K Agbessi

N Saintier

T Palin-Luc

High cycle multiaxial fatigue crack initiation : experimental observations and microstructure simulations

Keywords: high cycle multiaxial fatigue, crack initiation, extreme value statistic

Cette tude propose une analyse des modes d'amorage de fissures en fatigue multiaxiale grande dure de vie en se basant principalement sur des observations MEB. L'tude statistique des sites prfrentiels d'amorage de fissures montre que les grains glissement multiple prsentent une forte probabilit d'amorage de fissures. L'application du critre de Dang Van l'chelle des grains grce des calculs par lments finis (lasticit cubique avec/sans plasticit cristalline) sur des microstructures synthtiques 3D semi-priodiques montre une forte htrognit de la contrainte hydrostatique et du cisaillement. L'volution de cette htrognit en fonction du type de comportement introduit dans le calcul est discute. Enfin, une mthode base sur la statistique des valeurs extrmes est propose pour dpouiller les calculs sur agrgats et est applique la contrainte quivalente associe au critre de fatigue de Dang Van. Les effets de surface libre et du modle de comportement ont t analyss.

Introduction

After several decades of research in the field of fatigue of materials and structures [START_REF] Schijve | Fatigue of structures and materials in the 20th century and the state of the art[END_REF][START_REF] Fatemi | Multiaxial fatigue : An overview and some approximation models for life estimation[END_REF], the current criteria have reached a certain maturity and allow scientists and engineers to understand many service situations efficiently. Although there is no unified approach for all real situations that may occur on a structure or for all classes of materials, current approaches have a wide spectrum of applications. In most cases, such approaches allow efficient structure design against fatigue crack initiation on metallic materials. The vast majority of methods for high cycle fatigue (HCF) life assessment are based on mechanical quantities calculated at macroscopic or mesoscopic scale. Most of these methods found in the literature are based on assumptions at the mesoscopic scale of the material's polycrystalline plasticity among which Dang Van [START_REF] Dang-Van | Criterion for high-cycle fatigue failure under multiaxial loading[END_REF] or Papadopoulos [START_REF] Papadopoulos | A new criterion of fatigue strengh for out-of-phase bending and torsion of hard metals[END_REF] approaches widely used in industry. Although different levels of damage mechanisms involved in the process of HCF crack initiation are relatively well identified (grain scale and slip bands [START_REF] Trochidis | Formation and evolution of persistent slip bands in metals[END_REF]) very few models take into account explicitly the exact nature of these mechanisms. The aims of this study is to better understand the role of microstructure in the process of HCF crack initiation under complex loading conditions so as to allow their integration into multiaxial HCF criteria sensitive to microstructure.

Experimental procedures

An OFHC (Oxygen Free High Conductivity) pure copper (99.99%) has been used in this study. The material is supplied in the form of rectangular hot rolled plate. An annealing relaxation treatment of 230 • C during 1h was performed. The grains are equiaxed with an average diameter of 35 µm. The material has a isotropic texture with a texture index of 1.05. The fatigue tests were carried out in air, at room temperature, under fully reversed loadings (tension, torsion and combined tension-torsion) at a frequency f = 20 Hz on cylindrical smooth specimens. The specimen surface was observed after 10 6 cycles without macroscopic failure under loading conditions corresponding to the median fatigue limit at 10 6 cycles (see Table 1). 3 Micro-mechanical modeling

Material mechanical behavior

In this work, the crystal plasticity constitutive model proposed by Meric and Cailletaud [START_REF] Meric | Single crystal modeling for structural calculations : Part 2 -finite element implementation[END_REF] is used for polycrystalline aggregates computations. The constitutive equations of the model are written in the framework of crystal plasticity taking into account both the isotropic and kinematic hardening on the slip systems. It is a time dependent crystal plasticity model with the assumption of small perturbations. The shear strain rate γs on the slip system s is described by a power function :

γs = |τ s -x s | -r s K n sign(τ s -x s ) with x = x if x ≥ 0 0 if x ≤ 0 (1)
where K and n are material parameters describing the viscosity. The evolution of the isotropic and kinematic hardening are non linear.

r s = R o + Q ∑ s h sr (1 -e -bυ s ) (2)
where τ o is the initial critical resolved shear stress, Q and b represent respectively the capacity and speed of isotropic hardening and h = describe the slip systems interactions. υ s is so that υs = | γs |.

x s = cα s with αs = γsdα s υs

where c/d and d describe respectively the capacity and speed of kinematic hardening. The material parameters (cf. table 2) are identified form experimental cyclic hardening curves under constant total strain rate of ε = 3.2.10 -3 s -1 using the half-life stabilized hysteresis loop. Parameters describing the viscosity and the interaction matrix between slip systems are not identified but chosen identical to those identified by Grard [START_REF] Gerard | Mesures de champs et identification de modèles de plasticité cristalline[END_REF] on pure copper. The anisotropic elasticity constants suggested in [START_REF] Meric | Single crystal modeling for structural calculations : Part 2 -finite element implementation[END_REF] have been used. Indeed, C 11 = 151 GPa ; C 12 = 121.9 GPa ; C 44 = 80.9 GPa (Voigt notation). Numerical and experimental stabilized cyclic behavior are compared on Figure 1. There is a good agreement between model and experiments especially for low strain amplitudes corresponding to HCF domain.

ε [%] Exp Nf/2 Simu Nf/2 (d) ε = ±0.8%
FIGURE 1 -Stabilized hysteresis loops numerical ("Simu") and half-life experimental ("Exp Nf/2") ones.

Strategy and finite element computations

The size of the representative element volume (REV) is defined in terms of the mechanical behaviour of a material in order to accurately reflect its macroscopic behavior. The size of the REV depends on the considered phenomenon. The definition of the REV size, in terms of it constitutive behaviour is different in terms of fatigue. The process of fatigue crack initiation is not a grain average type response. Cracks are initiated mostly in "critical" grains whose orientation, morphology and other parameters (neighbouring grain effects, free surface effects, etc.) create favourable conditions. Thus, the size of a REV in regards to fatigue can be very large, to the point that the computational tools currently available are not capable of undertaking the calculation [START_REF] Przybyla | Microstructure-sensitive modeling of high cycle fatigue[END_REF].

