Aggregation in an expanding cloud: experiments and numerical simulations

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Aggregation in an expanding cloud: experiments and numerical simulations

Françoise Dziedzinl, Robert Botet

To cite this version:

Françoise Dziedzinl, Robert Botet. Aggregation in an expanding cloud: experiments and numerical simulations. Journal de Physique II, EDP Sciences, 1991, 1 (3), pp.343-352. �10.1051/jp2:1991172�.

�jpa-00247522�

J

Phys

II1 (1991) 343,352 MARS 1991, PAGE 343

Classification

Physics

Abstracts

82 70R 05 40

Aggrigation _in

an

expanding cloud

_:

experiments and numerical simulations

Frangoise

Dzledzini and Robert Botet

CE B, BP n° 3, 91710, Vert-Le-Petit, France

Laboratoire de

Physique

des Sohdes, Bit 510, Unlversitd

Pans-Sud,

Centre

d'orsay,

91405

Orsay,

France

(Received14 November 1990, accepted17 December1990)

Rksumk. Nous _avons construit un dispositlf

expdnmental

et _un moddle numdnque _pour dtudier

l'agrbgation

d'un abrosol (oxyde de

titane)

_{en expansion} dans

l'atmosphdre.

A partir ^de

l'analyse

de

photographles

_{pnses au microscope}

blectronlque

I

balayage,

on montre que l'abrosol

agglombrb

forme des amas fractals de dimension fractale de l'ordre de 1,75. Les simulations

numbnques

confirment quantitativement cette

particulantb gbombtnque

Nous montrons

comment nos rbsultats

numbnques

peuvent

complbter quelques

points qui ne sont pas accessibles

I

l'expbnence.

Abstract. We have set-up an

expenmental

device and _a numencal model _to

study

aggregation of an aerosol (titanium

oxide) expanding

_m the

atmosphere

By mean of scanning microscopic

analysis,

it is shown that

agglomerated

aerosol forms fractal clusters of fractal dimension of about 75. The nurnencal simulations

quantitatively

confirm this

geometncal

feature. We show how

our numencal results _can complete _some points which _are not available _m expenments

In order _to

study aggregation

^{of solid}

atmosphenc

aerosols _in standard ambient cond1tlons of

temperature,

pressure and

humidity,

_We have chosen _as

test-products

titanium

compounds

formed

by

chemical reaction of

TiCl4

With Water Vapor

[I].

These

compounds

_are used _m such

matters _as

metallurgy [2]

or

photodissociation

of Water

[3].

These _are also Well-known at

smoke obscurants The

elementary particles

_are

spherical

and therefore _easy to

study

1.

Experhnental

device.

The

expenmental,device

_is based _{on a} 50

sphere

made of

glass

Thls rather small Volume has been chosen because it _was much easier _to

sample

and control the initial conditions. There _are several inlets _on the surface of the

sphere

which makes it

possible

to add filters

(for example

to check the _pressure inside the

sphere)

and different apparatus which will be descnbed below

As _a first

tnal,

we have

dispersed

the

TiC14

in

liquid phase by

_mean of _a nebulizer connected to one of the inlets of the

sphere

But this idea _was given up because the chemical

reaction between

Ticl~

and _{water vapor}

happens

_very

quickly

and takes

place

in the _pipe where

consequently

_a

large

amount of

product

settles without

having enough

time to reach the inside of the

sphere

itself. To

disperse TiC14 directly

inside the

chamber,

_we have

developed

_a small metallic device which is able to hold and break _a little

phial (glass) containing

⁵ _~Ll ^of

Ticl~.

Note that

by

this _way, the device allows the

expenments

to be

quite reproducible,

_as for the

quantity

^of

product generated

m the

sphere.

Then _we have first remarked that the

just-formed particles

^tended to be attracted

by

the metallic

parts avoiding

main part of the diffusion inside the

sphere

That is the _reason

why

we have added _a thin tube which

bnngs

_a

nitrogen

^flux

(nitrogen

does not react with

TiC14) just

at the

top

^{of the broken}

phlal leading

to a

quasi-sphencal dispersion

of the

product.

Moreover it _is quite easy to tune the expansion

velocity

of the cloud

by

mean of this

nitrogen

_pressure

The

experiments

^{have taken}

place

at the ambient

temperature (23-24 °C).

The

humidity

_is tunable

by

the

quality

of _air

filling

the

sphere

at the

beginning

of the

expenment,

and it is

controlled in permanence

by

a

humidity probe.

A

sampling probe penetrates

at about

2/3

the radius of the

sphere (8.5

cm from the

center)

and _is connected to a thermal

precipitator.

