F Schoefs
email: franck.schoefs@univ-nantes.fr

F Aury

P Vilvoisin

N Menard

E Bastidas-Arteaga

EXPERIENCE FEEDBACK OF REPAIR TECHNIQUES FOR WHARVES IN THE TIDAL AREA: THE MAREO PROJECT

Keywords: reinforced concrete, repair, wharves, tidal area, reliability, sutainability

Ageing of concrete structures located in coastal environments is very actual challenge since a lot of them have been built from 30 to 50 years ago. Three main challenges are faced by companies and owners: diagnosis, reassessment and repair. The paper focuses on the last one and more precisely, on the repair of concrete beams of wharves in the tidal area. In fact, this zone is the point of confluence of a modeling challenge (non saturate conditions), a repair technique application (repair during the low level tide) and a real stake (most of the beams of concrete wharves located in this area).

INTRODUCTION

Concerning repair of concrete for marine structures, european standards offer the basis in terms of requirements in predefined and standardized conditions. However, repair of wharves is performed in harsh conditions: access, humidity, position of the operator for instance. These conditions move the real works away from the standardized ones. By accounting for these requirements, the repair technique consists in: (1) to rebuild the concrete cover and in some cases (2) to use protective coating for some beams. We focus here on the first point. The aim of Mareo project (French project of competitiveness group for civil engineering) is to compare several repair techniques carried out in the most difficult area for repair: the tidal zone. The project focuses on four criteria: initial performance, sustainability, durability, cost. This paper focuses on the experience feedback of three repair techniques carried out in real conditions of works. The comparative study is carried out on twelve beams that were part of the structural system of a port built in 1927 at Lorient, France and demolished in 2006. The comparative test consists in to subject the repaired beams to controlled tidal cycles during the following years. Since the study is principally addressed to compare the repair techniques independently of products it is preferable to use products of similar composition. This requirement is also important to calibrate the future destructive and non-destructive tests. The selected repair techniques should be applicable to largescale projects; therefore the local "patch" repairs are beyond the scope of the study. The choice of the repair techniques/products should consider that the structural components to be repaired are located in the splash and tidal zones (e.g., beams and piles of quays). The selected techniques were: wet shotcrete, dry shotcrete, formed concrete and manual repair. This paper focuses on three criteria to compare these techniques: sustainability (including water consummation, transport of materials, …), cost, efficiency.

MAIN STAKES OF REPAIR OF WHARVES IN FRANCE

Social and economical stakes: Why this project? 564 various types of ports can be found along the French coastline, 60% of which were built before 1955 (see Figure 1). Various techniques have been used during the last century. Boero et al (2009) have shown that the most current one in France in the seventies and eighties was the on pile supported wharf made on steel piles and a concrete platform supported by concrete beams. Figure 2 presents the distribution, in linear kilometres, of the construction techniques used for the berthing and mooring's structures. They are classified from right to left in order of chronological appearance but, after the six first ones, techniques coexist. The great majority of the structures are masonry, followed by concrete blocks, reinforced-concrete caissons, steel piles supported wharves, steel sheet piles seawalls and diaphragm walls. This figure illustrates the inventiveness of engineers confronted with the challenges of designing maritime structures; however it also means that owners are faced with delicate management decisions with regards to the principles of mechanical functioning and multiple failure modes. In this context, the patrimony to maintain is larger than the patrimony to build. Environmental considerations have raised important questions regarding the new developments and decommissioning of existing infrastructure. Maintenance planning optimization is therefore a major challenge, with multiple constraints imposed by tourism, economical and environmental considerations.

Infrastructure owners must optimizing actions even if the choice of multiple technical solutions is complex, due to the lack of accurate and reliable knowledge on the performance of these techniques and their evolution with time.

Taking these considerations as its basis, the MAREO project (2008( -2010, French project , French project

Scientific stakes:

The research is performed at several levels: -Identify performance indicators of structures with a view to prescribing condition based criteria. -Investigate exhaustively evaluation chains, direct or indirect, of these indicators and compare them in terms of risk analysis. This implies particularly to consider modeling uncertainties and to propose other models based on non-deterministic approaches (for example probabilistic: physical or analytical response surfaces) and to identify uncertainties on performance indicators or influential factors and thereby to characterize the so-called intrinsic hazards.

-Outline experimental devices and protocols, in laboratory or on site (pilot building sites), with controlled conditions of measurement; identify failure mechanisms.

-Suggest computational methods coming from previous works for re-engineering. Develop a methodology for semi-probabilistic analysis through pilot studies (20 x 80 years old beams repaired and tested, repaired structures from harbors (PANSN) and general council (CG44), repaired beam with composite sticking).

Economical expectation from the project results:

Currently the rehabilitation market represents half of the building trade activity and a significant percentage of public works. Companies consider that as in the building trade, the rehabilitation activity will develop in the public works field and may reach a volume near that of new works. Owners of structures are waiting for validated methodologies, service societies require tested models and services and contractors need guidance on optimized techniques.

SELECTION OF MATERIALS IN THE MAREO PROJECT

Since the study is principally addressed to compare the repair techniques independently of products it is preferable to use products of similar composition. This requirement is also important to calibrate the destructive and non-destructive tests. The selected repair techniques should be applicable to large-scale projects; therefore the local "patch" repairs are beyond the scope of the study. The choice of the repair techniques/products should consider that the structural components to be repaired are located in the splash and tidal zones (e.g., beams and piles of quays). That means that work conditions are very difficult: in some cases the repair should be applied in 6 hours with a very quick efficiency.

French standards:

The series "special products for hydraulic concrete constructions" encompasses the French standards in this field. These standards deal the following issues: Wet shotcrete

FIRSTS RESULTS

The study case concerns the Agri-foodstuffs terminal of the port of Nantes Saint-Nazaire, a structure subjected to chloride-induced corrosion. The Autonomous Port of Nantes Saint-Nazaire is the government agency managing the harbor activities of this structure. This agency observed a generalized problem of corrosion affecting mainly the RC beams and decided to perform a large-scale repair [START_REF] Rosquoet | Chloride propagation in concrete harbor[END_REF].

This terminal is part of the port of Nantes Saint-Nazaire (fourth largest port in France) which is linked to 400 ports worldwide. With a maximal draught of 14 m, the Agri-foodstuffs terminal plans to receive big tonnage ships as container carriers (50,000 Ton). The Port of Nantes Saint-Nazaire is the French market leader for cattle feed imports with nearly 60% market share. There are four berths at the Agri-foodstu

P1 P2 P3 Po1 Po2 Po3 P4 Po4 Po5 P5 Po6 Po7 470 cm
s terminal, which also handles fertilizers, peat, cement and other miscellaneous industrial bulk products. This wharf was built in 1971 and is located at the west of France (Montoir de Bretagne) in the estuary of the Loire River.

Assumptions for random variables and stochastic processes

Let us now compare the techniques within an optimization framework in the context of sustainable maintenance. A computational comprehensive model of chloride ingress in un-saturated environment has been developed and is presented in Bastisdas et al. 2010. Table 2 gives the description of random variables and the probabilistic model based on the review of publications [START_REF] Saetta | Analysis of chloride diffusion into partially saturated concrete[END_REF] and the first results of the project [START_REF] Villain | Protocoles de mesures non destructives et profils en chlorure: premiers résultats de marnage accéléré sur dalles en laboratoire[END_REF]. The environment -temperature, humidity and environmental chloride concentration -is modeled as a stochastic process. Trajectories are plotted on figure 6 (see Bastisdas et al. 2010 for details): they are based on classical assumptions along european Atlantic coasts. The stochastic process of chloride ingress is simulated and coefficients of Markov Matrix are assessed. The corrosion is initiated and repair is performed if, respectively the thresholds C th = 2.56 kg/m3 with COV[C th ]=0.20 and C rep = 1.2 kg/m3 are exceeded. Note that imperfect inspection is modeled too but is not discussed in the paper (see [START_REF] Bonnet | Statistical study on total chloride profiles[END_REF].

