Domain structures in thin layers of a ferrocolloid

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Submitted on 1 Jan 1993

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Domain structures in thin layers of a ferrocolloid

Yu. Buyevich, A. Zubarev

To cite this version:

Yu. Buyevich, A. Zubarev. Domain structures in thin layers of a ferrocolloid. Journal de Physique

II, EDP Sciences, 1993, 3 (11), pp.1633-1645. �10.1051/jp2:1993223�. �jpa-00247930�

Classification Physics Abstracts

75.50M 75.70K 82.70

Domain structures in thin layers of

_a

ferrocolloid

Yu. A.

Buyevich

(~,~) and A.Yu. Zubarev (~)

(~) Mod41isation

en M4canique, Universit4 Pierre et Marie Curie, 75252 Paris, France

(~) Department of Mathematical Physics, Urals State University, 620083 Ekaterinburg, Russia

(Received

16 March 1993, revised 15 July1993, accepted 5 August

1993)

Abstract We consider the problem offormation ofstructures composed ofidentical domains

in _a ferrocolloid that fills _{a space} between two closely situated parallel planes under action of

a normal external magnetic field. Those structures begin to arise under conditions favouring the phase separation of the ferrocolloid. An inhomogeneous system

including

two equivalent

colloidal phases would ultimately evolve due to Ostwald ripening, but for

a scaling effect of the

planes that stops ^the evolution at _a certain level and makes for the resulting domain _structure being stable. A method to find out main characteristics of the domains, such _as their size and

concentration, is developed for both chaotic and regular

(ordered)

structures.

1 Introduction.

Numerous

experiments,

_among which those in [1-5] ^{should be}

cited,

evidence _a ferrocolloid

layer

between

parallel

boundaries to lose its

thermodynamic stability

if the

strength

of _an

externally applied

normal

magnetic

field exceeds _some critical value. As _a

result,

_a system ^of

cylindrical

domains

develops,

the concentration of

magnetic particles

inside the domains

being

considerably higher

than that in the ambient parent colloidal

phase.

Such _a transformation from _an

original homogeneous

state to that with _a stable domain arrangement ^resembles to all appearances a

phase

transition of the first type. ^{A similar}

phenomenon

of the _emergence of domains is also

specific

to thin

layers

of _an

electrorheological suspension

under action of

a constant electrostatic field [6].

Visually,

the mentioned domains _are like domains in solid

magnetics

_or vortice structures that _are sometimes

generated

in

superconductors

of the second type.

The authors _{are aware} of the

only

attempt to

theoretically explain

the

origination

of the domains

by assuming

the _reason of this

phenomenon

_to be due to the break of

thermodynamic stability

of ferrocolloids in

sufficiently

strong ^{fields and} to _an

ensuing

_process of

spinodal

decomposition

_{[7, 8].} The

theory

of these _papers

leads, however,

to _a conclusion that the

particle

concentration has to vary

smoothly,

almost

sinusoidally, along

the

layer plane.

Such _a conclusion does not

correspond

in the least to available

experimental evidence,

accord-

ing

to which all the domains _are

practically

^of _a

cylindrical shape

and _possess

easily

discernible

sharp

boundaries. This cannot be

explained

with the aid of almost

sinusoidally varying

_concen-

trational fields.

Moreover,

it has been shown _{in [4]} that there is

merely

_a

comparatively

small number of

randomly positioned

domains if the external field is either weak _or of

reasonably

moderate

intensity,

_so that the domain

assemblage surely

reminds of _a two-dimensional ideal gas. The domains become

capable

of

forming

_an ordered lattice

only

at a condition of their concentration

being sufficiently high,

what demands _a rather

strong

field. At

last,

it has been

rigorously proved

_[9] that the

layer

boundaries cannot exert a

stabilizing

influence _{on an}

early

stage of

spinodal decomposition

which

happens

to be relevant to the

theory

of [7, 8].

A main

physical

_reason for the

origination

^of domain systems seems to be clear

enough.

The

dipole interparticle

interaction in

magnetic dispersions

results in _an effective attraction between the

particles

which

significantly

increases _as the

strength

of _an external

orienting magnetic

^field a

dispersion

is

exposed

to grow [10, 11]. ^{The attraction tends} to _{cause con-}

densation of the

particles

whereas the

particle

thermal motion opposes this process. If the

dispersion

concentration is

sufficiently high and/or

its temperature ^is

sufficiently low,

the dis-

persion thermodynamic stability

cannot be

maintained,

and _{a process} of

phase separation

of the

dispersion

into two

phases

with different concentrations of the

particles

evolves. Such _a process is

quite

^similar to that

commonly

^observed in molecular systems

undergoing

_a first type

phase

transition

[10-13]. Examples

^of transformations of the _same nature in colloids in

which the

interparticle

attraction is of another

physical origin

_can be found in

[14, 15].

