Volume 99A, number 6,7 PHYSICS LETTERS 12 December 1983

NONLINEAR PHENOMENA IN KINETICS OF PHASE TRANSITIONS

A. GORDON Department o f Physics, Technion-Israel lnstttute o f Technology, Haifa 32000, Israel

Received 11 July 1983 Revised manuscript received 17 October 1983

The kink-type solution of the time-dependent Gmzburg-Landau equation is obtained. It represents the moving interface at first-order phase transitions. The estimate of the velocity of the interface at the ferroeleclxlC phase transition in BaTIO 3 shows a satisfactory agreement w~th the experiment.

Nonlinear parabolic equations of the reaction-dif- fusion type or the time-dependent Glnzburg-Landau equations are widely applied in different fields. They describe chemical reactions and nonequilibrium phase transitions (PTs) [1-6] , propagation of nerve pulses [7-10] , motion of domain walls [11], dynamics of fluctuations of the order parameter at PTs [12-14] , self-organization phenomena I15], and kmetics of PTs in liquids [16-19] .

In this letter the kink-type solutions of the time- dependent Ginzburg-Landau equation are obtained to describe the kinetics of solid-solid first-order PTs. This case applies for instance, to ferroelectric PTs. These solutions represent the moving interfaces, sepa- rating the regions of paraelectrlc and ferroelectric phases.

The time-dependent Glnzburg-Landau equation for evolution of the order parameter of the PT 11 is given by [20-23]

0111bt = - F6FI611, (1)

where P is the Landau-Khalatnikov damping coeffi- cient which is assumed to depend noncritlcally on a temperature [21], F is the free energy which is given for the first-order PT by

F = F 0 +ka112 -~b114 +kc116 +D(a11/Ox) 2 , (2)

where a, b, c > 0 and D is the coefficient of the in- homogeneity term [23], 6F/611 is the variational de- rivative o f f in respect with 11 [11 ].

Then we obtain

~11/~t + F(a11 - b113 + c115) - 2FD a211/~x 2 = 0 . (3)

The original partial different equation in the indepen- dent variables of a coordinate x and time t (3) can be reduced to ordinary differential equation in the van- able s by rewriting s = x - ot, where v is a velocity in the direction x. After the transformation in (3) we ob- tain

2FD d211/ds 2 +vd11/ds - F(a11 -b113 +c115)= 0. (4)

The solutions which we seek are stationary in the moving coordinate system and they are of wave front type. We assume that these solutions satisfy the fol- lowing boundary conditions: d11/ds--" O, where s ~ --+ ~o and 77 ~ 111 for s --r + 0% 77 ~ 112 for s ~ - 0% where 111,112 are extrema of the free energy as a function of 11.

The solution of eq. (4) for these boundary con&- tions is given by

11 = 111/[1 + e x p ( - - s / A ) ] l / 2 , (5)

where 111 and 112 are here the minimum and maxi- mum of free energy, respectively,

112 = (b/2c) I1 +(1 - 4ac/b2) 1/2 ] , (6)

A =]01/2 ( a ' ( T c - T0)[1 +(1 - - ]6 ) 1/2 --~5] ) -1 /2 ,

o = 2FD 1/2 [a'(T c - TO)] t/2 (7) 2 --~ [1 +(1 _~5)1/21

X [1 + (1~- ~ ~ - ] 6)1/211/2' (S)

0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland 329

Volume 99A, number 6,7 PHYSICS LETTERS 12 December 1983

19 T,

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The author expresses his sincere gratitude to Pro- fessor J. Genossar for valuable enhghtenmg discussions. He is indebted to Professor M. Revzen, Drs. J. Adler,

L. Bengulgul, S. Flshman for useful comments. The support of the Wolf Foundation is gratefully acknowl- edged.

References

Fig. 1. The kink-type solution of the time-dependent G1 nzburg- Landau equation describing the shape of the profile of the order parameter r~ In a first-order PT (A = 1, r/1 > 0).

a=a'(T-To), 6=(T-To)/(Tc-To), (9)

where T c is the PT temperature, T O is the Curie-Weiss temperature.

One can check by direct substitutxon that eq. (4) has the solution (5). The stabihty of solutions of this type under small perturbations has been proved m the general case, for instance, in ref. [1].

The shape of the profile of the order parameter gwen by eq. (5) is shown in fig. 1. This profile describes

the interface between the ferroelectric phase 0/1 > 0) and the paraelectric phase (7? 2 = 0). Consequently, v

is the propagation rate of the interface, which is pro- portional to velocity of a first-order PT.

We esttmate the velocity u and the width A of the interface between the ferroelectrlc and paraelectrxc

phases in BaTiO 3 and compare these results with the experiment [24]. We use the following data. a ' = 6.66 X 10 -6 K -1 [ 2 5 ] , T c - T O : 1 5 K [ 2 5 1 , D : 3 . 3 5 × 10 -16 cm 2. The estimate of D was carried out ac- cording to ref. [23]. The damping coefficient P is taken

as. F = 0.6 × 108 s -1 at T = T c +2 K [26]. We sup- pose that the slow thermal motion of the interface contributes to the central peak width. The data for the coefficient P were taken for SrT10 3 because the analogous measurements in BaT10 3 have not been carried out. We obtain the following results, u = 4.5 X 10 -3 cm/s and A = 1.4 X 10 -6 cm. In ref. [24] the experimental value of u is equal to (0 6 - 9 . 5 ) × 10 -3 cm/s. There is a satisfactory agreement between the theory and the experiment.

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