^This

apparatus

^{allows the}

sampling

of _very small volumes

(e

_g 6

cm~/min)

which _is

indispensable

_{in our} small chamber. Particles taken in the flow _come to _a hot wire and

precipitate

_on small

glass

lamellas situated _on each side of the wire. These lamellas _are then coated with

gold

for observation _{m a} scanning electron

microscope

This coat _is 4 _nm thick which _is much smaller than the

typical

diameter of the

individual

particles (~

40

nm).

At the

magnification

used to

study

the aggregates, ^the surface of the

spheres

_appears ^smooth.

Figure

^I is a

simplified

sketch of this device.

Fillei

Thernial Humidity-

PhiJl

§f~

Filter'

, i<iiicr

~~§Pump

Fig

[.-Sketch of the

expenmental

device

2. Some

experimental

results.

The

nitrogen

pressure, the relative

humidity

and the temperature can

easily

be varied. In this paper we are

only

concemed _in the influence of the radial

velocity

via the

nitrogen

_pressure.

The influence of the relative

humidity

will be

published

^elsewhere

[4]

and the influence of the temperature has not

yet

been done

experimentally.

Quahtatively

_we observe _a radial expansion of a white cloud of titanium

compound.

Of

course the

larger

_is the _pressure of nitrogen ^{flux the}

larger

_is the

velocity

of this

expansion

Typically,

the

edge

of the

sphere

_is reached after about ten seconds Observation of

samples

M 3 AGGREGATION IN AN EXPANDING CLOUD 345

by

the thermal

precipitator

^shows

elementary particles (spherical

_in

shape

with _{a mean-size} of about

0.l~Lm)

_as well _as

aggregates

^{of these}

particles. Figure2

is

typical

of such

configuration.

To obtain

quantitative results,

_we

d1gltalize

the

photographs

taken from the electron microscope. In such _a way, we have

directly

_an idea of the size-distnbution of the part of the

sample

which has been

photographed,

_as well _as the fractal

dimension,

if any, of the

largest aggregates belonging

to the field. The method to _measure this dimension is the

nesting-circles

method _as

explained

_now for each

cluster,

_a standard _program transforms its

d1gltahzed

image ^into a binanzed _{image m} which each

point

_is I _or 0 whether _{it is}

occupied by

matter or not

Then,

the center of _mass of the black

pixels

_is calculated and _one chooses the closest black

pixel

to be the

origin

of the local coordinates. One draws circles centered _on this ongln

and of _successive radii _:

4, 6, 8, pixels,

and _one counts all the black

pixels

inside each circle.

This

gives

_an estimation of the total _mass of the

cluster,

within _a

given

distance from the ongln. The fractal dimension of each individual cluster could be extracted from _a

Log-Log

plot

of the _{mass versus} this distance since the formula _:

M(R)

~

R~

holds for fractal

objects

of fractal dimension D

~ 2 We shall

verify

_{a posteriori} that all the fractal dimensions founded here _are smaller than 2 But since _{we are}

working

with many

if

Fig.

2

Typical

_scanning macroscopic

photographs (nitrogen

pressure ₌

01bar,

relative humidi- ty ₌ ^{73 5} 9b, time ₌ mln 30 s) The two largest aggregates of the field have _a diameter of order 3 _~Lm

objects

of

relatively

small

sizes,

_we have to be very careful about the way ^to estimate the

averaged

value of this fractal dimension. A crude average of all values of D estimated for each cluster would _{give a}

quite

^{bad estimate of} its

averaged

value _since the fractal dimension acts as an

exponent

m the formula

relating

the _mass and the radius. So _we have realized the _average

of all masses

M(R)

contained inside a circle of

given

radius R for all the clusters of the

collection for which the estimated diameter _was

larger

than I _~Lm. Then _we have calculated the fractal dimension with the

Log,Log plot

of

(M(R))

_versus R

We have obtained very nice

straight

lines _on these double

logarithmic plots,

and this makes believable the

interpretation

of the

photographs

in term of fractal

objects (see Fig 3).

In table

I,

_we

present

two series of results : in the first _one the fractal dimension _is

given

for

a nitrogen pressure of 0.I bar

(and

relative

humidity

73.5

fb).

In the second _one the

nitrogen

pressure is 0.2 bar

(and

relative

humidity

87.4

9b).

