New phase nucleus appearance on a twin boundary, which is with an invaribe velocity or oscillating

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New phase nucleus appearance on a twin boundary, which is with an invaribe velocity or oscillating

A. Bulbich, P. Pumpyan

To cite this version:

A. Bulbich, P. Pumpyan. New phase nucleus appearance on a twin boundary, which is with an invaribe velocity or oscillating. Journal de Physique I, EDP Sciences, 1993, 3 (5), pp.1175-1186.

�10.1051/jp1:1993263�. �jpa-00246788�

J. Phys. I France 3 (1993) l175-l186 _MAY 1993, _PAGE l175

Classification

Physics

Abstracts

61.70N 64.70K

New phase nucleus appearance

on a

twin boundary, which is

moving with

_an

invariable velocity

_or

oscillating

A. A. Bulbich

(I)

and P. E.

Pumpyan (2)

(')

Laboratoire des Transitions de Phases, FaculI£ des Sciences, Universitd de Picardie, 80039 Amiens, France

(2) Research Institute of

Physics,

Rostov

University,

344104 Rostov-on-Don, Russia

(Received 12 March 1992, revised 18 December1992, accepted 26 January 1993)

Abstract. The _occurrence and

dynamics

of _a nucleus of _{a new}

phase

_{on a}

moving

twin

boundary

are described. The

phase diagram

of _a nucleus _{on a}

moving

twin

boundary

is obtained. It is shown that the nucleus _can arise _on the

moving

twin

boundary only

if its

velocity

_v is smaller than _a

certain value v~. The existence of _two kinds of

low-symmetry phase

trails (unstable and metastable) is described. The friction force,

resulting

from _a trail and

acting

_{on a} twin

boundary

is shown to arise. It is of _a viscous type ^{in the} case of _an unstable trail, and of _a

«

dry

_» friction force type in the _case of metastable trail. The oscillation of the twin

boundary

with the nucleus in the

extemal

periodic

field,

conjugated

to the order parameter ^{is described.}

The

phase

transition

(PT)

_{process was} first considered

by

Frenkel in the framework of

liquid-

vapor transformation.

According

to Frenkel's mechanism the

heterophase

fluctuations

give

rise to nuclei

of,

_{say, vapor}

phase

on the

background

of the

superheated liquid phase. Being

at

nonequilibrium,

this nucleus either

disappears (if

the size of the nucleus is smaller than the critical

value),

_or

rapidly

increases

unlimitedly (in

the

opposite case)

and thus the

crystal

transforms into the vapor

phase [Il.

Frenkel's mechanism _was

generalized

for

solids, taking

into account the elastic

properties

and

anisotropy

and

making

it

possible

to describe _some

phenomena

^{of PTS in}

crystals [2].

However,

another mechanism of PTS in

crystals

has attracted considerable interest of late.

Real

crystals always

contain defects _: dislocations and dislocation

clusters, cracks, inclusions,

twin

(domain)

boundaries and _{so on.} Even the

perfect monocrystal

has at least _a dislocation of

growth

as well _as

edges,

botli of which _can be

regarded

_as defects in the _sense considered below. These

inhomogeneities change

the local value of _a transition

temperature.

^{Thus the}

nucleus of _{a new}

phase

must appear in the

vicinity

of the defect under _a temperature ^different from that of _a bulk PT. This nucleus is stable in contrast to Frenkel's nucleus. It is stabilized

by

the

inhomogeneity.

Stable nuclei _were observed

experimentally

in various

crystals

under PTS of different nature in the

vicinity

of defects of different types.

New

phase

nucleation _{on a} crack

tip

was observed

indirectly

because of existence of _{« a} trail _{» a} thin

layer

of _{a new}

phase

on a fracture

surface, differing

from that in the bulk of _a

crystal.

The monoclinic and orthorombic trails _were observed _on

tetragonal Zr02

^fracture

surface

[3, 4],

a

graphite

trail under the thermal fracture of diamonds

[5],

_a martensitic trail under the

fatigue

steel fracture

[6]

and ferroelectric

phase

trail under the

Pb(Ti, Zr)O~

fracture

[71.

The nucleation _on dislocations _were observed

directly

in

gadolinium

iron garnet

[8],

in

NH~Br [9],

in

Ni~mo alloy [10]

and under

afe->4~e

PT

[11].

