Thibaut Jonckheere

Jérôme Rech

Benoît Grémaud

Thierry Martin

Anyonic statistics revealed by the Hong-Ou-Mandel dip for fractional excitations

Anyonic statistics revealed by the Hong-Ou-Mandel dip for fractional excitations T. Jonckheere, J. Rech, B. Grémaud, T. Martin Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France (Dated: July 14, 2022)

The fractional quantum Hall effect (FQHE) is known to host anyons, quasiparticles whose statistics is intermediate between bosonic and fermionic. We show here that Hong-Ou-Mandel (HOM) interferences between excitations created by narrow voltage pulses on the edge states of a FQHE system at low temperature show a direct signature of anyonic statistics. The width of the HOM dip is universally fixed by the thermal time scale, independently of the intrinsic width of the excited fractional wavepackets. This universal width can be related to the anyonic braiding of the incoming excitations with thermal fluctuations created at the quantum point contact. We show that this effect could be realistically observed with periodic trains of narrow voltage pulses using current experimental techniques.

Fractional quantum Hall effect (FQHE) is an important example of a many-body system where electronic correlations have an essential impact. [1] When a fraction ν of the states of the lowest Landau level is occupied, the system reaches a state which cannot be understood without electronic interactions. The well-known Laughlin wavefunction describes the highly correlated ground state of the FQHE when ν = 1/(2n + 1) for n integer. The fundamental excitations of the FQHE are anyons: quasiparticles which bear a fractional charge, and obey fractional statistics. [2,3] In a given Laughlin state, when two anyons are exchanged, the system acquires a phase exp(iπν), to be contrasted with the ±1 of bosonic/fermionic statistics. More complex fractions exist beyond the Laughlin series. They are described with more involved states than the Laughlin wavefunction, with correspondingly more complex statistics. In particular, some states are believed to obey non-Abelian statistics, which could have important consequences for applications in quantum computing [4].

Experimental demonstration of these exceptional properties have lead to an intense activity. A fractional charge e/3 was observed for the Laughlin state with ν = 1/3 more than twenty years ago, by measuring the shot noise across a quantum point contact (QPC) in the tunneling regime, where individual fractional quasiparticles can tunnel between opposite edge states. [5][6][7][8] Fractional statistics, however, has proved more difficult to observe. Only very recently, two different experiments have been able to clearly show specific signatures directly associated with the fractional statistics of anyonic quasiparticles. [9,10] Electronic transport in FQHE occurs only through chiral edge modes, traveling at the boundary of the system, which can be used as 1d electron beams, equivalent to the light beams for photons. The growing field of electronic quantum optics aims at using these edge states to realize transport experiments inspired from setups involving photons in quantum optics. [11] An important example is the Hong-Ou-Mandel (HOM) interference experiment, where two identical photons are sent with a controlled time delay on a beam-splitter. There, the intensity correlations gives useful information on the photon coherence properties. [12] It has been performed a few years ago for electron collision in the integer QHE, where it was shown to give precious information on the electronic wavepackets and on the many-body electronic state. [13][14][15] In this work, we show that using narrow periodic pulses of voltage, periodically exciting fractional charges, and measuring the HOM noise at the output of a QPC, one obtains a signal which is directly related to the anyonic statistics. To this aim, we first explain the unique properties of the time-dependent tunneling current at a QPC when a single fractional quasiparticle is incident, which are associated with braiding of the fractional quasiparticle with the thermal anyonic excitations occurring at the QPC. Our quantitative predictions, obtained with perturbative calculations performed using the nonequilibrium Keldysh Green function formalism, could be checked with current experimental techniques, providing a relatively easy path for the study of fractional statistics.

We consider a FQH bar, with Laughlin filling factor ν = 1/(2n + 1) for n integer, and describe the edge states in terms of the bosonic Hamiltonian

H 0 = v F 4π dx µ=R,L (∂ x φ µ ) 2
where φ R and φ L are the bosonic fields describing the right-and left-moving edge states [16] and v F is their propagation velocity along the edges. The quasiparticle (QP) operator is related to the bosonic field through a bosonization identity ψ R/L (x) = U R/L /(2πa)e ±ik F x e -i √ νφ R/L (x) with a a small cutoff parameter and U R/L a Klein factor. The presence of a QPC (located at x = 0), in the weak backscattering regime, allows the tunneling of individual QP of charge e * = νe between the two edges. This is described by the tunneling Hamiltonian H T = Γψ † R (0)ψ L (0) + H.c. See Fig. 1 for a sketch of the setup.

To better understand the importance of the anyonic statistics for tunneling at the QPC, let us first consider the somewhat simpler situation where a single QP of charge e * is incoming on the R edge. To this aim,

V R (t) V L (t) e * e * I T ϕ R ϕ L FQHE FQHE FIG.
1. Schematic view of the setup. A Hall bar in the Laughlin series, whose edge states are described by the bosonic fields φR and φL, is equipped with a QPC at position x = 0. The right-and left-moving edges are driven respectively by the time-dependent potential VR(t) and VL(t), resulting in a tunneling current IT in between edges at the position of the QPC.

when computing physical quantities (current, etc.), we replace the ground state by a prepared state |ϕ = ψ † R (-x 0 , -T )|0 where a single QP has been added at an initial time -T < 0. Without loss of generality, we choose x 0 = v F T , such that the QP reaches the QPC position at t = 0. We now proceed with the perturbative calculation of the mean tunneling current I T (t) at the QPC, using standard Keldysh Green function formalism. The tunneling current operator is given by I T (t) = ie * (Γψ † R (0, t)ψ L (0, t) -H.c.). To lowest order in Γ, the mean current is given by [17]

I T (t) = - i 2 dt η,η η ϕ T K I T (t η ) H T t η ϕ , (1)
where T K is time-ordering along the Keldysh contour, and η, η = ± are Keldysh indices. Using the bosonized form of the quasiparticle operators, and keeping in mind that x 0 = v F T , we have

I T (t) = Γ 2 e * 2 dt η,η η G σ ηη tt (t -t ) 2 × G(-t )G(t) G(t )G(-t) - G(t )G(-t) G(-t )G(t) , (2)
with

G(t) = 1 2πa sinh(iπa/(βv F )) sinh(iπa/(βv F ) -πt/β) ν , (3)
where

G(σ ηη tt (t -t )) = G(0; t η , t η ) = 0|T K ψ † (0, t η )ψ(0, t η )|0 , with σ ηη tt = sign(t - t )(η + η )/2 + (η -η)/2
accounting for the effect of time-ordering along the Keldysh contour. G is the quasiparticle Green function (identical for right and left movers), directly obtained from its bosonic counterpart, with β the inverse temperature. Note that the power ν leads to a slow decay of this Green function at long times since ν < 1. In the limit of vanishing cutoff a → 0, it is easy to check that G(t)/G(-t) = exp(-sign(t) × iπν). This directly arises from the nontrivial exchange properties of anyonic quasiparticles, exploiting their linear dispersion along the edge.[17] It follows that the last factor of Eq. ( 2) can be simplified as

G(-t )G(t) G(t )G(-t) = exp -i ν t t dτ 2πδ(τ ) . (4)
The current can thus be written as

I T (t) = 2ie * Γ 2 t -∞ dt sin 2πν t t dτ δ(τ ) × G(t -t ) 2 -G(t -t) 2 . ( 5
)
One readily sees from Eq. ( 5) that the tunneling current has remarkable properties, which are unique to fractional charge tunneling in the FQHE (see supplemental material for the complete analytical expression). It is of course zero for t < 0, i.e. before the arrival of the e * QP. On the other hand, for t > 0, the t integration is restricted to the negative portion of the real axis, and the current is simply proportional to sin(2πν). This, in turn leads to a non-zero current even for a time t taken long after the e * QP has reached the QPC position, as a consequence of the slow decay in time of the Green function G(t -t ). Indeed, the current slowly decreases at first, as

∼ β 1-2ν Cte -ν 1-2ν [1 -exp(-2πνt/β)] 1-2ν ,
for times t smaller than the thermal time scale τ Th = β, before vanishing exponentially for times larger than τ Th . The mean current thus remains finite for a long time interval, set by the thermal time scale. It is important to stress out that this is in sharp contrast with the case of an electron charge incoming on the QPC, since even for fractional edge states, the mean tunneling current is nonzero only at the specific time that the electron reaches the position of the QPC. [17] This nontrivial behavior of the tunneling current after the arrival of a single QP of charge e * can be directly linked to the anyonic statistics of the fractional excitations. The phase 2πν occurring for t < 0 < t can be understood qualitatively from Eq. ( 1) by considering the time-ordering of the right-moving edge operators (ψ R , ψ † R ). From the expressions of |ϕ , I T and H T , one readily sees that the average current in Eq. ( 1) involves a contribution of the form

T ψ R (0)ψ † R (t)ψ R (t )ψ † R (0)
, as the prepared state ensures that the QP reaches the QPC at time 0. For t > 0 and t < 0, the time ordering thus requires to bring both ψ R (0) and ψ † R (0) between the operators at t and t , yielding twice a phase πν. On the opposite, if t and t have the same sign, one can easily see that the exchanges needed for the ordering now contribute with opposite phases, thus giving a zero net result. An equivalent point of view, developed in Refs. [18][19][20] is to see the expression of Eq. ( 1) as the interference between a process where a quasiparticle/quasihole (QP/QH) excitation is created at the location of the QPC at time t , before the passage of the e * QP, and another where the QP/QH is created at time t, after the passage of the e * QP. In both points of view, the tunneling current is non-zero because of the braiding of the incoming fractional QP with a thermal QP/QH excitation created at the QPC. This braiding results from the anyonic statistics, giving a nontrivial phase πν when two quasiparticles are exchanged.

As there is currently no experimental way to emit a single fractional QP in a controlled manner, [21] the prepared state used here is not directly achievable in practice. However, this is not a fundamental obstacle, as we now show that the same tunneling current, with the same signature of fractional statistics, can be obtained by applying a short voltage pulse which excites a fractional average charge. This is a highly nontrivial statement, as it is known that such a voltage pulse does not create the same many-body state as the one obtained by adding a single quasiparticle on top of the ground state. [START_REF] Rech | [END_REF] The presence of an external time-dependent voltage bias leads to an extra term in the total Hamiltonian, of the form

H V = -2e √ ν v F V (x, t)∂ x φ R .
The voltage can be taken into account by using the following transformation:

φ(x, t) = φ (0) (x, t) + e √ ν t -∞ dt V (x , t ), (6)
with x = x -v F (t -t ) and where φ (0) (x, t) is the equilibrium bosonic field. [START_REF] Rech | [END_REF] Assuming that the voltage is applied on a long contact, we can simplify

t -∞ dt V (v F (t -t), t ) t -∞
dt V (t ). This leads to a time-dependent tunneling amplitude at the QPC Γ

(t) = Γexp[ie * t -∞ dt V (t )].
Proceeding with the perturbative calculation of the tunneling current, one gets [17]

I T (t) = 2ie * Γ 2 t -∞ dt sin e * t t dt V (t ) × G(t -t ) 2 -G(t -t) 2 . ( 7
)
One can thus readily recover the result of Eq. ( 5), provided that one chooses a voltage pulse with the shape V (t) = 2π e δ(t), which excites a mean charge e * = νe. The tunneling current I T (t) is the same for a single QP of charge e * arriving on the QPC, or when applying a very short voltage pulse V (t) exciting a mean charge e * .

