Sarah Hettal

Sébastien Roland

Konsta Sipila

Harri Joki

Xavier Colin
email: xavier.colin@ensam.eu

A new analytical model for predicting the radio-thermal oxidation kinetics and the lifetime of electric cable insulation in nuclear power plants. application to silane cross-linked polyethylene

Keywords:

A new analytical model for predicting the radio-thermal oxidation kinetics and the lifetime of electric cable insulation in nuclear power plants. application to silane cross-linked polyethylene

Introduction

Since the early 1980s, in order to meet the urgent need of nuclear power plant operators, the polymer aging community is looking for a non-empirical approach for lifetime prediction of polymeric materials exposed in nuclear environments. A practical case that mobilized renowned research teams around the world is the radio-thermal ageing of electric cable insulations under normal operating conditions in a nuclear reactor. According to Khelidj et al. [START_REF] Khelidj | A simplified approach for the lifetime prediction of PE in nuclear environments[END_REF] , at room temperature in air, the lifetime t F of electric cable insulations made of unfilled and unstabilised polyethylene (PE) varies with dose rate I according to the general trend depicted in Fig. 1 . This almost universal long-term behavior of PE seems to be completely independent of its macromolecular structure (i.e. linear, branched or chemically cross-linked) and its crystallinity.

This curve displays a negative curvature in the domain of low dose rates where it tends towards a horizontal asymptote (when I → 0) corresponding to the "ceiling lifetime" t F0 due to "pure" thermal oxidation. Schematically, it can be decomposed into three distinct domains, each one corresponding to a different kinetic regime:

-"Pure" thermal oxidation predominates at very low dose rates (typically for I < I T ≈ 1.6 × 10 -7 Gy.s -1 ) and is characterised by very long lifetimes:

t F 0 ≈ 18 . 5 years (1)
-"Pure" radio-oxidation predominates at high dose rates (typically for I > I R ≈ 5.0 × 10 -1 Gy.s -1 ) and is characterized by a lethal dose independent of the dose rate. As a result, the lifetime is proportional to the reciprocal dose rate:

t F = 1 . 6 × 10 5 I -1

(2)

Fig. 1. Shape in logarithm co-ordinates of the lifetime t F versus γ dose rate I in air at room temperature for unstabilised and unfilled PE [START_REF] Khelidj | A simplified approach for the lifetime prediction of PE in nuclear environments[END_REF] . The shaded area (in red) corresponds to the normal operating conditions of electric cables in a nuclear reactor.

the other during the exposure [START_REF] Colin | Lifetime prediction of polyethylene in nuclear plants[END_REF] : polymer radiolysis (1R) and thermal decomposition of hydroperoxides (1T):

1R) PH + h ν → P • + 1 / 2 H 2 r i = 10 -7 G i I 1T ) δPOOH → λP • + μPO • 2 ( k 1 )
where r i is the rate and G i is the radical yield (expressed in number of radicals P • per 100 eV absorbed) of the radiochemical initiation. k 1 is the rate constant and δ, λ, and μ are stoichiometric coefficients describing the molecularity of thermal initiation such as: k 1 = k 1u , δ = 1, λ = 2 and μ = 0 for unimolecular decomposition, and k 1 = k 1b , δ = 2 and λ = μ = 1 for bimolecular decomposition.

To our knowledge, there is still no analytical equation for describing the entire shape of the curve t F = f(I) reported in Fig. 1 .

In the second past half century, the vast majority of studies were devoted to the rigorous identification and understanding of the oxidation mechanisms, but exclusively at high dose rates where the duration and cost of aging experiments under irradiation can be significantly reduced [START_REF] Colin | Determination of thermal oxidation rate constants by an inverse method. Application to polyethylene[END_REF][START_REF] Niki | Aging and degradation of polyolefins. -I. Peroxide-initiated oxidations of atactic polypropylene[END_REF][START_REF] Decker | Aging and degradation of polyolefins. -II. γ -initiated oxidations of atactic polypropylene[END_REF][START_REF] Decker | Aging and degradation of polyolefins. -III. Polyethylene and ethylene-propylene copolymers[END_REF][START_REF] Carlsson | Polyolefin oxidation: Quantification of alcohol and hydroperoxide products by nitric oxide reactions[END_REF][START_REF] Carlsson | On the structures and yields of the first peroxyl radicals in gamma-irradiated polyolefins[END_REF] . When I > I R , it was noted that the induction period is really too short to be measured. It was thus concluded that the oxidation kinetics already starts in steady regime from the early periods of exposure. In addition, it was also observed that the effect of dose rate is a simple kinetic consequence of the radio-oxidation reaction. Schematically, the formation rate of oxidation products is proportional to I 1/2 when they result from the propagation step (of the mechanistic scheme), whereas it is proportional to I when they result from the initiation or termination step.

Based on only two assumptions: steady-state (for radical species) and thermal stability of hydroperoxides throughout the exposure, analytical equations were proposed in the literature to account for these experimental trends. As an example, the two following general equations were determined from a kinetic analysis of the mechanistic scheme for the formation rate r Y (expressed in mol.L -1 ) and the radiochemical yield G Y (dimensionless) of a given chemical event Y, respectively [START_REF] Khelidj | A simplified approach for the lifetime prediction of PE in nuclear environments[END_REF] :

r Y = 10 -7 G Y I (3)
with G Y = 10 7 a Y + 10 7 b Y I -1 / 2 (4)

where a Y and b Y I -1/2 are initiation/termination and propagation terms, respectively. However, even at low temperature close to ambient, the second assumption (thermal stability of hydroperoxides) can only be valid for sufficiently short durations. Indeed, it should be recalled that the rate of the thermal decomposition of hydroperoxides is an increasing function of their concentration [START_REF] Carlsson | Degradation and stability of polymers subjected to high energy radiation[END_REF] . Consequently, thermal initiation is expected to gradually gain in importance with the accumulation of hydroperoxides throughout the exposure, to finally become competitive with the radiochemical initiation. It is obvious that the global oxidation kinetics should be even more impacted in the case of an industrial application for which a very long lifetime of several decades is expected.

To our knowledge, the pioneering researchers in this field, to have propose a kinetic model in which irradiation generates hydroperoxides which decompose thermally to give radicals initiating new oxidation chains, are Gillen & Clough [START_REF] Gillen | A kinetic model for predicting oxidative degradation rates in combined radiation-thermal environments[END_REF] . Their model predicts an auto-acceleration of the oxidation kinetics during the exposure whereas, in all the previous models (for instance, see Eq. ( 3) ), oxidation occurred at a constant rate. Unfortunately, Gillen & Clough chosen questionable initial conditions:

[ POOH ] = [ POOH ] 0 = 0 at t = 0 . However, polymers (especially industrial grades having undergone processing operations) contain always thermally unstable species such as hydroperoxides, otherwise it would be difficult to explain their thermal or photochemical oxidizability. As a consequence of these unrealistic initial conditions, the model developed by Gillen & Clough diverges in the domain of very low dose rates. Indeed, it predicts that t F → ∞ when I → 0.

In most recent publications [ 11 , 12 ], Gillen tried to get around this problem by relying on the research works of Ito [START_REF] Ito | Application of chemorheology to radiation damage of polymer -III. Synergism of heat and radiation on the chemorheology of ethylene-propylene rubber[END_REF][START_REF] Ito | Application of chemorheology to radiation damage of elastomer[END_REF][START_REF] Ito | Degradation of elastomer by heat and/or radiation[END_REF][START_REF] Ito | Methodology study of qualification of electric wire and cable used in nuclear power plant[END_REF] and Mares et al. [START_REF] Mares | Determination of the lifetime of ethylene-propylene rubber under the simultaneous action of heat and ionizing radiation[END_REF] . He considered an extremely simplified radiothermal oxidation mechanistic scheme composed of only three first-order reactions: "pure" radio-oxidation, "pure" thermal oxidation, and their coupling, which is assumed to lead to synergistic effects. To take into account these synergistic effects, he defined a global time-dependent rate constant k R+T as follows: [START_REF] Decker | Aging and degradation of polyolefins. -II. γ -initiated oxidations of atactic polypropylene[END_REF] where k R and k T are effective rate constants to be determined experimentally from the changes in mechanical properties (conventionally, elongation at break) in pure radiochemical (i.e. under high dose rate at low temperature) and pure thermal environments (i.e. in the absence of irradiation), respectively. The effective rate constant k S of the synergistic reaction is deduced from the changes in mechanical properties measured in radio-thermal environment.

k R+T ( t ) = k R ( t ) + k T ( t ) + k S ( t )
Finally, Gillen defined the synergistic effects as follows:

q ( t ) = k R+T ( t ) k R ( t ) + k T ( t ) = 1 + k S ( t ) k R ( t ) + k T ( t ) (6)
If the ratio q > 1, the synergistic effects are clearly indicated. However, as the rate constants k R , k T and k S are time dependent, q varies non-monotonically throughout the exposure [START_REF] Gillen | Importance of synergism for degradation of elastomers in combined radiation plus temperature environments[END_REF] .

However, at its current state, this new approach is not yet completely convincing for the following reasons. From a practical point of view, it does not allow avoiding the issue of the choice of initial conditions. Instead of an initial concentration ( [ POOH ] 0 ), it is now necessary to determine an initial rate constant for thermal oxidation ( k T0 ), but also for radio-oxidation ( k R0 ) and for synergistic reaction ( k S0 ).

From the perspective of a non-empirical lifetime prediction, this new approach raises a series of fundamental issues that have never been addressed by its author, for instance: i) Eq. ( 5) was postulated but never justified. Thus, the assumption of a simple linear combination of the contributions of the radio-and thermal oxidations and their synergistic effects remains to be demonstrated. ii) Even if Eq. ( 5) was valid, it would remain to determine nonempirical equations for the three rate constants: k R , k T , and k S = f(t) . iii) After having fully developed the kinetic model of radio-thermal oxidation, it will also remain to predict the consequences of chemical changes on the mechanical properties.

In our opinion, the two first issues could be solved by a careful kinetic analysis of the radio-thermal oxidation mechanistic scheme. In contrast, the last issue requires the rigorous identification and understanding of the embrittlement mechanisms and their kinetics (for instance, the relative predominance of chain scissions over cross-linking), but also the establishment of a series of structure/property relationships in order to fully describe the causal chain presented in Fig. 2 , as already done, for instance, in references [START_REF] Fayolle | Mechanism of degradation induced embrittlement in polyethylene[END_REF][START_REF] Fayolle | Review: Degradation-induced embrittlement in semi-crystalline polymers having their amorphous phase in rubbery state[END_REF][START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. -I. Chemical aspects[END_REF][START_REF] Rodriguez | Effect of UV-aging on the mechanical and fracture behavior of low density polyethylene[END_REF] .