The nucleation _on twin boundaries

(TB)

_were observed within

examples

of the

superconduc- ting

PTS in Sn

[12],

In

[13],

Nb

[141,

Re and Tl

[151

and

organic p-(ET)2X

metals

(where

X

=13, IBr2

or

AuI2,

whereas ET is

(ethylenedithio)tetrathiofulvalone) [161.

The

example

of

nucleation _{on a} TB

during

structural PT _was observed under PTS into ferroelectric and

antiferroelectric

phases

in

NaNbO~ [171.

The details of _a PT process

taking place

in the framework of this mechanism _were observed within the

example

of the

magnetic

^{PT in}

gadolinium

iron gamet

[81.

It

appeared

to be different with respect to Frenkel's mechanism. The stable nucleus of _a

low-temperature phase

arised

near the dislocation under _a

higher

temperature than that of the PT. The size of the

equilibrium

nucleus

(being

the function of the

temperature)

was

increasing

under

decreasing temperature,

while the

crystal

was almost

completely occupied by

the

low-temperature phase.

Meanwhile the

high-temperature phase

nucleus remained _near the

dislocation,

its size

decreasing

with

decreasing

temperature

[81.

Thus nucleation _on defects is _a

general phenomenon taking place

^for all PTS in

crystals independently

of the _nature of PT. This kind of nucleation demnnstrates

specific

features

which _are

important

to understand in order _to describe _a PT process and

macroscopic

properties

of

crystals

in the

vicinity

of _a PT temperature.

Possibility

of existence of _a stable nucleus _on dislocation _was first

proposed by

Cahn

[181.

Later it _was shown

(within

the

example

of _a

superconducting

transition in the framework of the

Ginzburg-Landau theory)

that the stable nucleus _can arise at an

edge

dislocation at a

temperature ^T _> T~

[19].

The TB _can also be the

preferential place

for nucleation. In order to describe the

superconductivity

localized on a TB the

Ginzburg-Landau

functional _was

supplemented

^with

the term which contains

singularity

F

=F~~+ iAJ~~8(x)Sdx

[15, 23-271,

where F is _a

crystal

free

energy, _F~~

is the

Ginzburg-Landau functional,

_J~ is the order parameter

(OP) (in

the _cases considered in

[15, 20-251

it describes the

superconductivity,

S is the TB _area.

A,

the TB power, is the

only phenomenological

_parameter

describing

the

interaction of _a TB and _an order parameter field. The

advantage

of this

simplified approach

in the TB

description

is that it makes it

possible

to obtain some exact solutions of the nucleation

theory

which _are the basis for the

description

of nucleation on more

complicated

defects. This way of

taking

into _account the TB is

obviously

valid if the nucleus size is much greater ^{than the} TB width. This

approach

allowed to describe the appearance of localized

superconductivity

and its

disappearance

under either

magnetic

field _{or current}

[20],

_{to construct} the

phase

H-T

diagram

of _a nucleus _{on a} TB of

tin,

to calculate the

diamagnetic susceptibility

of _a

superconducting

TB

[211,

to describe the

specific

heat

anomaly

in

high-temperature superconductors [221,

_to describe the

superconductivity

with _a discontinuous order parameter

on a TB

[23],

to describe

a nucleation in _a

crystal containing periodically

distributed TBS

[241

and

containing

close TBS

[15]

thus the

experimental phenomena

^mentioned above _were

described

[15].

N° 5 NUCLEUS ON A TWIN BOUNDARY l177

Nucleation _on a TB under structural

PT,

observed in

NaNbO~ [17],

_was described in papers

[17, 25].

Besides the features of nucleation

itself,

the nucleation influence _on

crystal macroscopic properties

_was studied. It _was shown that

during

a first order structural

phase

transition the

nucleus

markedly

decreases the linear tension of dislocation which results in _a decrease of the elastic

point (the

stress of Frank-Reed _source

initiation)

and in the intemal friction

peak arising

near the transition

point [201,

in

instability

of _a dislocation segment ^form

[21]

and in the

generation

of

loops

and helicoids

[221,

which

improve

the

plasticity

of _a

crystal.

It is similar to the influence of _a TB nucleus _{on a}

crystal macroscopic properties.