The methodology consists of constructing a set of numerical elementary volumes (representative in terms of the monotonic/cyclic macroscopic behavior) so as to reproduce a statistical response that is representative of the structure with respect to the phenomenon of crack initiation. Each of these elementary volumes is a microstructure sample (in terms of statistics) and will be called SEV (statistical elementary volumes). Each SEV contains 200 grains (cf. figure 2), which allows the calculations to be done in sequential mode without parallel processing. Several calculations were performed in order to have statistically representative results using 7 different semi-periodic SEV and 5 sets of random orientations (i.e. isotropic texture). Periodicity is ensured in 2 directions (X and Y ) where semi-periodic stress boundary conditions are applied. For each SVE, the macroscopic behavior is stable. Two material behaviors are considered : cubic elasticity (EL) with 30 calculations per loading conditions and cubic elasticity + crystal plasticity (PC) with 20 calculations per loading conditions. The applied loadings are those applied experimentally and corresponding to the experimental median fatigue limit of the material at 10 6 cycles (see Table 1).

FIGURE 2 -3D semi-periodic microstructure with 200 grains (about 50 grains on the free surface). About 500,000 elements and 300,000 degrees of freedom.

Results et discussions 4.1 Fatigue crack initiation

Figure 3a represent the proportion of grains according to the different modes of crack initiation for each loading conditions. In tension, we note that the cracks often initiate at the grain boundaries (about 57 % of cases). Grain boundaries seems to facilitate the localization of plastic deformation under tension loading. Strain incompatibilities at grain boundaries seem to play a significant role in the development of HCF damage in tension. In the case of torsion, where PSB grow more evenly in several grains, the proportion of crack initiation at grain boundaries is 40% (17 points less compared to tension). In this case, plastic activity inside the grain dominates the activity at the grain boundary. For all the loading cases investigated, the majority of observed cracks are initiated within the grains with a single slip system activated visible. The proportion of cracks initiated at grain boundaries is about 40%. An amount of about 10% of the grains with at least two visible activated slip systems is observed. It is clear from this analysis that the HCF crack initiation is mainly intragranular on OFHC copper studied. The percentage of crack initiation in grains with multiple slip is relatively low. However, the effect of grain boundaries (including twin boundaries) is important. Focusing only on intragranular crack initiation, we can also establish a link between the plastic activity in terms of number of activated slip systems and the modes of crack initiation in the grains. The proportion of intergranular crack initiation over the percentage of grains with single slip and multiple slip is analyzed. Let us note P ss the percentage of grains with single slip and P ms with the multiple slip. One can define the percentage of grains with intragranular crack initiation reported to the percentage of grains with PSB (plastic slip) by P is/ss = P is P ss and P im/ms = P im P ms where P is et P im are respectively the percentage of grains with intragranular crack initiation in single slip and multiple slip grains. Intragranular crack initiation occurs mostly in grains with single slip (see Figure 3b) for all the loading conditions investigated. However, non-proportional multiaxial and torsion loading conditions induce high probability of crack initiation on multiple slip grains. These grains are also vulnerable to crack initiation in the same way as grains with single slip for these non-proportional multiaxial loading conditions.

Extremes values statistic

Critical grain of the microstructure (in which cracks are likely to initiate) according to the Dang Van criterion are those that maximize the fatigue indicator parameter (FIP), noted I DV (eq. ( 4)). The Dang Van FIP is computed for each grain considering its local stress state (obtained by FE analysis) and its orientation. The planes and the directions on which the criterion is evaluated correspond to the slip systems of the grain. We propose in this section, an analysis of the extreme values of I DV .

I DV = max s=1,...,12 max t [ τ τ τ(s,t) +α DV σ H (t)] ≤ β DV (4)
where σ H (t) = 1 3 trace(σ σ σ ∼ (t)) represent the hydrostatic stress. τ τ τ(s,t) denotes the centered resolved shear stress for each slip system s, by constructing the smallest circle circumscribing the loading path. This resolved shear stress is computed using the averaged stress tensor σ σ σ ∼ in the grain. At the grain scale, significant variability was observed for the hydrostatic stress and shear, as it was recently shown in the case of 2D polycrystalline aggregates computations [START_REF] Robert | Micro-mechanical modelling of high cycle fatigue behaviour of metals under multiaxial loads[END_REF].

Generalized extreme values probability

As given a random variable X with distribution function F X (x), the overall n extreme values of the variable X can be defined as Y n = max (X 1 , X 2 , ...X n ) and the cumulative probability function

F Y n is linked to F X by F Y n (x) = [F X (x)] n .
As proposed by the Fisher-Tippet theorem, if there are two normalizing real sequences (a n ) n≥1 > 0 and (b n ) n≥1 and a non-degenerate distribution (not reduced to one point), G (x) so that

P Y n -b n a n ≤ x = [F (a n x + b n )] n -→ G (x) n →+∞ (5)
then the generalized extreme value (GEV) distribution function can be written as the following unique equation [START_REF] Jenkinson | The frequency distribution of the annual maxima (or minimum) values of meteorological elements[END_REF] :

G ξ (x) =    exp -(1 + ξ x) -1 ξ if ξ = 0, ∀x / 1 + ξ x > 0 exp (-exp (-x)) if ξ = 0 (6)
The sign of the extreme index ξ indicates the attractive domain of the extreme values (ξ = 0 for Gumbel, ξ > 0 for Frchet and ξ < 0 for Weibull). This allows us to determine the best extreme value distribution function represent the extreme values of X and not to choose a priori a given type of distribution function.

Free surface effects

The median thresholds of Dang Van criterion at the mesoscopic scale are compared in Figures 4a-b. The free surface causes a decrease of the mechanical quantities (shear and normal stress) associated to the criterion. In the case of cubic elasticity behavior only, the macroscopic criterion threshold is identical to the mesoscopic one considering the free surface only. For the grains in the volume of the microstructure, the mesoscopic criterion threshold is shifted upwards compared to the macroscopic one. This shift comes mostly from the strong variation of the hydrostatic stress. The extreme values of the volume grains gives a higher median value of I DV and in the ranges of greater hydrostatic stress compared to the free surface grains. The same results are obtained for cubic elasticity + crystal plasticity behavior. It should be noted that in this case, the median mesoscopic criterion threshold with the free surface grains is below the macroscopic one.

Material behavior effects

Whether the free surface or the volume of the aggregate, one can see on Figures 4c-d, that cubic elasticity alone leads to extreme values higher than when coupled with crystal plasticity. The elastic anisotropic has important effect compare to the crystal plasticity. Thus, taking into account the crystal plasticity constitutive law reduces shear and normal stress. This effect is also more pronounced in volume grains compared the free surface ones for the same material behavior. This seems to come from the important neighboring effect in grains located in the volume where the grains are confined. Due to higher hydrostatic stress, grains located in the volume seems to be more critical than those on the free surface. This is different from experimental observations but may be related to the definition of the free surface (for instance the number of grains in the thickness of the aggregates). Furthermore, roughness of the real surface is not considered in our calculations.

