Dislocation-mediated period adaptation in magnetic "bubble" arrays

HAL Id: jpa-00246907

https://hal.archives-ouvertes.fr/jpa-00246907

Submitted on 1 Jan 1994

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Dislocation-mediated period adaptation in magnetic

”bubble” arrays

M. Seul

To cite this version:

M. Seul. Dislocation-mediated period adaptation in magnetic ”bubble” arrays. Journal de Physique

I, EDP Sciences, 1994, 4 (2), pp.319-334. �10.1051/jp1:1994140�. �jpa-00246907�

Classification Physics Abstracts

61.50K 61.708 82.70 75.70K

Dislocation.mediated period adaptation in magnetic

_«

bubble

_»

arrays

M. Seul

AT & T Bell Laboratories, Murray HiII, New Jersey 07974, U-S-A-

(Received J4 June J993, accepted in final foi-m 2J October J993j

Abstract. Dislocation-mediated mechamsms of stress relaxation, generating disorder in magne- tic bubble arrays in response to

temperature-induced period

adaptation, are described. In _contrast to the response of assemblies of

rigid particles,

the stress-induced evolution of dense magnetic domain pattems ^is charactenzed by ^the mterplay ^{of lattice}

topology

and bubble demain size and shape. Such

coupling

of

topological

and geometrical degrees of freedom determines the local

transformations which mediate lattice expansion («

coarsening

») and govem dislocation

dyna-

mics : these _are shown to be predicated upon

adjustments

_in the number density ^of bubbles _via bubble collapse. Elementary processes

underlying

the translation of dislocations, ^the dissociation («

splitting

») of dislocation _cores and the formation of virtual pairs _as well _as mterstitials in the

forrn of dislocation clusters of _zero net Burgers vector are exammed. Later stages ^of coarsenmg are

shown ta permit pattem «

heabng

_{» via} annihilation of interstitials. Exammation of the close

connection between the elementary _processes of dislocation dynamics and the

topological

transformations of polygonal networks (« froths ») reveals the elimination of 5-sided cells _to play _a central rote.

1. Introduction.

Magnetic

demain pattems

composed

of

stripes

and bubbles may be viewed _as manifestations

of modulated

phases

which _are stabilized

by

competmg interactions

[1, 2].

This

simple

concept ^{has been advanced} to account for demain formation _{in a} wide

variety

of solids and fluids in bath two and three dimensions

[3, 4],

and thus tends support to the view that the appearance of stable demain pattems ⁱⁿ

magnetic

gainer ^films may be

regarded

_as the direct

analog

of pattem ^{formation in}

non-magnetic

systems ^such as monomolecular

amphiphihc

films

confined to an air-water interface where the

phenomenon

has _{in recent} years

generated

considerable interest

[5].

Within this

framework,

_{a new,} macroscopic

length

scale, the

modulation

penod,

d, _is set

by

the ratio of

pertinent

interactions and may be tuned

by altenng

that ratio via its

dependence [6]

_on

applied magnetic field, H,

and temperature, T. The

evolution of

uni-directionally

modulated

stripe

_pattems from well-defined initial states into disordered («

labynnthine »)

final _states has been

recently

examined in

quantitative

dotait,

employing digital

hne pattern

analysis

to douve robust structural attributes of

labynnths [6, 7].

Topological

defects _were shown to

play

a fundamental rote in the evolution of

disorder,

and related considerations

apply

to the

«

topological hysteresis

_» ^of

stripe

patterns

subjected

to

magnetic field-induced

cycling [8]

and to the defect-mediated

melting

^of bubble lattices

[9].

While _non central _to these instances of

disordering

and

melting

of patterns, a fundamental additional aspect ^{of domain} pattern ^{evolution resides} in the

interplay

^of collective

degrees

of freedom and the associated

topological

defects, and the

degrees

of freedom associated with

shape

transformations of individual domains,

including simple

dilatation.

Perhaps

the _most direct manifestation of this

interplay

is Ihe

morphological

transformation between bubbles and

stripes required

for the transition between the

corresponding

modulated

phases [3, loi.

The

inaccessibility

of the

triagonal

bubble

phase throughout

much of the

(H, n-phase diagram il,

II may well have ils

origin

in this

coupling only

in a limited range of temperatures near

the mean-field cntical temperature

il Tc

is the spontaneous ^{nucleation of} the

expected

bubble

phase expenmentally

observed

il Ii,

while outside of this range the

demagnetization

from the saturated _state leads _to the appearance ^not of _a bubble

lattice,

but of

isolated, elongating stripes [6,7].

The spontaneous

elongation («stnp-out»)

of _an isolated bubble,

generated by heterogeneous nucleation,

but, at the

requisite

_size

possibly

unstable _to this very

shape

transformation

[2, 12],

would account for this rather drastic deviation from the mean-field

phase

behavior. A _recent

investigation

of the

disordering

of bubble arrays and evolution of

networks

(«froths »)

of

polygonal

bubbles

during magnetic

field-induced

coarsening

has

revealed _a

special

rote of

topological

transformations

involving pentagonal

colis

il 3].

A further

instance of the

interplay

between

topological

and

geometrical degrees

of

freedom, involving

only

the

adjustment

in _area

(but

_non in

shape)

of circular

bubbles,

_is the inherent

couphng

between the

proliferation

of dislocations and the appearance of

polydispersity

_in

triagonal

bubble pattems

[14].

As _in the _case of

quenched

_size

disorder,

realized in

binary

mixtures of

colloids

[15]

and other

ngid particles [16],

the translational symmetry

characterizing

the ordered solid

phase

is

destroyed, particles

of mismatched _size

acting

_{as traps} for dislocations

il?].

