Simulation of Ostwald Ripening in Two Dimensions: Spatial and Nearest Neighbor Correlations

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Spatial and Nearest Neighbor Correlations

Norbert Masbaum

To cite this version:

Norbert Masbaum. Simulation of Ostwald Ripening in Two Dimensions: Spatial and Nearest Neighbor Correlations. Journal de Physique I, EDP Sciences, 1995, 5 (9), pp.1143-1159. �10.1051/jp1:1995188�.

�jpa-00247126�

Classification Physics Abstracts

64.60My 64.60Cn 64.75+g

Simulation of Ostwald Ripening in Two Dimensions: Spatial and Nearest Neighbor Correlations

Norbert Masbaunl

Mathematisches Institut, Universitàt _zu KôIn, 50931 KôIn, GerJnany

(Received

6 May 1995, ^received in final form 20 May 1995, accepted 2 June

1995)

Abstract. Recent experimental results _on fate stage coarsening

(Ostwald ripening)

in two di- mensions

were reinvestigated by _means of numerical siJnulation using the Cahn-Hilliard equation

(mortel B).

We determine the spatial partiele-partide and charge-charge correlation functions

according to the experimental results of Krichevsky and Stavans. We find that _our numerical results correspond well with these experiments. ^{We aise} investigate partiale-partiale correlations

between nearest neighbors

(defined

by the Voronoi diagram of partiale

centroids)

following the

experiments of Seul, Morgan and Sire and _{compare our} numerical results with their Maximum

Entropy mortel of Ostwald ripening.

1. Introduction

In the fate stage of

phase

separation ⁱⁿ a

binary

mixture the

large partides

of the nlinority

phase

_grow

by

material diffusion at the expense of the

shrinking

small _ones. This process,

generally

known _as Ostwald

ripening iii,

is driven

by

the reduction of the total interface between the

particle

and the nlatrix

phase.

Since 1961, when

Lifshitz, Slyozov

and

Wagner [2,3]

developed

the

nowadays

classical

LSW-theory,

the

physical

laws of Ostwald

ripening

have been

extensively

studied. While the result obtained

by

LSW is _correct in the hmit of _a

vanishing

volume fraction

ç§ of the

droplet phase,

the

understanding

of Ostwald

ripening

for _{a nonzero ç§}

is still

incomplete.

In _two dimensions the situation is _{even more}

diflicult,

_smce the

LSW-theory

tutus out to be not

applicable

due to _a

logarithmic singularity

_m ^{the diffusion}

equation

[4].

Moreover,

similar ta the situation in 3D [5], ^the

existing

2D-theories of Ardell

[6,7], Marqusee

[4],

Zheng

and

Gunton _[8] and Yao et ai. [9]

give quite

^{diiferent results for the}

#~dependence of,

for instance, trie asymptotic distribution function

f(p,

_ç§) of trie reduced

partiale

radii p =

R/(R)

_or trie

growth

rate

K(ç§)

in trie

generally

observed

growth

law

(R)~

₌ Ro +

K(ç§)

t.

From the

experimental

point of view,

however,

two-dimensional work has trie great ^advan- tage of

allowing

_a direct

imaging

of the

coarsening

_process.

Recently Krichevsky

and Stavans

(KS) _[10, iii

and

Seul, Morgan

and Sire

(SMS) [12,13] performed

experiments ^{of Ostwald} ripening at diiferent _area fractions of the

droplet phase

in thin

layers

of succionitril and

Lang-

muir filJns of

binary anlphiphilic

_systems. Bath groups

analyzed

their data of the

videotaped

Q Les Editions de Physique 1995

coarsening

_process in view of

spatial

correlations of the

coarsening partiales.

These correlations

are

generally

believed to

play

_an

important

rote in trie

understanding

of Ostwald

ripening.

In this work _we present trie

corresponding

results of numerical simulations _on trie basis of trie Cahn-Hilliard

equation.

In the last _years, modem computer ^{facilities have} made this nonlinear diffusion

equation popular

for _a simulation of

phase

separation.

Large

scale computations for the _case of Ostwald

ripening

have been

performed by Rogers

and Desai [14] ^and

Chakrabarti,

Toral and Gunton

(various

_{papers, see} Ref. [15] ^for a

review).

These works

mainly

deal with

a numerical determination of trie

particle

size distribution

functions,

trie

growth

law and trie structure function. Here _we focus _on trie

spatial partiale-partiale correlations,

1-e-, trie

prob- ability

of

finding

other

particles

in trie environment of _a

partiale,

_as well _{as on} trie

spatial

correlations between the

charges

of the

partiales

[16]. ^{Here the}

charge

_q

=

R~~

of _a

partiale

is defined

proportionally

to the _rate of

change

of the _area

occupied by

the

paÀlcle, _leading

to the

question

^how

probable

it is to find

growing (or shrinking) partiales

in the environment of

growing (or shrinking) particles.

