Houssni Lamkaddam

Aline Gratien

Manon Ropion

Edouard Pangui

Jean- François Doussin

Kinetic Study of the Temperature Dependence of OH-Initiated Oxidation of n-Dodecane

Reaction rate constants for the reaction of n-dodecane with hydroxyl radicals were measured as a function of temperature between 283 and 303 K, using the relative rate method in the CESAM chamber (French acronym for "Chamber for Atmospheric Multiphase Experimental Simulation"). The rate constants obtained at 283, 293 and 303 K are (1.27 ± 0.31)×10 -11 , (1.33 ± 0.34)×10 -11 , (1.27 ± 0.40)×10 -11 cm 3 molecule -1 s -1 , respectively. Rate constants measured were in excellent agreement with the few available data in the literature over the studied temperature range (283-340K). Rate constants estimated by the structure-activity relationship and transition state theory methods agreed with our experimental data within 14 %. From these data combined with previous literature measurement, the following Arrhenius expression; 𝑘 𝐷𝐷𝐶+𝑂𝐻 = (9.77 ± 6.19)10 -11 × exp [ -595 ±5580 𝑇 ] 𝑐𝑚 3 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 -1 𝑠 -1 , was found valid over a temperature range (283 -340 K) of the tropospheric interest.

Introduction

Alkanes are ubiquitous volatile organic compounds (VOCs) in the atmosphere. They are mostly emitted by human activities, including combustion sources, vehicle exhaust, and evaporation and comprise 40 -50% of the anthropogenic VOCs in urban areas [START_REF] Calvert | The Mechanisms of Atmospheric Oxidation of Aromatic Hydrocarbons[END_REF][START_REF] Schauer | Measurement of Emissions from Air Pollution Sources. 4. C1-C27 Organic Compounds from Cooking with Seed Oils[END_REF] . In particular, long-chain alkanes (C>10), which represent a significant fraction of the unresolved complex mixture of fossil fuel [START_REF] Gentner | Elucidating secondary organic aerosol from diesel and gasoline vehicles through detailed characterization of organic carbon emissions[END_REF][START_REF] Chan | Detailed chemical characterization of unresolved complex mixtures in atmospheric organics: Insights into emission sources, atmospheric processing, and secondary organic aerosol formation[END_REF] . For example n-dodecane (DDC) mixing ratio in polluted area can reach up to 16.47 µg/m 3 5-6 . Further, long chain alkanes are expected to exhibit during daytime a lifetime of 5-20 hours with respect to OH-initiated oxidation -their main removal process -making them good contributors to atmospheric reactivity at local to regional scale [START_REF] Atkinson | Kinetics of the gas-phase reactions of OH radicals with alkanes and cycloalkanes[END_REF] . Long-chain alkanes (C>10), have also been recognized to be high secondary organic aerosol (SOA) precursors [START_REF] Lim | Products and Mechanism of Secondary Organic Aerosol Formation from Reactions of n-Alkanes with OH Radicals in the Presence of NOx[END_REF][START_REF] Lim | Effects of Molecular Structure on Aerosol Yields from OH Radical-Initiated Reactions of Linear, Branched, and Cyclic Alkanes in the Presence of NOx[END_REF][START_REF] Presto | Secondary Organic Aerosol Formation from High-NOx Photo-Oxidation of Low Volatility Precursors: n-Alkanes[END_REF][START_REF] Tkacik | Secondary Organic Aerosol Formation from Intermediate-Volatility Organic Compounds: Cyclic, Linear, and Branched Alkanes[END_REF][START_REF] Lamkaddam | High-NOx Photooxidation of n-Dodecane: Temperature Dependence of SOA Formation[END_REF] .

While SOA impacts human health [START_REF] Pope | Lung Cancer and Cardiovascular Disease Mortality Associated with Ambient Air Pollution and Cigarette Smoke: Shape of the Exposure-Response Relationships[END_REF] and global climate [START_REF]Climate Change[END_REF] , chemical transport models have difficulties to predict ambient SOA concentration [START_REF] Heald | Exploring the vertical profile of atmospheric organic aerosol: comparing 17 aircraft field campaigns with a global model[END_REF][START_REF] Ensberg | Emission factor ratios, SOA mass yields, and the impact of vehicular emissions on SOA formation[END_REF][START_REF] Hodzic | Rethinking the global secondary organic aerosol (SOA) budget: stronger production, faster removal, shorter lifetime[END_REF] . To reduce the gap between measurement and modeled SOA, efforts have been recently continued to study the SOA formation from the oxidation of long-chain alkanes [START_REF] Lim | Products and Mechanism of Secondary Organic Aerosol Formation from Reactions of n-Alkanes with OH Radicals in the Presence of NOx[END_REF][START_REF] Lim | Effects of Molecular Structure on Aerosol Yields from OH Radical-Initiated Reactions of Linear, Branched, and Cyclic Alkanes in the Presence of NOx[END_REF][START_REF] Presto | Secondary Organic Aerosol Formation from High-NOx Photo-Oxidation of Low Volatility Precursors: n-Alkanes[END_REF][START_REF] Tkacik | Secondary Organic Aerosol Formation from Intermediate-Volatility Organic Compounds: Cyclic, Linear, and Branched Alkanes[END_REF][START_REF] Lamkaddam | High-NOx Photooxidation of n-Dodecane: Temperature Dependence of SOA Formation[END_REF][START_REF] Lim | Chemistry of Secondary Organic Aerosol Formation from OH Radical-Initiated Reactions of Linear, Branched, and Cyclic Alkanes in the Presence of NO x[END_REF][START_REF] Yee | Secondary Organic Aerosol Formation from Low-NOx Photooxidation of Dodecane: Evolution of Multigeneration Gas-Phase Chemistry and Aerosol Composition[END_REF][START_REF] Yee | Effect of chemical structure on secondary organic aerosol formation from C12 alkanes[END_REF][START_REF] Loza | Secondary organic aerosol yields of 12-carbon alkanes[END_REF][START_REF] Yeh | Alkyl Nitrate Formation from the Reactions of C8-C14 n-Alkanes with OH Radicals in the Presence of NOx: Measured Yields with Essential Corrections for Gas-Wall Partitioning[END_REF][START_REF] Yeh | Identification and Yields of 1,4-Hydroxynitrates Formed from the Reactions of C8-C16 n-Alkanes with OH Radicals in the Presence of NOx[END_REF][START_REF] Takekawa | Temperature dependence of secondary organic aerosol formation by photo-oxidation of hydrocarbons[END_REF][START_REF] Zhang | Role of ozone in SOA formation from alkane photooxidation[END_REF] . To date, the most studied alkane was n-dodecane (DDC), not only because of its potential contribution to SOA formation but also because -to its linear structure -it represents also a target molecule of choice to investigate the relative importance of the various radical pathways during the oxidation scheme. Fragmentation, functionalization and oligomerization are indeed key processes that determine both the product distribution and the related SOA yield. However, these studies were mainly performed at room temperature (295-298 K), while temperature is an important environmental factor, which both affect vapor pressure of the products, branching ratios and initiation rate constants. To date, only the study of Lamkaddam, et al. [START_REF] Lamkaddam | High-NOx Photooxidation of n-Dodecane: Temperature Dependence of SOA Formation[END_REF] has investigated the SOA formation over a range of temperature (283 -304.5 K). The latter showed a small effect of temperature on the SOA yields, indicating a chemical composition dominated by nonvolatile species such that a change in temperature would not change the partitioning. Therefore, in order to fully assess the impact of this class of compounds on the environment and to model its chemistry, a better knowledge of its oxidation kinetics is needed to more precisely represent the temperature dependence of the rate constant of n-dodecane with OH radicals and disentangle the kinetic effects from the phase partitioning effects on the SOA yield.

Only a few experimental studies have provided reaction rate constants of n-dodecane (kDDC+OH) with OH radical at various temperature using the relative rate technique [START_REF] Behnke | A smog chamber for studies of the photochemical degradation of chemicals in the presence of aerosols[END_REF][START_REF] Nolting | A smog chamber for studies of the reactions of terpenes and alkanes with ozone and OH[END_REF][START_REF] Phan | Kinetics Study of the Reactions of OH with n-Undecane and n-Dodecane Using the RR/DF/MS Technique[END_REF] . First determinations at atmospheric pressure from the Zetzsch's group have been reported to be kDDC+OH = (1.39 ± 0.01)×10 -11 cm 3 molecule -1 s -1 at 300 K and kDDC+OH = (1.50 ± 0.05)×10 -11 cm 3 molecule -1 s -1 at 312 K with n-octane and n-heptane as reference compound, respectively from Behnke, et al. [START_REF] Behnke | A smog chamber for studies of the photochemical degradation of chemicals in the presence of aerosols[END_REF] and Nolting, et al. [START_REF] Nolting | A smog chamber for studies of the reactions of terpenes and alkanes with ozone and OH[END_REF] .

By exploring these experimental studies, Atkinson (2003) provided a recommended value of (1.32 ± 0.26)×10 -11 cm 3 molecule -1 s -1 at 298 K. In addition, measurements made by the Zetzsch's group has been re-evaluated using recent rate data for the reference compound by Atkinson [START_REF] Atkinson | Kinetics of the gas-phase reactions of OH radicals with alkanes and cycloalkanes[END_REF] . The resulting re-evaluation led to reaction rate of kDDC+OH = (1.29 ± 0.02)×10 -11 cm 3 molecule -1 s -1 at 300 K and kDDC+OH = (1.40 ± 0.05)×10 -11 cm 3 molecule -1 s -1 at 312 K, respectively for the studies of Behnke, et al. [START_REF] Behnke | A smog chamber for studies of the photochemical degradation of chemicals in the presence of aerosols[END_REF] and Nolting, et al. [START_REF] Nolting | A smog chamber for studies of the reactions of terpenes and alkanes with ozone and OH[END_REF] .

Later, Phan and Li 28 performed a low-pressure flow tube reactor coupled with a quadrupole mass spectrometer at 240-340 K with n-nonane and n-decane as reference compounds. They determined a reaction rate constant of (1.83 ± 0.26)×10 -11 cm 3 molecule -1 s -1 at 298 K, which was only in fair agreement considering their uncertainties with previous studies. Interestingly, in this study, from 277 to 340 K, rate constants have been found to correlate positively with the temperature, while from 240 to 260 K a negative correlation has been observed.

