Giulia Ferri

Severine Humbert

Jean-Marc Schweitzer

Mathieu Digne

Veronique Lefebvre

Maxime Moreaud

Mass fractal dimension from 2D-microscopy images via an aggregation model with variable compactness

Keywords: fractal dimension, morphological model, image analysis

Microscopy-image analysis provides precious information on size and structure of colloidal aggregates and agglomerates. The structure of colloids is often characterized using the mass fractal dimension d f , which is different from the two-dimensional fractal dimension d p that can be computed from microscopy-images. In this work we propose to use a recent morphological aggregation model to find a relationship between 2D image fractal dimension and 3D mass fractal dimension of aggregates and agglomerates. Our case study is represented by scanning transmission electron microscopy-images of boehmite colloidal suspensions. The behaviour of the computed d f at different acid and base concentration shows a fair agreement with the results of Small Angle X-Ray Scattering and with the literature, enabling to use the d f vs d p relationship to study the impact of the composition of the colloidal suspension on the density of colloidal aggregates and agglomerates.

Introduction

Image-analysis is frequently used to characterize colloidal particles of silica 1 , alumina [START_REF] Chiche | Accurate determination of oxide nanoparticle size and shape based on X-ray powder pattern simulation: Application to boehmite AlOOH[END_REF] , soot or carbon. These particles can exist as a powder or within liquid suspensions, and they find applications in different domains like electronics, pharmaceutic and catalysis [START_REF] Brune | Nanotechnology: Assessment and Perspectives[END_REF] . Such applications depend on particles size and structure that are the result of the particles aggregation or agglomeration mechanism 4;5;6 . For this reason the control of aggregation and agglomeration within colloids is of high importance. In literature, the terms aggregate and agglomerate are often used interchangeably. When it is necessary to distinguish between them, authors refer to the type of cohesive force that can be either strong or weak [START_REF] Nichols | A review of the terms agglomerate and aggregate with a recommendation for nomenclature used in powder and particle characterization[END_REF] . With regard to the colloidal boehmite suspensions characterised in this work, assemblies of maximum diameter lower than or higher than 80 nm are called aggregates or agglomerates, respectively.

The mass fractal dimension d f is often used to characterize the structure of aggregates and agglomerates. Assuming that an aggregate/agglomerate consists of N identical spheres of radius a, the fractal dimension relates N to the gyration radius R gyr 8;9;10 .

N = k f R gyr a d f (1)
where k f is the fractal pre-factor whose value varies typically between 1 and 1.2, depending on the fractal dimension itself [START_REF] Christopher | The prefactor of fractal aggregates[END_REF] . The mass fractal dimension is close to unity for linear clusters and increases up to a maximum value of 3 for clusters with several folded branches. d f can be measured by experimental scattering techniques such as Small Angle X-Ray [START_REF] Mircea | Small-angle scattering from fractals: Differentiating between various types of structures[END_REF] or Neutron [START_REF] Teixeira | Small-angle scattering by fractal systems[END_REF] Scattering. A change in d f of a colloidal aggregate/agglomerate modifies its density, involving different dispersibility and settling behaviour [START_REF] Ibaseta | Ultrafine aerosol emission from the free fall of tio 2 and sio[END_REF] . This changes the workability of a colloidal suspension, as well as the textural properties of the solid obtained after a precipitation or drying process. In the context of γ-alumina catalyst carriers synthesis, the porosity of the final solid depends on the size and shape of the aggregates (meso-porosity ≃ 10 nm) and agglomerates (macro-porosity ≃ 100 nm) resulting from colloidal inter-particle interactions during different stages of preparation. By varying the chemical parameters of the colloidal suspension, it is possible to change the aggregation and agglomeration processes [START_REF] Lawrence M Anovitz | Effects of ionic strength, salt, and pH on aggregation of boehmite nanocrystals: Tumbler small-angle neutron and x-ray scattering and imaging analysis[END_REF] , leading to different size distribution and fractal dimension. Our interest is to use microscopy-image analysis to study this behaviour, in order to determine the physical-chemical parameters that lead to specific structure change within boehmite suspensions.

In this work, a characteristic fractal dimension d p is computed from microscopy images of boehmite aggregates and agglomerates. d p is then related to the mass fractal dimension d f . A similar purpose has been addressed many times in literature. The fractal dimension computed on two-dimensional images depends on the calculation method used 16;17 . For this reason, in order to compare several images it is necessary to use the same computation algorithm. Other studies on the relationship between cluster mass fractal dimension and their projection report that for 2 < d f < 3 the difference d f -d p increases with d f [START_REF] Tence | On the measurement of the fractal dimension of aggregated particles by electron microscopy: experimental method, corrections and comparison with numerical models[END_REF] . In this work the projected fractal dimension d p is measured using the relationship between area and perimeter [START_REF] Benoit B Mandelbrot | Fractal character of fracture surfaces of metals[END_REF] . This method is extensively used 20;21 , enabling to obtain directly a two-dimensional fractal dimension from the slope of the graph log(P 2 ) vs log(A), where P and A are respectively the perimeter and the area of the projections. The slope tends to 1 for circular objects and to 2 for linear objects. Since the fractal dimension of the projection d p is different from the mass fractal dimension d f , a relationship between these two parameters 22;23;24;25;26 is needed.

The aim of our work is to numerically provide such relationship, using a recent morphological aggregation model [START_REF] Ferri | Aggregation morphological model with variable compactness: application to colloidal system[END_REF] . The model is used to build clusters with a statistical average fractal dimension, as well as their opaque projections.

Determination of 3D-fractal dimension from 2D-microscopy images

In this section, we consider Scanning transmission electron microscopy (STEM) images of colloids of a boehmite suspension (on the left in Fig. 1) and we explain the step-by-step strategy to extract d f from such images.

Image processing and determination of d p

To compute the perimeter and area on STEM images, several operations are applied. The image noise is reduced 28;29 by preserving the sharp edges of objects. Morphological black TopHat operator [START_REF] Jean | Introduction to morphological filters[END_REF] is applied consisting on difference between the image and a morphological opening γ of the image, and a threshold operator tr s such that

tr s (I(x)) = 1|I(x) > s 0|I(x) ≤ s .
Extracted solid suspension can be expressed for an image I as tr s (I -γ B (I)) where s is obtained by inter-class variance maximization on the histogram of I 31 , and B is the structuring element which in our case is a disc of radius 30 pixels. In order to reduce the remaining noise, the morphological operations of opening γ and closing φ are applied, defined for the image I and the structuring element C as γ C (I) = δ C (ε C (I)) and as φ C (I) = ε C (δ C (I)) where δ and ε are respectively morphological dilation and erosion [START_REF] Jean | Introduction to morphological filters[END_REF] , C is a disc of diameter 6 pixels. The last step of image cleaning consists in a hole filling and in the removal of the objects touching the edges of the image [START_REF] Jean | Introduction to morphological filters[END_REF] . The holes are not always related to the existence of a real hole but rather to variations in thickness. While objects touching the borders would provide a misleading perimeter and area. Fig. 1 reports an example of an experimental image and of the final binary image. On the binary images it is possible to calculate the perimeter P and area A of each object. The d p is estimated from the slope of the line interpolating the experimental points on the plot log(P 2 ) vs log(A) [START_REF] Benoit B Mandelbrot | Fractal character of fracture surfaces of metals[END_REF] . To estimate the d f corresponding to a given d p it is necessary to pass through a morphological model. It is worth considering that the structures observed via STEM images are affected by the sample preparation process (drying under IR lamp). Furthermore, the choice of structuring element B can have a significant impact on the calculation of d p . For this reason, in order to compare the results, it is important to prepare all samples following the same procedure and to perform the image analysis using the morphological operations in the same order and with the same structuring elements.