Table 2. Random variables for modeling chloride ingress

Assumptions for costs and wastes

Assumptions concerning costs of works and relative costs (coefficients applied on C 0 ) are given on table 3.

Table 3. Computed agency costs and coefficients for cost models Parameters used for CO2 emission computation are given in table 4 and waste generation lies on the expert judgment: dry shotcrete requires a higher quantity of material (30% of waste generation) increasing the emissions of transportation and production.

Results of optimization

The multi-objective index (MOI) is used for research of the optimal maintenance strategy [START_REF] Lounis | Risk-based maintenance optimization of aging highway bridge decks[END_REF]. The procedure using Markov chains is presented in [START_REF] Sheils | Development of a two-stage inspection process for the assessment of deteriorating bridge structures[END_REF]. The assessment of the MOI assumes that the agency's policy gives weighting factors of 0.5 for costs (w C ), 0.25 for waste (w W )and 0.25 for CO 2 emissions (w E ). We focus here on one result mainly: the over-cost due to the accounting of environmental criterion. For that the cost factor varies between 0.2 and 1 and the factors for CO 2 and waste are equal and such that the sum equals 1. The cost weighting factor w C = 1 implies that the decision is controlled only by costs. In this case, environmental constrains are not considered, and then, the overcharges are zero. On the contrary, for w C = 0, the overcharges lead to a constant value determined at a larger inspection interval where the repair rate is minimum (i.e., Figure 7). For the range of cost weighting factors presented in Figure 7, the maximum overcharges vary from 10% to 12% depending on the characteristics of each the repair technique. If the agency decides to give equal importance to economical and environmental criteria (i.e., w C = 0.50, w W = 0.25 and w E = 0.25) the overcharges are lower than 6% for all the cases. This means that including environmental constrains in decision making does not generates larger overcharges.

CONCLUSIONS

The analysis of repair efficiency for concrete beams in unsaturated conditions is still a challenge. MAREO project suggest improvements in three main axis:

-the stochastic modeling of material properties and environmental conditions; -new data for repair techniques efficiency; -optimisation accounting for sustainable criterion.

The paper provides both the methodology and results concerning this project. Moreover the method is shown to be efficient when comparing repair techniques when environmental constrains are accounting for. It is shown that, within an optimal policy, these constrains lead to an over-cost less than 6% whatever the repair technique.

Figure 2 .

 2 Figure 2. Distribution of construction techniques of berthing and mooring structures in Autonomous Harbours (AH) and National Interest Harbours (NIH) according to Boero et al. 2009.

  -P18-821: Hydraulic binder based leveling and sealing products. -P18-822: Synthetic resin based leveling and sealing products. -P18-840: Synthetic resin or hydraulic binder based products or products systems intended for hardened concrete surface repairs. -P18-870: Synthetic resin or hydraulic binder products or products systems for structural bonding between two concrete elements. -P18-880: Products based on synthetic resins or binders for injections into concrete structures. -GA P18-902: Application guide -Products and systems for the protection and repair of concrete structures -Recommendations for the selection of surface protection systems for concrete intended for the works of civil engineering. are made in the CERIB laboratory by immersion of 2 to 3 centimeters depth of a slab during 6 hours and a drying with a ventilator during 18 hours. Thus 4 types of specimens are considered. -slabs (40cm x 40cm x 12cm) for accelerated tests. Their dimension allows performing all Non Destructive Tests (NDT); -slabs (50cm x 50cm x 15cm) for mechanical and physical characterization and calibration of NDT tools; -concrete beams placed in basins, in controlled tide; -real wharf in natural conditions. Concerning concrete beams (Figure 4 and 5, table 1), they were part of the structural system of a port built in 1927 at Lorient, France and demolished in 2006. Figures 1 and 2 present the beams before repair and indicates their designation. The comparative test consists in to subject the repaired beams to controlled tidal cycles during the following years. Twelve beams have been repaired: small beams Po have the size: 20x30*150 and large ones, 60*30*470.

Figure 4 .

 4 Figure 4. Beams after 80 years of exposure.

Figure 5 .

 5 Figure 5. Scheme of beam P1 before repair.

Figure 6 .

 6 Figure 6. Stochastic inputs of: (a) weather (temperature, humidity), (b) environmental chloride concentration.

Figure 7 .

 7 Figure 7. Overcharges produced when environmental criteria are considered.

Répartition de l'âge des quais -Ports d'Etat (PM, PA & PIN) 32 32 17 12 8 0 5 10 15 20 25 30 35 t ≤ 1905 1905 < t ≤ 1955 1955 < t ≤ 1980 1980 < t ≤ 1995 t > 1995 (année) (%)

	Figure 1. Date of building of wharves in France (Source: project GEROM (OXAND-GeM) 2005-2008).

  Compare the efficiency of repair techniques on concrete structures in coastal regions, within the global context of reassessment of repaired structures. The project therefore relies on global methodologies for risk analysis and assessment. -Identify and quantify uncertainties and hazard sources in assessing the performance functions.

	-
	The partners are: French laboratories (GEM, LCPC, IFREMER, CERIB, LMDC, CETE de l'Ouest),
	companies (SEMEN-TP, ETPO, ARCADIS-ESG), risk analysis societies (OXAND), owners (CG44 et
	Date of building of wharfs in France PANSN (associated members)), and international laboratories (Trinity College of Dublin (IRELAND)).
	Years

supported by PGCO (West Competitiveness Pole for Civil Engineering), French Government, and French west district 'Pays de la Loire') has been developed with objectives to:

Table 1 .

 1 Beams and corresponding repair techniques

	Beam	Repair technique
	P1	Formed concrete
	P2	Wet shotcrete
	P3	Dry shotcrete
	P4	Manual repair
	P5 (reference sample)	Without repair
	Po1a	Wet shotcrete
	Po1b

Table 4 .

 4 Assumptions for CO 2 emission

ACKNOWLEDGEMENTS

The authors acknowledge financial support of the MAREO project -contact: franck.schoefs@univnantes.fr.

European standards:

From 1st January 2009 the French standards concerning to products and methods to protect and to repair reinforced structures will be substituted by the European standard EN 1504 called "Products and systems for the protection and repair of concrete structures -Definitions, requirements, quality control and evaluation of conformity". This standard is divided into ten parts: Selection within the MAREO project:

By accounting for the requirements described in before paragraphs, the repair technique consists in: (1) to rebuild the concrete cover and (2) to use protective coating for some beams. For all the beams, the contaminated concrete was removed using high-velocity water jets (hydrodemolition). Figure 3 shows a beam after hydrodemolition and the different techniques used to rebuild and increase the cover (from 3 cm to 5cm). The selected techniques were: wet shotcrete, dry shotcrete, formed concrete and manual repair.

Beam after hydrodemolition

Wet/(dry) shotcrete

SELECTION OF SPECIMENTS AND EXPOSURE CONDITIONS

When considering natural exposures like marine environment ones must select with care the specimens and the testing protocol. First it was decided to perform both accelerated and non-accelerated tests. For non accelerated tests, a controlled tide (pure sinusoid) has been obtained in an external basin at IFREMER (Brest city) and a pure natural exposure is considered in the Nantes Harbour. Accelerated tests


Thomas De Larrard

Emilio Bastidas-Arteaga

Frédéric Duprat

Franck Schoefs

Effects of climate variations and global warming on the durability of RC structures subjected to carbonation

Keywords: carbonation, reinforced concrete, corrosion, reliability, climate change

Carbonation affects the performance, serviceability and safety of reinforced concrete (RC) structures when they are placed in environments with important CO 2 concentrations. Since the kinetics of carbonation depends on parameters that could be affected by climate change (temperature, atmospheric CO 2 pressure and relative humidity (RH)), this study aims at quantifying the effect of climate change on the durability of RC structures subjected to carbonation risks. This work couples a carbonation finite element model with a comprehensive reliability approach to consider the uncertainties inherent to the deterioration process. The proposed methodology is applied to the probabilistic assessment of carbonation effects for several cities in France under various climate change scenarios. It was found that climate change and local RH have a significant impact on corrosion initiation risks.