A constant external

magnetic

field intensifies attraction of

particles

of _a ferrocolloid and _so favours its

separation

into two colloidal

phases.

The _process of

phase separation

starts with the formation of initial critical nuclei of _{a new}

phase

that

give

rise to

dropwise

aggregates

which _are

elongated

in the external field direction

[12].

^{If the} ferrocolloid _were

unbounded,

the

aggregates

^would

incessantly

_grow ^till Ostwald

ripening begins

to dominate the _new

phase evolution,

when

larger

aggregate continue to increase in size at the _expense of smaller _ones.

A finite distance between the

layer

^boundaries sets,

however,

_a limit to such _{a process as}

compared

with that in the bulk. When the aggregate size _comes to

equal

this

distance,

_any aggregate can continue to grow

only

in the radial

direction,

that is, in the

layer

plane. Then the

aggregate

^radius

eventually

_grows under _a condition of constant

length,

and the aggregate

virtually

transforms into _a domain. This

results,

of _course, in

strengthening

of the aggregate

demagnetization

^{field and}

ultimately

_{causes an energy}

expenditure

to become _too

high

_to

provide

for the domain

growth

_{over a} certain critical size.

In what

follows,

this

qualitative picture

is

proved

_to be _correct and main

properties

of

resulting

stable domain arrangements _are ^found out and discussed at _some

length.

2» Chaotic

arrangements.

Consider _an

assemblage

of

nearly cylindrical

domains of radius _a in _a ferrocolloid

layer

of thickness 21 between

parallel plates,

The ferrocolloid is

subject

to _an external

magnetic

^{field of}

strength

Ho which is directed

normally

to the

plates,

^Let _us introduce the volume concentration

#d

of

magnetic particles

in each

domain,

also _a surface concentration of the domains themselves that represents ^{the fraction}

occupied by

domain crossections in the

layer plane

and

equals

s =

gra2n,

_n

being

understood _as the number of the domains per unit _area of this

plane.

In accordance with the

majority

of

experimental

data, _{we assume}

right

_away the domains to be

thin,

what _means the domain radius _a to be much smaller than the

layer

thickness I, a < I.

Our intended

goal consists, first,

in

identifying

conditions at which there exists _a ther-

modynamically equilibrium

domain arrangement ^{within the} ferrocolloid

layer and, second,

in

finding

out

properties

of the

arrangement.

^This

implies

that not

only

the domain

shape

and their surface concentration but also the

magnetic particle

concentration and the

magnetic

^field

strength

both inside the domains and in interstices between them have to be

simultaneously determined,

to

incorporate

the

findings

into _a

complete

statistical

thermodynamic

model of the system under

study.

Construction of the last model is also

by

_{no means an easy} task.

Although

fundamentals of such _a model have been

recently

disclosed in

[11],

one can

hardly

hope

to overcome all obstacles and to

get

a

complete

solution of the whole

problem

^without

some decisive

simplifications,

which _{we are now}

going

to enumerate.

First of

all,

_{we presume} the domains to be

regular

circular

cylinders,

and their surface _con- centration

(which evidently equals

the domain volume concentration within the

layer)

to be small _as

compared

with

unity.

Then each domain _can be

regarded

_as immersed in _a ferrocol- loidal medium whose properties, such _as concentration of

particles by

volume

ii,

_{Pressure pi}

and

magnetic

field

strength Hi approximately

coincide far _away from the domain with their actual values which _can be got

by averaging

_over

merely

the interstices. The last

assumption

is made

solely

to

simplify

needed calculation. If it _were wrong, an

approximation

based _on

the self-consistent field

theory

could be

used,

in which _case the mentioned

quantities

should be looked _{upon as} those obtained

by averaging

_over the whole

layer plane. Obviously, quantities ii,

_Pi and Hi _can be

regarded

_as ^uniform.

Even

though

the above

assumptions

hold true, _{we are}

again

faced with _{a very} difficult

problem.

A crucial

simplification

_can be

attained, nevertheless, by noticing

that the

magnetic

field within _a thin

cylinder

normal to the

layer

^boundaries

depends mostly

_on the field H2 _near its side

surface,

and that the

demagnetization

field h vanishes _as

all

tends to _zero. It _means

that _{we are} free to

take,

in the zeroth

approximation,

the

strength

of the

magnetic

field inside

a domain to be

equal

to

H2

"

Hi

This field induces _a domain

magnetization Md

that has to

be found with the

help

^of _some model of

macroscopically

uniform ferrocolloids

(e.g.,

_see

ill] ).

