A mixed Boolean and deposit model for the modeling of

metal pigments in paint layers

Enguerrand Couka, Francois Willot, Dominique Jeulin

To cite this version:

Enguerrand Couka, Francois Willot, Dominique Jeulin. A mixed Boolean and deposit model forthe modeling of metal pigments in paint layers. Image Analysis and Stereology, InternationalSociety for Stereology, 2015, 34 (2), pp.125-134. <10.5566/ias.1220>. <hal-01154597>

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Submitted on 22 May 2015

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Submitted to Image Anal Stereol, 10 pagesOriginal Research Paper

A MIXED BOOLEAN AND DEPOSIT MODEL FOR THE MODELING OFMETAL PIGMENTS IN PAINT LAYERS

ENGUERRAND COUKA, FRANCOIS WILLOT AND DOMINIQUE JEULIN

Center for Mathematical Morphology, MINES ParisTech, PSL Research University, 35 rue Saint-Honore, 77300Fontainebleau, Francee-mail: enguerrand.couka@mines-paristech.fr, francois.willot@mines-paristech.fr,dominique.jeulin@mines-paristech.fr(Submitted)

ABSTRACT

Pigments made of metal particles of around 10 µm or 20 µm produce sparkling effects in paints, due to thespecular reflection that occurs at this scale. Overall, the optical aspect of paints depend on the density anddistribution in space of the particles. In this work, we model the dispersion of metal particles of size up to50 µm, visible to the eyes, in a paint layer. Making use of optical and scanning electron microscopy (SEM)images, we estimate the dispersion of particles in terms of correlation functions. Particles tend to aggregateinto clusters, as shown by the presence of oscillations in the correlation functions. Furthermore, the volumefraction of particles is non-uniform in space. It is highest in the middle of the layer and lowest near the surfacesof the layer. To model this microstructure, we explore two models. The first one is a deposit model whereparticles fall onto a surface. It is unable to reproduce the observed measurements. We then introduce a “stack”model where clusters are first modeled by a 2D Poisson point process, and a bi-directional deposit model isused to implant particles in each cluster. Good agreement is found with respect to SEM images in terms ofcorrelation functions and density of particles along the layer height.

Keywords: heterogeneous media, optical properties, paints, random microstructure models..

INTRODUCTION

In automotive applications, most paints consistof multi-layered materials, where each layer hasa different purpose, ranging from anti-corrosionto visual aspects (Streitberger, 2008). The overallappearance of paints depends on the paint compositionbut also on the spatial distribution of particles, mostoften heterogeneous and multi-scale (Couka et al.,2015). At the largest scales, where geometrical opticsapply, metal particles produce specular reflection andsparkling effects. These effects depend on the chemicalnature of the particles, their surface roughness, but alsoon the dispersion of particles and on clustering effects(Dumazet, 2010).

The description of industrial paint layers is achallenging problem. The separation and identificationof chemical elements in paint coatings, used toestimate their dispersion in the material, is a complextask (Yang et al., 2010 a;b). At the nanometricscale, the effect of pigments on the optical propertieshas been studied experimentally and compared witheffective medium theories (Cummings et al., 1984).More recently, numerical works have been carried outto estimate the effect of the dispersion of pigmentsin a deposit model (Azzimonti et al., 2013). Torepresent the material and compute its properties usingnumerical means, a microstructural model is required.

In other contexts, microstructure models basedon hardcore processes have been proposed to forbidthe overlap of inclusions in the embedding medium.The effective elasto-plastic response of particle-reinforced composites has been studied in (Chawlaet al., 2006), making use of an optimized hardcoreprocedure to generate microstructures. In (Moreaudet al., 2012), a hardcore deposit model is proposedto simulate a boehmite material. However, thesimulation of microstructures using hardcore modelsbecomes difficult when approaching the close-packinglimit (Escoda et al., 2015). To overcome thisdifficulty, random walk stochastic procedures havebeen developed (Altendorf and Jeulin, 2011) togenerate fibrous microstructures, as an improvementon random sequential adsorption models (Feder,1980). Another problem concerns the generation ofmultiscale models, for which various methods havebeen proposed (Paciornik et al., 2002; Jean et al.,2011).

