L Desgranges

H Palancher

CEA/DEN/ M Gamaléri

J S Micha

Virgil Optasanu

Laura Raceanu

Tony Montesin

Influence of the U 3 O 7 domain structure on cracking during the oxidation of UO 2

Cracking is observed when a UO 2 single crystal is oxidised in air. Previous studies led to the hypothesis thatcracking occurs once a critical depth of U 3 O 7 oxidised layer is reached. We present some µLaue X-Ray diffraction results, which evidence that the U 3 O 7 layer, grown by topotaxy on UO 2 , is made of domains with different crystalline orientations. This observation was used to perform a modelling of oxidation coupling chemical and mechanical parameters, which showed that the domain patterning induces stress localisation. This result is discussed in comparison with stress localisation observed in thin layer deposited on a substrateand used to propose an interpretation of UO 2 oxidation and cracking.

Introduction

The oxidation of uranium dioxide is a key process involved at least in two important steps of nuclear industry: fuel fabrication and safety of spent fuel dry storage [ 1 ]. Oxidation induces cracking and spalling of the initial UO 2 material, which might lead to rod cracking in case of defective fuel in dry storage conditions. The morphological evolution of a UO 2 pellet during oxidation was studied by Bae [ 2 ], who evidenced two stages: first macro-cracking occurs when U 4 O 9 and U 3 O 7 are formed, then micro-cracking is associated with the formation of U 3 O 8 . This description is consistent with the one previously proposed by Tempest et al. [ 3 ] in which cracks appear in the uniform layer of U 3 O 7 formed on the surface of a UO 2 sample. In situ Environmental Scanning Electron Microscope experiments showed that a UO 2 single crystal is transformed in a powder during oxidation by the formation of series of macro-cracks [ 4 ]. The formation of these macrocracks had been observed as soon as the oxidised layer formed on the initial UO 2 sample reached a critical depth of about 0.4µm. This experimental result was used to propose a criterion for the safe handling of oxidized defective fuel rods [ 5 ]. This criterion assumes that the nuclear fuel will be safe as long as no macro-crack is formed. A numerical evaluation of this safe duration was also recently proposed by calculating the time needed to form a critical thickness of the oxidised layer [ 6 ].

Cracking of thin films has been widely studied both theoretically and experimentallybecause of its relevance for the semi-conductor industry. Although large elastic strains can be tolerated in very thin films, above some critical thickness the film will act to relieve the strains by generating dislocations, micro-twins, surface instabilities and cracks. The formation of cracks in oxidised UO 2 shares many similarities with cracking of thin film. Macrocracking of UO 2 is equivalent to the 2D periodic cracking of thin film witha regularspacing of similar cracks [see for example 7 , 8 ].In [4] it was shown that the crack which splits the oxidised layer also penetrates in the UO 2 substrate, which was observed on thin film deposited on brittle substrate [ 9 ]. But some discrepancies exist between UO 2 oxidation , which proceeds by topotaxy (growth of a new crystalline phase within the substrate by chemical reaction) and thin film growthwhich proceeds by epitaxy (growth of an external phase on a substrate by deposit).First the interface between a film and its substrate is defined by a sharp change of crystalline lattice at a monolayer scale; in the UO 2 on the contrary this interface rather corresponds to an interdiffusion layer whose thickness is at least 5 nm [ 10 ].Second, UO 2 oxidation proceeds at temperatures less than 400°C (in dry storage condition) by oxygen diffusion only [ 11 ], thus the formation of dislocations, which are usually invoked to describe thin film formation, is not likely to exist in UO 2 oxidation.

In order to achieve a better description of UO 2 oxidation, a new model was developed in which mechanical stresses are coupled to diffusion processes.The first results obtained with a one dimension version of this model [ 12 ] showed that the oxidation mechanism was better described with an oxidised layer with tensile stress, i.e. a lattice mismatch due to a smaller unit cell volume in the oxidised layer than in the bulk UO 2 .This model is only valid for isotropic crystalline phase, which is not the case for U 3 O 7 .

To better describe UO 2 oxidation a 3D model is needed. In this paper we describe how such a model was built. Because UO 2 oxidation is a complex phenomenon involving several crystalline phases, which change as a function of temperature, oxidation conditions were fixed to get a simple system. The oxidation of a UO 2 single crystal in air at 300°C was then chosen, because only UO 2 and U 3 O 7 phases are observed when cracking occurs [4]. Even in this case, oxidation modelling is still a complex problem because U 3 O 7 phase has a tetragonal crystalline symmetry and several crystalline orientations of U 3 O 7 on UO 2 are possible. In literature the existence of orientation relationships between UO 2 and its oxidation layer is reported [ 13 ],but the topotaxy relationships at the UO 2 -U 3 O 7 interface, at which cracking occurs, still needs to be determined.That is why we first studied them experimentally using X-Ray diffraction. Then our mechanical modelling of UO 2 oxidation was modified to take them into account. This modified modelling is then used to discuss whether the existence of a critical depth of U 3 O 7 is needed for cracking to occur.

characterisation of UO 2 -U 3 O 7 interface by X-Ray diffraction

Several X-Ray diffraction studies of UO 2 oxidation are reported in literature [see for example 14 , 15 ]. But no decisive information were gained on the topotaxy relationships at the UO 2 -U 3 O 7 interface, because U 3 O 7 appears as very broad peaks on the diffraction pattern as long as UO 2 can be observed. This can be explained by a continuous change of U 3 O 7 c/a ratio from 1 to 1.03, and also by the existence of micro-strains induced by the mismatch between UO 2 and U 3 O 7 unit cells. Therefore a new experimental approach is needed to get a more in depth characterisation of UO 2 -U 3 O 7 interface.