We _conlpare the

numerically

obtained correlation functions with trie

corresponding experimental

results of KS

[10, Iii.

In addition _we follow trie work of SMS and

analyze

_our numerical

coarsening

data in view of correlations between nearest

neighbors.

^SMS

developed

_a maximum entropy

theory (ME)

for these correlations

leading

to laws similar to the famous Lewis law and the Aboav-Weaire law of cellular patterns. We will _see that _our numerical results show _some

discrepancies

with this

theory.

A diiferent and

numerically

less expensive method for the simulation of Ostwald

ripening

bas been

developed by

Voorhees and Glicksman

Ii?i.

Instead of

solving

for trie time

dependent

diffusion field

according

to trie Cahn-Hilliard

approach

used in _our work

they developed

_an

algorithm

eut of the

monopole approximation

of the

quasistationary

diffusion

equation,

which allows _to calculate the individual

growth

rate of each

partiale provided

that trie sizes and locations of ail

partiales

_are

given. Recently

this formalism has been extended _to the so-called

dipole approximation,

which aise considers the

moving

of

partiales

in the _process of

coarsening (which

can be observed in

experiments [1Ii). Recently

this

approach

has been used

by

Akaiwa and Voorhees [181 ^and Akaiwa and Meiron [19] ^for a determination of

spatial

correlations in 3D and 2D,

respectively.

The 2D-results will be shown later _on in context with the correlation

functions _we obtained.

The _{paper is} structured _as follows: the second section

gives

_a

description

of the simulation

technique.

In Section

3.1,

_{we compare} ^the results of the numerical simulations

concerning

the

spatial partiale-partiale

^and

charge-charge

correlation functions with the

experimental

results of KS. In Section 3.2, we discuss the simulations

analyzed

in view of nearest

neighbor

correla- tions

followmg

SMS and _compare the results with their Maximum

Entropy

model.

Finally,

in Section

4,

we summanze our results and _{give a} conclusion.

2. Trie Model and trie Numerical Method

In the framework of the Cahn-Hilliard model [20] ^the

phase

separation _process in _a

binary

system is formulated

by

the evolution equation

where

c(r,t)

represents the conserved order parameter ^{of the} concentration of _one of the two

species.

M is _a

mobility

factor and F denotes the total free energy of the system

generally

assumed in the

Ginzburg-Laudau

form

F =

/lflc)

^{+ ~}

^{l?Cl~ldr (2)}

Here

f(c)

is _a double-well

potential

and _{~ a}

positive

parameter ^related to the interfacial _energy.

The thermal noise term

~(r,t)

in the

Langevin equation (1),

aise known _as model B in the

literature,

has been

developed by

Cook [21]. ^{Since the numerical simulation of the thermal} noise is very time

consuming,

_we will trot consider it in this work.

Generally,

the noise is

believed trot to affect the

growth

of the

droplets

in the fate stages ^of

phase separation,

at least at low temperatures

[14,15].

It should be

noted, however,

that _a strict validation of this

assumption

is still

outstanding [25].

It has been shown

by

Grant et ai. [23] ^{that for the Mass of} double-well

potentials f(c)

=

~c~ rc~

_after

a suitable

rescaling

of

concentration,

_space and time to ~fi, x, and 2

r,

respectively, equation (1)

_can be put in the dimensionless form

)

=

V~(1fi~

-1fi

V~1fi), _{ifi =} 1fi(x,

T), (3)

which is

generally

used for simulations of the Cahn-Hilliard model. The solutions of equa- tion

(3)

exhibit

coarsening

domains with values of the rescaled concentration _~b close _to 1 and -1,

respectively,

which _are identified with the two

phases.

^At the interfaces _a steep

gradient

of the concentration _cari be observed.

We salve equation

(3)

on a square lattice

using

an

expirait

Euler method _as described

by Rogers

et ai. [22]. ^{In order to reduce}

boundary

eifects

periodic boundary

conditions _were

employed

in that scheme. As discussed in reference [22] ^{the mesh} size hz has to be chosen small

enough

to reflect the

sharp

concentration

gradient

at the

phase

boundaries. As _a consequence, the time step ^{ht has} to be chosen

carefully

_m order to avoid instabilities of the numerical

scheme

[22].

^We worked

successfully

with the values hz

= 1 and AT

= 0.04 in ail simulations

reported

in this work.

As

usual,

_we

identify

the

partiales

with the clusters of

gridpoints

with _a concentration _~fi < 0.