The latter observations

[14]

_were facilitated

by subjecting

_an

initially ordered, triagonal

bubble pattern to _«

parametric

_{coarsening » in near-zero magnetic} field, _a condition

ensuring

constant overall

magnetization

and thus _{a constant area} fraction

occupied by

bubbles

il 8].

Penod

adjustment

in response to

cooling requires

lattice expansion

[6] qualitatively speaking,

as T decreases below

Tc,

the domain watt energy increases _more

rapidly (with

_an exponent related to that of the bulk correlation

length)

than does the

demagnetizing

_energy

(with

the

square of the order parameter

exponent)

so that the relative

predommance

^of the former

impedes

the formation of interface. The consequent ^reduction in the

density

of domain watts

implies

an increase in the modulation

penod, d=d(H,T);

that _iS, the temperature

dependence

of d in _zero

magnolia

field is such that

d~~o(T)

incieases with

decreasing

temperature

[6].

^The

accumulating

stress may be accommodated

by

the

collapse

of individual bubbles

[19],

_{or, as}

descnbed, by

the formation and motion of dislocations

[14].

The focus of the present ^{article is} on the mechanisms

underlying

the _motion of dislocations and the creation of interstitials and virtual _{pairs in} the form of dislocation clusters of _{zero net}

Burgers

vector. It

is shown that the

collapse

^of five-fold coordinated bubbles

plays

a fundamental rote, thus

introducing

an element into the dislocation

dynamics

which is non manifest in assemblies of

ngid partiales,

but may well be relevant in

dense-packed

^{assemblies of deformable} bodies such

as flexible latex

spheres [20],

_or concentrated emulsions

[21].

2. Materials and

experimental procedures.

The observations

reported

here _were made _via

polanzation

_{microscopy on} thin, transparent

films of _a magnetic gainer ^{of uniaxial}

magnetic

anisotropy, ^{and of} composition

(YGdTm)3(FeGa)~)Oj~,

_{grown on}

single crystal

substrates of

gadolinium gallium

gainer

(GGG)

of

Il1)

orientation to a thickness of about 13 _~Lm. The

experimental

arrangement ^is

described elsewhere

[6].

An ordered initial state was

produced by cooling

^the

sample

^{from the}

paramagnetic

state in _a small

(Hi1Oe),

constant

magnetic

field

[6, 8, 14].

The ordered bubble lattice _so

produced

was

slowly cooled,

_over a

penod

of

approximately

two hours, from

approximately

190°C _{to room} temperature,

24°C,

to generate

increasingly

disordered patterns.

Video

images

were recorded _on tape or disk

[6, 7],

to

permit analysis

of frame sequences in

single

_steps

(of 1/30 s), required

to resolve details of bubble

shape

distortions and

collapse mediating

the motion of dislocations. Values for the modulation

period, d,

_were obtained from the

position

of the first harmonic _in

azimuthally averaged

Fourier spectra

computed

from each pattem.

Following high

pass

filtering

and

simple global

binarization based _on

identifying

the threshold with the _mean of the gray scale

histogram,

bubble _area

histograms

were constructed

on the basis of _a

simple region filling algonthm

which facilitates

counting

of interior

pixels

and

simultaneously

generates a point pattem

composed

of bubble centroids.

Topological

defects

were located in the Voronoi

diagram

denved from this

point

pattem

[14, 22].

3. Dislocation.mediated stress relaxation _m bubble patterns.

Cooling

from the

paramagnetic

state in a constant, small field

produces

bubble paiiems ^Such as

those in

figure

1,

typically composed

of

large

ordered

regions (Fig. IA) separated by

_grain boundanes. As with conventional

polycrysialline materais,

these grain boundanes _are forrned

by

chains

(or arrays)

of dislocations

[23],

_{as is} apparent ⁱⁿ the Voronoi

diagram (Fig. lB).

and

they

_may thus _{act as sources} and sinks of dislocations.

Fig.

I. -Grain boundanes observed _at T =184.5 °C (10.99

Tc)

in ordered initial bubble domain

pattem (A),

generated

by coolmg ^from the paramagnetic state in a small magnetic ^field [6], consist of chains of dislocations. The Voronoi diagram (B) _contams 5-sided and 7-sided polygons, shaded hght gray and dark gray,

respectively,

which mark the positions ^of 5-fold and of 7-fold coordinated bubbles

m

the original _pattem. The penod is approximately 9.5 _~m, the horizontal dimension of the field of _view

in

each panel _is approximately 195 _~m.

Stress may be induced in _an

initially

ordered demain pattem

by

tuning the modulation

penod, d~(T),

_i>ia ils temperature

dependence along trajectones

of constant overall magnetiza-

lion in the

(H, T)-phase diagram il, 6]. Figure

2 contains _a

plot

of the

experimentally

deterrnined temperature

dependence

^{of d} for the material under consideration here. The

analysis underlying

the fils of the data in _zero field in

figure

2A is based _{on a}

scaling

argument

[24] yielding

_a

prediction

of the form

d~ ~o(t) t~",

where _t

m

Tc

T denotes _a

reduced temperature. ^To

parametrize

the

experimental

data, fils to the

suggested

_power law with functional form

d~(T)