This picture is further generalized by considering a voltage V (t) composed of several short pulses of charge e * . There, the phase of the sine term counts the number of fractional charges e * that have passed through the QPC, each of them contributing a phase 2πν. This then has important consequences for the tunneling current. For example, at filling factor ν = 1/3, when two short fractionally charged pulses arrive at the QPC with a time delay much smaller than the thermal scale, the main contribution to the current in Eq. ( 7) comes with a factor sin(4π/3) < 0, making it negative. As an illustration, Fig. 2 shows the current for an ensemble of short pulses (each with a charge e * , and a width δt β). The dashed lines show the arrival times of the pulses at the QPC. We see that the current decreases slowly in absolute value after each pulse reaches the QPC, reflecting the slow decrease of the QP Green function G. More interestingly, the value of the current depends on the history of the pulses applied at earlier times. In particular, as argued above, the current can be negative when two pulses arrive at closely separated times (e.g. for t between 0.2 and 0.4β). The inset of Fig. 2 shows the equivalent picture when similar pulses, but carrying a charge e rather than e * , are incident on the QPC. One can see that the current is non-zero only when the pulse is precisely at the QPC position, and that there is no effect related to the pulses at earlier times. Note that the same expression for the tunneling current, Eq. ( 7), can also be used to describe a random stream of pulses, which allows one to recover exactly the results of Ref. [23] describing the collision between two Poissonian streams of charges e * .

While the use of voltage pulses is routinely performed, the measurement of time-dependent currents still constitutes an experimental challenge in quantum Hall junctions. We now propose a simpler alternative, within grasp of modern experiments, in order to reveal the effect of anyonic statistics. This relies on the measurement of the HOM noise, i.e. the current correlations resulting from two individual voltage pulses of fractional charge colliding at the QPC with a controllable time delay.

Let us first consider two narrow pulses of charge e * , each conveniently represented as a delta function in time, incoming on the two inputs of the QPC. The tunneling current noise is defined as

S(t, t ) = T K δI T (t -) δI T (t + ) , (8)
with δI T (t) = I T (t) -I T (t) , and ± are Keldysh indices. The HOM noise is the zero-frequency tunneling noise, when two pulses are incident on the QPC with a given time delay δt. It serves as a measure of the interference between the colliding excitations at the QPC. It can be written as [17]

S HOM (δt) = 1 2S HBT ∞ -∞ dtdt G(t -t) 2 × {cos [2πνf δt (t, t )] -1} , (9)
where the function f δt (t, t ) is 1 if only one of the times t or t is in the interval [-δt/2, δt/2], and 0 otherwise, and normalization is given by twice the value of the Hanbury-Brown Twiss (HBT) noise S HBT [24].

A very good approximation of Eq. ( 9), which turns out to be exact in the limit δt/β → 0, is given by the formula

S HOM (δt) δt τ Th -→ 1 -exp -2πν |δt| β . (10)
This result shows a behavior typical of a HOM dip for long and short time delays. For very large |δt|, it saturates to 1 as the two incident charges e * reach the QPC at very distant times and do not interfere at all, therefore reproducing twice the amount of the HBT noise. For δt = 0, the HOM dip drops all the way to 0, as a result of perfect interference between the two identical incoming charges. The most important result, however, is the behavior at intermediate δt: Eq. (10) shows that the width of the HOM dip is ∼ β, set by the thermal time scale τ Th , independently of the width of the incoming pulses. This is in sharp contrast with the conventional HOM dip, for example between electronic wavepackets in the integer QHE [14,25], where the dip width is directly proportional to that of the incoming wavepacket. This striking result can be understood from our discussion of the tunneling current above. Indeed, we showed that, as a consequence of anyonic statistics and the braiding with thermal excitations, a single charge e * reaching the QPC creates a nonzero current up to times ∼ β after the tunneling event occurred. Two charges incident on both inputs of the QPC thus interfere up to times set by the thermal time scale, which explains the width of the HOM dip.

The observation of a HOM dip of width ∼ β, when sending two fractionally charged short pulses (temporal width τ Th ) can thus provide a direct proof of the anyonic statistics of these fractional charges. Experimentally, however, it is not yet possible to measure the noise from individual charges, and one has to resort to using a periodic signal. We now show how a realistic periodic voltage bias with frequency ω, sending pulses of charge q e (with non-integer q), can be used to observe the HOM dip of width ∼ β. For illustrative purposes, we consider a periodic voltage V (t) consisting of Lorentzian pulses, also known as levitons, [START_REF] Lee | Orthogonality catastrophe in a mesoscopic conductor due to a time-dependent flux[END_REF][START_REF] Keeling | [END_REF][28][29] but the results are independent of the actual shape of the voltage potential, as long as the pulse width is small compared to β. We use the Floquet formalism, where the essential ingredients are the coefficients p l , which correspond to the Fourier coefficients of the phase φ(t) = e * t -∞ dt V AC (t ) created by the AC part of the time-dependent voltage V (t). The DC part of the voltage leads to a mean charge qe injected per period, with q = e * V DC /ω. We consider that the voltages V R (t) and V L (t), which are applied on the right and left edge respectively, differ by a time-shift δt only, so that

V L (t) = V R (t -δt) = V DC π k η η 2 + (t/T 0 -k) 2 , ( 11
)
where T 0 = 2π/ω is the period of the drive, and η is the finesse, which describes the width of the Lorentzian relative to the period of the drive [17].

The results for the normalized HOM noise for a periodic Lorentzian drive, in the case of a filling factor ν = 1/3, are shown in Fig. 3. Here, we have chosen realistic values for the experimental parameters, with a frequency ω = 1 × 2π GHz, and a finesse η = 0.01. [30] The black dotted line shows the shape of the narrow Lorentzian pulse over one period, thus highlighting the width of the incoming wavepackets. The full curves show the HOM dip as a function of the time-shift δt. In panel a), the average charge per pulse is fixed to qe = e/3, and the temperature T is varied from 250mK down to 25mK. One can readily see that, while the width of the HOM dip is close to that of the Lorentzian pulse at T =250mK, it significantly increases as the temperature is lowered, ultimately being much larger at T =25mK. Since it is rather easy experimentally to modulate the injected charge per period qe, we consider, in panel b), a fixed temperature T =25mK, and an injected charge per period which varies from qe = e down to qe = e/3. There, the width of the HOM dip is similar to that of the incoming pulse for q = 1 (corresponding to the injection of a full electron per period on each edge), before increasing substantially as q is lowered, recovering a wide HOM dip for q = 1/3. The thick dashed line corresponds to the analytical prediction of Eq. ( 10) for T = 25mK. This shows a very good agreement with the full numerical result obtained for q = 1/3, with only a small underestimation of the width of the dip associated with the assumption of infinitely sharp pulses.

In conclusion, we have shown that the anyonic statistics of quasiparticles in the FQHE has direct consequences on the HOM interference of excitations created by narrow voltage pulses. Contrarily to the usual picture, 10) for two infinitely narrow pulses at temperature T =25mK. The dotted curve shows the shape of V (t) over one period for the chosen value of the finesse η.

where the width of the HOM dip is trivially related to the temporal extension of the incoming excitations, here the width of the dip is fixed by the thermal scale, which can be much larger at low temperature. We have shown how this can be explained by the anyonic braiding of the incoming quasiparticles with thermal excitations naturally occurring at the QPC. Our proposal could be realized with current experimental techniques, and could lead to an original and relatively simple way to observe directly the consequences of anyonic statistics in the FQHE. A natural extension of this work would be to consider more exotic fractions of the FQHE, where several edge states carrying different QP are present, like ν = 2/5 or ν = 2/3 [31,32], and which can even obey non-Abelian statistics as in ν = 5/2 [4,[33][34][35].

We are grateful to G. Fève, C. Glattli, B. Plaçais and U. Gennser for enlightning discussions. This work was carried out in the framework of the project "ANY-HALL" (Grant ANR No ANR-21-CE30-0064-03). Centre de Calcul Intensif d'Aix-Marseille is acknowledged for granting access to its high performance computing resources for early parts of this work.

Supplemental material

GREEN FUNCTIONS AND THEIR PROPERTIES

The quasiparticle Green function is defined as

G R/L x, x ; t η , t η = T K ψ † R/L (x, t η ) ψ R/L x , t η . (S1)
Using the properties of time ordering, and the linear dispersion along the edge, this can be recast under the simplified form

G R/L x, x ; t η , t η = G R/L σ ηη tt t -t ∓ x -x v F , (S2)
where

σ ηη tt = sign(t -t )(η + η )/2 + (η -η)/2 and G R/L (t) = ψ † R/L (0, t)ψ R/L (0, 0) . (S3)
Invoking the bosonization identity, this is further reduced as

G R/L (t) = 1 2πa e i √ νφ † R/L (0,t) e -i √ νφ R/L (0,0) = 1 2πa e νG R/L (t) , (S4)
where we introduced the bosonic Green function G R/L (t) = φ † R/L (0, t)φ R/L (0, 0) . From the free Hamiltonian H 0 , one can readily extract the corresponding Green function for the bosonic modes as

G R/L (t) = -log   sinh i πa βv F -πt β sinh i πa βv F   , (S5)
so that the quasiparticle Green function ultimately reads

G R/L (t) = 1 2πa   sinh i πa βv F sinh i πa βv F -πt β   ν (S6)
One can easily show that this Green function is identical for right-and left-movers, so that we can safely drop the R/L subscript from this point onward.

As anyons obey fractional statistics, they show nontrivial exchange properties which ensure that, at equal time, one has

ψ † R (0, t)ψ R (x, t) = e -iπνSign(x) ψ R (x, t)ψ † R (0, t) (S7)
Making use of the linear dispersion along the edge, this is rewritten as

ψ † R (0, t)ψ R 0, t - x v F = e -iπνSign(x) ψ R 0, t - x v F ψ † R (0, t) (S8)
Since this is valid for any set of parameters (x, t), one can choose x = v F t, without loss of generality. Taking then the quantum average, this yields

ψ † R (0, t)ψ R (0, 0) = e -iπνSign(t) ψ R (0, 0)ψ † R (0, t) G(t) = e -iπνSign(t) G(-t) (S9)
It follows that the value of the ratio G(t)/G(-t) can be viewed as a direct consequence of the exchange statistics of anyons.