In view of _a determination of

dR/dr,

_{1-e-, the} variation of the

partiale

size, the radius R of _a

partiale

has to be

carefully

^defined. It tumed eut to be _{necessary net}

only

to count the cluster size but aise to look for the concentration _~fi at the

grid points belonging

_{to a} cluster. We

found the definition

~j~2

~

~

~fi

~

z

an

appropriate

_measure and calculated the variation of size

by dR(r)/dr

GÎ

(R(r

+

5Ar) R(r)) /5Ar.

The location of _a

partiale

_was defined

by

the _center of _mass

rcm =

£

~~

~

_~fiz(rz)rz

~

For the

study

of

partiale

_coarsenmg each simulation _was started

by

_an initial concentration field ~fi(x, r =

0), consisting

of circular clusters with concentration values _~b

= -1 distributed

on the

grid,

and

setting

_~fi to +1 at ail

grid points

outside the

partiales. By varying

the number of

partiales

diiferent values of the _area fraction _ç§ of the

partiale phase

could be studied. The

advantage

of this method

(see [14,15]

for _a diiferent

approach)

is the

possibility

of

creating

an

intial condition with _a

given

distribution of

partide

radii. As mentioned

above,

_an

asymptotic

partiale

size distribution

(depending

_{on ç§) is} reached in the fate stage of

coarsening.

Here _we

Table I. Parameters ~lsed in the sim~llations.

ç§ d _furax size

0.13 0.3 23.000 2048 x 2048

0.25 0.3 30.000 1024 _x 2048

0.40 0.15 23.000 1024x2048

follow _an idea of Akaiwa and Voorhees [18]

starting

_our

large

scale simulations with

partiale

size distributions _near to the

asymptotic

form

(known

from

previous

smaller _ruas at the _same

area

fraction).

It is

generally

found that

thereby

the

scaling regime

is reached much faster than

by starting

with _an

arbitrary

distribution.

Another important parameter for the initial condition of the simulation is the _mean

partiale

radius

(Ro).

On the _one

hard,

_a small

(Roi

is desired in order _to be able _to start with _a

large

number of

partiales

to reduce the scatter in the data. On the other

hard,

it is well known [24]

that the Cahn-Hilliard model reflects the

principle

of _a critical

partiale

radius in the _sense that smaller

partiales

_are trot stable in the matrix and tend to dissolve. A reasonable value _was found to be

(Ro)

" 6.2, which is used

throughout

^ail simulations.

Using

this _mean

particle radius,

_we

successively

tilt _up the

grid

with

partides according

to the distribution function at random locations until the

prescribed

_area fraction is reached.

Following

_a method of Akaiwa and Voorhees [18], we do trot

only

exdude

partide overlap

but _more than that create _a

depletion

_{zone, 1-e-, a}

spherical

shell of thickness d R with _a

parameter d around _every

partide

of radius

R,

which is forbidden for other

partides.

In agreement ^with

[18,19]

we did trot find _an influence of these

depletion

_zones in the initial

condition _on the fate stage

scaling

functions. The

advantage

of

using depletion

zones is,

however,

that

they

hinder

partiales

to

instantly

coalesce _so that _{we cari} start with _a

partide

number _as

large

_as

possible.

For

larger

values of the _area fraction _ç§ the parameter d is

naturally

limited to smaller

values,

otherwise the

particles

could trot be

successfully placed.

In Table I ail parameters for the diiferent simulations _are summarized. The main

problem

in

studying

trie statistical laws of Ostwald

ripening

in trie

asymptotic

fate stage

by

numerical simulation _is the

decreasing

number of

partides, causing problems

in the statistics of the

analyzed

_quantity.

One method _{to overcome} this

problem

is to average over diiferent _ruas with initial random

configurations [14,15].

Since _we had the

opportunity

of

using large

computer memory, we

preferred

in this work _a second method of

performing single

_ruas ^of _very

large

systems

(Tab. I).

The

advantage

of this method

is,

of _course, the reduction of finite size

eifects,

which tumed out to be _a

problem

in the

study

of

spatial

correlations at small system sizes. The simulations

performed

in this

work, however,

start with

+~ 2600 3100

partides (slightly diifering

^with the

area fraction ç§) and end _up with

+~ 600 850

particles

at furax,

allowing

_a precise

analysis

of the diiferent correlation functions _even in the fate stage.

The calculations have been carried _{out on}

partitions

of 128 _or 256 processors of the 1024 node

parallel

_computer

Gcel-3/1024

of PARSYTEC

using geometric parallelisation,

_1-e-,

dividing

the

grid

in parts

assigned

to the dilferent processors. It turned out that with the

grid

sizes and the number of _processors that _we used the finite diiference scheme could be calculated with

an

eflicieny

of _more than

90il,

where

dividing

into squares does _a

significantly

berner

job

than

dividing

into

stripes [25]. Finally,

ii should be mentioned that due to technical

provisions

for data reduction the cluster identification for the calculatiolî of

partiale

sizes and locations has also been

performed

in

parallel, using

_a

technique

similar to trie host model of Hackl [26] ^based

on the

Hoshen-Kopelmann algorithm

_[27].