= A

(Te T)~

_were carried _{ouf ;} A and

Tc

were vaned in ail fils, while the exponent, z, was varied _{as an} additional parameter, or held fixed _at

z = IN in the fits

actually

shown in

figure

2. Fils with fixed _{z were} based _{on a} subdivision of the

probed

temperature

innervai,

_as indicated in the

figure, leading

to results far superior to those obtained from fils of the entire data set with

floating

_or with fixed _z. Of _most immediate interest _to the

present exposition is the obvious faon that

d~(T)

exhibits the

expected

increase with

decreasing

temperature. ^In

addition,

while _no claim is made _{as to} the uniqueness of the power law

representation,

_{or to} the values for

Tc

extracted from it, distinct regimes of temperature

O.is

j

~~~

1

a

~

~

j o 1

- _j

0.05

~~ ~----~

.04 ~

184 ¹⁸⁸

140 155 170

mperature

(°C)

0.30

o o

0.20 _o

E ~

à

o °

o o

~° o

a ^°

O.lO o ^°

~20 _{50 80 l10 140} 170 200

Temperature (°C)

Fig. 2. -Temperature dependence of modulation penod,

d~(Tj,

determmed for stnpe pattem at

H 0 Oe IA) and for bubble pattem at H

= I Oe (B) _on the basis of azimuthally averaged ^2d Fourier spectra computed from the pattems. ^The temperature dependence shown here _is

completely

reversible, while the accompanymg pattem

morphology

is

generally

net [3, 6]. As discussed in the text, solid fines _m (Al were derived from fits to the functional form

d~(T)

₌ A

(Tc

T)~, with _z fixed _at Il and

respective

optimal values for

Tc

of 209 °C and 192 °C for the _two separately analyzed temperature ^intervals

mdicated in the

figure.

The msert to (A) also mdicates that, as T

- Tc, d~ (T) approaches a fimte limiting value

de

> 0.

dependence

of

d~(T)

in the

vicinity

of the

(mean-field)

critical point are

clearly

identified. The

origin

of these

regimes

is _an issue which will be revisited in the Discussion below. The lower temperature regime,

probed by

the-data for bubble patterns in

figure 28,

exhibits obvious deviations from the

scaling

behavior

just

discussed _; this is non

surprising

in view of the

garnet's ferrimagnetic

nature. Pertinent _to the present ^{discussion is the}

continued,

_now

evidently super-linear

increase of

d~(T)

with

decreasing

temperature.

Dislocations _are forrned in response to the

accumulating expansive

strain _to facilitate the

requisite

^reduction in the number

density

of domains. In the _case of

expanding stripe

pattems, the

ejection

of

stripes by

dislocation chmb

[6] actually

suffices to

maintain,

albeit _ma

disordered

transients,

the ordered lamellar

stripe configuration.

For the _case of

expanding

bubble patterns,

subjected

to the condition of

maintaining

constant area fractions of the two

coexisting magnetization

states, the ordered initial _{state is} non

preserved

under the

imposed

lattice

expansion.

While

possible

in

principle

_via simultaneous

ejection

of _{excess rows in} the three symmetry

directions,

this would

require

_a directional correlation between dislocations

existing

in the pattern, ^{and this is} non

observed,

dislocations instead appeanng with random

orientation

(modulo w/3).

The ensuing evolution of disorder _via

prohferation

of dislocations

and _an

inherently

_ansing

polydispersity

_were the

subject

of _recent

quantitative analysis [14].

The condition of _constant «coverage»

imphes simple proportionality

between

(A~),

A6 denoting

the _area of bubbles

decorating

6-fold sites, and the

hexagonal

unit cell _area of the

ordered bubble pattem, A~~~, so that

(A~) ~A~~~ q~~,

where q m

[q~(T)[

_represents the

modulation _wave number.

Figure

3 demonstrates that the _mean domain _area,

(A),

denved

from _{an area}

histogram [14],

also scales _as

(A) ~q~~

_This

_{implies (A)} (A~),

_an

io3

~

ÎÉ

ù9

~~

~

~b Z

iol

L

j

.~ g i~

<

~

~ oe

~

~~

~

Ii

fi

o

0 1000 2000 3000 4000

q2 arb. units

Fig. 3. Modulation _wave number (q-) dependence of _various geometric quantifies describing magnetic bubble domain patterns. ^{Sobd fines} m the plots of

(A)~~

and of AN

w

N -N

[P'i-s,

q~ connect the respective rightmost point,

corresponding

ta the ordered initial state, with the engin. Solid and dashed fines _m the plot ^of (f~ q)~^~ vs, q~ serve ta

guide

the eye. Values for q were determmed from azimuthally

averaged

2d Fourier spectra.

observation consistent with that of _constant coverage

il 8]. Explicit inspection actually

reveals that, _to better than 5

fé, (A~)

=

(A),

_a condition which appears

surprising

in view of the asymmetry ^{of the domain} area

histogram [14],

manifest in the difference between relative

magnitudes

of _{successive mean areas,}

(A~) (A~)

_~

(A~) (A~),

_as well _{as in} the

systematic

increase, ar5 < ar~ w ar~, of the

corresponding

standard deviations. This

point

will be further considered in the Discussion.

Dislocations, generated

in response to the

temperature-induced

strain, move to the

edges

or to

nearby

grain boundanes to facilitate trie

requisite

reduction _in the number

density

of domains. As the

plot

in the lower portion of

figure

3 reveals, the disordered pattem ^of

(fixed)

hnear dimension

Lo

maintains a

roughly

_constant,

positive

_excess, AN

