2 April 2001

Physics Letters A 281 (2001) 357–362www.elsevier.nl/locate/pla

Finite-size effects in dynamics of paraelectric–ferroelectricinterfaces induced by latent heat transfer

Alex GordonDepartment of Mathematics and Physics, Faculty of Sciences and Science Education, University of Haifa at Oranim, 36006 Tivon, Israel

Received 28 November 2000; accepted 1 February 2001Communicated by J. Flouquet

Abstract

A theory of the thermally induced dynamics of interphase boundaries in the case of latent heat transfer is presentedfor confined ferroelectrics. Two distinct types of interface motion, thermally induced, have been observed in experimentin ferroelectric perovskites: (a) slow motion usually governed by the polarization relaxation mechanism and presented bypolarization kink migration, and (b) rapid motion that takes place over short periods of time and has not been sufficientlyclarified. We show that latent heat transfer may cause the observed rapid motion of the interphase boundary during the phasetransition temperature passage. On cooling the heat generated during the phase transition accelerates the interphase boundary.The dependence of the interface velocity on the crystal size is calculated for the latent heat transfer mechanism of the interfacemotion. The theoretical result is in agreement with the experiment according to which the interface velocity decreases withincreasing crystal size. 2001 Elsevier Science B.V. All rights reserved.

PACS: 77.80.-e; 77.80.Bh; 77.90.+kKeywords: Ferroelectrics; Phase transitions; Interfaces; Dynamics

In recent years there has been increasing interestin the rates and mechanisms of interfacial kineticsfor first-order phase transitions [1]. In particular, in-terface motion in ferroelectrics and antiferroelectricshas been the object of extensive experimental and the-oretical studies [1–15]. This topic of current inter-est is important as a means of probing multi-stabledynamic systems and also as a challenge exampleof nonlinear behavior. The appearance of the inter-phase boundaries is usually a result of the first-orderphase transition occurrence. The most common ap-proach to the interphase boundary dynamics involvesthe use of the time-dependent Ginzburg–Landau the-

E-mail address: algor@techunix.technion.ac.il (A. Gordon).

ory for first-order phase transition [16]. The model as-sumes that the interfacial dynamics are entirely con-trolled by the evolution of the order parameter, whichis polarization. It has been shown that the movinginterphase boundary is described by a moving po-larization kink that is an exact solution of the time-dependent Ginzburg–Landau equation [17]. Accord-ing to the model, the paraelectric and ferroelectricphases are in contact with each other at a well-definedboundary. Relaxation of the metastable system to thethermodynamically stable state takes place due tomotion of an interphase boundary. This slow inter-phase boundary dynamics can be described by usingthe above-mentioned model [9,13,17]. Our aim is toshow that the fast motion of the interphase boundary[5,18,19] measured in the experiment on ferroelectric

0375-9601/01/$ – see front matter 2001 Elsevier Science B.V. All rights reserved.PII: S0375-9601(01)00076-7

358 A. Gordon / Physics Letters A 281 (2001) 357–362

perovskites can be due to the latent heat generation atthe phase transition temperature. In addition, we con-sider the size dependence of the interface velocity dur-ing the comparatively fast motion induced by thermalconductivity. Thus, we present here a theory of thethermally induced dynamics of interphase boundariesin confined ferroelectrics.

The moving interphase boundary is a source or sinkof heat, and in general it may thus alter the temperaturedistribution in the region of the crystal within theboundary [20,21]. On cooling and approaching thephase transition temperature the latent heat of phasetransition releases. As a result, a local heating occursand the interphase boundary accelerates. Thus, themoving interface acts as a heat source with strengthproportional to the latent heat of phase transitionand the forward rate of motion. The heat generatedduring the interphase boundary motion accelerates theinterface, which in turn increases the heat productionrate. A system involved in this avalanche-like processis stabilized by heat removal via heat conductivity andheat exchange with the thermal bath. This interfacevelocity is controlled by heat flow. Then the interfacevelocity is determined by the rate at which the latentheat generated during the phase transition is removedfrom the interface. The velocity of the interface causedby the latent heat transfer may be found from thefollowing considerations.

The heat flowing across an interphase boundary is

(1)Q= kdT

dnS∆t,

wherek is the thermal conductivity coefficient,n iscoordinate normal to the interface,dT/dn is the com-ponent of the temperature gradient in then direction,S is the square of the area of the interface and∆t =∆R/vn. ∆R is the change of size of the new phaseduring the interface motion with the velocityvn, whichis the normal velocity of the interface. The above-mentioned heat per unit mass is the latent heatl,

(2)l = kdT

dn

∆V

∆mvn;

∆m/∆V is the densityρ of the metastable phase.Thus, the interface velocity is determined by

(3)vn = k∇T · nρl

.

