Suliman Al Shammari

Guilherme C Almeida

Maria Luiza

M Rocc

Jean-Luc Le Garrec

J Brian

A Mitchell

The Dissociative Recombination of Acetone Fragments, Adducts and Dimer Ions

The room temperature dissociative recombination of ions formed in an acetone/argon plasma has been studied using the Flowing Afterglow Langmuir Probe-Mass Spectrometer method. By changing the concentration of acetone density, it is possible to have a plasma dominated by different types of ions (fragments, adducts and dimer ions). The application of these measurements to astrophysical plasmas is discussed.

Introduction:

Dissociative recombination (DR) is the reaction in which a molecular ion recombines with an electron and dissociates into neutral fragments. General reviews on this reaction have been given by several authors such as Florescu-Mitchell and Mitchell 1 , Larsson and Orel 2 and Adams and co-workers 3 . This process plays a major role in industrial processing and nuclear fusion plasmas as well as in the evolution and composition of interstellar gas clouds and planetary ionospheres.

Neutral acetone (CH 3 COCH 3 ) was first detected in the interstellar medium (ISM) by Combes et al, 4 inside a hot core of the Sgr B2 molecular cloud and few years later by Friedel et al, 5 in the Orion-KL star-forming region. The first suggested mechanism to explain interstellar acetone involved a radiative association reaction followed by a DR reaction as shown below:

CH ଷ ା + CH ଷ COH → (CH ଷ ) ଶ CHO ା + hv (I) (CH ଷ ) ଶ CHO ା + e → CH ଷ COCH ଷ + H (II)
Herbst, Giles and Smith 6 showed that, based on laboratory measurements and theoretical calculations, this mechanism was in fact too slow to explain the high abundances of acetone observed in these star-forming regions. Further work [7][8] suggested that an ice phase chemistry involving carbon, oxygen and hydrogen on the surface of CO ice coated interstellar grains could produce molecules such as formaldehyde (CH 3 COH) and methanol which, when protonated, can react by alkyl cation transfer, followed by a DR reaction, to produce neutral acetone according to reactions III and IV.

CH 3 OH 2 + + CH 3 COH → (CH 3 ) 2 COH + + H 2 O (III)

(CH 3 ) 2 COH + + e -→ CH 3 COCH 3 + H (IV)
Although there is no consensus as to whether the acetone observed in interstellar clouds is produced directly from gas phase components or requires the evaporation of ice phase components seeded by the interstellar grains, there is no doubt that DR is the key process for obtaining large neutral organic molecules like acetone in the ISM.

In order to provide basic data for a better understanding on the formation of interstellar and circumstellar acetone, we have performed a series of measurements to determine the DR rates of its ions. A key feature of the Rennes FALP-MS apparatus is that because it has a movable mass spectrometer, the decay of several different ions can be followed as they drift through a flow tube and their individual recombination rates can be determined. This is different from other Flowing afterglow apparatuses [9][10][11][12] which require that the afterglow contains only a single ion species and where the recombination rate is derived simply by studying the decay of the afterglow electrons using Langmuir Probe Measurements. Thus in this case we have been able to study the electron recombination of the parent ion (CH 3 COCH 3 + ) as well as that of the major fragment CH 3 CO + and of the adduct ions CH 3 COCH 3 .CH 3 + , CH 3 CO CH 3 .CH 3 CO + and the dimer ion (CH 3 COCH 3 ) 2 + .

These adduct ions should also be present inside the hot cores of dense interstellar clouds where acetone is abundant. Almeida et al. 13 derived the destruction cross section of ice phase acetone in a synchrotron irradiation experiment. The authors concluded that if the photon flux is sufficiently high in an ISM environment, the acetone ice phase can be effectively decomposed and this will seed the gas phase environment with acetone molecules and its derivatives such as its adducts, which will provide a path for the development of a gas phase organic chemistry leading to the complex organic molecules that have been measured inside these hot cores.

Experimental Method

Overview of FALP technique

The Flowing Afterglow (FA) technique was first developed for the study of ion-molecule reactions by Ferguson and co-workers 14 . The basic concept of this apparatus is that the concentration of an active species can be tracked as a function of distance along a tube through which gas flows with a constant velocity. This distance can thus be converted to time and so rate coefficients for chemical reactions can be determined. A significant development of the FA was the introduction by Adams and Smith 15 of a movable Langmuir Probe (LP) for measurements of electron density and temperatures along the flow tube 16 . The probe method provides absolute values of electron concentration at a given position. Thus the Flowing Afterglow with Langmuir probe (FALP) technique could be used for the direct measurement of the dissociative recombination (DR) rate coefficients of studied ions by tracking the decay of the electron concentration along the flow. Generally speaking, the major limitation of this implementation has been that only one ion could be present in the afterglow during the measurements as it was considered that the concentration of this positive ion is equal to the concentration of the negative electrons so that plasma neutrality is maintained . In a recent article however, by Fournier et al. 17 DR rate coefficients have been measured in a flowing afterglow apparatus (VENDAMS) where the plasma contained many ions, using a method based on plasma modelling to obtain the individual recombination rate coefficients. In our case, the single ion limitation was overcome with the development of the flowing afterglow apparatus by Rowe and co-workers at the University of Rennes [START_REF] Lehfaoui | The dissociative recombination of hydrocarbon ions: I Light alkanes[END_REF][START_REF] Mostefaoui | The dissociative recombination of NO + : the influence of the vibrational excitation state[END_REF] which employs a movable mass spectrometer to track the concentration of individual ions along the flow. Thus measurements of DR rate coefficients, α, in those situations where several ions are simultaneously created, become possible. The apparatus used in this technique is called the flowing afterglow Langmuir probe mass spectrometer (FALP-MS) and this is illustrated in Figure 1. In the present version of the FALP-MS, pure helium buffer gas flows through a 4.2 cm diameter glass tube (G1) and is ionized in a microwave cavity operating at 2.45 GHz to form a plasma containing of ݁ܪ ା , ݁ܪ ଶ ା ions, metastable helium ݁ܪ and electrons. The electrons are quickly thermalized down to room temperature by helium buffer gas collisions [START_REF] Rowe | FALP and CRESU studies of ionic reactions Int[END_REF] . The plasma is then driven by the buffer gas through a stainless steel flow tube with an inner diameter of 4 cm. There are several reactant gas inlets at different positions along the flow tube (G2, G3, and G4). The ݁ܪ are destroyed by introducing some Argon gas though the second entry port (G2) into the afterglow, thus creating ݎܣ ା ions and more electrons by Penning ionization:

He m + Ar → He + Ar + + e (V)

The He + ions react with He neutral atoms to produce He 2 + ions which also react with the Ar gas to produce Ar + ions which, in turn will be used to produce the ion to be studied, by reaction with its parent gas. It should be noted that He ା ions react very slowly with argon 15 and that the likelihood of forming argon metastable atoms in any of these reactions is very small. The great advantage of helium as a buffer gas is to ensure that hot electrons created in the microwave discharge, are quickly cooled by electron-atom collisions so that one obtains the same temperature in the afterglow for both ions and neutrals (T gas = T ions = T electrons ).