3. Results of Numerical Simulation

3.1. SPATIAL CORRELATIONS. In this

chapter

_we present ^{trie results of}

spatial particle- partiale

and

charge-charge

correlations

analogously

to the

experimental

work of

Krichevsky

and Stavans

(KS) [loi.

Like

Krichevsky

and Stavaus _we

study

the _{two area} fractions _ç§

= 0.13

and ç§ ⁼ 0.40. For the calculation of the

spatial

correlation functions _we follow the definition

given by

Akaiwa and Voorhees [18], 1-e-, the radial distribution function

Gir)

is defined _as the

ratio of the number of

partiales

in _a circular shell of radius _r and thickness dr

surrounding

a

partiale

to the number of

partiales

in trie shell

expected

from the

partiale density

in the system. ^The two type correlation functions _are defined in the _{same mariner.}

Gir, LS),

for

example,

_measures the

probability

of

finding

_a small

partiale (S)

at the distance _r of _a

large partide IL).

The idea of

dividing

the

partides

in two classes of

large (R

_>

(R))

and small

partiales (R

_<

IRI)

has been introduced

by

KS. We follow this concept in order to allow _a direct

comparison

with their

experimental

results.

In

Figure

1 the radial distribution function

G(r)

is shown for the two values of trie _area fraction

ç§.

Throughout

this

work,

trie distance _r between the

particles

is measured in units of the _mean radius

IRI.

We

generally

found _a

scaling

in the correlation functions in trie

fate stages ^of the simulations

(Fig. l), allowing

to follow KS in

averaging

_over 10 times

(r

₌ 5000,

7000,..., 23000)

in order to reduce the scatter in the data. We

performed

this _av-

eraging throughout

^this work.

Figure

1 shows that trie correlation

functions, diifering strongly

for trie two _area

fractions,

could be well

reproduced by

trie numerical simulation of trie Cahn- Hilliard model.

Very

similar results bave been obtained

by

Akaiwa and Meiron

(AM) [19], calculating

_a

multiple approximation

of the

quasistationary

^diffusion equation. Since their

monopole

and

corresponding dipole

approximation results _are very close [19] we

replot only

the

dipole

results in _our

figures.

There _are two

typical

values _m the radial distribution function

G(r).

That is first trie maximum

defining

trie _mean distance to trie _next

particle, being

_more distinct _at

higher

_area

fractions with

partiales packed

_more

densely,

and second trie size of the

depletion

_{zone sur-}

rounding

_a

partide.

Both

typical

^distances cari be

reproduced

_very

precisely by

simulation.

A doser look _can be taken

by studying

trie correlations between trie

partiales

of diiferent

or of _same Mass _as shown in

Figure

^2.

Again

_we observe _a

good

agreement of _our simulation with trie

experiment

and the simulation method of AM

[28].

^{At trie lower} area fraction trie two correlation functions _are

qualitatively

similar

showing slightly higher

values for

particles

of diiferent size

compared

with

partiales

of trie _same

size,

while at the

high

_area fraction of

ç§ ⁼ 0A this diiference _{is very} strong,

showing

_a

high probability

of

finding

small

particles

in trie environment of

large

_ores and ~ice _versa. This of _course is _a consequence of trie fact that

large partiales

_grow at the expense of small _ones in trie

neighborhood.

According

to the definition of the correlation function _it froids that

G(r, LS)

=

G(r, SL)

=

G(r,

SL

ILS).

Like AM _we found _a strong

diiference, however,

in the correlations of two

large partides

and _two small

partides,

_1-e-,

G(r, LL)

and

G(r, SS).

While _we

averaged

these two functions in

Figure

2 in order to compare them with trie

corresponding

results of

KS,

_we show both functions in

Figure

3

together

with trie simulation results of AM.

Qualitatively

both methods

give

the _same

results, especially

in view of trie

position

of the maximum and trie

size of the

depletion

_zone. On trie other

haud,

the

quantitative

diiferences in trie correlation values exceed the statistical _errors

significantly,

which is

probably

_a consequence of trie diiferent

approximation

^{methods for}

studying

the

multiparticle

^diffusion

problem

^{of Ostwald}

ripening.

