Temperature at constant force

Since (df/dL)pj is always positive, the sign of df/dL)p L is opposite to that of (dL/dT)pj. Both coefficients are zero at the same length or force. The dependence of the length on the temperature at constant force is schematically illustrated in Fig. 8.5a for an idealized homogeneous fiber. At large L a small, positive thermal expansion coefficient typical of a crystalline solid is indicated. The melting... 

One of the primary features of the Gay-Berne potential is the presence of anisotropic attractive forces which should allow the observation of thermally driven phase transitions and this has proved to be the case. Thus using the parametrisation proposed by Gay and Berne, Adams et al. [9] showed that GB(3.0, 5.0, 2, 1) exhibits both nematic and isotropic phases on varying the temperature at constant density. This was chosen to be close to the transitional density for hard ellipsoids with the same ellipticity indeed it is generally the case that to observe a nematic-isotropic transition for Gay-Berne mesogens the density should be set in this way. The long range orientational order of the phase was established from the non-zero values of the orientational correlation coefficient, G2(r), at large separations and the translational disorder was apparent from the radial distribution function. 

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

This version of the thermodynamic equation of state for elasticity is most useful for interpretation of the experimental data discussed below. By measuring the force as a function of temperature at constant pressure and elongation a, one may readily derive (dE/dL)T,v from Eq. (22) and dS/dL)T,v from Eq. (20). 

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). 

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. 

Rusakov 107 108) recently proposed a simple model of a nematic network in which the chains between crosslinks are approximated by persistent threads. Orientional intermolecular interactions are taken into account using the mean field approximation and the deformation behaviour of the network is described in terms of the Gaussian statistical theory of rubber elasticity. Making use of the methods of statistical physics, the stress-strain equations of the network with its macroscopic orientation are obtained. The theory predicts a number of effects which should accompany deformation of nematic networks such as the temperature-induced orientational phase transitions. The transition is affected by the intermolecular interaction, the rigidity of macromolecules and the degree of crosslinking of the network. The transition into the liquid crystalline state is accompanied by appearence of internal stresses at constant strain or spontaneous elongation at constant force. 

These fluctuations, which are referred to as order-parsmeter fluctuations in studies of critical phenomena (3). comprise the driving forces for transport in the system. For liquid mixtures near a critical mixing point, the order parameter is concentration, and for pure gases near the vapor-liquid critical point, the order parameter is density. For gas mixtures such as supercritical solutions near the critical line, the order parameter is again density, which is a function of composition and temperature compared to a pure gas where density is a function of only temperature at constant pressure. 

In general, an increase in temperature (at constant density) should decrease lifetimes. Vibrational relaxation is caused by fluctuating forces in... 

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.)...

In plastics, as in many solids, increasing temperature at constant elongation results in a strain level drop. To phrase this differently, the amount of force required to elongate these materials decreases. Fig. 15. This reduction in elasticity does not, however, follow a uniform curve. On the other hand, nor are the sudden changes in state in evidence that are observed when low molecular substances change from one state of aggregatiiMi to another. 

Expansion due to the heating of a gas, or indeed of anything at aU, produces a force. This case is of special interest to us, because the work done by expansion or contraction of systems due to a change in temperature at constant pressure cannot be avoided. We can choose to eliminate other forms of work, but not this one (unless we consider only constant volume systems, which is useful at times, but not very practical). It is treated in more detail below. 

The ideal theory of rubber elasticity explains the linear dependence of the force on the temperature at constant extension, but the dependence of the force on extension at constant temperature is not well represented by Equation 4.15. The change in force with extension, df/da)j, is usually underestimated in the initial part of the expansion. The effective number of chains in Equation 4.15 appears to exceed the number of network chains determined by the synthesis of the network. 

At constant force, heating increases disorder, forcing the sample in the direction of the state of disorder it had at a lower temperature and smaller deformation. The result is therefore a decrease in length. 

In contrast to the pure one-component polymer system, AS, AL, and AH are now dependent on the composition. Therefore, the force-temperature derivative depends on A. The force in this instance is not uniquely determined by the temperature, and total melting does not occur at constant force. 

When the supernatant phase is multicomponent, the system is no longer univariant. Although the conditions of Eq. (8.68) must still be satisfied, this does not ensure that the composition of the amorphous phase will remain fixed with changes in A. At constant pressure the equilibrium force need no longer depend solely on the temperature. Consequently, total melting does not have to occur at constant force, in analogy to the behavior of a closed system. 

An important variable is pressure as under pressure interatomic distances in crystals show larger variations than those induced by temperature. At constant temperature T pressure P is related to the rate of energy change with the unit-cell volume V by relation P = — y)j, including the first-order derivative of total energy with respect to the cell volume. Such observables as the bulk modulus, the elastic and force constants depend on the second-order derivatives of the total energy. 

As mentioned above, the assvunption of a pixrely entropic elasticity leads to the prediction, Eq. (1.14), that the stress should be directly proportional to the absolute temperature at constant a (and V). The extent to which there are deviations from this direct proportionality may therefore be used as a measure of the thermodynamic non-ideality of an elastomer [9, 68-74]. In fact, the definition of ideality for an elastomer is that the energetic contribution /e to the elastic force / be zero. This quantity is defined by... 

FIGURE 9.17 (a) Force versus temperature at constant values of a for sulfur-cross-linked... 

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