Figure 2 shows the mass spectrum taken in the present experiment after injection of argon into the helium plasma. The absence of He 2 + in the mass spectrum (in the measurement area, about 1 cm after the argon injection through the second entry port G2) indicated that reaction (VII) had gone to completion, prior to arriving at the measurement region and from this, knowing the rate for reaction (V), one could be sure that all helium metastable atoms had also been converted.

Fig. 2 Mass Spectrum taken 1 cm after the eight-needles port with ݎܣ gas injected into the ݁ܪ afterglow. The electron density in the plasma was typically 3.20 × 10 ଽ ܿ݉ ିଷ for this measurement performed at 0.54 Torr.

Acetone vapor ܪܥ ଷ ܪܥܱܥ ଷ was introduced into the flow at different flow rates via a third port (G4, needles entry) that consists of eight hypodermic needles that inject gas into the center of the plasma without disturbing the aerodynamics of the flow. As will be seen later, a variety of ions were formed with their relative concentrations depending upon the flow rate of the mixture and thus the number density of acetone molecules in the flow.

Data analysis

Under these experimental conditions, i.e. 0.54 Torr, room temperature (300 K) and typically 10 10 electrons per cm 3 , ݎܣ ା ions are reacted away within a short distance downstream of the ܪܥ ଷ ܪܥܱܥ ଷ injection. The measurements of the electron-ion recombination rate coefficient can be performed using one of three methods [START_REF] Prabhudesai | Dissociative electron attachment studies on acetone[END_REF] . In this paper, we shall investigate these three methods and compare the results obtained using acetone as the parent molecule.

For a plasma containing a single positive ion that does not react with the neutral gas, the recombination process dominates and the electron density decay with time, ,ݐ can be written as: where ߙ is the recombination rate coefficient and ݊ , ݊ ା are the electron and ion densities.

ௗ ௗ௧ = -ߙ ݊ ݊ ା (1)
Assuming that the ݊ = ݊ ା (the plasma is neutral and does not contain negative ions), the equation ( 1) can be re-written as:

ௗ ௗ௧ = -ߙ ݊ ଶ (2)
By using the Langmuir Probe (LP), the electron density at several positions in the flowing afterglow can be measured. Equation (2) can then be written in terms of the downstream distance, ,ݖ and the flow velocity, v :

ݒ ௗ ௗ௭ = -ߙ ݊ ଶ (3)
Integrating equation (3) from the origin, ݖ = 0 (taken as the position of the needles entry port G4), to some downstream position z, yields the following expression for ݊ :

ቀ ଵ ቁ ௭ = ቀ ଵ ቁ + ఈ ௩ ݖ (4)
The plot of 1 ݊ ⁄ vs ,ݖ allows us to determine the recombination rate coefficient ߙ if the flow velocity ݒ is known. In the following, this method for measuring recombination rate coefficients shall be referred to as the single-ion method.

This method has been used during the course of these experiments with ܱ ଶ ା ion as a systematic, standard measurement to test the FALP-MS apparatus. The recombination rate coefficient for ܱ ଶ ା was found to be 2.01 x 10 -7 cm 3 s -1 and this value is in very good agreement with accepted value 1 for this ion (2.0 x 10 -7 cm 3 s -1 ).

The second method to determine the recombination coefficient rate is when the plasma contains several ions, as in our case, during the measurements of the recombination coefficient rate for acetone ions. One can write down the equation for the change in the density ܯ[ ା ] of the ions under study, as a function of time, t, and therefore of distance, along the flow tube:

ௗ[ெ శ ] ௗ௧ = ݒ ௗ[ெ శ ] ௗ௭ = ݊݅ݐܿݑ݀ݎܲ ݏ݉ݎ݁ݐ -ߙ ܯ[ ା ]݊ -ܯ[݇ ା ][ܺ] - ಲ [ெ శ ] Λ మ … .. ( 5
)
where ݒ is the flow velocity which was 1.85 × 10 ସ ݏ/݉ܿ in our measurements, [ܺ] denotes the density of reactant gas present in the plasma reacting at the rate constant, k, ܦ is the coefficient of ambipolar diffusion and Λ is the characteristic diffusion length. The flow velocity can be measured by pulse modulating the microwave discharge and determining the time of flight of the electron pulse so produced, between the source and a downstream detector (Langmuir probe).

The production terms have generally been ignored in our past measurements [START_REF] Prabhudesai | Dissociative electron attachment studies on acetone[END_REF] considering that the ions we studied were produced during the titration process in which the argon ions dominating the plasma were totally converted into monomer ions whose recombination was then studied. Thus in this case one could ignore the production terms and so by integrating equation (5) from some initial starting point, ݖ to some position, ,ݖ one obtains the following solution:

ݒ ln

[ெ శ ] [ெ శ ] బ = -ߙ  ݊ ݖ݀ - ௭ ௭ బ ቄ݇[ܺ] + ಲ Λ మ ቅ ݖ( -ݖ ) …. (6)
where ܯ[ ା ] ௭ and ܯ[ ା ] ௭ బ are the ion densities at position ݖ and ݖ , respectively. As can be seen, a plot of the left-hand side of equation ( 6) versus  ݊ ݖ݀ yields ߙ.

It can also be seen from equation ( 6) that when ion-molecule reactions and diffusion losses are small compared to those due to recombination, one can measure both ܯ[ ା ] and ݊ , at various ݖ values from some initial starting point, ݖ to the point where the measurement is made for a given initial electron density ݊ (0) . It should be noted that this analysis supposes that the parent gas does not undergo electron attachment reactions. This is the case for acetone [START_REF] Macneil | Ion-molecule reactions in gaseous acetone[END_REF] . This method is referred to in the following as the multiple ion/ z variable method.

If ion-molecule reactions are not negligible, M + ions react with the parent gas to produce other ions. M + ions may also be continuing to be produced by ion-molecule reactions. We shall see that this is the case in the present study and we can extend equation 6 to include the production term for a particular case, assuming that the ratio [A + ]/[M + ] remains approximately constant, thus: ݒ ln

[ெ శ ] [ெ శ ] బ ~ ݇ ଵ [ శ ][] [ெ శ ] (z -ݖ ) -ߙ  ݊ ݖ݀ - ௭ ௭ బ ቄ݇ ଶ [ܺ] + ಲ Λ మ ቅ (z -ݖ ) …. (7)
Where [A + ] is the ion responsible for the formation of M + with reaction rate k 1 and k 2 is the rate of destruction of M + by ion molecule reaction.

It is still possible however to use equation ( 7) to determine α, by measuring the ratio on the Left Hand Side (LHS) but at a fixed value of z for different electron densities. Indeed, the ion molecule reactions which produce and destroy M + do not depend on the electron density but only on the density of the reactant gas and the time, i.e. the position z in the reaction zone. Varying the electron density in the flow can be achieved in practice, by varying the position of the microwave cavity with respect to helium inlet port (G1). In this method, the production and loss terms on the RHS involving ion-molecule reactions are now constant (being independent of electron density) and only reactions involving electrons are taken into consideration. (In practice the electron density is not varied greatly and so one can ignore any change in the diffusion term). Therefore, assuming that electron attachment does not occur in this chemical reaction network, only dissociative electron recombination plays a role in the evolution of both the ions and electron density. As a consequence, this analysis is well suited to the determination of the dissociative recombination rate coefficient of ions which are not terminal, electron density being the only variable parameter. This method is referred to in the following as the multiple ion/fixed z method.