We _now tum to the

charge-charge correlations,

_{1-e-, to} the

spatial

correlations of

growing

and

i-o a

O 1

~ _~'

a fl~is work

/

o KS

é

i AM

~'~

O

~ ~

^~~~~

u

, ~

o i

1

~ ^,

o /

~

a)

^~'~ 0 2 4 6 r

&4.40

0

,

o ~=5000

° ~=7000

o ~=9ooo

6 ~=I1000

0.5 ~ ~=13000

V ~=15000

~ ~=17000

+ ~=19000

x ~=210W

~ ~ ^*

w23000

b)

^{0 2 4 6 r}

Fig. 1.

a)

Numerical siJnulation of trie radial

partiale-partiale

correlation function

G(r) (r

in units of < R

>)

for the _area fraction # ₌ 0.13

(empty

symbols and dashed fine) and # ₌ 0.40

(fuit

symbols

and solid

une)

in the fate stage. The experiJnental results by KS [10] as well

as the numerical results by AM [19] are also shown; b) The _salue correlation function

G(r)

Jneasured at ter times _m trie fate

stage ^of simulation, showmg _a dynaJnical scaling. The solid une corresponds to the averaging shown

m

a).

shrinking partides.

Here the

growing partides

have _a positive

(+) charge,

_q

= R

~~,

_while

dr the

shrinking partides

have

negative charge (-).

In order to make trie results

comparable

_we

follow KS and derme trie correlation of the

charges by G(r,...)

=

(q(0)q'(r)), assuming

trie

charge q'

to be

uniformly

distributed

along

the

boundary

of the

particles.

The numerical results for the correlations of _two

particles

of _same

sign

and two

partiales

of diiferent

sign

_are shown _m

Figure

4,

together

with the

experimental

results of KS

[29].

For bath _area fractions there is _a strong correlation of

partiales

of opposite

charge, especially

&4.13

j

i.o

,j'

OE ©fl~is work

0.5 _{> ~ KS}

' - fl~is work

-- KS

AM

a)

0 2 4 6 r

2.0 w a dûs work

~'~~

~Î

~ls

_work

1.5 ~~j

i o

o.5

b) ~~0

2 4 6 r

Fig. 2. The radial correlation function

G(r, LL/SS) (empty syJnbols)

for two partides of the _same dass and the correlation function

G(r,SL/LS) (fuit syJnbols)

for two partiales of different class _m

comparison with the expenJnental results of KS [10]. ^{The nuJnerical results of AM} [19] are aise shown.

interesting

in the _case of _ç§

= 0.13 where such _a distinct maximum could _net be observed for the S-L correlation.

Obviously

there is _a strong

screening

^eifect _even for small _area

fractions,

1-e-,

growing (shrinking) partides

_ar surrounded

by

_a shell of

shrinking (growing) particles.

Trie

good correspondence

between trie numerical simulation and trie

experiment

indicates that trie Cahn-Hilliard model is also

appropriate

to describe trie

charge-charge

correlations.

Again

trie size of trie

depletion

_{zone as} well _as trie

position

of the maximum, _1-e-, trie first cell of interactions, are

reproduced

_very

precisely. Finally,

trie four diiferent

charge-charge

correlation functions

G(r, ++), G(r, -), G(r, -+), G(r,

+- are shown _m

Figure

5. Note

that,

unlike

G(r,LS)

=

G(r, SL),

the correlation functions

G(r, +-)

and

G(r, -+)

_are diiferent due to the

unsymmetric

^definition

given by

KS.

However,

it is

interesting

to _see that the correlations of _two

growing

and two

shrinking partides

_{are very} close, which _was not the _case with trie L L

&4.40

1.5 çq

pf

~

Ù~,

» ~

,

l-O ,

' w

~ --a fl~'sw rk

, °

~'~

j

~ ~

~lwork

'

-- AM

i i

'W

~ r

~~

~~

0 2 4 ~

0m0.13

,a

?'~' ',

' S,

,~-~ ~,

i-o

w B

a ~ÎÎ~Î~~~~

'

- fl~is work

Î

i

-_ AM

j i

5

1 w

i i

i i

i w

i i

i~

~,

~

°.°

o 2 4 ~ ~

Fig. 3. The radial large-large correlation function G(r, LL)

(empty

symbols) and the small-small correlation function

G(r,

SS) (fuit syJnbols). The numerical results of AM [19] _are ^{also shown.}

or S S correlations.

3.2. CORRELATIONS _oF NEAREST NEIGHBORS. In this section _we present ^{results of} numer-

ical simulation

according

to the correlations of

adjacent partides during coarsening,

motivated

by

recent

experimental

and theoretical results of

Seul, Morgan

and Sire

(SMS) [12]. Following

SMS,

two

partiales

_are defined _as nearest

neighbors

if

they belong

to

neighboring

cells of trie cellular pattern

given by

the Voronoi

diagram

of the partiales centroids. An

example

of this construction [30] is shown in

Figure

6. As in the experiments ^of

SMS,

_we simulated the _area fraction _ç§

= 0.25, but _we also looked for _{an area} fraction

dependence

of ail measured quantities

by analyzing

the simulations

concerning

_{ç§ =} 0.13 and

ç§ ⁼ 0.40 studied in the last section _now in view of _nearest

neighbor

correlations.