Introduction

The performance, serviceability and safety of reinforced concrete (RC) structures are affected by corrosion-induced deterioration when they are placed in environments with important CO 2 concentrations. Carbonation is a chemical evolution of the cementitious materials, induced by the reactive diffusion of CO 2 in the porous media. Indeed, atmospheric CO 2 reacts with cement hydrates, dissolves portlandite and forms calcite. The main consequence of this chemical reaction is a significant decrease in pH in the porous solution that induces corrosion of the reinforcing steel. The kinetics of this degradation depends on temperature, atmospheric CO 2 pressure and relative humidity (RH), all parameters related with time and climate change.

Natural and anthropogenic actions will increase environmental CO 2 concentrations in the future and produce changes in temperature and humidity. For instance, following the current patterns, an increase in CO 2 concentration from 379 ppm in 2005 over 1000 ppm by the year 2100 is expected (IPCC 2007). Therefore, these new environmental conditions will modify the kinematics of carbonation by increasing or reducing the consequences of corrosion-induced damage. Besides climate change, there are other uncertainties related to the carbonation process. Consequently, probabilistic approaches are suitable tools for assessing structural risk and reliability for this problem.

Various studies have focused on the effect of climate change on the durability of corroded RC structures. [START_REF] Bastidas-Arteaga | Influence of Weather and Global Warming in Chloride Ingress Into Concrete: A Stochastic Approach[END_REF] proposed a stochastic approach to study the influence of global warming on chloride ingress for RC structures. They found that chloride ingress could induce reductions of the corrosion initiation stage varying from 2% to 18%. Concerning corrosion propagation until failure, Bastidas-Arteaga et al. (2013) found that global warming could reduce the time to failure by up to 31% for RC structures subject to chloride ingress. Recent work also focused on the assessment of climate change on the durability of concrete structures in specific locations. Stewart, Wang, and Nguyen (2011) found that the temporal and spatial effects of a changing climate can increase current predictions of carbonation-induced damage risks by more than 16% which means that one in six structures will experience additional and costly corrosion damage by 2100 in Australia and presumably elsewhere. [START_REF] Wang | Impact of Climate Change on Corrosion and Damage to Concrete Infrastructure in Australia[END_REF] studied the impact of climatic change on corrosion-induced damage in Australia. They proposed a probabilistic approach to assess corrosion damage taking into account the influence of climate change on areas characterised by different geographical conditions. [START_REF] Talukdar | Carbonation in Concrete Infrastructure in the Context of Global Climate Change: Part 2, Canadian Urban Simulations[END_REF] estimated the effects of climate change on carbonation in Canadian cities (Toronto and Vancouver). They found potential increases in carbonation depths over 100 years of approximately 45%. However, this work did not consider the uncertainties related to climate, materials and models.

This study uses a probabilistic approach to compare and quantify the effect of different climate change scenarios on the structural safety of an RC structure exposed to a risk of corrosion induced by the cement paste carbonation. In comparison with previous studies, this paper integrated a relevant carbonation finite element model that takes into account all these climate evolutions (Section 3). Probabilistic assessment using this finite element model requires a significant computational effort. Therefore, Section 4 presents the adopted reliability approach that is used in this work. Finally, Section 5 presents an application to the assessment of climate change effects to RC structures placed in various cities in France.

Climate change effects

Increase in carbon dioxide emissions

Following the current patterns, studies have estimated that environmental CO 2 concentration could increase from 379 ppm in 2005 over 1000 ppm by the year 2100 (IPCC 2007). Consequently, the long-term management of RC structures should consider the effect of atmospheric CO 2 concentration change and global warming. The increase in atmospheric CO 2 concentration is difficult to estimate because it depends on several socio-technical and political factors. These factors have been considered by the Intergovernmental Panel on Climate Change (IPCC 2007) that defined four scenarios families of global warming (i.e. A1, A2, B1 and B2). These scenarios integrate different assumptions in terms of population growth, economical development, transfer of clean technologies, etc. The A1 scenario describes a future world of very rapid economic growth, global population that peaks in mid-century and declines thereafter, and the rapid introduction of new and more efficient technologies. This scenario family is divided into three groups distinguished by different technological emphasis: fossil-intensive (A1FI), non-fossil energy sources (A1T) or a balance across all sources (A1B). The A2 scenario describes a very heterogeneous world with continuously increasing population. The economic development is primarily regionally oriented and technological change more fragmented and slower. The B1 scenario is similar to the A1 scenario with respect to population growth, but with rapid change in economic structures towards a service and information economy, with reductions in material intensity and the introduction of clean and resource-efficient technologies. The B2 scenario describes a world in which the emphasis is on local solutions to economic, social and environmental sustainability. The population growth pattern is similar to the scenario A2 but with a lower rate. It also includes more diverse technological solutions. Figure 1 presents a projection from 2000 of environmental CO 2 concentrations the A2, A1B and B1 global warming scenarios. These scenarios were selected in this study because they represent pessimistic, medium and optimistic climate change effects on climate, respectively (Section 2.2). These values were computed using the MAGICC software (Model for Assessment of Greenhouse-gas Induced Climate Change) [START_REF] Wigley | Economic and Environmental Choices in the Stabilization of Atmospheric CO2 Concentrations[END_REF]. Every scenario includes three versions for the CO 2 emissions: a lower one, a medium one and a higher one.

For this study, it has been decided to take into account only the medium one (assessing the higher the CO 2 pressure, the faster the carbonation). Taking into account the description of the scenarios presented previously, the highest projections for CO 2 concentration for 2100 correspond to the A2 scenario.

Effects of climate change on temperature and relative humidity

The effect of climate change on weather will change depending on the geographical and meteorological characteristics of the studied location. The impact of climate change on the future weather of the selected locations was estimated by using data computed by the French general circulation model SCRATCH-ARPEGE-V4-RETIC [START_REF] Déqué | The ARPEGE-IFS Atmosphere Model: A Contribution to the French Community Climate Modelling[END_REF]. This model was selected because it is able to account for climate projections at a regional scale with a 8 km grid. For instance, Figure 2 presents the yearly projections of temperature and RH for the city of Nantes and the selected climate change scenarios. It is noted that climate change projections announce a temperature increase and RH decrease for all scenarios. The most important changes in temperature and RH are related to the larger emission scenarios described in the previous section. Concerning consequences on climate change, it is possible to define A2 as a pessimistic scenario, A1B as a medium impact scenario and B1 as an optimistic scenario.

Carbonation model

This section presents the model used herein for describing the coupled drying and carbonation of concrete, which is detailed in de Larrard et al. (2013a). This is a simplified version of the model proposed in [START_REF] Bary | Coupled Moisture-Carbon Dioxide-Calcium Transfer Model for Carbonation of Concrete[END_REF], which assumes that the main phenomena involved in the carbonation are the water migration through the connected porosity, and the diffusion of carbon dioxide in the gaseous phase and its subsequent interactions with the hydrated phases initially present to produce calcium carbonate (calcite). Accordingly, it is governed by two coupled mass conservation equations for water (Equation ( 1)) and carbon dioxide in gas phase (Equation ( 2)):

∂(ρ l φS r ) ∂t = ∇ K(φ) ρ l η k r (S r )∇(P l ) + W H 2 O , (1)
∂((1 -S r )φP c) ∂t = ∇ f (φ, S r )D c∇ (P c) + W CO 2 .
( 2 )

Equation ( 1) implicitly assumes that the main transport process for moisture results from gradients of partial pressure of water in the liquid phase P l (permeation), i.e. the migration of vapour in gas phase is neglected [START_REF] Mainguy | Role of Air Pressure in Drying of Weakly Permeable Materials[END_REF][START_REF] Bary | Coupled Moisture-Carbon Dioxide-Calcium Transfer Model for Carbonation of Concrete[END_REF]. Moreover, the pressure of the gas phase is neglected with respect to the liquid phase, such that P c ≈ -P l , where P c is the capillary pressure. Equation ( 1) is also driven by P c in the desorption isotherm that accounts for the relationships between the saturation degree S r and P c (van Genuchten 1980):

