So, the processes are not quick enough to make the states deviate from the corresponding stationary states in all the time. The region III where the phase transition will occur is in equilibrium and has no dissipation. In the region I where the phase transition has occurred, there is no external power supply, and in the region II i.
According to the former analysis in two cases, we may conclude that they are in stationary states except for the very narrow intervals of time after the sudden lose of phase stability. In the paraelectric-ferroelectric interface dynamics induced by the latent heat transfer [ 6 , 7 ], the normal velocity of interface v n was obtained.
The temperature gradients can be studied from the point of view that a ferroelectric phase transition is a stationary, thermo-electric coupled transport process [ 8 ]. In the thermo-electric coupling case, the Gibbs equation was given as the following [ 8 ]. And there, it was assumed that the crystal system is mechanically-free i. Differentiating Equation 15 and using the following relations. J u should consist of three parts: the energy flux caused by the heat conduction, the energy flux caused by the charge transport and the energy flux caused by the matter transport. Equation 22 can be written as.
This is the local entropy balance equation. We know, the system is in the crystalline states before and after a phase transition so that there is no diffusion of any kind of particles in the system. The local entropy production can be reduced as. We know the existence of ferroics is due to the molecular field. It is an internal field. So we must take it into account. Here, the electric field should be the sum of the external electric field E e and the internal electric field E i.
There is no harm in letting the constant equal zero. Then the entropy production equals.
According to the crystal structures of ferroelectrics [ 2 ], we know the polarization current J P originates from the displacement or ordering of ions in ferroelectrics. We may consider it as the transport of charges influenced by the internal electric field. Assume the external electric field is not applied. They should satisfy the generalized Onsager relations [ 14 ]. Because to a first-order ferroelectric phase transition the electric displacement changes suddenly and so does the internal electrical potential, the force X P of the region where the phase transition is occurring can be regarded as a large constant roughly in the characteristic times of phase transition i.
For the ferroelectric phase transition may be regarded as a stationary states process, the principle of minimum entropy production must be satisfied [ 15 ]. According to Equations 30 - 35 , we have. If there is no any restriction on X q and X P , according to the conditions on which the entropy production is a minimum.
We know the stationary states are equilibrium ones actually. Then, a first-order ferroelectric phase transition can be described by the second paradigm. This states clearly that the pure heat conduction and the heat conduction induced by the thermo-electric coupling cancel out each other so as to release or absorb the latent heat. It is certain that the latent heat passes through the region where the phase transition has occurred at the outside of the region where the phase transition is occurring and exchanges itself with the thermal bath.
This states clearly that the electric displacement of the region where the phase transition has occurred will not change but keep the value at Curie temperature or zero until the phase transition finishes. Differently, the region where the phase transition will occur should be described by the first paradigm for there is no restriction on the two forces X P , X q.
The states of this region are equilibrium ones. According to Equation 39 , we calculate the corresponding temperature gradient to be However, in [ 19 ] it is reported that the experimental temperature gradient varies from 1. Considering the model is rather rough, we may conclude that the theory coincides with the experiments. In the experiments, the latent heat and the spotaneous polarization are measured often for first-order ferroelectric phase transitions.
So in the follows, we will establish the relation between latent heat and spontaneous polarization in the realm of non-equilibrium thermodynamics. In the region where the phase transition has occurred,.
Introduction to the Phenomenological Theory of Ferroelectricity (Monographs in Natural Philosophy)
In the region where the phase transition is occurring,. The heat which is transferred to the region where the phase transition is occurring is absorbed as the latent heat because the pure heat conduction and the heat conduction induced by the thermo-electric coupling cancel out each other. According to Eqations 41 - 43 , we work out. The relation between latent heat and spontaneous polarization are obtained. In the cooling processes of phase transition,. The comprehensive thermo-electro-mechanical coupling may be found in ferroelectric phase transition processes.
To a first-order ferroelectric phase transition, it occurs at the surface layer of system firstly, then in the inner part. So, the stress may be found in the system. Since one aspect of the nature of ferroelectric phase transitions is the thermo-electro-mechanical coupling, we take the mechanics into account on the basis of Section 3, where only the thermo-electric coupling has been considered. This may lead to a complete description in the sense of continuum physics. In terms of. Generally, they are the sums of real and nominal volume force or stress.
The nominal volume force and stress are not zero until the eigen or free deformation of system finishes in phase transitions. If they are zero, the eigen or free deformation finishes. The Gibbs equation was given as the following [ 9 ]. If the external electric field is not applied, the quantity E is the internal electric field E i only [ 8 ]. Make the material derivative of Equation 59 with t , then obtain.
