Aggregation kinetics of macroscopic particles

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Aggregation kinetics of macroscopic particles

Jean-François Roussel, Robert Blanc, Christian Camoin

To cite this version:

Jean-François Roussel, Robert Blanc, Christian Camoin. Aggregation kinetics of macroscopic parti-

cles. Journal de Physique, 1989, 50 (21), pp.3269-3283. �10.1051/jphys:0198900500210326900�. �jpa-

00211142�

Aggregation kinetics of macroscopic particles

Jean-François Roussel, Robert Blanc and Christian Camoin

Laboratoire de Physique ^des Systèmes Désordonnés, URA 857 du CNRS, Université de Provence, ^{Centre de} Saint-Jérome, ^Case 161, ^avenue de l’Escadrille Normandie Niémen, ¹³³⁹⁷ Marseille Cedex 13, ^France

(Reçu le 24 avril 1989, accepté ^sous forme définitive ^{le 10} juillet 1989)

Résumé.

On étudie la cinétique d’agrégation ^de particules macroscopiques ^en suspension, ^en

fonction du taux de cisaillement et de la concentration. La cinétique ^{obéit aux lois d’échelle} classiques. ^En introduisant des temps caractéristiques, les résultats peuvent ^être exprimés ^sous

une forme unique indépendante des conditions expérimentales.

Abstract.

We present ^a study ^{of the} aggregation kinetics of macroscopic particles ^{in a} suspension, ^{as a} function of the shear rate and of the surface concentration of particles. ^The

kinetics are found to obey ^classical scaling ^laws. By introducing ^{a set of} characteristic times, ^the results can be expressed ^{in a} universal form independent ^{of the} experimental conditions.

Classification

Physics ^Abstracts

05.40

05.60

This paper is the second of ^two articles devoted to the study ^of aggregation kinetics of

particles. ^{The first one} (referred ^as [I] ^{in the} text) ^{describes the} experimental procedure ^and

the image-analysis methodology, while this one reports ^{on the} experimental results and their

interpretation.

1. Introduction.

Aggregation, ^{which we define as the formation} of clusters starting from isolated particles, ^{is a} phenomenon ^of great importance ^{in such fields as} physics, chemistry, biology, meteorology,

etc.

The principles ^of aggregation ^were originally established by Smoluchowski [1], ^who derived, ^{for the case of Brownian} particles, ^an equation giving the evolution of the number of clusters containing, ^at time t, s particles (Ns ⁱⁿ Eq. (1)) :

where kij represents ^the probability ^{of a collision between an i-sized cluster and a} j-sized ^one,

with i + j

^{s. The second term on the} right hand side of the equation represents the loss of s sized clusters by sticking ^{with i-sized ones.}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500210326900

Since then, ^the progress made ⁱⁿ the theory ^{of critical} phenomena has allowed the

description ^{of the} system ^{in terms of} scaling laws, leading ^{to the} general ^{form for the} variations, with s, ^{number of} particles ^{in a} cluster, ^and time t, of the number of s particles

clusters N (s, t ) :

where g(x) ^{is a} scaling function of the variable x, and 0 and z are scaling exponents.

N (s, t ) represents ^the asymptotic solution, ^for large ^values of t, ^of equation (1).

The theory of fractals describes the phenomena ^{under a more} geometrical aspect. ^{In the}

case of our experiments, ^where macroscopic particles ^are used, hydrodynamic ^and interactive effects are preponderant ^{over Brownian ones} (the ^{Peclet number} being ^much larger ^than unity, ^see [1]). ^{In this} paper, ^we describe two types ^of experiments.

In the first set of experiments, ^we study ^the aggregation kinetics of an assembly ^of

interactive macroscopic spheres ^{in a} two-dimensionnal suspension for different values of the surface concentration. We observe, ^{as a} function of the concentration, ^{either a} flocculation process ^{or a} gelation process. In these experiments, aggregation ^{can be} considered as an

irreversible phenomenon : ^two spheres, ^after coming into contact, remain stuck together. ^In

the second set of experiments, ^the suspension, ^{for a} given concentration value, ^is subjected ^to

a shear : in these conditions, aggregation ^{is no} longer ^an irreversible phenomenon ^and aggregate restructuring ^{can occur,} giving ^{a different} geometrical shape than in the latter case.

