Trouton’s coefficient

When a material is subjected to a tensile or compressive stress, Eqs. (5.63) through (5.74) should be developed with the shear modulus, G, replaced by the elastic modulus, E, the viscosity, rj, replaced by a quantity known as Trouton s coefficient of viscous traction, k, and shear stress, r, replaced by the tensile or compressive stress, a. It can be shown that for incompressible materials, k = 3r], because the flow under tensile or compressive stress occurs in the direction of stress as well as in the two other directions perpendicular to the axis of stress. Recall from Section 5.1.1.3 that for incompressible solids, E = 3G therefore the relaxation or retardation times are k/E. 

For the case of continuous deformation under a constant tensile stress, we can write, by analogy to Newton s Law of Viscosity [Eq. (4.3)], a relationship between tensile strain rate, e, tensile stress, a, and Trouton s coefficient, k, as... 

For incompressible materials, Trouton s coefficient can be related to X = 3tj, so that... 

Some good measurements by this method were made by Louginine,6 who used a small metal boiler connected by a glass tube, sloping back to the boiler, with a special metal condenser consisting of a spiral tube and a receiver below, immersed in water in a calorimeter. A table of some of his results is given below, including the values of Trouton s coefficient ( 14.VIIIL) MljTb (M=mol. wt). 

Cederberg s formula holds very well for the observations here used, as may be seen from a comparison of the two columns before the last the following remarks may be made about certain cases in the above table. The molecular heat of evaporation for NO has been calculated from the vapour-pressure measurements by Adwentowski, but the different measurements give values so discordant that the figure so obtained can only be approximate in any case, it is certain, if we take into consideration also the values of Trouton s coefficient obtained according to Cederberg, that... 

The relations between chemical equilibrium and Trouton s coefficient are therefore unmistakable, and may be employed with advantage. We must not, however, lose sight of the fact that the equations with which we are dealing are only approximately true. The exact treatment of the problem, which depends on the use of the specific heats of gases down to very low temperatures, will be learned in Chapters XIII and XIV, where it will also be seen that these specific heats are mostly inaccessible to experiment at the present time. This is the reason why I have thought it necessary to devote so much space to the treatment of the approximation formula. 

The problem is to be attacked in quite another manner when our concern is simply to calculate chemical equilibria in this case it will of course be best to derive the chemical constants from actual chemical equilibria. I have naturally worked in both directions in my numerous calculations, of which I have published, of course, only a small fraction. In my publications I have laid less stress on the accurate calculation of equilibria than on the remarkable fact, that quantities like Trouton s coefficient, and in particular certain coefficients in my vapour-pressure formula, bore a dose relation to chemical equilibria. Those who are not very practised in thermodynamical calculations will hardly have recognized this distinction, and for this reason a repetition of the calculation of the ammonia equilibrium will be desirable. 

Table 4.1 The structuredness of solvents, measured by their Trouton s constant, the entropy deficit, the dipole orientation correlation coefficient, and the heat capacity density...

Van Aubel found that Trouton s rule held for metals, the constant being about 20, but Baur and Brunner found values from 16-5 to 29-7 (see l.VIII L). Sonagha concluded that Trouton s rule holds best at a temperature T— Tb. It is quite generally found that for gases of low b.p. the Trouton coefficient is abnormally low, and Kleeman deduced that 4 and d4/dr are both zero at r=0. Some observed molecular heats of evaporation... 

Kireev said Trouton s rule applies to each component of mixtures of normal liquids if p//mj=const., where p/=partial pr sure, w/=fractional amount. The Trouton coefficient MlJTb—Ks is related to the critical point coefficient S =0-447 log r6+2-562 by the equation 5=0-195 Ki/i -t-l-71. 

This relation was found by some obscure reasoning by Pictet,5 and was stated as an empirical result by Trouton 6 it is generally called Trouton s rule. The average value of the constant ( Trouton coefficient ) for normal liquids... 

Hence, for Newtonian liquids, the tensile stress growth coefficient is a constant quantity and the extensional viscosity is three times the shear viscosity. This result was verified experimentally in 1906 by TroutonOD and is known as Trouton s rule. In addition, the ratio of the extensional viscosity to the shear viscosity is known as Trouton s ratio. 

The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

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