In conclusion, despite these experimental studies, the atmospheric oxidation rate of n-dodecane is still subject to uncertainties ranging from 20% to 40%. Therefore, to improve our knowledge of the atmospheric reactivity of n-dodecane, we present in this paper an additional investigation of the rate constant as a function of temperature (283 -303 K) with the objective to converge toward a better-established temperature dependence of the rate constant of the reaction between OH radical and n-dodecane. In this study, reported reaction rate constants were obtained with the relative rate method, using isoprene (ISP) as reference compound.

Material and Methods

CESAM chamber.

Experiments were performed in the CESAM chamber which has been described in detail elsewhere [START_REF] Wang | Design of a new multi-phase experimental simulation chamber for atmospheric photosmog, aerosol and cloud chemistry research[END_REF] . Briefly, it is an evacuable chamber comprising a stainless steel cylindrical vessel of 4.2 m 3 , equipped with three high-pressure xenon arc lamps (4 kW, MH Diffusion ® , MacBeam™ 4000) and 6.5 mm thick Pyrex filter to filter out radiation below 300 nm. During this set of experiments, NO2 photolysis frequency (JNO2) was experimentally verified to be equal to (1.74 ± 0.14)×10 -3 s -1 . The reactor contains a double wall which allows the circulation of a coolant liquid using a thermostat (LAUDA Integral T 10000 W). Thanks to this set-up, temperature was controlled at 283, 293 and 303 K with a maximum variation of1.5K during the course of an experiment. At the bottom of the reactor a stainless steel fan allowed a fast mixing of the gaseous species introduced within 60 s. [START_REF] Wang | Design of a new multi-phase experimental simulation chamber for atmospheric photosmog, aerosol and cloud chemistry research[END_REF] 2.2. Measurement. Temperature and relative humidity were measured using a Vaisala® probe (HMP234). Studied and reference compounds were monitored using an in situ Fourier transform infrared (FTIR) spectroscopy (Bruker® Tensor37™) coupled to a multipath path cell for a path length of 192 ± 4 m. Infrared spectra were obtained at a resolution of 0.5 cm -1 and derived from the co-addition of approximately 200 scans collected over 5 min. Concentrations of the interest species were determined by subtracting pure reference spectra from spectra of reaction mixtures using homemade software based on matrix algebra. To guarantee the performance of the automatic routine, selected spectra for each experiment were subtracted manually and results were compared. Integrated band intensities (IBI, base log 10) of the main infrared absorption bands of the mixture compounds are (1.03 ± 0.02)×10 -17 cm.molecule -1 and (3.97 ± 0.13)×10 -17 cm.molecule -1 for isoprene (830 -950 cm -1 ) and n-dodecane (3020 -2820 cm -1 ), respectively. Ozone was measured by a commercial UV absorption monitor (Horiba®, APOA-370). A commercial chemiluminescence NOx analyzer (Horiba®, APNA-370) was used to monitor NO and NO2.

Experimental procedure of the kinetic study (relative rate technique). All experiments

were carried out at atmospheric pressure. The chamber was filled with clean dry air by mixing 800 mbar of nitrogen produced from the evaporation of a pressurized liquid nitrogen tank (Messer, purity > 99.995 %, H2O < 5 ppmv) and 200 mbar of oxygen (Air Liquide, ALPHAGAZ™ class 1, purity 99.9 %). Temperature was set and kept constant throughout the experiments at 283 ± 1 K, 293 ± 1 K and 303 ± 1.5 K. n-Dodecane (≥99 %, Sigma Aldrich) is a semi-volatile specie that may be difficult to introduce quantitatively. In order to inject it into the chamber without being off the volatility equilibrium and generate unwanted phase exchange, DDC was introduced by evaporating a known mass quantity through a gently heated glass tube at 333 K via N2 stream into the chamber. To accurately quantify the n-dodecane introduction, the glass tube was weighed before and after the injection with the help of a balance (METTLER TOLEDO, XA205DU) providing a precision of 0.1 mg. Isoprene (≥99 %, Sigma Aldrich), the reference compound, was injected into the chamber by flushing a known amount of vapor via a calibrated glass bulb. OH radicals were produced by photolysis of nitrous acid (HONO) added to the chamber: HONO + hν  OH + NO HONO was synthetized from the drop-wise addition of sulfuric acid (10 -2 M) into a solution of NaNO2 (0.1 M) and was carried into the reactor with a flow of N2. In some experiments, HONO was produced from NO2 (99%, Air Liquide) wall reaction [START_REF] Wang | Design of a new multi-phase experimental simulation chamber for atmospheric photosmog, aerosol and cloud chemistry research[END_REF][START_REF] Pitts | An investigation of the dark formation of nitrous acid in environmental chambers[END_REF][START_REF] Ramazan | The photochemical production of HONO during the heterogeneous hydrolysis of NO2[END_REF] , which was injected with a gastight syringe and a septum valve. NO2 injection was followed with an introduction of water through a bubbler and resulted in 5 % relative humidity (RH) in the chamber. The production mechanism of HONO during the heterogeneous hydrolysis of NO2 in the dark is given as follow:

2NO2 + H2O 𝑤𝑎𝑙𝑙 → HONO + HNO3 Before turning on the lamps, initial concentrations were systematically measured for a minimum of 15 minutes in order to derive first order reactant loss rates. Initial conditions of the reference and target compounds at each temperature are given in Table 1. Indeed, as we are using a relative rate method to determine the reaction rate constants (kDDC+OH) of n-dodecane with OH radical, it was crucial that OH was the only oxidant responsible for the decay of the target and reference compounds. Rate constants for the reaction of OH can be reliably determined by monitoring the simultaneous decay of the studied and reference compound in presence of OH radicals:

n-Dodecane + OH  products (1) Isoprene + OH  products (2)
Consequently, a flow of NO diluted in N2 from a 8 ppm gas cylinder (8 ppm in N2, Air Liquide, Alphagaz 1) was adjusted between 0.5 LPM and 1 LPM to systematically minimize the O3 production (NO + O3  NO2 + O2). Results of the O3 titration allowed us to keep O3 mixing ratio below 20 ppb so that ozonolysis remains always negligible. Maximum O3 concentrations during the experiments are given in Table 1.

As the CESAM chamber is rigid, instrumental sampling flows results in a decrease of the pressure, which need to be compensated by an addition of synthetic air to keep the pressure constant during the whole experiments. This procedure induces a continuous dilution loss of the reaction mixture, ranging from 5.9 ×10 -6 to 1.4×10 -5 s -1 . n-Dodecane being semi-volatile has also been sensitive to deposition on the wall of the chamber, but only during the experiments carried out at 283 K. The wall loss rate of n-dodecane has been assessed during 1-2 h in dark period. In consequence, the integrated equation of the chemical system yields to the following:

Ln ( [DDC] t 0 [DDC] t ) -(𝑘 𝑤 + 𝑘 𝑑𝑖𝑙 ) × (t -t 0 ) = k DDC+OH k ISP+OH × [Ln ( [ISP] t 0 [ISP] t ) -𝑘 𝑑𝑖𝑙 × (t -t 0 )] (3)
where [DDC] t 0 and [ISP] t 0 are the initial concentration of n-dodecane and isoprene (at time t0);

[DDC] 𝑡 and [ISP] 𝑡 are the corresponding concentration (at time t) ; kw and kdil are rate constants to account for vapor wall-and dilution-losses respectively. Therefore, plotting the first term of Eq. ( 3) as a function of the second term should yield a straight line with zero intercept and slope k DDC+OH k ISP+OH (see figure 1). The Arrhenius expression of isoprene with OH radical used was equal to (2.7 ± 0.1) × 10 -11 exp(390 ± 100/T) (in cm 3 molecule -1 s -1 ) in the temperature range 240 -340 K [START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume II -gas phase reactions of organic species[END_REF] . Reaction rate constants of isoprene with OH radical kISP+OH at different temperature are given in Table 2. The Arrhenius expression from Atkinson, et al. [START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume II -gas phase reactions of organic species[END_REF] has been chosen as it represents the recommendation of IUPAC Sub-committee. Note the reported kinetic rate constant from IUPAC was in excellent agreement with the following available literature for isoprene reaction with OH. [START_REF] Vimal | Experimental and Ab Initio Dynamical Investigations of the Kinetics and Intramolecular Energy Transfer Mechanisms for the OH + 1,3-Butadiene Reaction between 263 and[END_REF][START_REF] Singh | Kinetics Investigation of OH Reaction with Isoprene at 240-340 K and 1-3 Torr Using the Relative Rate/Discharge Flow/Mass Spectrometry Technique[END_REF]

Results and discussion

Obviously, the photooxidation of n-dodecane and isoprene under high-NOx condition might produce O3 molecules, which could react further with isoprene, the reference compound. This reaction might lead to error in determination of the reaction rate constant of n-dodecane with OH radical. In consequence, a continuous injection of NO was manually adjusted to minimize the O3 production. In order to assess and make sure that O3 did not bias our measurements, lifetime with regards to OH and O3 has been evaluated with the following calculation:

𝑘 𝑂 3 +𝐼𝑆𝑃 [𝑂 3 ] 𝑘 𝑂𝐻+𝐼𝑆𝑃 [𝑂𝐻]
with 𝑘 𝑂 3 +𝐼𝑆𝑃 = (1.13±0.09)×10 -17 cm 3 molecule -1 s -1 at 293 K [START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume II -gas phase reactions of organic species[END_REF] and 𝑘 𝑂𝐻+𝐼𝑆𝑃 = (1.02 ± 0.15) ×10 -10 cm 3 molecule -1 s -1 at 293 K [START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume II -gas phase reactions of organic species[END_REF] . In the worst-case scenario, when O3 reached it maximum mixing ratio, i.e. 20 ppb, OH reaction with isoprene was still being the highest sink in comparison to O3, since the estimated lifetime ratio was 40. In other words, this procedure has guaranteed that OH-oxidation of isoprene was always and by far the main oxidative process (99.86% OH-oxidation vs 0.14% ozonolysis).