m N

(T) N(P~(T)

_~

0,

in trie number, N

(T),

of domains relative _to trie

«

optimal

_» number,

N(P~(T) L~/A~,

^for the

corresponding perfectly

ordered pattern,

composed

of six-fold coordinated domains of _area

A~. ^{AS will be described} below, this _{excess is} in fact accommodated _via the formation of

mterstitials. The

roughly

_constant AN suggests ^{that the} pattern remains at trie

yield

threshold,

forrning

dislocations and

eliminating (rows off particles

at a sufficient rate to track trie _increase

in modulation

period.

The average

(lateral)

distance between

dislocations, f~ né

'~~, deterrnined

simply by

their

density,

_nD, sets the scale for the

decay

^of translational correlations

il 6, 17].

The

dependence

of the related dimensionless

quantity (#~q)~~ ~nD/q~

_{on q~ suggests} the emergence of

asymptotic

level

corresponding

to

fD

_- 4 d.

The motion of _an isolated dislocation

through

_an ordered pattem, ^driven

by

_expansive strain

(and

the

resulting

Peach-Kohler

force),

is

depicted by

the

snapshots

_in

figure

4. It is apparent

~

a' ô~~

~-'~_~éW

Fig. 4. Snapshots m (A) and (B), separated in time by 1/6 _s, depict the translational _motion of _an isolated dislocation through _an ordered bubble pattern (Ai captures a transient state. The period is approximately 10 _~m. A unit translational step of _a dislocation _is recorded in (C) and (Dl, separated in

time

by

1/15 _s. The motion involves climb of

one of the

partial

dislocations marked

by

_{excess rows} termmatmg _in the dislocation _core. For each panel, the _area of interest _is marked by _a box of which _a two- fold

magnification

_is aise displayed _; the black dot marking the center of each 7-fold coordinated bubble

serves to

guide

the eye. The horizontal dimension of the field of _view for each panel is 570 _~m.

that the dislocation _moves via chmb of _one of its constituent

partial

dislocations

[23]

_in the direction of _one of the _{excess rows}

terminating

in its _core. Careful

inspection

of the

elementary

translation, shown in

figure

5, ^reveals this stop to involve the elimination of the 5-fold coordinated bubble of the

original

dislocation _core.

Employing unpriJJied

and

primed

numerals to

refer, respectively,

to domains of the indicated coordination

befoie

and

after

the unit

(a) Translation of Dislocation

o

b=<iio>

O

o

o o Il

~

°

o ~

o

~

Î~Î _°@Î ~ll

o

,, ~ ~ o ~',,

~

o /~ _o ^° _o ^°

^'

<oik-àoo> ^~

u=<ioo>

(b) NetworkTransformations

~

@'

s

'~

d

'~ 6

-1

+1

Elimination of 5-sided cell

-i (Ae ₌ -1)

Fig. 5. Schematic

representation

m (A) _{serves to} illustrate the process

underlymg

climb of _a

partial

dislocation, resulting _{m a}

displacement along

the vector _u. Indicated in the

original configuration

_are the

core

(striped

and hashed circles) of the dislocation of Burgers vector b, and the hashed cell surrounding a

6-fold coordinated bubble which will _{tum into a} 7-fold coordinated bubble. The requisite elementary topological transformation for this translational dislocation motion,

applied

to the Voronoi

diagram

associated with the bubble pattem, ^involves elimination of _one 5-sided cell, _as sketched in (B). This

topological

_{process is} detailed in the inset : sohd fines represent polygonal cell edges _: of these, the _one marked _« X

» will be elimmated, leading ta appearance of the _{two new} edges mdicated by dashed fines.

Numerals mdicate the change m the number of

edges

of

adjacent

cells [29].

translation, and

defining

the

symbols

« Î

», « % » and _« 4 » to

signify, respectively,

the operations

«collapses»,

_«contracts into» and

«expands into»,

the process may be

schematically

summanzed _as follows

51; _{7%6'; 647'; 6%5',}

In

regular crystals,

dislocation climb generates ^either vacancies or

interstitials,

and the requisite mass transport

by

^diffusion

generally

_represents the

rate-limiting

step

[25].

The spontaneous elimination of bubbles from

magnetic

_pattems

by collapse

removes the constraint of conserved interstitial transport,

thereby facilitating

dislocation climb. Pure dislocation

glide

has in fact not been observed under the

experimental

conditions

pertinent

here.

Dislocations first appear m any finite field of observation

containing

_an ordered pattern ^of

typically

0.5 _x 0.5 ~Lm2 area via translational motion

(Fig. 5) originating

at sources,

possibly grain

boundaries, externat _to the observed

patch.

The

Subsequent

dissociation of dislocation

cores is a

frequent

_{occurrence in} the

early

stages ^of

coarsening.

As

figure

6

demonstrates,

the

net

Burgers

vector is conserved

by

this

splitting

_process whose

elementary

_{stops are} shown in

figure

7 to involve the

consumption

of three bubbles.

~Ç ^~&

o

w-

O

' -

4w w~

*W ~

~* *W MW

W+ _{*% -}

*W _,~ ~

- ~M M+

Ww - ~w

++ %- W4

Fig.

6. Dissociation («

sphtting

») of dislocation _core IA) yields two dislocations (Bj whose

Burgers

vectors sum to that of the original ^{defect. The} penod ^{of the} pattem, recorded _at 132 °C (= 0.87

Tc

j, is

approximately

10 _{~m ;} the horizontal dimension of each field of view _m (A) and (B) is 190 _~m. The _area marked in the original pattem (Aj/(B) is

