Classification of phases

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. 

Classification of Phase Boundaries for Binary Systems. Six classes of binary diagrams have been identified. These are shown schematically in Figure 6. Classifications are typically based on pressure—temperature (P— T) projections of mixture critical curves and three-phase equilibria lines (1,5,22,23). Experimental data are usually obtained by a simple synthetic method in which the pressure and temperature of a homogeneous solution of known concentration are manipulated to precipitate a visually observed phase. 

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. 

CLASSIFICATION OF PHASE TRANSFORMATIONS CONTINUOUS VERSUS DISCONTINUOUS TRANSFORMATIONS... 

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. 

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. 

An interesting classification of phase states is that based on two parameters the degree of order (long- and short range) and the time dependence of stiffness (long- and short time). Each state can be characterised by a matrix of the following form ... 

Figure 3. Classification of phase-behaviour for systems carbon dioxide-water-alcohol...

An old alchemist maxim, similia similibus solvuntur" ( like dissolves like"), is the oldest rule for selecting suitable solvents, meaning that the nature of the solute determines the nature of the solvent. Due to the classification of phase systems in Section 4.2.2, organic and aqueous solvents are distinguished. Non-polar to medium-polar substances show best solubility in typical organic solvents while medium-polar to polar substances show best solubility in aqueous solvents. 

In a number of publications [12], classification of phase transitions in small systems has been presented. This scheme is based on the distribution of zeroes of the canonical partition function in the complex temperature plane. Among others. Gross has suggested a microcanonical treatment [13], where phase transitions of different order are distinguished by the curvature of the entropy 5 = In According to this scheme, a back-bending in the micro-... 

On the basis of analysis of experimental facts Ehrenfest has introduced the following classification of phase transitions a conversion is called the phase transition of nth order if successive derivatives of a thermodynamic function U up to and including (n — 1) are continuous functions, whereas the nth derivative has a step discontinuity at the transition point the... 

The Ehrenfest classification is not too well suited for the description of real phase transitions occurring in nature. The above remark concerns most of all the phase transitions which are not first-order. Better suited for an examination and classification of phase transitions is the Landau classification. Landau s idea is based on an assumption that in the case of many phase transitions one may always find a quantity, called the parameter of order, whose small change (with respect to the value q — 0) causes a qualitative changes in the parameters of a body (this implies that for q = 0 the system is in the sensitive state). 

The initial classification of phase transitions made by Ehrenfest (1933) was extended and clarified by Pippard... 

Figure 1. Classification of phase-transfer catalytic reaction.

Typical phase morphologies of block copolymers are illustrated in Figs. 5.38 0 and 5.79. In the classification of phases, the phase-separated block copolymers are considered to be amphiphilic liquid crystals despite the fact that inside the phase areas the typical liquid-crystalline order is missing (see Sect. 5.5). In this section the question will be addressed what happens when the usually micro- or nano-phase separated block copolymers show solubility, i.e., when the amphiphilic liquid crystal becomes thermotropic, i.e., dissolve at a given temperature. 

This section starts with a classification of phase-contacting principles according to the type of catalytic bed. Advantages and disadvantages of the reactor types are explained, followed by a discussion of criteria for reactor selection and an overview of purchasable microreactors for catalytic gas-phase reactions. 

The phase portraits of the model represented by Eq. (7.133) provide us with useful information on the dynamic behavior of this model. These phase portraits are calculated by numerical integration for a given set of parameters at different initial conditions. Every parametric domain, which is determined by the mutual rearrangement of the lines and L , relates to a special type of phase portrait. Eor a model with two variables, it is possible to present all types of phase portraits. However, for many models with three variables, the number of phase portraits is enormous and the corresponding classification of phase trajectories is quite difficult. 

OreSki, S., Zupan, J. and GlaviC, P., 2001, Neural network classification of phase equilibrium methods, Chem. Biochem. Eng. Q. 15, 3. 

---