Classification of phase transitions

Lest us assume that the physical system being considered may be described by means of a family of potential functions U(x c), depending on 

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). 

Landau assumed that near the phase transition a thermodynamic function, dependent on the order parameter q and on some other variables X, may be expanded in a power series in the order parameter q 

At equilibrium, the condition (8U/dq)(0) = 0 must hold. According to the Landau concept, qualitative changes in a system take place when the parameter q passes through zero hence, the critical point of potential (with respect to q) must be a singular point for the state q = 0 to be sensitive. 

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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 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. 

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. 

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-... 

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

The discontinuity in the first derivatives of function thus appears as the most suitable for an idealized classification of phase transitions [3,297,365]. The characteristic value of a variable, at which a phase transition occur, is termed the phase transition point fT , /cq). The changes in the derivatives can be then expressed according to Ehrenfest classification and give the limit for the first-order phase transitions... 

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