TO ascertain that there is _a

dynamical scaling

in trie

configurations

of

partiales

in trie fate

2.° OE#0.40

5

1.o

0.5 ^{* ~} fl~is work

> ~ KS

- fl~is work

-- KS

a)

^{~ ~} 0 2 4 6 8 r

0m0.13

2.0

1.5

1.0

@~

i

'

~

OE B ~s work

0.5 _{> ~ KS}

- fins work

-- KS

b) ~~0

2 4 6 8 r

Fig. 4. The radial charge-charge correlation functions

G(r,

+ +

/ -)

and

G(r,

+

/ +)

for

partides with charges of _salue sign

(eJnpty syJnbols)

and opposite sign

(fuit symbols)

in comparison with the experimental results of KS [10].

stages ^{of Ostwald}

ripening

_we measured trie

probability

densities

P(d

_{N N}

/ (d

_{N N} of the normed distances dNN between trie boundaries of

adjacent partiales.

In agreement with yet

unpublished

results

by

SMS [12] we found that the distribution becomes time

independent

in the fate stage

(Fig. 7).

We observed that the

scaling

functions showed _no

significant dependence

_on the _area

fraction with _a characteristic maximum at

approximately 0.8(dNN),

and almost _no pairs of

partiales

with distances

larger

than

2.5(dNN).

A _more detailed _{view on} the _structure of the relations between nearest

neighbors

will be

given

ⁱⁿ

Figure 8,

where the distribution

Pin)

of the coordination number _{n, le-,} the _mean number of

neighbors

of _a

partiale,

_are

plotted. Especially

for the coordination number _n

= 6

we found _a

significant

lower

probability

in trie simulation _(ç§

=

0.25)

than in the experiments ^of SMS. Moreover, we obtained the value 0.84~0.01 for the second moment ~12 "

L(n _6)~ pin),

_a

OEA.40

~ a G(r,+ +)

~'~

> ~ G(r,- -)

- G(r,- +)

2 o ^-- G(r,+ -)

1.5

o

o.5

a)

0 2 4 6 8 r

OEA.13

~ a G(r _++~

~'~

> ~G(rÎ- -)

- G(r,-+)

~ ~ ^-- G(r,+-)

1.5

0

aa~~od

o.5

b)

O.O o 2 4 6 8 r

Fig. 5. The different radial charge-charge correlation functions in the fate stage.

measure for the deviation from the

hexagonal

order

pin)

_e 6, while SMS report ^{smaller values}

ranging

^from 0.64 to 0.85. The _reason for this

higher (hexagonal)

order in the

coarsening

experiments ^{of SMS} has to be _seen in additional

long-range repulsive

electrostatic interactions between the

droplets

in the

amphilic

_system. We will _come back to this point ^{later. Trie}

distribution of coordination numbers for the other _area fractions _are also shown in

Figure

8.

We observe _a lower

hexagonal

order for the low _area fraction _ç§

= 0.13, while there is _no

significant

diiference in the distribution between _ç§

= 0.25 and _ç§

= 0.40. For the second

moment we found _~12

" 1.08 ~ 0.02 for

ç§ ⁼ 0.13 and

~12 ^" 0.86 ~ 0.02 in the _case of _area

fraction _ç§

= 0.40.

We _{now tum to} the maximum entropy

theory (ME)

for

partiale coarsening developed by

Sire and Seul

(SMS)

_[13]. Since the pattem formed

by

the

polydisperse droplets

_are connected via the Voronoi construction to _a cellular pattern with each cell

containing

_one

droplet,

trie

authors had the idea to translate _{a maximum} entropy

analysis

^of cellular patterns

by

Rivier

Fig. 6. Determmation of the nearest neighbors for _a

droplet configuration

with _{an area} fraction

#

= 0.40. Trie Voronoi diagram

(bold fines)

and trie dual graph, ^the Delaunay triangulation

(fine unes)

_are shown. Two partides _are ^defined _as nearest neighbors if they belong to adjacent cells, _or

equivalently, if they _are connected by _an edge of the Delaunay triangulation.

et ai. [31] ^{into the}

droplet

_case.

Conceming

correlations of

coarsening droplets,

SMS stated two laws well known from the

theory

of cellular pattems, ^{that is the Lewis law}

zn = b +

Ain 6),

_xn

=

jAnj /jA) j4)

and the Aboav-Weaire law

arum)

= 16

a)in 6)

+ 36 _{+ t12.}

là)

Here the Lewis law describes _a linear relation between the

topological charge

C

= n 6 of _a

partiale

and the normed _{mean area} An of

particles

with coordination number _n.