COMPUTING THE TUNNELING CURRENT

Tunneling current when injecting a single quasiparticle

The tunneling current operator reads I T (t) = ie * (Γψ † R (0, t)ψ L (0, t) -H.c.). Here, we consider the situation where a single quasiparticle is incoming along the right edge, described by a prepared state of the form |ϕ = ψ † R (-x 0 , -T )|0 . To lowest order in Γ, the mean current is thus given by

I T (t) = - i 2 dt η,η η ϕ T K I T (t η ) H T t η ϕ = e * 2 dt , η,η η 0|T K ψ R (-x 0 , -T -) Γψ † R (0, t η )ψ L (0, t η ) ( ) × Γψ † R (0, t η )ψ L (0, t η ) ( ) ψ † R (-x 0 , -T + )|0 (S10)
where = ± is used to include the Hermitian conjugated terms, such that for = +, one has for any operator O,

O (+) = O while for = -, one has O (-) = O † .
Here, T K ensures the time-ordering along the Keldysh contour, and η, η = ± are Keldysh indices. Note that we consider the injection of QP to have happened in the distant past. The Kelsdysh indices added to the times -T have been chosen to ensure that the ψ R (-x 0 , -T -) and ψ † R (-x 0 , -T + ) operators remain in the same position after time ordering, independently of the values of t and t , for T large enough. In particular, keeping in mind that x 0 = v F T (corresponding to a quasiparticle reaching the QPC at t = 0), this allows us to simplify some of the resulting Green functions as

G(-x 0 , 0; -T -, t η ) = G(-t) (S11) G(0, -x 0 ; t η , -T + ) = G(t), (S12)
independently of η and t, provided that t T . Using the bosonized form of the quasiparticle operators, we have

I T (t) = Γ 2 e * 2 dt η,η η G σ ηη tt (t -t ) 2 G(-t )G(t) G(t )G(-t) (S13)
Using the properties of the Green function derived in Eq. (S9), this then becomes

I T (t) = 2ie * Γ 2 t -∞ dt sin 2πν t t dτ δ(τ ) × G(t -t ) 2 -G(t -t) 2 (S14)
Changing the integration variable to τ = -t , and using the expression of the Green function, we get:

I T (t) = θ(t)2ie * Γ 2 (2πa) 2 sin(2πν) ∞ 0 dτ sinh(iπT τ 0 ) sinh(πT (iτ 0 -t -τ ) 2ν - sinh(iπT τ 0 ) sinh(πT (iτ 0 + t + τ ) 2ν (S15)
where τ 0 = a/v F , T = 1/(k B β) is the temperature, and we use k B = = 1. Defining the reduced variables α = πT τ 0 , u = πT τ and z = πT t, the first term in the integral can be written as

∞ 0 du sinh(iα) sinh(iα -z -u) 2ν = ∞ 0 du e iα -e -iα -e -iα e z 1 1 -e 2iα e -2z e -2u 2ν e -2ν u (S16) = e -z 1 -e 2iα 2ν ∞ 0 du 1 -e 2iα e -2z e -2u -2ν e -2ν u (S17) = 1 2 e -z 1 -e 2iα 2ν 1 ν 2 F 1 2ν, ν, ν + 1, e 2iα-2z (S18)
where 2 F 1 is the hypergeometric function. Using this result, the current can eventually be recast as

I T (t) = θ(t) 2e * Γ 2πv F τ 0 2 sin(2πν) 2πνT e -2νπT t (2 sin(πT τ 0 )) 2ν × 2Im 2 F 1 2ν, ν, ν + 1, e -2νπT t e -2iπT τ0 e iπν(1-2T τ0) (S19)
Performing a perturbative expansion in the tunneling amplitude Γ, this gives up to second order

I T (t) = e * 2 Γ 2 η,η η ∞ -∞ dt exp i e * t -∞ dt V (t ) T K ψ † R (0, t η )ψ R (0, t η ) T K ψ L (0, t η )ψ † L (0, t η ) (S27)
Using the expression for the quasiparticle Green function, and performing explicitly the sum on the Keldysh indices η and η , one eventually gets

I T (t) = 2ie * Γ 2 t -∞ dt sin e * t t dt V (t ) G(t -t ) 2 -G(t -t) 2 . ( S28
)
where the Keldysh summations end up restricting the t integral from -∞ to t.

COMPUTING THE NOISE

General expression

The current noise is defined as:

S(t, t ) = T K δI T (t -) δI T (t + ) (S29)
with δI T (t) = I T (t) -I T (t) , and ± are Keldysh indices.

In the presence of a voltage bias applied to both edges, the tunneling part of the Hamiltonian can be written as

H T (t) = Γ exp ie * t -∞ dt (V R (t ) -V L (t )) ψ † R (0, t)ψ L (0, t) + H.c. ( S30
)
where we applied a standard gauge transformation in order to reabsorb the effect of the voltage drives into the tunneling amplitude. In this situation, the tunneling current operator reads

I T (t) = ie * Γ exp ie * t -∞ dt (V R (t ) -V L (t )) ψ † R (0, t)ψ L (0, t) -H.c. . (S31)
Substituting this back into Eq. (S29), one readily obtains, up to lowest order in the tunneling amplitude Γ

S(t, t ) = 2 e * Γ 2πa 2 cos e * t t dt (V R (t ) -V L (t )) G(t -t ) 2 . (S32)
In what follows, we focus on the Hanbury-Brown Twiss (HBT) and the Hong-Ou-Mandel (HOM) setups, corresponding respectively to applying a single voltage drive, or to applying both of them.

HOM noise for two narrow pulses of average charge e *

We consider here the case of two infinitely short pulses so that both V R (t) and V L (t) are composed of a single delta function, with a time-shift δt between them. Focusing on pulses of average charge e * , one can thus write

V R (t) = 2π e δ t + δt 2 V L (t) = 2π e δ t - δt 2 . ( S33
)
The cosine factor entering the expression for the noise in Eq. (S32) then simply reduces to either cos(2πν) or to 1, depending on the values of t and t , so that we write it as cos [2πνf δt (t, t )]. The newly defined function f δt (t, t ) is 1 if one of the times t or t is in the interval [-δt/2, δt/2] while the other one is not, and reduces to 0 otherwise. The HOM noise is defined as the zero-frequency noise due to the collision of these two excitations, as a function of the time-interval δt. Focusing on the zero-frequency contribution, and filtering out the equilibrium thermal noise (by subtracting the value in the absence of voltage drives), one has for the un-normalized HOM noise

S HOM = S(V R , V L ) -S(0, 0) = 2 e * Γ 2πa 2 ∞ -∞ dt ∞ -∞
dt {cos [2πνf δt (t, t )] -1} G(t -t ) 2 (S34)

HOM noise in the Floquet formalism

The applied voltages on the right and left edges are now given by periodic Lorentzian pulses. They are identical except for a time-shift δt, so that

V L (t) = V R (t -δt) = V DC π k η η 2 + (t/T 0 -k) 2 (S44)
In the Floquet formalism, the essential ingredients are the coefficients p l , which are the Fourier components of the accumulated phase φ(t) = e * t -∞ dt V AC (t ) created by the AC part of the time-dependent voltage. In practice, it is convenient to introduce the time-dependent voltage V diff (t) = V R (t) -V L (t) which naturally appears in the expression of the noise.

Starting back from the general expression of Eq. (S32), and inserting the p l coefficients associated with a generic drive V (t) (this allows us to replace V with V R , V L or V diff ), one can write S(t, t ) = 2 e * Γ 2πa p * l p m e ie * V DC (t-t ) e ilωt e -imωt + e -ie * V DC (t-t ) e -imωt e ilωt G(t -t ) 2 (S45)

where ω = 2π T0 is the frequency of the drive. In this Floquet formalism, the zero-frequency noise is now defined as S = dτ where we introduced the average charge q = e * V DC ω injected by the drive over one period. Introducing the coefficients p diff,l for the voltage difference V diff (t), as well as the coefficients p L,l and p R,l corresponding to V L (t) and V R (t) applied individually, and noticing that V R,DC = V L,DC = qω e * , while V diff,DC = 0, one finally has for the HOM noise ratio with

F (p l , q) = |p l | 2 Γ ν + i l + q 2πθ 2 cosh l + q 2θ (S49)
and θ = k B T / ω is the reduced temperature. Note that this expression is very general and can describe any kind of periodic potentials, provided that one uses the correct corresponding expressions of the p l coefficients.

t

  FIG.2. Mean current IT (t) (in units of e/β) as a function of t (in units of β) corresponding to Eq. (7) with ν = 1/3, for a random ensemble of short pulses of width β/100, each carrying a charge e/3. The arrival times at the QPC are shown as dashed vertical lines (pulses for t < -0.4β are not shown). Inset: same figure for pulses carrying a charge e.

FIG. 3 .

 3 FIG.3. HOM noise as a function of δt for a filling factor ν = 1/3, for V (t) made of voltage pulses with Lorentzian shape of finesse η = 0.01, with ω = 2π/T0 = 2πGHz. a): each pulse carry a fractional charge e/3, and the temperature in mK is indicated near each curve. b): temperature is 25mK, and the charge of each pulse (in units of e) is shown near each curve. The thick dashed line shows the theoretical prediction of Eq. (10) for two infinitely narrow pulses at temperature T =25mK. The dotted curve shows the shape of V (t) over one period for the chosen value of the finesse η.

2 = 2 e * Γ 2πa 2 l |p l | 2 dτ

 222 p m e ie * V DC τ e ilω( t+ τ 2 ) e -imω( t-τ 2 ) + e -ie * V DC τ e -imω( t+ τ 2 ) e ilω( t-τ 2 ) G(τ ) cos [(l + q)ωτ ] G(τ ) 2 (S47)

S 2 l 2 (

 22 HOM (δt) = S HOM 2S HBT = l F (p diff,l , 0) -|Γ(ν)| [F (p L,l , q) + F (p R,l , q)] -2 |Γ(ν)| S48)

Taking then the leading order in the cutoff parameter τ 0 leads to

2 sin(πν) sin(2πν) ν e -2νπT t (2πT ) 2ν-1 2 F 1 2ν, ν, ν + 1, e -2νπT t (S20)

We see that this is a function of 2νπT t = 2νπt/β, which implies that the typical length scale for this function is ∼ β.

The behavior of the current in the two limits t β and t β is obtained by using the asymptotic behavior of the hypergeometric function:

Tunneling current when injecting a single electron

It is instructive to repeat the same kind of derivation, only this time considering the situation where a single electron is incoming along the right edge. The prepared state now takes the form |ϕ = Ψ † R (-x 0 , -T )|0 , where the electron operator Ψ R satisfies the bosonization identity

. Following a similar derivation to the one above, one obtains instead of Eq. (S13), the following expression for the tunneling current

From the properties of the quasiparticle Green function, Eq. (S9), one readily sees that for t = 0

so that the tunneling current vanishes at all times t = 0 and is nonzero only at the specific time that the electron reaches the QPC.

Tunneling current in the presence of a time-dependent voltage

In the presence of a voltage bias, the tunneling part of the Hamiltonian can be written as

where it now contains the effect of the applied votlage V (t). The tunneling current operator now reads

Taking the quantum average, the mean tunneling current is given in full generality by

where the sum on = ± is used to represent the Hermitian conjugate, and η, η = ± are Keldysh indices.

Similarly, one can work out the expression for the corresponding noise when only one of the drives is present. The resulting HBT noise reads

The standard HOM noise ratio is then defined as the ratio of the un-normalized HOM noise to twice the HBT noise, so that

Substituting the actual value of f δt (t, t ), this can be further rewritten as

where we introduced

with the reduced variable δ = π |δt| /β, and the infinitesimal α = πτ 0 /β. This integral can be worked out as

where one clearly sees that for δ 1, the exponential prefactor dominates, so that

It follows that, in the regime where |δt|/β → 0, one has

1 -e -2πν |δt| β (S41)

HOM noise for two narrow pulses of average charge qe

The previous results can be easily extended to the case of pulses carrying a different charge. We now define

Following the lines of the previous calculation, one can similarly obtain an expression for the HOM noise ratio as

Interestingly, while the resulting integrals are finite for different domains in time, they always contain a prefactor cos(2πq) -1. For q / ∈ Z, this prefactor simplifies between numerator and denominator, leaving us with the same expression as Eq. (S37), independently of q. This, however, is specific to the very short pulse situation, as a finite extent leads to slightly different contributions for the numerator and denominator, which depend on q in a nontrivial way.


