## 1. Introduction

The thermodynamics of ferroelectric phase transitions is an important constituent part of the phenomenological theories of them, as well as the interface dynamics of them. In particular, if we confine the thermodynamics to the equilibrium range, we can say that the Landau-Devonshire theory is a milestone in the process of the development of ferroelectric phase transition theories. This can be found in many classical books such as [1,2]. Many studies centering on it, especially the size-effects and surface-effects of ferroelectric phase transitions, have been carried out. For the reason of simplicity, we just cite a few [3,4]. But we think these are a kind of technical but not fundamental progress.

Why we think so is based on that the Landau-Devonshire theory is confined to the equilibrium range in essene so it can’t deal with the outstanding irreversible phenomonon of first-order ferroelectric phase transitions strictly, which is the „thermal hysteresis“. The Landau-Devonshire theory attributes the phenomenon to a series of metastable states existing around the Curie temperature

This contribution are organized as the follows. In Section 2, we will show the unpleasant consequence caused by the metastable states hypothesis, and the evidence for the non-existence of metastable states, i.e. the logical conflict. Then in Section 3 and 4, we will give the non-equilibrium ( or irreversible ) thermodynamic description of ferroelectric phase transitions, which eliminates the unpleasant consequence caused by the metastable states hypothesis. In Section 5, we will give the non-equilibrium thermodynamic explanation of the irreversibility of ferroelectric phase transitions, i.e. the thermal hysteresis and the domain occurrences in ferroelectrics. At last, in Section 6 we will make some concluding remarks and look forward to some possible developments.

## 2. Limitations of Landau-Devonshire theory and demonstration of new approach

The most outstanding merit of Landau-Devonshire theory is that the Curie temperature and the spontaneous polarization at Curie temperature can be determined simply. However, in the Landau-Devonshire theory, the path of a first-order ferroelectric phase transition is believed to consist of a series of metastable states existing around the Curie temperature. This is too difficult to believe because of the difficulties encounted ( just see the follows)

### 2.1. Unpleasant consequence caused by metastable states hypothesis

Basing on the Landau-Devonshire theory, we make the following inference. Because of the thermal hysteresis, a first-order ferroelectric phase transition must occur at another temperature, which is different from the Curie temperature [5]. The state corresponding to the mentioned temperature ( i.e. actural phase transition temperature ) is a metastable one. Since the unified temperature and spontanous polarization can be said about the metastable state, we neglect the heterogeneity of system actually. In other words, every part of the system, i.e. either the surface or the inner part, is of equal value physically. When the phase transition occurs at the certain temperature, every part of the system absorbs or releases the latent heat simultaneously by a kind of action at a distance. ( The concept arose in the electromagnetism first. Here it maybe a kind of heat transfer. ) Otherwise, the heat transfer in system, with a finite rate, must destroy the homogeneity of system and lead to a non-equilibrium thermodynamic approach. The unpleasant consequence, i.e. the action at a distance should be eliminated and the lifeforce should be bestowed on the non-equilibrium thermodynamic approach.

In fact, a first-order phase transition process is always accompanied with the fundamental characteristics, called the co-existence of phases and the moving interface ( i.e. phase boundary ). The fact reveals that the phase transition at various sites can not occur at the same time. Yet, the phase transition is induced by the external actions ( i.e. absorption or release of latent heat ). It conflicts sharply with the action at a distance.

### 2.2. Evidence for non-existence of metastable states: logical conflict

In the Landau-Devonshire theory, if we neglect the influence of stress, the elastic Gibbs energy

The long-standing, close correlation between analytical dynamics and thermodynamics implies that Equation (1) can be taken as a scleronomic constraint equation

where

In the Landau-Devonshire theory, the scleronomic constraint equation, i.e. Equation (1) is expressed in the form of the power series of

where

Equivalently, imposed on the generalized displacements

So, the possible displacement

where

for certain

How can this difficulty be overcome? An expedient measure adopted by Devonshire is that the metastable states are considered. However, do they really exist?

Because the metastable states are not the equilibrium ones, the relevant thermodynamic variables or functions should be dependent on the time

For the same reason as was mentioned above, Equation (8) can be regarded as a rheonomic constraint on the generalized displacements

In this case, the possible displacements

Comparing Equation (3) with Equation (10), we may find that the possible displacements here are not the same as those in the former case which characterize the metastable states for they satisfy the different constraint equations, respectively. ( In the latter case, the possible displacements are time-dependent, whereas in the former case they are not. ) Yet, the integral of possible displacement

What are the real states among a phase transition process? In fact, both the evolution with time and the spatial heterogeneity need to be considered when the system is out of equilibrium [6-9]. Just as what will be shown in Section 2.3, the real states should be the stationary ones, which do not vary with the time but may be not metastable.