The purpose of this paper is the study ^of concentration and shear effects on the kinetics of

aggregation.

2. Expérimental procedure.

We recall here the main aspects ^{of the} experimental procedure, ^described in detail in [I]. ^In

the two sets of studies, ^the particles ^are placed in the gap of a special Couette-type apparatus.

For the studies with variable concentration, ^{the inner and outer} cylinders ^{are at rest ; for the}

studies with variable shear, ^{the inner} cylinder ^{rotates. The} propylene spheres ^{used in this}

work have a 3.17 mm diameter and the viscosimeter gap is about 25 sphere ^diameters.

Spheres ^are located in a viscous fluid layer, the thickness of which is smaller than their diameter, resulting ⁱⁿ attractive forces between spheres ^{due to} capillarity ^effects [2].

For a given situation, fixed concentration or fixed imposed shear, spheres ^are dropped randomly ^{on the} viscous fluid surface. From this instant, ^{chosen as time t}

0, twenty pictures

of the suspension ^{are taken,} fifteen every five seconds, ^{and the} remaining ^{ones at} exponentially growing ^time intervals. The twentieth frame, ⁱⁿ general, corresponds ^{to a} stationary ^{state. For} each concentration or shear value, fifty experiments ^{were carried out and}

the results cumulated, ^{the frames are} analysed by ^an image analysis procedure, ^{so that we can}

get ^{the different moments of} the cluster distribution.

3. Models.

Aggregation ^{studies can be} considered under two aspects :

-

a geometrical aspect ^whose principal object ^{is the} investigation of fractal structure of aggregates ;

-

a kinetic aspect considering ^{the evolution} of the different distribution moments with time.

Aggregation ^{has been studied} experimentally [3] ^{but most of the} existing ^{work is based on}

numerical simulations. In the Witten and Sander model [4], particles ^{are allowed to diffuse on}

a regular lattice and to stick to a static cluster. In the cluster-cluster model [5], diffusing

clusters are allowed to coalesce ; the hierarchical model [6] ^{is a variant of the} cluster-cluster model : particles ^{are isolated in a first} stage, ^then they ^can randomly ^{move and form two-} particle clusters which in turn, ^form four particles clusters, ^{and so} on, the final stage consisting

of one cluster uniting ^{all the} particles. ^{In these} models, ^{there is no} interaction between

particles. ^{In a first} stage these simulations did not take into account the microscopic ^{details of}

the aggregation process. To give ^{a more realistic} description ^{of the} aggregation process, ^one

can introduce a diffusion coefficient [7] representing ^the mobility of clusters of the form :

D,

Do s y (D, diffusion coefficient of a cluster of size s, Do ^{is a} constant), ^so that various mobilities can be studied by varying ^{y. The} reactivity ^{can be simulated} by ^an exponent u [8] : aggregation, during the collision of two clusters, ^{can take} place ^or not, ^the sticking probability Pij ^{of two} clusters of sizes i and j ^is expressed by ^{the relation} Pij

P o (1 j )" (Po is ^a constant).

Other situations such as aggregation of molecules with dipole interactions [9], ^or coagulation

with fragmentation [10] ^or restructuring [11] ^{have been simulated.}

Viksek et al. [12] have derived an expression representing the variations of s-sized clusters with time :

where f(x) ^{is a} cut-off function

By writing ^the conservation law for the total number of particles, ^{one can derive a relation}

between exponents :

and then write the expression (3) ^{under the form :}

where the scaling function g (x ) obeys ^the conditions :

thus, ^{we can define a new set} of relations between exponents :

The expressions ^for N (t ), the total number of clusters, ^and S2 (t ), ^{second moment of the}

cluster distribution, ^{are :}

These relations can be applied ^{to the} different models described above, the values of the

scaling exponents being functions of _y and [13].

