Christel Pierlot

Mélanie Menuge

Marianne Catté

Olivier Devos

Jesús F Ontiveros

Visible light backscattering monitored in situ for transitional phase inversion of BrijL4-isopropyl myristate-water emulsions

The transitional phase inversion of BrijL4/isopropyl myristate/water emulsions is detected by monitoring the visible Light BackScattering (LBS) using an optical fiber. For comparison, electrical conductivity is simultaneously followed during the heating-cooling cycle. The LBS signal exhibits always a minimum near the Phase Inversion Temperature value determined with conductivity. The LBS results are also analyzed in terms of Winsor type for equilibrated systems and particle size of O/W emulsions. LBS signal points out transparency area for inversion through Winsor IV type and allows thus to determine the X point of fish diagram for equilibrated systems more quickly. The LBS signal presents a maximum for average diameter of O/W emulsions around 1 micron, and depends on the particle size distribution. The use of a commercial visible light

backscattering optical fiber exhibits thus several advantages over the classical conductivity measurement for transitional inversion phase emulsification process monitoring.

INTRODUCTION

A temperature increase tends to trim down the interactions between water molecules and polyether chains of polyethoxylated nonionic surfactants; in the same way, these surfactants allow the emulsion to inverse during heating from O/W to W/O morphology, at the so-called Phase Inversion Temperature PIT [START_REF] Shinoda | The Correlation between Phase Inversion Temperature in Emulsion and Cloud Point in Solution of Nonionic Emulsifier[END_REF] . Reverse direction phase inversion from W/O to O/W takes place upon cooling at a temperature more or less close to the PIT. Such phase inversions driven by a phase behavior transition often produce fine droplets emulsions with a low energy expense, even with high viscous phases. This has been used for decades for such purpose [START_REF] Esquena | Highly Concentrated W/O Emulsions Prepared by the PIT Method as Templates for Solid Foams[END_REF] and is called the PIT emulsificationmethod [START_REF] Shinoda | Stability of O/W [Oil/Water] Type Emulsions as Functions of Temperature and the HLB [Hydrophile-Lipophile Balance] of Emulsifiers: The Emulsification by PIT [Phase Inversion Temperature]-Method[END_REF] .

Any in situ emulsion characterization method able to detect changes in drop size, electrical properties or viscosity is likely to be a candidate to monitor the phase inversion process, and to pinpoint the PIT with accuracy. The classical way to detect the emulsion inversion temperature consists in measuring the conductivity of the emulsion. Conductivity is typically high for the O/W morphology and low for the W/O one, particularly if the aqueous phase contains some electrolytes as often the case in practice. Conductivity of O/W emulsions can be calculated through mathematical models [START_REF] Lira | Electrical Conductivities of Concentrated Emulsions and Their Fit by Conductivity Models[END_REF] ; equations have also been established for three-phase systems considering volume fractions of each dispersed phase [START_REF] Lee | Morphologies of Three-Phase Emulsions of the Ternary Nonionic Amphiphile/Oil/Water Systems and Their Determination by Electrical Method[END_REF] . Electrical properties of emulsions under sinusoidal constraints have been studied and the phase inversion temperature was detected from impedance at low frequencies by use of parameters such as double layer capacitance and charge transfer resistance. [START_REF] Araujo | Phase Inversions in Emulsions Probed by Electrical Impedance Spectroscopy[END_REF][START_REF] Lehnert | Investigation of Thermal Phase Inversion in Emulsions[END_REF][START_REF] Sjoblom | Emulsions Characterized by Means of Time Domain Dielectric Measurements (TDS). Technical Applications[END_REF][START_REF] Asami | Characterization of Heterogeneous Systems by Dielectric Spectroscopy[END_REF] . However, the conductimetric method has some drawbacks. First of all, it needs to add an electrolyte such as NaCl in the aqueous phase to get a clear-cut detection of the phase inversion, and is not totally reliable with very high viscous systems for which creation of stagnant unmixed zones is favored [START_REF] Galindo-Alvarez | Viscous Oil Emulsification by Catastrophic Phase Inversion: Influence of Oil Viscosity and Process Conditions[END_REF] . Moreover, conductivity is inefficient to detect multiple emulsions of the o/W/O type. Finally, conductivity does not give any valuable information about the drop size, which is often an important issue in practical cases.

For in situ inversion phase process monitoring, optical fibers can be used to measure Light BackScattering signal (LBS). As conductivity, this is a non destructive and non invasive emulsion characterization technique. Fiber in Near-InfraRed area (NIR) has been used to study transitional phase inversion of water-heptane/toluene emulsions [START_REF] Charin | Transitional Phase Inversion of Emulsions Monitored by in Situ Near-Infrared Spectroscopy[END_REF] . More recently, the same team followed NIR spectra during catastrophic inversion of emulsions [START_REF] De Oliveira Honse | Emulsion Phase Inversion of Model and Crude Oil Systems Detected by Near-Infrared Spectroscopy and Principal Component Analysis[END_REF] . The NIR backscattering at 850 nm has also been used to study on line transitional [START_REF] Pizzino | Light Backscattering as an Indirect Method for Detecting Emulsion Inversion[END_REF][START_REF] Pizzino | On-Line Light Backscattering Tracking of the Transitional Phase Inversion of Emulsions[END_REF] (technical-C12E4/decane/water emulsion) and catastrophic (Tween 80 or Span 20/petroleum ether/water emulsion) inversions. However, evaluated emulsion was circulating outside the reactor through the cylindrical cell of the used apparatus (Turbiscan-online) [START_REF] Pizzino | On-Line Light Backscattering Tracking of the Transitional Phase Inversion of Emulsions[END_REF] . The authors have noted a variation of the LBS near the catastrophic inversion point, neat when increasing the oil content in a O/W emulsion and more gradual when adding water into an W/O emulsion. Surprisingly no study considering emulsion phase inversion monitored with backscattered light in the visible range has been published.

In the present article, we analyze the visible part of the backscattering signal transmitted by an optical fiber during phase inversion of ethoxylated alcohol BrijL4/isopropyl myristate/water systems at different concentrations of surfactant. The purpose of this work is to study if on line LBS in visible range can be used as a method to detect the phase inversion temperature with accuracy. For each emulsion, both conductivity and visible light backscattering measurements were monitored during heating-cooling cycle. Moreover, obtained light backscattering signals are analyzed with respect to the phase diagram (temperature vs. surfactant concentration) of equilibrated studied systems and also with respect to the diameter and polydispersity of the O/W emulsions at 20°C.

MATERIALS AND METHODS

Materials

Technical grade polyoxyethylene lauryl ether with an average of 4 ethylene oxide groups per molecule was supplied by Sigma-Aldrich as BrijL4. Isopropyl myristate (IPM) was purchased from Cooper. Sodium chloride NaCl (purity ≥ 99.5%) was obtained from Acros Organics. Demineralized water (conductivity 1.34 µS.cm -1 at 21°C) was used in the aqueous phase.

Definitions

The percentage of BrijL4 and the water weight fraction fw are calculated below where mBrijL4, mO and mW are the weights of BrijL4, IPM and water respectively.

Preparation of emulsions by magnetic stirring at room temperature

9% BrijL4/IPM/water system (fw =0.5) was prepared by pouring in a 25 ml glass bottle 22.75 g water, 22.75 g IPM, and then 4.5 g of BrijL4. The system is mixed at room temperature by continuous magnetic stirring at 800 rpm. 0.5 g samples are withdrawn at different stirring time for size analysis of formed emulsions.

Preparation of emulsions by phase inversion temperature process

A 50 mL double jacketed reactor that permits introduction of both conductivity cell and optical fiber was used. To ensure homogeneity inside the reactor, mechanical stirring at 400 rpm is performed using a turbine. Heating-cooling cycles are achieved by circulation of water in the double jacket of the reactor. Temperature changes are achieved thanks to a cc3 Huber Ministat bath. This experimental setup is different from that already described for rheological studies [START_REF] Pierlot | Cone-Plate Rheometer as Reactor and Viscosity Probe for the Detection of Transitional Phase Inversion of Brij30-Isopropyl Myristate-Water Model Emulsion[END_REF] .

The first 9% BrijL4/IPM/water emulsion is prepared by pouring in the reactor 22.75 g of water, 22.75 g of IPM, and then 4.5 g of BrijL4 (fw =0.5). This mixture is stirred for 15 min at 20°C. A heating-cooling cycle (20-65-20 °C) under stirring is then applied with a gradient of temperature of 1 °C.min -1 . The next emulsions with 11, 13, 15 and 17% surfactant are formulated, by adding an aliquot of 2% BrijL4 between each heating-cooling cycle. Schema 1 lists the successive steps of the process and specifies the names used for each of the obtained O/W emulsions. and so on Schema 1. Fluxogram of successive steps of the process managed at 400 rpm and specific names used for each of the obtained O/W emulsions.

In place of the protocol of scheme 1, we could have made 5 emulsions with 9, 11, 13, 15 and 17% of surfactant and apply for each emulsion a cooling heating cycle. However, for high concentrations of surfactants (15 and 17%) it is perfectly necessary to apply several cycles of heating cooling so that the system is stabilized (ie when there is practically no difference between PIT in heating ramp heat and PIT in cooling ramp. So we preferred to add fractions of 2% surfactant, to realize only one cycle of heating-cooling. In addition, this allows us to reduce the amount of oil and surfactants used.

It is to notice that the first 9% BPI emulsion is particular because not obtained like the others after two transitionnal phase inversions, but only by a simple stirring of the system at 20°C.

Conductivity measurements

The conductivity and temperature are measured by a Radiometer Analytical CDM 210 conductimeter fitted with a CDC741T platinized platinum probe. The conductimeter and temperature signals are sent to a data acquisition system. The software used was custom written in the Labview 7.1 National Instruments platform software.

Light backscattering measurements

The Light BackScattering (LBS) intensities are measured with an UV/VIS/NIR optical reflection and backscattering probe form Ocean Optics. The light source end of the probe is connected to a tungsten halogen light source (HL-2000-FHSA by Mikropack). The spectrometer end fiber is connected to a USB2000+ miniature spectrometer configured for visible and near-IR measurements (wavelength range: 327.0-1105.5 nm) from Ocean Optics controlled with the SpectraSuite software.

Droplet mean size and size distribution measurements

The emulsion average droplet size and the size distribution were determined by the laser diffraction instrument MasterSizer3000 (Malvern Instrument). The emulsion sample (0.5 g) was diluted (1/1000 or 1/100 depending of the obscuration level) with distilled water to until correct 10% obscuration was obtained. The size distribution is characterized with Dv(10), Dv(50), and Dv(90) parameters which are particle sizes below which 10, 50 or 90 % respectively, of the sample falls; 100% being the total droplets volume. The width of the distribution is characterized by a polydispersity index called span that is a dimensionless number calculated as followed: span = (Dv(90) -Dv( 10)) / Dv(50)

RESULTS AND DISCUSSION

Several emulsions were formulated with different surfactant percentages. For each emulsion, both conductivity and LBS measurements were monitored during heating-cooling cycle.