In the papers mentioned above the statics of _an

equilibrium

nuclei state on the motionless defects _were

investigated, though

all

plastic

_processes mentioned above take

place by

the way of motion of defects. Thus it is

important

_to

study

the

dynamics

of _a nucleation process, First of

all,

the nucleation _on the

moving

defect and the nucleation process in presence of _a

periodic

extemal field.

In this paper nucleation _{on a} TB

moving

with the invariable

velocity

is described. It is shown that in the

phase diagram region

which

corresponds

to the existence of _a nucleus on a TB the trail follows the

moving

TB. This trail is the _trace

(permanent

or

temporary)

of the

phase

of the nucleus. This

region

of nucleus existence is divided into _two parts :

(I)

the trail is unstable,

(it)

the trail is metastable. The unstable trail _causes the viscous friction force which _{acts on a}

TB,

while the metastable trail generates ^the

dry

friction force. The

phenomenon

of the TB

swinging by

the

periodic

field

conjugated

to the order parameter

(OP)

is described.

1. Nucleation _{on a} twin

boundary

which is

moving

with _an invariable

velocity.

Consider _a

crystal undergoing

several structural PTS. Under the

high-temperature

PT the

crystal

is divided into twins

(domains).

Thus the twin structure in

crystal already

exists when the

low-temperature

PT takes

place,

TBS

being

the defects which _can favour the nucleation. If the difference between the temperatures ^of

high-

and

low-temperature

PTS is not small the

action of the nucleus _on the TB _structure is weak. In this _case the existence of the TBcan be taken into account

by

_a

singular

term which is added into the free energy

describing

the low-

temperature PT. Consider the

simplest

case of _a

low-temperature

^{PT which is described}

by

one-component ^OP _J~. The

equations

of motion

describing

the PT

dynamics

can be obtained with the

help

of _a

Lagrangian

and _a

dissipative

function of

crystal.

Its

Lagrangian

has the forrn

L

=

I( @~)

dV F

(1)

v 2

where the free energy F of

crystal

has the form

F

=

f(J~

A_J~ ^~ 8

(x

_xo

(t)))

dV

(2)

v

where V is the

crystal

volume.

Considering

the flat TB _{one can} take dV

= S dx, where S is the

TB area. m is the _mass of _a soft mode,

m a_J~ lot, t is the

time, f(J~

is the free energy

density

in the

homogeneous crystal (the crystal

^without the

TB)

f(1J)

₌

jgo(VIJ)~ +ja1J~+(PIJ~+ yJ~~ (3)

go, a, p, _y are the

phenomenological

constants, go, y

being positive,

while _a and p

signs

can be

changed

_{a =}

a(T T~).

T is the temperature, T~ is the Curie temperature, ^{A is} the TB power. ^{The values A} > 0

correspond

the _case where the TBS _to be the

preferential

nucleation

places. xo(t)

is the TB coordinate. Under the constant

velocity

v one can take xo = vt.

The

dissipative

function of tile

crystal

should be taken in _a form

Q=~ii~dv (4)

2

v

where _x is the kinetic

factor, being

_{x >} 0.

The

equation

of motion

following

from

(1-4)

takes the form

mi

+

xi

=

~o($

a~

-fl~~- Y~~+2A~8~x-vt~.

_~5~

If A =

0, equations (1-5)

describe PT in _a

homogeneous crystal.

Under fl _> 0 the

phase

J~ = 0 transforms into the

phase

_J~

= Const.

by

^the _way of _a second order PT _on the line

a =

0

(Fig. I).

And under fl _< ⁰

by

the way of _a first order PT _on the line of

equality

of

energies

of

phases

_a

=

3fl~/16

_y

_{(line I, Fig. I).}

In this _case the

phases

_J~ =0 and J~ =

Const. coexist between the lines _a

=

fl~/4

y

(line 2, Fig, I)

and

a =

0.

~

-~

-O-O-O-3

£~) ~OO-,,-OO-~

~

~,~

~ ~

o,

~

"o- oo-

>Ii

i)

Fig.

I. The

phase

diagram of _a nucleus _{on a}

moving

twin boundary in the

vicinity

of the tricritical

point

_{a =} fl ^=0. I) The line _a

=

3fl~/16y

_{of energy}

_equalities

_{of the}

homogeneous

phases

~ = 0 and _~

=

Const. 2) The

stability

failure line _a

= fl~/4 _y of the

phase

_~

= Const. 3) The nucleation line in the case of the

moving

twin

boundary.