Control experiments under irradiation have shown negligible loss of isoprene (<10 -6 s -1 ) at all temperature investigated. For n-dodecane, these losses were also below <10 -6 s -1 at 303 K and 293 K while at 283 K we were able to measure a loss of (1.57 ± 0.09)×10 -6 s -1 (or τ=177 h, τ=1/kloss, with kloss the rate constant of loss processes) that were due to wall adsorption.

From the n-dodecane mixing ratio decay and n-dodecane + OH rate constant (kDDC+OH), a mean concentration of OH radicals of 2ⅹ10 6 molecules cm -3 has been evaluated. Considering this value, typical lifetimes (τ=1/kX+OH [OH], with X = DDC or ISP) in our experiments of ndodecane and isoprene are 10.5 and 1.3 h, respectively. In consequence, within the duration of the experiments, roughly an hour, wall and photolysis losses were clearly negligible in comparison to OH-oxidation in this study, except at 283 K where a slight correction from the n-dodecane wall deposition was accounted.

Plots of the relative rate measurements at 283, 293 and 303 K for the reaction of n-dodecane with OH radical using isoprene as reference compound are shown in Figure 1. As seen in the later plot, two to three isothermal experiments were carried out and are in very good agreement.

As expected from Equation 3, for all decay data points, a linear correlation is obtained with intercept close to zero within the uncertainties. The rate constant ratios (kDDC+OH/kISP+OH) was obtained by applying a linear regression, taking into account both the uncertainties on x-and y-axis as recommended by Brauers and Finlayson-Pitts [START_REF] Brauers | Analysis of relative rate measurements[END_REF] (cf. Table 2 and Figure 1). These latter were evaluated from injection calibration of both studied and reference compounds.

Uncertainties on kDDC+OH/kISP+OH correspond to 2σ (with σ the standard deviation of the linear regression). kISP+OH (from Atkinson, et al. [START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume II -gas phase reactions of organic species[END_REF] ) and kDDC+OH/kISP+OH, kDDC+OH are given in Table 2 at different temperature. Error on kDDC+OH (i.e. given in Table 2) were determined from the quadratic sum of the error of kISP+OH and kDDC+OH/kISP+OH. Average of the rate constants at 283, 293 and 303 K yielded to (1.27 ± 0.31)×10 -11 , (1.33 ± 0.34)×10 -11 , (1.27 ± 0.40)×10 -11 cm 3 molecule -1 s -1 , respectively. Errors on the average rate constants correspond to the mean standard deviation. A plot of the average rate constants obtained from our work is given in Figure 2 with the available literature data at different temperatures. As we can see from this figure, reported rate constants at 300 K by Behnke, et al. [START_REF] Behnke | A smog chamber for studies of the photochemical degradation of chemicals in the presence of aerosols[END_REF] (as reevaluated by Atkinson 7 ) and at 312 K by Nolting, et al. [START_REF] Nolting | A smog chamber for studies of the reactions of terpenes and alkanes with ozone and OH[END_REF] (as reevaluated by Atkinson 7 ) showed a good agreement within 2 and 9 %, respectively, with our determination at 303 K considering the error domain. Please note reaction rates by Phan and Li 28 has been re-evaluated with the recommended 298 K rate constant for the reference compound, leading to a reaction rate of (1.57 ± 0.24)×10 -11 cm 3 molecule -1 s -1 . As a consequence, comparison of our work at 303 K with the re-evaluated 298 K value from Phan and Li 28 agreed within 19 % considering the uncertainties. In the same way, recommended Arrhenius parameters for the reference compound provided by Atkinson (2003) has been used to estimate rate constants for the temperature ranges (298 -1110 K) over which these recommendations are applicable. As shown in Figure 2, the effect of this re-evaluation decreased slightly the reaction rate constant by 6 % at 320 K and 2 % at 340 K. The theoretical study from Sivaramakrishnan and Michael [START_REF] Sivaramakrishnan | Rate Constants for OH with Selected Large Alkanes: Shock-Tube Measurements and an Improved Group Scheme[END_REF] and the recommendation from Atkinson (2003) are in excellent agreement with our work, which is coherent since they were determined from the studies of Nolting, et al. [START_REF] Nolting | A smog chamber for studies of the reactions of terpenes and alkanes with ozone and OH[END_REF] and Behnke, et al. [START_REF] Behnke | A smog chamber for studies of the photochemical degradation of chemicals in the presence of aerosols[END_REF] .

The rate constants were also plotted with the Arrhenius expression (283 -340 K)

𝑘 𝐷𝐷𝐶+𝑂𝐻 = (9.77 ± 6.19)10 -11 × exp [ -595 ±5580 𝑇 ] 𝑐𝑚 3 𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 -1 𝑠 -1 ,
which was determined by linear least-square analysis of ln(k) against 1/T. Note that the Arrhenius fit was weighed by error bars and was determined from our work combined with rate constants reported by Behnke, et al. [START_REF] Behnke | A smog chamber for studies of the photochemical degradation of chemicals in the presence of aerosols[END_REF] , Nolting, et al. [START_REF] Nolting | A smog chamber for studies of the reactions of terpenes and alkanes with ozone and OH[END_REF] and Phan and Li [START_REF] Phan | Kinetics Study of the Reactions of OH with n-Undecane and n-Dodecane Using the RR/DF/MS Technique[END_REF] . The study of Phan and Li [START_REF] Phan | Kinetics Study of the Reactions of OH with n-Undecane and n-Dodecane Using the RR/DF/MS Technique[END_REF] that has led to somewhat outlying results is not without any potential issue. First, the use of a quadrupole mass spectrometer with electronic impact ionization source and unit mass resolution coupled with a flow tube might have induced uncertainties in the determination of reaction rate constants since the fragmentation pattern of secondary oxidation products from long chain alkane would recurrently present the same ion of the parent precursor due to the simple loss of stable neutral species (e.g. H2, OH, H2O, H2O2, (CH2)n, etc.) [START_REF] Mclafferty | Interpretation of mass spectra[END_REF] . Second, oxidation products from both DDC and reference compounds used in their study, i.e. n-nonane and n-decane, might also have contributed to the DDC peaks in the mass spectra. Therefore, the effect of secondary reactions within the flow tube reactor would potentially bias the determination of the reaction rates. In addition, despite the low pressure system (P=1-5 Torr), the saturation vapor pressure of n-dodecane -i.e 7.6 ⅹ10 -4 Torr at 240 K and 5.32 Torr at 340 K estimated with the prediction method by Nannoolal, et al. [START_REF] Nannoolal | Estimation of pure component properties: Part 3. Estimation of the vapor pressure of non-electrolyte organic compounds via group contributions and group interactions[END_REF] -decreases by several order of magnitude with the temperature. Consequently, the negative correlation from 240 to 277 K might be explained by the additional deposition of the n-dodecane vapor to the flow tube wall, which is not considered in their integrated kinetic equations. In other words, the reported negative correlation with temperature from 240 to 260 K might be explained by the increase of the vapor wall loss rate of n-dodecane. Since we can not rule out these measurement biases from Phan and Li 28 's, non-re-evaluated rate constants from 240 to 277 K were excluded from the Arrhenius fit. Uncertainties of the Arrhenius parameters (A, the pre-exponential factor, and Ea/R, the ratio of activation energy over ideal gas constant) are ±2σ of the fit.

As shown in Figure 2 from 283 to 340 K, the reactivity between the n-dodecane and OH radicals shows a relatively low positive temperature dependence. This observation is in good agreement with the well-known hydrogen-atom abstraction mechanism. [START_REF] Calvert | Mechanisms of atmospheric oxidation of the alkanes[END_REF] In addition, this low temperature dependence observed, in the typical tropospheric temperature range, is consistent with low activation energy reactions such as a long chain alkane with OH radicals ( 7 and references therein).. It would be interesting to investigate further the temperature effect of the reaction rate constant with particular care of the vapor wall deposition of the target compound to validate/invalidate observation from Phan and Li [START_REF] Phan | Kinetics Study of the Reactions of OH with n-Undecane and n-Dodecane Using the RR/DF/MS Technique[END_REF] .

The obtained rate constants for the reaction of n-dodecane with OH radicals can be estimated with prediction methods. These later are based upon the group-additivity transition-state theory (TST) described in Sivaramakrishnan and Michael [START_REF] Sivaramakrishnan | Rate Constants for OH with Selected Large Alkanes: Shock-Tube Measurements and an Improved Group Scheme[END_REF] and the structure activity relationship (SAR) method detailed by Kwok and Atkinson [START_REF] Kwok | Estimation of hydroxyl radical reaction rate constants for gas-phase organic compounds using a structure-reactivity relationship: an update[END_REF] . As seen in Figure 2, application of these prediction methods shows a nice agreement with our data and represents very well the low temperature dependence on the reactivity.

Conclusion

In this work, reaction rate constants of n-dodecane with OH radicals as a function of the temperature were determined between 283 and 303 K, using the relative rate method with the CESAM chamber. Rate constants measured are in excellent agreement with the literature and those estimated by the SAR and TST methods agreed reasonably, i.e. within 14%. Arrhenius expression were developed over a fair range of temperature (283 K -340K), representative of the troposphere. These kinetic data can be used to assess the atmospheric chemical lifetimes (τ=1/kDDC+OH[OH]) of the n-dodecane with the OH oxidants. Using this relation and a typical tropospheric OH concentration of 2ⅹ10 6 molecule cm -3 41 , the atmospheric lifetime was found to be 10.7 h at room temperature (298 K). Therefore, it appears that n-dodecane is likely to be removed rapidly and degraded relatively near to its emission sources. It may hence participate to the ozone and SOA burden at local/regional scale downwind urbanized/industrial area.

Corresponding Author

*Email: aline.gratien@lisa.u-pec.fr

Tables

FiguresFigure 1 .

 1 Figures

Figure 2 .

 2 Figure 2. Reaction rate constants of n-dodecane with OH radical as a function of temperature compared

Table 1 .

