Free energy ferroelectric

Since P must remain normal to z and n, the polarization vector forms a helix, where P is everywhere normal to the helix axis. While locally a macroscopic dipole is present, globally this polarization averages to zero due to the presence of the SmC helix. Such a structure is sometimes termed a helical antiferroelectric. But, even with a helix of infinite pitch (i.e., no helix), which can happen in the SmC phase, bulk samples of SmC material still are not ferroelectric. A ferroelectric material must possess at least two degenerate states, or orientations of the polarization, which exist in distinct free-energy wells, and which can be interconverted by application of an electric field. In the case of a bulk SmC material with infinite pitch, all orientations of the director on the tilt cone are degenerate. In this case the polarization would simply line up parallel to an applied field oriented along any axis in the smectic layer plane, with no wells or barriers (and no hysteresis) associated with the reorientation of the polarization. While interesting, such behavior is not that of a true ferroelectric. 

It is now instructive to ask why the achiral calamitic SmC a (or SmC) is not antiferroelectric. Cladis and Brand propose a possible ferroelectric state of such a phase in which the tails on both sides of the core tilt in the same direction, with the cores along the layer normal. Empirically this type of conformational ferroelectric minimum on the free-energy hypersurface does not exist in known calamitic LCs. Another type of ferroelectric structure deriving from the SmCA is indicated in Figure 8.13. Suppose the calamitic molecules in the phase were able to bend in the middle to a collective free-energy minimum structure with C2v symmetry. In this ferroelectric state the polar axis is in the plane of the page. 

The free energy density terms introduced so far are all used in the description of the smectic phases made by rod-like molecules, the electrostatic term (6) being characteristic for the ferroelectric liquid crystals made of chiral rod-like molecules. To describe phases made by bent-core molecules one has to add symmetry allowed terms which include the divergence of the polar director (polarization splay) and coupling of the polar director to the nematic director and the smectic layer normal ... 

Figure 1.8 Free energy of a ferroelectric with a second-order phase transition (left) and with a first-order phase transition (right) at different temperatures. Tc is the phase transition temperature and is the Curie temperature. ...

Fp, as observed. At lower T, non-linearity in the free energy functional is expected to favor sharp domain walls [21], The transition to the monodomain ferroelectric phase F1 has been predicted by treating the ferroelectric as a semiconductor, in which carriers can be created by the field effect [16]. This transition can be calculated to occur at Tc - T = 325 K for d = 10 nm [4], in qualitative agreement with our observations. The rich phase diagram we have observed in this simple system makes it an excellent test case for more quantitative development of these concepts. 

The study of ferroelectrics has been greatly assisted by so-called phenomenological theories which use thermodynamic principles to describe observed behaviour in terms of changes in free-energy functions with temperature. Such theories have nothing to say about mechanisms but they provide an invaluable framework around which mechanistic theories can be constructed. A.F. Devonshire was responsible for much of this development between 1949 and 1954 at Bristol University. 

Recently, a quantitative model for the size-dependent Curie temperature has been established based on mechanic and thermodynamic considerations using the Landau-Ginsburg-Devonshire (LGD) phenomenological theory by equalizing free energies of the ferroelectrics and paraelectrics phases [1-4],... 

A more complete description of smectic A needs to take into account the compressibility of the layers, though, of course, the elastic constant for compression may be expected to be quite large. The basic ideas of this model were put forward by de Gennes. > We consider an idealized structure which has negligible positional correlation within each smectic layer and which is optically uniaxial and non-ferroelectric. For small displacements u of the layers normal to their planes, the free energy density in the presence of a magnetic field along z, the layer normal, takes the form... 

With d being different from a" for both polymers XII and XIII, the relationship between P and 0 is nonlinear. Such behavior is typical of ferroelectric liquid crystal materials with high P, and can be explained on the basis of the generalized Landau model for the free energy density. A complete treatment is available for polymers XII and XIII and the different calculated coefficients. 

