Ostwald ripening on a substrate : modeling local interparticle diffusion

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Ostwald ripening on a substrate : modeling local interparticle diffusion

Xin Zheng, Bernard Bigot

To cite this version:

Xin Zheng, Bernard Bigot. Ostwald ripening on a substrate : modeling local interparticle diffusion.

Journal de Physique II, EDP Sciences, 1994, 4 (5), pp.743-750. �10.1051/jp2:1994161�. �jpa-00247997�

Classification Physic-s Absiracts

05.70F 82.65J

Ostwald ripening

_{on a}

substrate

_:

modeling local interparticle

diffusion

Xin

Zheng (*)

and Bernard

Bigot

Laboratoire de Chimie

Th60rique,

Ecole Normale

Supdrieure

de Lyon, 46 Al16e d'Italie, 69364 Lyon Cedex 07, France

and Institut de Recherches _sur la

Catalyse,

CNRS, 2 Av. A. Einstein, 69626 Villeurbanne Cedex, France

(Received I December 1993, received in finalform 7 February 1994, accepted16 February 1994)

Abstract. A model with local interparticle diffusion is considered, in _contrast with the classical

model of Ostwald

ripening

(the _mean field model) and its

multiparticle

extensions which have long range interactions. Simulations of the evolution of the system ^{show that the} asymptotic ^behavior

obeys

_{a power} law. It is also found that the scaled

asymptotic

distribution of

particle

radii is

broader than in the

previous

models, even at low initial coverage where the

multiparticle

models

have the _{same narrow} scaled distribution _as the _mean field model. The

scaling properties

are

independent

of the initial

particle

size distribution.

1. Introduction.

The

sintering

of

particles (grains consisting

of

atoms)

on a substrate is of

technological

and fundamental interest.

Among

the

technological applications,

_we recall the formation of solid thin films _{on a} substrate

[I]

and the

ageing

of

supported

^metal

catalysts [2].

The

challenging

theoretical

question

of

sintering

is how to describe the evolution of the

disperse droplet-like phase

at late stages

[3]. Ordinarily,

whether

sintering

_occurs in three dimensions _or in two

dimensions,

it _can be

qualitatively explained by

the _curvature

dependence

of the chemical

potential

small

panicles

emit

single

atoms to the low

density phase (called

the vapor

phase),

the _atoms then

diffuse,

and _are

finally captured by large particles.

Thus the

large particles

_grow

at the expense of the small _ones,

reducing

the total surface energy. The process is often called Ostwald

ripening.

The models for Ostwald

ripening

have evolved from _mean fields models

(classical theory)

to

multiparticle

models

(modem theory) [4]. Early quantitative descriptions

for Ostwald

ripening

in three dimensions _were the models

by

Lifshitz and

Slyozov [5]

and

by Wagner [6].

,

(*) To whom correspondence should be addressed.

744 JOURNAL DE PHYSIQUE II N° 5

Chakraverty

extended the models _to the

coarsening

of

particles

_on a substrate

[7].

The

Chakraverty

model, ^{like its}

original

three-dimensional

versions,

is based _on the _mean field

assumption.

In the _mean field

formalism,

the vapor concentration reaches its uniform _mean field value at a

given

distance

(called screening distance)

from the

particles

each

particle

influences the vapor

phase

around

it,

but it is _not influenced

directly by

the relative location and size of any other

particle.

These _mean field models

predict

a

temporal

_power law of the

particle growth

and _a scaled form for the distribution of

particle

radii.

Despite

the _success of these models, there remains _a

major discrepancy

between

theory

and

experiment

the scaled

particle

size distributions obtained

by experiments

_are broader and

more

symmetrical

-than the _ones

predicted by

the classical theories. Recent studies have tried to

explain

this

disagreement by taking

into _account the effect of

interparticle

diffusional interaction. As _a

result, multiparticle

models have been

proposed [4]. According

to these

models,

the effect of

interpartide

diffusional interaction is related _to the volume fraction

(in

three

dimensions)

_or surface coverage

(in

two

dimensions)

of the

coarsening particle phase.

The model

developed by Dadyburjor

et al.