Morphological aggregation model

The morphological aggregation model 27 builds clusters consisting of N non-overlapping elementary objects arranged according to a given statistical average fractal dimension d f . Clusters are built by sequential addition, assigning a different sticking probability to the points on the cluster. In particular, "concave points", leading to compact clusters, are distinguished from "non-concave" points, leading to loose structures. In addition the concave points closest to the center of mass of the cluster are distinguished as well, in these points the maximum compactness is reached. The model enables to obtain clusters with a fractal dimension d f between 1.3 and 3. A fast scheme is available in case of spheres [START_REF] Ferri | Aggregation morphological model with variable compactness: application to colloidal system[END_REF] and is used in this present work. A new object P is stuck to a cluster C either in a point on the cluster dilation

X P = F (δ P (C)), (2)
either on a concave point of the dilation

X CV P = F (δ P (C)) ∩ F (φ r (δ P (C))/δ P (C)) (3)
or in the concave point closest to the center of mass of the cluster C

X M P = x ∈ X CV P |infd(x, M ). ( 4
)
The expression φ r (δ P (C))/δ P (C) means the difference between the two sets. F (X) is the set of points at the border of X, F (X) = {z : B z ∩ X ̸ = ∅}, with B unit ball, M is the center of mass of the cluster C and d(x, M ) is the euclidean distance between the position x of the concave point and M . Fig. 2 illustrates how the morphological operations enable to select the concave regions of a cluster's boundary. An implementation of this model is available on the plugim! platform [START_REF]plugim![END_REF] .

𝑃 𝑃 ∈ 𝑋 𝑃 𝑀 𝑃 ∈ 𝑋 𝑃 𝑃 ∈ 𝑋 𝑃 𝐶𝑉 Figure 2:
The dilated set X P (in gray), the set of concave regions X CV P (in blue) and X M P are illustrated for a cluster C during the simulation. P is the primary particle.

Construction of d f vs d p chart

In order to compute d p of a cluster generated by the model, it is necessary to generate the projections of numerous clusters for a given d f in the largest range for N (the number of elementary spheres). The mass fractal dimension d f of each cluster is determined with Eq.(1) using the plots reported in Appendix A. In the present study a fractal pre-factor k f of 1.2 is used, in order to establish reference values for d f of the clusters generated [START_REF] Ferri | Aggregation morphological model with variable compactness: application to colloidal system[END_REF] . This value has been chosen because it provides a mass fractal dimension less than or equal to 3 for simulated clusters consisting of spherical particles.

The model inlet parameters were chosen in order to obtain a target statistical average d f . Only the clusters with d f -d f ≤ 0.01 were used to compute the projection. For each cluster, one orthogonal projection is computed, since a large number of simulations is performed it is possible to assume that area and perimeter computed on the projection are anisotropic. Fig. 3 represents three-dimensional clusters with their projection for different number of elementary spheres N . A hole filling operation is performed to obtain projection comparable to experimental images [START_REF] Jean | Introduction to morphological filters[END_REF] . The computation of area and perimeter enables the construction of the log(P 2 ) vs log(A) plot, whose slope is d p [START_REF] Benoit B Mandelbrot | Fractal character of fracture surfaces of metals[END_REF] .

The coefficients of the polynomial minimize least-squares fit. Eq.( 5) is valid for the morphological aggregation model described in a previous work [START_REF] Ferri | Aggregation morphological model with variable compactness: application to colloidal system[END_REF] , using spherical primary particles. Since there is no one-to-one correspondence between the fractal dimension of a cluster d f and the dimension of its projection d p , it is not possible to conclude whether this relationship can also be valid for other random aggregation models.

Validity region

To compare experimental and model projections, the perimeter-area plots are represented as log(P 2 ) vs log(A/A 0 ), where A 0 is the projection of the elementary object forming the cluster.

Aggregation simulations were performed with a sphere of radius 10 pixels, corresponding to an opaque projection of area A 0 of 380 pixels. This is important so that the relationship shown in Fig. 4(a) does not depend on the size scale and unit of measurement. The Eq.( 5) is valid within limits, in terms of projected log(A/A 0 ), for a given d f . The lower and the upper limit (respectively l A,min and l A,max ) are reported as reachable limit ranges by the aggregation model. The upper limit is also due to the maximum simulation volume, which in our case is of 10 9 voxels for time computing constraints. The lower and the upper limits can be described as a function of d f fitting respectively the lowest and the highest log(A/A 0 ) of the opaque projections of the simulated clusters. 5) in (a) and its validity in terms of surface area A in (b).

l A,min = 0.67 • (d f ) 4 -5.12 • (d f ) 3 + 14.37 • (d f ) 2 -17.12 • d f + 7.49 l A,max = -7.17 • (d f ) 4 + 68.19 • (d f ) 3 -241.09 • (d f ) 2 + 377.70 • d f -220.91. (6)
The validity region and its limits are represented in Fig. 4(b). A 0 does not influence the results for d p , but only the domain of validity of the relationship in Eq.( 5) on the log(P 2 ) vs. log(A/A 0 ) plot.

Experimental application to boehmite suspensions

In this section, we use the relationship between d f and d p to compute the mass fractal dimension from STEM images of different boehmite suspensions. Results are compared to Small Angle X-Ray Scattering and literature.

Boehmite suspensions preparation

Boehmite suspension are prepared with a highly dispersible powder from Sasol GmbH whose crystallites have the geometry of a rectangular beam with dimensions 5.2, 8.1 and 2.4 nm. These values are obtained from the X-Ray diffractogram using generalized Debye equation 2 . The colloidal suspensions are prepared with different concentration of 1M Fisher Scientific nitric acid (J/5550/PB15) and 1M Chem-Lab ammonia (CL05.0101.1000). The powder concentration is kept at 0.02 g/L in order to minimize overlapping in STEM images. The composition of the samples is shown in Tab.1. The table also reports the measured pH (Mettler-Toledo Seven2Go pH meter). A2 and B0 are the same suspension, which represents the initial conditions before ammonia addition (samples B1-3).