Systematic errors in the measurement arise primarily due to two factors. Firstly, the determination of the exact length of the Langmuir probe (and thus of the electron density) is subject to an uncertainty of a 10% and secondly the measurement of the flow velocity is accurate to within ± 15%. Other errors associated with the measurement of the ratio of the ion density at different distances along the flow are primarily statistical in nature and are quoted in the results below. Taking these factors into account, one can say that the combined systematic and statistical uncertainty in the quoted values for the rate coefficients presented in this paper is ± 30%.

Results and discussion

The ion spectra obtained depend on acetone density in the flow because the ions initially formed can react with the parent gas to form other ions, adducts and at high acetone concentration, the dimer. Thus, unlike in previous experiments performed with this apparatus, the ions in the flow are generally not terminal ions and so the analysis has to suit the appropriate situation as discussed above. The following subsections describe the physical chemistry processes, as a function of acetone density, introduced into the plasma.

Low acetone density

In the experiment, acetone vapor was diluted with helium gas in a ratio of 1:20, and it was introduced into the argon plasma, via the eight-needle entry port. Figure 3 shows the mass spectrum obtained 30 mm downstream from the entry port. In this example, the acetone was injected with a rather low flow rate (50 cm 3 min -1 ) into the argon plasma so as to produce an acetone density of 3.2 x 10 12 molecules cm -3 , the initial electron density being 7.91 x 10 9 cm -3 . It can be seen that there are two principle ions appearing in the reaction zone of the FALP-MS apparatus, these being CH 3 + (m/z =15) and acetyl cations CH 3 CO + (m/z = 43) formed by reactions VIII [24], along with a remnant of Ar + . Thus, it is seen that reaction VIII is the dominant ion formation process at this acetone density:

Ar ା + CH ଷ COCH ଷ → CH ଷ CO ା + CH ଷ + Ar (VIII) → CH ଷ ା + CH3CO + Ar
It appears that the direct formation of the acetone parent ion is negligible. It should be noted here that other smaller peaks also appear at higher mass numbers indicating the presence of limited further reaction. Neither these nor evidence of residual Ar + were seen further downstream, indicating that at this acetone density, production and ion-molecule loss processes can be neglected.

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The Journal of Physical Chemistry Fig. 3: Mass spectrum taken 30 mm downstream from the final injection port following injection of the helium/ acetone mixture into the Ar + dominated plasma to produce a neutral acetone density of 3.2 x 10 12 cm -3 . Chamber pressure 0.53Torr;

Although CH 3 CO + is highly dominant in the plasma, however, there are also other molecular ions and so the multiple ion / z variable method as described by equation ( 6), was used to derive the rate coefficient. The measurements were repeated several times and a typical result is shown in figure 4.

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The value obtained for the DR rate coefficient for CH 3 CO + using this method, was 1.29 ± 0.15 x 10 -6 cm 3 s -1 . This is certainly a high value for what is not a very complex ion but can be compared with other polyatomic ions 1 such as CH 3 NH 3 + (1.4 x 10 -6 cm 3 s -1 ), or CH 3 SSHCH 3 + (1.48 x 10 -6 cm 3 s -1 ). One of the major problems with dissociative recombination, is that there is no real understanding of the effect of complexity on the recombination rate and there seems to be no real trend from one ion to another.

Medium acetone density

When the flow rate of the He/acetone mixture was increased to yield an acetone molecule density in the flow of 2.3 x 10 13 cm -3 , a much more complex spectrum was found (Figure 6) with the acetone ion CH 3 COCH 3 + , the adduct ions (CH 3 COCH 3 .CH 3 + and CH 3 COCH 3. CH 3 CO + and the cluster ion (CH 3 COCH 3 ) 2 + . This is because the two primary ions seen to be formed at low acetone densities, go on to react with the acetone molecules once the latter are present in higher concentration. It is interesting to note that the acetone parent ion is now seen to be present in the spectrum. CH 3 CO + is still in a significant proportion but CH 3 + has disappeared. Indeed, measurements [START_REF] Glosik | The recombination rate coefficient of a protonated acetone dimer with electrons: indication of a temperature dependence[END_REF] have shown that CH 3 + reacts with acetone via the reactions (IX):

CH 3 + + CH 3 COCH 3 → CH 3 CO + + C 2 H 6 (46%) (IX) → CH 3 COCH 3 + + CH 3 (10%) → CH 3 COCH 3 .H + + CH 2 (10%)
The mass resolution of the spectrometer used in this experiment is not sufficient to distinguish between the parent ion and protonated acetone so it must be assumed that both are present in the spectrum shown in figure 6.

The adduct ions are formed via the three-body association reactions (X) ion densities respectively, measured downstream at z, to the corresponding densities at the origin ݖ = 30 mm, plotted at various z values between 30 and 100 mm, against the integral of the electron density divided by the flow velocity. The experiment was repeated several times and using this method, the following average values of the dissociative recombination rate coefficients for these ions were obtained. CH 3 COCH 3 + (2.6 ± 0.3 x10 -6 cm 3 s -1 ), CH 3 COCH 3 + .CH 3 + (2.25 ± 0.1 x10 -6 cm 3 s -1 ) and CH 3 COCH 3 + .CH 3 CHO + (2.3 ± 0.35 x10 -6 cm 3 s - 1 ).

Fig. 8:

The ratio of ܪܥ( ଷ ܪܥܱܥ ଷ ). ܪܥ ଷ ܱܥ ା ion density measured downstream to that at origin plotted against the integral of the electron density divided by the flow velocity.

The measurements just described are valid under the conditions where ion-molecule reactions, leading to the production or destruction of these ions can be ignored. Indeed these processes would contribute to there being a difference in the measured number densities as a function of distance z along the flow and given the complexity of the situation here (aggregation, adduct formation), an alternative approach was used to eliminate this problem. Ratios of the number densities for these ions at a fixed given point z, to those at an arbitrarily chosen point z=0, were measured for different electron densities. This is the multiple ion/fixed z method. Since the Page 12 of 32

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The Journal of Physical Chemistry change in the ratio of these number densities due to ion-molecule reactions is independent of the electron density, then the latter can be ignored, whether they are responsible for production or destruction.

The electron density in the plasma could be changed by varying the position of the microwave cavity with respect to the position of the helium gas inlet tube, G1 (see figure 1). The measurements illustrated in figures 9, 10, 11 were made in this way. With the multiple ion / fixed z method, the following values were found: CH 3 COCH 3 + (2.7x10 -6 cm 3 s -1 ), CH 3 COCH 3 + .CH 3 + (2.4x10 -6 cm 3 s -1 ) and CH 3 COCH 3 + .CH 3 CHO + (2.9 x10 -6 cm 3 s -1 ), i.e. very close to those obtained with the multiple ion/z variable method.