The Aboav-Weaire law states that the

product

of the coordination number _n times the mean

coordination number

min)

of the _n

neighbors

of _a

partiale depends linearly

_on the

topological charge

C

= n 6.

Investigating

^these laws

by

numerical

simulation,

_we observed

scaling

in both _cases. The

averaged

fate stage

dependencies

_are shown in

Figure

9. Our simulation results _are

precisely

described

by

the

hnearity according

to the Aboav-Weaire law. While SMS measured the constant to a +~ I.I for _ç§

= 0.25, _we obtained _a

= 1.15 ~ 0.07, not

significantly depending

on the _area fraction _ç§.

Moreover,

the

validity

of the Aboav-Weaire law _seems not to be _a characteristic of the fate stage

spatial partide

correlations _as assumed

by

SMS. We found that

even a random

configuration

of

partiales

with the _same

partiale

size distribution

(like

trie initial

configuration

used in trie

simulations)

follows trie Aboav-Weaire law with

high precision.

In contrast to trie Aboav-Weaire law _a strong

dependence

_on trie _area fraction _can be _seen

in the _case of trie Lewis

law, leading

to _a

higher

variation of trie number of

neighbors

with the

partiale

size at

higher

_area ^fractions.

Moreover, significant

deviations from the

linearity

~i'* ^-- 0Em0.13

o.8 !." _{. .} 0Em0.25

iÎ _+----+ 0Em0.40

0.6

0.4

0.2

1 t 1

~~0.0

_{1-ù 2.0} dNN/<~j~~~>

Fig. 7. Probability densities of trie distances between trie boundaries of nearest neighbors _in the fate stage of coarsening for different values of _area fraction #.

P(n)

o.5 o

o.4 ,"~, -- &4.13

' "

. . &4.25

o.3 +----+ &4.40

o SMS

0.2

o-1

~'~

2 4 5 6 7 8 9 n

Fig. 8. Probability densities of the coordination numbers

Pin)

_m the fate stage of coarsening. The

experiJnental values of SMS [12] ^{for the} area fraction # ₌ 0.25 _are aise shown.

could be observed. For that _{reason we} did _trot try to determine the _constant in

equation (4).

Figure

⁹

^shows,

however, that the

experimental

value of SMS could be confirmed in trie _case of the _area fraction _ç§

= 0.25.

Analogous

to the

study

of

spatial

correlations of the

partiales

sizes in Section 3.1 it is also

possible

to

study

these _size correlations between nearest

neighbors

_as has been

proposed by

SMS. In

Figure

10 the _mean normalized

partide

_area

(ANN) /(A)

of the

neighbors

is shown

o

i.o a

O ₁

~ _~'

a MS Work

'

o KS

é i~

Î AM

. fl~is work g

0.5 1

~~

o

j AM

~ i

o i

i ,

o

),/

~

~°

o 2 ~ ~

&4.40

~ l.0

,

o ~=5000

° ~=7000

o w9000

6 ~=I l0lX~

0.5 ~ ~=13000

V ~=150lX~

~ ~=170lX~

+ W190lX~

x W2IOIX~

* w230lX~

b)

^{0 2} 4 6 r

Fig. 9. Mean coordination nuJnber

m(n) (Aboav-Weaire

law

(5))

and the _mean particle _area

< An _>

/

< A >

(Lewis

law

(4))

of the nearest neighbors of _a partiale with topological charge

C = n 6 in the fate stage [32]. ^{The dotted unes reflect trie} experimental values of trie slope À(0.2 < < 0.25) determined by SMS

(#

=

0.25).

as a function of the normalized

partide

_area

A/(A) (under averaging

_over ail

partides

of the

system). Strong

anticorrelations of these

quantities

could be

observed,

that is small

partides

are surrounded _more

likely by large neighbors

than

by

small

neighbors (and

~ice

~ersa),

_a

phenomenon

that _was also found in the experiments of SMS.

According

to trie numerical

simulation this screemng eifect is _more pronounced at

higher

_area fractions with

particles packed

_more

densely (Fig. 10).

Combining

trie Aboav-Weaire law

(5)

and the Lewis law

(4),

SMS deduced the

analytic expression

(ANN

1

a(1 ~)

+ _À~12 A

IA)

_z ^~ 6 +

lx ₁₎ là

^{' ~}

IA)

^~~~

<ANN>/<A>

-- &4.13

",,

. . &4.25

~',,

+----+ &4.40

w

",~

Î",-~_

".

