Image analysis and kinetics of aggregation

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Image analysis and kinetics of aggregation

Jean-François Roussel, Christian Camoin, Robert Blanc

To cite this version:

Jean-François Roussel, Christian Camoin, Robert Blanc. Image analysis and kinetics of aggrega-

tion. Journal de Physique, 1989, 50 (21), pp.3259-3267. �10.1051/jphys:0198900500210325900�. �jpa-

00211141�

Image analysis ^and kinetics of aggregation

Jean-François Roussel, ^{Christian Camoin} and Robert Blanc

Laboratoire de Physique ^des Systèmes Désordonnés, URA 857 du CNRS, Université de

Provence, Centre de Saint-Jérome, ^Case 161,

de l’Escadrille Normandie-Niémen, 13397 Marseille Cedex 13, ^France

(Reçu le 24 avril 1989, accepté

forme définitive ^{le 10} juillet 1989)

Résumé.

Dans le but d’étudier les effets du taux de cisaillement et de la concentration

particules

^la cinétique d’agrégation,

réalisé des expériences

^monocouche

de sphères macroscopiques ^flottant

fluide. Nous décrivons la procédure expérimentale ^{et la}

méthode d’analyse d’images que

mise

point pour effectuer, de manière

automatique, les études statistiques ^des agrégats ^et leur évolution dans le temps.

Abstract.

In order to study ^the hydrodynamic ^{shear rate} and concentration effects

the kinetics of aggregation,

^have performed experiments ^with

monolayer ^of macroscopic spheres floating

fluid. We describe the experimental procedure ^{and the} image analysis ^method

used in order to obtain, ⁱⁿ

automatic way, the statistical description ^{of the} aggregates ^{and their} evolution with time.

Classification

Physics ^Abstracts 05.40201305.60201306.60

1. Introduction.

This paper is the first of two articles devoted to the description ^{of results we have obtained in}

a study ^of concentration and shear effects on the kinetics of an aggregation process. In this

work, ^we have studied how macroscopic spherical particles, compelled ^to stay ^{in a horizontal} plane, progressively ^build aggregates under the simultaneous action of attractive and

hydrodynamic ^forces. In many experimental situations, such processes, in which large aggregates ^are built from small particles suspended ^{in a sheared} fluid, ^{are observed.} So, ^it

appears interesting ^to try ^to understand the influence of the shear on the aggregation ^kinetics

The effect of the shear on the collision process between ^two particles ^{has been studied} by

various authors [1], ^{after a first} approach by Smoluchowski [2]. Generally, ^{two limit cases are}

considered : orthokinetic and perikinetic collisions, according ^as the Peclet’s number is much

larger ^or lower than one. Let us recall that this number is the ratio between Brownian and

hydrodynamic characteristic times. Then, ^{the two limits} respectively correspond ^{to a} purely hydrodynamic ^{or a} purely ^diffusive approach ^{of the two} particles.

In the case of interest for us, the hydrodynamic ^{effects are} fully dominant in regard ^{to the}

Brownian ones : the Peclet’s number, ^{in our} experiments, ^{is about 101°. On the other} hand, ^it

will be very useful ^to compare the hydrodynamic and attractive forces between two particles.

This point ^{will be} developed ^more precisely ^{in the} companion paper (referenced ^as [II] ^{in the}

present article). ^In [II], ^we shall describe in details the theoretical framework of our study, ^the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500210325900

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results of experiments ^we made and their interpretation. In this paper, ^we shall focus our

interest on the description ^{of the} experimental method with special ^{attention on the} image analysis ^{device we} used in order to follow in time, ^{in an} automatic way, the aggregation

process.

2. Description ^{of the} experiments.

The experimental apparatus ^{has two} principal parts : ^{on one} hand, ^a cylindrical Couette-type

cell between the cylinders ^{of which the} suspension ^{is set} and, ^{on the} other, ^a picture-taking

and image-analysis ^{device. This set} is shown in a schematic _{way in} figure ^la.