Conclusion

Different modes of high cycle fatigue microcracks initiation were observed under complex loading conditions at stress levels close to the median fatigue limit of the material at 10 6 cycles. SEM observations showed that, for the studied material, non-proportional loading conditions induce an increase of the probability of crack initiation in multiple slip grains. The multiple slip plays a considerable role in the process of crack initiation under these loading. However, in general for all loading cases studied, the probability of microcracks initiation in the single slip grains is the highest. From the finite element computations of polycrystalline aggregates, it can be conclude that the heterogeneity of the hydrostatic stress and shear is mainly due to the elastic anisotropic behavior. The small effect of plasticity is probably due to the low stress levels applied (corresponding to the median fatigue limit at 10 6 cycles) with small plastic strain amplitude. The sensitivity of the microstructure in multiaxial fatigue was explored by performing simulation of multiple statistic volume elements and showing

FIGURE 3 -

 3 FIGURE 3 -(a) Proportion of intragranular crack initiation (in grains with one and at least two visible activated slip systems) and intergranular ( at grain boundary). (b) Proportion of intragranular crack initiation with regard to the percentage of grains with single and multiple slip. Approximately 60 grains were considered for each loading conditions.

FIGURE 4 -

 4 FIGURE 4 -(a,b) Comparison of extreme values plotted with "dark blue", "red" and "light blue " corresponding respectively to the extreme values obtained by considering only the free surface, the entire microstructure and the volume. (c,d) Comparison of the extreme values obtained with cubic elasticity alone ("el") and cubic elasticity + crystal plasticity ("el+pc").

TABLE 1 -

 1 Different applied loading conditions corresponding to the median fatigue limit at 10 6 cycles.

	Loading conditions	σ a [MPa] τ a [MPa] σ a /τ a β [0 o ]
	Tension (σ a )	85	-	-	-
	Torsion (τ a )	-	58	-	-
	Tension-torsion (σ a /τ a = 2.0(0 o ))	68	34	2.0	0
	Tension-torsion (σ a /τ a = 0.5(0 o ))	24	48	0.5	0
	Tension-torsion (σ a /τ a = 2.0(45 o ))	68	34	2.0	45
	Tension-torsion (σ a /τ a = 0.5(45 o ))	24	48	0.5	45
	Tension-torsion (σ a /τ a = 2.0(90 o ))	68	34	2.0	90
	Tension-torsion (σ a /τ a = 0.5(90 o ))	24	48	0.5	90

TABLE 2

 2

				-Material parameters identified for pure OFHC copper.
	Viscosity	Isotropic hard.		Kinematic hard.		Interaction matrix
	K	n	R o [MPa] Q [MPa] b c [MPa]	d	h o h 1 h 2 h 3 h 4 h 5
	8	20	13	9	33	2000	960	1	1 0.2 90 3 2.5


Frédéric Gruy

Prof F Gruy

Origin and value of a limit size during shear aggregation revisited

Keywords: shear aggregation, limit size, aggregation kernel, fragmentation kernel

During the aggregation of fine particles in a shear flow a limit size value a L for aggregates is reached. Most researchers have related a L to the shear rate γ by means of a power law.

We examine in this paper the different ways in order to model the phenomena leading to a limit size. The main results in the field of drop-drop and bubble-particle systems are briefly reviewed to help us to propose a coherent description of phenomena occurring in particleparticle systems. Kernels for coalescence, aggregation, breakage and erosion are recalled.

An improvement of the aggregation kernel in the case of the collision between aggregates is proposed. We show that an analysis of the whole process in term of aggregationfragmentation competition will be preferred to a collision which would be less efficient between large aggregates. In this framework we present a modelling relating aggregation kernel and fragmentation kernel to a limit size value. As a consequence, the main result is the exponent value of the L a γ power law.

Introduction

Aggregation occurs in many biological, chemical and physical processes. It often concerns suspension of small particles in a liquid. Dynamics of aggregation mainly depends on the hydrodynamic conditions and on the particle size. In many practical situations, it is necessary to put the solid-liquid suspension on motion in order to homogenise or to convey it. In this case, whatever the nature (laminar or turbulent) of the flow, the role of the local shear flow in collisions becomes predominant. The collisions lead to the formation of aggregates. It has been observed that a limit size is reached for aggregates. The higher the shear rate γ , the smaller the limit size L a . The causes of the existence of a limit size value are not so clear. This can be due to two reasons : breakage or collision efficiency becoming zero beyond a critical size. Even if only partial results are available concerning the relation between the limit size and the shear rate, all researchers agree with a relation expressed as a power law. However, they propose different criteria (time, force or energy) to get it.

Another problem lies in the link between the exponent and other characteristic parameters of aggregation-fragmentation. This paper attempts to bring answers to these three questions.

It is organised as follows : after a brief presentation of the theoretical background connected with aggregation and fragmentation of solid particles, previous results in the field of other dispersed media (drop-drop system and particle-bubble system) will be reminded. Then, we will present a survey of experimental data and results of modelling for L a γ relation for solid particle system ; at last, we will propose a general expression for L a γ relation, which will be discussed.

Theoretical background concerning solid particle suspensions

Aggregation

Aggregation is the consequence of a collision between particles. The mechanism which brings particles into close proximity results from the hydrodynamics of the suspension. An aggregate is characterised by its number i of primary particles (supposed identical).

Aggregation between i-mer and j-mer may be represented by the quasi chemical equation :

i-mer + j-mer →(i+j)-mer

The corresponding reaction rate can be written as :

, d d i j agg i j i j N K N N t + = (1)
where , agg i j K is the kinetic constant, also known as kernel. , agg i j K contains two contributions :

the particle-particle collision frequency 0, , agg i j K and the aggregation efficiency , agg i j α : , 0, , ,

agg agg agg i j i j i j K K α = (2)
The particle-particle collision frequency function 0, , agg i j K is depending on the origin of the encounters between particles : Brownian motion, differential settling velocity, shear flow.

The collision efficiency, , agg i j α , depends on the different interactions between particles : physical forces and hydrodynamic resistance.

The morphology of the aggregates depends on the physicochemical and hydrodynamic conditions of their formation, as well as on their intrinsic mechanical properties. However the aggregation dynamics also depend on the morphology of the colliding particles.

Experiments have shown that aggregates have a fractal structure (see for instance [1,2]).