displayed

twofold magnified in the upper/lower portion ^{of the} central panel ^{the black dot}

marking

the center of each 7-fold coordinated bubble _{serves to} guide the eye.

As dislocations

proliferate,

clusters of dislocations

begm

to appear. Prevalent among these

are

pairs

of dislocations of _{zero net}

Burgers

vector, illustrated

by

^the pattem ^{and its} associated

Voronoi-diagram

_in

figure

8. These pairs are identified _in

figure

9 _as virtual pairs and interstitials

[9, 26].

This assignment may be venfied

by

_companng the number of bubbles

(a) Dissociation of

Dislocation Gore

O ~l

O O O O _/~

O O

_/~

O fJ

O fT O O

~ _O

~~j~

^-

° °

o ~ÎÎ~ O

O

° O ~O

O O

,

o ~Ç

O O

_, ,

~

^,

O

b

b'

(b) Network Transformations Mediating

Gore Splitting

~1

d d

d

^~~~

-1

3

~i

~

_~

~~

6

~

d

^~~

@

Fig.

^7. Schematic representation m (A) illustrates the process

underlymg

the dissocation (« splittmg ») of _a dislocation _core. Indicated in the

original configuration

_are the dislocation _core (striped and hashed

cirdes), the _two pairs of 6-fold coordinated bubbles from each of which will emerge one 7-fold coordinated bubble, and,

finally,

the hashed cells surrounding 6-fold bubbles which will emerge as 5-fold coordinated bubbles. The requisite elementary topological transformation for this _core splitting process,

applied

to the Voronoi diagram associated with the bubble pattem, _may be

represented

in forrn of the (successive) ehmination of three 5-sided cells, _as sketched in (B) _: step (I)

- (II) represents a translation, identical _to that _m figure 4 the _new dislocation is generated in step (II)

- (III) _; the final

configuration

is reached

by

another translation, (III)

- (IV), of the

newly

created dislocation.

enclosed

by

a circuit about the defect pair with the number of bubbles enclosed

by

the _same

circuit in the ordered reference _{state :} this

procedure yields

a

vanishing

differential for the

virtual pair and _{an excess} of _one bubble for the interstitial for _a vacancy, not observed in the patterns ^discussed here, _a deficit of _one bubble would result.

, .,

>

"~

i Î

~

~

~~ ^"'

'~

Jj

j

~,

O ~T

~

W ', ~)

-~

~l #

~t

Fig.

8. Disordered bubble pattern ^{and associated Voronoi}

diagram

[14], revealing a high

density

of dislocation pairs of _{zero net} Burgers vector in the forrn of virtual pairs (- vP) and interstitials (- I).

Several _instances of

incipient unbinding

of disclinations (-d) _are aise apparent. ^The

penod

of the pattem, ^recorded at 36 °C (= 0.66

Tc

), is approximately 22 _{~m ;} the horizontal dimension of the field of view is 850 _~m.

Virtual

pairs

were observed to nucleate

spontaneously

in

magnetic

bubble domain patterns

subjected

_to shear, their ionization generatmg pairs of

counterpropagating

dislocations

[27].

In the present

experiments,

virtual pairs formed _as the result of the recombination of dislocations of

opposite Burgers

vector, a process

facilitating

the ehmination of _excess

(half~

rows. The

core of the

resulting

virtual pairs represents a metastable

configuration.

As indicated in

figure 9,

_its elimination requires a redistribution of

edges

between

adjacent

5-sided and 7-sided

polygons

_in the associated Voronoi

diagram.

This

edge exchange

_is identical _to the

(edge

and hence cell

conserving)

Tl

elementary

transformation

mediating

^the coarsening of

polygonal

networks, _or froths

il

3,

28].

Virtual pairs were found _to persist

inspite

^of

being

unstable to this

process. The

requirement

for bubble size

adjustment,

a

geometncal

_response to be made

concurrently

with the

topological edge exchange,

may be cited _{as a}

hkely impediment

to the recombination of virtual pairs.

Interstitials accommodate the _excess of domains, AN, contained in disordered pattems

relative to the ordered reference state

[14].

The annihilation of these defect

configurations

_is

therefore

predicated

_upon the removal of domains via bubble

collapse.

As illustrated in

figure

10, trie annihilation of interstitials

by just

such _a

collapse

_process,

accompanied by

the

(non-conserved)

transfer of

magnetization

from the

vanishing

5-fold to an emerging 6-fold bubble, _is

actually

observed in the _« later _» stages ^of coarsenmg. Interstitial annihilation _is

readily accomplished by

the ehmination of _a 5-sided cell from the Voronoi

polygon

associated with the bubble pattem. ^This process, detailed in

figure

11, _is identical _to that

postulated [29]

for the elimination of 5-sided cells from magnetic ^{froths of}

polygonal

^bubbles

subjected

to increasing

applied

field where it appears to control the rate of coarsening

[13].

Given the

Virtual Pair Inierstiiial (ôN

=

0)

(ôN ₌

+1)

fÎl

o o O

o

Collapse

of 5-sided cell Tl

,

Ill

~

° o

lll~llll

-

o)o .(

-

o

°

~ ^°

~

°

Fig. 9. Schematic

representation

of virtual

pair

IA) and interstitial (B), each forrned by _a pair of dislocations of _{zero net} Burgers vector. Open, striped ^{and hashed circles} respectively _represent 6-fold, 7- foId and 5-fold coordinated domams. Aise indicated (« _- ») _are CCW _circuits about the defect pair employed to establish the mismatch, ôN, in the number of enclosed domams relative _to that expected for the ordered reference state. Sketched below each defect pair is the