We used X-ray diffraction with synchrotron radiation in Laue mode on a UO 2 single crystal, because this method allows an exploration of the reciprocal space with a better accuracy than conventional X-Ray diffraction.

Sample preparation

The 5*5*1mm UO 2 single crystal analysed here, was also used for a study of cracking [4] in which it was oxidised in air at 300°C and characterised at different oxidation times. This single crystal has only (111) oriented natural faces, on which characterisation was performed. After 1+1/2 hours of oxidation, a U 3 O 7 layer was observedby X-Ray diffraction; taking into account the penetration depth of X-Rays, its thickness was estimated around a few tenths of a µm. Its diffraction peaks in (111) direction were characterised by conventional Bragg-Brentano diffraction, and its (111) peak was recorded in a 2D (, 2) diffraction pattern. The spot corresponding to the (111) U 3 O 7 diffractionplan is 10 times wider than the UO 2 spot in the  direction. This suggests some disorientation of the U 3 O 7 crystallites grown on the UO 2 single crystal. In order to better characterise this disorientation, the single crystal was brought to ESRF to perform µ-XRD in Laue mode. µ-XRD in Laue mode µ-XRD has been performed at the BM32 beamline at the ESRF (Grenoble, France). There, a dedicated set-up enables a 2D XRD -mapping with micrometer spatial resolution [ 16 ]. Before the experiment, the incoming polychromatic X-ray beam has been carefully characterised: its energy band pass ranges from 5 to 27 keV and its size was 1*2µm² (horizontal and vertical full width at half maximum). During the measurement, the X-ray beam incidence angle has been kept fixed to 40°. XRD data have been collected with a two-dimensional MAR165 CCD camera at several locations on the oxidised surface. A typical image is shown on Figure 1;it is compared to that collected on a UO 2 single crystal previously annealed at 1700°C during 24hours, considered as a strain free reference sample.

Figure 1.Comparison of the µ-XRD Laue images collected on a strain free (A-) UO 2 and on an oxidised (B-) UO 2 single crystals

The shape of the Bragg spot,initially round on the reference UO 2 , became elongated along three different directions. These elongations were interpreted considering the crystalline orientations of the U 3 O 7 crystallites on UO 2 substrate.

A-B-

Data analysis

Considering the X-ray penetration depth of X-ray beams into UO 2 in the available X-Ray range (5 to 27 keV), both the oxidised layer and the preserved substrate are always probed. The collected image can thus be considered as the sum of two diagrams, the first representative of unstrained UO 2 (central round part of the Bragg spot) and the second of the oxidised layer (elongations). At a second step, the elongations were simulated as strained UO 2 using the Laue tool software [ 18 ]. These simulations allowed determining the directions and intensities of the measured strain.

Each elongation on Figure 2was successfully attributed to a distorted UO 2 lattice. The relationships between UO 2 and U 3 O 7 unit cell used in order to describe the U 3 O 7 as a distorted UO 2 are presented in Figure 3. Both phases share the same crystalline axis. Going for UO 2 to U 3 O 7 leads to a relative increase of 3% along one axis, the elongated axis becomes c axis of U 3 O 7 structure. Three choices for the U 3 O 7 c axis are possible, corresponding to the three elongations observed around the UO 2 round spot on Figure 2.

The measured distortion is consistent with the U 3 O 7 crystalline structure, whose c/a ratio is reported to be 1.03 [1].Moreover UO 2 and U 3 O 7 unit cell axis sharing the same orientation is consistent with the topotactic growth of U 3 O 7 on UO 2 . However the intensity of the three elongations shows some fluctuations that can not be explained if the three types of domains were given the same weight. These intensity fluctuations could be interpreted as the U 3 O 7 domain distribution within the sampling surface of the X-Ray beam, equal to 2 µm². Local variations of the domain microstructure due to crystal surface defects could also lead to intensity fluctuations.

. In the experimental part it has been shown that U 3 O 7 is formed with crystallites having different crystalline orientations on the (111) oriented natural face of a UO 2 crystal, related to the choice of the c axis when transforming the cubic UO 2 unit cell in the tetragonal U 3 O 7 unit cell. The choice of (111) surface for UO 2 implies three different orientations of U 3 O 7 crystallites. For the modelling purpose, these three different orientations would have led to a complicated geometry. So we decided to describe UO 2 oxidation on a (100) surface, because it leads to a simpler geometry.

hypothesis

We consider multi-domains of U 3 O 7 lying on a UO 2 The direction (001) inUO 2 is parallel to the direction (010) inU 3 O 7 . Thus, the c(U 3 O 7 ) direction lies in the plane of the interface. We excluded the case in which c(U 3 O 7 ) direction is perpendicular to interface because it corresponds to the highest mismatch between UO 2 and U 3 O 7 (Figure 3). This assumption is consistent with some previous X-Ray diffraction results [15]. TheU 3 O 7 domains have two possible directions for the axis c. The U 3 O 7 layer is made of alternate domains in which a and c axis contained in the interface permute. The boundaries between these domains are located on (101) planes for energy minimisation. In Figure 4we show the mesh of our FEM model. The U 3 O 7 domains are in red and green; the white mesh represents UO 2 substrate. This geometry was implemented in the Finite Element Model (FEM) code CASTEM, we used for our calculations.The U 3 O 7 domains were not modified during the calculations: no mobility of the interfaces was simulated. For a given thickness of U 3 O 7 layer, the stress distribution and the oxygen diffusion fluxes are successively computed.

