Transition temperature in second order transitions

In first order transitions, siich as melting, there is a discontinuity in the volume-temperature plot (see Fig. 2.19) or enthalpy-temperature plot at the transition temperature. In second-order transitions, only a change in slope occurs and there is thus a marked change in the first derivative or temperature coefficients, as illustrated in Fig. 2.20. The glass transition is not a first-order transition, as no... 

Some selected examples of the variation with pressure of the transition temperatures of second-order transitions are shown in Figure 2.6. 

Second-order phase transitions are those for which the second derivatives of the chemical potential and of Gibbs free energy exhibit discontinuous changes at the transition temperature. During second-order transitions (at constant pressure), there is no latent heat of the phase change, but there is a discontinuity in heat capacity (i.e., heat capacity is different in the two... 

In spite of these uncertainties we can derive some more general results from the above gap equations. With increasing temperature both condensates, 5 and 5, are reduced and eventually vanish in second-order phase transitions at... 

Another important contribution by Landau is related to symmetry changes accompanying phase transitions. In second-order or structural transitions, the symmetry of the crystal changes discontinuously, causing the appearance (or disappearance) of certain symmetry elements, unlike first-order transitions, where there is no relation between the symmetries of the high- and low-temperature phases. If p(x, y, z) describes the probability distribution of atom positions in a crystal, then p would reflect the symmetry group of the crystal. This means that for T> T p must be consistent with... 

Whether the phase transition is first- or second-order depends on the relative magnitudes of the coefficients in the Landau expansion, Eq. 17.2. For a first-order transition, the free energy has a discontinuity in its first derivative, as at the temperature Tm in Fig. 17.1a, and higher-order derivative quantities, such as heat capacity, are unbounded. In second-order transitions, the discontinuity occurs in the second-order derivatives of the free energy, while first derivatives such as entropy and volume are continuous at the transition. 

In second-order or nearly second-order phase transitions, the dielectric dispersion is observed to show a critical slowing-down a phenomenon in which the response of the polarization to a change of the electric field becomes slower as the temperature approaches the Curie point. Critical slowing-down has been observed in the GHz region in several order-disorder ferroelectrics (e.g. Figs. 4.5-8 and 4.5-9) and displacive ferroelectrics (e.g. Fig. 4.5-10). The dielectric constants at the Curie point in the GHz region are very small in order-disorder... 

In rst order transitions, such as melting, there is a discontinuity in the volnme-temperature plot (see Fig. 2.19) or enthalpy-temperature plot at the transition temperatnre. In second-order transitions, only a change in slope occurs and there is thus a marked change in the rst derivative or temperamre coef cients, as illustrated in Fig. 2.20. The glass transition is not a rst-order ttansi-tion, as no discontinuities are observed at Tg when the speci c volume or enfropy of the polymer is measured as a function of temperature. However, the rst derivative of the property-temperature curve, i.e., the temperamre coef cient of the property (e.g., heat capacity and volnmeUic coef -cient of expansion), exhibits a marked change in the vicinity of Tg for this reason it is sometimes called a second-order transition. 

In second order transitions, the first derivatives of the Gibbs energy are also continuous but higher derivatives are discontinuous. Thus the coefficient of expansion and heat capacity are discorrtinuous at these transitions. This situation is illustrated in Figure 1.3. However it is common to observe a comparatively rapid increase in enthalpy over a certain temperature range in the vicinity of a second order transition, as shown in Figitre 1.4. 

Figure B3.6.3. Sketch of the coarse-grained description of a binary blend in contact with a wall, (a) Composition profile at the wall, (b) Effective interaction g(l) between the interface and the wall. The different potentials correspond to complete wettmg, a first-order wetting transition and the non-wet state (from above to below). In case of a second-order transition there is no double-well structure close to the transition, but g(l) exhibits a single minimum which moves to larger distances as the wetting transition temperature is approached from below, (c) Temperature dependence of the thickness / of the enriclnnent layer at the wall. The jump of the layer thickness indicates a first-order wetting transition. In the case of a conthuious transition the layer thickness would diverge continuously upon approaching from below.

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