This work is concerned by the modeling ofmicroscopic metallic particles, responsible of thesparkling effects in a paint layer, by means of 3Drandom microstructures. Such virtual microstructuresare required to simulate the paint layers opticalresponse with a rendering engine. We first present thematerial, then segment and separate particle in SEMimages. We use the segmented images to measure

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COUKA E et al.: A mixed Boolean and deposit model

the dispersion, size, shape and orientation of particlesusing various morphological tools. We introduce andevaluate a deposit and “stack” model, we conclude ontheir ability to modelize the dispersion of the metallicparticles.

METHODS

OPTICAL AND ELECTRON MICROSCOPYIMAGES

This work focuses on a sparkling, grey-coloredindustrial paint layer used in automotive applications.The material color results from two types of pigments:several metal oxides of unknown composition, notlarger than a few micrometers, and, at a larger scale,aluminium metallic particles of up to 50 µm visibleto the eyes. The two types of pigments are encircledin red and orange respectively in Fig. 1a, an opticalmicroscopy image representing the surface of thelayer. The image, however, does not allow one toseparate all pigments. Instead, aluminium particles areclearly visible in SEM (scanning electron microscopy)images (Fig. 1b). This image represents a 2D sectionorthogonal to the surface layer. As shown in the twoviews, the average shape of the aluminium particles isflat and the particles are often aligned with the layersurface.

The aim of this work is to characterize and modelthe shape and dispersion of aluminium particles inthe paint layer, which produce its sparkling effects.We disregard the oxide pigments entering the paintcomposition at a smaller scale. Hereafter, we make useof 5 non-overlapping SEM images similar to Fig. 1b tomodel the aluminium particles spatial distribution.

The actual manufacturing process and exactcomposition behind this industrial paint is unknownand is therefore not modeled. Instead, we use 3Dmodels with a small number of parameters to mimicthe real material. The latter bear no relation with theactual manufacturing process.

(a)

(b)

Fig. 1. Surface of the paint layer (a) (taken by opticalmicroscopy on a raw surface of the paint), and its slice,orthogonal to the surface (b) (SEM image, taken in SEmode).

SEGMENTATION AND SEPARATION OFPARTICLES

We now segment the aluminium particles from thebackground in the grey-level SEM images. We alsoseparate individual particles. This step is necessary forcomputing statistics on particles. The black region atthe top of the image, outside of the paint layer, is firstsegmented by thresholding the images. The thresholdvalue is chosen manually. Furthermore, the paint layeris not exactly aligned with the borders of the SEMimages. We identify it with a parallelogram in eachimage and measure its disorientation with the imageborders.

We now apply an area opening of size 20 pixelswith a square structural element to remove noise inthe image and smoothen the background (Fig. 2b).The resulting image is segmented by an entropy-basedautomatic thresholding (Otsu, 1979) (Fig. 2c).

Some of the aluminium particles have mergedbecause of the low contrast in the gaps locatedin between two particles that are close to oneanother. The watershed algorithm is used to separatethe particles. Watershed markers are obtained inseveral steps. We first compute the distance functiongiven by successive erosions using a horizontalstructural element (Fig. 2d). The size of the structuralelements size is 3 pixels. The deepest markers areextracted from the distance map using the H-maximaalgorithm (Schmitt and Preteux, 1986) with size

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parameter set to 35 pixels. The histogram of thedistance map is stretched for more convenience. Usingthe markers, we compute the watershed partition on theinverse of the distance map previously computed. Thisleads to a slightly over-segmented image (Fig. 2e).Only about 6 boundaries per image are not welllocated. We correct them manually.

In the next sections, we use the 5 segmentedimages to measure various statistics on the segmentedimage χ(x,y) where χ = 1 in the particles and 0outside. Then the measures are used to build two 3Dmicrostructure models for the aluminium particles.

(a)

(b)

(c)

(d)

(e)

Fig. 2. Original (a) and opened (b) images. Segmentedimage (c). Distance by erosion using a horizontalstructural element (d). Result of the watershedalgorithm (e). Magnification; Resolution of the initialimage: 1280x77 pixels

Hereafter we compute the lineic fraction ofparticles along the height, the correlation functions,a distribution of size and orientations and statisticson particles aggregates. The area fraction, extractedfrom the segmented images, is about f = ⟨χ(x,y)⟩x,y =6.5%.