S r = 1 + (P c /P 0 ) 1/(1-m) -m , (3)
where P 0 and m are calibration parameters. Other parameters involved in Equation ( 1) are K(φ) is the intrinsic permeability coefficient that depends on the initial porosity φ and the initial intrinsic permeability K 0 (van Genuchten 1980); ρ l and η are the density and dynamic viscosity of water, respectively; k r (S r ) is the relative permeability for liquid that depends on the saturation degree S r (van Genuchten 1980) and W H 2 O is the rate of water formation. Equation ( 2) is driven by the partial pressure of CO 2 in the gaseous phase denoted as P c. f (φ, S r ) and D c are the reduction factor [START_REF] Millington | Gas Diffusion in Porous Media[END_REF]) and the diffusion coefficient of CO 2 in gas phase, respectively, such that their product can be considered as the effective diffusion coefficient of CO 2 through the porous material and W CO 2 is the rate of CO 2 dissolution. The coupling between Equations ( 1) and (2) appears through (i) the saturation degree S r , and (ii) the rates of water formation and carbon dioxide dissolution. These rates, W H 2 O and W CO 2 , respectively, derive from the molar formation rate of calcite ( Ṅcal , Equation ( 4)), and the molar dissolution rate of portlandite ( Ṅpor , Equation ( 5)) and of other hydrates ( ṄCSH , Equation ( 6)), mainly C-S-H. We have indeed considered separately the dissolution-precipitation processes of portlandite and of the other hydrates, due to the higher reactivity of portlandite [START_REF] Bary | Coupled Moisture-Carbon Dioxide-Calcium Transfer Model for Carbonation of Concrete[END_REF][START_REF] Thiery | Investigation of the Carbonation Front Shape on Cementitious Materials: effects of the Chemical Kinetics[END_REF].

Ṅcal = Ṅpor + ṄCSH , ( 4 )

Ṅpor = k sl (S r ) k H kp (V p )P c, (5)
ṄCSH = k sl (S r )k H kCSH (C cal )P c, (6)
where k sl (S r ) is a function introduced by [START_REF] Bazant | Nonlinear Water Diffusion in Nonsaturated Concrete[END_REF] to describe at a macroscopic scale the influence of the saturation degree on the chemical reaction processes (i.e. the reduction of water in the pores leads to decrease the reactions); k H is the Henry constant for carbon dioxide; and the functions kp (V p ) and kCSH (C cal ) are kinetic coefficients depending on the current volume fraction of portlandite V p and concentration of calcium in solid phase C cal , respectively. It should be stressed that the simplified chemical reactions relative to the hydrate dissolution-calcite precipitation as expressed by these equations correspond to first-order kinetic law with respect to the CO 2 pressure. The kinetic coefficient for the calcite formation from portlandite kp is assumed to evolve as proposed in [START_REF] Thiery | Investigation of the Carbonation Front Shape on Cementitious Materials: effects of the Chemical Kinetics[END_REF], i.e. the calcite is supposed to form a growing layer surrounding the portlandite crystals during the dissolution process. The consequence is that this layer progressively reduces the corresponding chemical reactions kinetics, which are driven by the diffusive phenomena through the calcite layer. The kinetic coefficient for the other hydrates is assumed to evolve linearly with the calcium concentration.

The main simplification of this numerical model in comparison with the previous version [START_REF] Bary | Coupled Moisture-Carbon Dioxide-Calcium Transfer Model for Carbonation of Concrete[END_REF]) concerns the calcium concentration calculation. In the present approach, no diffusion of calcium is considered and it is instead replaced by a local chemical equilibrium of calcium, meaning that the calcium dissolved from portlandite and the other hydrates is locally transformed into calcite. Formally, this is equivalent to considering that the whole calcite formation process (including CO 2 diffusion) is faster than calcium migration through the solution. de Larrard et al. (2013a) numerically demonstrated that this simplification has a very weak influence on the results in the case of accelerated carbonation. Further investigations, which are beyond the scope of this paper, would be necessary to analyse its effects in atmospheric carbonation conditions. The model presented above has been modified to take into account the influence of temperature variations on carbonation. The modifications of the carbonation model to take into account temperature effects are detailed in de Larrard et al. (2013b), based upon the studies proposed in de [START_REF] De Larrard | Modelling the Influence of Temperature on Accelerated Leaching in Ammonium Nitrate[END_REF]. These modifications include the influence on the following thermo-activated parameters/phenomena: (i) the CO 2 diffusion coefficient, (ii) the dissolution of portlandite, as well as other hydrates and (iii) the constant of Henry (Equation ( 5)), and (iv) the sorption-desorption isotherms. The dependency of temperature of all these parameters/phenomena is modelled with Arrhenius functions. According to [START_REF] Yuan | Prediction of Temperature Response in Concrete in Natural Climate Environment[END_REF], we assume that the temperature inside concrete is constant and equal to the atmospheric temperature. It is to be noted that the effective temperature of RC elements exposed to solar radiation may differ from the ambient temperature, but it is assumed that the time steps used for numerical simulations are large enough to work with a mean value for the temperature and the difference (as the kinetics for the heat transfer is important with regards to these for the aggressive species diffusion) involved by this assumption are neglected. This will be the subject for a future study, beyond the scope of the present paper.

This carbonation model was implemented in the finite element code Cast3M (www-cast3m.cea.fr), and the equations solved successively and iteratively with a fully implicit numerical scheme. To conclude, the present model turns out to provide results in terms of calcite and portlandite concentration profiles very close to the ones obtained with a previous and more complex version of the model. Furthermore, the phenolphthalein carbonation depths calculated in the case of an accelerated test appear in very good agreement with both experimental and numerical results (de Larrard et al. 2013a). We then consider that the model is relevant and suitable for the probabilistic simulations required to determine the reliability index of RC structures under the combined action of climate change and carbonation.

Reliability analysis

A common practical measure of the structural reliability is the Hasofer-Lind's reliability index denoted β [START_REF] Hasofer | An Exact and Invariant Second Moment Code Format[END_REF]. This index is defined in the standardised space of reduced normal and independent variables as the minimum distance from the origin to a point of the failure surface, the so-called design point P * . The determination of β is a constrained optimisation problem [START_REF] Hasofer | An Exact and Invariant Second Moment Code Format[END_REF][START_REF] Rackwitz | Structural Reliability Under Combined Random Load Sequences[END_REF]. The function to be minimised is the Euclidean distance u in the standardised space under the constraint G(u) = 0, where G is the limit-state function. If u * is the solution of the optimisation with β = OP * = u * , then:

u * = arg min( u ) G(u)=0 . ( 7
)
As the carbonation depth is not directly accessible numerically, the failure function for the carbonation model presented in this paper is based on the quantity of portlandite dissolved at the steel/concrete interface. The carbonation depth, as defined by the widely used phenolphthalein test, is simply estimated from the portlandite profile for a threshold value of dissolution. For instance, if the remaining volume fraction of portlandite is less than 33% of the initial value after dissolution at the cover depth, it is considered that corrosion starts. This threshold is among the values reported by [START_REF] Chang | The Experimental Investigation of Concrete Carbonation Depth[END_REF]Chen (2006), de Larrard et al. (2013a) and [START_REF] Park | Carbonation of Concrete in Relation to CO 2 Permeability and Degradation of Coatings[END_REF]. Thus, for this threshold value, the limit-state function becomes

G(u) = V p (u, c t (u), t fin ) -0.33 × V p (u, c t (u), t 0 ), ( 8
)
where V p is the volume fraction of portlandite, depending on c t , which is the concrete cover thickness from the exposed face, and the time. V p is calculated with the FE model presented above. t fin is the final time of the simulations (100 years) and t 0 is the initial time (0 year). It is clear that the choice of the dissolved portlandite threshold will affect the assessment of reliability index. However, for a comparative study of climate change effects on various climates, the global trends will be the same for all the studied cases if the threshold is fixed.

