Mohamed Dkhil
email: mohamed.dkhil@femto-st.fr

Aude Bolopion

Stéphane Régnier

Member, IEEE Michaël Gauthier

Modeling and experiments of high speed magnetic micromanipulation at the air/liquid interface

I. INTRODUCTION

Smart surfaces are well studied solutions to manipulate, convey, and position objects of a few millimeters [START_REF] Georgilas | Towards an intelligent distributed conveyor[END_REF], [START_REF] Delettre | Robust control of a planar manipulator for flexible and contactless handling[END_REF], [START_REF] Baz | Distributed part differentiation in a smart surface[END_REF]. For industrial tasks such as the assembly and packaging of products automatic feeding of objects is crucial. To perform planar displacements a variety of actuation methods have been developed including electrostatic actuators [START_REF] Dao | A novel micro transportation system with fast movement of a micro container based on electrostatic actuation and a ratchet mechanism[END_REF], electromagnetic actuators [START_REF] Khan | A long stroke electromagnetic xy positioning stage for micro applications[END_REF] and air jets [START_REF] Delettre | 2-dof contactless distributed manipulation using superposition of induced air flows[END_REF]. These methods enable to manipulate the objects with no direct contact which avoids damages of fragile components such as electronic cards [START_REF] Park | Contactless magnetically levitated silicon wafer transport system[END_REF].

The miniaturization of these smart surfaces in order to manipulate objects of a few micrometers would be an innovative solution with a wide range of application domains, from microassembly to cell manipulation. Several actuation techniques are available to perform 2D displacement of micrometer size objects, such as magnetophoresis [START_REF] Pawashe | Two-dimensional autonomous microparticle manipulation strategies for magnetic microrobots in fluidic environments[END_REF], [START_REF] Hagiwara | High speed microrobot actuation in a microfluidic chip by levitated structure with riblet surface[END_REF], [START_REF] Chen | On-chip magnetic separation and encapsulation of cells in droplets[END_REF]; dielectrophoresis [START_REF] Yue | Fabrication of microstructures embedding controllable particles inside dielectrophoretic microfluidic devices[END_REF], [START_REF] Nam | Dielectrophoresis in a slanted microchannel for separation of microparticles and bacteria[END_REF], [START_REF] Kharboutly | High speed closed loop control of a dielectrophoresis-based system[END_REF]; laser beams [START_REF] Wang | Automated laser-induced cell fusion based on microwell array[END_REF], [START_REF] Chowdhury | Automated manipulation of biological cells using gripper formations controlled by optical tweezers[END_REF] or acoustic waves [START_REF] Lei | Acoustic streaming in the transducer plane in ultrasonic particle manipulation devices[END_REF], [START_REF] Foresti | Acoustophoretic contactless transport and handling of matter[END_REF]. However, one of the greatest challenge consists in providing an ultra-rapid surface for transportation. Currently, the proposed systems are either in ambiant environments or in liquids. At micrometer scale adhesion forces are a major issue in ambiant environments, and the repeatability of the manipulations is not guaranteed [START_REF] Ivan | Nist micrororobotics challenge. magpier: The fastest mobile microrobots in the world[END_REF]. In a liquid medium the velocity is limited due to the viscous forces [START_REF] Gauthier | Analysis of forces for micromanipulations in dry and liquid media[END_REF].

Complementary to smart surfaces proposed in the literature, this paper proposes a planar magnetic micromanipulation principle. The micro-object is placed at the interface between two fluids, and the actuation is ensured by magnetic forces. The object is in equilibrium on the interface under the action of the surface tension and its weight and it moves on the surface with minimum friction. This approach enables to avoid adhesion issues and provides a high velocity motion. A model of the system is proposed, and the physical parameters, namely the drag coefficient and the magnetization, are identified. The repeatability and velocity are discussed and the dynamic model derived in this article can be used for closed loop control in future works.

This paper is organized as follows. In Section II, the dynamic behavior of the micro-object is modeled. The repeatability of the motion is evaluated in Section III based on experimental measurements. In Section IV, the physical parameters of the model are identified and its validity is discussed. The advantages and limitations of this approach are discussed in Section V. Conclusions and perspectives of this work are presented in Section VI.

II. DYNAMIC MODEL

Most of the magnetic actuation systems present similar designs. The modeling proposed in this paper is illustrated on a magnetic device composed of four electromagnets. The magnetic force produced induces the motion of the microobject on the air/liquid interface (see Figure 1). In this section a model of the motion of the object on a meniscus is derived. Some hypothesis will then be assumed to simplify the dynamic model and highlight the influence of the main physical parameters (drag coefficient, magnetization) on the micro-object behavior.

A. Model of the motion of a micro-object on a air/liquid interface

So that the object can be moved by magnetic fields it must be composed of a magnetic material. The microobject is then placed on the air/liquid interface. At this scale the workspaces are small (at most some millimeters). A meniscus appears on the reservoir, which must be taken into account while modeling the motion (see Figure 1). The shape of the meniscus can be concave surface depends on the material properties of the reservoir. It is defined by the function f :

z = f (x). (1)
It will be considered in the following that the motion of the object is on the (x, z) plane.

The forces applied to the micro-object are: (i) the magnetic force F mag generated by electromagnets, (ii) the weight P (iii), the resultant force of surface tension T and (iv) the drag force F v . The magnetic force F mag applied to a magnetic microobject is given by:

F mag = V.∇( M. B), (2)
where V is the volume of the object, M is its magnetization and ∇ B is the magnetic field gradient produced by the magnetic source. The fluid flow is considered as a Stokes flow. The drag force is given by:

F v = -h v, (3)
where h is the drag coefficient and v is the velocity of the object.

The weight of the micro-object P is compensated by the resultant force of the surface tension T (see Figure 2). The equilibrium condition is given by:

γCsin(φ) = mg, ( 4
)
where γ is the surface tension, C is the perimeter of the object, φ is the contact angle between the micro-object and the liquid, m is the mass of the object and g = 9.8 m.s -2 is the acceleration of the gravity. To evaluate the deformation of the meniscus due to the presence of the object the φ angle is evaluated numerically. A rectangular nickel object is considered (density: ρ = 8902 kg/m 3 , size: 100x90x25 µm 3 ) and the liquid is water (surface tension: γ = 72 10 -3 N/m).

Based on Equation (4) the φ angle is about 7.17 10 -4 rad which is negligible. The shape of the meniscus is not affected by the weight of the micro-object and Eq. ( 1) can be used despite the presence of the object. Using the second Newton's law, the motion of the microobject can be expressed as follows:

m a = P + F mag + F v + T , ( 5
)
where a is the acceleration of the micro-object. This equation is projected along (o, x) and (o, z):

               m ẍ = F magx -hv cos(α) -T sin(α), mz = F magz -P -hv sin(α) + T cos(α), z = f (x), tan(α) = d f (x) dx , (6)
where α is the angle between the vertical direction and the direction going from the center of the sphere tangent to the meniscus and the object (see Figure 1). The general behavior of the micro-object is given by Eq. ( 6). Some assumptions will now be made to highlight the influence of the physical parameters on its behavior.

B. Simplified dynamic model

In order to study the characteristics of this system a linear model is derived based on the following assumptions:

-The curvature of the meniscus is considered spherical, with a radius R c . The position x and z are related to R c and the angle α by:

   x = R c sin(α), z = R c (1 -cos(α)), (7)
-Only local displacements around the center of the workspace are considered. Thus α << 1 and the approximations sin(α) ≈ α and cos(α) ≈ 1 are satisfied. -Due to the small size of the object the inertial terms m ẍ and mz can be neglected in Eq. ( 6).

The validity of these assumptions will be discussed in Section IV.

Based on the above assumptions the dynamic model simplifies to:

h ẋ + mg R c x = F mag (x, I), ( 8
)
where I is the current in the electromagnets. To illustrate the motion of the object only one electromagnet will be considered in the following. It is considered that the magnetic force applied on the micro-object is directed along the (o, x) axis.

In order to define the expression of the magnetic force the magnetic field gradient must be determined. Experimental measurements are performed (see Figure 3). If the microobject is far from the electromagnet (further than 9 mm in this case) it experiences an almost constant magnetic field and magnetic field gradient. The magnetic field gradient ∇ B(I) and the magnetization M(I) depend only on the current I set to the electromagnet. In the following this approximation will be made since the object will remain further than 9 mm away from the electromagnet. Based on the above results the simplified dynamic model is:

h ẋ + mg R c x = V (M(I)∇B(I)) . (9)
The dynamic behavior of the micro-object is given by a first order differential equation whose solution depends on the shape of the meniscus. The magnetic force attracts the object toward the electromagnet. While it moves on the meniscus the projection along the (o, x) axis of the resultant force of surface tension and the magnetic force cancel out and the object reaches an equilibrium position noted X ∞ (see Figure 4). When the magnetic force is switched off the object goes back to the center of the workspace under the action of gravity. The permanent position X ∞ obtained when the gravity and the magnetic force cancel out is expressed as follows:

X ∞ = R c (M(I)∇B(I)) ρg , (10)
where m = ρV with ρ the density of the object. The response time τ to reach the permanent position is:

τ = hR c mg . ( 11
)
The determination of these two parameters (X ∞ and τ) allow the identification of physical parameters, namely the drag coefficient h and the magnetization of the material M. They will be identified based on experimental measurements. To ensure the validity of this identification the repeatability of the system is first evaluated in the next section.

III. EVALUATION OF THE REPEATABILITY FOR HIGH SPEED MICROMANIPULATION

Several factors can affect the repeatability of the system among which external disturbances such as dust and vibrations, or the shape of the meniscus. They are discussed in the following for objects of different sizes.