As far as we know, there have been no other measurements of recombination of these ions and indeed no measurements of the recombination of adduct ions. The latter are also known as "pseudomolecules" and are often found in the mass spectra of polyatomic ions. They are distinct from cluster ions as the attached functional groups are bound to the parent molecule by hydrogen bonds and thus are much more stable than cluster ions which are bound by polarization forces.

It is interesting to speculate why the two measurement methods have such close agreement even though the multiple ion/fixed Z approach should be more precise. In fact examination of equation ( 7) yields a clue to this since, if [A+] and [M + ] are not too different and if we can assume that the ion molecule reactions involved in the formation and destruction of [M + ] are not too different (granted that the branching ratios for the products are not well known), then the production and destruction terms will tend to cancel each other out. This argument could apply to CH 3 CO + which when there are no longer any Ar + in the plasma, is formed by the reaction of CH 3 + with acetone (reaction IX) but then reacts further with acetone to form the corresponding adduct (reaction X). For the CH 3 + -acetone adduct, the situation is different as it can be seen in figure 5 that under these conditions, CH 3 + is exhausted, having been used up in forming its adduct. As we shall see in the next section, the adducts themselves are not terminal ions in the sense that they can undergo further reaction if there is sufficient acetone density. It should be remembered however, that the formation process is a three-body reaction and the reaction of these adducts to form dimer ions is not known. It may just be that the high recombination rate found, dominates the ion chemistry under these acetone density conditions

High density of acetone

For a moderate acetone density, a large signal was found for the clusters ions (CH 3 COCH 3 ) 2 + but the decay of its signal along the flow was not seen. In fact the signal was found to rise with increasing z. This is because this ion is continually formed by the reaction of acetone ions with neutral acetone molecules, and indeed reactions of the adduct ions with acetone. Going to a much higher flow rate of acetone mixture (the acetone density was roughly 5.8 x 10 13 cm -3 ), yielded the spectrum shown ion fig. 12. It is seen that now there is only one ion in the plasma under this condition. Because of this, the rate coefficient could be determined by using the single ion method based on equation 4 (fig. 13) and an average value of 1.45 x 10 -6 cm 3 s -1 was obtained. Glosik and Plasil 26 did measure the recombination rate for the protonated form of the ion in a high pressure (5 Torr) flowing-afterglow apparatus. In their experiment, proton transfer reaction was used to create protonated acetone clusters. They found a value of 3.4 x10 -6 cm 3 s -1 at Te = 580K ± 150 K and 7 x 10 -6 at Te = 450 ± 100 K. These values are much higher than what we have found here and indeed seem to be unreasonably high, implying a value of about 10 -5 cm 3 s -1 at 300K.

Conclusion

Finally, table 2 presents a summary of the values obtained for these ions using these different methods.

Ion

Single ion method (10 -6 cm 3 s -1 )

Multiple ions/z-variable (Electron density fixed) (10 -6 cm 3 s -1 )

Multiple ions/z fixed Electron density variable (10 -6 cm 3 s - The unique ability of the FALP-MS apparatus, allows a plasma containing several ions to be treated and the recombination rates of the individual ions to be determined. Several methods have been used and it is interesting to note in Table 1 that the values obtained are consistent for a given ion. This indicates that under the conditions studied, the destruction of the ions by recombination dominates, with the exception of the dimer ion where this is only true at high acetone densities.

The values measured are generally rapid as seen for many other ions 1 but there is no systematic dependence of this value on ion complexity. Barring a couple of exceptions (He 2 + , ArH + ), dissociative recombination rates 1 general fall into a rather narrow range from about 1 x 10 -7 to 5 x 10 -6 cm 3 s -1 . (This can be compared to the sister reaction of electron attachment to neutrals where rates vary over about 5 or 6 orders of magnitude). The values obtained for the recombination rate for the adduct ions and the dimer, are in line with other measurements of hydrocarbon dimer ions 1 (e.g. (CH 3 COH) 2 .H + , α = 1.4 x 10 -6 cm 3 .s -1 ). Our value for the acetone dimer is much lower than that found by Glosik and Plasil 26 but their measurement was at a pressure an order of magnitude higher than ours and it is possible that three body recombination effects may have played a role in that experiment.

In this study, we have demonstrated that it is possible with our FALP-MS apparatus, to measure the recombination rate coefficients under plasma conditions where there are multiple ions and where these ions are not terminal ions, but where production and destruction via ion-molecule reactions play a role in the decay processes.

(

Fig. 1 :

 1 Fig. 1: Sketch of FALP-MS apparatus at University of Rennes 20 .

Fig. 4 .

 4 Fig. 4. Plot of the logarithm of the CH 3 CO + signal at position z over the signal at the origin, z 0 as a function of the integral over the distance of the electron density. The slope multiplied by the flow velocity (v=1.85 x 10 4 cm.s -1 ) yields the recombination rate coefficient.

Fig. 5 :

 5 Fig.5: Mass spectrum taken 30 mm downstream from the final injection port following injection of the He/acetone mixture into plasma dominated by ݎܣ ା a flow rate yielding an acetone density of (2.3 x 10 13 molecules.cm -3 . Chamber pressure was 0.55 Torr.

Fig. 6 :

 6 Fig. 6: The ratio of ܪܥ ଷ ܪܥܱܥ ଷ ା ion density measured downstream to that at origin plotted against the integral of the electron density divided by the flow velocity.

Fig. 7 :

 7 Fig. 7: The ratio of ܪܥ( ଷ ܪܥܱܥ ଷ ). ܪܥ ଷ ା ion density measured downstream to that at origin plotted against the integral of the electron density divided by the flow velocity.

Fig. 9 :

 9 Fig. 9:The ratio of the density of CH ଷ COCH ା at a single value of z = 100 mm downstream from the origin to the density at ݖ vs the integral of the electron density from ݖ to z, for three different initial electron densities, 2.8, 3.2, 5.7 x10 ଽ ܿ݉ ିଷ .

Fig. 10 :

 10 Fig. 10: The ratio of the density of ܪܥ( ଷ ܪܥܱܥ ଷ ). ܪܥ ଷ ା at a single value of z = 100 mm downstream from the origin to the density at ݖ vs the integral of the electron density from ݖ to z, for three different initial electron densities, 2.8, 3.2, 5.7 x 10 ଽ ܿ݉ ିଷ .

Fig. 11 :

 11 Fig. 11:The ratio of the density of ܪܥ( ଷ ܪܥܱܥ ଷ ). ܪܥ ଷ ܱܥ ା at a single value of z = 100 mm downstream from the origin to the density at ݖ vs the integral of the electron density from ݖ to z, for three different initial electron densities, 2.8, 3.2, 5.78 x 10 ଽ ܿ݉ ିଷ .

Fig. 12 :

 12 Fig.12: Mass spectrum taken 30 mm downstream from the final injection port following injection of acetone mixture (5.8 x 10 13 cm -3 ) into an Ar + dominated plasma.

Fig. 13 :

 13 Fig. 13: A plot of the inverse of the electron density vs distance downstream for an afterglow formed from ܪܥ ଷ ܪܥܱܥ ଷ for two initial electron densities. The rate coefficient is obtained directly from the slope, given by α/v.