The present research work aims at developing a robust analytical model for accurately simulating the radio-thermal oxidation kinetics of electrical cable insulations in nuclear power plants. To reach this goal, a new analytical method for solving the system of differential equations derived from the mechanistic scheme will be detailed. Then, the reliability and limitations of the corresponding analytical solution will be checked experimentally from the changes in the physico-chemical properties of a silane crosslinked polyethylene (Si-XLPE) exposed in various radio-thermal environments. All these results will allow identifying accurately the radio-thermal oxidation behavior of this new generation insulating material, and will help to compare it with other types of PE much better known in the literature [ 1 , 2 ], which will be naturally chosen as reference materials in this study.

Experimental

Materials

Free additive Si-XLPE films of about 500 μm thick were directly provided by Nexans NRC. These films were produced by extrusion of a linear low density polyethylene (LDPE) grafted with vinyl trimethoxy silane side-groups. The chemical cross-linking was then performed by immersion in water at 65 °C for 48 h [START_REF]Deliverable D2.2: Specifications of tests -accelerated ageing protocol, test strategy[END_REF] . The density and gel content of the resulting Si-XLPE are about 0.912 g.cm -3 and 71%, respectively.

Radio-thermal ageing conditions

Radio-thermal ageing was performed in the Panoza facility at UJV Rez, Czech Republic, with a 60 Co γ -ray source at different temperatures. All the exposure conditions are summarized in Table 1 .

It should be noted that the ageing experiments numbered 1, 3, and 4 were performed at three distinct dose rates (8.5, 77.8, and 400 Gy.h -1 ) at low temperature close to ambient in order to investigate the effect of dose rate on the oxidation kinetics. In contrast, ageing experiments numbered 1 and 2 were performed at almost the same dose rate (6.0 and 8.5 Gy.h -1 ) but at two different temperatures (47 and 86 °C) in order to investigate the effect of temperature on the oxidation kinetics.

Experimental characterization 2.3.1. Uniaxial tensile testing

After ageing, some films were characterized by uniaxial tensile testing to determine their lifetime t F and thus, compare the longterm durability of Si-XLPE with PE (see Fig. 1 ). The elongation at break ε R was measured with a Instron 5500K8810/4505H2190 machine at 23 °C, 50% RH and a 50 mm.min -1 crosshead speed on dumb-bell specimens according to standards ISO 527-1:2012 and ISO 527-2:2012 [ 23 , 24 ]. For each radio-thermal ageing condition, ε R was plotted as a function of the time of exposure and the corresponding lifetime t F was determined using the conventional endof-life criterion ε F for electric cable application in nuclear industry: t = t F when ε R = ε F = 50% .

FTIR spectroscopy

Other films were characterized by classical physico-chemical laboratory techniques to evaluate the progress of oxidation and to determine the consequences of oxidation on the structure of the macromolecular network. In particular, FTIR spectroscopy was used in transmission mode to monitor and titrate the oxidation products. FTIR spectra were recorded from 40 0 0 to 40 0 cm -1 with a Perkin Elmer FTIR Frontier spectrometer, after averaging the 16 scans obtained with a resolution of 4 cm -1 . For each film, at least three FTIR measurements were performed. As expected, a large amount of carbonyl and hydroxyl products was detected within the film samples from the early periods of exposure.

As an example, Fig. 3 shows the changes in the carbonyl region (i.e. between 1650 and 1800 cm -1 ) of the IR spectrum in air under the highest (i.e. 400 Gy.h -1 ) and lowest dose rates (8.5 Gy.h -1 ) at low temperature close to ambient. In both cases, it can be clearly observed that a broad IR absorption band centered at 1714 cm -1 rapidly builds up. According to literature [START_REF] Geuskens | Photo-oxidation of polymers -Part V: A new chain scission mechanism in polyolefins[END_REF][START_REF] Domke | Oxidative structures in polyolefins: FTIR method of quantitative determination[END_REF][START_REF] Lacoste | Gamma-, photo-, and thermally-initiated oxidation of linear low density polyethylene: A quantitative comparison of oxidation products[END_REF][START_REF] Cruz | Thermo-oxidative degradation of additive free polyethylene -Part I: Analysis of chemical modifications at molecular and macromolecular scales[END_REF][START_REF] Rapp | Influence of the physical state of a polymer blend on thermal ageing[END_REF][START_REF] Blivet | Thermal oxidation of cross-linked PE and EPR used as insulation materials: multi-scale correlation over a wide range of temperature[END_REF] , this band is mainly attributed to the stretching vibration of the C = O bond of carboxylic acids. However, due to its large width, the presence of many other secondary carbonyl products such as: unsaturated (at 1690-1705 cm -1 ) and saturated ketones (1720-1725 cm -1 ), aldehydes (1730-1735 cm -1 ), linear (1740-1750 cm -1 ) and cyclic esters (i.e. γ -lactones, 1780-1790 cm -1 ), is highly probable [START_REF] Geuskens | Photo-oxidation of polymers -Part V: A new chain scission mechanism in polyolefins[END_REF][START_REF] Domke | Oxidative structures in polyolefins: FTIR method of quantitative determination[END_REF][START_REF] Lacoste | Gamma-, photo-, and thermally-initiated oxidation of linear low density polyethylene: A quantitative comparison of oxidation products[END_REF][START_REF] Cruz | Thermo-oxidative degradation of additive free polyethylene -Part I: Analysis of chemical modifications at molecular and macromolecular scales[END_REF][START_REF] Rapp | Influence of the physical state of a polymer blend on thermal ageing[END_REF][START_REF] Blivet | Thermal oxidation of cross-linked PE and EPR used as insulation materials: multi-scale correlation over a wide range of temperature[END_REF] .

In the same way, it was clearly observed that an even broader IR absorption band in the hydroxyl region (i.e. between 3100 and 3700 cm -1 , see Fig. 4 ) also quickly grows. According to literature [ 25 , 27-30 ], this band is attributed to the stretching vibration of the O-H bond of hydroperoxides, alcohols, and carboxylic acids. The concentrations of carboxylic acids and hydroxyl products [Prod] were determined by applying the classical Beer-Lambert's law:

[ Prod ] = OD ep ε ( 7
)
where OD is the optical density of the IR absorption band (dimensionless), ε is the molar extinction coefficient for the corresponding oxidation product (expressed in L.mol -1 .cm -1 ), and ep is the film thickness (in cm).

The orders of magnitude of ε were determined in a previous publication [START_REF] Cruz | Thermo-oxidative degradation of additive free polyethylene -Part I: Analysis of chemical modifications at molecular and macromolecular scales[END_REF] . It was found that ε = 680 and 70 L.mol -1 .cm -1 for carboxylic acids and hydroxyls, respectively.

Differential scanning calorimetry

Differential scanning calorimetry (DSC) was used under pure nitrogen to monitor and titrate the hydroperoxides (POOH), but also to highlight the changes in crystalline morphology during the exposure. DSC thermograms were recorded with a TA instruments DSC Q10 0 0 calorimeter beforehand calibrated with an indium reference. Film samples with a mass ranged between 5 and 10 mg were introduced in a closed standard aluminum pan to be analyzed between -50 °C and 250 °C, with a heating rate of 10 °C.min -1 under a nitrogen flow of 50 mL.min -1 .

As an example, Fig. 5 shows the changes in the DSC thermogram in air under the highest (i.e. 400 Gy.h -1 ) and lowest dose rates (8.5 Gy.h -1 ) at low temperature close to ambient. It can be clearly observed that an exothermic peak starts to grow above the melting zone of the crystalline phase. This new peak is typically ranged between 130 and 230 °C. As shown in a previous publication [START_REF] Cruz | Thermo-oxidative degradation of additive free polyethylene -Part I: Analysis of chemical modifications at molecular and macromolecular scales[END_REF] , it results from the thermal decomposition of POOH in the DSC cavity.

The concentration of hydroperoxides [POOH] was determined as follows:

[ POOH ] = H POOH H theory [START_REF] Carlsson | On the structures and yields of the first peroxyl radicals in gamma-irradiated polyolefins[END_REF] where H POOH is the area under the exothermic peak between 130 and 230 °C on the DSC thermogram and H theory is the theoretical value of the decomposition enthalpy of POOH, which can be calculated from the classical theoretical concepts of thermochemistry or determined experimentally from POOH model compounds: H theory = 291 kJ.mol -1 [START_REF] Cruz | Thermo-oxidative degradation of additive free polyethylene -Part I: Analysis of chemical modifications at molecular and macromolecular scales[END_REF] .

In addition, the global crystallinity ratio X C of the polymer was determined with the common equation:

X C = H m H m0 × 100 ( 9
)
where H m is the sum of the areas under the endothermic peaks observed between 50 and 125 °C on the DSC thermogram, and H m0 is the melting enthalpy of the PE crystal:

H m0 = 292 J.g -1 [31] .

Swelling measurements

Film samples were swelled with a good solvent to determine their swelling ratio and deduce the concentration of elastically active chains in the macromolecular network. A sample having an initial mass m i was introduced into xylene previously heated at 130 °C for 24 hours until reaching its equilibrium mass m S after complete swelling. This sample was then dried under vacuum at 80 °C for 48 hours in order to determine the gel mass m d . The swelling ratio Q was calculated as follows:

Q = 1 + ρ pol ρ sol . ( m S -m d ) m d ( 10
)
where ρ pol and ρ sol are the respective densities of Si-XLPE ( ρ pol = 0 . 912 ) and xylene ( ρ sol = 0 . 761 ).

The Flory-Rhener's theory [START_REF] Flory | Statistical mechanics of cross-linked polymer networks -II. Swelling[END_REF] was then used to determine the concentration of elastically active chains ν between two consecutive chemical cross-links in the macromolecular network:

ν = ln ( 1 -V r0 ) + V r0 + χV 2 r0 V sol V 1 3 r0 -2 V r0 f ( 11
)
where V sol is the molar volume of xylene ( V sol = 139 . 3 c m 3 . mo l -1 ), χ is the Flory-Huggins' interaction parameter between xylene and Si-XLPE ( χ = 0.31 [ 33 , 34 ]), f is the functionality of chemical crosslinks (f = 4 for Si-XLPE) and V r0 is the volumic fraction of polymer in the swollen macromolecular network, of which expression is given below:

V r0 = 1 1 + ( Q -1 ) ρ pol ρ sol (12)

Rheometry in solid state (DMTA)

Rheometry in solid state (DMTA) was used in tension mode to determine the storage modulus E' at the rubbery plateau and check the concentrations of elastically active chains determined by swelling in xylene. DMTA thermograms were recorded with a TA instruments DMA Q800 viscoelastic analyzer. Parallelepipedic film samples of about 30.3 mm long and 6.3 mm large were attached to the device clamps to be analyzed between 25 and 200 °C, with a heating rate of 3 °C.min -1 , a frequency of 1 Hz, and a strain amplitude of 0.2% under nitrogen.