 1 Summary of the kinetic experiments.

			OH	[DDC]t0	[ISP]t0	[NO2]t0	[H2O]	[O3]max
	Exp	T (K)
			source	(ppb)	(ppb)	(ppb)	(molecule.cm -3 )	(ppb)
	E1.283	283 ± 1	NO2	471 ± 16	93 ± 3	66	2.10 ×10 16	12
	E2.283	283 ± 1	NO2	592 ± 21	76 ± 3	18	2.00 ×10 15	15
	E1.293	293 ± 1	NO2	408 ± 14	76 ± 3	24	3.21 ×10 16	12
	E2.293	293 ± 1	HONO	600 ± 21	86 ± 3	41	3.93 ×10 16	4
	E1.303	303 ± 1.5 HONO	559 ± 20	63 ± 2	67	3.51 ×10 15	20
	E2.303	303 ± 1.5 HONO	520 ± 18	57 ± 2	69	4.68 ×10 15	18
	E3.303	303 ± 1.5 NO2	641 ± 22	93 ± 3	34	6.58 ×10 16	20

Table 2 .

 2 Measured rate constants for the OH-oxidation of n-dodecane (DDC) with isoprene (ISP) as reference compound[START_REF] Atkinson | Evaluated kinetic and photochemical data for atmospheric chemistry: Volume II -gas phase reactions of organic species[END_REF] in the temperature range 283 -303 K.

	Exp	T (K)	This work kDDC+OH/kISP+OH	This work kDDC+OH (cm 3 .molecule -1 . s -1 )	kISP+OH (cm 3 .molecule -1 . s -1 ) 32
	E1.283 283 ± 1	0.129 ± 0.010	(1.38 ± 0.24)ⅹ10 -11
	E2.283		0.108 ± 0.008	(1.16 ± 0.19)ⅹ10 -11	(1.07 ± 0.16)ⅹ10 -10
				Average : (1.27 ± 0.31)ⅹ10 -11
	E1.293 293 ± 1	0.152 ± 0.017	(1.55 ± 0.29)ⅹ10 -11
	E2.293		0.108 ± 0.009	(1.10 ± 0.19)ⅹ10 -11	(1.02 ± 0.15)ⅹ10 -10
				Average : (1.33 ± 0.34)ⅹ10 -11
	E1.303 303 ± 1.5 0.146 ± 0.008	(1.43 ± 0.23)ⅹ10 -11
	E2.303		0.124 ± 0.014	(1.21 ± 0.23)ⅹ10 -11
					(0.98 ± 0.15)ⅹ10 -10
	E3.303		0.119 ± 0.016	(1.16 ± 0.24)ⅹ10 -11

Average : (1.27 ± 0.40)ⅹ10

-11

Acknowledgments

This work was supported through the project "Atmospheric oxidation of n-dodecane: fundamental studies and parameterization of SOA formation (Dodec-AOS)" within the PN-LEFE CHAT program (French National Program of Atmospheric Chemistry). In addition, CNRS-INSU is gratefully acknowledged for supporting the CESAM chamber as a national facility as well as the AERIS data center for distributing and curing the data produced by the CESAM chamber. This work was also supported by the European Community within the 7 th Framework Program, section "Support for Research Infrastructure -Integrated Infrastructure Initiative": EUROCHAMP-2, by the European Union's Horizon 2020 research and innovation program through the EUROCHAMP-2020 Infrastructure Activity under grant agreement n°730997.

Present address: ‡ Paul Scherrer Institute, Laboratory of Atmospheric Chemistry, CH-5232 Villigen-PSI, Switzerland


Giulia Lisarelli
email: giulia.lisarelli@dm.unipi.it

Samuele Antonini
email: samuele.antonini@unipv.it

Anna Baccaglini-Frank
email: anna.baccaglinifrank@unipi.it

Capturing 'time': characteristics of students' written discourse on dynagraphs

Keywords: Covariation, dynagraph, dynamism, function, time

We study high school students' written discourse about an experience in a dynamic interactive digital environment in which functions were represented in one dimension, as dynagraphs. In a dynagraph of a real function one variable can be acted upon while the other varies as a consequence of the movement induced on the independent variable. Students were asked to write about their experience with the dynagraphs, and their written productions were collected and analyzed using a classification, that emerged a posteriori, in "snapshots, live-photos and scenes". This classification is presented as a tool of analysis that allows to put discourse about dynagraphs in relation with discourse about functions. Examples of excerpts analyzed through this tool are given.

Functions, covariation, graphs, and dynagraphs

Functions and their graphs have always had a leading role in mathematical practice, including school practices. Being able to interpret the Cartesian graph of a function and to construct a graph starting from the function's properties are essential processes in mathematical thinking. However, these processes include an understanding of the meaning of variable and of the relation between variations of the variables, that is covariation; and several kinds of difficulties that students encounter when grappling with these ideas are widely reported in the literature (e.g. [START_REF] Tall | The Transition to Advanced Mathematical Thinking: functions, limits, infinity and proofs[END_REF][START_REF] Tall | Dynamic mathematics and the blending of knowledge structures in the calculus[END_REF][START_REF] Kaput | Patterns in students' formalization of quantitative patterns[END_REF][START_REF] Monk | The case of Dan: Student construction of a functional situation through visual attributes[END_REF][START_REF] Carlson | Applying covariational reasoning while modeling dynamic events: A framework and a study[END_REF][START_REF] Thompson | Variation, Covariation, and Functions: Foundational Ways of Thinking Mathematically[END_REF].

A Cartesian graph is defined as the set of points (x, f(x)) with real coordinates on the Cartesian plane, where x belongs to the domain of the function and f(x) is its image; these constitute a curve incorporating the functional relation between the two variables. Students encounter various difficulties when dealing with graphs, one being in recognizing that each point on the graph is a coordinated presentation of two pieces of information [START_REF] Colacicco | Funzioni e grafici in ambienti digitali dinamici[END_REF]. This leads them to consider the curve to be the function itself and to identify a point on the curve only as its f(x) value. Both the asymmetric relation between the variables and their covariation are lost.

In this paper we focus on the use of a particular technological artifact to represent functions, realized through a dynamic interactive environment (DIE). Many research studies support the use of technology in the teaching of functions [START_REF] Healy | If this is your mathematics, what are your stories[END_REF][START_REF] Falcade | Approaching functions: Cabri tools as instruments of semiotic mediation[END_REF]. Here we are interested in representing functions through DynaGraphs [START_REF] Goldenberg | Dynamic representation and the development of an understanding of functions[END_REF], dynamic representations in which the domain variable is dynamically draggable on a line and it is presented separately from its image. Both the x-and y-axes are horizontal; originally, they were referred to as "the x Line" and "the f(x) Line". DynaGraphs cannot be constructed without using a DIE, where objects can be moved on the screen.

The independence of the x variable is realized by the possibility of freely dragging a point, bound to a line (the x Line), and the resulting movement visually mediates the variation of the point within a specific domain. Whereas the dependence of the f(x) variable is realized by an indirect motion: the dragging of the independent variable along its axis causes the motion of a point, bound to another line (the f(x) Line), that cannot be directly dragged. The use of dragging can visually mediate the discourse on functional dependency, since it can help students interpret the exploration in terms of logical dependency between two different types of motion: direct and indirect motion. The authors, together with Nathalie Sinclair, built on the original idea of DynaGraphs to realize functions as in Figure 1, within the dynamic geometry software Sketchpad (a similar design was adopted also in [START_REF] Lisarelli | How dragging mediates discourse about functions[END_REF] which will be called dynagraphs (no capitals, to distinguish them for the original ones) in this paper. Many properties of a function can be recognized in a dynagraph. For example, looking at the movements of the two variables that either follow the same direction or move in opposite directions, an expert can recognize when a function is increasingif both variables have the same direction of movementor when it is decreasingif the variables move in opposite directions. The movement and the identification of invariants are central in the exploration of dynagraphs, since the DIE enables students to deal with the dependence relation in terms of possible or impossible movements; we are interested in how this dynamic and temporal experience can be narrated on a static sheet of papera process that is frequent in mathematics.

Many researchers have described the relations between movement, time and mathematics, in particular in calculus. [START_REF] Tall | Dynamic mathematics and the blending of knowledge structures in the calculus[END_REF] describes calculus as the mathematical field that begins with the desire to quantify how things change, the function, the rate at which they change, the derivative, and the way in which they accumulate, the integral. So, this field is fundamentally dynamic: even the calculation of static quantities, such as areas or volumes, involves dynamic processes of adding up a large number of very tiny elements. At a certain historical moment, a transformation took place and calculus turned into rigorous definitions, developing into the formal theory of mathematical analysis which is used today by textbooks and by many teachers. This transformation led to the teaching of calculus based on the definition of limits, which satisfies mathematicians' logical needs, but it proves to be rather complicated for students [START_REF] Tall | The Transition to Advanced Mathematical Thinking: functions, limits, infinity and proofs[END_REF]. However, even if formal mathematical discourse aims at eliminating time and dynamism, this does not imply that mathematicians engage in purely a-temporal modes of thinking. Indeed, they seem to frequently communicate in ways that suggest they think of mathematical objects in motion (Sinclair & Gol Tabaghi, 2010).

Theoretical background and research focus

For this study we make use of discourse analysis; in particular, we adopt the commognitive theoretical perspective [START_REF] Sfard | Thinking as communicating: Human development, the growth of discourses, and mathematizing[END_REF]. According to Sfard, communication and cognition are two manifestations of a same phenomenon and mathematics can be considered as a special type of discourse, or communication. Mathematical learning is the process by which students become able to communicate about mathematical objects, that, unlike others, are purely discursive objects that can have different realizations (p. 165). Moreover, mathematical discourse, as a particular kind of discourse, involves a frequent use of visual mediators, visible objects realizing the object of the discourse (p. 133).

In this study, we use dynagraphs as particular realizations of the mathematical object 'function' within a DIE, in order to allow students to experience the dependency between two covarying quantities in terms of motion. This approach brings temporality into play. We intend to study how students deal with the temporal dimension that characterizes their experience in the DIE, when they are asked to describe it through a written explanation. Moreover, we are interested in the mathematical aspects expressed in these written explanations, especially in how variation of variables and covariationthe relation between variationsare conveyed.

