Deborah Peruzzo

Mathieu Drelon

Clement Dumont

Andre Mortreux

Isabelle Suisse

Mathieu Sauthier

Clément Dumont

André Mortreux

Palladium catalyzed telomerization of bio-based polyols with atmospheric pressure bubbling butadiene

Keywords: Telomerization, butadiene, glycerol, isosorbide, solketal

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

The concept of green chemistry has a great impact on research developments as it takes into account the environmental and health impacts of chemicals production. [START_REF] Anastas | Design through the 12 principles of green engineering[END_REF][START_REF] Zimmerman | Designing for a green chemistry future[END_REF][START_REF] Sheldon | The road to biorenewables : carbohydrates to commodity chemicals[END_REF][START_REF] Hülsey | Sustainable routes for the synthesis of renewable heteroatomcontaining chemicals ACS Sustainable[END_REF] These green chemistry concepts foster research priorities such as reducing the consumption of non-renewable resources, using environmentally friendly solvents, decreasing wastes from the use of atom economical reactions for the synthesis of targeted products and using catalytic reactions that allow smooth conditions and high selectivities. As example, the palladium catalyzed telomerization of dienes with bio-based polyols is an efficient synthesis of ethers that fulfills these criteria. [START_REF] Behr | Telomerization: Advances and Applications of aVersatile Reaction[END_REF][START_REF] Faßbach | The Telomerization of 1,3-Dienes -A Reaction Grows Up[END_REF] The telomerization reaction corresponds to the metal catalyzed dimerization of dienes with addition of a nucleophile as water [7], alcohols [START_REF] Klinkenberg | Sterically encumbered and poorly electron-donating oxaphosphaadamantane ligands for the Pd-Catalyzed telomerization of butadiene with methanol[END_REF][START_REF] Jackstell | Efficient telomerization of 1,3-butadiene with alcohols in the presence of in situ generated palladium(0) carbene complexes[END_REF] or amines [START_REF] Grotevendt | Efficient catalyst for telomerisation of butadiene with amines[END_REF][START_REF] Suslov | Palladium(II)acetylacetonate complexes containing phosphine and diphosphine lignads and their catalytic activities in telomerisation of 1,3-dienes with diethylamine[END_REF] in most cases. In case of alcohols as nucleophiles, linear or branched octadienylethers are obtained (Scheme 1). This reaction has been largely investigated in literature [START_REF] Benvenuti | Telomerization of 1,3-butadiene with alcohols catalyzed by homogeneous palladium(0) complexes in the presence of mono-and diphosphine ligands[END_REF][START_REF] Jackstell | A highly efficient catalyst for the telomerization of 1,3-dienes with alcohols: first synthesis of a monocarbenepalladium(0)-olefin complex[END_REF][START_REF] Clement | Development of palladium-carbene catalysts for telomerization and dimerization of 1,3-dienes: from basic research to industrial applications[END_REF][START_REF] Faßbach | Telomerization of 1,3-butadiene with highly substituted alcohols using Pd/NHC-catalysts -Structure-reactivity-relationship of the O-nucleophile[END_REF][START_REF] Briggs | Palladium-catalyzed 1,3-butadiene telomerization with methanol. Improved catalyst performance using bis-o-methoxy substituted triarylphosphines[END_REF][START_REF] Herrmann | Homogeneously Catalyzed 1,3-Diene Functionalization -A Success Story from Laboratory to Miniplant Scale[END_REF] ever since its discovery back in the Sixties [START_REF] Smutny | Oligomerization and dimerization of butadiene under homogeneous catalysis. Reaction with nucleophiles and the synthesis of 1,3,7-octatriene[END_REF][START_REF] Takahashi | The dimerization of butadiene by palladium complex catalysts[END_REF] and important improvements have been done in the last decades to generalize this reaction to polyhydroxylated derivatives. [START_REF] Faßbach | Telomerisation of Renewables. In: Homogeneous Catalysis with Renewables[END_REF][START_REF] Bouquillon | Palladium-catalyzed telomerization of butadiene with polyols: from mono to polysaccharides[END_REF][START_REF] Bruijnincx | Pd-Catalyzed Telomerization of 1,3-Dienes with Multifunctional Renewable Substrates: Versatile Routes for the Valorization of Biomass-Derived Platform Molecules[END_REF][START_REF] Drelon | Towards Selective Syntheses of Octadienylethers from the Butadiene Palladium-catalyzed Telomerization with Polyols in Aqueous Biphasic Medium[END_REF][START_REF] Hausoul | Pd/TOMPPcatalyzed telomerization of 1,3-butadiene: from biomass-based substrates to new mechanistic insights[END_REF] Scheme 1 : Telomerization of alcohols with butadiene From polyols, mono to polyethers can be obtained and efforts have been mainly devoted to produce selectively monoethers, the objective being to produce compounds with amphiphilic structures. In view of sustainable chemistry, this catalytic transformation of polyols that are generally biobased is 100% atom-economical and no wastes are produced. Moreover, water could be used as solvent providing that hydrosoluble catalysts are employed.

In that frame, our group has proposed the efficient palladium catalyzed telomerization reaction of glycerol and isosorbide under biphasic media in a batch mode and under butadiene pressure.

Depending on the choice of the organic co-solvent and nature of the catalyst employed to promote the reaction, mono-to fully substituted ethers could be obtained with high selectivities and yields (Scheme 2).

Scheme 2 : Telomerization of butadiene with polyols

The use of a low pressure of butadiene is of interest as it allows the set-up of more simple glass reactors and thus avoids the need of specific and less widely accessible high-pressure reactors.

However, very few examples of telomerization reaction at atmospheric pressure have been reported. We had proposed the telomerization of glycerol and isosorbide in aqueous medium under gaseous atmosphere of butadiene [START_REF] Bigot | Telomerisation of 1,3butadiene with glycerol under aqueous biphasic conditions: Influence of the reaction conditions on the products distribution[END_REF][START_REF] Lai | Telomerisation of 1,3-butadiene with 1,4:3,6-dianhydrohexitols: an atom-economic and selective synthesis of amphiphilic monoethers from agro-based diols[END_REF]. In that latter case, the reaction rate is lower than with batch conditions and butadiene pressure. This low catalytic activity was attributed to a difficult mass transfer of butadiene in the catalytic phase. We now propose the implementation of the palladium catalyzed telomerisation of glycerol, solketal and isosorbide in biphasic media under atmospheric butadiene pressure by bubbling the gas continuously in the liquid phase via a sintered glass thanks to a home-made designed glass reactor.

Experimental

Materials

Chemicals were purchased from Alfa Aesar, Tokyo Chemical Industry, Janssen Chimica, Sigma Aldrich, Linde (Butadiene) and Strem Chemicals. Isosorbide was generously given by Roquette Frères and used without further purification. The reactor used for the study is given in the SI.

2.2.Telomerization of butadiene with glycerol

All experiments were performed under a nitrogen atmosphere using standard Schlenk techniques.

The catalytic precursor [Pd(acac)2] and the ligand (TPPTS = TriPhenylPhosphineTriSulfonate) were introduced in the glass reactor. Glycerol was mixed in another Schlenk tube together with methyl ethyl ketone (MEK), dodecane as internal standard (1 mL) and a solution of KOH 1 M. This solution was transferred into the reactor which was then heated at the desired temperature and magnetically stirred in a thermostated bath.

Butadiene at a flow of 2 mL/min was then bubbled in the liquid phase through a frit (see the S.I.).

The pressure of butadiene was kept constant by means of a back-pressure regulator fitted at 1.05 bar. During the reaction, samples of the upper phase were taken at defined time slots and analyzed with GC.

During the whole reaction, a biphasic liquid / liquid system is observed. The hydrosoluble catalyst is located in the lower phase that contains glycerol and aqueous base (please see the SI). The upper phase contains the organic solvent (MEK), the internal standard used for GC analysis (dodecane) and the products of reaction.

Analyses and quantification of the products

Conversions and yields were determined by gas chromatography using dodecane as internal standard.

The GC analyses were performed on a GC Shimadzu® 2010 equipped with a HT 5 column (30 m, 0.25 mm i.d., 0.25 m thickness), equipped with a flame ionisation detector (FID).

Conditions: injector and detector = 280°C, gas = nitrogen at 0.64 mL/min, temperature = 150°C, ramp of 10 °C/min up to 270°C.

Telomers were observed as main products. In all analyzed samples, less than 2 % of butadiene dimers were observed.

Results and discussion

For this work, we decided to work at atmospheric pressure of butadiene by bubbling the gas into the liquid aqueous phase (See the SI for the reactor). Butadiene was delivered through a frit in order to increase the contact surface of the gas with the water/polyol solution that contains the catalyst.

Moreover, thanks to this approach, aliquot samples of the reaction mixture could be taken and analyzed by GC at regular time intervals, allowing to follow the products distribution during the course of the reaction.