FRACTION OF PARTICLES ALONG THEHEIGHTThe lineic fraction of aluminium particles is not

usually constant along the paint layer height. Toestimate it, we first correct the angular deviation ofthe paint layer in each image by a rotation. Thefraction of particles ft(y) = ⟨χ(x,y)⟩x is then obtainedby averaging over lines parallel to e2 (vertical axis).For each sample, the latter function is represented in

Fig. 3 as a function of y/Ly, where Ly is the paintlayer height. We emphasize that this is a relative,not an absolute, distance. The fraction of particles, ofabout 17%, is highest in the middle of the paint layer,and nearly 0 close to the outer layers. Accordingly,the microstructure is not stationnary in the verticaldirection.

0 20 40 60 80 100Height (%)

0

0,05

0,1

0,15

0,2

0,25

Lineic fraction (%)

Sample 1Sample 2Sample 3Sample 4Sample 5Average

Fig. 3. Lineic fraction of particles along the paintheight, for the 5 samples (colored curves) and theirmean (black line).

CORRELATION FUNCTIONThe covariance function is defined as the

probability :

C(r) = P{x ∈ H ,x+ r ∈ H }, (1)

where H is the union of the aluminium particles, andx is a point. At large distances |r| ≫ 30 µm the twoevents x ∈ H and x + r ∈ H become uncorrelatedand C(r=∞)≈C(0)2. Thus, we define the normalizedcovariance, or correlation function, as:

C(r) =C(r)−C(0)2

C(0) [1−C(0)]. (2)

We take r = re1 (r ≥ 0), aligned with the paint layersurface (Fig. 4) or r = re2 along the paint height(Fig. 5). The slope at the origin for C(re1) is nearlythe same for each sample (Fig. 4). The tangent atthe origin, represented in the same figure, intersectsthe r-axis at about r ≈ 12 µm. This value is theaverage length of the particles along the direction e1.The “average” here is not an average in number (ofparticles), but in number of chords, i.e. it is weightedby the particles thickness.

Along the layer height, the correlation functionC(re2) (Fig. 5) instead shows local minima that

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COUKA E et al.: A mixed Boolean and deposit model

are typical of repulsion effects in stacks of non-overlapping inclusions. Here, the tangent at the originintersects the r-axis at about 3 µm, the mean particlethickness. As expected, this length is much smallerthan the average size along e1.

8.8 17.6 26.4 35.2 r (µm)0

0.2

0.4

0.6

0.8

1C(r)

Sample 1Sample 2Sample 3Sample 4Sample 5Average

Fig. 4. Correlation function C(re1) along a directionparallel to the paint surface: each of the 5 samples(colors), mean (black line).

0 4.4 8.8 13.2 17.6 22 r (µm)0

0.2

0.4

0.6

0.8

1C(r)

Sample 1Sample 2Sample 3Sample 4Sample 5Average

Fig. 5. Correlation function C(re2) along a directiontransverse to the paint surface: each of the 5 samples(colors), mean (black line).

PARTICLES SIZE AND ORIENTATION

In the following, we model aluminium particlesas cylinders and measure the distribution of lengthsalong the principal direction of the particles to inferetheir size distribution in 3D. We first identify twoextremal points in each particle, with maximum andminimum coordinate along e1. The thickness of the

particle is estimated as the diameter of the inscribedcircle centered in the middle of the two extremalpoints (Fig. 6). Knowing the centre of each particle,we compute the size of the inscribed circle from thedilation by a rhombicuboctahedron. The orientationand length along the main direction of the particle isestimated by simple geometrical formulae (Fig. 6).

Fig. 6. Representation of the identification of theparameters of a segmented cylinder.

Assume first that all particles are identicalcylinders with the same basis diameter D, andunspecified distribution of orientations. The particlesappear as elongated rectangles on a 2D section. Thecumulative distribution of the length L of the largestside of the rectangles reads:

P1(ℓ) = P{L ≤ ℓ}= 2π

sin−1 ℓ

D. (3)

The cumulative distribution function for the measuresis concave whereas it is convex for the theoreticaldistribution P1(ℓ), as seen in Fig. 7, which invalidatesour hypothesis. Therefore, we seek for an alternative3D model for the particles size distribution.

We notice the existence of small particles inoptical images, and consider a uniform distribution ofdiameters in the range [0;D] for the cylinders basis.The resulting cumulative distribution is derived byintegrating on Eq. 3. We find:

P2(ℓ)=2π

(tan−1 ℓ√

D2 − ℓ2+

ℓ

Dlog

D+√

D2 − ℓ2

ℓ

).

(4)

Choosing D = 26.4 µm, a good agreementis found between the distribution above and themeasures (Fig. 7).