Gaëlle Lebrun

Bilal El Mokdad

Claude Le Men

Véronique Pimienta

Christophe Coudret

Clément Roux

Gilles Hébrard

Nicolas Dietrich
email: dietrich@insa-toulouse.fr

LUMINESCENT PROBE SYNTHESIS FOR OXYGEN VISUALIZATION TECHNIQUE: APPLICATION TO THE EFFECT OF SURFACTANT STRUCTURE ON OXYGEN MASS TRANSFER

Keywords: PLIF-I, fluorescence quenching, surfactants, oxygen transfer, gas/liquid interface, adsorption

Planar Laser Induced Fluorescence with Inhibition (PLIF-I) is a powerful technique for studying the local gas/liquid oxygen mass transfer from a single rising bubble. However, to track oxygen in the liquid phase, it is necessary to use an oxygen sensitive luminescent probe, which needs to be unreactive toward the liquid phase. This article presents the synthesis of a fluorophore, [Ru(dpp-diSO3)3]Na4, to be used in liquid phases containing anionic molecules. To avoid electrostatic interactions, the fluorophore and the component present in the liquid phase need to have the same charge. Thanks to the use of this new luminescent probe in PLIF-I application, the influence of the chain length of anionic surfactants (dodecyl sulfate sodium salt and tetradecyl sulfate sodium salt) on oxygen mass transfer has been studied at different concentrations (1.3×10 -3 -2.5×10 -7 mol.L -1 ). Results show that, for a given bulk concentration, the longer the hydrophobic chain, the greater the decrease in velocity and mass transfer coefficient.

These results are compared with those of a previous paper dealing with length of cationic and nonionic surfactants and a correlation is proposed, taking account of the intrinsic properties of surfactants and their concentration in the liquid phase.

Introduction

Mass transfer at gas-liquid interfaces is of great interest in many industrial applications [START_REF] Roustan | Transferts gaz-liquide dans les procédés de traitement des eaux et des effluents gazeux[END_REF]2], such as chemical and biological processes, and in environmental systems. For wastewater treatment, the water aeration through air bubbles is determinant for the efficiency of pollutant degradation [START_REF] Gillot | Predicting Oxygen Transfer of Fine Bubble Diffused Aeration Systems-Model Issued from Dimensional Analysis[END_REF]. The high industrial demand concerning decreases of power consumption during the aeration step led to intensive studies about mass transfer in gas-liquid contactors, but this issue becomes more complex in the presence of contaminants, especially surfactants, which are amphiphilic molecules, coming from all kinds of domestic detergents, and are also excreted by microorganisms present in wastewater treatment plants.

As a result, surfactants are found in a large amounts in wastewater and predicting mass transfer in such configurations requires a good understanding of all the phenomena occurring during mass transfer in the presence of contaminants.

If the medium in which the bubble rises contains surfactants, this has an impact on the hydrodynamic parameters of the bubble, such as shape, velocity and interfacial area, [START_REF] Clift | Bubbles, Drops, and Particles[END_REF][START_REF] Guet | Fluid Mechanical Aspects of the Gas-Lift Technique[END_REF][START_REF] Vázquez | Application of the Danckwerts Method in a Bubble Column: Effects of Surfactants on Mass Transfer Coefficient and Interfacial Area[END_REF]. The velocity of the bubble is a key parameter in mass transfer phenomena because it affects (i) the residence time of the bubble in the liquid phase and also (ii) buoyancy at the vicinity of the interface. As has been described by Palaparthi et al. [START_REF] Palaparthi | Theory and Experiments on the Stagnant Cap Regime in the Motion of Spherical Surfactant-Laden Bubbles[END_REF], surfactants adsorb to the bubble surface and are swept from the nose to the rear of the bubble. The resulting high local concentration at the rear leads to a desorption of surfactants. A surface tension gradient appears between the nose and the rear of the bubble and leads to the appearance of a Marangoni flow, which opposes the advection flow and slows the bubble down. From this description of bubble contamination, the bubble can be split in two regions [START_REF] Sadhal | Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film -exact solution[END_REF]: at its nose, the bubble is clean and the flows satisfy slip conditions, while the rear of the bubble is fully contaminated and the interfacial velocity drops to zero. A contamination angle θcap can be used to describe the level of contamination of the rear of the bubble. Sadhal and Johnson [START_REF] Sadhal | Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film -exact solution[END_REF] proposed a calculation of this angle from the drag force acting on the bubble, which is highly dependent on the contamination. The large majority of the literature conclude on a decrease of velocity and mass transfer coefficient in the presence of surfactants, however it can be underlined that Àlvarez et al. and Gómez-Díaz et al. that at very low surfactant concentration, mass transfer can be enhanced thanks to the buoyancy created by Marangoni flow [START_REF] Gómez-Díaz | Mass-Transfer Enhancement or Reduction by Surfactant Presence at a Gas-Liquid Interface[END_REF][START_REF] Álvarez | Continuous Removal of Carbon Dioxide by Absorption Employing a Bubble Column[END_REF].

Taking this molecular consideration of bubble colonization by surfactants into account, numerous experimental studies have investigated mass transfer from a contactor point of view [START_REF] Bouaifi | A Comparative Study of Gas Hold-up, Bubble Size, Interfacial Area and Mass Transfer Coefficients in Stirred Gas-Liquid Reactors and Bubble Columns[END_REF][START_REF] Cockx | Modeling of Mass-Transfer Coefficient K-L in Bubble-Columns[END_REF][START_REF] Hebrard | Effect of Surfactants on Liquid Side Mass Transfer Coefficients: A New Insight[END_REF][START_REF] Sardeing | Effect of Surfactants on Liquid-Side Mass Transfer Coefficients in Gas-Liquid Systems: A First Step to Modeling[END_REF][START_REF] Vasconcelos | Gas-Liquid Mass Transfer to Single Bubbles: Effect of Surface Contamination[END_REF]. The study of bubble columns brings us closer to an industrial application. However, because of the numerous events taking place in the column at the same time (coalescence, breakage), and measurement of an averaged mass transfer in the whole column , it is sometimes difficult to identify and isolate mechanisms involved in mass transfer. Some research works have pointed out that some additional resistance having a physicochemical source should be taken into account [START_REF] Hebrard | Effect of Surfactants on Liquid Side Mass Transfer Coefficients: A New Insight[END_REF][START_REF] Sardeing | Effect of Surfactants on Liquid-Side Mass Transfer Coefficients in Gas-Liquid Systems: A First Step to Modeling[END_REF][START_REF] Ahmia | Impact of Cellulose and Surfactants on Mass Transfer of Bubble Columns[END_REF][START_REF] Painmanakul | Effect of Surfactants on Liquid Side Mass Transfer Coefficient[END_REF]. In response to this interrogation, local scale studies are necessary to investigate mechanisms that occur during mass transfer. This local scale can be reached by Direct Numerical Simulation (DNS), which is a powerful tool to investigate the local distribution of surfactants around the bubble and thus gain access to a better understanding of their link with hydrodynamics and physicochemical phenomena close to the interface [START_REF] Deising | Direct Numerical Simulation of Mass Transfer in Bubbly Flows[END_REF][START_REF] Pesci | Computational Analysis of Single Rising Bubbles Influenced by Soluble Surfactant[END_REF][START_REF] Weiner | Experimental and Numerical Investigation of Reactive Species Transport around a Small Rising Bubble[END_REF][START_REF] Dani | Effect of Spheroid Bubble Interface Contamination on Gas-Liquid Mass Transfer at Intermediate Reynolds Numbers: From DNS to Sherwood Numbers[END_REF]. From an experimental point of view, few techniques for direct visualization of mass transfer have been developed. These techniques use a fluorophore or colored molecule as the oxygen [START_REF] Weiner | Experimental and Numerical Investigation of Reactive Species Transport around a Small Rising Bubble[END_REF][START_REF] Wolff | A Fluorescence Technique to Measure Concentration Gradients near an Interface[END_REF][START_REF] Wolff | Instantaneous Concentration Profiles of Oxygen Accompanying Absorption in a Stratified Flow[END_REF][START_REF] Woodrow | Laser-Induced Fluorescence Studies of Oxygen Transfer Across Unsheared Flat and Wavy Air-Water Interfaces[END_REF][START_REF] Butler | Modelling of Mass Transfer in Taylor Flow: Investigation with the PLIF-I Technique[END_REF][START_REF] Roudet | Hydrodynamic and Mass Transfer in Inertial Gas-Liquid Flow Regimes through Straight and Meandering Millimetric Square Channels[END_REF][START_REF] Roudet | Experimental Investigation of Interfacial Mass Transfer Mechanisms for a Confined High-Reynolds-Number Bubble Rising in a Thin Gap[END_REF][START_REF] Bork | The Impact of Local Phenomena on Mass Transfer in Gas-Liquid Systems[END_REF] or carbon dioxide [START_REF] Hiby | Eine Fluoreszenzmethode zur Untersuchung des Stoffübergangs bei der Gasabsorption im Rieselfilm[END_REF][START_REF] Huang | Influences of Gas-Liquid Interface Contamination on Bubble Motions, Bubble Wakes, and Instantaneous Mass Transfer[END_REF][START_REF] Kong | Dual Emission LIF Technique for PH and Concentration Field Measurement around a Rising Bubble[END_REF][START_REF] Kováts | Influence of Viscosity and Surface Tension on Bubble Dynamics and Mass Transfer in a Model Bubble Column[END_REF][START_REF] Lacassagne | Ratiometric, Single-Dye, PH-Sensitive Inhibited Laser-Induced Fluorescence for the Characterization of Mixing and Mass Transfer[END_REF][START_REF] Someya | DeLIF Measurement of PH Distribution around Dissolving CO2 Droplet in High Pressure Vessel[END_REF][START_REF] Stöhr | Visualization of Gas-Liquid Mass Transfer and Wake Structure of Rising Bubbles Using PH-Sensitive PLIF[END_REF][START_REF] Valiorgue | Concentration Measurement in the Wake of a Free Rising Bubble Using Planar Laser-Induced Fluorescence (PLIF) with a Calibration Taking into Account Fluorescence Extinction Variations[END_REF] sensor. Among these techniques, Planar Laser Induced Fluorescence with Inhibition (PLIF-I) enables measurements of mass transfer from a single bubble rising in a quiescent liquid [START_REF] Jimenez | Oxygen Mass Transfer and Hydrodynamic Behaviour in Wastewater: Determination of Local Impact of Surfactants by Visualization Techniques[END_REF][START_REF] Francois | Direct Measurement of Mass Transfer around a Single Bubble by Micro-PLIFI[END_REF][START_REF] Dani | Local Measurement of Oxygen Transfer around a Single Bubble by Planar Laser-Induced Fluorescence[END_REF][START_REF] Xu | Fast Determination of Gas-Liquid Diffusion Coefficient by an Innovative Double Approach[END_REF][START_REF] Xu | Mass Transfer and Diffusion of a Single Bubble Rising in Polymer Solutions[END_REF][START_REF] Lebrun | Gas-Liquid Mass Transfer around a Rising Bubble: Combined Effect of Rheology and Surfactant[END_REF][START_REF] Jimenez | Mass Transfer in the Wake of Non-Spherical Air Bubbles Quantified by Quenching of Fluorescence[END_REF] and provides access to (i) its shape, velocity, and diameter, and (ii) the oxygen mass transfer coefficient and diffusion coefficient in the liquid phase, with good precision. This very powerful technique then allows hydrodynamic and mass transfer to be uncoupled. They are both knew with good precision, which makes it a very good technique for investigating mechanisms of oxygen transfer in the presence of surfactants.

