Diffusivity, phase transitions

Then, there are model Hamiltonians. Effectively a model Hamiltonian includes only some effects, in order to focus on those effects. It is generally simpler than the true full Coulomb Hamiltonian, but is made that way to focus on a particular aspect, be it magnetization, Coulomb interaction, diffusion, phase transitions, etc. A good example is the set of model Hamiltonians used to describe the IETS experiment and (more generally) vibronic and vibrational effects in transport junctions. Special models are also used to deal with chirality in molecular transport junctions [42, 43], as well as optical excitation, Raman excitation [44], spin dynamics, and other aspects that go well beyond the simple transport phenomena associated with these systems. 

In addition, many of the ferroelectric solids are mixed ions systems, or alloys, for which local disorder influences the properties. The effect of disorder is most pronounced in the relaxor ferroelectrics, which show glassy ferroelectric behavior with diffuse phase transition [1]. In this chapter we focus on the effect of local disorder on the ferroelectric solids including the relaxor ferroelectrics. As the means of studying the local structure and dynamics we rely mainly on neutron scattering methods coupled with the real-space pair-density function (PDF) analysis. 

Figure 1.16 (a) Change of the phase transition tempreratures for different dopants and (b) change of a sharp phase transition (BaTiC ) to a diffuse phase transition observed in a broadening of the dielectric peak... 

Figure 1.18 (a) Normalized polarization for first-order, second-order and diffuse phase transition in ferroelectric and relaxor materials and (b) dielectric behavior of relaxor-type... 

G. Helke, The Diffuse Phase Transition, unpublished manuscript, 1992. 

Fig. 7.9 shows the temperature dependence of the dielectric constant and dielectric loss at 1 kHz for the PMN-PT ceramics obtained by sintering the calcined powders from a soft-mechanochemical route at 1200°C for 2 h. A diffuse phase transition, being typical for a relaxor, is observed for each ceramics. As x increases from 0 to 0.2, the maximum dielectric constant, K, , increases from 13000 to 27000. The temperature correspondent to K ,... 

One of the main purposes of developing structural models of porous solids is to predict the effects of confinement on the properties of adsorbed phases, e.g., adsorption isotherms, heats of adsorption, diffusion, phase transitions, and chemical reaction mechanisms. Once a structural model for a particular porous solid has been chosen or developed (see Section 5.3), it is necessary to assume an interaction potential between the solid (adsorbent) and the confined fluid (adsorbate), as well as a fluid-fluid potential, and to decide on a theory or simulation method to calculate the property of interest [58]. A great many such studies have been reported in the literature, particularly for simple pore geometry models, and we do not attempt to review them here. Instead we present a few examples of such stuches, with emphasis on those involving more realistic pore models. 

DIFFUSE PHASE TRANSITIONS. SPECIFIC-HEAT ANOMALY. 

It is seen from Table 1.2 that the features in three upper rows are characteristic for dipole glasses and mixed ferro-glass phases (see Sect. 1.4), while the feature in the lowest row is intrinsic to relaxor ferroelectrics. The width of diffusive Curie region AT varies from AT 373 K for PMN to AT 313 K for completely disordered PST (see Fig. 1.13 [39]). One can see from Fig. 1.13, that AT decreases with increase of the degree of order so that for completely ordered material AT 0. We note here, that Smolenskii, who was the first to synthesize relaxor ferroelectrics [40] named these materials ferroelectrics with diffused phase transition. Their modern (and widely used) name relaxor ferroelectrics is attributed to their relaxation properties in upper row of Table 1.2. 

Li, S., Eastman, J.A., Newnham, R.E., Cross, L.E. Diffuse phase transition in ferroelectrics with mesoscopic heterogeneity Mean-field theory. Phys. Rev. B 55, 12067-12078 (1997)... 

Several authors (Leslie et al., 1961 Scheucher, 1969a,b Murphy and Ball, 1972 Hausmann, 1987) tried to explain this recrystallization behavior by the Avrami and Johnson-Mehl relation. This relation was originally derived by Avrami to explain the kinetics of diffusive phase transitions and later applied by Johnson and Mehl to recrystallization. Accordingly the quantity 1 — R, where R is normalized yield stress, should behave as a function of annealing time like the growth of the volume of a new phase as a function of time. The experimental curve can in fact be fitted by desired relation... 