2.1 THE SUSPENSION.

The suspension (Fig.1b) ^{is made of} spherical particles (B) ^of polypropylene (Hostallen PP) floating in pure vaseline oil (A). ^The spheres diameter is 2 a

3.17 mm, their specific ^mass (p = ^0.8 g. Cm -3) ^is approximately ^{the same as} that of the vaseline oil. The viscosity ^of this oil is about 6 x 10-2 Pa.s at the room temperature. ^{The set}

Fig. ^1.

Experimental ^device. la) ^{S :} suspension (monolayer ^of particules) ; E : water ; ^{D :} light

diffuser ; ^{F :} fluorescent

1b) Sectional elevation of the suspension. ^The polypropylene ^balls

(B), ^3.17

ⁱⁿ diameter, float in vaseline oil (A) which has the

density

^the spheres. ^The

monolayer is sustained by ^water (C).

« oil plus spheres » ^{floats on a volume of water} (C), the role of which is to reduce the

hydrodynamic influence of the cell bottom. In the immediate neighbourhood ^{of this} latter, ^the hydrodynamic velocity, when the cell is rotating, ^is proportional ^to the distance R to the axis.

The water’s level has been chosen high enough ^so that, in the oil layer, ^the velocity ^is independent ^of the presence of the bottom and given by ^v = AR + B /R. In this last

expression, ^{A and B} depend ^{on the radii} Ri ^and Ro of the internal and extemal cylinders ^and

on their angular velocities [3].

As one can see in figure 1b, ^{the thickness h of the oil} layer is lower than the sphere

diameter. The interfaces between water and oil, ^{on one} hand, and between oil and air, ^{on the} other, ^{are modified} by ^the balls. This gives ^{rise to} capillary attractive forces between spheres.

When two spheres ^{are involved} these forces depend ^{on the center to center distance}

and ^on

the oil thickness :

where À is a capillary length characterizing the range of interactions [4]. ^{In our} experimental conditions, ^{À is of the order of few} sphere diameters. The amplitude A ^{and the} characteristic

length À depend ^{on the oil} thickness h. When h is about 2 millimeters, A ^is of the order of

10- 5 Newton which is of the same order of magnitude ^{as the} purely hydrodynamic ^force

between two spheres ^{in contact in a} sheared fluid [5], when the shear rate value G is 1 s 1. Indeed, taking ^{into account the} sphere ^{diameter and the oil} viscosity, ^{such a force is} given numerically by ^{1.3 x} 10-5 G sin 2 0, where e is the angle between the centers line and the direction of the flow. Thus, acting ^{on the oil} thickness and/or the shear rate, ^{one can} easily

vary the ratio between hydrodynamic ^and capillary ^forces.

However this ratio cannot be known with accuracy ^{as, on one} hand, ^the given expression ^of

the hydrodynamic ^force corresponds ^{to an} infinite three dimensional fluid and, ^{on the} other,

the shear rate is not a constant between two cylinders ^but depends ^{on the} distance R to the axis. In order to get ^an approximate value for the ratio, ^{we shall} use a mean value of the shear rate between the two cylinders, ^defined by :

where G (R ) ^is the local value of the shear rate at the distance R of the axis.

Ro ^and R; ^are respectively ^{the outer} (Ro ^{= 18} cm ) ^{and the} inner diameter (R;

⁶ cm ). ^The

former cylinder, ^{made of} PVC, may ^rotate around its vertical axis with an adjustable angular velocity, using ^{a DC motor.} The latter is made of stainless steel. The values of these radii represent ^a compromise ^{between the} conditions of homogeneity ^{of the shear} (relative ^interval (Ro - R; )/Ro ^{as weak as} possible) ^{and those} of accuracy in the statistical study ^of aggregates

(number ^of particles and, then, ^area between the cylinders ^as large ^as possible) ^{for a} given (and reasonnable) value of the outer radius.