The example substance here may be SrTiOs, which is widely known as incipient ferroelectric with ferroelastic phase transition at Tfree energy difference of its 90°-domains is proportional to (833 — that corre-... 

It could be expected that the presence of built-in electric field in incipient ferroelectric films generates the ferroelectric phase in them despite the fact that latter phase is absent in corresponding bulk samples up to zero Kelvins. The ferroelectric phase has actually been observed in SrTiOs thin films (see [17] and references therein). However the theoretical description in Ref. [17] is laeking consideration of the surface effects and depolarization field, which are extremely important for nanomaterials description [18]. All these factors have been taken into account in Ref. [11] in the framework of aforementioned theory. Namely, it was taken into account there, that as low temperature region plays an important role in the properties of incipient ferroelectrics, the parameter a T) in the free energy (3.5a) has to be written with respect to Barrett formula [19] a T) = quantum vibration temperature. It is seen that at... 

To estimate the temperature evolution of the entire domain structure we express its energy at T = 0 (i.e. at f = —1) in the Kittel approximation, assuming a kink-like structure of the domain wall and a flat profile of a polarization inside domains. The calculations are similar to those for the domain structure of the uniaxial ferroelectric film surrounded by paraelectric dead layers and embedded into short circuited capacitor [27]. The free energy is superimposed from the domain walls energy and electrostatic contributions ... 

The free energy functional for antiferroelectric films can be written similarly to that of ferroelectrics. Namely, such functional should incorporate the bulk AGV and the surface AGs parts. For the monodomain antiferroelectric films of thickness / (—1/2 < z < 1/2) having sublattices a and b with polarizations, respectively, Pa(z) and Pb(z), the free energies AGV and AGs can be written in the form... 

Fig. 3.21 Free energy dependence on electric field. Solid curves correspond to antiferroelectric phase, dotted ones to paiaelectric phase and dashed ones to the regions of ferroelectric phase stability. The parameters values = 0.9, Pp/t) =0.9, g/p,2A = 0.1, /8 = — 1, /fr = 3, / /, = 0.5, X = 1, h = 3, 10, 30, 100 (curves 1, 2, 3, 4) lattice constants, temperatures T/Ta = 0.25 (a) and 0.5 (b) [70]...

Since all physical quantities are determined primarily by the renormalization of the coefficient Ar in the free energy (the other coefficients are approximately similar to those in the bulk ferroelectrics [93]), its minimization gives following re-lations between observable properties and this coefficient P(T, / ) ... 

The results of specific heat measurements were reported earlier in the Sect. 2.2.1.4 (Chap. 2) for nanogranular BaTiOj ceramics. As in this case the ferroelectric phase transition is of the first kind, the Td R) in Eq. (3.85) is a boundary of paraelectric phase stability, while the real transition temperature is shifted from it by some R-independent value [93]. With respect to this statement it is possible to write specific heat Cp on the base of Eq. (3.85) with the only /(-dependent free energy coefficient Ar T). Since Cp = -7 (d /dr )(4> is a free energy) one can find, that the difference between specific heat in ferroelectric and paraelectric phases reads ... 

Glinchuk, M.D., Eliseev, E.A., Stephanovich, V.A., FarW, R. Ferroelectric thin film properties - depolarization field and renormalization of a bulk free energy coefficients. J. Appl. Phys. 93, 1150-1159 (2003)... 

It is rather important that the FME effect (4.56b) can induce the spontaneous polarization (i.e. improper ferroelectricity) in the antiferromagnets. Equation of state for polarization vector follows from the variation of the free energy (4.33) 8Fp/8P, — 0. Variation of the Eq. (4.56) leads to the built-in field appearance in the right-hand side of the equation for polarization... 

We now consider the properties potential barrier of polarization reorientation for uniaxial and perovskite ferroelectric spherical nanoparticles. For uniaxial ferro-electrics the barrier between the states Po can be estimated on the basis of the free energy ... 

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