According to design codes, G(u) defines a limit-state function, which is expected to be continuous, at least in the neighbourhood of the design point P * . Most part of optimisation algorithms used in structural reliability are useful for differentiable constraint functions. However, in the carbonation model considered in this study, the limit-state function G is computed by using the finite element method. Therefore, the reliability index is computed according to the gradient projection algorithm in the standardised space [START_REF] Duprat | The Projection Gradient Algorithm with Error Control for Structural Reliability[END_REF]. The result of the reliability analysis could therefore provide β over the lifetime t fin .

Numerical example: application to RC structures placed in France

Problem description

The objective of this example is to estimate the effects of several scenario of climate change on the carbonation of an RC structure placed in several cities in France. The material considered is a CEM I concrete with a water/cement ratio equal to 0.42% and 68% aggregate volume fraction. The material is initially set to saturation degree corresponding to the year before 2000, and exposed to drying, carbonation and temperature variations according to climate change scenarios for each city. It is also supposed that carbonation takes place in one dimension. The material properties as well as the model parameters were chosen equivalent to those proposed in de Larrard et al. (2013a).

Table 1 presents the five random variables considered in this study. Concrete cover corresponds to a global structural uncertainty related to a variability introduced by the building process. The initial porosity of the cement paste has a significant influence on the kinetics of the diffusion processes studied here, as it appears in every term of the governing equations. Two other variables concern the drying process: the intrinsic permeability K 0 and the m parameter from the van Genuchten model, which govern the desorption isotherm. The last random variable is the diffusion coefficient for CO 2 . These four last parameters are materials properties. Their 'randomness'stands for the natural uncertainty and spatial variability of the materials properties, even for the same concrete formulation. These random variables follow independent normal distributions with a coefficient of variation (COV) equal to 5%. These means and COVs are among the values observed for a given concrete formulation [START_REF] De Larrard | Concrete Calcium Leaching at Variable Temperature: Experimental Data and Numerical Model Inverse Identification[END_REF](de Larrard et al. , 2013a)). It is to be noted that the choice of a normal distribution may not be the most relevant, even though it is in good accordance with the data observed in de Larrard et al. (2013). Besides, these values for the COVs are within the lower values experimentally observed. The COVs depend on the mix design and the implementation process of the concrete. However, this study aims mainly at comparing the different scenarios for different locations; therefore it is mostly important to keep the same hypothesis for all cases.

This work focused on the study of the effects of climate change on specific locations in France: Marseille, Paris, Toulouse, Strasbourg, Clermont-Fd. and Nantes (Figure 3). These cities correspond to different types of climate: Nantes is close to the Atlantic Ocean in the Northern part of the country and has a temperate oceanic climate; Toulouse is in the South and its climate has continental, oceanic and Mediterranean influences; Marseille in the South-East on the Mediterranean coast and has a Mediterranean climate rather hot and dry; Strasbourg in the North-East, close to the German frontier with a northern continental climate; Paris is more in the centre of the country with an intermediate continental climate; and Clermont-Fd. is in the mountains of 'Massif Central', in elevated South-Central France and has rather a continental climate.

Results and discussions

As mentioned in Section 2, climate change effects on weather will be different for each location. By using the outputs of the French atmosphere model, Table 2 describes the mean temperature and RH for the first decade of the study (2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010). It is observed that Marseille is the most hot and dry city, Strasbourg is the most cold and Nantes is the most humid. Tables 3 and4 summarise the effects of climate change by comparing the mean values of the first (2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010) and the last decade (2091-2011) for each climate change scenario. Figure 2 presents an example of evolution of T and RH for one of the studied locations, whereas Tables 3 and4 summarise these evolutions for the different cities and for various periods through the elevation of their mean values. Table 3 indicates that temperature could increase from 1.14 • C up to 3.63 • C with respect to current values for all the locations over the period 2001-2100. It is noted that Strasbourg could be more affected by climate change. Table 4 shows that climate change could reduce the current RH by -0.21% up to -4.92% with respect to current RH. The major changes could be expected for Paris. It is also confirmed that the A2 scenario is the most pessimistic with larger temperature increases and RH decreases. The B1 scenario predicts lower changes in temperature and RH and the A1B scenario is intermediate between A2 and B1. Table 5 presents the reliability indexes for these climate change scenarios over a period of 100 years. It is observed that there is a large variation of the reliability index from β < 0 to β > 8. Let us specify here that a reliability index negative refers to a situation where the probability of corrosion initiation is higher than 50%. This implies a larger difference in the safety issues for the extreme cases. This larger variability is due to the fact that the same material has been tested for cities with different climates (Table 2). For instance, the climate is very different in Nantes and Marseille. Since the exposure conditions are not the same, we should design a specific formulation for a concrete or select a different concrete cover adapted for each city. Considering that the same concrete mix design for such different exposure conditions was required only for the needs of the comparison within the framework of this study. The reliability index for Marseille was β < 0, indicating that the probability of corrosion initiation is very high in a period of 100 years. The lower values of β for Marseille are explained, on the one hand, because the temperature is more important and carbonation is thermo-activated. On the other hand, RH varies between 65% and 70% by accelerating carbonation. These results could justify the implementation of an adaptation measure for this type of climate that could consist of increasing the concrete cover or the strength grade (Bastidas and Stewart 2013;[START_REF] Stewart | Climate adaptation engineering and risk-based design and management of infrastructure[END_REF]).

On the opposite, corrosion initiation risks are very low for the cities of Strasbourg, Clermont-Fd. and Nantes. Even considering the effects of climate change, the lower temperatures and higher RH for these cities decelerate the carbonation process. For example, RH in Nantes remains between 80% and 85% which corresponds to a 'retarding effect', modelled by the function presented in Equation ( 2) [START_REF] Millington | Gas Diffusion in Porous Media[END_REF]): when RH is larger (RH > 80 %), there is not enough gaseous phase in the pores for the diffusion process to occur efficiently. This 'natural safety' due to a higher RH appears mainly because only carbonation is studied here, but the opposite effect would appear if the corrosion process was considered instead (because its kinetics is driven by water saturation).

Climate change could have important effects on structures placed in Paris and Toulouse. Although Paris has lower temperatures than southern cities, the value of RH for the decade 2001-2010 is within the optimal interval for accelerating carbonation (Table 2). Climate changes also accelerate carbonation in Paris by a significant decrease in RH (Table 3). Toulouse has a similar behaviour to that of Paris. On the one hand, the higher temperature in Toulouse accelerates the carbonation effects; on the other hand, the larger RH reduces them. It is therefore possible to conclude that, for the studied locations, carbonation is more sensitive to RH than temperature. It is to be noted that, except for Marseille, the reliability index is higher than the 1.5 value given for a serviceability limit state in Eurocode 0. The reliability index calculated here only takes into account the carbonation process, and not the oxygen transfer for instance which is linked to the corrosion process kinetics. Therefore, no conclusion concerning actions required on the RC elements considered here can be drawn directly from these results.

Concerning the effects of different climate change scenarios, it appears that the reliability index decreases when temperature increases and RH decreases. The impact of the scenario is significant for Paris and Toulouse. For instance in Paris, according to B1-scenario, the reliability index is close to 6.6, which corresponds to a rather safe area, but according to A2-scenario, it is reduced to 1.7. This means that, under A2-scenario, the RC structures should be surveyed with care or adaptation measures implemented, because the risk of carbonation around the steel rebar (and corrosion initiation) is significant.

This discussion will end with some comments about the coordinates of the design point in the standardised space. These coordinates are presented in Table 6 for the A1B (intermediate scenario). These coordinates indicate how many standard deviations separate the design point from the origin of the standardised space (being the mean value for each input parameter). A high value for a coordinate indicates that the design point is located in the distribution tail for the considered parameter. Therefore, it implies that this parameter has a significant influence on whether the mean point is in the security domain or in the failure area.

These results indicate that the most influent random variable on the reliability analysis is the concrete cover thickness. The other random variables have a similar order of importance. One can also discuss the physical sense of these coordinates. It is obvious that increasing cover thickness increases the structural reliability as it lengthens the path from the external environment to the steel rebar. A reduction of the coefficient of diffusion or the permeability, as well as the porosity, decreases the transfer ability of the porous material (either for CO 2 or water or even both). The last parameter is the m factor from the desorption isotherm (Equation (3)). A decrease in this parameter rises the desorption isotherm curve in a (RH; S r ) plot, meaning that for the same hydric state (governed by the water pressure, and therefore the RH), the saturation degree S r is higher. This reduces the gaseous phase in the pores, where the CO 2 diffuses, and thus decelerates the carbonation process.