As an example, Fig. 6 shows the DMTA thermogram of Si-XLPE before its radio-thermal ageing. It can be clearly observed that the melting of the crystalline phase leads to sudden decrease of the storage modulus of more than one decade around 120 °C. The rubbery plateau of the completely amorphous Si-XLPE is clearly visible for temperatures higher than 140 °C.

The Flory's theory [START_REF] Flory | Principles of polymer chemistry[END_REF] was used to determine the total concentration of elastically active chains ν t from the values of E' measured at 140 °C:

E = 3 ν t ρ pol RT (13)
where ρ pol is the polymer density at 140 °C ( ρ pol = 897 kg . m -3 ), R is the perfect gas constant ( R = 8 . 314 Pa . m 3 . mo l -1 . K -1 ), and T is the absolute temperature (expressed in K).

When the macromolecular network is not swollen by a solvent, in addition to chemical cross-links, it is expected that E' depends also on a second type of cross-links, which are not of a chemical but physical nature, namely entanglements. In a first approach, ν t can be written as the sum of these two contributions: [START_REF] Ito | Application of chemorheology to radiation damage of elastomer[END_REF] where ν e would be the concentration of elastically active chains resulting only from entanglements in the macromolecular network (i.e. in the total absence of chemical cross-links). The value of ν e can be simply deduced by comparing the rheometry and swelling data.

ν t = ν + ν e
For the unaged Si-XLPE, it was found that: ν t = 0 . 14 ± 0 . 01 mol . L -1 and ν = 0 . 12 ± 0 . 02 mol . L -1 , so that: ν e = 0 . 03 ± 0 . 02 mol . L -1 . In other words, it was found that ν e is very low and almost of the same order of magnitude than the experimental scattering of ν, i.e. almost undetectable. It was thus concluded that there are too few entanglements to significantly influence the concentration of elastically active chains. Consequently, entanglements were neglected in Si-XLPE and the concentration of elastically active chains ν resulting only from chemical cross-links was simply deduced as follows:

ν = E 3 ρ pol RT (15)

Rheometry in molten state

For the radio-thermal ageing conditions under consideration, chain scissions were found very efficient in Si-XLPE. Indeed, a very advanced damage of the macromolecular network was observed for moderate to high conversion ratios of the oxidation reaction, as the film samples were almost completely dissolved in xylene and their rubbery plateau was no longer detected by DMTA (the polymer passed directly from semi-crystalline to molten state). Therefore, when swelling and DMTA measurements were no longer possible, rheometry in molten state was performed at 140 °C to determine the shear modulus G' and thus, access the concentration of elastically active chains. Strain sweep curves were recorded with a TA Instruments ARES viscoelastic analyzer. Disc film samples of 25 mm diameter and 1 mm thick were introduced in a coaxial parallel plate geometry to be analyzed at 140 °C with an angular frequency of 1 rad.s -1 and a strain amplitude ranging between 0.01 and 100% under nitrogen flow.

As an example, Fig. 7 shows the strain sweep curve after 167 h of radio-thermal exposure in air under the highest dose rate (i.e. 400 Gy.h -1 ) at room temperature. As expected, it can be observed that G' reaches a constant value in the domain of low strains (typically for ε < 10%) where linear viscoelasticity is valid. This value Fig. 7. Sweep strain curve of Si-XLPE after 167 h of exposure in air under 400 Gy.h -1 at 21 °C.

was thus chosen for calculating the concentration of elastically active chains with Eq. [START_REF] Ito | Degradation of elastomer by heat and/or radiation[END_REF] . Knowing that: E = 3 G , it comes finally:

ν = G ρ pol RT (16)

Theory

Radio-thermal oxidation mechanistic scheme

As said in introduction, in the domain of practical interest for nuclear power plant operators (i.e. for 1.6 × 10 -7 < I < 5.0 × 10 -1 Gy.s -1 ), oxidation is initiated by both polymer radiolysis (1R) and thermal decomposition of hydroperoxides (1T). The corresponding radio-thermal oxidation mechanistic scheme can be thus written as follows:

Initiation:

1R) PH + hn → P • + 1 / 2 H 2 r i = 10 -7 G i I 1T) δPOOH → λP • + μPO • 2 ( k 1 )
Propagation:

2)

P • + O 2 → PO 2 • (k 2 )
3) PO 2

• + PH → POOH + P • (k 3 )
Terminations:

4) P • + P • → Inactive products (k 4 ) 5) P • + PO 2 • → Inactive products (k 5 ) 6) PO 2 • + PO 2 • → Inactive products + O 2 (k 6 )
where PH, POOH, P • , and PO 2

• designate an oxidation site, an hydroperoxide, alkyl and peroxy radicals, respectively. δ, λ, and μ are stoichiometric coefficients. r i , G i , and k j (with j = 1, …, 6) are a rate, the radical yield, and rate constants, respectively. In most cases of "pure" thermal oxidation, POOH decomposition is exclusively unimolecular at high temperature (i.e. k 1 = k 1u , δ = 1, λ = 2, and μ = 0) or mainly bimolecular at moderate to low temperature (k 1 = k 1b , δ = 2, and λ = μ = 1), but rarely a combination of both modes. In fact, it can easily be shown that there is a hyperbolic shape boundary delimiting the two domains in which each thermal initiation predominates over each other in a map (T, t). This boundary corresponds to a critical hydroperoxide concentration [ POOH ] C reached when the respective rates of the two thermal initiation modes are equal [START_REF] Colin | Determination of thermal oxidation rate constants by an inverse method. Application to polyethylene[END_REF] . Since the activation energy is higher for the unimolecular than bimolecular mode: E 1u > E 1b , [ POOH ] C is an increasing function of temperature. In PE, it writes [START_REF] Khelidj | Oxidation of polyethylene under irradiation at low temperature and low dose rate -Part II: Low temperature thermal oxidation[END_REF] :

[ POOH ] C = k 1u k 1b = 2 . 9 × 10 3 Exp - 35 0 0 0 RT ( 17
)
The numerical application of Eq. [START_REF] Mares | Determination of the lifetime of ethylene-propylene rubber under the simultaneous action of heat and ionizing radiation[END_REF] gives that [ POOH ] C would be ranged between 10 -3 and 10 -2 mol.L -1 at low temperature close to ambient for PE. This is the same order of magnitude of the initial concentration of hydroperoxides [ POOH ] 0 usually reported in the literature for many hydrocarbon polymers [ 28 , 37 ]. That is the reason why, in the present study, it will be considered that "pure" thermal oxidation is directly initiated in bimolecular mode:

1b) 2POOH → P • + PO 2 • (k 1b )

System of differential equations

At the low conversion ratios of the oxidation reaction (i.e. for [PH] ≈ [PH] 0 ≈ constant), the following system of differential equations can be derived from the previous radio-thermal oxidation mechanistic scheme:

d [ P • ] dt = r i + k 1b [ POOH ] 2 -k 2 C [ P • ] + k 3 [ PO • 2 ] [ PH ] -2 k 4 [ P • ] 2 -k 5 [ P • ] [ PO • 2 ] (18) d [ PO • 2 ] dt = k 1b [ POOH ] 2 + k 2 C [ P • ] -k 3 [ PO • 2 ] [ PH ] -k 5 [ P • ] [ PO • 2 ] -2 k 6 [ PO • 2 ] 2 (19) d [ POOH ] dt = -2 k 1b [ POOH ] 2 + k 3 [ PO • 2 ] [ PH ] ( 20
)
where the oxygen concentration C is related to the oxygen partial pressure P O2 in the exposure environment by the classical Henry's law:

C = S . P O2 ( 21
)
and S is the coefficient of oxygen solubility into the polymer. Typical values of S reported for low density polyethylene (LDPE) in the literature are about 1.8 × 10 -8 mol.L -1 .Pa -1 regardless the temperature [START_REF] Seguchi | Diffusion and solubility of oxygen in γ -irradiated polymer insulation materials[END_REF] . Knowing that in air under atmopspheric pressure: P O2 = 0 . 21 × 10 5 Pa , it comes finally:

C = 3 . 8 × 10 -4 mol . L -1 .
There is no rigorous analytical solution for this system of differential equations, which can be solved numerically without any simplifying assumption [START_REF] Colin | Lifetime prediction of polyethylene in nuclear plants[END_REF] . However, an approached analytical solution would be very useful because it would allow appreciating the behavioral trends of the reactive species

[P °], [PO 2 °], [POOH]
with key exposure parameters, in particular: γ dose rate I, temperature T, and oxygen partial pressure P O2 . In addition, this analytical solution would then allow calculating key physico-chemical properties, from a practical point of view, because they can be checked experimentally, such as: oxygen consumption (Q), carbonyl buildup ([P = O]), damage state of macromolecular network ( ν), etc. (see section 3.4 ). Finally, if a structural end-of-life criterion was identified, it would be then possible to propose an equation for predicting the lifetime of the insulating material (see section 3.5 ).

Analytical solving is only possible through the use of simplifying assumptions. As said in introduction, the two most widely used simplifying assumptions are: steady-state (for radical species) and thermal stability of hydroperoxides. If the first assumption seems to be well founded and allows explaining the constancy of the oxidation rate in the first periods of exposure in radio-thermal environments, in contrast, the second one prevents accounting for the auto-acceleration of the oxidation kinetics in the rest of the exposure. This acceleration is due to the decomposition of hydroperoxides above a critical concentration [ POOH ] F , which is assumed to be a fraction of the steady concentration of hydroperoxides [ POOH ] ∞ reached in the case of "pure" thermal oxidation [START_REF] Colin | Lifetime prediction of polyethylene in nuclear plants[END_REF] .

Next sections are devoted to the proposal and validation of a sound analytical kinetic model for radio-thermal oxidation taking into account the hydroperoxide decomposition.