In the previous paper [START_REF] Pierlot | Cone-Plate Rheometer as Reactor and Viscosity Probe for the Detection of Transitional Phase Inversion of Brij30-Isopropyl Myristate-Water Model Emulsion[END_REF] the authors have shown that because of the high solubility of the surfactant in the MIP, it was not possible to obtain a phase inversion for Brij30 concentrations lower than 7% surfactant. For excessive concentration of surfactant with concentrations greater than 16%, the viscosity of the medium becomes too high to ensure good homogenization of the emulsion. This is the reason why we worked with Brij30 concentrations between 9 and 17%."

Conductivity curves versus temperature and BrijL4 concentration

Figure 1 presents the variation of conductivity with temperature for BrijL4/IPM/water (NaCl 10 -2 M) systems for surfactant concentrations ranging from 9 to 17%. The phase inversion occurs during heating and cooling for the five tested concentrations of BrijL4. ) and cooling ( ) ramps (1°C.min -1 ; 400 rpm) for 9 to 17% BrijL4.

The conductivity decreases sharply from a very high value to a low one within a small interval of temperature. The temperature at which the conductivity is half of the maximum conductivity value on an arithmetic scale is used to define the Phase Inversion Temperature named PITcond. Figure 1 shows that phase inversion temperature determined from conductivity curves during heating (PITcond,heat) and cooling (PITcond,cool) ramp temperature are practically the same. Preliminary trials have shown that for a slow rate of temperature change only a small hysteresis phenomenon is exhibited between heating and cooling. Some authors [START_REF] Pizzino | On-Line Light Backscattering Tracking of the Transitional Phase Inversion of Emulsions[END_REF] proposed to repeat the procedure on a same system: second and third temperature cycles lead to some PIT differences if the temperature variation is relatively rapid [START_REF] Marquez | Hysteresis Behavior in Temperature-Induced Emulsion Inversion[END_REF] .

Transitionnal phase inversion temperature with oligomeric surfactant varies depending on the surfactant concentration [START_REF] Wade | Interfacial Tension and Phase Behavior of Surfactant Systems[END_REF][START_REF] Graciaa | The Partitioning of Complex Surfactant Mixtures between Oil/Water/Microemulsion Phases at High Surfactant Concentrations[END_REF][START_REF] Graciaa | The Selective Partitioning of the Oligomers of Polyethoxylated Surfactant Mixtures between Interface and Oil and Water Bulk Phases[END_REF] . Figure 1 shows lower PIT for higher concentrations of BrijL4. An increase of surfactant concentration tends to decrease the selective partitioning of less hydrophilic EON (low Ethylene Oxide Number oligomers) species to the oil phase, the remaining oligomer mixture at interface becoming thus more lipophilic. A lower temperature is then required for O/W to W/O phase inversion during heating.

The small peaks of conductivity at 57, 39.5°C and 38.5°C observed for 9, 15 and 17% BrijL4 respectively (Figure 1) could correspond to liquid crystals as suggested by Saulnier et al. [START_REF] Saulnier | Liquid Crystals and Emulsions in the Formulation of Drug Carriers[END_REF] that gradually appear with surfactant concentration. For all the samples the raw light backscattering spectra have globally the same "shape" with changes in the overall intensity. The spectral shape is mainly due to the light source intensity, the fiber transmittance and the spectral sensitivities of the detector for the different wavelengths.

Effectively the LBS maximum observed at 520, 570, 585, 610, 620, 675 and 750 nm, correspond to local maximum in the spectral sensitivities of the detector combined with the intensity of the source at these wavelengths. In term of overall intensity an important increase of LBS signal is observed between 9% BrijL4 emulsions before and after phase inversion. This confirms that LBS data are linked to the emulsification process. On the other hand, successive additions of 2% BrijL4 and heating-coooling cycles lead to a decrease of LBS intensity. The phase inversion process was followed at a wavelength of 675 nm for which the raw signal is relatively high and constant. The LBS curves in heating and cooling mode are always characterized by a minimum in the vicinity of the PIT, that can extend over a rather large temperature interval (TIminLBSheat and TIminLBScool)

reported in table 1. The PIT values obtained through conductivity (figure 1) are also reported. The conductivity defining the PIT (PITCond) is chosen as the temperature at which the specific conductivity of the emulsion is half of the maximum PIT value (on a arithmetic scale). In our experimental conditions, preliminary trials have shown that if the temperature ramp is lower than 1°C/min (heating and cooling) only a small hysteresis between PITCondheat and PITCondcool phenomenon is exhibited.

From Figure 1, at 17% BRIJL4 several temperatures correspond to half of maximum conductivity, due to a second extremum in conductivity linked with apparition of liquid cristals at high surfactant concentration. This secondary pic was not considered to determine PITCond . The temperature interval over which this minimum extends is more important as the BrijL4 concentration increases. For 9% BrijL4, the LBS signal whether in heating or cooling mode has a sharp minimum that spreads over one degree and includes the PIT values determined by conductivity [START_REF] Charin | Transitional Phase Inversion of Emulsions Monitored by in Situ Near-Infrared Spectroscopy[END_REF][START_REF] Pizzino | Light Backscattering as an Indirect Method for Detecting Emulsion Inversion[END_REF][START_REF] Pizzino | On-Line Light Backscattering Tracking of the Transitional Phase Inversion of Emulsions[END_REF] . For 13, 15 and 17% BrijL4, the minimum LBS temperature interval grows from 2 to 11 degrees and matches with observation of transparency emulsions. These temperature intervals also include the PIT values determined by conductivity.

For both O/W emulsions before and after phase inversions the LBS signal is the same, excepted for 9% surfactant emulsion (figure 3A: 6265 BPI and 23139 API). The 9% BPI emulsion is the only one obtained with mechanical agitation at 20 °C; for all others the emulsification process consists in two transitional phase inversions. Figure 3A to 3E shows that for emulsions after phase inversions (API), the LBS signal decreases from 23139 AU to 7765 AU when surfactant % increases from 9 to 17%.

From the conductivity curves one often speaks of phase inversion temperature due to the mathematical method used to obtain a single temperature value (method of the tangents, or halfmethod as in this study). But the decay of the conductivity curves also extends over several degrees, and a phase inversion temperature interval would be more suitable. If the graphical determination of the PITcond is somewhat arbitrary, a single value of PIT can easily be detected by LBS for 9% BrijL4.

For temperatures higher than the phase inversion zone, the LBS signal in heating mode is always greater than in cooling mode on the contrary to the conductivity profiles. In fact, since the continuous phase of the emulsion is oily, the conductivity signal is constant and equal to 0 µS.cm -1 .

For temperatures below the PIT the most striking difference is obtained for 9% BrijL4 (figure 3A).

In the 20-45°C temperature range, the conductivity values in the heating and cooling ramps are identical, whereas the LBS curves are sometimes different, particularly for starting emulsion with 9% BrijL4 (figure 3A at 20°C: 6265 BPI much lower than 23139 API). In fact, additional cooling/heating cycles (data not shown) leads to LBS signals practically similar to those obtained during the first cooling ramp.

In situ light backscaterring with respect to equilibrated phase diagram

The phase diagram based on visual observation of Winsor types (WI, WII, WIII and WIV) [START_REF] Pierlot | Cone-Plate Rheometer as Reactor and Viscosity Probe for the Detection of Transitional Phase Inversion of Brij30-Isopropyl Myristate-Water Model Emulsion[END_REF] However we did not want to indicate the calculated value of (X, T) in dynamic mode with the optical fiber, because even if it is close to the one published in literature [START_REF] Pierlot | Cone-Plate Rheometer as Reactor and Viscosity Probe for the Detection of Transitional Phase Inversion of Brij30-Isopropyl Myristate-Water Model Emulsion[END_REF] with the fish diagram, we could not work with the same lot of surfactant (it was Brij30 in 2016, and BrijL4 in 2019).

Figure 5 presents a zoom on the LBS minimum versus temperature obtained for cooling ramp with 17% BrijL4. Slight rises of the LBS signal can be observed (black arrows) which match exactly with the liquid crystals peak [START_REF] Saulnier | Liquid Crystals and Emulsions in the Formulation of Drug Carriers[END_REF] visible on conductivity curve. These small increases of LBS signal in the TIminLBS,cool temperature interval could be due to particulate sprouts that appear in the beginning and at the end of liquid crystal formation. This indicates also that liquid crystals formation occurs as the LBS signal is very low, what is compatible with the transparent appearance of WIV observed emulsions. These slight variations of LBS are so small that they do not correspond to a detectable phenomenon by visual observation or by optical microscopy. µm) with the largest distribution (span=2.9). As expected, first heating-cooling cycle permits to reduce both size (1.25 µm) and span (0.787). Figure 7 shows that supplementary 2% BrijL4 addition and heating-cooling cycle leads to regular decrease of mean diameter (1.27 to 0.565 µm) and to more monodisperse size distribution (span = 0.787 to 0.555). phase inversions versus %BrijL4.

The addition of 2% BrijL4 by simple stirring at 20°C (change from red dots to black dots through horizontal dashed lines) does not significantly change the size of the droplets (figure 7A), but expands the size distribution (figure 7B). For a given BrijL4%, the transitional inversion by changing the temperature (change from black dots to red dots through vertical dashed lines) leads to a reduction of both size and span. The minimum size is thus obtained for 17% Brijl4 (0.628 µm). But even with several phase inversions cycles, this minimum droplet size is not the only condition needed to get a stable emulsion and after several hours most samples show a creaming separation at room temperature. This confirms that the PIT process certainly reduces the size of the droplets, but is in no way sufficient to ensure the preservation of the emulsion if droplet size are still too important (superior to 0.628 µm in our case) [START_REF] Brochette | Creation and Study of Emulsions[END_REF] .

