Third order transitions

The initial classification of phase transitions made by Ehrenfest (1933) was extended and clarified by Pippard [1], who illustrated the distmctions with schematic heat capacity curves. Pippard distinguished different kinds of second- and third-order transitions and examples of some of his second-order transitions will appear in subsequent sections some of his types are unknown experimentally. Theoretical models exist for third-order transitions, but whether tiiese have ever been found is unclear. 

The single residue related to third-order transition moments is... 

The three-photon absorption cross section is related to the fifth order susceptibility which clearly forms a challenging computational task. Most often we are interested in the resonant absorption to a particular state, f, in which case the cross section can be expressed in terms of the third-order transition moment to this state, which in turn is much easier to compute. Hence, for 3PA it is sufficient to evaluate the third-order transition moment, T ", as... 

The microscopic origin of the three-photon absorption can be traced to the fourth order hyperpolarizability. Despite the high order of this property, it is still attainable due to two decisive steps of simplification. First, at resonance, i.e. under excitation with a frequency equal to one-third to the excitation energy of the final state, Wy/3, a fraction of the terms will dominate the summation completely, which leads to a formulation where the fourth order hyperpolarizability can be expressed as a product of third order transition dipole moments ... 

Even though the response approach represents a significant simplification, it is still computationally demanding for three-photon absorption, and the third order transition moments are therefore obtained at the RPA level, i.e. cubic response... 

Table 2. Excitation energies of the final state, /ioj, the third order transition moment, 7",. and the three-photon probability, calculated at SCF level with the basis sets 6-31G and 6-310. Geometry is optimized by DFT/B3LYP with 6-3IG ...

We recalculated the three-photon absorption coefficient, y, making use of the formulas which relate the third order transition dipole moment, three-photon transition probability, 8 and 7 ... 

In inflection-point analysis, we refer to it as a third-order transition. 

The discontinuity in dCp/dT is characteristic of an apparent third order transition. A possible physical interpretation in terms of a third order transition is discussed. is interpreted as a basic molecular process related to the breaking of weak secondary bonds, with an accompanying drop in the dynamic elastic modulus. 

This general question of observing secondary transitions by thermal means was considered by one of us (RFB) on a prior occasion. We came to the conclusion, from existing litraature data, that Cp - T plots exhibited not a discontinuity in Cp but a change in slope at these secondary relaxations specifically, at T < Tg for atactic polystyrene and PVC, and at Tn for several elastomers. In each case precision measmements of heat content, AH, were available. A change in slope of a - r curve is a discontinuity in d HIdT, and hence a third order transition in the sense of Ehrenfest.22... 

Flory results, but found evidence for a third order transition near 443 K. This is consistent with the DSC results of this paper, More recent data by Rudin and by Richardson S fail to show any 7 /-related change in slope of - 7 plots above Tg. We examined by computer the numerical data from the thesis on which the Rudin paper was based. In some cases there were slope increases, but they were not related to 7 /. 

PHYSICAL MEANING OF AN APPARENT THIRD ORDER TRANSITION... 

The existence of an apparent third order transition above Tg was noted independently by Boyei and by Hocker et a1. No attempt at a physical explanation was made in either case. It was noted in the first instance that there was a discontinuity in d HIdT in the second case, a discontinuity in d VIdP. This current DSC data on polystyrene, as well as similar results on a number of polymers, when coupled with the data discussed in ref. 21 and Fig. 1, suggests that at least preliminary discussions of the physical meaning of a third order transition are in order. 

All of this represents a phenomenological, and essentially ad hoc, interpretation (tentative hypothesis) for an apparent third order transition (relaxation). In addition, one should keep in mind the various parallels between Tn and Tg, i.e.,... 

Computer analysis of National Bureau of Standards adiabatic calorimeter data on polyisobutylene shows that the increase in Cp above Tg = 200 K can be best represented by three straight lines intersecting at 253 and 292 K, which intersections we identify with T / and Tn. Such behavior signifying a discontinuity in dCp/dT and hence in (fiHIdfi, corresponds to that expected for a third order transition. For the time being we refer to this event as an apparent third order transition. 

The endothermic change in slope at Tjj is indicative of a third order transition. A preliminary discussion of the physical significance of an apparent third order transition is presented in terms of the breakup of weak secondary forces and the resulting increase both in molecular motions and in thermal expansion. 

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