Six

samplings

have been done at _six different _times for each

series,

from I _mm 30 _s to 18 _mm. The number of

analyzed

clusters _are indicated below the fractal

dimensions,

_so that _{one can} estimate

roughly

the confidence in the

resulting

^value The _error bars _on the values of the fractal dimension _are estimated of about

± 0.05. The results of Table I

appeal

_some comments. We observe first that there _{are no}

significant

variations of this fractal dimension with time. As

aggregation

goes on, the fact that

D _remains constant _means that it is _a

good quantity

to charactenze the internal

geometry

of

the clusters

independently

of their _sizes.

Moreover,

_our results for the smallest time _{we can} do

(1.5 mini

show that

aggregation

process is

already

well

engaged

at this time

(see Fig. 2) 1ihls

is conforted

by

the

analysis

of

experimental

size,distribution the _mean diameter _increases

only slightly

^with time

(an example.

0.306 _~Lm, 0.308 _~Lm, 0.388 _~Lm, 0.378 _~Lm for

respective

times. I _mm 30 _s, 6 _mm, 13 _mm, 18 _mm, for

nitrogen

pressure 0.I bar and relative

humidity

73 5

fb,

the _{errors on} the diameter _are of order 0.I ~Lm). ^In the _{two series} of experiment, we

have found D

=

75 _± 0.08 for the value of the fractal dimension

Log(M) ⁴

D 1.68

O,5 1,0 1,5 2,0

Log(R)

Fig

3.-One

typical

experimental result

(nitrogen

_pressure

= 02bar, M is the _mass

averaged

_over

13 clusters of _{a same} sample) The saturation for Log (R) _~ l 5 _is due to the finite _sizes ofthe clusters The scales _are

arbitrary

M 3 AGGREGATION IN AN EXPANDING CLOUD 347

Table I.

Experimental fractal

dimensions

for d#jferent

times and two pressures

of

_nitrogen

flux.

The relative

humidity

_ts indicated below the _pressure

for

the _two experiments. The small

numbers below the values

of

the

fractal

dimensions _are the number

of

clusters used _to extract the value

of

the dtmensions.

PN~)

0.I bar 0.2 bar

time

(RH)

(73.5 fb) (87.4 fb)

mln 30 1.62 70

(23j (6j

6 _mm 83 79

('21 (61

13 min 1.70 75

(1 14) (8)

18 _mm 1.67 1.79

(26) (18)

A

striking

result is that the fractal dimension does not seem to

depend

_on the radial

velocity

of

expansion

of the cloud

(see

Table

I).

In fact _a

rapid sight

_on this

problem

could lead to the

erroneous conclusion that the clusters should be less compact ^{than clusters} grown in

homogeneous

conditions _since, because of the radial drift which tends to

keep

the clusters far the _ones from the

others,

the number of

possible

collisions between two of them _is

considerably lower,

_so that

just sticking by

the tips is

predominant.

As _we shall _see, this

simple

_{reasoning is wrong} for fractal

objects,

but

the

_reason of this failure _is not immediate.

Apart

an

explanation

of the value of the fractal dimension the _reasons

why

_we have tried _to

explain

these features

by

a numencal simulation of this sort of

physical aggregation

within this radial

geometry

are that this numerical

approach

_{gives access} to the first times of

coagulation,

which cannot be reached

experimentally

and which _is

certainly important,

_as

quoted

before

3. Numerical model.

Contrary

to colloids

dispersed

_{in a} viscous

liquid

like water, and for which it _is easy to avoid

coagulation by

agitation, in most

applications

aggregation ^of aerosols takes

place

_{in an}

homogeneous

medium

(steady

state gas of low

viscosity)

but with

inhomogeneous starting

cond1tlons. The

expenment

described _in Part I _{is a}

typical example

of such conditions where _a competition exists between expansion of the aerosol cloud and

aggregation

^{of the aerosol.}

To

study

the influence of _a radial drift _on the

geometrical

structure of

agglomerated

aerosols _we have

imagined

_a numencal model which _we

present

now a set of N

sphencal

particles

of _same diameter d _is put ⁱⁿ a continuous infinite space This set simulates the imtal

phlal

of the

experiment

and _is defined _as follows _a cluster of N

particles

_is constructed

according

to _a variant of the Bennett-scheme

[5] (see Appendix)

which realizes very compact

close-packed

random

objects (see figure

of the

Appendix)

At this stage ^the diameter of the

particles

is set

equal

to I and it will be the definition of the unit of

length

_m the

problem.

Then the diameter of all the N

particles

is decreased _to the desired value d. This initialization allows

to obtain _a disordered _set of ind1vldual

disjointed particles

and to

adjust

the value of the initial concentration to any value between 0 and the most compact

package

of iso-diameter

spheres (about

0 61

by

this

way).