Analytical solving

Denote r 1 the global initiation rate of the oxidation reaction. It writes as the sum of the radiochemical and thermal initiation rates: [START_REF]Deliverable D2.2: Specifications of tests -accelerated ageing protocol, test strategy[END_REF] In all the developments which follow, it will be assumed that oxidation is mainly initiated by the polymer radiolysis throughout the exposure (i.e. r i 2 k 1 [ POOH ] 2 ) even if, at long-term, hydroperoxide decomposition is added as additional (but secondary) source of radicals. In other words, only the right part of the domain of intermediary dose rates will be the subject of a kinetic analysis (see Fig. 1 ).

r 1 = r i + 2 k 1 [ POOH ] 2
Applying the steady-state assumption to radical species leads to:

d [ P • ] dt + d [ PO • 2 ] dt = 0 (23) i . e . k 4 [ P • ] 2 + k 5 [ P • ] [ PO • 2 ] + k 6 [ PO • 2 ] 2 - r 1 2 = 0 ( 24
)
whose positive root is:

[ P • ] = k 5 [ PO • 2 ] 2 k 4 -1 + 1 + ψ [ PO • 2 ] 2 0 [ PO • 2 ] 2 -1 1 / 2 (25)
where is a kinetic parameter quantifying the competition between the different terminations of radical species:

ψ = 4 k 4 k 6 k 2 5 ( 26
)
and [ PO • 2 ] 0 is the steady concentration of peroxy radicals in oxygen excess:

[ PO • 2 ] 0 = r 1 2 k 6 1 / 2 (27) i . e . [ PO • 2 ] 0 = ( r i 2 k 6 ) 1 / 2 ( 1 + 2 k 1 [ POOH ] 2 r i ) 1 / 2 (28) i . e . [ PO • 2 ] 0 ≈ r i 2 k 6 1 / 2 (29)
As expected, [ PO • 2 ] 0 takes approximatively the same value as in a "pure" radiochemical environment [START_REF] Khelidj | Oxidation of polyethylene under irradiation at low temperature and low dose rate -Part I: The case of "pure" radiochemical initiation[END_REF] .

If, as shown by Gillen et al. [START_REF] Gillen | General solution for the basic autoxidation scheme[END_REF] , << 1, then there is a more or less wide range of oxygen concentrations for which:

1 ≥ [ PO • 2 ] 2 [ PO • 2 ]
2 0 > 5 ψ 1 + 5 ψ [START_REF] Blivet | Thermal oxidation of cross-linked PE and EPR used as insulation materials: multi-scale correlation over a wide range of temperature[END_REF] In this range of oxygen concentrations, Eq. ( 25) can be simplified as follows:

[ P • ] = 2 k 6 k 5 [ PO • 2 ] 0 -[ PO • 2 ] (31)
If introducing Eq. ( 31) into Eq. ( 19) and if applying the steadystate assumption to peroxy radicals, it comes finally: [START_REF] Hendra | The morphology of linear polyethylenes crosslinked in their melts. The structure of melt crystallized polymers in general[END_REF] where β -1 corresponds to the critical value of the oxygen concentration C C above which oxygen excess is reached:

[ PO • 2 ] = [ PO • 2 ] 0 βC 1 + βC (32) i . e . [ PO • 2 ] = r i 2 k 6 1 / 2 βC 1 + βC
β = 1 C C = 2 k 6 k 2 k 5 k 3 [ PH ] + ( 2 r 1 k 6 ) 1 / 2 (34) i . e . β = 2 k 6 k 2 k 5 k 3 [ PH ] + ( 2 r i k 6 ) 1 / 2 1 + 2 k 1 [ POOH ] 2 r i 1 / 2 (35)
i . e . β ≈

2 k 6 k 2 k 5 k 3 [ PH ] + ( 2 r i k 6 ) 1 / 2 (36)
As also expected, β takes approximatively the same value as in a "pure" radiochemical environment.

Finally, Eq. ( 31) leads to: [START_REF] Seguchi | Diffusion and solubility of oxygen in γ -irradiated polymer insulation materials[END_REF] It is now possible to integrate Eq. [START_REF] Colin | Aging of polyethylene pipes transporting drinking water disinfected by chlorine dioxide. -I. Chemical aspects[END_REF] . The introduction of Eq. ( 33) into Eq. ( 20) leads to: [START_REF] Khelidj | Oxidation of polyethylene under irradiation at low temperature and low dose rate -Part I: The case of "pure" radiochemical initiation[END_REF] This equation can be rewritten such as:

[ P • ] = 2 k 6 k 5 [ PO • 2 ] 0 1 - βC 1 + βC (37) i . e . [ P • ] = ( 2 r i k 6 ) 1 / 2 k 5 1 - βC 1 + βC
d [ POOH ] dt = -2 k 1b [ POOH ] 2 + k 3 [ PH ] r i 2 k 6 1 / 2 βC 1 + βC
d [ POOH ] [ POOH ] 2 -k 3 [ PH ] 2 k 1b r i 2 k 6 1 / 2 βC 1+ βC = -2 k 1b dt (40) i . e . Ln ⎧ ⎪ ⎨ ⎪ ⎩ k 3 [ PH ] 2 k 1b r i 2 k 6 1 / 2 βC 1+ βC 1 / 2 -[ POOH ] k 3 [ PH ] 2 k 1b r i 2 k 6 1 / 2 βC 1+ βC 1 / 2 + [ POOH ] ⎫ ⎪ ⎬ ⎪ ⎭ = -Kt + a (41)
i . e .

k 3 [ PH ] 2 k 1b r i 2 k 6 1 / 2 βC 1+ βC 1 / 2 -[ POOH ] k 3 [ PH ] 2 k 1b r i 2 k 6 1 / 2 βC 1+ βC 1 / 2 + [ POOH ] = b Exp ( -Kt ) (42) i . e . [ POOH ] = k 3 [ PH ] 2 k 1b r i 2 k 6 1 / 2 βC 1 + βC 1 / 2 1 -b Exp ( -Kt ) 1 + b Exp ( -Kt ) (43)
where a and b are constants to be determined from boundary conditions, and K is a first-order rate constant expressed by: [START_REF] Mikdam | A kinetic model for predicting the oxidative degradation of additive free polyethylene in bleach desinfected water[END_REF] Imposing initial conditions (i.e. [ POOH ] = [ POOH ] 0 when t = 0 ) to Eq. ( 43) gives finally:

K = 2 2 k 3 [ PH ] k 1b r i 2 k 6 1 / 2 βC 1 + βC 1 / 2
[ POOH ] = [ POOH ] ∞ 1 -b Exp ( -Kt ) 1 + b Exp ( -Kt ) (45) with [ POOH ] ∞ = k 3 [ PH ] 2 k 1b r i 2 k 6 1 / 2 βC 1 + βC 1 / 2 (46) and b = [ POOH ] ∞ -[ POOH ] 0 [ POOH ] ∞ + [ POOH ] 0 ( 47
)
For weakly pre-oxidized samples, [ POOH ] ∞ [ POOH ] 0 . Experimental measurements [ 28 , 37 ] show that: [ POOH ] ∞ > 10 [ POOH ] 0 , therefore it can be reasonably considered that: b ≈ 1 .

According to Eq. ( 45) , triggering thermal initiation will completely slows down hydroperoxide build-up during the exposure to finally lead to a maximum concentration of hydroperoxides [ POOH ] ∞ , in the same way as for "pure" thermal oxidation [START_REF] Colin | The classical kinetic model for radical chain oxidation of hydrocarbon substrates initiated by bimolecular hydroperoxide decomposition[END_REF] . If this is really the case, it would be interesting to consider this asymptotic value in order to define a more relevant end-of-life criterion in this study, given that this latter now depends on all the exposure conditions (i.e. γ irradiation dose rate, temperature, and oxygen partial pressure).

Calculation of physico-chemical properties

From these analytical solutions for [ POOH ] , [ PO •

2 ] , and [ P • ] , it is now possible to calculate several key physico-chemical properties, from a practical point of view, because these properties can be checked experimentally. Undoubtedly, hydroperoxide concentration [POOH] and oxygen consumption Q are the most relevant properties because their mathematical expressions can be derived from the radio-thermal oxidation mechanistic scheme without using any additional adjustable parameter.

Recall that the rate of oxygen consumption writes:

r ( C ) = - d [ O 2 ] dt = k 2 C [ P • ] -k 6 [ PO • 2 ] 2 (48) i . e . r ( C ) = 2 k 6 k 2 C k 5 [ PO • 2 ] 0 -[ PO • 2 ] -k 6 [ PO • 2 ] 2 (49) i . e . r ( C ) = k 3 [ PH ] r i 2 k 6 1 / 2 βC 1 + βC + r i βC 1 + βC 1 - βC 2 ( 1 + βC ) (50)
It can be easily checked that, when βC 1 , Eq. ( 50) leads to the expression of the rate of oxygen consumption in oxygen excess [START_REF] Khelidj | A simplified approach for the lifetime prediction of PE in nuclear environments[END_REF] :

r 0 = k 3 [ PH ] r i 2 k 6 1 / 2 + r i 2 ( 51
)
The integration of Eq. ( 50) between 0 and t gives access to the concentration of oxygen consumed by the chemical reaction Q:

Q = ∫ t 0 r ( C ) dt (52) i . e . Q = k 3 [ PH ] r i 2 k 6 1 / 2 βC 1 + βC + r i βC 1 + βC 1 - βC 2 ( 1 + βC ) t (53)
Other key physico-chemical property in the literature are the concentrations of carbonyl products [P = O] and chain scissions S. In the radio-thermal oxidation mechanistic scheme, carbonyl products can only be formed in thermal initiation (1T) and termination [START_REF] Decker | Aging and degradation of polyolefins. -III. Polyethylene and ethylene-propylene copolymers[END_REF] through specific chemical events (e.g. β scission, disproportionation, etc.) which generally compete with many other chemical events, in particular with hydrogen abstraction which gives alcohols. Consequently, the calculation of the formation rate of these oxidation products requires the use of adjustable parameters:

d [ P = O ] dt = γ 1 CO k 1b [ POOH ] 2 + γ 6 CO k 6 [ PO • 2 ] 2 ( 54
)
where γ 1 CO and γ 6 CO are the respective formation yields of carbonyl products in thermal initiation (1T) and termination [START_REF] Decker | Aging and degradation of polyolefins. -III. Polyethylene and ethylene-propylene copolymers[END_REF], see section 3.1 .