3

U 3 O 7 UO 2 1 2
The first calculation step computes the stress distribution induced by the connection of differentU 3 O 7 domains and UO 2 substrate.The second step computes theoxygen chemical diffusioncoupled to the previously calculated mechanical state. The background theoryfor coupling chemical diffusion to mechanical statewas widely exposed in [ 19 ], [ 20 ], [ 21 ],and is based on previous works [ 22 ]. The diffusion coefficient we used depends onboth the mechanical stress and the stress gradient:

D = D 0 1- M 0 h ij c RT s ij c + ¶s ij ¶c ae è ç ö ø ÷ é ë ê ù û ú
whereM 0 is the molar mass of the UO 2 ,  ij is the chemical expansion coefficient, c is the oxygen concentration which is dissolved in UO 2 lattice,  is the mechanical stress, T is the temperature and R is the universal constant of gases.

Table 1 shows the parameters used in our simulations.

D 0 [cm 3 /s] h ij 1 [m 3 /kg] T °C E GPa  Lattice parameters Å a b c
Ref. The lateral size of each domain of U 3 O 7 is 100 nm. Their thickness was fixed at 50 nm, 100 nm, 200 nm, 400 nm successively while the thickness of UO 2 is considered as infinite.

 23   24   25   26 , 13  UO 2 0.0055 exp -26.3 RT ae è ç ö ø ÷ -1.248.10 -5 300 200 0, 32 5,47 (cubic) U 3 O 7 5,363 (tetragonal) 5,

Results

The stresses calculated in the first computation step is presented in Figure 5 on a section parallel to the UO 2 /U 3 O 7 interface at 1 nm depth inside UO 2. The second step computation evidenced that heterogeneities in stress led to different oxygen fluxes in UO 2 . This produces a transversal oxygen concentration gradient. Moreover, differences in concentration induce differences in chemical expansion, which generates additional mechanical stresses, from -70 MPa to 220 MPa, having a strong feedback on the diffusion. The oxygen concentration distribution is shown in Figure 6in a transversal section parallel to the UO 2 /U 3 O 7 interface at 10 nm depth in UO 2 . Thisoxygen concentration leads to changes of chemical composition in the UO 2+x phase with x varying 0.01 to 0.11.

Discussion

Our experimental results evidenced that the formation of a U [11]. In a uniformly oxidised layer, the formation of U 3 O 8 is likely to induce an expansion in a direction perpendicular to the surface,which minimises the strain state at the interface with the substrate. But the compression stress in U 4 O 9 makes the structural transformation in U 3 O 8 more energetically costly.

The minimisation of the mechanical energy of an oxidised layer on a substrate can also be achieved by stress localisation, as in the case of thin layer deposited on a substrate.In this case, Asaro & Tiller [ 27 ], and Grinfeld' [ 28 ] showed that a sinusoidal deformation of the surface may decrease the mechanical energy of the system. This sinusoidal deformation creates hills and valleys with different mechanical states, stress relieved at the top of the hills and stress increased at the bottom of the valleys. In the case of UO 2 oxidation, stress localisation would be achieved by the formation a patterned U 3 O 7 layer, which minimises the mechanical energy by a reorganisation of crystalline structure rather than by surface deformation. Thus the formation U 3 O 7 domains with different crystalline orientations on a UO 2 substrate should be understood as a consequence of the minimisation of mechanical energy. Domain formation is a known energy minimisation phenomenon, also for example in magnetism [ 29 ].

In the case of thin layer deposited on a substrate, stress localisation leads to periodical cracking of the sample at a critical depth because the stress level increases with increasing depth of the layer. In the case of UO 2 oxidation we performedcalculations with several thickness of U 3 O 7 layer in order to check the stress intensity as a function of U 3 O 7 thickness.

In .Figure 7 11 stress is shown as a function of the distance to the UO 2 /U 3 O 7 interface. The maximal value of the stress increases as a function of the U 3 O 7 thickness.

.Figure 7 11 stress for several thickness of U 3 O 7 layer.

The increase of traction stress with U 3 O 7 layer depth justifies that this layer cracks when its thickness reaches a critical value. No exact value of the critical thickness can be derived from our calculation, because the numerical values we used are, for some them, crude approximations. The formation of cracks changes the mechanical state of the layer and would allow the formation of U 3 O 8 , as it is observed experimentally. Considering a martinsitic type formation of U 3 O 8 on UO 2 as proposed by Allen [13], these U 3 O 8 crystallites should have a size smaller or equal to the one of the U 3 O 7 grain, they were formed from.