First experiments have been performed with glycerol. The polyol was dissolved in a basic aqueous phase and MEK as organic medium was added allowing the extraction of the telomers. The catalytic system was formed from a palladium precursor and the hydrosoluble TPPTS ligand. We initially used catalytic conditions similar to those described in batch mode : 10 mL MEK, 3.6 mL KOH 1 M, 0.03 g Pd(acac)2 (0.098 mmol), 0.279 g TPPTS (0.475 mmol). [START_REF] Bigot | Telomerisation of 1,3butadiene with glycerol under aqueous biphasic conditions: Influence of the reaction conditions on the products distribution[END_REF] Butadiene at a flow of 2.2 mL/min was bubbled in the aqueous phase and the reaction was heated at 80°C. Applied to glycerol with three OH functions, the formation of mono-, di-and tritelomers could be expected. As the butadiene is introduced continuously, we were interested in the production of telomers and their selectivity along time (Figure 1). During the first 8 hours, the conversion of glycerol and production of ethers remained extremely low. In these reaction conditions, this rather long induction period could correspond to the time necessary to generate Pd(0) catalytic species by in situ reduction of the metallic precursor by the phosphine. As the reaction began, the production of monotelomers was constant during the first 30 hours. Consistent with the low glycerol conversion and the efficient extraction of the ethers from the catalytic phase to the organic one, the selectivity in monotelomers was very high within this time period with no ditelomer formed. Surprisingly, longer reaction times led to increasing reaction rates and thus higher products formation along with lower selectivity in monotelomers. 37 % yield of monotelomers (12 g) and 5 % yield of of ditelomers (2.5 g) are obtained (overall 42% glycerol conversion) after 55 h of reaction. The increase of reaction rates that follows the conversion of glycerol appears as related to the composition of the reaction crude that changes with the formation of the ethers. In the case of a butadiene mass transfer limitation, one can expect that the increase of the activity is due to the amphiphilic character of the glycerol monotelomers which contributes to increase the solubility of butadiene in the water/glycerol catalytic phase. In order to better assess mass transfer limitations and delineate the origin of the rate increase in the course of the reaction, we further studied the impact of other reaction parameters on the formation of glycerol telomers.

The effect of catalyst loading was first studied and catalytic reactions conducted with respectively 0.03, 0.06 and 0.12 mol % of palladium-based catalyst were setup using the same reactor and reaction conditions as in the first experiment (Figure 2). The activation time and, very unexpectedly, the reaction rates were very similar whatever the catalyst loading. The telomerization reactions under butadiene pressure are commonly under chemical control and the reaction rates are classically related to the catalyst loadings in homogeneous or biphasic conditions. The independency of reaction rates with catalyst loadings is characteristic of a mass transfer controlling the reaction rate. In the present case, the rate of solubilization of butadiene in the polyol/water phase limits the overall process. Interestingly, after 30 hours of reaction, the three curves clearly diverge and show different reaction rates that follow the expected trend (higher rates for higher catalyst concentrations). As the monotelomers are produced, these amphiphilic compounds are partitioned between the catalytic aqueous phase and the organic one, thus allowing a better butadiene solubilization into the catalytic phase and favoring the production of the monotelomers together with the subsequent formation of ditelomers. In that case, the butadiene solubilization is no longer a limiting process and the reaction rate is chemically controlled. The reaction rate is increasing with higher catalyst loadings.

We anticipate then that reducing the amount of co-solvent, an important parameter to obtain a high monotelomers selectivity, would allow to observe this acceleration effect more rapidly, while maintaining the telomers in the catalytic aqueous phase. Starting from 10 mL, the volume of MEK was decreased to 5 mL and 1 mL (Figure 3). Conditions : Glycerol = 0.163 mol ; butadiene flow rate= 2.2 mL/min ; KOH 1 M = 3.6 mL ; [Pd(acac)2] = 0.06 mol% ; TPPTS/Pd = 5 ; T = 80°C.

Whereas using 5 and 10 mL of MEK, the average reaction rates appear to be very similar until 25 h, a spectacular activating effect is observed using a lower amount of co-solvent, giving a strong increase of the reaction rate after 8 h. Using a low amount of organic phase, the amphiphilic monotelomers concentration at the interface of this biphasic system is a key parameter favoring an increase of the butadiene solubilization earlier in respect to the reaction time. A precise analysis of the composition of the reaction mixture was performed at 28 h (Table 1). Monotelomers were major products and the ratios between linear/branched structures were similar in all cases. The reaction with 1 mL of MEK is therefore remarkable since it combines a very high selectivity into monotelomers (95%) and a satisfying conversion of 43% for a reaction using butadiene at atmospheric pressure.

The optimization experiments clearly show that the catalytic system used in this study is rather slow but elsewhere stable for several days and thus suitable for glycerol ethers. The atmospheric pressure reactor used in this study is particularly handy for semi-continuous experiments under constant atmospheric butadiene pressure and would thus allow a selective and continuous production of ethers along with catalytic phase recycling. An experiment was thus performed starting from the reaction conditions used in Figure 1 with 10 mL of MEK in order to facilitate the extraction procedure. This amount of solvent is particularly suitable to perform the extraction of the products formed during the reaction: the phases are rapidly and well separated, the viscosity is low and thus facilitates the solvent extraction with a simple syringe equipped with a fine needle.

The telomers containing the organic phase were removed and a new organic phase of MEK containing glycerol was introduced after 24 h of reaction. The amount of added glycerol corresponds to the amount of converted glycerol (determined by GC analysis of the extracted organic phase). This protocol was further iterated several times and the reaction lasted 110 h (Figure 4).

a mis en forme : Surlignage The first 24 h of reaction showed the lowest catalytic activity which is explained by the long activation time. The results obtained after 4 recycling steps (Figure 4) evidence a faster and stable productivity of ethers over 110 h of reaction. As expected from the optimization results, the reaction proved to be very selective with 99% of monotelomers and allowed the synthesis of up to 20 g of product (TON=1000) simply recovered after MEK evaporation. The results obtained from this experiment showed that a reaction performed under atmospheric pressure and continuous flow of butadiene with a simple glassware equipment is beneficial for the production of high quantity of pure glycerol monotelomers.

The experimental implementation was extended to solketal and isosorbide (Figure 5). The aim of this approach was to evaluate the impact of the polyol hydrophilicity on the reaction rates. Solketal is a monoalcohol obtained from the acetalization of glycerol with acetone [START_REF] Da Silva | Water-tolerant zeolite catalyst for the acetalisation of glycerol[END_REF]. In contrast to Glycerol addition glycerol, solketal is less polar with two hydroxyl groups masked by the acetal function. With this alcohol, the ligand TPPTS, insoluble in apolar solvent, can be advantageously replaced by the most commonly used TPP (triphenylphosphine). Moreover, no extracting co-solvent is necessary as only monotelomers are obtained. Isosorbide is a diol with two secondary hydroxyl groups industrially produced from the double dehydration of D-sorbitol [START_REF] Dussenne | Synthesis of isosorbide: an overview of challenging reactions[END_REF]. Isosorbide has been successfully used as telogen in the telomerization reaction thus leading selectively to any of the two possible monoethers depending on the chosen experimental conditions [START_REF] Lai | Telomerisation of 1,3-butadiene with 1,4:3,6-dianhydrohexitols: an atom-economic and selective synthesis of amphiphilic monoethers from agro-based diols[END_REF]. In that latter case, the addition of toluene as organic co-solvent, was performed in order to extract the monotelomers from the catalytic phase.

In line with the experiments that involved glycerol as reactant, the evolution of the mass of telomers during the reaction course are described in Figure 5. The reaction with solketal didn't show any induction period and was clearly much faster than with glycerol. The absence of activation time is coherent with most of catalytic reactions performed under butadiene pressure with the TPP ligand in homogeneous conditions or the TPPTS ligand in biphasic conditions [5,|6]. The different nature of the ligands used thus doesn't explain this difference of reactivity. In all cases, long activation times have been observed under low butadiene pressure and it is thus striking that in a lipophilic alcohol such as solketal that easily solubilizes butadiene, no activation time is observed under atmospheric pressure. This indicates that not only the alcohol nor the phosphine are involved in the key catalyst activation (palladium reduction), butadiene is also involved and one can expect that this is related to a higher stabilization of transient palladium(0) species by the diene. The higher solubility of butadiene in solketal additionally explains the higher reaction rate obtained with this alcohol in comparison to glycerol. In the case of isosorbide, which necessitates the use of the Pd/TPPTS catalytic system and the addition of a co-solvent (toluene), an induction time is observed. Nevertheless, as the reaction began, the observed rate is higher than with glycerol probably because of the lesser hydrophilic character of isosorbide in comparison to glycerol. After 22 h of reaction, the conversion of isosorbide is complete and isosorbide monoethers are obtained with a 94 % yield (22 g of product).

Conclusion

Telomerization of various bio-based alcohols as glycerol, isosorbide and solketal with butadiene at atmospheric pressure has been successfully performed with a non-pressurized glassware reactor that allows bubbling of butadiene in the catalytic phase. Very high selectivities toward monotelomers could be achieved with glycerol by using water/MEK biphasic media and the Pd/TPPTS catalytic system. The reaction necessitates a long activation time and the study of the reaction parameters clearly shows a low mass transfer-controlled reaction rate. The rate is more particularly limited by the low solubility and slow solubilization of butadiene in the catalytic phase.