If replacing [ POOH ] and [ PO •

2 ] by their analytical solutions in Eq. (54) , it comes:

d [ P = O ] dt = γ 1 CO k 1b [ POOH ] 2 ∞ 1 -b Exp ( -Kt ) 1 + b Exp ( -Kt ) 2 + γ 6 CO r i 2 βC 1 + βC 2 (55) i . e . d [ P = O ] dt = γ 1 CO k 3 [ PH ] 2 r i 2 k 6 1 / 2 βC 1 + βC 1 -b Exp ( -Kt ) 1 + b Exp ( -Kt ) 2 + γ 6 CO r i 2 βC 1 + βC 2 (56) i . e . d [ P = O ] dt = γ 1 CO k 3 [ PH ] 2 r i 2 k 6 1 / 2 βC 1 + βC + γ 6 CO r i 2 βC 1 + βC 2 -γ 1 CO k 3 [ PH ] r i 2 k 6 1 / 2 × βC 1 + βC 2 b Exp ( -Kt ) ( 1 + b Exp ( -Kt ) ) 2 (57)
The integration of Eq. (57) between 0 and t leads finally to:

[ P = O ] = γ 1 CO k 3 [ PH ] 2 r i 2 k 6 1 / 2 βC 1 + βC + γ 6 CO r i 2 βC 1 + βC 2 t + 2 γ 1 CO k 3 [ PH ] K r i 2 k 6 1 / 2 βC 1 + βC × 1 1 + b Exp ( -Kt ) - 1 1 + b (58)
Similarly, chain scissions can only be formed in thermal initiation (1T) and termination (6) through β scissions. Consequently, their mathematical expressions are identical in form to those previously established for carbonyl products, only their formation yields are different:

dS dt = γ 1S k 1b [ POOH ] 2 + γ 6S k 6 [ PO • 2 ] 2 (59)
i . e . dS dt

= γ 1S k 3 [ PH ] 2 r i 2 k 6 1 / 2 βC 1 + βC + γ 6S r i 2 βC 1 + βC 2 -γ 1S k 3 [ PH ] r i 2 k 6 1 / 2 βC 1 + βC 2 b Exp ( -Kt ) ( 1 + b Exp ( -Kt ) ) 2 (60) and S = γ 1S k 3 [ PH ] 2 r i 2 k 6 1 / 2 βC 1 + βC + γ 6S r i 2 βC 1 + βC 2 t + 2 γ 1S k 3 [ PH ] K r i 2 k 6 1 / 2 βC 1 + βC 1 1 + b Exp ( -Kt ) - 1 1 + b (61)
where γ 1S and γ 6S are the respective yields of chain scissions in thermal initiation (1T) and termination [START_REF] Decker | Aging and degradation of polyolefins. -III. Polyethylene and ethylene-propylene copolymers[END_REF], see section 3.1 .

In the case of PE oxidation, it is clear that embrittlement results essentially from chain scissions [ 18 , 19 ]. Neglecting, in a first approach, an eventual chemical cross-linking process, the changes in the concentration of elastically active chains in Si-XLPE throughout the exposure can simply be calculated as follows [START_REF] Colin | Chemo-mechanical model for predicting the lifetime of EPDM rubbers[END_REF] :

ν = ν 0 -S ( 62
)
where ν 0 and ν are the respective concentrations of elastically active chains before and after radio-thermal ageing. Swelling measurements and rheometry in solid state (DMTA) allowed showing that ν 0 = 0 . 12 ± 0 . 02 mol . L -1 for Si-XLPE.

Lifetime prediction

Embrittlement is expected to occur at very low conversion ratio of the chain scission process. On one side, the corresponding critical ratio S F is related through Eq. (62) to a critical concentration of elastically active chains ν F to be determined experimentally. On the other side, as explained at the end of section 3.2 , S F would also be related to a critical hydroperoxide concentration [ POOH ] F corresponding to the onset of the rapid auto-acceleration of the oxidation reaction when triggering thermal initiation. As shown in a previous publication [START_REF] Colin | Lifetime prediction of polyethylene in nuclear plants[END_REF] , the definition of this last structural endof-life criterion is quite simple:

[ POOH ] F = [ POOH ] ∞ q ( 63
)
where q is an arbitrary criterion, higher than unity but not so far, to be determined experimentally. In a first approach, it was decided to use [ POOH ] F for predicting the lifetime t F of the insulating material. It was thus assumed that t = t F when [ POOH ] = [ POOH ] F .

In this case, introducing Eq. (63) into Eq. (45) leads to:

q = 1 + b Exp ( -K t F ) 1 -b Exp ( -K t F ) (64) i . e . Exp ( -K t F ) = q -1 b ( 1 + q ) (65) i . e . t F = 1 K Ln b ( 1 + q ) q -1 (66)

Results and discussion

Long-term behavior of PE insulating materials

Fig. 8 shows the changes in the elongation at break ε R of Si-XLPE in air under the three dose rates under study at low temperature close to ambient. As expected, γ irradiation induces a catastrophic embrittlement of Si-XLPE. The horizontal dashed line plotted for ε R = ε F = 50% was used to graphically determine the lifetime under these various exposure conditions. It was found that t F is about 30, 50, and 260 days under the highest (i.e. 400 Gy.h -1 ), medium (77.8 Gy.h -1 ), and lowest dose rates (8.5 Gy.h -1 ), respectively. These t F values were plotted as a function of dose rate in Fig. 9 where they compared to those previously reported in the literature for other types of PE [START_REF] Khelidj | A simplified approach for the lifetime prediction of PE in nuclear environments[END_REF] . It should be note that the t F value determined at 25 °C for "pure" thermal oxidation is also plotted, but for a dose rate of 1 . 5 × 10 -10 Gy . s -1 corresponding to natural radioactivity.

The almost universal character of the long-term behavior of PE is confirmed. It appears to be completely independent of the chemistry (grafted with chemical group or not), macromolecular

Table 2

Experimental values of the lifetime and steady and critical hydroperoxide concentrations determined experimentally for Si-XLPE in air in the various radio-thermal environments under study. structure (i.e. linear, branched or chemically cross-linked) and crystallinity of PE. It can be clearly seen that all the t F values determined for Si-XLPE are well positioned on the master curve. In addition, these values are located in the right part of the domain of intermediary dose rates where radio-thermal oxidation proceeds. Consequently, all the physicochemical data determined in this study will be used to check the reliability of the analytical equations established in previous sections. due to the depletion of oxidation sites (PH) can be observed, but this second part of the kinetic curves cannot be correctly described by Eq. ( 45) because this latter is based on the assumption of low conversion ratios.

I (Gy

Reliability of the analytical kinetic model

[ POOH ] F was determined graphically in Fig. 10 (i.e. when t = t F ) for the various radio-thermal exposure conditions under study. Its values are also reported in Table 2 . It was found that [ POOH ] F is independent of the radio-thermal exposure conditions, which completely validates the choice of this critical quantity as structural end-of-life criterion in this study. The average value of ≈ 1.6 × 10 -1 mol.L -1 was chosen for predicting the lifetime of Si-XLPE. This is exactly the same [ POOH ] F value that was retained in a previous publication for other types of (linear and chemically cross-linked) PE, but without making real demonstration, just based on mechanistic considerations [START_REF] Colin | Lifetime prediction of polyethylene in nuclear plants[END_REF] .

In the early periods of exposure, [P = O] and ν vary linearly with time (absence of induction period). Then, when thermal initiation gains in importance, a small loss in linearity (progressive slow- down) can be observed for both physico-chemical quantities. The whole kinetic curves are satisfyingly simulated with Eqs. ( 58) and (62) , respectively. In a second stage, Eqs. ( 45) , (58) , and (62) were used to respectively simulate the physico-chemical changes of Si-XLPE in air under 6.0 Gy.h -1 at 86 °C. Simulations are also reported in Fig. 10 , Fig. 11 , and Fig. 12 , respectively. It can be clearly seen that the impact of the thermal initiation on the global oxidation kinetics is much more important at 86 °C than at 47 °C. Unfortunately, the three analytical equations are not able to accurately account for the whole kinetic curves, presumably due to the starting assumption, which was deduced from Eq. [START_REF]Deliverable D2.2: Specifications of tests -accelerated ageing protocol, test strategy[END_REF] . In order to reach this goal, it would be necessary to take into account that thermal initiation could be of the same order of magnitude as (if not greater than) radiochemical initiation. This further kinetic analysis will be the subject of a future publication. At this stage of investigations, it can be seen that the three analytical equations allow accounting for the general trends of the oxidation kinetics under these critical exposure conditions (i.e. under low dose rate at moderate temperature) where, obviously, a stronger coupling between radiochemical and thermal oxidations operates.

The values of the kinetic parameters used for all these simulations, i.e. radiochemical yield G i , rate constants k j (with j = 1, …, 6) and the formation yields γ 1 CO and γ 6 CO of carbonyls products,

Table 4

Values of the rate constant K and criterion q used for predicting the lifetime of Si-XLPE in air under the three dose rates under study at low temperature close to ambient. and γ 1S and γ 6S of chain scissions, have been reported in Table 3 .

I (Gy.h

These values call for the following comments. The orders of magnitude previously determined on linear PE (i.e. LDPE, MDPE and HDPE) for almost all these kinetic parameters were kept for these simulations [ 1-3 , 20 , 36 , 39 , 43 , 44 ]. Only k 5 was slightly reduced at low temperature close to ambient (i.e. at 47 and 21 °C) because it had already been proposed to allocate a small activation energy to this termination rate constant in order to better simulate the oxidation kinetics initiated by chlorine disinfectants in PE pipes carrying drinking water at ambient temperature [ 20 , 43 , 44 ]. Thus, the importance of this activation energy is completely checked in this study. In addition, if considering all the k 5 values published to date for PE [ 2 , 3 , 20 , 43 , 44 ], it is found that its activation energy would be around 14 kJ.mol -1 .

As there was no notable change in the values of all the kinetic parameters, it can be concluded that the reliability of the analytical kinetic model was checked in this study. Its great heuristic value has been demonstrated in Fig. 10 , Fig. 11 , and Fig. 12 , even if some improvements are still possible, in particular at the highest temperature (86 °C).

The last stage of this study was to check the reliability of the last Eq. (66) . Rate constant K and criterion q were determined with Eq. ( 44) and (63) , respectively. Then, their values were introduced into Eq. (66) in order to predict the lifetime t F of Si-XLPE in air under the three dose rates under study at low temperature close to ambient. The resulting t F values are reported in Table 4 . It can been seen that they are of the same order of magnitude than those determined experimentally, thus confirming in turn the great heuristic value of Eq. (66) .