Correlation between LBS and O/W emulsions droplets size

It is well known [START_REF] Bordes | Online Characterization of Particle Size during an Ultrafine Wet Grinding Process[END_REF][START_REF] Mengual | Multiple Light Scattering Measurement for Concentrated Emulsion and Suspension Instability Analysis[END_REF] that LBS of heterogeneous dispersions presents a maximum intensity for a mean particule size around 1 µm is also affected by the particules size distribution [START_REF] Ulloa | Effect of the Particle-Size Distribution on the Backscattering Ratio in Seawater[END_REF][START_REF] Jacob | Fundamentals of Atmospheric Modeling[END_REF][START_REF] Xu | Particle Characterization: Light Scattering Methods[END_REF] . Figure 8 presents The size (Dv(50) was calculated using the laser granulometer software, indicating the refractive index of the continuous (Water) and the dispersed (MIP) phase and the absorption coefficient of the dispersed phase. However, we also made simulations with BrijL4 (through its refractive index), and considering that the surfactant could modify the refractive index of the aqueous and dispersed phase, depending on whether it is totally or partially dissolved in each of both phase. These different scenarii have not shown a significant change in size, as far as the Dv(50) is concerned.

CONCLUSION

The contribution of light backscattering (LBS) measurement with an optical fiber to follow in situ the transitionnal phase inversion was studied on BrijL4/Isopropyl Myristate/Water emulsions with fw= 0.5 and 9 to 17% BrijL4. Obtained results were compared to the usual conductimetric method and analysed in regards to the phase diagram of equilibrated systems and O/W emulsions particle size. The 675 nm lengthwave was chosen corresponding in the spectra at a maximum for which the LBS signal is relatively constant. During transitional phase inversion, a minimum of LBS appears always near the phase inversion temperature detected by conductivity. This minimum is relatively sharp for low surfactant concentrations (9% and 11%) corresponding to inversion through WIII type, but becomes larger with higher surfactant concentrations corresponding to inversion trough WIV type. For these higher surfactant concentrations, minimum LBS signals corresponding to transparent emulsions allow to determine the X point (C*,T*) of the system. In situ visible LBS can thus be envisaged for transitional phase inversion detection as well as a rapid way for X point determination.

The measurement of the light backscattered by the optical fiber is finally a complementary technique to the conductivity to follow the evolution of emulsification processes. The conductivity makes it possible to determine the nature of the continuous phase of the emulsion (unlike the optical fiber). For W / O emulsions the conductivity is zero so it is difficult to extract other information than the oily nature of the dispersed phase. On the other hand, the optical fiber indicates a backscattered light signal caused by the presence of the drops of the dispersed aqueous phase, this signal itself being correlated with the size of these droplets. The evolution of LBS versus O/W emulsions average diameter Dv(50) is a curve (Figure 8) with a maximum of about 1 micron, which moreover depends on the particle size distribution (span index).

Thus it is necessary to distinguish 2 size classes. For droplets greater than one micron, the backscatter signal is dependent on the average size as well as the distribution profile (span), so it is not possible to obtain an accurate measurement of the size with only the signal at a minimum.

wavelength (675 nm). We have not studied the possible contribution of light retrodiffusion signals to other wavelength, because we did not find qualitatively spectrum profile variation (Figure 2)

according to the emulsions.

For droplets smaller than one micron, the particle size profiles are narrower, and the span that remains constant is no longer an influencing factor in the size correlation, that varies almost linearly (rather affine) with the intensity of backscattered light.

Although it remains difficult to accurately determine the emulsion size from LBS measurements, this signal is a considerable contribution to size control throughout the emulsification process.

Finally, the use of a commercial visible light backscattering optical fiber exhibits several advantages over the classical conductivity measurement for transitional inversion studies: it does not imply any electrolyte addition in the aqueous phase; is adapted for highly viscous media; and brings informations during the process particularly for W/O emulsion.

concerning the emulsion size (mean diameter).

Finally, an optical fiber user can use it by controlling the evolution of the signal to control the reproducibility of an emulsification process. The user could also have information on the size of the emulsion (provided if he knows that the size is smaller than 1 micron). The user could also realize a standard range of emulsions between 50 and 500 nm which are generally stable for calibrating the optical fiber.

Figure 1 :

 1 Figure 1: Conductivity of BrijL4/IPM/water (NaCl 10 -2 M) systems at fw =0.5 versus temperature during heating () and cooling ( ) ramps (1°C.min -1 ; 400 rpm) for 9 to 17% BrijL4.

3. 2 .Figure 2

 22 Figure 2 presents the light backscattering (350-1100 nm) raw spectra at 20°C for O/W emulsions

Figure 2 .

 2 Figure 2. Raw light backscattering spectra between 350 and 1100 nm at 20°C, 400 rpm, of BrijL4/IPM/water (NaCl 10 -2 M) O/W emulsions at fw =0.5 after phase inversions for 9% to 17%

3. 3

 3 Comparison between light backscattering at 675 nm and conductivity signals during heating-cooling ramp LBS and conductivity signals during heating-cooling ramps are presented on Figure 3. Since conductivity curves during heating and cooling are relatively similar, only the cooling conductivity signal is represented for clarity reasons (dashed line).

Figure 3 .

 3 Figure 3. Transitionnal phase inversion for BrijL4/IPM/water (NaCl 10 -2 M) systems at fw =0.5 for

4 .

 4 of equilibrated BrijL4/isopropyl myristate/water (NaCl 10 -2 M) systems with fw = 0.5 appears in figure It is to notice that such observations required a very long time (several weeks) attributed to the slow kinetics of formation and destruction of liquid crystals 21 . The usual fish contour plot of this diagram gives access to the characteristic X point of the system defined by the minimum surfactant concentration C * and the temperature T * at which aqueous and oily excess phases vanish and a single transparent WIV microemulsion phase is observed. To analyze LBS signal with respect to equilibrated systems, Temperature Intervals corresponding to minimum LBS at 675 nm during cooling (TIminLBS,cool) for transparent systems (13, 15 and 17% BrijL4) are superposed to the phase diagram in figure 4. These temperature intervals are represented in green. The transparency of emulsions formulated in the WIV area leads certainly to this minimum LBS temperature interval.

Figure 4 :

 4 Figure 4: Determination of X point (C*,T*) based on minimum LBS at 675 nm during cooling for

Figure 5 . 3 . 5 .Figure 6

 5356 Figure 5. Conductivity and LBS at 675 nm versus temperature during cooling ramp at 1°C.min -1with 400 rpm stirring rate for 17% BrijL4/IPM/water (NaCl 10 -2 M) systems at fw=0.5. Black arrows

Figure 6 . 7 .

 67 Figure 6. Volumic density versus class size for BrijL4/IPM/water (NaCl 10 -2 M) O/W emulsions at

Figure 7 .

 7 Figure 7. Mean diameter Dv[50] (7A) and polydispersity index span (7B) at 20°C of BrijL4/IPM/water (NaCl 10 -2 M) O/W emulsions at fw=0.5 before (black dots) and after (red dots)

Table 1 .

 1 Phase inversion temperatures determined for conductivity during heating (PITcond,heat) and

	cooling (PITcond,cool) ramps and Temperature Intervals for minimum value of light backscattering at
	675 nm during heating (TIminLBS,heat) and cooling (TIminLBS,cool) ramps .
	BrijL4	PITcond,heat	PITcond,cool	TIminLBS,heat	TIminLBS,cool
	(%)	(°C)	(°C)	(°C)	(°C)
	9	54.5	54.9	54.5 -55.5	54.7 -55.8
	11	49.3	49.7	47.4 -48.5	48.0 -48.8
	13	44.4	44.8	42.7 -44.9	43.1 -45.0
	15	40.6	41.0	36.6 -42.7	36.6 -42.9
	17	34.9	35.2	29.8 -40.6	29.8 -40.8
	.

  675 nm light backscattering versus mean diameter Dv[50] for BrijL4/IPM/water O/W emulsions at 20°C before (black dots) and after (red dots) phase inversions. For these diameters less than 1 micron, LBS is an increasing function of Dv[50] which appears clearly sensitive to the Span index. For example, emulsions 9% API and 11% BPI have the same mean particule size (1.25 and 1.27 μm) but rather different span index (0.787 and 0.857) leading to different LBS values(23 139 and 21 282). For a same mean size, a smaller Span index corresponds to a higher LBS signal.

		30 000
	nm (AU)	25 000	BPI	API	Magnetic stirring (span)
	at 675	20 000
	Light Backscattering	5 000 10 000 15 000			(2.9) (1.9) (2.5) (2.6)	(2.1)	(1.5)
		0
		0,2			2	20
				Mean diameter Dv[50](μm)
	To complete Figure 8 for particule sizes between 1 and 10 microns, 9% BrijL4 emulsions were
	simply formulated at room temperature with magnetic stirring for different times. For these Figure 8. Light backscattering at 675 nm versus mean diameter Dv[50] for 9 to 17%
	emulsions (yellow squares) the span index is added in italics next to the points. The Dv[50] and span BrijL4/IPM/water (NaCl 10 -2 M) O/W emulsions at 20°C, obtained before () and after () phase
	values are of course not sufficient to characterize a particle size profile. Nevertheless, the same trend inversions; and for 9% BrijL4 emulsions magnetically stirred at 800 rpm and room temperature for
	seems to emerge regards to the three points with same Dv[50] (4.42, 4.22, 4.6 μm) but different span different times ( ) with span values in parenthesis.
	index values (1.9; 2.5; 2.9 respectively). For a same mean size, the smaller the Span index, the higher
	the LBS signal.

ACKNOWLEDGMENTS

Chevreul Institute (FR 2638), Ministère de l'Enseignement Supérieur et de la Recherche, Région Hauts-de-France and FEDER are acknowledged for supporting and funding partially this work.

AUTHOR INFORMATION

Corresponding Author : Christel Pierlot Email: christel.pierlot@univ-lille.fr

Notes

The authors declare no competing financial interest.


domain: scco math

text:
Olivier Lebreton

Changement de point de vue sur des problèmes mathématiques et inhibition au cycle 3 : impact de la structure des énoncés

come

Changement de point de vue sur des problèmes mathématiques et inhibition au cycle 3 : impact de la structure des énoncés

Résumé

Il existe un lien entre inhibition et résolution de problèmes élémentaires [START_REF] Houdé | Apprendre à résiter[END_REF]. Son implication pour les problèmes complexes, en revanche, reste peu étudiée. Pourtant, pour un problème tel que "Madame Agricola a planté sur son terrain 45 manguiers et 73 papayers. Malheureusement les vents d'un cyclone ont arraché 23 manguiers et 21 papayers. Combien d'arbres sont toujours debout après le cyclone?", la capacité des élèves à changer de perspective dépendrait de l'inhibition de l'heuristique "arracher = soustraire" [START_REF] Lebreton | Compréhension des problèmes arithmétiques additifs à plusieurs étapes et stratégies de résolution chez les élèves de cycle 3 : Impact de la formulation des sous-buts et du niveau hiérarchique des items[END_REF]. Si les élèves initient la Procédure de Traitement Groupé (PTG) pour traiter la phrase 1 ((45+73) -(23 + 21)), elle est abandonnée lors du traitement de la phrase 2 au profit de Procédures privilégiant un Traitement Séparé (PTS) : (54 -23) + (73 -21) ou ((45 + 73) -23) -21) [START_REF] Lebreton | Changement de point de vue sur des problèmes arithmétiques additifs complexes et inhibition au cycle 3[END_REF]. Notre objectif est de savoir s'il est possible de favoriser le maintien de PTG en modifiant les énoncés. Méthodologie: 52 élèves de CM1 et 42 de 6ème ont eu à résoudre 3 problèmes à plusieurs étapes sous 4 formulations distinctes croisant les variables Item (2 vs 4) et Forme (active vs passive). Chaque énoncé est composé de 3 phrases et a été proposé phrase par phrase avec pour consigne d'anticiper une question et son calcul. Chaque phrase est codée 1 si PTG et 0 sinon. 32 élèves de CM1 et 24 de 6ème ont réussi au moins deux problèmes et leurs données ont été traitées selon une ANOVA à mesures partiellement répétées : S < Item * Forme> * Phrase (P1, P2, P3).