The less hydrophilic solketal as better butadiene solvent than water is rapidly converted without any initial activation time while isosorbide reacted more slowly after an activation period. In the case of glycerol, the formation of monotelomers that favors the solubilization of butadiene in the aqueous catalytic phase increases the rate of glycerol etherification in the course of the reaction.

Monotelomers can be synthesized selectively albeit with a low reaction rate thanks to the removal of the MEK/monotelomers phase after 24 h of reaction. This could be repeated in this study four times every 24 hours using the same catalytic mixture. We believe that this system can be easily setup and up-scaled, the excess of unreacted butadiene being recycled or looped back to the reactor.

Figure 1 :

 1 Figure 1 : Telomerization of butadiene with glycerol Conditions : Glycerol = 15 g (0.163 mol) ; butadiene = 2.2 mL/min ; KOH 1 M = 3.6 mL ; MEK = 10 mL ; [Pd] = 0.06 mol% ; TPPTS/Pd = 5 ; T = 80°C.

Figure 2 :

 2 Figure 2: Variation of the catalyst amount during the telomerization of butadiene with glycerol. Conditions : Glycerol = 0.163 mol ; butadiene flow rate = 2.2 mL/min ; [Pd] = X mol%; KOH 1 M = 3.6 mL ; MEK = 10 mL ; TPPTS/Pd = 5 ; T = 80°C.

Figure 3 :

 3 Figure 3: Variation of the MEK volume during telomerization of butadiene with glycerol.

Figure 4 :

 4 Figure 4 : Telomerization reaction of glycerol. The red arrows indicate the organic phase removal and addition of new reactants (fresh glycerol and organic solvent) Initial conditions : Glycerol = 0.163 mol; butadiene flow rate = 2.2 mL/min ; KOH 1 M = 3.6 mL ; MEK = 10 mL ; [Pd(acac)2] = 0.06 mol% ; TPPTS/Pd = 5; T = 80°C. The amount of added glycerol corresponds to the amount of converted glycerol (determined by GC analysis of the extracted organic phases).

solketalFigure 5 :

 5 Figure 5 : Telomerization reaction with solketal and isosorbide with continuous butadiene flow.

Table 1 :

 1 Effect of MEK volume during glycerol telomerization with butadiene. a

			Glycerol	Monotelomers
	Entry	MEK (mL)	Conversion (%)	Yields (%)	Mono (l/b) b	Mono/Di (%/%)
	1	10	11	11	79/21	98/2
	2	5	14	12	73/27	89/11
	3	1	43	42	72/28	95/5

a Reaction conditions : Glycerol = 15 g (0.163 mol); Pd(acac)2 = 0.03 g (0.098 mmol); TPPTS/Pd = 5; KOH 1 M = 3.6 mL; T = 80 °C, t = 28 h; Butadiene flow rate = 2.2 mL/min. b Measured at t = 10 h.

Acknowledgements

This work (M. Drelon's fellowship) was financially support by the State as part of the "Programme d'Investissement d'Avenir" under reference ANR 15-CE07-0018-01 H2CAT. We acknowledge the Ministère de l'Enseignement supérieur, de la Recherche et de l'Innovation, the Institut Universitaire de France and the CNRS for their financial support. We also would like to thank Céline Delabre for her technical support (GC analysis of the crudes) and Maïa Matsakis for the construction of the glass reactor.


Minghao Wang

Mehdi Stiti

Hadrien Chaynes

Simon Becker

Edouard Berrocal

Fabrice Lemoine

Guillaume Castanet

• Edouard Berrocal

Two-photon fluorescence lifetime imaging applied to the mixing of two non-isothermal sprays: temperature and mixing fraction measurements

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

Introduction

Temperature is a particularly relevant parameter for studying heat and mass transfer phenomena within droplets that are heating, cooling, undergoing transformations such as vaporization, icing or drying. The transient phenomena occurring in these droplets are the focus of extensive research in various applications. In icing wind tunnels, water droplets are cooling in the air flow carrying them, but the transport duration is usually too short for the droplets to reach the temperature of the air flow. It is therefore essential to know the droplet temperature in order to study the icing under the conditions prescribed for the tests. Measuring the droplet temperature is also of great interest in research studies related to spray cooling for applications in the steel processing industry or in the heat management of electronics. Temperature measurement of the droplets provide insight into the sensible heat removed by the liquid sprayed over the hot surface. In spray combustion, droplet heating and evaporation requires advanced experimentally obtained knowledge of the droplet behavior to understand the formation of pollutants. Among the measurement techniques that are commonly used in the study of sprays, phase Doppler instruments [START_REF] Bachalo | Phase/Doppler Spray analyzer for simultaneous measurements of drop size and velocity distributions[END_REF] or LIF/Mie imaging [START_REF] Domann | Quantitative measurement of planar droplet Sauter mean diameter in sprays using planar droplet sizing[END_REF] have considerably progressed over the last few decades. These techniques are dedicated to characterizing droplet size, mass flux or droplet concentration, but with respect to scalar measurements like droplet temperature or chemical composition, measurement techniques are still in development and remain difficult to implement. Several approaches relying upon a wide variety of physical principles have been proposed in recent decades [START_REF] Lemoine | Temperature and chemical composition of droplets by optical measurement techniques: a state-of-the-art review[END_REF]. A first group of measurement techniques rely on light scattering by the droplets or by the molecules of certain chemical species present in the droplet. For instance, global rainbow thermometry was used to quantify the refractive index as a function of the angular position of the rainbow produced by the scattering of an extended light beam by the droplets in the spray [START_REF] Van Beeck | Global rainbow thermometry for droplet-temperature measurement[END_REF]. Raman scattering was exploited to characterize the chemical composition and, in some cases, also the temperature using both the surface area and the position of the peaks in the Raman spectrum [START_REF] Carns | Time-resolved raman spectroscopy from reacting optically levitated microdroplets[END_REF][START_REF] Schweiger | Raman scattering on single aerosol particles and on flowing aerosols: a review[END_REF][START_REF] Müller | High-precision measurement of the temperature of methanol and ethanol droplets using spontaneous Raman scattering[END_REF]. A second group of measurement techniques relies on photoluminescence processes like fluorescence [START_REF] Stiti | Fluorescence lifetime measurements applied to the characterization of the droplet temperature in sprays[END_REF] or phosphorescence [START_REF] Strizhak | Application of the laser induced phosphorescence method to the analysis of temperature distribution in heated and evaporating droplets[END_REF] which are the consequence of the spontaneous relaxation of molecular species after these have been excited by a type of energy. Laser-induced fluorescence (LIF) offers unprecedented temperature sensitivity and versatility in its applications with a wide variety of fluorescent dyes excitable at various wavelengths and soluble in different solvents. Temperature measurement with this approach have been reported in many different configurations, including droplets spreading onto solid surfaces [START_REF] Chaze | Spatially and temporally resolved measurements of the temperature inside droplets impinging on a hot solid surface[END_REF], mono-and multi-component fuel droplets combusting and evaporating in high-temperature environments [START_REF] Perrin | Characterization of the evaporation of interacting droplets using combined optical techniques[END_REF], supercooled droplets in icing processes [START_REF] Stiti | Temperature measurement and state determination of supercooled droplets using laserinduced fluorescence[END_REF](Stiti et al. , 2021a)). In these works, temperature measurements are usually based on the intensity of the fluorescence signal.

The fluorescent dye, usually an organic compound, emits a fluorescence signal which varies in intensity with the temperature, when it is excited by a laser tuned on its absorption spectrum. The intensity measured by an instrument is not an absolute quantity as it depends on the instrument used and its settings. Moreover, any disturbance (in the light source, in the background light, in the light transmission through the detection optics, or due to the dye photobleaching and degradation) is likely to modify the signal intensity. Within the spray itself, the variations of the droplet size and number density also directly affect the signal intensity during the measurements. For this reason, it is generally unavoidable to use a ratiometric approach, i.e., to calculate the ratio of the intensities of the fluorescence signal collected in two spectral bands. Properly used, intensity ratios cancel out most of the adverse effects mentioned above. Nevertheless, some measurement limitations and biases may remain, as evidenced by the pioneering works of [START_REF] Labergue | New insight into two-color LIF thermometry applied to temperature measurements of droplets[END_REF]. Scattering of the laser light by the droplets can induce a significant out-of-field fluorescence, which extends outside the focal plane of the collection optics and contributes to the overall fluorescence signal. Droplets normally localized well outside the laser beam emit a substantial amount of parasitic fluorescence because of laser light scattering. Problems related to the achromaticity of the optics with the depth of field are thus becoming critical [START_REF] Labergue | New insight into two-color LIF thermometry applied to temperature measurements of droplets[END_REF][START_REF] Labergue | Study of the droplet size effect coupled with the laser light scattering in sprays for two-color LIF thermometry measurements[END_REF]. Another issue inherent to intensity-based measurements in sprays relates to the dynamic range of the detectors, which are not able to detect with the same settings (gain voltage, laser power…) individual droplets when they have very different sizes. This is due to the fact that the fluorescence signal varies with the diameter d of the droplets as d , where is a parameter close to 3. The use of a detection threshold in the acquisition processing, to improve the signal-tonoise ratio by limiting the detection to the most intense drop events, unavoidably contributes to bias the measurements. The larger droplets tend to be more prominently represented in the measured temperature, as the signal from the smaller droplets falls to nearly the same level as the noise and these smaller droplets are discarded from the acquisition.