Fig. 4 .Fig. 5 :

 45 Fig. 4. Plot of the logarithm of the CH3CO+ signal at position z over the signal at the origin, z0 as a function of the integral over the distance of the electron density. The slope multiplied by the flow velocity (v=1.85 x 104 cm.s-1) yields the recombination rate coefficient. 152x129mm (150 x 150 DPI)

Fig. 6 :

 6 Fig. 6: The ratio of 〖CH〗_3 〖COCH〗_3^+ ion density measured downstream to that at origin plotted against the integral of the electron density divided by the flow velocity. 151x129mm (150 x 150 DPI)

Fig. 7 :

 7 Fig. 7: The ratio of 〖(CH〗_3 〖COCH〗_3).〖CH〗_3^+ ion density measured downstream to that at origin plotted against the integral of the electron density divided by the flow velocity. 150x127mm (150 x 150 DPI)

Fig. 8 :

 8 Fig. 8: The ratio of 〖(CH〗_3 〖COCH〗_3).〖CH〗_3 〖CO〗^+ ion density measured downstream to that at origin plotted against the integral of the electron density divided by the flow velocity. 151x127mm (150 x 150 DPI)

Fig. 9 :

 9 Fig. 9: The ratio of the density of CH_3 COCH^+at a single value of z = 100 mm downstream from the origin to the density at z_0 vs the integral of the electron density from z_0 to z, for three different initial electron densities, 2.8, 3.2, 5.7 x〖10〗^9 〖cm〗^(-3). 153x128mm (150 x 150 DPI)

Fig. 10 :

 10 Fig. 10: The ratio of the density of 〖(CH〗_3 〖COCH〗_3).〖CH〗_3^+ at a single value of z = 100 mm downstream from the origin to the density at z_0 vs the integral of the electron density from z_0 to z, for three different initial electron densities, 2.8, 3.2, 5.7 x 〖10〗^9 〖cm〗^(-3). 154x127mm (150 x 150 DPI)

Fig. 11 :

 11 Fig. 11: The ratio of the density of 〖(CH〗_3 〖COCH〗_3).〖CH〗_3 〖CO〗^+ at a single value of z = 100 mm downstream from the origin to the density at z_0 vs the integral of the electron density from z_0 to z, for three different initial electron densities, 2.8, 3.2, 5.78 x 〖10〗^9 〖cm〗^(-3). 152x129mm (150 x 150 DPI)

Fig. 13 :

 13 Fig. 13: A plot of the inverse of the electron density vs distance downstream for an afterglow formed from 〖 CH〗_3 〖COCH〗_3 for two initial electron densities. The rate coefficient is obtained directly from the slope, given by α/v. 148x119mm (150 x 150 DPI)

  Table1lists the reactions (and their corresponding reaction rate coefficients) involves in the initial plasma formation and its conversion to an argon plasma. The rates are taken from ref.22.

	He 2	+ ions are removed by reaction (VII):
			He + + He + He → He 2	+ + He	(VI)
			He 2	+ + Ar → He + He + Ar +	(VII)
			Reaction	Rate Coefficient
				(cm 3 s -1 )
		He + + He + He → He 2	+ + He	1 x 10 -31
		He m + He m → He + + He + e -	5 x 10 -9
			→ He +	2 + e -	5 x 10 -9
			He +	2 + e -→ 2He	< 3 x 10 -10
				Ar
		He + + Ar → Ar + + He	1 x 10 -13
		He m + Ar→ He + Ar + + e	7 x 10 -11
		He 2	+ + Ar → He + He + Ar +	2 x 10 -10

Table 1 .

 1 Summary of measured recombination rates determined using the three different methods discussed in the text.

	1 )

  2) Larsson, M. and Orel, A. E. Dissociative Recombination of Molecular Ions, Cambridge University Press, Cambridge, 2008. (3) Adams, N.G. and Smith. D. Measurements of the dissociative recombination coefficients for several polyatomic ion species at 300 K, Chem. Phys. Letters 1998, 144, 11-14. Millar, T. J.; Herbst, E.; Charnley, S. B. The formation of oxygen-containing organic molecules in the Orion compact ridge, Astrophys. J. 1991, 369, 147-156. (8) Ehrenfreund, P.; Charnley, S. B. Organic Molecules in the Interstellar Medium, Comets, and Meteorites: A Voyage from Dark Clouds to the Early Earth, Annu. Rev. Astron. Astrophys. 2000, 38, 427-483. (9) Smith, D. and Spanel, P. Swarm Techniques in Experimental Methods in the Physical Sciences, Academic Press, Orlando, 1995, Vol. 29A, p. 273. Mahdavi M. R.; Hasted, J.B.; Nakshbandi, M.M. Electron-ion recombination measurements in the flowing afterglow, J. Phys. B: Atom. Molec. Phys. 1971, 4, 1726. (17) Fournier J.A.; Shuman N.S. ; Melko J.J.; Ard S.G.; Viggiano A.A. A novel technique for measurement of thermal rate constants and temperature dependences of dissociative

	(16) recombination: CO 2	+ , CF 3	+ , N 2 O + , C7H 8 + , C 7 H 7 + , C 6 H 6 + , C 6 H 5 + , C 5 H 6	+ , C 4 H 4	+ and C 3 H 3	+ , J.
	(4) Combes, F.; Gerin, M.; Wootten, A.; Wlodarczak, G.; Clausset, F.; Encrenaz, P. J. Acetone in Chem. Phys. 2013, 138, 154201-7.
	interstellar space, Astron. Astrophys. 1987, 180, L13-L16.
	(5) Friedel, D. N.; Snyder, L. E.; Remijan, A. J.; Turner, B. E. Detection of interstellar acetone
	towards the Orion-Kl hot core, Astrophys. J. 2005, 632, L95-L98.
	(6) Herbst, E.; Giles, K.; Smith, D. Is interstellar acetone produced by ion-molecule chemistry?
	The Astrophysical Journal, 1990, 358, 468-472.
	(7) (10) Gougousi,T.; Johnsen, R. and Golde, M.F. Recombination of H 3	+ and D 3	+ ions in a
	flowing afterglow plasma, Int. J. Mass Spec. and Ion Proc. 1995, 149-150, 131-151.
	(11) Adams, N.G. Afterglow techniques with spectroscopic detection for determining the
	rate coefficients and products of dissociative electron-ion recombination, Int. J. Mass Spec.
	and Ion Proc. 1994, 132, 1-27.
	(12) Novotny, O.; Plasil, R.; Pysanenko, A.; Korolov, I.; Glosik, J. The recombination of
	D +	3 and D +	5 ions with electrons in deuterium containing plasma J. Phys. B: At. Mol. Opt.
	Phys. 2006, 39, 2561-2544.
	(13) Almeida, G.C.; Pilling, S.; Andrade, D.P.P.; Castro, N. S.; Mendoza, E.; Boechat-Roberty,
	H. M.; Rocco. M. M. Photodesorption and photostability of acetone ices: Relevance to solid
	phase astrochemistry, J. Phys. Chem. C 2014, 118, 6193-6200.
	(14) Ferguson, E.E.; Fehsenfeld, F.C. and Schmeltekopf, A.L. Flowing afterglow measurements
	of ion-neutral reactions, Adv. Mol. Phys. 1969, 5, 1-56.
	(15) N.G. Adams, N.G.; Smith, D.; Alge, E. Measurements of dissociative recombination
	coefficients of H +	3 , HCO + , N 2 H + , and CH + 5 at 95 and 300 K using the FALP apparatus, J. Chem.
	Phys. 1984, 81, 1778-1784.