4) The nucleation line in the motionless thin

boundary

_case.

Shaded is the

region

of nucleus existence. Twice shaded is the

region

in which metastable trail is realized.

a) The order parameter distribution for the unstable trail _case b) for the metastable trail _case.

N° 5 NUCLEUS ON A TWIN BOUNDARY 1179

The solution of

(5)

^should be tried in _a form _J~

= J~

((),

where ( ₌ x vt. The

equation

takes the form

g1J"+ «v1J'-

_a1J

-fl1J~- YIJ~+2A~&(t)=

_o

₍₆₎

with the

boundary

condition _J~

(±

_co

=

0, corresponding

to the solution which describes the nucleus. Here

J~'m

a

J~la(

_{; g}

= go

mv~.

Equation (6)

_can be examined

by

tile methods of the bifurcation

theory [291.

Under

large positive

_a values

equation (6)

has the

single

solution _J~

=

0.

According

to the bifurcation

theory [291

in order

study

the

stability

of tliis solution _one should consider the

equation

for small

J~ value. This

equation

is the linear part ^of

equation (6)

_:

lgwi+""wi-"owe+2Awo8(<)=0 po<±co)=0.

^~~~

The

stability

of tile solution _J~

= 0 fails when _a becomes

equal

to ao where ao is the first

eigenvalue

of

equation (7).

As _{a «} ao, the small solution

~ =

i~ro(<)

₊

o(13) ₍₈₎

branches off the solution _J~

= 0. Here

$ro(()

is the

eigenfunction (7) corresponding

to

eigenvalue

_ao. The

amplitude f

should be determined later _on.

The

single

discrete

eigenvalue (7)

and its

eigenfunction

_are

A

x~v~

"°

g

4g

The solution

(7, 9)

is

displayed

in

figure 2a,

b.

The

equation

for the

amplitude f (the branching equation)

should be deduced from

equation (6) [291.

It takes the form

(a ao)(4ii) _f

₊

fl (4i() ^f~

_{+ y}

(4i() ^f~

= o

(lo)

where

(#'$)

"

it#'$(f)df

₌

^(~~

~ ~

~~ _n(4A (ii)

x v

Thus the bifurcations of the differential

equation (6) correspond

to the bifurcations of the

algebraic equation (lo).

The solutions of

equation (lo)

_are

f

= 0 under _a

» ao

(fl

»

0)

and _a

» ao + 3

p

^~ 8/16 _y

(fl <0) (line3, Fig. I),

here 8 =3/4 and

f#0

in the

opposite

_case. The solutions

f

₌ 0 and

f

# 0 coexist under ao < a < a o +

fl~

8/4 _y

(fl

<

0). Evidently

the

phase diagram dependence

_on the TB

velocity

is

only generated by

tile ao =

ao(v) dependence.

Under tile

increasing velocity

line 3

(Fig. I)

is lowered and the

region

of nucleus existence _on the

phase diagram

becomes smaller. Under tile

velocity

value

v~~ =

~ ~

(12)

the

region

of nucleus existence vanishes. If _v

» v~~ the TB nucleus cannot exist.

If fl _> ⁰ the term

f~

in

(lo)

can be

neglected

and the solution takes the form

f2 ~~" "0~fl

,

" ~ "0

0,

_{a m a~,} ~~~~

Iffl

~0

~

3 p +

(9 ^p~

48

y(a ao))

^~~~

f

~ ~

(14)

The solution

(14)

_as well _as the

equations

of the

phase diagram

lines in the

p

_< ⁰

region

_are

valid

only

in the _case of small

[p

values.

Note that the first

eigenvalue

of the linearized

equation gives

the exact value of the

instability point, though

the overcritical solution

(8)

is

asymptotically

exact

[29].

The solutions

(9)

of

equation (7)

for _ao and $ro are exact. Thus the

expressions

for the

phase diagram

lines _are

exact and the solution for the order parameter ^is

asymptotically

exact.

In

[25]

the

equation

for line 3 of

figure

I _was obtained from the exact solution of the state

equation

of nucleus _{on a} motionless TB. In that _case the

expression

for the line 3 has the form

a = ao ⁺ 3

PSI16

_y

equal

to the form _on the

moving

TB with 8

= 0.75 if ply ~

[0,

2.2

and 8

= 0.82 if

PI

_y ~

[-

4,

5.51.