Résultats:

Le choix PTG diffère selon la phrase traitée : 85.6% (P1), 51.3% (P2) et 52.3% (P3), F(2, 104) = 32.3, p.< .001.

Il existe une interaction significative entre Phrase, Item et Forme : F(2,104) = 4.98, p =.009. L'utilisation de 4 items et de la forme passive pour P2 permettent de résister à l'abandon de PTG : 78,7% (P1), 72,7% (P2) et 72,7% (P3).

Discussion:

Des travaux ont montré la pertinence d'une pédagogie de l'inhibition [START_REF] Lubin | Apprendre à inhiber en classe : une pédagogie innivante[END_REF]. Nos résultats suggèrent qu'il est possible aussi de favoriser l'inhibition d'une stratégie surapprise en modifiant l'énoncé. Selon Houdé (2000), il existerait 3 systèmes cognitifs : système heuristique, système algorithmique et système d'inhibition. L'implication de 4 items et l'utilisation de la forme passive pour la phrase 2 permettraient au système d'inhibition de bloquer le système heuristique au profit du système algorithmique.


Eric André
email: eandre@em-lyon.com.

Antoine Bommier
email: abommier@ethz.ch

François Le Grand
email: legrand@em-lyon.com.

The Impact of Risk and Ambiguity Aversion on Annuity and Saving Choices

Keywords: recursive utility, lifecycle model, ambiguity aversion, risk aversion, saving choices, annuity puzzle. JEL codes: D91, G11, J17

We analyze the impact of risk and ambiguity aversion using a lifecycle recursive utility model. Both risk and ambiguity aversion are shown to reduce annuity demand and enhance bond holdings. We obtain this result using an intertemporal framework in which we can vary both risk and ambiguity aversion, while preserving preference monotonicity.

Introduction

Risk and ambiguity aversion are two central behavioral traits affecting economic lifecycle problems, such as saving and portfolio choices. A natural question is whether ambiguity and risk aversion have different impacts, or whether they generate qualitatively similar effects, especially when risk and ambiguity aversion are considered simultaneously. Although these traits appear to have certain similarities, with both traits aiming to model attitudes towards (objective or subjective) uncertainty, they have generally been studied in two separate strands of the economic literature. As pointed out in [START_REF] Guetlein | Comparative Risk Aversion in the Presence of Ambiguity[END_REF], the reason for the lack of joint analysis is that the problem is complicated and may in general lead to non-clear-cut results. Some insights have been given in static setups for portfolio choice problems (see e.g., [START_REF] Dow | Uncertainty Aversion, Risk Aversion, and the Optimal Choice of Portfolio[END_REF]Werlang 1992, Gollier 2011), and for self-insurance and self-protection questions (see among others, [START_REF] Treich | The Value of a Statistical Life under Ambiguity Aversion[END_REF][START_REF] Snow | Ambiguity Aversion and the Propensities for Self-Insurance and Self-Protection[END_REF][START_REF] Alary | The Effect of Ambiguity Aversion on Insurance and Self-protection[END_REF]. However, the question has never really been theoretically addressed in intertemporal problems.

In this paper, we succeed in deriving non-ambiguous results regarding the joint role of risk and ambiguity aversion in a lifecycle model. A lifecycle model with uncertain lifetime is a natural workhorse for such an analysis. Not only is mortality a large risk in life but life expectancies are highly heterogeneous, as measured by so-called life disparity. Furthermore, even if life disparity can be partly explained by socio-economic factors, or differences in health systems, significant unexplained factors remain [START_REF] Shkolnikov | Gini Coefficient as a Life Table Function: Computation from Discrete Data, Decomposition of Differences and Empirical Examples[END_REF][START_REF] Edwards | Inequality in Life Spans and a New Perspective on Mortality Convergence Across Industrialized Countries[END_REF][START_REF] Shkolnikov | Losses of Expected Lifetime in the United States and Other Developed Countries: Methods and Empirical Analyses[END_REF]. Mortality can therefore be seen as both risky and ambiguous.

Formally, we focus on a smooth ambiguity model à la [START_REF] Hayashi | Intertemporal Substitution and Recursive Smooth Ambiguity Preferences[END_REF], which nests some standard models such as that of [START_REF] Klibanoff | Recursive Smooth Ambiguity Preferences[END_REF] or the recursive specification of [START_REF] Epstein | Substitution, Risk Aversion, and the Temporal Behavior of Consumption and Asset Returns: A Theoretical Framework[END_REF]. We analyze the saving decisions of an agent who is both ambiguity and risk averse, and who may invest both in risk-free bonds and in annuities. Mortality is the sole risk faced by the agent and she can live at most for two periods. The key feature of our approach is that we additionally assume that the agent's preferences are monotone with respect to first-order stochastic dominance. This property enables us to jointly characterize the respective roles of ambiguity and risk aversion.

What does the monotonicity property imply? In a nutshell, monotonicity prevents agents from opting for dominated choices. More precisely, if two choices are available, with the first one yielding preferred outcomes in all circumstances, then an agent with monotone preferences will always prefer the first choice to the second. This property has already been studied in a risk setting. Bommier et al. (2017b) show that the only [START_REF] Kreps | Temporal Resolution of Uncertainty and Dynamic Choice Theory[END_REF] monotone preferences able to disentangle risk aversion and intertemporal elasticity of substitution are the socalled risk-sensitive preferences introduced by [START_REF] Hansen | Discounted Linear Exponential Quadratic Gaussian Control[END_REF]. These preferences have proved to be useful for characterizing the role of risk aversion for various consumption-saving problems. For instance, in a very general infinitehorizon setting, Bommier and Le Grand (2019) show that once monotonicity is imposed, risk aversion unambiguously increases precautionary savings.

While the impact of monotonicity has already been explored in a risk setting, we are not aware of any extension to an uncertainty setting, apart from the representation results provided in Bommier and Le Grand (2014a) and Bommier et al. (2017b). In particular, no application has been developed, leaving the practical implications of using monotone preferences in an ambiguity setting unclear. The current paper fills this gap, showing that monotone preferences can be used to derive clear-cut and intuitive predictions regarding the impact of risk and ambiguity aversion on savings and annuity purchases. More precisely, we prove that a higher ambiguity aversion, while maintaining risk aversion constant, leads to higher holdings of riskless bonds but smaller holdings of annuities. 1 There are different ways to define what is an increase in risk aversion. An increase in risk aversion in the sense introduced by Guetlein (2016) -which does not preserve ambiguity attitudes -typically leads to non-clear-cut results. However, a compensated or "net" change in risk aversion, where the ambiguity parameter is modified together with the risk aversion parameter so as to keep ambiguity attitudes unchanged, is shown to have unambiguous implications, with an impact on asset demands similar to that of an increase in ambiguity aversion. Our results are consistent with intuition, since in the presence of mortality uncertainty alone, the annuity -which only pays off in good states of the world, when the agent remains alive -is a risk-taking device. Unsurprisingly, agents exhibiting greater ambiguity aversion or greater (net) risk aversion are less willing to take a bet on their survival through annuity saving, and therefore invest fewer resources in annuities.

Interestingly, our results tend to indicate that ambiguity and risk aversion have, qualitatively speaking, very similar impacts. In other words, what matters is the degree of uncertainty aversion (encompassing both risk and ambiguity aversion), rather than whether uncertainty is objective or subjective. Of course, quantitative results are likely to depend on whether uncertainty is objective or subjective, and on whether agents exhibit greater or smaller aversion for objective and subjective uncertainty. Nonetheless, the overall message is that greater uncertainty aversion (be it greater risk aversion or greater ambiguity aversion) tends to reduce annuity purchases and enhance investments in bonds. Uncertainty aversion in particular appears to be a natural candidate for explaining the low annuitization level observed in the data (see [START_REF] Johnson | Annuitized Wealth at Older Ages: Evidence from the Health and Retirement Study[END_REF] for empirical evidence in the US). This confirms the early findings of Bommier and Le Grand (2014b), who examined risk aversion, or those of d [START_REF] Albis | Ambiguous Life Expectancy and the Demand for Annuities[END_REF], who focused on ambiguity aversion in a static, one-period model.

To the best of our knowledge, this is the first paper to derive clear-cut results regarding the joint impact of risk and ambiguity aversion in an intertemporal framework. Our article obviously connects to the literature that discusses the roles of risk and ambiguity aversion separately. In the risk setting, there is an abundant literature on precautionary savings (see, among others, [START_REF] Drèze | Consumption Decisions under Uncertainty[END_REF][START_REF] Kimball | Precautionary Savings in the Small and in the Large[END_REF][START_REF] Bleichrodt | Saving under Rank-Dependent Utility[END_REF][START_REF] Courbage | Precautionary Saving in the Presence of Other Risks[END_REF][START_REF] Kimball | Precautionary Saving and Consumption Smoothing Across Time and Possibilities[END_REF][START_REF] Jouini | On Multivariate Prudence[END_REF][START_REF] Nocetti | Robust Comparative Statics of Risk Changes[END_REF]) and on annuity choices [START_REF] Yaari | Uncertain Lifetime, Life Insurance, and the Theory of the Consumer[END_REF][START_REF] Davidoff | Annuities and Individual Welfare[END_REF][START_REF] Lockwood | Bequest Motives and the Annuity Puzzle[END_REF][START_REF] Pashchenko | Accounting for Non-Annuitization[END_REF], among many others). The role of risk aversion is, however, not often studied because most of the literature relies on the standard time-additive setup, which makes it impossible to isolate the role of risk aversion, since intertemporal elasticity of substitution and risk aversion are intertwined. Among the exceptions, we find papers that use recursive frameworks, such as van der Ploeg (1993), [START_REF] Weil | Precautionary Savings and the Permanent Income Hypothesis[END_REF], [START_REF] Kimball | Precautionary Saving and Consumption Smoothing Across Time and Possibilities[END_REF], and several others. Nevertheless, most of these contributions rely on non-monotone preferences (notably those using the most popular Epstein-Zin specification with an intertemporal elasticity of substitution different from 1), with recursivity and monotonicity being combined in very few papers, including van der Ploeg (1993), [START_REF] Tallarini | Risk-Sensitive Real Business Cycles[END_REF] and Bommier andLe Grand (2014b, 2019). None of these articles feature ambiguity aversion.

