Phase transition loop

Figure 17.21 The two basic binary diagram elements. In the phase transition loop (left diagram) solution 1 and solution 2 can be solid and solid, solid and liquid, or liquid and vapor, respectively. and are melting temperatures, boiling temperatures, or polymorphic phase transition temperatures for pure A and B respectively. Three representative tie-lines are shown. In the solvus (right diagram), solution 1 and solution 2 can be two solids or two liquids. Two representative tie-lines are shown.

The Fowler-Guggenheim equation exhibits phase-transition loops when Z(o/RT 4. The values of the coverage limits of the isotherm step and the corresponding step position, p/K, have been tabulated. 

This local isotherm has been discussed at length by de Boer, Ross and Olivier and also Broekhoff and van Dongen. Phase-transition loops occur when 2a/fcT 3 >6.75 and the transition step position (i.e. coverage limits and corresponding pJKi) have been tabulated. ... 

In the microcanonical ensemble, the signature of a first-order phase transition is the appearance of a van der Waals loop m the equation of state, now written as T(E) or P( ). The P( ) curve switches over from one... 

FIG. 4 Qualitative phase diagram close to a first-order irreversible phase transition. The solid line shows the dependence of the coverage of A species ( a) on the partial pressure (Ta). Just at the critical point F2a one has a discontinuity in (dashed line) which indicates coexistence between a reactive state with no large A clusters and an A rich phase (hkely a large A cluster). The dotted fine shows a metastability loop where Fas and F s are the upper and lower spinodal points, respectively. Between F2A and Fas the reactive state is unstable and is displaced by the A rich phase. In contrast, between F s and F2A the reactive state displaces the A rich phase. 

Crystals with one of the ten polar point-group symmetries (Ci, C2, Cs, C2V, C4, C4V, C3, C3v, C(, Cgv) are called polar crystals. They display spontaneous polarization and form a family of ferroelectric materials. The main properties of ferroelectric materials include relatively high dielectric permittivity, ferroelectric-paraelectric phase transition that occurs at a certain temperature called the Curie temperature, piezoelectric effect, pyroelectric effect, nonlinear optic property - the ability to multiply frequencies, ferroelectric hysteresis loop, and electrostrictive, electro-optic and other properties [16, 388],... 

Phase equilibrium requires that A2 = Al and hence that the integral vanish. All conditions are satisfied if the points 1 and 2 are located such that the areas A = B. This geometry defines the Maxwell construction. It shows that stable liquid and vapour states correspond to minima in free energy and that AL = Ay when the external pressure line cuts off equal areas in the loops of the Van der Waals isotherm. At this pressure that corresponds to the saturated vapour pressure, a first-order phase transition occurs. 

Fe(btr)2(NCS)2]-H20 undergoes a complete ST centred at 134 Kwith a hysteresis loop of width 21 K. This derivative represents the first example of a 2D ST compound and has become a model material in SCO research. The presence of a crystallographic phase transition to account for the observed hysteresis was first proposed since crystal cracking was regularly observed when the sample was cooled through the temperature region of the spin transition [59]. Recent X-ray data recorded at 95 K, where the com-... 

A characteristic manifestation of the coexistence of two gel phases and hence of the first-order phase transition in a swollen network consists of the van der Waals loop which appears in the dependence of the swelling pressure P (or of the chemical potential of the solvent plf see Eq. (1)) on 0. The composition of coexisting gel phases at the collapse (values

2) is given by the condition of equality of the chemical potentials of the solvent px and polymer p2 in both phases... 

The NLDFT predicts the critical point for capillary condensation phase transition (capillary critical pore size) at ca. 2 nm, which is approximately the minimum pore size in which capillary condensation is experimentally observed [21,27], However, the theory fails to predict the disappearance of the hysteresis loop for pores smaller than ca. 4 nm (hysteresis critical point) [20,15], It should be noted that the theory of Broekhoff and de Boer fails to predict both critical points unless some additional semi-empirical corrections are made [16]... 

There is a large body of experimental work on ternary systems of the type salt + water + organic cosolvent. In many cases the binary water + organic solvent subsystems show reentrant phase transitions, which means that there is more than one critical point. Well-known examples are closed miscibility loops that possess both a LCST and a UCST. Addition of salts may lead to an expansion or shrinking of these loops, or may even generate a loop in a completely miscible binary mixture. By judicious choice of the salt concentration, one can then achieve very special critical states, where two or even more critical points coincide [90, 160,161]. This leads to very peculiar critical behavior—for example, a doubling of the critical exponent y. We shall not discuss these aspects here in detail, but refer to a comprehensive review of reentrant phase transitions [90], We note, however, that for reentrant phase transitions one has to redefine the reduced temperature T, because near a given critical point the system s behavior is also affected by the existence of the second critical point. An improper treatment of these issues will obscure results on criticality. 

More generally, the dynamic behavior of domain walls in random media under the influence of a periodic external field gives rise to hysteresis cycles of different shape depending on various external parameters. According to a recent theory of Nattermann et al. [54] on disordered ferroic (ferromagnetic or fe) materials, the polarization, P, is expected to display a number of different features as a function of T, frequency, / = iv/2tt, and probing ac field amplitude, E0. They are described by a series of dynamical phase transitions, whose order parameter Q = uj/2h) Pdt reflects the shape of the P vs. E loop. When increasing the ac... 

Fig. 4. The first 1992 Garching frequency chain for the measurement of the IS — 2S transition in atomic hydrogen (0 phase-locked loop, SHG second harmonic generation)...

Phase transitions may be detected by the corresponding anomaly in the temperature dependence of the quadrupole coupling constant. For example, in monomethylamine a phase transition occurs at 80 °K and a hysteresis loop has been observed in the frequency vs. temperature curve with a broadening of the lines just before the transitions 24> This transition has been confirmed by the N.M.R. study of the proton line width and by the dielectric behavior, but no precise explanation has been given. 

---