A. Experimental setup

The magnetic device is composed of four in-plane electromagnets (see Figure 5). Only one electromagnet will be activated to move the micro-object. To minimize the mechanical noise the magnetic device is placed on an antivibration base (Newport SG Breadboard). Control of the current in the electromagnet is performed by a custom software that communicates with an electronic amplifier through a National Instrument card (PCI 6733). A rectangular microobject composed of electrodeposited nickel (see [START_REF] Wang | Simulation and experiments on magnetic microforces for magnetic microrobots applications[END_REF] for the fabrication process) is placed on an air/water interface (distilled water is used). An image-processing algorithm able to detect the micro-object position in real time has been developed based on thresholding techniques. Image acquisition is done using a Photon focus camera with a frame rate of about 70 fps.

B. Repeatability with a constant meniscus

To determine the impact of the environmental conditions the same experiment is performed five times. The electromagnet is powered with the same actuation signal (step of current of magnitude I = 0.47 A during 2 s). The size of the micro-object used is 100x90x25 µm 3 , and the radius of curvature of the meniscus R c is estimated to 10.5 mm from a side view of the setup. Figure 6 presents the position response of the system for each of the five trials. The microobject moves under the action of the magnetic force to reach a permanent position at around X ∞ = 300 µm and comes back to the center of the meniscus under the action of the gravity. The standard deviation of the response time σ (τ) is a few milliseconds while the one of the permanent position σ (X ∞ ) is a few micrometers. The behavior of the micro-object is thus repeatable, which offers the possibility to use the air/water interface for planar magnetic micromanipulation.

C. Impact of the meniscus variations on the repeatability

The curvature of the meniscus is highly sensitive to the filling operation. It is thus not possible to guarantee a constant radius of curvature since the reservoir has to be re-filled periodically due to the evaporation of the water. To limit the inaccuracies the following procedure has been defined: the workspace is completely filled with distilled water. The micro-object is placed on the center of the workspace. A heating element is placed 15 minutes under the workspace to evaporate partially the water and generate

D. Impact of the micro-object size on the repeatability

To evaluate the impact of the size of the object on its behavior experiments are performed with a smaller object (60x50x25 µm 3 ). The same experimental conditions than in Sec. III-B are used. Table I compares the results for both the large and the small objects. The permanent position X ∞ is similar in both cases. Indeed from Eq. ( 10) it can be seen that this parameter does not depend on the size of the micro-object. Its standard deviation does not vary significantly between large and small objects. On the contrary the response time τ is larger for smaller objects as expressed in Eq. [START_REF] Yue | Fabrication of microstructures embedding controllable particles inside dielectrophoretic microfluidic devices[END_REF]. It is also less repeatable for small micro-objects. One explanation can be that their motion is more sensitive to the mechanical vibrations because of the reduction of the inertia. Additional experiments have been performed on a 30x20x25 µm 3 size object but its behavior is not repeatable.

The reduction of the object size decreases the repeatability of the dynamic behavior. Thus the approach of the micromanipulation using air/liquid interface and magnetic actuation will be limited to objects larger than 50 µm.

Size Meniscus

τ σ (τ) X ∞ σ (X ∞ ) (µm 3 ) - (ms) (ms) (µm) (µm

IV. IDENTIFICATION AND VALIDATION OF THE MODEL

To control the position of the micro-object x, the adequate current I must be sent to the electromagnet to produce a given force F mag (see Figure 8). The analytical relations between I and F mag and between F mag and x are known (Eq. ( 2), ( 8)). However the value of several key parameters, namely the magnetization of the micro-object and the drag coefficient, must be determined to get a numerical model that could be used for closed loop control. This section presents a method for the identification of these parameters based on experimental measurements. Data from the previous section are used and additional experiments are performed to obtain measurements for several values of I.

A. Identification of the magnetic behavior

The magnetization M of the micro-object is a key parameter, which varies a lot depending on the properties of the object, such as its fabrication process. It is thus not easy to obtain its value from the literature. However it can be identified based on Eq. ( 10) and on the experimental measurements. Table II presents its value for several magnitude The identified values depend on I but remain below the saturation of the nickel material (4.7 10 5 A/m according to [START_REF] Mulyokov | Magnetic hysteretic properties of submicron grained nickel and their variations upon annealing[END_REF]). The magnetic force is non linear with respect to I and can be expressed by the following expression for the typical range of current used for this magnetic device 0.26 A < I < 0.47 A:

F mag (I) = V (16200I 2 -2900I + 95) (12)
This relation enables to derive the analytical relation between the input current I and the magnetic force applied to the micro-object (first bloc of Figure 8). In the case this model is used for closed loop control this relation can be easily inverted.

B. Identification of the dynamic behavior -drag coefficient

The dynamic behavior of the micro-object is ruled by the response time expressed by Eq. ( 11) and the static gain expressed by the ratio K = R c mg . These relations involve several physical parameters, namely R c , m and h. The radius of curvature R c can be estimated with side view images of the setup. In addition the mass of the object m can be calculated through the value of the density of the material founded in literature and the dimensions of the object measured under a microscope. The drag coefficient is however an unknown parameter in the dynamic model, which is not easily found in the literature since it varies greatly depending on the shape of the object. The value of this parameter is calculated in Table III for several values of the input current.

A predicted by Eq. ( 11) its value does not depend on the input current. The average value of the drag coefficient is h = 3.39 10

C. Validation of the model

Using the parametric approximation of the magnetic force determined in Sec. IV-A and the drag coefficient h = 3.39 10 -7 kg/s identified in Sec. IV-B the dynamic model (Eq. ( 9)) of the micro-object behavior is simulated. Figure 9 provides a comparison between the simulation and the experimental measurements of the position response for several current values. The trends of the simulated and experimental plots are similar. The differences that can be noted might come from the approximation of the magnetic force in Eq. ( 12). The assumptions assumed in Sec. II-B are indeed verified: -From side view images of the reservoir it has been verified that the shape of the meniscus is spherical (see Figure 5). -The magnetic force induces only local displacements around the center of the workspace. From the experimental measurements it has been estimated that the displacement to reach the permanent position X ∞ is around 35 times smaller than the value of the radius of curvature R c . -The plots of the position measurements (see Figure 6 for example) are typical of a first order system, for which the velocity is not continuous. They confirm that the inertia of the object can be neglected.

V. DISCUSSION

This approach of micromanipulation is limited to planar manipulation. In addition, the repeatability of the dynamic behavior is not ensured for objects smaller than 50 µm, most probably due to mechanical vibrations and possible dust particles that can be present on the liquid. In future works the source of this limitation should be identified and this issue should be addressed to generalize this approach to smaller micro-objects.

On the other hand, compared to current approaches in ambiant environments, the proposed magnetic micromanipulation principle avoids the issues related to adhesion forces. To compare the velocity that can be reached on the air/liquid interface to the one in liquid a numerical simulation has been performed using Comsol Multiphysics. The viscous coefficient of the micro-object of size 100x90x25 µm 3 immersed inside water is equal to 2.210 -6 kg/s. This value is about 10 times larger than the identified value h which means (see Eq. ( 9)) that the velocity of the micro-object on the air/water interface should be 10 times larger than inside water. The approach proposed in this paper is a promising solution for high speed conveying of micro-objects.

VI. CONCLUSION

This paper demonstrates the ability to move a microobject on a air/liquid interface using magnetic actuation. A model of the dynamic behavior of the micro-objects on the meniscus has been derived, and the physical parameters that rule the behavior of the objects (their magnetization and their drag coefficient) have been identified. Experimental measurements demonstrate the repeatability of the motion for objects larger than 50 µm (the variation of the permanent position of an object of 100x90x25 µm 3 for a constant meniscus represents about 1% of its size). This good repeatability is mainly due to the absence of adhesion forces. In addition the velocity of the micro-object at the air/water interface can be 10 times higher than in water. This work represents a promising solution for the design of magnetic device for high throughput conveying of micro-objects. However, if a precise positioning is required, a closed loop control must be implemented due to the sensitivity of the system to the radius of curvature of the meniscus.

This work is only the first step towards high speed/precise planar magnetic micromanipulation at microscale. In future works closed loop control will be implemented based on the model derived in this paper. The velocity of the micro-object will be increased by decreasing the distance between the electromagnets and the workspace, which will also require a more accurate modeling of the magnetic force that could not be considered as constant anymore.

Fig. 1 .

 1 Fig.1. Example of a planar magnetic device. The electromagnets control the motion of the micro-object on the air/liquid interface. A meniscus of radius of curvature R c is formed at the interface. α is the angle between the micro-object and the vertical direction. The magnetic force F mag , the weight P, the resultant force of surface tension T of the air/liquid interface and the drag force F v are applied to the micro-object.

Fig. 2 .

 2 Fig.2. Equilibrium of the magnetic micro-object on the meniscus. The resultant force of surface tension T compensates the weight P. It can be shown that the contact angle φ between the micro-object and the liquid is negligible, therefore, the shape of the meniscus in the reservoir is not deformed by the weight of the micro-object.

Fig. 3 .

 3 Fig.3. Magnetic field gradient produced by one electromagnet along the (o, x) axis. Measurements are made using a teslameter (F.W Bell, Model 7010) based on the use of an Hall probe. The magnetic field gradient is considered constant further than 9 mm away from the electromagnet. Thus, it is constant along the trajectory of the micro-object.

Fig. 4 .

 4 Fig. 4. Trajectory of the micro-object on the air/water interface. Only the first phase, when the magnetic force is activated, is shown here. The electromagnet is situated at the bottom of the images.

Fig. 5 .

 5 Fig. 5. Experimental setup. The inlet represents the cylindrical workspace used (diameter: 4 mm). The meniscus formed is spherical. Its radius is measured on the image (R c = 10.5 mm).

Fig. 6 .

 6 Fig. 6. Position response measured for five different trials in the same conditions. The object size is 100x90x25 µm 3 , the radius of curvature of the meniscus is 10.5 mm and a current I = 0.47 A is applied for 2 s. The average values of the permanent position and the response time are respectively X ∞ = 300 µm and τ = 187 ms and the standard deviations are σ (X∞) = 1 µm and σ (τ) = 6 ms.

Fig. 7 .

