is a set of master curves, one for the sol (Fig. 23a) and one for the gel (Fig. 23 b). 

During our early experiments on chemical gels, when first observing the intermediate state with the self-similar spectrum, Eq. 1-5, we simply called it viscoelastic transition . Then, numerous solvent extraction and swelling experiments on crosslinking samples showed that the viscoelastic transition marks the transition from a completely soluble state to an insoluble state. The sol-gel transition and the viscoelastic transition were found to be indistinguishable within the detection limit of our experiments. The most simple explanation for this observation was that both phenomena coincide, and that Eqs. 1-1 and 1-5 are indeed expressions of the LST. Modeling calculations of Winter and Cham-bon [6] also showed that Eq. 1-1 predicts an infinite viscosity (see Sect. 4) and a zero equilibrium modulus. This is consistent with what one would expect for a material at the gel point. 

Plots of G at 0.5 Hz and the reduced stress ore(j obtained from stress-strain measurements at small strains against temperature, give almost identical straight lines (Figure 5). This similarity was expected because no frequency dependence of G had been observed. Hence G equals the equilibrium modulus G G moreover equals the reduced stress ore(j, if the latter is measured in the vicinity of X= 1. The measurements were always performed at X = 1.02 - 1.04, so that this requirement is fulfilled. 

In this contribution, we report equilibrium modulus and sol fraction measurements on diepoxidet-monoepoxide-diamine networks and polyoxypropylene triol-diisocyanate networks and a comparison with calculated values. A practically zero (epoxides) or low (polyurethanes) Mooney-Rivlin constant C and a low and accounted for wastage of bonds in elastically inactive cycles are the advantages of the systems. Plots of reduced modulus against the gel fraction have been used, because they have been found to minimize the effect of EIC, incompleteness of the reaction, or possible errors in analytical characteristics (16-20). A full account of the work on epoxy and polyurethane networks including the statistical derivation of various structural parameters will be published separately elsewhere. 

The phantom network behaviour corresponding to volumeless chains which can freely interpenetrate one through the other and thus to unrestricted fluctuations of crosslinks should be approached in swollen systems or at high strains (proportionality to the Mooney-Rivlin constant C-j). For suppressed fluctuations of crosslinks, which then are displaced affinely with the strain, A for the small-strain modulus (equal to C1+C2) approaches unity. This situation should be characteristic of bulk systems. The constraints arising from interchain interactions important at low strains should be reflected in an increase of Aabove the phantom value and no extra Gee contribution to the modulus is necessary. The upper limit of the predicted equilibrium modulus corresponds therefore, A = 1. 

From a theoretical point of view, the equilibrium modulus very probably gives the best characterization of a cured rubber. This is due to the relationship between this macroscopic quantity and the molecular structure of the network. Therefore, the determination of the equilibrium modulus has been the subject of many investigations (e.g. 1-9). For just a few specific rubbers, the determination of the equilibrium modulus is relatively easy. The best example is provided by polydimethylsiloxane vulcanizates, which exhibit practically no prolonged relaxations (8, 9). However, the networks of most synthetic rubbers, including natural rubber, usually show very persistent relaxations which impede a close approach to the equilibrium condition (1-8). 

The adjustable parameters, G (o), t0, m, AG, a and p were determined by a least-squares analysis. In physical terms G (o) Is the equilibrium modulus, AG represents the Increment to the entanglement plateau modulus, t0 Is a characteristic time, related, for the last three equations, to the frequency In the point of Inflection, and the exponents m, a and p relate to the slope at the characteristic frequency. 

Figure 18 Calculated stress-re la at ion curves for styrene-butadiene and silicone rubbers, both uncross-linked (from Figure 17) and cross-linked to vr - 50 x 10 6 mol/cm, and for SBR additionally cross-linked to v, = 100 and 200 x 10 mol/cml. The horizontal bars show the location of the equilibrium modulus for SBR. M - 200.1100. T - 29 K.

The measurement of the equilibrium modulus offers another possibility to compare the branching theory and experiment. 

However, in doing so one tests two theories the network formation theory and the rubber elasticity theory and there are at present deeper uncertainties in the latter than in the former. Many attempts to analyze the validity of the rubber elasticity theories were in the past based on the assumption of ideality of networks prepared usually by endllnklng. The ideal state can be approached but never reached experimentally and small deviations may have a considerable effect on the concentration of elastically active chains (EANC) and thus on the equilibrium modulus. The main issue of the rubber elasticity studies is to find which theory fits the experimental data best. This problem goes far beyond the network... 

Figure 6. Expected change in the equilibrium modulus, Gg, with respect to its ideal value for a perfect network, Gg produced by a 3% change in conversion, AC, functionality, Af the molar ratio [OH]/[NCO], Ar, and cyclization, AC in dependence on conversion. The data refer to a system composed of. trifunctional telechelic polymer and difunctional coupling agent. (Reproduced with permission from Ref. 42. Copyright 1987 CRC Press.)...

Diluent added during crosslinking has two main effects it Increases the population of elastically Inactive cycles and it weakens the interchain constraints. Studies of poly(oxypropylene) triol-diisocyanate networks in the presence of diluent have shown that the effect of diluent on the equilibrium modulus is much stronger than would correspond to the effect of cycles (Figure 10) (32) which again corroborates the concept of permanent interchain constraints. 

Figure 9. Reduced equilibrium modulus of polyurethane networks from POP trlols and MDI in dependence on the sol fraction. networks from POP triol Mjj - 708, o networks from POP triol Mjj = 2630. C-) calculated dependence using Flory junction fluctuation theory for the value of the front factor A indicated. (Reproduced from Ref. 57. Copyright 1982 American Chemical Society.)...

Figure 10. Dependence of the reduced equilibrium modulus of POP triol - MDI networks prepared in the presence of diluent. POP triol Mu= 708 stress-strain measurements in the presence of diluent (o) and after evaporation of the diluent ( ). Flory theory for the values of the front factor A indicated, theoretical dependence including trapped interchain constraints Numbers at curves Indicate the value of ry.

Thus, the simplified Two-Network experiment shows by a direct comparison of forces at constant length that the trapped entangled structure of a well cross-linked elastomer contributes to the equilibrium modulus by an amount that is approximately equal to the rubber plateau modulus. The modulus contribution from the trapped entangled structure will be less for lower molecular weights and especially at low degrees of cross-linking (14). 

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