'---x-__

1

~'~~o.o _{o.5 o .5}

Fig. 10. Numerical results for the correlations of the _area A of _a partide and the _{mean area}

< ANN _> of its nearest neighbors _m the fate stage [32].

for these correlations shown in

Figure

10. However, _{we were} unable to confirm this expression

on the basis of _our numerical

results, probably

_a consequence of the fact that

already

trie

linearity

in trie Lewis law _was violated [34].

Finally

_we

analyzed

our simulations

according

to _a

question

raised

by

SMS

[13],

^{that is} whether the _von Neumann law of cellular patterns is also

appropriate

to describe the

partiale

coarsening in the fate stage. This law states that the

growth

of _a cell is

proportional

to the number of its

adjacent

cells minus 6, _1-e-, cells with _more than 6

neighbors

_grow, cells with less than 6

neighbors shrink,

and cells with 6

neighbors

maintain their _area. In trie _case of

two dimensional _soap froth

growing by

_gas diffusion

through

trie watts of trie

cells,

this law bas been deduced

by

_von Neumann [33] ^{aud it is also} very

popular

for

describing grain growth

in

metallurgic

cellular structures. Translated into trie framework of trie ME

theory

of

SMS,

it

gives

trie

growth

law

~j"

=

k(n 6) (7)

for trie _area A of trie

partiales,

thus _a

proportionality

of

charge ^~~"

and

topological charge

dr

C = n 6. In

particular

the average

growth

rate of _a

particle

^would be determined

only by

a

topological quantity

of the number of

neighbors. Figure

II shows trie

corresponding

results of numerical simulation. Indeed _we found _a

scaling

of ~~

with

topological charge

and could confirm _a

linearity

between these

quantities

with _a

strolldependence

on the _area fraction. It should be noted that there is _a

significant

_non-zero intercept of the _curves,

leading

to a more

appropriate

formulation ~~"

= const +

k(ç§)(n 6)

of equation

(7).

With this

completion

dr

of the

growth law, however,

the coarsening

dynamics

_seem to be well described.

4. Discussion

In this work _we have studied the

spatial

correlations of

partiales

in the fate stages ^of coars-

ening by

numerical simulation of the Cahn-Hilliard equation

(model B).

A _comparison with

recent experiments on 2D Ostwald

ripening [10, iii

shows that within this model the

spatial

particle-partiale,

_as well _as the

charge-charge

correlation

functions,

_can be well understood and

reproduced

with

high

_accuracy.

Especially

_{we were} able to confirm _a

dynamical scaling

^of

these correlation functions in the fate stage ^of

particle coarsening. By comparing

the obtained

scaling

functions in the _case of

particle-particle

correlations with simulation results of Akaiwa and Meiron [19] we found that the

dipole approximation

of the

quasistationary

^diffusion _equa-

tion that

they

used

gives quite

similar results. Seen from the

viewpoint

of the numerical

effort,

this simulation method is less _expensive than _ours, however

accepting

restrictions of circular

partides,

for instance.

Especially

at

higher

_area

fractions,

where

partides

tend to

change

their

shapes,

trie

importance

^of this

approximation

is undear. We believe that _a direct

compari-

son of both

methods, possibly following

_a

ripening

of trie _same initial

condition,

^could

help

to

darify

this

question.

^Another

major

part ^of this work _was concemed with trie correlation of nearest

neighbors, following

_an

experimental study

of

Seul, Morgan

and Sire

[12].

As in

these experiments, _we found

pronounced

anticorrelations in trie sizes of

adjacent partides.

We _were

unable, however,

to confirm _a

theory given by

SMS for these

anticorrelations,

based

on maximum

entropy principles.

Trie numerical simulation showed that _even trie

postulated

Lewis

law,

_{1-e-, a} linear

dependence

of

particle

size and number of

neighbors,

_was violated in contradiction _to the

experimental

results of SMS. A _reason for this difference

might

be _seen in _an additional repulsive electrostatic interaction of the

partides

in the

amphilic

system, as

supposed by

SMS. This conjecture is

supported by

the

significautly higher

values of

topologi-

cal

defects,

_{i e.,}

partides

with coordination number _n

#

6 in the simulation. It is well known

~ l'

~ ~~ ~

~Î~~Ù. Î ~

'

'~

~

~Î~~.~~ ~'

/

~ ~ ''

:"

~Î~~Ù.~~ ~

~'

Ô-ÔÔ~ ,' ~'

' '

'

.~

~,'

~Ô-ÔÙS ,'

#

' ' / / ' /

~~~~ ~

-3 -2 -1 0 2

n-6

Fig. Il. Relation between the variation of the _area

(dAn/dT)

and the topological charge C _{= n} 6

of _a particle _m the fate stage corresponding to the _van Neumann law (7) [32].