In geometrical studies, ^{authors are} interested in the determination of a fractal dimension ;

the fractal dimension varies, according ^{to the} type ^of interaction, for instance in 2 D-

experiments ^{realized on} regular ^{lattices :}

-

without interaction : the clusters move according ^{to a Brownian} model, D f ^{= 1.44} [19] ;

-

with dipole interactions : attractive interactions are created between the aggregate extremities [9] (three dimensionnal simulations) ;

-

with restructuring : ^{after the} formation of clusters some parts ^{are allowed to « rotate »} with respect ^{to a} « weak » site 1.475 D f ^{1.50 [11] ;}

-

Eden model : all the surface sites are filled with equal probability and this leads to

compact clusters, D f

^{2 [14].}

4. Experiments ^{with a} suspension at rest : concentration effects.

In this case, there are only attractive forces between particles leading ^to aggregate ^formation

[2].

4.1 DEFINITION OF A CRITICAL CONCENTRATION : CPsc.

^We define the surface concen-

tration Os as the total ^area occupied by ^the spheres ^divided by the gap ^area.

In the experimental conditions described above, ^we define the critical concentration

cP sc’ ^{as the} smallest value of the surface concentration 0, for ^which only ^one aggregate exists, this aggregate ^{touches the inner and outer} cylinders ^of the viscometer (or ^more precisely ^the

inner and outer repulsive ^{zones, see} [I]), it is then contained in ^a circle, the diameter of which is equal ^to the gap width. Taking ^{into account} the fractal dimension determined experimen- tally (see ^the experimental article) ^we obtain CPsc

^0.20 (the O,c ^{value is} dependant ^{of the}

size of the « box » used : when the size tends to infinity, Os,, goes ^to zero).

In this part ^we report results obtained with concentrations below and above es,.

4.2 RESULTS.

Figures ^{1 to 4} represent ^{on a} log-log plot the variations of the total cluster number and the second moment of the cluster distribution as a function of time for different

Qsc ³ CPsc ⁵ CPsc

values of 03A6s, respectively 2 ’ -- , CPsc and 5 Osc ^{2 4 5 sc 4 These curves have the same general}

aspect for every concentration :

-

first, ^{a decrease} (growth) ^of N(t)(S2(t)) accelerating ^{with time}

-

then, variations in power laws in t for N (t ) ^and S2(t)

-

finally, ^a region where decrease (growth) ^{is slower.}

The transition region between short and long ^{times is not} exactly ^{linear : to define the}

exponent ^{we use the} slope ^{of curves at the} inflection point and thence determine the exponents ^of N (t ) ^and S2(t).

For a given concentration these exponents ^have opposite ^{values, z and - z. In table 1 we}

report ^{the z values for the} different surface concentrations used in this work. It is observed that z is an increasing function of the concentration ; ^an interpretation ^{can be} given by saying

that when Os grows ^the interparticle ^distance is lower and aggregation ^forces bigger ^{so that} aggregation is accelerated.

From the experimental values of z and N (s, t ) ^{we can then} plot ^the variations of Ln S2N(s, t ) ^versus Ln s and thus display ^{the function} g (Fig. 5). ^For adequate values of the

tz

couple (s, t ), associated to those for which N (t ) ^and S2 (t ) vary ^{as a} power law of _t, we can

observe a linear variation of g (x ) ^{with x} giving ^the exponent ^{5 which is} nearly ^{the same for}

every concentration : 8

1.4 ± 0.1. The values of s for which linear dependence ^{is observed}

t

are smaller than 0.05. For long times, i. e. s smaller than 0.01, fluctuations caused by ^the

t small number of results appear.

Fig.1. Fig. ^2.

Fig. ^3. Fig. 4.

Variations of the number of clusters N(t) and of the second moment of the cluster distribution

S2(t ) ^{versus t :}

Fig. ^5.

Variations of N (s, t ) number of clusters of size ^s at time t as a function of S for tz

qs

Qsc ^and (G)

^0.

Table 1.

Variations of ^the exponents z, b and w with Qs.

We have plotted ^{in a} log-log coordinates system ^the variations of N (s, t) :

-

with _s, for given QS ^{and t} (Fig. 6)

-

with t, ^for given es ^{and s} (Fig. 7)

for adequate values of s and t, ^{we observe a linear} variation and we can determine the exponents w ^{and T.}

Table 1 gives the variations of w with 0,.