STEM Images Acquisition

A droplet of 10 µL of the suspension is deposited on copper grids with carbon membranes without holes and dried under IR lamp. After 2 min, the excess water is drawn off with a tissue paper. The samples are characterized with Scanning Transmission Electron Microscope (STEM), using a FEI Nova NanoSEM microscope operating at 15keV.

The different chemical parameters of the suspensions (acid and base concentration) of the samples causes the formation of structures ranging within a wide size range (10 nm -10 µm). For this reason, the magnification level for a given sample is chosen in order to visualize the largest objects in their entirety. In particular we use a magnification of X 50000 for the samples A0, A1, A2 and A3, and a magnification of X 1500 is used for B1, B2 and B3.

Small Angle X-Ray Scattering

The analyses were performed at the SWING beamline at synchrotron SOLEIL in Saint Aubin, France. The boehmite suspensions are placed in a stirred reactor, from which the sample is pumped continuously through a quartz capillary of 1.5 mm, crossed by the X-ray beam, whose incident energy is 10 keV. The experiments are realized on an initial boehmite suspension analogous to the A2-B0 at 1 g/L. The SAXS curves measured 30 min after the base injection are fit with Beaucage model 33 in order to compute the mass fractal dimension. The model is detailed in Appendix B.

Monitoring of boehmite dispersion upon acid addition

The aggregates (observed in the samples A0, A1, A2 and A3) have an average diameter D between 20 and 40 nm (Tab.1). This diameter is calculated by averaging the diameters of objects on binary microscopy images. The diameter of each object is calculated as the average of the minimum and maximum diameters. For these objects A 0 is the one of an elementary crystallite. The average surface of the opaque projection of the crystallite is 38.25 nm 2 , obtained by averaging on 1000 projections considering random 3D orientations 34 with a precision of 0.1 nm/voxels. Such estimation accounts for discretization issues, and it is close to the one computed using the Cauchy average projected area theorem, which is 37.02 nm 2 . We report the STEM images of the aggregates at different acid concentrations and their corresponding log(P 2 ) vs log(A)/ log(A 0 ) plots in Fig. 5.

The grey regions in these graphs refer to the validity limits of the d f -d p relationship (Fig. 4). The results of d p and d f (computed with Eq.( 5)) are reported in Tab.1. The dispersion in acid leads to an increase in d p , indicating the formation of loose structures. This is in agreement with d f , and quite visible from the STEM images. The objects present in A3 are more elongated than the ones in A0-1. We observe a change of around 0.3 units of d f . The difference between A3 and A2 is less noticeable, but the image processing enables to identify it. In conclusion, the acid content decreases fractal dimension of the boehmite suspension as it has been observed in other research works [START_REF] Speyer | Peptization of boehmites with different peptization index: An electron microscopy and synchrotron small-angle x-ray scattering study[END_REF] . 5), and measured with Small Angle X-Ray Scattering at different pH.

Monitoring of boehmite agglomeration upon alkaline addition

In a similar way, the relationship between d f and d p is used to characterize objects of a bigger scale, the agglomerates (in samples B1-3), obtained by ammonia addition to the suspension A2-B0.

The average diameter D is of several hundreds of nanometers (Tab.1). In this case, the elementary particle is the aggregate, and A 0 was taken as the average area of the dispersed sample A2-B0, which is 453 nm 2 . We now focus on the STEM images in Fig. 6. The effect of the base content seems to favour the formation of more compact structures with a decrease in d p . Between samples B1-2 and B3, we notice an evident change in the structure. Sample B3 presents highly compact objects Fig. 6(e), for which the mass fractal dimension is 2.9. For B3 the pH is close to the Point of Zero Charge (PZC) of the suspension [START_REF] Kosmulski | ph-dependent surface charging and points of zero charge. iv. update and new approach[END_REF] . In these conditions the agglomeration is typically favoured [START_REF] Johannes | Theory of the stability of lyophobic colloids: the interaction of sol particles having an electric double layer[END_REF] . We were able to observe an increase in the mass fractal dimension towards the PZC also with in-situ Small Angle X-Ray Scattering. Details on SAXS data are reported in Appendix B. The fractal dimension of SAXS agglomerates is compared to the one obtained from microscopy images in Fig. 7.

The behaviour observed in SAXS results suggests that the proximity to PZC, besides favouring faster kinetics, leads to more compact structures. Since the operating conditions required by each analysis technique are different, a comparison can only be made on the basis of a trend. Even if the concentration and the sampling is different, the STEM image processing also showed that a pH close to PZC lead to more compact structures.

For a further improvement of our approach, it would be interesting to investigate in the future the effect of drying (under IR lamp) on the fractal dimension of boehmite agglomerates. Or to compare the fractal size estimated from scattering data with that obtained from liquid phase microscopy techniques.

Conclusion

The presented work provides a strategy to compute the mass fractal dimension from two-dimensional microscopy images. A relationship between d p and d f is numerically found using a recent morphological aggregation model. The mass fractal dimensions we obtained are in fair agreement with a visual estimate, enabling to use STEM images to determine the impact of different operating conditions on the structure of aggregates and agglomerated formed within a colloidal suspension. This allows a better insight of the physical-chemical aspects that produce such structures In particular, we observed that an increase in acid concentration, causes the formation of more loose structures, with a decrease in d f . While the increase of base leads to more compact structures, leading to higher d f . A complete validation of our approach is achieved by comparison of our results for boehmite suspension at different acid and base contents, with the literature and Small Angle X-Ray Scattering experiments.

Appendix A log(P) vs log(A) plots

This section shows the plots produced by relating the perimeter and area of the morphological model projections for

d f = 2, d f = 2.3, d f = 2.

B SAXS data B.1 Beaucage model

The general formula of the model for n structural levels is

I(q) = n i=1 G i • exp - q 2 R 2 gyr,i 3 + B i - q 2 R 2 gyr,i+1 3
•      erf q • k i • Rgyr,i √ 6 3 q      Pi (7)
• n equals to 1,2 or 3 for agglomerates, aggregates and crystallites respectively.

• G i is the Guinier pre-factor depending on the number concentration of the structural unit

N d • r 2 e • ∆ρ 2 • V i (N d
is the particles number density, r e is the electron radius, ∆ρ is the electron density difference between the sample and the environment, V is the average particle volume).

• R gyr,i is the gyration radius.

• B i is a constant pre-factor depending on the type of power-law scattering and on P i 38 .

B i = G i • P i R ( gyr, i) Pi • Γ P i 2 • 6 • P 2 i (2 + P i ) • (2 + P i ) P i 2 . (8)
• k i is an empirical constant that depends on P i 38 , its value is close to 1.

• P i is the slope of the curve in the structural level i, corresponding to the fractal dimension.

In our case, the SAXS curves present three scales. A nonlinear Least-Square Algorithm DN2FB 39 has been used for the curve fitting.