In order to obtain identical wetting conditions on each wall, ^the cylinders ^were coated with

a layer ^of paint ^{chosen so that} repulsive forces take place between the walls and the spheres.

That way, ^we avoid the sticking ^{of the} spheres ^{on the} cylinders. ^{But this} repulsive ^effect

reduces the effective gap between the two cylinders, its actual value is about 8 cm which

corresponds ^{to about 25} spheres ^diameter.

2.2 THE SOWING.

The initial repartition ^{of the} spheres ^{in the} oil layer ^{has to be random}

with a mean size of the aggregates ^{as weak as} possible, the ideal situation being ^{one with all}

the balls isolated. ^{In order to} achieve these conditions, it is necessary that all the spheres ^are

simultaneously dropped ^{on the} oil surface. For that, ^we use ^{a set of two} grids, ^{each one} being

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a square ^network (mesh ⁸ mm) ^of square holes (5 x 5 mm2). ^{One of this} grid ^{is 6 mm thick,}

the other 1 mm. This latter is put under the thick grid, ^{in contact with} it, ^{and shifted of half a} mesh so that it blocks the holes of _{the upper} grid. ^The spheres ^{which are} going ^{to sow the oil} layer ^are randomly distributed in these holes. Then, ^{the two} grids ^{and the balls are} put ^{on the} Couette cell and the thin grid is shifted so that the two networks are in coincidence : the

spheres simultaneously ^{fall in the oil. This instant is taken as} origin ^of time for the aggregation

process. The heigth of fall of the spheres ^{has to be} sufficiently large ^{so that the balls sink into}

the oil, giving ^a good wetting ^{and then a} good reproducibility ^of capillary ^{forces. However, a}

too high ^{fall is to avoid as the} spheres may burst into the ^water and get jamed ^at the oil-water interface. When the aggregation process takes place ^{in a sheared} fluid, ^{the inner} cylinder ^is rotating ^{since a time} long enough ^{so that} hydrodynamic regime ^{is well} established when the

spheres ^are dropped.

2.3 TAKING OF PICTURES AND DIGITIZER BOARD. - A snapshot ^{of the state of the} suspension ^{is taken at different} times, using ^a video-camera (vidicon type) ^{and a} microcompu-

ter with a plug-in digitizer ^board. This board samples ^{the video} signal given by ^{the camera} and, using ^an analog-digital converter, ^turns this signal ^{into an} integer taking ^{a value between}

0 (black) ^{and 255} (white). ^{So, to the} image ^{on the camera} photocathode, corresponds ^{a table}

with 512 elements (pixels) ^{on each one of the} 512 lines. In other words, ^the image ^is replaced by ^a mosaic, ^{each element} of which is a rectangle. ^{The ratio} width/height ^{of this} rectangle ^is

4/3. A one-byte binary number, representative ^{of the} illumination of a pixel, ^{is then}

associated to each one of these 262 144 elements.

The camera is put ^{on the} cell, ^{its axis} vertical, ^{at a distance such that the} image ^{of the cell}

take the whole photocathode ^{area, in order to} get the maximum geometrical resolution. The cell has a translucent bottom lighted by ^two concentric circular luminescent tubes of 22 and 32 W. The room where this apparatus is located is dark. With these conditions, ^a good spheres-fluid ^contrast is obtained.

As we have said previously, ^the origin ^{of the time for an} experiment ^{is taken when the two} grids ^{used for the} sowing ^are put ⁱⁿ coincidence. From this instant, ^the micro-computer ^allows

us to take and to store twenty pictures ^{of the} suspension ^{at well defined} times. The fifteen first

images ^{are taken at} regular intervals, ^{5 s} each, ^{this time} being ^{that of} writing ^a digitized image

on the hard disk. The last five views are taken at exponentially growing ^time intervals. The last view is taken at a time equal ^to 12 min and 30 s. Pictures of some states of the suspension

at different times are shown in figure ^2.