### 2.3. Real path: Existence of stationary states

The real path of a first-order ferroelectric phase transition is believed by us to consist of a series of stationary states. At first, this was conjectured according to the experimental results, then was demonstrated reliable with the aid of non-equilibrium variational principles.

Because in the experiments the ferroelectric phase transitions are often achieved by the quasi-static heating or cooling, we conjetured that they are stationary states processes [8]. The results on the motion of interface in ferroelectrics and antiferroelectrics support our opinion [10-12]. From Figure 2, we may find that the motion of interface is jerky especially when the average velocity _{3} are alike [11]. This reveals that in these segments of time ( i.e. characteristic time of phase transition ) the stationary distributions of temperature, heat flux, stress, etc. may be established. Otherwise, if the motion of interface is continuous and smooth, with the unceasing moving of interface ( where the temperature is

The non-equilibrium variational principles are just the analogue and generalization of the variational principles in analytical dynamics. The principle of least dissipation of energy, the Gauss’s principle of least constraint and the Hamiltonian principle etc., in non-equilibrium thermodynamics play the fundamental roles as those in analytical dynamics. They describe the characteristics of stationary states or determine the real path of a non-equilibrium process.

For the basic characteristics of non-equilibrium processes is the dissipation of energy, the dissipation function

where

where

where

Both the real paths in the two cases reveal that the deviations decrease exponentially when the system regresses to the stationary states. Stationary states are a king of attractors to non-equilibrium states. The decreases are steep. So the regressions are quick. It should be noted that we are interested in calculating the change in the generalized displacements during a macroscopically small time interval. In other words, we are concerned with the determination of the path of an irreversible process which is described in terms of a finite difference equation. In the limit as the time interval is allowed to approach zero, we obtain the variational equation of thermodynamic path.

So, if the irreversible process is not quick enough, it can be regarded as the one that consists of a series of stationary states. The ferroelectric phase transitions are usually achieved by the quasi-static heating or cooling in the experiments. So, the processes are not quick enough to make the states deviate from the corresponding stationary states in all the time. In Figure 3, three types of regions and their interfaces are marked with I, II, III, 1, 2 respectively. 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.e. the paraelectric-ferroelectric interface as a region with finite thickness instead of a geometrical plane ) where the phase transition is occurring, there exists the external power supply, i.e. the latent heat ( per unit volume and temperature ). 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.

## 3. Thermo-electric coupling

In the paraelectric-ferroelectric interface dynamics induced by the latent heat transfer [6,7], the normal velocity of interface

where

### 3.1. Local entropy production

In the thermo-electric coupling case, the Gibbs equation was given as the following [8]

where

we have

where

where

Then we deduce the following

If we define a entropy flux

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. So,

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

Correspondingly, there are the external electrical potential

If the external electric field is not applied,

According to the crystal structures of ferroelectrics [2], we know the polarization current

### 3.2. Description of phase transitions and verification of interface dynamics

Assume the external electric field is not applied. Here are the thermodynamic fluxex

where

Because to a first-order ferroelectric phase transition the electric displacement changes suddenly and so does the internal electrical potential, the force

According to Equations (30)-(35), we have

If there is no any restriction on

We know the stationary states are equilibrium ones actually. If we let

Then, a first-order ferroelectric phase transition can be described by the second paradigm. Since the force

Considering that

In order to compare it with the experiments, we make use of the following values which are about PbTiO_{3} crystal: ^{3} [16], ^{5}erg/cm﹒s﹒K [18]. The value of the velocity of the interface’s fast motion, which has been measured in the experiments, is 0.5mm/s [11]. According to Equation (39), we calculate the corresponding temperature gradient to be 57.35K/cm. However, in [19] it is reported that the experimental temperature gradient varies from 1.5 to 3.5K/mm while the experimental velocity of the interface’s motion varies from 732 to 843μm/s. Considering the model is rather rough, we may conclude that the theory coincides with the experiments.

### 3.3. Relation between latent heat and spontaneous polarization

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.

All the quantities of the region where the phase transition has occurred are marked with the superscript “I”; all the quantities of the region where the phase transition is occurring are marked with the superscript “II”; and all the quantities of the region where the phase transition will occur are marked with the superscript “III”. Let’s consider the heating processes of phase transition firstly. In the region where the phase transition has occurred,

where we have ignored the difference between the mass density of ferroelectric phase and that of paraelectric phase ( almost the same ) and denote them as

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. So,

According to Eqations (41)-(43), we work out

where the superscript “-1” means reverse. While

where we utilized the boundary condition of

The relation between latent heat and spontaneous polarization are obtained. In the cooling processes of phase transition,

Repeating the above steps, we obtain

Then we find that

## 4. Thermo-electro-mechanical coupling

The comprehensive thermo-electro-mechanical coupling may be found in ferroelectric phase transition processes. Because there exists not only the change of polarization but also the changes of the system’s volume and shape when a ferroelectric phase transition occurs in it, the mechanics can not be ignored even if it is mechanically-free, i.e. no external force is exerted on it. 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.