Now,

let _us find the

demagnetization

field h caused

by

_one domain that _can

justly

be

regarded

as

uniformly magnetized

with _a

specific magnetization

Md. This _can be done

by summing

_up field contributions created

by

infinitesimal parts ^{of the domain. If} one

neglects

effects due to possible refraction of h at the

layer boundaries,

which have to be of little

significance

because of the

accepted inequality

_{a <} I, one gets ^for _any point _{ro near} the domain surface

fit ^I-zo a 2W ~2 1

hz izo,

To) =

T _{/~~~ dz£}

_r dr d@

13v v)

R~

=

r~

₊

r(

_2rro cos@ ₊

z~. (2.1)

This

expression

defines _a component of h

parallel

to the symmetry axis of the

domain,

and _zo and _{To are} axial and radial (To >

a)

coordinates of _ro in the

cylindrical

^coordinate system ^with

an

origin

at the intersection

point

of the donlain symmetry axis and the middle

plane

of the ferrocolloid

layer. Quantity ii

is the

magnetic permeability

of the ambient ferrocolloid. It is easy ^to show

that,

at _a < I, radial component hr of the

demagnetization

^field is much smaller than

hz

almost

everywhere,

_so that there is _no need to take it into account at

all,

and _vector h _may be

regarded

_as directed

normally

to the

layer plane.

It is convenient to _use

further,

instead of

hz(zo,

_To

),

_a value of this

quantity

obtained

by averaging

_over the

layer

thickness

hi (T0) _"

j /

_hz

_{(20,T0) dZ0, (2.2)}

without noticeable loss of _accuracy. Then _one is

always

able _to write

hi (To) " -MdL

(all, roll), (2.3)

where L is _a known function that _can be

readily

written out when _necessary, in

conformity

with the definitions in

(2.I)

and

(2.2).

Consider,

next, a

cylinder

in _a field of

strength H2

= Hi + hi directed

along

the

cylinder

axis and introduce

#2

and _{p2 as} the

particle

volume concentration and _pressure at the side

cylinder

surface outside the

cylinder.

Similar

quantities pertaining

to the interior of the

cylinder

_are

marked with _a

subscript

d. All the

thermodynamic potentials

of the uniform ferrocolloidal

phases

_are

admittedly

functions of the above-introduced variables. Furthermore _we shall _con- sider chemical

potentials pip, #, H)

and

po(P, #, H)

of the

magnetic particles

and of molecules of the

suspending liquid, respectively.

Conditions of

thermodynamic equilibrium require

the chemical

potentials

inside and

just

outside the

cylinder

to coincide.

Also,

mechanical

equilibrium

demands the _pressures at both sides of the

cylinder

to be

distinguished by

the surface _pressure

~la,

where _~ is understood _as the surface tension coefficient. Thus _we have

@d " @2 _, @0d " @02

, Pd P2 " ~

la,

Pj =

(vj, #j,

Hi + hi

, Poj = Po

(vj, #j,

Hi + hi

,

j ₌ d, 2.

(2.4)

Simple

^{evaluation shows} _{pd P2} to be

relatively

small.

By accounting

for this and for

thermodynamic

identifies

dp/dp

= u and

dpo lap

_{= uo, u} and _uo

standing

for the volumes of

a

particle

and of _a

molecule, respectively,

_{we get} from equation

(2.4) (PI, ~d, lfl

+

~dhl

_"

(PI, ~2, HI

+ ~2

hi

_~U

la,

PO

(Pi, ~d,

Hi +

godhi

" @o

(Pi, ~d,

Hi ₊

go2hi

_~uo

la,

gj =

~/~

Pi Ii,

^Hi

, goj =

~/~°

~ji(J,

^Hi

,

j

=

d,

~,

hi

«

Hi. j~.~~

Further,

the ambient ferrocolloid

phase

between the

cylinders

is at

equilibrium

if the chemical

potentials

_are uniform

throughout

interstices. Hence _we arrive at new equations

yielding

necessary conditions of the

equilibrium,

in much the _{same manner as} when

getting equation (2.5),

@(i'l,~2,lfl)+~2/l1"@(Pl,~l,lfl)+U(pl ~P2),

@0

(PI,

_{~2 HI + ~02}hi

" @0

(PI,

~l, HI _{+ U0}

(PI P2) (2.6)

Equations (2.5)

and

(2.6) together

with that

defining

the _pressure difference in equation

(2.4)

form _a set of five equations to determine

id, #2,

_{Pi P2, a} and

hi

_{at any} fixed

ii.

It

can be

solved,

after which all the indicated variables become known

functions,

of

ii,

within _a certain _range of the last

quantity

where _a solution

exists,

if _ever. In such _{a case, a} also

changes

when

ii

is

permitted

to vary. The

particle

chemical

potential

in the interstitial _space

depends only

_on

ii,

_Pi and Hi and is

independent

of _a, whereas that of the

particles

inside the domains

may now be

regarded

_{as a} function of _a under otherwise identical conditions.

The situation is

schematically

illustrated in

figure

I. If the domain radius is smaller than _al, say, a domain is bound to

eventually disappear.