B.2 SAXS results

The scattering curve of a boehmite suspension at 1 g/L at pH close to the PZC (9.4) is reported in Fig. 9. The three contributions of the simulated scattered intentity by agglomerates, aggregates and crystallites are illustrated. The experimental data are fit within a range of 0.001255 Å-1 ¡ q ¡ 0.1998 Å-1 for the scattering vector. Corresponding to a maximum diameter ξ l between 3 and 500 nm. According to the model, the agglomerates scatter in a range of 0.0078 Å-1 ¡ q ¡ 0.1998 Å-1 , for a maximum diameter 80 nm ¡ ξ l ¡ 500 nm according to the relationship

ξ l = 2π q . ( 9
)
Due to the size limit of SAXS results, the comparison with microsopy images is made under the assumption that the fractal dimension is the same even for objects larger than 500 nm in diameter. This assumption can be confirmed in terms of d p by the microscopy images analysis. Fig. 10 shows a microscopy image of our boehmite suspension, its binary image and the relative plot log(P 2 ) vs log(A) in which the points relative to the objects with a maximum diameter higher and lower than 500 nm are highlighted in different colours. The points can be represented by the same line, and therefore by the same dimension d p .

  μm

Figure 1 :

 1 Figure 1: Experimental image (on the left) and its binary version (on the right).

Figure 3 :

 3 Figure 3: Illustration of three-dimensional clusters and their projections for α = β = 0.1 and N = 20, 40, 160. α and β are the compactness parameters of the morphological aggregation model, more details are in Ferri et al.27.

Figure 4 :

 4 Figure 4: d f vs d p relationship described in Eq.(5) in (a) and its validity in terms of surface area A in (b).

  STEM image of A0 (b) log(P) vs log(A) plot of A0 500 nm (c) STEM image of A1 (d) log(P) vs log(A) plot of A1 500 nm (e) STEM image of A2-B0 (f) log(P) vs log(A) plot of A2-B0 500 nm (g) STEM image of A3 (h) log(P) vs log(A) plot of A3

Figure 5 :

 5 Figure 5: STEM images and log(P 2 ) vs log(A) plots of boehmite suspensions for varying acid content.

  STEM image of B1 (b) log(P) vs log(A) plot of B1 30 µm (c) STEM image of B2 (d) log(P) vs log(A) plot of B2 30 µm (e) STEM image of B3 (f) log(P) vs log(A) plot of B3

Figure 6 :Figure 7 :

 67 Figure 6: STEM images and log(P 2 ) vs log(A) plots of boehmite suspensions for varying base content.

  5 and d f = 3 in Fig.8. The d p is obtained from the slope of these plots.

Figure 8 :

 8 Figure 8: Log(P ) vs log(A) plots obtained with the morphological model.

Figure 9 :

 9 Figure 9: Experimental and simulated scattered intensity of a boehmite suspension at 1 g/L and pH close to the PZC. The simulated curve is given by the sum of three terms corresponding to agglomerates, aggregates and crystallites.

Figure 10 :

 10 Figure 10: (a) STEM image of boehmite agglomerates obtained with a magnification of X 3000. Binary STEM image in (b). In (c) log(P ) vs log(A) plot where the blue and green dots refer to the objects whose maximum diameter is below and above 500 nm respectively.

Table 1 :

 1 Composition, fractal dimension and average diameter of the boehmite suspension samples Sample Name HNO 3 [mM] NH 3 [mM] pH d p

	d f	D [nm]

Acknowledgment

SAXS experiments were performed on the SWING beamline at SOLEIL Synchrotron, France. We are grateful to Thomas Bizien for assistance and to the SOLEIL staff for smoothly running the facility.


Giulia Ferri

Severine Humbert

Mathieu Digne

Maxime Moreaud

Jean-Marc Schweitzer

Boehmite agglomeration through experimental and model approaches : from colloidal system to porous solid

Keywords: Colloidal agglomeration, Fractal dimension, Catalyst carrier

The agglomeration dynamics within colloidal boehmite suspensions is crucial to understand the formation of a porous boehmite network during the manufacturing of γ-alumina catalyst carriers. Such carriers are frequently used in petroleum hydrotreating processes and must have specific textural characteristics. The method described in this work enables to model the three-dimensional morphology of colloidal agglomerates of boehmite for different conditions of pH in the colloidal mixture. These agglomerates are then used to generate a boehmite grain, whose textural properties can be numerically estimated. The coagulation kinetics has been studied with experimental Dynamic Light Scattering and Lagrangian model including Brownian dynamics and DLVO interaction potential. The adjustment of the Brownian aggregation kernel of a population-balance model enables to estimate the agglomerates size distribution and fractal dimension.

Introduction

Tuning the porosity of catalytic supports is attracting significant interest for the control of reaction kinetics in many industrial processes. Most of the used catalytic supports are in γ-alumina, prepared from boehmite powder which undergoes several processing steps where its water content varies. Boehmite (the γ-alumina precursor) passes from the powder, to the paste and finally to the porous solid state. During these processes, structural changes occur at the meso-and macro-scale, which have an impact on the final porosity of the solid and its performance as a catalytic support. The structure of γ-alumina catalyst carrier is due to the organisation of primary boehmite crystallites, of a few nm [START_REF] Chiche | Accurate determination of oxide nanoparticle size and shape based on X-ray powder pattern simulation: Application to boehmite AlOOH[END_REF], formed during a precipitation phase. The primary crystallites assemble to form aggregates and agglomerates. The shape of crystallites, aggregates and agglomerates determines the type of porosity at micro-, meso-and macro-scale [START_REF] Euzen | Alumina[END_REF]. 1 Present address: Politecnico di Milano, Dipartimento di Energia, Via La Masa 34, 20156, Milano; Italy Corresponding author: giulia.ferri@polimi.it

In order to perform a tailored synthesis of γalumina supports in the coming years, it is necessary to develop an experimental and numerical method to study the effect that the operating con-25 ditions of the manufacturing process have on the structure of a porous solid. In this study we focus on the effect that the chemical composition of a colloidal suspension of boehmite has on the fractal dimension and on the size distribution of boehmite 30 agglomerates. Our strategy (see Fig. 1) enables to describe the morphology of the solid structure according to synthesis parameters. To this purpose, ideal and low concentration boehmite suspension have been studied under Brownian conditions. In 35 these conditions, we are in an ideal case where the physical interactions are fully theorized. Therefore, no unknown parameters are used to fit the experimental data set. These systems have been described using theoretical models of colloidal agglomeration 40 (like DLVO theory [START_REF] Verwey | Theory of the stability of lyophobic colloids: the interaction of sol particles having an electric double layer[END_REF][START_REF] Hamaker | The london-vander waals attraction be-500 tween spherical particles[END_REF] and Smoluchowski equation [START_REF] Smoluchowski | ber brownsche molekularbewegung unter einwirkung uerer krfte und deren zusammenhang mit der verallgemeinerten diffusionsgleichung[END_REF]).

The experimental characterisation of coagulating colloidal suspension via DLS (dynamic light scattering) provides data for the adjustment of the ag-45 gregation kernels of a population-balance model.

