Retractive force at constant

Figure 6. Retractive force at constant strain, A corrected for thermal expansion, plotted against temperature. Behavior is completely reversible in range of 263 to 373 K. Serious degradation takes place above 393 K.

Both copolymers were cross-linked by copolymerization with (1-2%) ethylene dimethacrylate the effect of irradiation was followed by measuring the change of elastic retractive force at constant elongation. In the first case the contraction upon irradiation of the gel is mainly due to changes of chain conformation and swelling equilibrium induced by trans-cis photoisomerization. In the second case by comparing the retractive force for irradiated and unirradiated samples the heat effect was evaluated, and the photoinduced contraction estimated to 1%. 

This means that we have essentially true rubber elasticity in the region of the maximum in the relaxation times. The thermod3mamic theory of rubber elasticity says that the retractive force at constant strain will increase with increasing temperature. This is exactly what is observed in the time dependent stress curves, T5Q Tioo sec F g- 19. This data adds confirmation to the model. 

In connection with the stress-strain relations discussed here, it is interesting to observe that Moonev has shown how the retractive force in a stretching process is related to the shearing force in pure shear. We have seen on p. 107 that the shear modulus x is defined by the ratio between shearing force and shear. If x remains constant in finite deformation, i.e., if the shearing force is simply proportional to the shear, then the retractive force at finite stretch cannot be proportional to the elongation, and Mooney finds that the retractive force is then related to the degree of stretch v as follows... 

For the purpose of illustrating the application of the thermodynamic equation of state to experimental data, consider the plot given in Fig. 84 for the retractive force, measured at fixed length, against the absolute temperature for a hypothetical elastic substance. The slope at any temperature T gives the important quantity —(dS/dL)T,p according to Eq. (12) an increase in / with T at constant L shows immediately, therefore, that the entropy decreases with increase in length... 

Fig. 84.—The retractive force / of a hypothetical elastic body plotted against the absolute temperature at constant pressure.

Fig. 89.—The total force of retraction at 25°C and dE/dL)T,v obtained from the force-temperature intercepts at constant elongation for natural rubber gum-vulcanized using an accelerator. (Wood and Roth. )...

Fig. 90.—The force of retraction at 25°C and its internal energy component for gum-vulcanized GR-S synthetic rubber. Upper curve, total force / middle curve, dE/dL)T,p from the intercepts of force-temperature plots at constant length lower curve, dE/dL)T.v from the intercepts of stress-temperature plots at constant elongation. (Roth and Wood. )...

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... 

Fig. 88.—The force of retraction and its components dE/dL)T,v (curve A) and —T(dS/dL)T,v (curve B) as obtained through the application of Eqs. (20) and (22) to force-temperature plots at constant elongation, such as are shown in Fig. 87. (Anthony, Caston, and Guth. )...

From the dynamic mechanical investigations we have derived a discontinuous jump of G and G" at the phase transformation isotropic to l.c. Additional information about the mechanical properties of the elastomers can be obtained by measurements of the retractive force of a strained sample. In Fig. 40 the retractive force divided by the cross-sectional area of the unstrained sample at the corresponding temperature, a° is measured at constant length of the sample as function of temperature. In the upper temperature range, T > T0 (Tc is indicated by the dashed line), the typical behavior of rubbers is observed, where the (nominal) stress depends linearly on temperature. Because of the small elongation of the sample, however, a decrease of ct° with increasing temperature is observed for X < 1.1. This indicates that the thermal expansion of the material predominates the retractive force due to entropy elasticity. Fork = 1.1 the nominal stress o° is independent on T, which is the so-called thermoelastic inversion point. In contrast to this normal behavior of the l.c. elastomer... 

Fig. ISa-c. Effects of radiation on retractive force f (in Newtons) at constant length of a sample of poly MAH-STY-AAB) swollen in diethylphthalate and on temperature T inside the sample [3. ...

Figure 2b. A schematic of a typical AFM force-distance plot using a PEO modified tip. At (a), there is no interaction between tip and surface. As the chains begin to compress (b) a repulsive steric exclusion force is observed. At (c), the chains are compressed even more producing an even larger repulsive force that dominates the attractive van der Waals force. At (d), the chains arc so much compressed that the cantilever spring constant is much weaker than the spring constant of the PEO chains and the cantilever continues to bend upward the same amount as the sample has been moved due to the large repulsive force gradient. Upon retraction, no adhesion is observed (provided there is no bridging) and the curve coincides with the approach curve.

A hydrocarbon chain is in a constant thermal motion, and without external force field, the chains fluctuate around the most stable position given by the distribution of possible conformations at the temperature. The action of external forces at the ends of a molecule causes displacements of chains from their equilibrium conformations and evokes retractive forces. For a hydrocarbon chain of M = 14,000, extended length 125.5 nm, and the end-to-end distance r = 1 mn, the maximum exerted force is 10 MPa. The level of forces exerted by the random coil macromolecules are much lower than the theoretical strength of the primary bonds. The presence of strong intermolecular interactions, such as hydrogen bonds in polyamides, affects the retractive force substantially, causing a restriction of the number of possible chain conformations. In addition, the transitions... 

Another possibility, perhaps more appropriate in the case of a continuous phase-separated network, is that the retractive force of the gel arises from a minimization of interfacial energy. For example, if the gel structure is imagined to be cellular, consisting of approximately spherical holes in the polymer-rich phase entrapping the dilute phase, then small deformations at constant volume (illustrated in two dimensions in Fig. 8) would result in an increase in interfacial area and, thus, an increase in free energy. The retractive force in such a... 

A deformation dL at constant pressure and temperature induces a retractive force... 

Equation (9.15) implies that you can get the entropic component of the force, dSld )T, from a very simple experiment. Hold the rubber band at a fixed stretched length T (and constant pressure) and measure how the retractive force depends on the temperature (see Figure 9.1). The slope of that line, (df jdT)(, will give - (dSld )r. The positive slope in Figure 9.1 indicates that the entropy decreases upon stretching. 

Figure 9.1 The retractive force / of a rubbery polymer, amorphous polyethylene, held at constant length, as a function of temperature T. The slope is (dfldT)i. Source JE Mark,...

Rubber bands are entropic springs. Experiments show that the retractive force / of polymeric elastomers as a function of temperature T and expansion H is approximately given by f(T,C) = aT ( - tn) where a and l ) are constants. 

While the tip is oscillating in the transverse or lateral direction, the cantilever is brought into contact by extending the Z-piezoceramic transducer at constant speed by a known distance and is then retracted to its initial position. The friction force is proportional to the amplitude of the tip s lateral displacement analyzed by the lock-in technique. Measuring the lateral component while varying the load achieves in one step the measurement of the friction versus load. 

In an amorphous glassy polymer, work hardening is considered to correspond to stretching of the entanglement network invoked to account for the rubbery plateau above Tg (see Section 14.3.3). This explains the recoverability of the deformation above Tg (in the absence of an applied force, the network retracts to its equilibrium conformation) and it is borne out by the observed correlation between the value of X in the neck and the maximum extensibility of the network, X ax In semicrystalline polymers, the evolution of the crystalline texture may also contribute to work hardening, because the resolved shear stress on activated slip systems tends to decrease as deformation proceeds at constant stress, as will be discussed further (see Section 14.4.3). 

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