If the radius exceeds another critical

value,

_a2,

the domain also dissolves, but

merely

down to _{a =} a2, when the dissolution _ceases. Domains

/Ld

lLe '

'

0 al ae a2 a

Fig. I. -Sketch of dependence of particle chemical potential in both domains and interstices _on domain radius. Explanations _are given in the text.

where radii lie within the _range from _{ai to a2 may} either _grown but

only

till the value of _a2 is

reached,

_or diminish _{to a}

= ai

and,

next, ^vanish at _a =

0, depending

_on what

phase trajectory corresponds

to _a monotonous

approach

of _{Ad to pi} Characteristic trends of the

development

of _a

single

domain _are

conventionally

indicated in

figure

I with _arrows, from which it is not difficult _{to see} that the domains under

question

_can be stabilized

merely

at a = a2.

Spontaneous

fluctuational

origination

^of _new domains results

inevitably

in _a decrease in the

particle

concentration within the interstitial space

and, consequently,

in _a fall of _pi This is

why

the whole system ^evolution causes a line _p

= pi = Cte. to _come

gradually downward,

until the line touches the _curve of _p

=

pd(a)

at _a

point

_{a = a~} where pi = Ad " @e. The last

equation

determines also _a critical concentration

iie

of

particles

outside the domains. The state that

corresponds

to this critical

point

_can be shown to be stable since _a further increase in the total amount of

magnetic particles

within the domains reduces

ii

below

iie,

and _so the

whole line _p

= pi = Cte. is

displaced

down with respect to p =

pd(a).

It

compels,

in its turn,

an excessive amount of the domain

phase

to dissolve

and, thereby,

leads to

reestablishing

of the critical

particle

concentration

specific

to the stable

equilibrium

situation. This concentration has to be

found,

in

compliance

with the above arguments, ^from an

equation

dpd(a) Ida

= 0.

j2.7)

Equations (2.5)

and

(2.6) supplemented

with

equation (2.7)

_{serve now} to find

id, #2, lie,

pi P2, ae and

hi

In order to close the

formulation, however,

_we need _a

relationship defining

Hi in terms of the other variables. An evident relation _can be put ^forward

Ho =

(I s)liHi

₊ sld

(Hi

₊

h), (2.8)

in which id is the

magnetic permeability

of the

domains,

the surface

(or

volume within the

layer)

domain concentration to be derived from _a condition of conservation of the overall

number of the

magnetic particles

which is

certainly

not affected

by

the domain formation.