The value 8

= 0.75 in the case of small p ^{obtained for}

the motionless TB is

equal

to the value for

moving

^{TB mentioned} above. In the _case

v ₌ 0 the line of energy

equality

of the TB with nucleus and without nucleus

(line 3, Fig. I)

intersects with the line of energy

equality

of the bulk

phases

_J~

=

0 and _J~

=

Const.

(line I, Fig. I).

The

point

of intersection P has coordinates which _{are ap}

= 5.7

a o,

flp

= 5.5

(ao

_y

)'/~

[251.

For small values of

[fl[

the size of the

phase diagram depends

_{on u} due _to

a

(v) dependence only.

It is

possible

to

hope

that such

dependence

also takes

place

for

large

values of

[fl

and in the _case of _a

moving

TB _:

a~

= 5.7

ao(v), pp

₌ ^5.5

(ao(v)

_y

)'/~

The

phase diagram

is

displayed

in

figure

I. The

region

of TB nucleus existence is shaded. This

region,

however, is divided into two different parts in which two different sorts of trail _can be observed.

2. The trail.

The OP distribution in the TB

vicinity

is

displayed

in

figure

2. In the _case of _a motionless TB it is

symmetrical (with

respect to the TB

plane).

In the _case of _a

moving

TB the distribution is

compressed

in front of the TB and stretched _at its back.

The nucleus size ahead of _a TB is

I((

> 0

=

io(I

₊

_v/v~~)~

^'

_though

behind the TB it is

I((

<

0)

=

lo(I v/u~~)~ ',

where

to

=

g/A

is the size of _a nucleus _{on a} TB.

Thus the trail follows

moving

TB it is the

phase

_J~ # 0 nucleus part ^which is situated behind the TB. Its size

depends

_on the TB

velocity

and increases under u _- u~~ 0. This trail is

unstable. It is related to OP relaxation time

locally slowing

down in the

region

situated behind the TB. The trail

decay

time is _r~~

=

I((

~

0)/u.

It minimum value

r$,~

=

4 I~/u~~ can be obtained under v

=

v~/2.

The trail _causes the friction force

acting

_on the TB. In order to obtain the

expression

for this force _{one can use} the fact that the free energy

(2)

is the function of the TB coordinate xo. Thus _{one can} define the force

acting

on the TB _as

aF

(xo)

~

axe

N° 5 NUCLEUS ON A TWIN BOUNDARY l181

I

-

2

-.-.- .

3

~ ~

_ ^.

,~ ~

_~

o-o

~ _~

~~

~

' o

x-~i

, ^'

,

.

i ^'

' ,

'

1)

Fig.

2. Order parameter distribution in the

vicinity

of I) moving ^twin boundary (the unstable trail case) _: 2) motionless twin

boundaly

3) moving twin boundary (the metastable trail case). a, b) The free energy

density

versus ~

dependence.

Point

displays

the realized state a) in front of _a twin

boundary

b) in the metastable trail, c) the trail.

Using

this

general expression

one can obtain in the case of the free energy

(2)

G

=

~~

l( ^°'~~

+

°'~~ (15)

2 ax x=x~+o ^ax x=x~-o

where

(aJ~

~/ ax

)

~ means the derivative value taken in front _{(« +}

»)

_or behind («

»)

the

T xo+

TB.

Substituting

the solution

(8, 9)

into

(15)

_{one can} obtain

G

=

I _vf~

(16)

g

JOURNAL DE PHYSIQUE -T ~ N'5 MAY 199~

where the

amplitude f

is

given by

_one of the

expressions (13, 14)

and

depends

on the TB

velocity.

In the _case

fl

_» 0 it

gains

the form

_2xAS

j(A~ ~j

~

x~~~j ~2~4g

jA~ ~j

G =

~ ~

~~

^~

"~

_~

(17)

2

4g

~'

~

'x~ jA~

g

~

in other words the nucleus

generates

a viscous friction force which _{acts on} the TB in the _case of unstable trail.

However in _the first order PT _case another kind of trail can take

place

^{the trail} can come

into metastable state. Let J~~~ be the OP value

J~$~~= (-fl (fl~-4ay)1/~)/2

_y

corresponding

to the maximum of the free energy

density (Fig. 2).