 7 Fig. 7. Position response for five different trials. The workspace is re-filled between each trial, thus the shape of the meniscus is slightly different for each trial. The object size is 100x90x25 µm 3 , a current I = 0.47 A is applied for 2 s. The average values of the permanent position and the response time are respectively X ∞ = 283 µm and τ = 174 ms and the standard deviations are σ (X∞) = 12 µm and σ (τ) = 3 ms.

Fig. 8 .

 8 Fig. 8. Relation between the input of the system (current I set to the electromagnet) and its output (position x of the micro-object).

Fig. 9 .

 9 Fig. 9. Comparison between the simulated and the experimental responses. The numerical values used in the simulation are the following: R c = 10, 5 mm, ρ = 8902 kg/m 3 .

TABLE I IMPACT

 I OF THE SIZE OF THE OBJECT ON THE REPEATABILITY. I = 0.47 A, R c = 10.5 mm, Constant SIGNIFIES "REPEATABILITY WITH CONSTANT MENISCUS", Variable SIGNIFIES "IMPACT OF THE MENISCUS VARIATIONS ON THE REPEATABILITY".

TABLE II VALUES

 II OF THE PERMANENT POSITION X ∞ , THE MAGNETIC FIELD GRADIENT ∇B moy IN THE WORKSPACE AND THE MAGNETIZATION M FOR DIFFERENT VALUES OF THE CURRENT. R c = 10.5 mm, ρ Nickel = 8902 kg/m 3 .

	I(A) X ∞ (µm) ∇B moy (T/m) M(A/m)
	0.26	52	0.26	1661
	0.32	98	0.33	2467
	0.40	157	0.39	3344
	0.47	283	0.48	4898
	of current.

  -7 kg/s. It can be noted that this coefficient does not depend on the velocity of the micro-object. The computation of the Reynolds number R e indeed confirms the assumption of Stokes flow since R e < 1. The identified value

TABLE III EXPERIMENTAL

 III MEASUREMENTS OF THE RESPONSE TIME τ FOR DIFFERENT INPUT CURRENT I. THE DRAG COEFFICIENT h AND THE REYNOLDS NUMBER R e ARE COMPUTED BASED ON THESE MEASURES FOR R c = 10.5 mm, ρ Nickel = 8902 kg/m 3 .

	I(A) τ(ms)	h(kg/s)	R e
	0.26	194	3.63 10 -7 0.15
	0.32	178	3.32 10 -7 0.17
	0.40	181	3.37 10 -7 0.16
	0.47	174	3.24 10 -7 0.17
	of the drag coefficient gives access to the numerical model
	of the dynamic behavior of the micro-object.

Acknowledgements

This work has been supported by the French Agence Nationale de la Recherche, through the LEMA project (contract "ANR 12 BS03 007 01"), by the Labex ACTION project (contract "ANR-11-LABX-01-01"), by the "Région Franchecomté" and by the French RENATECH network and its FEMTO-ST technological facility.


Julie Simeon

Florent Lebreton

Laure Ramond

Florian La Lumia

N Clavier

Guillaume Bernard-Granger
email: guillaume.bernard-granger@cea.fr

Sintering of a UO 2 -PuO 2 freeze-granulated powder under reducing conditions

Keywords: Sintering, oxides, microstructure, MOX

A freeze-granulated powder made of UO 2 and PuO 2 , containing 15 mol% of Pu/(U+Pu), was sintered under reducing conditions (oxygen potential of -468 kJ/mol at 1700 °C). Constructing trajectory, using the constant rates of heating and the master sintering curves approaches and calculating the diffusion coefficients by exploiting the results of the sintering runs enabled to propose that densification was probably controlled by grain boundary diffusion and grain growth by the grain boundaries. An activation energy around 525 kJ/mol was obtained for densification, which was close to what was reported for grain boundary diffusion of plutonium cations in U 0.55 Pu 0 . 45 O 2-x polycrystalline materials. The sintered microstructure appeared homogeneous regarding the plutonium and uranium cations spatial distribution. By combining the master sintering curve approach for anisothermal and isothermal conditions, it was possible to predict the evolution of the relative density over time for any type of thermal path.

Introduction

Mixed OXides (MOX, UO 2 -PuO 2 ) nuclear fuels are used in light water reactors and are also candidates for fast neutron ones. The advantage of such fuels is that it allows the use of plutonium oxide recovered from the recycling of spent UOX (Uranium OXide) fuels, as well as depleted uranium oxide resulting from the UOX fuel enrichment protocol.

The MOX fuel takes usually the shape of cylindrical pellets (typically 10 mm of diameter and height) that are industrially manufactured using a powder metallurgy dry-route process. In a first step, the UO 2 and PuO 2 raw powders are dry ground/milled together. Then the pellets are shaped using automatic uniaxial pressing and sintered at high temperature under a specific atmosphere by controlling the oxygen partial pressure. Depending on the application, the sintered pellets may also encounter an additional rectification step of their lateral surface by dry centerless-grinding. Then, they are introduced into tubes made from an appropriate material, referred to as the cladding. Theses tubes are sealed and loaded onto an assembly, with specific configurations, that will be positioned in the core of a given nuclear reactor.

The Pu content incorporated in a MOX fuel formulation depends on the type of reactor where it will be burned. For light water nuclear reactors, the Pu/(U+Pu) ratio typically ranges between 3 and 11 mol%. For fast neutron nuclear reactors it is not clearly specified up to now, but it should range between 15 and 35 mol% [START_REF] Noyau | Influence of the oxygen potential on the sintering of UO 2 -45% PuO 2[END_REF][START_REF] Berzati | Controlling the oxygen potential to improve the densification and the solid solution formation of uranium plutonium mixed oxides[END_REF]. If the oxygen over metal ratio, named O/M, is 2.00 for MOX fuels devoted to light water nuclear reactors, it should be strictly comprised between 2.00 and 1.94 for the ones dedicated to fast neutron nuclear reactors mainly to avoid pellets/cladding interaction and to be able to control the thermal conductivity of the fuel at high temperature. Therefore, MOX fuels dedicated to fast neutron nuclear reactors are sintered under more reducing conditions than those for light water nuclear reactors.

Sintering at high temperatures of MOX fuels has been investigated in the past regarding the plutonium oxide content incorporated and in different kinds of atmospheres (oxidizing, inert and reducing) [START_REF] Noyau | Influence of the oxygen potential on the sintering of UO 2 -45% PuO 2[END_REF][START_REF] Berzati | Controlling the oxygen potential to improve the densification and the solid solution formation of uranium plutonium mixed oxides[END_REF][START_REF] Kutty | Sintering studies on UO 2 -PuO 2 pellets with varying PuO 2 content using dilatometry[END_REF][START_REF] Kutty | Densification behaviour of UO 2 -50%PuO 2 pellets by dilatometry[END_REF][START_REF] Nakamichi | Effect of O/M ratio on sintering behavior of (Pu 0.3 U 0.7 )O 2-x[END_REF]. Despite experiments perfectly well conducted, the results were not conclusive in identifying the mechanisms controlling densification and development of the microstructure.

In comparison to the traditional powder-metallurgy process, the freeze-granulation route has been shown to be an advantageous approach to manufacture MOX fuel pellets [START_REF] La Lumia | Dense and homogeneous MOX fuel pellets manufactured using the freeze granulation route[END_REF]. Highly flowable, dustless and easy-to-press MOX granules were elaborated using such a process [START_REF] La Lumia | Dense and homogeneous MOX fuel pellets manufactured using the freeze granulation route[END_REF]. Then, the purpose of this paper is to rigorously investigate the sintering behavior in reducing conditions of powder compacts made of such granules.

Experimental procedure

Freeze granulated powder

A UO 2 -PuO 2 powder, containing 15 mol% Pu/(U+Pu), was prepared using the freeze-granulation protocol defined by La Lumia [START_REF] La Lumia | Dense and homogeneous MOX fuel pellets manufactured using the freeze granulation route[END_REF]. The UO 2 powder used was synthesized through a dry route by reduction of UF 6 in gas phase. It contains also 8.5 wt% of the U 3 O 8 phase and has an overall O/M ratio of 2.20.

The PuO 2 powder was obtained by oxalic precipitation of plutonium (IV) nitrate and calcination in air. Its O/M ratio is 2.00. More details about both powders are available in the literature [START_REF] La Lumia | Preparation and co-dispersion of TiO2-Y2O3 suspensions through the study of their rheological and electrokinetic properties[END_REF]. Fig. 1 shows the morphology of some individual granules observed using scanning electron microscopy (SEM, Supra 55 VP, Zeiss). They have a spherical shape and are devoid of any central cavity. Laser particle size distribution measured in dry mode (Mastersizer 3000, Malvern Panalytical, Mie configuration, compressed air pressure in the venturi fixed to 0.1 bar) is also shown in Fig. 1. The most important point is that the median diameter for the number distribution ([D n (50)]) is around 100 µm and that there are no granules . The Pu/(U+Pu) content in the freeze-granulated powder was determined to be 14.4( 0.4) mol% using (temperature between 20 and 1700 °C) because the impact on the theoretical density value would be marginal.

Green compacts

Cylindrical samples (diameter of 5.4 mm and a height around 7 mm) were obtained by uniaxial pressing the freeze-granulated powder with a compaction pressure of 450 MPa (manual hydraulic press, Atlas 25, Specac). The geometric relative green density was around 57( 0.4)% for all the different compacts prepared.

It has to be emphasized that all samples underwent a debinding thermal treatment before sintering to remove the organic compounds (dispersant and binder) which were incorporated during the preparation of the freezegranulated powder [START_REF] La Lumia | Dense and homogeneous MOX fuel pellets manufactured using the freeze granulation route[END_REF]. Thus, samples were exposed at 600 °C for 2h in an Ar/4 vol% H 2 atmosphere humidified with 1200 vpm of water (oxygen partial pressure and oxygen potential set to 1.3x10 -27 bar and -450 kJ/mol at 600 °C, respectively).