A discrete-element model, relying on DLVO and Langevin [START_REF] Langevin | Sur la thorie du mouvement brownien[END_REF] theories, enables to fill the experimental gaps and to estimate the evolution of average radius and fractal dimension along the agglomeration process. The fractal dimension d f is here considered as a structural parameter to characterise individual agglomerates. The Lagrangian approach and the population balance can represent agglomeration under Brownian motion and a validation is possible between these two modeling approaches. These models both depend on the Fuchs stability coefficient, which is a function of the pH and ionic strength of the colloidal suspension. The Fuchs stability ratio reaches a value close to unity for strongly unstable suspensions which have a high tendency to agglomerate [START_REF] Hiemenz | Principles of colloid 510 and surface chemistry[END_REF], its theoretical value can be computed from the DLVO interaction potential

DLS
U tot W ≈ 1 2κa exp U tot,max k B T (1)
assuming spherical particles of radius a, homogeneous surface charge distribution and negligible solvation force. r is the distance between the centers of the two particles, k B is the Boltzmann constant and T is the temperature.

κ = k B T 2e 2 A v -0.5 √ I (2)
is the inverse of the Debye length, the length of the layer of ions that are strongly bond to the particles and screen the electrical field of the particle. The Lagrangian model input parameters are the 50 size of the primary particles, the boehmite concentration and W . The model results is a dependence of the fractal dimension on the gyration radius. This dependence justifies the consideration of a size-dependent fractal dimension and is an input 55 of the population balance, where the Smoluchowski kernel is considered. The population balance also depends on the size distribution of the primary dispersed boehmite particles (in our case a Dirac function), on W and on the boehmite concentration.

60

The result of the population balance is the size distribution of the boehmite agglomerates at the end of the agglomeration process.

The relationship between the fractal dimension and the agglomerate gyration radius, and the ag-65 glomerate size distribution are th input parameters of a morphological agglomeration model [START_REF] Ferri | Aggregation morphological model with variable compactness: application to colloidal system[END_REF], whose result is the final morphology an agglomerates assembly. This last model is based on a random sequential addition algorithm and on morphological operators. It does not describe the coagulation physics, but simulates a stochastic assembly of particles. It is suitable for the simulation of a realistically large morphology, which makes possible the estimation of textural properties.

Experimental methods

Boehmite suspensions were prepared using a highly dispersible powder (Pural SB3, from Sasol GmbH) with a loss on ignition of 26%. The suspension was prepared with a concentration of boehmite powder of 0.04 g/L. The boehmite concentration was chosen according to the measurement techniques. Indeed, the coagulation time scale must be of the same order of magnitude of the acquisition time. The nitric acid (1M Fisher Scientific J/5550/PB15) concentration has been chosen in order to peptize the boehmite powder at a pH around 3, 0.0015 M. In order to control the pH, ammonia has been injected (1M Chem-Lab CL05.0101.1000).

The pH was measured using a pH-meter (Seven2Go Mettler Toledo). The zeta potential measurement have been performed with Malvern Zetasizer Nano ZS with 1 mL sample using capillary cells with electrodes. A study of the titration curve and the corresponding zeta potential vs pH curves was realized to tune the zeta potential via NH 3 injection in the initial peptized boehmite suspension. DLS measurement were performed with Malvern Zetasizer Nano ZS, using square polysyrene cell. 1 mL of initial peptized suspension was taken, filtered with a syringe filter at 5 µm. This procedure was necessary to remove any dust that could falsify the measurement and to eliminate the few grains of un-peptized powder that would not be visible with DLS. The initial suspension is characterized. Then, 0.2 ml of ammonia (and/or NaCl) solution was injected in the measurement cell and the insitu analysis was started. The average diameter was computed with the cumulant method [START_REF] Pecora | Dynamic light scattering: Applications of photon correlation spectroscopy[END_REF]. SAXS analyses were performed at the SWING beamline of SOLEIL synchrotron in Saint Aubin, France. The incident energy of the X-ray beam was 10 keV. 200 mL of the initial suspension were placed in a stirred reactor, the sample was pumped continuously through a quartz capillary of 1.5 mm. The injection of 40 ml of the ammonia was achieved via remote control of two syringes. The average gyration radius and fractal dimension was computed using the Beaucage model [START_REF] Beaucage | Structural studies of complex systems using small-angle scattering: a 520 unified guinier/power-law approach[END_REF].

Numerical methods

Three numerical approaches were used to model the colloidal agglomeration process and the morphology of the agglomerates. Concerning the ag-glomeration process, the two methods were Brown-125 ian dynamics and the population balance equation. The first enabled to simulate fast agglomeration kinetics, which cannot be observed experimentally, giving insight into the evolution of the size and fractal dimension of agglomerates in a system under 130 Brownian conditions and according to the DLVO interaction potential. The drawback was, however, the computation time, which became very high for the simulation of large systems. The second, the population balance, enabled to simulate the evo-135 lution of a system consisting of a large number of particles. The third approach allowed to build the morphology of colloidal agglomerates using stochastic packings of elementary objects.

Lagrangian Brownian dynamics 140

The Lagrangian model was described in a previous work [START_REF] Ferri | Aggregation morphological model with variable compactness: application to colloidal system[END_REF]. The initial system was composed by 2048 identical spherical particles of radius a =40 nm, with homogeneous surface charge within a cube of length 3800 nm. A random sequential addition algorithm was used to generate the initial positions of the spheres using particle a concentration of 1%. The dynamics of each sphere i was computed via the momentum balance

m i • a i = i =j F ij + F R i + F D i , (3)
where m i and a i are the mass and the acceleration of the particle. F ij is the DLVO interaction force deriving from DLVO potential [START_REF] Verwey | Theory of the stability of lyophobic colloids: the interaction of sol particles having an electric double layer[END_REF], the model accounts for the interaction between the particle i and all the other particles j contained within an 145 interaction sphere of radius 6a. Indeed, the interaction forces between particles at a larger distance are negligible. This leads to a significant reduction in calculation time. F R i is the random Langevin force (accounting for Brownian motion)

F R i = 2 • k B • T • W ( 4
)
where W is a random vector whose components are 150 independent Gaussian random numbers with zero mean and unit variance. F D i is the viscous drag force (Stoke's law)

F D i = -6 • π • µ • a • v i (5)
which depends on the viscosity of the medium µ, on the velocity of the particle v i and on the particle radius a.