Hence

s =

~°

~~~

,

(2.9) Id lie

where

io

stands for _an initial

particle

volume concentration in the _same ferrocolloid without domains.

Thus all the

equations

_necessary to resolve the

problem

_are stated.

They suffice,

in

principle,

to determine all

equilibrium properties

of _a ferrocolloid

layer

with domains _at

arbitrary io,

I and Ho What

solely

remains to be done in order to

bring

the

analysis

to a

completion

consists in

specifying expressions

for both

magnetostatic properties

and chemical

potentials

_on the basis of _some reliable model of ferrocolloids.

Judging by

_a

good _agreement

with

experiments,

the

model of

ill]

_can be

successfully

used in such _a

capacity.

Then

M =

~~

(cth

^a

(l

⁺

^8~fi j)

U ° ° s a ,

~ ~

~~'

°

~~'

_'~

i~~'

_~~'~~~

and,

further,

P(P, #, H)

₌

P° Iv)

+

VII, H), PolP, #, H)

=

PilP) lUo/U) Uo(#, H), v(j, H)

=

kT# in # #

in

(sh ala)

+

#(8 5#)(1 #)-2 _{jj2 #)GjT,} H)j

,

vo(#, H)

=

kT# jji

₊

#

+ #~

#~j (1 #)-3 _#GjT, H)j

,

G(T, H)

=

4~t(cth

_a

ila)2

₊

_{(4/3)~t2. (2.ii)}

Here

p°

and

p(

_are the standard chemical

potentials

of

particles

and

molecules, respectively,

m is the

particle dipole magnetic

moment and kT is the temperature in energy units.

Although

the relations in

equations (2.10)

and

(2.ll)

have been

meticulously

discussed in

ill],

it would be worth

while, perhaps,

to

give

_a short

explanation

of their

origin. Equations (2.10)

_were obtained _on the basis of _an approximate version of the hard

sphere perturbation

theory

which

happened

to describe

experimental

data

by

various authors in

quite

_an excellent way. The second contribution within the brackets in

equation (2.10)

is called _to allow for _an effect of the

interpartide dipole

interaction _on the

equilibrium magnetization

of _a crowded

assemblage

of

magnetic particles being subject

to a constant external

magnetic

^field. In the dilute limit

ii

<

I),

this

specific magnetization

turns to that

by Langevin

of _an ideal _super-

paramagnetic

_gas. Parameters _a and _~f reflect the interaction of the

particles

with the external field and between

themselves, respectively.

The _sum of the first three addendumda in the expression of _u in

equation (2.ll)

presents the chemical

potential

of

particles

of _an ideal

paramagnetic

_gas in _an external field of constant

strength

H. The forth addendum takes into account the steric interaction of hard

spheres.

It has been evaluated with the

help

of the equation of state of _{a gas} of hard

spheres suggested by

Carnahan and

Starling

[16]. ^{The fifth addendum takes} care of the

dipole dipole

interaction between the

particles

in the external field which is estimated

by

_means of the method

proposed previously by

Brout [17]. ^It can be _seen,

by

the _way, that interaction parameter ^G increases with

H,

what is indicative of _an intensification of the

interparticle

attraction due to the

magnetic

interaction between the

particles

^when _an ^external

magnetic

field is present.

Quantity

_uo coincide with the pressure of _a dense gas of hard

magnetic spheres

evaluated within the frames of the method of

ill],

_a deviation of the first term within the brackets

from

unity being

^caused

by

the steric component ^{of the}

interparticle

interaction, whereas the emergence of the second _term is due _to the

dipole dipole

part ^{of the} interaction. It is worth

noting

^that

equations (2.10)

and

(2.ll)

have been derived in [11]

by perturbation theory

methods which

ought

to be

adequate

if

~fi

is of the order of

unity

_or smaller. In _an

opposite

case of

large ~fi, equations (2.10)

and

(2.ll)

have _to be

thought

of _as estimates in _an order of

magnitude

^of

thermodynamic properties

^of a

highly

concentrated ferrocolloid. Such estimates

are

proved

in

[11], nevertheless,

to be

sufficiently

accurate.

Conditions at which the ferrocolloid under consideration loses its

thermodynamic stability

in the bulk and

ought

to

undergo phase separation

^{have also} been

investigated

in

ill]

at

some

length

_on ^the basis of

equations (2.10)