At time t one gJves ^to each cluster

(or

individual

particles)

its actual radial

velocity

v~ whose modulus _is

r/t [6] (this

_{is a} solution of the Namer-Stokes equation of _a ball of viscous gas

expanding radially

_{in an}

empty space).

Here _r is the distance of its center of _{mass to} the

center of

explosion (center

of _mass of the inital

configuration).

One gives also to each cluster its dilTusion coefficient D

= D

o/k

_[7~ where

Do

is the coefficient for _an individual

particle

and k _is the number of

particles (that

is _: its

sue)

of the considered cluster. At this

point

one tnes

to increase the time of _an amount At, and all the clusters _move at the _same time. To achieve

these movements _one calculates the individual

displacement

of each cluster _as the _sum of the radial

displacement (which

is just the

product

of the

velocity

_v~ and

At)

and the Brownian

displacement

which _is

equal

to

fi

_in

a random direction of the _space. The trial value of At is chosen such that _no

displacement

of _a cluster _can exceed the value

d/2

_in modulus.

Overlaps

of

particles

_can

happen.

If it _is the case, we search the

largest

value of At which avoids

overlap

but allows contacts, and

by

this way :

aggregation.

Th1s value _is given

solving

_as many 4th

degree

_{equations as} there _are

overlaps

The direction of the Browman motion of each cluster _is

kept

constant

throughout

this search

(only

the modulus of this

displacement

_is affected

by

the

change

of

Ail.

This

sticking

_on contact _is irreversible _m the

sense that, _once

stuck,

two

particles

_{can never} unstick. Clusters _are

ngld

and _{so we} do not

study

here the influence of the

temperature

^{in this} process.

The fractal geometry ^{of these clusters} are

investigated

_{as a} function of time and of the adimensional

parameter

p =

Do/vod,

where vo is the initial dnft

velocity

Thls

parameter

characterizes the relative

importance

^of the radial dnft and the Brownian motion, and it _may be estimated

lying

_in between I and 10 in _our

ordinary physical

conditions. Note that the

larger

_{is p,} the smaller is the radial

expansion

of the cloud.

4. Numerical results.

At the

beginning,

the fractal dimension of the clusters have been calculated _{in two} different ways. The first _one consists _in

plotting

_image of all the

largest

clusters

projected

_on three

orthogonal planes

and to

digitallze

then binanze these

pictures

We treat them with

the

_same program as described _in the

expenmental _part.

This _insures

exactly

the _same treatment for the

experimental

^{clusters and for the simulated} ones The second _{one consists in}

plotting

the

loganthm

of the number of

particles

of each cluster _versus the

logarithm

of its radius of gyration for all clusters

present

at _a

given

time. The

slope

of the

straight

line

(if any)

_gives a

value of the fractal dimension of the collection of

clusters,

_{since we} know the formula

N~Rf'

(this

is

equivalent

to the formula given in section II but for _a collection of clusters of _same fractal dimension

D).

In table

II,

the fractal dimensions obtained

by

the two methods _are

shown and _can be

compared

for two numencal simulations. One _sees that

agreement

is

satisfying.

Th1s result gives confidence in the method

(the

first

one)

used _in

expenmental

set- up, and

explains why,

for all the

following,

numencal results have been obtained with the

second method

(easier

to _use on a

computer).

We have

checked, by

^this method that the fractal dimensions do not

depend

on the time.

But the poor statistics due to the small number of clusters _at small

times,

lead _{us to} take _in fact all the clusters any ^time

they

_{appear in} the simulation to have better statistics to extract the

common fractal dimension. Th1s

independence

of the fractal dimension with respect to time _is

M 3 AGGREGATION IN AN EXPANDING CLOUD 349

Table II. -Numerical

fractal

dimensions

for

two

diameters,

with _two methodY the nesting- circles with three

different projections

on

orthogonal planes,

and the methods

of

the radius

of

gyration

(N

=

2

000,

_p

=

5).

Diameter o.90 0.91

j~ projection

1.74 1.67

along

X E

~

~

projection

(

_{1.70 1.67}

£

along

Y

c- o

i ^projection

~ l.60 1.67

( _along

_Z

Method of the radius

of

gyration

71 1.72

well-known for

closely

related models _as standard Cluster-Cluster aggregation

[8].

This _is due to the scale-invanance of the system.

We have

systematically

studied the influence of the parameter p on the value of the fractal

dimension,

for p

belonging

to the _range 1-10

(see Fig. 4)

We have found the value D = 1.72 ± 0.10. One remarks _on

figure

4 that the first points are well

supenmposed

_over

about _one decade

(in radius).