The research question that led the design and the development of this study is the following: What are characteristics of students' written discourse on dynagraphs related to their management of the movement and time experienced during the exploration that can be put in relation with discourse about functions (as mathematical objects)?

Methodology

We designed 7 dynagraphs of functions to be explored. The markers realizing the two variables had no labels, because we wanted to allow students to decide which words or symbols to use. We consider this to be an important process related to distinguishing the two variables and the asymmetric relation between them. The two points 0 and 1 were marked on the lines with ticks in order to provide the unit segment and to highlight that the lines visually realize two copies of the real numbers.

Students belonging to four different 10 th grade classes of Canadian high schools worked in pairs on as many dynagraphs as they had time for during one lesson (60 minutes). Their task was to "explore the dynagraphs on the iPad and write down their observations about them" on the whiteboard and/or on paper. We collected these written productions. This task was designed to "force" the students to describe in writing an experience which was completely time-and-motion-immersed.

Tools of analysis: snapshots, live-photos, and scenes

From analyses of the high school students' written productions we reached a characterization that was later refined through further rounds of analysis of the excerpts. Coherently with our research interest in gaining insight into how movement (the variation of position in time) enters students' written discourse we introduced a first possible characterization of written accounts of the experiences in the DIE, using a terminology that refers to images in the context of photos and videos. In Table 1 we define the main types of accounts and describe characteristics of the discourse it characterizes, both in the case in which the discourse is about the dynagraph (which may or may not be related to the notion of function for the students) and when it is about the function realized through the dynagraph. 1 shows the main types of accounts (snapshot, live-photo and scene) and their further characterization with respect to ways in which more than one could be used (as albums or clusters) in the excerpts of written discourse that we will now analyze.

We observe that live-photos may also be found in albums and in clusters. A live-photo album is set of live-photos, each showing something in particular; it expresses properties in selected positions, which constitute a finite discrete set; each property is shown in one live-photo. From a mathematical point of view, these correspond to local properties in neighborhoods of a particular set of points. A live-photo cluster is a set of live-photos showing relationships between them.

Analysis of students' written products

This section contains examples of analyses of students' written products using the characterization. "When the top number was zero, the bottom number was between 4 and 5. When the top was at 1, the bottom was about 5 or 6. I'm guessing that the equation would be b(t)=t+5."

Excerpt 1 contains discourse about the dynagraph of the function f(x) = x+5; Figure 2a and 2b depict a scene through snapshots presented using written text and visual mediators. The first property in Figure 2a is described through a snapshot taken 'when the top number was zero'. The first drawing in Figure 2b presents this specific instance. The second property is described analogously as a snapshot taken "when the top number was at 1"; the second drawing in Figure 2b presents this specific instance. The third property in the scene is described through the algebraic formula "b(t) = t+5". The dynagraph in excerpt 2 realizes the nearest integer function. The students describe the jumping of the red line as A is dragged across the midpoint of the interval [0,1]. This part of their discourse appears to be a live-photo depicted through two visual mediators that are particular snapshots showing an 'initial' and a 'final' state of the dynagraph as A moves in a neighborhood of 0.5. The dynagraph in excerpt 3 realizes the nearest integer function. The students produce a single visual drawing realizing the dynagraph's interval [-1,3] on both axes (Figure 4), and describe its behavior written in words. Each live photo shows A near the midpoint of a unit interval. We see these as live-photos because each 'sudden change' is shown by a position of the segment AB immediately before and immediately after the transition of A across the midpoints (the ones realized are -0.5, 0.5, 1.5, 2.5) of the unit intervals. The dynagraph discussed in Figure 5 realizes the function f(x)=x+5. The students who author the discourse in excerpt 4 emphasize the invariance of the distance of A from B, conveyed through the words "no matter what direction" and through the visual mediation of two congruent parallel segments in the drawing with in between a double-headed arrow and two arrows pointing to the words "same distance". The written text and the visual mediators appear to be a cluster of snapshots depicting the invariant property identified. In excerpt 5 (Figure 6) the students use only written words to outline the behavior of dynagraph of f(x)=2x. They consider a whole interval of time (and space): this is suggested by their use of the adverb "always", and by the fact that they make several references to possible movements and to the reciprocal position of the two variables ("forwards", "ahead", "backwards", "behind").

Excerpt 2: Live-photo depicted through two snapshots showing 'initial' and 'final' states

Excerpt 3: Album of live-photos

Excerpt 5: Scene album depicting two properties of a dynagraph

Moreover, the students start with the present tense of the verb "to be" ("B is always ahead […] B is behind"), which gives their statements the form of an absolute observable truth, but then they add details on the quality of movement, passing to the "-ing form" of other verbs, indicating spatial dynamic features ("always […] gaining", "keeps falling behind"). This could suggest that the students see the movement as not ending even when it cannot be seen anymore within a large interval.

Conclusions

The analytical tool we have introduced to describe written accounts of the DIE experience allows to characterize different ways used by students to realize movement and time experienced during the exploration of dynagraphs, and puts them in relation with mathematical discourse about functions. In particular, it highlights how the choice of the time interval being described, that can be identified in students' discourse by looking at the verb tenses, adverbs and at their use of visual mediators to realize movement, corresponds from a mathematical point of view to different properties of the functions realized through the dynagraph. The classification has value at the cognitive, didactical and epistemological levels. From a cognitive point of view, the tool is important because the classification proposed was identified a posteriori, through empirical analysis of the data collected, and it seems to be a powerful tool for analyzing all the data collected. Epistemologically, the types of accounts can be put in relation with different mathematical properties of functions in formal mathematical discourse, as shown in Table 1.

Thinking about properties that are pointwise, local or global, the discourseeven that of expertsthat can emerge about these may have characteristics such as those we have found in students' discourses, focusing on "instants" that capture the behavior of isolated points, systems of points, sequences, neighborhoods of points, points as limits, intervals… Moreover, experts' discourse can capture properties at certain points, variations and invariants. For example, a description of what happens in a neighborhood of a point can be conveyed through a live-photo, while the behavior at infinity requires a scene, since a neighborhood of infinity can be seen as a scene "from a certain point on", that is, as a live-photo at infinity. From a didactical point of view the classification is important, assuming that we value promoting various forms of students' discourse, and therefore students' flexibility in constructing such discourse. The teacher can interpret students' discourse from a mathematical point of view, seeing it as a 'mirrored image' (with some possible distortions) of experts' discourse about pointwise properties, local or global properties, variations, and invariants. Doing this, the teacher can then promote these forms of discourse through appropriate tasks (e.g., by using the behavior of dynagraphs in certain positions, intervals, etc.), knowing what to expect and how to gradually foster the transition to more formal mathematical discourse. One particularly fruitful type of discourse seems to be that involving transitions from pointwise to local properties, and thus discourse involving live-photos. However, this needs to be studied in future research currently under development.

Figure 1 :

 1 Figure 1: Screenshot of a dynagraph used in this study

Excerpt 1 :

 1 Figure 1a,b: Scene of snapshots

Figure 2 :

 2 Figure 2: Live-photo

Figure 3 :

 3 Figure 3: Album of live-photos

Excerpt 4 :

 4 Figure 4: Scene

Figure 5 :

 5 Figure 5: Scene album

Type of account of the DIE experience

				Discourse about dynagraph	Discourse	about
					function
	snapshot snapshot: photo at a	provides an instantaneous shot in which	pointwise properties
		certain position showing	movement is stopped at a certain position
		one or more properties
		snapshot-album: set of	expresses properties in selected positions;	pointwise properties
		snapshots, each showing	these constitute a finite discrete set; each	at a particular set of
		something in particular	property is shown in one snapshot	points
		snapshot-cluster: set of	one or more properties are described	properties that refer
		snapshots	showing	through a comparison of snapshots of the	to variation and/or to
		relationships	between	dynagraph; the focus is on the	invariants in specific
		them		relationships between snapshots; it may	intervals identified
				be in a sequence coherent with the order	by analyzing and
				on the line or not.	comparing different
					points
	live-	live-photo:	animated	one or more properties that depend on the	local properties of
	photo	photo showing what	nearby positions are described at a certain	functions,	at	a
		happens a bit before and	position; certain positions of the	specific point and the
		a bit after a certain	dynagraph are shown close to a specific	variations in any
		instant.		position	neighborhood of the
					point.
	scene	scene: video showing	description of one or more properties of	properties	of
		movement over a period	the movement over an arc of time, in an	functions that are
		of time; it can be realized	interval of real numbers, or in the space	global, general, or
		through snapshots, or	covered by the point moving along the	present in a whole
		live-photos, in an album	line; it includes various positions. The	interval
		or cluster.		intervals can be limited or unlimited; the
				scene can be described statically or
				dynamically

Table 1 : Characterization of students' written discourse Table

 1

Acknowledgments

We graciously thank Nathalie Sinclair for her support in the realization of the activities described in this paper, and for all the fruitful conversations she has engaged in with us. This study has been partially supported by GNSAGA of INdAM.


Jasmina Milinkovic
email: milinkovic.jasmina@yahoo.com

Aleksandra Mihajlovic
email: mihajlovic@gmail.com

Mirko Dejic
email: mirko.dejic@uf.bg.ac.rs

Effective choices of representations in problem solving

Keywords: Representations, problem solving, argumentation, solution representation

This paper investigates the types of representations used by high-achieving eleven-year-old pupils in their argumentation in the process of mathematical problem solving. The literature review addresses pedagogical and psychological aspects of knowing representations and using them in the process of problem solving. In our empirical study we examine a sample of 109 eleven-year-old pupils who took part in the final round of a national mathematical competition in Serbia. A total of 656 problem solutions was analyzed. The results show that these pupils used and often combined inventive iconic and conventional numerical/symbolical representations in their argumentation. We discuss pupils' choice of representations which lead to correct solutions and adjusted representations corresponding to the type of problem (algebraic or logical-combinatorial). The aim of the paper is to draw attention to the importance of studying representational approaches pupils take when solving problems.