Conclusions and perspectives

In this paper, we proposed a probabilistic finite element model for carbonation under variable conditions of temperature, RH and CO 2 concentration. This model allows us to determine the reliability index of RC structures exposed to a risk of corrosion induced by the cement paste carbonation and climate change. We calculated this reliability index for six cities in France corresponding to different kinds of climate (oceanic, Mediterranean and continental) according to several scenarios of climatic changes.

This study has shown that carbonation is very sensitive to local climate. There are locations with a large carbonation risk for which current environmental conditions could accelerate corrosion initiation (Marseille). A specific design improvement and/or adaptation strategy should be considered for these climates. There are also other climates for which carbonation risks are lower for an optimistic climate change scenario but could be significantly increased for the most pessimistic scenario (Paris and Toulouse). These results highlight the importance of considering the climate evolution in lifetime assessment to predict, as precisely as possible, carbonation risks. This study has also shown that in first approach, the most influent parameter on the structural safety remained the concrete cover thickness, more than materials properties (such as porosity, permeability, diffusion coefficient or desorption isotherm). This work appeals in-depth studies about a comprehensive characterisation of the probability distributions of the random variables. Also the correlations between the materials properties involved in the degradation processes could be considered. Another final improvement is the selection of a threshold defining the carbonation depth and its relationship with effective corrosion initiation.

Figure 1 .

 1 Figure 1. CO 2 emissions for the studied scenarios.

Figure 2 .

 2 Figure 2. Temperature and RH projections for Nantes.

Figure 3 .

 3 Figure 3. Studied locations in France.

Table 1 .

 1 Statistical parameters of the random variables[START_REF] De Larrard | Concrete Calcium Leaching at Variable Temperature: Experimental Data and Numerical Model Inverse Identification[END_REF](de Larrard et al. , 2013a)).

	Variable	Unit	Distribution	Mean	COV (%)
	Concrete cover, c t	cm	Normal	2	5
	Initial porosity, ϕ 0 Intrinsic permeability, K 0	% m 2	Normal Normal	30 1.8 × 10 -22	5 5
	Parameter, m (Equation (3)) Diffusion coefficient, D c	m 2 /s	Normal Normal	0.532 1.9 × 10 -8	5 5

Table 2 .

 2 Mean temperature and RH over the period 2001-2010.

	Scenario	Marseille	Paris	Toulouse	Strasbourg	Clermont-Fd.	Nantes
	Temperature	14.75	13.04	14.08	11.53	11.66	12.76
	RH	68.78	74.22	77.36	77.02	75.44	80.65

Table 3 .

 3 Change in temperature (in • C) over the period 2001-2100.

	Scenario	Marseille	Paris	Toulouse	Strasbourg	Clermont-Fd.	Nantes
	A2	+3.04	+3.59	+3.23	+3.63	+3.43	+3.22
	A1B	+2.21	+2.56	+2.29	+2.58	+2.39	+2.23
	B1	+1.16	+1.33	+1.18	+1.40	+1.28	+1.14

Table 4 .

 4 Change in RH (in %) over the period 2001-2100.

	Scenario	Marseille	Paris	Toulouse	Strasbourg	Clermont-Fd.	Nantes
	A2	-2.61	-4.92		-4.79	-4.64		-4.59	-3.96
	A1B	-1.10	-3.44		-2.79	-2.87		-2.33	-2.75
	B1	-0.21	-1.14		-0.93	-1.11		-0.79	-0.59
			Table 5. Reliability indexes for the dif-
			ferent climate change scenarios.
			City		A2	A1B	B1
			Marseille		< 0	< 0	< 0
			Paris		1.70	3.68	6.55
			Toulouse		2.09	3.90	6.91
			Strasbourg		7.51	> 8	> 8
			Clermont-Fd.	> 8	> 8	> 8
			Nantes		> 8	> 8	> 8

Table 6 .

 6 Design point coordinates in the standardised space for A1B scenario.

	City	c t	φ 0	K 0	D c	m
	Marseille	3.05	-1.58	-0.28	-1.62	-4.49
	Paris	-3.09	0.72	0.38	0.91	1.58
	Toulouse	-3.48	0.87	0.38	1.03	1.07
	Strasbourg	-8.26	1.29	0.76	1.64	2.37
	Clermont-Fd.	-10.68	1.46	0.69	1.73	2.55
	Nantes	-14.29	1.26	0.63	1.53	1.81

Acknowledgements

The authors would like to thank Météo-France for providing the climate change projections used in this work.

Funding

The authors would like to thank the French Alternative Energies and Atomic Energy Commission (CEA) for its financial and technical support to implant the numerical model in CAST3M.


domain: spi phys

text:
Nicolas Locatelli

Damir Vodenicarevic

Weisheng Zhao

Jacques-Olivier Klein

Julie Grollier

Damien Querlioz

Vortex-based spin transfer oscillator compact model for IC design

I. INTRODUCTION

The recent development of spin transfer torque magnetic random access memories (STT-MRAMs) unveiled the potential of CMOS integration of spintronic devices into innovative architectures [START_REF] Khvalkovskiy | Basic principles of STT-MRAM cell operation in memory arrays[END_REF]- [START_REF] Zhang | Spintronics for low-power computing[END_REF]. Among the large diversity of spintronic elements, other devices also hold potential for applications [START_REF] Prenat | Beyond MRAM, CMOS/MTJ integration for logic components[END_REF], [START_REF] Locatelli | Spin-torque building blocks[END_REF]. Magnetic tunnel junctions (MTJs), the basic cell of the MRAM, can also be engineered to achieve self-sustained magnetic oscillations under DC current bias, and the spintransfer effect thereof. These magnetization oscillations are then transcribed into resistance oscillations through the tunnel magneto-resistance (TMR) effect, generating oscillations of the device voltage at frequencies typically ranging from 100MHz to 10's of GHz depending on the device geometry and magnetic parameters.

The small size of the so-called spin-transfer nanooscillators (STNOs) and their ability to tune their frequency with the supply current amplitude are strong assets for their integration as radio-frequency sources in communication systems [START_REF] Villard | A GHz spintronic-based RF oscillator[END_REF]- [START_REF] Choi | Spin nanooscillator-based wireless communication[END_REF]. Also recently, those STNOs were proposed as key-elements for the design of associative memories architectures based on highly connected oscillators networks [START_REF] Levitan | Associative processing with coupled oscillators[END_REF], [START_REF] Csaba | Spin torque oscillator models for applications in associative memories[END_REF], expected to yield high computing capability notably for pattern recognition. This proposal is based on the demonstrated ability of the STNOs to achieve mutual synchronization with one or more interacting neighbors under adequate conditions [START_REF] Kaka | Mutual phase-locking of microwave spin torque nano-oscillators[END_REF]- [START_REF] Belanovsky | Numerical and analytical investigation of the synchronization of dipolarly coupled vortex spin-torque nano-oscillators[END_REF]. For all these purposes, the development of compact SPICE-compatible models for integration in CMOS design platforms is a prerequisite.

Previous works on compact modeling of spin transfer oscillators have focused on a certain type of oscillators based on the excitation of a uniform magnetic mode [START_REF] Stan | Nano-pattemed coupled spin torque nano oscillator (STNO) arrays ; a potentially disruptive multipurpose nanotechnology[END_REF], [START_REF] Csaba | Modeling of coupled spin torque oscillators for applications in associative memories[END_REF], and models were integrated under the strong assumption of macrospin behavior. The present work focuses on STNOs based on the excitation of a magnetic vortex. This type of oscillators has experimentally demonstrated high quality factors as compared to other types of STNOs [START_REF] Pribiag | Magnetic vortex oscillator driven by d.c. spin-polarized current[END_REF], [START_REF] Dussaux | Large microwave generation from current-driven magnetic vortex oscillators in magnetic tunnel junctions[END_REF], as well as a capabilities to tune its frequency with the bias current amplitude and to synchronize to an external excitation from an AC-source [START_REF] Dussaux | Phase locking of vortex based spin transfer oscillators to a microwave current[END_REF], [START_REF] Lehndorff | Injection locking of the gyrotropic vortex motion in a nanopillar[END_REF] or to mutually-synchronize to coupled oscillators [START_REF] Belanovsky | Numerical and analytical investigation of the synchronization of dipolarly coupled vortex spin-torque nano-oscillators[END_REF].