This confirms that the fractal dimension is not _even the _same,

but_

_{we are in presence} of the _same statistical

objects.

The value of the fractal dimension _must be

compared

with the value measured in the expenments D

= 175 _± 0.08. So either

experimental

_or numencal results _agree to say that there is _no influence of the radial dnft _on the geometry of the

clusters,

even if the _size-

distnbut1on is indeed _very sensitive to this parameter.

The _reason of this absence of influence _is that two clusters close _one to the other have _a

small relative dnft

velocity (in regard

to their relative

distance)

_so that Brownian motion

remains

predominant

_as for the standard

agglomeration

_process. This

exp1alns

also

why

the

experimental

or numencal values of the fractal dimensions _are identical

(within

_error

bars)

to the standard cluster-cluster _one.

We have

changed

the value of another parameter ^d

(and

_so the 1nltial concentration, as

exp1alned before).

In table

III,

the fractal dimensions for parameter p =

5,

N

= 1000 and d

TableIII.-Numerical

fractal

dimensions _versus the diameter

of

individual

particles (i.e.

versus the initial

concentration) for

the parameters N

= 2

000,

_p

= 5

Diameter 0 90 0 91 0 93 0.95 0 97

Fractal

Dimension 1.71 1.72 1.74 71 74

Log(N)

a

a

$f°

~a

~

~

~f

^a

~

~

# all

a.

m

»

«

« w

»

I

M 3 AGGREGATION IN AN EXPANDING CLOUD 351

Acknowledgements.

It is _a

pleasure

to thank M.

Cabane,

C.

Tiret,

R Julhen and Ph. Adam for

permanent help

and discussion. Th1s work _was

supported by

contract n°

25073/89/ETCA/CEB/DED.

Appendix.

The Bennett-scheme

[5]

is an

algorithm

used to construct very compact structures. The idea is to start with _a set of three

adjacent particles forming

_an

equilateral tnangle.

Then _one adds

successively

individual

particles

of _same diameter in such _{a way} that the added

particle

touches three other _ones

(in

_a d-dimensional space, this _maximum condition writes.

tangent

to d other

particles).

An _economic way ^to achieve

this,

is to make _a list of all the

triplets

of

particles

which have the two properties : each

particle

of the

triplet

touches the two

others,

and _{one can} put at least a fourth

particle touching

the three others. The number of such

triplets

_in this list _grows

just

as the surface of the cluster

(so

it _is a

N~'~-process,

where N _is the total number of

particles

of the

aggregate).

To _insure

good

randomness of the

cluster,

_one adds _{a new}

particle

at _a site chosen

randomly

_among all the

positions

of the available

tnplets.

This _is the _version used here One reactuahzes then the list of

triplets

and _{so on.} In his work

Bennett choose

systematically

the site the closest to the center of _mass of the

aggregate

A

typical

cluster of 1024

particles

_grown

by

this _way is shown in

figure

5.

Fig ⁵ Typical ⁰²⁴

particles

cluster obtained

by

the Bennett-like scheme _as descnbed _m the next

JOURNAL DE PHYSIQUE II T I,M3, MARS [WI 16

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d'Adrosol.

Expdnences, Analyse

Fractale et

Simulations, Thdse de Doctorat, Unlversitb Pans 7

(1990)

DzIEDzINL F and BOTET R _m preparation

[5] ^BENNETT C H, J

Appl. Phys.

43

(1972)

2727

[6] ^The geometry ^{of the} aggregates ^{do not} seem to

depend

_on the precise law giving

v~ ^as

long

_{as v~ remains} centro-symmetnc

[7] ^Here we suppose a priori

tiat

_{the clusters have}

so tenuous structure that the fnction force _is

proportional

to the total number of

particles

of the cluster This

happens

for

example

with fractal clusters of fractal dimension

< 2. For more compact

objects,

_one has to

replace

this formula

by

D

=

Do/R

where R _{is a}

typical

radius of the cluster

(m

the Stokes

regime)

Once again the

geometncal

features do not

depend

_on the _precise vanation of D with k _or R, as

long

as D _{is a}

decreasing

function of k, cf BOTET R

,

JULLIEN R and KOLB M

,

J

Phys

A Letters 17 (1984) L75

[8] For a recent _review BOTET R and JULLIEN R. Phase transitions 24

(1990)

691.

[9] ^{The elastic}

light

scattering of iso-fractal

polydisperse

_{aggregates is} proportional to the second

moment of the size-distribution. See for

example

DJORDJEVIC Z B, Ph. D. Thesis MI T

(1984) for _a detailed presentation of this result