In an ambiguity setting, intertemporal problems are typically addressed using recursive extensions of standard static ambiguity models, though without having enough flexibility to change both risk and ambiguity aversion. We note, among others, the analyses of [START_REF] Osaki | Precautionary Saving and Ambiguity[END_REF], [START_REF] Berger | Precautionary Saving and the Notion of Ambiguity Prudence[END_REF], and [START_REF] Kajii | Precautionary Saving with Changing Income Ambiguity[END_REF], who examine the precautionary savings of ambiguity averse agents, or the study of [START_REF] Collard | Ambiguity and the Historical Equity Premium[END_REF], who investigate whether ambiguity aversion can explain historical values of the equity premium. All of these papers focus on ambiguity aversion, while restricting their analyses to models that reduce to the standard time-additive model when uncertainty is purely objective (in cases where there is no ambiguity). In other words, they retain the lack of flexibility of the standard time-additive model, which makes it impossible to explore the role of risk aversion. An exception is [START_REF] Peter | Revisiting Precautionary Saving under Ambiguity[END_REF], who considers a more flexible framework, relying on a non-separable, two-period utility function, although the author does not investigate the role of risk aversion. The current paper introduces flexibility by using the recursive framework of [START_REF] Hayashi | Intertemporal Substitution and Recursive Smooth Ambiguity Preferences[END_REF], a route also followed by more quantitatively oriented papers (see e.g., [START_REF] Backus | Risk and Ambiguity in Models of Business Cycles[END_REF].

Our contribution is notable for its use of monotone preferences, which in addition to being an intuitive assumption, affords great tractability and the ability to derive formal results.

Risk and ambiguity aversions in intertemporal frameworks

The recursive smooth ambiguity model

To study the effect of risk and ambiguity aversion on the optimal decisions of an agent who faces an ambiguous mortality risk, we adopt the recursive smooth ambiguity utility model axiomatized by [START_REF] Hayashi | Intertemporal Substitution and Recursive Smooth Ambiguity Preferences[END_REF], in which Seo (2009)'s construction of a static smooth ambiguity model using [START_REF] Anscombe | A Definition of Subjective Probability[END_REF] acts is embedded into an infinite horizon setting.

Let Ω be the finite state space and let X be the set of consequences. The set of Anscombe-Aumann acts is G 0 ≡ ∆ (X ) Ω , where ∆ (X ) denotes the set of Borel probability measures defined on the Borel subsets of X . We refer to [START_REF] Hayashi | Intertemporal Substitution, Risk Aversion and Ambiguity Aversion[END_REF] for the formal construction of the domain of compound lottery-acts, which are defined as the subset of dynamically coherent acts in the product space ∞ t=0 G t , where the spaces G t are defined by the recursion G

t = ∆ (X × G t-1 ) Ω .
The representation result of [START_REF] Hayashi | Intertemporal Substitution and Recursive Smooth Ambiguity Preferences[END_REF] states that the following exist: an aggregator W : X × R → R, continuous and increasing in its second argument; two continuous and strictly increasing functions ψ A , ψ R : R → R; and a probability measure µ ∈ ∆ ∆ (Ω) over the set of objective distributions, such that the utility of (x, g) ∈ X × G is given by:

V (x, g) = W   x, ψ -1 A ∆(Ω) ψ A • ψ -1 R ω∈Ω π (ω) (1)
× C×G ψ R V (x , g ) g (ω) (dc , dg ) µ(dπ)      .

Preferences exhibit ambiguity aversion if ψ

A • ψ -1 R is concave.

Monotone preferences

For our analysis, we will additionally require preferences to be monotone with respect to first order stochastic dominance. Loosely speaking, monotonicity property means that if an agent prefers the outcomes of a given action to those of another action in all states of the world, she should always prefer the former action to the latter. This property is a consistency requirement between preferences for deterministic outcomes, subjective beliefs, and preferences in the presence of risk and ambiguity.

Monotonicity has already been analyzed in risk settings. It has straightforward implications for saving behaviors. Consider, for instance, an agent who is likely to live for one or two periods. With monotone preferences, her savings in the presence of mortality risk will be bound by her savings when she is sure to live for either one or two periods. These savings will, in particular, be smaller than her savings when she is sure to live for two periods. With non-monotone preferences, and for certain parameterizations, the agent is likely to save more in the presence of mortality risk than she would save knowing that she is sure to live for two periods.2 Aside from this implication regarding saving behaviors, monotonicity imposes strong restrictions on preference representations, as shown in Bommier et al. (2017b). This latter contribution shows, in particular, that risk-sensitive preferences are the only [START_REF] Kreps | Temporal Resolution of Uncertainty and Dynamic Choice Theory[END_REF] monotone preferences able to disentangle risk aversion and the elasticity of intertemporal substitution. Because of monotonicity, these preferences may yield clear-cut insights into the role of risk aversion (see Bommier and Le Grand 2019 for precautionary savings).

In the setting of [START_REF] Hayashi | Intertemporal Substitution and Recursive Smooth Ambiguity Preferences[END_REF], monotonicity requires the representation in (1) to have an affine aggregator W and functions ψ R and ψ A to be of the "constant absolute risk aversion" kind. See Bommier and Le Grand (2014a) for a formal proof. The specification (1) can then be written as:

V (x, g) = f (x) - β k A log   ∆(Ω) exp k A k R log ω∈Ω π (ω) e -k R V µ(dπ)   , (2)
where f : X → R + is the instantaneous utility function. The parameters k A and k R drive ambiguity and risk aversion. Formally, two agents are comparable in terms of ambiguity aversion if they only differ by the parameter k A , the agent with greater k A being more ambiguity averse. Two agents are comparable in terms of risk aversion if they only differ by the parameter k R , the agent with greater k R being more risk averse. It is therefore natural to refer to k A as being the ambiguity aversion parameter, and to k R as being the risk aversion parameter.

However, as emphasized by [START_REF] Guetlein | Comparative Risk Aversion in the Presence of Ambiguity[END_REF], the smooth model does not offer a straightforward separation between ambiguity and risk attitudes. In particular, whether the agent is ambiguity averse or ambiguity lover depends on whether the ratio k A k R is greater or smaller than one. Ambiguity neutrality is obtained when this ratio is equal to one. It follows that increasing the risk aversion parameter k R while keeping k A constant will change ambiguity attitudes. For example, an agent who is more risk averse than an agent who is ambiguity neutral will necessarily be ambiguity lover (see discussion in Section II of Guetlein).

To avoid such shortcomings, [START_REF] Guetlein | Comparative Risk Aversion in the Presence of Ambiguity[END_REF] suggests considering a simultaneous increase in both risk aversion and ambiguity aversion that leaves ambiguity attitudes unchanged (see Section III of her paper). In the general framework of [START_REF] Guetlein | Comparative Risk Aversion in the Presence of Ambiguity[END_REF], such joint increases can be rather complex, involving different transformations of the risk aversion and ambiguity aversion functions. Under the assumption of preference monotonicity considered in the current paper, the situation is, however, much simpler. Increasing uncertainty aversion -to use Guetlein's terminology -while preserving ambiguity attitudes, is obtained by increasing both k A and k R , while keeping the ratio k A k R constant. To avoid confusion, we will introduce the terminology "net" vs. "gross" to distinguish two possible forms of the increase in risk aversion. By definition, an increase in "gross risk aversion" involves increasing k R while keeping k A constant. An increase in "net risk aversion" involves increasing k A and k R , while keeping keeping the ratio k A k R unchanged, so as to preserve ambiguity attitudes. Here, the adjective "net" refers to an approach where risk aversion would be defined residually by looking at the difference (in fact the ratio here) between uncertainty aversion and ambiguity aversion.

3 Results in a two-period framework

Preference specifications

We use the above setting to study the effect of risk and ambiguity aversion on the optimal annuitization of an agent who faces an ambiguous mortality risk.

There are two periods, indexed by t = 1, 2. The agent is alive in the first period, and can be either dead or alive in the second period. When alive, the agent consumes, when dead she may bequeath wealth to her heirs. The formal connection between our two-period setting and the infinite horizon setting of [START_REF] Hayashi | Intertemporal Substitution and Recursive Smooth Ambiguity Preferences[END_REF] can be made by assuming that the first two periods are always followed by an infinite sequence of periods where the agent is dead and bequeaths nothing.

In the second period, the state of the agent can be described by two variables:

(i) a dummy variable d ∈ {alive, dead}, indicating whether the agent is alive or dead, and (ii) a real number indicating how much she consumes (if alive) or bequeaths (if dead). Thus, formally, the set of consequences for the second period is X = {alive, dead} × R + . Because of our assumption that the agent is always alive in period 1, the first-period set of consequences is simply {alive} × R + and the domain of choice is {alive} × R + × ∆ (X ) Ω , where Ω is a finite subjective state-space.

This can be simplified since we assume that mortality is the only source of uncertainty. The agent's subjective prior can then be described by a probability distribution over survival probabilities, i.e. a list of pairs

(m i , p i ) ∈ [0, 1] 2 for i = 1, . . . , n such that n i=1 m i = 1.
The interpretation is that there are n subjective states and that the agent (subjectively) associates the probability m i with the state i ∈ {1, . . . , n} that corresponds to a survival probability equal to p i (and a mortality probability equal to 1 -p i ). A setting with no ambiguity is one in which m i = 1 for some i and m j = 0 for all j = i: The agent is certain to survive with probability p i . Conversely, ambiguity occurs whenever there are two states i and j with p i = p j and m i , m j = 0.

The instantaneous utility function f that appears in specification (2) is formally defined on X = {alive, dead} × R + . In other words, f :

(d, x) ∈ {alive, dead} × R + → f (d, x) ∈ R.
To simplify the notation, rather than introducing the dummy variable d to indicate whether the agent is alive or dead, we will simply use different letters c and w to indicate whether it refers to a level of consumption or a bequest level. A number c ∈ R + must therefore be interpreted as an element (alive, c) ∈ X , while a number w ∈ R + must be interpreted as an element (dead, w) ∈ X . We will introduce the notation u(c) = f (alive, c) for the instantaneous utility of consumption (if alive) and v(w) = f (dead, w) for the instantaneous utility of bequest (if dead). As is standard, functions u and v are both assumed to be twice continuously differentiable, strictly increasing, and concave. In representation (2), the instantaneous utility f is defined up to a constant and we can assume without loss of generality that v(0) = 0. Note that once this normalization has been adopted, adding or subtracting a constant to u cannot then be seen as mere normalization.

