Second-order thermodynamic transition

The Landau theory predicts the symmetry conditions necessary for a transition to be thermodynamically of second order. The order parameter must in this case vary continuously from 0 to 1. The presence of odd-order coefficients in the expansion gives rise to two values of the transitional Gibbs energy that satisfy the equilibrium conditions. This is not consistent with a continuous change in r and thus corresponds to first-order phase transitions. For this reason all odd-order coefficients must be zero. Furthermore, the sign of b must change from positive to negative at the transition temperature. It is customary to express the temperature dependence of b as a linear function of temperature ... 

Figure 4.14 Behavior of thermodynamic variables at Tg for a second-order phase transition (a) volume and fb) coefficient of thermal expansion a and isothermal compressibility p.

The work on iron-nickel alloys has described shock-compression measurements of the compressibility of fee 28.5-at. % Ni Fe that show a well defined, pressure-induced, second-order ferromagnetic to paramagnetic transition. From these measurements, a complete description is obtained of the thermodynamic variables that change at the transition. The results provide a more complete description of the thermodynamic effects of the change in the magnetic interactions with pressure than has been previously available. The work demonstrates how shock compression can be used as an explicit, quantitative tool for the study of pressure sensitive magnetic interactions. 

Note that while the power-law distribution is reminiscent of that observed in equilibrium thermodynamic systems near a second-order phase transition, the mechanism behind it is quite different. Here the critical state is effectively an attractor of the system, and no external fields are involved. 

An extensive treatment of the thermodynamic properties of second-order phase transitions in magnetic crystals has been given by K. P. Belov, Magnetic Transitions, Consultants Bureau, Enterprises, Inc., New York, 1961. 

Thermodynamic arguments [23] indicate that the transition from the normal to the superconducting state at zero field does not involve a latent heat and therefore must be a higher-order transition. Experimental evidence indicates that it is second-order transition. 

A different type of phase transition is known in which there is a discontinuity in the second derivative of free energy. Such transitions are known as second-order transitions. From thermodynamics we know that the change in volume with pressure at constant temperature is the coefficient of compressibility, /3, and the change in volume with temperature at constant pressure is the coefficient of thermal expansion, a. The thermodynamic relationships can be shown as follows ... 

Thermodynamic representation of transitions often represents a challenge. First-order phase transitions are more easily handled numerically than second-order transitions. The enthalpy and entropy of first-order phase transitions can be calculated at any temperature using the heat capacity of the two phases and the enthalpy and entropy of transition at the equilibrium transition temperature. Small pre-tran-sitional contributions to the heat capacity, often observed experimentally, are most often not included in the polynomial representations since the contribution to the... 

Second-order structural transitions are less frequently represented in applied thermodynamic calculations. Still, the Landau approach for determination of... 

When d > 0 the expansion describes a thermodynamic second-order transition. The equilibrium condition neglecting higher order terms is... 

The Greek indices a,j3= II, B,G) count colors, the Latin indices i = u,d,s count flavors. The expansion is presented up to the fourth order in the diquark field operators (related to the gap) assuming the second order phase transition, although at zero temperature the transition might be of the first order, cf. [17], iln is the density of the thermodynamic potential of the normal state. The order parameter squared is D = d s 2 = dn 2 + dG 2 + de 2, dR dc dB for the isoscalar phase (IS), and D = 3 g cfl 2,... 

Both kinetic and thermodynamic approaches have been used to measure and explain the abrupt change in properties as a polymer changes from a glassy to a leathery state. These involve the coefficient of expansion, the compressibility, the index of refraction, and the specific heat values. In the thermodynamic approach used by Gibbs and DiMarzio, the process is considered to be related to conformational entropy changes with temperature and is related to a second-order transition. There is also an abrupt change from the solid crystalline to the liquid state at the first-order transition or melting point Tm. 

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