In recent decades, numerous studies have shown the strong impact of the structure of surfactants and their properties on the mass transfer coefficient. Jia et al. [START_REF] Jia | Effect of Surfactant Type on Interfacial Area and Liquid Mass Transfer for CO2 Absorption in a Bubble Column[END_REF] demonstrated that mass transfer coefficient decreases with decreasing Critical Micellar Concentration (CMC). The charge of the head has been demonstrated to have an impact on the mass transfer coefficient [START_REF] Sardeing | Effect of Surfactants on Liquid-Side Mass Transfer Coefficients in Gas-Liquid Systems: A First Step to Modeling[END_REF][START_REF] Orhan | Effects of Surfactants on Hydrodynamics and Mass Transfer in a Co-Current Downflow Contacting Column[END_REF][START_REF] Chen | Effects of Surfactant Contamination on Oxygen Mass Transfer in Fine Bubble Aeration Process[END_REF]. Garcia-Aubuin et al. [START_REF] García-Abuín | Effect of Surfactant Nature upon Absorption in a Bubble Column[END_REF] showed that, the longer the hydrophobic chain length of cationic surfactant is, the more the mass transfer coefficient decreases. Similar results highlighted in our previous study [START_REF] Lebrun | Effect of Surfactant Lengths on Gas-Liquid Oxygen Mass Transfer from a Single Rising Bubble[END_REF], conducted with the PLIF-I technique, showed that the length of the tail of cationic surfactants has a strong impact on the oxygen mass transfer coefficient, although the hydrophilic length of nonionic surfactants does not impact mass transfer for a given bulk concentration. This previous study could not conclude on the effect of anionic surfactant length because the fluorophore used as the oxygen tracer interacted with anionic surfactants.

This study proposes a luminescent probe for use in the PLIF-I technique; [Ru(dpp-diSO3)3]Na4, in order to make possible the use of this technique in the presence of anionic surfactants. This paper will first present the synthesis of this fluorescent molecule, and characterize its fluorescence properties. Then, its use as an oxygen sensor in the PLIF-I technique will be validated in pure water conditions.

Once the use of [Ru(dpp-diSO3)3]Na4 is validated in pure water, the study shows its compatibility with anionic surfactants, and presents results of PLIF-I experiments run with sodium dodecyl sulfate (SDS) and sodium tetradecyl sulfate (STS), which are anionic surfactants presenting the same polar head and hydrophobic chains of 12 and 14 carbons, respectively. This work is then compared with the results of Lebrun et al. [START_REF] Lebrun | Effect of Surfactant Lengths on Gas-Liquid Oxygen Mass Transfer from a Single Rising Bubble[END_REF] and the parameters that impact the mass transfer are investigated. Thus, a model correcting Higbie's [START_REF] Higbie | The Rate of Absorption of a Pure Gas into Still Liquid during Short Periods of Exposure[END_REF] and Frössling's [START_REF] Frössling | Uber Die Verdunstung Fallender Tropfen[END_REF] correlations is proposed, by adding a new dimensionless parameter that describes the equilibrium between the surfactants adsorbed and in the bulk.

Material and method

Physico-chemistry

Surfactant solutions

All solutions were prepared with ultra-pure water having a conductivity of 0.054 mS.cm -1 . The surfactants chosen to study their effect on oxygen transfer were dodecyl sulfate sodium salt (SDS, CAS: 151-21-3, Sigma-Aldrich, USA) and tetradecyl sulfate sodium salt (STS, CAS: 1191-50-0, Sigma-Aldrich, USA). Surfactant solutions were prepared at concentrations between 1.3×10 -3 and 2.5×10 -7 mol.L -1 . Surface tension of solutions was measured according to the Du Nouÿ ring method [START_REF] Du Noüy | An interfacial tensiometer for universal use[END_REF] (tensiometer: K6, Krüss, Germany).

Fluorophore synthesis

A fluorophore for the PLIF-I technique was synthetized in this work. The complex [Ru(dpp-diSO3] 2- has been selected, since it is known to exhibit O2 sensing properties [START_REF] Castellano | A Water-Soluble Luminescence Oxygen Sensor[END_REF]. It was readily prepared using reported procedures but using microwave oven (Anton-Paar-300 monomode oven) heating [START_REF] Castellano | A Water-Soluble Luminescence Oxygen Sensor[END_REF][START_REF] Babak | Heteroleptic Ruthenium(II) Complexes with Bathophenanthroline and Bathophenanthroline Disulfonate Disodium Salt as Fluorescent Dyes for In-Gel Protein Staining[END_REF] such as an oven is known to reduce side reactions and to accelerate the transformation [START_REF] Beves | 4′-Chloro-2,2′:6′,2″-Terpyridine (L): Ethyl Sulfate Salts of [H2L] 2+ and the Single Crystal Structures of[END_REF]55]. The reagents used were Ruthenium (III) chloride, RuCl3.xH2O (x: ca 6) (CAS: 10049-08-8, Sigma-Aldrich, USA), and the 2,7 bis-(sulfonatophenyl)-1,10-phenanthroline sodium salt trihydrate (CAS:52746-49-3, Sigma-Aldrich, USA); they were used as received. The protocol followed was as follows: a solution of RuCl3.xH2O (113 mg, 0.546 mmol), 2,7 bis-(sulfonatophenyl)-1,10phenanthroline sodium salt trihydrate (1 g, 1.69 mmol) in argon-degassed ethylene glycol (20mL) was heated in a microwave oven for 2 hours at 220°C. After cooling, the dark orange reaction mixture was added to an excess of diethyl ether. The resulting precipitate was filtered then redissolved in a minimum amount of methanol before being applied in a reverse phase C18-column (ca 1x15cm) eluted with a MeOH: water mixture. Dark orange fractions were freeze dried. The yield obtained was 400 mg (42%).

Experimental setup

The experimental setup of Planar Laser Induced Fluorescence with Inhibition (PLIF-I) is depicted in Figure 1. It is composed of a column (1) (10x10x30 cm 3 ) filled with an aqueous solution. A millimetric single air bubble is generated in this solution through a needle 75 µm in internal diameter connected to a syringe pump, which can be activated remotely (Harvard apparatus PHD, 22/2000 Programmable, USA) (2). A CMOS camera (3) (Basler ace 12 bits, 1920x1200 pixel 2 , Basler, Germany) was placed next to the column, and recorded the rise of the bubble at a rate of 250 images.s -

Figure 1: Experimental setup

This camera was positioned in such a way as to measure the diameter and velocity of the bubble; the viewing window was then around 39x10 mm 2 (the highest visualization axis in the vertical direction).

The amount of oxygen transferred in the wake of the bubble was measured by fluorescence inhibition of a ruthenium complex. The ruthenium complex generally used is ruthenium dichlorotris(1,10phenanthroline)ruthenium(II) hydrate (CAS: 20782-45-7, Sigma-Aldrich, USA), which is very suitable in the presence of non-charged or positively charged molecules. However, in order to perform measurements in the presence of anionic molecules, the ruthenium tris(diphenylphenanthroline disulfonate) sodium salt (CAS: 301206-84-8) was synthetized and will be more extensively described in the Results section. The fluorophore was then excited by a horizontal laser sheet (4) (Nd:YAG, laser: Dualpower 200-15, 15Hz, 2x200 mJ, Dantec Dynamics, Denmark) at 532 nm, in a plane perpendicular to the bubble wake, 50 mm above the injection point of the bubble (the bubble crossed this plane between 0.2 and 0.5 seconds after its generation, depending on its velocity). Fluorescence intensity was recorded by a CCD camera (5) (FlowSense, 12 bits, 2048x2048 pixels, Dantec Dynamics, Denmark) synchronized with the laser frequency and focused on the laser sheet. This camera was placed at the bottom of the column with a sampling rate of 4.37±0.04 µm.pixel -1 (viewing window about 9x9 mm 2 ).

The camera was equipped with a 570 nm high pass filter. Before each experiment, a calibration curve was built by using an optical oxygen probe (multimeter: HQ40D, probe: Intellical LDO101, Hach, Germany) in order to determine the Stern Volmer [START_REF] Beves | 4′-Chloro-2,2′:6′,2″-Terpyridine (L): Ethyl Sulfate Salts of [H2L] 2+ and the Single Crystal Structures of[END_REF] constants according to equation (1)

! " = ! " ! + # "# " ! [𝑂 $ ] (1)
where G is the grey level, G0 is the grey level in the absence of the quencher, [O2] mg.L -1 is the oxygen concentration and Ksv is the Stern Volmer constant (L.mg -1 ). This calibration curve is built by injection of a mix of nitrogen and compressed air to obtain different concentration of oxygen. After mixing, the concentration of oxygen is supposed to be homogeneous and 100 images are taken from the bulk and related to an oxygen concentration, measured by the probe. To avoid spatial fluctuation, the calibration curve is built pixel by pixel. Once the calibration curve was applied to the images obtained in the bubble wake, the oxygen concentration was displayed on each pixel.

It was assumed that, far from the bubble, in its wake, convection could be neglected and all the matter motion was due only to diffusion of oxygen in the plane perpendicular to the bubble wake [START_REF] Jimenez | Oxygen Mass Transfer and Hydrodynamic Behaviour in Wastewater: Determination of Local Impact of Surfactants by Visualization Techniques[END_REF][START_REF] Francois | Direct Measurement of Mass Transfer around a Single Bubble by Micro-PLIFI[END_REF][START_REF] Xu | Mass Transfer and Diffusion of a Single Bubble Rising in Polymer Solutions[END_REF][START_REF] Lebrun | Effect of Surfactant Lengths on Gas-Liquid Oxygen Mass Transfer from a Single Rising Bubble[END_REF]. In previous studies [START_REF] Francois | Direct Measurement of Mass Transfer around a Single Bubble by Micro-PLIFI[END_REF][START_REF] Dietrich | Fast Measurements of the Gas-Liquid Diffusion Coefficient in the Gaussian Wake of a Spherical Bubble[END_REF], it was observed that, for spherical and quasi-spherical bubbles, the diffusion spot was circular and presented a Gaussian profile. As a result, it was possible to fit the oxygen field by equation (2), where the concentration [O2] in each pixel xp, yp was estimated.

[𝑂 ! ](𝑥, 𝑦) = 𝐴𝑒𝑥𝑝 "#$ ! "%& " '#( ! ")& " * + 𝐶 (2)
Parameters were found by fitting the equation with the raw image using the "fminsearch" solver (Matlab R2017a). Note that A and B are parameters (mg.L -1 and pixel 2 respectively), and (X, Y) is the centre of the spot, the parameter C representing the background.

It was possible to calculate the flux of oxygen (mg.s -1 ) transferred by the rising bubble with equation

(3), where Vb is the velocity of the bubble (m.s -1 ) measured by the side camera [START_REF] Gillot | Predicting Oxygen Transfer of Fine Bubble Diffused Aeration Systems-Model Issued from Dimensional Analysis[END_REF].

𝐹 + " = 𝑉 , ∬ [𝑂 ! ](𝑥, 𝑦)𝑑𝑥𝑑𝑦 (3)
The mass transfer coefficient was then deduced from equation ( 4), where Sb is the surface area of the bubble, deduced from the equivalent diameter measured with the side camera (3); [O2] is the concentration at the beginning of the experiment (measured with the optical oxygen probe, close to 0 mg.L -1 ); and [O2] * is the saturation concentration of oxygen, also measured by the optical oxygen probe (close to 9.10 mg.L -1 ).