The utility gap u(c) -v(w) in fact measures how much the agent values being alive and consuming c versus being dead and bequeathing wealth w. The larger the utility gap, the more attractive being alive and consuming becomes compared to being dead and bequeathing. The baseline situation corresponds to the one where the agent exhibits no altruism towards her heirs, that is to the case in which v is constant (and thus equal to 0). More generally, the utility gap u -v closely relates to the value of life concept, which quantifies how much an agent is willing to pay for increasing her survival probabilities. As shown in Bommier et al. (2017b), the sign of the value of life is crucial for understanding the role of risk aversion on savings. There is a wide empirical literature on value of life estimates, both from academics (see for instance Using the above notation and representation (2), the utility of an agent who consumes c 1 in period 1 and consumes c 2 or bequeaths w in period 2 (depending on whether she survives or not) is given by:

U (c 1 , c 2 , w) = u(c 1 ) - β k A log   n i=1 m i p i e -k R u(c 2 ) + (1 -p i )e -k R v(w) k A k R   . (3)
The utility specification U (c 1 , c 2 , w) in equation ( 3) is the objective function that the agent maximizes, subject to the budget constraints that will be introduced later on. We will derive comparative statics results related to the ambiguity and risk aversion parameters k A and k R , while assuming that individuals' beliefs (the m i and p i ) are unchanged.

Particular cases

The general preference representation (3) embeds several particular cases that we describe below.

No ambiguity. The no-ambiguity case corresponds to m i 0 = 1 for some i 0 and m i = 0 for i = i 0 . The preference representation (3) becomes:

U (c 1 , c 2 , w) = u(c 1 ) - β k R log p i 0 e -k R u(c 2 ) + (1 -p i 0 )e -k R v(w) , ( 4
)
which corresponds to the risk-sensitive preferences of [START_REF] Hansen | Discounted Linear Exponential Quadratic Gaussian Control[END_REF].

In particular, the parameter k A no longer plays a role and risk aversion is driven by the parameter k R , where more risk averse agents correspond to larger values of the parameter k R .

The case k R = 0, can be obtained by taking the limit of expression (4) when k R → 0, and corresponds to the the standard additive model where

U (c 1 , c 2 , w) = u(c 1 ) + β p i 0 u(c 2 ) + (1 -p i 0 )v(w) .
Ambiguity neutrality. Ambiguity neutrality corresponds to k A = k R . The utility expression (3) becomes:

U (c 1 , c 2 , w) = u(c 1 ) - β k R log pe -k R u(c 2 ) + (1 -p)e -k R v(w) , (5)
where p = n i=1 m i p i is the average mortality probability. Formally, this setup is similar to the no-ambiguity case and expression (5) looks very much like expression (4). Again, when k R = 0, we fall back on the additive model.

Temporal risk neutrality.

The case k R = 0 implies temporal risk neutrality.

By continuity, for k R → 0 from expression (3), we deduce that the utility expression becomes:

U (c 1 , c 2 , w) = u(c 1 ) - β k A log   n i=i m i exp -k A p i u(c 2 ) + (1 -p i )v(w)   .
This fits into the [START_REF] Klibanoff | Recursive Smooth Ambiguity Preferences[END_REF] recursive framework and exactly corresponds to the specification in [START_REF] Collard | Ambiguity and the Historical Equity Premium[END_REF].

No objective probabilities. The last case corresponds to a setup where all information is subjective. In other words, a situation where, for all i, we have either p i = 0 or p i = 1. This situation reduces to a two (subjective)-state model,

where p 1 = 0 occurs with subjective probability m 1 and p 2 = 0 with subjective probability m 2 = 1 -m 1 . We then have: w) , similar to expressions (4) and ( 5), with k A instead of k R .

U (c 1 , c 2 , w) = u(c 1 ) - β k A log m 1 e -k A u(c 2 ) + (1 -m 1 )e -k A v(

The agent program

We consider an ambiguity-and-risk-averse agent, whose preferences are assumed to be represented by the utility function defined in equation ( 3). In the first period, the agent is endowed with an initial wealth W > 0. She has no other source of income during the two periods but can transfer consumption from the first period to the second through savings in a bond and in an annuity. These savings are denoted by s and a, respectively, and the first period budget constraint is simply

W = c 1 + s + a.
The bond pays off the riskless gross interest rate r > 0 in the second period, regardless of whether the agent is alive or not. Bond payoffs, in particular, can be bequeathed to the agent's heirs. The annuity pays off the gross rate r a > r, which is assumed to be higher than the riskless rate. Annuity payoffs are only made in cases of survival. This means that the second period budget constraints are c 2 = r a a + rs when the agent lives for two periods and consumes the outcomes of her savings and w = rs when the agent dies and bequeaths her bond savings.

We make two observations regarding the annuity return r a . First, in our ambiguity setting, the notion of an actuarially fair annuity is not a very relevant concept. An objective survival probability can still be defined, as can an objective definition of actuarially fair annuities. However, from the agent's perspective, what actually matters is a subjective notion of fairness, which is not generally unequivocally defined due to the ambiguity setting. We therefore define the annuity return simply by using the gross rate r a , higher than r, instead of defining it using the riskless interest rate and some survival probability. The second observation is that the annuity return rate r a is not affected by ambiguity and is assumed to be perceived as constant by the agent. This implicitly means that the return r a actually comes from pooling the heterogeneous mortality risk, such that the agent cannot infer her objective survival probability from the gross rates r a and r. Another (nonexclusive) possibility would be to assume that the annuity return embeds a fee that cannot be observed by the agent.

Using the previous notation, the agent's program can therefore be written as:

max (s,a)∈R 2 u(W -a -s) - β k A log   n i=1 m i p i e -k R u(raa+rs) + (1 -p i )e -k R v(rs) k A k R   , (6) s.t. W -a -s ≥ 0, s ≥ 0, a ≥ 0, ( 7
)
where we exclude negative consumption levels, borrowings, and annuity shortsellings.

The role of risk and ambiguity aversion

The impact of k A and k R on saving choices is summarized in the following proposition, whose proof is given in the Appendix.

Proposition 1 Consider the consumption-saving program of equations ( 6) and ( 7)

with k A ≥ k R > 0. We denote by a(k A , k R ) ≥ 0 and s(k A , k R ) ≥ 0 the optimal
savings in annuities and bonds, respectively.

We assume that the optimum is characterized by a positive value of life, i.e., formally, u(r a a(k

A , k R ) + rs(k A , k R )) > v(rs(k A , k R )).
We distinguish three cases (besides the trivial case a(k

A , k R ) = s(k A , k R ) = 0). 1. Case a(k A , k R ) > 0 and s(k A , k R ) > 0.
(a) Ambiguity aversion. We have:

∂a(k A ,k R ) ∂k A k R < 0, ∂s(k A ,k R ) ∂k A k R > 0, as well as ∂(raa(k A ,k R )+rb(k A ,k R )) ∂k A k R < 0.
(b) Net risk aversion. We have:

∂a(k A ,k R ) ∂k R k A /k R < 0, ∂s(k A ,k R ) ∂k R k A /k R > 0, ∂(raa(k A ,k R )+rb(k A ,k R )) ∂k R k A /k R < 0.

Case a(k

A , k R ) > 0 and s(k A , k R ) = 0: ∂a(k A ,k R ) ∂k A k R < 0 and ∂a(k A ,k R ) ∂k R k A /k R < 0.

Case a(k

A , k R ) = 0 and s(k A , k R ) > 0: ∂s(k A ,k R ) ∂k A k R and ∂s(k A ,k R ) ∂k R k A /k R have the same sign as v rs(k A , k R ) -u rs(k A , k R ) .
The proposition characterizes the role of ambiguity and net risk aversion for an ambiguity-and-risk averse agent (k A ≥ k R > 0). It distinguishes three cases depending on whether saving is constrained to one instrument. The interior case, corresponding to unconstrained annuity and bond holdings, is reported in case 1.

Varying the ambiguity aversion parameter k A -while keeping the risk aversion parameter k R fixed -has an unambiguous impact on security holdings. A larger ambiguity aversion diminishes the demand for annuities but increases the demand for bonds. The overall impact is such that second period consumption (equal to the quantity r a a(k A , k R ) + rb(k A , k R )) also diminishes when ambiguity aversion rises.

The only condition required to derive these results is that, at the optimum, the utility of being alive in the second period and consuming is greater than the utility of being dead and bequeathing one's wealth. In other words, the condition simply means that life is worth living, or that the value of life is positive. As already mentioned, despite some measurement heterogeneity, there is plenty of empirical evidence that the value of life is large and positive. This assumption is therefore reasonable and not very restrictive.

The role of the "net" risk aversion parameter is obtained by increasing k R and k A while maintaining a constant ratio k A /k R . We find that the impact of an increase in net risk aversion is qualitatively similar to that of an increase in ambiguity aversion.

Case 2, which corresponds to constrained bond savings, is very similar to case 1. Since only one saving instrument is available and no portfolio choice is possible, the result is much simpler to state. A higher ambiguity aversion -or a higher net risk aversion -implies smaller annuity holdings.

Case 3 corresponds to constrained annuity holdings. The result, be it for the role of ambiguity aversion or the role of net risk aversion, depends on the sign of the difference between the marginal utility of survival and the marginal utility of bequest. The intuition is as follows. An increase in ambiguity aversion -or in net risk aversion -translates into the willingness to increase lifetime utility in the bad state (i.e., death). With monotone preferences, this necessarily reduces the lifetime utility in the good state (i.e., survival) and diminishes the dispersion in lifetime utilities. In the absence of an annuity holding, the dispersion in lifetime utilities amounts to β(u(rs) -v(rs)), which is positive because the value of life is positive.

The way in which the riskless saving s should be varied to reduce the dispersion in lifetime utilities therefore depends on whether u(rs) -v(rs) is increasing or decreasing with s and thus on the sign of the derivative u (rs) -v (rs). When the marginal utility in the good state is higher than the marginal utility in the bad state (u (rs) -v (rs) > 0), reducing the dispersion in lifetime utilities will involve smaller savings. Conversely, if u (rs) -v (rs) < 0, an increase in ambiguity aversion or in net risk aversion will increase savings.