As an alternative to intensity-based measurements, [START_REF] Stiti | Fluorescence lifetime measurements applied to the characterization of the droplet temperature in sprays[END_REF] proposed a new approach based on the fluorescence lifetime. Contrary to the fluorescence intensity, the value of the fluorescence lifetime is not dependent on the measurement method and system. It is closely dependent on the quenching rate of the fluorescence and therefore on the temperature for some dyes. The Time Correlated Single-Photon Counting method (TCSPC) was employed by [START_REF] Stiti | Fluorescence lifetime measurements applied to the characterization of the droplet temperature in sprays[END_REF] to measure the temperature of water droplets in a polydisperse spray. An ultrashort laser beam focused on the spray, and a fast-responding photodetector are used to detect the arrival times of the fluorescence photons. The Poisson distribution of these times is reconstructed through repeating the laser pulses as much as necessary. Although the TCSPC is not suitable for measuring the temperature of individual droplets (the reconstruction of the fluorescence decay typically requiring several seconds), it was possible to obtain the average volume temperature of the liquid with limited measurement errors. As a benefit of the technique, it operates on single photons regardless of their origin (large or small droplets), which makes it possible to characterize the average temperature of the droplets with a higher degree of accuracy (i.e., with no bias related to their size). Given the very high signal-to-noise ratio of TCSPC, measurements were possible in relatively low-density regions of the spray simply increasing the registration time of the photons. The present work is built upon this previous effort to improve temperature measurement in sprays. Compared to this previous work, a two-photon excitation of fluorescent molecules is implemented. A key benefit of two-photon absorption is its ability to restrict the fluorescence excitation to a tiny focal volume around the laser beam waist [START_REF] Ulrich | Effects of liquid properties on atomization and spray characteristics studied by planar two-photon fluorescence[END_REF][START_REF] Berrocal | Two-photon fluorescence laser sheet imaging for high contrast visualization of atomizing sprays[END_REF]. The problem of the out-of-field fluorescence, encountered when fluorescence is induced by the onephoton absorption of the laser light scattered by the droplets, is mitigated or even completely removed (Fig. 1). Photons undergoing multiple scattering processes, spread in space and time with a very low probability of having two-photon simultaneously absorbed. In the present study, TCSPC is applied to the mixing of two sprays: one hot and one cold. The aim is not only to measure the temperature in this situation of spray mixing, but also to show that the characterization of the fluorescence decay in the time domain can extend the capabilities of measurement to the mixing fraction and the temperature of the two individual sprays.

Two-photon absorption and measurement volume

The simultaneous absorption of two photons is used to excite the fluorescent molecules from the ground state energy to the excited state. Several fluorescent dyes including rhodamine B (RhB), eosin Y (EY) and rhodamine 6G (R6G) are dissolved into water to conduct these experiments. The fluorescence is induced by a femtosecond Ti:Sapphire laser (Coherent Chameleon Ultra II) whose emission wavelength is tunable from 680 to 1080 nm and generates light pulses with a width of about 140 fs. As shown in Fig. 2, the same microscope objective is used for focusing the laser beam inside the spray and for collecting the fluorescent photons emitted by the droplets in return.

Since the fluorescence excitation depends on the simultaneous absorption of two photons, its probability is proportional to the square of the laser light intensity I laser (r, z) .

Considering a Gaussian laser beam (Fig. 3), the intensity profile of the laser beam is given by: where I 0 is the incident intensity, r is the transverse radius, and z is the path length. At the beam waist ( r = w 0 and z = 0 ), the intensity I laser drops to 1∕e 2 of the incident inten- sity I 0 . The evolution of the beam radius w(z) along the z direction is determined by: where the Rayleigh length z 0 = w 2 0 ∕ is a critical distance for the intensity distribution in the z direction. Parameters (1)

I laser (r, z) = I 0 ⋅ w 0 w(z) 2 ⋅ exp - 2r 2 w 2 (z) , (2) w 2 (z) = w 2 0 ⋅ 1 + z z 0 2
, z 0 and w 0 depend on the magnification of the microscope objective lens used to focus the laser beam.

Considering a small liquid volume dV containing the fluorescent dye, the number of photons absorbed dN abs can be written as:

In this expression, denotes the molecular two-photon absorption cross-section of the fluorophores in the liquid solution. It is usually quoted in (GM), where 1 GM is equal to 10 -50 cm 4 s photon -1 . Equation (3) implies a sufficiently moderate laser fluence (well below the saturation threshold of the two-photon fluorescence) to be a linear regime of fluorescence excitation. A high magnification objective comes with a very small fluorescence spot size and thus a reduced fluorescence signal due to the low probability of intercepting droplets. Additionally, a reasonable working distance is recommended to limit the deposition of droplets onto the lens surface. Due to these requirements, a microscope objective with a long working distance of 25 mm (50X Mitutoyo Corrected Infinite Plane Apo NIR B) was chosen. Figure 4 shows images of the fluorescence spot recorded in a cuvette filled with an aqueous solution of EY. The images were obtained using a camera (xiQ SuperSpeed) which has a 2/3″ sensor size. The zoom lens OPTEM 125C mounted on this camera allows to obtain a field of view of 1.02 × 1.35 mm when the magnification is maximum. A chromatic bandpass filter [570-620 nm] is placed in front of the zoom lens to eliminate the parasitic light which is not fluorescence. Due to the smallness of the beam waist w 0 , the fluorescence profile along the transverse direction r cannot be extracted accurately from these images. However, it is possible to determine z 0 and thus w 0 . Based on Eq. ( 3), the fluorescence signal can be written as:

Here is related to the collection efficiency and is the fluorescence quantum yield. In a section of the fluorescence spot located at z: Using Eqs. (1) and (2), the profile of F normalized by its maximum value at z = 0, can be expressed as:

(4) dF ∝ N abs .

(5)

F(z) ∝ 2 C ∫ +∞ r=0 rI 2 laser (r, z)dr.
The same integration has been performed with the experimental images. The corresponding curves are shown in the right-hand side of Fig. 4. A good agreement with Eq. ( 6) is obtained when the laser power P is low. For the high val- ues of P , a saturation of the fluorescence can be evidenced near the center of the fluorescence spot (z = 0) . Among the three dyes tested, EY proved to be the most sensitive to this phenomenon. For an excitation wavelength = 700 nm and P = 2.2W , the decrease in the florescence intensity F around z = 0 indicates that photobleaching is occurring (Fig. 4D). While the ratio F∕F 0 is not expected to vary with the laser fluence, the estimated value of z 0 has to be increased with P to maintain an acceptable match between the experimental data and the theoretical model based on Eq. ( 6) due to the effects of fluorescence saturation. In the followings, the experiments with RhB are performed tuning the fs laser on the wavelength of 800 nm at which the output power is 3.5 W. For EY, a wavelength of 700 nm was preferred as it is more efficiently absorbed, and the laser power was fixed at P = 1.5 W. Satura- tion and photobleaching are encountered for these high values of the laser powers ; however, the fluorescence lifetime is not affected contrary to the signal intensity. The Rayleigh length z 0 can be estimated to be about 60 µm from Fig. 4A. Consider- ing that z 0 = w 2 0 ∕ , this yields to a beam waist w 0 of about 3.9 µm. This clearly small value is a bit of a limitation for the measurement as it limits the interception of the excitation volume with the droplets and thus the count rate. At the same time, this allows to have a reduced probability of having two droplets illuminated at the same time. ( 6) is estimated using the least mean square method and comparison to the theoretical model (Eq. 6). Saturation of the fluorescence and photobleaching can be evidenced in B) and D) when the laser power P is high

F(z) F(z = 0) = z 2 0 z 2 + z 2 0 .