We propose a Verilog-A implementation of the physical model describing the self-sustained vortex oscillations in a pillar STNO for a use in IC design of hybrid MTJ/CMOS systems. To correctly describe the oscillator behavior at working temperature, our model implementation includes parameters non-linearities [START_REF] Dussaux | Field dependence of spin-transfer-induced vortex dynamics in the nonlinear regime[END_REF] as well as the phase and amplitude noise effects under thermal fluctuations [START_REF] Tiberkevich | Temperature dependence of nonlinear auto-oscillator linewidths: Application to spin-torque nano-oscillators[END_REF], [START_REF] Grimaldi | Response to noise of a vortex based spin transfer nano-oscillator[END_REF]. These factors can indeed be of major importance to accurately describe the synchronization phenomenon.

After introducing the device and the associated magnetic and electrical equations, we present the model implementation under the Cadence platform and confront the obtained simulations to previously obtained experimental results. Finally, we demonstrate the compatibility of our model with a CMOS design kit for IC design by simulating an integrated current source for biasing and synchronization of the STNO.

II. THE SPIN-TRANSFER VORTEX OSCILLATOR MODEL

The device under consideration is a nanopillar hybrid spin transfer vortex based oscillator, as introduced in [START_REF] Dussaux | Large amplitude spin torque vortex oscillations at zero external field using a perpendicular spin polarizer[END_REF] and presented in Figure 1(a). It is made of a layered stack, first composed of a metallic spin-valve structure including a perpendicularly magnetized ferromagnetic layer, a non-magnetic spacer and a central ferromagnetic layer. The latter is also part of a magnetic tunnel junction, composed of an insulating barrier and an in-plane magnetized ferromagnetic layer. The magnetizations of both perpendicular and in-plane magnetized ferromagnets are considered pinned. Dimensions of the pillar and particularly the aspect ratio of the central ferromagnet are chosen so that its remanent magnetic state is a vortex [START_REF] Guslienko | Magnetic vortex state stability, reversal and dynamics in restricted geometries[END_REF]. Such magnetic state corresponds to the magnetization curling in the layer's plane and popping out-of-plane in the center region called the vortex core (see Figure 1). The studied mode of the vortex, called "gyrotropic mode", corresponds to a circular displacement of the vortex core around the center of the magnetic dot [START_REF] Guslienko | Magnetic vortex state stability, reversal and dynamics in restricted geometries[END_REF], [START_REF] Thiele | Steady-state motion of magnetic domains[END_REF] as illustrated in Figure 1(b).

As current flows through the device, the two pinned layers act as spin-current sources generating a spin-transfer effect on the vortex. As demonstrated in previous works, a source of perpendicular spin polarization is mandatory to start and sustain the vortex precession [START_REF] Mistral | Current-driven vortex oscillations in metallic nanocontacts[END_REF], [START_REF] Khvalkovskiy | Vortex oscillations induced by spin-polarized current in a magnetic nanopillar: Analytical versus micromagnetic calculations[END_REF]. On the other hand, the MTJ with in-plane magnetized layer is needed to convert the magnetization oscillations into resistance oscillations [START_REF] Dussaux | Large amplitude spin torque vortex oscillations at zero external field using a perpendicular spin polarizer[END_REF], and plays a major role in the synchronization process by providing an in-plane component to the current spin polarization [START_REF] Khvalkovskiy | Nonuniformity of a planar polarizer for spin-transferinduced vortex oscillations at zero field[END_REF]. Under sufficient current bias, the device then demonstrates permanent resistance oscillations and a resulting AC voltage appears across the device.

Because the vortex gyrotropic mode is isolated from other higher frequency modes, the self-sustained oscillations under spin-transfer effect can be properly described by a collectivevariable equation, called the Thiele equation [START_REF] Guslienko | Magnetic vortex state stability, reversal and dynamics in restricted geometries[END_REF], [START_REF] Thiele | Steady-state motion of magnetic domains[END_REF], where the global dynamics is simply described through the evolution of the position of the magnetic vortex core. This equation was recently extended to include the spin-transfer terms [START_REF] Dussaux | Field dependence of spin-transfer-induced vortex dynamics in the nonlinear regime[END_REF], [START_REF] Mistral | Current-driven vortex oscillations in metallic nanocontacts[END_REF], [START_REF] Khvalkovskiy | Vortex oscillations induced by spin-polarized current in a magnetic nanopillar: Analytical versus micromagnetic calculations[END_REF]. The device is then modeled according to the following equations, where X = ρe iχ is the vortex core complex coordinate (see Figure 1(b)):

iG Ẋ = -D(ρ) Ẋ -k(ρ)X + iκ ⊥ IX + κ I + F noise (t) (1) R = R min 1 + ξ C TMR % Im(X) r ( 2
)
Equation 1 is the modified Thiele equation describing the dynamics of the vortex core, accounting for conservative and dissipative terms. The left hand side gyrotropic term is equal to the sum of the effective forces acting on the vortex core : the effective damping force, the spring-like confinement force, and the effective spin-transfer forces induced respectively by the perpendicular and the in-plane spin polarizations of the current. Equation 2 relates the value of the resistance to the position of the vortex core. All the terms appearing in these equations are derived from geometrical and magnetic parameters. Their has been added, describing the action of thermal fluctuations according to fluctuation-dissipation theorem, and derived in the framework of the classical nonlinear oscillators theory [START_REF] Tiberkevich | Temperature dependence of nonlinear auto-oscillator linewidths: Application to spin-torque nano-oscillators[END_REF], [START_REF] Grimaldi | Response to noise of a vortex based spin transfer nano-oscillator[END_REF], [START_REF] Kamppeter | Finite temperature dynamics of vortices in the two dimensional anisotropic heisenberg model[END_REF]. Finally, oscillations amplitude limitation is ruled by the existence of a critical velocity v c for the vortex core that will trigger a polarity switching and subsequently stop the self-sustained oscillations [START_REF] Khvalkovskiy | Vortex oscillations induced by spin-polarized current in a magnetic nanopillar: Analytical versus micromagnetic calculations[END_REF].

III. MODEL IMPLEMENTATION

The described physical model was implemented in Ver-ilogA language under the Cadence Spectre simulator. Simulations were conducted for a 2r = 180nm diameter device, with a t = 10nm thick free N iF e layer whose magnetic parameters are gathered in table I. The minimum resistance was chosen to be R min = 130Ω with a TMR ratio TMR % = 70% for the tunnel junction, in agreement with previously fabricated devices. A p ⊥ = 60% perpendicular polarization (from the the spin-valve) and an ideal p = 100% in-plane polarization (from the MTJ) were assumed. These parameters correspond to a critical 220µA current to start the self-sustained oscillations.

The model was tested under constant current bias for transient simulations, in both cases of noise-free and noisy oscillator. The vortex core position was initiated in a random position in the vicinity of the ferromagnet center, allowing the observation of a transitory regime where the oscillations amplitude increases up to its stable value, starting the steady regime of oscillations.

A major criteria for the validation of our model was the stability of the noise-free auto-oscillations frequency and amplitude in the steady-state regime and their independence to the simulation time step. Due to different time-scales for the evolution of the phase and the amplitude of oscillations, we found that the compact model is largely improved by implementing equation ( 1) under polar coordinates instead of Cartesian. We then evaluated that to ensure a steady-state frequency stability, a reasonable criteria was ensuring that the time step for the simulation verifies ∆t < T (ρ)/50 where T (ρ) = 2π/ω(ρ) is the instantaneous oscillation period of the vortex. We enforce this minimum time step thanks to a bound step VerilogA call within the compact model.