𝑘 . = / # " 0 $ ([+ " ] * "[+ " ]) (4)
Finally, the diffusion coefficient of oxygen in the bulk was measured by a method developed by Xu et al. [START_REF] Xu | Fast Determination of Gas-Liquid Diffusion Coefficient by an Innovative Double Approach[END_REF] assuming that the surface area of the spot Sspot increased with time following equation [START_REF] Guet | Fluid Mechanical Aspects of the Gas-Lift Technique[END_REF].

𝑆 5678 = 2𝜋𝐷 + " 𝜂𝑡 (5)
where η is a constant that can be fixed following the procedure described by Xu et al. [START_REF] Xu | Fast Determination of Gas-Liquid Diffusion Coefficient by an Innovative Double Approach[END_REF]. Then the surface area of the spot was plotted versus time and the diffusion coefficient was extracted from the slope. Each condition was repeated 6 times in order to ensure the repeatability of measurements. All measurements were performed at room temperature (294±1K) and atmospheric pressure.

Results and discussion

Fluorophore synthesis

The goal was to synthetize a fluorescent molecule presenting (i) fluorescence inhibition in the presence of oxygen (ii) a negative global charge to avoid interaction with anionic molecules. This is made possible ). Fluorescence inhibition by the oxygen was detected in both configurations. The important feature is the slope of the curve, which corresponds to the Stern Volmer Constant Ksv, as this constant expresses the ability of the oxygen to inhibit the fluorescence. As a result, the higher the constant is, the more the fluorescence is inhibited by oxygen, and thus, the easier it is to detect an oxygen variation.

Concerning the new anionic fluorophore [Ru(dpp-diSO3)3] 4-, the constant found was 9.20.10 -5 L.mg -1 , the constant found for [Ru(Phen)3] 2+ (the cationic fluorophore used previously) being 4.32.10 -5 L.mg -1 .

It is thus possible to conclude that the synthetized fluorophore was rather well inhibited by oxygen, and it would thus be possible to use it in the PLIF-I configuration to track the oxygen field in the presence of anionic surfactant.

Once the calibration curve had been plotted and the assumption that the fluorescence inhibition followed a Stern Volmer law with a high Ksv constant was validated, a PLIF-I experiment to measure oxygen transfer from a single millimetric bubble in pure water was run and compared with results obtained using [Ru(Phen)3] 2+ . The results are presented in Table 1. [START_REF] Clift | Bubbles, Drops, and Particles[END_REF]. The diffusion coefficient of oxygen is calculated from equation (5) (procedure described in the word of Xu et al. [START_REF] Xu | Fast Determination of Gas-Liquid Diffusion Coefficient by an Innovative Double Approach[END_REF] with η=1) and is found to be the same in both solution (2.0×10 - and it is thus possible to conclude that the synthetized fluorophore [Ru(dpp-diSO3)3] 4-can be used to track the oxygen transferred by bubbles to pure water solutions.

Fluorophore [Ru(Phen)3] 2+ [Ru(dpp-diSO3)3]
The ability of [Ru(dpp-diSO3)3] 4-to serve as a good oxygen sensor in PLIF-I has been shown, but the compatibility with anionic molecules needs to be checked. For that purpose, measurements were made in a solution containing 2.95×10 -5 mol.L -1 of SDS. An example of the results obtained is shown in Figure 3, where some raw images are presented versus time. The dark spot in the center of the image demonstrates the presence of oxygen. When the Stern Volmer calibration is applied to these images, the oxygen concentration field is obtained as shown in Figure 3 A and B. The signal to noise ratio is almost 100 at 0.20 seconds after the bubble's passage and 37 after 1.58 seconds. This ratio is good enough for the signal of interest to be clearly detected. The integral of this oxygen field corresponds to the total mass transferred in the plane perpendicular to the bubble wake. In the absence of vertical convection, this integral should be constant. This mass conservation is observed in Figure 3 on the graph representing the integral according to time, which is constant with a standard deviation of 2.5% over a period of 1.65 seconds. With the same protocol as described in Section 2, it is thus possible to determine the mass transfer coefficient of oxygen by the PLIF-I technique using [Ru(dpp-diSO3)3] 4-as the oxygen probe.

Figure 3: Example of oxygen field concentration obtained in the wake of a bubble rising in an aqueous solution of SDS

Finally, the PLIF-I technique allows the oxygen diffusion coefficient in the liquid phase to be determined by following the spreading of the surface spot with time. The figure presents the spreading of a surface inside the oxygen concentration field, in such a way that the mass in this surface area is constant. The linearity observed between the surface area and time gives access to the diffusion coefficient of oxygen.

This linearity can be observed in Figure 4, and the diffusion coefficient in the SDS solution is then calculated to be 1.95×10 -9 m 2 .s -1 .

Figure 4: Expansion of the oxygen concentration field with time to measure diffusion coefficient of oxygen in an aqueous solution of SDS

Mass transfer measurements

The accuracy of [Ru(dpp-diSO3)3] 4-as an oxygen probe in the presence of an anionic surfactant has been shown. This new fluorophore will be applied below to measure the mass transfer from a single, isolated bubble in the presence of anionic surfactants presenting different chain lengths and at different concentrations. The impact of the chain length of cationic and nonionic surfactants has been discussed in a previous paper [START_REF] Lebrun | Effect of Surfactant Lengths on Gas-Liquid Oxygen Mass Transfer from a Single Rising Bubble[END_REF] but it was not possible to consider the question in the presence of anionic surfactant due to the lack of a compatible fluorophore. Now this issue has been cleared up, measurements can be made with anionic surfactants and compared to the previous study. m.s -1 ) at the high concentration but, for intermediate concentrations, the surfactant with the longest chain (STS) is impacted at lower surfactant concentration than SDS, which has a shorter chain. This effect is the same as observed for the effect of hydrophobic chain length of cationic surfactants [START_REF] Lebrun | Effect of Surfactant Lengths on Gas-Liquid Oxygen Mass Transfer from a Single Rising Bubble[END_REF].

The diffusion coefficient of oxygen in the different surfactant solutions studied is plotted in Figure 5 (d). It can be observed that, in this range of concentrations, far from the interface, the diffusion coefficient of oxygen DO2 is not impacted by the presence of surfactant and keeps a value close to the one in water (around 2×10 -9 m 2 .s -1 ).

Comparison with cationic and nonionic, modeling purpose

These new data on mass transfer from a single bubble rising in a solution containing anionic surfactant complement a previous study that considered cationic and nonionic surfactants. The data set obtained now covers the large diversity of surfactants and is big enough to propose some models. Current predicting models take the Schmidt number and the Reynolds number into account to predict the Sherwood number. It is shown in [START_REF] Jimenez | Oxygen Mass Transfer and Hydrodynamic Behaviour in Wastewater: Determination of Local Impact of Surfactants by Visualization Techniques[END_REF][START_REF] Lebrun | Gas-Liquid Mass Transfer around a Rising Bubble: Combined Effect of Rheology and Surfactant[END_REF][START_REF] Jimenez | Etude Du Transfert de Matière Gaz/Liquide En Milieux Complexes: Quantification Du Transfert d'oxygène Par Techniques Optiques[END_REF] that, for very concentrated solutions, the Frössling model, used for fully contaminated bubbles, overestimates the mass transfer. To explain the overestimation, the authors propose that another effect, taken into account neither by the Reynolds number nor by the Schmidt number in the bulk, inhibits mass transfer.

To show the effect of addition of surfactants on the Reynolds, Schmidt and Sherwood numbers, the ratio between the dimensionless number in the presence of surfactant and the one obtained in a pure solution has been calculated for all operating conditions. The θcap angle, calculated from the Sadhal & Johnson [START_REF] Sadhal | Stokes flow past bubbles and drops partially coated with thin films. Part 1. Stagnant cap of surfactant film -exact solution[END_REF] model, describes the ratio for the bubble contaminated by surfactant. This contamination angle is calculated from a normalized drag coefficient C * D, presented in equation ( 6)

𝐶 9 * = ; & "9 & ' ; & (' "; & ' = < != (2:𝜋 -𝜃 >?6 = + sin:𝜃 >?6 = + sin:2𝜃 >?6 = - < @ sin:3𝜃 >?6 =) ( 6
)
where CD is the experimental drag coefficient. CD im and CD m represent the drag coefficients of a fully contaminated and a clean bubble, respectively. These coefficients are calculated from the Schiller &

Naumann correlation [START_REF] Schiller | Ueber Die Grundlegenden Berechnungen Bei Der Schwerkraftaufbereitung[END_REF] (equation ( 7)) and Mei et al. correlation [START_REF] Mei | A Note on the History Force on a Spherical Bubble at Finite Reynolds Number[END_REF] (equation ( 8)).

𝐶 9 AB = !C DE (1 + 015𝑅𝑒 F.HIJ ) (7)
𝐶 9 B = <H DE (1 + DE I'F.K(DE'@.@<KDE ).+ ) ) (8)
We will then use the coverage ratio Rcap calculated using equation [START_REF] Gómez-Díaz | Mass-Transfer Enhancement or Reduction by Surfactant Presence at a Gas-Liquid Interface[END_REF].

𝑅 >?6 = <IF"L ,-! <IF ( 9
)
The ratio is reported versus the coverage ratio Rcap in Figure 6.

Figure 6: Comparison between Sherwood, Reynolds, and Schmidt numbers for clean and contaminated bubbles

It can be observed on Figure 6 that the Schmidt number is not significantly impacted by the increase of coverage ratio; the ratio of Schmidt numbers in the presence of surfactant to Schmidt number in pure water stays close to 1 even at high surface coverage. This makes sense, as the surfactant concentrations under test do not affect the diffusion coefficient (Figure 5(d)) nor the viscosity. The Reynolds number, however, is highly impacted by the presence of surfactant, as has already been highlighted by the study of the velocities (Figure 5(b)). The ratio decreases with increasing coverage ratio until it reaches a plateau, where the Reynolds number reaches around 30% of its value in clean water. With this in mind, it is obvious that the Reynolds number plays an important role in the decrease of mass transfer. However, while the Reynolds number is decreased by around 70% by the presence of surfactant, the Sherwood number undergoes a greater decrease from a coverage ratio of 0.4, and the decrease in Sherwood number at high surface coverage is 15% greater than that in the Reynolds number.

Thus a large part of the decrease of Sherwood number is explained by the decrease of Reynolds number, but an additional effect would still be needed to explain why the Sherwood number is impacted 15% more than the Reynolds number.

To identify what parameter could be responsible for this additional effect, it is important to have a better view of the behavior of surfactants in the liquid phase and their affinity toward the interface.