If the

inequality f~

» J~$~~ ^is

valid,

the

crystal

part ^{situated behind the TB transforms from} the state f # 0

(Fig. 2a)

to the _state

J~j

=

(-

fl ₊

(fl

^~ 4 _{a y}

)1/~)/2

y which

corresponds

to the lateral minimum of the free energy

density

in the

homogeneous crystal (Fig. 2b) (but

not to the state _J~

= 0 _as it took

place

in the unstable trail

case).

The free energy of this state is

higher

than that of the state _J~

=

0,

however the

decay time,

which

depends

_on the barrier

height

Af~,

_can be rather

large.

A metastable trail _was observed in

Zr02 13, 41,

and in

Pb(Ti, Zr)O~ [71.

In both _cases the trail followed the nucleus

generated by

the stress concentrations _on the

tip

of

moving

crack. In

both _cases the

decay

time _{was more} than _a

day.

The

inequality i~

>

J~$~ _together

with the

reality

condition of

J~j

is fulfilled in _a

phase diagram region singled

out

by inequalities _fl

< 0 3

fl ~/16

_{y w a w}

fl ~/4

_{y ;}

a w ao + 3

p

^~ 8/16 _y

(twice

shaded

region

in

Fig, I)

in this

region

the metastable trail is realized.

In order _to obtain the force

acting

_on the TB in this _{case one} has _to

multiply equation (6) by J~'

and

integrate

it

by (

from _{co to + co.}

Taking

into account the conditions _J~

(-

_co

)

= J~o,

J~

(+

_co

)

=

0, J~'(±

_co

)

= 0, _one can obtain

«UJ

+f(11(- m)) +i _{II ~i~} ^~=~~~~

+

ii~ l~~~~)

^{= 0}

⁽¹⁸⁾

+ m

here J

=

(J~')~ d(

_J~

(Ii ₍₍

» 0

f(J~

is

given by expression (3). Taking

into _account

-

m

that the last _term in

(18)

is the force

acting

_on a TB

(15),

_{one can} obtain

G/S = xJv

f (J~o). (19)

The first _term in the

right-hand

_part of

equation (19)

is the viscous friction force related _to the energy

dissipation

in the nucleus. The second terra

f(J~o)

_> 0 is the difference between the free energy densities of

phases

_J~ # 0 and _J~

= 0. It is the energy which is necessary ^to

expand

in order to create the unit volume of metastable trail.

Note that the

equations

for the

amplitude

f

(lo, 13, 14)

and the

phase diagram

lines

equations depend

_on the structure of

potential (3) (particularly

its

polynomial

form and the fourth or the sixth order term limitation in this

polynom)

and _{as a} result are valid

only

for small values of

[f[.

For

expression (14)

it _means that it is valid

only

_near the tricritical

point

a =

p

= 0. However

expression (19)

is valid in _a

general

_case,

including

the _case of _a PT of _a manifest first order.

N° 5 NUCLEUS ON A TWIN BOUNDARY l183

3. The tlvin

boundary

nucleus

pulsation

in _an external field.

In the

previous chapters

the interaction of _a TB with _a nucleus under their _movement with _an invwiable

velocity

_was examined.

Under the

periodic

TB and _a nucleus movement this interaction _{can cause some additional}

phenomena.

Under

application

of _a

pulsing

field

h,

which is

conjugated

to

OP,

TB oscillations

can arise.

In order to describe OP

dynamics

in field

h,

the _term J~h has _to be added _to the free _energy

density (3).

Here h

=

ho

_exp

(I (wt qx)).

The

dynamic equation resulting

^from

expressions (1-4)

takes the form in _a linear

approximation

_:

~q

₊

xi

= g~

~~

_{« ~ +} 2

A~

s

(x)

+

ho

exP

Ii (Wt qx)1 ~~°~

Consider the

crystal

at _a temperature ^{which is}

higher

than the nucleation temperature To. ^{The solution of}

(20)

should be tried _{as an}

expansion

in the

eigenfunction $r~(x)

of _an

equation

a~~r~

g0

q ^{An ~n}

^{~ 2}

^A~n

⁸

^{(X) (21)}

J~(x, t)

=

£b~(w) $r~(x)exp(iwt).