and

(2.ll).

The _same conditions

pertain

to _a domain arrangement

beginning

to occur in _a thin

layer

of the ferrocolloid _as well.

Equations (2.4)

and

(2.5) supplied

with expressions

(2.8-2.ll)

make _sense, of _course, if the

point

that

specifies

_an actual state of the ferrocolloid in the

parametrical

_space before separation ^lies within the

instability region,

that

is,

above the

phase diagram

_curve obtained in

[11],

in which

case those

equations

_can well be solved _to

yield

all the desired

equilibrium

characteristics of the ferrocolloid

displaying

the domains.

Calculation of dimensionless radius and surface

(volume

concentration of the domains with allowance for formulae

(2.10)

and

(2.ll)

is

quite straightforward,

however

tedious,

and _so

hardly

merits _a detailed discussion. Its results _are illustrated in

figure

2. The surface tension coefficient has been evaluated with the

help

of _a

special

model

developed

in [18],

according

to which

~ ~-

7~ (id Ii

_)~

u~~/~kT. (2.12)

The radius of stable domains decreases and the concentration of domains increases when either the external field

strength

_or the

particle magnetic

moment grows, in _a fair agreement with the

experimental

conclusions in [1-6].

ae ~e~2

1~3

0.04 0.07 ~ ~~

° l

mHo /kT

0 ₁

mHo /kT

a) b)

Fig. 2. Dependence of dimensionless radius

la)

and of number concentration 16) ^{of domains} on

dimensionless external magnetic field strength. Figures at _{curves on} b give Values of lo-

3. Ordered arrangements.

Chaotic domain structures

commonly

arise in ferrocolloid

layers

in

comparatively

weak _or mod-

erately

intensive

magnetic

fields. When the field

strength increases,

_a transition to _an ordered

hexagonal

arrangement of the domains _can be

foreseen,

which reminds _us, in its outward _ap- pearances, of _a disorder order

phase

transition. Such _a transition has been

actually

observed in [4]. ^An

elementary

cell of the

hexagonal

arrangement ^{is shown in}

figure

3. Below _{we are}

going

to demonstrate how the

theory developed

for chaotic structures is to be

generalized

to include ordered structures. It should be

pointed

out that experiments of [1-6] ^{confirm the} concept of the ordered domain arrangements

corresponding

to a

regular hexagonal

two-dimensional

lattice.

rc

Fig. 3. Representative cell of hexagonal arrangement.

Particle concentration, pressure and

strength

^of the external

magnetic field, ii,

_Pi and

Hi,

in the interstitial _{space can} be

presented

_as Fourier transformations with respect to coordinates

x and y chosen in the

layer plane.

Since the harmonic that possess the symmetry ^of a

hexagonal

lattice under

study

must be of maximal amplitude, _{we are} free to take, _as a first approximation,

2 2

~bl " ~b~ + _~bk

~

C°S

(~nr),

_PI

" P~ ⁺ ilk

£

CDS

(knr)

,

n=0 n=0

where quantities marked with _a

degree

as well _as

amplitudes

#k, _Pk and hk are so far unknown.

Let _us suppose that

~$~ $ ~$k, P~ $ pk,

lf~

$

/lk. (3.2)

The

validity

of these strong

inequalities

_can be checked later _on. It should also be noted that this

assumption

is not of

principal significance

and _can be avoided at the cost of

making

calculation somewhat _more cumbersome and

unwieldy.

The chemical

potentials

of both

particles

and molecules must be uniform

throughout

the interstices.

By using equation (3.2)

and all the former considerations _{we can}

consequently

write out

2

p =

p° (p()

+ kTu

(#[, Hi

+ kT

(upk

+

uhhk

+

u~#k) £

_cos

_(knr)

= Cte.

n=0

@0 " @(

(P~)

^kT _(U0

IV)

_U0

($, lf~)

kT _(U0

IV)

(U0Pk + "0h/lk + "0@~k) X

2

x

£

cos

(knr)

=

Cte., (3.3)

n=0

are evaluated at #

= #( and

H

=

Upk + Uh/lk _{+ U~~$k}

" 0, _U0i'k + U0h/lk + U0@~k " 0.

(3.5)

Further,

the

magnetic

field

strength

at the side surface of _a domain _can be

expressed

_as

2

Hs =

H(

+ hk

£ _cos[a(@)k

_cos(@

_grn/3)], (3.6)

n=0

where is _an

angular

coordinate of the

cylindrical

coordinate system ^with an

origin

in the

centre of symmetry ^and a(@) means the distance from this centre to the domain

boundary

along

the line drawn at