The thermal fluctuation term was implemented through a random Gaussian drawing at each integration step of the differential equation. Compatibility of this model with fluctuationdissipation theory was then verified by evaluating the average fluctuating energy at equilibrium. At zero bias, this energy = 1 2 k(ρ)ρ 2 was verified to be equal to the thermal energy k B T .

10µs simulations of a single oscillator on a CPU were obtained in respectively 9.43s and 9.81s for the noisefree and room-temperature noisy oscillator. In Figures 2(a) and (b), we present the temporal evolution of the device voltage under a constant 260µA current bias in both cases. The observed auto-oscillation frequency is 617.8MHz, in good agreement with the expected frequency for the chosen device dimensions [START_REF] Guslienko | Magnetic vortex state stability, reversal and dynamics in restricted geometries[END_REF]. As it appears on the corresponding spectrum on Figure 2(c), fluctuations in the noisy oscillator translate into phase noise that broadens the spectrum peak, accounting for the non-zero peak linewidth observed experimentally [START_REF] Dussaux | Large amplitude spin torque vortex oscillations at zero external field using a perpendicular spin polarizer[END_REF]. A 14.6MHz linewidth is measured, independent on the time step, in good agreement with the 14.0MHz theoretically predicted linewidth [START_REF] Tiberkevich | Temperature dependence of nonlinear auto-oscillator linewidths: Application to spin-torque nano-oscillators[END_REF].

IV. COMPATIBILITY WITH CMOS DESIGN KIT

We have tested the compatibility of our model with a commercial 28-nm CMOS design kit by building a MOSFETbased supply for the STNO (see Figure 3). The circuit is a voltage controlled current source. A cascode architecture is implemented to reduce the impact of voltage oscillations across the STNO on the current bias. The frequency of the autooscillator is then controlled by the gate voltage of T 1 transistor, as summarized in Figure 3.

The circuit is designed so that a zero command voltage corresponds to below threshold current bias. Self-sustained oscillations appear when a V cde > 10mV voltage is applied, and their amplitude keeps increasing as V cd is swept up to 180mV above which the critical vortex speed is reached and oscillations are terminated. Because both current bias and resistance oscillations amplitude are different for every frequency value, it is important to note that the AC voltage amplitude is also varying as the command voltage is swept. As it appears on Figure 3, the voltage amplitude varies between 0 and 15mV.

Spin transfer nano-oscillators show the ability to respond to an AC-current excitation by synchronizing to the input signal for sufficient amplitude and if its frequency is close enough to the auto-oscillation frequency [START_REF] Dussaux | Phase locking of vortex based spin transfer oscillators to a microwave current[END_REF], [START_REF] Lehndorff | Injection locking of the gyrotropic vortex motion in a nanopillar[END_REF]. To test the ability of the implemented model to describe this ability, we simulated the response of the STNO when a harmonic AC-voltage is added to the command voltage. The simulation was run for a V cde = 120mV corresponding to a 261µA current bias and a frequency 619MHz, to which an AC-signal at a frequency 609MHz was added. We then found that a minimum 100mV amplitude is necessary for the AC-input for the STNO to shift its frequency and synchronize, corresponding to an AC-current amplitude i AC = 27.2µA.

V. CONCLUSION AND PERSPECTIVES

We proposed and tested a Verilog-A implementation of a vortex-based spin transfer oscillator compact model for CMOS integration. This model accounts for the non-linear behavior of the oscillator as well as the thermally induced fluctuations of its phase and amplitude. It has been found to describe the frequency tunability with bias current as well as the synchronization ability of the oscillators, in accordance with experimental results.

After being tested under ideally constant current bias, the device model was successfully inserted in a MOSFET-based circuit for supply and frequency control of the STNO using a commercial CMOS design kit. We then demonstrated the compatibility of the proposed model for integration in IC design platforms.

This model can readily be used for of STNOs which have been proposed for telecommunications. Additionally, the use of STNOs and of their ability to synchronize have been brought up as a serious candidate for the design of innovative processing architectures, especially for data recognition [START_REF] Levitan | Associative processing with coupled oscillators[END_REF], [START_REF] Csaba | Spin torque oscillator models for applications in associative memories[END_REF]. As vortex-based spin transfer oscillators are promising candidates for the study and proof of concept for these architecture, this model will be useful for IC design of such future hybrid MTJ/CMOS systems.

Fig. 1 .

 1 Fig. 1. (a) Sample description. Arrows illustrate the magnetization distribution in ferromagnets. (b) Illustration of the vortex magnetization dynamics. Arrows and color scale respectively describe the in-plane and out-of-plane magnetization distribution.

Fig. 2 .

 2 Fig. 2. (a) Simulated voltage versus time for a noise-free oscillator under a constant 260µA bias. (b) Simulated voltage versus time and (c) associated spectrum for a noisy oscillator under a constant 260µA bias. Full Width at Half-Maximum is determined by fitting the spectrum peak with a Gaussian function.

Fig. 3 .

 3 Fig. 3. (a) MOSFET-based control circuit for the spin transfer oscillator, simulated using a commercial 28nm-node design kit. (b) Frequency and peakto-peak amplitude of the output voltage Vout versus command voltage.

TABLE I .

 I GEOMETRICAL AND MAGNETIC PARAMETERS FOR THE FREE FERROMAGNETIC LAYER AND SPIN POLARIZED CURRENT.

	Symbol	Expression	Description
	r	90 nm	Pillar radius
	t	10 nm	Free layer thickness
	µ0M S	1.0 T	Free layer saturation magnetization
	α	0.01	Free layer Gilbert damping
	Le	6nm	Free layer exchange length
	p ⊥	60%	Current spin polarization along perpendicular z-axis
	p	100%	Current spin polarization along in-plane x-axis

TABLE II .

 II PARAMETERS FOR THE THIELE EQUATION MODELING THE VORTEX GYROTROPIC MOTION. γ IS THE ELECTRON GYROMAGNETIC RATIO, µ B IS THE BOHR MAGNETON, µ 0 IS THE VACUUM PERMEABILITY, g = 2 IS THE LANDE FACTOR AND e IS THE ELECTRON CHARGE.

	Symbol	Expression			Description	Units
	G	2πtM S /γ			Gyrovector amplitude	N.s/m
	η	1 2 ln	r 4Le	-1 8	1 st order damping	unitless
	η	1/12					2 nd order damping	unitless
	D(ρ)	αG η + η ρ r	2	Non-linear damping	N.s/m
	kms	10 9 µ0M S	2 t 2 /r		1 st order magneto-static	N/m
							confinement
	k ms	kms/4				2 nd order magneto-static	N/m
							confinement
	k Oe	0.85µ0M S t/(πr)	1 st order Oersted field	N.m/A
							confinement
	k Oe	-k Oe /2				2 nd order Oersted field	N.m/A
							confinement
	k(ρ)	(kms + k Oe I)		Non-linear confinement	N/m
		+ k ms + k Oe I	ρ r	2	coefficient
	a J	gµ B / (2teM S )		Spin transfer efficiency	m 2 /A/s
	κ ⊥	a J p ⊥ G/ 2πr 2		Orthogonal spin transfer	N.m/A
	κ	a J p GLe ln(2)/ πr 2	In-plane spin transfer	N.m 2 /A
	ξ	2/3					Average magnetization to	unitless
							vortex displacement ratio
	ω(s 2 )	k(ρ)/G				Instantaneous frequency	rad/s
		+αη(ρ)κ ⊥ I/G	(amplitude dependent)
	vc	1 3 Leγµ0M S			Critical vortex velocity	m/s

expressions are gathered in table II. Additionally, on the right-hand side, a phenomenological stochastic force F noise (t)

ACKNOWLEDGMENT

The authors acknowledge E. Grimaldi and J.-V. Kim for fruitful discussions. This work was supported by the ANR COGNISPIN (ANR-13-JS03-0004-01) and the FP7 ICT BAMBI (FP7-ICT-2013-C) projects, and by Labex NanoSaclay (Chaire fiabilité des nanos).