The mechanism described here is consistent with all reported results on UO 2 at temperature less than 400°C. It could also be applicable to a wider class of materials which are submitted to cracking during a chemical reaction, a phenomenon sometimes named chemical fragmentation [ 30 ].

Conclusion :

UO 2 oxidation in air at temperature less that 400°C shares some common features with thin film deposited on substrate. The topotactic formation of U 3 O 7 oxidised layer can indeed be compared to the epitaxial formation of a thin layer. In both case the mismatch in unit cell parameter between the layer and the substrate drives the mechanical evolution of the system. Because uranium oxide does not allow surface deformation, stress localisation, which minimises mechanical energy, is achieved because of the formation of domains of the metastable U 3 O 7 phase. This original mechanism can explain the existence of a critical depth for crack formation and could be applied to materials subject to "chemical fragmentation".

  At a first step, using the XMAS software[ 17 ], the peak search parameters were set to only take into account the central part of the Bragg peak (i.e. the unstrained one). Based on the obtained peak list, an indexation was performed and the orientation matrix calculated. This indexation confirmed that the UO 2[111] axis was perpendicular to the single crystal surface within several degrees. The position of the (-7,7,7) diffraction spot at the centre of the diffraction pattern in Figure2confirms this orientation.

Figure 2 :

 2 Figure 2 : Analysis of the image collected on the oxidised UO 2 single crystals. Indexation (A-)and simulations of images implying a strained UO2 structure

Figure 3 :

 3 Figure 3 : Schematic description of the crystalline relationships between UO 2 and U 3 O 7 experimentally observed and modelled.

Figure 4 :

 4 Figure 4 : An example of mesh. Multi-domain of U 3 O 7 on UO 2 substrate

U 3 O

 3 7 material exists mainly in powder form for which mechanical parameters are difficult to obtain. Thus we assumed that the U 3 O 7 Young modulus and Poisson'sratiowere the same as those of UO 2 . Chemical expansion coefficientwas not considered in U 3 O 7 .

Figure 5 :

 5 Figure 5 : 11 and  33 stresseson a section parallel to the UO 2 /U 3 O 7 interface at 1 nm depth inside UO 2 .

Figure 6 :

 6 Figure 6 : Concentration after diffusion in a transversal section UO 2 -side.

Table 1 :

 1 531

parameters used for computations

  3 O 7 layer on a UO 2 single crystal is associated with the formation of U 3 O 7 domains having different crystalline orientations related to the choice of U 3 O 7 c axis. Moreover our model evidenced that the existence of periodical U 3 O 7 domains induces variations in stresses and oxygen concentration at the UO 2 /U 3 O 7 interface. In the following we discuss why U 3 O 7 is formed with a domain microstructure, and how this microstructure can justify the existence of a critical depth needed for the formation of cracks in the oxidised layer. U 3 O 7 is a metastable phase: in thermodynamic equilibrium conditions without stresses, it is transformed in a mixture of U 4 O 9 and U 3 O 8 . One reason for the stabilisation of U 3 O 7 can be the stress state generated by the formation of an oxidised layer on the UO 2 substrate. U 4 O 9 has a cubic crystalline structure related to the one of UO 2 ; their unit cell parameter differs from 5.47 for UO 2 to 5.44 for U 4 O 9 . As U 4 O 9 is formed by topotaxy on UO 2 , this difference in cell parameter induces stresses that can be accommodated by an elastic deformation in which U 4 O 9 would be stretched in the interface plane and compressed perpendicular to it (assuming a Poisson coefficient =0.3). U 3 O 8 derives from UO 2 by a ~ 1/3 expansion along one UO 2 (111) axis

Acknowledgements:The authors would like to acknowledge the financial support from Electricité de France.


Nicolas Gouvy
email: nicolas.gouvy@lifl.fr

Essia Hamouda

Nathalie Mitton

Dimitrios Zorbas

Energy Efficient Multi-Flow Routing in Mobile Sensor Networks

Controlled mobility is one of the most complex challenges in Wireless Sensor Networks (WSN). Only a few routing protocols consider controlled mobility in order to extend the network lifetime. They are all designed to optimize the physical route topology from a source to a destination. However, there is often more than one sensor which reports an event to the sink in WSN. In existing solutions, this leads to oscillation of nodes which belong to different routes and their premature death. Experiments show that the need of a routing path merge solution is high. As a response we propose the first routing protocol which locates and uses paths crossing to adapt the topology to the network traffic in a fully localized way while still optimizing energy efficiency. Furthermore the protocol makes the intersection to move away from the destination, getting closer to the sources, allowing higher data aggregation and energy saving. Our approach outperforms existing solutions and extends network lifetime up to 37%.

