Phase transitions classification

In this section several general properties of phase transitions are considered, as well as a phase transition classification system. The discussion and results of this section are applicable to all phase transitions (liquid-solid, solid-solid, vapor-solid, vapor-liquid, etc.), although special attention is given to vapor-liquid equilibrium. 

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. 

Differential scanning calorimetry (DSC) is a common technique for the classification of individual phase transitions in liquid-crystalline materials and has been applied for the phase characterization of alkyl-modified chromatographic surfaces. Hansen and Callis [187] applied DSC to investigate phase changes in Cig and C22... 

The mysteries of the helium phase diagram further deepen at the strange A-line that divides the two liquid phases. In certain respects, this coexistence curve (dashed line) exhibits characteristics of a line of critical points, with divergences of heat capacity and other properties that are normally associated with critical-point limits (so-called second-order transitions, in Ehrenfest s classification). Sidebar 7.5 explains some aspects of the Ehrenfest classification of phase transitions and the distinctive features of A-transitions (such as the characteristic lambda-shaped heat-capacity curve that gives the transition its name) that defy classification as either first-order or second-order. Such anomalies suggest that microscopic understanding of phase behavior remains woefully incomplete, even for the simplest imaginable atomic components. 

Paul Ehrenfest suggested a widely used classification of thermodynamic transition phenomena according to the lowest derivative of Gibbs free energy that exhibits a mathematical discontinuity at the phase transition. 

To differentiate between the variety of phase equilibria that occur, Ehrenfest proposed a classification of phase transitions based upon the behavior of the chemical potential of the system as it passed through the phase transition. He introduced the notion of an th order transition as one in which the nth derivative of the chemical potential with respect to T or p showed a discontinuity at the transition temperature. While modern theories of phase transitions have shown that the classification scheme fails at orders higher than one, Ehrenfest s nomenclature is still widely used by many scientists. We will review it here and give a brief account of its limitations. 

Another important classification of particle dark matter rests upon its production mechanism. Particles that were in thermal equilibrium in the early Universe, like neutrinos, neutralinos, and most other WIMPs (weakly interacting massive particles), are called thermal relics. Particles which were produced by a non-thermal mechanism and that never had the chance of reaching thermal equilibrium in the early Universe are called non-thermal relics. There are several examples of non-thermal relics axions emitted by cosmic strings, solitons produced in phase transitions, WIMPZILLAs produced gravitationally at the end of inflation, etc. 

The Ehrenfest17 classification of phase transitions (first-order, second-order, and lambda point) assumes that at a first-order phase transition temperature there are finite changes AV 0, Aft 0, AS VO, and ACp VO, but hi,lower t = hi,higher t and changes in slope of the chemical potential /i, with respect to temperature (in other words (d ijdT)lowerT V ((9/i,7i9T)higherT). At a second-order phase transition AV = 0, Aft = 0, AS = 0, and ACp = 0, but there are discontinuous slopes in (dV/dT), (dH/<)T), (OS / <)T), a saddle point in and a discontinuity in Cp. A lambda point exhibits a delta-function discontinuity in Cp. 

We do not give a detailed classification of the types of liquid crystals, since the systems under consideration mainly form nematic or in some cases cholesteric phase. The latter phase belongs, in principle, to the same range of liquid crystals as the nematic phase, since there is no phase transition between them (unlike smectic-nematic phase transition). 

For adsorbents with sufficiently laig e porosities (often referred to as mesoporous systems) isotherms can exhibit hysteresis between the adsorption and desorption branches as illustrated schematically in figure 1. A classification of the kinds of hysteresis loops has also been made. It is generally accepted that such behavior is related to the occurrence of capillary condensation - a phenomenon whereby the low density adsorbate condenses to a liquid like phase at a chemical potential (or bulk pressure) lower than that corresponding to bulk saturation. However, the exact relationship between the hysteresis loops and the capillary phase transition is not fully understood - especially for materials where adsorption cannot be described in terms of single pore behavior. 

A characteristic feature associated with pore condensation is the occurrence of sorption hysteresis, i.e pore evaporation occurs usually at a lower p/po compared to the condensation process. The details of this hysteresis loop depend on the thermodynamic state of the pore fluid and on the texture of adsorbents, i.e. the presence of a pore network. An empirical classification of common types of sorption hysteresis, which reflects a widely accepted correlation between the shape of the hysteresis loop and the geometry and texture of the mesoporous adsorbent was published by lUPAC [10]. However, detailed effects of these various factors on the hysteresis loop are not fully understood. In the literature mainly two models are discussed, which both contribute to the understanding of sorption hysteresis [8] (i) single pore model. hysteresis is considered as an intrinsic property of the phase transition in a single pore, reflecting the existence of metastable gas-states, (ii) neiM ork model hysteresis is explained as a consequence of the interconnectivity of a real porous network with a wide distribution of pore sizes. 

According to Erenfest s classification, there are first-order and second-order phase transitions. The first-order transitions are discontinuous, and the second-order transitions are continuous. The KDP-type phase transitions for many years were regarded as purely continuous ones. Then it was shown by Kobayashi et al... 

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