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Acknowledgments: S. Al Shammari would like to thank the King Abdulaziz City for Science and Technology for funding his thesis work which underlies this paper.

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AUTHOR INFORMATION

Author Contributions mitchell@univ-rennes1.fr

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. The authors declare no competing financial interest.

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N Fourny

C Lan

F Kober

M Bernard

M Desrois

28 Intolerance to glucose and abdominal obesity in a diet-induced meta-bolic syndrome model were associated with modification of cardiac morphology and impaired myocardial function

Introduction

Metabolic Syndrome (MetS) is defined by multiple risk fac-tors that predict type 2 diabetes and cardiovascular complications, such as myocardial infarction, especially in women. Consequently the aim of this pre-liminary study was to investigate in vivo and ex vivo the effects of a high-fat-highsucrose diet (HFHSD) on the development of metabolic syndrome (MetS), cardiac morphology and function of female Wistar rat.

Material and methods

Female Wistar rats, subjected to HFHSD (FHFD) or Normal Diet (FND) during 5 months, were explored every month with multi-modal cardiovascular magnetic resonance (CMR) to determine in vivo cardiac function, morphology and triglyceride (TG) content. Liver TG content was # evaluated with 1H Magnetic Resonance Spectroscopy (MRS). Then, rats underwent an IPGTT to determine glycemic status, and ex vivo experiments on isolated perfused heart were performed to study cardiac function and energy metabolism with 31P MRS under baseline conditions.

Results

In FHFD vs. FND, CMR showed an increase of systolic wall thick-ness over time (p<0.05) and diastolic wall thickness at 3 and 5 months (p<0.01); 1H MRS showed that hepatic TG content was increased (p<0.01) but myocardial TG content was not different. IPGTT showed a significant glucose intolerance (p<0.001) and plasma free fatty acids were increased (p<0.05) in FHFD vs. FND. At 5 months, weight was not different between groups but FHFD exhibited an abdominal obesity with increased visceral adipose tissue (p<0.05), % fat (p<0.05) and% visceral fat (p<0.05) compared with FND. Under baseline conditions, ex vivo myocardial function was impaired in FHFD vs. FND (p<0.01).

Conclusion

HFHSD-induced MetS was characterized by glucose into-lerance, abdominal obesity, hepatic fat deposit which were associated with modification of cardiac morphology and impaired myocardial function. These results may be related to higher risk of cardiovascular complications among type 2 diabetic obese female.

The author hereby declares no conflict of interest


Rainer Buckdahn
email: rainer.buckdahn@univ-brest.fr

Ying Hu
email: ying.hu@univ-rennes1.fr

Shanjian Tang
email: sjtang@fudan.edu.

Uniqueness of solution to scalar BSDEs with L exp µ 2 log (1 + L) -integrable terminal values

Keywords: AMS Subject Classification: 60H10 Key Words Backward stochastic differential equation, L exp µ 2 log (1 + L) integrability, uniqueness

, the existence of the solution is proved for a scalar linearly growing backward stochastic differential equation (BSDE) if the terminal value is L exp µ 2 log (1 + L)integrable with the positive parameter µ being bigger than a critical value µ 0 . In this note, we give the uniqueness result for the preceding BSDE.

Introduction

Let {W t , t ≥ 0} be a standard Brownian motion with values in R d defined on some complete probability space (Ω, F, P), and {F t , t ≥ 0} its natural filtration augmented by all P-null sets of F. Let us fix a nonnegative real number T > 0. The σ-field of predictable subsets of Ω × [0, T ] is denoted by P.

For any real p ≥ 1, denote by L p the set of all F T -measurable random variables η such that E|η| p < ∞, by S p the set of (equivalent classes of) all real-valued, adapted and càdlàg processes {Y t , 0 ≤ t ≤ T } such that

||Y || S p := E sup 0≤t≤T |Y t | p 1/p
< +∞, by L p the set of (equivalent classes of) all real-valued adapted processes {Y t , 0 ≤ t ≤ T } such that

||Y || L p := E T 0 |Y t | p dt 1/p < +∞,
and by M p the set of (equivalent classes of) all predictable processes {Z t , 0 ≤ t ≤ T } with values in R 1×d such that

||Z|| M p := E T 0 |Z t | 2 dt p/2 1/p < +∞.
Consider the following Backward Stochastic Differential Equation (BSDE):

Y t = ξ + T t f (s, Y s , Z s )ds - T t Z s dW s , t ∈ [0, T ]. (1.1)
Here, f (hereafter called the generator) is a real valued random function defined on the set Ω × [0, T ] × R × R 1×d , measurable with respect to P ⊗ B(R) ⊗ B(R 1×d ), and continuous in the last two variables with the following linear growth: By BSDE (ξ,f ), we mean the BSDE with generator f and terminal condition ξ.

|f (s, y, z) -f (s, 0, 0)| ≤ β|y| + γ|z|, (s, y, z) ∈ [0, T ] × R × R 1×d with f 0 := f (•, 0, 0) ∈ L
It is well known that for (ξ, f 0 ) ∈ L p × L p (with p > 1), BSDE (1.1) admits a unique adapted solution (y, z) in the space S p × M p if the generator f is uniformly Lipschitz in the pair of unknown variables. See e.g. [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF][START_REF] Karoui | Backward stochastic differential equations in finance[END_REF][START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF] for more details. For (ξ, f 0 ) ∈ L 1 × L 1 , one needs to restrict the generator f to grow sub-linearly with respect to z, i.e., with some q ∈ [0, 1),

|f (t, y, z) -f 0 (t)| ≤ β|y| + γ|z| q , (t, y, z) ∈ [0, T ] × R × R 1×d
for BSDE (1.1) to have a unique adapted solution (see [START_REF] Briand | L p solutions of backward stochastic differential equations[END_REF]) if the generator f is uniformly Lipschitz in the pair of unknown variables.

In [START_REF] Hu | Existence of solution to scalar BSDEs with L exp 2 λ log (1 + L)integrable terminal values[END_REF], the existence of the solution is given for a scalar linearly growing BSDE (1.1) if the terminal value is L exp µ 2 log (1 + L) -integrable with the positive parameter µ being bigger than a critical value µ 0 = γ √ T , and the preceding integrability of the terminal value for a positive parameter µ less than critical value µ 0 is shown to be not sufficient for the existence of a solution. In this note, we give the uniqueness result for the preceding BSDE under the preceding integrability of the terminal value for µ > µ 0 .

We first establish some interesting properties of the function ψ(x, µ) = x exp µ 2 log (1 + x) . We observe that the obtained solution Y in [START_REF] Hu | Existence of solution to scalar BSDEs with L exp 2 λ log (1 + L)integrable terminal values[END_REF] has the nice property: ψ(|Y |, a) belongs to the class (D) for some a > 0, which is used to prove the uniqueness of the solution by dividing the whole interval [0, T ] into a finite number of sufficiently small subintervals.