I. INTRODUCTION

Miniaturization, cost decrease and advances in low-power electronics and radio communication technologies have made possible the emergence of Wireless Sensor Networks (WSN). WSN are cooperative sets of battery-powered nodes which communicate through radio links and can be employed for a wide kind of applications, gathering data from their environment on-request or on event occurrence. WSN are intended to be used in vast applications in hostile environments such as battlefield or surveillance systems. The importance of the applications makes essential to propose routing algorithms which take the battery issue into account in order to improve the network lifetime. A common solution to tackle the fast energy exhaustion is to increase sensor density in order to decrease communication range. A more recent approach is to introduce mobility-enabled sensors, that can move to areas where resources are most needed in order to efficiently route packets. Results in [START_REF] Wei | Extending the lifetime of wireless sensor networks through mobile relays[END_REF] show that deploying resourcefull mobile devices in a WSN provides better results than increasing the network density. Still, only a few works consider the use of controlled mobility in order to optimize route topology. Moreover, even if they differ in the relocation patterns they use, they all aim to relocate nodes on the source to the destination route. Nevertheless, in event-based WSN, there are often multiple sources emitting at the same time to the sink. It turns out that existing routing protocols which relocate nodes do not work well in this case. They provoke useless zig-zag movements when routes are close and consequently create holes in the network topology when multiple routes cross each other [START_REF] Liu | Localized mobility control routing in robotic sensor wireless networks[END_REF].

In this paper we propose PAMAL (PAth Merging ALgorithm), a new geographic routing algorithm for WSN with mobile nodes, that takes advantages of paths crossing and even favors it in the following respect:

• It guarantees network connectivity at each step.

• It is energy efficient as it takes both radio and relocation costs into account when relocating nodes. • It is localized as routing decisions are based only based on neighbors and destination locations. • It detects path crossing and achieves better network lifetime than existing solutions. Simulations show that PAMAL stops oscillation and makes close routing paths merge ahead to the destination from time to time, since it moves the intersection point close to the sources. This behavior and data-aggregation makes PAMAL improve network lifetime up to 37% compared to the best of our knowledge routing protocol which uses controlled mobility [START_REF] Elhafsi | Energy efficient mobile routing in actuator and sensor networks with connectivity preservation[END_REF]. The remainder of this paper is organized as follows. In Section 2, we provide a review of existing works on routing algorithms which use controlled mobility and data-aggregation. We detail the models needed for the rest of the paper in Section 3. The preliminaries are exposed in Section 4, while Section 5 describes our motivation. PAMAL routing algorithm and how path crossing is managed is introduced in Section 6. In Section 7, we describe our simulation methodology and present our simulation results. Finally Section 8 concludes the paper.

II. RELATED WORK

A. Position-based routing in static networks.

Position based routing algorithms for static sensor networks have been widely studied in the literature. In distance-based greedy routing a node currently holding a packet forwards it to the closest neighbor to the destination. This method has been extended to support energy efficiency [START_REF] Kuruvila | Progress and location based localized power aware routing for ad hoc sensor wireless networks[END_REF], guaranteed packet delivery [START_REF] Bose | Routing with guaranteed delivery in ad hoc wireless networks[END_REF] or both [START_REF] Elhafsi | End-to-end energy efficient geographic path discovery with guaranteed delivery in ad hoc and sensor networks[END_REF]. The Cost Over Progress (COP) based routing introduced in [START_REF] Kuruvila | Progress and location based localized power aware routing for ad hoc sensor wireless networks[END_REF] is a localized metric-aware greedy routing scheme. A node forwards a packet to its neighbor in the direction of the destination such that the ratio of the energy consumed for forwarding packet (any cost metric can be used) to the progress made (measured as the reduction in distance up to the destination) is minimized.

B. Routing in mobile networks

Node mobility is generally regarded as a hazard for geographic routing, causing a degradation of performances or even persistent routing failures. Therefore, little work has been done in controlled mobility assisted routing in WSN. It is generally used to distribute traffic load and consequently optimize energy consumption [START_REF] Goldenberg | Towards mobility as a network control primitive[END_REF]. However those protocols do not alter the network topology. Other approaches adopt existing routing protocols to find an initial route, and iteratively move each node to the midpoint of its upstream and downstream nodes on the route. These routing protocols may not be efficient. For example moving strategy in [START_REF] Goldenberg | Towards mobility as a network control primitive[END_REF] may cause useless zigzag movements. In MobileCOP [START_REF] Liu | Localized mobility control routing in robotic sensor wireless networks[END_REF], next hop on the path is selected based on COP metric [START_REF] Kuruvila | Progress and location based localized power aware routing for ad hoc sensor wireless networks[END_REF]. Once a path is computed, its nodes are moved and placed equidistantly on the straight line connecting the source to the destination. Such a strategy may induce a memory overhead on nodes and a high transmission delay, since they have to store the full path. Moreover, the network may be disconnected (a node may move out of range of its neighbor). CoMNet [START_REF] Elhafsi | Energy efficient mobile routing in actuator and sensor networks with connectivity preservation[END_REF] or [START_REF] Gouvy | Minimising energy consumption through mobility with connectivity preservation in sensor networks[END_REF] is the first fully localized COP-based geographic routing protocol that takes the cost of moving into account while it guarantee network connectivity. Nevertheless, to the best of our knowledge, none of existing solutions handles paths crossing. In all those protocols, an Intersection Node (IN ) between two different crossing paths will be relocated continuously on each source-destination line at each routing. We propose PAMAL which takes intersection into account, and handles it dynamically and provokes merging of different close routing paths to the same destination in order to make energy savings and data-aggragation.