4.3 SCALING LAW AND INTERPRETATION.

From the experimental ^results obtained above

we can then compare N(s, t), N (t ) ^and S2(t) with the theoretical equations (3), (7) ^and (8) by using ^the corresponding exponents ^values.

We have also verified that f(x ) ^{== 1 when x « 1 and} f (x ) ⁼ 0 when x > 1, ^{as well as the} relations (5) ^and (6) ^between exponents.

Exponents w, ^z and ^T depend ^on Os, & ^{is constant} within the concentration range studied.

For es values higher than the critical value, ^{at the final} stage ^{of the} study, ^{we observe} only

one cluster containing ^{all the} spheres : ^{we can} then say that ^we have a ^« gelation ^» regime (in

the case of our experiments, ^{the term «} gelation ^{» means} that, above the critical value, ^there is

a cluster much bigger than the other ones, this cluster cannot connect one side of the

container to the other owing ^{to the} experimental conditions [I]). ^For es ^{lower than}

Fig. 6. Fig. 7.

Fig. 6. - Variations of N (s, t ) ^{versus s} for e, S =Qsc ^and G > ^{= 0.}

Fig. 7. - Variations of N (s, t ) ^{versus t} for et> s = et> sc and G >0.

es,, ^we observe in the stationary ^{state more than one cluster : we can} define this state as a

« flocculation ^» regime (the remaining clusters in the suspension having ^{about the same} size).

We observe for long time values a decrease in the aggregation kinetics : the higher ^the concentration, the later the decrease takes place. ^{We can} suggest ^{several causes :}

-

when the cluster size is about the same than the gap dimension, ^the aggregation ^is

allowed in the orthogonal direction, ^{in such a case} aggregation ^{tends to become one-}

dimensional [15]

-

an ^« infinite » aggregate appears and we are in a gelation ^state

-

the interaggregate distance becomes larger ^{than the} capillary forces range

-

one of the referees suggested ^{that we} invoke the interpenetration of clusters at large

concentrations [16], ^{but such a} phenomenon ^{has never been observed in our} experiments : ^the aggregation occuring preferably ^{at clusters} extremities as shown by ^Cloitre [20].

5. Kinetics of aggregation ^{in a sheared} suspension.

5.1 INTRODUCTION.

In this case two effects are to be considered :

-

first, the effects of capillary ^forces, short range [2], which tend to aggregate particles,

-

second, the shear effects which either bring particles ^{into contact or separate them far}

from each other, ^{the force} depending ^{on the fluid} viscosity, ^on the shear rate, ^on the relative

position ^{of the} particles ^{in the} fluid, ^{and on the} particle characteristic size [17].

Then if the force between two spheres ^{in contact,} associated with the shear, is smaller than the force due to capillarity, ^{the two} particles will remain in contact, otherwise, ^the aggregate will be broken. This situation has been studied in the case of particle-particle interaction, ^the behaviour of aggregates containing ^{more than two} particles ^{has not been} studied, ^{but we}

assume that results obtained with two particles ^{can be extended to} bigger aggregates.

In the experiments ^described below, ^the concentration is kept ^{constant and is} equal ^to cP sc (gelation ^at 0-shear) ^{the mean shear rate} (G) ^takes values in the range 0 ^to 0.118 s-1: 0 s- l, ^{0.058 s-1 1 0.088 s-1 and 0.118 s-1.}

5.2 EXPERIMENTAL RESULTS.

Figures ^{3 and 8 to 10 show} N (t) ^and S2 (t ) variations with t

(in ^a log-log plot) for the different values of the mean shear rate, the ^{curves have} the same

aspect ^{as the ones} obtained in 4.2, giving ⁱⁿ their linear parts ^two opposite values for the

N (t ) ^and S2(t) exponents. ^{Table II} gives ^the variations of z with (G), ^z presents ^an

extremum around ( G )

^0.058 s-1, for this shear value there is a balance between the shear and capillary forces which act in the same sense for aggregation. ^{Below this value the shear}

enables aggregation ^effects by bringing ^{clusters near to each other ; above it acts in an} opposite ^sense by breaking ^clusters.