2 here should be understood _{as a sum} taken _over the discrete part and _an

integral

taken over the continuous part ^of spectrum ^of

equation (21).

Equation (21)

^{has the}

single

discrete

eigenvalue Ao =A~/g>0 corresponding

_to the nucleation temperature

To

on a motionless TB. Its

eigenfunction

is $ro =

11

^~/~ _exp

(- [x [lie)

where

to

=

g/A.

Under A <0

equation (21)

has _a continuous spectrum ^of

eigenvalues

A ₌ k~ _a and

eigenfunctions

$r~ (x)

=

(2 arro)~

^~/~

~

_~~~

~~

~~~°~

'

~ '~

(22)

exp

(ikx/ro)

+ R exp

(- ikx/ro)

,

x _< 0

~_

(x)

=

(2 arro)~

^~/~

~~~ _~~~°~

^{~ ~ ~} ~~~

~~~~~°~

'

~ '~

(23)

P * exp

(-

_i

kx/ro)

,

x < 0

where P

=

k(k

+ iD

)/(D~

₊

k~),

R

=

D

(-

D ₊

ik)/(D~

+

k~),

_ro

=

(gla

)~/~ is the correlation

length,

^D ₌

A(ag)~'~~

=

Aola, k/ro

^is the _wave vector. The solution

(20)

takes the forrn

'l

(x)

=

ho exp(I wt) Xo

wo(x)

+

j~ ^{jX+ (k) p~ (x)}

+ X-

(k)

p_

(x)i

dk/2

art(24)

where xo, x~, X- are the

generalized susceptibilities corresponding

to the modes

#'0'#'+,

#'-

xo=

~~~~

~

((a -~~) _-mw~+ixwj (25)

A +g q g

x~(k)=

^~°

~~~

^~~~ _+R*

ar8(k-qro)I[a(I +k~)-mw~+ixwl~~ ₍₂₆₎

2 _ar

I(k2-q2r()

r~ ^1/2 ~ ~~

l~

_~

~~~~~~ ~2gr ~i(k~-q~r()~ ^~~ ~~~"~~~ ~~°~ ~~~~ ~~~

_~°~

~~"°~~

(27)

The first item

(24) corresponds

to the mode

$ro(x generated

in _a

crystal.

If the condensate of this mode arises, it describes a nucleus _{at a} motionless TB. Two other items describe the OP

wave

dispersion

_{on a} TB. The function $r~

(x)

describes the

dispersion

ⁱⁿ _a

positive

^OX axis direction and $r_

(x)

in _a

negative

direction.

Susceptibility

_xo has _an

anomaly

if T

=

To (in

other words _{at a}

temperature

^{which is}

equal

to the

temperature

^{of nucleation} on the motionless TB).

4. The twin

boundary

oscillation due to the nucleation

pulsation.

OP distribution in field h

(24)

^is

nonsymmetrical.

Because of

expression (14),

it

gives

rise _{to a}

periodical

force

©

_which

acts on the TB.

The whole

expression

for has the forrn

i~

=

~~

))j _i~~ ^{~ ~~~}

^~

^{~~~~~ ~~}

⁺

^{4~~ ID}

^TW

^{iD (42}

⁺

^D2)

^rw

ⁱ

"

(D~

₊

4~)~ i(A2

+

12)2

+

r2

w212

(28)

where

A~

= I _w

~/wj,

_w~

=

(a/m)~/~

is the soft mode

frequency, (

= qro, r =

«la is the time of OP relaxation. In the limit of

large wavelength

and small

frequency low

q~ ^and

w «

(a/m)"~,

_force

© _expression

_{takes the forrn}

~2 G

=

/

qro S

(qro

D

rw

)

_exp

(2

I wt

) (29)

ar a

The force

represented by equation (28)

_generates the TB oscillations and transforms the energy of field h into the energy of mechanical oscillations.

Consider the TB oscillations in the framework of _a fixed membrane model

Ml

+

2

_4~

~

+

i

=

(30)

B

ax~

~~

where M is the effective TB _mass, B is the TB

mobility

and _4~, the TB surface energy.

The

swinging

work of force

during

each

period

^is

v/w /j

A

=

Re dt Z dx

dy (31)

0 s

~

where Z

=

Z(x,

_y, t

)

is the solution of

equation (30).

