RESUMEN
A survey of published experimental work on the modulus of natural rubber cross-linked by dicumyl peroxide permits a comparison with the results and molecular interpretations obtained in recent NBS work [J. Res. NBS 76A, No. 1, 51 (1972), 77A, No. 1, 171 (1973) and 80A, No. 3, 451 (1976)]. Excellent agreement was found among values of the shear modulus G at the same cross-linking when the cross-linking is calculated from the amount of decomposed dicumyl peroxide. The types of deformation included torsion as well as uniaxial extension and compression. G increases linearly with cross-linking (except at the lowest degrees) with a slope from 5 to 15 percent greater than that predicted by the simple statistical theory. Data of Mullins demonstrated that at each degree of cross-linking the value of G is intermediate between 2C 1 and 2(C 1 + C 2) where C 1 and C 2 are the Mooney-Rivlin constants. Measurements of equilibrium swelling at a given degree of cross-linking are in reasonable agreement with each other. However the entropy components of the modulus and the sub-chain density calculated from swelling measurements are appreciably greater than those calculated from cross-linking or from direct mechanical measurements. They increase linearly with cross-linking. It is concluded that the number of sub-chains effective in limiting swelling is greater than that effective in direct mechanical measurements.
RESUMEN
A survey of experimental data from the literature in cases where the deformation of a specimen is varied continuously from uniaxial compression to tensile deformation shows that Young's Modulus M, defined as the limit of stress to strain in the undeformed state, is independent of the direction of approach to the limit. The normalized stress-strain relation of Martin, Roth, and Stiehler (MRS, 1956) is F/M = (L-1 - L-2) exp A (L - L-1) where F is the stress on the undeformed section, L is the extension ratio, and M and A are constants. Values of M and A are obtained from the intercept and slope of a graph of experimental observations of log F/(L-1 - L-2) against (L - 1-1) including observations of uniaxial compression if available. They found the value of A to he about 0.38 for pure-gum vulcanizates of natural rubber and several synthetics. In later work several observers have now found that the equation is also valid for vulcanizates containing a filler, but A is higher, reaching a value of about 1 for large amounts of filler. In extreme cases A is not constant at low deformations. The range of applicability in many cases now is found to extend from the compressive region where L = 0.5 up to the point of tensile rupture or to a point where A increases abruptly because of crystallization. Taking A as a constant parameter in the range 0.36 to 1, graphs are presented showing calculated values of (1) F/M as a function of L and (2) the normalized Mooney-Rivlin plot of F/[2M(L - L-2)] against L-1. Each of the latter graphs has only a limited region of linearity corresponding to constant values of the Mooney-Rivlin coefficients C1 and C2. Since this region does not include the undeformed state, where L = 1, or any of the compression region, the utility of the Mooney-Rivlin equation is extremely limited, since it can not be used at low elongations. The coefficients are dramatically altered for rubbers showing different values of the MRS constant A. For rubbers showing the higher values of A, the coefficients are radically altered and the region of approximate linearity is drastically reduced.
RESUMEN
The shear modulus G = 5.925 × 10 - 3(fp - 0.45)T+G* (Part I), its energy component G* = 0.0684 (fp - 0.45)+ 2.70 (Part II), and the number of effective suh-chains per unit volume ve = (G - G*)/RT are given detailed molecular consideration. G is given in Mdyn cm-2 for rubber cross-linked by adding p parts of dicumyl peroxide per hundred of rubber, and heating until a fraction f of the peroxide is decomposed. ve is found to be approximately twice the density of cross-links, after a correction for impurities and chain ends is made. It can not be computed as G/RT since only the entropy component of modulus is related to ve. The sub-chains for the most highly cross-linked rubbers studied had a molecular weight of about 575 g mol-1, corresponding to about 8 isoprene units. The modulus corresponding to no added cross-links is not zero. It is determined chiefiy by the energy component of the modulus; it does not arise from entanglements. The "front factor" is found to be unity. An extensive literature survey yields values of the quantity RTΨ(v 2), where Ψ (v 2) is the Flory- Rehner equation function of v 2, the equilibrium volume fraction obtained by swelling the cross-linked rubber. RTψ (v 2) is found to be greater than G - G* but not as large as G itself.
RESUMEN
Long-time creep of natural rubber cured with a sulfur-accelerator recipe containing no filler can be conveniently represented by a plot of (E - E 1)/E 1 = ΔE/E 1 with a double-abscissa scale showing log t and t. E is the elongation at any time t, after application of the load, and E 1 its value at unit time. Experimental data conform to the equation except for a more rapid rise preceding rupture. The constants A and B can be evaluated from only three observations-at the longest time (about 70 days), at one minute, and at an intermediate time. ΔE/E 1 is approximately linear with log t when t is less than 0.1(A/B) and approximately linear with t when t is greater than 4.343(A/B). The observed modulus was about 1.4 MPa and A was about 2.4 percent/(unit log t) when the atmosphere was a vacuum, dry N2, or dry air. The modulus was lowered very slightly and A became about 4 percent/(unit log t) when the air was saturated with water. B was raised from about 2 × 10-5 percent/min to about 20 × 10-5 percent/min when the vacuum or dry N2 was replaced by dry air and to about 50 × 10-5 percent/min when the air was saturated with moisture. A is considered to be related to physical relaxation, while B corresponds to a chemical reaction, probably oxidative degradation.
RESUMEN
Thermodynamics and molecular considerations are applied to an examination of the equation G = S(fp + B)T + H(fp + B) + A = 5.925 × 10-3(fp - 0.45)T + 0.0684(fp - 0.45) + 2.70, found experimentally in Part I. G is the shear modulus in Mdyn cm-2 at a temperature T for natural rubber cross-linked by adding p parts of dicumyl peroxide per hundred of rubber (phr) and heating until a fraction f of the peroxide is decomposed. G*, the energy component of the modulus, is H(fp + B) + A. The ratio G*/G decreases from 1.00 at the gel point (fp = 0.45 phr) to 0.5 near 2 phr and to 0.09 at 23.8 phr. The modulus G is related to ν e , the number of moles of effective sub-chains per cm3, by the equation G - G* = v e RT where R is the gas constant. If each molecule of decomposed dicumyl peroxide of molecular weight M d produces one cross-link in the rubber of specific volume υ ¯ r , then it is predicted that S = 2 R ( 100 M d υ ¯ r ) - 1 = 5.5535 × 10 - 3 Mdyn cm-2 phr-1 K-1, as compared with the experimental value 5.925 × 10-3. Theory gives no prediction of the values of A, or of H. The gel point may be located experimentally as the point where the slope of the modulus-temperature relation is zero. The value of G at the gel point is the energy component G*. The experimental value of fp at the gel point permits a calculation of the molecular weight of the rubber before cross-linking as 193,000. The results afford a very satisfactory confirmation of the essential validity of the statistical theory of rubber elasticity in its simplest form, if due regard is paid to G*, the energy component of the modulus.