Mooney constant

Because the constant C2 decreases when the rubber is swollen by solvents, this extra term is deduced to be caused by the topological entanglements of the subchains. The entangled parts serve as the delocalized cross-links which increase the elasticity. Networks are disentangled on swelling, and the Mooney constant C2 decreases. 

Figure 4.16 shows the detailed experimental data on the depression of the sttess. The extrapolation to a oo gives the first Mooney constant 1C, which turns out to be independent of the polymer volume fraction 4>. In contrast, the slope of the lines decreases... 

OtherRota.tiona.1 Viscometers. Some rotational viscometers employ a disk as the inner member or bob, eg, the Brookfield and Mooney viscometers others use paddles (a geometry of the Stormer). These nonstandard geometries are difficult to analy2e, particularly for an infinite bath, as is usually employed with the Brookfield and the Stormer. The Brookfield disk has been analy2ed for Newtonian and non-Newtonian fluids and shear rate corrections have been developed (22). Other nonstandard geometries are best handled by determining iastmment constants by caUbration with standard fluids. 

Development of the Mooney viscometer gave compounders an indication of the processibiUty of different lots of the uncompounded polymer. This machine measures the torque resistance encountered by a rotor revolving in a chamber surrounded by polymer at a constant temperature. The resulting Mooney number describes the toughness of the polymer and is an indirect measure of molecular weight. 

Considerably better agreement with the observed stress-strain relationships has been obtained through the use of empirical equations first proposed by Mooney and subsequently generalized by Rivlin. The latter showed, solely on the basis of required symmetry conditions and independently of any hypothesis as to the nature of the elastic body, that the stored energy associated with a deformation described by ax ay, az at constant volume (i.e., with axayaz l) must be a function of two quantities (q +q +q ) and (l/a +l/ay+l/ag). The simplest acceptable function of these two quantities can be written... 

This is Mooney s equation for the stored elastic energy per unit volume. The constant Ci corresponds to the kTvel V of the statistical theory i.e., the first term in Eq. (49) is of the same form as the theoretical elastic free energy per unit volume AF =—TAiS/F where AaS is given by Eq. (41) with axayaz l. The second term in Eq. (49) contains the parameter whose significance from the point of view of the structure of the elastic body remains unknown at present. For simple extension, ax = a, ay — az—X/a, and the retractive force r per unit initial cross section, given by dW/da, is... 

In order to check this prediction, stress-strain measurements were made up to moderate strains at room temperature. The obtained data are plotted in the usual manner as a versus 1/X in Figure 8. Table V gives the Mooney-Rivlin constants 2C and 2C calculated from these plots and also the ratio C./Cj. 

Mooney-Rivlin constants obtained from strain dependent measurements at 298 K... 

Figure 9. Dependence of the Mooney-Rivlin constant on extent of cross-linking. Mn is 21,600 PDMS. Key , 2C, X, 2CS (3).

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. 

The results of stress-strain measurements can be summarized as follows (1) the reduced stress S (A- A ) (Ais the extension ratio) is practically independent of strain so that the Mooney-Rivlin constant C2 is practically zero for dry as well as swollen samples (C2/C1=0 0.05) (2) the values of G are practically the same whether obtained on dry or swollen samples (3) assuming that Gee=0, the data are compatible with the chemical contribution and A 1 (4) the difference between the phantom network dependence with the value of A given by Eq.(4) and the experimental moduli fits well the theoretical dependence of G e in Eq.(2) or (3). The proportionality constant in G for series of networks with s equal to 0, 0.2, 0.33, and 0. Ewas practically the same -(8.2, 6.3, 8.8, and 8.5)x10-4 mol/cm with the average value 7.95x10 mol/cm. Results (1) and (2) suggest that phantom network behavior has been reached, but the result(3) is contrary to that. Either the constraints do survive also in the swollen and stressed states, or we have to consider an extra contribution due to the incrossability of "phantom" chains. The latter explanation is somewhat supported by the constancy of in Eq.(2) for a series of samples of different composition. 

In Figure 3, o/(X—X-z) is plotted against 1/X to obtain the constants 2Cj and 2C2 in the Mooney-Rivlin equation. The intercepts at 1/X = 0 and the slopes of the lines give the values of 2Cj and 2C2, respectively, listed in Table I. If these plots actually represent data accurately as X approaches unity, then 2(Cj + C2) would equal the shear modulus G which in turn equals E/3 where E is the Young s (tensile) modulus. An inspection of the data in Table I shows that 2(Cj + C2)/(E/3) is somewhat greater than one. This observation is in accord with the established fact that lines like those in Figure 3 overestimate the stress at small deformations, e.g., see ref. 15. 

The Mooney arrangement of a bob with a conical base is an attractive design as it is relatively easy to fill and uses the base area to enhance the measurement sensitivity. However the cone angle must be such that the shear rates in both the cone and plate and concentric cylinder sections are the same. This means that the gap between the cylinders must be very slightly larger than the gap at the edge of the cone and plate if a constant shear rate is required. Unfortunately the DIN standard bob is poor in this respect. 

A rather complete survey of the entire field of viscometry, including the mathematical relationships applicable to various types of instruments, has been made by Philippoff (P4). The problem of slip at the walls of rotational viscometers has been discussed by Mooney (M15) and Reiner (R4). Mori and Ototake (M17) presented the equations for calculation of the physical constants of Bingham-plastic materials from the relationship between an applied force and the rate of elongation of a rod of such a fluid. ... 

Again it has been found experimentally that (Cx + C2) does not correspond with the Mooney-Rivlin constants in extension and compression. One is thus faced with the problem of deciding which Cx to identify with the Gaussian constant. Several authors consider Ct in extension to be the Gaussian constant, but in view of the above this position cannot be maintained. One could instead identify the compression modulus with the Gaussian constant 3(AvkT/Lt) Kr2)4/(r2 0) because in compression Cg is always found to be very small or zero. There is, however, no theoretical justification for this procedure either. 

Fig. 22, The same data as in Fig. 21 plotted as force versus deformation ratio. If one identifies the compression modulus with the Gaussian constant 3 (A vk T/Lt) (( >o) the experimental curve in extension lies below the Gaussian curve. The C part of the Mooney-Rivlin curve in extension lies again below the experimental curve...

Some further remarks concerning the Mooney-Rivlin equation are in place (14, 112). In dry rubber networks Ca in extension is often of the same order of magnitude as Cx, so that we are by no means confronted with a minor correction. In unilateral compression C2 is almost zero, and perhaps slightly negative. The constant Cx increases with the crosslinking density and with the temperature the ratio C2/C( in extension seems... 

---