Eq. (3) is obtained within the framework of theapproach dealing with the internal source of heatappearing in a homogeneous medium. However, thissource arises at the boundary between two phases withtwo different coefficients of thermal conductivity [22].The thermal gradient ahead the interphase boundaryis distinct from that behind the interphase boundary.Thus, the interface velocity is determined by a jumpin the thermal gradient at the interface and by thetemperature anomaly in the thermal conductivity. Inorder to obtain the velocity of the interphase boundarythe continuity condition for heat flow at the interfaceshould be written. Thus, the propagation rate of theinterphase boundary due do the latent heat transport isgiven by

(4)vn =(

1

lρ

)[kfer(∇T )fer − kpar(∇T )par

] · n.

The index “fer” denotes the properties related to theferroelectric phase and the index “par” denotes theproperties related to the paraelectric phase. Takinginto account the equation for latent heatl [16], l =(3/8)TcA′B/C, we obtain

(5)vn = 8C

3TcA′Bρ[kfer(∇T )fer − kpar(∇T )par

] · n.

The above velocity is therefore the rate at which thelatent heat is conducted away. Eq. (5) is also validfor the semi-finite with finite thickness: the interphaseboundary propagates in thex direction, while in theperpendicular direction (z, for example) the samplemay be finite. The sample is infinite in the directionof the interface propagation.

The theory of the interface dynamics due the po-larization relaxation is based on the expansion of thefree energy density in a power series of the polariza-tion [16,17,23]:

(6)F =∫ [

f + K

2(∇P)2

]dV.

f is the free-energy density given by

(7)f (T )= f0(T )+ 1

2AP 2 − 1

4BP 4 + 1

6CP 6,

f0 is the free-energy density for the paraelectric phase.A = A′(T − T0), T0 is the stability limit of the para-electric phase. For coefficientsB > 0 and C > 0,Eq. (7) describes a first-order phase transition.Tc is

A. Gordon / Physics Letters A 281 (2001) 357–362 359

the phase transition temperature in which the minimaof the free energy density of the paraelectric and fer-roelectric phases are at the same depth. The elasticenergy is not represented explicitly in (7). However,we may consider that it has been taken into account,since this only leads to a renormalization of the co-efficients [16]. It should be noted that the interphaseboundary motion has been modeled within the frame-work of the time-dependent Ginzburg–Landau theory,where the spontaneous polarization is coupled to astress wave [9,13,17,23].K is the positive coefficientof the gradient, inhomogeneous term. The correspond-ing equation of motion in thex direction is as follows:

(8)

∂P

∂t+Γ

(AP −BP 3 +CP 5) − ΓK

(∂2P

∂x2

)= 0.

Its steady-state solution for the interface boundaryconditions is known [17,23]. Eq. (8) has solutionshaving the kink form [17,23] in the moving referenceframe (s = x − vt),

(9)P = P0

[1+ exp

(± s

∆

)]−1/2

,

which clearly represents the interphase boundary be-tween the ferroelectric phase(s → −∞,P = P0), ifthe power of the exponent is equal to+s/∆ in (9), andthe paraelectric phase(s → +∞,P = 0). The nega-tive sign in the exponent of Eq. (9) corresponds to thesituation, in which the paraelectric phase is located tothe left and the ferroelectric phase is located to theright.P0 is the equilibrium value of polarization:

(10)P 20 = B

2C

(1+

√1− 4AC

B2

).

The interphase boundary has the width∆ and veloc-ity v:

(11)∆= 1

2

{3K

BP 20 −A

}1/2

,

(12)v =(

2

3

)∆Γ

[4A−BP 2

0

].