angle

@.

Dependence

of Hs _on

through

cos(@

grn/3)

and a(@) violates the

cylindrical

symmetry ^of any domain in favour of the

hexagonal

symmetry, ^{and this} fact makes calculation

extremely complicated.

In order to

simplify

the matter, ^{in what follows}

we shall make _use of _a

quantity

fl

=

H(

+

hkf, f

_"

i~ /~

cos

(a(@)k

^cos

@

^~"

)j ^{do, (3.7)}

2gr

_~ o 3

in the

capacity

of _an effective field

strength

at the domain surface. This amounts to

neglect

of deviations of the actual domain form from that of _a circular

cylinder.

Within the scope of the

same

approximation,

we are now able _{to use an}

averaged

domain radius _a instead of a(@). ^It is worth

noting right

away that if

equation (3.6)

_were used but _not its

angle-averaged

version

(3.7),

_we should infer domain cross-sections to be

hexagonal.

This inference agrees with the

findings

of [4].

It has also to be underlined that _we have used in the above consideration _a conclusion that

hk

is almost normal to the

layer plane,

and thus have _not retained vector notation while

dealing

with this and other vector

quantities.

The

period

of the

hexagonal

lattice _cannot

apparently

be smaller than the domain radius

(see Fig. 3),

_so that

surely

ak _{< gr.} This

inequality

becomes stronger ^when the concentration

decreases. When

expanding

^the

integrands

of

equation (3.7)

into _a

Taylor

series in powers of

ak,

_we obtain

f

=

j§(_1)qc2q ^{(~~)~~ ~~}

_~

f

/~

^~~~2q

^@ ~")

^~@

^j~

~)

~

(2q)!

^{' ~} 2gr

~ o 3

q" n=

If _a < I, _a variation of the field

strength

inside _a domain _can be

neglected,

and

fl

_of

_equation (3.7)

with

f

from

equation (3.8)

_can be identified with the

strength

of _an

approximately

uniform

magnetic

^field

Hd

inside the domains.

The chemical

potentials

of

particles

and fluid molecules in the domain _{interior are now to} be

expressed

_as

Ad =

P° lpi

+ u

lvd vi

+ kT Iv

lid, HI

+ Uhhk

ii

,

@0d " @~

(P~)

+ U0(Pd

P~) (U0/U)

^~i~("0 (~bd,

lf~)

+

"0hhkfj (3.9)

By equating

these

potentials

to those in

equation (3.3)

_we get ⁱⁿ an

equilibrium

state, after

averaging

_over the domain side

surface,

u

(pd PI Pkf)

+ kT

iv (id, H[

₊

uhdhkf]

_" kT

iv (#[, H[

₊

uhihkf)

,

U0(Pd P~

Pkf)

_(U0

IV)

kT _(U0

(~d, H~)

₊

"0hd/lkfj

_"

luo

Iv)

kT _lUo

Ill, HI

+

Uohihk ii

_13.

lo)

Here additional

subscripts

d and I at uh and _uoh

signify

that the derivatives with respect to H in

equations (3A)

have to be estimated at

#

₌

id

^and

#

=

ii, respectively.

Thus _we have arrived _at four

equilibrium equations (3.5)

and

(3.10)

_to be

employed

when

determining

nine unknown variables:

k,

_pd

Pi,

_Pk,

Hi, hk, Id, Ii, #k

and _a. The

physical meaning

of these

equations

is

entirely

similar to that of

equations (2.5)

and

(2.6),

_so that there is _no need to dwell _upon it in _more detail. It is evident that five

supplementary equations

must be

brought

into action for the _purpose of

completely defining

the mentioned

variables,

much in the _same way as

equations (2.7-2.9)

have been put ^forward in the

preceding

section.

One

supplementary equation immediately

follows from _a usual condition of mechanical

equi-

librium

imposed

_{upon a pressure}

jump

at the domain surface. We get

Pd

Pi Pkf

"

ala, (3.ll)

wich is

nothing

_more than another representation of the similar

equation

in

equation (2A).

A

requirement

of conservation of the total number of

magnetic particles

leads to _an

equation

s =

jio it) (#d 41)~~

,

13.12)

whose

meaning entirely

coincides with that of

equation (2.9).

On the other

hand,

we can derive _an expression ^for s from

considering

the unit

hexagonal

cell of

figure

3. The whole _area that is

occupied by

cross-sections of the central domain and of the other domains with centres at apexes of the

hexagon

which fall inside the cell

equals

^3gra~.

It _means that _s

= 3gra~

IS,

S ₌

(3/fl) _{r] being}

_{the total}

area of the cell.

By definition,

r~ =

2gr/k

whence _s

=

(2v5gr)~~ (ak)~. By comparing

^{this with}

equation (3.12)

_we

get

~

_~jQ