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0 22 44 66 88l (µm)

0

0,2

0,4

0,6

0,8

1P(l)

Uniform radiusConstant radiusMeasures

Fig. 7. Cumulative distribution of the lengths ofthe particles along their main direction, on a 2Dsection: measurements (red), theoretical distributionfor cylinders with constant (green) or uniformdiameter (blue).

The orientation of the particles, defined as theangle between their main directions and the surfaceof the paint, is extracted using previously computedpoints (Fig. 6). The cumulative distribution function(Fig. 8) as a result of a uniform distribution is linearin the range [−3.5◦;21.5◦]. Many of the misorientedparticles are small, for which no apparent orientationis visible. Neglecting the latter, and centering thedistribution around 0◦, we model the orientation ofthe particles as a uniform distribution in the range[−12.5◦;12.5◦].

-90º -60º -30º 0º 30º 60º 90ºOrientations

0

0,2

0,4

0,6

0,8

1Proportion of particles

Cumulative distributionLinear approximation

Fig. 8. Cumulative distribution of the orientation of theparticles, with respect to the paint layer surface (blue).The latter is quasi-linear in the range [−3.5◦;21.5◦](red).

DISTRIBUTION OF AGGREGATES

As seen in SEM images (Fig. 1), particles tendto aggregate in the direction transverse to the surfaceof the paint layer, as “stacks”. We define stacks aslines transverse to the surface layer and construct themrecursively. Each stack initially contains one particle.We identify them with the line passing by the particlecenter, parallel to the direction e1. We merge twostacks if the distance between them is smaller thanthe mean radius of all particles, about 7.24 µm. Onceagain, the new stack is defined as a line transverse tothe surface layer. Its e1 coordinate is the average ofthat of the merged stacks, weigthed by the number ofparticles each of them contain. The process is stoppedwhen stacks can not be merged anymore. An exampleof the set of stacks is shown in Fig. 9. The numberof particles per stack does not exceed 6. Its statisticis given in Tab. 1. The mean number of particles perstack is 1.7. Accordingly, the mean distance betweentwo particles in one stack is 5.85 µm.

Fig. 9. Distribution of particles along “stacks” (redlines).

Table 1. Proportion of stacks (column 2) per numberof particles (column 1).

Particles per stack Proportion of stacks1 56.9%2 28.8%3 7.84%4 3.27%5 1.96%6 1.31%

RESULTS

We compare now the measures obtained on theSEM images to the ones taken on numerical randommicrostructure models. For the sake of simplicity, wemodel the particles as cylinders in the rest of this work.Their diameters follow a uniform distribution in therange [0;D] with D = 26.4 µm. The angle betweenthe cylinder basis and the surface follows a uniformdistribution as well, in the range [−12.5◦;12.5◦].Furthermore, we fix the cylinders’s aspect ratio (basisdiameter over height) to 4. This value is consistent withthe particles surface density, related to the slopes at theorigin of the correlation functions (Figs. 4 and 5).

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COUKA E et al.: A mixed Boolean and deposit model

We emphasize that the models are general. Withminimal changes, they allow one to incorporate morecomplex particle shapes. This holds in particular forflakes which ressemble cylinders.

DEPOSIT MODEL

In this section, we model the dispersion of particlesby a 3D deposit model. This model is loosely inspiredby the actual fabrication process of the material.Cylinders fall along direction e2 from the top of thesimulated domain to the bottom. If they encounteranother cylinder, or the bottom, they are moved in theopposite direction, from bottom to top, by a distanced. Their initial position is taken randomly on the topof the domain. If a cylinder is not entirely contained inthe domain, it is rejected and a new initial position ischosen for another cylinder.

The simulation is stopped after N failed attempts.This number controls the density of particles at thetop of the layer. This model has two parameters:the repulsion distance d and the number of failedattempts N. The repulsion distance is chosen as arandom variable in the range [0;dmax] to allow contactpoints between particles as seen in the SEM images(Fig. 1). We choose dmax/2 = D/2. This value is closeto the mean distance between two particles in a stack(5.85 µm). Furthermore, we move cylinders by steps of2 voxels. This low value is necessary in order to makesure that no cylinder passes through another.

We let N vary to recover the measured surfacefraction of particles, and find N = 30. A 2D sectionof the model is represented in Fig. 10. The fractionof particles along lines parallel to the surface layeris represented in Fig. 11 with respect to the heightcoordinate e2, and compared with that of SEMimages. For the deposit model, we average over100 configurations of size 1024× 1024× 104 voxels.Although the fraction of particles is 0 along the bottomand top of the model as in the real material, the fractionof particles is overall much more regular in the depositmodel than in the paint layer. In the deposit model,the variation of the particle fraction near the surfacesis controlled by the size of the cylinders and theirorientation. However, these two parameters are unableto reproduce the fluctuations observed in the SEMimages.