Conclusion

A new analytical model was developed for predicting the radiothermal oxidation kinetics and the lifetime of electric cable insulations made of unfilled and unstabilised Si-XLPE in nuclear power plants. This model was derived from a mechanistic scheme in which the oxidation reaction is initiated both by the polymer radiolysis and the thermal decomposition of hydroperoxides, with-out making the usual assumption of thermal stability of hydroperoxides. After an initial period where the oxidation kinetics occurs with a constant rate, this model allows also predicting the autoacceleration of the oxidation kinetics when hydroperoxide decomposition is no longer negligible. A structural end-of-life criterion: [ POOH] F ≈ 1. 6 × 10 -1 mol. L -1 , corresponding to the onset of this auto-acceleration, was proposed for predicting the lifetime of Si-XLPE in the various radio-thermal environments under study.

A satisfying agreement was obtained between theory and experiments as long as thermal initiation remains a secondary source of radicals relatively to radiochemical initiation, i.e. under the three dose rates under study (from 8.5 to 400 Gy.h -1 ) at low temperature close to ambient (i.e. 47 and 21 °C). However, under the lowest dose rate (i.e. 6.0 Gy.h -1 ) at the highest temperature (86 °C), the analytical model does not correctly simulate the changes in concentration of the three physico-chemical properties investigated experimentally, i.e. hydroperoxides, carbonyls products and elastically active chains. In fact, in these critical radio-thermal exposure conditions, it would be necessary to take into account that thermal initiation could be of the same order of magnitude as (if not greater than) the radiochemical initiation. This further kinetic analysis will be the subject of a future publication. The final objective will be the proposal of an analytical equation allowing to simulate the whole master curve: t F = f(I) reported in Fig. 9 , whatever the dose rate.

However, industrial polymers are rarely used pure, in particular they contain antioxidants. That is the reason why the next longer term objective will be to extend the kinetic model to Si-XLPE stabilized by the most common families of antioxidants, namely hydroperoxide decomposers (i.e. sulphides and phosphites) and radical scavengers (i.e. hindered phenols and secondary aromatic amines). To reach this objective, the main action mechanisms of these antioxidants will be introduced into the radio-thermal oxidation mechanistic scheme in order to derive a new system of differential equations to be solved. It should be pointed out that the main role of these antioxidants is to increase the induction period. Consequently, when these latter have been completely consumed, the Si-XLPE matrix should oxidize according to almost the same kinetics and thus, the structural end-of-life criterion established in this study should remain valid for predicting its lifetime.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

CRediT authorship contribution statement

Fig. 2 .

 2 Fig. 2. Consequences of oxidation on the fracture properties of semi-crystalline polymers.

Fig. 3 .

 3 Fig. 3. Changes in carbonyl region of the FTIR spectrum of Si-XLPE before and after radio-thermal ageing in air under 400 Gy.h -1 at 21 °C (a) and under 8.5 Gy.h -1 at 47 °C (b).

Fig. 4 .

 4 Fig. 4. Changes in the hydroxyl region of the FTIR spectrum of Si-XLPE before and after radio-thermal ageing in air under 400 Gy.h -1 at 21 °C (a) and under 8.5 Gy.h -1 at 47 °C (b).

Fig. 5 .

 5 Fig. 5. Changes in the DSC thermogram of Si-XLPE before and after radio-thermal ageing in air under 400 Gy.h -1 at 21 °C (a) and under 8.5 Gy.h -1 at 47 °C (b).

Fig. 6 .

 6 Fig. 6. DMTA thermogram of unaged Si-XLPE.

Fig. 8 .

 8 Fig. 8. Changes in the elongation at break of Si-XLPE in air under the three dose rates under study at low temperature close to ambient.

Fig. 9 .

 9 Fig. 9. Plot in logarithm co-ordinates of lifetime t F versus γ dose rate I in air at room temperature for unstabilised and unfilled linear PE. Comparison of the t F values determined for Si-XLPE in this study with those reported for various other types of linear (i.e. LDPE, MDPE and HDPE) and chemically cross-linked LDPE (i.e. XLPE) in the literature [1] . The general trend of the curve is shown using a dotted line.

  In a first stage, Eqs. (45) , (58) , and (62) were used to respectively simulate the changes in concentration of hydroperoxides [POOH], carbonyls [P = O], and elastically active chains ν of Si-XLPE in air under the three dose rates under study (i.e. 8.5, 77.8, and 400 Gy.h -1 ) at low temperature close to ambient. Simulations are reported in Fig. 10 , Fig. 11 , and Fig. 12 , respectively. These results call for the following comments. At short term, [POOH] increases with time until reaching an asymptotic value [ POOH ] ∞ which depends on the radio-thermal exposure conditions. All this first part of the kinetic curves is satisfyingly simulated with Eq. (45) , the resulting values of [ POOH ] ∞ have been reported in Table2. As expected, [ POOH ] ∞ is an increasing function of irradiation dose I, but a decreasing function of temperature T. Indeed, this asymptote is the consequence of the sudden acceleration of the thermal initiation (1T), which counteracts completely the build-up of hydroperoxides formed in the propagation step of the oxidation reaction.At longer term, for certain radio-thermal exposure conditions (e.g. under 77.8 Gy.h -1 at 47 °C), a beginning of decrease in[POOH]

Fig. 10 .

 10 Fig. 10. Changes in the hydroperoxide concentration of XLPE in air in the various radio-thermal environments under study. Comparison between simulation with Eq. (45) (solid lines) and experimental data (symbols).

Fig. 11 .

 11 Fig. 11. Changes in the carbonyl concentration of XLPE in air in the various radio-thermal environments under study. Comparison between simulation with Eq. (58) (solid lines) and experimental data (symbols).

Fig. 12 .

 12 Fig. 12. Changes in the concentration of elastically active chains of XLPE in air in the various radio-thermal environments under study. Comparison between simulation with Eq. (62) (solid lines) and experimental data determined by swelling (solid symbols) and rheometry (empty symbols).

Sarah Hettal:

  Investigation, Writing -original draft. Sébastien Roland: Validation, Writing -review & editing, Supervision. Konsta Sipila: Writing -review & editing, Supervision, Project administration. Harri Joki: Investigation. Xavier Colin: Conceptualization, Methodology, Software, Validation, Writing -review & editing, Supervision.

Table 1

 1 Radio-thermal ageing conditions.

	Ageing no.
	1	8.5	2.36 × 10 -3	47	12800	109
	2	6.0	1.67 × 10 -3	86	16267	98
	3	77.8	2.16 × 10 -2	47	3830	298
	4	400	1.11 × 10 -1	21	668	269

Dose rate (Gy.h -1 ) Dose rate (Gy.s -1 ) Temperature ( °C) Maximum duration (h) Maximum dose (kGy)

Table 3

 3 Values of the kinetic parameters used for modeling the oxidation kinetics of Si-XLPE in the various radiothermal environments under study. Papers in which the same orders of magnitude were previously reported are referenced in the last column on the right.

	T ( °C)	21	47	47	86
	I (Gy.h -1 )	400	77.8	8.5	6.0	References
	G i	8	8	8	8	[1,2,39]
	k

1b (L.mol -1 .s -1 )

  5.0 × 10 -9 2.5 × 10 -7 2.4 × 10 -7 1.0 × 10 -5 [ 2 , 3 , 20 , 36 , 39 , 43 , 44 ] k

2 (L.mol -1 .s -1 )

	10 8	10 8	10 8	10 8	[ 2 , 3 , 20 , 36 , 39 , 43 , 44 ]

Acknowledgments

The TeaM Cables project leading to this application has received funding from the Euratom research and training program 2014-2018 under grant agreement No 755183 .


Cristel Chandre

Michel Vittot

Guido Ciraolo

Local control of area-preserving maps

come

Introduction

Chaotic transport arises naturally in Hamiltonian systems with mixed phase space. Achieving the control of these systems by restoring local conserved quantities is a long standing and crucial problem in many branches of physics (in particular, in plasma physics and fluid dynamics). A method for controlling continuous Hamiltonian flows has been developed based on the following idea: to find a small control term f for the perturbed Hamiltonian H = H 0 + V (where H 0 is integrable), in order to have a more regular dynamics for the controlled Hamiltonian H c = H 0 + V + f . Two approaches have been developed : A global control aims at making the controlled Hamiltonian H c integrable; A local control restores a particular invariant torus (local integrability). Both approaches give a control term of order V 2 .

Let us stress that this method of control differs from other methods by the fact that the controlled dynamics is Hamiltonian : This makes it relevant to the control of inherently Hamiltonian systems such as beams of charged test particles in electrostatic waves, two-dimensional Euler flows or the geometry of magnetic field lines.

These two control methods have been developed for continuous time flows. [START_REF] Ciraolo | Control of Hamiltonian chaos as a possible tool to control anomalous transport in fusion plasmas[END_REF][START_REF] Chandre | Control of stochasticity in magnetic field lines[END_REF] In Ref., [START_REF] Chandre | Controlling chaos in area-preserving maps[END_REF] the global control for symplectic maps has been proposed. In this article, we explicit the local control method for symplectic maps.

In Sec. 2, we derive the expression of the control term for area-preserving maps generated by a generating function in mixed coordinates. In Sec. 3, we apply the local control of area-preserving maps to two examples : the standard map and the tokamap.

Derivation of the control term

We consider two-dimensional symplectic maps (A, ϕ) → (A ′ , ϕ ′ ) = F (A, ϕ) on the cylinder R × T which are ε-close to integrability. [START_REF] Meiss | Symplectic maps, variational principles and transport[END_REF] In this section, our aim is to find a small control term f such that the controlled symplectic map F + f has an invariant curve. We consider area-preserving maps obtained from a generating function of the form

S(A ′ , ϕ) = A ′ ϕ + H(A ′ ) + εV (A ′ , ϕ).

The map reads

A = A ′ + ε∂ ϕ V (A ′ , ϕ), ϕ ′ = ϕ + H ′ (A ′ ) + ε∂ A V (A ′ , ϕ).
Here ∂ A V (A ′ , ϕ) denotes the partial derivative of V with respect to the action (first variable) and ∂ ϕ V (A ′ , ϕ) denotes the partial derivative of V with respect to the angle (second variable).