topological

transformation which,

when

applied

to the Voronoi

polygon

associated with each bubble pattem, ^{would eliminate that} defect.

Required

for the annihilation of the virtual

pair

_in (A) is the TI process of

edge

exchange, while the simplest local transformation _{ta remove} the interstitial in (B) is the elimination of _one 5-sided cell j29]

(see also Fig, 10).

temporal

resolution

(of

1/30

s)

afforded

by

^standard video

recording,

the

collapse

of _one 5-fold

bubble _is

virtually mdistinguishable

from the

«symmetrical»

altemative,

namely

the

coalescence of _two 5-fold coordinated bubbles _in the interstitial.

Apphed

to the associated Voronoi

diagram,

coalescence

corresponds

to the inverse of

mitosis,

or

polygon

_(« colt

»)

division

[28].

The elimination of

topological defects,

_{e-g- i,ia} the annihilation of interstitials described

here,

would facilitate _a retum to an ordered state of trie pattem. ^A necessary condition for trie

occurrence of this

particular

process appears to be the requirement ^{that the} area,

A~,

of the _one 6-fold bubble

replacmg

the two 5-fold bubbles not exceed the maximal _area

consistent with 6-fold coordination. This condition

apparently

prevents the annihilation of

interstitials at

high

number densities of bubbles _in the initial stages ^of

disordenng.

4. Discussion.

The formation of ordered pattems ^of given modulation

penod,

d

m

d(H, T),

in a

sample

of hnear dimension,

Lo,

_requires the realization of trie correct number

density,

n

Lo/d,

of stnpes

or _n

(Lo/d)~

of bubbles whose correct size, i e. width in the _case of infinite stnpes, and _area,

Fig.

10.

Snapshots,

separated in time by 1/30 _s,

recording

the elimmation of _an interstitial from _a disordered bubble pattern ^the middle

panel

captures a transient state. The _area marked _in the original pattem is

displayed

twofold magnified ⁱⁿ the

right

hand portion ^{of each} panel. ^The

period

of the pattern, recorded at 36 °C (= 0.66

Tc),

is

approximately

19 _~m the horizontal dimension of the left hand portion, contaming the original pattern, is 285 _~m.

Annihilation of Interstitial (AN

=

~l,

Ae ₌

-1)

O O O

O O O O O

~ O

~ O

O O O

O O

_-

O

0~~o -

O O o

o o o

O O O O O O O

~

O O O

Fig.

I1. Schematic representation

illustrating

the

topological

transformation

mediating

the annihilation of _an mterstitial. The process relies _on the elimination of _a 5-sided cell from the Voronoi diagram

associated with the bubble pattem (see Fig. 4), resulting in a net change _m the number of domains of

AN ₌ I, and in the number of polygon

edges

of Ae ₌

A~,

^{in the} case of circular bubbles, _is determined

by

the

requisite

value of the overall

magnetization. Consequently,

the evolution of ordered

triagonal

bubble

pattems

under the

global

constraint of _{constant net}

magnetization imphes

A~

d~,

because each bubble occupies

a constant fraction of the

hexagonal

unit cell

[14].

Under these

conditions, penod adjustment

introduces stress m

dense-packed

stnpe ^{and bubble} arrays, and this may be accommodated i,ia

formation and motion of defects. A

simple

_process ^of stnpe rupture and dislocation climb

mediates relaxation in lamellar

stripe

domain pattems,

facilitating preservation

of the state of lamellar

ordering

via formation of

transiently

disordered

stripe

pattems

[6].

In contrast, bubble patterns

generally

^{evolve into disordered} states of hexatic order via the

proliferation

of

dislocations

[14].

The pattem ^{evolution of interest here} occurs in response ^to

adjustments

in the modulation

period,

d

=

d~(T),

_in a manner

following

the

dependence

descnbed in connection with

figure

2. Fits _to the functional form

d~(t

m

Tc T)

t~~~, shown _in that

figure,

confirm the

expected

increase of

d~

with

decreasing

temperature over most of the accessible range

pertinent

to the pattem ^evolution described here. However, in the

vicinity

of the

(mean-field)

critical point,

striking

deviations from this behavior manifest themselves.

First,

it is apparent ^that

d~ (T)

-

de

_~ 0 _as T _-

Tc.

^{This behavior} was in fact

predicted [24]

on the basis of

considering

the interaction between

increasingly

diffuse domain watts, also

expected

in the

regime

of weak

segregation

_near the consolute point ^of

binary

mixtures. While

a

quantitative

ireatment remains _io be

given,

^the existence of _a finite

limiting

value for ihe

modulation

penod

as T-

Té

suffices _to lead _{one to} suspect a

peak

in the

suscepiibiliiy

x~ ^{of the disordered}

phase,

1-e- above

Tc,

at the finite q =

2

gr/de,

as observed in low

temperature dipolar Ising ferromagnets [30].

The existence of such _a

peak

for _a

planar

system of azimuthal symmetry ^would

imply profound

fluctuation-induced modifications of the _mean-

field critical

behavior, including

the elimination of the critical

point,

and the

suppression

of the critical temperature

[31].

Such _a

lowering

^of

Tc

could in fact _account for the existence of distinct regimes of temperature

