Ehrenfest second-order

Such an Ehrenfest second-order transition, although formally imaginable, is not believed to describe any real phase transition.)... 

Because these transitions are associated with a mechanism in which one phase gradually evolves into the other, they are also often referred to as continuous or cooperative transitions. The terms second order , lambda , and continuous transitions have often been used interchangeably to refer to the same transition even though a true Ehrenfest second-order heat capacity does not have a lambda shape. We shall use the designation continuous transition (in preference to second order or lambda) for all transitions in which the discontinuity occurs in the second derivative of G. 

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. 

For a second-order transition Eq. (4.53) is the analog of Eq. (4.5), which is useful for first-order transitions. Equation (4.53) and the first- and second-order terminology are due to Ehrenfest. 

The transition from a ferromagnetic to a paramagnetic state is normally considered to be a classic second-order phase transition that is, there are no discontinuous changes in volume V or entropy S, but there are discontinuous changes in the volumetric thermal expansion compressibility k, and specific heat Cp. The relation among the variables changing at the transition is given by the Ehrenfest relations. 

When the free energies F of the two crystal structures are identical, the system is at a critical point. The identity of F does not imply identical fimctions (otherwise the two phases would be indistinguishable). Therefore, at the critical point first derivatives of F might differ and therefore enthalpy, volume, and entropy of the two phases would be different. These transformations are first-order phase transitions, according to Ehrenfest [105]. A discontinuous enthalpy imphes heat exchange at the transition temperature, which can easily be measured with DSC experiments. A discontinuous volume is evident under the microscope or, more precisely, with diffraction experiments on single crystals or powders. Some phase transitions are however characterized by continuous first derivatives of the free energy, whereas the second derivatives (specific heat, compressibility, or thermal expansivity, etc.) are discontinuous. These transformations are second-order transitions and are clearly softer. 

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. 

For example, the heat capacity CP in a second-order Ehrenfest transition might be expected to behave as follows ... 

The transitions between the bottom five phases of Fig. 2 may occur close to equilibrium and can be described as thermodynamic first order transitions (Ehrenfest definition 17)). The transitions to and from the glassy states are limited to the corresponding pairs of mobile and solid phases. In a given time frame, they approach a second order transition (no heat or entropy of transition, but a jump in heat capacity, see Fig. 1). 

In case of second order transformation (Fig. 3 b), where the G(P, T) surfaces are in contact over a range of P and T the entropy changes continuously across the transformation. This transformation must obey the well-known Ehrenfest equations... 

According to Ehrenfest, a second-order transition occurs when p and its first derivatives are continuous across the transition region, but the second derivatives, Cp and compressibility, k are discontinuous. This behavior is illustrated in the second column of Figure 13.1. For these transitions, the enthalpy, in addition to the entropy and volume, is continuous across the transition. 

The transitions between phases discussed in Section 10.1 are classed as first-order transitions. Ehrenfest [25] pointed out the possibility of higher-order transitions, so that second-order transitions would be those transitions for which both the Gibbs energy and its first partial derivatives would be continuous at a transition point, but the second partial derivatives would be discontinuous. Under such conditions the entropy and volume would be continuous. However, the heat capacity at constant pressure, the coefficient of expansion, and the coefficient of compressibility would be discontinuous. If we consider two systems, on either side of the transition point but infinitesimally close to it, then the molar entropies of the two systems must be equal. Also, the change of the molar entropies must be the same for a change of temperature or pressure. If we designate the two systems by a prime and a double prime, we have... 

Ehrenfest s concept of the discontinuities at the transition point was that the discontinuities were finite, similar to the discontinuities in the entropy and volume for first-order transitions. Only one second-order transition, that of superconductors in zero magnetic field, has been found which is of this type. The others, such as the transition between liquid helium-I and liquid helium-II, the Curie point, the order-disorder transition in some alloys, and transition in certain crystals due to rotational phenomena all have discontinuities that are large and may be infinite. Such discontinuities are particularly evident in the behavior of the heat capacity at constant pressure in the region of the transition temperature. The curve of the heat capacity as a function of the temperature has the general form of the Greek letter lambda and, hence, the points are called lambda points. Except for liquid helium, the effect of pressure on the transition temperature is very small. The behavior of systems at these second-order transitions is not completely known, and further thermodynamic treatment must be based on molecular and statistical concepts. These concepts are beyond the scope of this book, and no further discussion of second-order transitions is given. 

The derivation of quadratic response proceeds analogous to the linear response case. Thus, we collect second-order terms from the BCH expansion of the Ehrenfest theorem as given in Eq. (74) and Fourier transform the resulting expression. In matrix form, this leads to... 

In contradiction to the melting point, the glass-rubber transition temperature is not a thermodynamic transition point. It shows some resemblance, however, to a second order transition. For a second-order transition, the following relationships derived by Ehrenfest (1933) hold ... 

The first theoretician of the vitrification process was Simon (1930), who pointed out that it can be interpreted as a "freezing-in" process. Simon measured specific heats and entropies of glycerol in the liquid, crystalline and glassy state below Tg the entropy of the supercooled liquid could, as a matter of fact, only be estimated. Linear extrapolation would lead to a negative entropy at zero temperature (paradox of Kauzmann, 1948) which would be in contradiction with Nernst s theorem. So one has to assume a sharp change in the slope of the entropy, which suggested a second order transition as defined by Ehrenfest. 

Ehrenfest s classification (see [11]) into first-order and second-order transitions is based on thermodynamic criteria. First-order transitions have discontinuities in the first derivatives of the Gibbs energy with respect to temperature (= entropy) and... 

Apart from temperature, hydrostatic pressure is the other intensive thermodynamical parameter that can be modified with high-pressure cells to build (T, P) phase diagrams. By changing the relative distances between the atoms and molecules, the strength of the interactions are modified, thereby modifying the transition temperature or even inducing new phases. The change of the transition temperature as a function of pressure depends whether the transition is continuous (second order) or discontinuous (first order). The Clausius-Clapeyron (dTc/dP) and Ehrenfest (dTc/dP)2 relationships apply to first- and second-order phase transitions, respectively,... 

The thermodynamics of the I-N phase transition has been extensively investigated for resolving the issue concerning the order of the transition. Following the Ehrenfest scheme, a phase transition is classified into a first-order transition or a second-order one, depending upon the observation of finite discontinuities in the first or the second derivatives of the relevant thermodynamic potential at the transition point. An experimental assessment of the order of the I-N transition has turned out to be not a simple task because of the presence of only small discontinuities in enthalpy and specific volume. It follows from high-resolution measurements that I-N transition is weakly first order in nature [85]. 

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