Sintering

Kato proposed the following expression for the O/M ratio of U 1-y Pu y O 2 x MOX formulations taking into account the thermal and atmosphere conditions they are submitted to [START_REF] Kato | Oxygen potentials, oxygen diffusion coefficients and defect equilibria of nonstoichiometric of (U, Pu)O 2 x[END_REF]:

7/1/2020
where is, as before, the Pu/(U+Pu) molar content, is the oxygen partial pressure of the atmosphere, is the universal gas constant and the absolute temperature. The oxygen partial pressure is related to the oxygen potential by the well-known expression:

where is the unit of the oxygen partial pressure (1 bar).

The green pellets were sintered from room temperature to 1700 °C in a high temperature dilatometer (DIL402C, Setaram) under Ar/4 vol% H 2 . According to Kato, the oxygen partial pressure required to obtain a O/M ratio around 1.96 for a MOX composition containing 14.4 mol% Pu/(U+Pu), at 1700 °C, is around 4.1x10 -13 bar, which gives an oxygen potential of -468 kJ/mol.

Accordingly, the atmosphere was monitored and controlled with an oxygen etnag) in order to have a constant water concentration of 100 vpm (calculated using the Wheeler approach [START_REF] Wheeler | Thermodynamic and composition changes in UO 2 x (x < 0.005) at 1950 K[END_REF] to obtain the target oxygen partial pressure and oxygen potential at 1700 °C) during the sintering runs.

Heating rates fixed to 2, 3 and 4 °C/min were used for the sintering runs. The isothermal duration was ranging from 0 to 4h and the target temperature from 1600 to 1700 °C (for the only test completed at 1600 °C, the oxygen partial pressure and the oxygen potential were adjusted regarding equations ( 3) and ( 4) to obtain a O/M ratio around 1.96). The equipment contribution was subtracted based on a blank measurement made with an Al 2 O 3 reference sample for each experiment.

Because the mass of the samples does not change significantly between the debinded and fired states (maximum weight loss of 1% resulting from reduction of the oxide-based material), the following relation is obtained [START_REF] Bernard-Granger | Apparent activation energy for the densification of a commercially available granulated zirconia powder[END_REF][START_REF] Benameur | Sintering analysis of a fine-grained alumina magnesia spinel powder[END_REF]:

where: is defined as the sample instantaneous relative density, is the relative green density, is representative of the shrinkage anisotropy of the sample and is given by with the final diameter, the final height, the initial diameter, the initial height, the instantaneous diameter and the instantaneous height, (<0) is the sample height variation, is the linear thermal expansion coefficient, is the instantaneous absolute temperature and the absolute room temperature.

From a practical point of view, is determined from the cooling steps of the dilatometer runs. An average value of 11.6( 0.6)x10 -6 is retained. All the sintering tests carried out have shown that the parameter is very close to 1.

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The density of the sintered samples was measured using the Archimedes method with bromobenzene (dry, immersed and humid weights, three series of measurements were carried out for each sample).

Grain size measurement

For green compacts, the grain size is determined from a fracture surface observed using SEM (Supra 55 VP, Zeiss) in secondary electron mode. For sintered samples, each complete sample is embedded in a resin and a mirror longitudinal polished cross section is prepared including a final mechanical-polishing step using 250 and 40 nm colloidal silica suspensions that reveals sufficiently the grain boundaries of the polycrystal using the backscattered electron mode during SEM (Mira 3, Tescan) observations. Then, for both kinds of samples, it is easy to determine the average grain size by the intercept method (at least 300 grains are taken into account, correction factor set to 1.56) from SEM images [START_REF] Mendleson | Average grain size in polycrystalline ceramics[END_REF].

Plutonium distribution in the sintered samples

The longitudinal polished cross sections of some sintered samples were characterized using electron probe microanalysis (EPMA, SX100, Cameca).

Measurements performed at 20 kV at the U-M and Pu-M lines allowed the U-Pu distribution to be assessed. The Pu content in the mappings was estimated based on quantitative measurements along 500 µm lines obtained by applying the ZAF method and using UO 2 and PuO 2 samples as standards for U, Pu and O.

Results and discussion

Fig. 2a shows the variation of the relative density as a function of temperature for the different heating rates. Whatever the heating rate, samples start to densify around 600 °C. The lower the heating rate, the higher the relative density for a given temperature. Such a behavior is not new and was 7/1/2020 previously reported for numerous non-radioactive oxides [START_REF] Bernard-Granger | Apparent activation energy for the densification of a commercially available granulated zirconia powder[END_REF][START_REF] Benameur | Sintering analysis of a fine-grained alumina magnesia spinel powder[END_REF][START_REF] Sato | Yttria doping and sintering of submicrometer-grained -alumina[END_REF][START_REF] Wang | Estimate of the activation energies for boundary diffusion from rate-controlled sintering of pure alumina and alumina doped with zirconia or titania[END_REF]. Sato argued that the lower the heating rate, the longer the exposure time and the higher the shrinkage and relative density reached at a given temperature [START_REF] Sato | Yttria doping and sintering of submicrometer-grained -alumina[END_REF].

Fig. 2b shows how the instantaneous densification rate is changing as a function of temperature for the three heating rates retained. The higher the heating rate, the higher the instantaneous densification rate whatever the temperature. This trend was also previously reported for non-radioactive oxides [START_REF] Bernard-Granger | Apparent activation energy for the densification of a commercially available granulated zirconia powder[END_REF][START_REF] Benameur | Sintering analysis of a fine-grained alumina magnesia spinel powder[END_REF][START_REF] Bernard-Granger | Sintering of an ultra purealumina powder: I. Densification, grain growth and sintering path[END_REF]. More interesting is the fact that the curves do not exhibit the traditional bell-shape with a clear maximum densification rate. If it might seem that a maximum densification rate around 5.7x10 -5 s -1 is positioned at 1680 °C when the heating rate is fixed to 2 °C/min, no maximum of the densification rate does appear for the two other heating rate values. This type of sample requires a significant residence time at high temperature to reach high densities under the atmospheric conditions applied. The temperature rise phase alone does not allow materials with closed porosity to be obtained (Fig. 2a). Similar trends have been reported by Kutty [3] in the past and more recently by Nakamichi [START_REF] Nakamichi | Effect of O/M ratio on sintering behavior of (Pu 0.3 U 0.7 )O 2-x[END_REF].

Fig. 2c is a zoom of Fig. 2b Fig. 2d shows the variation of the absolute instantaneous densification rate in function of for the different heating rates retained. The three curves overlap meaning that the densification mechanism is independent of the heating rate used.

Fig. 3 shows examples of the green and sintered microstructures observed using SEM (sample relative density increases from left to right for each row of micrographs). When observed in two dimensions and as soon as densification proceeds, the residual pores becomes isolated, switching from a vermicular morphology to a more rounded shape. For all sintered samples, the elemental grains constituting the polycrystalline material are well faceted with an equilibrium shape.

Fig. 4 shows results from EPMA observations concerning the plutonium spatial distribution and its quantification in sintered samples having relative densities between 84.2 and 97.9%. The Pu repartition inside all pellets is homogenous, notably compared to samples manufactured by conventional powder metallurgy processes [START_REF] Gervais | O 2 x MOX pellet for Astrid reactor project[END_REF][START_REF] Oudinet | Characterization of plutonium distribution in MIMAS MOX by image analysis[END_REF]. However, small and local enrichments, with Pu contents between 40-60 mol%, are still visible. MOX fuels with a very homogeneous U/Pu distribution are thought to have a better behavior under irradiation in reactors by limiting/preventing the formation of the typical high burnup structure [START_REF] Noirot | Detailed characterizations of high-burnup structures in oxide fuels[END_REF][START_REF] Noirot | High burn-up structure in nuclear fuel: impact on fuel behavior[END_REF].

Knowing the relative density and the grain size of each sample (green and sintered ones) trajectory named the sintering path , as shown on Fig. 5. The experimental points seem to belong to a single trajectory (dashed curve, first level fitting using the Lab Fit Curve Fitting Software) that is independent of sintering temperature, soak time and heating rate. Accordingly, the grain size can be described as a monotonous function of the relative density. A multitude of sintering experimental conditions led to the same microstructure, as was previously reported for other non-radioactive oxides [START_REF] Bernard-Granger | Apparent activation energy for the densification of a commercially available granulated zirconia powder[END_REF][START_REF] Benameur | Sintering analysis of a fine-grained alumina magnesia spinel powder[END_REF][START_REF] Bernard-Granger | Sintering of an ultra purealumina powder: I. Densification, grain growth and sintering path[END_REF].

The experimental points constituting the sintering path shown on Fig. 5 were fitted with theoretical expressions linking the grain size to the relative density [START_REF] Bernard-Granger | New relationships between relative density and grain size during solid-state sintering of ceramic powders[END_REF]. The best agreement coefficient value amongst these fits, almost 1, is obtained when grain growth is controlled by the grain boundaries and densification is controlled by grain boundary diffusion (see the fit on Fig. 6a).

Nonetheless, other mechanism combinations are leading to determination coefficient values above 0.99. At this point, we must remain careful regarding the mechanisms controlling densification and grain growth.

Let us now focus on the determination of the apparent activation energy for the mechanism controlling densification. The equation for the densification rate can be separated into temperature-dependent, grain size-dependent and density-dependent quantities as follows [START_REF] Benameur | Sintering analysis of a fine-grained alumina magnesia spinel powder[END_REF][START_REF] Wang | Estimate of the activation energies for boundary diffusion from rate-controlled sintering of pure alumina and alumina doped with zirconia or titania[END_REF][START_REF] Wang | Activation energy for the sintering of two-phase alumina-zirconia ceramics[END_REF]:

where is a constant, is a function only of density, is the apparent activation energy for the mechanism controlling densification, is the universal gas constant, is the instantaneous absolute temperature, is the 7/1/2020 grain size, is the grain size exponent whose value depends on whether the densification rate is controlled by lattice diffusion (n = 3) or grain-boundary diffusion (n = 4).