Introduction

There has been rising interest among mathematics educators in the usefulness of representations in the process of learning and problem solving. Competent use of representations in problem solving appears to be a significant indicator of level of mathematics literacy [START_REF] De Lange | Mathematics for Literacy[END_REF]. Then, it is a plausible assumption that investigations on how high-achieving pupils use representations in problem solving could lead to the development of strategies for choosing adequate representations in problem solving process. Some of the principal objectives of mathematics education are to enable pupils to be good problem solvers (Schoenfeld, 1992). Solving problems in various ways on one hand and "decoding, encoding, translating, distinguishing between, and interpreting different forms of representations of mathematical objects and situation" on the other hand are recognized as some of the key competences needed for mathematical literacy as it is defined by national curriculums worldwide (De Lange, 2003, p.77). Representations, as tools for presenting problems, are considered to be means for understanding. It is also advocated that using multiple representations in problems may contribute to higher achievement [START_REF] Cai | Mathematical Problem Posing as a Measure of Curricular Effect on Pupils' Learning[END_REF]. Role of representations in problem solving is recognized as important issue (Janvier, 1987;Cuoaco & Curcio, 2003;[START_REF] Shoenfeld | Learning to think mathematically: problem solving, metacognition, and sense making in Mathematics[END_REF]. Two important issues are 1) which representations pupils invent when solving non-routine tasks and 2) which representations more often assure successful argumentation.

Theoretical background

In numerous conceptualizations of mathematical competences and mathematical thinking, ability to use representations (symbolic, numerical, visual, or verbal expressions) is identified as important parameter of mathematical abilities. [START_REF] Tchoshanov | Representation and Cognition: Internalizing Mathematical Concepts[END_REF] discusses types of representations on the concrete to abstract continuum as they reflect different modes of representation: concrete (real object, physical model, manipulative), pictorial (photograph, picture, drawing, sketch, graph), abstract (sign, symbol, written, verbal language). [START_REF] Matteson | Middle school pupils' representational understandings and justification schemes: gleanings from cognitive interviews[END_REF] identified five categories of mathematical representations in problem solving: numerical, iconic, verbal, symbolic, and dual. Numerical representation focuses on specific numerical values in a variety of formats, such as decimal, fraction, percentage, or a numerical list (such as a list of numbers appearing as outcomes of probability). According to Matteson, iconic representations encompass different visual representations, from pictorial, (realistic) models, horizontal charts, vertical charts to graphs and coordinates graphs. Pictorial representations are pictures of real-world objects such as toys, dice, etc. As we slightly depart from Matteson's classification, we will consider iconic representation as distinct from pictorial (which closely resembles real world objects). For example, an apple may be "iconically" represented as a dot. Verbal representations require application of written (or spoken) language to express understanding, to describe, analyze, explain, or reflect upon numerical, algebraic, or graphic representation (which does not include brief phrases such as directions for solving the problem). Symbolic representations involve symbolic notation and include usage of variables and formulas such as: equation, expression, algebraic equation, algebraic expression, and formula [START_REF] Matteson | Middle school pupils' representational understandings and justification schemes: gleanings from cognitive interviews[END_REF]). Yet pupils may not use only one representation in problem solving. Under the term "dual representation" Matteson considers dealing with two categories of representations in problem solving (e.g. numerical and verbal). Matteson claims that "problems' (solutions) incorporating multiple representations generally result in more incorrect solutions" (Matteson, 2007, p. 60). Some researchers pointed out that visual representations play an important role in supporting reflection and as a tool of communicating mathematical ideas [START_REF] Arcavi | The role of visual representations in the learning of mathematics Educational studies in mathematics[END_REF]Gagatsis et. all, 2010;[START_REF] Sfard | On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin[END_REF]. The main functions of visual representation are to illustrate symbolic representation and to resolve conflict between intuition and symbolic solution. Successfully arriving at solutions to mathematical problems utilizes a combination of problem representation skills and symbol manipulation skills [START_REF] Brenner | Learning by Understanding: The Role of Multiple Representations in learning Algebra[END_REF]. The first mentioned function involves skills which "include constructing and using mathematical representations in words, graphs, tables, and equations" (ibid). Lesh and colleagues argue that besides the significance of each representational system, attention should be directed toward translation among representational systems as well as to transformation within them [START_REF] Lesh | Representations and Translations among representations in Mathematics Learning and Problem Solving[END_REF]. They point their research shows that the act of representing tends to be "plural, unstable and evolving". Along the line, Duval discusses changes of register of representation in terms of cognitive operations: 1) transformations within the same register, like for example in case of algebraic operations or numerical computation, 2) translation of representation into different register, which is more cognitively challenging [START_REF] Duval | Representation, Vision and Visualization: Cognitive Functions in Mathematical Thinking. Basic Issues for Learning[END_REF](Duval, , 2003)).

Several researchers attended to multiple representations in problem solving [START_REF] Lowrie | The Influence of Visual Representations on Mathematical Problem Solving and Numeracy Performance[END_REF][START_REF] Hegarty | Types of visual-spatial representations and mathematical problem solving[END_REF][START_REF] Matteson | Middle school pupils' representational understandings and justification schemes: gleanings from cognitive interviews[END_REF]. [START_REF] Lowrie | The Influence of Visual Representations on Mathematical Problem Solving and Numeracy Performance[END_REF], in a study of the relationship between different forms of problem representation and pupils' performance in problem solving, has found that high-achieving middle school pupils tend to use nonvisual methods when solving problems. He pointed out that most studies examining representational preferences along a visual-nonvisual continuum have considered preference alone and did not attend to effectiveness of method (ibid).

Lowrie found that those "pupils' who were able to use visual representations regularly and in an efficient manner, were able to solve mathematics problems in a more effective manner than pupils' who were more inclined to use nonvisual aids on a regular basis" (Lowrie, 2001, p. 360). [START_REF] Stylianou | The role of representations in advanced mathematical problem solving: an examination of expert-novice similarities and differences[END_REF] examined differences in using representations between mathematicians and undergraduate students (as novices). They found that experts construct visual representations more frequently and use qualitatively better way to understand problem situation. School teachers are traditionally inclined to rely on symbolic, analytical representations. Consequently, pupils tend to do the same. The role of different types of representation in problem solving, particularly the effects of the usage of multiple representations is not fully explored.

Methodology

We investigated types of representations used by high-achieving eleven-year-old pupils in solving problems. The main research questions were: 1. what representations pupils most frequently use when solving problems; 2. what representations are associated with correct solutions; and 3. how many representations used at the same time in argumentation are most often associated correct argumentation. Our questions were proposed based on previous research from Matteson, Lowrie, and Brenner and colleagues.

The data was collected in the final round of a national mathematics competition "Mislisa", similar to "Kangaroo without borders" In this competition, mathematics relevant for solving problems is not strictly limited to the national curriculum. Successful problem solving in this competition requires critical thinking and applying knowledge in non-standardized situations. The problems range from the simplest computation problems, to algebraic, logical, and combinatorial problems most often set in realistic context. The competition promotes the value of mathematics literacy in pupils.

The study sample consists of 109 pupils who qualified for the finals of the competition by achieving the maximum score in the preliminary round of competition. We examined problem solutions from all finalists. The subset of 8 problems in which competitors were explicitly asked to provide explanation was analyzed in our study. Descriptive statistics was done for those eight problems.

Solutions for two of the problems are analyzed qualitatively in greater details. Our preliminary analysis of all solutions, resulted in selection of The Bottle problem and The Castle problem as exemplars for the points we make. They are the following:

The Bottle problem There is three times more milk in the first bottle than in the second one. If 3l of milk is added in the first bottle, and 5l of milk in the second, than there would be two times more milk in the first bottle than in the second one. How many litres of milk has been in the bottles at the beginning? Explain your answer.

The Castle problem A Castle has 6 towers. Each tower has a room. Each room has one door. Each door has a lock. All rooms are locked and keys are mixed up. What is the maximum number of trials a guard needs to check, in order to open the doors?

In the Bottle problem, finding way(s) to represent relationships between amount of milk in the first and the second bottles is the key for finding the solution. Problems similar to this are commonly solved in schools with the strategy of forming an algebraic expression (symbolic representation). But there are alternative strategies for solving this problem such as 1) the "method of segments" which relies on expressing relationships between quantities on line segments (iconic representation) or 2) the "method of trials" (involving number representation) or 3) simply explaining the solution in words (verbal representation) or 4) some other creative approach.

The Castle problem is logical-combinatorial. The maximum number of trials in the first cycle to get to the door that can be unlocked is obviously equal to the number of rooms. In the following cycle of trials, one door is already unlocked so five doors remained to be checked in. There is no one "school" strategy for solving this problem.

Results and Discussion

Here we present descriptive analysis of pupils' solutions. A total of 656 problem solutions are taken as units of analyses. Out of them, 355 items were solutions to algebraic problems and 301 items solutions to logical-combinatorial problems.

Table 1 shows that choice of representations for algebraic problems did not match representations used in combinatorial problems. Since pupils often used more than one representation, the percentages in each column do not add to 100. Iconic representations are considered to be more intuitively comprehensible and closer to reality than symbolic representations which involve abstraction and generalization in mathematics domain [START_REF] Liu | Survey of primary pupils' mathematical representation status and study on the teaching model of mathematical representation[END_REF]). Yet sufficient experience with such representations is a prerequisite for their' appreciation. The high-achieving pupils in our study had a tendency to use symbolic representations in algebraic problems. Almost half of the solutions involved using of symbolic representation in those problems. The test of independence showed that there is significant association between correctness of problem solution and use of symbolic representation ( 2 =62,417, df=2, p<0,001). The strength of relationship is medium (Cramer's V=0,419, p<0,001).

Many pupils obviously assumed that they were expected to use pictures when explaining their solutions. But often pictures were neither sufficient nor effective aid in argumentation. Some pupils combined pictorial with numerical representation (e.g. counting up number of attempts in the Castle problem). The others combined pictorial with symbolic and verbal representations by creating number sentences and written explanations of steps.