Uniqueness

Define the function ψ:

ψ(x, µ) := x exp µ 2 log (1 + x) , (x, µ) ∈ [0, +∞) × (0, +∞).
We denote also ψ(•, µ) as ψ µ (•)

The following two lemmas can be found in Hu and Tang [START_REF] Hu | Existence of solution to scalar BSDEs with L exp 2 λ log (1 + L)integrable terminal values[END_REF].

Lemma 2.1 For any x ∈ R and y ≥ 0, we have

e x y ≤ e x 2 2µ 2 + e 2µ 2 ψ(y, µ). (2.2) Lemma 2.2 Let µ > γ √ T . For any d-dimensional adapted process q with |q t | ≤ γ almost surely, for t ∈ [0, T ], E e 1 2µ 2 | T t qsdWs| 2 F t ≤ 1 1 -γ 2 µ 2 (T -t) . (2.3) Proposition 2.3
We have the following assertions on ψ: Proof. The first assertion has been shown in [START_REF] Hu | Existence of solution to scalar BSDEs with L exp 2 λ log (1 + L)integrable terminal values[END_REF]. It remains to show the Assertions (ii) and (iii). We prove Assertion (ii).

(i) For µ > 0, ψ(•, µ) is convex. (ii) For c > 1, we have ψ µ (cx) ≤ ψ µ (c)ψ µ (x)
ψ µ (cx) = cx exp µ 2 log (1 + cx) ≤ cx exp µ 2 log [(1 + c)(1 + x)] = cx exp µ 2 log(1 + c) + 2 log(1 + x) ≤ cx exp µ 2 log(1 + c) + µ 2 log(1 + x) = ψ µ (c)ψ µ (x).
We now prove Assertion (iii). We have

(ψ b • ψ a ) (x) = ψ a (x) exp b 2 log (1 + ψ a (x)) = x exp a 2 log (1 + x) exp b 2 log 1 + xe a √ 2 log (1+x) ≤ x exp a 2 log (1 + x) exp b 2 log (1 + x)e a √ 2 log (1+x) = x exp a 2 log (1 + x) exp b 2 log (1 + x) + 2a 2 log (1 + x) .
In view of the following elementary inequality:

2a 2 log (1 + x) ≤ a 2 b 2 c 2 + 2c 2 b 2 log (1 + x),
we have

(ψ b • ψ a ) (x) ≤ x exp a 2 log (1 + x) exp b 2 log (1 + x) + a 2 b 2 c 2 + 2c 2 b 2 log (1 + x) ≤ x exp a 2 log (1 + x) exp b 2 log (1 + x) + b a 2 b 2 c 2 + b 2c 2 b 2 log (1 + x) .
Therefore,

(ψ b • ψ a ) (x) ≤ x exp a 2 log (1 + x) exp b 2 log (1 + x) + ab 2 c + c 2 log (1 + x) ≤ xe ab 2 c exp (a + b + c) 2 log (1 + x) .
Consider the following BSDE:

Y t = ξ + T t f (s, Y s , Z s ) ds - T t Z s dW s , (2.4)
where

f satisfies |f (s, y, z) -f 0 (s, 0, 0)| ≤ β|y| + γ|z|, (2.5)
with

f 0 := f (•, 0, 0) ∈ L 1 , β ≥ 0 and γ > 0.
Theorem 2.4 Let f be a generator which is continuous with respect to (y, z) and verifies inequality (2.5), and ξ be a terminal condition. Let us suppose that there exists µ > γ √ T such that ψ(|ξ|

+ T 0 |f 0 (t)| dt, µ) ∈ L 1 (Ω, P). Then BSDE (2.4) admits a solution (Y, Z) such that |Y t | ≤ 1 1 -γ 2 µ 2 (T -t) e β(T -t) + e 2µ 2 +β(T -t) E ψ µ |ξ| + T t |f 0 (s)| ds F t .
Furthermore, there exists a > 0 such that ψ(Y, a) belongs to the class (D).

Proof. Let us fix n ∈ N * and p ∈ N * . Set

ξ n,p := ξ + ∧ n -ξ -∧ p, f n,p 0 := f + 0 ∧ n -f - 0 ∧ p, f n,p := f -f 0 + f n,p 0 .
As the terminal value ξ n,p and f n,p (•, 0, 0) are bounded (hence square-integrable) and f n,p is a continuous generator with a linear growth, in view of the existence result in [START_REF] Lepeltier | Backward stochastic differential equations with continuous coefficient[END_REF], the BSDE (ξ n,p , f n,p ) has a (unique) minimal solution (Y n,p , Z n,p ) in S 2 × M 2 . Set f n,p (s, y, z) = |f n,p 0 (s)| + βy + γ|z|, (s, y, z) ∈ [0, T ] × R × R 1×d . In view of Pardoux and Peng [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF], the BSDE (|ξ n,p |, f n,p ) has a unique solution ( Ȳ n,p , Zn,p ) in

S 2 × M 2 .
By comparison theorem, |Y n,p t | ≤ Ȳ n,p t . Letting q n,p s = γ sgn(Z n,p s ) and

P q n,p = exp{ T 0 q n,p s dW s - 1 2 T 0 |q n,p s | 2 ds}P,
we obtain,

|Y n,p t | ≤ Ȳ n,p t = E q n,p e β(T -t) |ξ n,p | F t + T t e β(s-t) |f n,p 0 (s)| ds ≤ e β(T -t) E q n,p |ξ n,p | + T t |f n,p 0 (s)| ds F t ≤ 1 1 -γ 2 µ 2 (T -t) e β(T -t) + e 2µ 2 +β(T -t) E ψ µ |ξ| + T t |f 0 (s)| ds F t .
Since Y n,p is nondecreasing in n and non-increasing in p, then by the localization method in [START_REF] Briand | BSDE with quadratic growth and unbounded terminal value[END_REF], there is some Z ∈ L 2 (0, T ; R 1×d ) almost surely such that (Y := inf p sup n Y n,p , Z) is an adapted solution. Therefore, we have for a > 0, using Jensen's inequality and the convexity of ψ a (•) := ψ(•, a) together with Assertion (ii) of Proposition 2.3, we have

ψ a (|Y n,p t |) ≤ ψ a e β(T -t) E q n,p |ξ n,p | + T t |f n,p 0 (s)| ds F t ≤ ψ a e β(T -t) ψ a E q n,p |ξ n,p | + T t |f n,p 0 (s)| ds F t ≤ ψ a e β(T -t) E q n,p ψ a |ξ n,p | + T t |f n,p 0 (s)| ds F t ≤ ψ a e β(T -t) E exp T t q n,p s dW s ψ a |ξ n,p | + T t |f n,p 0 (s)| ds F t .
For b > γ √ T , applying Lemma 2.1, we have

ψ a (|Y n,p t |) ≤ ψ a e β(T -t) E exp T t q n,p s dW s ψ a |ξ n,p | + T t |f n,p 0 (s)| ds F t ≤ ψ a e β(T -t) E exp 1 2b 2 T t q n,p s dW s 2 F t +e 2b 2 E ψ b • ψ a |ξ n,p | + T t |f n,p 0 (s)| ds F t .
Using Lemma 2.2 and Assertion (iii) of Proposition 2.3, we have for any c > 0,