We expand the map around a given value of the action denoted K. The generating function after the translation is :

S(A ′ , ϕ) = A ′ ϕ + H(K + A ′ ) + εV (K + A ′ , ϕ).
We rewrite the generating function as :

S(A ′ , ϕ) = A ′ ϕ + ωA ′ + εv(ϕ) + w(A ′ , ϕ), (1)
where

ω = H ′ (K), (2)
v(ϕ) = V (K, ϕ), (3)
w(A ′ , ϕ) = H(K + A ′ ) -H(K) -ωA ′ + εV (K + A ′ , ϕ) -εV (K, ϕ).( 4
)
We notice that w(0, ϕ) = 0 for all ϕ ∈ T. Without loss of generality, we assume that 2π 0 v(ϕ)dϕ = 0. Our aim is to modify the generating function with a control term f of order ε 2 such that the controlled map has an invariant curve around A ′ = 0. We consider the controlled generating function

S c (A ′ , ϕ) = A ′ ϕ + ωA ′ + εv(ϕ) + w(A ′ , ϕ) + f (ϕ),
where we notice that the control term f we construct does only depend on the angle ϕ. The controlled map is given by

A = A ′ + εv ′ (ϕ) + ∂ ϕ w(A ′ , ϕ) + f ′ (ϕ), (5)
ϕ ′ = ϕ + ω + ∂ A w(A ′ , ϕ). (6)
We perform a change of coordinates generated by

X(A 0 , ϕ) = A 0 ϕ + εχ(ϕ),
which maps (A, ϕ) into (A 0 , ϕ 0 ), and (A ′ , ϕ ′ ) into (A ′ 0 , ϕ ′ 0 ). The mapping becomes

A 0 = A ′ 0 + ε (χ ′ (ϕ ′ 0 ) -χ ′ (ϕ 0 ) + v ′ (ϕ 0 )) + ∂ ϕ w (A ′ 0 + εχ ′ (ϕ ′ 0 ), ϕ 0 ) + f ′ (ϕ 0 ), ϕ ′ 0 = ϕ 0 + ω + ∂ A w (A ′ 0 + εχ ′ (ϕ ′ 0 ), ϕ 0 ) . (7)
We choose the function χ such that

χ(ϕ + ω) -χ(ϕ) = -v(ϕ).
By expanding v in Fourier series, i.e. v(ϕ) = k∈Z v k e ikϕ , this reads

χ(ϕ) = k s.t. ωk / ∈2πZ v k 1 -e iωk e ikϕ .
The control term is constructed such that the mapping in the new coordinated has A 0 = 0 as an invariant curve. In order to do this, we define the function Φ implicitly by

Φ(ϕ) = ϕ + ω + ∂ A w (εχ ′ (Φ(ϕ)), ϕ) .
The angle Φ(ϕ) is obtained when A ′ 0 = 0 in Eq. ( 7) The expression of the control term is such that

f ′ (ϕ) = εχ ′ (ϕ + ω) -εχ ′ (Φ(ϕ)) -∂ ϕ w(εχ ′ (Φ(ϕ)), ϕ). ( 8
)
¿From the expression of w given by Eq. ( 4), it is straightforward to check that if A ′ is of order ε then

∂ ϕ w = εV (K + A ′ , ϕ) -εV (K, ϕ) is of order ε 2 . Since Φ(ϕ) -(ϕ + ω) is of order ε (again if A ′ is of order ε) then εχ ′ (Φ(ϕ)) -εχ ′ (ϕ + ω) is of order ε 2 . Thus, f ′ is of order ε 2 .
The controlled mapping becomes

A 0 = A ′ 0 + ε (χ ′ (ϕ ′ 0 ) -χ ′ (Φ(ϕ 0 ))) + ∂ ϕ w (A ′ 0 + εχ ′ (ϕ ′ 0 ), ϕ 0 ) - ∂ ϕ w (εχ ′ (Φ(ϕ 0 )), ϕ 0 ) , (9)
ϕ ′ 0 = ϕ 0 + ω + ∂ A w (A ′ 0 + εχ ′ (ϕ ′ 0 ), ϕ 0 ) . ( 10
)
It is straightforward to see from Eq. ( 9) that if A ′ 0 = 0 then ϕ ′ 0 = Φ(ϕ 0 ) by definition of Φ and hence A 0 = 0 . Since we assume that the mapping is invertible, the curve A 0 = 0 is preserved by iteration of the map. Consequently, the controlled map ( 5)-( 6) has the invariant curve with equation

A = εχ ′ (ϕ). ( 11
)
Next, we derive an approximate control term by only keeping the order ε 2 . The expansion of f ′ gives the expression of f mix,2 :

f mix,2 = - ε 2 2 H ′′ (K) (χ ′ (ϕ + ω)) 2 -ε 2 ∂ A V (K, ϕ)χ ′ (ϕ + ω). (12)
Remark 1: If we assume that w is only a function of the actions, the control term f is

f (ϕ) = χ(ϕ + ω) -χ(Φ(ϕ)) + χ ′ (Φ(ϕ))w ′ (εχ ′ (Φ(ϕ))) -w(εχ ′ (Φ(ϕ))), ( 13
)
where Φ is defined implicitly by Φ(ϕ) = ϕ + ω + w ′ (εχ ′ (Φ(ϕ))).

Remark 2: If the time step of the map is equal to τ , i.e., if we consider controlled maps generated by

S c (A ′ , ϕ) = A ′ ϕ + τ ωA ′ + τ εv(ϕ) + τ w(A ′ , ϕ) + τ f (ϕ),
the generating function is given by

χ(ϕ + τ ω) -χ(ϕ) = -τ v(ϕ).
We define the operator

H τ = 1 -e -τ ω∂ϕ τ ,
and Γ τ as the pseudo-inverse of H τ given from H 2 τ Γ τ = H τ . The projector R τ is defined accordingly. Hence the solution for χ is

χ = (1 -Γ τ -R τ )v. ( 14
)
The control term is

f ′ (ϕ) = τ -1 ε (χ ′ (ϕ + τ ω) -χ ′ (Φ τ (ϕ))) -∂ ϕ w(εχ ′ (Φ τ (ϕ)), ϕ),
where Φ τ (ϕ) = ϕ + τ ω + τ ∂ A w(εχ ′ (Φ τ (ϕ)), ϕ). We expand the expression of the control term and we neglect the order τ :

f ′ (ϕ) = -εχ ′′ (ϕ)∂ A w(εχ ′ (ϕ), ϕ) -∂ ϕ w(εχ ′ (ϕ), ϕ) + O(τ ).
Since ωχ ′ = -v, we have

f (ϕ) = -w(-εΓ 0 v ′ , ϕ) + O(τ ),
where Γ 0 is the pseudo-inverse of ω∂ ϕ . This expression of the control term corresponds to the one obtained by the local control of Hamiltonian flows in Ref. [START_REF] Chandre | Control of stochasticity in magnetic field lines[END_REF]

Numerical examples

Application to the standard map

The standard map S is

A ′ = A + ε sin ϕ, ϕ ′ = ϕ + A ′ mod 2π.
After a translation of the action A by ω, the map becomes

A ′ = A + ε sin ϕ, ϕ ′ = ϕ + ω + A ′ mod 2π.
A phase portrait of this map for ε = 1.5 is given in Fig. 1. There are no Kolmogorov-Arnold-Moser (KAM) tori (acting as barriers in phase space) at this value of ε (and higher). The critical value of the parameter ε for which all KAM tori are broken is ε std ≈ 0.9716.

The standard map is obtained from the generating function in mixed coordinates

S(A ′ , ϕ) = A ′ ϕ + ωA + A ′2 2 + ε cos ϕ, i.e. v(ϕ) = cos ϕ and w(A) = A 2 /2.
The generating function χ given by Eq. ( 14) is thus

χ(ϕ) = - sin(ϕ -ω/2) 2 sin(ω/2) .
The control term given by Eq. ( 13) is where Φ(ϕ) = ϕ + ω + εχ ′ (Φ(ϕ)). We notice that the equation for Φ is invertible for ε ≤ 2 sin(ω/2).

f (ϕ) = χ(ϕ + ω) -χ(Φ(ϕ)) + 1 2 (χ ′ (Φ(ϕ))) 2 , (15)
The dominant control term is given by

f mix,2 (ϕ) = -ε 2 cos 2 (ϕ + ω/2) 8 sin 2 (ω/2) , (16)
and the resulting map S mix,2 generates the phase portrait displayed on

Application to the tokamap

The tokamap 5 has been proposed as a model map for toroidal chaotic magnetic fields. It describes the motion of field lines on the poloidal section in the toroidal geometry. This symplectic map (A, ϕ) → (A ′ , ϕ ′ ), where A is the toroidal flux and ϕ is the poloidal angle, is generated by the function

S(A ′ , ϕ) = A ′ ϕ + H(A ′ ) -ε A ′ A ′ + 1 cos ϕ.

It reads

A = A ′ + ε A ′ A ′ + 1 sin ϕ, ϕ ′ = ϕ + 1 q(A ′ ) - ε (A ′ + 1) 2 cos ϕ,
where q(A) = 1/H ′ (A) is called the q-profile. In our computation, we choose H ′ (A) = 1/q(A) = π(2-A)(2-2A+A 2 )/2 and ε = 9/(4π). A phase portrait of this map is shown in Fig. 4.

We select a given value K of the action A for the localization. The tokamap is then generated by a function S of the form given by Eq. (1)

v(ϕ) = - K K + 1 cos ϕ, w(A, ϕ) = H(K + A) -H(K) -ωA -ε K + A K + A + 1 - K K + 1 cos ϕ.
The generating function is given by Eq. ( 14) :

χ(ϕ) = K 2(K + 1)
sin(ϕ -ω/2) sin(ω/2) .

The expression of f mix,2 is given by Eq. (12) :

f mix,2 = ε 2 K 2(K + 1) 2 cos(ϕ + ω/2)
sin(ω/2) cos ϕ K + 1 -KH ′′ (K) cos(ϕ + ω/2) sin(ω/2) .

For K = 1/2 the controlled tokamap is

A = A ′ + ε A ′ 1 + A ′ sin ϕ -
ϕ ′ = ϕ + 1 q(A ′ ) - ε (1 + A ′ ) 2 cos ϕ, (17)
where α = 11π/32 We notice that this control term is the same as the one obtained by performing the global control and then by expanding the control term around a given value of the action. The main advantage here is that the whole series of the control term can be computed which was not the case with the global control.

Fig. 1 .

 1 Fig. 1. Phase portrait of the standard map S for ε = 1.5.

Fig. 2 .

 2 Fig. 2. Phase portrait of the controlled standard map S mix with the control term (15) for ε = 1.5 and ω = π. The bold curve is the invariant curve created by the control term.

Fig. 3 Fig. 3 .

 33 Fig. 3.

Fig. 4 .

