Temperature at constant length

Ic. The Results of Stress-Temperature Measurements.—Hysteresis in the stress-strain behavior of rubber and rubberlike materials has presented the most serious problem encountered in the execution of otherwise simple experiments on the change of stress in stretched rubber with temperature at constant length (L) or at constant elonga-... 

Figure 2.29- - An analysis of the thermodynamic equation of state [Eq. (2.69)] for rubber elasticity using a general experimental curve of force versus temperature at constant length. The tangent to the curve at T is extended back to 0°K. For an ideal elastomer, the quantity (dU/df)r is zero, and the tangent goes through the origin. The experimental line is, however, straight in the ideal case. (After Flory, 1953.)...

More rigorous consideration suggests that this is not correct and that there is an anisotropic compressibility in the strained state. Gee s experimental result that the internal energy contribution is negligible, based on measurements of the change in stress with temperature at constant length, is therefore capable of another interpretation, which leads to the conclusion that the internal energy contribution may not be zero. 

Fig. 85.—Force-temperature curves at constant length obtained by Anthony, Gaston, and Guth for natural rubber vulcanized with sulfur for elongations from 3 percent to 38 percent (at 20°C), as indicated.

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. )...

Figure 5. Stretching force at constant length and a temperature of 283 K for the same polymer as in Figure 1. The sample is uncross-linked for the first 1000 s and then rapidly cooled to the glassy state and cross-linked with high energy electrons. The force f is obtained by subsequent heating to the stress relaxa-...

Thus from a knowledge of the amount adsorbed at various temperatures at constant pressure the alteration in length of life of the adsorbed molecules may be computed. 

In Eq. (1), a is the equilibrium stress (Nm 2) supported by the swollen specimen a is the stretched specimen length divided by the unstretched length (extension ratio) v2 is the volume fraction of dry protein and p is the density of dry protein. In the common case of tetrafunctional crosslinks, the concentration of network chains n (mol network chains/g polymer) is exactly one-half the concentration of crosslinks, so that n = 2c. The hypothesis that a specimen behaves as if it were an ideal rubber can be confirmed by observing a linear relation with zero intercept between a and the strain function (a — 1/a2) and by establishing a direct proportionality between a and the absolute temperature at constant value of the extension ratio, as stipulated by Eq. (1). 

Experimental determination of the components of the elastic force thus requires measurements of the changes in force with temperature at constant volnme and length. The constant volume requires the application of hydrostatic pressure during measurement of the force-temperature coefficient. This experiment is extremely difficult to perform 22-i3). 

Instead of measuring the force-temperature dependence at constant volume and length, one can measure this dependence at constant pressure and length but in this case it is necessary to introduce the corresponding corrections. The corrections include such thermomechanical coefficients as iso-baric volumetric expansion coefficient, the thermal pressure coefficient or the pressure coefficient of elastic force at constant length 22,23,42). 

Although traditionally the thermodynamic treatment of the deformation of elastomers has been centered on the force, the alternative condition of keeping the force (or tension) constant and recording the sample length as a function of temperature at constant pressure is even simpler 23,271. 

Since in the elastic region (0 In L/0 In Ot.p is always positive, the force-temperature coefficient at constant length and pressure must be of opposite sign to PL. For rubbers, at some extensions the isometric inversion [(0 In f/6T), P = 0] must occur since pL of the isotropic sample is always positive. For solids, such measurements correspond to the determination of the coefficient (5 of an elastically stretched sample which, however, does not differ from the usual coefficient of thermal expansion. 

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... 

The relative effects of supercitical carbon dioxide density, temperature, extraction cell dimensions (I.D. Length), and cell dead volume on the supercritical fluid extraction (SFE) recoveries of polycyclic aromatic hydrocarbons and methoxychlor from octadecyl sorbents are quantitatively compared. Recoveries correlate directly with the fluid density at constant temperature whereas, the logarithms of the recoveries correlate with the inverse of the extraction temperature at constant density. Decreasing the extraction vessels internal diameter to length ratio and the incorporation of dead volume in the extraction vessel also resulted in increases in SFE recoveries for the system studied. Gas and supercritical fluid chromatographic data proved to be useful predictors of achievable SFE recoveries, but correlations are dependent on SFE experimental variables, including the cell dimensions and dead volume. 

Heat capacities can be defined for processes that occur under conditions other than constant volume or constant temperature. For example, we could define a heat capacity at constant length of a sample. However, regardless of the nature of the process, the heat capacity will always be positive. This is ensured by the zeroth law of thermodynamics, which requires that as positive heat is transferred from a heat reservoir to a colder body, the temperature of the body will rise toward that of the reservoir in approaching the state of thermal equilibrium, regardless of the constraints of the heat-transfer process. 

Equations (54) and (56) suggest that measurements of the force required to keep a rubber at constant length as a function of temperature would determine the thermodynamic properties of the rubber. However, due to thermal expansion, very large changes in pressure would be required to keep the polymer volume constant as its temperature is varied. Thus, measurements of (df /dT)l are usually made at constant pressure. Extending Eq. (10) of Appendix A to an additional variable gives... 

This gives the entropy change per unit extension, (dS/dT)j, which occurs in Eq. (4-32), in terms of the temperature coefficient of tension at constant length (9//9T )/, which can be measured. With Eq. (4-38), Eq. (4-32) becomes... 

Figure 3.4 Stress at constant length as a function of temperature. Elongations as indicated (5,6). (From Ref 5.)...

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. ...

Polyester is not a fast crystallising polymer. The rate of crystallisation is maximum for polyester at around 180 C. The maximum crystallisation rate for polyester is about 0.016/sec, while it is 0.14/sec for nylon 6. The following changes take place as a function of increasing temperature in polyester when heat-set under free to relax conditions (i.e. free annealing) and when held taut at constant length (i.e. taut annealing) ... 

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