Likewise, we compare the model in terms ofcorrelation function (Fig. 12). The slope at the originis correctly reproduced. Repulsion effects show upas a local minimum on the correlation function forthe deposit model around a size of 5.85 µm. Thissize is comparable to that of the bump observed onthe SEM images. This bump is itself an effect of

local minima in the correlation functions of individualsamples (Fig. 5). Nevertheless, repulsion effects aremuch more pronounced in the deposit model thanin the paint layer. Thus, both measures investigatedhere, the fraction of particles along lines parallel tothe surface and the correlation function, invalidateour deposit model. In the next section, we consider adifferent model where we directly use statistics on thenumber of particles per stack.

Fig. 10. 2D section of the deposit model. White:particles.

50 100Height (%)

0

0,05

0,1

0,15

0,2Lineic fraction (%)

Deposit modelAverage

Fig. 11. Mean fraction of particles ⟨χ(x,y)⟩x as afunction of the height y: SEM images (blue) anddeposit model (red).

0 4.4 8.8 13.2 17.6 22 r (µm)0

0.2

0.4

0.6

0.8

1C(r)

Deposit modelIEM images

Fig. 12. Correlation functions over 2D sections of thedeposit model (red) and SEM images (blue).

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STACK MODELIn the “stack” model developed hereafter, we

proceed in two steps. We first simulate stacks by a 2DPoisson point process on the surface layer. The Poissonpoint process is the most simple random point process.The validity of this assumption will be examined lateron. Second, we associate each stack with a randomnumber of particles. This random number follows thedistribution in Tab. 1. Next, particles are recursivelyadded in the simulated domain, along each stack. Eachparticle is a cylinder with shape, size and orientationdefined as in the deposit model. As in the depositmodel, the particles do not intersect. Initially, we placethe center of each cylinder at a random initial positionalong the stack axis. This initial position follows agiven distribution. Cylinders are moved along the stackaxis to the top of the domain if the initial position iscloser to the top than it is from the bottom. Otherwisewe move the cylinder in the opposite direction. Welet the cylinder move as long as it intersects anothercylinder.

The distribution of initial positions stronglyinfluences the final density of particles along e2,as computed in Fig. 3. We choose a triangulardistribution, symmetrical around the middle of thestack axis. More precisely, we take:

f (y) =

(1/δ ) [1− (1/δ )|y−L2/2|]if |y−L2/2|< δ

0 otherwise,(5)

where L2 is the width of the domain, i.e. the distancebetween the bottom and top. The density of probabilityf (y) is highest at the center y = L2/2 and decreaseslinearly with y. It is zero at y = L2/2±δ . We optimizeon the parameter δ to approach the density fractionof the SEM images, in terms of least squares. Thedensity fraction of the cylinders in the stack model isnarrow for small values of δ and wider for large values(Fig. 13). We find δ = 0.05L2 as optimal value.

Two random 2D sections of the optimal stackmodel are shown in Fig. 14, together with onesegmented SEM image for comparison. Overall, thestack model is much more heterogeneous than thedeposit model. On the one hand, particles are stackedtogether as clusters. On the other hand, isolatedparticles appear, although no repulsion effect has beenintroduced in the model. We remark that both featuresare present in the SEM images. Furthermore, thereexists vertical areas joining the bottom and top whichare free of particles. This last feature is present in thedeposit model but is difficult to control. It presumablyhas implications on the optical rendering. Hereafter,we compare our model with SEM images in quantitiveways.

The density fraction of particles along e2 is shownin Fig. 15vand compared with measures on the SEMimages. Measures for the stack model are averagedover 100 configurations of size 1024×1024×104. Asexpected, the higher concentration of particles in themiddle is recovered in the stack model. Overall, thedensity fraction of particles is very close to that of thereal material.

50 100Height (%)

0

0,05

0,1

0,15

0,2Lineic fraction (%)

δ=40δ=20

Fig. 13. Effect of the parameter δ on the densityfraction of particles in the stack model (over 100realisations): δ = 20 (blue), δ = 40 (red). Dottedlines: probability density as defined in Eq. 5, governingthe initial position of the cylinders.