3. The image analysis ^device.

If all ideal requirements (perfectly ^uniform lighting, opaque spheres, very high geometrical resolution) ^were satisfied, ^{one should be able to} determine the number of spheres belonging

to a given ^cluster by simply dividing ^{the total area} occupied by ^the spheres ^{in that cluster} by

the area of one sphere. Actually, this method cannot be used for several reasons and, specially, ^{on account of the} inhomogeneities ^{in the} lighting ^{and of the weak} size of the spheres which, ^on average, occupy ^{an area} of twenty pixels. ^{It is} then necessary ^to proceed ^{to a set of} operations ^{in order to isolate each} aggregate ^{and to} determine how _many spheres ^are belonging ^{to it. But, before, the} image ^{has to} undergo ^{a numerical treatment.}

3.1 NUMERICAL TREATMENT. - An image ^{of the} cell without spheres ^{is used as a reference.}

One subtracts, pixel by pixel, from that reference view, ^the image ^{to treat. In} principle, ^after

this operation, ^the pixels corresponding ^{to the fluid should} take the value zero. But the video

camera has an gain ^control adjust ^which keeps ^{to a constant level the mean value of the} signal

Fig. ^2.

Views of the aggregating suspension ^at different times.

given by ^{the whole} photocathode. ^{As the mean} illumination is lower for a given image ^than

for the reference view, ^{the numerical value} corresponding ^{to the} pixels located in the fluid is

larger ^when spheres ^are present ^{on the view.} Then, the subtraction _{may have} a negative

result. We have reduced to zero all the negative values of these results.

Due to the existence of a liquid meniscus around each ball, ^the transition between the levels 0 (corresponding ^to the fluid far from this sphere) and that inside the sphere ^{is not} sharp ^but progressive. This effect is even more increased by ^the small size of the images ^{of the balls on}

the photocathode ^{and also because the} polypropylene ^{is not} perfectly opaque. Then, ^about half the pixels ^the level of which is influenced by the presence of the sphere ^{are located on the} periphery ^{of that} sphere. ^{We have} tightened ^the transition between the fluid and the balls

using ^a convolution. ^In this operation, ^{the level of each} pixel ^{is modified,} taking ^{into account}

the level of the eight surrounding pixels. ^{If i} and j ^{are the} coordinates (row ^{and column}

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numbers) ^{of a} given pixel ^and ifp (i, j) is ^{its level, the new value of the} intensity ^{for this} pixel

after the convolution is given by :

where m and n take the values -1,0 ^{and 1. The} matrix of the k(m, n ) coefficient we used is :

Such an operation allows the tightening ^{of the} transition between two areas with different illuminations.

3.2 LOCALIZATION OF THE SPHERE CENTERS.

After the above treatment, ^one proceeds ^to

the localization of the centers of the spheres. ^{With this end in view, one defines a threshold} So (taken ⁱⁿ general equal ^to 30) ^{and a set of eleven} particles, ^called

^fictif sphere », ^{shown in} figure ^{3. This set has been defined} taking ^{into account} the size of the balls and the aspect ^ratio of the pixels. ^{In a first} stage, ^{one reads the value of the} pixels ^{from left to} right and up ^to down. For each pixel ^{with a} level larger ^than So, ^{one looks if the ten near} pixels (fictif sphere

centered at the pixel ^under consideration) ^{have also a} level larger ^than So ^and (see below)

lower than 254. If these conditions are not satisfied, ^{the next} pixel is read. If they ^{are, one}

goes into the second stage that is the centering ^{of one} sphere. ^{For this} purpose, ^one adds the values of the eleven pixels. ^{Then, one} compares this sum to the three other sums obtained the

same way after a shift of the center of the fictif ball one step ^{on the} right ^{or one} step downwards or one step ^{on the} right ^{and one} downwards. That of these four points ^{for which}

the sum takes the larger value is chosen as the center of the restored sphere. ^{One times this}

choice made, ^one assigns ^{to the center the value 255 and the value 254 to each one of the other}

ten pixels of the fictif sphere. ^This way prohibits ^{two restored} spheres overlapping since, ⁱⁿ

the first stage, ^{one considers} only pixels ^{with a level} larger ^than So ^and lower than 254.