C. In-Network data aggregation

Data-aggregation algorithms combine gathered data coming from different sources to eliminate redundancy or lower the routing overhead in order to minimize the size and/or number of radio transmissions and consequently make possible energy saving. For a complete survey, readers should refer to [START_REF] Rajagopalan | Data-aggregation techniques in sensor networks: a survey[END_REF].

III. SYSTEM DESCRIPTION

A. Assumptions

We consider a sensor network where nodes are randomly and uniformly scattered. It can be modeled as an undirected simple finite graph G(V, E), with V the set of nodes in the network and E the set of links/edges. (u, v) ∈ E if u and v are in transmission range of each other. We denote by N (u) the set of neighbors of u. Let δ(u) = |N (u)| be the cardinality of N (u), also called the degree of node u. We also define N D (u) the subset of neighbors of node u which are closer to the destination node D than u:

N D (u) = {v ∈ N (u) ∧ |vD| < |uD|} .
We assume that every node v ∈ V is aware of its geographical location. Moreover each node can be either a mobile or stationary sensor. The latter assumption can be further relaxed by making either all nodes static or mobile.

B. Energy models

Although our approach is model independent, we use the following widely employed radio and mobility cost models as a proof of concept.

1) Transmission cost: We denote by C s (.) the energy consumption or cost for radio transmission between two nodes [START_REF] Rodoplu | Minimizing energy mobile wireless networks[END_REF]:

C s (r) = q * (r α + c) if r = 0, (1)
where c represents the energy overhead due to radio device, α is a real constant (> 1) that represents the signal attenuation, and q represents the size of the packet transmitted in bits.

According to this model, the optimal radio transmission radius for routing is r * = α c α-1 [START_REF] Stojmenovic | Power-aware localized routing in wireless networks[END_REF]. 2) Mobility cost: We denote by C m (.) the cost to relocate a node. We use the model adopted in previous similar works [START_REF] Liu | Localized mobility control routing in robotic sensor wireless networks[END_REF]:

C m (|vv |) = a * |vv | (2)
where v is the initial position of v (before it is moved) and v is its new position. a is a constant defining the units of energy consumed per units of distance traveled.

IV. PRELIMINARIES PAMAL relies on the CoMNet (COnnectivity preservation Mobile routing protocols for actuator and sensor NETworks) [START_REF] Elhafsi | Energy efficient mobile routing in actuator and sensor networks with connectivity preservation[END_REF] routing protocol. CoMNet uses a COP approach: current node u chooses v ∈ N D (u) which minimizes the ratio of the global cost (packet transmission cost plus node relocation cost) to the progress made towards the destination. Indeed, v satisfies the following optimization problem:

v = arg min v∈N D (u) C s (|uv|) + C m (|vv |) |uD| -|v D|
The connectivity of the network is guaranteed using a Connected Dominating Set (CDS) and v = v if v belongs to the CDS.

In order to adapt to various cost models CoMNet comes with three different variants -more details can be found in [START_REF] Elhafsi | Energy efficient mobile routing in actuator and sensor networks with connectivity preservation[END_REF]-:

• CoM N et-ORouting on the M ove which aligns nodes on a straight line from S to D (SD), • CoM N et -M ove (DSr) which aligns nodes on (SD) line with all hop lengths to be equal to the optimal transmission distance r * , • CoM N et-M ove r in which next hop node v is relocated on the intersection of the circle C r * of radius r * centered at u and the (vD) line.

V. MOTIVATION

The motivation of this work is to take advantage of node mobility and path crossing to extend the network lifetime. Currently even though the idea of using mobile sensors to optimize network topology begins to be recognized, all current approaches still rely on the strong hypothesis of a unique source-destination pair. Available solutions iteratively move each node of the Source-Destination (S,D) route on a (SD) line. However in eventbased WSN, a same event oftenly triggers multiple sensors in a close area, provoking competition and useless zig zag for nodes which belong to the different (S 1 ,D)... (S n ,D) routes. In order to demonstrate the need for path crossing management, we perform simulation on random topologies of various densities δ (δ = 15, 20, 25, 30) with only two source S 1 and S 2 and one sink D. Routing is performed on those networks between each pair of source-destination (S 1 ,D) and (S 2 ,D) using CoMNet. Figure 1 shows that intersections frequency vary from 13% and up to 20% of network topologies depending on network density. Those results vindicate the necessity of dynamic path crossing management as we can infer a higher intersection rate in real environment where sources are close.

VI. PAMAL

We propose PAMAL which takes advantage of paths crossing. We argue that making the different routes merge -in a fully localized way-provide energy saving since less nodes are sollicited. When a node detects that it is an intersection node (IN), i.e. a node which receives packets from two different sources for a same direction, it broadcasts the information to its 1-hop neighborhood. Each of its neighbors v will stop considering the sink D as final destination for its relocation pattern computation but will use the (v,IN ) route instead. From time to time, route merging can be done closer to the sources up to a stable state in a purely localized way. Note that for the sake of clarity we focus on the case of two different sources with a unique sink, however PAMAL could easily be extended to n-sources. 3). An IN node turns back to standard node as soon as it receives already aggregated packets. Its neighbors stop to target it as destination in their relocation pattern computation. As we propose a general purpose routing protocol, the data aggregation mechanism we use is minimalistic. A more advanced data aggregation could be performed with respect to the data and/or application. Aggregated and non aggregated packets are routed using Algorithm 2. Routing process is detailled in Section VI-B.