[8]

is the two-dimensional

example

of this

approach. According

_to the formulation of the

multiparticle model,

the emission _or

absorption

of

single

atoms from

growing

or

decaying particles

are modeled

by placing

_a

point

_{source or}

sink of the vapor

phase

at the center of each

particle.

The

strengths

of these

point

sources or

sinks _are determined

by

a

general

^{solution of} the diffusion

equation

with

appropriate boundary

conditions. Since the diffusion field is of

long-range

nature, the diffusional interaction between

particles

is also of

long-range

nature. In other words, each

particle

interacts with all the other

particles.

The usefulness of this model is based _on the

postulate

that the

strength

of each

point

source must converge when the number of

particles

in the system ^under consideration tends to

infinity.

This

assumption

is

probably

not fulfilled.

In the present

work,

_we take the

opposite point

of view. We model

coarsening

_{on a} substrate

by

a local model _a

given particle

interacts

diffusively only

with its _nearest

neighbors.

As this

model is

short-ranged,

it is called local

interparticle

diffusional interaction model. The interest in this model is _{not so} much that it is _more realistic than the _mean field

models,

but that it

represents ^the

opposite

limit _{to mean} field

theory

infinite range i's. short range. Its

temporal asymptotic

behavior and the scaled distribution of

panicle

radii will be

investigated by

use of

computer simulations. The results _are

compared

with _mean field and

multiparticle

models.

2. Model.

First, let _us consider _a system ^of two fixed

hemispherical particles (labelled by

I, I

= 1,

2)

on a surface.

They

are far apart ^{and coexist with} a common two-dimensional vapor

phase.

At

long times,

the

quasistationary approximation

_may be

employed

to describe the diffusional flux.

Therefore,

the concentration _c, of the vapor

phase

at the interface of the

particles (I,e.,

the dissolution surface

concentration)

is

given by

the Gibbs-Thomson

equation

_:

L

c, = c~ exp 2

(I)

R<

where c~ is the surface

equilibrium

concentration of the vapor

phase,

_R~ the radius of

particle

I and L~ ^the

capillary length.

In the time interval At, ^the mass

change

_Am~ of the

particle

is

expressed approximately by

Am~ ₌ s, ^At

(2)

where _s, is the total diffusion flux _across the

particle perimeter

on the substrate. In the

following,

_we

proceed

_to determine s, for the _two

particle

system.

When the _two

particles

are treat6d _as

point

_{sources or}

sinks,

the

quasistationary

diffusion field

c(r)

satisfies the

following equation

V~c(r)

2

=

2 _ar

£

_B~ (r

r,) (3)

1=1

where _r locates _a field

point,

_r,

signifies

the

position

of the

particle

I, and

B,

is the

strength

of

the _source

representing

the

particle. By

_use of Fick's law and the

divergence

theorem, _one

obtains

s, =

2

grDB~. (4)

There is a

general

solution for

equation (3)

of the form

c(r)=Bo+B,log [r-r,[ +B~log [r-r~[. (5)

Substituting

the

boundary

conditions

given by

the Gibbs-Thomson relation into

equation (5) yields

c, =

Bo

+

Bj log R,

+

B~ log d,~ (6a)

c~ =

Bo

+ B

log d,~

+

B~ log R~ (6b)

along

^with the constraint :

B,

+

Bj

₌ o.

(7)

It follows from

equation (6a), (6b), (7)

and

(4)

that _:

c~ c,

sj =

2 grD

~~

(8a)

~°~ R, i~

c, c~

s~ =

2 grD

~ ⁼ s,

(8b)

~°~ /~i~

The above system ^of two

panicles

_can be extended to N

hemispherical particle system

at

time _{t on a} substrate. The

position

of the

particle

I _on the substrate is denoted

by

its center, r,, and the

spatial

distribution of

particles, f(r),

is described

by

the summation of Dirac

functions _:

f(r)

=

( (r

_r,

) (9)

where _r locates the

position

on the

planar

surface of the substrate.

For any

particle

I we determine the closest

particle j

in the system. ^{For each such}

pair (I, j

we compute ^{the flux} _{s,~ = s~,}

according

to the

pair approximation, equation (8).

