Isothermal volume contraction

It has to be noticed that no isothermal volume contraction on cooling or volume expansion on heating is associated with Tg, contrary to crystallisation or melting. The latter is a true first-order transition exhibiting a discontinuity in... 

The expressions in brackets are the expansivities above and below Tg. The constant K3 is a function of bond type in chains and is really constant for every class of polymers. The physical interpretation of this equation may be consistent with the iso-free-volume concept. However, we believe that the introduction of this equality is in practise a denial of the concept. There are also other arguments against this concept. Kastner56 found, for example, that dielectric losses diminish during the isothermal volume contraction, which indicates a dependence of relaxation times on free-volume. However, if we assume that relaxation time depends exclusively on free-volume, the calculated reduction factor differs from the experimental one. 

Figure 7.2 (a) Variation of the isothermal volume contraction. V(t) is the volume at... 

The decrease in volume that accompanies physical aging is known as volume recovery or volume relaxation. Dilatometry (Dil) can be used to follow the volume relaxation in glasses by monitoring the time-dependence of the volume change on aging. The material is either cooled from above Tg to the aging temperature 7], (down-jump) and the isothermal volume contraction is measured or the sample is heated in the glassy state (up-jump), in which case an expansion follows. 

For many liquid mixtures, it is assumed that there is equivolumetric transport, and hence molecular volume contraction is negligible. For isothermal conditions and without external forces, the pressure gradient vanishes, and we have... 

Fig. 11 Degree of crystallization a as a function of crystallization time during an isothermal crystallization process. Shrinkage V = const, (memory effect), not to confuse with volume contraction where V / const...

With thermomechanical analysis (TMA), the expansion or shrinkage of the sample under constant stress is monitored against time or temperature. It is also possible to measure dilatometric changes [53]. Near vitrification, the change in volume contraction versus isothermal reaction time will be reduced due to diffusion control. 

We now examine more closely the details of the volume contraction as portrayed in Fig. 11-7. The isotherms at different temperatures are closely superposable by horizontal translations just as are those for shear deformation in Fig. 11-1. Moreover, if one particular temperature in the transition region is arbitrarily designated as Tg. the translations of logarithmic time expressed as log Qt follow equation 21 closely. This result, obtained by Kovacs for several polymers, indicates that the temperature dependences of the molecular processes in bulk compression and in shear deformation are identical. Moreover, it permits calculation of the effect... 

Figure 5.16 Isothermal volume recovery curves of glucose by contraction and expansion at T = 30°C after quenching the samples to 25 °C and 40°C, equilibration of the samples at those temperatures and reheating to 30°C. The equilibrium volume at 30°C is denoted Uqo. Drawn after data from Kovacs (1963).

It follows that the efficiency of the Carnot engine is entirely determined by the temperatures of the two isothermal processes. The Otto cycle, being a real process, does not have ideal isothermal or adiabatic expansion and contraction of the gas phase due to the finite thermal losses of the combustion chamber and resistance to the movement of the piston, and because the product gases are not at tlrermodynamic equilibrium. Furthermore the heat of combustion is mainly evolved during a short time, after the gas has been compressed by the piston. This gives rise to an additional increase in temperature which is not accompanied by a large change in volume due to the constraint applied by tire piston. The efficiency, QE, expressed as a function of the compression ratio (r) can only be assumed therefore to be an approximation to the ideal gas Carnot cycle. 

Non-isothermal measurements of the temperatures of dehydrations and decompositions of some 25 oxalates in oxygen or in nitrogen atmospheres have been reported by Dollimore and Griffiths [39]. Shkarin et al. [606] conclude, from the similarities they found in the kinetics of dehydration of Ni, Mn, Co, Fe, Mg, Ca and Th hydrated oxalates (first-order reactions and all values of E 100 kJ mole-1), that the mechanisms of reactions of the seven salts are probably identical. We believe, however, that this conclusion is premature when considered with reference to more recent observations for NiC204 2 H20 (see below [129]) where kinetic characteristics are shown to be sensitive to prevailing conditions. The dehydration of MnC204 2 H20 [607] has been found to obey the contracting volume... 

Rising temperature measurements, supplemented by isothermal studies, gave values of E, based on obedience to the contracting volume equation [eqn. (7), n = 3], ranging from 96 to 123 kJ mole"1 and increasing in the sequence Zn < Fe(III) < Co < Ni < Cu. 

The kinetics of the contributory rate processes could be described [995] by the contracting volume equation [eqn. (7), n = 3], sometimes preceded by an approximately linear region and values of E for isothermal reactions in air were 175, 133 and 143 kJ mole-1. It was concluded [995] that the rate-limiting step for decomposition in inert atmospheres is NH3 evolution while in oxidizing atmospheres it is the release of H20. A detailed discussion of the reaction mechanisms has been given [995]. Thermal analyses for the decomposition in air [991,996] revealed only the hexavanadate intermediate and values of E for the two steps detected were 180 and 163 kJ mole-1. 

The contraction of solids on heating seems anomalous because it offends the intuitive concept that atoms will need more room to move as the vibrational amplitudes of the atoms increase. However, this argument is incomplete. Figure 11.9 plots schematically the variation of A with V at two temperatures, for both positive and negative thermal expansion. The volumes marked explicitly on the E-axis give the minima of each A vs. V isotherm. These are the equilibrium volumes at temperatures T and T2 respectively (J2 > 7j) and zero pressure. 

Figure 13.3. A P- V-T surface for a one-component system in which the substance contracts on freezing, such as water. Here Tj represents an isotherm below the triple-point temperature, 72 represents an isotherm between the triple-point temperature and the critical temperature, is the critical temperature, and represents an isotherm above the triple-point temperature. Points g, h, and i represent the molar volumes of sohd, hquid, and vapor, respectively, in equilibrium at the triple-point temperature. Points e and d represent the molar volumes of solid and liquid, respectively, in equihbrium at temperature T2 and the corresponding equilibrium pressure. Points c and b represent the molar volumes of hquid and vapor, respectively, in equilibrium at temperature and the corresponding equihbrium pressure. From F. W. Sears and G. L. Sahnger, Thermodynamics, Kinetic Theory, and Statistical Thermodynamics. 3rd ed., Addison-Wesley, Reading, MA, 1975, p. 31.

Here V is the crystal volume, k-p and ks are the isothermal and adiabatic compressibility (i.e., the contraction under pressure), P is the expansivity (expansion/contraction with temperature), Cp and Cv are heat capacities, and 0e,d are the Einstein or Debye Temperatures. Because P is only weakly temperature dependent,... 

In reality, the data on isothermal contraction for many polymers6 treated according to the free-volume theory show that quantitatively the kinetics of the process does not correspond to the simplified model of a polymer with one average relaxation time. It is therefore necessary to consider the relaxation spectra and relaxation time distribution. Kastner72 made an attempt to link this distribution with the distribution of free-volume. Covacs6 concluded in this connection that, when considering the macroscopic properties of polymers (complex moduli, volume, etc.), the free-volume concept has to be coordinated with changes in molecular mobility and the different types of molecular motion. These processes include the broad distribution of the retardation times, which may be associated with the local distribution of the holes. 

We calculated the fractional free-volumes/gl and/g2 according to the method developed by Covacs from the curves of the isothermal contraction82. The values obtained are not really the fractional free-volumes of the components at the corresponding temperatures, since in the calculation from contraction curves it is impossible to exdude the contributions of both components to the total free-volume. 

The activation energy of isothermal contraction in polymer blends calculated in 9 is considerably lower than for pure components, this pointing to the appearance of the free-volume as well, which facilitates the relaxation processes and diminishes the activation energy. 

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