(a)

(b)

(c)

Fig. 14. Segmented SEM image (a) compared with two2D sections orthogonal to the layer’s surface of thestack model (b and c).

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COUKA E et al.: A mixed Boolean and deposit model

0 50 100Height (%)

0

0,05

0,1

0,15

0,2Lineic fraction (%)

Stack modelIEM images

Fig. 15. Density of particles over lines parallel to thesurface layer, as a function of the coordinate along theheight e2: stack model (red), SEM image (blue).

We now evaluate our stack model in termsof the correlation function C(re2) along e2. Thecorrelation functions of individual SEM images showfluctuations characteristic of repulsion effects (Fig. 5)or hardcore phenomena due to contact points in 3D.This effect disappears when averaging the correlationfunctions measured over all the images of the sample(black curve). Accordingly, we compare separately thecorrelation functions of one 2D section of randomconfigurations of the stack model with individual SEMimages (Fig. 16), and that of averages (Fig. 17).For the latter, we average over all SEM images andover 10 realisations of the 3D stack model. Overall,the correlation functions for the stack model followthe same patterns as in SEM images. Averagingsuppresses the fluctuations observed on individualsections for r > 5µm, for both SEM images and thestack model. Furthermore, these fluctuations are ofthe same order of magnitude in the SEM imagesand 2D sections of the stack model. However, thecorrelation functions of SEM images show slightlylarger differences between individual images thanwas found for random configurations of the stackmodel. Good agreement is found between the meansof correlation functions as well (Fig. 17): the slopesat the origin are nearly the same, and the correlationfunctions show a regime change at about 4 µm. Thesmall discrepancy observed at length scale larger than4 µm is hardly representative, given the small numberof available SEM images.

Finally, we compare the distribution of lengths ofthe particles along their main directions, as observedin 2D sections of the stack model with that of SEMimages and the theoretical formula seen in Eq. 4(Fig. 18). We compute this length by the same

methodology as described previously (see Fig. 6).Again, the stack model reproduces the general trendmeasured on the SEM images.

0 4.4 8.8 13.2 17.6 22 r (µm)0

0.2

0.4

0.6

0.8

1C(r)

IEM imagesStack model

Fig. 16. Correlation function C(re2) along e2 along theheight: individual 2D sections of random realizationsof the stack model (red) and individual SEM images(blue).

0 4.4 8.8 13.2 17.6 22 r (µm)0

0.2

0.4

0.6

0.8

1C(r)

Stack modelSample

Fig. 17. Correlation function C(re2) along e2 alongthe height: mean over random realizations of the stackmodel (red) and mean of SEM images (blue).

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0 22 44 66 88l (µm)

0

0,2

0,4

0,6

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1P(l)

Uniform radiusMeasuresStack model

Fig. 18. Cumulative distribution function of the lengthof particles along the main directions as observed onSEM images (blue) and stack model (red). Theoreticalformula seen in Eq. 4 shown in green.

DISCUSSION

In this work, we studied micrography imagesof a paint layer. The 2D distribution of sizes andorientations of metallic particles, measured alongsections in the material, were used to determine similar3D parameters required by the modeling. Furthermore,the dispersion of the particles showed strong clusteringeffects and inhomogeneous dispersion along the heightof the paint layer. The particles density is highest alongthe middle of the layer and zero along its borders.

A basic deposit model where particles fall alongthe direction normal to the paint layer was unableto qualitatively reproduce the observed clusteringeffects and fluctuations of density over the height.Accordingly, we modelled the particles dispersion bya combined Poisson point process for the clustersand a hard-core process for implanting particles ineach cluster. We use statistics on the number ofparticles per cluster and density of clusters in thepaint layer, that were determined on the SEM images.The resulting “stack model” showed good agreementwith the SEM images in terms of particles size andcorrelation functions. The density of particles alongthe height is also well reproduced. This stack modelis straightforward to implement and requires only oneparameter that is easy to identify. It may also takeinto account particles with any distribution of shapesand orientations, and could be adapted to other similarpaints.

ACKNOWLEDGEMENT

This study was made with the support of A.N.R.(Agence Nationale de la Recherche) under grant 20284(LIMA project). The authors are grateful to PhilippePorral and Christian Perrot-Minnot (PSA, France)for providing the material samples, and to MonaBen Achour, Anthony Chesnaud and Alain Thorel(Center of Material, MINES ParisTech, PSL ResearchUniversity) for contributing images of the material.

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