Prototypical representations in pupil's solutions

Pupils created diverse representations for their argumentation. In the Castle problem a significant number of pupils used numerical representation. Yet, others supported argumentation with pictorial representation in combination with other representations. Figure 1 shows an example of argumentation based on dual representation (pictorial and numerical). When using iconic representation, most commonly the pupils relied on number line and the method of segments. This method is commonly taught in Serbian schools. Figure 2 illustrates an example of argumentation based on number line representation in solving the Bottle problem. In her problem solution, Ana used the method of segments (number line representation) to show the relationship between elements in the initial state and the final state (so this pupil represented how the situation changed by drawing two pairs of number line segments). Ana switched from the number line representation to symbolic ones as she wrote a chain of equivalent equations.

An example of the solution of the Castle problem, solely based on iconic representation is presented in Figure 3. Marko put a key for his iconic presented argumentation on the right, where "x" meant unsuccessful attempt, " ", successful attempt, and"-" that the door cannot be opened. Formally, without (absent) supplementary explanation the record was not mathematically correct as it stated e.g. I = 6 and V = 1. Sara used different numeration to distinguish between number of attempts and number of trials in the cycle, overlooking meaning of the equality sign. Her solution was assessed to be too sketchy and although the answer was correct, she did not earn points for argumentation.

Pupils often combined different representations in solving problems. In those cases they usually started with pictorial representation, than they turned to verbal or symbolic representation. Using pictorial representation, enabled them to provide simpler explanation. An effective presentation of thought process with no verbal explanation is presented in Figure 1. In Figure 5 is Luka's solution of The Castle problem with multiple representations. He initially used pictorial and numerical representations along with verbal explanation and finalized the solution with numerical calculation. The realistic picture of doors itself hardly helped in solving the problem. Nevertheless Luka used it as a tool in argumentation. He probably meant to indicate with that picture that all 6 doors were checked with one key. The other cycles of tries were not visually presented in Luka's solution. But his verbal explanation presented on the right side of the picture was accurate. Luka wrote: "Since there were 6 keys and 6 doors, with the first key there could be no more than five failed attempts, in the next step 4, than 3, and so on, in each successive step 1 less".

Conclusion

Successful problem solvers construct significantly more representations than unsuccessful problem solvers. Our results do not endorse previous arguments that multiple representations are associated with more incorrect solutions. Using combined representations often results in correct solution. Typically, pupil's argumentation starts with pictures, than turns to verbal or symbolic representation. Using pictures frequently enables pupils to provide sound explanation.

We found that talented pupils had a tendency to use symbolic representations. Pupils' choice of representation could be a consequence of limited experience in using different representations in the past. Ultimately teachers need to help pupils to develop flexibility in using different representations and ability to combine representations (or move in argumentation from one to another one).

In particular, future research should test whether pupils should be encouraged to provide argumentation by 1) combining numerical and verbal representations in the case of logical combinatorial problems and 2) numerical pictorial and iconic representations in the case of algebraic problems. More generally, we need to focus on developing methods for improving pupils 'abilities to visualize and verbalize thought process and to be flexible in combining representations in the process of problem solving.

Figure 1 :

 1 Figure 1: Dual representation -pictorial and numerical (left) Figure 2: Dual representation -number line and algebraic expressions) (right)

Figure 3 :

 3 Figure 3: Iconic representation Figure 4 shows Sara's solution, assessed to be a creative argumentation involving Roman and Arabic numerals in solving the Castle problem. The solution consisted of five equalities. Sara counted the cycle of attempts by Roman numerals. She used Arabic numerals to show number of attempts in the particular cycle.Formally, without (absent) supplementary explanation the record was not mathematically correct as it stated e.g. I = 6 and V = 1. Sara used different numeration to distinguish between number of attempts and number of trials in the cycle, overlooking meaning of the equality sign. Her solution was assessed to be too sketchy and although the answer was correct, she did not earn points for argumentation.

Figure 4 :

 4 Figure 4: Combined numerical representations (left) Figure 5: Multiple representations -pictorial, verbal and numerical (right)

Table 1 : Percentages of problem solutions with particular type of representation

 1 Thus, successful problem solvers constructed significantly more representations than unsuccessful problem solvers. Our finding confirm Lesh's observation about plural and evolving nature of the act of representing. The finding does not reinforce Matteson's statement that "multiple representations result in more incorrect solutions"[START_REF] Matteson | Middle school pupils' representational understandings and justification schemes: gleanings from cognitive interviews[END_REF]

	Type of representation Algebraic problems	Logical-combinatorial problems
	Numerical	31,9%	57,8%
	Iconic	22%	20,8%
	Verbal	21,8%	61,2%
	Symbolical	45,9%	8,3%
	In accordance to Duval's claim, pupils tended to use non-visual representations. To explain
	algebraic problem solutions, pupils most frequently used symbolical representation. It is not
	surprising since traditionally solving algebraic problems in a way other than writing down
	equivalent equations is undervalued in classrooms. In contrast, the logical-combinatorial problem
	solutions pupils primarily relayed on verbal and numerical representations. Particularly, verbal
	representations were often used for argumentation in solving logical combinatorial tasks, rarely for
	algebraic problems.
	We examined whether there is an association between number of used representations (1, 2 or 3)
	and correctness of solution (False/Partially correct/Correct). The percentages are presented in the
	Table 2. Percentage of correct solutions including more representation, were bigger than of correct
	solutions with smaller number of representations used. We performed the χ 2 test of independence
	for variables Number of used representations (1, 2 or 3) and Correctness of solution (False/Partially

correct/Correct) on problems. There is a significant relationship between "number of used representations (1,2,3)" and "Correctness of solution (false, partially correct, correct)". (χ 2 =67,478, df=4, N=656, p<0,001). Cramers's V coefficient of 0,227, p<0,001 suggests moderate correlations.

Table 2 : Percentages of number of used representations according to correctness of solution

 2


F Da Silva

S Heuraux

E Ricardo

A Silva

T Ribeiro

Modelling reflectometry diagnostics: finite-di erence time-domain simulation of reflectometry in fusion plasmas

Keywords: Nuclear instruments and methods for hot plasma diagnostics, Simulation methods and programs, Plasma diagnostics -interferometry, spectroscopy and imaging

A

: Reflectometry simulations are particularly important since they allow to assess the measuring capabilities in experimental fusion devices and to predict the performance of future ones. We present a brief overview of reflectometry and introduce the family of REFMUL* codes for timedependent reflectometry simulation. REFMUL* codes are Finite Di erence Time Domain (FDTD) that allow to set up synthetic diagnostics to assess the behaviour of reflectometry diagnostics. This is illustrated in the current manuscript using the example of the Plasma Position Reflectometers of DEMO.

Introduction

Microwave reflectometry has become one of the most important techniques to diagnose the electronic density n e of fusion plasmas [START_REF] Laviron | Reflectometry techniques for density profile measurements on fusion plasmas[END_REF][START_REF] Mazzucato | Microwave reflectometry for magnetically confined plasmas[END_REF]. It has its origins in the techniques used to probe the ionosphere [START_REF] Breit | A test of the existence of the conducting layer[END_REF]. Reflectometry simulations are particularly important since they allow to assess the measuring capabilities of existing experimental systems and to predict the performance of future ones. We will introduce the reader to the family of REFMUL* codes. These time-dependent codes allow to set up synthetic diagnostics that not only consider the wave propagation in a given time-dependent plasma, but also take into account the system's location within the vacuum vessel, together with a characterisation of its access to the plasma (wave guides and antennas). We illustrate the use of reflectometry synthetic diagnostics in the assessment of the Plasma Position Reflectometers (PPR) of DEMO.

Microwave reflectometry

This diagnostic relies in the propagation of waves in a plasma. A probing signal s src = A src cos(!t) is launched through a plasma where it propagates until being reflected at a cuto surface x c , where the refraction index is zero N(x c ) = 0. The received reflected signal s rcv = A rcv cos(!t + '), di ers from the emitted one in amplitude and phase. This phase shift ' = ' µ +' 0 +' p results from di erent contributions: propagation in the microwave circuitry, ' µ ; in a vacuum, ' 0 ; and in the plasma ' p . A careful calibration of the system is done to determine ' µ and a good approximation to ' 0 can be inferred. This leaves us with the phase di erence ' p , that reflects the propagation of the wave along a path described by a refractive index N(r) and contains information on the electronic density n e . The Altar-Appleton equation (aka Appleton-Hartree or Appleton-Lassen) describes the refractive index of a wave propagation in a cold magnetised plasma [START_REF] Lassen | Über die Ionisation der Atmosphäre und ihren Einfluß auf die Ausbreitung der kurzen elektrischen Wellen der drahtlosen Telegraphie[END_REF], where a magnetic field B 0 exists. Of interest to reflectometry is the particular case of propagation perpendicular to the external magnetic field k ? B 0 which gives two expressions. One for the ordinary mode (O-mode) with N 2 = 1 X and another for the extraordinary mode (E-or X-mode) with N 2 = 1 X(1 X)/(1 X Y 2 ) [START_REF] Hutchinson | Principles of Plasma Diagnostics[END_REF]. Note the normalisation X = ! 2 p /! 2 and Y = ! c /!, being ! p , ! c and ! the plasma, the electron cyclotron and the wave angular frequencies, respectively. In the WKBJ framework ignoring the scattering e ects, the O-mode describes a linear polarised wave with its electric field parallel to B 0 (E//B 0 ) while the X-mode describes an elliptical polarisation with the wave electric field parallel to B 0 (E ? B 0 ). For the O-mode, the phase due to plasma propagation ' p is a function of the wave frequency f and the plasma electronic frequency ! p , ' p [ f , N O (n e )] while for the X-mode also depends on B 0 , ' p [ f , N X (n e , B 0 )].

Why the need for complex simulation codes?