B. Routing process

Unlike previous solutions, the next hop selection does not simply use one relocation pattern for the next hop computation. On the contrary, current node a holding a packet the COP of the relocation of every node v in N D (a) with each relocation pattern presented in Section IV. This requires higher computing complexity but reduces energy cost. A node a holding a packet execute Alg. 2. First, a checks whether it has been notified for an IN node for this source/destination pair. If true (Alg. 2, line 4) a will use the IN location (Alg. 2, line 5) instead of D location (see Figure 3) in its relocation pattern. Then a computes for each v in N D (a) the COP and relocation according to the three different CoMNet relocation pattern (Alg. 2, lines 11-14). a stores the next hop v which minimizes the COP and its relocation (Alg. 2, lines 15-20) and return it (Alg. 2, line 22). In any case, routing fails if there is no neighboor closer to D (Alg. 2, lines 7-9). PAMAL relies on the CDS introduced by CoMNet to guarantee network connectivity.

VII. PERFORMANCE EVALUATION

In this section, we evaluate the performance of PAMAL compared to CoMNet. We do not compare to other routing protocols which use controlled mobility as CoMNet already outperfomed them [START_REF] Elhafsi | Energy efficient mobile routing in actuator and sensor networks with connectivity preservation[END_REF]. 14:

v mdsr ← CoMNet -MoveDSR(a, v, D) 15: vCOP ← min    COP (a, v, v or ) COP (a, v, v mr ) COP (a, v,

23: end if

A. Simulation Setup

We perform simulations using the last release of WS-Net [START_REF] Fraboulet | Worldsens: Development and prototyping tools for application specific wireless sensors networks[END_REF] network simulator on randomly generated maps where two different sources target one destination and their routing path crosses. Maximal radio range of nodes is set to 50 m. The MAC header size is set to 40Bytes and packet size is either 200Bytes or 400Bytes. Aggregation factor is of 1:2. Node speed is set to 1 m.s -1 . To compute C s (.), we use common values, i.e. c = 3 × 10 -9 J. and α = 2, which lead to an r * = 22.36m. Regarding the mobility model parameter a (see Eq. 2), we run simulations with the following options :

1) if sending cost equals the moving, C s (.) = C m (.), a is solution to the equation C s (r * ) = C m (r * ). 2) if sending cost is much higher than moving cost, C s (.) >> C m (.), then a is solution to C s (r * ) = 10 2 C m (r * ). 3) if moving cost is much higher than sending cost, C s (.) << C m (.), then a is solution to C s (r * ) = 10 -2 C m (r * ).

B. Topology changes

Figures 4 and 5 illustrate the average length of the common part of the two paths in terms of number of hops as a function of time. Results show that PAMAL provides effective path merging and those merged parts are significantly longer than in CoMNet. More precisely, the path merging at the beginning is significantly quicker and higher in PAMAL thanks to the IN advertising regardless the energy cost model or packet size, until death of nodes. When nodes begin to die, path merging turns to be low while it raises again as the IN selection process takes some time in order to adapt to the network topology. With respect to network density, results show that the lower the network density is, the lower gap between our PAMAL proposal and CoMNet is. This happens because most of the nodes are members of the CDS and thus cannot move. When the density increases (δ = 20), more and more nodes are free to move, and thus the energy gain is much higher. With respect to packet size we observe that merged path length fluctuation is higher with a packet size of 400Bytes than with a packet of 200Bytes. It is reasonable since the larger the packets are the more energy they require. Consequently nodes die quicker and topology changes quickly. With respect to cost model, results show that the improvement is minimal when C s (.) << C m (.) compared to the other models as it tends to limit node movements.

C. Energy cost

Figures 6, 7 and 8 compare the average network lifetime as a function of network density for each of the studied algorithms with an initial packet size of 400Bytes. The network lifetime is defined as the average between reception time of the last packet of route (S1,D) and (S2,D). Simulation results show that PAMAL outperforms the other protocols when C s (.) = C m (.) (Figure 6), when C s (.) >> C m (.) (Figure 7) and when C s (.) << C m (.) (Figure 8). Based on our results we can infer that when a sufficient density is reached (δ >= 20) PAMAL achieves a significantly higher network lifetime. The network lifetime extension is maximized when the C s (.) << C m (.) model is used.

VIII. CONCLUSION AND FUTURE WORKS

In this paper we proposed a new kind of routing protocol which takes mobility into account with path crossing in WSN. The mechanism of route merging we exposed provides significant energy savings regardless of the network density or the packet size. Through extensive simulations we show that our approach increases up to 37% the network lifetime compared to the best of our knowledge routing protocol relying on controlled mobility. Future work will focus on more complex environment and network topologies and address the hole bypassing problem.