An aggregate containing i primary particles of radius a 1 is characterised by : its fractal dimension D f , its outer radius a i and its hydrodynamic radius a Hi ; as the structure of the aggregates is non-uniform, their volume density φ a (r) depends on the distance r from the centre of mass of the aggregate ; the average volume density is written φ a . These different characteristics are linked by the following relations [3] :

a a S i D f =       1 1 i (3)
φ a f D r S D r a f ( ) = 3 1 3       - (4)
φ a i D S a a f =       - 1 3 ( 5
)
where S is a structure factor, which is a function of D f [3].

Fragmentation of aggregates

In the aggregation processes, aggregates usually reach a maximum size. This is due to two reasons : a breakage or a collision efficiency becoming zero beyond a critical size.

Breakage

The occurrence of breakage depends on the balance between the desaggregation effects due to the action of the fluid, and the overall cohesion of the aggregate due to the interactions between primary particles. The hydrodynamic effects are of different natures depending if the aggregate is larger or smaller than the Kolmogorov microscale.

Fragmentation of i-mer into two fragments (i-j)-mer and j-mer may be represented by the quasi chemical equation :

i-mer → j-mer + (i-j)-mer
The corresponding reaction rate can be written as :

d d frag i i i N K N t = (6)
where

frag i K
is the kinetic constant, or kernel, for fragmentation.

frag i α : 0, frag frag frag i i i K K α = (7)
A topic still under discussion is relative to the size of the fragments produced by breakage.

Two cases are currently envisaged :

-the erosion of single or small groups of primary particles from the aggregate surface [4] ;

-the production of equisized fragments [START_REF] Kusters | The influence of turbulence on aggregation of small particles in agitated vessel[END_REF][START_REF] Sonntag | [END_REF].

In all cases, the breakage rate depends on the hydrodynamic conditions of the flow and on the characteristics of the aggregates : outer radius, fractal dimension, primary particle radius and cohesion force between two primary particles.

The competition between aggregation and fragmentation leads to a steady particle size distribution (PSD). The corresponding mean particle size a L depends on shear rate, according to :

1 m L a a γ - ∝ (8)
with 0 1 m < <

Zero collision efficiency

This approach was especially developed by Brakalov [7]. The collision efficiency between two spherical particles of the same size decreases with their particle size. The decrease is sharper when the particles are porous which is the case for the aggregates. Otherwise one aggregate, which results from two smaller aggregates, can be too loose to survive. Brakalov showed that it exists a limit value for the aggregate size. However, the assumption of an additional short range interaction force was necessary to interpret the experimental results.

Restructuring of aggregates

Restructuring occurs during aggregation. Restructuring, that is a dynamic process, leads to a densification of aggregate, i.e an increase of contact numbers in aggregate. Two mechanisms are possible : the rolling of primary particles into aggregates due to the motion of the fluid or the rupture into fragments followed by reaggregation. Restructuring is characterized by an increase of the fractal dimension with time. At long time, fractal dimension value reaches a plateau. Selomulya et al. [8,9] proposed an empirical law for the change of fractal dimension versus time. Their main result was that no restructuring (constant low fractal dimension) occurs at low shear, whereas very fast restructuring (constant high fractal dimension) occurs at high shear, while restructuring competes, in a complex way, with aggregation and fragmentation at intermediate shear. But intensity of restructuring depends on the primary particle size. As the aggregate limit size is measured at long term, this corresponds to the maximum fractal dimension. So the latter increases with shear rate, as observed by [9][10][11][12].

Previous results for other dispersed media

Before studying the behaviour of solid particle suspension under shear flow it is interesting to examine two related topics :

the coalescence and breakage of emulsion drops in turbulent medium -the collection of particles by large bubbles in flotation process This preliminaries will be followed by the study of aggregation-fragmentation in solid particle suspension.

Drop-drop system

Drops (with diameter d) undergo coalescence and fragmentation in an emulsion submitted to turbulence. The emulsion is characterized by the interfacial tension σ , by the density and the dynamic viscosity of the continuous phase ( ,

c c
ρ µ ) and of the dispersed phase ( ,

d d ρ µ ).
Turbulence is characterized by the turbulent dissipation rate ε and the Kolmogorov scale K λ .

Coalescence

The coalescence Kernel can be written, following Ross' equation [13] similar to equation (2), by :

0 coal coal coal K K α = (9)
with exp

coal coal contact t t α   = -     (10)
The term 0 coal K is coming from the classical work of Saffman and Turner [14] for

K d λ < or
Abrahamson [15] for the contact time and the coalescence time. The contact time depending on hydrodynamics only is given by : ( )

2 2 / 2 contact t d u d ∝ ( 11
)
where ( )

2
u λ is the mean square velocity difference between two points separated by the distance λ.

The coalescence time is expressed depending on the drop deformability :

For a deformable drop :

( ) 2 0 2 , c coal c F t f h h d µ σ ∝ ( 12
)
where F is the hydrodynamic force acting on the two colliding drops :

( )

2 2 2 d F u d d ρ ∝ (13)
( ) 0 , c f h h is an expression related to the drainage of the liquid film, of which thickness varies between the initial value 0 h and the critical or final value c h .

For a rigid drop :

2 c coal t d F µ ∝ (14)
Recently, Narsimhan [16] proposed a modelling of drops coalescence in a turbulent medium.

Drops were considered as rigid particles. He wrote the kinetic constant of coalescence as in equation ( 9).

Because repulsive and attractive forces were considered, the two-drop system presented an energy barrier which had to be overcome by turbulence to conduct to coalescence. By using the theory of stochastic processes he showed that :

( )

2 2.7 / sep k u d d = ( 16
)
Association was studied in a similar way as the trajectory analysis [17], where the drop-drop distance h obeyed an ordinary differential equation :

( )(
)

2 int / 8 / 3 ' c dh dt h d F F F πµ = - + + (17)
with ( )

2 2 / 4 d F u h d ρ π = (18)
and

( ) int 0 F h h = at 0 t = int
, ', F F F were respectively the mean turbulent (attractive) force, the fluctuating turbulent force and the interaction forces. Narsimhan considered that coalescence was instantaneous when the two drops were so close that the attractive interaction force became stronger than the repulsive one ( int 0 F < ). He deduced from (17) the expression of the mean association time 1 ass k -. So he showed that association or coalescence time became dramatically long when : int / 0.5 F F < The two forces are calculated at the distance h corresponding to the force barrier.