In the absence of a bequest motive (v = 0), the sign of the derivatives in case 3 is clear, and an increase in either net risk aversion or ambiguity aversion diminishes both savings and second period consumption. With non-trivial bequest motives, a very common specification (up to a normalization constant) is v(w) = θ(u(w + w) -u(w)), where θ ≥ 0 reflects the intensity of the altruistic motive, and w > 0 makes a bequest a luxury good. Furthermore, the intertemporal elasticity of substitution is also often assumed to be constant, with u (c) = c -σ .3 In such a case, the sign of u (rs) -v (rs) is not clear-cut: it is typically positive for small values of s and becomes negative for larger values. The impact of ambiguity and net risk aversion will then depend on this sign, as shown in the proposition.

Discussion

We have proved that the comparative statics of ambiguity and net risk aversion yield non-ambiguous results in an ambiguity setting à la Anscombe-Auman. A larger ambiguity aversion or a larger net risk aversion means a smaller demand for annuities and a larger demand for riskless bonds. To obtain the intuition, note that annuities are uncertain assets that pay off in the good state of the world (i.e., when the agent lives a long life) and pay nothing in the bad state (i.e., in case of an early death). By analogy with finance, we can therefore consider that an annuity has the fundamental properties of a pro-cyclical asset. It is then quite natural that uncertainty aversion, be it ambiguity or risk aversion, reduces investment in such an asset. We would of course have found a converse result if we had considered hedging assets (such as life-insurance products that only pay in the case of death).

While our results may look intuitive, we emphasize that they go against the common claim that annuities being insurance products, the demand for annuities should increase with uncertainty aversion. Since mortality risk makes preferences state-dependent, the link between insurance demand and risk aversion is not that obvious. As our results indicate, there are cases where the demand for insurance may decrease (and not increase) with risk aversion.

Preference monotonicity is the key ingredient as it enables us to highlight the fundamental assumptions required for our analysis, and in particular the assumptions related to the value of life. Because of the monotonicity assumption, we may view the agent's decision as being a trade-off between welfares in different states (without monotonicity the agent may want to reduce welfare in all states).

Uncertainty aversion then has a fairly clear role, dictating how much weight to put on bad states compared to good ones. To derive the impact of an increase in uncertainty aversion it is therefore necessary to properly identify good and bad states -or in our framework to identify whether survival should be viewed as a good or a bad realization. In the tradition of revealed preferences, this involves looking at the literature on endogenous mortality, which clearly indicates that the value of life is positive.

A final take-home message is that to make predictions regarding the impact of uncertainty aversion in the presence of an exogenous, non-monetary background risk (mortality risk in our paper, although this could also be health risk, for instance), it is necessary to use information that can only be gleaned from decisions where the degree of background risk is endogenous.

Appendix: Proof of Proposition 1

A Proof: Interior case a, s > 0

The program can be written as:

max (s,a)∈R 2 + u(W -a -s) - β k A log   n i=1 m i p i e -k R u(raa+rs) + (1 -p i )e -k R v(rs) k A k R   .
Assuming an interior equilibrium, the first-order conditions are:

u (W -a-s) = βr a u (r a a+rs) n i=1 m i p i p i + (1 -p i )e k R( u(raa+rs)-v(rs)) k A k R -1 n i=1 m i p i + (1 -p i )e k R( u(raa+rs)-v(rs)) k A k R , (8) u (W -a-s) = βru (r a a+rs) n i=1 m i p i p i + (1 -p i )e k R( u(raa+rs)-v(rs)) k A k R -1 n i=1 m i p i + (1 -p i )e k R( u(raa+rs)-v(rs)) k A k R (9) +βrv (rs)e k R( u(raa+rs)-v(rs)) n i=1 m i (1 -p i ) p i + (1 -p i )e k R( u(raa+rs)-v(rs)) k A k R -1 n i=1 m i p i + (1 -p i )e k R( u(raa+rs)-v(rs)) k A k R
.

We introduce the following notation:

η i = p i + (1 -p i )e k R( u(raa+rs)-v(rs)) , ( 10
)
as well as the expectation E m [•] and the covariance Cov m (when well-defined):

E m [X] = n i=1 m i η k A k R i X i n i=1 m i η k A k R i , ( 11
)
Cov m (X, Y ) = E m [XY ] -E m [X]E m [Y ], (12)
Using this notation, the first-order condition (8) and ( 9) become:

u (W -a -s) = βr a u (r a a + rs)E m p η , ( 13
) 1 - r r a u (W -a -s) = βrv (rs)e k R( u(raa+rs)-v(rs)) E m 1 -p η . ( 14
)

A.1 Impact of k A

A.1.1 First-order condition (13)

Computing the derivative of ( 13) with respect to k A yields, using the notation η i defined in (10):

-u (W -a -s) ∂a ∂k A + ∂s ∂k A = βr a r a ∂a ∂k A + r ∂s ∂k A u (r a a + rs) n i=1 m i p i η k A k R -1 i n i=1 m i η k A k R i + βr a u (r a a + rs)   u (r a a + rs) r a ∂a ∂k A + r ∂s ∂k A -v (rs)r ∂s ∂k A   e k R( u(raa+rs)-v(rs)) ×     (k A -k R ) n i=1 m i p i (1 -p i )η k A k R -2 i n i=1 m i η k A k R i -k A n i=1 m i (1 -p i )η k A k R -1 i n i=1 m i p i η k A k R -1 i ( n i=1 m i η k A k R i ) 2     + βr a k R u (r a a + rs)     n i=1 m i p i log (η i ) η k A k R -1 i n i=1 m i η k A k R i - n i=1 m i p i η k A k R -1 i n i=1 m i η k A k R i n i=1 m i log (η i ) η k A k R i n i=1 m i η k A k R i     .
Introducing the operator E m of equation ( 11), we obtain after dividing both sides of equality by βr a u (r a a + rs) and using ( 13):

-E m p η u (W -a -s) u (W -a -
  e k R( u(raa+rs)-v(rs)) ×   (k A -k R )E m p(1 -p) η 2 -k A E m 1 -p η E m p η   + Cov m p η , log(η) k R .

A.1.2 First-order condition (14)

We now compute the derivative of equation ( 14) with respect to k A . The computation is similar to the one for (13) and we get:

- u (W -a -s) u (W -a -s) E m 1 -p η ∂a ∂k A + ∂s ∂k A = r ∂s ∂k A v (rs) v (rs) E m 1 -p η (16) -   u (r a a + rs) r a ∂a ∂k A + r ∂s ∂k A -v (rs) r ∂s ∂k A   ×   (k A -k R )E m p(1 -p) η 2 -k A E m 1 -p η E m p η   + Cov m 1-p η , log(η) k R .

A.1.3 Conclusion

With matrix notation, equations ( 15) and ( 16) can be rewritten as: (17) with M = (M ij ) 1≤i,j≤2 and:

M     r a ∂a ∂k A r ∂s ∂k A     = 1 k R     Cov m p η , log(η) Cov m 1-p η , log(η)     ,
M 11 = - 1 r a E m p η u (W -a -s) u (W -a -s) -E m p η u (r a a + rs) u (r a a + rs) +e k R( u(raa+rs)-v(rs)) × u (r a a + rs)   k A E m 1 -p η E m p η -(k A -k R )E m p(1 -p) η 2   , M 12 = - 1 r E m p η u (W -a -s) u (W -a -s) -E m p η u (r a a + rs) u (r a a + rs) +e k R( u(raa+rs)-v(rs)) × u (r a a + rs) -v (rs)   k A E m 1 -p η E m p η -(k A -k R )E m p(1 -p) η 2   , M 21 = - 1 r a E m 1 -p η u (W -a -s) u (W -a -s) -u (r a a + rs)   k A E m 1 -p η E m p η -(k A -k R ) E m p(1 -p) η 2   , M 22 = - 1 r E m 1 -p η u (W -a -s) u (W -a -s) -E m 1 -p η v (rs) v (rs) -u (r a a + rs) -v (rs)   k A E m 1 -p η E m p η -(k A -k R )E m p(1 -p) η 2   .
We also define:

κ = k A E m 1 -p η E m p η -(k A -k R ) E m p(1 -p) η 2 = k R E m p(1 -p) η 2 -k A Cov m 1 -p η , p η . ( 18
)
Since r a ≥ r, we deduce that det M > 0 and that the matrix M is invertible.

We deduce from ( 17) the expression of partial derivatives ∂s ∂k A and ∂a ∂k A .

    r a ∂a ∂k A r ∂s ∂k A     = 1 k R det M     M 22 Cov m p η , log(η) -M 12 Cov m 1-p η , log(η) -M 21 Cov m p η , log(η) + M 11 Cov m 1-p η , log(η)     . (19) Partial derivative ∂s ∂k A . Equality (19) yields: k R det M r ∂s ∂k A = κu (r a a + rs) (20) ×   Cov m p η , log(η) +e k R( u(raa+rs)-v(rs)) Cov m 1 -p η , log(η)   - u (r a a + rs) u (r a a + rs) E m p η Cov m 1 -p η , log(η) - 1 r a u (W -a -s) u (W -a -s)   -E m 1 -p η Cov m p η , log(η) +E m p η Cov m 1 -p η , log(η)   .
From the definition (10), we deduce:

p η = e k R( u(raa+rs)-v(rs)) e k R( u(raa+rs)-v(rs)) -1 1 η - 1 e k R( u(raa+rs)-v(rs)) -1 , (21) 1 -p η = 1 e k R( u(raa+rs)-v(rs)) -1 - 1 e k R( u(raa+rs)-v(rs)) -1 1 η . ( 22
)
These two values imply the following equality:

Cov m p η , log(η) + e k R( u(raa+rs)-v(rs)) Cov m 1 -p η , log(η) (23) = e k R( u(raa+rs)-v(rs)) e k R( u(raa+rs)-v(rs)) -1   Cov m 1 η , log(η) -Cov m 1 η , log(η)   = 0
and the determinant expression (10) simplifies into:

k R det M r ∂s ∂k A = - u (r a a + rs) u (r a a + rs) E m p η Cov m 1 -p η , log(η) (24) - 1 r a u (W -a -s) u (W -a -s)   -E m 1 -p η Cov m p η , log(η) +E m p η Cov m 1 -p η , log(η)   .
Definitions ( 21) and ( 22

A.2 Impact of k R

We will compute the derivatives of ( 13) and ( 14) with respect to k R , while keeping the ratio k A /k R unchanged. For the sake of clarity, we will denote such a derivative of a function f as ∂f ∂k R instead of   e k R( u(raa+rs)-v(rs))

κ k R u(r a a + rs) -v(rs) e k R( u(raa+rs)-v(rs)) .