Fig. 3. - Fictif sphere with eleven pixels.

Doing that, ^{it is} possible, ^{in an} entirely ^automatic way, ^to find the centers of at least 99.5 % of the spheres ^{in the} suspension. ^{The weak} percentage ^of non-recognized spheres corresponds

to particles ^{which are not} located in the plane ^{of the} suspension ⁱⁿ consequence of bad wetting

conditions or, more often, ^because they ^are partly overlapped by ^their neighbours.

3.3 THE CLUSTERS.

When the previous operations ^{have been} performed, ^a catalogue ^of

the coordinates of the sphere ^{centers is} available. The next stage ^{is the} separation ^{of the}

aggregates ^{one after the} other, ^{the final} stage being ^{the count} in each cluster of the number of

centers and, then, ^of particles belonging ^{to it. In order to} distinguish ^{the different} aggregates,

one proceeds ^{first to a} binarization of the image ; then, ^{one uses a criterion to decide if a} given pixel belongs ^{or not to a} given cluster. ^{Due to} the subtraction made at the time of the image

treatment, the pixels ⁱⁿ correspondance ^{with the fluid have a} nul value. So, ^{it is} possible, ^with

a very low threshold Sl, ^{to binarize the} image, giving the value 254 to each pixel ^{with an} intensity larger ^or equal ^to Sl ^{and a nul value for each} intensity ^{lower than} that threshold. On the so-binarized image, ^{one looks for the} pixels ^{in contact,} using ^the following criterion : two

pixels ^{are said in contact, i.e.} belong ^{to the same} aggregate if, ^and only ^if, they ^{have a side in}

common (Fig. 4). This very simple ^{criterion has been chosen after a} comparison ^{between the} digitized images ^and photographic ^{views taken at the same time. It} appeared ^{that the contacts}

determined visually ^{on the} pictures ^{and those obtained} using this criterion were in very good agreement. Then, ^{it is} easy ^to separate ^{the different} aggregates. Therefore, it may happen, ^at

the end of this set of operations, ^{that two} spheres ^{which are not} trully ^{in contact} satisfy ^the

criterion. This makes slightly wrong the clusters statistics. But if, ^{at a} given time, ^two spheres

are very close each other, they ^{come into contact after a} very short time interval, ^{so that the} previous ^{error is not} important and may be interpreted ^{as a} slight ^{error on} the time when one

is interested by ^{the kinetic} aspects ^{of the} aggregation process.

Fig. ^4.

Contact criterion. Connected pixels (CM); ^{Non connected} pixels (BB)’

In order to determine the size (that is the number of spheres) ^{in each one of these}

aggregates, ^{one needs} only ^to put again ^{on the whole} image ^{the centers of the} spheres ^{with a}

255 value. A count of such points in each cluster gives the number of particles ^{it contains.}

That way, the size distribution of clusters, ^the coordinates of all the sphere centers, their

repartition ⁱⁿ the different aggregates ^{are known} for each view. From these data, ^other informations _as, for instance, ^the geometrical dimensions of the clusters, ^{the different mean}

sizes (number-averaged ^{mean size,} mass-averaged ^mean size,...), their fractal dimension D,... may be obtained.