A2+(bM+)03, a2+(B2+B 5+p A2+(B2+Bf+)03, A(B, B, B 003, or (A, A0(B, BOO3. Among the complex perovskite-type oxides, most of the Pb(B, BOO3-type oxides show a diffuse phase transition such that the transition point is smeared out over a relatively wide temperature range and exhibits a characteristic dielectric relaxation these materials therefore are called relaxors . 

The relaxation (or exponential decay) of the dielectric constant takes place over a range of temperature. One explanation of this DIFFUSE PHASE TRANSITION is that the materials have a structure comprising an assembly of microregions, each with its own Curie temperature, but there is a distribution of Curie temperatures from different individual microvolumes. Thus the transition from ferroelectric to paraelectric behaviour takes place over a wide temperature range, within which some of the... 

AH of these materials have at least two phase transitions that can be described in terms of thermodynamic functions with two ordering parameters (see Appendix B). Ferroic materials are operated near an instabiUty to make domain walls with their associated dipoles and strains moveable, as encountered in PZT or Terfenol . On the other hand, a second type of material involves a partially ordered phase, as in PMN or the shape memory alloys. These materials are operated near a diffuse phase transition with two coexisting phases, a high-temperature austenite-like phase and a low-temperature martensite-Uke phase. A third type of smart... 

Figures 1 and 2 respectively show the temperature dependence of the relative permittivity and loss tangent of relaxor ferroelectric PLZT (9.5/65/35). As the temperature increases from -60°C to 100°C, the relative permittivity generally increased due to the unfreezing of domains. Between 0°C and 10°C, a broad peak can be seen in the lower frequency curves. This peak corresponds to the diffuse phase transition in this relaxor ceramic from the ferroelectric to the paraelectric state (also called the relaxor phase). Further heating continued to increase the relative dielectric permittivity until a maximum was achieved, at which point, the crystal s structure became cubic. This maximum in the permittivity, which is frequency dependent, occurs at the Curie temperature. Evidence of these phase transitions can also be seen in the loss tangent graph in figure 2.

On the other hand, for the KSBN75 film, the dielectric maximum is observed at around 50-70°C, which depends upon the frequency (Sakamoto, 1997). The Curie point of the KSBN75 thin film is a little higher than that for SBN75 single crystals (Huang, 1994). This films also show diffuse phase transition as a relaxor dielectrics. 

The oriented niobate ferroelectric thin film exhibits high transparency and refractive index. Those niobate thinfilms with preferred orientation show diffuse phase transition as a relaxor dielectrics, which is characteristic of single crystals along a specified direction. 

The Pb(Znj/3Nb2/3)03 (PZN) materials are cubic above the transition temperature Tc, 140°C, and undergo a diffuse phase transition to rhombohedral as they are cooled below T(-. PT is cubic above its Curie temperature of 490°C and undergoes a diffuse phase transition to tetragonal as it is cooled below The resulting solid solution of these materials... 

Fig. 44. - Left The dependence of the degree of disorder, Q on temperature, T, for (a) ideal limiting phase transitions of the l and 11" order, (b) diffuse phase transition of the and II " order and (c) an approximate function — the degree of assignment of ZO) ands Z(H). Right the approximation of phase transitions using two stepwise Dirac function designated as L and their thermal change dUdT. (A) Anomalous phase transitions with a stepwise change at the transformation point. To (where the exponent factor n equals unity and the multipfication constant has values -1, -2 and oo). (B) Diffuse phase transition with a continuous change at tlic point, T with the same multiplication constant but having the exponent factor 1/3. ...

It is known that the displacive ferroelectries have quite perceptible dielectric nonlinearity at the temperature range of 50°C 100°C above T [4]. Marked tunability takes place near the phase transition point, which decreases with the increasing temperature. In particular, a wide temperature range of nonlinearity can be obtained in material with a diffused phase transition. 

An effective approach to modulate the Tc is chemical substitution in such films. Moreover, it is also known that adding more kinds of cations to the -site in perovskite structure (ABO3) usually results in diffuse phase transition, which can broaden the transition... 

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