Similarly, computing the derivative of ( 14) yields:

- In matrix notation, we deduce from ( 26) and ( 27): The matrix M is defined in ( 17) and det M > 0, κ > 0 and u(r a a + rs) -v(rs) > 0.

A.2.1 Partial derivative ∂s ∂k R .

Using the expressions of M 11 and M 21 , we obtain from (28):

Λr ∂s ∂k R = 1 r a E m 1 -p η - u (W -a -s) u (W -a -s) e k R( u(raa+rs)-v(rs)) (29) + E m p η - 1 r a
u (W -a -s) u (W -a -s) -u (r a a + rs) u (r a a + rs) , which unambiguously yields: ∂s ∂k R > 0.

B.1 Role of k A

Computing the derivative of (31) with respect to k A (with k R constant) yields after some manipulation:

- -κ βr k R u (rs) -v (rs) u(rs) -v(rs) e k R( u(rs)-v(rs)) , which implies that ∂s ∂k R has the same sign as u (rs) -v (rs).

C Proof: Corner case s = 0

The annuity choice is determined by the following first-order condition -which is (8) with a = 0: )

u (W -a) = βr a u (r a a) n i=1 m i p i p i + (1 -p i )e k R( u(raa)-v(0)) k A k R -1 n i=1 m i p i + (1 -p i )e k R( u(raa)-v(0)) k A k R . ( 32

  Viscusi and Aldy 2003 for a survey of value-of-life estimates throughout the world) and from institutions (see the US Environmental Protection Agency Office of Air and Radiation 2011 report, for instance). Despite the heterogeneity in estimates, there is a strong consensus that the value of life is positive and large. For instance, a reasonable estimate for the US is around 7 million US dollars.

  ) imply that Cov m p η , log(η) < 0 and Cov m 1-p η , log(η) > 0. Since det M, k R > 0, we obtain from (24) that: ∂s ∂k A > 0.Partial derivative ∂a ∂kA . Using equation (23) stating that Cov m p η , log(η) + e k R( u(raa+rs)-v(rs)) Cov m 1-p η , log(η) = 0, equality (19) yields: k R det M r a ∂a ∂k A = u (r a a + rs) u (r a a + rs)As for (24), Cov m p η , log(η) < 0 and Cov m 1-p η , log(η) > 0, which implies with k R det M r a > 0 that: ∂a ∂k A < 0.Sum of partial derivatives r ∂s ∂kA + r a ∂s ∂k A . We deduce from (20) and (25):η , log(η) > 0. We deduce: r ∂s ∂k A + r a ∂a ∂k A < 0.

  rs)e k R( u(rs)-v(rs)) (η) < 0, this proves that ∂s ∂k A has the same sign as u (rs)-v (rs).B.2 Role of k RComputing the derivative of (31) with respect to k R (with k A /k R constant) yields: k R( u(rs)-v(rs))

Electronic copy available at: https://ssrn.com/abstract=3769365

This result holds for interior solutions, where agents purchase positive quantities of bonds and annuities. We also derive slightly different results for corner solutions, where either annuity or bond holdings hit non-negativity constraints and are equal to zero.

See Bommier et al. (2017a) for a detailed example.

Such calibrations can be found, for instance, in De Nardi (2004), De Nardi et al. (2010),[START_REF] Ameriks | The Joy of Giving or Assisted Living? Using Strategic Surveys to Separate Bequest and Precautionary Motives[END_REF][START_REF] Lockwood | Bequest Motives and the Annuity Puzzle[END_REF], or[START_REF] Pashchenko | Accounting for Non-Annuitization[END_REF], among many others.

Le Grand gratefully acknowledge support from the Swiss-Re Foundation and the ETH Zurich Foundation.

We compute the determinant of the matrix M :

A.2.2 Partial derivative ∂a ∂k R .

The expressions of M 12 and M 22 imply using ( 28)

which is negative and implies that: ∂a ∂k R < 0.

A.2.3 Sum of partial derivatives r

Summing equations ( 29) and ( 30) leads to:

which implies since r a ≥ r that: r a ∂a ∂k R + r ∂s ∂k R < 0.

B Proof: Corner case: a = 0

The annuity choice is zero and the riskless saving choice is determined by the following first-order condition (which is ( 9) with a = 0):

C.1 Role of k A

Computing the derivative of (32) with respect to k A (with k R constant) yields:

which implies since κ > 0 and Cov m p η , log(η) < 0 that ∂a ∂k A < 0.

C.2 Role of k R

Computing the derivative of (32) with respect to k R (with k R /k A constant) yields:

-u (W -a -s) ∂a ∂k R = βr a r a ∂a ∂k R u (r a a)E m p η (34)

-κβr a u (r a a) 2 r a ∂a ∂k R e k R( u(raa)-v( 0))

-κ βr a k R u (r a a) 2 e k R( u(raa)-v( 0)) , which implies since κ > 0: ∂a ∂k R < 0.


Diego Ferrando
email: diego.ferrando@coria.fr

Chetankumar S Vegad

Saïd Idlahcen

Aalexis Vandel

Bruno Renou

Gilles Godard

Gilles Cabot

Jean-Bernard Blaisot

Benjamin Duret

Julien Reveillon

François-Xavier Demoulin

Curvature-surface analysis to characterize the early stage of the atomization process of the pressure swirl injector

Keywords: Curvature, Drop-Size-Distribution, Surface-Interface, LIF, PDA, VOF, LES

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

Droplet size has a significant impact on reacting and nonreacting processes. A sequence of complex physical phenomena that breaks the continuous liquid structure results in a wide range of droplets. They evolve in space and time in the near field regions, where such primary breakup occurs. This early stage of the atomization process significantly impacts the final spray. Thus, characterizing the spray in the near-field zones could be used to improve the accuracy of reacting simulations and our understanding of the spray evolution. In the present work, the n-heptane hollow cone spray at its early stage of atomization process is characterized using curvature-surface analysis.

There are many techniques to image optically dense regions of spray [START_REF] Linne | Imaging in the optically dense regions of a spray: A review of developing techniques[END_REF]). However, it is difficult to describe the entire spray using a single method. The choice of diagnostic technique is based on the region under investigation. Here, 2p-LIF technique is used to capture dense regions to reduce multi scattering effect. An analysis technique of the curvature-surface interface [START_REF] Palanti | An attempt to predict spray characteristics at early stage of the atomization process by using surface density and curvature distribution[END_REF]) is able to characterize the spray in near-field regions. This work aims to adapt this analysis from numerics to experiments and compare their results.

Experimental Methods

Since the primary breakup of n-heptane hollow cone spray took place in the first 1.5 mm, the experiments were begun with microscopic shadowgraphy. The near injector spray was captured using a microscope (LDM, QM-100, Questar) with a CMOS camera (1280 × 800 pixel 2 resolution, Phantom V2512) was used [START_REF] Durst | Light Sheet Fluorescence Microscopic Imaging for the Primary Breakup of Diesel and Gasoline Sprays with Real-World Fuels[END_REF].

The microscopic shadowgraphy provided the overall spray topology. However, it could not deliver clear insights into near-field regions because of line-of-sight imaging. Hence, the current analysis employed two-photon laserinduced fluorescent (2p-LIF) technique to characterize the primary atomization [START_REF] Berrocal | Two-Photon Fluorescence Laser Sheet Imaging for High Contrast Visualization of Atomizing Sprays[END_REF] of n-heptane spray. A schematic of the 2p-LIF arrangement is shown in Fig. 1. A more detailed description of the experiments can be consulted in [START_REF] Vegad | Experimental Study of Near Field Regions of n-Heptane Spray[END_REF].

For the same experimental arrangement and flow conditions, a two-color phase Doppler anemometry (PDA) system (Dantec make) was used to measure droplet size and velocity at near field locations.

Numerical Methods

The modelling of the simplex swirl atomizer studied is reported in (Ferrando et al. 2021[START_REF] Ferrando | Experimental Measurements and Numerical Simulation of Dense Spray at the Outlet of Pressure Swirl Injector[END_REF]). The OpenFOAM v6 toolbox was used and in particular the incompressible multiphase solver, interFoam. Three transport equations are solved: the continuity equation, the momentum transport equation and the liquid volume fraction (α) transport equation. Knowing the α field, the curvature (κ) and the surface interface density (Σ) fields are derived as:

𝜅 = ∇ ⋅ ( ∇α |∇α| ) (1) Σ = ∇α
(2) where, Σ represents the surface interface by total volume.

Thus, the total surface in function of the curvature (𝑆(𝜅)) is calculated. From this parameter, the DSD is extracted and also the velocity of the droplets [START_REF] Palanti | An attempt to predict spray characteristics at early stage of the atomization process by using surface density and curvature distribution[END_REF], Ferrando et al. 2021). For this particular injector, the DSD and the Sauter Mean Diameter (SMD) obtained from CFD analysis matched with PDA data from experiments [START_REF] Ferrando | Experimental Measurements and Numerical Simulation of Dense Spray at the Outlet of Pressure Swirl Injector[END_REF] as is shown in Fig 3.

Results and Discussion

Figure 2 shows a threshold image obtained from 2p-LIF technique. The white and black colors represent liquid and gas, respectively. The interface is represented with a grey scale. A correlation can be done with the α field from the VOF method. Considering 1 for white (liquid) and 0 for black (gas). Then, Eq. ( 1) and Eq. ( 2) can be applied to obtain the curvature and the surface interface density. The curvature derived from the 2p-LIF analysis is represented on the right in Fig. 2.

Conclusion

The DSD and mean droplet diameter (D10) obtained from the curvature-surface analysis of numerical simulations are in close agreement with the PDA measurements at Z = 3 mm axial distance. Further, these numerically obtained droplet statistics are also in the same order of magnitude as those determined from the 2p-LIF analysis. It is to be concluded that the curvature-surface methodology proposed in this work is able to provide the droplet information from the initial spray.

Figure 1 :

 1 Figure 1: Schematic of 2p-LIF experimental setup.

Figure 2 :

 2 Figure 2: Left: 2p-LIF image. Right: Curvature κ [1/m]. The DSD obtained from experimental measurements (PDA) and CFD simulation are represented Fig. 3. The red line DSD obtained from experiments using PDA at 3mm from the nozzle. This point was the nearest point to the orifice with over 90% of validation. Both distributions have a good agreement between them.

Figure 3 :

 3 Figure 3: On black: DSD from CFD. On golden line: DSD from PDA.

Acknowledgments

Authors thankfully acknowledge the computer resources at IDRIS, TGCC and CINES under the allocation A0112B06153 and A0092B06153 made by GENCI (Grand Equipement National de Calcul Intensit) and also for the computing time at CRIANN (Centre Régional Informatique et d'Applications Numériques de Normandie) under the scientific project No. 2006011.