The knowledge ^of the fractal dimension of the aggregates will be useful in [II], ^{in order to}

know the regime (flocculation ^or gelation) ^{of the} growth process. To determine D, ^we followed the classical method : a point ^{of the} aggregate ^is randomly ^{chosen as the center of a}

disk of radius r and the number N (r) ^of sphere ^centers (belonging ^{to this} cluster) ^{inside this}

disk is counted for increasing r ^values. Doing this many times, for various centers and various aggregates, ^one gets ^a relation between the mean value of N (r) ^{and the radius r} (cf. Fig. 5)

In the log-log plot ^{of the} figure 5, ^{D is the} slope ^{of the} straight ^{line. For} aggregates growing

when no external shear is imposed ^{to the} fluid, ^{the value obtained is D} = 1.48 ± 0.05. This value is in reasonable agreement ^{with that} (1.55 ^± 0.05 ) given by Allain and Cloitre [6] ^for

very similar experiments using ^{the same} spheres ^{and a} square box of large ^dimension (225 sphere ^{diameters on} the square side). ^{These two} experimental ^{values are} in rather good

agreement ^{with that} (1.51 ^± 0.03 ) ^obtained by ^{Ball and Jullien} [7] in the numerical simulation

of the cluster-cluster aggregation ^{with a balistic} approach.

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Fig. ^5.

Log-log plot ^{of the}

^{of the} aggregates

their linear size.

4. Conclusion.

We have developed ^an experimental method which allows us to determine, ^{in an} entirely

automatic way, from ^a snapshot ^{of an} aggregating suspension ^of spheres, the number and the size of the growing clusters. The number of particles ^{used in these} experiments is between 750 and 2 000, depending ^{on the} initial concentration of the suspension. ^Different experiments

are made in the same conditions : identical total number of spheres, height of fall, ^{thickness of}

the oil layer, angular velocity ^{of the inner} cylinder, temperature, ^the only difference being ^the

initial disposition ^{of the} spheres ^{which is random.} Snapshots ^{are taken at fixed} instants, ^the origin ^{of time} being ^{defined in the same} way for all the experiments. ^{When the results}

obtained from different views made in the same conditions and at equal ^{times are} compared,

one observes that the results are fluctuating ^{around a mean value. This is not} very surprising owing ^to the random initial disposition ^{of the} spheres. ^The relative fluctuations are all the

more important ^{as the} size of clusters under consideration is larger. ^{In order to minimize the}

influence of these fluctuations, ^{we made numerous} experiments ^{in the same} conditions. Thus,

for each value of the total number of spheres ^N (and then of the initial concentration of the

suspension) ^{and of the} angular velocity f2 ^{of the inner} cylinder (and ^{then of the shear rate} imposed ^{to the} aggregating suspension), ^{we made 50 identical} experiments, analysing ^thus

1 000 views of the suspension. ^{Such a number of} images ^{has been} studied for seven couples ^of

the variables N and f2. This accounts for the development ^{of an automatic} analysis. Using ^an

IBM PC AT micro-computer, ^the analysis ^{of one} image ^{takes a} time which is about 5 min.

The results we obtained by ^{this method are} developed in another paper [II].

References

[1] ^{DE GENNES P. G.,} Phys. ^Chem. Hydr. ^{III 2} (1981) ^31.

VAN DE VEN T. G. and MASON S. G., Colloid and Polymer ^{Sci. 255} (1977) ^468.

[2] ^VON SMOLUCHOWSKI M., Physik ^{Z 17} (1916) ^585.

VON SMOLUCHOWSKI M., ^Z. Phys. ^{Chem. 92} (1918) ^129.

[3] ^{See, f.i. LANDAU and LIFSHITZ,} Mécanique des fluides. Editions Mir, ^Moscou.

[4] ^{CAMOIN C., RoussEL J. F.,} FAURE R. and BLANC R., Europhys. ^{Lett. 3} (1987) ^449.

[5] ^{GOREN S.} L., ^{J. Colloid} Interface ^{Sci. 36} (1971) ^94.

[6] ALLAIN C. and CLOITRE M., in Fractals in Physics. ^L. Pietronero, E. Tossati Eds., ²⁸³ (1986).

[7] ^{BALL R.} and JULLIEN R., ^J. Phys. ^{France 45} (1984) ^L1031.

[II] ROUSSEL J. F., ^BLANC R., CAMOIN C., ^J. Phys. ^{France 50} (1989) ^3269.