The instantaneous densification rate may be also written as:

where , the heating rate, is held constant during anisothermal sintering experiments and is named . By combining relations ( 6), ( 7) and rearranging, the following expression is obtained:

A plot of the left-hand side of relation ( 8) versus would give a value for provided that the data points are taken at a constant value of and . Fig. 5 shows that each value of relative density corresponds to a single value of grain size, whatever the sintering conditions. Points for a constant value of are then generated by changing the heating rate. The measurements led to values of at different values of . This formalism is known as the CRH method (Constant Rates of Heating). Using this approach leads to an almost constant apparent activation energy of 528 7 kJ/mol for the mechanism controlling densification, as shown on Fig. 6b.

A second method to determine the apparent activation energy for the mechanism controlling densification is to construct the Master Sintering Curve (MSC). By extending the analysis of sintering beyond the confined segments described by the individual stage models, it was shown for an isotropic shrinkage (true for the freeze-granulated powder investigated here because the value of in relation ( 1) is very close to 1) and if there exists only one dominant diffusion mechanism (either volume or grain boundary diffusion), that the densification rate may be expressed as [START_REF] Hansen | Combined-stage sintering model[END_REF][START_REF] Su | Master sintering curve: a practical approach to sintering[END_REF][START_REF] Johnson | Finding and utilizing the master sintering curve[END_REF]:

where is the solid/vapor surface tension, 0 = v0 and n = 3 for volume diffusion, 0 = b b 0 with b the grain boundary thickness and n = 4 for grain boundary diffusion ( 0 is the pre-exponential factor of the diffusion coefficient), and (the grain size) are functions only of the relative density, is the temperature at which densification is effectively starting (assumed to be 1000 °C, at the end of solid solution formation) and the other parameters were described previously. Equation ( 9) is similar to Equation ( 6)

and also to densification rate expressions established in the past for a control by volume or grain boundary diffusion when sintering fine-grained ceramics [START_REF] Bernard-Granger | New relationships between relative density and grain size during solid-state sintering of ceramic powders[END_REF]. Su obtained the well-known relationship after rearranging and integrating expression (9) [START_REF] Su | Master sintering curve: a practical approach to sintering[END_REF][START_REF] Johnson | Finding and utilizing the master sintering curve[END_REF]:

Stating gives:

Because the left-hand site of Equation ( 11) is a function only of relative density, the MSC is defined as the relationship between relative density and [START_REF] Su | Master sintering curve: a practical approach to sintering[END_REF][START_REF] Johnson | Finding and utilizing the master sintering curve[END_REF]. To construct the MSC curve under anisothermal sintering 7/1/2020 conditions, we used the results from the dilatometer runs, the trapezoidal approximation and a curve-fitting procedure to determine the value of by minimizing the residual sum of squares (RSS) between the experimental and calculated values, assuming that all points must line up to form a single curve.

Such an analysis leads to the results shown in Figs. 6c and6d. The apparent activation energy for the mechanism controlling densification is then calculated to be around 510 10 kJ/mol (the uncertainty is only estimated).

Thereby, an apparent average activation energy of 525 ± 9 kJ/mol, associated with the densification mechanism, is determined from the CRH and MSC methods (average value and standard deviation calculated by taking into account the four values obtained from the CRH method and the single value calculated from the MSC analysis).

By expanding the concept of the MSC approach to isothermal sintering conditions (use of the same value of activation energy as that previously determined under anisothermal conditions), it is possible to predict the evolution of the instantaneous relative density as a function of temperature for any thermal cycle including anisothermal and isothermal steps. The results obtained are visible in Figure 7 where calculated (Calc) cycles are compared to experimental (Exp) ones. The agreement between the calculated and experimental values is very good. The use of such an approach makes it possible to design an ideal sintering cycle from the point of view of its speed (the cooling step is not taken into account because it depends on the type of furnace used) and its consumption of resources while making it possible to achieve a final relative density greater than 95% (MOX fuel criterion for the fast neutron reactors). In the future, transmission electron microscopy observations will be completed on some of the sintered samples. Particular attention will be paid to the analysis of grain boundaries (nature, defects, chemical segregation...), to the identification of a possible space charge zone in their vicinity and to the potential variation of chemical composition when moving from a grain boundary to the core of the surrounding grains.

Additional sintering runs will be also completed on freeze-granulated powders incorporating 25 and 33 mol% Pu/(U+Pu) and the results will be compared to the ones reported in this paper.

Conclusions

Sintering investigations on a UO By combining the master sintering curve approach for anisothermal and isothermal conditions, it was possible to predict the evolution of the relative density over time for any type of thermal path (heating rate, sintering soak temperature and holding time).

Figure captions

  Now let's try to specify in more detail what could be the densification mechanism involved. Noyau measured the volume and grain boundary (relative density below 69.2%) another mechanism may control densification. Such a possible change was not identified using the CRH and MSC approaches. More experimental results are then needed to conclude regarding this point.

Fig. 1 :Fig. 2 :

 12 Fig. 1: a) SEM micrograph of granules constituting the freeze-granulated powder; b) and c) SEM micrographs of the inside of an individual granule; d) and e) U and Pu EDS mapping (arbitrary color scale) of the inside of an individual granule (same area as that shown on c); f) Volume/number distributions from laser granulometer tests in dry mode on the freezegranulated powder.

Fig. 3 :

 3 Fig. 3: Microstructure of a green sample and different sintered samples. For each sintered microstructure are given in order of appearance the heating rate value, the sintering temperature, the time spend at the sintering temperature, the relative density D and the grain size G of the sintered sample.

Fig. 4 :

 4 Fig. 4: Sintering path. Plot of the grain size as a function of relative density.For each experimental point, the values given are the heating rate, the sintering temperature and the time spent at the sintering temmperature.

Fig. 5 :

 5 Fig. 5: Colored pseudo-quantified Pu mapping by EPMA on different sintered samples having a relative density ranging from 84.2 to 97.9%.For each map are given in order of appearance the heating rate value, the sintering temperature, the time spend at the sintering temperature, the relative density D and the grain size G of the sintered sample.

Fig. 6 :

 6 Fig. 6: a) Fitting of the experimental points for a scenario based on densification being controlled by grain boundary diffusion and grain growth by the grain boundaries; b) Determination of the apparent activation energy for densification using the anisothermal CRH method. Different relative density values have been fixed for the three heating rates applied during the sintering tests; c) Determination of the apparent activation energy for densification using the anisothermal MSC method Residual sum of squares (RSS), d) Determination of the apparent activation energy for densification using the anisothermal MSC method MSC curve.

Fig. 7 :

 7 Fig. 7: Extension of the concept of the MSC approach to predict the evolution of the instantaneous relative density as a function of temperature for any thermal cycle including anisothermal and isothermal steps. Calculated values are compared to experimental ones.

Fig. 8 :Fig3 4 C

 84 Fig. 8: Calculation of the grain boundary diffusion coefficient from sintering experiments. Comparison to the experimental values obtained by Noyau for a U 0.55 Pu 0.45 O 2-x polycristalline material [30]. The pre-exponential and activation energy values are indicated

Fig8

  Fig8

  in the 600-1200 °C temperature range. Whatever the heating rate, the densification rate exhibits a maximum value around 775 °C. It signifies that densification starts around 600 °C and slows-down from 775 °C. Similar results were reported by several authors investigating the sintering behavior in different kinds of atmospheres of UO 2 -20 wt% PuO 2[START_REF] Kutty | Sintering studies on UO 2 -PuO 2 pellets with varying PuO 2 content using dilatometry[END_REF], UO 2 -30 wt% PuO 2[START_REF] Manzel | Fabrication of UO 2 -Gd 2 O 3 fuel pellets[END_REF] fuels and UO 2 -30 wt% CeO 2 surrogates[START_REF] Dorr | Study of the formation of UO 2 -PuO 2 solid solution by means of UO 2 -CeO 2 simulate[END_REF]. the solid solution is formed by the interdiffusion of Pu 4+ cations into the UO 2 lattice and U 4+ cations into the PuO 2 lattice it decreases the densification rate

	until completion and then densification restarts with a polycrystalline sample
	made of U 1-y Pu y O 2-x grains. Accordingly, when densification resumes around
	1000°C, the samples are more homogeneous and can be described as a
	polycrystalline body constituted by pores and grains of U 1-y Pu y O 2-x solid
	solutions, rather than UO 2 and PuO 2 grains alone.
	According to them, the densification slow-down is correlated with the onset of
	a UO 2 -PuO 2 or UO 2 -CeO 2 solid solution formation. Even if additional
	experiments are required (X-ray diffraction experiments at high temperature

will be completed in the future), we postulate that a solid solution forms in the densifying samples around 775 °C, whatever the heating rate used. Because 7/1/2020

Acknowledgements: The authors warmly thank Gauthier Jouan, Romain Lauwerier, Jean-Robert Sevilla, Romain Vauchy and Patrice Signoret for their precious help during the experiments carried out within the framework of this study.

7/1/2020

diffusion coefficients of plutonium cations in an U 0.55 Pu 0.45 O 2-x polycrystalline material, for an oxygen potential set to -395 kJ/mol [START_REF] Noyau | plutonium dans les céramiques de type U 1-y Pu y O 2 x[END_REF]. He obtained:

where and are the volume and grain boundary diffusion coefficients.

The activation energy for the grain boundary diffusion of plutonium cations obtained by Noyau (around 506 kJ/mol), is very close to the one (525 kJ/mol) that was determined from the CRH and MSC methods for densification of the freeze-granulated powder investigated in the study reported in this paper.

Let us now calculate the grain boundary diffusion coefficient involved in densification by exploiting the sintering experiments completed on the freezegranulated powder and by considering the theoretical expression for the densification rate.