The

total flux away from _or towards _a

particle

I then is

approximated by

the _sum over all the flux contributions

s~~ of the

particles

which _are

paired

with I _:

~l

i

~<J ~~~~

j,pa>red^w>,h

746 JOURNAL DE PHYSIQUE II N° 5

If is

paired

with

j

and

j

is

paired

with I, the

contribution

s,~ = s~, is

only

counted _once.

In the simulations the flux defined above is calculated for each

particle.

The time step ^{then is} determined in the

following

_way

I/At

= Max

[- s,/m, (I I)

, i,. N

where m, is the _mass of the

particle

I. If At is

larger

than _a fixed cutoff AT, we set At

equal

to AT.

Finally,

the _mass of every

particle changes

as indicated in

equation (2),

and the size of every

particle

is

changed correspondingly.

If the _mass of _a

particle

after the _mass

exchange

is less than _a

given

small value M, the

particle

is

supposed

to

disappear.

Time then is increased from _{t to t +} At. The above process is

repeated

until

only

a

single particle

remains.

For convenience,

length,

time, concentration and _{mass are} made dimensionless. The

lengths

will be scaled with the

capillary length L~.

2

«V~

L~ =

~g

~

(12)

where _« is the surface energy ;

V~

the molar

volume, R~

^the gas constant and T the

temperature.

Similarly,

the concentrations _{c are} scaled

by

_c~ and the _{masses m}

by

2

grpL(/3 (p

is the volume

density

of

particles). Finally

time _t is rescaled

by pL(/Dc~

_to make it

dimensionless.

Hereafter,

all

quantities

are

expressed

in dimensionless form. The flux

s,~ between the _two

particles

I and

j

then is

exp

(I/R~ )

_exp

(I/R, )

s,~ ⁼ _~

(13)

log

d,~

~i ~J

The total flux around

particle

I

(Eq. (10)),

its _mass

change (Eq. (2))

and the time step

(Eq. (I

I

))

are

unchanged

in dimensionless units. The

particle

radius _now varies with _{mass as} m, ~

RI-

3. Simulations and results.

On _a

rectangle

^{of size} L~ x L~ ^with

periodic boundary,

the centers r of

hemispherical particles

are

placed

at random. The initial distribution of

particle

sizes is uniform between

R~,~

^and

R~~~,

^{I.e. it is} a box distribution. The ratio of the radius of _a

particle

to the distance

between the

particle

and its _nearest

neighbor

is

always

less than 0,I.

The simulations of the present ^model use the

following

_parameters the total

particle

number is

20000,

the size of the

rectangle

is

L, =L~

=6000, the maximum time increment AT

=

100 and the minimal

particle

mass M

= 0.001. The initial distribution has

R~,~

₌ ^{0,I and} R~~~ ₌ ^2.

Let _us call the

monotonically decreasing

number of

particles

N

(t)

and define the average radius

(R) by

N

(R)

=

£ R, (14)

~,=,

The scaled distribution of

particle radii, n(z),

then _can be defined

by n(z)

=

N~/N/Az (15)

where _z

= RI

(R ),

Az is _a small interval around _z and

N~

the total number of the

particles

in the

interval

(z- Az/2,

_{z +}

Az/2).

The interval Az is chosen small

enough

such that the size

distribution

n(z)

varies

smoothly

between _z and _{z +} Az but

large enough

to avoid

large

statistical fluctuations.

We have measured the evolution of _n

(RI (R ))

at

long

times. In order _to reduce the statistical fluctuation, ten

samples

with different initial conditions _are

averaged.

The average

n

(RI (R))

of the _ten

samples

is

presented

in

figure

la. It is found that the system ^reaches an

asymptotic regime

where _n

(RI (R))

is invariant. We have also measured the increase of the

average

(R)

and the decrease of the total

panicle

number N with time t. These _{curves are}

called

sintering

_curves. It is found that these _curves take the form of _a power law _:

(R )

_cc t~

(16a)

N _cc

t~~ (16b)

where _a and b _are exponents. ^An

example

is

given

in

figure

lc. The average ofthe exponents a

and b obtained from each of the ten

samples

is _a

=

0.233 ±0.003 and b

=

0.696 ±0.008,

respectively.