The motion of kink (9) caused by the polarizationrelaxation describes the interphase boundary motion,when one of the phases is metastable, while the otherone is stable. Eqs. (10)–(12) are valid not only for the

bulk case but also for the semi-infinite plate with fi-nite thickness because the interphase boundary prop-agates in thex direction (infinite size in this direc-tion) for which the bulk boundary conditions are suit-able: the plate is of finite thickness in thez directionand infinite extent in thex–y plane. They describe theslow propagation of the interphase boundary causedby the polarization relaxation. The kinetics of relax-ation of the polarization and the conduction of heat aretwo competing processes in the dynamics of the inter-phase boundary. The propagation rate is determinedessentially by the slower of the two processes. How-ever, at the phase transition temperature the velocityof the interphase boundary is equal to zero because ofthe equality of the free energy densities of the para-electric and ferroelectric phases at this point, at whichthe two phases are in equilibrium. Consequently, theinterphase boundary should stop at the phase transi-tion temperature. As a result, the polarization relax-ation mechanism does not work. Since the interphaseboundary continues to move at the phase transitiontemperature, it occurs due to a new mechanism of mo-tion. It may be the above-mentioned mechanism of thelatent heat release. After the local heating the sampleis stabilized by heat removal via heat conductivity andheat exchange with the thermal bath. On approachingthe phase transition temperature from above the heatreleases therefore leading to the local heating and tomotion of the interphase boundary faster than that isdue to the polarization relaxation.

Using the measured coefficients of the Landauexpansion in PbTiO3 [24], we estimate the inter-phase boundary width. Then we obtain the follow-ing width at the phase transition temperature:∆ =1.25× 10−7 cm (Eq. (11)). For comparison with theexperiment [5] we use the following values:ρ =7.1 g/cm3 [25], l = 15.9× 107 erg/g [16], k = 8.8×105 erg/cms K [26],A′ = 2.86× 10−5 1/K [24], B =1.27× 10−12 esu [24],C = 5.73× 10−23 esu [24].In the environment of the phase transition the thermalconductivity coefficient has an approximately samevalue in the phases [26]. The velocity 0.57 mm/s mea-sured in the ferroelectric perovskite PbTiO3 for thefast regime of motion [5] is obtained by using Eq. (5),if the difference in the temperature gradient at thephase transition is 731 K/cm. What change of tem-perature can cause the above change of the temper-ature gradient? To make this calculation we should

360 A. Gordon / Physics Letters A 281 (2001) 357–362

estimate the interphase boundary width by using (11).The change in the temperature gradient at the phasetransition produced within the interphase boundaryis therefore caused by the temperature change equalto 10−4 K:

[kfer(∇T )fer − kpar(∇T )par

] · n≈ kfer∆T

∆.

Therefore the observed fast motion of the interphaseboundary in PbTiO3 [5] can be caused by a very smalltemperature change produced by the local heater of theinterphase boundary at reaching the phase transitiontemperature during the cooling process. The effectunder consideration is analogous to that occurringat first-order liquid–solid phase transitions. The rateof solidification is governed by the rate at whichthe latent heat at the liquid–solid interface can beconducted away from the growing solid [27–30].

It is of importance to take into account the sizedependence of the velocity of the fast motion of theinterphase boundary described by (4) and (5). Forthis purpose, the size dependence of the stability limitof the paraelectric phaseT0 and the phase transitiontemperatureTc should be found. The temperatureT0,being the stability limit of the paraelectric phase andappearing inA (Eq. (7)), is the temperature which ischaracteristic of an infinite sample. The same fact isthe case regarding the phase transition temperatureTc .In what follows we designate the stability limit of theparaelectric phase in the bulk asT0∞ and the phasetransition temperature in the bulk asTc∞. We considerthe thickness dependence of the stability limit of theparaelectric phase and the phase transition temperatureat a first-order phase transition; i.e., we find thedependenceT0(L), Tc(L) and the connection betweenT0 and T0∞, Tc and Tc∞. Afterwards, we studythis size influence on the velocity of an interphaseboundary in the fast regime. We examine a slab offinite thicknessL and infinite extent in the plane.

We suppose that the ferroelectric material is anideal dielectric, uniaxial with a ferroelectric axis di-rected along thez axis perpendicular to the surfaceof the plate. To reduce the depolarization energy inthe ferroelectric state, the crystals tend to break upinto domains of different polarization. Adjacent do-mains have opposite polarization, aligned the ferro-electric axisz. Such domains are separated by 180◦walls, which lie parallel toz. It was shown that a

rectangular, ideally uniaxial plate with the polariza-tion direction perpendicular to its thicknessL breaksup into domains of alternating slices of up polariza-tion and down polarization with a domain widthD,the depolarization energy per unit volume is reducedto 1.7P 2D/L [31]. Meanwhile, the breakup createsdomain walls with a surface energyσ . Includingthe domain-wall contribution, the total depolarization-energy density of a multiple-domain plate is then

(13)1.7P 2D

L+ σ

(L

D− 1

)1

L.