When densification is controlled by grain boundary diffusion, the densification rate in the secondary and tertiary stages of sintering is given by [START_REF] Bernard-Granger | New relationships between relative density and grain size during solid-state sintering of ceramic powders[END_REF]:

which turns to:

By using a typical value of 1 nm for and 1 J/m 2 for , the grain boundary diffusion coefficient variation in function of temperature was calculated from the sintering experiment conducted on the freeze-granulated powder with a Accordingly, for temperatures above 1525 °C (relative density above 69.2%), densification of the freeze granulated powder is most probably controlled by the grain boundary diffusion of plutonium cations. In the same time, it is also likely that grain growth is controlled by grain boundaries (best fit of theoretical expressions, result shown on Fig. 6a). For temperatures lower than 1525 °C


Giulia Braghini

Francesco Salvarani

ON THE DETECTION OF HIDDEN OPINION MANIPULATION IN MICROBLOGGING PLATFORMS

This article studies a new model for describing opinion manipulation effects on a microblogging directed network system. The network can evolve in time, by means of the creation and the deletion of some connexions. When a connexion is created, the individuals have access to the whole history of posts written by the corresponding author and, when a connexion is destroyed, all the posts written by the corresponding author become invisible. The agents' opinions are described by a set of continuous opinion variables in the closed interval Ω = [-1, 1]. They represent the agreement (or disagreement) of the corresponding agent with respect to a binary question (such as a referendum or an election with two candidates). The model takes into account the effects on public opinion caused by the sign and the intensity of the initial opinions of the agents, their activity in microblogging platforms and the possible manipulations of the visibility of the posts by the microblogging platform owner. We show that hidden manipulation can have an important impact on the public opinion formation and that very mild interventions of the network owner may induce major effects on the population. We moreover suggest a strategy for detecting whether the blogging platform is neutral or not.

Introduction

In the last years, the use of internet, social media and social networks has become very popular because of its immediacy and ease of use. For this reason, much attention has recently been paid to the power of social media and social networks in the dynamics of collective choices, especially in the political and commercial frameworks.

On the other hand, the recent COVID-19 disease has caused an explosion of web-based contents related to the pandemic. The intrinsic interest of the subject and the social distancing due to the prevention health policies have given an unexpected contribution to the online dissemination of verified or unverified contents [START_REF] Depoux | The pandemic of social media panic travels faster than the COVID-19 outbreak[END_REF][START_REF] Li | Characterizing the propagation of situational information in social media during covid-19 epidemic: A case study on weibo[END_REF].

The authors of web-based contents may have several reasons which explain their actions. Many writers simply aim to inform the audience about some facts and the author's interpretation and opinion about them, other simply forward news and opinions which seem important to the sender and some others aim to persuade the audience and drive the public opinion towards a given direction.

In many cases, the influencers are known, but sometimes they are not. The explicit strategy is the easiest to be detected: the publisher of a content explicitly declares his viewpoint and develops all the arguments which support his thesis. However, when the public opinion is driven by means of non-declared manipulation techniques, many individuals may be the object of hidden manipulation, with possible serious consequences for the collective behaviour in democratic societies.

A very well-known case is the so-called Facebook-Cambridge Analytica data scandal [START_REF] Isaak | User data privacy: Facebook, cambridge analytica, and privacy protection[END_REF], which highlighted some possible consequences of social media sharing, especially in connexion with non-apparent opinion manipulation techniques related to elections or referendums.

The goal of this article is to understand and to contrast the effect of hidden influencing through social media. We focus on microblogging and social networking services on which users may post messages and interact with other users. The sociological literature agrees that one of the main dynamics of opinion formation is consensus [START_REF] Parsons | Values, motives, and systems of action[END_REF][START_REF] Thomas | Toward a sociological model of consensus[END_REF] and, consequently, many mathematical models of social interaction, such as the well-known Hegselmann-Krause model [START_REF] Hegselmann | Opinion dynamics and bounded confidence: models, analysis and simulation[END_REF] and the Cucker-Smale model [START_REF] Cucker | Emergent behavior in flocks[END_REF] are based on consensus dynamics.

Opinion formation being mainly driven by consensus, it is clear that the neutrality of the blogging platform is crucial for democracy. Indeed, if the platform is not neutral and ranks the posted opinions in a deliberated way, then the process of opinion formation is biased.

In this article we focus on a particular technique of manipulation, which does not hide any posts, but rather ranks them in such a way that only some tendencies are highlighted (for example, by means of an ordering algorithm based on AI semantic algorithms). Note that this technique is more sophisticated than the usual strategy based on opinion manipulation bots, because it works with trusted posts of real individuals, with no doubt about the authors, which are freely chosen by the follower, and which can influence him.

This technique requires that the owner of the microblogging platform network is involved in the manipulation dynamics.

We provide a model which shows in a quantitative way the evolution of the opinions in a closed community by taking into account two main features of the agents, their post productivity and their opinions, as well as the policy of the blogging platform in visualizing the posts among the interested users. We study how the platform policy may modify the public opinion and we compare the results with the unbiased situation, in which all the opinions are treated in an equal way.

The structure of the article is the following. In Section 2, we describe our mathematical model. We discuss some mathematical questions on the model in Section 3 and then we implement the model, test it on some relevant cases, and analyse its main consequences in Section 4. Finally, we conclude our analysis by suggesting a strategy for detecting such type of hidden manipulation.

Description of the mathematical model

We consider a population composed of N ∈ N * interacting individuals, described -at the individual level -by N time-dependent functions

x i : R + → Ω = [-1, 1] i = 1, . . . , N.
The functions x i (i = 1, . . . , N ) represent the opinion of the agent labelled with the index i with respect to a binary question (which can be, for example, a referendum, an election with two candidates or the opinion with respect to a commercial product). When x i = 1, the i-th agent completely agrees with the underlying question whereas, when x i = -1, the i-th agent is in full disagreement with the underlying question. All intermediate values belonging to the open interval (0, 1) denote partial agreement with the binary question, with a conviction proportional to the magnitude of x i and, symmetrically, the intermediate values belonging to the open interval (-1, 0) denote partial disagreement with the binary question, with a conviction proportional to the absolute value of x i . When x i = 0, the agent has no preference about the binary question. We suppose that the individual opinion is publicly available under the form of posts in the blogging platform and that it evolves only through reciprocal influence. The individuals of the population are totally or partially interconnected by means of an oriented graph, represented by a set of time-dependent functions of binary type σ i,j : R + → {0, 1} for all i, j = 1, . . . , N.

If the i-th agent is following the j-th agent at time t ∈ R + , then σ i,j (t) = 1, otherwise σ i,j (t) = 0. The matrix whose entries are the quantities σ i,j will be denoted, in the whole article, as the interaction matrix.

In what follows, we suppose that the population is interconnected in such a way that, for all i = 1, . . . , N and for all t ∈ R + , there exists at least an index j = i such that σ i,j = 1 (it means that no agent is fully isolated). We suppose moreover that each agent has a total access to his own posts: σ i,i = 1, for all t ∈ R + and for all i = 1, . . . , N . It is important to underline that, like in real social media, the interaction matrix is often sparse.

We moreover denote with b i = b i (t) the number density, with respect to t, of microblogs posted by the i-th individual.

Our model aims to forecast the opinion evolution on a short-time horizon (for example the dynamics of a referendum campaign). Consequently, we can assume that there is no loss of attention about the underlying question.

The set of ordinary differential equations of our model describes the opinion evolution through a consensus dynamics, and takes into account the activities of the agents as microbloggers. Its precise form is the following:

(2.1)

               db i dt (t) = µ i (γ i -b i (t)) N j=1 σ i,j (t) t 0 b j (θ) dθ, γ i , µ i > 0 dx i dt (t) = α i (x i (t))(Φ i (t) -x i (t)),
where

(2.2) Φ i (t) =                                      N j=1 σ i,j (t) t 0 b j (θ)x j (θ)ψ i,j (θ)dθ N j=1 σ i,j (t) t 0 b j (θ)ψ i,j (θ)dθ t > 0 N j=1 σ i,j (0)b j (0)x j (0)ψ i,j (0) N j=1 σ i,j (0)b j (0)ψ i,j (0) t = 0 and (2.3) ψ i,j = ψ i,j (θ, x j (θ), b j (θ)) > 0 i = j 1 i = j.
The model is coupled with suitable initial conditions: for all i = 1, . . . , N

(2.4) b i (0) = b 0 i ∈ (0, γ i ), (2.5)
x i (0) = x 0 i ∈ Ω. We now describe each term of the model.

The equations satisfied by the functions b i (t) are of logistic type. We indeed suppose that the activity of the i-th microblogger is proportional to the number of total microblogs seen by him up to a saturation phenomenon, with saturation constant γ i > 0. We are indeed considering a short-time model, which means that we do not expect any interest decrease about the underlying question.

In our model, when σ i,j = 0, the i-th agent loses all the posts sent by the j-th agent, but he will see again all the posts of the j-th agent as soon as σ i,j = 1, as customary in many microblogging platforms.

The equations describing the time behaviour of the opinions x i (t), i = 1, . . . , N , are of consensus type. We suppose that the i-th agent modifies his own opinion through a consensus dynamics by taking into account the opinions of the individuals he is following. In (2.1), the time evolution of the functions x i is governed by the joint contribution of two terms. The functions α i : Ω → R + is somehow the analogous of the admissible functions defined in [START_REF] Boudin | A kinetic approach to the study of opinion formation[END_REF] (Definition 2.6): they may be agent-dependent and translate the idea that individuals with a stronger opinion are more stable in their convictions. In general, we suppose that all the α i are even functions (because of the symmetry under the exchange of the underlying question with its opposite) and of class W 1,1 (Ω).

The variation of the opinion for the i-th agent with respect to time, at time t, is given by the difference (Φ i -x i ), weighted by the term α i .

The functions Φ i describe a weighted average opinion of the posts seen by the i-th agent. We suppose that all the posts of the followed individuals are available and that a post of an agent at a given time strictly reflects his opinion at the same time.

We suppose that the weighted opinion deduced by the set of posts available to the i-th agent is given by the integral in time of all the posts sent by all the individuals followed by the i-th agent, weighted with suitable highlighting functions ψ i,j .

The highlighting functions describe the possible manipulation induced by the owner of the platform. These quantities are then normalized by the total number of posts, weighted by the highlighting functions. Note that the quantities σ i,j in front of this weighted average (which guarantee that only the agents followed by the i-th individual are taken into account in this average) are considered at time t. It means that, when a user follows another agent, he has a complete access to all his posts, and, when a user decide to eliminate another individual from his set of contacts, he looses the access to all his comments.