~~~~~

~~~#[ #l'

^~~'~~~

To derive other relevant

equations,

let _us present ^the

specific magnetization

Md inside _a domain in the form

Md

(Id, Hi

+

hkf)

= M

(id, H[)

+

(0M/0Hi) hkf. (3.14)

Now, by using equations (2.1-3.3)

^{that define} the domain

demagnetization field,

_we arrive next at _a formula

hi

₌

-MdL(r), L(r)

₌

(~/8) (a~/lr). _(3.15)

Furthermore,

the nearest

neighbour approximation

_appears to be

applicable

since the domain arrangement ^is two-dimensional. Then the field at the domain surface is _a vector sum of the inner field

Hi,

that is directed

normally

to the

layer plane,

the _own field hi of the domain and _a field

hi

caused

by

its nearest

neighbours.

It is not difficult _to evaluate the latter field in _a

general

_case. Within the

accepted

_accuracy of the

approximation being used, however,

it suffices to

replace

the

strength

of this field

by

that of the _same field _at the symmetry axis of the central

domain,

that is also normal to the

layer plane

due to symmetry. In this _case,

h[

=

-6MdL (r~), (3.16)

r~

being

identified in

figure

3,

whence,

with allowance for

equation (3.7),

hkf

_" -Md

[L(a)

+ 6L

(r~)]

,

(3.17)

what defines Md in accordance with

equation (3.14)

and

leads,

_next, to _an equation

~~f

_"

l ₊

(d~/(it _)jj~(

6L

jr~)j

^~

i~~' ~i _(~'~~~

A further

equation

is to be deduced from _a

continuity

^condition of the

magnetic

induction at the boundaries of _a domain with the

layer plates.

It resembles

equation (2.8)

and is of the form

Ho

=

(I s)liH[

₊ sld

(Hi

₊

hi

+

hkf)

,

(3.19) hi,

hk and

f being

^defined

by equation (3.15,

3.16 and

3.7), respectively.

Thus _we have succeeded in

deriving

additional equations which _are necessary to

supplement equations (3.5)

and

(3.10) stemming

^{from the}

thermodynamic equilibrium

conditions. All these

equations

when

being

closed with the aid of

equations (2.10)

and

(2.ll)

enable _us to express

variables

k,

_pd

Pi,

_Pk,

H[, hk, Id, Ii

and

ik

_as functions of

averaged

^domain radius _a.

After

that,

the last variable that

only

remains unknown _can be found without much

ado,

from

equation (2.7).

The set of

equations (3.5)

and

(3.10-3.19)

_can be

greatly simplified

ⁱⁿ view of the fact that

normally Ho considerably

exceeds fields caused

by

the

magnetic particles,

_so that

Hi

m

Ho Further,

concentrations

ii

and

id

_may also

justly

be looked _{upon as}

quantities

which

insignificantly

differ from their

equilibrium

values which have to be established _{at a}

plane phase boundary,

that

is,

with _no account for the domain side surface curvature.

Besides, by

virtue of

f deviating

from

unity only slightly,

it may be taken

f

m I. As _a

result,

_we obtain _a

simplified

version of the

equations

which _can be resolved _even in _an

analytical

form. It is not our

intention, however,

to write out either the

simplified

set or its solution in _an

explicit

form because of them

being

rather

unwieldy.

Suffice it to say that results

concerning properties

of equilibrium states with

hexagonal

domain arrangements, ^which can be got ^from this set when

making

_use of the equations

(2.10)

and

(2.ll),

are

basically

of the _same kind _as those derived in the _case of chaotic domain structures.

4

Concluding

remarks.

Because of _a number of

comparatively

^bold

assumptions

made while

evaluating properties

of the

equilibrium

system under

study,

^the above results should be

ultimately regarded

_as

substantially approximate, pertaining

to rather _a crude attempt to

theoretically

describe those

properties.

The

assumptions help, nevertheless,

to avoid inessential calculation difficulties and

are

surely

not of _a

principal

nature, whereas the said attempt to

explain

the _very existence of stable domains in thin ferrocolloid

layers

is

seemingly

the first _{one, not} to mention that all

the results _can

readily

be

improved

_at the _expense of

making

calculation _more tedious. At _any rate, ^these assumptions do not violate in the least the

principal

conclusions that have _to be

drawn from the above _treatment.

The main inference consists in that the

phase separation

of _a ferrocolloid _or of _a

magnetic

fluid does not

necessarily implies

the terminal _occurrence of two massive colloidal

phases,

_as it is

commonly specific

to most of colloidal systems, but leads _to

establishing

_a stable state with domains of finite size.

Moreover,

this has been

proved

_{to occur,} due

exclusively

to _a

peculiar scaling

effect that is caused

by

the finite distance between the

layer

boundaries. It is

just

this effect that is

evidently responsible

for the indicated deviation of

phase separation

patterns in thin

layers

of _a ferrocolloid from those observed in unbounded systems.

It appears ^to be

important

^for

generation

of the stable domain arrangements that

growing droplets

of the _new ferrocolloidal

phase

must be

noticeably elongated

in the direction normal to the

layer plane.

This

happens

in _a system ^of

dipole particles

under action of _a

corresponding

normal external

field,

_no matter whether the

dipoles

_are of

magnetic

_or electric

origin.

Such _a

point

of view is

surely

confirmed

by

the _mere fact that stable domain structures establish them- selves not

only

in thin

layers

of ferrocolloids with permanent

particulate magnetic

moments but also in

layers

of electrocolloids when

particle dipoles

_are induced

by

_an

externally applied

normal electric field [6]. ^{For the} same reasons, the _occurrence of similar domain patterns

ought

to be

expected

in various other

dispersions

in which

dipole

interactions prove ^to be

significant,

such _{as new} ferrosmectics

recently synthesized

in [19]. ^{It should be noted}

that,

if the

layer

boundaries _are

pliant,

the formation of domains of

particles

with either permanent or induced

dipole

moments have to be

accompanied by

distortions of the form of these boundaries.

However, if would be unreasonable

and, perhaps,

somewhat preposterous to

envisage

the emergence of domain arrangements of the discussed type ^{in diverse latex}

layers

and in thin colloidal films of different

origins

in _cases when there _{are no}

particulate dipoles.

Domain patterns ^{in such}

layers

and

films,

if _{any, are}

hardy unlikely

to be

explained

_on the basis of the

same considerations _as those

exploited

above.

In

conclusion,

_we have to

emphasize that,

while

key

_reasons of formation and main

properties

of both chaotic and ordered domain arrangements can be understood and evaluated with sufficient _{accuracy, a} crucial question ^of

why

and at what exact condition _an ordered structure

suddenly replaces

_a chaotic _one remains obscure.

Similarly,

there is _{no answer to a} question:

why

the ordered structure

corresponds

to _a

hexagonal

lattice?

This

phenomenon

of disorder-order transition in _a two-dimensional system ^of domains is

certainly

not of the type ^of _a

liquid-crystal phase

transition in _a system ^of

attracting

molecules

or

particles.

It is far _more

likely

_to resemble

generation

of the

particle packing

with _a

long-

range order in _a system ^{of identical hard}

spheres

of

high

concentration due to

purely geometrical

reasons [20]. ^The

ordering

of the domains

really

_seems to be caused

by

domain

repulsion

at a

condition of the overall volume accessible _to the ferrocolloid under consideration

being

limited.

In contrast to

[20], however,

^{both size and} interaction

potential

of the domains that _are

going

fo form _an ordered lattice _are unknown, and this

hampers

to a great extent the

development

of _an

analysis

to describe the

phenomenon.

It _may be

expected,

nonetheless, that _a domain structure

is

likely

to be chaotic if the _mean distance between the symmetry axes of _two

neighbouring

domains

happens

to

considerably

exceed the domain

diameter,

and _to be ordered otherwise.

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Proofnot corrected

by

the authors