Fluorescence lifetime measurements in the time domain

After an excitation by an ultrashort pulse of light, fluorescent molecules emit a signal that can be written as follows:

Here is related to the collection efficiency of the measurement system. Fluorescence typically follows a first-order kinetics. The term (t) expresses the fluorescence decay resulting from the progressive decrease of the number of molecules in the excited state. In general, it follows a mono exponential evolution:

where is the fluorescence lifetime, which is on the order a few nanoseconds for organic dyes such as RhB, EY and Rh6G. In the literature, biexponential decays were sometimes used to describe the decay of some fluorophores including RhB [START_REF] Boens | Fluorescence lifetime standards for time and frequency domain fluorescence spectroscopy[END_REF][START_REF] Kristoffersen | Testing fluorescence lifetime standards using two-photon excitation and timedomain instrumentation: rhodamine B, coumarin 6 and lucifer yellow[END_REF][START_REF] Stiti | Fluorescence lifetime measurements applied to the characterization of the droplet temperature in sprays[END_REF]. In this case, the decay can be described using the following expression:

where a is a parameter between 0 and 1. According to Casa- devall I Solvas et al. ( 2010), an average fluorescence lifetime for this type of decay can be evaluated by:

The rate of non-radiative pathways k nr and radiative path- ways k f modifies the number of molecules in the excited state. The total decay rate is the sum over all the radiative and non-radiative decays rate:

The fluorescence lifetime depends on solvent effects and temperature through the non-radiative rate constant k nr . An Arrhenius law has been widely accepted to describe the effect of temperature on k nr for many solvents like rhodamine B [START_REF] Mendels | Quantitative comparison of thermal and solutal transport in a T-mixer by FLIM and CFD[END_REF]. For simplicity, this law is also considered for the temperature dependence of the fluorescence mean lifetime [START_REF] Mendels | Quantitative comparison of thermal and solutal transport in a T-mixer by FLIM and CFD[END_REF][START_REF] Mercadé-Prieto | Fluorescence lifetime of Rhodamine B in aqueous solutions of polysaccharides and proteins as a function of viscosity and temperature[END_REF]:

where E a is the activation energy, A is a pre-exponential factor, and R is the gas constant ( R = 8.314J K -1 mol -1 ).

To characterize the fluorescence decay, the TCSPC (Time Correlated Single Photon Counting) technique is used. The key elements of the optical setup and fluorescence detection ( 7)

F(t) ∝ (t) N abs . (8) F(t)∕F(t = 0) = e -t∕ , (9) F(t)∕F(t = 0) = ae -t 1 + (1 -a)e -t 2 , (10) = a 1 + (1 -a) 2 , (11) = k f + k nr -1 . (12) = Aexp E a RT ,
system are shown in Fig. 2. A fast hybrid photomultiplier tube (R10467, Hamamatsu) with a Peltier cooler to reduce the dark count rate, is used for photon counting. The output of the detector is sent to a time-correlating counter (Hydra-Harp 400 controller, PicoQuant GmbH) which is operating in forward start-stop mode to measure the arrival times of single photons. A laser pulse causes a photon event at the detector and the electronics measures the delay between the laser pulse and the subsequent photon event. This measurement implies a sharp timing of the laser pulse, which is obtained from a fast photodiode (TDA 200 from PicoQuant GmbH) connected to the timing electronics. As illustrated in Fig. 2, this photodiode receives the reflected light from a beam splitter (99% T, 1% R) placed at the laser output and allows detecting the laser pulses with a small-time jitter due to a very sharp falling edge (Fall time typ. 250 ps).

After its collection by the microscope lens, the fluorescence signal is diverted by a dichroic plate toward to a fast hybrid photomultiplier tube (R10467 from Hamamatsu) sensitive to single photon. A bandpass filter [570-580 nm] is placed in front of the photomultiplier tube to limit the detection of unwanted photons. The three dyes considered in the present study (RhB, EY and Rh6G) emit with a good efficiency in this spectral band. The basic principles of TCSPC have been discussed with great detail in the literature (Lakowicz 2006) The acquisition method illustrated in Fig. 5 is based on the repetitive and precisely timed registration of single photons using the previously described system. By counting a high number of photon events to achieve statistical convergence, a histogram of the photon arrival times distribution can be built up. The width of the time bins in this histogram depends on the required time resolution (typically a few tens of ps).

A nonlinear least squares analysis is used to test whether the model is consistent with the data and estimate the parameters of the fluorescence decay model. Basically, parameters are varying until the goodness-of-fit parameter 2 determined by: In the above expression, N(t k ) is the measured counts at time t k , N c (t k ) is the modeled number of counts at time t k assuming a mono (Eq. 8) or a biexponential (Eq. 9) function for the fluorescence decay, and n is the number of data points. A good fit is characterized by a value of χ 2 close to 1 and the weighted residual between the measured and calculated decay curves randomly distributed around 0. In Fig. 6, a few examples of fluorescence decays are presented for the three dyes considered in this study. The fitting by a monoexponential (Eq. 8) or a biexponential (Eq. 9) function takes into account the temporal convolution with the instrument

(13) 2 = 1 n n ∑ k=1 N t k -N c t k 2 N t k ,
rate of the detector makes it possible to carry out measurements in spray regions where the volume fraction of the liquid reduces to a few thousandths, but with a substantial increase of the acquisition time.

Temperature calibration of the fluorescence lifetime

Before measuring the droplet temperature in a spray, it is necessary to perform a calibration of the fluorescence lifetime as a function of temperature. The fluorescence lifetime is measured in a cell for different temperatures. The liquid is heated by means of a thermal resistance while it is continuously stirred to ensure a uniform temperature. A thermocouple is positioned a few mm from the LIF measurement volume. Figure 7 shows the fluorescence decay of RhB for different temperatures in the considered detection band. Since the monoexponential character of the fluorescence decay is not perfect and may change with the temperature, it is preferred to use the mean lifetime τ calculated from Eq. ( 10) as an indicator of the temperature of the liquid, rather than 1 and 2 . Figure 8 shows the evolution of the lifetime τ in function of the temperature for the three fluorescent dyes. As observed, Rh6G has a long lifetime of approximately 3.7 ns, which is not sensitive to temperature as already noticed by [START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF]. EY has a short lifetime of about 1.1 ns whose temperature variation is very weak. Based on Eq. ( 12), the activation energy E a and the parameter A can be evaluated from the curves presented in Fig. 8. In the case of RhB, this leads to E a = 19.890 kJ/mol and = 4.852 ⋅ 10 -4 s. The fluorescence decay of RhB was also examined in more details to determine its biexponential evolution with the temperature. Temperature calibrations, conducted both in a cell and in individual sprays (no mixtures), allowed

Laser pulse Single photon event

No photon event

A B

Fig. 5 The principle of the lifetime measurement with TCSPC. A the TCSPC device typically measures the time between one START event and one STOP event (like a stopwatch). B A histogram of startresponse function (IRF) of the TCSPC measurement system. Rh6G exhibits quasi-monoexponential decay has shown by the difference between the 2 value while using monoand biexponential decay. However, EY and RhB exhibit a biexponential decay. In the case of RhB, the biexponential nature of the decay changes with the temperature, meaning that parameters a, 1 and 2 in Eq. ( 10) are temperature dependent for this dye. For all the tested conditions, the fit by a biexponential function allows to perfectly reproduce the experimental fluorescence decay.

If more than one photon frequently reaches the active surface of detector per laser pulse, the TCSPC technique fails to reconstruct the fluorescence decay due to the deadtime of the photodetector. This problem referred as pile-up effect, is difficult to handle in spray measurements, because its occurrence must be verified beforehand in the conditions of the measurements (laser power, dye concentration, spray density). In general, the pile-up effect is most likely to occur a few millimeters downstream of the spray nozzle, as the liquid fraction is high at this location. Before performing measurements in a systematic way in the spray, it is ascertained that measurement value of the lifetime is not changing with the laser power at this location. In practice, the power of the laser is adjusted at a location in the spray where the liquid fraction is close to its maximum in the spray. With our spraying system, a suitable position is a few mm after the breakup of the liquid sheet into droplets. The laser power is gradually increased to determine the critical power for which the histogram of the photon arrival times change in shape due to the pile-up effect. Measurements are performed with a laser power below this critical value. The laser power is then maintained all along the measurements, even if the count rate is getting very low due to the expansion of the spray and a lower liquid volume fraction. The low dark count stop times is reconstructed by counting a high number of photon events to reach statistical convergence to establish that the biexponential decay of RhB can be well described using the following formula:

w h e r e a = 0.9901 ⋅ 10 -1 ⋅ τ(ns) + 0.04371 , t 2 (ns) = -0.411765 ⋅ τ(ns) + 1.4594 , a n d t 1 (ns) = a -1 ⋅ τ(ns) + 1 -a -1 ⋅ t 2 (ns) .
In this expression, is the mean fluorescence lifetime defined in Eq. (10). The above correlation is valid in the range 0.6 ns ≤ ≤ 2 ns. droplet density is of the order of 5.10 4 particles per cm 3 (at about 1 cm after the breakup zone). The temperature of the two sprays can be adjusted independently. A K-type thermocouple inserted in the inlet pipe of each nozzle is used to determine the injection temperature of the sprays. The spraying system is mounted on translations stages which allows moving the two sprays relatively to the optics and perform measurements at different positions. The probability of collision between droplets originating from the two sprays is very low according to [START_REF] Labergue | Study of the thermal mixing between two non-isothermal sprays using combined threecolor LIF thermometry and phase Doppler analyzer[END_REF]. In other words, there are almost no drops formed from the merging of droplets of the two sprays. The interaction between the sprays is expected to occur mostly via the air entrained by each spray and the subsequent convective heat transfer. Distinct fluorescence decays from the two sprays' droplets are expected in the following situations:

The temperatures of the two sprays are different, and a temperature-dependent dye (here RhB) is used.

