Apparent third order transitions

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. 

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. 

Liquid helium presents an interesting case leading to further understanding of the third law. When liquid 4He, the abundant isotope of helium, is cooled at pressures of < 25 bar, a second-order transition takes place at approximately 2 K to form liquid Hell. On further cooling Hell remains liquid to the lowest observed temperature at 10 5 K. Hell does become solid at pressures greater than about 25 bar. The slope of the equilibrium line between liquid and solid helium apparently becomes zero at temperatures below approximately 1 K. Thus, dP/dT becomes zero for these temperatures and therefore AS, the difference between the molar entropies of liquid Hell and solid helium, is zero because AV remains finite. We may assume that liquid Hell remains liquid as 0 K is approached at pressures below 25 bar. Then, if the value of the entropy function for sol 4 helium becomes zero at 0 K, so must the value for liquid Hell. Liquid 3He apparently does not have the second-order transition, but like 4He it appears to remain liquid as the temperature is lowered at pressures of less than approximately 30 bar. The slope of the equilibrium line between solid and liquid 3He appears to become zero as the temperature approaches 0 K. If, then, the slope is zero at 0 K, the value of the entropy function of liquid 3He is zero at 0 K if we assume that the entropy of solid 3He is zero at 0 K. Helium is the only known substance that apparently remains liquid as absolute zero is approached under appropriate pressures. Here we have evidence that the third law is applicable to liquid helium and is not restricted to crystalline phases. 

Amorphous materials usually exhibit an apparent second-order thermal transition, the glass transition, at about two-thirds of the crystalline melting temperature (measured in Kelvin)... 

Finally, we like to mention that equivalent to the conventional energy frame KHD formulation, the time-dependent theory of Raman scattering is free from any approximations except the usual second order perturbation method used to derive the KHD expression. When applied to resonance and near resonance Raman scattering, the time-dependent formulation has shown advantages over the static KHD formulation. Apparently, the time-dependent formulation lends itselfs to an interpretation where localized wave packets follow classical-like paths. As an example of the numerical calculation of continuum resonance Raman spectra we show in Fig. 6.1-7 the simulation of the A, = 4 transitions (third overtone) of D excited with Aq = 488.0 nm. Both, the KHD (Eqs. 6.1-2 and 6.1-18) as well as the time-dependent approach (Eqs. 6.1-2 and 6.1-19) very nicely simulate the experimental spectrum which consists mainly of Q- and S-branch transitions (Ganz and Kiefer, 1993b). 

Melting is normally driven by an entropy gain, then AS > 0. With the decrease of temperatures from T, the integral at the right-hand side of (6.53) decays gradually from zero to —AS, as demonstrated in Fig. 6.16b. However, a linear extrapolation to AS = 0 reaches a finite temperature rather than zero absolute temperature, which can be defined as T. This result implies that below Ts, Si < Sc-Apparently, the amorphous liquid state could not be more ordered than the crystalline solid state, which is against the third law of thermodynamics. Early in 1931, Simon pointed out this problem (Simon 1931). In 1948, Kauzmann gave a detailed description, and proposed that there should exist a phase transition such as crystallization before extrapolation to to avoid this disaster (Kauzmann 1948). Therefore, this scenario is also called the Kauzmann paradox. 

RMD Simulation of Chemical Nucleation (22). A series of microscopic computer experiments was performed using the cooperative isomerization model (Eq. 2). This system was selected for the trial simulations for several reasons First, only two chemical species are involved, so that a minimal number of particles is needed. Second, the absence of buffered chemicals (e.g., A and B in the Trimolecular reaction of the next section) eliminates the need for creation or destruction of particles in order to maintain constant populations (19., 22j. Third, the dynamical model of the cooperative mean-field interaction can be examined as a convenient means of introducing cubic or higher nonlinearity into molecular models based on binary collisions. Finally, the need for a microscopic simulation is most apparent for transitions between multi -pie macroscopic states. Indeed, the characterization of spatially localized fluctuations is of obvious importance to the understanding of nucleation phenomena. As for the equilibrium vapor-liquid and liquid-solid transitions, detailed simulations at the molecular level should provide deep physical insight into chemical nucleation processes whkh is unattainable from theory, higher-level simulation, or experiment. 

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