By using a dimensionless form of ( 17), the characteristic time 2 / c d F µ appears ; then the association time and the coalescence efficiency may be written as :

( ) int / 2 / F F ass c t d F e µ ∝ ( 19
) int int / / 2 sep F F F F coal ass sep ass c sep t k F F e e k t d F α µ γ - - = = ∝ ∝ (20)

Fragmentation

The fragmentation kernel can be written following Ross equation in [13] by :

0 frag frag frag K K α = (21)
with

0 frag i turb P K E = ( 22
)
where i P and turb E are respectively the power input and the turbulence kinetic energy at the drop scale. i P and turb E are expressed by :

3 6 i c P d π ρ ε = ( 23
)
and

3 2 11/ 3 2 / 3 turb c c E d u d ρ ρ ε ∝ ∝ ( K d λ > ) (24)
Ross (in [13]), Tavlarides [13] and Luo [18] introduced into the fragmentation efficiency the ratio of the cohesion or the surface energy coh E to the turbulence kinetic energy :

exp frag coh turb E E α   = -     (25)
with

2 coh E d σ ∝ (26)
Kostoglou et al. [19] used the Luo' formalism except that the cohesion energy was replaced by a threshold turbulence kinetic energy.

On the other hand, Sarimeseli [20] et al. proposed a rigorous modelling for drop fragmentation based on comparison between two characteristic times : the time frag t needed for fragmentation and the contact time contact t between drop and eddies. The modelling leads to the following expressions :

1 0 frag contact K t - ∝ (27)
and exp

frag frag contact t t α   ∝ -     (28) with ( ) 2 / contact t d u d ∝ (29)
and

drop surface frag c c p t d σ ρ ε ρ ε ∆ ∝ ∝ (30)
This modelling was close to Shamlou's approach [4], for whom the fragmentation efficiency might be written using the cohesion strength over fluid stress ratio : exp exp( )

drop surface frag frag contact contact p t p t α ∆   ∝ - = -   ∆   (31)
with

( ) 2 / contact c contact c p t d u d ρ ε ρ ε ∆ ∝ ∝ (32)
The same authors [18,20] proposed fragmentation kernels for drops in turbulent flow in the inertial sub-range. However, all the expressions contain the same dimensionless parameter :

the Weber number of the drop d We :

2 2 / 3 5 / 3 c c d u d d We ρ ρ ε σ σ = = (33)
max d critical We d We = for K d λ < or K d λ
> , which corresponds to the well-known empirical relation [13] for emulsion in a stirred tank : The limit size for colliding rigid drops d min obeys the relation :

0.6 32 / 0.05 a d D We - = (34)
1/ 2 d γ - ∝ K d λ < ( 35
)
As a summary, modelling of emulsion dynamics involves at once fragmentation kernel and zero collision efficiency to explain the limit drop size. The occurrence of a limit size without using fragmentation modelling is due to a coalescence efficiency including an exponential function. The characteristic time or force ratio seems the more appropriate one to represent coalescence or fragmentation.

Bubble-particle system

We consider a large bubble rising in a suspension of solid particle. This is equivalent to the motion of small particles towards the bubble. In the following d p and d b are respectively the particle diameter and the bubble diameter, u b is the rising bubble velocity. The particles move along the streamlines, go around the bubble, slide on the surface of the bubble and are captured. So the whole process, called collection, is divided into three successive steps : collision or approach, attachment and bubble-particle set evolution, i.e stability. We might define a collection or aggregation kernel, but investigators prefered to introduce quantities such as probability or efficiency. The collection efficiency contains the efficiencies for each step :

s a c E E E E = (36)
E c , E a , E s are respectively the efficiencies for collision, attachment and stability. The collision efficiency is proportional to the collision kernel 0, , agg i j K . As collision efficiency is strongly depending on hydrodynamics of rising bubble, we will stop the analogy with particle suspension in other hydrodynamic conditions. At the contrary, attachment efficiency is exactly the aggregation efficiency , agg i j α . The stability efficiency is another way to consider fragmentation in an aggregation process, as already done by Brakalov. Hence, we will focus our attention on attachment and stability efficiencies.

Dai [21] and Yoon [22] defined the contact time as the sliding time sl t of the particle on the bubble surface and compared it to an induction time i t , which is the time needed for rupture of the liquid film and the formation of the G-L-S contact line.

The induction time corresponds to the drainage of the liquid film due to macroscopic forces.

When the film thickness reaches a critical value, (short range) interaction forces can lead to a very fast rupture of the film. Simple expression for i t is available :

6 . 0 ) ( p i d A t ψ ∝ ( 37
)
where ψ is the wetting angle of the G-L-S system.

As efficiency is related to cross section in this case, attachment efficiency obeys the relation :

2 sin sin         = t a a E ϑ ϑ (38)
where t ϑ is the maximal angle measured from the vertical axis for particle capture by bubble (i.e.

+ Ζ = = ∫ ( 39
)
tan g u is the velocity along a streamline and Z is the stream function for the flow around the bubble. Hence, we may write Dai's theoretical result for the attachment efficiency as :

( ) ( ) / / n a p b coal contact E f d d t t = (40)
with ( ) ( ) ( )

2 3 / 4 sin 2 atan x f x e π - = ; coal i t t = ; ( ) ( ) / 4 contact p b b t d d u π = +
The exponent n is respectively equal to 0 for a large bubble (d b > 1mm) and to 1 for a small bubble (d b <0.1mm).

In turbulent medium, Li [23] proposed an expression similar to Ross's for drop coalescence (Eq. 11):

i contact t t a E e - with (
)

2 p b contact p b d d t u d d + ≈ + (41)
Thus, as seen in ( 40) and ( 41), the attachment efficiency, i.e. aggregation efficiency, is a function of the ratio of coalescence and contact times.

It exists another approach considering short range interaction forces (attractive and repulsive).

In most cases, the total interaction potential ( )

T V h presents a maximum ,max T V
for a separation distance value denoted max h and a primary minimum denoted a W -, a W being the adhesion work. Song [24] suggested that the potential barrier was linked to the attachment efficiency, whereas the energy gap between primary minimum and maximum was related to the stability efficiency. Then, attachment efficiency was expressed as :

,max T c V W a E e - = (42)
where c W is the kinetic energy of the particle at max h h = ; its value was coming from the analysis of the particle trajectory.

The stability efficiency was expressed as :

,max '

1 a T c W V W s E e + - = - (43)
where ' c W is the kinetic energy of the particle attached to the bubble. The adhesion work was written by means of macroscopic quantities :

2 2 (1 cos ) 4 p a d W π σ ψ = - ( 44
)
where σ is the surface tension for liquid-gas system.