Consider K

=

AIR

_where

W

_{is an average} energy of field h in

crystal

: W

=

h(

V/16 _ar. V

=

L~L~L~

^{is the}

crystal volume,

_L~ is the

crystal

size in _a

corresponding

direction. K

gives

the estimate of the

efficiency

of transformation of the field energy ^{h into the acoustic} _energy

4 096

Nh(

w

q~ r( _(qro

_D

r w )~S~

~

QT~

a

~

M~

_BV ^~

^~~

^~~~~

where

£(w)= jj ((2k+1)2(2n+1)21(w]n-4w2)2+(2w/MB)21j-~

wj~

₌

ar~((2

^k + I

)~Lj~

₊

₍₂

_{n +}

I)~Lj~)

4~/M-are the TB resonant

frequencies,

N is the number of TBS in the

crystal

^and k, n are whole numbers. If the field h

frequency

_w is much

N° 5 NUCLEUS ON A TWIN BOUNDARY i185

smaller than the basic

frequency

_woo

= ar

(4~ (Lj

^~ +

L~~)/M)

^'~~ one can obtain 4 096

Nh( wq~ r((qro

D

rw )~S~

K

= ~ ~ ~ ~

(33)

ar a M woo BV

Considering

_{q w one can} obtain k

w ~.

5. Discussion.

The results

given

above show that the nucleus _can arise _{on a}

moving

TB under

T~ w T _w T~ +

A~lag.

_{At a} fixed temperature

belonging

to this interval the nucleus _can arise if the TB

velocity

satisfies the

inequality

v w v

j ⁼

(2

AIM I _a

(T

_T~

g/A~)

^'~~

⁽³⁴⁾

(under

the

velocity

value v

= vi the

amplitude

I

(12, 13)

tums to

zero).

Under _u

m u~~

(I I)

the nucleus _cannot arise _at any

temperature.

One _can estimate

a~l0~Jm/C~K _[301, g~l0~~Jm~/C~ _[31],

OP time relaxation

r ~10~ ^'° _s

[321,

_x

= ra x I K

10~~ Jsm/C~.

_In

experiments [12, 13]

it _was observed that

To T~ 0. I K and in

[171

that

To

T~ ^{5 K.}

Considering _To

_T~ l K estimate the TB

power A

=

[a (To T~) gl'~~

³ x 10~ ^~

Jm~/C~.

Using

this values _{one can} obtain the values _u~~,

to

and _{r~,~. u~~}

~10~ _m/s,

_{the motionless}

nucleus size

to ~10~?

m =

0.I _~Lm, the minimum trail

decay

time _r~~~

~10~~

s.

The maximum value of viscous friction force

(17)

is arrived _{at u}

=

v~/ /

:

8

A4

_{~ ~} 312

~~~~

3

/ Pg~

~ ~~~ ~~~

A~

^~~~~

Estimating p 10~ 10~ Jm~/C~ _[301,

a

(T T~) g/A~

« I _{one can} obtain

G~~~/S lo-10~

_Pa.

Maximum

dry

friction force value

(18)

can be arrived _near the line _a

=

p~/4

_y of

phase diagram (line 2, Fig. I).

In this _case

f(J~o)= [p[~/48y~.

For the _case of

PbTi03

p

=

-1.9 _x

10~ Jm~/C~

y =

0.8 _x

10~ Jm~/C~ _[301,

_{whence it follow8 G/S 3}

x

10~

_Pa.

Mention that the mechanical _stress value which _tears the TB off the

locking

devices in

PbTi03

is _« l MPa

[301,

the spontaneous deformation in

PbTiO~

_s 10~ ^~

[30]

thus the maximum

force value related to the unit TB _area

acting

from the

locking

devices _on the TB is

us

10~

_Pa.

Therefore,

the viscous friction force

acting

_{on a} TB in _an unstable trail _case is

comparable

with the

TB-locking

device interaction force,

though

the value of the

dry

^{friction force} can be

more than that force.

One _can estimate the energy transformation factor K for

PbTiO~, considering

B

10~

_m/Js,

Wo/S 10~~-10~~ J/m~ _[30]

_and

considering

the

crystal

size L~

L~ 10~~

m and

frequency

w lo MHz one can obtain K

~10~~ 10~~

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