Fig. 1 :

 1 Fig. 1: Path crossing frequency (in %) with regards to network degree and cost model.

Figure 2 Fig. 2 : 11 :

 2211 Fig. 2: Source nodes S 1 and S 2 route data up to sink D. Routing paths intersect on IN node. from both sources. It becomes an IN that will aggregate data from both sources. Starting from IN , the (IN ,D) route belongs to both (S 1 ,D) and (S 2 ,D). We can observe that source node S1 routes packet towards D by relocating successively nodes A, B and C on the (S 1 D) line. At the same time source node S 2 routes packet towards D by relocating successively nodes Y and Z on the (S 2 D) line. However, nodes C and Z belong to different routing paths provoke oscillation. Node C aims to relocate the IN node on the IN 1 location while node Z asks IN to relocate itself on the IN 2 location. Both C and Z are in competition for the same node, IN . In existing solutions IN would be relocated successively on (S 1 ,D) in IN 1 or (S 2 ,D) in IN 2 for each received packet and would provoke oscillation. In PAMAL, the IN will: • be re-localized on the iso-barycenter of its original, IN 1 and IN 2 locations and refuse to move while it is an IN, • advertize its one-hop neighbors that they should target it as the destination for their relocation pattern. • aggregate packet from both sources and route them towards the destination. Consequently, (S1,D) and (S2,D) tend to be reformed as (S1,IN ,D) and (S2,IN ,D) respectively. On path (IN , K, L, D) aggregated packets are routed. Depending on the distance between the sources and between each source and the destination, the IN moves toward the sources, extending the length of the shared path (IN ,D). In the next subsection we detail how an IN is elected.A. Intersection Node election process

Fig. 3 :

 3 Fig. 3: The IN broadcasts its existence to its 1-hop neighborhood. As a consequence they do not consider D location for their relocating pattern anymore neighbors in their relocation patterns computation. An IN appears dynamically when a node has noticed that it has recieved packets to route from different sources (Alg. 1, Line 7). It advertises its neighbors (Alg. 1, Line 9) to use it as final destination in their relocation pattern computation (Figure 3) instead of D. IN starts aggregating the packets (Alg. 1, Line 8) and forwards them. From time to time, the paths will tend to merge and a new IN may appear ahead to the destination (node C or Z Figure3). An IN node turns back to standard node as soon as it receives already aggregated packets. Its neighbors stop to target it as destination in their relocation pattern computation. As we propose a general purpose routing protocol, the data aggregation mechanism we use is minimalistic. A more advanced data aggregation could be performed with respect to the data and/or application. Aggregated and non aggregated packets are routed using Algorithm 2. Routing process is detailled in Section VI-B.

Algorithm 2

 2 SelectNextHop(a,D) -Run at node a toward destination node D. 1: next ← -1 // next hop node 2: next ← {0,0,0} // next hop relocation 3: minCOP ← +∞ // best COP over N D (a) 4: if isAnIN (p) then 5: D ← intersectionN ode 6: end if 7: if N D (a) = ∅ then 8: return NULL // routing has failed. 9: else 10: for all {v ∈ N D (a)} do 11: v or , v mr , v mdsr ← {0,0,0} 12: v or ← CoMNet -ORouting(a, v, D) 13: v mr ← CoMNet -MoveR(a, v, D)

  part (in Hops) Time (in Second) CoMNet, C s (.)=C m (.) PAMAL, C s (.)=C m (.) CoMNet, C s (.)>>C m (.) PAMAL, C s (.)>>C m (.) CoMNet, C m (.)>>C s (.) PAMAL, C m (.)>>C s (.) (a) δ = 15, Packet size is 200Bytes.

  part (in Hops) Time (in Second) (b) δ = 20, Packet size is 200Bytes.

  part (in Hops) Time (in Second) (c) δ = 25, Packet size is 200Bytes.

  part (in Hops) Time (in Second) (d) δ = 30, Packet size is 200Bytes.

Fig. 4 :

 4 Fig. 4: Length of merged part of routes with regards to time. Packet size is 200Bytes. Caption is the same for all figures.

  part (in Hops) Time (in Second) CoMNet, C s (.)=C m (.) PAMAL, C s (.)=C m (.) CoMNet, C s (.)>>C m (.) PAMAL, C s (.)>>C m (.) CoMNet, C m (.)>>C s (.) PAMAL, C m (.)>>C s (.) (a) δ = 15, Packet size is 400Bytes.

  part (in Hops) Time (in Second) (b) δ = 20, Packet size is 400Bytes.

  part (in Hops) Time (in Second) (c) δ = 25, Packet size is 400Bytes.

  part (in Hops) Time (in Second) (d) δ = 30, Packet size is 400Bytes.

Fig. 5 :

 5 Fig. 5: Length of merged part of routes with regards to time. Packet size is 400Bytes. Caption is the same for all figures.

Fig. 6 :

 6 Fig. 6: Length of merged part of routes with regards to time. Packet size is 400Bytes. C s (.) = C m (.)

Fig. 7 :

 7 Fig. 7: Length of merged part of routes with regards to time. Packet size is 400Bytes. C s (.) >> C m (.)

Fig. 8 :

 8 Fig. 8: Length of merged part of routes with regards to time. Packet sie is 400Bytes. C s (.) << C m (.)

  v mdsr

	16:	if (vCOP < minCOP ) then
	17:	next ← v
	18:	next ← associated relocation for this COP
	19:	minCOP ← vCOP
	20:	end if
	21:	end for
	22:

return {next, next } // routing has succeeded.

This work was partially supported by CPER Nord-Pas-de-Calais/FEDER Campus Intelligence Ambiante and the ANR BinThatThinks project.