This equation is valid if the domain-wall width is neg-ligible compared to the domain widthD. We alsosuppose thatL � D. Adding Eq. (13) to Eq. (7),we determine the equilibrium polarization for a giventhicknessL by minimizing the free-energy densitywith respect toP andD. We consider only 180◦ do-main walls. We deal with a first-order phase transi-tion. Therefore the surface tension should be calcu-lated by using the profile of 180◦ domain walls de-rived for a first-order phase transition [32]. At first-order phase transitions the polarization profile is dif-ferent from that obtained for second-order phase tran-sitions, which is a hyperbolic tangent function [33].We use the polarization shape characteristic of a first-order phase transition [32]. By calculating the surfacetension, substituting it into (7) and (13), and minimiz-ing the free-energy density∂f (P,D,L)/∂P = 0 and∂f (P,D,L)/∂D = 0 we derive

(14)T0 = T0∞ − 6

A′

(K

Lδ

)1/2

,

whereδ is the domain wall width determined in [32]for the first-order phase transition case.

The phase transition temperature of the size-drivenphase transition determined from the equality of thefree-energy densities of the two phases can be approx-imately written as

(15)Tc = Tc∞ + 3B2

16A′C− 6

A′

(K

Lδ

)1/2

.

The ferroelectric state of the slab under study is a po-larization-up and polarization-down multiple-domainphase. It is distinct from the ferroelectric state in thesingle domain state. The difference in the phase tran-sition temperature is described by (15). This approach

A. Gordon / Physics Letters A 281 (2001) 357–362 361

differs from that applied in [34], in which the polar-ization profile used for a first-order phase transition, isobtained for a second phase transition. The describedrenormalization of coefficientA is caused by the ef-fect of the depolarization and the domain-wall energyon the transition temperature of the slab. CoefficientA

is no longerA=A′(T −T0∞), butA=A′(T −T0∞),whereT0∞ is given by (14). This means that as theslab thickness is reduced, the stability limit of the para-electric phase and hence the transition temperatureare lowered. Consequently,T0 andTc are the stabil-ity limit of the paraelectric phase and the phase tran-sition of the multiple-domain phase. These results arein qualitative agreement with experiment; i.e., the fer-roelectric phase transition temperature is shifted to alower temperature with decreasing film size. To ourknowledge, Eqs. (14) and (15) presenting the temper-ature of the stability limit of the paraelectric phaseand the phase transition temperature at a first-orderferroelectric phase transition, have been for the firsttime derived as functions of both the sample size anddomain structure size — the domain wall width. Forthe infinite sample(L → ∞), Eq. (15) becomes theusual relation between the phase transition tempera-ture and the stability limit of the paraelectric phaseat the paraelectric–ferroelectric phase transition [16].The same thing is the case when the domain struc-ture disappears(δ → ∞). Thus, Eq. (15) includes theknown Landau theory equation giving the relation be-tween the phase transition temperature and the stabil-ity limit of the paraelectric phase for the infinite sam-ple [16] as a limiting case. The ferroelectric phasetransition temperature is therefore shifted to a lowertemperature with decreasing sample thickness. To ourknowledge, only one paper has concerned the first-order transition problem [34], but by using the polar-ization profile characteristic of a second-order phasetransition.

After substituting Eq. (15) into (5) we obtain

vn = 1

3A′Bρ

× 8C

T0 + 3B2/16A′C − (6/A′)(K/Lδ)1/2

(16)× [kfer(∇T )fer − kpar(∇T )par

] · n,and we see that the interface velocity due the la-tent heat release (fast regime) decreases with increas-

ing crystal thickness in according with experiment inPbTiO3 [19]. It should be noted that we actually con-sider the dynamics of two interfaces. It follows fromthe fact that in the multiple-domain phase the do-main wall splits into two paraelectric–ferroelectric in-terfaces — wetting phenomenon — near the first-orderphase transition (see, for instance, [35]). Thus, the fer-roelectric phase near the first-order phase transition isa heterogeneous structure consisting of three states:the paraelectric state and the two domain states —polarization-up and polarization-down. The shapes ofthe two interphase boundaries between the paraelec-tric and+P states and paraelectric and−P states aredescribed by Eq. (9) [35]. Therefore the motion of thetwo interfaces may occur.

Acknowledgements

The author is grateful to S. Dorfman and D. Fuks forvaluable discussions. He is indebted to P. Wyder forhis participation in the research on the magnetic fieldinfluence on ferroelectric perovskites and membranesand for his interest in this work.

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