A semi-implicit finite differences method was used for numerical integration [START_REF] Volpe | Simulation of a brownian particle in an optical trap[END_REF]. In order to ensure robustness when the particles are close to each other (the DLVO potential tends to -infinity when the particles are in contact), a contact force was added when two particles were at a distance lower than 2a which was modeled as a Hertz contact force [START_REF] Cundall | A discrete numerical model for granular assemblies[END_REF]. In order to approach the behaviour of an infinite medium, periodic boundary conditions were implemented [START_REF] Yang | Periodic boundary conditions for discrete element method simulation of 530 particle flow in cylindrical vessels[END_REF]. Identical spheres of 40 nm radius 165 were considered within a cubic volume of 3800 nm side. The volume fraction occupied by the spheres is 1 %, leading to 2048 spheres. At time 0, a minimum interparticle distance of 1 nm is imposed. An ionic strength of 0.005 M and a surface potential of 1 mV, which is characteristic of conditions close to the point of zero charge, are considered.

Gyration radius and fractal dimension

When the positions of all the spheres at a given time instant are known, it is possible to compute the gyration radius R gyr and the fractal dimension d f of the agglomerates [START_REF] Gmachowski | Mass-radius relation for fractal aggregates of polydisperse particles[END_REF]

d f = log ξ k f log Rgyr R , ( 6
)
where ξ is the number of primary particles (spheres of 40 nm radius) that constitute the aggregate and k f is the fractal pre-factor [START_REF] Sorensen | The prefactor of fractal aggregates[END_REF] fixed at 1.2 [START_REF] Ferri | Aggregation morphological model with variable compactness: application to colloidal system[END_REF]. R gyr is given by the geometrical average distance between the spheres'centers and the agglomerate center of mass. The formula used by Filippov et al. [START_REF] Filippov | Fractal-like 540 aggregates: Relation between morphology and physical properties[END_REF] contains also the radius of the primary particle a, so that for ξ → 1, R gyr → a:

R 2 gyr = 1 ξ ξ i=1 (x i -x g ) 2 +(y i -y g ) 2 + +(z i -z g ) 2 + a 2 (7)
where the coordinates of the centers of the spheres are x i , y i and z i and the coordinates of the agglomerate center of mass are x g , y g and z g :

x g = ξ i x i ξ ; ( 8
)
y g = ξ i y i ξ ; (9)
z g = ξ i z i ξ . (10)

The population balance model

The population balance equation was used to describe the evolution of large colloidal boehmite systems in terms of size distribution and number of primary particles 10 21 . The governing equation was solved in its discrete form

dN k dt = 1 2 k-1 i=1 β agg i,k-i N i N k-i -N k ∞ i=1 β agg ik N i (11)
where

• N k is the concentration number of clusters of 180 mass ξ k , where the mass means the number of primary particles constituing the aggregate k;

• the first term represents all possible collisions leading to the formation of an aggregate of mass ξ k ;

185

• the second term represents the rate of disappearance of the aggregates of mass ξ k due to aggregation with aggregates of any mass;

• the β terms are Brownian agglomeration kernels

β i,j = 2 3 k B T ηW (ξ 1/d f i +ξ 1/d f j ) ξ -1/d f i -ξ -1/d f j ; ( 12
)
The number of primary particles in each agglomerate was related to the agglomerate fractal dimen-190 sion via the relationship in Eq.( 6).

The population balance parameters that were adjusted with DLS data are the Fuchs coefficient W and the parameters of the curve d f = f (R gyr ). For this latter, an arctangent function was used to re-195 produce the shape obtained from the Lagrangian simulation. However, the parameters of this function were adjusted with experimental data, because Lagrangian simulation is impacted by the low number of particles and by the assumption of periodic 200 conditions involving agglomerates breakage.

Before adjusting the model kernels with DLS data, it was necessary to consider the maximum radius that could be measured with DLS as well as the type of weighting to compute the average agglomerate radius. Indeed it was necessary to "cut" the population distribution to a maximum agglomerate radius R lim gyr of 1.5 µm. This entailed a limit mass number of

ξ lim = k f • R lim gyr a d f (13)
where k f is the fractal pre-factor (assumed equal to 1.2 [START_REF] Ferri | Mass fractal dimension from 545 2d-microscopy images via an aggregation model with variable compactness[END_REF]) and a is the radius of the primary particle. The cut population is composed by all the agglomerates that have

ξ ≤ ξ lim . ( 14
)
Afterwards it was necessary to estimate a weighted average radius considering that DLS measures an effective diffusion coefficient based on how quickly the intensity of the scattered light changes. The diffusion coefficient is then related to the average radius R via the Stokes-Einstein law

D = k B • T 6π • µ • R . ( 15
)
Considering the effective scattering coefficient described by [START_REF] Sun | Investigating diffusion coefficient using dynamic light scattering technique[END_REF]

D ef f ∞ 0 R 6 i • D i • G i • dR ∞ 0 R 6 i • G i • dR , (16)
where D ef f is the effective diffusion coefficient, R i , D i and G i are the radius, the diffusion coefficient and the number distribution of particles within the class i. By substituting D i with the Stokes-Einstein law. Eq.( 16) in a discretized form becomes

D ef f n class i=1 R 6 i • D i • N p,i R 6 i • N p,i . (17)
According to the Stokes-Einstein law the average radius R is related to the effective diffusion coefficient

R 6π•µ k B •T D ef f . (18)
Substituting Eq.( 17) in Eq.( 18) leads to R

n class i=1 R 6 i • N p,i n class i=1 R 5 i • N p,i (19)
which is the average weighted radius consistent with the DLS principle. The algorithm of Levenberg-Marquardt was used to adjust W and the parameters of the d f = f (R gyr ) relationship.

Morphological model

The morphological aggregation model [START_REF] Ferri | Aggregation morphological model with variable compactness: application to colloidal system[END_REF] relies on a sequential aggregation of primary objects. It enables to tune the compactness of the final assemblies according to the probability of sticking the primary objects ether in the concave points or in concave points close to the center of mass of the assembly. In the case of identical spherical primary objects, a specific numerical scheme enables to build assemblies of several millions of primary objects.

215

In order to build the agglomerates within the size range predicted by the population balance, a resolution of 35 nm 2 /voxel was considered. A discretized agglomerates mass distribution was built from the solution of the population balance equation. The 220 agglomerates belonging to such distribution were used as primary agglomerates to build an agglomerate of agglomerates, respecting the distribution [START_REF] Pereira De Oliveira | A monte carlo modeling methodology for the simulation of hydrotreating processes[END_REF].

Results

225

Experimental results

The evolution of the average gyration radius measured with DLS i reported, for different pH, in Fig. 2(a). These results are qualitatively in agreement with DLVO theory [START_REF] Verwey | Theory of the stability of lyophobic colloids: the interaction of sol particles having an electric double layer[END_REF] since the closer the pH 230 is to the PZC, the more unstable the suspension, and therefore the more favoured is the agglomeration.

In Fig. 2(b) we find the value of Fuchs W using the theoretical formula in Eq.( 1) and the experimental 235 values computed according to Holthoff at al. [START_REF] Holthoff | Coagulation rate measurements of colloidal particles by simultaneous static and dynamic light scattering[END_REF] (details on the computation are in Appendix A).