### 4.1. Deformation mechanics

For a continuum, the momentum equation in differential form can be written as

where

where

we can deduce the following balance equation of mechanical energy basing on Equations (51) (52)

which is in differential form and in Lagrangian form. Or in Eulerian form

where

To a ferroelectric phase transition,

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.

### 4.2. Local entropy production and description of phase transitions

The Gibbs equation was given as the following [9]

where

Make the material derivative of Equation (59) with

where

After the lengthy and troublesome deduction [9], the local entropy balance equation in Lagrangian form can be obtained

with the entropy flux

and the rate of local entropy production

where

Here are the thermodynamic fluxes

where

So the rate of local entropy production can be written as

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

We may describe a ferroelectric phase transition by using the two paradigms above similarly as we have done in Section 3. To a first-order ferroelectric phase transition, the forces

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. For

The region where the phase transition will occur should be in equilibrium because there are no restrictions on the forces

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.

## 5. Irreversibility: Thermal hysteresis and occurrences of domain structure

### 5.1. Thermal hysteresis

The “thermal hysteresis” of first-order ferroelectric phase transitions is an irreversible phenomenon obviously. 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. The system in which a first-order ferroelectric phase transition occurs is heterogeneous. The occurrences of phase transition in different parts are not at the same time. The phase transition occurs at the surface layer then in the inner part of system. According to the description above, we know a constant temperature gradient is kept in the region where the phase transition has occurred. The temperature of surface layer, which is usually regarded as the temperature of the whole system in experiments, must be higher ( or lower ) than the Curie temperature. This may lead to the thermal hysteresis.

No doubt that the shape and the area of surface can greatly affect the above processes. We may conclude that the thermal hysteresis can be reduced if the system has a larger specific surface and, the thermal hysteresis can be neglected if a finite system has an extremely-large specific surface. So, the thermal hysteresis is not an intrinsic property of the system.

The region where the phase transition will occur can be regarded as an equilibrium system for there are no restrictions on the forces

The region where a first-order ferroelectric phase transition will occur is stressed. This reveals that the occurrences of phase transition in the inner part have to overcome the bound of outer part, where the phase transition occurs earlier. This may lead to the delay of phase transition in the inner part.

### 5.2. Occurrences of domain structure

Though the rationalization of the existence of domain structures can be explained by the equilibrium thermodynamics, the evolving characteristics of domain occurrences in ferroelectrics can not be explained by it, but can be explained by the non-equilibrium thermodynamics.

In the region where the phase transition is occurring, the thermodynamic forces

Now, we are facing a set of complicated fields of

There are always several ( at least two ) symmetry equivalent orientations in the prototype phase ( in most cases it is the high temperature phase ), which are the possible orientations for spontaneous polarization ( or spontaneous deformation or spontaneous displacement ). Therefore, the spontaneous polarization, the spontaneous deformation and the spontaneous displacement must take an appropriate orientation respectively to ensure

where

It seems that the picture of domain occurrences for first-order ferroelectric phase transition systems should disappear when we face second-order ferroelectric phase transition systems. This is true if the transition processes proceed infinitely slowly as expounded by the equilibrium thermodynamics. But any actual process proceeds with finite rate, so it is irreversible. Then the above picture revives.

In [21], the domain occurrences in ferromagnetics can be described parallelly by analogy. And the case of ferroelastic domain occurrences is a reduced, simpler one compared with that of ferroelectrics or ferromagnetics.

It is well known that the Landau theory or the Curie principle tells us how to determine the symmetry change at a phase transition. A concise statement is as follows [22]: for a crystal undergoing a phase transition with a space-group symmetry reduction from

After all, for the domain structures can exist in equilibrium systems, they are the equilibrium structures but not the dissipative ones, for the latter can only exist in systems far from equilibrium [24].

## 6. Concluding remarks

In order to overcome the shortcoming of Landau-Devonshire theory, the non-equilibrium thermodynamics was applied to study the ferroelectric phase transitions. The essence of transitions is the thermo-electro-mechanical coupling. Moreover, the irreversibility, namely thermal hysteresis and domain occurrences can be explained well in the realm of non-equilibrium thermodynamics.

The non-equilibrium thermodynamic approach utilized here is the linear thermodynamic one actully. In order to get the more adequate approaches, we should pay attention to the new developments of non-equilibrium thermodynamics. The thermodynamics with internal variables [25] and the extended ( irreversible ) thermodynamics [26] are two current ones. They all expand the fundamental variables spaces to describe the irreversible processes more adequately. Whereas, the relevant theoretical processing must be more complicated undoubtly. This situation needs very much effort.

## Acknowledgments

The work is supported by the Natural Science Foundation Program of Shandong Province, China ( Grant No. ZR2011AM019 ).## References

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