ψ a (|Y n,p t |) ≤ ψ a e β(T -t) 1 1 -γ 2 b 2 (T -t) + e 2b 2 + ab 2 c E ψ a+b+c |ξ n,p | + T t |f n,p 0 (s)| ds F t .
For µ > γ √ T , we can choose a > 0, b > γ √ T , and c > 0 such that a + b + c = µ. Then, we have

ψ a (|Y n,p t |) ≤ ψ a e β(T -t) 1 1 -γ 2 b 2 (T -t) + e 2b 2 + ab 2 c E ψ µ |ξ n,p | + T t |f n,p 0 (s)| ds F t ≤ ψ a e β(T -t) 1 1 -γ 2 b 2 (T -t) + e 2b 2 + ab 2 c E ψ µ |ξ| + T t |f 0 (s)| ds F t .
Letting first n → ∞ and then p → ∞, we have

ψ a (|Y t |) ≤ ψ a e β(T -t) 1 1 -γ 2 b 2 (T -t) + e 2b 2 + ab 2 c E ψ µ |ξ| + T t |f 0 (s)| ds F t ≤ ψ a e βT 1 1 -γ 2 T b 2 + e 2b 2 + ab 2 c E ψ µ |ξ| + T 0 |f 0 (s)| ds F t .
Consequently, we have ψ a (|Y |) belongs to the class (D). Now we state our main result of this note.

Theorem 2.5 Assume that the generator f is uniformly Lipschitz in (y, z), i.e., there are β > 0 and γ > 0 such that for all (y i , z

i ) ∈ R × R 1×d , i = 1, 2, we have |f (t, y 1 , z 1 ) -f (t, y 2 , z 2 )| ≤ β|y 1 -y 2 | + γ|z 1 -z 2 |.
Furthermore, assume that there exists µ > γ √ T such that ψ(|ξ| + T 0 |f (t, 0, 0)| dt, µ) ∈ L 1 (Ω, P ). Then, BSDE (2.4) admits a unique solution (Y, Z) such that ψ(Y, a) belongs to the class (D) for some a > 0.

Proof. The existence of an adapted solution has been proved in the preceding theorem. It remains to prove the uniqueness.

For i = 1, 2, let (Y i , Z i ) be a solution of BSDE (2.4) such that ψ a i (Y i ) belongs to the class (D) for some a i > 0. Define

a := a 1 ∧ a 2 , δY := Y 1 -Y 2 , δZ := Z 1 -Z 2 .
Then both ψ a (Y 1 ) and ψ a (Y 2 ) are in the class (D), since ψ(x, µ) is nondecreasing in µ, and the pair (δY, δZ) satisfies the following equation

δY t = T t [f (s, Y 1 s , Z 1 s ) -f (s, Y 2 s , Z 2 s )] ds - T t δZ s dW s , t ∈ [0, T ].
By a standard linearization we see that there exists an adapted pair of processes (u, v) such that

|u s | ≤ β, |v s | ≤ γ, and f (s, Y 1 s , Z 1 s ) -f (s, Y 2 s , Z 2 s ) = u s δY s + δZ s v s .
We define the stopping times

τ n := inf{t ≥ 0 : |Y 1 t | + |Y 2 t | ≥ n} ∧ T, n = 1, 2

  for any x ≥ 0. (iii) For any triple (a, b, c) with a > 0, b > 0 and c > 0, we have ψ(ψ(x, a), b) ≤ e ab 2 c ψ(x, a + b + c).

  1 , β ≥ 0 and γ > 0. ξ is a real F T -measurable random variable, and hereafter called the terminal condition or terminal value. By a solution to BSDE (1.1) we mean a pair {(Y t , Z t ), 0 ≤ t ≤ T } of predictable processes with values in R × R 1×d such that P-a.s., t → Y t is continuous, t → Z t belongs to L 2 (0, T ) and t → f (t, Y t , Z t ) is integrable, and P-a.s. (Y, Z) verifies (1.1).

	Definition 1.1

  On the other hand, since ψ a is nondecreasing and convex, we have thanks to Proposition 2.3 (ii)ψ a (|δY τn |) ≤ ψ a (|Y1 τn | + |Y 2 τn |) = ψ a ( From (2.7) it now follows that, for t ∈ [T -a 2 4γ 2 , T ], the family of random variables e τn t∧τn vs dWs |δY τn | is uniformly integrable. Finally, letting n → ∞ in inequality (2.6), we have δY = 0 on the interval [T -a 2 4γ 2 , T ]. It is then clear that δZ = 0 on [T -a 2 4γ 2 , T ]. The uniqueness of the solution is obtained on the interval [T -a 2 4γ 2 , T ]. In an identical way, we have the uniqueness of the solution on the interval [T -a 2 2γ 2 , T -a 2 4γ 2 ] . By a finite number of steps, we cover in this way the whole interval [0, T ], and we conclude the uniqueness of the solution on the interval [0, T ].

	and, thus, the family of random variables e	1 2a 2 ( τn t∧τn vs,dWs )
												1 2	× 2|Y 1 τn | +	1 2	× 2|Y 2 τn |)
			≤	1 2	ψ a (2|Y 1 τn |) +	1 2	ψ a (2|Y 2 τn |) ≤	1 2	ψ a (2)[ψ a (|Y 1 τn |) + ψ a (|Y 2 τn |)].
	Therefore,
			|δY t∧τn | ≤ E e	τn t∧τn us ds+ τn t∧τn vs,dWs |δY τn | F t
						≤ e βT E e	τn t∧τn vs,dWs |δY τn | F t .	(2.6)
	Now we show that the family of random variables e	τn t∧τn vs,dWs |δY τn | is uniformly integrable. For
	this note that, thanks to Lemma 2.1,
		e	τn t∧τn vs,dWs |δY τn | ≤ e	1 2a 2 ( τn t∧τn vs,dWs )	2	+ e 2a 2 ψ a (|δY τn |).	(2.7)
	For t ∈ [T -a 2 4γ 2 , T ], we have from Lemma 2.2,
	E e	1 2a 2 ( τn t∧τn vs,dWs )	2 2	= E e	1 a 2 ( τn t∧τn vs,dWs )	2	≤	1 a 2 (T -t) 1 -2γ 2	≤	√	2,

, • • • , with the convention that inf ∅ = ∞. Since (δY, δZ) satisfies the linear BSDE δY t = T t (u s δY s + δZ s v s ) ds -T t δZ s dW s , t ∈ [0, T ], we have the following formula δY t∧τn = E e τn t∧τn us ds+ τn t∧τn vs,dWs -1 2 τn t∧τn |vs| 2 ds δY τn F t . 2 is uniformly integrable.

Partially supported by Lebesgue center of mathematics "Investissements d'avenir" program -ANR-11-LABX-0020-01, by ANR CAESARS -ANR-15-CE05-0024 and by ANR MFG -ANR-16-CE40-0015-01. cn). Partially supported by National Science Foundation of China (Grant No. 11631004) and Science and Technology Commission of Shanghai Municipality (Grant No. 14XD1400400).