Second-order disorder transition

Order-disorder transitions Different atoms that statistically occupy the same position become ordered or vice versa. Usually this is a second-order transition. 

Fig. 35 Phase diagrams AB miktoarm-star copolymers for n = 2, n = 3, n = 4 and n = 5. mean-field critical point through which system can transition from disordered state to Lam phase via continuous, second-order phase transition. All other phase transitions are first-order. From [112]. Copyright 2004 American Chemical Society...

In the paramagnetic regime, the evolution of the EPR line width and g value show the presence of two transitions, observed at 142 and 61 K in the Mo salt, and at 222 and 46 K in the W salt. Based on detailed X-ray diffraction experiments performed on the Mo salt, the high temperature transition has been attributed to a structural second-order phase transition to a triclinic unit cell with apparition of a superstructure with a modulation vector q = (0,1/2, 1/2). Because of a twinning of the crystals at this transition, it has not been possible to determine the microscopic features of the transition, which is probably associated to an ordering of the anions, which are disordered at room temperature, an original feature for such centrosymmetric anions. This superstructure remains present down to the Neel... 

The Inden model [20] is frequently used to describe second-order magnetic order-disorder transitions. Inden assumed that the heat capacity varied as a logarithmic function of temperature and used separate expressions above and below the magnetic order-disorder transition temperature (TtIS) in order to treat the effects of both long- and short-range order. Thus for z = (T/TtIS) < 1 ... 

Choice of initial condition. To avoid the slow growth of ordered domains out of disordered initial states and to avoid that the system gets trapped in metastable states ", it is preferable to start the simulation in the appropriate perfectly ordered configuration. The various perfectly ordered states that a model can have are usually found from an analysis of the ground state which may be tedious but in most cases is rather straightforward (for examples, see Refs. 20,22,66). If for the chosen system parameters (temperature T, chemical potential /i) the system is in a disordered phase, the system will relax towards this state smoothly, even if it started out fully ordered, if the order-disorder transition is of second order. In the case of... 

The compounds BajInjOj, Agl and PbFj illustrate ionic conductivity in stoichiometric compounds. The first is a fast oxide-ion conductor above a first-order order-disorder transition at 930 °C that leaves the Bain array unchanged the second is a fast Ag -ion conductor above a first-order transition at which the I -ion array changes from close-packed to body-centred cubic and the third exhibits a smooth transition to a fast F ion conductor without changing the face-centred-cubic array of Pb " ions. 

Coefficients a, B, and C in equation 5.175 have the usual meanings in the Landau expansion (see section 2.8.1) and for the (second-order) displacive transition of albite assume the values = 1.309 cal/(mole X K) and B, = 1.638 kcal/mole (Salje et al., 1985). is the critical temperature of transition = Bla = 1251 K). The corresponding coefficients of the ordering process are = 9.947 cal/(mole X K), B = -2.233 kcal/mole, = 10.42 kcal/(mole X K), and = 824.1 K. With all three coefficients being present in the Landau expansion relative to substitutional disorder it is obvious that Salje et al. (1985) consider this transition first-order. A is a T-dependent coupling coefficient between displacive and substitutional energy terms (Salje et al., 1985) ... 

A second-order phase transition is one in which the enthalpy and first derivatives are continuous, but the second derivatives are discontinuous. The Cp versus T curve is often shaped like the Greek letter X. Hence, these transitions are also called -transitions (Figure 2-15b Thompson and Perkins, 1981). The structure change is minor in second-order phase transitions, such as the rotation of bonds and order-disorder of some ions. Examples include melt to glass transition, X-transition in fayalite, and magnetic transitions. Second-order phase transitions often do not require nucleation and are rapid. On some characteristics, these transitions may be viewed as a homogeneous reaction or many simultaneous homogeneous reactions. 

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. 

These hole-rich droplets represent a ferromagnetic second phase having a 7c < 7n the volume fraction of this minority phase grows with increasing x, and in the range 0.1 < x < 0.13 the droplets are condensed into large clusters having 7c > 7n. The crossover from 7c < 7n to 7c > 7n occurs at about x = 0.10 near a point where the 2D orbital order-disorder transition (or onset of orbitally disordered clusters) at T (same T as in fig. 14) crosses Tn. 

Further X-ray diffraction work revealed a second-order phase transition at 150 K, attributed to an orientational order-disorder transition of the K+-NH3 pair at the octahedral site [36]. The low-temperature orthorhombic structure (space group Fddd) is derived by doubling the lattice constants of the high-tem-perature phase along all three axes, with the K+-NH3 pairs orienting along the [110] direction in an antiferroelectric fashion. 

The third type of phase change of the second order is fortunately easy to treat theoretically, at least to an approximation, and it, is the one which will be discussed in the present chapter. This is what is known as an order-disorder transition in an alloy, and can be better understood in terms of specific examples, which we shall mention in the next section. 

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