However, Bloom [25] expressed the stability efficiency by means of a Bond number, which was defined as a ratio between repulsive force due to inertia in turbulent flow and attractive force due to capillarity :

Analysis of the different modelling shows a great disparity in the expressions (40)(41)(42) of the attachment efficiency. Equation ( 40), contrary to ( 41) and ( 42), takes into account the geometry of the system. Conversely ( 41) and ( 42) consider attachment as a stochastic process ; they are more convenient in turbulent medium. However ( 40) and ( 41) involve time whereas (42) involves energy. Stability efficiency uses either energy ratio [START_REF] Hounslow | International Symposium Industrial Crystallization[END_REF] or force ratio (45).

Particle-particle system

Many investigators have studied aggregation of micronic particles under shear flow, i.e in a

Couette flow or in a turbulent flow. Generally chemical conditions are such that only attractive forces act between particles, and that the size of formed aggregates is smaller than the Kolmogorov scale for turbulent aggregation. Aggregation occurs in the smallest eddies which are characterized by a shear rate expressed by : ( )

1/ 2 / γ ε ν ∝ .

Aggregation kernel

The kernel 0, , agg i j K is currently written as ( [14], [26]) :

3 0, , 4 ( ) 3 agg i j i j K a a γ = + (47)
Experimental and theoretical results are known about :

-the aggregation efficiency for collisions between primary particles

∝

),

-the characteristic time of aggregation expressed as ( )

1 B γφ -or ( ) 1 1,1 agg B α γφ -
, where φ is the solid volume fraction in suspension and B a constant. This corresponds to the time needed for the aggregate to reach the limit size L a .

For instance, aggregation efficiency for two equally sized micronic spheres is expressed as a function of the ratio of contact time and aggregation time :

3 36 A A C a π µ γ = ( 48
)
where A is the Hamaker constant. This approach is similar to this one for coalescence of rigid drops. A C can be seen as the ratio between attractive Van der Waals force and hydrodynamic resistance at surface-surface distance equal to particle radius.

But contrary to coalescence of drops, it has been theoretically shown [27] that the aggregation efficiency is better represented by a power law of A C instead of an exponential law :

1,1

agg n A C α ∝ with 0 1 n < < (49)
The aggregation efficiency for two aggregates is more difficult to estimate. ,

agg i j α is a function of 1/ 2 i i a κ , 1/
agg i i i a α κ -       (50)
Gmachowski ([3], fig. 2) indicated that 1/ 2 i i a κ was a single function of the fractal dimension which we will represent as :

( )

1.75 1/ 2 i 6.6 3 i f a D κ - - (51)
Following the procedures of Kusters [2] and Vanni [28], the permeability can be evaluated.

According to these authors,

1/ 2 i / i a κ
presents a weak dependence with the number of primary particles in aggregate :

( ) 0.19 1/ 2 i i f a f D i κ (52)
This expression will be preferred to the older one [START_REF] Sonntag | [END_REF] :

( ) ( ) ( ) 1/ 2 1 / 2 1 1/ 2 i 0.6 / Df i f i a SD a a κ - then, ( 50
) is reduced to ( ) 0.082 , 1 / f D agg i i i a a α - ∝ (53)
so for all researchers ([2], [3], [28]) ,

agg i i
α is a very weak function of aggregate size.

Kusters [2] suggested that only the flows outside and inside the aggregates determined the aggregation efficiency. However, Van der Waals forces and hydrodynamic resistance could contribute to aggregation efficiency especially at the beginning of aggregation. So by analogy with (49) Kusters proposed to use : , '

agg n i i A C α ∝ (54)
as much as the value of , agg i i α was higher than the one given by (53).

Kusters [2] mentioned that the contribution of the two opposite primary particles, each one in each colliding aggregate, to Van der Waals forces was the most important . Hence, '

A C was expressed by ([2], [7], [29]):

1 1 3 ' 36 A A i i i A a a C C a a a π µ γ = = (55)
However we think that a more rigorous approach is possible. By considering Van der Waals interactions between all the primary particles of aggregates and the hydrodynamic radius of aggregates (see Appendix 1), we obtain : , then e is equal to 0.2 for large aggregate or equal to 0.73 for small aggregate. This small value for exponent e is unable to explain the limit size reached by aggregates under shear.

( ) ( ) ( ) 2 7 5 6 2 0.15 / 1/ 1/ 1 ' 2 / 3 / f f f f f D D D D D A A f i C C D S a a - - + = (
By means of trajectory analysis Brakalov [7] calculated the collision efficiency from the following hypothesis :

-monosized, spherical and impenetrable aggregates -interaction between the two opposite primary particles (as Kusters)

-existence of interparticle short-range repulsive forces.

The total force was expressed as :

( )

8 0 (1 / ) VW F F h h = - (58)
VW F was the Van der Waals force between two primary particles. h and h 0 were respectively the distance between the two particles and a fitting parameter. By using this force law, the limit size reached by aggregates corresponded to a vanishing collision efficiency without considering breakage.

Fragmentation kernel

Fragmentation kernel contains at once the fragmentation frequency and the fragmentation efficiency. Shear rate is often chosen as fragmentation frequency. However this is amplified by a surface term if fragmentation mechanism is erosion. So fragmentation frequency can be written as :

( ) 0, 1 / r frag i i K a a γ = ( 59
)
with 0 1 r < < for breakage and 2 r = for erosion. The more evoked mechanism is breakage (Table 2).

Two kinds of expressions were proposed by investigators for fragmentation efficiency :

-exponential law : frag R i e α - = (60)
-power law : Equations ( 59) and ( 61) are consistent with the fragmentation kernel expressed as :

frag q i R α - ∝ ( 0 q > ) ( 61
( ) 64) is often used by researchers. whereas B value equals 10. Main theoretical results concerned the limit size for aggregates and the fragmentation kernel. The limit size was obtained either from comparison between aggregate cohesion and fluid motion ( [START_REF] Sonntag | [END_REF], [46][47][48]) or from competition between aggregation and fragmentation dynamics ( [START_REF] Kusters | The influence of turbulence on aggregation of small particles in agitated vessel[END_REF], [9], [50]). Two different criteria were used :

1 / p frag b i i K a a γ ∝ (64) with 1 b q = + and ( ) 2.2 3 p q Df r = - + Equation (
one was based on energy, the other one on stress or force. Investigators did not bring out reasons or proofs about their choice. Another uncertainty concerns the use of fragmentation and its kernel in order to describe a whole aggregation process under shear flow. Modelling of Shamlou [4] and Subbanna [49] rested on equation (60). Conversely, modelling of Serra [33][34][35], Spicer [36] and Lu [50] used equation ( 61) with q respectively equal to 0.75, 0.6 and 1. The same authors used equation ( 64) with p respectively equal to 1, 1 and 4.