3 3

4h t.13mo

$ _~~~~$

+ t.m2m

4~ t.«loo - t.5smo

~ 2

(

( ~i

a

b

o 2 o 2

RllR> w/R>

Ian idn

m a <R>

. N . N

loo

z z

I

£

« ~

~ Jo ^v Jo

c

d

i

iaX0 10oom iooooot Jan iaX0 10oom iooooot

t t

Fig. I.

-Scaling

behavior of the system at low initial coverage

(20000particles

_on the 6 000 _x 6 000 square) ^{the scaled}

particle

^radius distribution for the box initial distribution (a) and the Gaussian initial distribution (bl the

sintering

curves of the average ^radius

(R)

and the total

particle

number N, respectively, for the box initial distribution jc) and the Gaussian initial distribution (d).

748 JOURNAL DE PHYSIQUE II N° 5

We also considered _a Gaussian initial distribution with average I and variance 0.5

(cutoff

at 0. I and

2).

Ten different

samples

are also used. The results show that the

asymptotic

_power

laws have

nearly

the _same exponents as the simulations with _a box initial distribution _:

a =

0.228 _± 0.002 and b

=

0.680 _±

0.006,

the

sintering

curves of _one

sample

_are demonstrat- ed in

figure

ld. Their scaled distributions _are also

given

in

figure16.

Furthermore _we also simulated _a system ^with a lower initial coverage. We increase the size of the square to 70 000 _x 70 000, and

keep

all other

quantities unchanged.

Ten

examples

are

tested

respectively

^{for both the box} distribution and the Gaussian distribution. The average scaled

particle

size distributions _are measured after transient processes have

disappeared.

The

distributions _are

clearly

invariant in time

(Figs.

2a and

b).

In the

asymptotic regime,

the

sintering

_curves of the average of the radius and the total

particle

number

satisfy

_power

laws,

the _curves for the _two

examples

are

given

in

figures

2c and 2d. The average of the exponents ^of

the ten

examples

_{are a}

= 0.234 _± 0.002 and b

= 0.701 _± 0.005 for the box initial

distribution,

and _a

=

0.233 _± 0.003 and b

=

0.696 _± 0.008 for the Gaussian initial

distribution,

_respec-

tively.

3

» t-mm ⁺ t-144M

+ t.3XW ⁺ t-mm

+ t.85138 ⁺ t-Mm

~

2

~ ~

~

~

~

b

0 2 ~ 2

RllR> RllR>

m <R>

m <R> _o N

o N

z

4 I

A «

« v

v lo

C

d

lam 10oom iannJ 10oom ioxon

t _t

Fig. ^2.- Scaling behavior of the system at very ^low initial coverage (20000 particles _on the 70 000 _x 70 o00

square)

the scaled

particle

radius distribution for the box initial distribution la) and the Gaussian -initial distribution 16) the sintering _curves of the average radius

(R)

and the total

particle

number N,

respectively,

for the box initial distribution (c) and the Gaussian initial distribution id).

4. Discussion.

Theoretically,

the Ostwald

ripening

_process is

usually

studied in two extreme cases reaction controlled _or diffusion controlled

[7].

In the reaction controlled _case,

single

atoms diffuse very fast _on the

substrate,

such that the

growth (or decay)

of

particles

is dominated

by

reactions _at the surface of the

particles.

In the diffusion controlled _case, the surface process

happens

so fast that it _can be

neglected,

hence the Ostwald

ripening

is controlled

by

the diffusion of

single

atoms on the substrate.

Evidently,

the model

proposed

above describes the diffusion controlled

process. In the _mean field model

by Chakraverty [7],

the

sintering

curve of the average

particle

radius has _a power law with the exponent

I/4

when the

coarsening

is controlled

by

diffusion.

The simulations of the present ^{model show that the}

sintering

_curves also follow _a power law but with _a

slightly

different exponents : about 0.23 for _our model.