Fig. ^8. Fig. 9.

Fig. 10.

Fig. 8. - Variations of N(t) ^and S2(t) ^{versus t for} (G)

^0.058 s - ’ 0,

0 ,.

Fig. 9. - Variations of N (t ) ^and S2(t) ^{versus t for} (G)

^0.088 s - ’ 0, = lPsc.

Fig. 10. - Variations of N (t ) ^and S2(t) ^{versus t for} G>

^0.118 s-1 Os

0..

Table II. - Variations of the exponents z, 8 and w with (G).

Figures 11, ^and 12, represent respectively, ^{for a} given ^{value of the mean shear rate}

-

the variations of N (s, t ) ^{with s, for t} given

-

the variations of N (s, t) ^{with t, for s} given

giving in their linear part the values of the exponents ^{T and w.}

With the experimental determination of z and N (s, t), ^{we have} plotted ^Ln S2 N (s, t) ^{as a}

function of

Ln i t (Fig. 13). ^In the linear regime, ^we determine the exponent 5 which is shear tz

independent.

Fig.11. Fig. 12.

Fig. ^11. - Variations of N (s, t ) ^{versus s,} els s = el> sc ^and G >

^{0.058 s-1.}

Fig. 12. - Variations of N (s, t) ^{versus t,} Qs = e . ^and G >

^{0.058 S-l.}

Fig. ^13.

Variations of S2 N (s, t ) versus s ^{for (G)}

^0.058 s-1 and CPs = 0..

tz

From the values of ^T and w determined above, ^{we can} then write :

or

where f(x) ^and g(x ) ^are the cut-off and scaling functions defined in 3.

From the experimental results, ^{we can also} observe that for mean shear rates lower than 0.058 s-1, ^at long times, only ^{one cluster} containing ^{all the} spheres, ^{remains in the} suspension ; the cluster having ^{a fractal} structure. We can thus _{say that} we are in the gelation

state. For higher ^{values of} (G), ^at long times, ^we observe in the suspension several clusters

(flocculation state), each cluster containing ^{about the same number of} spheres, the clusters have in this case an hexagonal ^structure [18]. The number of clusters is an increasing ^function

of (G) .

Considering the values of the exponents ^of N (t ) ^and N (s, t ), ^{we can conclude} that, ^when the shear is weak (G> ^0.058 s-1 ), ^the aggregation is favoured by the shear. For higher

values of G. , ^{the clusters, as} they ^{reach a critical size, are broken : under the shear effect,}

clusters rotate around a vertical axis and in turn, their external parts ^{come in the} vicinity ^of

the inner cylinder ^{where the shear value is} maximum. If this value is higher ^{than the critical}

value corresponding ^to the balance shear effect-capillary effect, spheres ^{can be} pulled ^{out of}

the clusters and so cluster size decreases and their number grows.

6. Universality of the results.

6.1 NORMALIZATION BY INTRODUCING ^« NATURAL TIME UNITS ».

On the curves 14, ^it

appears that g (x ) curves can be reduced to a single ^one by ^a translation parallel ^{to the}

abscissae axis. This corresponds ^{to a} change in the unit of time. This unit, ^{the same for all the} experiments, ^{has been choosen without} taking ^{into account the} physical conditions governing

the aggregation phenomena :

-

capillary ^{forces in the case} of kinetics versus concentration studies,

-

capillary ^and hydrodynamic ^{forces in kinetics versus shear} studies,

in the first case, one can think that unit capillary ^{time can be} considered as constant : viscous fluid thickness being ^constant whereas, ^{in the second case the} hydrodynamic ^{unit time is shear} dependent.

We define the ^« unit capillary ^{time » :} Tc ^as the time necessary, for two spheres ^{located at a}

given ^{distance in a} suspension ^{at rest, to come in contact. We} determine this time by

observing ^{the movement of a} free sphere subjected ^{to the} attraction of a second one which is fixed : the resolution of the equation ^of motion leads to a characteristic time of a few seconds,

for instance for a distance equal ^to the characteristic length ^{À, of the} capillary force between two spheres (cf. [I] ^and [2]) ^{we find :}

by taking ^a distance 3 À between spheres ^we determine :

We have experimentally determined that inertial forces could be neglected.

