Linear transformations, equilibrium phase

The cross products in T and V may be eliminated by a linear transformation to a set of new (normal) co-ordinates Q in which each nucleus executes a simple harmonic motion about its equilibrium position with frequency vt and in phase with all other nuclei, v is determined from ... 

In the field-off state the macroscopic polarization of the antiferroelectric phase is zero. With increasing field, the induced polarization, at first, increases linearly with field and then, at a certain threshold, the antiferroelectric (AF) structure with alternating molecular tilt transforms in the ferroelectric one (F) with a uniform tilt, see Fig. 13.24a. Correspondingly, the macroscopic polarization jumps from a low value to the level of the local polarization Po [34]. The process is quite similar to that observed in crystalline antiferroelectrics. With a certain precaution we can speak about a field-induced AF-F non-equilibrium phase transition . The magnitude of the switched polarization in some antiferroelectric materials can be quite... 

This equation gives the dynamics of the quantum-classical system in terms of phase space variables (R, P) for the bath and the Wigner transform variables (r,p) for the quantum subsystem. This equation cannot be simulated easily but can be used when a representation in a discrete basis is not appropriate. It is easy to recover a classical description of the entire system by expanding the potential energy terms in a Taylor series to linear order in r. Such classical approximations, in conjunction with quantum equilibrium sampling, are often used to estimate quantum correlation functions and expectation values. Classical evolution in this full Wigner representation is exact for harmonic systems since the Taylor expansion truncates. 

Nonlinear optimization techniques have been applied to determine isotherm parameters. It is well known (Ncibi, 2008) that the use of linear expressions, obtained by transformation of nonlinear one, distorts the experimental error by creating an inherent error estimation problem. In fact, the linear analysis method assumes that (i) the scatter of points follows a Gaussian distribution and (ii) the error distribution is the same at every value of the equilibrium liquid-phase concentration. Such behavior is not exhibited by equilibrium isotherm models since they have nonlinear shape for this reason the error distribution gets altered after transforming the data... 

Above relation (1) between cr and y is exact in linear response, where nonlinear contributions in 7 are neglected in the stress. The linear response modulus (to be denoted as g (f)) itself is defined in the quiescent system and describes the small shear-stress fluctuations always present in thermal equilibrium [1, 3]. Often, oscillatory deformations at fixed frequency co are applied and the frequency dependent storage- (G (m)) and loss- (G"((u)) shear moduli are measured in or out of phase, respectively. The former captures elastic while the latter captures dissipative contributions. Both moduli result from Fourier-transformations of the linear response shear modulus g (f), and are thus connected via Kramers-Kronig relations. 

Douglas and Victor [58DOU/V1C] found that, at constant temperature, the heat content was a linear function of the hydrogen content of the hydride at temperatures below 823 K. Similarly, Yamanaka et al. [2001YAM/YAM] found that the heat capacity was also linearly dependent on the hydrogen content. To temperatures between 773 and 823 K, [58DOU/V1C] indicated that the zirconium hydrides used in their experiments was a mixture of two phases. This mixture transformed to a third phase above about 823 K, however, a considerable lack of phase equilibrium was observed at temperatures from the transition temperature to 1173 K. Conversely, [2001 YAM/YAM] found only a single phase in their experiments (8-zirconium hydride) and that the heat capacity of the phase studied in their experiments could be described by ... 

Since the water content of a hydrated mixture in equilibrium conditions is a linear function of the water content of the single hydrated phases, thermal analysis, more specifically TG, provides a good opportunity to resolve hydrated mixtures, provided the single components, the water content of which should be known, display sufficiently distinct dehydration ranges. Unfortunately, this circumstance is very uncommon, since the mixture components normally present wide overlaps in the thermal transformation effects, which prevents an easy resolution of the TG traces. 

As we discussed in Section 6.2, a thermodynamic driving force, quantified by AG, must be present in order for a phase transformation to occur. In general, the magnitude of this driving force increases linearly with the degree of deviation from equilibrium, as expressed, for example, via... 

In T.MDSC, a perturbation in the form of an oscillating sine wave of known frequency is applied to the linear temperature control program. This variation can be applied in principle to heat-flux DSC and to power compensation DSC instruments. The thermal response is analyzed using Fourier transformation, with the component in-phase with the temperature oscillation thought to be caused by reversible or equilibrium changes in the sample, and the out-of-phase component associated with non-reversible changes. 

Initially the pseudo-elastic material is in its austenitic phase at room temperature. Initially the material in the austenitic phase deforms like a conventional material linear elastic under load. With increasing loads a stress-induced transformation of the austenitic to the martensitic phase is initiated at the pseudo-yield stress Rpe- This transformation is accompanied with large reversible strains at nearly constant stresses, resulting in a stress plateau shown in Fig. 6.53. At the end of the stress plateau the sample is completely transformed into martensite. Additional loading passing the upper stress plateau causes a conventional elastic and subsequently plastic deformation of the martensitic material. If the load is decreased within the plateau and the stress reaches the lower stress level a reverse transformation from martensite to austenite occurs. Since the strains are fully reversible the material and the sample respectively is completely recovered to its underformed shape. These strains are often called pseudo-elastic because the reversible deformation is caused by a reversible phase transformation and is not only due to a translation of atoms out of their former equilibrium position [74]. 

Phase and chemical equilibrium calculations are essential for the design of processes involving chemical transformations. Even in the case of reactions that cannot reach chemical equilibrium, the solution of this problem gives information on the expected behaviour of the system and the potential thermodynamic limitations. There are several problems in which the simultaneous calculation of chemical and phase behaviour is mandatory. This is the case, for example, of reactive distillations where phase separation is used to shift chemical equilibrium. Also, the calculation of gas and solid solubility in liquids of high dielectric constants requires at times the resolution of chemical equilibrium between the different species that are formed in the liquid phase. Several algorithms have been proposed in the literature to solve the complex non-linear problem however, proper thermodynamic model selection has not received much attention. 

---