As said before, the highlighting functions ψ i,j (t) describe the manipulation effect. These terms are supposed to be under the control of the microblogging network's provider. When there is no manipulation, ψ i,j (t) = 1 for all i, j = 1, . . . , N . In the case of hidden manipulation, we have that ψ i,i (t) = 1 for all i = 1, . . . , N for preventing the individual to see any manipulation effect on his own posts. Moreover, we suppose that ψ i,j (t) ∈ (0, 1] for all t ∈ R + (hence, ψ i,j ∈ L ∞ (R + ) for all i, j = 1, . . . , N ). We underline that the highlighting functions may depend on time, on the opinion of the agents followed by the i-th agent and of the number of posts seen by him. Timing is a crucial factor in opinion formation dynamics (see [START_REF] Boudin | Conciliatory and contradictory dynamics in opinion formation[END_REF] for a discussion about this point): for this reason we allow that the manipulation strategy may vary in time. We do not allow complete censorship; consequently, we suppose ψ i,j > 0 in (2.3). A possible implementation of a manipulation technique consists in simply promoting, in the ranking of posts, those which are favourable to the thesis supported by the manipulator and by postponing in the ranking the unfavourable posts. The effectiveness of this technique is the consequence of the information overload of microblogging platforms [START_REF] Gomez Rodriguez | Quantifying information overload in social media and its impact on social contagions[END_REF]: usually not every post is read, especially when their number is high, and the reader limit himself to the first ones. As we will see in the next sections, the effect of the highlighting functions may result in a modification of the asymptotic state of the system. Note that, if ψ i,j = 1 for all i, j = 1, . . . , N , and all the α i are constant, we obtain a linear system of ODEs for the unknowns x i of Hegselmann-Krause type [START_REF] Hegselmann | Opinion dynamics and bounded confidence: models, analysis and simulation[END_REF].

Without manipulation phenomena, the evolution of the population is the consequence of several factors inside the population. The number of individuals with opinion of the same sign is, of course, important and is the goal of the majority of polls. However, it is not enough for explaining the evolution of the population's global opinion, because at least two other factors are of paramount importance: the activity in sharing their opinions -here measured by means of the individual number of posts b i -and the conviction degree of each agent, which corresponds to the absolute value of his/her opinion, |x i (t)|. The implementation of polls with multiple answers about a binary question, modulated on a scale, is hence very useful for producing accurate forecasts (see, for example, [START_REF] Boudin | Opinion dynamics: Kinetic modelling with mass media, application to the Scottish independence referendum[END_REF]).

Basic mathematical properties of the model

This section is devoted to the mathematical analysis of our model. In particular, we will discuss the existence and uniqueness of the solution, which are two key features of any good mathematical model. Our model is composed of two sets of weekly coupled unknowns: the functions representing the number of posts written by the i-th individual (denoted b i ) and the the functions representing the opinions x i .

It is possible to decouple the equations satisfied by the unknowns b i in Equation (2.1): the first family of equations in Equation (2.1), i.e., (3.1)

             db i dt (t) = F i (t, b 1 , . . . , b N (t)) := µ i (γ i -b i (t)) N j=1 σ i,j (t) t 0 b j (θ) dθ b i (0) = b 0 i ∈ (0,
γ) can be solved as a coupled system for all the b i , by applying the standard Cauchy-Lipschitz theory, all the F i being of class C ∞ ((R + ) N +1 ). We can hence deduce that there exists one and only one vector

b = (b 1 , . . . , b N ) ∈ C ∞ ((R + ) N ), solution of system (3.1) with initial data b i (0) = b 0 i . Consider the second half of system (2.1) with known vector b ∈ C ∞ ((R + ) N ): (3.2)          dx i dt (t) = α i (x i (t))(Φ i (t) -x i (t)) i = 1, . . . , N x i (0) = x 0 i ∈ Ω.
All the Φ i satisfy the following bound for all T > 0:

0 ≤ Φ i ≤ max j=1,...,N sup t∈[0,T ] |x j (t)|.
As a consequence of the binary character of the interaction matrix and the regularity hypotheses of the highlighting functions ψ i,j ∈ L ∞ (R + ), the terms Φ i -which are linear with respect to the opinion variables -cannot be more regular than L ∞ functions with respect to time. Moreover, the equations are not sufficiently regular with respect to the unknown x = (x 1 , . . . , x N ) for applying the Cauchy-Lipschitz theory because of the low regularity of the vector α = (α 1 , . . . α N ).

For this reason, we need to base our study on a more general theory (we refer to the lecture notes [START_REF] Ambrosio | Well posedness of ODE's and continuity equations with nonsmooth vector fields, and applications[END_REF] and to the references therein for a complete introduction to the theory of flows associated to non-smooth vector fields).

The Cauchy problem (3.2) has the following structure:

(3.3) ẋ(t) = η(t, x(t)) x(0) = x 0 ,
where x : [0, T ] → Ω N is the opinion vector for the whole population and η : [0, T ]×Ω N → R N is the associated vector field, which may have no Lipschitz regularity. Let L N be the N -dimensional Lebesgue measure and consider a map

X : [0, T ]×R N → R N . The fact that X(t, •)#L N ≤ LL N for all t ∈ [0, T ],
where the symbol # represents the pushforward of a measure, means that there exists L > 0 such that, for all t ∈ [0, T ] and for all

φ ∈ C 0 c (R N ) ≥ 0, R N φ(X(t, x))dx ≤ L R N φ(x)dx.
For a given vector field η, we consider as admissible solutions to the system the maps called regular Lagrangian flows (see [START_REF] Ambrosio | Well posedness of ODE's and continuity equations with nonsmooth vector fields, and applications[END_REF]):

Definition 3.1. A regular Lagrangian flow is a map X : [0, T ] × R N → R N such that: i) for L N -a.e. x ∈ R N , the function t → X(t, x
) is a solution of the ODE in the integral sense, i.e. such that:

X(t, x) = x 0 + t 0 η(s, X(s, x))ds for all t ∈ [0, T ];
ii) there exists a constant L > 0 such that X(t,

•)#L N ≤ LL N , for all t ∈ [0, T ].
The constant L is called compressibility constant. The following theorem holds [START_REF] Ambrosio | Well posedness of ODE's and continuity equations with nonsmooth vector fields, and applications[END_REF]:

Theorem 3.2. Consider η ∈ (L 1 [0, T ]; W 1,1 (R N ; R N )), such that η ∈ L ∞ ([0, T ] × R N ; R N ) and [div x η] -∈ L 1 ([0, T ]; L ∞ (R N )).
Then there exists a unique regular Lagrangian flow X associated to the field η, solution of the Cauchy problem (3.3).

By using the notation of this article, we immediately deduce:

Theorem 3.3. Consider the Cauchy problem (2.1)-(2.5) for t ∈ [0, T ], T > 0. Let σ i,j : R + → {0, 1} for all i, j = 1, . . . , N be a set of functions of class L ∞ (0, T ). Let ψ i,j (t) ∈ (0, 1] for all t ∈ R + for all i, j = 1, . . . , N . Suppose moreover that the field α

∈ (L 1 [0, T ]; W 1,1 (Ω N ; R N )), α ∈ L ∞ ([0, T ] × Ω N ; R N ) and [div x α] -∈ L 1 ([0, T ]; L ∞ (Ω N )). Let b 0 i ∈ (0, γ i ), γ i > 0, µ i > 0 and x 0 i
∈ Ω for all i = 1, . . . , N . Then, there exists one and only one solution of (2.1)-(2.5). The opinion vector x is a regular Lagrangian flow and b ∈ C ∞ ((R + ) N ).

Moreover, if α is a Lipschitz field with respect to the opinion vector x, uniformly in time, then existence and uniqueness of the solution hold in the classical sense for both b and x.

Numerical results

Because of the weak coupling of the model, already described in the previous section, the numerical simulations have been produced by decoupling the problem in two sub-problems.

We introduce the functions (4.1)

B i = t 0 b i (θ) dθ, i = 1, . . . , N,
which represent the total number of microblogs posted at time t by the individual labelled with the index i. Thanks to the regularity of b, proved in the previous section, we deduce immediately that (3.1) can be written as a pure differential system:

(4.2)                    db i dt (t) = µ i (γ i -b i (t)) N j=1 σ i,j (t)B j (t) dB i dt (t) = b i (t) b i (0) = b 0 i ∈ (0, γ) B i (0) = 0.
We have first solved the Cauchy problem (4.2) and then we have stocked the results of the problem. We have subsequently used them as input data for solving the Cauchy problem (3.2). Both systems have been discretized by means of a standard fourth-order Runge-Kutta routine.

In what follows, we always suppose that

α i (s) = β(1 -s 2 ), β > 0, for all i = 1, . . . , N.
This specific form of the nonlinear fields α i is coherent with the hypothesis that individuals with extreme opinions are more stable in their convictions. We separately treat three network geometries. The first geometry describes a fully connected system; the second one considers a strongly connected network (i.e. there exists a path linking each pair of agents of the population) and the third one describes a partially interconnected network, composed of separate clusters of agents.

The time evolution of the opinions in each geometry is then analyzed by looking at three situations: the first one without hidden manipulation, and the other two with two types of hidden manipulation, which may be active for the whole duration of the simulation or not.

In all the numerical simulations, we consider a population composed of N = 100 totally or partially interconnected individuals. The choice of this value for N allows to produce readable figures. Of course, simulations with a greater number of agents are possible and do not induce major difficulties, at least when N is not too big. We moreover choose the following numerical values: for all i, µ i = µ * = 10 -4 (relaxation constant for the first set of equations), γ i = γ * = 15 (maximum number of daily posts for each agent) and β = 2 (relaxation constant for the second set of equations). The time step of the Runge-Kutta algorithm is ∆t = 5 × 10 -4 and the simulations have been displayed for t ∈ [0, 5], t being measured in weeks.