The "Sa" software [60] has been used to determine the adsorption parameters of SDS and STS according to the Frumkin isotherm [START_REF] Frumkin | Die Kapillarkurve Der Höheren Fettsäuren Und Die Zustandsgleichung Der Oberflächenschicht[END_REF] and the results, accompanied by previous ones on cationic and nonionic surfactants, are reported in Table 2. This software allows to fit experimental parameters with a solver baser on the Powell alogithm [START_REF] Powell | An efficient method for finding the minimum of a function of several variables without calculating derivatives[END_REF]. In the case of surface contamination, knowing that the surface tension is linked to the surface concentration according the Gibbs equation [START_REF] Gibbs | On the equilibrium of heterogeneous substances[END_REF] :

𝛾 = 𝛾 F + 𝑛 M 𝑅𝑇Γ N [ln(1 -𝜃) + 𝑎 / 𝜃 ! ] (10)
Where aF (-) is a parameter that takes electrostatic interactions between surfactants into account and θ is the surface coverage, defined as the ratio between the surfactant concentration 𝛤 (mol.m -2 ) and the maximal surfactant concentration 𝛤ꝏ (mol.m -2 ). The adsorption parameters obtained are fitted from the Gibbs equation and the Langmuir [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids II. Liquids.1[END_REF] and Frumkin [START_REF] Frumkin | Die Kapillarkurve Der Höheren Fettsäuren Und Die Zustandsgleichung Der Oberflächenschicht[END_REF] isotherms presented in equations ( 11) and [START_REF] Cockx | Modeling of Mass-Transfer Coefficient K-L in Bubble-Columns[END_REF].

𝐾 .M 𝐶 = L <"L (11)
𝐾 / 𝐶 = L <"L exp (-2𝑎 / 𝜃) ( 12
)
where C is the surfactant concentration (mol.m -3 ), KF and KLG are the adsorption constants (m 3 .mol -1 ).

Kind of surfactant Name

Frumkin KF (m The results obtained for anionic surfactants show the same trend as cationic surfactant: the longer the hydrophobic chain, the lower the maximal surface concentration: 3.55×10 -6 mol.m -2 for SDS (the shortest chain), and 2.48×10 -6 mol.m -2 for STS (the longest chain). Moreover, this longer chain led to an adsorption at lower bulk concentrations. This effect is shown by the constant KF. This value of KF (and KLG for nonionic surfactants, modelled by the Langmuir isotherm [START_REF] Langmuir | The constitution and fundamental properties of solids and liquids II. Liquids.1[END_REF]) describes the equilibrium between surfactants in the liquid phase and at interface, according to the chemical equation :

𝑆 !" + * # O ⇔ 𝑆 !$%
where Saq represents the number of surfactants in the liquid phase, * the number of vacant sites, and Sads the number of surfactants adsorbed. In fact, if the constant KF increases, the equilibrium is shifted toward the right (i.e., adsorbed surfactants).

For the following development, we note the adsorption constant as K, which represents KLG for nonionic surfactants and KF charged surfactants.

It has been suggested that the Sherwood number is not only impacted by the Schmidt and the Reynolds numbers, so, in this part, we chose to work with points that have the same Reynolds (100<Re<107) and Schmidt numbers (500<Sc<562). The corresponding Sherwood number varies between 98 and 30. This first observation again stresses that, for the same Schmidt and Reynolds conditions, the Sherwood number can vary by about 70%. The purpose is to find a third dimensionless number that could represent the additional impact of contamination on the mass transfer. The factor KC (adsorption constant and concentration, respectively in m 3 .mol -1 and mol.m -1 ) seems to be a good candidate, since it represents the intrinsic adsorption properties of the surfactant and the state of the equilibrium between the bulk and the interface with bulk concentration. In order to take the electrostatic interactions into account, this dimensionless number can be divided by exp(-2aFθ). This term is present in the Frumkin isotherm and weights the equilibrium by taking electrostatic interactions into account. If interactions are neglected, as is the case for nonionic surfactants, the term aF is equal to 0 and exp(-2aFθ) takes the value of 1 (i.e.

does not impact KC). As a result, to take in account the previously mentioned parameters, the new dimensionless number should be KC/ exp(-2aFθ).

Figure 7: Variation of the Sherwood number according to KC/exp(-2aFθ) for the same range of Reynolds and Schmidt numbers

Figure 7 plots the Sherwood number against this new dimensionless number KC/exp(-2aFθ), for 100<Re<107 and 500<Sc<562. On this figure, we can observe that the Sherwood number undergoes a decrease when KC/exp(-2aFθ) increases. This means that, the more the equilibrium of surfactants between the bulk and the interface favors the adsorption (i.e., high KC/exp(-2aFθ)), the lower is the Sherwood number. To propose a model to fit these experimental points, we built our physical model using the scheme presented in Figure 8.

Figure 8: Scheme of the physical phenomenon occurring during the rise of a bubble in a liquid containing surfactants.

The bubble rises in the liquid containing surfactants. Surfactants are adsorbed and swept from the nose to the rear of the bubble because of convection as described in Palaparthi et al. [START_REF] Palaparthi | Theory and Experiments on the Stagnant Cap Regime in the Motion of Spherical Surfactant-Laden Bubbles[END_REF]. As a result, the bubble can be split into two parts: the nose, almost clean of surfactant, and the rear, full of surfactants.

The part covered by surfactant can be estimated by Rcap (cf. equation ( 9)). Therefore, the mean Sherwood number on the bubble is calculated from equation ( 13):

𝑆ℎ >7PPEQ?8ER = :1 -𝑅 >?6 =𝑆ℎ >QE?S + 𝑅 >?6 𝑆ℎ >7S8?BAS?8ER

By taking the third, previously mentioned, dimensionless number into account to model the effect of surfactant adsorption on the bubble we have:

𝑆ℎ >7PPEQ?8ER = S:1 -𝑅 >?6 =𝑆ℎ >QE?S + 𝑅 >?6 𝑆ℎ >7S8?BAS?8ER T S T; E ."-/ 0 T U ( 14
)
where 𝜓 is a constant that needs to be determined. The clean and contaminated Sherwood number can be calculated from the hydrodynamic thickness of each film and is expressed as equation [START_REF] Vasconcelos | Gas-Liquid Mass Transfer to Single Bubbles: Effect of Surface Contamination[END_REF].

𝑆ℎ = R 12 V 3 9 # " (15)
The mass transfer coefficient of oxygen in the liquid side is

𝑘 . = 9 # " W (16)
where δ is the thickness of the diffusion film. By combining ( 15) and ( 16):

𝑆ℎ = R 12 W (17)
Finally, by replacing ( 17) in ( 14) the correlation becomes:

𝑆ℎ >7PPEQ?8ER = (𝑅 >?6 R 12 W ,41-5 + (1 -𝑅 >?6 ) R 12
W ,657-'(5-718

) S

T;

E ."-/ 0 T U (18)
The thickness of each diffusion film is determined by the hydrodynamic conditions as equations [START_REF] Pesci | Computational Analysis of Single Rising Bubbles Influenced by Soluble Surfactant[END_REF] and ( 20) ( [START_REF] Frössling | Uber Die Verdunstung Fallender Tropfen[END_REF] and [START_REF] Higbie | The Rate of Absorption of a Pure Gas into Still Liquid during Short Periods of Exposure[END_REF]):

𝛿 >QE?S = 𝑑 EX 𝑅𝑒 "F.K 𝑆𝑐 "F.K (19)
𝛿 >7S8?BAS?8ER = 𝑑 EX 𝑅𝑒 "F.K 𝑆𝑐 "F.@@ (20)

The solver of excel® was used to determine the value of 𝜓 giving the best fit for the experimental points with this correlation, the method used were the least square method. The value converged toward -0.4 and the averaged deviation was about 20%. Figure 7 compares the experimental and correlated points.

The correlated points follow the trend of experimental points well. The goal is thus to extend this correlation to all the experimental points. However, since the bubble is rising and undergoes convection, the amount of surfactants around the bubble will depend on (i) the convection, (ii) their adsorption kinetics. The contribution of the contamination term KC/exp(-2aFθ) to the Sherwood number will depend on these two parameters. In order to compare the effect of adsorption and convection, it is useful to calculate a scale, defined as:

>7SYE>8A7S EZZE>8 >7S8?BAS?8A7S EZZE>8 = E ."-/ 0 DE T; (21)
If this ratio is high, the convection is strong, and the effect of contamination can be neglected. When the ratio is weaker, contamination by surfactant takes on more importance. This ratio varies between 10 -2 and 10 The mean deviation is 24%. The Sherwood numbers are well predicted by our correlation for the whole range of Sherwood numbers obtained experimentally. It seems that the correlation can slightly overestimate low Sherwood numbers and underestimate higher ones. This gap between the correlation and the model may be corrected with a better comparison between convection and contamination of the bubble. In order to compare the convection flux and the flux of surfactants contaminating the bubble in a more rigorous way, we would need the diffusion coefficient of surfactants (D, m 2 .s -1 ), and the adsorption and desorption constants (ka and kd respectively in m 3 .mol -1 .s and s -1 ). As described in Palaparthi et al. [START_REF] Palaparthi | Theory and Experiments on the Stagnant Cap Regime in the Motion of Spherical Surfactant-Laden Bubbles[END_REF], the adsorption rate (kinetic rate) and the diffusive rate of surfactants may be compared with the convection rate to give information about the contamination state of the bubble, as proposed in equations ( 22) and ( 23):

RAZZ[5AYE P?8E >7SYE>8A7S P?8E = V 8 R 12 V -!\ 9 𝑃𝑒 "F.K (1 + V -; V 8 ) (22)
VASE8A> P?8E

>7SYE>8A7S P?8E = V 8 R 12 !] $ S1 + V -; V 8 T 𝑃𝑒 "F.K (23)
where the Peclet number represents the ratio between the convective and the diffusive transport as Pe=Vbdeq/D. However, accessing these parameters is not easy and requires additional experimentation, so this method would not be easily usable for industrial applications. The ratio representing the regime as it is presented in equation 12 can be used to calculate Shcorrelated and gives an easy way to predict the mass transfer coefficient in the presence of surfactants, with an average error of 24%. This correlation gives interesting perspectives for its use in industrial application such as in wastewater treatment plants, and it would be interesting to investigate the use of this correlation in more complex media where rheology can as well impact the mass transfer [START_REF] Bajón Fernández | Gas to Liquid Mass Transfer in Rheologically Complex Fluids[END_REF][START_REF] García-Abuín | Oxygen Absorption in Polymer+surfactant Aqueous Solutions[END_REF][START_REF] Jamnongwong | Experimental Study of Oxygen Diffusion Coefficients in Clean Water Containing Salt, Glucose or Surfactant: Consequences on the Liquid-Side Mass Transfer Coefficients[END_REF].

Conclusion

In this paper, a luminescent probe [Ru(dpp-diSO3)3] 4-, has been synthetized in order to make measurement by Planar Laser Induced Fluorescence with Inhibition possible for the determination of oxygen mass transfer from a single isolated bubble in the presence of anionic molecules. In the first part, after a description of the synthesis protocol, the feasibility of using this fluorophore in PLIF-I experiments was checked, as was its performance in the presence of anionic surfactants. The fluorophore, [Ru(dpp-diSO3)3] 4 , was found to be a very good oxygen sensor and compatible with anionic molecules. Then, thanks to its use, PLIF-I experiments were run with anionic surfactants: SDS and STS, at different bulk concentrations. These two surfactants were chosen because they both present the same polar head, a sulfonate, and have linear chains with different numbers of carbons on their carbonated chain: 12 for SDS and 14 for STS. The study of hydrodynamic and mass transfer showed that mass transfer is highly impacted by the tail length of the anionic surfactant. The longer the hydrophobic chain is, the greater are the decreases in velocity and mass transfer, for a given bulk concentration. This effect is found to be the same as the effect of chain length variation found in our recent publication and can be explained by an equilibrium in favor of adsorption for lower bulk concentrations when the chain is longer and, as a result, the surfactant more hydrophobic.

