Thermodynamic Fragility

Scaled excess entropy versus Tg-scaled inverse temperature. The excess entropy corresponds to the entropy of the liquid minus that of the corresponding crystal, and it is scaled by the value of excess entropy at Tg so that all curves converge at Tg. (Reproduced from [21].] 

This connection is not fully unexpected since greater structural mobility at higher temperatures should result in faster diffusion (lower viscosity) but should also generate new degrees of freedom that contribute to the heat capacity and raise the entropy of the system. Other connections between transport properties and thermodynamic parameters have also been identified. For example, the activation energy for viscous flow and enthalpy relaxation appear to be strongly correlated, as will be discussed below [53,54]. 

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As mentioned in Section I, the melting temperature has been probed as a correspondence states parameter for viscous flow [68]. This idea has been pushed further in defining a thermodynamic fragility as [37,88]... 

An alternative and perhaps more accurate definition of thermodynamic fragility that can be determined experimentally for almost all liquids would be to determine the fraction of entropy of fusion lost by T/Tm = 0.8 [37,88]. In both the definitions, the property of the crystal phase enters this disadvantage is missing in the kinetic measures of fragility [37,88]. 

Figure 5. The horizontal axis shows the value of fragility as computed from the thermodynamics by the RFOT theory, and the vertical axis contains the fragility directly measured in kinetics experiments. Here m is the so-called fragility index, defined according to m = [(ilogjoi(7 )/(i(l/7 )]j,. m is somewhat more useful than the fragility D, because deviations from...

Volume swelling measurements have produced erratic results even under the most carefully controlled conditions. One important contribution in this regard is the work of Bills and Salcedo (8). These investigations showed that the binder-filler bond could be completely released with certain solvent systems and that the volume swelling ratio is independent of the filler content when complete release is achieved. Some thermodynamic problems exist, however, when such techniques are used to measure crosslink density quantitatively. First, equilibrium swelling is difficult to achieve since the fragile swollen gel tends to deteriorate with time even under the best conditions. Second, the solubility of the filler (ammonium perchlorate) and other additives tends to alter the solution thermodynamics of the system in an uncontrollable manner. Nonreproducible polymer-solvent interaction results, and replicate value of crosslink density are not obtained. 

An explicit expression relating kinetic fragility to thermodynamic behavior of supercooled liquids was accomplished for the first time by Mohanty and coworkers [55,56] and independently by Speedy [54], These authors derived an expression for the steepness parameter, a measure of kinetic fragility, from the temperature variation of the relation time or viscosity, with the ratio of excess entropy and heat capacity changes at the glass transition temperature [54-56]. A detailed description of this work will be provided later in the review chapter. 

In parallel with Eq. (2.12), the E 3/ 4 tllermo metric is defined as the value of ( Tg/T) for which Sexc(Tg)/Sexc(T) With this definition, a correlation was observed between kinetic F1/2 and thermodynamic F3/2 thermo fragilities [38]. 

A comparison of Eqs. (2.20) and (2.18) leads to a remarkable expression that relates the fragility index, a kinetic property, with measurable thermodynamic quantities... 

Figure 2.2. A plot of the predicted fragility of several glass-forming polymeric liquids versus average experimental fragility index (adapted from Ref. 56). The solid line denotes the points for which the y-axis and the x-axis are equal, is a guide to the eye. The predicted correlation indicates an intricate connection between thermodynamic and kinetics of supercooled polymeric liquids [56],...

It is well established that structural relaxation within the glass transition region is nonlinear. Can the degree of nonlinearity be connected with fragility If so, the nonlinearlity of structural relaxation would have a thermodynamic foundation. 

Lennard-Jones binary mixture of particles is a prototypical model that describes glass-forming liquids [52,53,158,162-165]. The temperature and the density dependence of diffusivity D(T, p) have been obtained by computer simulations for the Lennard-Jones binary mixture in the supercooled state. To relate fragility of binary Lennard-Jones mixture to thermodynamic properties necessitates determination of the configurational entropy SC(T, p) as well as the vibration entropy Sv,h(T, p) at a given temperature and density. 

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