We define the « unit hydrodynamic ^{time » :} Th ^as the inverse of the mean shear rate

(approximate value).

For the different shear values choosen :

we obtain

On figure ^{14 we have} reported ^the 9i(X) ^{curves for} (01)

^0.058 s-1, G2>

^0.088 s-1,

and G3>

^{1.118 s-1. Let us call £12 and e23} the translations values between the linear parts of the curves 91 (x) - 92 (x) ^and 92 (x) - g3 (x) respectively, ^{we find}

Fig. ^14.

Set of the s2 N (s, t ) versus s ^showing the translations _E12 and _E23 0,

e. ^{from left to} right

tz

Initially, the unit time is the shortest interval between two views : 5 s, then, ^{if we choose as}

unit the hydrodynamic ^{time we} get ^{a new variable :}

The absolute translation ei ^{on a} g (x) curve, characterized by ^an exponent Zi and ^an

hydrodynamical ^time Thh for a given s, ^{is then :}

and we can calculate the relative translation eij ^{between curves}

With the ^z values determined above (see ^Tab. II)

We observe a good agreement ^{between the} calculated values and the measured ones : for the higher ^{shear rates we can then assume} that the natural time unit is the hydrodynamic time, hydrodynamical ^effects being preponderant ⁱⁿ comparison ^{with the} capillary ^{one. For lower}

shear rate values, ^{the two effects are in the same} range and if ^we introduce the capillary ^time

unit in the relative translation expression

we obtain as Tel about 15 _s, which is in a rather good agreement ^{with the value} defined above.

With these quantities, ^{we can} reduce the g {x ) ^{curves to a} single ^one (Fig. 15) ^{which can be}

considered as « universal », in the limits of our experimental range.

Fig. ^{15. - «} Universal » g (x ) versus s ^{for all} G> ^values.

tz

6.2 NORMALIZATION BY THE MEAN CLUSTER SIZE.

The mean cluster size :

can be written :

and by assuming :

we obtain :

From the experimental ^curves giving N (t ), ^we obtain to ^{for the different shear rate values} by extrapolation ^{of the} tangent ^{to the curves in their linear} part.

On figure 16, ^{we have} plotted ^the normalized g (x ). ^{We should} point ^{out that a} better fit of the curve would lead to an adjustment of the values of the exponent 0.

To summarize this section, ^{we reiterate} that, ^{in the case} of variable shears or concen-

trations, ^{we have} been able to give ^{an «} universal » description ^{of the} phenomena by introducing ^{« natural »} time units.

Fig. 16.2013«Universal» g (x ) versus s s> for all (G) ^values.

7. Conclusion.

From the distribution moment study, ^{we have showed that} the number of s-sized clusters, ^for

kinetics versus concentration or shear studies, ^{could be} expressed by ^{the law :}

were g (x ) ^{is a} scaling function, ^or by ^an equivalent ^{form :}

where f(x ) ^{is a cut-off function.}

We have experimentally ^{verified the} properties of the functions :

and

The exponents ^{z, w, T and 6} verify scaling relations w

z5 and ^T

2 - 5 w, ^z and 5 are

experimental values ; the value of the exponent ^{T has been deduced from the} scaling ^relation

written above, ^its experimental determination being ^difficult owing ^{to the} important

fluctuations in the curves giving N (s, t) ^{versus s} for different values of t.

We might ^{remark that we can} justify ^the legitimacy ^of extracting scaling ^{behavior in a small}

system ^{because the} exponents ^{are deduced in} regions ^{of the curves where the relation} (2) ^is

verified. In these conditions, ^at intermediate times, ^{clusters are «} small » and they ^{do not}

« see » the container limits ; ^{thus we can consider the} system ^{as infinite.}

These results confirm those obtained in the case of numerical simulations and show that results usually ^obtained in the low concentrations limit can be extended to concentrations _up to 0.25.

We have to point ^out that the kinetic exponents ^{we have defined are related to the}

homogeneity exponent ^{of the kernel} kij ^{of equation (1) and their} variations with shear or

concentration can give ^{access to the behaviour of this} homogeneity exponent.