Aggregation dynamics and final size of aggregate

Aggregation dynamics can be modelized by three ways.

i.) The first one uses equations ( 2) and ( 7) for aggregation and fragmentation kernels. A steady state has been experimentally observed and theoretically [51] showed for long time.

The relation between the m-exponent and the two kernels can be obtained by the following arguments.

Assuming the inequality , , agg agg i j i i α α , let us choose a hierarchical model to describe aggregation :

2 A 1 ↔ A 2 2 A 2 ↔ A 4 2 A j ↔ A 2j j L ≤
A j is an aggregate with j primary particles. The steady state is characterized by :

1 2 ... 0 j W W W = = = with 2 , 2 2
agg frag j j j j j j W K N K N = - +
The maximum of steady state PSD corresponds to aggregates with k primary particles :

/ 2 2 k k N N and / 2 0 k k W W = = thus, ( ) ( ) 2 / 3 1/ 3 2 , / 2, / 2 / / frag agg frag agg k k k k k k k N K K K K = as 1,0 i k i N iN kN = ∑
Hence by using equations ( 47), ( 57) and (64) :

( ) 3 / 1 / / / e Df agg frag d b p Df k K K k k γ γ - - ∝ ∝ and
( ) ( ) 1 / 3 b d Df e p L k a a γ -- -+ + ∝ ∝ (65a) then ( ) ( )( ) ( )
/ 3 1 2.2 m d q Df q e r = + - - + + (65b)
The condition expressed as / agg frag k K K ∝ can be interpreted as the comparison of aggregation time agg T to fragmentation time frag T .

Kostoglou [51] used a similarity method in order to study the steady state for the case where 0 d e = = . He deduced that :

-the steady state exists if 3 0 Df p -+ > -the standard deviation of the PSD, assumed as lognormal, is a function of Df, p, n F (fragments number after fragmentation). The PSD is not depending on γ [36].

-

( ) ( ) 1 / 3 b Df p k a γ - -+ ∝ (66)
Equations ( 65a) and ( 66 ii.) Alternately to the first modelling, the second modelling uses alternately a stability efficiency ,

s i j E + without fragmentation : , 0, , , ,
agg agg agg i j i j i j s i j K K E α + = and 0 frag i K = (67)
, s i j E + can be a step function or a continuous decreasing function of i+j-aggregate size. This approach was already used for bubble-particle systems [START_REF] Hounslow | International Symposium Industrial Crystallization[END_REF]46) and for particle-particle systems to a certain extent by Brakalov [7].

So applying Bloom's approach (equation ( 46)) in the case of shear aggregation and using equations (62) and (63) for repulsive and attractive forces one may write :

1 / rep att F F R - ∝ The limit size L a corresponds to : 1 R = (or , 0 s i j E + = )
Thus, from equations ( 8), ( 62) and ( 63) :

( ) ( )

1/ 2.2 3 m Df = - and
( ) 1/ 1 / , 1 m i j L a a s i j E e - + - + = - i j L + ≤ (68)
It should be noted that the stability efficiency as expressed in ( 46) and ( 68) is related to the fragmentation efficiency by : ,

1 frag i j s i j E α + + + (69)
This can also be in Table 2 ( [4], [49]).

iii.) The third modelling (Kusters) uses stability efficiency , s i j E + in the aggregation kernel and the fragmentation kernel (equation ( 69)).

Discussion and conclusion

One may compare the different expressions for the m-exponent established by Sonntag [START_REF] Sonntag | [END_REF],

Mills [47][48], Bache [46] and the author ( (65b) and ( 68 Equation (65b) contains several other parameters : n, r (characterizing erosion or breakage) and q. Their standard values can be taken as : n = 0.3 ; r = 1 ; q = 1 (q is in the range [0.6;1.2];see Table 1). The first one can be associated to weak forces between primary particles, i.e due to small size or small Hamaker constant. In this case, erosion or small fragments loss from the surface predominate [START_REF] Sonntag | [END_REF]. The second one corresponds to stronger interaction between primary particles and to larger aggregates. The different dynamics of emulsion and particle suspension are probably due to the higher sensitivity with the size of collision efficiency and fragmentation efficiency in the case of emulsion.

We described two ways in order to modelize the whole aggregation process, i.e. either by using an aggregation kernel (2) and a fragmentation kernel (7) or by introducing a stability efficiency (67) into the aggregation kernel (2). The two modelling contain the same ingredients. However the L a γ dependences are different. The analysis of the behaviour of other dispersed media and the agreement of the modelling of particles aggregation with experiments show that the best representation involves fragmentation.

f f f f f = + + + ( ) ( ) 2 2 2 1 1 2 1/ 2 f r r h - = -- ( ) ( ) 2 2 2 2 1 2 1/ 2 f r r h - = - -+ ( ) ( ) 3 2 2 2 3 1 1 2 2 / 3 f r r r h - = - -- ( ) ( ) 3
) ( )( ) ( ) ( ) 2 6 5 2 6 1/ 1 1 , , / 3 / f f f D D D VW i f VW i f i F a a D F a D S a a - - 1.6 3 f D < < (A2) ( ) VW i
F a corresponds to the Van der Waals force between non porous spheres with the same outer diameter i a .

The hydrodynamic resistance for fractal aggregates has been studied by Vanni [28] and Gmachowski [3]. It is expressed by using a corrective drag coefficient i Ω , which is a function of D f and a i /a 1 . By using Neale and Veerapaneni's work, Vanni [28] shows that a good approximation for

Ω i (D f > 2) is : 2 2 tanh 2 (1 ) tanh 2 3(1 ) i β β β β β β - Ω = + - with 1/ 2 i i a β κ = (A3)
where i κ is the aggregate permeability at the aggregate surface.

It is not possible to find such a simple expression for i Ω for D f < 2 ; Some authors assume that i Ω only depends on the fractal dimension. Thus, Gmachowski, [3], from different considerations, suggests the following expression :

1/ Df i S Ω = (A4)
The difference between the two approaches is a weak dependence on a i /a 1 for the expression of Vanni. From the Vanni's work (figure 7), one can approximate the corrective drag coefficient by the simple expression :

( )

0.15 / 1/ 1/ 1 2 / f f f D D D i i S a a - Ω 3 1 10 / 10 i a a < < (A5) Hence, ( ) ( ) ( ) ( ) (
) ( ) ( )

2 7 5 6 2 0.15/ 1 1 1/ 1/ 1 , , , , ' 2 / 3 / 6 f f f f f D D D VW i f VW i f D D A A A f i i i i VW i i F a a D F a a D C C C D S a a a a F a πµ γ - - + = = = Ω Ω (A6)

  efficiency coal α contains two characteristic times :

  collision governed only by the fluid motion leading to the formation of a doublet ; then the doublet might separate with a kinetic constant sep k thanks to turbulence, or associate with a kinetic constant ass k due to both turbulence and interaction forces. The association was followed by a very fast coalescence. So the coalescence efficiency could be

  are respectively the Sauter diameter of the drop, the impeller diameter and the stirrer speed.

