Thermoelastic inversion

No thermoelastic inversion should appear in the force-temperature coefficient at constant elongation a, inasmuch as the effect of ordinary thermal expansion is eliminated by fixing a instead of the length L as the temperature is varied. As the elongation approaches unity, both the force and its temperature coeffi.cient df/dT)p,a must van-... 

These expressions show that a deformed polymer network is an extremely anisotropic body and possesses a negative thermal expansivity along the orientation axis of the order of the thermal expansivity of gases, about two orders higher than that of macromolecules incorporated in a crystalline lattice (see 2.2.3). In spite of the large anisotropy of the linear thermal expansivity, the volume coefficient of thermal expansion of a deformed network is the same as of the undeformed one. As one can see from Eqs. (50) and (51) Pn + 2(iL = a. Equation (50) shows also that the thermoelastic inversion of P must occur at Xim (sinv) 1 + (1/3) cxT. It coincides with F for isoenergetic chains [see Eq. (46)]. 

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... 

Results of force-temperature experiments for a PAA solution on rubber are shown in Figure 2. For the first cycle, the initial load is that of the rubber since the coating is still in the liquid state and cannot support a load. At this load the rubber is just below its thermoelastic inversion point and its contribution to the force change is negligible. 

Note that since (8H/dL)TP 0, an ideal rubber has a thermoelastic inversion point. 

The stress-temperature behavior of natural rubber at various extension ratios has been measured by Shen et al. (1967) and shown in Figure 14.12. Compare trends between data at varying extension ratios with those shown in Figure 14.3 and provide an explanahon for the changes at low elongations (a phenomenon that is termed thermoelastic inversion). 

Experiments performed at lo i/er elongations (or compressions) belovi/ the so-called thermoelastic inversion point lead to decreasing o-values i/ith increasing temperature because the thermal expansion of the samples predominates the effect of the retractive force. 

Rg. 4.4 (a) Magnification of the small deformation region. Thermoelastic inversion is seen around... 

Thus, the Gough-Joule effect can be understood as the manifestation of the thermoelastic inversion when seen from a different viewpoint. 

We can understand the thermoelastic inversion by noticing the difference between the two distinct concepts of elongations one defined relative to the reference state (A.), and the other defined relative to the initial state (a). 

Fig. 7.4. Observation of a thermoelastic inversion point for natural rubber The temperature dependence of the force at constant extension exhibits a reversal in slope. Measurements by Anthony et al.[74]...

One might think at first that the energetic part of the force, could be derived also from a temperature dependent measurement of the force on the basis of Eq. (7.6). In fact, direct application of this equation is experimentally difficult since the volume does not remain constant under the normally given constant pressure conditions. Indeed, thermal expansion is observed and this is also the reason for the occurrence of a thermoelastic inversion point . It shows up in temperature dependent measurements on rubbers which are kept at a fixed length. Figure 7.4 shows a series of measurements which were performed at different values of A. For high extensions, we find the signature of ideal rubbers, i.e. an increase f T. For low extensions, on the other hand, thermal expansion over compensates this effect, and then even leads to a decrease of the force. 

The particular value of X at which a becomes zero is of interest. It is the stretch associated with the so-called thermoelastic inversion and has the value ... 

The linear expansion of (71) reveals that the position of the thermoelastic inversion is insensitive to internal energy effects and depends principally upon thermal expansion. 

We see that initially the ratio of energy stored internally to work done on the body is infinite. This means that heat must diffuse into the body to maintain it isothermal. Contrariwise, if the stretch is performed adiabatically, this means that there is initially a temperature drop prior to a later rise. The stretch at which the temperature is exactly zero for adiabatic stretch is, of course, related to the thermoelastic inversion, and can easily be calculated from (21). 

These observations are confirmed in Fig. 14.4. The negative slope at low elongations arises from the predominance of thermal expansion when elongation, and hence f, is low. Note that there is an intermediate elongation, the thermoelastic inversion point, at which force is essentially independent of temperature, where thermal expansion and entropy contraction balance. 

The data in Fig. 5.18 were obtained at a fairly high extension (350 percent). When the experiment is repeated at lower extensions the slope of the curve decreases and below about 10 percent extension it becomes negative. This is caused by a reduction in stress due to thermal expansion of the rubber as the temperature is increased and it is known as the thermo-elastic inversion effect. If the effective change in extension due to thermal expansion is allowed for then the thermoelastic inversion effect disappears and the stress increases proportionately with temperature at low extensions as well as high extensions. 

---