dependence

of

d~(T)

_apparent in

figure

2A. This conclusion _is

suggested by

the

fils, displayed

in that

figure,

of

d~(t)

_{to a power} law with _a fixed

exponent

^of

z = IN _: this

analysis

^{would indicate that} the

primary

distinction between the _two

regimes

of

figure

2 is _a sudden

drop (of approximately

15 °C

according

to the

fits)

in the effective transition iemperaiure,

Tc.

There _is,

however,

an alternative

explanation, invoking

a transition between Bloch and

Ising

domain watt _structures

[24, 32].

Laser

Bragg scaitering experimenis

would offer _a

possible experimental approach

to ascertam the effects of fluctuations above

Tc

and _to

help

resolve the

origin

of the observed behavior.

The evolution

dynamics

^of

magnetic

bubble pattems ^is charactenzed

by

_a close

interplay

between

topology

and geometry. The

topology

of

magnetic

bubble pattems is

faithfully

represented by

the

Delaunay triangulation,

_a

planar graph containing only

vertices of

degree

three centered _on the bubble centroid

positions,

and

representing

the dual of the Voronoi

diagram,

associated with _a

given

bubble pattem

[22, 28].

The _set of bubble _areas represents a

scalar field defined _on the

triangulation

and _must therefore

respect

^the constraints _so

imposed.

Magnetic

bubble patterns arrange in such _{a way} that each domain

adjusts

so as to occupy ail

locally

available space while maintaining a finite, minimal separation between domam

boundanes

[33],

the latter condition

representing

_a manifestation of

repulsive dipolar

interactions between domains

il, 2].

An immediate

implication

of this observation

[33]

is the

fact that the

availability

of

dilatation,

1-e- _area

adjustment,

and

eventually

of

shape

distortions

[3],

as

additional, geometrical degrees

of freedom avoids the open areas associated with

«

jammed

_»

configurations charactenzing

random

packings

of

ngid

disks

[34].

The _area of _a given bubble _cannot be

adjusted

without

eventually adjusting

bubble

coordination. Thus, for

given

net

magnetization,

or bubble _« coverage », bubble

coordination,

n, and bubble _area, A, _are net

mdependent

variables, ^but are instead

subject

to conditions goveming static pattem

configurations

as well _as evolution

dynamics. Polygonal

froths

display

an

interdependence

of

shape

and _size of

polygonal

cells descnbed

by

_a

joint probabihty

distribution,

p(n, A)

of cell coordination and _area

[28, 35].

A

corresponding

distribution should

apply

to the Voronoi

diagram

associated with _a bubble pattern, ^{with the} consequence

that

larger

bubbles

display higher

coordination. In support ^{of this view} we have the trimodal bubble _area

histogram characterizing

disordered patterns

[14], restricting

the form of the

requisite

distribution

p(n, A)

so that

(A~) (A~)

~

(A~) (A~)

^for the _{mean areas,}

(A,,)

=

Ap (n,

A

dA,

of n-fold coordinated bubbles, and

OE~ ^< ar~ w ar~ for the

correspond-

ing vanances. In

addition,

the pattems ^{of interest here conform} to the conditions

N~

=

N~

and

N~

₊

N7

+

N~

=

N

expressing

the fact that

topological

defects appear, ^over the range of disorder

investigated

here,

exclusively

in the form of

pairs

of

(±1) disclinations, respectively

marked

by

5-fold and 7-fold coordinated bubbles.

Invoking

these

conditions,

it _is

readily

shown

[36]

that

(A ) (A~)

= c~~

(2 (A~) (A~)

₊

(A~) )).

That is, the difference

between

(A~)

and

(A)

is determined

by

the number

density

of defects, _c-D

m

N~/N

=

N~/N

and

by

the curvature of the function

(A~)

=

(A,,) (n ).

The

sign

of the asymmetry ^{in the} mean

areas cited above

implies

that the latter is _in fact finite and

positive

for the distribution

p(n, A) pertinent

to the patterns

investigated

here. With

typical

numbers extracted from the

most disordered pattern encountered here

[14],

_{1-e- CD}

4

0.25, (A~)/(A~)

-0.593and

(A~)

/

(A~) 10.774,

_one indeed finds that

( (A) (A~) )/ (A~)

_< 0.03. The

explicit

form of

an

appropriate p(n, A)

for _a disordered bubble pattern ^{would have} to take into _account the

expected n-dependence

of the ratio

(A,~)

^/

(A$°~)

^{of the} mean area of _an n-fold bubble and that

of the n-sided Voronoi

polygon, although

such _a

p(n, A) only specifies near-neighbor

correlations and will

hkely

not be

unique [36].

A

topological

characteristic of the evolution

dynamics

of magnetic ^bubble pattems

mvestigated

here is the conservation of the

triangulation.

More

specifically, only topological

excitations of unit

charge (±1)

_appear in the temperature range accessed

by

the pattern

evolution

reported

here. These

topological

restrictions

fundamentally

limit the set of

possible

pattem

configurations, eliminating

in

particular configurations

otherwise observed in

planar

patterns ^such as the

droplet

_pattems

produced by

_vapor

deposition

_on surfaces

[37]

and

notably

the disordered bubble pattems

frequently

encountered in

Langmuir

films

[4, 38]

whose

topology

cannot be

represented by

_a

planar graph [33].

Within the framework of _a

recently proposed