The two sprays are seeded with two dyes having significantly different fluorescence lifetimes like EY and Rh6G.

These two situations will be examined in the following.

6 Fluorescence signal in the mixing region of two sprays

Two dyes with a monoexponential decay

In this section, it is assumed that the two sprays are seeded with two dyes that have a monoexponential decay. A volume V 1 of liquid coming from the first spray and V 2 from the second spray are crossing the measurement volume during the acquisition period of the fluorescence signal.

Based on Eqs. ( 3), ( 7) and ( 8), the detected fluorescence signal is expected to have the following time evolution:

Here, indices 1 and 2 refer to the two dyes used for the seeding of the sprays. Normalizing the signal by F(t = 0), the previous evolution can be rewritten as follows:

where the average coefficient is given by:

Introducing the volume fraction = V 1 V 1 +V 2 : (15) F(t) = 1 C 1 I 2 laser exp - t 1 V 1 + 2 C 2 I 2 laser exp - t 2 V 2 .
(16)

F(t) F(t = 0) = exp - t 1 + (1 -)exp - t 2 , ( 17
) = 1 V 1 C 1 1 V 1 C 1 + 2 V 2 C 2 . RhB Rh6G EY

Application to temperature measurement in sprays

The experimental system allows to generate two full cone sprays downward with an adjustable tilt angle (Fig. 9). The two sprays are produced by a pressure swirled atomizer (Danfoss oil nozzle, angle 30° type S, 0.65 USgal/h). Indications about the size and velocity distributions of the droplets can be found in the study by [START_REF] Labergue | Study of the droplet size effect coupled with the laser light scattering in sprays for two-color LIF thermometry measurements[END_REF] using the well-established Phase Doppler technique to obtain joint size/velocity PDFs. The liquid flows across a heat exchanger, which allows to control the liquid temperature in the nozzle. At an inlet pressure of 6 the droplet mean diameter D 10 is roughly 20 µm, while the where parameter = 1 C 1 ∕ 2 C 2 . The latter expression establishes a relationship between the mixing fraction and the average coefficient . In the following, the absorption cross-section 1 and 2 and thus the value of are assumed to be unsensitive to the temperature. This is consistent with our knowledge of the fluorescent dyes used in this study (EY, RhB and Rh6G). Equation ( 15) is based on the assumption that the fluorescence decay can be expressed as the sum of two independent decays (one for each spray). For that, there must be no interaction of the two sprays in the measurement volume. If two droplets (one from each spray) are present at the same time in the measurement volume, the photons coming from the droplet with the short-lived fluorescence could reach the PMT systematically before the photons from the droplets with the long-lived fluorescence. Because of the dead time of the detector, this could bias the histogram of the photon arrival times in favor of the spray with the shortlived fluorescence. However, the problem is very limited here because of the densities of the two sprays, the small size of the 2P excitation volume and the optical configuration.

When two droplets are present in the excitation volume at the same time, the droplet in front of the objective deflects most of the photons coming from the backside droplet. Thus, we expect the result to remain about the same as with a single droplet. The analysis of the fluorescence decay measured in the sprays makes it possible to evaluate parameters , 1 and 2 . Then, using Eq. ( 18), the volume fraction can be derived provided that the value of is known. The individual temperature of the two sprays (namely T 1 and T 2 ) can be also retrieved from the lifetimes 1 and 2 if they are temperature dependent. The fluorometric temperature T LIF of the mixing can be defined by: where the average lifetime LIF is defined as:

It is interesting to note that T LIF is also the temperature that would be obtained from an intensity-based measurement which is not affected by any perturbation of the excitation and the transmission of the fluorescence signal in the sprays. Indeed, using Eq. ( 16), the time-integrated intensity F can be evaluated by:

(18) = + (1 -) , (19) T LIF = R E a ⋅ ln LIF ∕A , ( 20
) LIF = 1 + (1 -) 2 (21) F = ∫ +∞ 0 F(t)dt = F(t = 0) ⋅ 𝛼𝜏 1 + (1 -𝛼)𝜏 2 = F(t = 0) ⋅ 𝜏 LIF
F varies the same manner as LIF with the temperature. Therefore, Eq. ( 19) relates F to the temperature T LIF pro- vided a linear response of the dyes to the laser excitation (no saturation of the fluorescence signal).

T LIF differs from the mixing temperature T m given by the conservation of energy and the volume fraction : Even in the case of a single dye ( = 1 ), the analysis shows that T LIF deviates from T m . In fact, T LIF is closer to the temperature within the spray with the lowest values of fluorescence lifetime. This deviation between T LIF and T m is further quantified in Fig. 16 for the case of the measurements described in Sect. 5.3.

Two dyes with a biexponential decay

In the general case, the fluorescence decay in each of the two sprays differ from a monoexponential decay but can be modelled by a biexponential decay (Fig. 6). The methodology described above remains mostly valid, but Eq. ( 15) must be replaced by a combination of two biexponentials:

where i a i , t 1i , t 2i , t = a i exp -t∕t 1i + 1a i exp -t∕t 2i , 0 ≤ a i ≤ 1 and the index i = 1 or 2 refers to the dye used to seed the ith spray. Normalizing the fluorescence signal by F(t = 0), Here, the expression for the average coefficient is exactly the same as Eq. ( 18). Parameters a 1 , t 11 , t 12 for spray 1, and a 2 , t 21 , t 22 for spray 2, must be determined prior to the measurements to determine the volume fraction and the temperature of the two sprays if the dyes are temperature dependent.

Measurement of the mixing fraction alone

To illustrate the principle of measuring the mixing fraction , the previous model is applied to the mixture of a hot liquid spray injected at 60 °C and a liquid spray at room temperature (19 °C). The hot spray is seeded with Rh6G and the cold spray with EY. Since EY and Rh6G are not temperature sensitive, these measurements are intended to provide only the average coefficient and subsequently the volume fraction . The mean lifetime of EY being very different from that of Rh6G (1.1 ns against 3.7 ns), the errors made in considering a monoexponential decay for both EY ( 22)

T m = T 1 + (1 -)T 2 .
(23)

F(t) = 1 C 1 I 2 1 a 1 , t 11 , t 12 , t V 1 + 2 C 2 I 2 2 a 2 , t 21 , t 22 , t V 2 , ( 24
)
F(t) F(t = 0) = 1 a 1 , t 11 , t 12 , t + (1 -) 2 a 2 , t 21 , t 22 , t .
and Rh6G can be accepted. Hence, the method described in Sect. 5.1a is applied. The weak temperature dependence of both dyes allows to fix 1 = 1.1 ns and 2 = 3.7 ns in Eq. ( 16) for the analysis of the fluorescence decays. Since the absorption cross-section 1 and 2 are unknown for Rh6G and EY, a calibration is necessary to establish the relationship between the average coefficient and the volume fraction . This calibration is performed in a cell where the two dyes are mixed in a perfectly controlled proportion. As an illustration, Fig. 10 shows the fluorescence decay for several values of ranging between 0 and 1.

For each case, the fluorescence decay was characterized. A biexponential law was assumed for the fitting and the mean lifetime was determined. Finally, assuming 1 = 1.1 ns and 2 = 3.7 ns, the average coefficient was evaluated as followed:

Figure 11 shows the evolution of the average coefficient as a function of the volume fraction . Calibrations were carried out for three concentration ratios C Rh6G ∕C EY =0.1, 0.3 and 1.05 between the two solutions mixed in the cuvette. An acceptable agreement with the prediction of the model (Eq. 18) is obtained by adjusting the value of to 0.33, 1.0 and 3.5. From these observations, it can be deduced that the ratio of the two-photon absorption cross sections Rh6G ∕ EY of EY and Rh6G is approximately equal to 3.3 at an excitation wavelength of 700 nm.

Finally, measurements in the sprays were carried out with the following concentrations C Rh6G = 3 × 10 -6 mol/L and C EY = 10 -5 mol/L. Based on Fig. 11, = 1 and the average coefficient is equal to the volume fraction . The spraying system (the nozzles and the mixing chamber) is shifted by two motorized moving stages relatively to the microscope objective by steps of 0.5 mm in the x

(25) = - 2
1 -2 0.0001 0.01 0.1 1 Fig. 10 Fluorescence decay of the couple Rh6G/EY when mixed in different proportions. These results were obtained in a cuvette at room temperature of 20 °C ( ψ = 0 when there is no Rh6G in solution). The initial concentrations of the solutions before the mixing were C Rh6G = 3 × 10 -6 mol/L and C EY = 10 -5 mol/L. Different volume fractions ψ were obtained by changing the volume of liquid from the two solutions added in the cuvette Fig. 11 Evolution of the average coefficient a as a fu nction of t h e volume fraction ψ for different concentrations of Rh6G and EY. The experimental data (in dot) are compared to the prediction of Eq. ( 18) (in red) adjusting β to 0.33, 1 and 3.5. Measurements correspond to a concentration of Rh6G equal to 10 -6 mol/L, 3 × 10 -6 mol/L and 1 × 10 -5 mol/L, while the concentration of EY is fixed at 10 -5 mol/L and z directions. An acquisition time of 30 s is applied at each measurement location, which proved more than sufficient to ensure a converged histogram of the fluorescence decay and assess the value of the mixing fraction . Several methods can be used to evaluate the average coefficient and thus the mixing fraction :

Method 1: The fluorescence decay is fitted by a biexponential without any constraint imposed on the estimation of parameters , 1 and 2 . The value of obtained from this, is converted into the volume fraction using the calibration curve in Fig. 11.