F

  σ were respectively the shear stress and the cohesion strength. They obeyed the relations[4] : was the adhesion force between primary particles in aggregate. Equation (60) is similar to(31) for drop fragmentation.

  ) are equivalent for 0 d e = = . By comparing the calculations with the experiments Kostoglou et al. found out 1.5 p = .

  )). According to all investigators, as the fractal dimension increases in the range [0;2.6], m increases in the range [0;1]. Expression of Mills and (68) contain only one parameter : the fractal dimension. On the contrary, Sonntag's and Bache's expressions contain the exponents r 1 or r 2 which appear in φ-dependence of aggregate mechanical properties. r 1 and r 2 are close linked by Mills' equation corresponds to 2 2 r = . High (=5) or low (=1) value of 1 r corresponds respectively to a strong or a weak sensitivity of mechanical property with the solids volume fraction. As the solids volume fraction has the smallest value at the surface of aggregates, the strong sensitivity corresponds to an important weakness at the surface, i.e leads to an erosion or to small fragments loss from the surface. Fresh prepared aggregates are characterized by intermediate values of r 1 ( , whereas aged aggregates have higher values of r 1 ( 1 4 r > ). The former are formed during a fragmentation-aggregation process whereas the latter are made by restructuring of the former.

Figure 1

 1 Figure 1 represents the curves of m-exponent versus the fractal dimension from Sonntag

Figure 2

 2 Figure 2 represents the curves of m-exponent versus the fractal dimension from Sonntag ( 1 4.5 r = ), equation (65b) for small aggregates ( 2 r = ) and equation (65b) for large aggregates ( 2 r = ). For shear aggregation ( 2.4 f D = ), m is found in the range [0.3-0.34].

(

FIGURESFigure 1 :

 1 FIGURES

Figure 2 :

 2 Figure 2 : m-exponent versus fractal dimension from different modelling Case 2 : high values for r 1 and r.

  in φ φ φ φ-dependence of volumic cohesive energy * agg X is differently defined in the Bache's paper.

Figure 1 0

 1 Figure 1

  1

			a E E = = ) and a s ϑ with a ϑ ϑ t < is the actual angle considering attachment phenomenon.
	a ϑ is such that the sliding time between a ϑ and t ϑ equals the induction time :
	t	i	t a ϑ ϑ	tan d d 2 p g u	b	d and u ϑ	tan	g	d	b	2 sin	ϑ	d dr

  and an equivalent radius. For instance, the procedure proposed by Kusters et al.[2] (shell-core approach) has been successfully applied for aggregation of polystyrene latex, alumina, titania and silica suspensions. Kusters showed that aggregation

	permeability of aggregates between equally sized aggregates was favoured ([5], fig. 5.11) ; the corresponding
	aggregation efficiency can be approximated by :
			0.43
	,	1.55	1/ 2
			i
			j a κ	j	2	and	C	A	'	=	36 π µ γ A	3 , , i j eq a	. i κ and , , i j eq a	are respectively the

Table 1

 1 gathers a few representative experimental results about characteristics of aggregation-fragmentation. Materials were either micronic polymer latex or metallic oxide particles. Experiments were carried out in a Couette cell where the flow was laminar or in a stirred tank where the flow was turbulent. Ranges of shear rate were similar except for aggregation in viscous liquid[30]. Reported values are n, Df, B and m. Expressions for fragmentation kernel are also reminded. Thus we can see that the average values of n, Df, B and m are respectively about 0.3, 2.4, 10 and 0.5. Unfortunately due to difficulties for measuring aggregate size by optical methods all experimental results were not accurate. Table 2 gathers the corresponding theoretical results. Values for n are in the range [0.1-0.2]

  Equation (65b) rests on aggregation-fragmentation dynamics, i.e the comparison of two times : collision-aggregation time and fragmentation time. Equation (68) rests on stability of a freshly formed aggregate, i.e the comparison of two stresses or forces. Thus there is no reason to get the same γ -dependence of limit size a L . Equation (65b) is coming from an It can be emphasized that the ε-dependence of the drop limit diameter in the inertial range of turbulence is not so different for the two cases :

	accurate analysis of aggregation kernel which does not appear when deriving (68). Forces involved in Equation (68) only appear in fragmentation kernel. It can be noted that Scurati et al. [53] gave a similar L a γ -law with exponent ( ) 1/ 3 m Df = -in the case of aggregates coming from the fragmentation of dry aggregates in viscous fluid. The modelling of steady state for emulsion and particle suspension can be compared. Applying the hierarchical model to emulsion and considering efficiencies expressed as exponential function one obtains : / 8 / exp 1 frag frag coal coal frag V coal contact t t T K N K C T t π -  = =     (70) where C V is the drop volume fraction in emulsion. The characteristic time coal T coming from the population balance equation is different from the time issued from individual collision coal t . However, both ( coal T , frag T ) are related to ( coal t , frag t ) by an unique way (Eq. 70). If efficiencies are very weak, then 1 frag frag coal coal t T T t . This approach would be correct only if coalescence and fragmentation could occur at the same time. However it seems that it is not the case for emulsion where depending on the initial state either coalescence or fragmentation would occur. Thus only the ratios coal cont t t or frag cont t t d ε -∝ determine the limit size. 2 / 5 for fragmentation and 5 /17 d ε -∝ for coalescence-fragmentation.

Table 1 : experimental results for aggregation-fragmentation

 1 TABLES

	L : laminar (Couette flow) T : turbulent (stirred tank)
	PS : polystyrene latex

Table 2 : theoretical results for aggregation-fragmentation

 2

LA : laminar aggregation TA : turbulent aggregation LF : laminar fragmentation TF : turbulent fragmentation β : percentage of broken links between primary particles in aggregate E : binding energy between primary particles σ: attractive force between primary particles per area unit in aggregate r1 :exponent in φ-dependence of elastic shear r2 : exponent in φ-dependence of volumic cohesive energy * agg X is differently defined in the Bache's paper.

APPENDIX

Following the procedure of Hamaker for the calculation of Van der Waals interaction potential 12 U between two equally sized porous macroscopic bodies (denoted 1 and 2), one derive :

( ) ( ) ( )