From the previous introduction it would seem that the solid mathematical description of wave propagation in plasmas provides the necessary tools to interpret reflectometry experimental results obtained from experiments in fusion devices. Why all the investment in the development of complex simulation codes? It turns out that the analytic solutions describing propagation in plasmas must forcibly include some simplifications and the resulting ideal situation, such as the one depicted in figure 1 (left) exists, in fact, only in books as in [START_REF] Hutchinson | Principles of Plasma Diagnostics[END_REF]. Propagation in a thermonuclear plasma is far from trivial since the plasma is an extremely complex, inhomogeneous, non-stationary, anisotropic medium, where waves su er the e ects of turbulence, magnetohydrodynamics, Doppler shifts, absorption, di usion, mode conversion, to name a few. Thus the need of a numerical full-wave treatment based on a simplified model which retains the main mechanisms. Several Computational ElectroMagnetic (CEM) techniques have been used to tackle this problem, being finite-di erence time-domain (FDTD) [6] using Yee's scheme [START_REF] Yee | Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media[END_REF] the one most up to the job. As an example we point to figure 1 (right), where a demonstration of the impact of forward scattering on reflectometry is shown. We can observe a multitude of phenomena that shatters the coherence of the probing beam. In general, an analytical description of such e ects is not possible, leaving numerical simulation as the only means to evaluate them.

The REFMUL* code family

The REFMUL* family includes REFMUL, a 2D O-mode simulation code [START_REF] Silva | Unidirectional transparent signal injection in finite-di erence time-domain electromagnetic codes -application to reflectometry simulations[END_REF], REFMULX-/REFMULXp: 2D X-mode (serial/parallel) simulation codes [START_REF] Silva | Numerical advances in kernels for FDTD Maxwell codes using the Yee algorithm[END_REF] and two recent additions, REF-MULF and REFMUL3. REFMULF, is a 2D code able to cope with full polarization waves, treating all components of E = E x e x + E y e y + E z e z and H = H x e x + H y e y + H z e z of the wave. It supports a generic external magnetic field B 0 = B 0 x e x + B 0 y e y + B 0 z e z , coupling the Transverse-Electric Mode (TE, X-mode) with the Transverse-Magnetic Mode (TM, O-mode) via a linear vector di erential equation for J, which is solved using a modified Xu-Yuan kernel [START_REF] Silva | Unidirectional transparent signal injection in finite-di erence time-domain electromagnetic codes -application to reflectometry simulations[END_REF][START_REF] Xu | FDTD formulations for scattering from 3-d anisotropic magnetized plasma objects[END_REF], with extended long-run stability. The components of the external magnetic field B 0 lying on the propagation plane are responsible for linking the TE and TM modes, enabling the description of oblique propagation. For a B 0 purely perpendicular to the propagation plane, the code describes simultaneously O-mode and X-mode propagation. This code enlarges the possibilities of simulation of microwave reflectometry o ering capabilities unavailable in present day 2D reflectometry codes such as coupling O/X, Faraday rotation, and Cotton-Moutton e ect, for instance. REFMUL3, presently in an advanced stage of development, is a 3D full-wave code. Its numerical kernel shares a similar structure to REFMULF but involves more equations to be solved, since all field components are included in a volume. Furthermore, its 3D nature translates into increasing the grid-count by two or three orders of magnitude when compared with 2D codes, making the computational burden more stringent and the memory demands discouraging. This implied the usage of a parallel approach from the beginning, since only with a highly scalable parallel code can useful simulations be made. The REFMUL* family of codes aims primarily at improving the understanding of microwave reflectometry, which constitutes a standard diagnostic for current and future tokamaks. In this framework it has already been used to support physics studies in ASDEX Upgrade and Tore Supra. REFMUL has been used in the simulations for the design of Plasma-Position-Reflectometer for ITER [START_REF] Heuraux | Final Report on Procurement Package 1, Diagnostic design for ITER-EFDA contract[END_REF] and also to assess its measurement behaviour [START_REF] Silva | Assessment of the measurement performance of the in-vessel systems of gaps 4 and 6 of the ITER Plasma Position Reflectometer using a FDTD Maxwell full-wave code[END_REF]. This code was also used for the first assessment of microwave diagnostics for DEMO [START_REF] Silva | First assessment of microwave diagnostics for DEMO[END_REF]. New simulations of DEMO to assess its PPRs' measurement performance are being made using REFMULF [START_REF] Ricardo | Assessment of the measurement performance of DEMO Plasma Position Reflectometry gaps perpendicular to the separatrix[END_REF].

REFMULF and REFMUL3 geometry and equations

REFMULF, like the usual 2D O-and X-mode codes (such as REFMUL and REFMULX), also considers propagation in a 2D plane (x-y plane) with no gradients along z (@/@z = 0). Maxwell equations are solved using the classic Yee schema on two sets of staggered grids, one for the transverse magnetic mode (TMz) with field components E z , H x and H y , and another for the transverse electric mode (TEz), with field components H z , E x and E y . The major di erence from classic 2D codes is that a generic external magnetic field B 0 = B 0 x e x + B 0 y e y + B 0 z e z is now allowed. The TMz and TEz planes are not coupled through Maxwell equations but coupling through the equation for J = J x e x + J y e y + J z e z is possible through the product ! c ⇥ J, with ! c = ! x e x + ! y e y + ! z e z , being ! i = eB i /m e , i = x, y, z. If the co-planar B 0 x (! x , 0) and B 0 y (! y , 0) are present, the TMz (J z ) plane is coupled to TEz and the TEz (J x , J y ) plane to the TMz. For ! x , ! y = 0, we revert to the uncoupled simultaneous description of O-mode and X-mode. REFMULF uses a variation of the Xu-Yuan schema [START_REF] Xu | FDTD formulations for scattering from 3-d anisotropic magnetized plasma objects[END_REF] with the modifications proposed in [START_REF] Silva | Stable explicit coupling of the yee scheme with a linear current model in fluctuating magnetized plasmas[END_REF] for extended long-run stability in time-dependent problems. A more detailed description of this schema can be found in [START_REF] Silva | Introducing REFMULF, a 2D full polarization code and REFMUL3, a 3D parallel full wave Maxwell code[END_REF].

REFMULF not far from a full polarization 3D code. In fact if we overlay N TEz planes intermediated by N 1 TMz planes, instate dependency on z, e.g., J = J(x, y, z), and allow for variation along the z axis (@/@z , 0), we obtain the bases of REFMUL3. For a more comprehensive description refer to [START_REF] Silva | Introducing REFMULF, a 2D full polarization code and REFMUL3, a 3D parallel full wave Maxwell code[END_REF]. One major di erence while implementing REFMUL3 is that one must start thinking parallel. In order to have an usable 3D code we need a parallel kernel, parallel Input/Output (I/O) and parallel visualization of the results. This goal is achieved in REFMUL3 by employing a hybrid MPI/OpenMP parallelization with a 3D domain decomposition. This implementation shows a good strong scaling behaviour on the MARCONI cluster at Cineca Italy [START_REF] Silva | Introducing REFMULF, a 2D full polarization code and REFMUL3, a 3D parallel full wave Maxwell code[END_REF][START_REF] Hatzky | HLST Core Team Report[END_REF].

Examples of applications using REFMUL* codes

REFMULF is being used to assess the performance of the future PPR of DEMO reactor and to evaluate how many views are necessary for a reconstruction of the plasma shape [START_REF] Ricardo | Assessment of the measurement performance of DEMO Plasma Position Reflectometry gaps perpendicular to the separatrix[END_REF][START_REF] Marchiori | Study of a plasma boundary reconstruction method based on reflectometric measurements for control purposes[END_REF]. In figure 2, on the left, a poloidal section of DEMO appears with several possible positions of the PPRs' access to the plasma. The rectangular region marks the region-of-interest (ROI) for a simulation of a reflectometer on the high field side at the equatorial plane. On the right, a snapshot of the electric field taken at a frequency of f = 43 GHz is displayed for this ROI (G40). Multi-reflection structures can be clearly seen. The red line shows the position of the separatrix. As an example of the development status and usability of REFMUL3 we present the simulation of a plasma probing using O-mode fixed-frequency continuous-wave (FFCW) at a frequency f = 30 GHz probing a slab plasma with a distance from the edge to cuto of L = 6.43 35 GHz . The vacuum distance between the horn mouth and the plasma edge is d v = 2.188 35 GHz . A 3D Unidirectional Transparent Source (UTS) is implemented in the waveguide [START_REF] Silva | Unidirectional transparent signal injection in finite-di erence time-domain electromagnetic codes -application to reflectometry simulations[END_REF]. The waveguide and the standard horn are modelled as Perfect Electric Conductor (PEC). The simulation was performed on a volume of 800 ⇥ 450 ⇥ 450 grid points with 8 MPI processes and 36 OpenMP threads each, for a total of 288 cores. In figure 3, on the left, we visualise the 3D field structure of E z being reflected at the plasma and returning to the antenna. On the right, a contour of a cut done through the H-plane of the antenna (X-Y plane taken on the antenna's line of sight) is presented. Such simulations will improve the density profile reconstruction by giving a relevant evolution of the amplitude of the reflected wave and the e ects of the open cavity one can find in the case of ITER PPRs.

Conclusions

Being numerical simulation of key importance to use reflectometry as a robust diagnostic for electronic density in plasmas, we have introduced the REFMUL* family of full-wave codes as a set tools devised for that purpose. These consist of several 2D codes, with di erent degrees of approximations, as well as a new 3D code. They allow to set up synthetic reflectometry diagnostics in order to assess their behaviour. An example of such a study was provided, in which an assessment of the Plasma Position Reflectometers of DEMO was made.

Figure 1 .

 1 Figure 1. On the left, without turbulence, an analytical approach can still provide useful informations. With a turbulent plasma, on the right, the impact of forward scattering in a probing beam showing e ects such as: 1-multi-scattering; 2-local resonances; 3-secondary cuto s; 4-beam splitting and 5-wave trapping. These e ects can only be assessed using a fullwave code.

Figure 2 .

 2 Figure 2. On the left a poloidal section of DEMO with a HFS equatorial reflectometer ROI is marked. A snapshot of the electric field in this ROI (G40), for a frequency of f = 43 GHz, appears on the right.

Figure 3 .

 3 Figure 3. Parallel simulation of a plasma probing using a FFCW on a volume 800 ⇥ 450 ⇥ 450. The 3D rendered volumes appear on the left while on the right a 2D contour of taken on the plane x-y (antenna's H-plane).

Acknowledgments

This work has been carried out within the framework of the EUROfusion Consortium and has received funding from the Euratom research and training programme 2014-2018 and 2019-2020 under grant agreement No 633053. The views and opinions expressed herein do not necessarily reflect those of the European Commission.