The DLS results for pH within the range 8.8 -9.7 are similar, leading to a quite fast growth of the average hydrodynamic radius. Indeed these condi-240 tions are close to the point of zero change (PZC) of the boehmite which is at pH 9 [START_REF] Kosmulski | ph-dependent surface charging and points of zero charge. iv. update and new approach[END_REF]. At pH 10.5 and at pH 11 the agglomeration is slower but still visible within 20 minutes.

For a pH within 8.8 and 10.5, all the experimen-245 tal points show an initial rapid growth, followed by a slower one. The high variability of the mean radius can be due to several factors. First, the low boehmite concentration, necessary to observe slow kinetics, decreases the signal-to-noise ratio.

250

In addition, the necessity of maintaining Brownian conditions imposed that the suspensions were not mixed after the ammonia injection, leading to polydisperse systems. The experiments at pH 11 show different linear behaviour on Fig. 2(a), this can 255 be attributed to the fact that local pH variations have a strong influence on the agglomeration in the first few seconds and then, as the suspension evolves slowly, there is less noise. Additional information can be extracted from the 260 results of the SAXS analysis, where the scattering curves were interpreted with the Beaucage model using the method adopted by Speyer et al. [START_REF] Speyer | Peptization of boehmites with different peptization index: An electron microscopy and synchrotron small-565 angle x-ray scattering study[END_REF]. [START_REF] Verwey | Theory of the stability of lyophobic colloids: the interaction of sol particles having an electric double layer[END_REF], experimental [START_REF] Holthoff | Coagulation rate measurements of colloidal particles by simultaneous static and dynamic light scattering[END_REF] and the population balance value.

These analyses offer insight about the fractal dimension of the agglomerates, and how it changes

with gyration radius, as can be seen in Fig. 3.

These results are related to the size limits of the analysis and the approach used for the interpretation of the SAXS curves. In this regard, it is to be noted that our results refer to agglomerates whose 270 gyration radius is between 50 and 150 nm.

A quasi-linear trend can be observed in the phase diagram. This result enables us to assume that the fractal dimension changes as a function of agglomeration size. In the bottom right plot, it can be 275 observed that for pH 8 and 9.5 (near the point of Zero Charge) the fractal size has a fairly rapid ini- tial growth, followed by stabilisation. In contrast, the results at pH 10 and 11 show slower growth.

A similar behaviour to that of the DLS gyration 280 radius is thus observed for the SAXS fractal dimension. This may be due to a different agglomeration mechanism in the initial instants of the process, since for all pH values, after an initial growth phase (fast or slow) the size and fractal dimension tend to 285 stabilize.

Comparison between population balance and Brownian dynamics model

During the first instants of the agglomeration process, particles tend to form linear agglomerates, 290 before undergoing a rearrangement that increases their fractal dimension. This is illustrated by the Brownian-DLVO simulations (Fig. 4), showing the evolution of the agglomerates arrangement with time. The fractal dimension evolution with respect 295 to the average agglomerate diameter is reported in the phase diagram obtained from the Brownian-DLVO simulations in Fig. 5, in the corner it is also reported the evolution of the average fractal dimension, which increases linearly with time in the first 300 instants and stabilizes afterwards.

It can be assumed that the fractal dimension and the gyration radius respect a relationship of the type 20). The evolution of the average fractal dimension over 0.3 s is reported in the right corner.

d f = d f,1 + (d f,2 -d f,1 ) • atg a a • Rgyr-Rcut Rcut + π 2 π (
The fact that the fractal dimension evolves with the size of the agglomerate during agglomeration has also been observed with SAXS analysis as it is reported in Fig. 3. The experimental curve cannot represent the two plateaus since the initial pri-315 mary particle and the finale agglomerate are out of the measurement size limits. In addition, the non-spherical geometry of the primary particle may have an impact on the agglomerate fractal dimension, which can explain the difference between the

340

The two models are in fair agreement for W = 1.7. This value is close to 1 when the pH of the suspension is close to the Point of Zero Charge (PZC), and it is highly sensitive to small varitions of pH. Therefore the Fuchs coefficient is still in the order 345 of magnitude estimated for these conditions. It is experimentally challenging to measure a Fuchs stability ratio and small variations are tolerable if W remains within the right order of magnitude. The population balance respects the time scale of the 350 coagulation simulated via Brownian dynamics. The error between 0 and 0.05 s can be explained by the fact that the Lagrangian simulation accounts only for 2048 particles. Consequently the Brownian dynamics is less statistically representative of a real

Adjusting population balance parameters with DLS results

The population balance result is reported in Fig. 2(a). The Tab.1 reports the parameters obtained for the different DLS experiments. All the data were compared with the population balance model using a d f,1 of 1.5. d f,2 and R cut vary within a narrow range. The parameter that has the highest impact is the Fuchs stability ratio W P BE . Indeed for the data set at pH 11, for which W P BE = 31.14, the population balance result is independent of the values of all the other parameters. W P BE appears to be between the theoretical and the experimental Fuchs stabiity ratio as shown in Fig. 2(b).

The comparison between the population balance and the DLS data obtained at low boehmite concentration enables to set ranges for the parameters of the model, which make it capable to represent evolution of suspensions at higher concentration. In 375 order to approach the boehmite concentration of the manufacturing process the population balance was extrapolated at a concentration of 1 g/L and at PZC. Fig. 7(a) shows the mass cumulative distribution after an agglomeration process of 1 hour, when 380 it is assumed that the colloidal systems has reached the equilibrium size distribution.

This result has been used to adjust a morphological model of a powder obtained by drying the suspension.

385

In order to build the morphological model of a boehmite grain formed after Brownian agglomeration, the primary aggregates are considered as spheres with radius 35 nm in agreement with DLS measurement. Keeping the aim of reducing the 390 computation time, we considered a discretized cumulative distribution (Fig. 7(a)). Seven agglomerates have been built with a number of primary particles ranging from 25 to 81•10 5 in order to cover the entire distribution. The fractal dimension of 395 these agglomerates is computed with the adjusted d f (xi) relationship, and it takes a value of 1.70 for the smaller agglomerates and of 2.82 for the larger ones.

In Fig. 7(a) the red points represent the discrete 400 cumulative distribution obtained assembling 100 agglomerates. The coagulation of these agglomerates has been simulated via the morphological model [START_REF] Ferri | Aggregation morphological model with variable compactness: application to colloidal system[END_REF] where the maximum compactness parameters have been set. An illustration of the morphol-405 ogy of the final dry powder is shown in Fig. 7(b).

Conclusion

Boehmite agglomeration phenomenon is extremely chaotic and involves assembly at multiple size scales, from a few tens of nanometers to sev-410 eral microns. The use of multi-technique characterizations highlights structural properties at different scales. A strategy has been developed for gener- ating the 3D morphology of a boehmite grain after agglomeration in a colloidal suspension. Our contribution consists in describing a system that evolves according to physical-chemical parameters as a function of structural parameters like size distribution, fractal dimension and order of assembly. The final microstructure is generated with 420 an aggregation morphological model where different sticking probabilities are assigned on concave and non-concave points of a cluster. In order to rely these probabilities to physical-chemical parameters, two physical models have been used: a La- The experimental DLS results are qualitatively in agreement with the DLVO theory, indeed the closer the pH to the point of zero charge, the faster the agglomeration process is. The resistance to agglomeration has been quantified via the Fuchs stability 445 coefficient. A good agreement has been found between the experimental, the theoretical and the adjusted Fuchs coefficient of the Smoluchoswki kernel.