For each numerical experiment, we systematically show two figures. The first one describes the time evolution of the individual opinion with respect to time for the whole number of individuals and the second one shows the time evolution of the quantity

S + = 1 N N i=1 1 x i (t)>0 ,
which represents the fraction of individuals which favour the underlying binary question at a given time t ∈ R + . Of course, from S + is possible to deduce the fraction of individuals which do not approve the underlying binary question at a given time t ∈ R + :

S -= 1 -S + = 1 N N i=1 1 x i (t)≤0 .
When needed, we add the visualization of the interaction matrix and of the evolution of the number of posts.

Fully interconnected population.

In this first series of cases, we suppose that σ i,j = 1 for all i, j = 1, . . . , N . At the individual level, we suppose that all the agents of the population have identical features: they post the same number of comments on the social network (the initial condition is b 0 i = 1 for all i = 1, . . . , N ), and differ only with respect to their initial opinion. Figure 1 (left) describes the interaction matrix, whose entries are all equal to one. Figure 1 (right) summarizes the time evolution of the number of posts of each agent. Because of the initial conditions, all the curves are superposed.

The initial condition for the opinion functions is the following: The results of the first simulation, without hidden manipulation (i.e., ψ i,j = 1 for all i, j = 1, . . . , N ), are shown in Figure 2. The average opinion is constant in time and the system relaxes to the equilibrium x ∞ = 0 (as expected because of the symmetry of the problem); moreover, individuals with extreme opinions tend more slowly to the equilibrium than individuals with opinions close to the average (see Figure 2, left). The symmetry of the problem and of the initial conditions, together with the other conditions chosen for this test, lead to a perfect equilibrium between the subpopulation favorable to the underlying question and the fraction of the population which is against the underlying question. 4.1.2. Case 2. We introduce an opinion manipulation technique described by the highlighting functions

(4.3) x 0 i = -0.9999 + 1.9998 × i -1 N -1 , i = 1, . . . , N.
ψ i,j (t, x i (t), x j (t)) =      1 if x i (t) ≥ x j (t) 1 2 otherwise.
This choice induces a dynamics towards negative opinions. The other settings stay unchanged with respect to Case 1. In Figure 3 and 4 we show the corresponding numerical results on the time evolution of the opinions and on the time history of S + . Note that the interaction matrix and time evolution of the number of posts are the same as in Case 1 (see Figure 1).

ψ i,j (t, x i (t), x j (t)) =      1 if x i (t) ≥ x j (t) or t ≤ 4 9 10 otherwise.
We underline that ψ i,j = 1 for all i and for all j when t ≤ 4. In practice, this family of highlighting functions is a mixing of the previous choice, and induces a dynamics towards negative opinions. The other settings stay unchanged with respect to Case 1. In Figure 4 we show the corresponding numerical results on the time evolution of the opinions and on the time history of S + . The interaction matrix and time evolution of the number of posts are the same as in Case 1 (see Figure 1). The presence of a mild manipulation, active only in the last week, induces a non-negligible modification of the population S + .

Strongly connected population.

In this second series of tests, the form of the interaction matrix is the following. We first impose that σ i,i = 1 for all i = 1, . . . , N ;

σ i,i+1 = 1 for all i = 1, . . . , (N -1) σ N,1 = 1.
σ i+1,i = 1 for all i = 1, . . . , (N -1) σ 1,N = 1. These conditions guarantee that the network represented by the interaction matrix is strongly connected. Moreover, we add some extra non-zero entries to the interaction matrix by means of a sampling from the uniform distribution. The explicit form of the interaction matrix is described in Figure 5 In Case 4, we suppose that the microblogging network's provider is neutral. We see that, even if the initial condition is the same as in Case 1, the geometry of the network representing the connexions between the individuals strongly modifies the behaviour of the system. In particular, the system tends, slower than in Case 1, to an equilibrium. This equilibrium is however different from zero (see Figure 6). This behaviour is the consequence of the non-symmetric interactions in the network representing the interactions between the members of the population. The evolution of the population S + starts from 0.5 at time t = 0 and reaches in a non-monotone way, at time t = 5, the value S + (5) = 0.12. In this case, we study the effect of an hidden manipulation on the same population studied in Case 4. We suppose that the highlighting functions have the form

ψ i,j (t, x i (t), x j (t)) =      1 if x j (t) ≥ x i (t) 9 10
otherwise.

We underline that this form of the highlighting functions drives the system towards positive opinions. This feature of the highlighting functions is confirmed by the numerical experiments. We see that the subpopulation S + is weakly oscillating, but reaches in five weeks the value 0.65, starting from the value S + (0) = 0.5 (Figure 8) and with an interaction matrix which clearly favours the negative opinions, as shown in Case 4. 4.2.3. Case 6. In this test, we slightly modify the form of the highlighting functions, in the sense that we turn off the manipulation effect in the simulation after two weeks. The precise form of the highlighting functions, which drive the system towards positive opinions, is the following

ψ i,j (t, x i (t), x j (t)) =      1 if x j (t) ≥ x i (t) or t ≥ 2 9 10 otherwise.
We observe that a short-time manipulation (see [START_REF] Boudin | Conciliatory and contradictory dynamics in opinion formation[END_REF] for a discussion about the timing of a perturbation on an opinion formation process) is enough for strongly modifying the collective behaviour of the population: we pass from the non-manipulated situation of Case 4, in which S + (5) = 0.12, to a final result in which the population is favourable to the underlying question, with S + (5) = 0.56. The initial condition -which is randomly chosen from the uniform distribution -and the interaction matrix are detailed in Figure 9. The total number of connexions of the network is equal to 849. Initially, the population has average opinion equal to 0.3069. The fraction of the population with positive opinion at time t = 0 is S + (0) = 0.48. Even if S -(0) > S + (0), we see that the average opinion has a major effect on the time evolution of the population, as underlined in Section 2. This indicator can be more important than the fraction of the population having opinions of the same sign.

We will study two types of manipulation. In Case 8, the network highlights opinions which have negative sign whereas Case 9 treats a possible structure of highlighting functions which put in light, for the i-th individual, all the opinions which are closer to -1 than x i (t).

Being interested in studying the manipulation effects, we suppose that all the agents have the same blogging activity (i.e. b 0 i = 1 for all i = 1, . . . , N ).

4.3.1. Case 7. We first treat the situation without manipulation. In Figure 10 we reproduce the individual opinion evolution and the evolution of the fraction of the population with positive opinions. We observe that the agents aggregate themselves in several clusters and that S + grows from S + (0) = 0.48 to S + (5) = 0.75. The four detected clusters are consistent with the four interconnected sub-populations of the network. We underline that this form of the highlighting functions has a decisive effect in pushing the system towards negative opinions. In Figure 11, we note that the subpopulation S + decreases from S + (0) = 0.48 to S + (5) = 0.25. Moreover, the whole population reduces itself to four clusters, three of them are centred below zero. The result is very sensitive to the wights of the opinion in the highlighting functions. If we replace the value 4/5 with 9/10, the manipulation effect would be much weaker and the system would converge to a fifty-fifty equilibrium. 4.3.3. Case 9. The manipulation effect of this simulation is obtained thanks to highlighting functions of type ψ i,j (t, x i (t), x j (t)) =

     1 if x i (t) ≥ x j (t) 4 5
otherwise.

Figure 12 shows that this strategy is less efficient than the strategy used in Case 8, even if this strategy could help in decreasing the value of the opinion variable of individuals which are exclusively in contact with individuals of positive opinion. However, this strategy is enough

Conclusion

We have studied some dynamics of opinion manipulation. We have considered a fixed network, but the model allows to treat, in the same way, an evolutionary network. The model takes into account the effects on public opinion caused by the sign and the intensity of the initial opinions of the agents, their activity in microblogging platforms and the possible manipulations of the visibility of the posts by the microblogging platform provider. We have shown that hidden manipulation can have an important impact on the public opinion formation and that very mild interventions of the network owner can have major effects on the population.

A way for detecting such manipulation activities could be the design of an artificial isolated cluster of accounts, with initial conditions and blogging activity identical to those described in Case 1. If the behaviour of the system is not coherent with the dynamics of Case 1, it is possible that a hidden manipulation can be in progress. The fact that even temporary manipulation can be detected, as in Case 3, shows the sensitivity of this strategy for discovering manipulation attempts.

Figure 1 .

 1 Figure 1. Interaction matrix and time evolution of the number of posts (Cases 1, 2 and 3).

4. 1 . 1 .

 11 Case 1.

Figure 2 .

 2 Figure 2. Time evolution of the opinions and of S + (Case 1).

Figure 3 . 4 . 1 . 3 .

 3413 Figure 3. Time evolution of the opinions and of S + (Case 2).

Figure 4 .

 4 Figure 4. Time evolution of the opinions and of S + (Case 3).

Figure 5 .

 5 Figure 5. Interaction matrix and time evolution of the number of posts (Cases 4, 5 and 6).

4. 2 . 1 .

 21 Case 4.

Figure 6 .

 6 Figure 6. Time evolution of the opinions and of S + (Case 4).

Figure 7 .

 7 Figure 7. Time evolution of the opinions and of S + (Case 5).

Figure 8 . 4 . 3 .

 843 Figure 8. Time evolution of the opinions and of S + (Case 6).

Figure 9 .

 9 Figure 9. Interaction matrix (left) and initial condition (right), (Cases 7, 8 and 9).

Figure 10 .

 10 Figure 10. Time evolution of the opinions and of S + (Case 7).

Figure 11 .Figure 12 .

 1112 Figure 11. Time evolution of the opinions and of S + (Case 8).

& Dipartimento di Matematica "F. Casorati", Università degli Studi di Pavia, Via Ferrata 1, 27100 Pavia, ItalyEmail address: giulia.braghini.1995@gmail.com Email address: francesco.salvarani@unipv.it

Acknowledgements. This work has been carried out in the framework of the projects Kimega (ANR-14-ACHN-0030-01). This research was moreover supported by the Italian Ministry of Education, University and Research (MIUR), Dipartimenti di Eccellenza Program -Department of Mathematics "F. Casorati", University of Pavia.