So, n i as the local molar quantity per unit mass does not change with t , i. After the lengthy and troublesome deduction [ 9 ], the local entropy balance equation in Lagrangian form can be obtained. According to the condition on which the local entropy production is a minimum, from Equation 74 we can deduce the following. This reveals that if the k forces among those are kept constant, i. We may describe a ferroelectric phase transition by using the two paradigms above similarly as we have done in Section 3. This states clearly that the pure heat conduction and the heat conduction induced by the thermo-electric coupling and the thermo-mechanical coupling cancel out each other so as to release or absorb the latent heat.
The phase transition occurs at the surface layer firstly, which is mechanically-free. The region where the phase transition is occurring, i. This has been predicted and described with a propagating stress wave [ 20 ]. It is certain that the latent heat passes through the region where the phase transition has occurred at the outside of the region where the phase transition is occurring and exchange itself with the thermal bath.
According to [ 20 ], we know the region where the phase transition has occurred is unstressed, i. This reveals that the eigen or free strain or deformation of system induced by the thermo-electro-mechanical coupling of phase transition is complete and the change of it terminates before the phase transition finishes. The two deductions coincide with each other. Whereas, according to [ 20 ], the region is stressed, i. To the heating process of phase transition, this may lead to a change of the spontaneous polarization of this region because of the electro-mechanical coupling piezoelectric effect.
An immediate result of the above irreversible thermodynamic description is that the action at a distance, which is the kind of heat transfer at phase transitions, is removed absolutely. The latent heat is transferred within a finite time so the occurrence of phase transition in the inner part is delayed. Of course, another cause is the stress, just see Section 5 In other words, the various parts absorb or release the latent heat at the various times.
The action at a distance does not affect the phase transition necessarily. But it was treated by using the equilibrium thermodynamics for ferroelectric phase transitions, the well-known Landau-Devonshire theory [ 2 ]. So, there is an inherent contradiction in this case.
Viscosity, electric relaxation and thermal conduction remain described by Eq. The material coefficients are introduced by performing an expansion of the free energy. With the above and in the absence of any plastic deformation, an electric version of the celebrated Drucker inequality can be obtained in the form. Proceeding as in Bassiouny et al. One gets:. Several examples of the loading functions, the influence of stresses on these functions via electromechanical couplings and both isotropic and kinematical hardening have been studied by Bassiouny et al.
One dimensional model for poling of ceramics: As an application of the model presented in the previous sections we consider the case where the vectorial internal variable has the physical interpretation of a polarization density defined by is a unit vector in the direction of the poling field, P i is an elementary electric dipole and the summation is carried out over each subpart of the sample in comparison with Chen, , Chen and Montgomery, ; Bassiouny et al. The norm of the vector N gives us an indication to the number of the aligned dipoles.
Accounting for Eq. As the constitutive Eq. These equations constitute a fully coupled, three-dimensional thermodynamical set of equations describing the poling of ferroelectric ceramics. As in Eq. Consequently, the starting point for the experimental study of ferroelectric is the preparation of single crystals which are both single domains electrically and untwined crystallographically. Thus it seems natural that we seek certain simplifications when need arises and at the same time not destroy the essence of our present approach.
The purpose of this study is to consider the simplest non trivial constitutive relations and rate laws which are sufficient to yield results comparable to experimental observations. To shed some light and be more specific, let us consider the following one-dimensional form of the free energy. Considerations shall restrict to a fairly thin specimen which is large in lateral extent and whose surfaces are stress free.
Phase transitions and soft modes in ferroelectric superlattices
Expanding Eq. Now turn to the problem of determining a rate law governing the internal variable N, For this purpose, we turn to Eq. Accounting to the fact that the electric internal variable is nothing but the residual polarization up to a constant Eq. Equation 24 - 27 and Eq. Determination of the material coefficients: All the material coefficients introduced in the modeling are defined at two different states of the sample, the virgin state and the poled state. It is assumed that the virgin state corresponds to a natural reference state in which the material behaves isotropically.
Mathematically, a virgin, or thermally depoled specimen is one for which and. The vector can be measured experimentally. In this manner sets of material coefficients obtain in both one-and three-dimensional form for the changes due to the poling process.
For instance, for the effect of poling on pyroelectricity we obtain. Equation 31 can be useful in the observation of domain switching through the pyroelectric measurement, by determining the rate of variation of the reversible polarization with temperature in the poled state Lines and Glass, Subscribe Today. Science Alert. All Rights Reserved.
Research Article. Poling of Ferroelectric Ceramics. Similar Articles in this Journal. Search in Google Scholar. How to cite this article: E. Bassiouny , Journal of Applied Sciences, 6: DOI: Bassiouny, E. Damage mechanics in ferroelectric ceramics-model and application. Applied Sci. An brand 's specific for this high-resolution.
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