In order to relate the morphology of an agglomerate to the conditions of agglomeration, experimen-450 tal methods and models of agglomeration dynamics are often compared and inter-combined [START_REF] Lattuada | Estimation of fractal dimension in colloidal gels[END_REF][START_REF] Sorensen | Light scattering by fractal aggregates: A review[END_REF] or a stochastic agglomeration model is parameterised using experimental data [START_REF] Nakouzi | Impact of solution chemistry and particle anisotropy on the collective dynamics of oriented aggregation[END_REF]. In this paper, it is proposed to use all these approaches, and in par-455 ticular to use the population balance as a bridge between the experimental results and the morphological model of agglomeration.

This work has made it possible to highlight many prospects for improvement, like implementing a 460 Brownian model with non-spherical particles, this may have an impact on the parameters of the ξ(d f ) function. In addition, the natural perspective of this work is to consider boehmite crystallites as primary particles rather than aggregates. This would 465 enable the simulation of the textural properties like pores distribution and tortuosity [START_REF] Chaniot | Tortuosimetric operator for complex porous media characterization[END_REF]. Then, taking into account the inclusion of shear-induced agglomeration and breakage to higher concentrations, will approach the industrial operating conditions of the 470 manufacturing process of the support. In order to innovate catalytic materials, it would be useful to simulate the impregnation process on the modelled morphology, with the possibility of relating the ag-the final solid [START_REF] Catita | Mathematical modeling and magnetic resonance imaging experimental study of the impregnation step: A new tool to optimize the preparation of heterogeneous catalysis[END_REF], and use the morphological model to feed a pore network model [START_REF] Ledezma Lopez | Generation of gamma-alumina digital twins using a nitrogen porosimetry simulation[END_REF] to relate manufacturing process and textural and transport properties within a catalytic support.

Acknowledgement

DLS experiments were realized with the resources and equipment provided by IFP energies nouvelles. SAXS experiments were performed on the SWING beamline at SOLEIL Synchrotron, France. We are grateful to Thomas Bizien for assistance and to the

Figure 1 :

 1 Figure 1: Strategy to estimate textural properties of a boehmite porous solid according to the physical-chemical parameters of Brownian agglomeration.

  205

Figure 2 :

 2 Figure 2: (a)Evolution of average hydrodynamic intensity radius, measured with DLS on a Pural SB3 suspension of 0.04 g/L and different pH. Comparison between experimental results (dots) and the adjusted population balance model (lines). (b)Fuchs stability ratio: comparison between theoretical[START_REF] Verwey | Theory of the stability of lyophobic colloids: the interaction of sol particles having an electric double layer[END_REF], experimental[START_REF] Holthoff | Coagulation rate measurements of colloidal particles by simultaneous static and dynamic light scattering[END_REF] and the population balance value.

Figure 3 :

 3 Figure3: Experimental SAXS data acquired following the agglomeration process of a boehmite powder at 5 g/L which was brought to a pH 10 stanting from and initial pH of 3.

Figure 4 :

 4 Figure 4: Configuration of the agglomerates system at different time instants, simulated with Brownian dynamics model.

Figure 5 :

 5 Figure 5: Illustration of the phase diagram and of the parameters of the Eq.(20). The evolution of the average fractal dimension over 0.3 s is reported in the right corner.

dFig. 6 (

 6 Fig.6(a) shows the comparison between the evo-

Figure 6 :

 6 Figure 6: Comparison between the Lagrangian simulation and the population balance in terms of average diameter evolution (a) and final mass distribution (b) after 0.278 s.

Figure 7 :

 7 Figure 7: (a)Cumulate probability density function obtained from a population-balance simulation. The operating conditions are pH = 9.6 and 1g/L, the time simulated is of one hour. The seven agglomerates of the discrete distribution are also represented. (b)Simulated morphology of a boehmite dried powder after agglomeration within a colloidal suspension at 1 g/L ad PZC.

425grangian

  Brownian dynamics model and a population balance model. The first one provided information about the relationship between fractal dimension and agglomerate mass ξ. The second one enables to simulate a realistically large colloidal sys-430 tem. Since the relationship found with the first model can also be confirmed by SAXS analysis results, the function ξ(d f ) has been implemented in the Smoluchowski kernel of the population balance model. DLS experimental results have been used to 435 adjust the kernel parameters. The size distribution and the fractal dimension obtained from the population balance was finally used to parameterize the morphological model and to build an agglomerate of agglomerates.

  440

Table 1 :

 1 Adjusted parameters of the population balance model.

	pH	8.8	9.3	9.7 10.5	11
	d f,2	2.75 2.75 2.75 2.85	/
	R cut	600 596 607 600	/
	a a	3.96 3.97 2.69 1.8	/
	W P BE 1.32 1.31 1.17 2.50 31.14

Table A .

 A 2: Results of the estimation of the initial dimerization constant nd of the Fuchs stability ratio. The lowest acquisition time with DLS device is 20 s, we assume that at this time only 620 monomers are present. Therefore we call R 2 = R(20s). Tab.A.2 reports the results for the experiments at 0.04 g/L of Pural SB3

	pH	8.8	9.3	9.7 10.5	11
	k 11 • ξ 0 139 135 161	47	6
	W exp	1.16 1.19 1.00 3.45 26.75
	•

Appendix A. Computation of experimental 600

Fuchs coefficient

According to [START_REF] Holthoff | Coagulation rate measurements of colloidal particles by simultaneous static and dynamic light scattering[END_REF], the Fuchs stability coefficient can be computed at a time t close to 0 assuming that only dimers have formed at that instant. For dynamic light scattering the dimerisation kinetic 605 constant can be approximated

1) where

• R 1 and R 2 are the hydrodynamic radius of the monomer and dimer repsectively;

• I(q, t) is the scattered intensity at scattering 610 vector q and at time t;

• a is the radius of the primary particles;

• ξ 0 is the initial particle number concentration.

The stability ratio can be estimated by dividing the fast coagulation rate constant k f by a slow coagulation rate constant k s [START_REF] Snoswell | Colloid stability and the influence of dissolved gas[END_REF]

From the experimental data the average fitting curves shown have been calculated using the fol-615 lowing functions

• from pH 8.8 to pH 10.5

• for pH 11

k 11 has been estimated with the following hypotheses:

• Since the concentration is low we assume that the contriubution of the scattered intensity is negligible with respect to the hydrodynamic radius one dI(q,t) dt

dt | t→0 R(0) .

(A.5)