By choosing ^{a « natural time} unity » : to, ^{we can} express, for every shear or concentration value (included ^{in our} study domain), the cluster size distribution under the universal form :

where g (x ) ^{is the} scaling function defined above and to is characteristic of the experimental

conditions.

It appears also, experimentally, ^{that the} quantity : S s> ^{gives, for a given set of} experiments, ^the following universal form for the cluster size distribution :

where 0 is an exponent, the value of which bigger ^or equal ^to 2, ^{is shear} independent ^but

grows with the concentration value.

Our results do not deal with the behaviour at long times ; however, ^{we can mention that in}

this case, it seems that a power law could describe the aggregation ^{kinetics. Such a} study

would need _many more experiments ^{in order to lower} the fluctuations.

Other experiments allowing ^the determination of attractive forces between more than two

spheres ^{would define more} precisely the notion of capillary time ; ^{moreover, we think that} they would enable the determination of other exponents ^{such as those of} mobility ^or sticking probability.

Acknowledgements.

We thank C. D. Mitescu for a critical reading ^{of the} manuscript.

References

[1] SMOLUCHOWSKI M. V., ^Z. Phys. ^{Chem. 92} (1917) ^129.

[2] ^{CAMOIN C.,} ROUSSEL J. F., ^FAURE R., ^BLANC R., Europh. ^{Lett. 3} (1987) ^449.

[3] ^{FORREST S.} R., ^{WITTEN T.} A., ^J. Phys. ^{A 12} (1979) ^L109.

ALLAIN C., ^JOUHIER B., ^J. Phys. ^{Lett. 44} (1983) ^L421.

WEITZ D. A., ^{LIN M.} Y., SANDROFF C. J. , Surf. ^{Sci. 157} (1985) ^147.

[4] ^{WITTEN T.} A., SANDER L. M., Phys. ^{Rev. Lett. 47} (1981) ^1400.

[5] ^MEAKIN P., Phys. ^{Rev. Lett. 51} (1983) ^1119.

KOLB M., ^BOTET R., ^JULLIEN R., Phys. ^{Rev. Lett. 51} (1983) ^1123.

[6] ^{BOTET R., JULLIEN} R., ^KOLB M., J. Phys. ^{A 17} (1984) ^L75.

[7] ^MEAKIN P., ^VICSEK T., ^FAMILY F., Phys. ^{Rev. B 31} (1985) ^564.

[8] ^FAMILY F., ^MEAKIN P., ^VICSEK T., J. ^Chem. Phys. ⁸³ (1985) ^4144.

[9] ^{MORS P. M., BOTET R., JULLIEN} R., ^J. Phys. ^{A 20} (1987) ^L975.

[10] ^{KOLB M., J.} Phys. ^{A 19} (1986) ^L263.

[11] ^{MEAKIN P., JULLIEN R., J.} Phys. ^{France 46} (1985) ^1543.

[12] ^{VIKSEK T., FAMILY} F., Phys. ^{Rev. Lett. 52} (1984) ^1669.

[13] ^{FAMILY F.,} Kinetics of Aggregation ^{and Gelation, eds. F.} Family ^{and D. F. Landau} (North Holland, Amsterdam 1984).

[14] ^EDEN M., Proceedings of the Fourth Berkeley Symposium ^on Mathematical Statistics and

Probability, ^eds. Neyman-Berkeley ^{and Los} Angeles (Univ. of California Press 1961).

[15] ^{REDNER S., KANG} K., Phys. ^{Rev. A 30} (1984) ^3362.

[16] ^{KOLB M.,} HERRMANN H. J., ^J. Phys. ^{A 18} (1985) ^L435.

[17] ^{GOREN S. L., J.} Colloid and Interf. ^{Sci. 36} (1971) ^94.

[18] ^{CAMOIN C., BLANC R., J.} Phys. Lett. France 46 (1985) ^L67.

[19] ^{JULLIEN R., KOLB M., BOTET} R., ^J. Phys. ^{France 45} (1984) ^L211.

[20] ^CLOITRE M., ^9e Congrès Français ^de Mécanique-Metz-Septembre ^1989.