Method 2: The fluorescence decay is fitted, as for method 1, by a biexponential without any constraint. Then, the mean lifetime is evaluated. Finally, Eq. ( 25) is used to obtain a value of assuming 1 = 1.1 ns and 2 = 3.7 ns. This value is converted into the volume fraction using Fig. 11.

Method 3: The fluorescence decay is fitted by a biexponential with the constraint to have 1 = 1.1 ns and 2 = 3.7 ns. Only the parameter is estimated using the least squares minimizing 2 in Eq. ( 13).

Method 4: The approach described in Sect. 5.1.b is applied to account for the biexponential nature of the decays of both fluorescent dyes. The decay parameters (a 1 , t 11 , t 12 ) and (a 2 , t 21 , t 22 ) for the two dyes are fixed and only the average coefficient is adjusted as in method 3. For this method, the values of the fitting parameters chosen area 1 = 0.03, t 11 = 1.04, t 12 = 4, a 2 = 0.91, t 12 = 0.75 and t 22 = 3.87.

The results of the above methods are compared in Fig. Since the two sprays are identical and arranged symmetrically from a vertical plane, it seems rather consistent to obtain a left/right symmetric distribution of the volume fraction when the two sprays cross each other and emerge diagonally on the opposite side. Method 1 clearly suffers from measurement biases. It is apparent that the volume fraction does not decrease down to 0 as expected. The advantage of methods 2 and 3 is mainly that they are based on the estimation of a single parameter ( in method 2 and in method 3), which limits the propagation of errors. These two methods give roughly equivalent results. Deviations are marginal with method 4, which accounts for the true biexponential nature of the dyes' fluorescence decay.

Measurement of both the mixing fraction and the temperatures of the two sprays

An even more valuable type of measurement is the combined determination of the temperatures of both sprays and the mixing fraction. This can be achieved using RhB as unique fluorescent dye. As the absorption cross-section of RhB is not dependent on the temperature [START_REF] Coppeta | Dual emission laser induced fluorescence for direct planar scalar behavior measurements[END_REF][START_REF] Chaze | The saturation of the fluorescence and its consequences for laser-induced fluorescence thermometry in liquid flows[END_REF], the value of in Eq. ( 18) is equal to 1 which means = . An accurate assessment of both the temperature and the mixing fraction requires a sufficiently high temperature difference between the sprays. In the present work, the injection temperature of the two sprays was set to 19 °C and 59 °C, respectively. To ensure a good data convergence, the acquisition time was fixed at 100 s and measurement points were discarded from the acquisition when the photon count rate is below 2000 cps, since the measurements require more time of acquisition for a good statistical convergence.

As previously mentioned, the fluorescence decay of RhB deviates significantly from a monoexponential but can be well described by a biexponential (Fig. 6). If this biexponential behavior is not considered, the same approach as method 1 in Sect. 5.2 can be used for the data analysis. The result is illustrated in Fig. 13A. It is apparent that never falls to 0 nor reaches 1, in the regions where only one spray is present in the measurement volume. The estimation of the average coefficient is affected by a large error.

To avoid this problem, it is necessary to consider the combination of two biexponentials and adopt the method described in Sect. 5.1.b. Denoting 1 and 2 the mean fluorescence lifetimes in the two sprays, the fluorescence signal in the mixing region can be rewritten as:

Here, the function ϕ is determined from the temperature calibration of the RhB temporal responses. Its expression is given in Sect. 4 by Eq. ( 14). For each measurement in the sprays, the triplet ( = , 1 , 2 ) is evaluated using the least squares method, i.e., minimizing the value of 2 in Eq. ( 13). The obtained distribution of the volume fraction is shown in Fig. 13.B. It does not appear to be affected by the same biases as in Fig. 13A, since values of are getting close to 0 and 1 in the regions where the two sprays are separated.

Parameters 1 and 2 are related to the temperature of both sprays following Eq. ( 12). Hence, the temperatures T 1 and T 2 can be determined as well. Using the values of , T 1 and T 2 , it is also possible to calculate the mixing temperature T m based on Eq. ( 22). As observed in Fig. 14A, the temperature distribution follows a rather similar evolution compared to that of the volume fraction. However, the temperature does not fully recover its initial value after the mixing region. The hot spray ( T inj = 59°C) appears to get slightly cooler after the mixing, while the cold spray ( T inj = 19 • C ) appears to be heated up by a few °C. Figure 14B shows experimental ( 26) 16)16, or the combination of two biexponential decays B using Eq. ( 26)26 measurement in the mixing zone obtained by using a more refined step of 0.1 mm for the scanning. Data are in good agreement with Fig. 14A and allow to observe more precisely the evolution of the mean temperature near the center of the mixing region. Figure 14C shows the evolution of the volume fraction along the diagonal lines which are drawn as violet dots in Fig. 14.A. As expected, the value of approaches 1 in the hot spray region and 0 in the cold spray region but a deviation (about 5%) is sometimes observed. Between z = 3 mm and z = 5 mm, is varying due to the mixing of the sprays. It is decreasing for the hot spray diagonal down to about 0.5, and it increases for the cold spray diagonal up to about 0.5. Downstream to the mixing zone (z > 5 mm), the mixing fraction returns progressively to a value close to its initial value.

F(t) F(t = 0) = 1 , t + (1 -) 2 , t .
Figure 15 shows the evolution of the temperature in the hot and cold sprays (respectively T 1 and T 2 ), the tempera- ture T LIF derived from the average fluorescence lifetime (Eq. 19) and the mixing temperature T m based on Eq. (22) as a function of the vertical position z on the hot and cold spray diagonals drawn in Fig. 14A. It can be observed that T m is systematically a few degrees higher thanT LIF , and the largest differences can be pointed out in the center of the mixing zone. This agrees well with the fluorescence model presented in Sect. 5.1. Considering a temperature T 1 = 20 °C for spray 1 and T 2 = 56 °C for spray 2, the evolution of T LIF and T m as a function of the volume fraction are shown in Fig. 16. The difference between T LIF and T m reaches about 5 °C when is close to 0.4. To obtain Fig. 16, Eq. ( 14) was first used to evaluate the fluorescence lifetimes t 1 and t 2 , and then Eq. ( 20) was as an approximation to estimate LIF for ranging from 0 to 1.

In Fig. 15A, T 1 and T 2 are found identical and close to 56 °C at the beginning (z < 0.5 mm) because there is only one spray. When spray 2 (cold spray) starts to be present in sufficient quantity (in the mixing zone), the temperature T 2 decreases to take a value close to 20 °C, which is com- parable to the injection temperature of the cold spray. At the center of the mixing zone (approximately for z between 2.5 and 5.5 mm), T 1 and T 2 are consistent with the expected values. After the mixing zone ( z > 5.5 mm), most of the liquid remaining in the measurement volume is from the Even if the temperatures T 1 and T 2 are reasonably well recovered in Fig. 15, the measurement method suffers from limitations when one of the sprays is weakly present in the measurement volume. In this case, the temperature of the weakly present spray cannot be reliably determined. Based on the results presented in Figs. 14 and 15, it seems that the volume fraction of a given spray must exceed a certain level in order to determine its temperature with an accuracy. To further investigate this issue, additional tests were conducted. Figure 17 provides further insights into the accuracy of the temperature measurements of the sprays. In Fig. 17A, the measurement position is situated in a region of the cold spray is dominant ( x = 1.5 mm and z = 7 mm). It is evidenced that the temperature T 1 of the hot spray can be changed in a large range while the fitting of the experimental fluorescence decay remains quite acceptable. The confidence interval for the hot spray temperature T 1 extends from about 44 to 72 °C, which is very large. In contrast, the temperature T 2 of the cold spray ranges between 21 and 22.5 °C, which is rather good for a measurement accuracy. Further investigations demonstrate that T m and T LIF are measured with an accuracy better than 1 °C even when T 1 is measured with the largest error in Fig. 15. Both T m and T LIF can still be measured correctly while the mixing fraction T 1 or T 2 are significantly wrong as errors compensate.

Figure 17B corresponds to a position in the center of the mixing zone, where the volume fraction of the two sprays is comparable ( x = 4.5 mm and z = 4 mm). It can be clearly observed that both temperatures T 1 and T 2 , as well as the mixing fraction , are estimated with a good accuracy in that case. Further investigations, presented in Appendix, hot spray, and it is observed that the temperature T 1 of the hot spray has slightly decreased from the injection temperature. The temperature T 2 continues to be evaluated with good accuracy despite the small amount of liquid from spray 2. Similar conclusions can be drawn in Fig. 15B for the results obtained along the axis of the cold spray. The hot spray has a low volume fraction except in the central region (2.5 mm < z < 5.5 mm). Fluctuations affecting the temperature T 1 of the hot spray can be observed everywhere else.