 4 Fig. 4. Phase portrait of the tokamap for ε = 9/(4π).

  with


Delphine Martire

Sarah Garnier

Sébastien Sagnol

Annick Bourret

Stéphane Marchal

Norbert Chauvet

Amandine Guérin

Dominique Forgues

Dominique Berrebi

Christophe Chardot

Phenotypic switch of smooth muscle cells in paediatric chronic intestinal pseudo-obstruction syndrome

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Ludovic Bellot-Gurlet
email: bellot-gurlet@glvt-cnrs.fr

Sandrine Pagès-Camagna
email: sandrine.pages@culture.fr

Claude Coupry

Raman Spectroscopy in Art and Archaeology

Keywords: Archaeometry, cultural heritage materials, biomaterials, minerals, SERS applications

The contributions present an extended view of works in the field, from technical developments, special analytical procedures to various applications.

INTRODUCTION

Since the first conference of this series in London [START_REF]Raman Spectroscopy in Archaeology and Art History[END_REF] , 2001, the number of studies around cultural heritage involving Raman spectroscopy is in continuous increase as shown for example by the Journal of Raman Spectroscopy publications (20 articles between 2004 issue 10 and 2006 issue 7) or other journals but also some specialised journals or publications in archaeometry, cultural heritage or conservation studies [31][32][33][34][35] . As already mentioned 36 the instrumental improvements and the commercial availability of various devices are some keys of this success.

Among the devices abilities some are significant in these applications, as the opportunity to record spectra at very low laser power which respect sample integrity or the developments of mobile spectrometers. The latter seems actually to reach a maturity with at the technical point of view a clear feasibility demonstrated and the treatment of some specific investigations.

Since the first conference the audience continuously increases with in Paris about 120 participants and 88 abstracts selected for presentation representing 18 countries. This wide international representation is currently also notable. This underlines the Raman spectroscopists growing interest for ancient materials and the need of Raman characterisation ability in cultural heritage analysis.

Materials and questions diversify, furthermore Raman analyses are integrated in global analytical strategies while established studies (as around pigments and minerals) continue their developments. This special issue introductory paper will present a brief overview of the papers collected after the "3 rd International Conference on the Application of Raman Spectroscopy in Art and Archaeology" which reflect the present trends in the field. These 33 papers are classified in four topics chosen to underline these developments. In fact, some articles cover more than one topic with a certain degree of overlap, especially because studies are not first classified by materials, but rather by methodological originality or thematic investigations which advance the research in the field of cultural heritage.

TECHNICAL DEVELOPMENTS

Fluorescence emissions shading the Raman signal is a general limitation for Raman spectroscopy especially for studies dealing with "non-model materials", which is chiefly the case with ancient materials. Hence different directions are explored, as the search for optimal excitation wavelength 37 or more non classic approaches with some mathematical spectra treatments 38 .

An alternative way could be the increase of the Raman signal through Surface-Enhanced Raman Scattering (SERS). Actually widely used for weak Raman signal or low concentrated samples (especially in biology) 39 , the developments of various SERS substrates and experimentalist's skilfulness offer the opportunity to obtain Raman spectra from highly fluorescent materials and also from micro samples, allowing the application to work of art dyes or lakes 40,41 . It could thus offer an alternative to established techniques as High Performance Liquid Chromatography (HPLC), with the set up of specific non-extractive sample preparation and even direct SERS analysis on a single fibre sample 40 . These experiments both exploit SERS specificities for low concentrated substances (and/or micro-sampling) and weak Raman signal against luminescence emission.

All these improvements also highlight the increasing interest for organic material studies. This uses the versatility of Raman spectroscopy among materials allowing to explore the large diversity of organic materials encountered in cultural heritage as presented by various papers of this special issue, with supports such as wood, textiles, paper, but also dyes, binders, colorants, etc. Previously the applications were limited by device efficiency (with degradation under laser irradiation) and/or material preservation state (low signal and high fluorescence signals).

Another specificity of the field is to develop analytical procedures which could be used by spectroscopy non-specialists. The devices and software offered by Raman companies are a first step, after to enlarge the community being able to manage spectra identification or treatment some automated or supervised procedures are developed. An example for spectra recognition is given by Castanys Tutzó et al. 42 .

MOBILE MEASUREMENTS

Examples given in this issue present the use of various commercial or prototype Raman devices. This technical opportunity reached by Raman spectroscopes offer particular applications in the field of cultural heritage. The connection between Raman companies and end-users also generate some specific adaptations for our applications, like deported microanalysis by optic fibre, visual control for sample and analysis, Laser power modulation, coupling with other technique, etc., or the development of prototypes/homemade devices for specific applications.

Mobile Raman spectroscopy is useful in several situations. On one hand, because each artefact is unique and because to achieve a meaningful level some problematic needs the examination of large sample collections, thus it is needed to reach some specific samples even in their environment. Sometimes it is easier to reach each object than to gather all in one lab. On the other hand, on-site measurements are requested for many objects which cannot be sampled and moved outside museums 43 , or are none moving by nature because these are part of architectural monuments 44 . In addition, progress in the studies of degradation processes will tend to propose alteration diagnosis procedures recurrently asked by conservators. It requires a simple operation which could be based on Raman spectroscopy because of its easy implementation and its mobile availability.

An important perspective for mobile Raman devices is offered by all other techniques actually developed to be also performed on site. It opens a larger object understanding and Raman spectra interpretation often not obvious with material complexity or heterogeneity. Additional to molecular information the opportunity to obtain chemical composition could be obtain with the coupling of X-Ray Fluorescence measurements (XRF) currently available as mobile. The work by Andrikopoulos et al. 45 present an original coupled XRF-Raman mobile device and its performances on Byzantine icons. To support the significance of on-site measurements Fremout et al. 46 show that conclusions obtained with on-site non destructive measurements could be compared with destructive laboratory experiments.

As a confirmation of its feasibility, even if it is not their main topic, other papers of this issue call upon onsite experiments.

COLOURS, PIGMENTS AND TECHNOLOGIES

The characterisation of pigments, binding media, etc., or generally the question of "colour" and techniques to obtain it, is a large field of investigation where Raman spectroscopy is regularly used. Some work focuses on some pigments with the study of their synthesis and properties, as Sánchez del Río et al. 47 about the famous Maya blue pigment composed by a clay and indigo mixture with the question of its structure and stability.

Examples of studies concern a wild time span and diverse cultures, from rock art 48,49 and wall paintings [50][51][52][53] (which could also call for the use of mobile measurements), to various supports (paper [54][55][56] , wood 57 ). For numerous of these studies Raman analysis is involved in a general analytical strategy with complementary measurements as SEM-EDS, XRF, LIBS or XRD. Each technique with its performances or specific implementation will bring part of the answers.

The questioning about colour could also concern ceramics where aspects and colours are linked with their manufacturing technologies. From Mediterranean Antiquity 58 , North America prehistory 59 to renaissance 60 and contemporary 61 , Raman spectroscopy could document technical history and production processes.

STUDIES DEALING WITH DEGRADATION PROCESSES

This field is under development, strongly linked with the conservation of cultural heritage, a technical and economical challenge (and probably social) for present and future societies. It is also on request for previsions over multi secular periods, with the use of ancient materials as analogues to provide experimental data for ageing numerical models. The next motivation is for partly or highly damaged materials and the attempt to get a better idea of the original object, as it is illustrated with a pigment identification 62 .

Degradation of numerous materials is explored from organic to mineral. Here are presented studies about waterlogged wood degradation and their conservation treatments optimisation 63,64 . The diagnostic of degradation state is tackled for paper 65 or textile 66 . The formation mechanism of patinas from long term natural alteration processes is explored with a case study on a stone object where raw material identification could be erroneous 67 . With copper artefacts, patinas could not only come from natural processes but can be also the result of intentional artificial treatments for appearance modification which then emphasize technological performances. It is then necessary to make the share between these scenarios 68 .

The questioning of alteration mechanisms through ancient artefacts and complementary techniques is presented for amber 69 with the complementarity of vibrational spectroscopies (IR and Raman) for a recovered molecular description of this fossil resin.

An additional specificity of Raman spectroscopy is exploited to revise iron atmospheric corrosion mechanisms 70 . The Raman structural imaging gives access to a sharp description of material microheterogeneity and organisation. Because these features are closely related to degradation mechanisms these images, even obtained for low crystallised phases, bring statements to improve mechanisms descriptions.

FUTURE PROSPECTS

This field of research essentially multidisciplinary call for cooperation between historians, museum workers, conservation scientists and archaeologists on the one hand and spectroscopists, chemists, geologists, biologist, etc., on the other. Fruitful results are then obtain about cultural heritage but also some original by-products for other specialities via unusual experimental procedures/devices (because ancient material often offers challenges), weathering processes or material properties.

The role of Raman spectroscopy, thanks to instrumental improvements and its versatility among materials, will continue to develop in ancient materials studies. Its expansion is accompanied by devices which go out of the spectroscopy laboratory to museum conservation labs. Present researches also have to set up analytical strategies based on complementary methods where Raman spectroscopy is setting its role.

Acknowledgements

We are extremely grateful to the institutions and individuals who assisted in making the conference possible. In particular we would like to thank: University Pierre et Marie Curie-Paris6 which accommodates the conference, and for their financial support: Mission de la Recherche et de la Technologie of the Ministère de la Culture et de la Communication, Département Sciences Chimiques of CNRS, Eu-ARTECH, UFR926 Chimie UPMC. The organising committee also expresses its thanks to the following companies who are present at the conference exhibition: HORIBA Jobin Yvon, Kaiser Optical Systems and Renishaw.


Ana Magalhães

Yannick Rossez

Catherine Robbe-Masselot

Emmanuel Maes

Joana Gomes

Anna Shevtsova

Jeanna Bugaytsova

Thomas Borén

Celso A Reis

Muc5ac gastric mucin glycosylation is shaped by FUT2 activity and functionally impacts Helicobacter pylori binding

Supplementary Information

Supplementary Methods

Quantitative real-time PCR analysis of Muc5ac and Muc6

Total RNA was extracted from 7-9 week old wild-type and Fut2-null mice stomachs using TRI reagent LS (Sigma-Aldrich), according to manufacturer's protocol. RNA yield and quality were determined spectrophotometrically and 5.0µg of total RNA was reverse transcribed using Superscript III (Invitrogen), according to manufacturer's instructions. Expression of Muc5ac and Muc6 was quantified using Taqman probes, acquired as pre-developed assays from Applied Biosystems (Mm01276718_m1 and Mm00725165_m1) and normalized to the expression of the endogenous control Hprt (Mm03024075_m1). Each sample was amplified in triplicate in an ABI Prism 7500 (Applied Biosystems). Relative transcript levels were determined using the ΔΔCTmethod.

Supplementary Figure S1