Hysteresis equilibrium transition pressure

Sinee no f eleetrons ean eontribute to these transitions in Y, earlier suggestions (Wittig 1978) whieh eonsidered eontributions from f eleetrons as essential driving faetors for these transitions in the regular lanthanides, are not eonfirmed. However, more reeently, in a eomparison of the transition pressures given in the literature (whieh are forward transition pressures) some anomalies were pointed out and interpreted as a possible f eontribution (Gschneidner 1985). If, on the other hand, best estimates of the equilibrium transition pressures from studies of kineties and hysteresis (Kruger et al. 1990) are used, no special contributions from f electrons seem to be substantiated, since the critical radius ratios r,= at the different phase transitions for Y... 

In general, the Bl-type actinide compounds show a large difference between the transition pressure on pressure increase and the transition pressure on pressure decrease (Dabos-Seignon and Benedict 1990). This hysteresis is probably due to delayed establishment of thermodynamic equilibrium. It has been shown in some lanthanide metals (Kruger et al. 1990) that a slight temperature increase reduces the width of this hysteresis zone and that the equilibrium transition pressure has some intermediate value. 

Figure 4. Schematic showing a hysteresis loop for the CdSe nanociystals with the smearing of the thermodynamic transition pressure caused by the finite nature of the nanocrystal particle. The thermodynamic transition pressure is offset from the hysteresis center to emphasize that in first-order solid-solid transformations, this pressure is unlikely to be precisely centered. The lower plot shows the estimated smearing for CdSe nanocrystals as inversely proportional to the number of atoms in the crystal, at two temperatures, as discussed in the text. Note that nanocrystals are not ordinarily synAesized or studied in sizes smaller than 20 A in diameter. This figure shows that this thermal smearing is insignificant compared to the large hysteresis width in the CdSe nanociystals studied (25-130 A in diameter), such that the transition is bulk-like from this perspective. This means that observed transformations occur at pressures far from equilibrium, where there is little probability of back reaction to the metastable state once a nanociystal has transformed. In much smaller crystals or with larger temperatures, the smearing could become on the order of the hysteresis width, and the crystals would transform from one stmcture to the other at thermal equilibrium.

Hygroscopic behavior has been well characterized in laboratory studies for a variety of materials, for example, ammonium sulfate (Figure 14), an important atmospheric material. When an initially dry particle is exposed to increasing RH it rapidly accretes water at the deliquescence point. If the RH increases further the particle continues to accrete water, consistent with the vapor pressure of water in equilibrium with the solution. The behavior of the solution at RH above the deliquescence point is consistent with the bulk thermodynamic properties of the solution. However, when the RH is lowered below the deliquescence point, rather than crystallize as would a bulk solution, the material in the particle remains as a supersaturated solution to RH well below the deliquescence point. The particle may or may not undergo a phase transition (efflorescence) to give up some or all of the water that has been taken up. For instance, crystalline ammonium sulfate deliquesces at 79.5% RH at 298 K, but it effloresces at a much lower RH, 35% (Tang and Munkelwitz, 1977). This behavior is termed a hysteresis effect, and it can be repeated over many cycles. 

Fig. 38. Isothermal sections at 25°C of (a) intra-lanthanide and (b) intra-actinide generalized binary phase diagrams, showing equilibrium phase boundaries [with estimated hysteresis for (a)] as full hnes (Benedict et al. 1986). The broken line in (a) indicates the interpolated boundary for the volume collapse transition of the lanthanides. The atomic radius of Ce at room temperature as a function of pressure is shown in (c) (Franceschi and Olcese 1969), with the Kondo-volume collapse transition at about 7 kbar. This transition can be traced to negative pressures by alloying (Lawrence et al. 1984), as seen in (d) via the temperature dependence of the resistance.

In accordance with equilibrium thermodynamics, at constant pressure in a one-component system phase transition occurs at a sp>ecific temperature and it should be accompanied by a sudden change in heat release or absorption, i.e. phase transition has no extension in time or hysteresis. It should be noted, that although interphase border is an integral part of any system where phase transition takes place, classical equilibrium thermodynamics does not pay any attention to the possible contribution of this border to phase transitions. 

An adsorption/desorption isotherm below Tec, exhibits a hysteresis loop. In between condensation and evaporation metastable pressures, there is an equilibrium pressure for which both the gas and liquid phases coexist. In the case of cylindrical pores of any dimensions, there are theoretical arguments to indicate that no first order transition can exist because at the critical point, the correlation length can diverge only in the direction of the pore axis a cylindrical pore can be considered as a one-dimensional system whatever its diameter. However, DFT [4], simulations [5] and experiments suggest that fluids in both confining geometries (slit and cylinders with size of several nm) behave similarly as far as condensation and evaporation are concerned the hysteresis loop shrinks as temperature increases and eventually disappears. We note that in a van der Waals picture of gas-liquid transitions in such simple systems, the temperature of hysteresis disappearance is the capillary critical temperature, i.e.. Tec-... 

In the case of equilibrium liquid drops and menisci (see Section 2.3), they are supposed to be always at equilibrium with flat films with which they are in contact with in the front. Only the capillary pressure acts inside the spherical parts of drops or menisci, and only the disjoining pressure acts inside thin flat films. However, there is a transition zone between the bnlk liquid (drops or menisci) and the thin flat film in front of them. In this transition zone, both the capillary pressure and the disjoining pressure act simultaneously (see Section 2.3 for more details). A profile of the transition zone between a meniscus in a flat capillary and a thin a-fllm in front of it, in the case of partial wetting, is presented in Figure 2.5. It shows that the liquid profile is not always concave but changes its curvature inside the transition zone. Just this peculiar liquid shape in the transition zone determines the static hysteresis of contact angle (see Chapter 3)... 

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