A B

were carried out using the Monte Carlo method to evaluate the measurement uncertainties of the different parameters. They reveal that the measurement errors on the average temperatures T m and T LIF are always acceptable (less than 1 °C) under the measurement conditions of the experiments (a minimum count rate of 2000 Cps and an acquisition time of 100 s). The uncertainties on the volume fraction and the temperature T 1 of the hot spray increase when tends toward 1 (i.e., when little liquid of the hot spray passes through the measurement volume). The error on T 1 can exceed several °C when exceeds about 0.8, whereas the error on the cold spray temperature T 2 does not exceed 1 °C for values of as low as 0.1. The error on the volume fraction remains lower than 1% when the hot spray is predominant in the measurement volume (more precisely for < 0.4).

Conclusions

A new method for measuring the local temperature in sprays using two-photon absorption was developed and tested. The method is based on measuring the fluorescence lifetime of certain dyes for which the quantum yield is sensitive to temperature such as RhB. One of the main advantages of using the fluorescence lifetime (instead of the fluorescence intensity) relates to the fact that it is an intrinsic molecular property, which is not affected by fluctuations

A B

Fig. 17 Estimations of the temperatures T 1 and T 2 of the hot and cold sprays in a region where the cold spray is predominant (A) ( x = 1.5mm; z = 7mm ) and in a region situated near the center of the mixing zone (B) ( x = 4.5mm; z = 4mm)

over conventional approaches based either on the signal intensity or on the average fluorescence lifetime ( LIF ). The latter only permit to determine the temperature T LIF , which can deviate from T m by several °C. The temperatures T LIF and T m can be accurately measured using TCSPC, even when the estimates of the hot spray temperature can reach a few °C in the mixing zone.

Appendix

A Monte Carlo method was implemented to estimate the measurement uncertainties. It consists in generating synthetic histograms of the photon arrival times, while taking into account the photon count rate, the acquisition time, the noise level, the IRF, and the time-channel width. Such method has been already employed by [START_REF] Stiti | Fluorescence lifetime measurements applied to the characterization of the droplet temperature in sprays[END_REF] for assessing the precision of temperature measurements in a single spray. In this study, a monoexponential function was used to describe the fluorescence decay. Presently, since two sprays are considered, the fluorescence model has been extended to a two-exponential decay to follow the theoretical model presented in Sect. 5.1a. Results obtained with the method are presented hereafter for T 1 = 60 °C and T 2 = 20 °C. Two count rates, 2000 Cps and 15000 Cps, are also considered. The count rate of 2000 cps corresponds to the worst-case scenario (no measurements were performed with a lower count rate in the experiments). The count rate of 15,000 cps is a median value among the measurements. For some given values of the volume fraction and the count rate, more than 2000 synthetic histograms are generated using the Monte Carlo method. These histograms are analyzed the same manner as their experimental counterparts in order to estimate the temperatures T 1 and T 2 , the volume fraction , the mix- ing temperature T m and the LIF temperature T LIF . Based on the thousands of generated data, the average and the standard deviation can be calculated for those quantities, which allows to evaluate the uncertainties of their measurements.

Figure 18 shows the influence of the acquisition time on the estimates of the volume fraction and the mixing temperature T m . The colored areas on the graphs correspond to the confi- dence intervals of half-width equal to the standard deviations or T m . It can be observed that an acquisition time of 100 s is a good compromise. It allows having a limited estimation error while maintaining an acceptable acquisition time. This value of the acquisition time was thus retained in the experiments and in the tests performed in the following for evaluating the uncertainties with the Monte Carlo method.

Using the same approach, uncertainties were also quantified for different values of the volume fraction at a fixed acquisition time of 100 s in Fig. 19. Significant differences can be observed between T 1 and T 2 . These can be explained by the fact that the hot spray has a much shorter fluorescence lifetime than the cold spray (Fig. 7). When approaches 1, there is very little of the hot spray passing in the measurement volume in comparison with the cold spray. Therefore, the presence of the hot spray is difficult to detect based on the fluorescence decay. Not only it is hardly visible at the long times but also at the short times. So T 1 is large while T 2 is small. When gets close to 0, the hot spray has a stronger influence on the fluorescence decay especially at the short times, but the cold spray keeps a significant influence at the long times. The uncertainty on the temperature T 2 of the cold spray does not increase that much. The value of T 2 remains smaller than 1 °C for values of as low as 0.1. Finally, it is interesting to note that the uncertainties on the parameters T m and T Lif remain always lower than 1 °C, while the uncertainty on the temperature of the hot spray T 1 reaches several °C. This is due to the fact that the different parameters are correlated, and errors can compensate each other. The error on the volume fraction remains lower than 1% when the hot spray is predominant in the measurement volume (more precisely for < 0.4).

Fig. 1 Fig

 1 Fig. 1 Scheme of the fluorescence emission and scattering when using (A) a monophoton excitation and (B) a biphoton excitation

Fig. 3 Fig. 4

 34 Fig. 3 Illustration of the key parameters used to describe the Gaussian beam propagation and to model the size of the fluorescence spot

Fig. 6

 6 Fig. 6 Fluorescence decays of A RhB at 60 °C and B at 19 °C, EY at 20 °C C and Rh6G at 20 °C D. The decays are fitting by a monoexponential and a biexponential functions

Fig. 8 Fig. 9

 89 Fig. 8 Effect of the on the average fluorescence lifetimes of RhB, EY and Rh6G

Fig. 12

 12 Fig. 12 Distribution of the volume fraction in the mixing region of the two sprays. The liquid flow rate of the two sprays is about 0.6 L/min. A map of the distribution of is constructed to obtain a 2D image of the mixing fraction in the mixing region of the two sprays using several methods

Fig. 13

 13 Fig. 13 Spatial distribution of the volume fraction = . The values are obtained by assuming either a biexponential decay A based on Eq. (16)16, or the combination of two biexponential decays B using Eq. (26)26

Fig. 14 C

 14 Fig. 14 Evolution of the mixing temperature T m and the mixing fraction . A 2D map of the spatial distribution of the mixing temperature T m and B data in the mixing zoom with a spatial step of 0.1 mm.

Fig. 15 Fig. 16

 1516 Fig. 15Evolution of the fluorometric temperature T LIF , the mixing temperature T m and the temperatures T 1 and T 2 of the two sprays along the diagonals of the hot and cold sprays plotted in Fig.14A. A represent the result along the hot spray and B along the cold spray

  in excitation of the laser source or in the transmission of the fluorescence signal across the spray and the optics. Time-correlated single-photon counting (TCSPC) is well suited to carry out such measurements because of a high signal-to-noise ratio and its capability to characterize the fluorescence decay in the time domain. This last feature can be very useful to study the mixing of two sprays, when the fluorescence emissions of the droplets originating from the two sprays fluoresce with a sufficiently different fluorescence lifetime. In such situation, the time decay of the fluorescence makes it possible to retrieve the temperature of the two sprays as well as their mixing fraction. The possibility of measuring the mixing fraction alone was first tested in the case of a mixture of two sprays seeded separately with EY and rh6G, which have very different fluorescence lifetimes but both insensitive to temperature. The combined measurement of temperature and mixing fraction using RhB alone was carried out for a thermal contrast of approximately 40 °C between the two sprays. The biexponential behavior of the fluorescence decay of RhB must be considered to obtain acceptable values of the mixing fraction. It was found that the volume fraction of a spray must exceed about 10% to make it possible to derive its temperature with an acceptable accuracy of about 2-3°C. Measuring the volume fraction and the mixing temperature weighted by the liquid volume of the sprays ( T m ) is a clear advantage of the proposed multi-exponential analysis

Fig. 18

 18 Fig. 18 Confidence intervals of the estimations of the volume fraction and the mixing temperature T m when the acquisition time is data point is the result of the computation of 2000

Acknowledgements

The authors are grateful to the China Scholarship Council (CSC) for the scholarship attributed to Mr. M. Wang, who was hosted by LEMTA at Université de Lorraine, France. The authors also gratefully acknowledge the financial support of the ENERBATIN CPER program and the European FEDER program.

Author Contributions WM performed the measurements; SM and CG were involved in planning and supervised the work. CH and BS were involved in the implementation of the optical system and the measurement methods. MW, MS and GC processed the experimental data, performed the analysis, drafted the manuscript, and designed the figures. FL and EB provided critical feedback and helped shape the research. GC conceived the original idea and was in charge of overall direction of the project. All authors discussed the results and commented on the manuscript.

A B

Fig. 19 Evolution of the standard deviation of the estimated temperature of the hot ( T 1 ) and cols ( T 2 ), the mixing fraction ( ), the mixing temperature ( T m ) and the LIF temperature ( T LIF ) as a function of the mixing fraction. Each point is the result of 2000 synthetic fluorescence decays using an acquisition time of 100 s, a noise count rate of