Finally, the results of mass transfer measurement in the presence of anionic surfactants and the results of a previous study on cationic and nonionic surfactants have been compared with classical Higbie and

Frössling correlations. The impact of modification of Reynolds and Schmidt numbers due to the presence of surfactants on the Sherwood number has been evaluated. It has been highlighted that, for every surfactant studied, the decrease of Reynolds and Schmidt numbers alone cannot explain the decrease of mass transfer at high surfactant contamination. A new correlation has been proposed, with a new dimensionless factor KC/exp(-2aFθ) that accounts for the effect of surfactants contamination. This factor represents the equilibrium between the adsorbed surfactants and the surfactant present in the bulk.

Promising perspectives for applications arise from this study, especially in wastewater treatment plants,

where predicting the mass transfer coefficient is key to ensuring performance levels and energy preservation. With this in mind, it would be interesting to apply this model to a larger range of Reynolds numbers with a mix of contaminants, to approach industrial applications more closely. Moreover, it would be interesting to gain access to the diffusion coefficient, the adsorption, and the desorption constant of surfactants to check whether knowledge of these parameters leads to a better estimation of the contamination state of the bubble and so to a more accurate prediction of the mass transfer coefficient.

SUPPLEMENTARY MATERIAL

Evaluation of errors: (with Δ[O2] given by the oxygen probe ±0.02 mg/L) It has to be noted that each measurement has been run 6 times to ensure the repeatability of measurements. The dispersion of results was always found to be closed to

:; $ ; $ .

Diffusion coefficient

The diffusion coefficient is calculated with a correlation coefficient higher than 97%, this suggest that the measured uncertainties are lower than the relative standard deviation. The error bar given on experimental results are then the standard deviation.

  Figure 2(a).

Figure 2 :

 2 Figure 2: Fluorophore (a) synthesis (b) absorption/emission spectrum (b) calibration curve for PLIF-I experiments

3. 2

 2 Fluorophore compatibility and mass transfer measurement 3.2.1 Fluorophore compatibility Once the fluorescence properties of the [Ru(dpp-diSO3)3] 4-had been verified, confirming that it could be used as an oxygen sensor in PLIF-I configuration, its fluorescence inhibition needed to follow the Stern Volmer law (equation (1)), with a Stern Volmer constant Ksv high enough to detect fluorescence fluctuation with variation of the oxygen concentration. A classical PLIF-I calibration curve was plotted with 70 mg.L -1 of [Ru(Phen)3] 2+ and compared with a calibration curve for 55 mg.L -1 of our new fluorophore, [Ru(dpp-diSO3)3] 4-. The concentration of the fluorophore was chosen to be close to the concentration of [Ru(Phen)3] 2+ , usually preferred for PLIF-I experiments (70 mg.L -1 ), and to use a minimum of product. The two curves are presented in Figure 2(c) and are in good agreement with the Stern Volmer law (R 2 =99%

Figure 5 (

 5 Figure 5 (a) presents diameters of bubbles generated in different surfactant solutions. A slight decrease

Figure 5 :

 5 Figure 5: (a) Diameters (b) velocities (c) liquid side mass transfer coefficient and (d) oxygen diffusion coefficients measured in aqueous solutions of anionic surfactants.

[ 6 .

 6 8×10 -1 -4.5×10-2 ] The first condition then represents a situation where the convection outweighs the contamination by surfactants which can almost be neglected. Conversely, the 6th condition represents a major contamination, compared to the convection. Each regime contains around 10 experimental results (i.e., experiments on 60 bubbles). A value of 𝜓 was determined for each regime (with excel® solver, least squared method). The value of 𝜓 for each regime considered is supposed to represent the weighting of contamination effect of mass transfer. It is presented in Figure9(a), in function of the ratio between convection and contaminaation. For a convection:contamination ratio lower than 100, 𝜓 takes negative values and for ratios higher than 100 (i.e., convection preponderant) 𝜓 takes positive values and tends toward 0.

Figure 9 :

 9 Figure 9: (a) Values of 𝝍 obtained numerically according to the operating condition (b) Value of the corrective factor obtained according to the operating condition.

Figure 10 :

 10 Figure 10: (a) Sherwood numbers of experimental results and correlations in function of the Reynolds number (b) Comparison between experimental points and correlationIn order to compare the experimental values of Sherwood number and the predictions of our correlation,

Velocity

  and b repectively the big and small diameters of the bubble. Δa and Δb are calculated from the sampling of the camera, we consider an uncertainty of 1 pixel = 20,31 µm) Integral of oxygen concentration ∬ [𝑂 # ](𝑥, 𝑦)𝑑𝑥𝑑𝑦 𝑖𝑛 𝑛𝑜𝑡𝑒𝑑 𝑚

Table 1 : Comparison between the PLIF-I results obtained in ultra-pure water with [Ru(Phen)3] 2+ and [Ru(dpp-diSO3)3] 4-. Table

 1

	4-

1 presents hydrodynamic and oxygen transfer from a single isolated bubble. The second column gives results obtained in an aqueous solution of [Ru(Phen)3] 2+ and the third column those found in an aqueous solution of [Ru(dpp-diSO3)3] 4-. The diameter and velocity of the bubbles generated are very close: diameter around 1 mm and resulting velocity around 270 mm.s -1 , as expected for a clean bubble of millimetric size

  3 .mol -1 )

			Langmuir KLG	aF	Γꝏ
			(m 3 .mol -1 )	(-)	(10 -6 m 2 .mol -1 )
	TX100	-	1053	-	2.76
	TX102	-	1194	-	2.43
	Nonionic [48]
	TX165	-	2240	-	1.90
	TX305	-	3294	-	1.36
	C8TAC	0.02	-	0.04	2.43
	C12TAC	2.58	-	0.63	2.48
	Cationic [48]
	C14TAC	1.05	-	0.91	2.60
	C16TAC	3.38	-	2.75	1.54
	SDS	0.43	-	1.37	3.55
	Anionic
	STS	0.72	-	2.73	2.48

Table 2 : Constants of Frumkin and Langmuir adsorption isotherm for nonionic, cationic and anionic surfactants.

 2

  6 . It was decided to split our results into 6 conditions for which Re and Sc are constant:

	-Condition 6 :	>7SYE>8A7S EZZE>8 >7S8?BAS?8A7S EZZE>8
	-Condition 1 :	>7SYE>8A7S EZZE>8 >7S8?BAS?8A7S EZZE>8 [7.7×10 6 -2.8×10 5 ]
	-Condition 2 :	>7SYE>8A7S EZZE>8 >7S8?BAS?8A7S EZZE>8 [2.0×10 5 -6.6×10 3 ]
	-Condition 3 :	>7SYE>8A7S EZZE>8 >7S8?BAS?8A7S EZZE>8 [6.6×10 3 -2.2×10 2 ]
	-Condition 4 :	>7SYE>8A7S EZZE>8 >7S8?BAS?8A7S EZZE>8 [2.2×10 2 -2.7×10 1 ]
	-Condition 5 :	>7SYE>8A7S EZZE>8 >7S8?BAS?8A7S EZZE>8 [2.7×10 1 -6.8×10 -1 ]

  With MO2 the dispersion on all values of the integral during the measurement (always <5%) and 𝛿 5.&(). is the uncertainty given by the sampling of the camera (4.37 µm))

					4#
	𝛥𝑚 4# 𝑚 4#	= 8 9 𝛥𝑀 4# 𝑀 4#	;
	Mass transfer coefficient
		Δ𝑘 6 𝑘 6	= 89	Δ𝑚 4# 𝑚 4#	; #	+ 4 @ Δ𝑑 (7 𝑑 (7	A + 9	Δ𝑉 ' 𝑉 '	;	#	+ (	([𝑂	Δ[𝑂 # ]

# + 9 𝛥𝛿 5.&(). 𝛿 5.&(). ; # (# ] * -[𝑂 # ] 9 ) )

Mass transfer coefficient (10 -4 m/s)

				SDS
	Concentration (mol/L)	1.3×10 -3		2.5×10 -3	2.5×10 -4	2.95×10 -5	2.5×10 -6
	Bubble velocity
		114±1		109±1 STS	124±2	117±1	284±8
	(10 -3 m/s)
	Concentration (mol/L)	1.3×10 -3			2.5×10 -6	9.1×10 -7	2.5×10 -7
	Bubble diameter
	Bubble velocity	0.96±0.02	0.92±0.02	1.03±0.02	0.93±0.02	1.16±0.02
	(10 -3 m)	104±1			177±3	284±8	294±9
	(10 -3 m/s)
	Equilibrium surface tension (10 -3
	Bubble diameter	52±1		44±1	68±1	71±1	71±1
	N/m)	0.87±0.02		1.17±0.02	1.28±0.02	1.20±0.02
	(10 -3 m)
	Integral of oxygen concentration
	Equilibrium surface tension (10 -3 field (10 -5 mg/m)	1.60±0.08 40±1	1.65±0.08	2.8±0.1 71±1	3.5±0.2 71±1	6.8±0.3 70±1
	N/m)
	Oxygen diffusion coefficient (10 -9
	Integral of oxygen concentration m 2 /s)	2.0±0.1 1.47±0.07	2.0±0.2	1.9±0.1 2.9±0.1	1.89±0.03 6.3±0.3	2.00±0.07 8.0±0.4
	field (10 -5 mg/m)
	Mass transfer coefficient (10 -4 m/s) 0.70±0.04	0.75±0.04	1.13±0.07	1.7±0.1	5.0±0.3
	Oxygen diffusion coefficient (10 -9
	m 2 /s) Relative standard deviation of mass	0.07 2.0±0.2		0.03	0.03 1.89±0.08	0.05 2.0±0.1	0.06 1.9±0.1
	transfer coefficient
		0.73±0.04		1.20±0.07	3.9±0.2	5.7±0.4
	Schmidt number	500±25		500±50	532±16	529±8	500±17
	Relative standard deviation of mass
	Reynolds number	109±3 0.05		100±2	128±3 0.05	109±3 0.03	329±11 0.05
	transfer coefficient
	Sherwood number	34±2		34±4	62±4	83±4	291±11
	Schmidt number	505±51			529±22	493±24	521±27
	Reynolds number	90±2			207±5	364±11	352±12
	Sherwood number	32±3			80±4	249±13	357±20

in a window of 1920x500 pixel -1 at

[START_REF] Weiner | Experimental and Numerical Investigation of Reactive Species Transport around a Small Rising Bubble[END_REF].31±0.04 µm.pixel -1 .

ACKNOWLEDGEMENT

The authors wish to thank the French National Research Agency (ANR) for the support received from the project MAMOTHS ANR-17-CE06-001.


S Arnaud

Engonga Edzang
email: arnauld.engonga.edzang@gmail.com

Eric Badel

Bruno Moulia

Rostand Moutou Pitti

Joseph J Gril

Arnauld Engonga Edzang

Wind safety of rubber trees in plantations: comparison of the resistance to breakage of two clones Clauvy'

come

Figure 1 :

 1 Figure1: Principle of a trunk-bending test. H: total tree height; Hc: crown-base height; HF: height of cable attachment to the tree; α: initial inclination of tree base; β: cable angle; Lhw: distance from tree base to cable anchorage; ε: surface deformation; θ: inclination of root-ground system; F: applied force; P: crown weight; ΔL cable shortening.

Figure 2 :

 2 Figure 2: Trunk breakage of a 6-year-old rubber tree after bending test.