Scaling

arguments ^would

suggest ^{that the} two models have the _same

asymptotic

_exponents. The

discrepancy

_may be related to

logarithmic

corrections

(see Eq. (13))

_or to the

changing

effective

panicle density

_:

the average shortest distance between

neighboring particles, id) (defined

_as the average

distance between the _center of _a

given particle

and the _center of the

particle

closest to

it)

scales with the average

particle

radius like

id )

cc

(R )

^3'~

(17)

for three-dimensional

particles

_{on a} two-dimensional substrate. The most

imponant

result of the local model is that it has _a broader scaled distribution of

particle

radii than the _mean field

model

(Fig. 3).

This is _not

surprising

_as in the present

model,

the

growth

or

decay

of _a

particle

is

only

determined

by

its

surroundings.

The diffusional interaction between

particles

is forced

- a

+ b

+ c

- d

+ e

0 2

R/<R~

Fig.

3.

Comparison

between the scaled

particle

radius distributions of the Chakraverty model (a) and of the present ^{model the box initial} distribution _on the 6000 _x 6000 square 16), the box initial

distribution on the

70000x70000square

(c), the Gaussian initial distribution on the

6 000 _x 6 000 square id), and the Gaussian initial distribution _on the 70 000 _x 70 coo square (e).

750 JOURNAL DE

PHYSIQUE

II N° 5

to be short range. In contrast, in the _mean field

model,

the effect of _a

given particle

on the

growth

_or

decay

of another

particle

is

independent

of the distance between them _; the diffusional interaction is of infinite range.

In the

multiparticle

model

by Dadyburjor

et al.

[8],

it is

supposed

that the system of

particles

consists of _a basis _set and its infinite

periodic

translations in _two dimensions. Thus it _can be resolved

numerically by

_use of the Ewald construction. Their results indicate that the _curves of the average radius _{as a} function of time vary from _{concave to convex as} the initial coverage

increases.

They probably

do not reach _a universal

asymptotic

_power law because of the limited simulation time. Their results also show that for low initial coverage the

particle

size distribution is _narrower than for

high

initial coverage. In

particular,

at very low coverage, the distribution is

only marginally

different from the _mean field distribution. As the

particles

are 3-

dimensional and the support ^is 2-dimensional, the

density

decreases with time. The

distribution of _a system ^with a

heigh

initial coverage

gradually changes

to a system ^{with low} coverage.

In _our

simulation,

_very different initial distributions of

particle

^radii

(such

_as box and

Gaussian) finally

reach the _same scaled distribution. The initial coverage in _our simulation is

always

_very low.

Nevertheless,

the scaled distribution of

particle

radii _at late

stages

^{is broader} than both the _mean field model and the

multiparticle

model

by Dadyburjor

et al.

[8]

at low

initial coverage. We conclude that the local

coarsening

mechanism is the _reason for the

broadening

of the scaled distribution.

According

to the

prediction

of the _mean field

model,

the

sintering

_curve of total

particle

number N

(t)

is _a power law with an exponent 3/4. In the

multiparticle

model

by Dadyburjor

et al., the evolution of the average

particle

number has _not been calculated. In the local model studied here, N is found to have _a power law form with _a

slightly

different exponent from 3/4.

The exponent ^ratio bla is close _to 3, _a consequence of the exact mass conservation.

References

[1] Eckertovh L.,

Physics

of thin solid films, second revised edition (Plenum Press, 1986) _p. 121.

[2] Ruckenstein E.,

Metal-support

interactions in catalysis, sintering and redispersion, ^S. A. Stevenson, J. A. Dumesic, R. T. K. Baker and E. Ruckenstein Eds. (Van Bostrand Reinhold, 1987)

p.187.

[3] Gunton J. D.,

Miguel

M. S. and Sahni P. S., Phase transitions and critical

phenomena,

Vol. 8, C. Domb and J. L. Lebowitz Eds. (Academic Press, 1983) _p. 269.

[4] Voorhees P. W., J. State

Phys.

38 (1985) 231.

[5] Lifshitz I. M. and Slyozov V. V., J. Phys. Chem. Solids19 (1961) 35.

[6] Wagner C., Z. Elektrochemie 65 (1961) 581.

[7] Chakraverty B. K., J.

Phys.

Chem. Solids 28 (1967) 2401.

[8]

Dadyburjor

D. B., Marsh S. P. and Glickman M. E., J. Catal. 99 (1986j 358.