model for the evolution

dynamics

of dense bubble patterns, ^{the existence of} a

topological

conservation law and the absence of spontaneous,

temporal

coarsening, a property associated with

magnetic

bubble patterns, are

intimately

related

[331.

The evolution of disorder in

dense-packed

bubble patterns is mediated

by

the

proliferation

of dislocations,

coupled

with the

polydispersity arising

from the

availability

of bubble dilatation

as an additional

degree

of freedom _to accommodate stress

il 4].

The disordered pattern _may ^be viewed as a « frustrated' _{» state} of the system ^for given field, H, and temperature, ^T : while _an ordered

ground

state

realizing

the _correct modulation

period, d~(T),

is

always

well-defined,

relaxation _to this »reference« state is

prevented by

an excess of

bubbles, AN,

and

by

the presence of

topological

defects

generated

in the _course of

penod adaptation.

As with the evolution of

labynnthine

_stnpe domain patterns ^from an initial lamellar state

(6, 7],

bubble pattern ^{evolution would} seem to be

pnmarily

determined

by

the requirement to realize the

correct modulation

penod

^for given H and T. That _is, the

emerging

pattem

morphology

_may be

regarded

to represent ^the constrained

optimization

of the pertinent ^froc energy functional

il, 2, 6, 7]

under the condition of fixed coverage and given number

density

of

bubbles,

and hence _in

the presence of _a preset concentration of

topological

defects.

The

intriguing possibility

of

enabling

_processes which contribute _{to a «}

healing

_» of _a given disordered pattem

by deletmg topological

defects and

thereby facilitating

a relaxation into the ordered reference state _is in marked contrast to the coarsening of

polygonal

froths from _an

initially

^ordered state in that

situation,

the _existence of _a

surgie topological

^{defect in} the form of _a 5-7 _pair of

polygons

_may

trigger

the destabilization of the entire pattern

[13,

28, 35, 36]. It

will be

interesting

to see whether suitable constraints may be

placed

_on model evolution

dynamics [33, 35]

_to

investigate

the

possibility

of re-entrance into ordered states. On the

experimental

side, examination of _more advanced states of disorder than those reahzed in the present

study

^{will have} to reveal whether such _reentrance does _in fact _{occur, or} whether

additional mechanisms of

topological

disorder, for

example

_m the form of disclination

unbinding (Fig. 8), predominate

the further evolution of disorder into _an

amorphous

state

il

4, 16,

17].

Acknowledgements.

The work

presented

here has benefited from conversations with R.

Seshadri,

C. Volkert and R.

Wolfe,

from _a

variety

of incisive observations offered

by

M.

Magnasco,

and from the

comments of _a

pair

of cntical reviewers.

References

[1] Garel T. and Doniach S., Phys. Rev. B 26 (1982) 325.

[2]

Cape

J. A, and Lehman G. W., J. Appt. Phys. 42 (1971) 5732.

[3] Seul M. and Andelman D., _m

preparation.

[4] Andelman D., Brochard F. and Joanny J.-F., J. Chem. Phys. 86 (1987) 3673 Môhwald H., Ann. Rei,. Phys. Chem. 41 (1990) 441;

Mcconnell H. M., ibid. 42 (I99I) I7I,

Knobler C. M, and Desai R. C., ibid. 43 (1992) 207.

[5]

Rosensweig

R. E., Ferrohydrodynamics

(Cambridge

Univ. Press,

Cambridge,

UK, 1983).

[6] Seul M. and Wolfe R., Phys. Rei,. Lent. 68 (1992) 2460

Phys.

Rei,. A 46 (1992) 7519 and ibid.

(1992) 7534.

[7] Seul M., Monar L. R., O'Gorrnan L. and Wolfe R., Science 254 (1991) 1557

Seul M., Monar L. R. and O'Gorman L., Philos. Mag. B 66 (1992) 471.

[8] Molho P., Portesefl J. L., Souche Y., Gouzerth J. and Levy J. C. S., J. Appl. Phys. 6t (1987) 4188.

[9] Seshadri R. and Westervelt R. M., Phys. Rev. Lett. 66 (1991) 2774 Phys. Rei. B 46 (1992) 5142 and 5150.

[loi Zavadskii E. A. and Zablotskii V. A., Phys. ^{Stat. Sol.} A tt2 (1989j 145.

Il Seul M. and Wolfe R.,

unpublished.

[12] Cebers A. O. and Maiorov M. M.,

Magnetohydiodynamics

t6 (1980) 21

Thiele A. A., J.

Appl.

Phys. 4t (1970) II 39 _; Mcconnell H. M., J. Phys. Chenl. 94 (1990) 4728.

[13] Babcock K. L. and Westervelt R. M., Phys. Rev. Lett. 63 (1989) 175

Babcock K. L., Seshadri R. and Westervelt R. M., Phys. Rei,. A 4t (1990) 1952.

[14] Seul M. and Murray ^C. A., Science 262 (1993) 558.

[15] Pusey ^P. N., J. Physique ^{Fiaiice 48} (1987) 709

Bartlett P., Ottewill R. H. and Pusey P. N., J. CheJJi. Phys. 93 (1990) 1299 _; Rodriguez B. E. and Kaler E. W., Langmuir 8 (1992) 2376 and p. 2382.

[16] Nelson D. R., Rubinstem M. and

Spaepen

F., Philos. Mag. A 46 (1982) 105.

[17] Rubinstein M. and Nelson D. R., Phjs. Rev. B 26 (1982) 6254 _; Barker G. C. and Grimson M. J., J. Phj,s. Cl (1989) 2779

Thorpe

M. F. and Cai Y., Phj's. Rei,. B 43 (1991) IIOI9 _; McRae R, and

Haymet

D. J., J. Chem. Phys. 88 (1988) II14.

[18] With the exception ^{of the} symmetry axis H

= 0 of the (H, T) phase diagram,

trajectones

with

constant H do _not preserve overall

magnetization.

However, the

trajectory

H

= I Oe, followed

in the present

experiments,

_was found, lia evaluation of histograms, to closely _preserve the

area fraction of mmonty (« white ») bubble phase at a value of 45 §b _± 2 %,

mdicatmg

the overall magnetization to remain

approximately

constant as weII.